ernst cassirer and a transcendental approach towards ......ernst cassirer and the philosophy of...

Ernst Cassirer and a transcendental approach

towards contemporary physics

Laurens Vanderstraeten

Dissertation submitted in fulfillment of the requirementsfor the degree of Master of Arts: Philosophy

Supervisor: Maarten Van Dyck

Academic year: 2016–1017

Foreword

In the first decade of the twentieth century Ernst Cassirer wrote Substance and Function, abook with the bold ambition of philosophically analyzing the complete conceptual status oftheoretical physics. Of course, around that time physics was a lot less diversified than it is now,so that a philosopher such as Cassirer could still have a pretty good overview of all importantdevelopments. Also, physics itself was a lot stronger connected to the philosophical literatureof its time, such that the bridge between the two could be crossed more easily. Nowadays,these two conditions for a fruitful interplay between philosophy and physics are no longer met.Physics has become too diverse to overlook its structure in the way Cassirer did, and physicistsare too immersed in the problems of physics to relate to the philosophical literature. As a result,philosophy of physics has become a discipline that works in the margins of both physics andphilosophy, without the hope of appealing to an audience outside its niche.

One could argue that this is the rightful place for the philosophy of physics, because physicshas become a scientific discipline for which all rules of the game have been decided a long timeago – around the time of Cassirer, I suppose. In fact, there have been many times this hasturned out as my only conclusion. Yet, the writing of this thesis requires the hypothesis thatsomething interesting can still be said, and, at the end, I finally believe this to be true. I believethat the philosophy of physics can still prove its worth in the future for philosophy and physicsalike, and I believe that the tools of Cassirer can be a great inspiration. I hope that this thesismight show a glimpse of what is possible in that respect.

I am indebted to Maarten Van Dyck for pushing me in the neo-Kantian direction; any failureto carry through his original suggestions are on my account. Also, I want to thank him for thepatience he has shown during the five years it took me to write this thesis. I am grateful toMatthias Bal for reading this thesis and learning me about renormalization.

Laurens VanderstraetenGent, May 28, 2017

Contents

1 Overview 1

2 Ernst Cassirer and the philosophy of physics 32.1 Theoretical physics in the 19th century . . . . . . . . . . . . . . . . . . . . . . . . 32.2 Ernst Cassirer and the Marburg school . . . . . . . . . . . . . . . . . . . . . . . . 62.3 Substance and function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.3.1 A new logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.3.2 The serial forms in mathematics . . . . . . . . . . . . . . . . . . . . . . . 92.3.3 The concepts of natural science . . . . . . . . . . . . . . . . . . . . . . . . 102.3.4 Physical theory and experience . . . . . . . . . . . . . . . . . . . . . . . . 132.3.5 The progress of scientific knowledge . . . . . . . . . . . . . . . . . . . . . 14

2.4 Cassirer on modern physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.4.1 The theory of relativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.4.2 Quantum mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.5 Michael Friedman . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.5.1 The dynamics of reason . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.5.2 The historicized a priori . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.5.3 Constitutive or regulative? . . . . . . . . . . . . . . . . . . . . . . . . . . 22

Coda. The transcendental philosophy of contemporary physics . . . . . . . . . . . . . 25

3 Renormalization in contemporary physics: a transcendental perspective 273.1 Theoretical physics in the 21st century . . . . . . . . . . . . . . . . . . . . . . . . 283.2 The theory of renormalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.2.1 The prehistory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333.2.2 Wilson’s intervention . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343.2.3 Renormalization and the many-body problem . . . . . . . . . . . . . . . . 38

3.3 Review of philosophical literature . . . . . . . . . . . . . . . . . . . . . . . . . . . 413.3.1 Empiricism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423.3.2 Physical understanding . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433.3.3 The question of emergence . . . . . . . . . . . . . . . . . . . . . . . . . . 453.3.4 More is different . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

3.4 Renormalization as a functional concept . . . . . . . . . . . . . . . . . . . . . . . 493.4.1 Energy in the work of Cassirer . . . . . . . . . . . . . . . . . . . . . . . . 503.4.2 The necessity of scale and effective degrees of freedom . . . . . . . . . . . 503.4.3 The importance of the computational approach . . . . . . . . . . . . . . . 533.4.4 The object of physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 543.4.5 Historicizing renormalization . . . . . . . . . . . . . . . . . . . . . . . . . 56

Bibliography 59

Chapter 1

Overview

In this thesis we will give a transcendental account of the theory of renormalization, one of thecornerstones of contemporary theoretical physics. This account is based on the work of ErnstCassirer, which is the subject of the first chapter. The second chapter contains the applicationto the theory of renormalization theory. Let us give a short overview.

Ernst Cassirer and the philosophy of physics

This chapter discusses Cassirer’s philosophy of physics as he wrote it down in his book Substanceand Function. For a good understanding of the goals of Cassirer’s writing, we first take a quicklook at the situation of theoretical physics around the turn of the century. In particular, wediscuss an exchange between Ernst Mach and Max Planck, two physicists, which clearly indicatesthe philosophical problems that surrounded nineteenth-century physics. Next, we trace thephilosophical roots of Cassirer back to the neo-Kantian revival of transcendental philosophyand, in particular, the so-called Marburg school as articulated by Cohen.

After these introductory sections, we dive into Substance and Function and show in detail howCassirer analyzes the conceptual structure of theoretical physics. In particular, he explains howthe use of relational concepts allows physics to give determinate meaning to physical phenomenaand to integrate different phenomena in an inclusive whole. In line with the Marburg school,Cassirer’s approach is historical in the sense that these concepts of physics are analyzed withintheir historical development, and point to an ideal end-point of physics where the object ofphysics is to be fully determined. After Substance and Function, this focus on the historicalprogression of physics forced Cassirer to apply his philosophical framework to the theory ofrelativity and quantum mechanics as well, and we shortly discuss his monographs on thesesubjects.

We conclude the chapter by bringing Cassirer to the 21st century through the work ofMichael Friedman. We will explain how Friedman believes that a neo-Kantian approach to thephilosophy of physics is still possible today through his notion of the relativized a priori. Weare particularly interested in his claim that he goes beyond the work of Cassirer by historicizingthe a priori and by safeguarding the constitutive function of a priori principles, because bothnotions will come back at the end of the next chapter.

Renormalization in contemporary physics: a transcendental perspective

In the second chapter we apply the ideas of the previous one to the theory of renormalization.In order to set the stage, we start by exploring some of contemporary theoretical physics, and,in particular, the many-body problem. We show that this problem reappears in many areas of

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Chapter 1. Overview 2

physics, and therefore constitutes an interesting case for the transcendental approach that cap-tures the conceptual structure of theoretical physics, in contrast to metaphysical or ontologicalapproaches.

We explain the historical development of renormalization theory from the early ideas of Lan-dau, over the seminal insights of Wilson, and to the place it has obtained in modern condensed-matter and high-energy physics. In a next section, we focus on the philosophical literatureconcerning renormalization, and we try to indicate where our approach diverges from the exist-ing frameworks. In particular, we are out to show that existing approaches (i) focus too muchon ontological commitments that cannot be found in the physics itself, and (ii) fail to integratethe crucial lessons from renormalization in a comprehensive philosophical framework. This isargued for by going back to the original writings of Anderson and others, which have put theidea of emergence on the map.

In the last section, we try to give our own account of renormalization in the spirit of Cassirer.Thereto, we shortly reiterate Cassirer’s account of the nineteenth-century concept of energy. Welay bare the constitutive functions of the use of effective degrees of freedom and the concept of ascale transformation. Though seemingly unrelated, we show that computational approaches intheoretical physics can be nicely integrated into our framework, which learns that computationalphysics deserves more attention from the epistemologist than it commonly receives. Next, weidentify the advent of the ideas of renormalization and emergence as installing a new ideal ofunification, similarly to the way in which Cassirer understood the energy concept. Finally, wetake up the critique of Friedman as we have left it in the previous chapter, and show thatCassirer’s conception is both richer and more limited than Friedman’s in capturing the case ofrenormalization.

Chapter 2

Ernst Cassirer and the philosophy ofphysics

We start this chapter with a quote of Ernst Cassirer that appears towards the end of Substanceand Function, a quote that nicely summarizes what, for Cassirer, is at stake in the philosophyof physics:

He who grants science the right to speak of objects and of the causal relations ofobjects, has thereby already left the circle of the immanent being and gone over intothe realm “transcendence”. [1, p.295]

Indeed, theoretical physics is taken to yield knowledge about nature that should be, in somesense, objective and independent of the particularities of the working physicist, any place or time,and even man itself. If we grant physics this claim to objectivity, the philosophical questionopens up as to how this is possible: How does physics arrive at an objective and determinatepicture of the world? How do we leave the circle of immanent being – the chaos of senseimpressions and subjective states of our ego – and transcend towards the scientific picture ofthe external world?

In this chapter we investigate Cassirer’s philosophy of physics as it was formulated in hisbook Substance and Function. As it was written in 1910, the book analyzes the structureof theoretical physics as it looked like around the turn of the century, so the first section ofthis chapter consists of a brief sketch of nineteenth-century physics and, in particular, theconceptual difficulties that faced physicists around that time [Sec. 2.1]. We go on by placing thephilosophical roots of Cassirer in neo-Kantianism [Sec. 2.2], and then discuss the argumentationof Substance and Function in some detail [Sec. 2.3]. In a next section, we briefly go over the laterwork of Cassirer on modern physics [Sec. 2.4]. In the following section, we make the jump tocontemporary philosophy of physics and, in particular, the work of Michael Friedman [Sec. 2.5].The discussion of the work of Friedman will show how Cassirer can still prove to be relevanttoday. We conclude the chapter by summarizing what we, based on the work of Cassirer andFriedman, believe contemporary philosophy of physics should consist of.

2.1 Theoretical physics in the 19th century

Nineteenth-century physics is often described as a rather dull episode in the development ofphysics, a century dominated by a mechanistic world-view as originally designed by Newton.The conceptual structure is deemed monolithic, in stark contrast with the ground-breakingconceptual revolutions that have shaken theoretical physics to its foundations at the beginningof the twentieth century.

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Chapter 2. Ernst Cassirer and the philosophy of physics 4

In this chapter we would like to start from a different view on this period. Although relativitytheory and quantum mechanics unmistakably reshuffled a lot of nineteenth-century beliefs on ourpicture of reality, and some fin-de-siecle physicists were overly optimistic about the approachingcompleteness of physics [2], it is nonetheless clear that “classical physics” has had its ownnumber of conceptual innovations. Moreover, and on this aspect we would like to focus, fromthe epistemological point of view, these innovations spurred lively debates on the status ofphysical knowledge and the physicist’s picture of reality. [3]

Before 1800, exact mathematical laws were exclusively used for mechanical phenomena withinthe Newtonian framework, whereas heat, electricity, and optics were described rather qualita-tively. In the nineteenth century this situation changed: On the one hand, the work of e.g.Laplace and Fourier on heat brought non-mechanical phenomena under the scope of mathemati-cal analysis, while, on the other, the works of e.g. Fresnel on the wave nature of light and of Jouleon the conversion of mechanical and thermal energy showed how optical and thermal processesare intimately connected with mechanics. Behind these developments, we can characterize thegoal of nineteenth-century physics as the search for unification of different fields of physics underthe strict validity of mathematical laws. At first, the unifying principles were mostly thought ofin mechanical terms, but later on world-views based on electromagnetism or energetics becameviable options as well. [3]

A paradigmatic example of this unification of physical phenomena guided by mathemati-cal laws is Maxwell’s formulation of the laws of electrodynamics, which instantly brought thefields of optics and electromagnetism into one mathematical theory. In order to consistentlyinterpret the wave equations, Maxwell felt forced to introduce mechanical models underlyingthe electromagnetic fields – the fields are identified with the motion or rotation of a mechanicalether; without this mechanical basis, the mathematical framework would be not intelligible. Still,Maxwell always refused to interpret the mechanical models as an explanation for the electromag-netic equations, and instead stressed the hypothetical status of these models, serving more as ananalogy or illustration than as a description of physical reality. Indeed, in later works Maxwellintroduced the field equations in a purely mathematical manner, without a specific mechanicalmodel, but retained the idea that mathematical physics should still keep dynamical concepts inmind as these are appropriate to the representation of physical reality. [4]

Maxwell’s struggles with the status of mechanical models reflect the dominant programof nineteenth-century physics, where all physical phenomena were explained by the structureand laws of motion of a mechanical system. Yet, the dominance of this mechanical ontologydid not imply that mechanical models were to be interpreted literally as representations ofphysical reality, but mostly should serve as hypothetical constructions that elucidate the physicalmeaning of the mathematical laws. As with the later work of Maxwell, it could even be enoughto formulate laws within the framework of Lagrangian dynamics, where these laws were stillsubsumed under the principles of mechanical phenomena, but speculation of a mechanical naturewas avoided altogether. [3]

As an excellent illustration of these epistemological debates we consider an interesting ex-change between Ernst Mach and Max Planck taking place in the years 1910-1911. The exchangeis about the status and goal of theoretical physics in an age of radically new conceptions andtheories, and is clearly taking place against a post-Kantian background where any naive realismis out of the question. [5]

Let us start with Mach, the scientist-philosopher who was one of the most influential figuresin the devaluation of the mechanical ontology through a historical and critical analysis of physicaltheories. For Mach, science can only be understood as a product of human evolution, whereabstract scientific theories are viewed as ever more complex ways of man to cope with his natural


environment. In The Science of Mechanics, his famous book on the historical development ofmechanics [6], Mach illustrates in detail how “the task of scientific knowledge now appears as:the adaptation of ideas to facts and the adaptation of ideas to one another” [7, p.31]. Thisimplies that there is a continuous transition from man’s pre-scientific coping with his naturalenvironment to scientific theorizing:

The attitudes and humble everyday skills of the artisan change imperceptibly intothe attitudes and devices of the physicist; and economy of action develops graduallyinto the intellectual economy of the scientist, which can also play its part in thepursuit of purely ideal goals. [7, p.33]

Mach is especially wary of metaphysical speculations in scientific theories; every element of ascientific theory should in the end be brought back to sensations or perceptions. These sensationsare no simple uninterpreted sense-data, but are determined as the “final link in a chain reachingfrom the environment to the central organ of sense” [7, p.39]. In fact, it is one of the goals ofscience to show the complex relation between sensations and the sense organs, a relation that isin no way interpreted within a “naıve-realistic view of the world” [7, p.38].

On the other side of the debate we have Planck, who characterizes, almost directly in re-sponse to the Machian position, the development of physics as a progressive unification of allphysical phenomena “achieved by emancipating the system from its anthropomorphic elements,in particular from specific sense impressions” [8, p.6]. He gives the example of the second law ofthermodynamics, which has, throughout its different formulations, been stripped of all humanassociations1; only in this way can this physical law be given “a firm basis in reality” [8, p.18].Planck defends the idea of a physical world-picture as reflecting real natural events that takeplace in a way that is completely independent from us, devoid of any arbitrary creations ofthe human intellect. The goal of science “is not the complete adaptation of our ideas to ourimpressions, but the complete liberation of the physical world-picture from the individualityof the creative mind” [8, p.26]. Interestingly, Planck concedes that the Machian conception isperfectly coherent, but claims that it misses the essence of natural science as it conceived byany working scientist. Indeed, instead of the sensationalism of Mach, Planck contests that

a constant, unified world-picture is, as I have tried to show, the fixed goal whichtrue natural science, in all its forms, is perpetually approaching. . . . This constantelement, independent of every human (and indeed of every intellectual) individuality,is what we call “the Real”. Or is there today a single physicist worthy of seriousconsideration who doubts the reality of the energy principle? [8, p.25]

Both views on the goal and structure of physics clashed when it came to the scientificstatus of the kinetic theory of gases, according to which the thermodynamic properties of agas were understood as probabilistic laws for the large number of atoms out of which the gassupposedly consists.2 Indeed, whereas Mach, because of his “dislike for hypothetico-fictitious

1In the case of thermodynamics, these antropomorphic elements are the ability to do work or the idea ofirreversible processes, which are, in Planck’s conception, both dependent on or affiliated to human technical skills.It is by Boltzmann’s probabilistic definition that entropy first gets a mathematical meaning independent of anyhuman association.

2In the kinetic theory of gases, associated with Maxwell and Boltzmann, the validity of irreversibility inphysical processes is understood to be probabilistic and thermodynamic entropy is defined in a probabilistic way.In particular, Boltzmann defined the entropy of a certain thermodynamic state as the logarithm of the numberof microscopic configurations that give rise to this state, entailing that a state has a higher entropy as it ismore likely to be realized by the microscopic particles (atoms). The underlying ontology of this theory is againof a mechanical and/or dynamical nature. In fact, it appears that, initially, Planck himself strongly opposedto this atomistic interpretation of the laws of thermodynamics, by questioning the intelligibility of probabilisticexplanation of entropy; it is only in later years that he accepted the Boltzmann definition. [3]


physics” [7, p.35], refuses to accept the existence of imperceptible atoms, Planck acknowledgesthe Boltzmann definition of entropy as the first giving it a “firm basis in reality” [8, p.18].Another point where Mach and Plank disagreed was the relation between electrodynamics andmechanics: whereas Planck suggests that electrodynamics should in the end be subsumed underthe unifying concepts of mechanics, it seems that Mach was a lot more inclined to drop themechanical ontology that dominated nineteenth-century physics. Ironically, around that sametime this particular issue was being resolved by the theory of relativity, and it was precisely thework of Mach who, amongst others, has proven a great inspiration for Einstein [9].

From this brief sketch we get a picture of the epistemological concerns of physicists at theturn of the century. We can say that nineteenth-century physics is characterized by the math-ematization and unification of a whole new range of physical phenomena, and that these higherlevels of abstraction confronted physicists with concerns about the status and methodology oftheoretical physics. These concerns played on the level of philosophical and physical theoriz-ing, and were deemed important both for philosophy and the content of the physical theoriesthemselves. Indeed, our discussion of the Mach-Plank dispute illustrates that this type of philo-sophical debate was not the pass-time of two retired physics professors who have decided todedicate some time to philosophical reflection, but that these discussions were followed by physi-cists and philosophers alike, and their conclusions had a great impact on physical theories.3

The Mach-Plank dispute also shows that the debates could be extremely polarized anduncompromising. On one side of the spectrum, we find an empiricism aiming to avoid any‘metaphysical speculations’ and reduce every physical concept to simple sense impressions, andon the other side, there is a realism for which physicists try to find the concepts that describenature, independently from any human particularities. Although arguments for both positionsare drawn from the history of physics and the experience of the working phycisist, the argumentslack a unified perspective and a specific philosophical perspective seems to be missing. As a result,Mach’s position seems outdated in the light of the theoretical aspirations of physics at the turnof the century, but Planck’s focus on anthropomorphisms is overly simplistic for capturing alldevelopments in theoretical physics. Moreover, both approaches fail to systematically accountfor the motors behind nineteenth-century physics, viz. the mathematization and unificationof nature. It would take a systematic philosophical analysis of the situation to clear up theseepistemological issues.

2.2 Ernst Cassirer and the Marburg school

These philosophical concerns and debates taking place within natural science exercised a greatinfluence on the development of academic philosophy as well. In a time where philosophy ofscience was dominated by speculative ‘Naturphilosophie’, some scientists took recourse to theoriginal work of Kant to look for a philosophical articulation of science. A central figure in thisdevelopment was Hermann von Helmholtz, who investigated to what extent our perception ofexternal reality is the result of a process whereby neural stimulations are made intelligible to thehuman mind. Helmholtz framed these investigations in Kantian terms, where transcendentalphilosophy was transformed into physiognomy. This scientific return to Kant was welcomed byneo-Kantian philosophers, but they proposed a much more systematic reappraisal of the workof Kant. [10, 11]

3The strategy of this section has been to illustrate this importance by highlighting one specific philosophicalexchange between well-known physicists. A complete argument would have to include a discussion of physicistssuch as Helmholtz, Hertz, Boltzmann, Poincare, Duhem, etc., who wrote important, essentially philosophical,works on similar issues.


The neo-Kantians inherited the fundamental Kantian insight that the object of knowledge isnot an external reality existing independently from our judgement – a transcendent realm of realobjects in the realist conception, or uninterpreted sense-data in the empiricist tradition – butthat every object is ‘constituted’ as it is conceptualized within a certain a priori logical structure.This structure is not to be understood in a psychological or physiological way – as Helmholtzdid – but should be pictured as a set of logical ‘faculties’ or functions, that allow to fix senseimpressions into a conceptual space. Without these faculties, sense impressions are devoid ofany objective meaning; they are a priori, in the sense that they come before experience. Thisapproach of thinking about knowledge is called ‘transcendental’, indicating that the conditionsof possibility for objective judgements are up for philosophical analysis.

