Evolving Foundations ofScienceE. Atlee Jackson

SFI WORKING PAPER: 1994-06-035

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SANTA FE INSTITUTE

Evolving Foundations of Science

E. Atlee Jackson

Department of Physics, and

Center for Complex Systems Research

Beckman Institute

University of Illinois at Urbana-Champaign

The world is a dynamic place, and much of the evolutionary activity going on in the foundations

of science is in response to desires to understand dynamic processes in Nature. These changes are both of a

technical nature, requiring a careful examination of our sources of information and their possible interrela­

tionships, as well as the exploration for the new unifying concepts that can be applied to all scientific efforts.

This is a big topic, and the present brief essay will only give a structural outline of some of the these initial

changes, in an attempt to clarify the relationships between various approaches to "understanding" dynamic

systems.

More specifically, the approach will be a "bottom-up" organization, starting with the most basic

operations of generating scientific information, proceeding to the less objective activities that relate ("en­

code/decode") these different forms of information, which can be used to produce various cyclic "scientific

methods" for validating our understanding of phenomena, and finally trying to discover the new concepts

that will produce a unifying approach for understanding phenomena in all of the natural sciences. This

ordering helps to eliminate many ill-defined "scientific" concepts, and clarifies these new basic aspects and

resulting challenges in this evolving scene.

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This bottom-up ordering of scientific activities will be organized into three "tiers," for lack of a

better name:

I. The three information-generating operations used in modern science, and the distinct characteristics

of their information.

II. The problems associated with the encoding/decoding of these different forms of information between

these operations, and the multiple types of cyclic routes that can be used to connect these forms of

information. Each cycle yields a different type of "scientific method," which is intended to validate

our understanding of some phenomenon.

III. Some of the questions that are related to the new organizing principles of science, which are needed

to replace the historic, nonscientific philosophy of microreduction-followed-by-deductive-syntheses

(or simply, the reduction/synthesis philosophy).

While each of these tiers involve a myriad of issues, I will select only a few of the most basic elements,

following the above order.

Tier I: Informational Methods

Presently there are three methods for generating information that are used in "natural sciences."

These three methods will be referred to as: (1) Physical Observations, (2) Mathematical Models, and (3)

Computer Experiments. It is essential for understanding the changing character of science that these methods

be defined with some care. Here I can only give a flavor of these defining issues:

1. Physical Observations (PO)

Physical observations produce the basic information for our knowledge about Nature. Scientific ob­

servations must satisfy various conditions related to the arrival of a concensus, and the ability to record

information, all of which requires the use of some instruments. There are presently many methods of record­

ing both ('qualatitive" and "quantitative" information, which historically have been viewed as disconnected,

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excluded-middle forms of information, while there are now possibilities for bridging these forms of informa­

tion, as will be discussed in (3). The quantitative sciences have historically held a special prestige, because

of their relationship with mathematics and its logical precision (forming the "hard" sciences, Le., relatively

simple sciences). Changes which tend to make qualitative descriptions an extension of quantitative factors,

will dramatically change this elitism, as will be discussed in Tier II.

In POs we attempt to find a set of observables S(P) = {t,Oi(P)li = 1, ... , n(P)}, related to a

specific physical phenomenon, P, which we believe are "correlated" with each other (a "discovery" process).

Here, t represents the observable ('time," as defined by an accepted set of clocks, and "phenomenon" refers

to a history of observations over some finite duration of time, T. By "observables" is meant some distin­

guishable set of characteristics associated with physical phenomena, detectable by some instruments that are

agreed upon by interested scientists, and which are either reproducible (as in experiments) or comparable

with other examples (as in Astronomy, Geology, and Meteorology). Quantitative observables always have

some associated accuracies, {dt, Ai}, related to the instruments employed. While obvious at one level, it is

important to emphasize that we only observe a limited set of different phenomena related to any system, not

the "entire physical system" (a reductionistic phrase). Thus each set of observables, S(P), simply represents

one ''window'' through which we have viewed the physical system, and a very limited colIection of these sets

represents our total knowledge about a system. Issues such as these will very likely be fundamental to any

hierarchical understanding of Nature (outlined in Tier III).

