The Cambridge History of Eighteenth-Century Philosophy

Volume II

EDITED BY

KNUD HAAKONSSEN

University of Sussex

" CAMBRIDGE ,:, UNIVERSITY PRESS

28

ARTIFICE AND THE NATURAL WORLD :

MAT HEMATICS, LO GIC, TECH NOLOGY

JAMES FRANKLI N

If Tahiti suggested to theorists comfortably at home in Europe thoughts ofnoble savages without clothes, those \vho paid for and w ent on voyages there were in pursuit of a quite opposite human ideal. Cook's voyage to observe the transit of Venus in 1769 symbolises the eighteenth century's commitment to numbers and accuracy, and its \villingness to spend J lot of public money on acquiring them. The state supported the organisation of quantitative researches, employing surveyors and collecting statistics to compute its power, J People volunteered to become more l1umerate; .l even those who did no t had the numerical rationality of the metric system imposed on them, ) There was an increase of two orders of magni tude or so in the accuracy of measuring instruments and the known values of physical comtants. 4 The graphical display of quantitative information made it more readily available and comprehensible. j O n the research fiunt, mathematics continued its advance, even. if with notably less speed than in the two adjoining centuries. The methods ofehe calculus plUved successful in more and more problems in mechanics, both celestial and terrestriaL Elasticity and fluid dynamics became mathematically tractable for the first time. 6 The central Limit theorem brought many chance phenomena within the purview of reason.

These successes proved of interest for ' low philosophy' , or philosophy- as­propaganda, as practlsed by the natural theologians and the Encydopedistes. Both had their uses for scientific breakthroughs, though sometimes not much in­terest in the de tails. For 'high philosophy'. as consti tuted by the great n~l.lnes, mathematics and science had a different importance. A feature common to the biographies of all the well-knm.,vl1 philosophers of [he eighteenth century is a mathematical youth. WolfF began as a professor of mathematics, and it \-vas in that subject that he first made the contributions to the intellectual vocabu­lary and style of German for which he is so universally loathed. Kant taught mathematics, and his Prize Essay begins with an analysis of the mathematical method. D ' Alembert , Condorcet. Lambert, even Diderot in a smaller way (and of course Leibniz earlier) made serious mathematical contributions. R eid also taught mathematics , and his first published vmrk was on quantity. Paley was

817

James Frank.fin

Senior Wrangler in the Cambridge Mathematical Tripos. Berkeley's Analyst is one of the most successful interventions ever by a philosopher in to mathem~ti cs .

Hume and Vico, though no mathematicians, used mathematical examples as the fi rs t illus trations of their theories. Adam Smith's 'invisible hand' and Malthus's model of population growth both belong to \",'hat is nm'\! called dynamical sys­ten15 theory.

N aturally, these philosophers did not all draw the same lessons fr0111 their mathematical experience. But philosophers have one ching in common in their attitude toward mathematics, in this last century before the surprise of non­Euclidean geometry undermined the pretcl15io ns of mathematics to infallibility. It is envy. What is envied, in parti cular, is the 'mathematical method' , whi ch apparently produced what philosophy \vished it could but had been unable to: certain trUtllS, agreed to by all , delivered by pure thought.

I. T HE ' MATHEM ATICAL METH OD ' PRAI SED

The eighteenth cenrury was the last to accept, fundam.entaily with out question, certain approved opinions of the ancients concerning the method and content of mathematics. The ideas were largely Aristotelian in origin but had survived the demise of scholasticism by being accepted almost in full by the Cartesians

and N ewton. The much admired 'mathematical method' is the deriving of truths by syJ10gisms from self-evident first principles; the method was believed to be instantiated by Euclid 's Elements. As to the content, mathematics is the science of 'quantity' , which is 'whatever is capable of increase, or diminution'. 7

N umbers arise fro m co nsidering the ratio of quantities to an arbitrar ily chosen unit. Quantity is of two kinds: discrete (studl ed by ar ithmetic) and continuous (studied by geom etry). However, geo metry is also the study of 'extension' , or real space. Q uantity in the abstract is studied by pure mathematics, w hile 'magnitude as subsisting in material bodies'S is the object of mixed or applied mathematics, which includes opti cs, astronomy, mechanics, naviga tion, and the like. Tendencies to regard mathematics as about some abstraction of reality did exist but were generally resisted. Euler, for example. says that in geometry one does not deal with an ideal or abstract tr iangle but with triangles in general , and that generality in mathematics is no different from generality elsew here;9 to the same purpose, d' Alembert defends an approximation theory, whereby the petfect circles of geometry allow us to 'approach' the truth, 'if not rigorously, at least to a degree sufficient for our use'. \0

There are several philosophical problems v.lith this complex of opinions, which are sufficiently obvious to keep surfacing in one form or another again

Artifice and the natural world 819

and again:

Why is the mathematical method found in nlm/tematics? • Why are the fIrSt principles in mathematics necessary, and ho\v are they knO\vn? • Is the reasoning in Euclid in fact all syllogisms? (in the strict sense, that is, of the form:

All A are B, All Bare C , so AJI A are C). If not, what kind of reasoning is it?

1. Wolff

Wolff at least had answers to these questions. The mathematical method, he thinb, is applicable everyvvhere; and there is no problem about the self-evidence of the first principles because there is only one of theIn, and it is the principle of non-contradiction . Geometrical demonstrations can aU be resolved into formal ~yllogisms, and discoveries in mathematics are made exclusively by syllogistic means. II His central place In eighteenth- century philosophy results iiom his at­tempt to derive all philosophical truths from th e principle of non-contradiction, by the 'mathematical method' .1 2 A look a t how he actually proposes to prove that everything has a sufficient reason, using only the principle of non- contradiction, reveals \vhy Wolff's 'method' achieved less than universal agreement:

Let us supposc A to b(! v ... ithou t a sufficient rcason why it is rather than is not. T herefore nothing is supposed by \vhich it can be under;tood why A is. Thus A is admitted to be, on the basis of an assum ed nothing; but since this is absurd, nothing is \-vithout a sufficient reason. 'J

Wolff's idea l differs from those of ochers essentially in lacking anything like Plato's djalectic, or Aristotle's inducti on, or Kant's analysis: the roundabout dis­cllssion and soning of experience which allows the intellect to come to an insight into first principles. It is unnecessary in Wolff's system because the prin­ciple of non- contradiction is the sole starting point. Any tendency to regard brute fact" as contingent and oucside the scope of explanation by necessary rea­sons is suppressed, in Wolff, by his acceptance of Leibniz's Best of All Possible Worlds theory. According to that theory, everything, however particular, has an explanation in principle.

2. iWathematics as philosophical propa .. ,!(fNda

Mathematics, because of its immense prestige, is always destined to be used in support of various philosophical positions. It was a natural as a prop for the Enlightenm ent motif that there should be more R eason all round. The Eucyclopedie says: 'M. Wolff ... made it clear in theory, and especiaUy in practice, and in the composition of all his works, that the mathematical method belongs

820 James Franklin

to all the sciences, is natural to the human spirit , and leads to discoveries of truths of all kinds. 1I4 Nor that the French needed Wolff to tell th em this, given the Cartesian ideals expressed by, for example, Fontenelle:

The geometrical spi rit is not so attached to geometry that it cannot be carried over to other know ledge as weU. A work of ethics, of policics, o f cr iticism, perhaps even of eloquence will be better, aU th ings else being equal, if it is made by the hand of a mathematician. The o rder, clarity, precision and exactitude which have re igned in the bener works recently, can well have had their first source i ll this geometrica l spirit which extends itself more than ever and "\vhich in some fashion cOllnl1unicltes i t~e1f even to those "vho have no knowledge of geometry. IS

Of course, there were counter-currents. There were the complaints common in all centuries from self-proclaimed 'practical men' like Frederick the Great and Jefferson, 16 w ho regarded the higher abstractio n .. of mathematics as use1ess, and from humanists like VieD and Gibbon, w ho abhorred the 'habit of rigid demonstration, so destructive of the fin er feeli ngs of moral evidence'. 17

Mathematics was also called to the aid of more particular philosophical theses. On the one side, there was the support allegedly given to natural theology by the various 'principles of least action'. On the other. the success of prediction in astronomy could be a support to determinism . [t was fowld th;,n many phe­no mena in p hysics could be derived from 'methods of maxima and minima' , or 'principles oflcast action', sllch as the one sta ting that the path ofl ight fi:om one point to another is the one which minimises the time of travel (even if the path is not straight, because of reflection or refrac tion). Maupertuis and Euler take this to be evidence of final causes, and for che existence of God. IR Their idea O\~tes something to the more general claim of Leibniz's T7Ieodicee that everything lS th e necessary resul t of a maximum principle, nanlely, chat th e goodness of this world is the best possible. D 'Alemben , to the contrary, warns of the danger of ' regarding as a primitive law of nature ",vhat is only a purdy mathematical consequence of some formulae ' . 19 H e believes the best hope for the countries

of Europe oppressed by superstition is to begin studying geometry, \vhich will lead to sound philosophy.20 Laplace also, in an image that haunts philosophy still, invites mathematics to assist an anti-religious 'Yvorldvlew:

Given for one instant lHl intelligence w h ich could comprehend all tht: fo rces by which n.Uure is animated and the respective situation of the beings who compose it ... it would embrace in the same formula the movements of the greatest bodies of the universe and those of the lightest atom. 21

History would thus be a subfield of the theory of differential equati ons .

Artifice and tile naturaiworld 821

3. D'Alcmbert versus Diderot

WoHrs ant-like progress through a farrago of equivocations and circularities is merely dispiriting and only brings rationalism into disrepute. D' Alembert is not so easily dismissed when he argues for essentially the same conclusions. His inunedia te claim, is not, indeed, that all subjects are entirely amenab1e to the geometric method, o nly that mechanics is. But mechanics is very inclusive, on a typical eighteenth-century view. If La Mettr ie and d ' Holbach were right about the nature of m an, fo r example, psychology would be a sub-branch of mechan­ics. D'Alembert argues, more convincingly than Descartes, that mechanics is a branch of mathematics , based like arithmetic on absolutely necessary first prin­ciples . In a kind of mathematical version of H"llme's scepticism about causes, he regards forces as 'b eings obscure and metaphysical': ' All we see distinctly in the movement of a body is that it crosses a certain space and that it employs a certain time co cross it. '22 Hence collisions are to be explained in terms of impenetrability, and the density of a body is merely 'the ratio of its mass (that is, the space it would occupy if it were absolu tely without pores) to its volume, that is, to the space it actually occupies'. 23 Tt might seem that there is no hope

of demonstrating the conservation of momentum purely geometrically:

However, if ·we consider the matter carefully, we shall see that there is one case in which equilibrium manifests i(self clearly and disti nctly; (hat is where (he masses of the tvvo bodi e.~ are equal, and the ir velocities equal and opposite.2.t

T he EI1C}'clopedie article 'Exphimentalc ', which one expects to be along Baconian lines, is in fa ct used by d'Alembert to propagate his extreme anti­experimental vie"\vs. H e regards collecting facts as a rather m edieval exercise, superseded by Newton's in troduction of geometry into physics. The laws of colliding bodies are demonstrable: nature could not be any other way. But there is an admission that how fast a body falls under gravity, and what the weight of a fluid is, must be measured; only after that do the relevant sciences become 'entirely or almost entirely mathematical':

No theory could have allowed us to find the law that heavy bodies follow in their vertical faU, but once this law is found through experience, all chat belongs to the movement of heavy bodies, whe{her rectilinear or curvilinear, whether inclined or vertical, i.~ found entirely by theory.1. j

W hile these cases appear as unfortunate weakenings of his original wish for purely deductive science, d 'Alembert's comm.cnts here are perhaps his most solid achievement. In the more mathematical sciences, experience does appear only in support of a few easily checkable synullctry principles and simple laws, while

822 James Franklin

most of the weight of explanation rests on the difficult mathematical derivations of more subtle phenomena £i-om these. And, as Leibniz points out ,26 a symmetry principle has a special logical status, being an application of the principle of (in)sufficient reason: in d'Alembert's example, if two bodies h ave equal and opposite velocities, theif mo menta must balance, as there is no reason why o ne should overcome the other. D ' Alembert was widely thought to have succeeded in showing that the principles of mechan ics had 'a necessity as rigorous as the first elementary truths ofgeometry' .27 Kant concurred in d'Alenlbert's unlikely conclusion . .28 Lagrange's MCcanique analytique of 1788 confirmed furth er that mechanics could look like a deductive system, managing almost to conceal the existence of forc es. 29

Of one mind on the iniquity of priestcraft, the inevitable progress of mankind, and o ther stich Enlightenmen t staples, the two priIne movers ofthe Encyclopedic feU OUt over mathematics. Diderot believed mathematics had reached its highest point and was now in decline. 30 He preferred sciences full of life and ferment, like chemistry and biology, criticising mathematics as abstract and ovt::r-simple. 31

D'Alembert, on the other hand, held that an abundance of experiential 'princi­p les' is 'an effect of our very poverty' Y Diderot's attack is not all invective; he has a philosophical argum ent to undermine rationalist pretensions about mathe­matics which L'i the same as the contention of cvvenricth-cenrury empiricists and positivists that mathematics is essentially triviaL Geometrical truths are merely

iden tities, saying the same thing in a tho usand diffe rent ways witho ut generat­ing any new facts. B D' Alembert allowed this argument to appear in the Di.scoUH preliminairc to the Encyclopedie, but replied that it just showed how powerful math ematics was to be able to get so much fr0111 so little. 34

II. THE 'MATH EMATICAL METH OD' DOUBTED

As in the twen tieth century, the success of science and mathematics attracted from professional philosophers not praise. but complaints, to the effec t thac they, the philosophers, could not see hmv so much knowledge could possibly be achievable. While very few were prepared to go as far as Diderot, much argument was undertaken to show that the claims of mathematical proof were not all they seemed. T he argument mostly centred on geometry. The problem. at its simplest, as Gauss put it, i ~, 'if number is entirely a product of our OV.r1l

minds, space has a reality outSide of our minds and we can not prescribe its laws a priori'.3) The century inherited what could be called the Euclid- Newton view of space and time. The essential features are these: Space and time are infinite in extent in all directions, homogeneous, fiat, and infinitely divisible. Truths about space and time may be proved with absolute certainty, in th e sryle

Artifice aHd the natural uJOrld 82 3

of Euclid. After t\VO thousand years of success. the fac ade seemed unbreakable. This prevented two developments which, at various times during the eighteenth century, seemed on the point of happening. The first is the discovery of non­Euclidean geom etry. The second is the adoption of the philosophical opinion that knowledge of space, \vhich is something real outside the mind, must be empirical and fallible. The problem with the Euclidean claims is that it is difficult to see how they could be knm';n, if true. Without the scholastic magic of an intellect equipped with a natural aptitude for truth, and with epistemological worries becoming more central to philosophy, empiricism and rationalism were in equal but opposi te quandaries. For the empiricist, the inflllitely large and the infinitely small are not available for inspection, so where is knowledge of them to come from? H ume will pursue chis thought to its limit. For me rationalist, the certainty of the deliverances of reason on space and time wi ll suggest a dependence of those concepts on the lrUnd. Kant will pursue chis idea to, or beyond, its limits.