Famously, Kant explored this approach with respect to Newtonian physics. Indeed, as hewas confronted with a confusion of different metaphysical interpretations of classical mechanics,he investigated how the Newtonian paradigm of objective knowledge was made possible by thefaculties of sensibility, understanding and reason that are involved in the act of judgement. Kantarrived at a tripartite structure with (i) the faculties of pure intuition (essentially the intuitionsof space and time), (ii) the faculties of understanding (logical structures or forms of judgement),and (iii) the faculty of reason (providing regulative principles or ideals). The principles ofphysics (the categories) then arise when the pure forms of judgement are given spatio-temporalcontent in relation to the pure forms of intuition (through the transcendental schematism of theunderstanding), whereas the regulative ideals serve as the non-determinate guiding principlesthat drive the progress of science.

Although the neo-Kantians reinvigorated the transcendental approach of Kant, they did nottake over this structure. In particular, they refused to accept a dualism between, on the onehand, a discursive (conceptual) faculty of understanding, and, on the other, an intuitive (non-conceptual) faculty of sensibility. Instead, they wanted to understand the structure that makesobjective knowledge first possible, in purely logical or conceptual terms alone.4 It is by applyinglogical concepts to experience, that experience is first constructed; talk about a reality (or puresensibility) existing before the logical faculties is non-sensible.

In the Marburg school of neo-Kantianism, ‘experience’ was understood exclusively in sci-entific terms; it was scientific experience for which they wanted to analyze the conditions ofpossibility.5 Indeed, whereas Kant seemed to posit something like a persistent self (a tran-scendental unity of apperception) as the fundament on which judgement was constructed, theMarburgers saw the body of science, with its rules, methods and procedures, as responsible forthe constitution of experience [10]. The strategy for exposing this function of science is containedin Cohen’s ‘transcendental method’, which takes the best physical theories of the day as startingpoint and seeks to explain the possibility of experience by identifying the a priori laws that arepresent in these theories. Put differently, (scientific) experience is given as a task to philosophy,in the sense that philosophy strives to articulate the principles of mathematical natural sciencethat generate objects of possible experience. This method is essentially historic, in that philoso-phers in different periods of the history of science will be faced with a different science, and will,consequently, arrive at different conceptions of what constitutes objective experience. [13]

4The reason for this refusal to allow an intuitive faculty of sensibility is, in part, due to the discovery ofnon-Euclidian geometries. Indeed, for Kant, the a priori structure of space that was generated by the faculty ofsensibility had an a priori Euclidean structure, and geometry derives from this faculty of intuition. The formulationof different, yet consistent, geometries suggested that this could no longer be true and led the neo-Kantians tobelieve that geometry (and mathematics in general) is due to the logical faculties of understanding alone.

5It was their claim that this was also the focus of Kant himself: the Critique of Pure Reason was supposedto expose the a priori structure of classical mechanics. This Kant interpretation has recently been revisited withgreat success [12].


To a large extent, Cassirer takes over the methodology of Cohen, and carries it further.Cassirer’s first major work, Das Erkenntnisproblem in der Philosophie und Wissenschaft derneueren Zeit, first published in 1906, traces the history of science and philosophy from theperspective of Marburg Neo-Kantianism. In particular, Cassirer discusses the ‘mathematizationof nature’, or the application of ideal mathematical structures to an empirically given nature, asthe decisive achievement of the scientific revolution. The book also contains a lengthy discussionof Kant, where, in the line of Cohen, Cassirer contests the separation of the faculties of intuitionand understanding and proposes to replace these faculties by a fundamental creative activity ofthought that progressively generates the object of natural science. Space and time then arise notas expressions of a separate, non-discursive intuition, but as the first products of this creativityof thought. Importantly, and in clear disagreement with Kant and the logicist tradition of Fregeand Russell, formal logic is not fundamental but appears as an abstraction from ‘transcendentallogic’, where the latter denotes the unitary process of constructing scientific knowledge. InSubstance and Function this transcendental logic will be worked out more systematically, aswell as the idea of the constitution of the object of science through a progressive determinationof the a priori principles of mathematical physics. [14]

2.3 Substance and function

A systematic philosophical study of theoretical physics by Cassirer appeared in 1910 in his workSubstance and Function.6 From the two previous sections we can deduce a number of problemsor concerns that Cassirer is aiming to address in this work. Firstly, there are the epistemo-logical difficulties that physicists faced in the light of the mathematization and unification oftheoretical physics in the nineteenth century. As an answer to the rather negative sensational-ism of Mach, the challenge is to give a positive account of this development that goes beyondPlanck’s account of the elimination of antropomorphuous elements. Secondly, in the line of neo-Kantianism, Cassirer sets out to provide a thoroughly systematic philosophy of science, wherehe ultimately wants to show how mathematical physics allows to give a fixed, determinate andunified meaning to objective knowledge. This should stand in stark contrast to metaphysicalor speculative accounts of science, but also to abstractionism, empiricism or logicism. Thirdly,this philosophy of science has to be historical, such that it could fix the rational progress ofscience throughout its historical development. Ideally, the progress of physics would confirm theprimacy of transcendental philosophy in the spirit of Kant.

2.3.1 A new logic

We have seen that Cassirer, in the line of Marburg neo-Kantianism, wants to give a purely logicalcharacterization of the a-priori structure of objective knowledge, but it appears that traditionallogic does not provide the tools to make this happen. Therefore, Substance and Function beginsby advocating an alternative methodology of logic.

He starts with a discussion of the basic structure of traditional logic, where different thingsor objects are collected in classes in virtue of some common feature, giving rise to a genericconcept which comprehends all the determinations in which things in the same class agree. Itis through the process of abstraction that these concepts can rise up from the multiplicity ofindividual things. At this point, however, Cassirer expresses his doubts whether this procedureof forming concepts through abstraction can lead to the sharp and unambiguous determinations

6The book appeared in 1910 under the title Substanzbegriff und Funktionsbegriff: Untersuchungen uber dieGrundfragen der Erkenntniskritik, and was translated to English in 1923 together with a monograph on thetheory of relativity (see Sec. 2.4).


we are used to in science. Indeed, it seems that in this rule of concept formation there is alwaysa tacit reference to another intellectual criterion. In the system of Aristotle, for example, theambiguity of the logical doctrine of abstraction is supplemented with a metaphysical theory,by which the formation of generic concepts ends in the discovery of the real essences of things.This form of logic has been transformed and refined7, but it remains that it is only through afixed thing-like substratum that logical concepts can obtain their application. It is precisely thisfixation on thing-concepts that Cassirer is out to contest and that he wants to replace with aform of logic based on relation-concepts.

The inspiration for this move comes from the nineteenth-century reshaping of mathematics.Indeed, in mathematics it is clear that the method of abstraction cannot characterize or justifythe necessary concepts, because in the definitions of pure mathematics another realm of objectsis created that is in no direct way connected to the world of ‘things’.8 Moreover, mathematicalconcepts or formulas have the feature that, as they become more general, they become moredeterminate, and that the more special cases of a given mathematical formula follow from thegeneral case. This relation between the universal and the particular stands in contrast to therelation of abstraction, where the more general concepts are stripped from sharp determinations.In mathematics, the most general concepts are also the richest.

This leads Cassirer to his fundamental idea of a logic based on relation-concepts, where (i)the individual is conceived as a determinate step under the rule of a more general concept, and(ii) these concepts are serial, in the sense that they generate a series of objects by successiveapplications of the same conceptual rule. Just as in mathematics, a logical concept “represents auniversal law, which, by virtue of the successive values which the variable can assume, containswithin itself all the particular cases for which it holds” [1, p.21]. This opens up a new paradigmfor a methodology of logic, where all concept formation is connected with a sequence generatedby functional relations between the members of this sequence – the form and meaning of theconcept are exhausted by this generating relation. In addition, every element falling under agiven concept only has meaning as an element within the series that is generated by the concept;an object has no independent ‘existence’ – not even in a logical or mathematical sense. AsCassirer sets out to show in the rest of the book, it is only through this logical methodologythat the determinate character of scientific concepts can be understood.

2.3.2 The serial forms in mathematics

Before embarking on the analysis of the natural sciences, Cassirer first considers the conceptualstructure of pure mathematics. In a parallel fashion he shows how the concepts of arithmeticand geometry have developed into forms that confirm the relation-based methodology.

In both cases it is not immediately clear that a founding in logic is actually needed for themathematical concepts. Indeed, both numbers and geometrical shapes could be taken to bearrived at by abstraction from experience. The response from Cassirer to this sensationalismprovides us with a clear statement of what is at stake in his analysis:

Thus what is here given is always only a temporally limited and determined reality,not a state which can be retained in unchanging logical identity. It is the fulfilment

7As an example of this metaphysical transformation of the same logical methodology, Cassirer discusses thepsychological epistemology of Berkeley: “While formerly it had been outer things that were compared and out ofwhich a common element was selected, here the same process is merely transformed to presentations as psychicalcorrelates of things.” [1, p.9]

8As we will see, the same is true for theoretical physics, since “these concepts of physics also are not intendedmerely to produce copies of perceptions, but to put in place of the sensuous manifold another manifold, whichagrees with certain theoretical conditions” [1, p.14]


of the demand for this latter, however, which constitutes all the meaning and valueof the pure numerical concepts. [1, p.33]

The ‘meaning’ and ‘value’ correspond to the universal applicability to every individual case,as a condition for judgements concerning individuals. As will become clear later on, thesemathematical concepts will indeed serve as conditions of possibility for the arrangement ofindividuals into an inclusive whole. In this sense, the logically determinate character as relation-concepts is a prerequisite for the role these mathematical concepts will play in mathematicalphysics.

The challenge of founding arithmetic in relation-concepts was met by the work of Dedekind,who founded all arithmetic definitions and propositions in the concept of progression. Indeed,starting from the concept of a series (i.e. a first a member and a relation of succession) theinteger and fractional numbers as well as addition and multiplication can be developed, withoutever taking recourse to the relations of concrete measurable objects. The framework can evenbe extended to include irrational, imaginary, and transfinite numbers. What is establishedby this logical construction is “a system of ideal objects whose whole content is exhausted intheir mutual relations” [1, p.39]. Indeed, the essence of number is exhausted by the conceptualrule that defines a structured manifold: no number is anything more than a place within thisconceptual whole.

The rationale behind Cassirer’s founding of arithmetic becomes clear when it is opposed tothe attempt by Frege and Russell to reduce number theory to logic through the use of classes.Although Cassirer admits that this reduction is a great advance over sensationalistic theories,it cannot be satisfactory given the function the number concept has to play in the whole ofknowledge. Again, we see that mathematical concepts are supposed to play a constitutive role,which the determination of number by the equivalence of classes cannot do. Instead, Cassireraims at defining numbers from “a purely categorical point of view” [1, p.54], without takingrecourse to thing-like concepts such as classes. Only in this way can the numerical concepts beapplied in the mathematical sciences.

The same motives drive Cassirer’s discussion of geometry, for which the development of apurely functional conceptualization has been more involved. This development is presented as aprogressive evolution starting with the geometry of the ancient Greeks, through Cartesian anddifferential geometry, and resulting in the formulation of projective geometry. In this form, forthe first time, “we start from an original unit from which, by a certain generating relation, thetotality of the members is evolved in fixed order” [1, p.88]. Importantly, Cassirer construes thishistorical development of concepts as a process with an inner necessity. Indeed, in this processthe formulation of group theory forms the final “conclusion to a tendency of thought, which wecan trace in its purely logical aspects from the first beginnings of mathematics” [1, p.94].9

In the last few paragraphs of the third chapter, Cassirer briefly discusses how the purelyfunctional concepts of geometry are to be applied to empirical reality, and, more specifically, howto decide between different geometries in their application to real space. With this discussion,however, we have left the realm of the pure functional concepts of mathematics and embarkedon the critical analysis of mathematical physics.

2.3.3 The concepts of natural science

The exposition of the conceptual structure of mathematics shows the clearest and most perfectexample of the “logical nature of the pure functional concept” [1, p.112]. Yet, it is only througha critical analysis of the natural sciences that we can find a definitive formulation of the problem

9This idea of interpreting the historical development of geometry as a progression towards an ideal end-pointwill be worked out further in Sec. 2.3.5.


of knowledge and, consequently, the basic meaning of the functional concept; it is only by layingbare the transcendental meaning of the functional concept that it finds its true import. Sothe real challenge that Cassirer takes up is showing that he can make sense of the historicalevolution of the natural sciences – in particular, mathematical physics – within the frameworkof the functional concept. And the ambitions towards this effect are quite high: Cassirer wantsto capture the whole structure of physics with only a small number of fundamental concepts;the most important are the concepts of space and time, substance and energy.

With respect to the physics of space and time this fundamental philosophical question hasbeen clouded by the metaphysical discussion on the absolute or relative nature of space-time.In the background, however, another question emerges that is of epistemological importance:

The question continually arises, whether in the foundation of mechanics we have toassume only such concepts as are directly borrowed from the empirical bodies andtheir perceptible relations, or whether we must transcend the sphere of empiricalexistence in any direction in order to conceive the laws of this existence as a perfect,closed continuity. [1, p.172]

Different attempts have been undertaken to ground the physical meaning of space-time deter-minations in purely empirical terms. In the system of Mach [6], for example, it is the influenceof the mass distribution of the universe – the fixed stars – that generates the law of inertia forthe earthly bodies. Looking at the meaning and function of the law of inertia in the system ofmechanics, however, no reference is made to these fixed stars. Indeed, we can easily transform toother frames of reference and lose the connection with the fixed stars, without the law of inertialosing its intelligibility. So the concept of uniform motion is only related to the “ideal schemataoffered by geometry and arithmetic” [1, p.175] and only functions as such in physical theory.The grounding of the law of inertia in empirical terms enters the system of physics only throughan external demand inspired by empiricism.10 This demand has inspired other approaches forfounding the existence of inertial frames in sensuous objects11, but always it appears that it isnot so much the existence of these objects but rather the assumption of their existence thatvalidates the use of mechanical concepts. But then it is clear that the meaning of these physicalconcepts was already established beforehand in an ideal, mathematical construction. The searchfor ‘things’ existing in the sensuous world for grounding e.g. the law of inertia involves a circle,because inertia and the other principles of mechanics are already tacitly recognized beforehandas universal mathematical principles.

This implies that the real philosophical problem with respect to space and time concerns theform and function these principles exhibit in the conceptual structure of theoretical physics. Inline with Cassirer’s basic logical convictions, the logical character of space and time is that of

10 At this point, it proves worthwile to further pinpoint Cassirer’s view on the philosophy of Ernst Mach. Indeed,when describing the “scientific ideal of pure description” Cassirer writes

The goal of this philosophy of physics would be reached, if we resolved every concept, which entersinto physical theory, into a sum of perceptions, and replaced it by this sum [. . . ]. [1, p.114]

The question Cassirer asks is whether this conception of physics is indeed a description of the actual status ofphysics or “confused with a general demand that is made of these theories” [1, p.115].

The answer to this question can only be won by following the course of physical investigation itselfand considering the function of the concept that is involved directly in its procedure. [1, p.115]

With this statement, Cassirer explicitly places himself in the debate for which we have taken the Mach-Planckdispute as an example [Sec. 2.1]. We see that Cassirer takes the philosophy of Mach as a demand on how physicsshould be structured, a demand that is out of touch with the way actual physics has evolved.

11Cassirer discusses the “fundamental body” of Streintz or the “body alpha” of Neumann as attempts to defineinertial frames through the introduction of some special body or object in empirical reality, with respect to whichinertial movements can be defined.


“systems of relations in the sense that every particular construction in them denotes always anindividual position, that gains its full meaning only through its connections with the totalityof serial members” [1, p.172]. Indeed, a particular position in space only gains meaning withreference to other positions, or more generally, a spatial manifold, and every moment of time isdetermined with reference to an earlier or later contrasted with it. Space and time thus appearas serial concepts, where individual space-time points only have meaning as elements within thespace-time manifold.

The same goes for matter and ether, the two physical ‘substances’ that are supposed tocapture all physical processes taking place inside the space-time framework. Cassirer describesa number of historical transformations, where the concept of matter has been stripped from allsensuous content and has evolved into a purely logical center of possible relations. Indeed, theidea of a point mass makes it possible for matter to be a subject of physical processes describedby purely mathematical relations (i.e., differential equations). The concept of the ether equallyexpresses the connections between different physical processes, and “all that physics teaches ofthe “being” of the ether can, in fact, be ultimately reduced to judgements about such connections”[1, p.163]. So again, the content of the physical concepts of matter and ether are exhausted byconsidering their logical place in the universal schemata, in which the relations of empiricalreality can be first represented in a scientifically determinate fashion.12

A last concept with special significance is that of energy. We have seen that an empiricalphenomenon only becomes an object for knowledge when it is ascribed a definite place in themathematical manifold of serial concepts, but, for Cassirer, the real task of knowledge consistsof placing the different series within a unified system. This requires a principle which enables toconnect different series according to an exact numerical scale and a constant numerical relationgoverning the transition from one series to the others. This scale is provided by the conceptof energy, which, starting from the famous equivalence of motion and heat, has progressivelyincluded more domains of physics. Energy represents a common series for all physical processes,making possible an objective correlation according to law in which all physical contents (light,heat, motion, etc.) stand. It signifies an intellectual point of view, “from which all thesephenomena can be measured, and thus brought into one system in spite of all sensuous diversity”[1, p.192].

Again, it would be a mistake to think that physics has discovered a new self-existent thing.Instead, energy simply appears as the expression of an exact numerical relation that pertainsto physical processes, and the meaning of the energy concept is exhausted by that numericalequivalence. In this respect, energy as a unifying concept seems to have an epistemologicaladvantage over the attempts at unification within the mechanistic world-view. Indeed, under theconcept of energy, two physical processes “are the “same” not because they share any objectiveproperty, but because they can occur as members of the same causal equation, and thus can besubsituted for each other from the standpoint of pure magnitude” [1, p.199]. Energism showsthat unification is not necessarily connected to analyzing things and processes into their ultimateintuitive parts, as a mechanical reductionism would do.13

12In this connection, Cassirer points to the fact that “the exactitude and perfect rational intelligibility ofscientific connections are only purchased with a loss of immediate thing-like reality” and that “it must appearas a genuine impoverishment of reality that all existential qualities of the object are gradually stripped off.” [1,p.164] These remarks echo the view of Planck [Sec. 2.1] associated with the banning of antropomorphous elementsin physical science, but gain a positive philosophical significance with Cassirer.

13Cassirer explicitly states that he does not favor energism, as “[t]he conflict between the two conceptions canultimately only be decided by the history of physics itself; for only history can show which of the two views canfinally be most adequate to the concrete tasks and problem” [1, p.202].


2.3.4 Physical theory and experience

In contrast to the purely mathematical concepts, the concepts of natural science should be insome way connected to experience or experiment. How does Cassirer characterize this relation?

At this point, the work of Duhem [15] is taken into account. Duhem shows how a scientificexperiment is mediated by a number of intellectual moves and that its interpretation depends ona number of fundamental theoretical assumptions. In fact, instead of the practical instrumentthe physicist substitutes an ideal instrument from which accidental defects are excluded. Itfollows that measurement is never a purely empirical procedure, but the result of conceptualoperations. Cassirer explains that it is the function of the scientific concept that makes possible“the transition from what is directly offered in the perception of the individual element, to theform, which the elements gain finally in the physical statement.” [1, p.143] More specifically, thesensuous qualities of things are brought under the serial concepts of mathematics: to measurea physical phenomenon is to transform it into a serial, numerical determination.

In this regard, physical concepts are conceived as apperceptive concepts that are necessaryfor empirical knowledge in general. Indeed, a theoretical structure provides a scheme into whichspecific observations can be fitted and, by this procedure, gain a fixed form and assume clearlydefined physical properties.

Even before its individual value has been empirically established within each of thepossible comparative series, the fact is recognized, that it necessarily belongs tosome of these series, and an anticipatory schema is therewith produced for its closerdetermination. [1, p.150]

Cassirer calls this “a type of transcendence” [1, p.281], where a particular given impressionbecomes a mathematical symbol and designates a fixed physical property in a larger theoreticalstructure. This shows that the body of physical concepts is constitutive for a scientificallydeterminate conception of reality, and that there are no ‘bare facts’ that any scientific theorycan compare to.

This also implies that physical concepts are not tested in isolation, but that their validity isevaluated by their function in a theoretical complex; it is these theoretical complexes that arejudged on their correctness as they unite the totality of experience into an unbroken unity. Theagreement between the observations and the system of deductions always remains an approxima-tion, since the mathematical structure of pure thought is always only postulated to correspondto physical reality.14 Indeed, “[w]e inscribe the data of experience in our constructive schema,and thus gain a picture of physical reality; but this picture always remains a plan, not a copy,and is thus always capable of change” [1, p.186].