An essential aspect of the information from physical observations is that it is finite, both in the sense

of the number of observables, n(P) + 1, their accuracies, {dt, Ai}, and the finite duration of observations,

T. The accuracies {dt, Ai} have a finiteness characterized by continuous intervals of real numbers. For

lack of a better description, I will describe such experimental numbers as "partition numbers," indicating

their association with continuous intervals of the real numbers. Such considerations are important in the

development of various scientific methods, to be discuss in Tier II.

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2. Mathematical Models (MM)

Historically the mathematical models (or "theories," as they were often called) that have dominated

science following Newton, are those that are based on differential equations. The introduction of calculus

into science has had a profound influence on the development of scientiflc ideas and philosophical attitudes,

some of which will be discussed in Tier II.

For the present, we simply want to outline the essential features of mathematical models. The

most important point is that mathematics is the "Science of the Infinite," as described by Leibniz. This is

characterized by the use of the real numbers, and such concepts as continuity, rational and irrational numbers,

dense sets, limiting processes, derivatives of functions, and more recently, fractals and fractional dimensions of

sets. This basis of information is quite distinct from the type of information obtained in physical observations,

as outlined in (1). This informational mismatch plays an important role in the evolving developments of many

new "scientific methods," to be discussed in (II).

The second essential aspect of mathematics is its use of Aristotelian logic for the development of

deductive processes. The various forms of analyses based on this deductive reasoning have been expanded

greatly over the past century. It has been found that there are basic limitations in some classic forms of

analyses, and at the same time new methods for generating other forms of information, particularly in the

area of dynamics. Many of these methods were initiated by Poincare , and involve topological, bifurcation,

and mapping concepts, which produced entirely new forms of information. One can, in fact, usefully draw

the analogy between these methods of analysis, related to MM, and the scientific instruments, related to PO.

The contribution of these deductive methods to various "scientific methods" will continue to be explored

and evolve in the future. Some of these issues wiil be discussed below.

3. Computer Experiments (CE)

The introduction of the digital computer around 1950 has fundamentally changed the opportunities

for generating scientific information. However, before exploring this in (II), we need to define more carefully

the nature of the information obtained by such computer operations (simulations, experiments, or analyses).

The point I want to make is that computer experiments are finite operations in several senses. All computer

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experiments involve finite algorithms, operating on a finite integer field of numbers, and performed for a

finite number of iterations. This finite character, while similar to physical observations, does not employ

the continuous number intervals (partitions) associated with experimental observations. The integers still

represent infinite precision, but the finite sets in CE differ fundamentally from the continuum of real numbers

in MM. On the other hand CE operate with the same logical manipulations that are used in MM, but

numerical round-offs add an "illogical" component, which may not be detrhnental in a scientific context (see

below). It is also important to emphasize that CE are generally quite distinct from many interests in the

field of Computer Science. In CE there are none of the infinite tapes and operations found in Turing-machine

considerations (which, in this sense, shares infinity with MM), which makes CE more closely akin to PO.

To stress the point again, the information contents of (PO, CE, MM) have the respective relationship

to sets of (partition, rational, real) numbers. This distinction plays an important role in (II). Perhaps at this

point it is also worth noting that, while the number of elements used in CE is finite, these sets of numbers

may he incomprehensible to scientists, and can only be rendered "comprehensible" (in any human sense)

with the use of computer graphics. Thus computer graphics has given us a method to begin to bridge the gap

between qualitative and quantitative observations-a practical issue of importance in data compression, used

in such fields as NMR images in medicine and elsewhere. However, this qualitativejquantitative connection

may have far more fundamental implications for the future in such areas as pattern recognition, and artificial

life research.

Before proceeding with the considerations in the next tier of this organization, it is worth noting

that this simple process of reviewing informational processes, points out the need to relate certain commonly

used terms to each of these forms of information. As a saying goes, "You know what you know when you

know how you know." Thus such terms as "determine," "predict," "reproducible," "understand/' which are

so commonly used in scientific discussions, are in fact very different concepts when viewed in terms of the

different sources of information which are used in their implementation. A review of these issues deserves

more discussion than can be given here, but some sense of what is involved can he conveyed by considering

the concept of ('determinism."