1. Bayle mid Saccheri: doubts on tltefollndations of geometry

The probJem as it appeared ac the beginning of the century can be seen in two widely known semi-philosophical works: Bayle's Dictionary (r697) and Saccheri's Euclid Cleansed from All Spo, ( ' 733). Baylc remarks tha, the certainty of the mathematical method is not all it is clajmed to be, since there are disputes even among mathematicians, for example, over infinitesimals. Y; He argues that space can consist neither of mathematical points nor of Epicurean excended atoffi'i, nor can it be infinitely divisible. H e takes this to exhaust all the possibilities, and concludes, in a remark that contains seeds of both Hume and Kant, that the

attempted geometrical proofs that space is infinitely divisible 'serve no other use but to shO\v that extension exists only in our understanding' Y

An essen tial claim of admirers of the 'mathemarical method ' vvas that Euclid's axioms were self-evident. But how true is this? Somewhere, Euclidean geometry must claIm that space is infinite, wh ich SeelTL'i a claim beyond the capacity of experience to know. Euclid's Fifth Postulate, in particular, asserts something that seems to require an intuition about arbitrarily distant space:

That if a straight line falling on two straight lines makes the interior angles on the same side less than rwo right angles, the t\VO scraight li nes, if produced indefinitely, meet on that side on which are rhe angles less tha n (he two right angles.

Saccheri undertook to derive the fifth postulate from the others by shO\ving that the first four postulaces, p lus [he neg.ulon of the fifth, led to a contradiction. This is in fact impossible to sho\\-: If true, it \yould have removed all doubts

824 James Franklin

about the self-evidence of E uclid 's axioms. H e proceeds well for some time, demonstrating what would later be called theorems in hyperbolic geometry, the non-Euclidean geometry in which the sum of the angles in any triangle is less than 180 degrees. Then he derives a 'contradiction' , but it is unconvincing, as it involves COJlUll0 11 perpendiculars to two straight lines 'a t infinity' . He makes another attempt and again claims success, but there is a mistake. He hin ts that the result is not as clear as it might be, and publication of his book was withheld in his lifetime (possibly entitling him to a footnote in the history of ethics).38

The problem became weU-known: d'Alenlbert caned it the 'scandal of the elements of geometry'. G. S. Khigel's dissertation of 1763 reviewed t\venty-eight attempts to prove the fifth postulate, concluding that they were all deficient. He gave his opinion [hat the postulate was nO[ provable, its truth thus resting on dle judgment of the senses. W Kant's only serious attempt to do work of his own in mathematics was an attempt to prove the fifth postulate.

Lambert came closest to thinking in terms of an actual alternative geometry, writing, '{ should almost conclude that the third hypothesis {of angle sum less than 180 degrees] holds on an imaginary sphere'.40 N evertheless, like Sac cheri, he incorrectly claims to derive a contradiction, and the genuine possibility of a non-Euclidean geolll.etTy ,,",'as not recognised unti l well after 1800. The philo­sophicaJ commitment to [he self-evidence of Euclid certainly stimulated im­portant mathematical work but at the same time delayed the discovery of the correct answer, which was not that desi red by philosophy.

2 . Berkeley's Analyst: calcu lus mid ilifmitesillIals

Berkeley's general philosophy of math ematics shows intellectual independence, to say the least. R ejecting completely views that mathematics is about either quantity or abstractions, he is the first formalist philosopher of arithmetic, main­taining that there is only the manipulat ion of symbols according to rules:u Ge­ometry, he believes, can only be about perceived extension. He is thus led to reject the infinite divisibili ty of space; like Hume after him (and this is where Hume's and Berkeley's philosophies come closest) he denies the meaningfulness of any talk about lengths less than the minimum visibile or l1Iinif1'111111 tangibileY·

Prepared by thc:=se non-standard speculations, Berkeley, in his AI/alys! of 1734,

attacked the mathematiciam' understanding ofehe fou ndations ofche calculus as hopelessly confused and contradictory. T he episode has a special place in the his­tory of philosophy, as one of the very few cases where a technical field eventually admitted that philosophy strictly so-called had won a victory over the technical practitioners. Berkeley intended the argument to serve a purpose in the philos­ophy of religion, by showing that there were mysteries as incomprehensible as

Artifice and the natural Ulorld 825

those of religion even in the paradigm of reason, mathematics.43 The argument itself, however, is quite independent of its purpose.

At issue is the meaning of a derivative, or rate of change of a variable quantity­the 'fluxion' of a 'fluent', in Newton's terminology. If we w ish to measurc the

speed of a moving object, that is, the rate of change of distance, we use a unit like

miles per hour. To fi nd the numerical value of an object's speed, therefore, we

d ivide the disrance it travd s in any rime interval by the length of the interval If the speed is constant, no problems arise: the an:iwer is the same w hatever interval

is taken: 12 miles divided by 3 honrs gives the same answer as 8 mi les divided by 2 honrs, namely 4 nules per hour. But if th e speed is itself variab le, conceptual

problems arise in trying to explain what the instantaneous speed is, at any given

instant. For the speed calculated from dividing any finite distance traversed by

the finite time taken to do so is no t an instantaneous speed but the average

speed over [he in terval. It is na tural ro approximate the speed at an instan t more closely by taking smaller and smaller intervals including that instant, but the problem remains that an instantaneous speed and an average speed are different,

both conceptually and numerically. N ewton used such doubtfully intelligible

language as

Fluxions are very nearly as the augments of the fll1cnt~ generated in equal, but very smail , particles of time; ,md, co speak accurately, they are in the first ratio of the nascen t augments ... -t4

I n calculating th e speed if the distance travelled in time x is x", he first finds the

distance travelled in the time between x and x + 0, divides it by the 'augment' of time 0, and finall y claims that vvhen the augm ent 0 vanishes, their 'ultimate ratio' is as I1X

II- r to !. Berkeley's criticism is perfectly correct:

Fo r when it is said, Jet the increments vani-,h, o r let there be no increments, the fo r­mer supposition that the increments were something, or that there were increments, is destroyed , and yet a consequence of that supposition, i.e., an expression got by virtue [hereof, is retained:H

Indeed, the division by 0 to find the average speed requires that 0 not be zero, while later 0 is taken to be zero. It is no use maintaining that 0 is small , since as

Berkeley again says, 'th e mi nutest errors are not to be neglected in mathemat­

ics' . Newton's arrempcs to speak of the augmen ts as ' nascent' and 'evanescent', and the ratios as 'first' Cl nd ' ultimate' attracts Berkeley's m ost famous piece of

ridicule : 'And w hat are these same evanescent increme nts? They are neither fi­ni te quantit.ies, nor quantities infinitely small , nor yet nothing. M ay we not call

them thc ghosts of departed quantities? '(§35 , 4: 89) . Berkeley also attacks with justice other parts of N ewton's calculus, notably the higher derivatives. A speed

Ja mes FraNklin

is itself a variable quan tity, so it has a fluxion, or rate at which it is changing, the acceleration. [ f it is hard to explain a fi rst derivative ill terlTIS of the ratios of 'evanescent' quantities, it is doubly so to explain in such terms what a second or third derivative is:

The incip ient celerity of an incip ient celerity·, rhe nascent auglnent of a nascent augment, i.e., of a thing which hath no magnitude. - take it in what light you please, the dear concepcion ofic will , if l m istake not, be fou nd impossible. (§4, 4: 67)

Bad answers to Berkeley began with his own,46 and a flood of them appeared from as far away as America. 47

There are other possible ways of expressing what it is of which ratios are being taken . On the continent, it was common to speak in terms of ' infinitesimal';'. These were conceived of as quantities smaller than any fi nite quantities, yet not zero. An instantaneous speed nught then be regarded as exactly the ratio of infinitesimal augments, though only approximately the ratio of any fini te augmencs.4~ 'The clear conception of them' proved no easier to achi eve than that of flu xions.

O ne of the more serious attempts to resolve the problems V-ias Maclaurin's Treatise ~f Ffuxio l1S of I742. It attempts to show that fluxions are a generalisation of the 'geometry of the Jntients ', by using Archimedes' method of exhaustion to replace infinltesimals, of which he complains, 'From geometry the infinities and in fini tesimals passed inca philosophy, carrying w ith them the obscurity and perplexity which carmot (.1il to accompany them '.49 H is idea is in pr inciple the same as the modern treatment using limits, but Maclaurin retains kinematic notions which would later be regarded as inadequate. In particular, he defmes a fluxion obscurely in term ... of a counterfac tual: 'the illcrel11_cnt or decrement that would be genera ted in a given time by this notion , if it was continued unifonnly. ' .5

0 D ' Alembert and L'Huilier tried to base calculus on lim its, in a

way that was essentially correct, but still lacked the precision achieved in the next century by the use of multiple quantifiers. 51 Lazare Carnot's Rij1exiol1s sur fa 1I11itaphysiql,le du (alcul il'ifjllitesil11al, of I797, achieved much greater popular Sllccess, by repeating all the. worst excesses of infi nitesimals. p The debates over \-vhether infinlcesimals are zero or not, whether they can be conceived, and \-vhether a liInit is or is not actually attained often read more like a Kantian antinomy than the real thing.

3. H ume 011 mathematics

Hume's philosophy of mathematics is a natural outgrowth of his combining the usual 'science of quan tity and extension' view \~.'ith his requirement that all

Art!ficc and the naturalluorld

concepts be explained in terms of impressions and ideas. Tn the division of truths between relations of ideas and matters of fact, mathematics falls entirely on the side of ideas. But whereas relations of ideas like resemblance and contrariety are

'discoverable at first sight', this is not so with 'proportions ~f quantity or number'. Though not different in kind from resemblance, because of their complexity

'their relations become intricate and involved', so that coming to know them

may need some 'abstract reasoning and reflexion', or demonstration (unless 'the difference is very great and remarkable'). The relations treated in arithruetic and algebra arc the best known, because of theif 'perfect precision and exactness'.

For example, tvvo numbers may be infallibly pronounced equal when 'the one

always has an unite answering to every unite of the other', since this is something

directly checkable. 5.1 Such complicated mathematical facts as that a number is

divisible by nine if the sum of its digits is also divisible by nine may at first appear

due to chance or design, but reasoning shovvs they result from 'the nature of these numbers'. 54 Hume thus does not agree that mathematics is syllogistic, or

in any other way 'analytic' in any trivial or vacuous sense. But he does hold that mathematical truths are known by subjecting ideas (of quantity) to some kind

of purely conceptual 'analysis' (not Hume's word).5)

Even demonstrated mathematical knowledge is in practice fallible, however.

For the certainty that results fiom discovering the relations is only an 'in princi­ple' one, since an actual reasoner can make mistakes. 'The rules are certain and

infallible; but "vhen we apply them, our fallible and uncertain faculties are very apt to depart from them.' 56

Geometry has less certainty than algebra and arithmetic, because it deals with continuous quantities, \\'hich cannot be measured exactly. The result is that

Hume becomes, with Berkeley, one of the few philosophers in history to reject

the infinite divisibility of space. The topic belongs more properly to his philos­

ophy of space than to his philosophy of mathematics - even granted that the

distinction is anachronistic. But his replies to the alleged mathematical detnon­strations of the infinite divisibility of space, approved by such good authorities as

the Port-Royal Logic and Isaac Barrow,57 are of some worth. The mathematical arguments simply consist in extracting the assumption of infinite divisibility that

is contained implicitly in Euclid, and cannot determine whether actual space is

infinitely divisible. Hume goes some way toward exposing this fla\v when he doubts the exact correspondence between the axioms of geometry and our ideas

of space: 'none of these demonstrations can have sufficient weight to establish such a principle, as this of infmite divisibility; and that because with regard to

such minute objects, they arc not properly demonstrations, being built on ideas,

which are not exact. '58 There is no way to be sure, for example, that two straight lines with a small angle between them meet in only one point (1.3.1.4, SBN 71).

828 James Franklin

Hut the errors of geometry 'are never considerable' , It would seem that Hume has been substantially vindicated by subsequent developments, which have re­vealed that deciding whether space is exactly Euclidean is an empirical question, altho ugh it is obvio usly approximately Euclidean in our region.

An aspect of Hume and Berkeley's wri ting on geometry that is central but that some commentators have found odd is their talk of 'part" of ideas'. T hey discuss, for example, into how m any indivisible parts an idea of extension 'as conceived by the imagination' can be divided. 59 If one takes this seriously, it

would appear that ideas or imaginatio n themselves have a quasi-spatial quali ty. Hume does no t develop this notion furth er, but R eid does, and goes so far as to say what exactly is the spatial structu re of the 'geometry of visiblcs' . Tt is the geometry of the surface of J sphere. (,0

4. Kant

The key to Kant's views on mathematics, and much else, is the notion of ( 011-

struct iOll in geometry. In Euclid, there are postulates, such as 'To draw a straight line from any point to any point ', w hich assert that certain things exist , or may be constructed . The fi rst thing Euclid proves is that an equilateral triangle may be col1structed on any line. In learning how to prove in geometry, as of course all

educated eighteenth-century persons did, one must spend a good deal of rime deciding which lines to prolong, when to draw a ne\v circle, and so on. From the point of view of modern formal logic, this can be regarded as a defect in

Euclid's treatment of geometry, bue from an earlier point of view it reillforces two convictions : that Euclidean geometry is not purely syllogistic, and that it is about real space.