Crucially, however, the meaning of the mathematical conceps and principles is not dependenton their application to physical reality. It is precisely because they are exactly determined asmathematical principles, that physical concepts such as space and time have the fixity andexactness that is required for them to function in physical theory. Indeed, space and time actas “pure functions, by means of which an exact knowledge of empirical reality is possible” [1,p.182]. They are first considered in intellectual abstraction and only then generate a “generalschema for possible changes in general” [1, p.182], and it is in this application to physical realitythat it is first decided whether real movements in fact conform to these determinations. In this

14Cassirer refers to Poincare, who has described these mathematical constructions as conventions when theyare introduced to survey physical facts more easily. For Cassirer, the characterization of the ideal conceptualcreations as conventions recognizes that “thought does not proceed merely receptively and imitatively in them,but develops a characteristic and original spontaneity” [1, p.187]. In the following section [Sec. 2.3.5] we willexplain that the concepts of theoretical physics are more than conventions.


respect, it is essential for scientific knowledge of nature that the empirical realization of themathematical concepts can shift, yet their logical meaning and necessity remain intact.15

Finally note that the scientific motive of unification seemingly confronts Cassirer with aparadox: It appears that every experimental observation will always demand for a growingnumber of natural laws in order to capture the observation in its peculiarity. Indeed, it will neverbe possible to isolate a physical process such that only one law will capture the phenomenoncompletely and exactly. It is at this point that the full power of the functional concept showsitself. In contrast to the generic concept of traditional logic, where every abstraction correspondsto a stripping of determination, the functional concept becomes more determinate in its contentand application to the particular as it becomes more universal. So in order for the physicalconcepts to capture the particularity of experience in growing exactness, a progression towardsmore universal concepts is needed. Indeed, every universal relation necessarily contains a growingnumber of more particular relations and “has a tendency to connect itself with other relationsto become more and more useful in the mastery of the individual.” [1, p.255]. Put differently,

[t]he advance of experiment goes hand in hand with the advancing universality ofthe fundamental law, by which we explain and construct empirical reality. [1, p.258]

2.3.5 The progress of scientific knowledge

Cassirer presents this progression towards more universal concepts in an explicitly historical way.Yet, if this historical process is to have an internal rationality behind it, it appears essential thatrelations that are progressively established be compatible with each other. According to Cassirer,“this compatibility is assured in principle by the fact, that the determination of the particularcase takes place on the basis of the determination of the general case, and tacitly assumes thevalidity of the latter.” [1, p.255] But, and this seems absolutely crucial, this process of growinguniversality can never be completed, and the fundamental laws of science, which at a certainpoint in time seem to represent the final form of all empirical processes, will at a later stageonly serve as the material for further consideration.

All scientific thought is dominated by the demand for unchanging elements, whileon the other hand, the empirically given constantly renders this demand fruitless.We grasp permanent being only to lose it again. From this standpoint, what we callscience appears not as approximation to any “abiding and permanent” reality, butonly as continually renewed illusion, as a phantasmagoria, in which each new picturedisplaces all the earlier ones, only itself to disappear and by annihilated by another.[1, p.266]

The sceptic might argue that this is the road to an epistemological relativity: every scientificpicture of the world, which makes an objective conception of this world first possible, will alwaysbe replaced by another picture that makes the previous conception worthless and arbitrary. Thisargument is refuted by Cassirer as he points out that this succession of scientific conceptionsmust “proceed according to a definite principle of methodic advance” [1, p.267]. Whenever anobservation is found not to agree with the body of scientific determinations, the scientist willfirst look for variations of the less universal laws. When this seems impossible, more fundamen-tal laws will have to be modified. But, “this transition never means that the fundamental formabsolutely disappears, and another absolutely new form arises in its place.” Indeed, this newform “must contain the answer to questions, proposed with the older form.” By this feature, alogical connection is established and a “common forum of judgement” is opened, “to which bothare subjected.” [1, p.268] In every transformation, a certain set of principles is always preserved,

15Cassirer refers to the work of Hertz [16] as the clearest expression of this relation of theory and experience.


because the reason for this transformation is precisely the preservation of these principles; with-out a fixed logical standard, it would make no sense to transform our scientific body in responseto some observations, because there would be no scientific observation. So, in order to makesense of the progression of scientific principles, we need to assume that there is “an ultimateconstant standard of measurement of supreme principles of experience in general.” [1, p.268] Itis the task of the critical theory of experience to search for this ‘universal invariant theory ofexperience’:

The goal of critical analysis would be reached if we succeeded in conceptually definingthose moments, which persist in the advance from theory to theory because they arethe conditions of any theory. At no given stage of knowledge can this goal be perfectlyachieved; nevertheless it remains as a demand, and prescribes a fixed direction tothe continuous unfolding and evolution of the systems of experience. [1, p.269]

These ultimate logical invariants or “invariants of experience” are called a priori by Cassirer,because they are contained as necessary premises in every judgement on empirical facts.

We have seen that the objects of physics arise as we transform experience to the demandsof theoretical concepts; through the different conceptualizations science gains different objecti-fications of physical reality. But these represent different stages in the fulfillment of the samefundamental demand of objectification. It is through the realization of this demand (i.e. theidentification of the invariants of experience) that the real meaning of the concept of the objectis established. So it is this fundamental demand or search to fix the object of physics in itsfull determination that, despite the impossibility to attain it in principle, drives the progress ofscience.

2.4 Cassirer on modern physics

We have seen that Cassirer was very ambitious in Substance and Function as he wanted to laybare the conceptual structure of all of theoretical physics in a unified way. The developmentsin physics, however, were soon to demand even more of his neo-Kantian approach. It speaks infavor of Cassirer’s framework – tailored originally to classical nineteenth-century physics – that itcan be applied to the revolutionary developments of twentieth-century physics without changingmuch of its basic convictions. Indeed, the revolutionary principles of relativity and quantummechanics are seen as confirmations of the central features of his epistemology, viz. (i) thefunctional, non-substantialistic (relational) meaning of physical concepts, (ii) the constitutivefunction of these concepts for a scientific experience of nature, and (iii) the continuity in theprogress of physics by the regulative ideals of unity under functional laws.16

2.4.1 The theory of relativity

In the beginning of the twentieth century the world of physics was revolutionized by Einstein’sspecial and general theories of relativity. Not only did these theories change the physicist’sconception of space and time drastically, but they also had great implications on the philosophyof science. In particular, the authority of Kant on the foundations of space-time physics wasseverely shaken by the general theory of relativity – it appeared that Euclidean geometry didnot provide a correct description of empirical space, a fact that Kant took to be true a priori.

16We will discuss two of Cassirer’s works that are explicitly focussed on modern physics. This exhausts all ofCassirer’s later writing on physics, because other later works follow the ideas of Substance and Function ratherclosely where the philosophy of physics is concerned. [17]


Because Cassirer had always placed himself in the philosophical tradition of Kant, it provedvital that he could incorporate Einstein’s theories within his project, a challenge that he met ina monograph on relativity theory.17 In this work, Cassirer argues that the theories of Einsteindo not provide a refutation of the philosophy of Kant, but, instead, are a new confirmationthat only transcendental philosophy can provide the correct explanation of the structure andmeaning of theoretical physics.

Cassirer begins by describing the advent of relativity theory as a critical revaluation of thesystem of physics. Indeed, after the experiments that had made a unified conception of physicalphenomena impossible using the laws of nineteenth-century physics – the Michelson-Morleyexperiment is the most famous – there was a need for a critical examination and correction ofe.g. the classical conceptions of space and time, the concept of matter in mechanics, and theether concept in electrodynamics. According to Cassirer, this intellectual process was continuousin the sense that the same demand for constancy and unity in nature had been at work indeveloping the old physics and overthrowing it. The result was, as always, a further liberationfrom the “presuppositions of the naively sensuous and “substantialistic” view of the world” [1,p.386] in favor of a unified system of functional space-time determinations, where space andtime themselves have been further stripped from their thing-like meaning.

What came in the place of the classical notions of space and time are the pure forms ofcoexistence and succession, which only have a meaning as serial concepts appearing in thedescription of physical phenomena. Indeed, in relativity theory physical processes are describedby world-lines in the four-dimensional space-time manifold, a manifold that presupposes theserial forms of space and time. Although the space and time coordinations are mixed up fordifferent observers, as dictated by the equations of relativity, the two functions of coexistenceand succession remain at work in every space-time description of a physical process. Indeed, thetheory of relativity proposes the epistemological insight that neither pure space nor pure timehave an existence in themselves, but only in their unified application under the mathematicallaws of relativity to physical phenomena do space and time retain empirical meaning.

Then, of course, the problem remains of making sense of the non-Euclidean structure ofthe space-time manifold. First of all, Cassirer opposes any empirical grounding of geometry,because the meaning of geometrical concepts is exhausted by their function in the ideal systemof geometry, and they possess no immediate correlate in the world of existence. Moreover, thegeometrical axioms are never to be regarded as concerning things or relations of things in reality;instead, they should be evaluated to what extent they, in their totality, constitute the physicalobject and make physical knowledge possible. But, Cassirer argues, this is exactly what relativitytheory has realized: The ontological meaning of geometry has lost all meaning, and the onlyquestion that remains is which geometrical system should be used for the interpretation of thephenomena of nature and their dependencies according to law.18 Indeed, the theory of relativityprovides a mathematical framework for space-time determinations, making possible the exactformulation of certain physical relations such as the laws of gravitation or electromagnetism,without attaching any existence to the space-time manifold itself.

17Cassirer published the monograph as Zur Einsteinschen Relativitatstheorie in 1920, which was translated toEnglish together with Substance and Function in 1923.

18At this point, Cassirer evokes the philosophy of Kant, and makes clear that pure intuition has no role to playin the realm of knowledge of the empirical and the physical. Indeed, it is only the rules of understanding thatgive the existence of phenomena their synthetic unity. In this regard, it is only a small step beyond Kant to alsotake into account non-Euclidean axioms.


2.4.2 Quantum mechanics

Possibly the conception of quantum mechanics revolutionized the physical way of thinking inan even more drastic way, and, in contrast to relativity theory, continues to confront physicistsand philosophers with interpretational problems. In his Determinism and Indeterminism inModern Physics [18]19 Cassirer takes up the challenge of incorporating quantum theory withinhis philosophical framework.

Cassirer first makes the important point that the idea that quantum theory implies a drasticdeparture from the classical idea of causality, is essentially misguided. He traces back the conceptof causality as it functioned in classical theory, characterizing it as a regulative principle of ageneral conformity to law. The Laplacean ideal of causal determination, often thought as thecanonical formulation of classical causality, is identified as a metaphysical fiction by Cassirer. Ifunderstood properly, the principle of causality functions both in classical physics and quantumtheory. [19]

The real departure from classical physics is situated on the level of the concept of a physicalstate. Classically, the “state of a thing in a given moment is completely determined in every wayand with respect to all possible predicates”, a conception that was thought of as “a definitionof what we are to understand by the “reality” of a thing” [18, p.189]. In particular, it isthe spatiotemporal determination of an empirical object that is considered as the true citerionof its existence. This classical notion is drastically transformed in quantum theory by thesuperposition principle and the uncertainty relations, where e.g. an electron no longer hasa determinate location in space or definite amount of energy; the classical (substantialistic)notion of “thing” loses meaning. In this respect, the formalism of quantum mechanics is a newstep in the progressive functionalization of physical concepts: the quantum formalism “was notcreated for the description of things and states but refers to the representation of the behaviourof physical systems” [18, p.192].

Just like relativity theory, quantum theory gives a physical realization of an essentiallyepistemological insight: By abandoning the notion of absolute determination of the classicalthing, quantum theory formulates mathematically strict conditions for physical knowledge ofnature. Of course, this is not a skeptical conclusion, in the sense of having only limited accessto an external reality, but leads to the realization that quantum theory “prescribes limits tothe being which we can ascribe to natural things, and not the reverse” [18, p.194]. Thus,quantum mechanics makes the explicitly transcendental conclusion that there are conditions ofaccessibility necessarily bounding the object of experience. [19]

2.5 Michael Friedman

Around the time that Cassirer wrote his monograph on the theory of relativity, a few moreradical philosophers such as Schlick and Reichenbach wrote down their own philosophical con-clusions based on Einstein’s revolutionary theory. Although these works were quite close toneo-Kantianism originally, the authors quickly diverged from the Kantian project and formed,under the influence of Russell and Wittgenstein, a new school of thought that would go underthe name of logical positivism. Mainly because of the dominance of logical positivism, the neo-Kantian philosophy of science of Cassirer has not received a lot of attention, although there havebeen a few exceptions [20].

Recently, however, the philosophy of Cassirer was revived in the work of Michael Friedman.Although the philosophical concerns and challenges have shifted considerably, Friedman’s work

19The book was first published in German as Determinismus und Indeterminismus in der modernen Physik in1936, but was translated into English only in 1956.


shows shat the project of Cassirer can still provide an important inspiration for contemporaryauthors. As Friedman puts it himself,

. . . I construct a narrative depicting both the development of the modern exact sci-ences from Newton to Einstein and the parallel development of modern scientificphilosophy from Kant through the early twentieth century. I use this narrative tosupport a neo-Kantian philosophical conception of the nature of the sciences in ques-tion – which, in particular, aims to give an account of the distinctive intersubjectiverationality these sciences can justly claim. [21, p.11]

2.5.1 The dynamics of reason

In this section, we will discuss how Friedman has incorporated Cassirer’s ideas in his Dynamicsof Reason [22], and assess to what extent Cassirer’s work can still provide viable insights incontemporary philosophy of physics.

Scientific philosophy after Kuhn and Quine

The first challenge to a contemporary neo-Kantian approach to the philosophy of science isthe epistemological holism as formulated by Quine in his Two Dogmas of Empiricism [23]. Inthis view, our knowledge should be described as a vast web of interconnected beliefs, whichimpinges on experience only along the edge, or “like a field of force whose boundary conditionsare experience” [23, p.39]. The body of scientific knowledge stands before the ‘tribunal ofexperience’ as a whole. In this web, some beliefs are closer to the periphery of experience thanothers, in the sense that they are more likely to be chosen for revision in the light of recalcitrantexperience. Simple statements about physical objects are of this kind, because they can beeasily revised in the light of experience without shaking up the whole system of beliefs. But,importantly, this does not imply that they are of another kind than the more entrenched beliefsabout e.g. arithmetic; it is only through pragmatic inclinations20 that these beliefs are held tobe more fundamental. On an epistemological level, the difference between analytic and syntheticstatements is no longer meaningful: all beliefs are equally empirical.

The second challenge consists of the conceptual relativism that has gained momentum inthe aftermath of Kuhn’s The Structure of Scientific Revolutions [24]. Based on the historicalstudies of Kuhn21 – showing the absence of rational rules governing the revolutionary transitionsbetween scientific paradigms or conceptual frameworks – it is argued that the only viable notionof scientific rationality is a local or contextual one, where non-rational factors such as persuasionand commitment within some particular social community determine the acceptance of a certainbody of scientific knowledge.

Friedman’s work can be read – and is presented as such in his Dynamics of Reason – asa direct response to these two challenges of contemporary philosophy of science. He proposesan approach that, on the one hand, doesn’t flatten out the conceptual structure of scientificknowledge, and, on the other, retains the inherent rationality of scientific progress. Let us seehow.

20The motivation of Quine for this view is much in the spirit of Mach:

As an empiricist I continue to think of the conceptual scheme of science as a tool, ultimately, forpredicting future experience in the light of past experience. [23, p.41]

21With respect to contemporary physics, the study of Pickering [25] on the historical development of particlephysics is particularly relevant.


The relativized a priori

In a direct response to Quinean holism, Friedman makes the case that the different parts ofphysical theories cannot be viewed as symmetrical elements of a larger conjunction, which canthen equally face the tribunal of experience. Instead, he works out the notion of relativizeda priori principles, which cannot be tested directly in experience, but rather define the spaceof empirical possibilities for a certain theory. This notion is illustrated for the case of threespace-time theories, i.e. Newtonian mechanics, special relativity and general relativity. Withinthese theories, three asymmetrically functioning parts can be distinguished. The first is thepart of the mathematical theories, representations or structures, describing the spatio-temporalframework in question (Euclidean space, Minkowski space-time and Riemannian space-timemanifolds, resp.). The physical or empirical part (universal gravitation, Maxwell’s equationsand Einstein’s equations, resp.) uses these structures in formulating precise physical laws forempirical phenomena. But, in order for these mathematical laws to acquire a precise empiricalmeaning a third part (the Newtonian laws of motion, the speed-of-light principle, the equivalenceprinciple, resp.) is needed to set up a general correspondence or coordination between themathematical and empirical part. This part consists of relativized, yet a priori principles ofcoordination.

Only under the form of this tripartite structure can the body of physical knowledge be relatedto sensory experience. Indeed, with the constitutive a priori framework in place, the physicallaws, expressed in the language of pure mathematics, obtain an empirical content and can beconfronted with empirical observation. The physicist can compare calculated values of variousphysical magnitudes – think of the advance of the perihelion of Mercurius, in the case of generalrelativity – with the values obtained through measurement, and make a quantitative assessmentof the correspondence between theory and experiment. But, crucially, such a correspondence isonly made possible by the constitutive framework that fixes the empirical content of the theory;the framework defines a set of empirical possibilities, and sensory experience decides which ofthese is actually realized.

This implies that the constitutive a priori principles cannot be tested directly by experience,like the properly empirical laws. Take the Michelson-Morley experiment, which, in retrospect,provided a very good reason to accept the speed-of-light principle of special relativity. Yet, theLorentz-Fitzgerald theory of electrodynamics equally provided an explanation of the experimentwithin the classical space-time structure. Einstein, however, elevated the result of the experimentto a new constitutive principle, whereas Lorentz and Fitzgerald retained absolute simultaneityand presented the experimental discovery rather as an empirical fact. At this point, the “decision”or “convention” between the two theories is essentially non-empirical. This decision is, of course,often motivated by empirical facts, but, from an epistemological point of view, is not decidedbefore the Quinean tribunal of experience.

The progress of scientific knowledge

This brings us to the Kuhnian challenge of making sense of scientific revolutions, without givingup the rationality of scientific progress. For, even though the previous section showed howscientific knowledge is structured through constitutive principles, it remains unclear how thetransition from one framework to the other can take place in a rational way.

Part of the answer to this question is provided by Friedman’s observation that successiveframeworks in physics are not independent, but rather provide ever greater expansions of thespace of empirical possibilities, such that a new framework contains the earlier one as a specialand/or approximate case. Typically, principles that count as constitutive at one stage shift tothe status of merely empirical laws at a later stage. From the point of view of the Einsteinian


constitutive framework, for example, the general relativistic field equations and the classicalNewtonian law of gravitation appear both as alternative empirical possibilities defined withina common empirical space, whereas in the old framework of classical mechanics the Einsteinequation could not even be formulated. From the Einsteinian perspective both gravitation lawscan be coherently formulated and their empirical meaning devised, such that, under a decisiveexperiment, one can be favored over the other. In this retrospective point of view, the transitionfrom Newton to Einstein seems to be perfectly reasonable.

In addition to this retrospective account, Friedman develops a prospective account of inter-paradigm rationality that explains how there can still be a rational route from the point of viewof the earlier framework leading to the later framework. This implies that new concepts andprinciples of a new constitutive framework develop out of, and as a rational continuation of,the old concepts and principles, and that, despite the incommensurability between frameworks,practitioners of a new framework can still appeal to the persons working within the old frameworkusing conceptual resources that are available for both sides.

This ambitious reply to conceptual relativism is argued for by Friedman’s detailed expositionof how Einstein, in writing down his special and general theory of relativity, made connectionwith (i) a long intellectual tradition of space-time theories going back to the seventeenth century,(ii) the debate on the foundations of geometry, (iii) the philosophical debate on the status andgoal of scientific knowledge, (iv) empirical evidence on the detectability of relative motion in elec-trodynamics and the equivalence of gravitational and inertial mass, etc. Indeed, by embeddingthis specific revolution of mathematical physics within a larger intellectual (philosophical, scien-tific, technological, experimental, etc.) tradition, it can be shown how relativity “could have everbecome a real possibility and thus a genuinely live alternative” [22, p.115], and, consequently,how the rational nature of the transition is laid bare.

2.5.2 The historicized a priori

At this point, it should already be clear that the neo-Kantian philosophy of Cassirer has beenof great influence for Friedman’s work. Let us therefore consider the correspondence betweentheir two conceptions of scientific philosophy in more detail.

We have seen that Friedman characterizes the development of mathematical physics as aprogression of ever greater intellectual possibilities, where new constitutive a priori frameworksfollow out of older ones and enlarge the space of empirical meaning, taking place on the back-ground of a common tradition of cultural change. Friedman finds himself now in “a positionto add, from a philosophical point of view, that we can thus view the evolution of succeedingparadigms or frameworks as a convergent series, as it were, in which we successively refine ourconstitutive principles in the direction of ever greater generality and adequacy” [22, p.63]. Thisidea of convergence is understood as a regulative ideal in the Kantian sense, as an ideal state ofcompletion never to be attained but always to be pursued. Our present constitutive principlesare thus taken to represent one stage in a convergent process, as an approximation to more gen-eral principles that will only be articulated at a later stage. At this point, Cassirer is broughtinto the story as an early defender of the same idea, as

our present scientific community, which has achieved temporary consensus based onthe communicative rationality erected on its present constitutive principles, as anapproximation to a final, ideal community of inquiry (to use an obviously Peirceanfigure) that has achieved a universal, trans-historical communicative rationality onthe basis of the fully general and adequate constitutive principles reached in the ideallimit of scientific progress. [22, p.64]


This regulative ideal is thoroughly Kantian because “we must view our present scientific com-munity as an approximation to such an ideal community, I suggest, for only so can the requiredinter-paradigm notion of communicative rationality be sustained” [22, p.64]. Yet, whereas Cas-sirer saw in relativity theory the culmination of Kantian philosophy as revised by the Marburgschool, Friedman believes that “we need a more far-reaching revision of Kantian transcendentalphilosophy than Cassirer has suggested” [26, p.250]. Indeed, Friedman suggests that it is neces-sary to “relativize the Kantian a priori to a given scientific theory in a given historical contextand, as a consequence, to historicize the notion of transcendental philosophy itself” [26, p.251].