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A mathematical system is deterministic provided the solution of the equations that determine the

dynamics of some function x(t) yields unique values of x(t) for all t > 0, when the value of x(t = 0) is given.

In other words, every initial state, x(t = 0), has only one future state. All "values" are infinitely precise real

nurubers, and the proof that a mathematical system is deterministic is based upon logical operations. This

then is "determinism" within the context of MM.

Determinism within the context of CE is likewise based on logical operations. Indeed, since algorithms

necessarily specify the succeeding values of a function at every time step (iteration), any CE is ipso facto

"deterministic" in this sense. However, CEs differ from MMs, in that the numbers being used are all bounded

integers, which requires machine-dependent ('roundoff errors" in terms of real numbers. This insertion of

an "illogical" informational influx may, in fact, better represent the sensitive response of a system to its

environmental perturbations, and its bounded determinism (particularly if the round-off is made machine

independent by stochastic means). Finally, the time interval over which the determinism is implied is finite.

Uniqueness is, of course, guaranteed for any properly functioning computer (but, did your computer function

"properly" at each step?), and will reproduce the same answer from the same initial state. Putting these

ideas together we see that, while MM-determinism does not refer to the mathematician, CE-determinism

does depend on the computer, as well as the operator of the computer. That is, on how high-tech the

computer may be, and the amount of time scientists are willing to invest in running the computer. Clearly,

CE-determinism is not as pure and abstract a concept as it is in MMs.

"Determinism" within the context of PO is even less precise than either of the two above concepts.

The initial states are now partition numbers, and so are our comparative states in the future, and of course

we only observe for limited periods of time. Repeating the experiment does not yield the same results

within these bounds, unless further specifications are made between the partition ranges and the temporal

bounds. Foremost, there is no logical basis for establishing any such "determinism" within the operations of

physical observations. In this sense "determinism" does not exist. For this logical basis, we must make an

association with either CE or MM (Tier II). Our only purely PO sense of determinism relies on our finding

some limited set of observabies, B(P), related to some phenomena, P, which remain quantitatively related'

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to each other within the bounds of the partition numbers, for some finite time, and for a limited number of

experiments-and are reproducible by other scientists. This is the reality of real-life science.

Historically, many of these issues did not arise because the phenomena studied were either equilib­

rium or steady-state properties of systems. With the extension of interests into the realm of the dynamics

of complex systems, such issues can no longer he ignored.

Tier II: Validating Cycles-"Scientific Methods"

A "scientific method" of validating our understanding of some set of physical phenomena ideally

involves a cyclic connection between PO and one or more of the information operations, MM and/or CEo

This is schematically represented by:

NATURE

~PO

~ \\c.CE ( ') MM)

<::

Much of the heart of scientific knowledge is contained within the loops at each of these vertices.

Unfortunately, to discuss this in any detail would greatly expand what is intended to be a brief introduction

to this subject. However, not to leave this totally obscure, consider the most familiar case of POs, which

involve Nature, the 'observations of natural phenomena, and the discoveries related to these observations. The

observations require various instruments, which compress what is going on "out there" into some recognizable

and recordable form of information. Any cyclic "scientific method" process to be discussed shortly, is required

to make contact with this information. However, at each of these vertices, there exists a different "Nature,"

which requires some "instruments," such as methods of analysis, or types of data compression, in order

to obtain the "observations" (information) relevant for these validating cycles. Much of the evolution of

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science has been, and will continue to be, in the development of these "instruments," which generates our

"information" from these sources. For the present brief discussion, only the connection of these forms of

information will generally be discussed.