A fascination with construction had already been evident in Vico. T he firs t statement of his l)crulll laclufII theory lS in the context of geometry: 'We demon­strate geometrical ehings because we make them.'GI Even when his 'New Sci­ence' of human things is fully developed and its contrast \vith the natural sciences empha..;ised, its links w ith geometry are retained . BOtll construct the world they study.6,2 Ie was realised that constructions did not fit well into the Wolffian vie\.v of sciences as demonstrating truths about universals from their definitions. Wolff was prepared CO assert that the dra\ving of a straight line betw"een two points flowed from the definition of a line, but this is not plausible. Andreas Rlidiger alleges that the Wolffian 'mathematical method ' is a travesty of rea] mathematics, by recalling the inspection of particulars in geometrical construc­tions and in counting.63 Johann Heinrich Lambert describes the experience of reading Euclid after Wolff and recognising that Euclid is nothing like Wolff says he should be. He notes that Euclid does not derive things fro m the definition

Artifice and the natural world 829

of space, but starts with lines as simples, and exhibits first the possibility of an equilateral triangle. 64

So, when the Berlin Academy posed the question, whether metaphysical truth could be equated with mathematical truth, Kant in his 'Prize Essay' replied:

mathematics has construction, or synthesis, while metaphysics does Dot. 65 In

Kant, construction in geometry is used to fill out the vaguer notions of the previous one hundred and fifty years along the lines that the possible is what can be clearly and distinctly conceived (in the 'imagination', conceived as a mental visualisation facility} Kantian 'intuition' is, like the scholastics' 'intelligible mat­

ter', a medium in vvhich can be dra\~Tn not just a few simple ideas to be compared

with one another, in the style of Locke, but whole geometrical diagrams. What

can be so drawn is more restricted than what merely does not contain a logi­

cal contradiction. For example, there is no contradiction in the concept of two

straight lines meeting in two points and enclosing a fIgure; nevertheless, no such figure is possible, since it cannot be constructed: 'That between two points there is only one straight line ... can [not] be derived fi~0111 some universal concept of

space; [it] can only be apprehended concretely, so to speak, in space itself'66

Similarl}~ that there is a plane passing through any three given points is evident because the intuition constructs the figure 'immediately'.67 These necessities

and possibilities are 'synthetic', in the sense that they do not follovv simply from

formal logical principles, and also in the sense that they involve 'synthesis', or construction. These truths are also a priori, since Kant is not prepared to com­

promise the absolute certainty of mathematics. So Leibniz, he thinks, cannot

be right about space arising out of relations between real objects because that would make geometry empirical, and there might be a non-Euclidean space,

which Kant takes to be impossible. 68 It is his own theory, that space is imposed

by the mind, that is needed to ensure the certainty of geometry: 'Assuredly, had not the concept of space been given originally by the nature of the mind ... then

the use of geometry in natural philosophy would be far from safe' (§ISE, Ak 2:

40 4-5). It is clear then how Kant's synthetic a priori, on which so much in his phi­

losophy depends, is the result of combining three pre-existing ideas: Euclidean construction, the reduction of concepts to ideas in 'imagination' or 'intuition',

and the certainty of geometry Kant finds construction also in arithmetic, in the thinking of how many

times a unit is contained in a quantity: 'this how-many-times is grounded on

successive repetition, thus on time and the synthesis (of the homogeneous) in it' (Kritik B 300). The concept of number is one which 'in itself, indeed, belongs

to the understanding but of which the actualisation in the concrete requires the

auxiliary notions of time and space (by successively adding a number of things

8J o James Franklin

and se tting them simultaneously side by side)' (De mundi, § I2 , Ak 2: 397). In the famous passage of the Kririk der rcinen VemU/!fi explaining why the proposition

'7 + 5 = 12' is synthetic, Kant writes:

The concept of twelve is by no means already thoughr merely by my thinking of that unification of seven and five, and no ma tter how long r analyze my concept of such a possible sum I will still not find twelve in it . ... For I take first the Humber 7, and , as I take the fingers of my hand 3S an intuition for assistance with the concept of 5, to that image of mine f now add the units tha t I have previously taken together in order w constitu te the number 5 one after another to the num ber 7, and thu.s see the number 12

arise. (B I5-16, sec also B 205 and B 299)

The construction here is with real fmgers, not 'in the imagination' , but Kant

m eans exactly to assimilate the mind's strucmring of experience while perceiving

fin gers to construction in the imagination: 'this very same formative synthesis, by means of which we construct a figure in imagination is e ntirely identical

w ith that which we exercise in the apprehension of an appearance, in order to

nuke a concept of experience of it' (B 271). Kant emphasises that he does not just mean reading off results from a picture; there is an intellectual operation

involved, which is responsible for the nece&..<;ity of the truth . Large numbers, for example, obviously cannot be counted by an immediate glance; what is

important is the 'schema' of successive addition of units that allows the aggregate

to be synthesised, that is, counted (13 16, B 179- 81). The essentials of this discovery, that the necessity in mathematica1 knowledge comes from assimilating an image or experience to construction or synthesis according to some rule, Kam attributes to the earliest Greek geometers (B xii).

Kant brings the same ideas to the problems of the infinite. H ow those problems appeared to mathematicians in Kant's time js apparent from the [enllS of the prize

set by the Berlin Academy of Sciences (Mathematical Section) for J786:

Jhere is needed a clear and precise theory '?/ what is called I'1fillite in lyfathcmarics . certain em.inent modern analyses admit that (he phrase infinite maR-HilI/de i ~ a COnlT3 -

dicciol1 in terms. T he Academy, therefore, desires an explanation of how it is that so m any correct theorems have been deduced from a contradictory supposition, cogether with enunciation of a sure, a clear, in short a truly mathematical principle that may be sllbsrituted for that of the illfinite.nry

Kant sees this problem too in terms of construction or synthesis: 'Since UlI­

representable and impossible are commonly treated as having the same m eaning,

the concepts both of the continuolis and of the !1!finite are frequently rej ected'

(De IlUl1Idi §r, Ak 2 : 388) . So the notion of a completed infmi ty contains no contradiction, but since it 'can never be completed through a successive syn~

thes is' (Kritik , B 454), it is not the obj ect of any possible experience, intuition,

Artifice and t!le natural world

or construction. But, o n the o ther hand, there seems no limit to space o r time either; for example, 'the beginning always presupposes a preceding time' (B SIS) . So to the question, 'But whae is the magnitude of the world we live in, finite or infmite?', Kant replies: n either; to demand an ansvver is to assume 'that the world (the whole series of appeara.nces) is a thing in itself. For (he world remains, even though I may rule out the infmite or the finite regress in the series of its

appearances' .70

Many of the same considerations apply to the infinitely small. To see the problems about the continuity of space in terms of 'infinite divisibility' already plays into Kant 's hands. Division is a human act, suggesting to Kant that the act of constructing a line by a continuous flow ing motion comes firs t, followed by the construction of its parts by a further act of division (rather than the part')

coming first and together forming space).7!

The demand for constructibility is, it appears , at the bottom of slich central Kamian themes as the ideality of the world. Also of the noumC110n, unreachable by e>"""perience, of which the infinite is, so to speak, the first example. Lest it seem that the problem of construction is an artifact of the eighteenth century's

primitive view of geometry, it may be noted chat the problem reCLIrs in the

modern foundations of mathematics. Th ere, on.e normally proves the consis­tency of a concept by constructing it Out of sets, but to do so requires an 'axiom of infinity'. which ensures that sufficiently many sets 'exist', in particular, that a completed infini ty of them exists.

Ill. NEW OBJE CTS OF MATHEMATICS

J. Algebra

Algebra tended to take over more and more of mathematics in the eighteenth

century. Where Newton had recast his reasoning in geometrical form for public consumption, Joseph-Louis de Lagrange's J\1ewflique al1alytiqtle of 1788 says:

No drawings are to be fou nd in this work. The methods which I present require neither constructions nor geometrical or mechanical arguments, bur only algebraic operations, subject to a regular ;md uniform progression. 7",

C ondorcet says chat Euler

sensed that algebraic analysis ,vas the most extensive and certain instrument one can em­ploy in all sciences, and he sought to render its usage universal. This revolution .. . earned him the honour, Lllli que so far, of having as many disc1ples as Europe has mathematicians.73

James Franklin

And in a rare moment ofagreemene w ith Euler, d 'Alembert says algebra

is the foundation of all possible discoveries concerning quantity ... . T his science is the farthest outpost to which the contemplation of the properties of matter can lead us, and we \,vo uld not be abJe to go further without leaving the material universe altogether. (Oeuvres I: 26, sec also 30-31; tra nsI. 20 and 26)

But, having established that algebra is a good thing, what exactly is it? Origi­nally, it was a method of solving problems by making letters stand for unknow n quantities, and manipulating the 1eners as if they were numbers. Even on this narrow view, algebra had philosophical significance, as it was a method for discovering answers , and tbus seemed on the side of 'analysis ' , as opposed to the 'synthetic' deriving of known truchs from axio ms in the style of Euclid.74

But by noo it seemed more than that. N oting that the letters could stand for geometrica l quantities as easily as for num erical ones, various thinkers proclaimed algebra to br;: the science of quantity in general, that lS, virtually the w hole of mathematics.75 Even if that were agreed, many things remained unclear. For example. what could the letters represent - complex numbers? Infinities? Infinitesimals? And if algebra was a general mathematics. where were its a. .... jonts?76 Another view of algebra was that of Wolff, who Sl 'W it as part of Leibniz's universal characteristic, that is, as a general method of reasoning symbolically.77 In the same vein, Condillac's idea of the mathematical method that ought to be imposed on philosophy was not so much Euclid as the solving of equations. manipulating known and unknown quantities until the knowns appeared by themselves.

EqlUltiollS, praposirio11S and jllrlgelUmts are basically the same thing, and . . . consequently ont! reasons in the same manner in all the sciences ... we have seen that, just as the equations x - I = Y + I , and x + I = 2 Y - 2, pass through different transformations to become y = 5 and x = 7, sensation passes equally through differe nt transformations to become the understanding. 78

R egarding French as a language lacking taste and precisio n, CondiUac pro­posed to reform it on the basis of the granunar of algebra.?9 Kant says that alge­bra p roceeds by manipulating uninterpreted symbols 'until eventually, w hen the conclusion is drawn, the m.eaning of the symbolic conclusio n is deciphered' . The simple pushing around of symbols is what gives ' the degree of assurance characteristic of seeing something with one's ow n eyes' (whereas with philos­ophy o ne must keep the meanings in mind all the time). Ro These are the same claims made around 1900 for fonnallogic. The possibiliry of manipulating sym­bols without attending to their meaning is not wi tho ut problems. One may end up with conclusions that do not m ean anything either. Euler is famou s for his

Artifice and the llatural world 83 3

lack of rigour in calcula ting \-vith in fini te seri es without worrying about their convergence; it is typicaJ not only of him but of the century to conduct long 'philosophical' debates about the true sum of the ser ies:

1 - I + I - r + T _ .•. 81

The case is even vmrse when manipulating symbols that :Ire explicitly stated to

have no meaning, such as those denoting the square roots of negative numbers. Though necessary for calculations, they do not satisfy the definition of quantities as being' capable of increase and decrease'. as they cannot be less than or greater than one anoth er. Euler describes them as ' impossible', but proceeds to calculate extensively with them. R2

Euler played a crucial role in emphasising the centrali ty of the notio n of junction in mathematics (its significance is indicated by the fact that about half of modern pure mathematics is 'functional analysis'). His aim was to replace vague geometrical notions and dynam.ical metaphors of'11uents' with something more precise and amenable co calculation. He initially defmed a fUIlction in algebraic terms as an expression involving variables: 'A fun ction of a variable quantity is an analytic expression composed in any way whatsoever of the variable quantity and numbers or constant quantities.' For example, a z + .J /1 2 - ;i1- is a fun ction of z (where IT is a constam).sJ But later, his debate "vith d'Alember t over the vibrating string convinced him this 11otion \vas too narrow, because of the need to consider more irregu lar functions, which might not be expressible by an algebraic formula. He had li ttle success in explaining what this notion should be.84 Lagrange also attempted a purely algebraic notion of function and tried to use it as a foundation for the calculus 'independent of all metaphysics'. He claimed to prove that every differentiable function could be expressed as a power series, that is, repn:scJ)[cd algebraically (except perhaps at isolated points). $j This is false, as Cauchy soon showed. When the best mathematicians in the \"lod d begin claiming to have proved what is fa lse - a rare event, much to the credit of mathematics - it is time to conclude thar rigour is not a luxury. The nineteenth century drew the corree[ conclusion, leading to the correct foundations of calculus, and to set theory.

Just visible in the work of Lagrange are the beginnings of modern abstract algebra. This is the subject that perhaps most obviously deals no t with quantity but with certain kinds o f abstract structure. Lagrange, inquiring why it had not been possible to find formulas for solving equations of degree 5 or higher, considers functions of the roots of the equation which do not change if the roots are permuted, or interchanged. He understands that some permutations may be 'independent' of others, th Lls thinking of the permutations as themselves entities w ith interrelationships. These perm.utations form the ftrst of a new kind of

834 J a ll lCS Frankliu

subject matter of mathematics, later [he object of modern group theory. 86 Paolo Ruffini's work of 1799 goes further, considering the totality of permutations (a group, in modern terms) and their composition. 1I7

Any or all of these developments in mathematics might have provided the philosophers of the eigh teenth century with perfect examples of the advance­ment a fknowledge through the analysis of ideas, had they informed themselves about them.

2. E xperimental ft'idellcf ill marhelnatics

While the eighteenth century admired the rigour ofElIclid. its own mathematics is famous for a lack of rigour. It may be that the philosophical emphasis 011 ideas as against fo rmal logic contributed to a disregard of formal rigour. 88 In any case, if m athematical conclusions are to be supported by anything less than complete

formal demonstration , there is a need to cnnsider how there can be a less than deductive logical support. Euler was the first, among either philosophers or mathematicians, to argue explicitly for the use of experimental, or probable, reasoning in mathematics.