Let us disentangle these two notions and discuss them separately. Firstly, Friedman seemsto claim that Cassirer did not endorse a fully relativized a priori, but this claim appears, fromour detailed discussion of Cassirer’s works, misguided. Indeed, we have identified one of theaims in Substance and Function as justifying the progression of scientific theories by a historicalreconstruction of the principles that make scientific experience possible at any stage. Thisinterpretation is confirmed by Cassirer in a letter to Moritz Schlick, where he states that thea priori “can assume the most various developments in the progress of knowledge”, and thatthe idea of unity in nature “can be specified in particular principles and presuppositions [...]depending on the progress of scientific experience” [27, p.50-51]. This supports our claim thatCassirer indeed elaborates a relativization of a priori principles connected with the progressof scientific knowledge and that Friedman underrates the extent to which Cassirer revised theepistemology of Kant in order to arrive at a relativized a priori. [28]

The notion of historicizing transcendental philosophy seems to be more to the point. Fried-man explains that, in Cassirer’s conception22,

[w]e have no way of anticipating a priori the specific constitutive principles of futuretheories, and so all we can do, it appears, is wait for the historical process to showus what emerges a posteriori as a matter of fact. How, then, can we develop a philo-sophical understanding of the evolution of modern science that is at once genuinelyhistorical and properly transcendental? [29, p.696]

We have seen that Friedman proposes to embed the development of natural science withina larger intellectual (philosophical, scientific, technological, etc.) tradition, showing how thereplacement of constitutive principles can be made intelligible. In particular, he shows thattranscendental philosophy exhibits its own historical transformation and provides the basis forthe constitutive principles of the natural sciences. The prime example of Friedman again servesas an excellent illustration: it is by tracing the development of transcendental philosophy troughKant, Helmholtz, Poincare, and, ultimately, Einstein that the relativity revolution can be madeintelligible. [29, 30]

Of course, Friedman’s idea of a historicized transcendental philosophy is a thoroughly post-Kuhnian philosophy of science. Indeed, the rationale behind this move is precisely to be ableto give not only a retrospective account of scientific progress, but also to provide a prospectiveone. By adding other intellectual dimensions, Friedman tells us a historical narrative that fixesthe rationality of science across scientific revolutions. Cassirer, as he was not faced with theKuhnian paradigm dynamics, did not share this concern, so that his account lacks Friedman’s‘broader intellectual perspective’.23 Indeed, Cassirer rests content with a retrospective accountof the conceptual development of mathematical physics, showing the internal rationality of itshistorical progression.

22In this quote, Friedman discusses the approaches of both Cassirer and Husserl.23One could interpret Cassirer’s philosophy of symbolic forms as a widening of his perspective in this sense. [28]


2.5.3 Constitutive or regulative?

The core distinction, however, between Cassirer and Friedman lies in the constitutive functionof a priori principles. Friedman explains how Kant consciously made a difference between con-stitutive principles (necessary conditions for the comprehensibility of the phenomenal world ofsensible experience) and regulative principles (providing the ideal ends or goals for seeking thecomplete science of nature). The former are due to the faculties of understanding and sensibility,and can be fully determined a priori, whereas the latter are due to the use of reason and judge-ment and have to remain indeterminate at any given stage of science [14]. Friedman believesthat “the Marburg tendency to minimize or downplay the role of the Kantian faculty of pureintuition or pure sensibility on behalf of the faculty of pure understanding represents a profoundinterpretive mistake” [26, p.247], and, Cassirer did no longer acknowledge the constitutive di-mension of the a priori, as it was precisely the function of the constitutive principles of Kantto bridge between the faculties of sensibility and understanding. Therefore, still according toFriedman, the a priori only retains its regulative function with Cassirer: the a priori obtains itsfull specification as the “ultimate logical invariants” that stand at the ideal completion of theprocess of science.24 Friedman concludes that

[a]s a matter of fact, Cassirer (and the Marburg school more generally) does notdefend a relativized conception of a priori principles. Rather, what is absolutely apriori are simply those principles that remain throughout the ideal limiting process.In this sense, [. . . ], Cassirer’s conception of the a priori is purely regulative, with noremaining constitutive elements. [22]

Since this will prove an important issue in the next chapter, let us discuss this conclusionin a bit more detail. Clearly Friedman likes to reconsider Cassirer’s rejection of the faculty ofsensibility, and would like to “preserve some kind of independence for a faculty of sensibilityconceived along broadly Kantian lines” [31, p.48]. This was attempted in the Dynamics of Rea-son by identifying, within the structure of physical theories, the level of coordinating principleswhose role was to relate mathematical concepts to empirical phenomena. In a later publication,Friedman makes his idea of coordinating principles as thoroughly constitutive in the Kantiansense more elaborate. His attempt at “a more structured reinterpretation of the Kantian facultyof sensibility [...] involves replacing the Kantian faculty of sensibility with what we now callphysical frames of references” [31, p.48]. The idea is that

[l]aboratory frames attached to the surface of the earth, for example, can be described,at least to a very high degree approximation, by Euclidean geometry and Newtonianphysics. So they are faithful, in this respect, to the independent a priori structure ofour faculty of sensibility according to Kant. In particular, any abstract theoreticalstructure we might then introduce (such as that of Minkowski space-time) must stillbe related to this prior perceptually given structure in order to have the empiricalmeaning that it does. [31, p.48]

24Friedman discusses this modification of the Kantian a priori in the context of the discussion between Cassirerand Schlick on the philosophic interpretation of relativity theory. Although the logical empiricism of Schlickcarries a lot of elements of a neo-Kantian approach, it is on this issue that the two philosophers clearly diverge.For logical empiricism (Carnap) a purely logical analysis of science provides a fully determinate constitution ofscience, whereas for Cassirer it is “transcendental logic” that characterizes philosophy of science, where only ageneric, at all times indeterminate, conception of the constitution of science is appropriate. A similar divergencebetween Cassirer and Reichenbach is discussed by Ryckman [20], where Reichenbach in the end also takes recourseto a logical analysis of physics with an unproblematic account of the empirical objects, whereas for Cassirer theconstitution of the physical object is precisely the problem for critical philosophy.


So Friedman characterizes the relation between abstract mathematical theories and observa-tional phenomena in a way that is surprisingly close to Kant’s original contentions: there aremathematical structures both at the observational level, which is a prior perceptually givenstructure, and the theoretical level, which is designed in abstract physical theories, and the twolevels are “coordinated with one another by a complex developmental interaction in which eachinforms the other” [31, p.49]. The first level is structured by Euclidean geometry and Newtonianphysics, and can be coordinated to Minkowksi spacetime by limiting procedures. In particular,“the familiar laboratory frames of classical physics play an essential role in relating the mathe-matical structure of Minkowski space-time to our actual perceptual experience of nature” [31,p.48]. According to Friedman, we find that empirical phenomena can only be generated inrelativity theory by virtue of coordinating it to a perceptual space with a Euclidan structure.25

In this respect, Friedman is correct in differentiating Cassirer’s conception of the constitutivefunction of a priori principles from his own. For Cassirer, there is only the space of physicaltheories as shaped by mathematical concepts on one side, and the unstructured chaos of senseperceptions on the other. Only the former space carries definite scientific meaning, and thereforecontains empirical phenomena in the scientific sense, whereas nothing definite can be said aboutthe latter. This is, indeed, the consequence of the Marburg school denying a faculty of puresensibility and only keeping the faculty of understanding in the game.

Yet, we believe that it is misleading to strip Cassirer’s a priori principles of their constitutivedimension. As Cassirer repeatedly claims, it is the proper task of critical philosophy to unravelthe different constitutive principles that make it possible for science to represent experienceas a determinate whole. Because he does not acknowledge a separate faculty of sensibility,Cassirer can no longer characterize the distinction between constitutive and regulative principlesin the way Kant did, but this does not imply that the Cassirer transformed the constitutive apriori into a purely regulative one. [28] Similarly, the fact that Cassirer does not identify adistinct level of coordinating principles does not mean that physical principles have lost theirconstitutive function. Instead, it just means that constitutivity does not necessarily follow thespecific meaning that Friedman attaches to it. In fact, we believe that Friedman’s insistence ona separate faculty of pure sensibility leads to a problematic characterization of the function ofphysical principles: theoretical physics does not need any bridges to a realm of pure sensibilityin order to give physical meaning to empirical phenomena. In the case of relativity theory, thereis only the four-dimensional curved space-time manifold. In particular, the physical meaning ofthe equivalence principle or reference frames has nothing to do with bridging the gap betweenabstract theories and empirical phenomena – their meaning is exhausted by the function of theseprinciples in the theory of general relativity, and it is the theory as a whole that gives physicalmeaning to the movements of planets in the curved spacetime. Similarly, we don’t need tocouple back to classical conceptions of the world in order to give empirical meaning to quantummechanics.

In conclusion: Just as an internal history of the development of theoretical physics is enoughfor laying bare its internal structure and evolution, we believe that the meaning of specificallyphysical concepts are exhausted by their function in physical theories. In particular, the prob-

25In the final chapter of Dynamics of Reason we find a preliminary attempt at applying the same idea ofcoordination to the case of quantum mechanics. Here Friedman assigns a central place to the correspondenceprinciple, according to which a quantum system behaves according to the laws of classical mechanics in the limitof large quantum numbers; it is supposed to explain why only classical behavior is observed in macroscopic phys-ical systems. As Friedman explains, “it performed this essential coordinating function by relating experimentalphenomena to limited applications of classical concepts within the new evolving theory of atomic structure” [22,p.122]. Just as in the case of inertial frames, we find here that coordinating principles are supposed to bridgebetween a classically structured perceptual space (pure sensibility in Kantian terminology) and a theoretical spacestructured by abstract mathematics. In the case of quantum mechanics this feels like a very suspicious move, notin the least because a principle is recovered that has not been found in physical theories for the last fifty years.


lematic way of relating abstract physical theories to some space of pure sensibility (put forwardby Friedman) can be avoided by realizing that there is no such relation, at least as theoreticalphysics is concerned. Therefore, Cassirer’s conception of the constitutive a priori is more tailoredto account for the conceptual structure of theoretical physics as far as the physical meaning andfunction of its concepts is concerned. In particular, it does not require us to take recourse toan a priori space with a different (read: classical) structure from the one we find in the mostadvanced theories of physics.

Moreover, we find that Cassirer’s conception allows for a more fruitful interplay between theconstitutive and the regulative dimensions of the a priori principles. Indeed, with Cassirer thesame concept can both have a constitutive function in a contemporary physical theory, and pointtowards the ‘invariants of experience’ that would function in an ideal stage of physical theorizing.Because the concepts and principles of physics develop in a historical progress, this interplay isnecessarily dynamic. In a domain of physics that is constantly evolving, the identification of astrict level of coordinating principles in the sense of Friedman would therefore underestimatethese dynamics.26 We will show in the following chapter that a philosophical account of thetheory of renormalization is better carried out without introducing the level of coordinatingprinciples.

Still, if we take Friedman’s concerns seriously, we must ask whether Cassirer’s account missessomething in relating physical theory to empirical observations. In a more recent publication[31] we find that Friedman starts focusing on the praxis of scientific observation and empiricaltesting. In the work of Cassirer we do not find an explicit account of how a physical theory isactually tested in practice. Friedman seems to suggest that this is, in the end, how Cassirer’sframework falls short:

And so Kants reliance on the a priori structure of the faculty of sensibility necessarilycommon to all human beings is replaced by the demand of the experimental (andtherefore technological) community for universally communicable (replicable) results.[...] The extremely abstract mathematical structure of general relativity, for example,thereby acquires a connection (via a reconfigured version of a schema connectingthe understanding to sensibility) with our actual perceptual experience of the worldaround us—now essentially including technologically enhanced perceptual experiencein engineered experimental contexts. Abstract (purely intellectual) mathematicalreasoning acquires a necessary and very productive relationship with the concretetechnological practice of experimenters and engineers. [31, p.50]

Thus Friedman opens up a dimension of constitutive principles that appeals to the experimentaltechnologies in which these principles are tested. This move is complementary to Friedman’snotion of historicizing the a priori, which we discussed in the previous section, and has thepotential of adding extra dimensions to the meaning and rationality of physical principles thatgo beyond the purely internal dimension. If understood in this way, we believe that Friedman’sideas can prove very fruitful in order to connect physical concepts to a larger intellectual andtechnological context. In the following chapter we will shortly hint at how this might go in thecase of the theory of renormalization.

26The reason why it does seem to work for the theory of relativity can then be explained by the fact that itconcerns a hundred-year old theory that, without the input from other theories such as quantum field theory, israther inert in describing real physical phenomena.


Coda. The transcendental philosophy of theoretical physics

Before making the transition to contemporary physics, let us first recapitulate what we be-lieve are the important aspects of a transcendental approach towards theoretical physics in thetradition of Cassirer, and as partly reinvigorated by Friedman.

• It traces the historical motives behind the development of principles in theoretical physics.The ultimate goal of this historical approach is finding the ‘ultimate invariants of expe-rience’ in the sense of Cassirer. The work of Friedman has shown that this goal is notrendered futile by the historiography of science in the aftermath of the work of Kuhn.

• It looks for the constitutive function of physical concepts and principles. With Cassirer weshould investigate how the concepts of mathematical physics make first possible a scientific,determinate and fixed experience of physical reality. The work of Friedman has shown thatthis goal is not rendered futile in the light of Quinean holism.

• Both of the above functions of the concepts of physics – the regulative and the constitutive– exhibit an interplay in the work of Cassirer; this interplay should be laid bare.

• The concepts of theoretical physics are functional (serial, structural) concepts. The historyof physics should be understood as a progressive elimination of thing-like concepts. In thatrespect, interpreting the concepts of physics in an ontological way is always misguided, andonly a thoroughly epistemological interpretation of theoretical physics by transcendentallogic is warranted.

• We want to understand the functional concept as an attempt at mathematization andunification of the scientific experience; the goal of theoretical physics is to bring all ofscientific experience into a whole structured by mathematical (serial) relations.

Chapter 3

Renormalization in contemporaryphysics: a transcendental perspective

Contemporary philosophy of physics is dominated by the realist/anti-realist debate. Indeed, inthe aftermath of Kuhn’s work on the history of science, the challenge is to show that thereis cognitive progress in science and that a realist conception of science can be maintained. Inorder to make this happen in the context of theoretical physics, there is needed a philosophicalarticulation of the fundamental ontology of physics. In addition, philosophy of physics typicallytakes place within the broader context of physicalism: a thorough analysis of fundamentalphysics is the first step to a naturalized metaphysics. In both cases, a reductive connotation isimplicit: the challenge is to lay bare the exact metaphysical nature of the fundamental ontologyof physics. Within that perspective, it is straightforward to turn the philosopher’s attention toelementary particle physics, with relativistic quantum field theory as its underlying theoreticalstructure, and the structure of space and time, for which the theory of relativity provides thebest source. Indeed, it is the standard model of particle physics and Einstein’s theory of gravitythat provide us with the best theories for describing the basic ontology of nature.

Yet, this primacy of high-energy physics1 is not found in theoretical physics itself. Certainly,high-energy physics retains a special place in the sense that it describes the physics at thehighest-energy scale that experiments can probe today. But this doesn’t mean necessarily thatit also contains the most interesting or the most fundamental physics, or that it stands at thebottom of a strict hierarchy. In a sense that will become clear throughout this chapter, the fieldof condensed-matter physics2 is equally interesting and fundamental, and requires no conceptualinput from high-energy physics.

If this picture is correct, the philosophical quest of understanding the epistemological struc-ture of theoretical physics should not focus on high-energy physics exclusively. Indeed, theprinciples and concepts that are used in e.g. condensed-matter physics should form the objectof philosophical analysis just as much as the concepts of particle physics. Once this is realized, afirst observation should already make clear that the field of condensed-matter physics containsjust as many concepts and principles that prove worthy of philosophical analysis. In fact, many

1We will use the terms ‘high-energy physics’ or ‘elementary-particle physics’ to denote the discipline of theoret-ical physics involved with the relativistic quantum field theories describing the elementary particles. All researchthat is specifically concerned with the general theory of relativity will not be taken into account in this thesis,whereas we will discuss string theory only in passing.

2Condensed-matter physics is the branch of physics that seeks to understand the behavior of condensed phasesof matter. The most familiar of these condensed phases are solid and liquids, but also contains superconductors,(anti)-ferromagnetic ordered phases, Bose-Einstein condensates, and many more. As it involves such a diversityof different subjects, methods, and concepts, it is hard to define condensed-matter physics without enumeratingall different subfields, but its import will become clear throughout this chapter.

27

Chapter 3. Renormalization in contemporary physics: a transcendental perspective 28

of the theoretical tools of high-energy physics have originated in condensed-matter theory, andvice-versa, so that it makes no sense to treat any of the two domains in isolation.

A possible reason for this misleading focus of philosophers could be the small interest fromprofessional physicists in philosophical questions. In the previous chapter we could clearly seea line from physicists such as Mach and Planck to the philosophical concerns of Cassirer, butno such lines can be discerned between contemporary physics and philosophy. Physicists aretypically not familiar with philosophical literature3, whereas philosophical accounts seem out oftouch with real-life physics.

In this chapter we will discuss the theory of renormalization, which has been a centraltheme in theoretical physics for the last fifty years across condensed-matter, high-energy andstatistical physics. Because it appears in so many subdisciplines, a philosophical account ofrenormalization is less likely to focus on ontological and more on epistemological issues, makingit the ideal subject for the transcendental approach that we have laid out in the previous chapter.We will start by situating this subject within the landscape of contemporary theoretical physics[Sec. 3.1], and then give a historical introduction to the theory of renormalization [Sec. 3.2]. Thephilosophical part starts with a review of the literature on renormalization [Sec. 3.3], so that wecan set the stage for our transcendental account following Cassirer [Sec. 3.4].

3.1 Theoretical physics in the 21st century

Whereas theoretical physics in the nineteenth century could be characterized by a few themesor motives, and the structure could be, more or less, captured in a small number of fundamentalconcepts, the discipline of theoretical physics seems to have become too diverse to discern anoverarching conceptual structure. This diversity makes it impossible to analyze the whole oftheoretical physics within one conceptual framework in the way that Cassirer did for nineteenth-century physics.

Yet we can still identify a few general problems or motives which pop up in different formsthroughout different sub-disciplines of physics. In this chapter we will focus on the many-body problem, the solution of which might be one of the most persisting challenges that runthrough the history of twentieth-century physics and still faces contemporary physicists withinsurmountable difficulties. The structure of a many-body problem always boils down to thefollowing elements: (i) we are faced with a physical system with a large number of constituents,(ii) for which we know how the individual constituents behave and interact with each other,but (iii) we are challenged to predict or understand the collective behavior of the system. Oneparticular example of this threefold structure is the problem of computing the chemical propertiesof large molecules starting from the laws of quantum mechanics. Indeed, we know that thechemical properties of a molecule can be deduced from the distribution of the electrons aroundthe nuclei, which, in turn, can be computed from the static Schrodinger equation

∑i

− ℏ2

2me∇2

i +∑j

e2

|xi − xj |−

∑k

Zke2

|xi − xk|

ψ(xi

)= E ψ

(xi

). (3.1)

Solving this eigenvalue equation for the electron distribution ψ is, however, practically impossibleeven for a simple one-atom system, because the complexity of this mathematical problem scalesexponentially with the number of electrons. This implies that, even though we know exactlyhow its constituents behave and interact, we cannot derive a molecule’s properties. Yet, despite

3Feyerabend would put it as follows: “The younger generation of physicists, the Feynmans, the Schwingers,etc., may be very bright; they may be more intelligent than their predecessors, than Bohr, Einstein, Schrodinger,Boltzmann, Mach and so on. But they are uncivilised savages, they lack in philosophical depth [...].” [32, p.386]


the impossibility of finding an exact solution, a variety of techniques has been proposed to findapproximate solutions that capture the correct physical properties of the molecule. Typically,these approximate methods assume that the electrons can be treated as independent particles –Hartree-Fock theory and density-functional theory are the most famous examples – and neglectthe collective phenomena (correlations) of the particles. Corrections to these mean-field meth-ods4 are typically treated in perturbation theory, a procedure that assumes that the correlationsdo not drastically change the properties of the system.