A cyclic, "scientific method" of validation, connecting PO with MM and/or CE, can be initiated

by some inspiration, drawn from any informational method (e.g., by proposing some mathematical model,

making some physical observation, or exploratory computer calculations). The connections between these

methods will be referred to as "encoding" process (leaving any source of information to a new source) and

"decoding" process (upon returning to any). This will involve both inductive processes and a transcription

of information between the forms obtained in the different methods. The process of inductive reasoning has a

long history, and is highly influenced by subjective considerations that will not be explored here. In contrast,

there are newly-found objective (technical) concerns about how these distinct forms of information can be

related to each other, both in the encoding and decoding processes. These issues are particularly important

in considering the dynamics of complex systems.

The encoding/decoding processes raise many interesting issues. One example is that we now know

that MM-determinism (as outlined above) can imply PO-indeterminism. Thus, paradoxically, the mathe­

matical model that is developed in order to explain some determinism observed in life, can also yield the

conclusion that other physical observations of the same system can not lead to PO-determinism. ll] What

is one to make of such results? Are they "real," or just some mathematical artifact, existing only in the

nonobservational world of THE INFINITE? Likewi$e, a CE of simple periodically forced damped pendulum

indicates that many physical observations cannot predict which of four (say) asymptotic stable oscillations

II] An example, which is the quintessential example of Newtonian determinism, involves three gravitationally attracting masses.For the "simplist" case, in which two large masses circle each other, while a small mass moves in this plane (the "reducedthree-body problem"), Poincare used a mapping technique to conjecture that chaotic dynamics is common to this system.Shortly after Poincare's death, Birkhoff proved Poincare's "geometric theorem," which led to the following results: In any (note,ANY!!) neighborhood of a certain type of periodic orbit (which occur densely in the phase space) there are an infinite numberof periodic orbits, with periods larger than any prescribed number. Now ask yourself, what if we tried to measure the period ofan orbit? What will we find? Do we believe the micro-prediction of a model originally established only from macro-information?Birkhoff, B. D. Dynamical Systems. Amer. Math. Soc. Colloquium Pub., 1927.Birkhoff, G. D. "Une generalisation a n dimensions du dernier theoreme de geometrie de Poincare." C.R. Acad. Sci. Paris192 (1931): 196-198.Birkhoff, G. D. "Proof of Poincare's Geometric Theorem." Trans. Amer. Math. Soc. 14 (1913): 14---22.

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will be obtained by this system.121 Perhaps more paradoxically, we can also show that this is not always

the case; that is, unobservable MM predictions can be used to predict other observable phenomena. 131 Thus

a case can be made for the idea that MM and CE, which contain unobservable information, nonetheless

represent a form of "understanding" that allows us to predict selective observations, based on this unobserv­

able information. Also, in encoding/decoding between MM and CE, there have been a number of studies

involving such concepts as computable numbers,4 and the "shadowing problem." [4] Some of these issues have

no relevancy to any scientific method, because they involve aspects of THE INFINITE that can not arise in

the encoding/decoding connection with PO (which is necessarily in the cyclic scientific method for natural

sciences). Indeed, all PO are only associated with collections (ensembles) of CE or MM solutions,5 and this

removes many technical considerations involved in any infinite process.

The variety of cyclic "scientific methods" that can be established within the framework the above

informational triad has hardly begun to be explored. To illustrate but a few examples of some new issues,

consider the cyclic connection between CE and PO. With the operational power of computers, it is possible

to take observational data, suitably encoded and used in algorithmic manipulations, then decoded back

to PO, yielding new physical predictions. These manipulations need not be based on any physical model

of the system, so that predictions can be obtained without any physical "understanding." This certainly

raises questions about the notion that predictions are a primary source of "validation" within the "scientific

method."6

[2] The system is a simple periodically forced damped pendulum, and the basins of attraction of the four stable fixed pointsappear to have the "basins of Wada" property, namely the boundary of any basin of attraction is the boundary of all basinsof attraction. See S. W. McDonald et al. Physica 17D (1985): 125. A yet more refined example of uncertainty, referred to as"riddled basin," is discussed in E. Ott et al. Phys. Rev. Lett. 71 (1993): 4134 and J. C. Sommerer and E. Ott. Nature 365(1993): 138.