It will seem not a little paradoxical [Q ascr ibe a great importance to obse rv:ttions in cllat part of the mathematical sciences vvhich is usually called Pure Mathemati cs, since the current opin ion is that obsenrations are restricted to physical object .. that make impression on the senses . As we mllst refer the numbers to the pure imellect alone, we can hardly undel·~tand how observations and quasi-experimen ts can bt~ of LIse in inve.~tigating the natu re of the numbers. Yet, in fac t, as I shall shm.v here ,.\,1th very good reasons, the propnties of the numbers known today have been m ostly discovered by observation, and discovered long before their truth has been confi rmed by rigid demoll5tracions. There are even many properties of the numbers with which we arc well acquainted, but which 'we are not yet able to prove; only observations have led us to their knovvledge. ~9

Euler's works contain a number of examples of how to reason probabilistically

in mathematics. He used, fo r example, some daring and obviously £1[ from

rigorous methods to conclude thar the infinite sum 1 + ~ + ~ + -16 + ~ + ... (" .. rhere the num bers o n the b o ttom of the £i-ac tions are the su ccessive squares of whole numbers) is equal to the prima facie unlikely value J[J. / 6. Finding that th e

two expressions agreed to seven decimal places, and that a similar argwnent led to

the already proved result I - ~ + i - ~ + ~ - ~ + .. . = ~, Euler concluded, 'For o ur method, w hich may appear to some as nor rdiable eno ugh , a great confirmation comes here to light. Therefore we shall not doubt at all of the other things which are reveaJed by the same m ethod. '90

Laplace and Gauss, who were in a position to know, ag reed casually that such reasoning was central to mathematicsY I Even WolfF \:\.,Tites that 'examples

A rtifice and lhe nalHrai world 835

of hypoth eses arc also found in arithmetic, which first influenced me to look upon philosophical hypotheses more favourably'. What he has in m ind is the calculation of answers by successive approximation, the ini tial guess being the hypQ(h ~l) is.92 Yet p hilosophers pronouncing on mathematics since have ra rely

given it a place. A different connection betvveen probability and pure mathematics was dis­

covered by Lambert. He understands that a series of digits produced by a random process, like thro"\ving a die, will be disordered or patternless, but that the same can be said of the digits of !f or of ../2, "vhich are completely determined. He is prepared to say that the probabili ty of the hundredth digit of ..[2 being five is I fraY3 Whether a notion of probability can be applied in such a deterministic case is stiU a cr ucial issue in the philosophy of probabili ty.

3 . 7i.1polo,gy

Topology provided the clearest example of an object of mathematics rhat would not fit under the old rubric, ' the science of qtlantity ' . The citizens of Konigsberg noticed that it seemed to be impossible to walk over all seven of the bridges connecting the two banks of the River Pregel and its islands, withou t walk ing over at least one of them twice. Euler proved they wefe right. This is a problem in [he area now called the ropology of netv.'orks . There is no quanti ty involved in the problem, only the arran,Rf11Jenr of the system of bridges and land areas. Euler writes:

T he branch of gconletry that deals \'lith magnitudes has been zealously studied through­out rhe past , but there i'i anorher branch that has been al most un known up to now; Lcibniz spoke of it £int, calling it the 'geometry of position'. This branch of geome­try dea ls \vith relations dependent on position alone, and investigate~ the properties of position; it does not take magnirudes into consideration, nor does it involve calcula tion with quantities. But as yet no satisfactory definition has been given of the problem~ that belong to this geometry of position Y.J-

What Leibniz said about the 'geometry of position' was both short and ex­tremely vague,9S but Euler \vas not the only one to fin d it suggestive. Buffon relates it br iefly [0 the folding of seeds and to symmetry in plai ting, and re­marks that 'the art of knO\ving the relations that result from the position of things would be as useful as and perhaps more necessary than that which has the magnitude of things on ly for its object' .90 Kant sees a connection betvveen it and his ideas on incongruent coun terparts. 97 The subject \:vas given some mOre definite con tent by Vandermo nde, who fi rst drew a graph, in the modern sense of a system of nodes connected by lin es . H e llsed it to solve the problem of the

James Franklin

knight's tour in chess, 'using numbers which do not represent quantities at all,

but regions in space'Ys

4. Social mathematics and /I1oral a{?ebra

Ifideas have parts, perhaps one should count them, or maybe \veigh them, if they ditTer in their force. There would result a 'moral arithmetic', or 'moral physics'.

Naturally, there are measurement problems: how are psychological units to be

measured and compared, or even identified? Hume suggests that the size of the smallest impression can be found from measuring the least visible dot en-ea­fise, 1.2.1-4, SBN 27-8); such measurements of the threshold of visibility were

carried out in Hume's lifetimeY9 Buffon suggests regarding the probability of sudden death in the next t\venty-four hours, for one in the prime of life, as a standard unit of 'moral impossibility', to which the reasonable man gives no

serious thought. roo Maupertuis reduces morality to prudence, and prudence to

a hedonistic calculus: 'The estimation of happy and unhappy moments is the

product of the intensity of the pleasure or pain by the duration.' Measurement of intensities may be difficult, but Maupertuis invites introspection on the in­

evitability of comparing, for example, the pain of an operation for the stone \vith the longer but lesser pain offorgoing the operation. W I The problem is urgent for

economics, which can hardly avoid being quantitative, when explaining prices, but seems to rely on a subjective 'utility' whose measurement is as dubious as that

of pleasure and pain. Adam Smith achieves the trick, so useful in these matters,

of claiming the right to speak quantitatively, \vhile avoiding the responsibility of commitment to any actual quantities or formulas. He \:>O'fites that the value of

any wealth to its owners 'is precisely equal to the quantity oflabour which it can

enable them to purchase or command .... Equal quantities oflabour, at all times and places, may be said to be of equal value to the labourer', but he undercuts

the apparent accuracy of his measure by adding:

It is often difftcult to ascertain the proportion between two different quantities oflabour. The time spent in two different sorts of \vork will not ahvays alone determine this proportion. The different degrees of hardship endured, and of ingenuity exercised, must likewise be taken into account. There Inay be more labour in an hour's hard \vork than in two hours easy business.102

He appeals to the market to coordinate different people's measures, but 'not by

any accurate measure'.

Benjamin Franklin advises, in cases of perplexity about a decision, the listing

of the reasons for and against in two columns:

When I have thus got thenl all together in one View, I endeavour to estimate their respective Weights; and where I find two, one on each side, that seem equal, I strike

Artifice al1d the l1atural world 837

them both out: If I find a Reason pro equal to some two Reasons con, I strike out the three ... and thus proceeding I find at length where the Ballance lies .... And tho' the Weight of Reasons cannot be taken \-'lith the Precision of Algebraic Quantities ... in fact I have found great Advantage from this kind of Equation, in what may be called jl;[orcll

or Prudential Algebra. 103

The special difficulties of measurement in the social and mental realm sug­

gested to Condorcet that social mathematics should rely chiefly on the theory of probability. 104 There, the equality of the beliefs one should have that a die will fall on any side is inferred from symmetry, or 'insufficient reason': there

is no reason to prefer any side to any other. He believed he had proved, using

probability, that decisions taken by majority vote were perfect for achieving the truth. 105 Before being hounded to death by a regime that exalted Equality

over Liberty and Fraternity, Condorcet had the opportunity to reconsider the

assumptions of his proof, and wonder if perhaps he did not mean that those voting had to reach some standard of Reason. 106

As long as there has been 'social mathematics', there have been explanations

of why it fails to work, or at least lacks anything like the success of mathematics

as applied to physics. The suggestion that lack of exact measurement is the problem was anticipated, and argued against, as the quotations above indicate.

Another idea was that of Reid, who thought the problem lay in the definition of quantity as 'whatever has increase or diminution'. This is too wide, he says, as

it allows in pleasure and pain, vvhich admit of degrees but cannot be measured in units. 107

Mathematical modelIing of social, as opposed to introspective, phenomena

was attempted qualitatively in Hume's and Adam Slnith's conception of the economy as a self-regulating system,108 but the most successful quantitative

project was that of Mal thus, whose conclusions abollt the poor laws are intended

to follow from a purely mathematical fact:

Population, when unchecked, increases in a geometrical ratio. Subsistence increases only in arithmetical ratio. A slight acquaintance \-'lith numbers will shew the immensity of the first power in comparison _'lith the second. 109

He takes these ratios to be evident and feels no need to support them with em­pirical evidence. This kind of a priori fitting of formulas has particularly afflicted economics, so it is interesting to see Condorcet criticising Verri's mathematical

economics on just this ground. It is true, Condorcet says, that more buyers mean a higher price, but what justification is there for Verri's assumption of a direct proportion bet\-veen the t\Vo, if no empirical data are considered?IIO A remedy

is to fit formulas to actual social statistical data. This is not a strong point of

James Franklin

eighteenth-century mathematics, but Lambert had some understanding of how to do it. III

IV. LOGIC

1. Textbook logic

Eighteenth-century writing on 'logic' is extensive, II:/. but neither the school of traditional Aristo telian logic nor theif opponents, the 'men of ideas' , produced much chat has commanded respect since.

As systems of thought go, Aristotelian logic was one of the great survivors. For severa] cencuries, ie was attacked vigorously, in almost identical terms, by almost all the thinkers remembered by history. It ""vas defended by nonentities. In each generation, when the dust settled, it was found to be still in control of the field (that is, of the undergraduate syllabus). T his was true in 1700, just after Locke had renewed the attacks of Bacon and Descartes . It was equally true in 18 0 0 ,

"""hen Aristotel ian logic was about to undergo a revival in Britain. Its contents are the traditional logic of terms, judgments, and inference by syllogism, largely unchanged since the logic textbooks of the thirteenth century.

The English representative of the old school was H enry Aldrich's Artis Logicae CompendiulII, the standard Oxford textbook for the whole century. First pub­lished in 1691, it appeared in many editions, epitomes, and expansions until the mid-nineteenth century, including an English translation by John Wesley. II3 On the continent, Wolff found traditional logic satisfactOry - so much so that he in effect tri ed, as we saw, to incorporate the rest of philosophy into it. It is a litde more surprising to find Kant largely on [he side of the syllogism. He has some minor criticisms of traditional arrangements of the four figures , which he thinks over-elaborate, )!4 but he accepts that the syllogism is not intended to be a method of discovery, and on the whole his logic teaching agrees with tradition. He is cJear about, and opposed to, psychologism in logic. II)

The opposing school produced voluminous works of 'logic', but they are fun of what would now be called cognitive psychology, epistemology, semiotics, philosophy of logic, and introspection. They are full also of invective, against 'scholastic headpieces', full indeed of everything except logic, in the mod­ern sense of formal logic. It is not that logic was confused with (not yet ex­istent) disciplines like psychology. On the contrary, the scholastics had been dear about formal logic, and it was in delib ~rate opposi tion to them that the

A rttfirc and ihe natflral UJorld

followers of the '''VAY of ideas' identified logic with 'psychologistic' notions instead.

Bacon, Descartes, and Locke, between them, had convinced most tha t the syllogism , and formal logic generally, "vas of no llse or interest.1I6 The essence of Locke's attack was that the syllogism \\.'<1$ not useful for the discovery of tru ths, and that it concealed the fact that inference consisted in the 'agreement or disagreement' or 'connexion' of ideas. 1I7 Traditional logic had certainly opened the way to such criticisms by holding that logic is abom 'thought' or 'judgment', and by concentrating on a single argument form, the syllogism, which has an air of being analytic and triviaL W hatever the justice of the Lockean cr iticisms, they failed to issue in anything better, either in new logical ideas or in textbooks. If adherence to the syllogism restricts logic, the 'agreement and disagreement of ideas ' is if anything an even worse straitjacket. It does nothing to encourage the discovery oflogical structure, and instead diverts logic in to vapidity. rt is all very well to offer advice like, 'Enlarge your general Acquaintance wi th Things daily, in order to attain a rich Fu rniture of Topies or middle Tenns' Hs, but how do you examine that? The logics ofCrousaz, Duncan, and W:ltts followed the Port­R oyal Logic in including enough of the traditional classifications, distin ctions, and so on to provide some content, and in simply adding cri tical observations in the style of Bacon and Descanes. 119 At the end of the century, however, R eid and C ampbell revert to a purely negative approach, speaking as if they have just discovered that the syllogism is not a logic of discovery. 120 But neither they nor any of their school have a replacement to offer. Campbell ventures the opinion that mathematical demons trations arc not syllogisms but does not suggest what their fonn is, if not syllogistic. R eid is closer to the trnth in holding that mathematical reasoning cannot usually be syllogistic, as it deals with relat ions of quantities, and the syllogism is not applicable to relations.

H ume takes to irs extreme the 'psychologising' oflogic, and so exhibits most dramatically the problems in doing $0. Plainly, there are tensions in 'natural­ising' logic by reducing it to manipulations the mind happens to perform on ideas, while relying on logic as normative for argument. Hume applies general principles, such as, 'like objects, placid ill like circumstances, will always produ.ce the same tjJects' , to particular cases, without apparently noticing that he is using a formal logical principle of instantiation. Much the same could be said of his use of 'not'.1.!r Hume exacerbates these difficulties by adding a sceptical project to his naturalising one. What he is sceptical about is the logical force of common inferences: causal inferences, inferences from ' is ' to 'ought', and so on. IU It is odd, certainly, to say that causal inference is not logically cogent but only an unavoidable habit, at the same time maintaining that all logical inference is only an unavoidable habit.