This example illustrates the general strategy for solving a generic many-body problem.Whereas taking into account the correlated behavior of all constituents in the many-body sys-tem leads to a far too complex mathematical problem, approximate methods are devised suchthat the constituents are somehow treated as if they behave independently. Although thesemethods have led to great successes in the past, there have always been many-body systemsfor which they do not predict the physical behavior correctly. These problems typically involvecollective behavior of the system’s constituents that go beyond the mean-field description. Forexample, in quantum chemistry there are many molecules for which density-functional theorygives wrong predictions. Other examples can be found in condensed-matter physics, where ma-terials have been discovered that do not exhibit the typical conductor/semi-conductor/insulatorbehavior that is expected from band theory5. These new phases of matter are characterized bycollective behavior of the material’s constituents, which cannot be captured by treating them asindependent particles because the correlations between the particles are too strong. It can evenhappen that the macroscopic properties of these materials are completely disconnected from themicroscopic constituents, in the sense that we cannot directly understand the collective behavioras somehow arising from the constituents. In that case, the macroscopic properties are said toemerge from its microscopic basis.

When this happens, the many-body problem takes on a new dimension: it involves the ques-tion of how the emergent properties of a many-body system can originate from its microscopicconstituents. Again, the structure of the many-body problem entails that we know the micro-scopic laws exactly, so we have, in principle, a complete description of the system. Yet, thisdescription does not lead to an understanding of the collective behavior of the system, becausequalitatively new macroscopic phenomena are seen to emerge in the system. We will discussthe issue of emergence in more detail later in this chapter, but, in order to make things moreconcrete, we will first discuss three examples in a bit more detail.

Example 1: The superconductor

The paradigmatic example of a condensed-matter system is the superconductor, which willserve as an instructive example throughout this chapter. It is the phenomenon where a materialexhibits a number of exotic properties such as zero resistivity or the Meissner effect when it iscooled below a certain critical temperature. Superconductivity is a typical example of a many-body phenomenon since it involves a large number of electrons moving inside a lattice6, forwhich we know how they interact with each other, but which display a strong collective effect

4In mean-field approximations the individual electrons are thought to move in an average potential that isgenerated by all the other particles. This is a drastic approximation because all correlations between the individualparticles are neglected.

5Band theory is a mean-field theory for the electrons that move through the crystal of a given material.6A superconducting material is typically a solid material with a certain lattice structure, where the atoms are

arranged in a periodic structure. Most of the electrons are closely bound to the atoms, but the electrons in theatom’s outer cells are delocalized and can move through the lattice. The lattice itself is not completely inert,because it can vibrate. The interactions between these lattice vibrations (phonons) and the electrons is crucialfor explaining superconductivity, because these phonons mediate the attractive interactions between the electronsand allow for the formation of Cooper pairs (see further).


Figure 3.1: Cartoon of superconductivity. On the left we see an uncorrelated state of electrons (repre-sented as black figures) while on the right we have a system where all electrons are bound into Cooperpairs, which are ’condensed’ into a correlated quantum state (the collective nature of the state is repre-sented by the correlated dance). The correlated motion of the electrons is said to be an emergent effectin this many-electron system. Figure taken from [33].

once the material is cooled.In the so-called conventional superconductors, the microscopic mechanism responsible for

these properties is more or less explained by the concept of Cooper pairing and the BCS theory.The idea is that the electrons in a superconductor are bound in pairs due to a attraction me-diated by lattice vibrations, and these Cooper pairs undergo Bose-Einstein condensation. ThisBose-Einstein condensation is a quantum-mechanical effect, and implies that all Cooper pairs‘condense’ in a strongly-correlated quantum state. As a result, the dynamics of the supercon-ductor cannot be understood on the level of the individual level of the electrons, but rather onthe level of the macroscopic quantum state in which the electrons settle. For that reason, thephenomenon of superconductivity is said to be an emergent property of a many-electron system(see Fig. 3.1). Note that a second group of superconductors (the non-conventional ones) cannotbe understood in this way and the physical mechanism giving rise to these high-temperaturesuperconductors is still not understood. The reason why this problem is so difficult is thatit involves even stronger quantum correlations between the electrons and, consequently, moreexotic collective (quantum) effects.

Example 2: Quantum chromodynamics

Another notoriously difficult problem of strongly-correlated many-body physics defying a sat-isfactory analytical or numerical treatment is provided by quantum chromodynamics. This isthe fundamental theory describing the interactions between quarks, and is a crucial part of thestandard model of elementary particle physics. It is a quantum field theory that is specified bythe QCD Lagrangian

LQCD = ψi

(i(γµDµ)ij −mδij

)ψj −

1

4Ga

µνGµνa ,

which, in principle, determines all the physical properties of quarks. When it comes to high-energy scattering experiments, perturbation theory gives us the tools to predict the experimentalresults starting from the fundamental Lagrangian. When computing quark-gluon properties atlower energies, however, perturbation theory fails because strong quantum correlations becomeimportant. In this regime, physicists have struggled the last decades to deduce the strong-correlation effects from the fundamental theory, but, in a lot of cases, have failed to do so.


Let us look at one particular issue in a bit more detail, viz. the determination of the hadronmasses such as the proton and neutron. This problem again has the threefold structure of amany-body problem: (i) we are faced with a system of a large number of microscopic constituents(the quarks described by a continuous field), (ii) we know the fundamental interactions for thefield (they follow directly from the QCD Lagrangian), and (iii) we want to understand howthe proton forms as the lowest energy state of two up quarks and one down quark. It appearsthat the quantum correlations and fluctuations inside the proton are extremely strong, makingperturbative calculations worthless. In fact, it is very hard to see how the proton can be thoughtto emerge from the microscopic quarks and gluons as described by the standard model.

Example 3: The Ising model

As a third example we introduce the Ising model, which serves as the paradigm of a systemof statistical mechanics exhibiting a non-trivial phase transition. The Ising model can be bestthought of as a toy model consisting of a lattice of spins si = ±1 arranged on a two-dimensionalsquare lattice, but other lattices are also possible (and can give rise to different properties).The model is further specified by a Hamiltonian, which dictates how the energy depends on theconfiguration of the different spins. In the ferromagnetic Ising model, the only terms in theHamiltonian are nearest-neighbor interactions,

H(si) = −∑⟨ij⟩

sisj .

Here the sum runs over all nearest-neighbor pairs ij and gives a negative energy contributionif the spins are aligned. The laws of statistical mechanics teach us that all properties of thesystem are determined by the partition function

Z =∑si

e−βH(si), β =1

kBT,

i.e. the sum over all spin configurations weighted by the Boltzmann factors, where T is thetemperature of the system.

We are for the third time confronted with a many-body problem: (i) we have a large numberof microscopic constituents (spins on a lattice), (ii) we have a full description of these constituentsand their interactions (everything follows from the partition function), and (iii) we are interestedin the collective behavior of the system. Here we are interested in the system’s magnetization,which is given by the averaged direction of the spin. In the limit of infinite temperature, all spinswill be uncorrelated and the average spin will be zero; in the zero-temperature limit all spins willsettle down in the same direction as energetically favored and the average spin will be one. Itappears now that in between these two limits there is a sharp phase transition (see Fig. 3.2), and,more interestingly, that the spin correlations become stronger around the transition point andcannot be understood from mean-field theory nor perturbative corrections. Again, the strongcorrelations at the phase transition originate from collective effects that seem to emerge fromthe microscopic constituents, and cannot be understood starting from the microscopics directly.

The relation between condensed-matter and high-energy physics

These three examples are taken from condensed-matter physics and high-energy physics alike,and illustrate the conceptual similarities between these two fields of physics. Yet originally


(a) (b)

Figure 3.2: The two-dimensional Ising model. (Left) A number of spins are arranged on a two-dimensionalsquare lattice, where every spin interacts with its four nearest neighbors. (Right) As the temperatureis increased, we go from a system where all spins point in the same direction (average magnetization isone) to a system where the spins are uncorrelated (average magnetization is zero). In between, there isa sharp phase transition.

the fields of particle physics and solid-state physics7 were involved with very different physicalphenomena, and not much overlap was found between the two. This has changed dramaticallysince the fifties, when methods of quantum field theory – originally the theory for describingelementary particles – were imported to describe condensed-matter systems as well. In theopposite direction the concepts of symmetry breaking, originally developed by condensed-matterphysicists, were applied to high-energy physics. Later ideas from the theory of phase transitionson the one hand, and divergences in quantum field theory on the other, were combined in thedevelopment of the renormalization group. Nowadays, much of the phenomena of high-energyphysics can be found back in condensed-matter systems, and the same concepts can be appliedto understand these physical phenomena. Famous examples include the observation of the Higgsmode in superconductors [34], and the emergence of gauge bosons and fermions in spin systems[35]. The defining difference between the two fields from a conceptual point of view is that inthe case of condensed-matter physics the microscopic constituents and laws are known and theobserved phenomena, exotic as they may be, are supposed to arise from a more ‘fundamental’level, whereas an underlying level is not known in the case of high-energy physics.

As will become clear in the rest of this chapter, we believe that the divisions between differentsubfields of physics, and a strong focus on one of these fields in particular, is unwarrantedfrom a philosophical point of view, at least if we want to map out the conceptual structure ofcontemporary theoretical physics. The focus on high-energy physics from the side of philosophyis rooted in the motive of grounding a metaphysical or ontological picture of the world, and isunwarranted from the epistemological point of view. Instead, it proves to be more rewarding toinvestigate concepts or motives that run across the different subdisciplines of physics. The many-body problem is one of these motives, and the theory of renormalization provides an extremelyelegant way of getting a grip on it. At the end of this chapter, we will see that the theory ofrenormalization gives us a way of understanding one of the key conceptual features that anyphysical theory shares, irrespective of the ontology the theory supposedly describes.

7Solid-state physics is, more or less, the old term for condensed-matter physics. Nowadays, the former alsorefers to the ‘old way’ of doing condensed-matter physics, where crystal structures and band theory were amongthe main subjects.


3.2 The theory of renormalization

Renormalization is not really a theory in the traditional sense of the word – the sense in whichrelativity or quantum mechanics are theories – but rather denotes a set of interrelated ideas,concepts, methods, and procedures. Therefore, the best option for describing renormalization isby a historical introduction, explaining the different aspects as they appeared in their historicaldevelopment.

The history of renormalization theory can be told in many ways, but two historical narrativesseem to present themselves. The first starts in the early days of quantum field theory, where theproblem of infinities in perturbative calculations were progressively solved with renormalizationprocedures. The second narrative takes off in the context of statistical physics, and, in particular,critical phenomena. We will follow the latter, because this is the road commonly taken incondensed-matter physics, and we will couple back to quantum field theory at the end of thissection.8

3.2.1 The prehistory

In our narrative9, the advent of the renormalization theory has its roots in the theory of clas-sical phase transitions. Traditionally, a phase transition is understood in terms of statisticalmechanics, which directly characterizes the link between the microscopic world of atoms, spins,and molecules, and the macroscopic world that can be directly observed; a phase transition isunderstood as a macroscopic system suddenly settling into a drastically different equilibriumconfiguration under the influence of an external parameter such as temperature or pressure.

It was the genius of Landau and his introduction of the order parameter that led to anotherperspective on phase transitions. He realized that, because a phase transition is a collectiveeffect of a many-body system, it cannot be characterized on the microscopic or macroscopiclevels alone, but is rather taking place across different scales inside the material10. The ideaof introducing an order parameter can be stated as providing a mesoscopic object that lives ondifferent length scales at the same time, capturing the system’s fluctuations on these differentscales as it approaches a phase transition. Indeed, the fluctuations and dynamics of the orderparameter describe the collective long-wavelength fluctuations, which eventually diverge exactlyat a critical point.11

Landau’s theory provided the first systematic understanding of phase transitions, but thetheory was soon proven wrong by the exact solution of the two-dimensional Ising model asprovided by Onsager. The problem was that Onsager’s computation of the partition functionand thermodynamic properties of the Ising model did not agree with Landau’s general theory ofphase transitions. Indeed, as we have seen, the Ising model exhibits a sharp critical point, wherethe expectation value of the order parameter vanishes according to a non-trivial power law asa function of temperature (see Fig. 3.2). Soon it was realized that this violation of Landau’stheory is a generic property of phase transitions, and a whole field of physics emerged trying to

8Our main source for writing this section was Ref. [36].9This is a rather technical section which aims at reviewing all important concepts leading up to Wilson’s

formulation of the renormalization group. The reader who is not familiar with these concepts is not required tofully understand this section.

10In this section, the concept of scale refers to the typical length of the fluctuations that we are talking about.We will have a lot more to say about scales later in this section.

11For later reference, we note here that Landau’s characterization of a continuous phase transition in terms ofan order parameter as a dynamic fluctuating object provides the first example of an effective field theory. Indeed,the fluctuations of the order parameter capture the dynamics of a macroscopic system as it goes through a phasetransition, without directly coupling back to the microscopic constituents: It captures the effective degrees offreedom that determine the system’s behavior across different scales.


understand the critical properties of many-body systems undergoing a phase transition.The assumption of the ‘classical’ theory of Landau was that all functions entering the the-

ory should have a smooth (analytic) character, even at the phase transition. Physically, thisamounts to a neglect of the interactions between fluctuations across different length scales, sothat the small-scale fluctuations can be safely incorporated into effective parameters that be-have smoothly on the larger scales. In the non-classical cases of criticality – the generic casefor interacting systems – this assumption proves to be wrong, and it is only by the advent ofrenormalization theory that the effect of fluctuations across scales has been incorporated in acorrect way.

On the road to the renormalization group, the theory of critical exponents, scaling behaviorand universality have proven to be crucial. Both experimentally and theoretically it was realizedthat, although the power laws that are observed around critical points are not those of theclassical theory of Landau, the same exponents systematically pop up in entirely different many-body systems. This points to the fact that phase transitions can be classified in a small numberof universality classes, characterized by a set of critical exponents. Moreover, the differentexponents within a certain class were shown to obey certain scaling relations.

In the sixties the existence of these phenomena related to phase transitions and criticalitywere firmly established experimentally and theoretically, but a real understanding that couldexplain all the different aspects was still lacking. Moreover, one would like to firmly relatethe phenomenology of criticality to statistical mechanics, which is supposed to describe themicroscopic constituents of the systems. This is a requirement that can be traced back to thework of Landau, where the order parameter was introduced without providing a direct link tothe underlying microscopic spins or atoms out of which a system is made. The crucial questionis: how do the order parameter and its fluctations arise from the microscopics, and how isit possible that an effective field theory can accurately describe the behavior of a many-bodysystem as it approaches a phase transition?

3.2.2 Wilson’s intervention

This is the point where the work of Wilson comes in.12 We will explain Wilson’s originalformulation of the renormalization group in some detail, based on his seminal review paper of1975 [37].

The first important point is that the renormalization group is about scales in many-bodysystems, entailing that dynamics, fluctuations, and correlations can be said to occur within acertain window of energy, length, or momentum.13 The basic assumption of Wilson is that thesescales are locally coupled:

The basic physical idea underlying the renormalization group approach is that themany length or energy scales are locally coupled. For example, the behavior offluctuations in a magnet with wavelengths from 1000 to 2000 A are assumed to beprimarily affected by fluctuations with nearby wavelengths, e.g., 500-1000 A or 2000-4000 A. Fluctuations with wavelengths much less or much greater than 1000 A areless important. [37, p.775]

Remember that the rationale behind Landau’s theory was that we don’t need the microscopicsof a system in order to understand the behavior at a much larger scale, but that this fails for

12The work of other people such as Widom, Kadanoff and Fisher are also important in this story, but we focushere on Wilson for pedagogical reasons.

13Here the notion of scale is left rather vague, because it can refer to energy, length or other dimensionalquantities. These different definitions of scale are typically related, in the sense that probing certain length scalesrequires probes with a certain characteristic energy. For example, resolving features on a certain length scale withlight probes requires the photons to have a certain energy or wavelength.


systems at criticality. In the end Wilson came up with a procedure of treating the connectionsbetween scales, explaining why Landau’s theory can make sense in some cases and why it failsin other.

Renormalization group flows

Instead of postulating the existence of an effective theory at a certain scale, Wilson went backto ask the question how such an effective description arises from the microscopics as they aredescribed by statistical mechanics. Suppose, thereto, that we have a system of a large numbersof spins s arranged on a regular lattice (the Ising model is an example of such a system). Aseach spin can take on two states, the partition function of this system consists of a sum over allpossible configurations,

Z =∑

s=±1

e−βH(s),

where H(s) is the energy of a particular configuration. One possibility of systematicallycomputing this huge sum – the number of configurations scales exponentially in the number ofspins – is by first dividing the spins into two groups. The first group s< we will keep in ouranalysis, but the second group s> we will sum over so that they drop out of the problem.After we have taken the partial sum, we will be left with an effective energy function Heff(s>)involving only the second group of untouched spins, but, if we do not want to change the physicsof the system, we should require that the new partition function

Z =∑s>

e−βHeff(s>)

remains the same. This implies that the following relation should hold:

e−βHeff(s>) =∑s<

e−βH(s<∪s>),

which gives us, in principle, a prescription for transforming the original model to a new effectivemodel, which acts only on a part of the spins of the lattice. Since only half of the spins areretained in the effective model, the spacings between the spins has doubled and the model canbe said to be defined on a larger scale. If one now rescales the spacings between the effectivespins to the original spacings, one ends up with the same basic constituents as in the originalmodel, but now with a different energy functional Heff(s>). Thus, we have designed a mapbetween models, which can be written down in full generality as

H(l)(s

)= R

(H(l−1)

(s

)).

In Fig. 3.3 we have summarized the renormalization procedure. This procedure can now be re-peated many times, which leads to a renormalization-group flow14 of effective models. Supposingthat we start from a spin system with only nearest-neighbor coupling, the renormalization-groupflow will introduce couplings that range further than only the one between nearest neighbors.Even worse, not only two-spin but also four-spin, six-spin, etc. interactions will be generated.Consequently, the renormalization-group flow can be pictured as a flow through the space of

14These ideas of renormalization are commonly grouped under the term ‘renormalization group’. In order toavoid confusion, we will avoid the use of this term, and rather refer to ‘renormalization theory’ or ‘theory ofrenormalization’, keeping in mind that this does not correspond to a physical theory in the strict sense, butrather to a set of interrelated ideas, concepts and procedures. We will use the term ‘renormalization-group flow’,however, since this term has a stricter meaning in the physics literature.


Figure 3.3: Schematic representation of a scale transformation. We start out with a set of spins sinteracting through a certain Hamiltonian H(s). We select half of the spins s< (yellow) and averageover them, while the other half s> (yellow) are kept as degrees of freedom. The yellow spins move awayfrom the picture, and, by imposing that the partition remain the same, we arrive at a new HamiltonianHeff(s>) for half of the spins. Finally, we rescale the spacings between these effective spins (and rotatethe lattice) such that we arrive at the same lattice structure, but with a different Hamiltonian. Thisrelation defines the map R(·) between Hamiltonians.

all possible spin couplings, where the flow dictates how these effective couplings change as thescale is tuned. The transformation that generates a renormalization-group flow – the above mapR(·) gives an explicit realization of such a transformation – is called a scale transformation asit maps a given model to another model living on a larger scale.

In most situations these renormalization-group flows terminate at a fixed-point model H∗,characterized by the fixed-point relation

H∗ (s) = R(H∗ (s)) .

Typically, one has a small set of fixed points, and different microscopic models can flow to thesame fixed-point models. Indeed, if a small interaction term is added to a certain microscopicmodel, it is typically expected that the model will still flow to the same fixed point; the perturba-tion is called irrelevant in that case. Given a certain model, one also has relevent perturbations,which have the effect that the fixed point of the model is changed. Some of these fixed pointscorrespond to trivial models, for which e.g. all the spins are frozen to point in the same direc-tion, or do not interact at all. Other fixed points, however, are more interesting and representcritical states. In fact, every one of these non-trivial fixed points corresponds to a universal-ity class: every model that flows to the same critical (non-trivial) fixed point belongs to thesame class. A critical fixed point has a defining set of properties such as scaling behavior andcritical exponents, which can be obtained by linearizing the above flow equations. So with theconcept of renormalization-group flows Wilson had found a natural explanation of universality:systems of different physical character may, nevertheless, flow to the same critical fixed point.The different incoming trajectories onto this same fixed point correspond to distinct irrelevantinteractions that are present in the microscopic models, but which are washed out by the scaletransformation.

Now the above renormalization procedure of averaging over degrees of freedom is particularlysimple in the case of spins, and is expected to become more difficult in the case of atoms,electrons, field theories, etc. Also, the effective degrees of freedom that arise in the courseof a renormalization-group flow can become rather different from the ones on the microscopiclevel. Therefore, the significance of Wilson’s work consists rather in the conceptual ideas of scaletransformations, renormalized couplings, effective degrees of freedom, and renormalization-groupflows. Fisher puts it as follows:

Indeed, the design of effective RG transformations turns out to be an art more thana science: there is no standard recipe! Nevertheless, there are guidelines: the general


Figure 3.4: Schematic representation of a renormalization-group flow. We can see that different micro-scopic models (l = 0) determine different itineraries through the space of effective models and lead to asmall set of fixed points. In this case the possibilities are simple: a microscopic model flows to one of thetwo trivial fixed points, unless it is at the critical point. In the latter case the model flows to the criticalfixed point (indicated with an asterix). Figure taken from Ref. [36]

philosophy enunciated by Wilson [...] is to attempt to eliminate first those micro-scopic variables or degrees of freedom of “least direct importance” to the macroscopicphenomenon under study, while retaining those of most importance. [36, p.672]

In the next section, we will give different examples of how this ‘general philosophy enunciatedby Wilson’ is realized in different physical contexts or theories.