[3l An unstable periodic orbit can not be physically observed, but in the neighborhood of this orbit the dynamics will be nearlyperiodic for a finite time, and this can be observed, and used for purposes of controlling chaotic motion. For example, see: T.Shinbrot, C. Gregbogi, E. Ott, and J.A. Yorke, "Using small perturbations to control chaos," Nature 363 (1993): 411. For amore general perspective: G. Chen and X. Dong, "From chaos to order: perspectives and methodologies in controlling chaoticnonlinear dynamic systems" Int. J. Bifurcation Chaos 3(6) (1993).

[4l For an introduction, see Jackson, E. A. Perspectives of Nonlinear Dynamics, Vol. 1 Cambridge, MA: CambridgeUniversity Press, 1990. For a recent reference see Sauer, T. and Yorke, J. Nonlinearity 4 (1991): 961.

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More generally, each vertex in the above triangular figure contains some form of mathematical,

computational, or experimental analysis or comparison before an encoding or decoding process is made to

another vertex. The variety of new mathematical methods of analysis that have been developed over the

past century has greatly expanded our appreciation of the amazing dynamic contents of many "simple"

equations-an insight that was not even suspected prior to Poincare's studies around 1890. This variety has

been joined by the entirely new potentials of computer experiments, which have already uncovered numerous

new features in the solutions of ODE and PDE's not revealed by any form of mathematical analysis. And, of

course, our ability to make new types of physical observations has enriched the potential of PO immensely.

Putting these all together, there is clearly a plethora of scientific methods to be developed in the future.

Tier III: Unifying Programs; Hierarchical Complexes

One of the most publicized philosophical positions taken by many influential physicists of this cen­

tury is that the objective of science is to understand the "fundamental" interactions between elementary

particles, from which all natural phenomena can be deduced, "in principle" at least, by pure logical reasoning.

This microreduction-followed-by-deductive-synthesis philosophy I will refer to simply as the RS (reduction­

synthesis) philosophy of science. The "deductive purity" of this program was particularly stressed by Einstein

on several occasions (1918, 1933),7 and has been recently supported by many people, with varying degrees

of ambiguity and qualifications. Among the most influential people are S. Weinberg (a "compromising re­

ductionist") and S. Hawking, who have swayed public opinion through congressional testimony, articles, and

books.s,9,lO,n,12 Not all eminent physicists have agreed that the RS program is viable, as notably illus­

trated in P. Anderson's early challenge,'3 L. Kadanoff14 who argued against reductionism and the concept

of a hierarchy of knowledge, and A. Leggett's critical examination of scientific knowledge, particularly as it

relates to quantum phenomena.15 Likewise, F. Dyson discusses many of these issues in his book "Infinite

in All Directions," whose title is derived from one of the early "diversifiers of physics," E. Wiechert (in

the 1890s),16 with whom Dyson associates Wheeler as a modern counterpart.l7,lS,19 While many of these

eminent physicists have been sensitive to certain limitations of the RS program, other scientists who have

been involved with the technical aspects of the dynamics of complex systems have been more informed and

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perceptive about other fundamental issues. Some have realized that the RS program was not only totally

unfounded by any scientific evidence, but that it had been dealt mortal blows at all levels (micro to macro)

of dynamic modeling by "newly" (> 1890!) discovered dynamic features that were deductively established

within the hallowed halls of mathematical analysis. [51 In addition, biology, the premier "qualitative" field

of science, began to acquire not only a quantitative voice in such areas as molecular biology, but also a

fuller appreciation of its unique contribution to our knowledge in the evolutionary field through the genetic

revolution:o,21,22 which brings the symbolic and qualitative domains into conjunction. Of course many of

these insights have already been encorporated into our exploration of the dynamics of nature. Nonetheless,

the relationships of these discoveries to the changing foundations of science has been recognized by very few.

There is a remarkably quite revolution going on, which has only begun to generate tremors during the last

decade. A reassessment of the "program of science" is barely in its initial stages.

With the demise of the RS program of science, we are challenged to find the new scientific concepts

that will unify at least some of the great variety of physical phenomena that we observe every day. It is only

in this changing scene that scientists are beginning to attempt to understand these complex phenomena.