James Franklin

Hume's views on ir!ferellce are seen (Q better advantage if they are thought of not in terms of formal logic, or even introspection, but as a research proposal to be implemented in, say, sili con chips. M odern Artificial Intelligence, like most eighteenth-centlllY writing, is concerned with the implementation of a system of inference, not j ust the fonnal structure of the system itself. From that point of view, it is necessary to J. l1swer questions that do not ar ise in formal logic, such as how the symbols become attached to the things they mean. O ne must consider, in shon, [he 'na tural history of the understanding'. It is then a matter for debate whether the syllogism needs to be explicitly represented internally, and whether one can replace an explicit generality with exemplars linked by 'custom', so that when one individual is activated, the bnked ones 'immediately crO\vd in upon us' (Treatise, I.1.7.7~8, SBN 20~1 ) . The links betvleen the exemplars are to be induced by the resemblance, constant conjunction, and like relations rhat hold between them. Hume's claim for his rules about causes that 'Here is all the LOGIC I think proper to employ in my reasoning' (1. 3.Is. H ,

SBN 175) is then a claim chac can be invesrigated empirically: will a mechanism equipped with only the principles of association H ume names be able to reason adequately'

Logic's place at the centre of the curriculum makes certain wider effects of eighteenth-century logic more interesring rhan the subject itsdf. O ne student of logic who took the natural undergraduate reaction again<;t the subject to an extreme was Swift, whose inversion of the stock logical examples, 'Man is a rat.ional animal; a horse is a v.,rhinnying animal' led co the satire of the Yahoos and Houyhnhnms. 12 3 Another who used its rhetoric to good effect was Thomas Jefferson .. r.24 The claim of the DeclarJtion of Independence, 'We hold these truths to be self- evident, that all men are created equal' , combines the logical theme of self-evidence with the mathematical one of deriving such sdf­evidence from a symmetry principle. A true logician wiU ask why, if a principle is indeed self-evident, it is necessary to 'hold' it to be so. The French were clearer that Equality is not a given but a goaL

The discrediting of logic in England had consequences in education that are still felt. While tradition- bound, High Church Oxford took no notice of the problem and continu ed to teach logic, in Latin from Aldrich's text, and examined by disputation, lZs Whig Cambridge did the opposite. It replaced logic by the only credible alrernative, mathematics, and produced the ancestor ofrhe modern written examinarion system , the Mathematical Tripos. By a happy feedback effect, mathematics permitted an ever fi ner objective grading of candidates, leading to ever m ore concentration on mathematics. Since mathematics was a substitute logic, however, the matter exanuned was confined largely to geometry, continental innovJtions like algebra being considered unpatriotic. 126 Geometry

A rtifice Oliff xhe lIatural world

is also m ore amenable to being considered in terms of 'ideas' than the formal manipubrio ns of algebra. I l 7

3- Symbolic lo ... ~i[ and logic diagrams

Leibniz's vision of a universal characteristic, allowing logical inference by cal­cuhtion, inspired some, but resulted in little of significance. The logical sym­bolism of Segner, Ploucquet, Holland, M aimon, and Castillon does not need much rc:: interpn:tation to yield vario us theorems in propositional and predicate calculus, but only the simplest ones. 12 8

Euler developed the traditional theory of the syllogism in a popular work, illustrating it with diagrams simih r to the lacer Venn diagrams. Particular (ex­istential) propositions have always posed problems for su ch diagrams, as there needs [0 be some way of indicating which of the regio ns are non-empty. E uler distinguishcs betv .. -ecl1 'Some A is B ' and 'Some A is not D' as follows:

Some A is B Some A is not B I29

If one believes that it is ;'i good thing for logic to become extensionalist, then logic diagrams \\'ill apPl;:ar one of the century's few advances in logic. Euler himself docs not mean to be taken this vIray. H e Ll ses o nly intentional vocabulary, su ch as 'If the notion C is entirely com ained in the notion A ... ' H e takes no no tice whatsoever of t\~lO centuries of criticism of the syllogism, and goes so fu r as to maintain that all truth arises from it.

A similar idea, but using lin es instead of circles. appears in Lambert. ]30 But this is only a small parr of a larger project for the mathemarising of logic. He proposes to give some precision to the analysis of concepts into simple ideas , thus doing for quali ty w hat geometers had done for quantity, and deducing everything from a flrm basis. 131 An aspect of the project was a symbolic logic of concepts; one analyses a concept ay into its genus. a and differentia as; the equation

ay = a - ao

then means that the genus of a is the result of abstracting the differentia from o.

Lambert is someti mes misled by f.:1.lse analogies to ordinary algebra, to the extent of considering the square root ofa relation. 13 .. In another attempt to bring logic

James Frarlklil1

and mathematics tOgether, he considers the valid argument: J3J

!.. of A are B, !... 4 A are C, $0 some Bare C. 4 3

Few logicians since have cared to follow him into such numerical territory.

v. TEC HNOLOGY

The subtitle of the Ellc}'cI(~pt:dje is Diaiollllaire Taisol1t1e des sciences, des arts, et des metiers. Th e prominence given to the 'arts and trades' is due to Diderot, \vho writes:

Let some man go O llt from the academje~ and down into the \vorkshops, and gather material on the arts to explain them in a work which will persuade artisans to read, philosophers to tJlink usefully, and the great to nu ke :'It last some .. vo rthwhile use of their luthori ty and wealrh.134

'Useful science' , as an idea, is Baconian, but science as an accepted route to profit, mili tary superiority, and progress is really an eighceemh- cemury devel­opment. While the French government and the Royal Navy were among the largest investors in research, the practical orientation of research was especially evident in peripheral regions, where abstract thought, including philosophy, survived at all only on the promise of its practicality. Boundless confldence in the usefulness of science was as characteristic of the America of Franklin and Jefferson l

)5 as it was of R ussia, where Euler and Lomonosov worked assiduously on 'improvements '. In England, the Tndustrial Revolution was associated less with London than with the provincial cities that wefe the homes of 'Philosoph­ical Societies' devoted to practical science. 1J6

Diderot found a significant fact about practical knO\:vledge: it could not be \.vritten down adequately in text. Asking the practition ers to clarify it produced simply a garbled mass of lloimelligibilitics and inconsistencies. It proved essen­tial to ask the tradesmen to shaUl what they were doing and present the result in pictures. Hence the EncydopCdie has eleven volmn es of plates (compared to seventeen oftext). 1J7 Though there were no large-scale encyclopedic projects in England, their place \vas to some extent taken by public lectures on science, es­pecially useful science. T he famous London lectures of John DesaguJiers taught by sbowing working machines. Science thus became accessible to those lack­ing mathematics; the Ne\vtonian philosophy, Desaguliers says, 'tho' its truth is supporred hy Mathematicks. yet its Physical Discourses may be communicated without. The great Mr Locke was the fi rst who became a Newtonian Philoso­pher without the help of Geometry' .138 Inventions like the steam engine, the

Art{fice and the !loiliral world

ligh tning rod and balloons were certainly spectacular and capable of conveying a message wi thout the need for supporting captions.

But what message? The relation of machines to abstract thought was a vexed o ne. The formula for gravity is not much usc, while the textile and steam engines \vere moscly invenced by practical engineers, not sc ientists. Still, inventions are 'efforts of the mind and understanding which are calculated to produce new effects from [he varied applications of the same cause, and the endless changes producible by different combinations and proportions' / 39 that is, intellectual products. Adam Smith, who recognises the importance of mach ine inventions in improving productivity (though he tends to subordinate it to his idee fixe of division of labour), speaks of 'philosophers or men of speculation, whose trade it is, not to do anything, but [0 observe every thing; and who, upon that account, are often capable of combining together the powers of the most distant and dissimilar objects ' . LIO The description is exactly true of James Watt, mathematical instrument maker to the University of Ghsgow, who analysed the heat losses in N ewcomen 's steam engine and realised that the condensation was a separable process that could be better situated somewhere else. I,p The skills involved are cognitive, but th ey are not so much the formal geometry of Euclid as the draughtsmanship or design of the engineer - Diderot's 'experien tial and manipulative mathematics', or the 'practical geometry' "\vhich Swift's Laputans 'despise as vulgar and mechanic'. And it was the eighteenrh century's advances in cast iron and sted making that meant any shape could be made cheaply and durably. The availability of arbitrary rigid shapes, cheap, long-lasting, and rehably resistant to high pressures, stimulated imaginations to fashion in tricate geometries of interacting pares. The iron machines are concrete realisations, so to speak, of several philosophical proj ect.;; at once: Bacon's useful science, Kant's constructions, Vico's ' maker's knowledge', and D escartes's dream of explaining the ""orId as the effect of interactions of rigid bodies.

There was some opposi tion to the idea of the beneficence of'lIseful' science. Swift 's satire attacked scientifiC research as either divorced from reality or pro­dllcdve of inventions chat did not actually work. 14·2 But in genera l, technology had a positive glow, like mathematics, sufficient to tempt philosophers of most persuasions to claim it as 011 their side. D erham's Physico-theology, (01' example, saw che advances in mechanical inventions as evidence for God 's providence.1+3

L'homme l1lachine may have seemed an idea of obviously atheist consequence in Paris, but Paley knt!w a good deal more about machines than La Mettrie, and convinced most, at least in the shor t term, that the teleological as pect of ma­chin es supported a theist interpretation of the man-machine analogy. 'Watches, relescopes, stocking-mills, steam-engines, &c.' are not the kind of things that can arise by chance - not even chance followed by selection. )44

James Franklin

The consequences fo r poJjtical philosophy of Diderot's pra ise of artisans are

generally left implic it in the Eucyclopedie, but Hume's essay OJ R~fil1elJ1elll ill the Arcs supplies the gap:

Wt: cannot reasonably expea, rhat a piece of woo llen doth will be wrought to perfection in a nation, wh ich is ignorant of astronomy, or where e th ics are negleaed .... Can we expect, that a government will be well Inodelled by a people, , .... ·ho know not hmv to m:lke a spinning-wheel, or to employ a loom to adv:lnuge? .. :l progress in the arts is rather f,woll rab.le to liberty, and has a natural tendency to preserve, if not produce a fi'ee

government. 14 j

The idea that machines create progress autonoruo usly has remained an attrac­

tive one for the .Enlightened. Citizen Gateau, administrator of ll1ilitary provi­

SiOllS, writes of the machine that has come to be most associated '\vith Liberty:

Saint Guillotine is most \vonderfully active, and the beneficent terror accomplishes in our midst, :IS though by a miracle, what a century or more or philosophy and reason

could no t hope to produce. q6

I\:OTES

1 Ian Hacking. TIle 7illnill<~ of Clulnce (Cambridge, 1990), ch. J. 2 Patricia C. Cohen, A Calmlaijllg Pe,'pit-: TIle Spread 4 j\ iumer<tcy ill Earl)' America (Chicago,

l L, t9Sl). 3 J. L. Heilbron, 'The Measure ofEnlightemnent' , in TIlc QUCll1Iffyin,!( Spiril illille 18rl! ee!!tll ry,

cds. T. Frangsmyr, J. L. H eilbron, an d rc E. [{.ider (Berkeley, CA, '(990). 207-42; R..onald

Ed,vard Zupko, RwollltiCl1/ in AIet1sIJrement: western BlfnJpl!{I/l H1~lglw (lIld iVleo.lllrrs Siu[(' the Age oj .'iciellce (Philadelphia, PA, T990), chs. 4-5.

4 Heilbron, ' Introductory E~~ay', in The QI!Cl nt!(yill,R Spirit, eds. Fdngsmyr et .11.. 1-23; Ivlaurice Dallmas, 'Precision of Measurement and Physical and Cht!mical Research in the Eighteenth Century' , in SdCll t!fir Chi/n.Rf: Histuri{al Stlldies in the llltdll.'ct!l(l/, Social amI Te{h­l1ical C{)lIdirit)lIsJi.)r Scientific Di5f()J!t'Ty and Tec/mical bll'elitiOlr, .Fo1l! Antiquity to rhe Prt'Senl, ed. A. C. CrombIe (London, 1963), 418-30.

5 L. Tilling, 'E.1rly Experi mental Graphs', Rriti.l/r j(llrT/wf I(ll 'he Hi."/(ITY ~f Scieuu, R (1975):

193-21 3. 6 H. J. M. Bo~, 'Mathematics and R ational Mechanics', in 71't' FU/II(tll if KIl t)wiNI,l!c: STlld­

it's i/l ,l,e H isforitwaphy 4 EI~/l ft'/,!Ilh - Cell t!/ry Sciell(C, eds. G. S. Rousseau and R . Porter (Cambridge, 1 9~O), ch. 8. John L. Greenberg, Ti'l' PT{!biCII/ <if tIll: Eartl/$ Shape/rom .".1('11'101/

/0 elm'raut: 77,1' Risl' <!f iVlnthrmalica{ Sciellu ill til l' E~l1h(eelll" Cell/llr), aud (h(' Fall vf 'Normal' Sdf'/l{t' (Cambridge, 1995).

7 Leonhard EllIeI', Vol/stiifldi}!1' .4111eitl/ll~ .i:tlr .4 (eebm mit dell L.IIsiirzt'1I !I(lI/Josel'lI Lillis Lagrallge, ed. H. Weber, Opera olHnia, 1. 1 (Leipzig. 19 T1 -), Pt. I, Sect. I, ch. I , p. 9; translated a~ HemeJlls 0/ Algehra (London, 1797), I . See aho Jean Le Rond d 'Alelllbert, 'Explica tion decaillee dt! systeme dl.'S co nnaissances humaines', appended co his DiSfOfm preliminairl' dl' 1']]IIcyc/opMic, ill OCIIVf CS, 5 vols. (Paris, 1821-2), T: 99- T14 at 105- 7; as translated in Prl'limi­Hary 1)i.'i(()HrSe to rhe Enryclopedia 4Didcwt, tram. R. N. Schwab and W. E. R ex (Indi:mapolis.

-

Artifice and the natural world

IN , 1963), 1'52- 4; article' AJgebra', in Ellcydopaedin Blifa/mim, 3 vol ~. (Edi nburgh, .1 771), I: 80.

H Article 'Mathematics'. in Encyclopnedia Bfi/anl1im, 3: JO .

9 Euler. L 'l/(i:.1 a ll/ It' P,incc$$f. rf"Allcmn.~rte SliT divers SI!iets de physique & dc pllilosop/Ji(' (1768), 1, letter r22 , 25 April 1761, in Opera omllia, 1lI.ll" : 2R 9; translated as Letters ~r Eulcr 0/1 D!lTcretl/ S lIbjects ill Natl/ral Phi/osop"y. Addressed lO a CJCrmall ['rhlccss, [rrans. H . H unter], eds. D. Brewster and ]. Gri!>com (facsim. of 1833 eun.), 2 vots. in 1 (New York, NY, 1975), 2 : 33 : see WA. Suchting, 'Euler's "Reflections on Space and Time"' , Srim lia, 104 (1969): 270-8.

10 Article 'Geometrie', in EllcyclopMie, 011 Diaio rmaire raisvllnf dl'5 ScieltU5, des arC5, el des mrtierJ·, eds. D. Diderot and]. d 'Alcmbert, f7 vol~. (Paris, f75 1---{)5), 7: 629-38, at 6)2 .