The numerical renormalization group

Still, there is one aspect of Wilson’s achievements that we have not discussed yet, one that is com-monly forgotten in historical overviews but, from our perspective, is crucial for understandingthe significance of renormalization theory in physics.

The fourth aspect of renormalization group theory is the construction of nondiagram-matic renormalization group transformations, which are then solved numerically,usually using a digital computer. This is the most exciting aspect of the renormal-ization group, the part of the theory that makes it possible to solve problems whichare unreachable by Feynman diagrams. The Kondo problem has been solved by anondiagrammatic computer method. [37, p.776]

Let us explain this ‘most exciting aspect’ in a bit more detail.15 The Kondo model describes amagnetic impurity coupled to the conduction band of a nonmagnetic metal; the crucial question,unsolvable by perturbation theory, is the low-temperature behavior of this impurity spin. Wil-son’s solution to the problem is to (i) discretize the conduction band to discrete energy levelswith a logarithmic spacing, (ii) transform the system to a half-infinite spin chain with the firstspin representing the impurity, and (iii) solving this spin-chain system iteratively. Starting fromthe impurity spin in every iteration a new site is added to the system and, in order to keepthe size of the Hilbert space tractable, the number of states is truncated by only keeping thelowest-energy states of the Hamiltonian for the current part of the chain.

So the general procedure of integrating out non-important degrees of freedom is motivatedfrom a very practical point of view. Because a computer can store only a finite number of

15Note that the details of Wilson’s numerical solution of the Kondo problem are not important to understandthe rest of this chapter.


states, Wilson needed a procedure to select only the important states in every iteration of therenormalization procedure. In this case, it appears that selecting only the lowest-energy statesleads to a numerical algorithm that predicts the impurity’s physical properties. But this mustmean that these states represent the effective degrees of freedom that capture the physics ofthe Kondo model at a given energy scale. Therefore, from the very start of the theory ofrenormalization, the numerical simulation and physical understanding of a system are two facesof the same coin, in the sense that both require to account for the effective degrees of freedomthat determine the physics of a many-body system at a certain scale. Indeed,

the solution of the Kondo problem is the first example where the full renormaliza-tion program (as the author conceives it) has been realized: the formal and scalingaspects of the fixed points, eigenoperators, and scaling laws will be blended withthe practical-aspect of numerical approximate calculations of effective interactionsto give a quantitative solution (the present accuracy is a few percent) to a problemthat previously had seemed hopeless. [37, p.805]

3.2.3 Renormalization and the many-body problem

In the previous section we have discussed three examples of many-body systems exhibitingemergent behavior which, although the constituents and their interactions are known exactly,cannot be understood starting from the microscopic description alone. From the insights ofWilson we can now understood how such emergent behavior can come about, and how it candisconnect from the microscopic description of the system.

One important concept is that of effective degrees of freedom determining a certain physicalphenomenon. In full generality, we can define the degrees of freedom for a certain system asdetermining the possible states or configurations that the system can take. In the case of theIsing model, the degrees of freedom are spins on a lattice, whereas we have quark and gluon fieldsfor quantum chromodynamics, and electrons moving through a crystal for the superconductor.Importantly, Wilson has shown that we can define scale transformations that average over a setof given degrees of freedom, giving rise to effective models with a new set of degrees of freedom.In the case of the simple scale transformation in Fig. 3.3, these new degrees of freedom are againspins, but scale transformations can also give rise to qualitatively different ones. For example,in the case of quantum chromodynamics we have seen that the effective degrees of freedom arenot the original quarks and gluons, but rather the protons and neutrons that emerge at lowenergies. Similarly, superconductivity is best described by identifying the symmetry breaking ofeffective gauge fields below a certain temperature.

This picture thus suggests that many-body systems are described by different effective de-grees of freedom depending on the scale at which the system is being probed. The renormali-zation-group flows determine how these effective degrees of freedom on a certain scale can beentirely disconnected from the degrees of freedom at a smaller scale. These effective degreesof freedom, and the physical phenomena they describe, can then be said to emerge from themicroscopic basis.

These insights and developments have led to a specific way of looking at a many-body system.Although the microscopic constituents and the elementary interactions are often well-known ina typical system, the crucial question for understanding a certain phenomenon is: What arethe effective degrees of freedom that determine the system’s behavior at this scale? The mostexciting research typically involves many-body physics for which these effective descriptionsinvolve exotic physical concepts such as collective excitations, gauge fields, effective particleswith anyonic statistics, etc. Let us discuss some examples in the fields of condensed-matter andhigh-energy physics.


Renormalization in condensed-matter physics

The field of condensed-matter theory has a few paradigmatic procedures of obtaining such effec-tive descriptions. We have already encountered the use of order parameters for characterizing e.g.the phase transitions in the Ising model, where the fluctuations of this order parameter occur onlength scales that are much larger than the scale at which we see the individual spins; the fluctu-ations of the order parameter are the degrees of freedom determining the low-energy features ofthe Ising model at criticality. The understanding of superconductivity follows a similar path, inthat the low-energy behavior of the superconductor can be described by quantum fluctuations ofthe order parameter. In the specific case of the superconductivity, the effective field theory forthese fluctuations has an electromagnetic gauge invariance, which is spontaneously broken. Thisbreaking of a gauge symmetry explains all characteristic superconducting behavior [38], so thatthe gauge field indeed describes the effective degrees of freedom on the energy scale at whichthe phenomenon of superconductivity is exhibited, whereas the microscopic degrees of freedom(the electrons and the lattice) do not allow for such a description.

Another example is fermi liquid theory, a theory that has been extremely successful indescribing the electronic properties of conductors. It was again Landau [39] who first came upwith the idea that the low-energy excitations of a gas of interacting electrons can be pictured as‘quasiparticles’. These are defined in the non-interacting limit as the free electrons, and acquirean effective mass and lifetime as the interactions are turned on. Most physical properties of afermi liquid at low temperature can be derived from the quasiparticle distribution function andtheir scattering cross sections. Only quite recently, this approach has been rigorously understoodfrom the viewpoint of renormalization theory: It has been shown that fermi liquid theory isobtained if one integrates out high-energy degrees of freedom, much in the spirit of Wilson.[40, 41] The conclusion from this renormalization analysis is that the quasiparticles are indeedthe effective low-energy degrees of freedom, capturing the phenomenology of the electrons in ametal at low energy or low temperature.

Together with the concept of an order parameter, the idea of quasiparticles remains to thisday one of the standard ways of understanding the low-energy behavior of a strongly-correlatedelectron system. Both paradigms of condensed-matter physics can be rightly viewed as effectivefield theories, and are both firmly rooted in renormalization theory16. In the case of thesetwo examples, this renormalization perspective can be made rigorous, but this does not alwayshave to be the case. Indeed, the use of a certain effective field theory is often motivated fromphysical intuition, experimental results, perturbation theory, mean-field approaches, numericalsimulations, etc. It is the power of an effective description that a direct link with the microscopicconstituents is not needed to understand a system’s behavior at a certain energy scale.

Also, we should stress that there is an extreme variety of renormalization approaches availablein the condensed-matter literature. Already in the paper of Wilson, we can find different scaletransformations: in real space, where e.g. groups of spins are averaged over; in momentum space,which is typically the approach in quantum field theories; or in energy levels, which leads tothe solution of the Kondo problem. Recently, it has been realized that one can also renormalizein entanglement degrees of freedom, which has led to effective parametrizations of quantummany-body states. One can also perform scale transformations on different objects: in the pathintegral of a field theory, in a partition function of statistical mechanics, in a quantum many-body wave function, on the level of correlation functions, etc. Finally, as we have illustrated withWilson’s solution of the Kondo problem, the ideas of renormalization can be implemented in aconceptual or analytic way, but also lead to computational methods for numerically simulating

16Another resource for motivating these approaches is the use of symmetries and symmetry breaking. Of course,the concepts of renormalization and symmetry are strongly connected, but we will solely focus on renormalizationin this text.


many-body systems.

Renormalization in high-energy physics

Above we have written the history of renormalization from the perspective of condensed-matterphysics, but it would be strongly misguided to claim that renormalization theory was inventedin the context of critical phenomena, and only subsequently applied to relativistic quantum fieldtheory. Indeed, preliminary notions of renormalization go back as early as the late 1940s withthe successful renormalization of the divergences in quantum electrodynamics.17 Yet it proveddifficult to generalize these successes to other field theories, and the whole project of quantumfield theories came under attack because of the non-renormalizability of e.g. the weak interaction.Although this particular issue was in the end solved around 1970, and the status of quantum fieldtheory as the theoretical framework for describing elementary particles and their interactionswas restored, it remained a matter of debate what ‘renormalizability’ was really supposed tomean and whether it should serve as a guiding principle or not. Arguably, renormalization inquantum field theory was fully understood only after Wilson’s insights on scale transformationsand renormalization flows. [43]

The ‘modern’ view on renormalization in relativistic quantum field theory is very similar tothe one that is at work in condensed-matter physics. As Weinberg has put it, the method in itsmost general form can be understood as “a way to arrange in various theories that the degrees offreedom that you’re talking about are the relevant degrees of freedom for the problem at hand”[44, p.15]. This has led to the formulation of effective field theories that are designed to describethe low-energy physics of another quantum field theory that is valid at higher energies. For agiven field theory, this might be implemented by deleting the ‘heavy fields’ from the theory, sincethese only have observable effects at high energies, and only keeping the low-energy fields withsuitably redefined masses and couplings. The prime example is again quantum chromodynamics,where at high energy the relevant particles (fields) are quarks and gluons, whereas, at lowenergies, the physics can be understood with e.g. massless pions or, at even lower energies,protons and neutrons. [44]

The logical conclusion of these developments is that the field theories for the elementaryparticles in the standard model are themselves effective field theories of another unknown fieldtheory, and that they are not fundamental in the traditional sense of the word. Also, becauseeffective degrees of freedom can be entirely different from the underlying theory, it seems to beprincipally impossible to deduce any properties of this more fundamental theory unless one cando experiments to sufficiently high energies. The situation is nicely summarized by Polchinskiin the following Q&A:

Q: Doesn’t all this mean that quantum field theory, for all its successes, is anapproximation that may have little to do with the underlying theory? And isn’trenormalization a bad thing, since it implies that we can only probe the high energytheory through a small number of parameters?

A: Nobody ever promised you a rose garden. [41, p.10-11]

From the previous paragraphs, one can very well imagine that the standard model of elemen-tary particle physics is an effective theory for other degrees of freedom living at a smaller scale.In fact, more and more physicists are actively working out the idea that the phenomenologyof high-energy physics arises from an underlying microscopic theory, just as in the case of acondensed-matter system. This ‘condensed-matter point of view’ can be stated as follows:

17The concept of a renormalization-group flow go back to the work on the running of coupling constants inquantum electrodynamics by Gell-Mann and Low [42], which served as an important inspiration for Wilson.


As we probe nature at shorter and shorter distance scales, we will either find in-creasing simplicity, as predicted by the reductionist particle physics paradigm, orincreasing complexity, as suggested by the condensed-matter point of view. Wewill either establish that photons and electrons are elementary particles, or we willdiscover that they are emergent phenomena—collective excitations of some deeperstructure that we mistake for empty space. [35, p.879]

This quote finally confirms that the ideas of renormalization and emergence are at the founda-tions of high-energy physics as well, and that the conceptual import of renormalization theory issimilar as in condensed-matter physics. Therefore, a philosophical account of the theory of renor-malization is needed that can capture its conceptual structure across the different subdisciplinesof physics.

3.3 Review of philosophical literature

In a seminal paper Cao and Schweber summarize the philosophical ramifications of renormaliza-tion theory as follows18:

Most notably, we found that the recent developments support a pluralism in theo-retical ontology, an antifoundationalism in epistemology and an antireductionism inmethodology. These implications are in sharp contrast with the neo-Platonism im-plicit in the traditional pursuit of quantum field theorists, which took mathematicalentities as the ontological foundation of physical theories and which assumed that,through rational (mainly mathematical) human activities, one could arrive at an ul-timate stable theory of everything. Also, contrary to the previous image of scientifictheories that was implicit in the mathematical structure of QFT, the new imagefostered by the EFT approach is that scientific theories are not to be conceived asnecessary products of scientific rationality, but rather should be seen as contingentdescriptions of nature, revisable in the course of changing circumstances. [43, p.69]

These drastic conclusions are drawn from a detailed historical analysis of the theory of renormal-ization in high-energy physics, and to a lesser extent, statistical physics. We can already notethat these conclusions seem at odds with the approach we take in this thesis, since for us thescientific rationality of physical theories is the starting point, the feature that a transcendentalanalysis should, in some sense, explain. Indeed, the idea that physical theories are contingentdescriptions of nature directly contradicts our goal of showing how the object of theoreticalphysics is ideally determined by a set of ‘ultimate invariants of experience’.

In the following four subsections we will discuss some of the issues that are raised in Cao andSchweber’s paper, and try to get a feeling of contemporary philosophical accounts of renormal-ization theory in physics. Since we aim at developing a transcendental account in the spirit ofCassirer in the following section, we will try to bring home the two important conclusions that (i)the specifically philosophical conclusions drawn by contemporary philosophers with regards toepistemology and/or ontology do not necessarily follow from the physics alone19, and (ii) these

18In this quote, EFT stands for effective field theory; the EFT program denotes the new approach in high-energyphysis, where every field theory is thought to describe the effective low-energy physics of another field theory thatlives at a higher energy scale. As we have discussed in Sec. 3.2.3, this is the consequence of the modern view onrenormalization in high-energy physics. Crucially, in the EFT program the most accurate field theories describingthe standard model of particle physics are themselves only effective field theories of a lower-lying level to whichwe haven’t had any experimental access so far.

19It is often suggested that, once one understands the physics thorougly – a vantage point that supposedly onlya small number of philosophers attain – these conclusions are the only viable option.


accounts lack a comprehensive view on the epistemological function of renormalization theory.Of course, the burden of proof lies with our own account of renormalization, rather than explic-itly disproving all other approaches. Therefore, this section should also be read as a setting ofthe stage for the following one, and as an indication of the challenges for a philosophical analysisof renormalization.

3.3.1 Empiricism

At multiple occasions, Cao and Schweber note that recent developments in renormalizationtheory support philosophical empiricism. The reason is that, according to the EFT program,an understanding of physical phenomena on a given scale needs to be supported by empiricaldata on that scale. This implies the fundamental importance of phenomenological approachesin physics. From that perspective, physical theories cannot be more than “effective instrumentsfor organizing the data by imposing local order and coherence, and they conceive and expresslocal causal regularities” [43, p.76]. This, in turn, supports a localist view of theory, which“characterizes physical (or more generally, scientific) theories as historically situated and context-dependent” [43, p.74].

Their position is further characterized by contrasting it with the idea that “the developmentof fundamental physics will end with the discovery of an ultimate, definitive, and conclusivemathematical formalism” [43, p.77]. Indeed,

the empiricist position in epistemology that is supported by the recent developmentsin renormalization theory is characterized by its antiessentialism and its antifounda-tionalism, its rejection of a fixed underlying natural ontology expressed by mathemat-ical entities, and its denial of universal, purely mathematical truths in the physicalworld. [43, p.77]

From our perspective this argument cannot carry any weight. In the previous section we haveshown that the physical motivation for the EFT program is not restricted to high-energy physics,but that the same commitment to ‘relevant degrees of freedom’ is present in e.g. condensed-matter physics. Moreover, condensed-matter physics is typically not driven by the dream of afinal theory describing some fundamental nature of the physical world. From that perspective,contrasting the rationale behind the EFT program with the dream of the string theorist20 servesnot as an argument for the claim that renormalization theory implies empiricism.

Moreover, we don’t agree with the claim that the inevitability of phenomonology implies anempiricism in the philosophy of physics. Indeed, Cao and Schweber observe from the develop-ments of renormalization theory that physics can only establish local regularities, but use thisobservation as follows:

The limited nature of our experience in producing knowledge of the world underminesthe universal claim of physical laws: it only allows ascertaining family resemblance(regularities) in local region of space and time. From local regularities we cannotconstruct physical theories that are unique and necessary. On the contrary, alltheories are context-dependent, culturally relative, and historically changeable. [43,p.75]

Again, the argument seems to be that, with the dream of a grand unified theory crushed byrenormalization theory, the only remaining option is that a physical theory is e.g. context-dependent. But the fact that the physical behavior of nature depends on the scale on which it

20String theory is currently the best option for developing an ‘ultimate’ theory of everything, see the footnotein Sec. 3.4.4 for more on string theory.


is probed, does not imply that our understanding of nature on that scale is culturally relative.Instead, renormalization theory has learned that a physical description of nature necessarilyinvolves a setting of the scale on which it is probed, but this stipulation of scale is an exact andwell-defined part of physical theory, and does not point to the “socially constructive nature ofphysical theories” [43, p.77].

Ultimately, it seems that Cao and Schweber’s ontological considerations are responsible fortheir empiricistic conclusions. In their view the EFT program implies a representation of thephysical world as “layered into quasi-autonomous domains, each layer having its own ontologyand associated ‘fundamental’ laws”, giving rise to a so-called “hierarchical pluralism in theoreti-cal ontology” [43, p.72]. Since every quasi-autonomous domain demands an empirical input thatis historically contingent, the conclusion is unavoidable that the ontologies that are discoveredare contingent as well. Then it is only a small step to the claim that “scientific theories arenot to be conceived as necessary products of scientific rationality, but rather should be seen ascontingent descriptions of nature” [43, p.69].

From the transcendental perspective, the conclusion that ontologies are historically contin-gent does not need to worry us that much, and, crucially, does not imply that scientific theoriesare contingent descriptions of nature. The reason is that Cao and Schweber assume that theontological commitments of scientific theories are inseparable from their scientific content, anassumption that we, following Cassirer, want to avoid at all costs. Instead, we would like tofocus on the conceptual structure of theoretical physics, and its evolution, where scientific ratio-nality is taken as a postulate and cannot be disproven by the content of physical theories. Inparticular, we will try to show that renormalization theory has redefined the “universal claim ofphysical laws” (see quote above), rather than it has undermined it.21

3.3.2 Physical understanding

Let us therefore focus on the epistemological ramifications of renormalization theory, where Caoand Schweber first lay their focus on the many fruitful interactions between quantum field theoryand statistical physics. Indeed, although a historical analysis shows the many intersectionsbetween the two fields in a coherent way (see Sec 3.2), this does not answer the question why itis, in fact, possible that physical insights from one domain are applicable to an entirely differentdomain – why is the formalism of critical phenomena applicable to field theory, and vice versa?

In the framework of Cao and Schweber, which is determined by the ontological commitmentsthat a certain theory contains, the apparent universality of physical concepts is a philosophi-cal problem: “Why are the physical insights obtained from one phenomenological domain (e.g.,spins in crystal lattices) relevant, translatable, and applicable to another entirely different do-main (e.g., continuous fields)?” [43, p.73]. According to Cao and Schweber, we can understandthis in terms of physical and mathematical analogies, as “different physical interpretations ofthe mathematical formalisms in different domains of phenomena are connected by metaphoricaltransformations of concepts involved in the formalisms” [43, p.76]. In Sec. 3.2 we have madeclear, however, that the universality of physical concepts involves more than metaphorical trans-formations, and rather point to a deep physical insight related to scale transformations andrenormalization. Indeed, the fact that the same field theory can be applied to different physicalsystems is explained through the idea of renormalization-group flows, and shows that differentsystems can exhibit the same physics at low energies or large length scales. Again, we come

21Note that another resource for Cao and Schweber’s empiristic conclusions is the realism/anti-realism debate,where the historical contingency of ontologies in physics is seen as a problem. One response is structural realism[45], for which ontological commitments in physics is not made with respect to the objects in physical theories,but rather with respect to the structures in these theories. Note that our approach does not commit to this lineof thinking either, as we are trying to avoid the realist/anti-realist debate altogether.


to the conclusion that connecting a physical theory with its ontological commitments ratherobscures the conceptual features of renormalization theory.

Let us therefore look at a philosophical analysis of physical understanding that is closer toour goals. In an interesting paper, Hartmann also discusses Cao and Schweber’s paper and comesto the conclusion that, “[l]eaving metaphysical questions aside, it seems to be philosophicallymore interesting to examine the formal relations between the theories, models and EFTs wehave already” [46, p.298]. From our perspective, this seems to be more to the point, indeed.