The operative word is "understanding," with all of the implications of our human limitations (physical

observations), and the appreciation of the interactive character inherent in complex adaptive systems. The

history of science to date points clearly in one direction: that our successes have only come about by our ability

to find models for specific phenomenon that consist of a few "components," "agents," "collective variables,"

or "quasi-particles," together with their "effective two-body interactions," which can be used to deductively

explain this phenomenon. All of the many successes in the "understanding" of phenomena in solid-state

physics has relied, at a basic level, on this lhnited form of reductionism.[B] That is, our understanding

[51 The results outlined in footnote [1] illustrates the impossibility of synthesizing knowledge about even this simple system, giventhe gravitational1aws. In the same period Poincare proved a theorem concerning the nonexistence of constants of the motion thatare analytic in the coupling constant, which is a basic assumption used in general methods of perturbation analyses. Poincare'sresult was emphasized by Brillouin in his critique of Born's approximation methods---see Brillouin, L. Scientific Uncertainty,and Information. Academic Press, 1964. Kinchin, A. 1. Mathematical Foundations of Statistical Mechanics. Dover,1949. Contopoulous, G. "A Classification of the Integral of the Motion." Ap. J. 138 (1963): 1297-1305. There are numerous"emergent" phenomena, generated by computer experiments, for which no deductive synthesis has been found, starting withthe relatively simple Lorenz strange attractor, and extending into complex adaptive systems.

[6]A. Leggett is one of the very few, if not only, eminent physicist to have emphasized this sever theoretical and philosophicallimitation. See Leggett, A. J. The Problems of Physics, 176-179, 183. Oxford University Press, 1987.

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has only been accomplished by identifying familiar composites, or complexes, of the physical system, and

their two-complex interactions, from which we could make deductive conclusions, and predictions (provided

that the component interactions is the sum of these two-complex interactions-a point that needs close

re-examination in complex systems). This was noted by both Anderson and Leggett, particularly within the

solid-state and quantum realm. What is clear is that this same form of "understanding" is all that can be

accomplished in virtually all physical phenomena, most of which is totally classical (nonquantum modeled)

in its origins. We thus arrive at one partially-defined program of the science that explores the complex,

adaptive, and "emergent" phenomena, which dominate our experiences: the discredited RS program must

be replaced by some hierarchical form of one-level-reductions-followed-by-deductive-syntheses, which will

give us our understanding of physical phenomena. I will call this a hierarchical-complexes program (or HC

program), and emphasize that this will be particularly important in the context of dynamic phenomena

(i.e., the "complexes" will not simply have structural, but also dynamic features). These hierarchies may be

dynamical, functional, or cognitive in Nature, or some mixture of these characteristics. Part of this program

is to see if there are unifying "dynamic motifs" associated with these levels. Presently it is not clear what

constraints need to be applied in the development of these HC, so that they will be generally accepted as a

basis of understanding of higher-order phenomena. Certain conservation laws obviously come to mind, but

these were established at a very low level in the HC, and the more general constraints are by no means

as clear. Moreover each "window of understanding" which we establish in relationship to some physical

phenomena, may generate its own He into more complex phenomena, so different windows can establish

parallel HC.

This picture of our understanding of Nature breaks completely with the RS program, not only in

relationship to deductive limitations, but perhaps more wrenchingly, in relation to the Judeo-Christian faith

associated with a "fundamental-law-driven" view of nature. The above program does not, of course, make any

judgements concerning such a faith, but it does recognize that we humans, with all of our finite limitations,

are quite incapable of verifying such a faith; that philosophically basing a science upon the primacy of any

set of physical "laws," which are inherently capable of only giving us a limited understanding of a few

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phenomena in nature, would be highly detrimental to the future development of scientific knowledge. What

we need to search for are those achievable methods that will help us explore and understand some parts

of our wondrous universe, and to discover those limited deductive connections between these hierarchical

complexes. So far we have glimpsed only a few enticing examples of such processes.f7l

REFERENCES

1. Birkhoff, B. D. Dynamical Systems. Amer. Math. Soc. Colloquium Pub., 1927.

2. Birkhoff, G. D. Une generalisation a n dimensions du dernier theoreme de geometrie de Poincare.

C.R.Acad.ScLParis, Vol. 192. (1931): 191>--198.