II Chri~tian Wolff. Vcmiil'!fJiigc Gednl/ckeu VOII dm Krqffren des mensclJiichclI Vmllln des lind i/,rem riclrr(~w GcbraJJche ill l;.·rkiirlllcnis der ~}illJYheir (r7 13), Wcrkc, I.I (1 965), ch , 4, §22 , I73; a~ tr.-Inslated in Lv]!ic, or RaliOrlnl T7lOrlghli 01/ the POUlers <if lire Human Unders((mdillg, with their U~e and Applicatio r-l iu the Knort'let{,<c nnd Search t1Trllth (London, 1770), 94; De, Anfimgsgriinde allcr A1atlrematisclierr vViss /' r1scfwfim enter 'nIta, welcher cim:n Ullfcrridlt von der jHatJuII!aiisc!u?I/ Lelu-Art, die R ecilcllkUllSt, Geometric, ·(;;go/lomelrie //lui Ball · Kllust in sidl m fIJiift (1710), esp. the 'Kurtzer Unterricht von der 1ll3thematischen Methode oder Lehr-Art' , prefaced to th is "york. in which Wolf( sets out his programme, Wake, 1.12 (1973) . See also Gottfried Wilhelm Leibniz, Secolld leller to Clarke, in 17,e Ldlmiz-Clarke COrTespolldCIICC, lo.~etlt ff lVilll EXlracrsJrom Newloll's Principia alld Op rieks. eli. H . G. Alexander (M anchester, 1956) , IS; W. Jentsch, 'Christian Wolff und d ie Mathen"J;l tik seiner Zeit '. in C/lristr·an Wolff aL. Philosoph der Aldkliinmg in Delltsrh/and, eds. H. -M. Gerlach, G. Schenk, and B. Thale r (Halle, 1980) , 173-llo; H:lns-Ji.irgen Engier, Phi/osop/lie als Analysis: Sir/ dier! z ur El1fWi(klrlM<~ pfrilosophisclu:r A llniysiskntlzepliollel1 Imler dem E il/fluss matl'('lIIa/ischer ,Her/lOdenmodellc im 17. uml irjjheu 1$. Jahrhllndcrt (Stuttgart-Bad C annstatt, 1982).

IZ Tore Friingsmyr, 'The Mathemat ical Philmophy' , in The QIIQHtifyifl.It Spirir, ctk Fr:ingsmyr ct aI. , 27- 44; ~ee esp. Wolff. DiswrsuJ pra'limillaris de plliio_,opl!in in ;:Cllere. Pt. I of hi~ PhiIcHop/lia raliO/w/is silJe logira, ed. J. Ecole. in Hhke , 11.1.1 (1983) , ch. 4, 53 - 71, e~p. §139. See aho {lrd imillary DiswlIrse 0 /1 Philosophy ill GC/lcral, trans. R . J. Bbck\-\rell (India napolis. IN, 1963), 76-,/.

T3 Wolff, Pfri/osophia pri llw :;ive olltologia, ed.. J. Ecole, in H0rke, 1I. 3 (11)62), §70, 47· '4 Article 'M ethode (Iogique)', in EllcydopCdie, 10: 445-6. 15 Berna rd Le Bovier de Fontenelle, Prlfa((, -, lIr /' uti/ite des matirematiqllf'£ 1'1 de la physiqlle I'i sur

II'S lravallx de I'Academie de)· Seiena's (1699), in Oellvres, ed. G.-B. Depping, 3 vok (Geneva, 191'18), J: 30-8 at 34; see Nod M. Swerdlow, 'Montuda's Legacy: The History of the E X3 Ct

Sciences' ,jmmlal (lfthe Hisrory ~fId/'as, 54 (r993): 299- )28 . 16 Floridan Cajori, ' Frederick the G rear on Mathem.:tcics :md Mathematicians ', Amrriralllvlal/t ·

ernatica! MOllthl)" 34 (1 927): 122- )0; Thoma~ Jeffcrson, letter 0( 1799 , quoted in David E u­gene Smith and Jekuthiel Ginsburg, History of Mnthemati(s ill Al/lerica Before 19M (Chicago, Ii . 1934),6,.

17 Edward Gibbon , Mernoirs ~fMy L!fe, ed. B. R adice (London, J984, repr. 1990), ch, lV, 99; Giambattista Vieo, Alltobiografia in Opcre, voL 5; tr,tOslated as n rc Autob r·o~raph)' of G i(lmh{lt . ri.lta Vico, trans . M. H. Fisch and T. C . Bergin (Ithaca, NY, 1944), 123-5 .

IS Euh:r, A.fet/i"dus irlV/'lIicndi linca:; Cilfl'(IS (1744), Additamenta I and II; Opera om nia, 1.24 : 23J-2, ).9~. The second pa~sage is translated in Her man H eine Goldstine, A H istory 0/ Inc Calcrdl!5 of Va riatiol/;; from rhe 171ft rlr rmlp.lr rlie 19th Cl'llwry (New York, NY, 19110), 10(i; See also Suzan ne Hachelard, Lt's polhlliqllcs (orrarnallf Ie pri1-!cipe dl: llIoirtdrt' arCion nil X VIf!" sih/r. (Paris, 1961); Pien e Brunet, ;V1ar/pe'/lIi.~ , 2 vols. (Paris, 1'929) , 2: ch. S; Joat:him Otto

jmlJes FrcHlklin

Fleckenstein , Introdu ction, in Eule r, OpfTa OtHilia, IlL S; Helmut Pulte, Dtls Prinzip der klcimlen Wirhmg IIlId die Krafik(>Nzcpfione/1 tifr ratio/wleJl Medwllik (Stuttgart, 1989).

19 D 'Alembcrt, ' Introductio n au Traite de dYIUUlliqlJC', in DeI/ life.', 1: 391-40(' ;I t 404. 20 Article 'Geometre', in ElIrydvpCdil', 7: 627-9. 2J Pierre Simon, lllarqui~ dl: Laplace, ESJtli phi/osvp/li'lue $IJr Its pfOoobifi res, translated as Philo­

sophical Essay on ProbalJlHtics, trans. E W Truscott aud F. 1. Emory (New York, N Y, 1951) , ch, 2, p. 4.

22 D'Alembert, ' lntroduction', 398 ; Thomas 1. Hankins,JeaH d'Alembcrl: Science alld the En ­l~~/llcmncJlI (Oxfo rd, 1970), 179-85; set' Jean C. D hmnbres and Patricia Raddet-de Crave, 'Contingence et necessitb en ll1ecaliique', Pllysis, 28 (1991): .15-114; Veronique Ie Ru ,Jran Ie R.olld d'Alcmbefi philMOpfic (Paris, 1994.).

23 Article ' DensitC', in ErIcycl(lpedie, 4: 833 . 24 D 'Alembert, 'I ntroduction', 398. 25 Article 'Experim.entale' , in Encyclopedic, 4: 300; see Thomas C hristensen, ' Music Theory as

Scientific Propaganda: The Case of d' Alembert's nlhIJem dl' IIIl1siq(u' ,jM!m al of {he His/ory if Urns, 50 ·3 (1989): 409-~7·

26 Leilmiz-Clarki' CO ffCsp oHr/cnu, 18 ; H crbert Breger, 'Symmetry in Leibnizi'lIl Physics', in TIre

Leiblliz R Cl1aissal1CC: International ~&rb'h\JP'" 1986 (Florcnce, 19119),23- 42. 27 Jean Etienne M ontucla, Histoirc rles malllbnaliqrlcs, ed. J. de la Lande, 4 vols. (Paris,

1799- 1802) , J: 628 . 28 Immanu el Kant, Kritik drr reiut:1I Vcmunfi (2nd edn., 1787, referred to as 'fl'), in Ak. 3:

'Einleitung' , B 17- 18; translated as CritiqH (~ ~r Pure Reasoll, trans. and cds . P Guyer and A. W Wood in Works (J998); Giorgio Tondli, 'La necessite des lois de 1a nature au XVIW siecle ct chez: Kant en 1762', R Cl'IJe d'llistoin: dC.1 ,,(ienus , 13 (J 959): 225- 41.

29 C liffo rd A. True~dell , 'A Program towa rd Redi.'icovering the Rational Mechanics of the Age of Reason', An/rivefor History ofEx(l(t Stiwces, j (1960h): 3- 36, c,~p. sections J4- 15.

30 Den is Didcrot, De l'Il1terpretatioli de la lIa tllre (1753), in Oelw1'I'sphi/osopliiql!es, ed. P. Vemi~re (paris, J956), §4, 180- 1: see trans. by D. Coltman in Diderot's Sc/l'Ctcd Wrifillgs. ed. L. C. Crocker (New York, NY, 1966), 71.

31 Hankins,Jean (/'Afembrrl, 74-5. 32 D 'Alemberr, Prelimillllr)' Dis[lJlIrse, 29 (Oellvres , T: 33), 33 D idcror, Lettte sur les aveugle$ (f749), in Oc/Wres pJzilosophiqllCS, 146; set· Didcrot's Early

Philosophical mJrks, tram. and ed. M . Jourdain (Chicago, IL , 19 16), 141, See also Jobann Gottfried Herder, Vers/tlnd l/11d Eljalmmg. Eille Mc /akri rik ZI/r Krifik der rcincH VemU/ift, Pt I

(1799), in Samllltliche l..fIuke, ed, Jl L Suphl n, 33 voIs (Berlin, 1877-1913),21: 36. 34 D 'Alemhert, Preliminary Diswu.rsc, 21\- 9 (OelJ11rCS, I : 32) . 35 Gauss, Brief an Bessel, 9 April 1830, in CirJ Fr iedrich Gauss, Werke, f2, vols. (Gottinge:n,

1863- 1929), 8: 200-- 1 ; selection transbted in The History o/ Marilw11lrir.1.' A R eader, ed~. J. FauveL and J. J. Gray (Basi ng;roke, 1987), 499·

36 Article 'Z~non [de Sidonl' , remark D, ill Pierre Bayle, Diaim/naire 1iistoriqlle et airiqlle (1696); translated in Histori(ailllul Critical J)iaioJ1ary: SelectiO!l>, trans. and ed. R . H. Popkin (Indi­anapolis, IN, 1965), 389-94.

37 Article 'Zeno d'Elce' , in Bayle, Diaionary, 359-72 at 366. 38 Girolamo Sacch~ri, EucJides lIil/((;((/tIlS, trans. and ed. G. B. Halstead, par;l.Jld Latin-English

text (Chicago, IL, 1920) ; Richard). Trudeau, '1'lie ./I·,flltJ-Elididcall Rcvo/mioll (B oston, MA, f987), 131-47; Boris Abramovic R.osenfeld, A I-lislllf)' of ,\ /o ll -EudideaJl Geometry: Ell()llIfiOI1 of til,· COJlCl:p l oj a Geoml.!tric Space, trans. A. Shenitzer (New York, NY, ly88), 98-9; Jeremy Gr:lY, IdellsofSp<uc: Euclidean, NOI1 -EII(]idclIlI alit! Rdativiitic, 2nd edn. (O xford, I989), ch . 4.

39 Trudeau, iVol/-Euclidean Rcrmlrllioll , 154·

Artifice and the natural world

40 j ohann Heinric h Lamhert, 'Theorie det Paulldli nien' , in F. Engd and P. Sdickel, Die T/u'Oric dr, Pamllellinim VCII Ellklid bis aI!(Callss (Leipzig, 1895), IP-207; Rosenfeld, Non-Euclideall Geomerry, TOO-I; Gray, !dens oj Space, ch. 5; W. S. Peters, 'Johann Heinrich Lamberts KOllzeption einer Geometrie allf ciner imaginiiren Kugel' , Kant-Studien, 53 (1961): 5r-<i7.

41 UOllglas M. jesseph, Befkeley~ Philosoph}' of A'la/hematin (C hicago, IL, 1993). ch. 3. 4.2 Jes:.eph, ch. 2 ; R o bert]. Fo~~lin , 'Hume and Berkeley 011 the Proofs oflnfinite D ivisibility',

Philwophical Review, 97 (1988), 47-69· 43 Geoffrey Cantor, 'Berkeley's 171i~ Analyst revisited' , Isis, 75 (t984) : 668- 83. 44 From Isaac Newton, QuadratI/ fa curvarulll, in David Eugene Smith, A SOl/ree Book if! ;Wlllhc­

lIIaliD; (New York, NY, r929) , 6 £4.- r8 at 0 14. The wu rk was published in N ew[Ou's Oplitk.:>, Of A -ITemise <!.f. . Lighl, A lso Iivo Trea/ises of Ihe Species (J lld M(~gl1itllde (if euroilinea, Figures (London, 1704), 165- 211. The: translation is slightly modified fwm that of John Stewan (London, 1745) .

45 Ge:orge Berkeley, TIle Analyst, §I3, in rVorks, 4: 72. 46 Berkeley. A t/aly.>t, §§2Q--S in Worb, 4: 76-8 1; Jesseph, Be,keley's PIJil050pIJ)' of Mmhefl1a/icr,

chs. 4- 7; Ivor Grattan-Cuinness, 'Berkeley 's C ri ticism of [he CalcuJu~ 3 S a Study in the Theory ofLimit~ ' ,Jamls, 56 (1909): 215-27.

47 G. C. Smith, 'Thomas Bayes and Fluxions', HiJ"loria Mathentatica, 7 (I980): 379-88; Roy N. Lokken, 'Discussions of Newton's lnfinitesimals in Eightee nth-Century Anglo- America " H is/oria Ala!/lf'ma,ic(J , 7 (1980): ' 41-55.

48 H .J. M. Bos, 'Differentials, Higher-Order Differentials and the Derivative in the Leibn izian Calculus', Ardlir,~ for Hi5tory oj Exact Scien(c~· , f4 (1974): 1- 90; Hide ishiguro, uilmiz's Philosophy ofuJ?j( (Jlld LmglltJSf! (Cambridge, 1990). ch . 5.

49 Colin Maclaurin, A '(reatise of FluxiotlS (Edinburgh, [742), 39; see Niccoi6 Guicciardini. Tlte D evdopJ11C1/f of Newloni(J1I CtJ/(JI/us il1 Bn·Mill, J70o-18M (C amb ridge, r989), ch. 3.

50 Macla min, ·lrcl1tise of Fluxions, 57. 51 Carlll. Boyer, The Ili.ltory of tile Calwills illld Its CO I1(Cptw!/ Dcvclopmwt: The Concepts ~r the

CtJ/CII/IIS (New York, NY, 1959) , ch. 6 . 52 Charles C. GilJi~pie. Luzare Camoi, Savn.lll: A MOIiORraph Trealitl}? Calt/ol's Scie1lfijic Wo,k

(Princeton, N), 197 1), ch. 5. 53 David Hume, A Treatise if Hamarl 1\,TatlJrc, eds. D. F N orton and M. J. Norton, ill the

Clare/lIloll Editi01I (2006), 1.3.1 .2- 5, SBN 69-'7 1; HUIl1e, An cnql,iry Concerning HIJman Ul/(lffSumdinj!" ed. T. L. Beauchamp, in the Clarendon EditiOll (2000), 7. J , SBN 60-'73 and 12.), SBN 161-5; R. E Atkinson , 'H ume on Mathematics' , Philosophical Quarterly. 10 (T96o): 127-37; Farhang Zabe:eh, Humr, P,ewrsor of }.JOdl!rrl Empirici5m: An Analysis oj His Opinioll s 011 M eal/ing, Ivietapilysics, Logic and MarhClI1atics (The Hague, 1960), 128-37.