Although Hartmann’s paper is mainly about differentiating the functions of theories, modelsand effective field theories in high-energy physics, we would like to focus on his idea of globaland local understanding. Whereas a local understanding is obtained by capturing a physicalphenomenon in terms of the degrees of freedom that are relevant at the energy scale underconsideration, a global understanding consists of showing how the phenomenon follows from themicroscopic equations governing the system.22 This difference in the theoretical understandingof physical phenomena is illustrated with a case study of quantum chromodynamics. As we haveseen in Sec. 3.1, the major difficulty with this theory is the fact that it is extremely difficult toactually compute, starting from the fundamental Lagrangian, observable consequences such asthe observed masses of the proton and neutron. In fact, this has only been realized recently withthe development of highly advanced computational methods and the use of huge computationalresources. According to Hartmann, this has produced a global understanding of the mass ofthe proton, since it is incorporated as a consequence of the fundamental equations of quantumchromodynamics, but fails to produce a local understanding, since this computational methodacts like a black box. On the other side, we have effective models such as asymptotic freedom,confinement, or dynamical chiral symmetry breaking, which provide a local understanding ofhow the proton arises as a particle, but these explanations lack a global understanding becauseno general principles are directly involved.

Although the differentiation of global and local understanding sheds light on the differentfunctions playing a role in explaining physical phenomena, there seems to be a tension betweenboth types of understanding. Both types of understanding are needed to fully understand aphysical phenomenon, but it is rather unclear how they relate to each other. The dichotomycan be analyzed further by reiterating the example of quantum chromodynamics. In the caseof the proton mass, for example, we believe that Hartmann misses the point of lattice gaugetheory as he downplays this computational method as providing just a black box. Instead, theefforts of lattice gauge theory and other computational approaches in determining the massof the proton from the fundamental Lagrangian should be understood as providing a physicalunderstanding of how the proton arises as an effective description of quantum chromodynamicsat low energies. Put more generally, physical understanding requires an understanding of howthe effective degrees of freedom at a certain scale – the energy scale at which the proton is probed– arise through the interactions of the degrees of freedom that live at other scales – the energyscale at which quark dynamics are important. In that respect, it is unclear what the added valueof the idea of global understanding could be, if not for the crucial requirement that the relevantdegrees of freedom and processes can be shown to arise from a more global standpoint. Thisis, of course, the place where renormalization theory comes in, so one of the challenges for thenext section will be to show how the concept of renormalization integrates the different levelsat which we can understand a physical phenomenon.

22According to Hartmann, whereas local understanding is produced by causal/mechanistic explanations of aphysical phenomenon, a global understanding is rather produced by an explanation that fits the phenomenonin a general framework. It is by combining these different accounts that one obtains scientific understanding:“science studies a given phenomenon from various theoretical perspectives, all of which reveal some explanatoryinformation about the phenomenon in question ” [46, p.300].


3.3.3 The question of emergence

Another case by Cao and Schweber that has proven to be influential, is the one for emergence:

Taking the decoupling theorem and EFT seriously would entail considering the re-ductionist (and a fortiori the constructivist) program an illusion, and would lead toits rejection and to a point of view that accepts emergence, hence to a pluralisticview of possible theoretical ontologies. [43, p.71]

This notion of ontological pluralism or emergence is typically seen in relation to reduction:“The claim that some phenomenon is emergent is usually understood as the claim that thephenomenon is in some sense not reducible to (i.e. deducible from) its base” [47, p.429]. Thisleads to the core distinction between ontological emergence (the failure of reduction in principle)and epistemological emergence (the failure of reduction in practice). The question of emergencethen becomes the articulation of which of the two types of reduction is not being exemplified ina certain physical theory.

This differentation of epistemological and ontological emergence takes a central place in acontribution by Morrison [48], where the focus is on emergent low-energy behavior in condensed-matter systems such as superconductors. Her argument starts from the demand that emergenceshould be distinguished from low-energy behavior that is merely ‘resultant’, because this rathercorresponds to the epistemological independence of a certain macroscopic description with re-spect to the microscopic details, and is a fairly common feature in physical explanation. In orderto have ontological emergence, something more is needed: there should be a complete autonomyof the low-energy physics from the microscopic theory, without an ontological link between thetwo.

This stronger sense of ontological emergence is exemplified in systems that exhibit universalbehavior, where the prototypical example is superconductivity. The standard explanation forthis phenomenon is the BCS theory (see Sec. 3.1), where (i) pairs of electrons are formed throughCooper pairing, (ii) the low-energy behavior of this collection of Cooper pairs is describedby introducing a bosonic field theory, and (iii) this boson theory exhibits symmetry breakingbelow a certain critical temperature. As Morrison explains, however, one can “derive the exact(emergent) properties of superconductors simply from the assumption of broken electromagneticgauge invariance without relying on the microphysics of Cooper pairing” [48, p.153]. This impliesthat we “do not need a microscopic story about electron pairing and the approximations thatgo with it to derive the exact consequences that define a superconductor” [48, p.155]. Thus,it is symmetry breaking that “provides the dynamical explanation of emergent phenomena”,whereas “the specific microphysical details are irrelevant; how the symmetry is broken is notpart of the account” [48, p.156]. The mechanism of symmetry breaking is universal, as differentphysical systems can exhibit exactly the same symmetry breaking pattern and give rise to thesame physical phenomena at low energies. Because universal phenomena “originate from vastlydifferent micro properties, there is no obvious ontological or explanatory link between the microand macro levels” [48, p.162], and, therefore, we have a clear example of ontological emergence.

The analysis of Morrison is guided by the goal of establishing some kind of ontological emer-gence, in contrast to a merely epistemological one. We believe, however, that this distinctionis nowhere to be found in theoretical physics itself, and can only be imported due to philo-sophical concerns. This conclusion can also be read in a paper by Crowther, where she showsthat understanding emergence in terms of reduction is “tied by the metaphysicians’ fixation onontological emergence, the need to sharply distinguish it from “merely” epistemological emer-gence” [47, p.430]. Our transcendental view on the philosophy physics is highly sympatheticwith Crowther’s claims that we should view emergence as it actually appears in physics, andnot base it on ontological and metaphysical conceptions that are imported from other branches


of philosophy. In particular, in contrast to the ontological pluralism of Cao and Schweber, this‘physics-first’ approach does not make any claims on the ontological commitments that the EFTprogram supposedly adheres.

Once we have realized this, we can see that this ontological dimension of emergence is un-called for in the case of superconductivity. The fact that we can use the same Nambu-Goldstonefield theory for describing the phenomenon of superconductivity in a variety of different ma-terials, irrespective of the microscopic properties, does not imply that we have found a newontology on that scale. The only conclusion is that, in order to understand the phenomenon ofsuperconductivity, we need to introduce new degrees of freedom (i.e. a bosonic field theory) thatlive on the energy scale at which superconductivity is observed. The fact that different materialsexhibit the same effective degrees of freedom is a very non-trivial physical fact, and has foundits explanation in renormalization theory as different microscopic systems can flow to the samefixed point under a renormalization-group flow. But this physical fact does not come with anyontological commitments: it is just a part of physical understanding that different degrees offreedom become important at different scales!

Let us therefore discuss Crowther’s account of emergence in more detail. She sees a tensionin the relation between an effective field theory and the underlying microscopic theory: on theone hand, it should be impossible to reduce the effective field theory from the more fundamentaltheory, but, on the other hand, there should be, in principle, a way to derive a low-energy theoryfrom the high-energy physics. This is essentially the same tension that we identified earlier in thepaper of Hartmann [46], and can again be illustrated with the example of QCD. There it should,in principle, be possible to derive the low-energy properties of the quarks (the hadrons) fromthe QCD Lagrangian, and physicists are actively pushing the numerical methods for makingthis happen. Nevertheless, it seems impossible to derive a theory describing the low-energybehavior from QCD only, and an external input is required for developing EFTs such as chiralperturbation theory.

At this point, Crowther notes that, although it is in principle possible to obtain quantitativelow-energy predictions from a high-energy theory, the EFT framework is often necessary in amore subtle sense:

An effective, low-energy theory is the only means of properly describing the low-energy behavior of a system. EFTs are formulated in terms of the appropriatedegrees of freedom for the energy being studied, and are necessary for imparting anunderstanding of the low-energy physics. Because the low-energy degrees of freedomdo not exist at higher energy, the high-energy theory is unable to present the relevantlow-energy physics. [47, p.428]

Based on this crucial insight, it is made obvious that hinging emergence on the notion of deriv-ability or reduction misses the point. Instead, Crowther proposes to focus on two positiveaspects of emergence, i.e. on the fact that the low-energy physics is novel and autonomous withrespect to the high-energy theory. Novelty means that new features appear in the low-energyregime that are not features of the high-energy theory, and autonomy captures the fact that alow-energy theory is impervious to changes in the high-energy system. This positive definitionhas the advantage that it is “naturally suggested by the physics”, whereas taking emergence asa failure of reduction “distracts from the lessons of the actual physics”: “It means developingan account true to the science rather than seeking to carry-over prior intuitions and conceptsfrom other branches of philosophy” [47, p.430].

Still, just as in Hartmann’s case, we believe that the tension is not taken care of. Crowtherrightly problematizes the differentiation between ontological and epistemological reduction from


the perspective of physical practice, and rightly emphasizes the fact that identifying the correctdegrees of freedom is necessary for understanding the physics at low-energy physics. In thatsense, the low-energy physics can be said to emerge from the high-energy system, and this ideaof emergence is indeed one of the conceptual innovations of modern physics. But, what is missedhere, is the fact that a real physical understanding of these low-energy degrees of freedom is onlyattained if it is understood how these arise from the high-energy system: What is the physicalmechanism that, starting from the high-energy physics, gives rise to the low-energy description?Indeed, whereas Crowther focuses on the novelty and the autonomy of the emergent physics,the question of how the emergent physics arises from the microscopic degrees of freedom isnot properly taken into account. The challenge here is, again, to show how both sides of thetension can be relieved in a positive way, i.e. how can we take the novelty and autonomy ofemergent physical behavior seriously, while, at the same time, not loosing the physicist’s goal ofunderstanding how these effective degrees of freedom emerge from the underlying microscopics.

In the next section we will show that it is precisely the idea of the renormalization groupthat gives us the conceptual framework for tackling this question. In our analysis, a crucialrole will be played by computational approaches: it is in the efforts of obtaining quantitativepredictions on low-energy behavior starting from the microscopic theory, that a physicist gainsunderstanding of the low-energy physics. Crowther downplays this aspect of theoretical physics:

Thus, we can distinguish between an EFT’s role in enabling quantitative predictionsin the low-energy regime—a role which, in principle, could be fulfilled by the high-energy theory—and its role in appropriately describing the behavior of a system atlow energy, and thereby facilitating an understanding of the low-energy physics—arole which could not be fulfilled by the high-energy theory. [47, p.428]

We will show that enabling quantitative predictions in the low-energy regime by numericalsimulations takes a crucial place in the structure of contemporary physics, and provides one ofthe keys for resolving the tension between reduction and emergence.

3.3.4 More is different

In Sec. 3.1 we have already explained how theoretical physics has developed into a diverse fieldof science, where the search for a fundamental theory of physical reality is no longer the onlygoal, nor the most interesting one. The debate on emergence is a direct consequence of thisdevelopment, as it has become clear that interesting physics can emerge in a physical system ona certain scale, without a direct connection to underlying microscopic degrees of freedom thatmake up the system. This realization was first made by condensed-matter physicists, as theywanted to make clear that their work is, at the least, equally fundamental and exciting as thework of the high-energy physicist. In this last subsection, we will investigate this side of thestory a bit more, specifically in order to argue for our claims above that a lot of the philosophicaldebates on ontological emergence do not have their basis in the works of physicists themselves.

The discussion on emergence in physics originates to a large extent from the seminal paperMore Is Different by Anderson from 1972 [49]. In this paper, Anderson opposes the commonview that “the only scientists who are studying anything really fundamental” [49, p.393] are thehigh-energy physicists working on the fundamental laws of elementary particles and cosmology.This seems to be an obvious corrolary of reductionism, where “the workings of our minds andbodies, and of all the animate and inanimate matter of which we have any detailed knowledge,are assumed to be controlled by the same set of fundamental laws” [49, p.393]. But, as Andersonis out to show, this reasoning is a fallacy, because reductionism does not imply the constructionisthypothesis: “The ability to reduce everything to simple fundamental laws does not imply theability to start from those laws and reconstruct the universe” [49, p.393]. In many-body physics,


this constructionism breaks down when confronted with difficulties that are related to scale andcomplexity, because it turns out that the behavior of large systems cannot be understood by asimple extrapolation from the properties of a few particles. Instead, at every level of scale andcomplexity “entirely new laws, concepts, and generalizations are necessary, requiring inspirationand creativity to just as great a degree” [49, p.393]. This breakdown of the constructionisthypothesis is illustrated by Anderson mainly through the concept of symmetry breaking, whichprovides a physical mechanism that explains how, at a certain scale, behavior can be observedthat is entirely new with respect to the underlying fundamental laws.

So, Anderson’s paper should, in the first instance, be read as a polemic against the high-energy physicist’s monopoly on the notion of fundamentality. “[A]t each level of complexityentirely new properties appear, and the understanding of the new behaviors requires researchwhich I think is as fundamental in its nature as any other” [49, p.393]. As Anderson explains,the research that is done for understanding the property of a system with a broken symmetryis “as fundamental as many things one might so label”, but “it needed no new knowledge offundamental laws and would have been extremely difficult to derive synthetically from thoselaws” [49, p.395].23

Secondly, Anderson does not oppose reductionism, but rather argues for “the breakdown ofthe constructionist converse of reductionism” [49, p.393]. What Anderson is claiming, is that, inorder to understand a physical phenomenon on a given energy scale, entirely new laws, concepts,and generalizations are necessary. Indeed, “the behaviour of large and complex aggregates ofelementary particles, it turns out, is not to be understood in terms of a simple extrapolation ofthe properties of a few particles” [49, p.393]. As Anderson accepts reductionism, he does not denythat low-energy behavior can be reduced to the more fundamental laws, but he emphasizes thatthis is (i) often extremely difficult or all but impossible, and (ii) not essential for understandingwhat is going on at the low-energy scale. We see the same tension appearing as the one we haveidentified earlier in the papers by Hartmann and Crowther, but taking on a more pragmaticform here. Anderson’s discussion of superconductivity is illuminating:

But sometimes, as in the case of superconductivity, the new symmetry—now calledbroken symmetry because the original symmetry is no longer evident—may be of anentirely unexpected kind and extremely difficult to visualize. In the case of supercon-ductivity, 30 years elapsed between the time when physicists were in possession ofevery fundamental law necessary for explaining it and the time when it was actuallydone. [49, p.395]

Thirdly, we note that no commitment to any form of ontological emergence is found. Unlesswe should interpret Anderson’s use of the word ‘fundamental’ as implying some kind of ‘ontology’appearing on different scales, there is no need to see any argument for ontological pluralism orontological emergence in his paper.

In a more recent paper [51], Laughlin and Pines have reiterated Anderson’s statements onthe status of reductionism in theoretical physics, but in a seemingly stronger sense. Parallel toAnderson’s distinction between reductionism and constructionism, they state that “[w]e havesucceeded in reducing all of ordinary physical behavior to a simple, correct Theory of Everythingonly to discover that it has revealed exactly nothing about many things of great importance”[51, p.28]. They seem to go beyond Anderson in identifying ‘higher organizing principles’ thatwork at a certain energy scale independently from an underlying microscopic theory. Theseprinciples are “transcendent”, and “would continue to be true and to lead to exact results evenif the Theory of Everything were changed” [51, p.28].

23The historical context [50] clearly shows that this reading of Anderson’s paper is the correct one.


The fact that emergent physical phenomena are regulated by higher organizing principlesimplies that these phenomena are insensitive to microscopics; they are determined by higherorganizing principles and nothing else. Examples include the quasiparticles of a Fermi liquid, orsuperconductivity, and are dubbed ‘protectorates’ as they are protected against changes in themicroscopics. This is relevant to “the broad question of what is knowable in the deepest senseof the term”, because the “the nature of the underlying theory is unknowable until one raisesthe energy scale sufficiently to escape protection” [51, p.29].

Let us again take the example of the superconductor, where the higher organizing principlewould be the breaking of a local gauge symmetry. Indeed, the field theory that is invokedfor explaining superconductivity cannot be reduced to the underlying equations governing theelectrons in the material: the field describes the collective, low-energy degrees of freedom, whichare entirely different from the microscopic constituents (i.e. the electrons). Moreover, they areinsensitive to the microscopic degrees of freedom as different materials can exhibit exactly thesame symmetry breaking pattern.

Drawing philosophical conclusions with respect to ontological emergence, however, againrequires something more.24 In line with Anderson, Laughlin and Pines do not rule out thepossibility of an explanation of how the field theory arises from the microscopic details; they juststate that this is, in general, extremely difficult – as the unsuccessful attempts at explaining high-Tc superconductivity from microscopic details show – and not necessarily essential or interesting.Indeed, the message of the paper is again polemic in trying to convince people that high-energyphysics is not more interesting than condensed-matter physics, and that the “deductive pathfrom the ultimate equations to the experiment without cheating” [51, p.30] is not necessarilythe path that theoretical physics should take.

We conclude, again, that the argument for ontological pluralism or emergence in physics –and any conclusions with respect to ontology – is not taken from the physics itself, but ratheris inspired from the metaphysical aspirations of philosophers. Ontological commitments are notmade by physicists, but are ascribed to physical theories by philosophers. In the next section, weshow that this is not necessary, and that we can make perfect sense of emergence (as physicistsunderstand it) without invoking any ingredients from metaphysics or ontology. As Cassirerwould put it: “Science at least knows nothing of such a transformation into substance, andcannot understand it” [1, p.192].

3.4 Renormalization as a functional concept

In the previous sections, we have tried to make clear that a physical description of nature de-pends crucially on the scale at which it is being probed. This is true in high-energy physics,where the program of effective field theories is an explicit realization of this insight, and incondensed-matter physics, where a physical description of a certain material always requiresthe identification of the relevant degrees of freedom at a certain scale. It is the development ofthe renormalization group that has provided a mathematical formalism and physical mechanismfor making this idea concrete and workable: renormalization-group flows show how parame-ters change under scale transformations, which of these parameters are relevant, what effectivemodels are obtained after a renormalization-group flow, etc.

The question that we are taking in this section, is how to make philosophical sense of this

24The philosopher Morrison acknowledges that something more is needed since “we need to differentiate ex-planatory from ontological claims since emergence is not simply about different organization principles beingimportant at different scales or laws not requiring specific micro details” [48, p.150]. For the physicists Laughlinand Pines, however, this is exactly what emergence is, and, as we will see in the next section, this is exactly whata transcendental analysis needs.


development in physics within a transcendental framework. We have seen that a realist phi-losophy of physics with a focus on ontological considerations leads to unacceptable conclusions,at least from the transcendentalist’s perspective. In particular, when taking the ontologicalcommitments of a certain physical theory seriously, there always looms the tension betweenthe emergence of new ontologies on a given energy scale, and the realization that these ontolo-gies should be a consequence of the underlying microscopic degrees of freedom, albeit only inprinciple.

3.4.1 Energy in the work of Cassirer

Before treating renormalization theory head-on, let us shortly reiterate how Cassirer incorpo-rated the concept of energy in his philosophical analysis of nineteenth-century physics. Wehave seen that in Cassirer’s time the goal of unification was an important motive in theoreticalphysics, and that the mechanical program was one of the prime options for establishing this –think about Maxwell’s efforts at reducing electromagnetism to a mechanical phenomenon.

According to Cassirer, this goal of unification is crucial in understanding the epistemologicalstructure of physics. Whereas the first step in obtaining physical knowledge is “the insertionof the sensuous manifold into series of purely mathematical structure”, this must remain “in-adequate as long as these series are separated from each other”. Indeed, the object of physicalknowledge means “more than the mere sum of properties; it means the unity of the properties,and thus their reciprocal dependency”. This postulate finds its expression in physics if a princi-ple is found, which “enables us to connect the different series, in which we have first arrangedthe content of the given, among themselves by a unitary law” [1, p.190]. According to Cassirer,such a principle is found in the nineteenth-century concept of energy, which allows to connectdifferent physical phenomena (heat, motion, electricity, light, chemical reaction, etc) into aninclusive system.

Cassirer suggests that the concept of energy is, from an epistemological point of view, prefer-able to mechanical reduction as a way of unifying physics. Indeed, energism, in contrast to themechanical program, permits to “relate two qualitatively different fields of natural phenomena,without having previously reduced them to the processes of movement, and thus having divestedthem of their specific character” [1, p.199].

Energism shows that this form of numerical order is not necessarily connected with-out analyzing the things and processes into their ultimate intuitive parts, and recom-pounding them from the latter. The general problem of mathematical determinationcan be worked out without any necessity for this sort of concrete composition of awhole out of its parts. [1, p.201]

So the demand of unification does not require that all of physics should be reduced to an inclusiveunitary picture for which every physical phenomenon is interpreted as an expression of the sameultimate substance. Instead, “the demands of the theory of knowledge are rather satisfied whena way is shown for [...] producing a complex of coordinations, in which each individual processhas its definite place” [1, p.203].