3. Birkhoff, G. D. "Proof of Poincare's Geometric Theorem." Trans. Amer. Math. Soc. 14 (1913): 14-

22.

4. Traub, J. F. & WozzniakowskL "Breaking Intractability." Sci. Amer. (1994): 102.

5. Brillouin, L. Scientific Uncertainty, and Information. Academic Press, 1964. Kinchin, A. 1. Mathemat­

ical Foundations of Statistical Mechanics. Dover, 1949. Contopoulous, G. "A Classification of the Inte-

gral of the Motion." Ap. J. 138 (1963): 1297-1305.

[71 The demise of the RS program and the hierarchical program suggested here, means that there is no hierarchy to knowledge(also see Kadanoff, L. P. From Order to Chaos, Essays: Critical, Chaotic and Otherwise, 399-429. Singapore: WorldScientific, 1993: Chaos: A View of Complexity in the Physical Sciences. Originally from The Great Ideas Today, 86.Chicago: Encyclopedia Britannica, 1986.). but a hierarchy of functional or organizational complexity (also see, my "Perspectivesof Nonlinear Dynamics," Vol. 2, PP 330). It is worth noting that, while this hierarchical organization Nature was noted in the1970's in a "scientific" context,13 it is an idea that arose 25 centuries ago, independently in both the cultures of Greeceand China. An interesting discussion of the concept of "The Ladder of Souls" can be found in J. Needham's "Science andCivilization in China," Vol. 2 , pp 21 (Cambridge Univ. Press, 1956). Thus water and fire is distinguished from, plants, thenanimals, and finally man, according as to their souls having "subtle spirits," plus life, plus perception, plus justice (in China),whereas Aristotle ordered plants, animals, and man according to vegative souls, plus sensitive souls, plus rational souls. Thedistinction between the souls of man in the two cultures was characteristic of basic differences in these two cultures, and relatedto to the lack of the development of science in ancient China. Another interesting hierarchical consideration is the developmentof "understanding" in science, beginning with the Newtonian paradigm involving inter-relationships of observables, to theinclusion of non-observables (such as fields, or wave functions), to nonintegrable mathematical models, to collective concepts("complexes" of some sort), and the various aspects of "emergence" (e.g., for a nice discussion see E. Mayr, lIThe Growth ofBiological Thought"; for an imaginative and more technical attach on llemergence," see: J. P. Crutchfield's liThe Calculi ofEmergence: Computation, Dynamics, and Induction," Santa Fe Institute Report 94-03-016).

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6. For a more thoughtful discussion of "scientific methods," see: Bauer, H. H. Scientific Literacy and the

Myth of the Scientific Method. Urbana: University of Illinois Press, 1992.

7. Einstein, A. Ideas and Opinions, 225-226. New York: Bonanza Books, NY: and in the 1933 preface to

Where is Science Going?, by M. Planck. Woodbridge, CN: Ox Bow Press, 1981.

8. Hawking, Stephen W. A Brief History of Time, 12-13. Bantam Books, 1988.

9. Weinberg, S. "Towards the Final Laws of Physics." In Elementary Particles and the Laws of Physics,

61-110. Cambridge, MA: Cambridge University Press, 1987.

10. Weinberg, S. "Newton's Dream." In Newton's Dream, edited by M. S.Stayer, 96-106. Queen's Quar­

terly, McGill-Queen's University Press, 1988.

11. Weinberg, S. "Newtonianism, Reductionism and the Art of Congressional Testimony." Nature 330

(1987): 433-437.

12. Weinberg, S. Dreams of a Final Theory: The Search for the Fundamental Laws of Nature. Pantheon

Books, 1992.

13. Anderson, P. "More is Different." Science 177 (1972): 393-396.

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