54 D:lvid Hutnt:, DitJ'o~«es Concerning ,\latlnal R el(,<ioll, Pt. 9, in TIle ,Vatural HislOfY 11 R eligioll and Dialoglfes Ctmcemillg NalUm f Ref(,(ion, eds. A. W Colver and J. V Price (Oxford , 1976), 21S.

55 Donald Gotterbarn, 'Kant, Hume and Analyticity-'. Kant-Str/dim, 65 (1974) : 274- 83; Dorothy P. C oleman, 'Is M athematics for Hume Synthetic a priori?' , SOlllhwestern jormla{ of Philosophy fO.2 (1 979): 113- z6 .

56 H unte, TfCCllise, 1.4. 1.1, SEN 180; W. E. Morris, 'H ume's Scepticism abOLlt Reason ', HI/me StLtdies, IS (1989): 3!)-60.

57 R . J. Fogelin , 'Hume and Berkeley on the Proofs of Infinite Divisibility' , Philosophical Review, 87 (1988) : 47-69; James Franklin, 'Achievements and Fallacies in Hume's Accotlllt of Infi nite Di\·i sib ili t ~·' , H ume Sllldies, 20 (H)94): 85-101.

S8 Hume. Trea;ise, 1.2.4. 17 , SBN 44- 5; Rosemary Newmall, 'Hume on Space and Geometry', Jilmle Sfudies, 7 (1981): 1-31.

Jam es Franklin

59 H ume, 1il'(ltiSI', 1.2.2 .1-2, SBN 29- 30; Berkeley, A 7rcatise Conccrning tllr Priwiples a/HI/man Kllowledge (17 10), §J24, in WOl-ks, 2.: I-H3 at 98-9.

60 Thomas R eid, An IlIquiry into tlJe HII/1l(1ll i'vfilJd (1764), ed . D. R . Brookes (Edin burgh, 1997), VI.9 , 12.2-5; Norman Daniels, Thomas R eid's 'lnqlliry': rlu: Gromnry of VisiMrs alJd the Case]or Realism (Stanfo rd , CA, 1989) ; see BerkeJey, Au Essay (IIfIJflfd.( a Nelli Theory of Vis ion (1709) , § 156 in Works, I: 141-239 at 234.

61 Vi co, Dc lIostri IWlporis stlldiormn ratiO/Ie (1709) in Opere, 1: ch. 4 , tramaated as On Metlwd in ell/ temporary Fields afStudy in Se/erted Writil/,~s, trans. and ed . L Pompa (Cambridge, I982) , 40- 1; see O n the Anriellf ~Visdom of the Itll/jam (De alltiqlliisillla Iia/orllff1 sapien/ia, 17 10) , Bk. I , eh. I, §.z, and ch. 3 in Stlu/cd VVrifill~s, 54-5 and 64-S; sec also G~orges-Loltis

Lt'cic:rc, COlllte de Buffon , His(oirc notHre lle, 'Premier di~col1rs: de la maniere d '~tudier et

de traiter l'Histoire natureJle', in Oeuvres phlla.((lp/liq rles, cd. J. Piveteau (P3ris, ' 954), 23-5 ; thi~ selection gives referen ces to the tmprimcrie Royale edn. of O euvres c(ll/Jph;teJ, 44 vols. (Pari .. , J749- 1804); see I: 3- 62 .

62 Vico, -nil: Third l'lew ScieHcr (&iel1z/1 III IO /ro, 3.rd cdn., J744), §349, in Selected Writings, 206; see Scw!ld New Science (Scimza /lIlOtJa, 1730), § 1I33 , in 5dectcd Writil1gs, 269.

63 Andreas Rudiger, De ,(C/IS/l wri [I,fa!.' i , 2nd cd n. (Leipzig, 1722) , §§R-T 2, see Lewis White Beck, Edrly German Philosoph)': KfIIlf .. md H is Predecessors (Cambridge, MA, Ty6y), 299; Engfer, 'Zur I3 edcutung WoltE fu r die Methoden-diskussion dt'r dcu tschen Aufklarun gs­ph ilosoph ic. Ana lytische und ~ynthctische M ethode bei Wolff lind bei m vorkritischen Kant', in C/zristi (l l1 VJ;o!/f 1679-1754. Interpretaliollen;:w seil1("f Pllilosophie IlIId dercII H'''irJwl1~i!, ed. W Schneiders (Hamburg, 19 83), 48-65; R affaele Ciafardone, 'Von der Kritik an Wolff zum vorkr itischen Kant: WoJIT-Kririk be j Ri.idiger lind Cru~i lls', in Schneid ers, 289- 305; D avid R apport Lachtennan, Tile Elhic.~ ojC;colIIl'try (New Yo rk , NY, 1989), ch. 2, Pt . 4 .

64 Lambert, AblulIIdlwlgcn VO/1l C riterium I.'eritalis, ed. K. Bopp, K(nl m udie l1 30 (T9 .T5); see letter to Kant, 3 February 1766, in Kant, WorkslCorrespolldewf, ed . A. Zweig (1999), 84-7 .

65 Kant, UJilerslldllmg iiber die Dell tlichkeit der G rii l1dsiirze der natiirlichell "J1lco/o,!?ie Imd dcr A10ral (1764), Fi r~t R eflection, §I, Ak 2: 276--8 , trambted as IIIquiry COl1fCfllillg the Distil/ctness of the Pril/dpfrs qf Na/lIral 'tllcolo,,!?y aud Morality, in WOTksI111~orl'l ical Philosoph)', '755-1770, tram. and eds. D. W.1.lfo rd and R . Mecrbote (1992); Krilik der 'eillell Verl/Illifr, B 74 1; see Michad Friedm an, Kal1t and the E,'act Sciellces (Cambridge, MA, 1992), ch. I ; Tonelli , 'D er Streit iibec d ie mathemarisch c:n Methode in d er Philosophie in dec ersten H alfte d i.,:s 18. J3hrhunder[s lind die En tsteh ung von Kan t~ Schrift tiber di e " Delltli chh:it"', Arrhiv fiir Philosophie 9 (1959): 37-66; Ted n. Humphreys, 'The H istorical and Conceptual Relations bet\vecn Kant's Metaphy,ics of Space and P hilosophy of Geometry' , j ,)//TJ/al qf the H istor), of Philosophy, Tl (1973) : 483- 5I2, Sect. 1; Gregor Bi.ichd, C e(lmelrie Wid Philosophic (Berli n, 1987) , ch. Tj W. R . De Jong, 'How Is Metaphysics as a Science Possible? Kant on the­Distinction between Phi losophical and MarheUl;Jtic;li Method ' , Re!!iew vI Melaphysics 49

(T995): 235-74 · 66 Kant, De ml//ldi sel/sibOis atqlle ime!hgibilisjorma et principii, (1770), §I5C, Ak 2: 402-3 ,

translated as Ott the Form and PrillClJllc$ of thc SCllsibie alld tflC 1I1tefl(,<iMe IVvr/d [If/al/gllml DiHc ftatioll] [1770], in ]li-iJrb ITlleorer i((li Pllifosophy 1755-1770; see Krili!'! dcr Telm'lI Verrml!/i, B 26R .

67 Kant, Kritik der reil len Vcr/IIJ1!fi, Ak 3: 480; Fr iedman , Kant 1111(/ tire J::'mcl SciCllfeS, ch. 2, §r; Introduction and several papl'rs ill Kalil's Philosophy of Mathelllil(ic;: ll.4odcm Essays, ed . Carl J. Posy (Dordrccht, 1992); Jaakko Hi lltikka, 'Kant's Theory of Mathem atics R.evi~­ited', ill Es~a)'s 0" Kwtf's C ritiqlle of PliTt: ReasolJ, eds. J. N. M o hanty :lnd R . W. Shahan (Norman, OK, 19SZ), 20 T-I5; R a ben E . Bum , 'R ules, Exam ples and Constructions: Kant's T heo ry of Math ematics' , SytJtllcsl! 47 (I~8 T ): 257-88 : Alfredo Ferrario, ' Constnlction and

Arfij1ce Q/1d the natural uJ()rld

Mathematical Schematism: Kant on the Exhibition ofa Concept in Intuition', Kilnt-Sl lldiclI 86 (1995): 1)1-'74 ·

68 Kam, Dc IIl1l/1di, § TSO, Ak 2: 40) - 4. 69 A. P. Yousc hkcvitch, ' LaZ:l.Ie C arnot and the Competition of the Berlin Academy in 1786

on the Mathematical Theory of the Infinite', ill G ilJispie, u z nre Cn fllol, Sarlflllt, '4y-68. 70 Krilik. 13 532; Anthony W interbo urne, Til l' iden! nlld tIll' Real: All Outline of Kant 3' Theory of

,Space, Time a/ld A1alhel!111tical Constructioll (D ordreeht, 1988) , 79- 89. 7f Arthur M dniek, Space, Time, t1I1I1 Thought ill Kallt (Dordrecht, 1. 989) , 5-20, 189--98; Fried­

man, K (J/ll (lIId lite Exa[f Sciell(eS, 74--9 .

72 Joseph-Louis de Lagrange, MCcoll ique nnalytiqlle (1788), 'Averti~semem', in OeIlVn'$, eds. J.- A Serret and G . Oarboux, 14 vok (Paris, 1867- 92), II : xi- xii

73 Marie J l:;!drl Antoine N icolas de Caritat, marquis de Condorcet, 'Discours sur le~ sciences mathbnatiques' , in OI'J!VfI~ .\", eds . A. C. O'Connor and E Arago, I2 vols. (Paris, rH 47- 9), I:

453 - R1 at 467. 74 Thomas 1. Han ki ns, SdcJI(e and the cl1iigltlelllllcllt (C ambridge, 1985), cn. 2. 75 Euler, All/f'il flf~t! z ltr A~ebro, Pt. I, Sect. r. eh. I , in Ope'" omnia, 1.1: 10; d'AJembert.

175M; )/1' /t>~. elements de phiJo$!)phie .. OI'C{ le5 f:d(lircJssemCIIs (1759-()7) . Pt. 14, §11 in OCllYres

wlllpleres, I: 263 ; K;ult, Kricik der reil1l'J1 Verl!lII!(r, B 745 : see R. E. Rider, Alathcmat;c$ in the EI11(~h tc/!llJeI!t: A 5wdy ~f A(eehra, 1685-1800, Ph D thesis, Univ. California, Berkeley, t 9S0.

76 Luho, Novy, Origills c:f Modern A (l;eilFlI, trans. J. Tauer (Leiden, 1973 ), eh. 2

77 Wolff, Psy(h%gia elilpifica (1 738), in f.k,kc, 11. 5, Pc I , Sect. 3, eh. 2, §294, :wS. 78 Etienne Bonnat de C ondillae, L I Logique (1780), P t. 2, eh_ 8, in Oeuvres pbilosophiqlJes, ed.

C. Le R oy, 3 vols. (Par is, 1947-51), 2.: 410, 4II , see La Lo~iqlle-Logic, tfans. and ed. w. R . Albury, with parallel French facs im. repro (New Yo rk, NY, 198 0), 3TT , }15 .

79 Condill ac, LA lan,R IJc des w ind.,' (179tl ), in OCllvres plrilvsvpmques, 2: +2 1- 558 , at 429; Isabel F. Knight, -nw GCGlllctric Spirir: The AMi: de COll dil/a( and the Frel1ch EH/ightenmcnt (New Haven, C T , 1968), 171-5.

80 Ka.n t, UIlfCfmdlfltfR iiber die DClitlidtkei( der Crtllldsii(;;:e, First R eflection, §2, and T hi rd R e­fleClion, § 1 (A k 2 : 278, .291 ).

:h M . KJine, 'Euler Jnd Infinite Series', Matlll:mati(s A1a~azi!l(" 56 (T 9S): 307- 14. 82 Euler, Vol/sfiilld(\(c .11lIeitll!1g Z ll f Al~ebra , Pt. T, Sect. 1, eh. 13, in Opera, Ui: 55; Novy, Moden!

A lgebra, ,}5-7, II2-I4; see also]. Playfair. 'On the Arithmetic of impossible Quantiti es' , Phi/osap/liml Tratlsadiolls 4 tilt' Rv)'ol Society C!.(wlldv!l , 68 (1778): JTH-43 , on which D. Sherry, The Logic of lmpos.~ible Qu.mtities' . Studies ill Hi;;/\Iry and PhilQsophy !)J Scienct', 22 (199 1): 37- 62.

83 Euler, IJ11 rodUClio ill (lfwlysill it!.fl lliIMllm, Bk. I, eh. I , §4 in Opera, 1. 8: IS; translated ;L~

IlIfrotiliCtioll 10 AI/alysis of t/Ie Iryil1ite, Bk I, trans. J. D. Blanton (New York, NY, 1988), ). tl 4 J. Lutzen, 'Euler's Vision of;) General Parti al Differential Calculus for a Generalized Kind of

Function', Mlltflclllarirs MIl,gazill.e, 56 (1983): 299-)06; C. Truesdell, Introduction, to Euler, R atiollal :VJec//(mi{S, in Opera, 11.11 , Sect. 2, especially 244-50.

85 l agrange, 'J11(!orie desjoflctiot15 tlflol)'tiqllC5 (paris, 1797), 2,7- 8, 1.2 ; a later edn. is incl uded in vol. 5 of the Ot/Mes. See also Judith V Grabiner, Thc Calm/us as Algebra: J. -L. 1AJ:rn/J~~t',

17]6-1813 (Ncw York, NY, 1(90): rvor Grauan-Guin ness, 171c Development oj the Fmll1 datio lls oIlv!atfll'lI/atical Analysis from Euler 10 Rit'lI1a1m (Cambridge, MA, 1970), ell. l.

86 Lagrange, 'R6fl. exions ~ur la resolution algebrique d l'S equations' (1 770/ r), in Oetlvrr~s, 20) ­

_4-2 [ ; Hans Wussing, '/Yle C cnl''(is C?,{ tl,e / l bstract Ct"()IIP COfl{ept: A Contributi!)ll to the History 4" t1le Or~~itl of Ab5fracl Croup Theory, trans. A. Shcn irzer (Cambridge. MA, J984).