3.4.2 The necessity of scale and effective degrees of freedom

The first important lesson from renormalization theory is that the physical understanding of acertain phenomenon takes recourse to degrees of freedom, physical concepts and mechanismsthat are only applicable at the scale at which the phenomenon takes place. Consequently, aphysical theory or description that conveys this understanding necessarily contains reference to


the scale at which the theory or description is supposed to hold, and it is generally impossibleto give a theory that describes a given system at all energy scales in a unified way.

The example of superconductivity illustrates why scale and effective degrees of freedomnecessarily enter into physical understanding. Suppose we had a way of actually solving allthe equations that describe the electrons on the microscopic level, but this solution acts as ablack box. Could we say that we have reached an understanding of why a certain material issuperconducting? The answer is obviously no, since we cannot see how the collective motionof the electrons gives rise to this low-energy behavior. We need the field theory describingthe effective degrees of freedom at this energy scale, in order to formulate the mechanism ofsymmetry breaking that gives rise to superconductivity. This implies that the need for aneffective description does not arise because of our limited intellectual or computational resourcesin deriving the observable consequences of the microscopic theory, but that the use of effectivedegrees of freedom is a necessary part of any physical theory.

How do we fit this feature of physical theories in a philosophical framework a la Cassirer?We begin by noting that for Cassirer a physical theory describes an observable phenomenonin the sense that the phenomenon undergoes a transition from what is offered in experienceto the form in which it appears in a physical statement. This transition or transformation ismediated by the physical concepts, and only after a phenomenon has been given a place in theconceptual structure does it gain a scientifically determinate meaning. In this sense the conceptsare constitutive for a scientific knowledge about the physical world. Importantly, these conceptshave a strict mathematical meaning independently from their application to physical reality; itis only because they have this fixed meaning beforehand, that they can constitute the exactnessthat is required for a scientific picture of the otherwise chaotic sensuous world.

The concepts of a field theory and symmetry breaking provide examples of such constitutiveconcepts. They have a strict mathematical meaning before they are supposed to explain any-thing: a quantum field is a well-defined mathematical object, the Lagrangian for the field canhave gauge symmetries, which the field can break to settle in a less symmetric configuration,this local symmetry breaking leads to massless excitations known as Goldstone bosons, etc. Allthese concepts and their consequences are derived within a strictly mathematical setting, and itis only by applying this field theory and its symmetry breaking to the degrees of freedom in anactual physical system that we aim to gain knowledge about the physical world.25 Because allthese physical concepts now take the role of effective degrees of freedom, it is all the more clearthat they are not just abstracted from empirical observation. Indeed, one does not ‘observe’ agauge field when a system becomes superconducting, but one applies the concept of a gauge fieldto a many-electron system in order to explain the observed superconducting properties. Thisis an operation that supposes, from the epistemological point of view, an active function fromthe side of theory. Furthermore, in the scheme of Cassirer there is no reference to ontology:understanding a many-body system by an effective field theory does not carry any ontologicalcommitments. This does not imply that we do not take the philosophical ramifications of physicsseriously, but, instead, our epistemological framework captures exactly in what way the physicistunderstands emergence.

So the procedure of theoretical physics is the following. A physical phenomenon can be un-

25This ‘capturing the degrees of freedom in a mathematical framework’ can take on different forms: One canwrite down a quantum field theory, where the fluctuations of the field correspond to the low-energy fluctuationsof the system; Or one writes down the Feynman diagrams for a quasiparticle propagator in a many-electronsystem; Or one can think of quantum states in an effective Hilbert space, where an effective Hamiltonian capturesthe interactions between the low-energy degrees of freedom; Or one writes down a path integral that acts as agenerating functional for computing the low-energy dynamical correlations; Or one comes up with a variationalwave function for the many-body system with the variational manifold encapsulating the low-energy subspace ofthe system; etc.


derstood if a way is found of identifying the degrees of freedom that live on the energy scale atwhich the phenomenon takes place, and formulating a mathematical theory that, (i) describesthe behavior of these degrees of freedom, and (ii) has the observed phenomenon as a mathe-matical consequence. The identification of effective degrees of freedom is an active operationby the theoretical physicist, an operation that is physically motivated from the concept of scaletransformations and an associated renormalization-group flow in the space of effective models.In fact, renormalization teaches us that this operation is a necessary step in understanding aphysical phenomenon: without specifying the scale at which a physical system is probed, itmakes no sense to refer to a certain description of the system.26

This is the first function of renormalization theory: it teaches us that effective degreesof freedom necessarily enter in a physical description of a physical system at a certainscale, and are only valid on that scale.

Thus renormalization theory learns us that a physical theory is necessarily only valid ata given energy scale, and cannot be naively extrapolated to give an accurate description of aphysical system across different scales. However, following Cassirer, this can only be the firststep in our physical understanding of the world: our physical picture of the world should be morethan the sum of successful theories for physical phenomena. Indeed, just as the energy conceptwas understood by Cassirer as a principle for connecting the different physical phenomena, “inwhich we have first arranged the content of the given, among themselves by a unitary law” [1,p.190], we need here a principle or mechanism of connecting the different energy scales, whichexplains why entirely different concepts are needed if the scale is changed, but also shows howwe can integrate these different conceptualizations into a unified whole.

It is, of course, the machinery of renormalization theory that accomplishes this demand.Indeed, renormalization-group flows give us a physical mechanism that explains why a physicaltheory cannot be extrapolated across different energy scales, and explains how effective degreesof freedom can arise that are qualitatively different from the underlying microscopic theory.

This determines the second function of renormalization theory: it teaches us howeffective degrees of freedom arise from lower-energy scales.

This function of renormalization theory is important in two ways. The first is that it gives us aphysical mechanism in principle that explains the disconnectedness of different scales. Indeed,as we have read in the papers by Anderson and Laughlin & Pines, the fully detailed mechanismthat gives rise to emergent physical phenomena is often not particularly interesting, and aphysicist can rest content with a theory on a certain scale without having reduced it to itsunderlying microscopics: as long as the physics on that scale is properly described by a certaintheory, the phenomenology can be said to be understood in a satisfactory way. No deeperinsights are necessary here. Still this procedure would be entirely unintelligible if the concept ofrenormalization would be absent: one still needs an understanding of how effective descriptionscan arise in general. Therefore, renormalization theory is crucial in the conceptual structure of

26This epistemological reconstruction of what it means to understand superconductivity for a given materialexplains how we should understand universality from a philosophical point of view. Indeed, the fact that it ispossible to apply the same mathematical formalism to different physical systems and at different energy scales doesnot require us to draw deep philosophical conclusions about the ontology of the physical world. The mathematicalformalism does not carry any ontological commitments: it only carries a constitutive function of giving physicalphenomena such as superconductivity a place within a conceptual structure, and, as such, yielding a theoreticalunderstanding of what is happening if a material becomes superconducting. The observation that entirely differentphysical systems can be understood with the same concepts is a feature of our physical picture of the world –universality is rightly understood as a deep physical insight! – but does not present us with any epistemologicaldifficulties.


theoretical physics, as it makes the approach of effective descriptions on a certain energy scaleintelligible.

Secondly, renormalization theory always guarantees the physicist that there should be amechanism or explanation for the emergent physics, and motivates physicists to look for thesemechanisms. But renormalization theory also shows what this explanation should consist of: thephysicist should come up with a physical mechanism of how effective degrees of freedom can arisefrom the microscopic constituents of the theory. In the case of superconductivity this is exactlywhat happened: the mechanism of Cooper pairing explains how an effective bosonic field theorycan arise in a system of electrons. In the case of topological systems it is the entanglementdegrees of freedom that explain how anyonic quasiparticles can emerge from an electronic orbosonic system. In fact, the rationale behind the whole field of strongly-correlated quantummany-body physics is determined by this second function of renormalization theory.

3.4.3 The importance of the computational approach

One particular aspect of many-body physics that is typically not taken seriously in a philosoph-ical analysis, is the importance of numerics in understanding physical phenomena. Yet, thereare hardly any papers published in theoretical physics that do not contain a numerical part.

The reason why philosophy does not take into account numerics is that it does not seemto provide any real insight into the physical mechanisms that are at work in nature. Indeed,the idea is that a numerical computation can produce quantitative predictions from a physicaltheory, but this does not add any understanding that was not already contained in the theory:the computer only acts as a black box that spits out numbers at the end of a computation. Ifthe numbers match the experiment, this counts as a confirmation that the theory is correct,but does nothing more than that. The prototypical example of this relation between a physicaltheory and numerics is the determination of the hadron masses by lattice gauge theory, wherethe fundamental Lagrangian provides the real physics of the system and lattice gauge theoryacts as a black box yielding the numerical value of the mass of the proton.27

Taking numerics seriously in physics starts by realizing why it is so hard to actually simulatea many-body system. Suppose one has a microscopic theory for the elementary particles out ofwhich the many-body system consists, and one has all the fundamental equations one needs todeduce, in principle, the microscopic behavior of the system – think of a gas of electrons insidea metal, moving in the static potential generated by the lattice and mutually interacting via thelaw of Coulomb. In principle, any quantum-mechanical problem should reduce to diagonalizingmatrices, for which there are very efficient algorithms available in any software package. Theproblem is, however, that the size of these matrices scales exponentially with the number ofparticles out of which the system is built! This implies that it is, as a matter of principle,impossible to simulate a quantum-mechanical system that contains a large number of degreesof freedom, just because the space of possible configurations (and the associated operators)explodes.

27In a paper by Hartmann, which we have discussed in Sec. 3.3.2, the situation is perceived as follows:

In the case of the strong interactions, QCD does indeed specify the overall dynamics of the system;there are quarks and gluons, and these entities interact in a very complicated way with each otheraccording to the Lagrangian density of QCD. But not much more can be said: the rest has to bedone numerically with the help of high-powered computers [...]. And computers function like a blackbox. All possible Feynman diagrams are summarised, although, perhaps, only a few of them (or acertain subclass of them) produce almost the whole effect under investigation. A knowledge of theseactually relevant processes would produce insight and understanding. Lattice gauge theory does notproduce this insight, and QCD is, therefore, effectively a black-box theory. [46, p.289]


This creates the situation that, in order to actually compute something, the physicist needsto have an idea what the relevant information is about the system she wants to store in thecomputer: she needs to find a way of only doing numerical operations on the degrees of freedomthat are important for simulating a given phenomenon. It is a priori not clear that is alwayspossible, but, as we have seen, it is precisely the theory of renormalization that guaranteesthat effective degrees of freedom can be identified at a certain scale that determine a system’sbehavior. This, in turn, opens up the possibility of simulating a system at that scale, as long asa way is found of implementing these degrees of freedom on a computer.

Through the formalism of renormalization, it appears that understanding a physical phe-nomenon and simulating it on a computer follow a conceptually similar path. This is not acoincidence, as this relation between understanding the low-energy behavior of a many-bodysystem and simulating it, is exactly what motivated Wilson in his formulation of the renormal-ization group. As we discussed in Sec. 3.2.2, it was the challenge of numerically simulating theKondo problem that made Wilson realize that one has to find a way of devising what the effec-tive degrees of freedom are on a given energy scale. This shows that the renormalization groupprovided, from the beginning, both a conceptual tool of understanding how theories changeunder scale transformations, and a numerical procedure for simulating the effective degrees offreedom on a certain scale.

In the forty years that followed simulating many-body physics has required computationalphysicists to come up with smart ways of capturing physical phenomena with limited numericalresources. This has led to lattice formulations for field theories, which can then be simulated withe.g. Monte-Carlo techniques, variational parametrizations that capture the essential features ofa many-body system at a given energy scale, mean-field approaches leading to self-consistencyequations, etc. All these examples of computational approaches to the many-body problemshow that physical understanding through numerical simulations has become essential in theway physicists work, and it would be a philosophical mistake to reduce numerics to a blackbox that does not lead to understanding of the physical phenomenon that is being simulated.Instead, in the context of computational physics the two functions of renormalization theoryflow naturally out of pragmatic concerns: (i) because of the limits on computational resources,the numerical simulation of a physical phenomenon necessarily requires an identification of therelevant degrees of freedom, and (ii) the theory of renormalization (in the broadest sense) pointsthe way towards an efficient simulation of the many-body problem. Therefore, incorporatingcomputational physics within our philosophical analysis confirms our picture on renormalizationtheory.

3.4.4 The object of physics

The two previous sections have showed that renormalization theory is constitutive for our phys-ical picture of the world. We have shown (i) that understanding a physical phenomena requiresthe physicist to write mathematical theories about effective degrees of freedom that are only rel-evant at a certain scale, (ii) that renormalization theory entails in a mathematically exact wayhow all these local understandings can be incorporated into an inclusive whole, and (iii) how bothanalytical and computational approaches can contribute to understanding physical phenomena.Therefore, renormalization theory provides us with the conceptual structure in which we canfurther build and order a unified understanding of the physical world: We have theories aboutphysical phenomena that are only valid on a certain energy or scale, and these different theoriesare connected through scale transformations as they appear in the renormalization group.28

28As an illustration of this view, the following quote by Leo Kadanoff (one of the co-founders of the theory ofrenormalization) is remarkable:


This picture of the physical world is different from the one theoretical physicists have longthought to be working towards. Indeed, another idea of unification has traditionally been themotor behind ‘fundamental’ physics, viz. the idea that we, in the end, want to understand allphysical phenomena starting from a fundamental theory of the elementary constituents of thephysical world. This was the drive behind the mechanistic program in the nineteenth century,or the hope in the early days of quantum mechanics, and remains, to this day, the goal thatstring theorists set themselves. In its place, another idea of unification has taken root in thestructure of theoretical physics. We believe that this is, from the philosophical point of view,the most interesting way of understanding the discussion concerning reduction and emergence.We can understand the writings of Anderson and others [see Sec. 3.3.4] as a renunciation of thisold idea of unification, and as an articulation of the new idea of what a unified physics consistsof.29

It is interesting to note that this view on physics is much in the line of Cassirer’s charac-terization of physical concepts as relational and non-substantialistic. In the beginning of thissection, we have already reiterated the views of Cassirer on the energy concept as preferablefrom an epistemological point of view, because it provides a way of relating qualitatively differentphenomena without reducing them both to some common substantial basis. Renormalizationtheory shows that there is no fundamental theory that explains all physical phenomena in onestretch, but that physics always needs effective descriptions valid on certain scales. Yet, allthese effective descriptions are connected through renormalization-group flows, which determinea strict mathematical relation between these descriptions. Therefore it would be a fatal mistaketo interpret the effective degrees of freedom on a given energy scale in a substiantialistic way –as the emergence of a new ontology – since they appear as elements in a renormalization flow.In the spirit of Cassirer, the development of renormalization theory is interpreted as yet anotherstep in the evolution towards less and less substantialistic conceptions of the physical world, andtherefore confirms the progression in the historical development of theoretical physics.

We have tried to make clear that ontology in philosophy of physics is uncalled for. Thereason why philosophers take up this notion time and again should be viewed in the light ofthe realism/anti-realism debate, and the fact that the ontological commitments of a certainphysical theory – the fundamental nature of the world that it lays bare – are important from a

After its modern construction by Wilson and others, the renormalization group has appeared inthousands of papers devoted to the development of the understanding of physical, social, biologicaland financial systems. However, renormalization is substantially more than a technical tool. It isprimarily a method for connecting the behavior at one scale to the phenomena at a very differentscale. It serves for example, to connect the physics at the scale of an atom with the observed macro-scopic properties of materials. One might argue, and I believe that argument, that the connectionamong “laws of nature” at different scales of energy, length, or aggregation is the root subject ofphysics. One would then argue that Wilson has provided us with the single most relevant tool forunderstanding physics. [52, p.2]

29In this chapter we have largely ignored all the efforts that theoretical physicists are investing in string theoryas the best option for a unified theory encompassing both quantum field theory and gravity. These efforts can beread as an articulation of the ‘old’ idea of unification, and could show that this idea has not at all disappearedfrom contemporary physics. We should note, however, that the endeavors of string theory are not necessarily incontradiction with the ‘new’ ideal that we have put forward. Indeed, in a paper by David Gross we read:

First this theory, used simply as an example of a unified theory at a very high energy scale, providesus with a vindication of the modern philosophy of the renormalization group and the effectiveLagrangian that I discussed previously. [...] String theory could explain the emergence of quantumfield theory in the low energy limit, much as quantum mechanics explains classical mechanics, whoseequations can be understood as determining the saddlepoints of the quantum path integral in thelimit of small ℏ. [53, p.66]

How to incorporate string theory in our philosophical framework, however, we leave for further study.


philosophical point of view. As a consequence, it is on the level of ontology that the scientificrationality of physical theories is to be maintained. In the framework of Cassirer, however,we have found other resources for grounding the rationality of physics: it is by showing howphysical concepts succeed in capturing more and more of the sensuous world in a mathematicallystructured whole, and, as such, giving physical phenomena an exact theoretical meaning. Byfitting renormalization theory within this framework, we have shown that the rationality ofcontemporary physics can be maintained without taking recourse to a realistic and/or ontologicalaccount of physics. Moreover, the work of Friedman has shown that this approach is not renderedfutile in the light of Quinean holism and Kuhnian paradigm dynamics, and our analysis can beread as a confirmation of this project.

3.4.5 Historicizing renormalization

In Sec. 2.5.3 we have discussed a crucial distinction between the frameworks of Cassirer andFriedman with respect to the constitutive function of a priori principles. Whereas for Cassirerthe conceptual structure of theoretical physics generates a scientifically determinate meaningfor empirical phenomena – there are no meaningful physical phenomena before the conceptsof physics – for Friedman there is a faculty of pure sensibility that generates a space of directperceptions, where coordinative principles are supposed to bridge between pure sensibility andabstract (mathematical) physical theories. Let us see how the insights from this chapter canshed more light on this distinction between Cassirer and Friedman.

In the previous subsections we have adequately shown that the approach of Cassirer is verywell suited to integrate the theory of renormalization into the conceptual structure of contem-porary theoretical physics. In particular, we have shown that the theory of renormalization,and the associated ideas of scale and effective degrees of freedom, are necessary to yield definitedescriptions of physical phenomena. In that sense, the theory of renormalization yields a setof principles that is constitutive for giving physical meaning to empirical phenomena. On theother hand, we have identified the framework of renormalization as yielding a new blueprint ofwhat a unified physics should consist of. Physicists no longer aim for one comprehensive theoryallowing to understand the physical world in one stretch, but rather look for a description ofthe effective degrees of freedom that determine a given physical phenomenon on a certain scale.This points to the regulative function of renormalization theory, because it teaches us what arethe conditions of any physical theory and which is the direction in which physical theories areevolving. Therefore, we believe that the theory of renormalization serves as an illustration of theinterplay between the constitutive and regulative dimensions of a priori concepts in theoreticalphysics, an interplay that is characteristic of a priori principles in the approach of Cassirer. Also,the history of the theory of renormalization clearly shows that this interplay is a dynamic one.

So what about the level of coordinating principles that Friedman takes to be essential forattaching empirical meaning to the abstract theories of mathematical physics? In the firstinstance, one is tempted to assign a coordinative function to the ideas of scale and effectivedegrees of freedom. Indeed, whereas mathematical concepts are entirely abstract and lack anyempirical meaning, their empirical content is gained by specifying the scale at which they aresupposed to apply and the effective degrees of freedom they are supposed to capture. Yet, onequickly realizes this is not what is happening: effective degrees of freedom do not live in thisperceptual space of pure sensibility, because they are elements of the mathematical frameworkand don’t have any meaning outside of it. Similarly, the concept of scale is a physical conceptthat only makes sense in the context of the renormalization group and scale transformations. Ingeneral we can say that it makes no sense to try to fit the principles of renormalization theory ina schematism that tries to bridge somehow between the space of abstract mathematical theorieson the one side, and a faculty of pure sensibility on the other.


However, we have seen that Friedman’s account aims at opening up other dimensions inwhich physical principles acquire meaning. In his notion of the historicized a priori, Friedmansets out to fix the rationality of scientific principles by placing them in a larger intellectual andtechnological development. In this chapter, we have focused on the internal dimension of theprinciples of renormalization theory, for which we found the framework of Cassirer to be ideallysuited. Yet, we believe that the theory of renormalization can provide an interesting case forexploring these larger dimensions that Friedman is aiming at. A few interesting directions couldbe

• the technological context: The advent of computational physics requires the introductionof computer technology.

• the intellectual context: The development of renormalization ideas takes place in a physicscommunity that focuses rather on widening the scope and solving problems in more diversefields of physics, than on redefining the ‘fundamental’ concepts.

• the political context: It should be noted that this new way of doing physics falls within theaftermath of World War 2 and, in particular, the Manhattan project, which have reshapedthe scope and funding of theoretical physics.

As an illustration of the intertwining of these three dimensions, we note that one of the firstapplication of computers in physical research was during the Manhattan project, where comput-ers (and physicists) were used to determine how much energy is released in an atomic explosion[54]. It remains a subject of further study to what extent these three dimensions can be workedout further, and could lead to a thoroughly historicized account of the development of postwartheoretical physics.

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