87 Paolo Ruffin i, Teoria ,Renero/e de/It: cqllflzioni ill wi .'il dimostra impossibif(' la so/uziolle a~i!e­

hmira delle r~q lJazi()!l i gCll crali di ,lfMtiO superiore al quarto (Bologna, 1799) ; Wmsing, 80-4:

85 0 James Frallklil1

R. Bryce, 'Paolo Ruffini and the Quintic Equation', Symposia iViafhcl1Iatiw, 27 (1986):

169-85. 88 Mary Tiles, Mathematics and the [mage of Reason (London, 1991), 10-24.

89 Euler, Specimen de HSII ohscr/!aliol1tll!l in mallil's! pllm (1761), in Opera, 1.2: 459-92 at 459; as translated in George Poiya, AJarfiematics and Pial/sible Reasoning, 2 vols. (Princeton, NJ,

1(54), I: 3· go Polya, r: 18-2(, see also 91-8.

91 Laplace, Philosophical Essay, eh. I, I; Gauss, Hhkc, 2: 3.

92 Wolff, Discur5lJs prC/iminaris, §127, trans. 68.

93 Lambert, Anla;\,'1' zlIr Arc/Ziieclollic, oder Theorie des EiI~racheJI rmd des Ersten ill der philosophisd/C/I twd !1IatliwJatisc/u.'I1 Erkel/lltllis, 2 vals. (Riga, 1771), Pt. 2, eh. 11, §§315-27, t3.csim. repr.

in Philosophisrhe Sc/Jrijten, ed. H. W. Arndt, 10 vols. (Hildesheim, 1965), 3: 306-19; 0. B. Sheynin, 'On the Prehistory of Probability', Arrlih)e for History ~I EXllct SCfences, I2 (1974):

97-141 at 1J('i-7. 94 Euler, 'Solutio problematis ad geometriam situs pertinentis', Opera, I.;: 1-10; translated as

'The Koenigsberg Bridges' in 5;cielltiji( Amrriwn 189.1 (1953): 66--'70, repro in The H·0rld ~f

iUallzemalics, ed. J- R. Newman, 4 vols. (New York, NY, 1956), 1: 573-80; see H. Sachs, M. Stiebitz, and R. J Wilson, 'An Historical Note: Euler's Konigsberg Letters' ,Jollrnal of Graph 77leory J2 (1988): 133-9.

95 Gottfried Wilhelm, Frciherr von Lcibniz, Phi!osopfliw! Paper:' mid Letfw: A Seh/ion, trans. and ed. L. E. Loemker, 2 vok (Chicago, IL, T956), J: 38T-96.

96 Buffon, I-jrstoire natllrcllc, eh. J J 'Histoire generale des ;tnimaux: Histoire naturdle de l'hommc', in Impr. Royale edn., 2: 373.

97 Kant, Hm dem (.",,-Im Gmflde des Untersc/tiedes der Gege/lden in-I Ralll1le (1768), in Ak 2: 375-83

at 377; translated as COI1CemillJ.: Ihe Ullimate GrOlmd of the Differentiatioll ~f Dirertiolls in Space, in vI/arks/Theoretical Philosophy, 1755-1770.

91l A.-T. Vandermonde, 'Remarques sur les prohlCmcs de situation', Histoire de !' AwdCrJ1ie Royale des Sciences, 1771 (Paris, 1774), 566-74, translated in Norman L Biggs, E. Keith Lloyd, and RobinJ. Wilson, Graph Theory 1736-1936 (Oxford, 1976),22-6.

99 0. J Griisser, 'Quantitative Visual Psychophysics during the Period of European Enlight­enment', Dowmenta Ophthalmolo,~r(a, 71 (191l9): 93-11I.

roo Button, 'Essai d'arithmetique morale', §8, in OClwfes pfrilosopfliques, 459; see Histoire Ilaturelfe: Supplement, IV, in Impr. Royale edn. 2: 46-148, at 50; Lorraine Daston, Classiwl Probability ill the El1b~Rhten/llcnt (Princeton, NJ, I988), 91-3.

101 Pierre-Louis Moreau de Maupcrtuis, b'ssai de philosophic morale (1749), ch. T, in Onwres (Lyons n61l and Berlin 1758 edns.), ed. G. Tonelli, 2 vols. (Hildcsheim, T974). T; 17T-252

at T95 T02 Adam Smith, All Inquiry into the j'\:afUrI' and Causes of the Hlcalth of ,'\Cations, eds. R. H.

Campbell, A. S. Skinner, and W B. Todd, 2 vols., in l>vorks (1976), I.v at 1: 4tl, So and 41l; see Condorcet, 'Tableau general de la science qui a pour objet l'application du caleul aux sciences politiques et morales' (J793), in OClJVfCS, J: 539--'73, at 558.

T03 Benjamin Franklin, Letter to Joseph Priestley, 19 September 1772, in Papers, ed. L W.

Labaree and \V J. Bell (Ne\v Haven, CT, 1959-), T9: 299-300. 104 Condorcet, 'Tableau', Ol'lll'res, I: 541: Dasron, Classical Prohability, 217-T8. 105 Condorcet, Essai sw' {'application de I'mwlyse ala probabilitf des decisions rt'Iu/lles a fa plumlire des

lJoix (Paris, 1785); Keith M. Baker, Condone!: From ;.,\latlJral Philosophy to Sodal )\1atliell1arin (Chicago, IL, J975), 225-44; Daston, Classical Pro/Jabifity, 345-55; R.. Rashed, COlldoreel: AlatliematiqHc et sociCtc (Paris, T974).

lao Baker, Ca1ldGreet, 340-1.

Atffrue mid the lIatllm! UJorld

107 Thomas Reid, All EssllY on QUdllfify (f74R) , in Phi/o$ophiwl VM11"k.(, ed . W Hamilton, '"" 'loIs.

in I (Edinburgh, [895),715-19. 108 Otto Mayr, 'Adam Smith and the Concept oCthe Feedback System', Tec1l11olOJtyand Cuilure,

I::! (1I}77): 1-22. 109 T homas R o bert Malthus, All Essa}' 011 ilw Principle c:.f Popul<ltioll, 1St cdn. (londo n , ' 79S) ,

13; the .EuI1Y was substantially re\vritten for th e influential edn. of 1803. 110 'Condorcetau comte Pierre Vcrri, 7 novembre 177I', in Oeuvres, I: 281-5 <It .2R] - 4; Baker,

COlld()fCCf, 337-8 . In 0. B. Sheynin. ] . H . Lambe rt's Work o n ProbabjJjry', Arcllivefor Hiswry oj Exact Scienc~·, 7

(I970h): 244-S6, Sect . 2. 11 2 Wilhelm Risse, Bibfio~,apltia fo,~ ica, 4 voh. (Hildes hcim, 1965), 1: J1)0-235 ; Ri~e, J)ie Logik

dN lVeuzeit, 2 \loIs . (Stuttgart- Bad Cannstatt, I96 4- 70) , voL 2 ; Anton Dumitriu, History of Logic, 4 '10k (Tunbridge Wells, T977), 3: 150-62 ; I. H . Am:llis, 'Theology against Logic: The O rigins of Logic in Old R ussia', History alld Philosophy vI L~ei( 13 ( 1~ 92) :

IS- 4:!. · 113 W ilbur S,lnlllci H owell, Eightrel1llt-Cctll1lry Brilisll UJ,!;i( atid R fll'to ric (Princeton, NJ, 197 1),

eh . 2, §5: see M. Feingold, 'The U ltim alC Pedagogue: Franco Petri Uurgersdij k and the English Speaking Academic Learning' , in FrclIl(O Blj~r.;ersdijk (159(r- 16J5): iVeo-Aristotelianism ill L 'idm, cd ~. E. P. Bas and H. A. Krop (Amsterdam, f993) , T51-65.

TI4 Kant, Die .fifl;·chc SpitzfilUligkeir der vier syf/ogisrisc llell F(,;ufC/l enpiescn von AI. 111I11f<llluel Kalil

(176 2), ~5 , in Ak 2: 45--61 at 55- 7, tramlated 3S "H,e False Subtlety q(the FOllr Syllogistic Figll re.~

dcmomtfa/ed by ;\1. Immat/llef Kaut in HII"hf"J1'l'ol"f!/iral Philosoph)' J755-J770' us Kanr,Ja~·d/e Lcgik, 'Einleitung', Sect. 1 (Ak 9: 1-150 at 14); as transla ted in IVorks ILer/rtrC5

Oil u~if, trans. and ed. J. M . Young (1992) , 529; Kritik dcr reiw~n i4.rmmft, 'Vorrede', B viii. II6 J. C. Buickerood, 'The Natural History of the U nderstanding: Locke and the Rise of

Facultative Logic in th e Eighteenth Century', History lind P/li/o"ophy of Logic, 6 (1985) : 157--90; John Passm ore, 'D escartl'\, the British Empi.icists and Formal Logic', Phi/osuphicaT Revif'w, 62 (1953 ): 545-53·

117 Joh n Locke, All Essay Concefllillg HUll/ali Undel·.'taIllIiI1J?, ed . P H . Njdditch (Oxford. ' 975), TV.xvii, esp. 070-6; H o\vell . Logic al/d Rhetoric, ell. 5, §2.

lIt! Isaac Watts, Logick: or, The Righi Usc c:.f R easol1 in the ./.;"nquilT fiji er 7hllh (London, I725),

Pt. 3. eh. 4 , (II Rule), 492. I19 H oweU, U'.I!i((ltId Rilctori(, eh. j, §3 . 120 R eid, A Brif'j A mm/lt if Ari5tot"~'s u,5!ic (1774), ch. IV, Sect. 5, in PllilMophicaf vf.ilrks, (HlI-? I4

at 701-2; George Campbell, Tile Phifmoplty oj Rhetori{, 2 vols. (Londo n, 1776), t'sp, Bk. I, ch. 6; see I-lowell, eh. 5, §4,

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127 H .. G. Olson, 'Scottish Philos()phy and Mathemati cs 1750-183° ' , jOlmwl of the Hi5/0fY (!(

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133 Lambert, /\i('lIes O'X(J/fOI', 2. vots. (Le ipzig. t764) 2: 355; Styazhkin, 123 _ 134 Diderot. article 'A rt' , in Encydopedic, T : 713-19. 135 I. Bernard Cohen, Benjamin Franklil/'s Sd/'/ICf (C ambridge. MA, 1990), eh. 3; Silvio A.

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136 R . E. Schofield , 'The Industrial Orientation of Science in the Lunar Society of B ir m­ingham', in .Siiel1(c, "1rdmo/ogy and &ol1om;;: Growth in the EigJIlCClltit Century, ed . A. E. Mus~on (London , 1972) , 136- 47; Roy Porter, 'Science , Provincial Culture and Public Opinion in Enlightenm en t England', British J Ollmal fi" J:;~I!JJfl'f:lltJJ CeJltllry Stlldies, 3 (J980) : 20-46.

[37 A Diderot Pirtoria/ J:!ncyclopedia ~fTradc, and Iml!H(ry, ed. C. C. Gillispie, 2 voh. (New York, NY, 1959); Gillispie, Samcr.' and Polit)' ill Fnmce lit {lie End ~flhe O ld ReJ!imc (Princeton, NJ, 1980), 337-56.

138 John Desagu!iers, A Coulse if Experimt'/tf(// Philosophy, 2 vols. (l ondon, 1734-44), Prdace; L .. rry Stew::trt, TIle Rise tif Public Sdel1re: Rlle/erir. TerJuw/ogy and l'\,'aittraf Philosophy ;11 1\;(,lI'to­l1ial1 Britaiu , 1660-1750 (Cambridge, 1!J92), ch . 7 ; Simon Schaffer, 'Natural Philosophy and Public Spect3de ill Eighteenth- C entury Sci ence' . His/ory 4 !Ximec 2J ( 1983): J-43; Mar­garet C. jacob, 'Sd enrifi c Culture in the early English Enlightenment' , in An/icipariOlls (1f the Enlighunillet/t ill Ellc-.; fand, Frll/ICe alTd GermallY, eds. A. C. Kors and P J. Korshin (ph iladel­ph ia, PA, J9H7), eh. 6; Barbara Maria Stallard , Arifill Sdl' llt'e: EI11(eh tcllmI'lH, Entertail1m ent an d the Eclipse C!. f Visllal Sri/w ee (Cambridge, MA, 1994) .

139 Joseph Bramah, A .Lettt'r t Il Ihe RI . H rJ fI. Sir james Eyre (London , 1797), quoted and discussed in Christine MacLeod , Invellting tl/ t' Il1dll.llr;a/ Revol/llion: The EII,,(/is/r Palm( SyS/CIIl, J6oQ-18oo (Cambridge, 19R8) , 2:20 .

140 A. Smith , '''''ealth 4 ,'\1atiolH". Bk. r. <.:h. I , I: Z T . 14 1 D. S. L Cardwell , 'Science, Technology and Industry' , in The Ferllll'llf of KllolI'ledgtt: Studies

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142 MarjOlie Nicolson w ith N. M . MoWer, 'The Scielltiflc Backgrou nd to Swift's kayage io LapH/a', in Marjorie N icohon, Scil'!t (/~ and imagitwflmt (Ithaca, N Y, '956), lJ o-54: R ich:uu G. Olson, 'Tory-High Church O ppositio n to Science ,IUd Scientism in the Eighteenth Century: The Works of Joh n Arbutb not,Jonathan Swift, and Samu eIJo hmon', in n,e Uses oj SeicHce ill the Age lif Newton, ed. J. G. Burke (Berkeley, C A. 1983) , T7l - zo4·

Artifice aHd the Ilatural UJorld 853

143 William Derham. Physi{(). J'}uology. or, A Dcmollstratit1ll of the Being al14 A ttrilmtCi of C'iOd (London, 1713) , Bk. 5, ch . I; W. Coleman, 'Providence, Capitalism and Environmental D t:gradation' ,JoIIYrlal f:!fflic HiHory of Idt:as 37 (1976) : 27-44·

T44 William Paley, j\la(ura( 71H'o'o,~}', ch. 5, in Works, a new c£ln., 7 vol~. (London, 1825), s: 39-52 at 46; N. C. Gillespil" 'Divine Design and the Industrial Revolution', Isis 8r (1990): 214-29.

(45 H llme, 'O f Refi nement in the Arts ', in Political Essay" ed. 1<. H aakonss(;:n (Cambridge, .1 994), t 05-J4 at 107 , 109 , Il l.

146 Quoted in H ector Fleisch mann. La Guillotine er/ 1793 (Paris, I 90S) , 255.