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Don't take me half the way: On Berkeley on mathematical reasoning
Don’t Take Me Half the Way: On Berkeley on Mathematical Reasoning David Sherry* has a portmanteau philosophy of mathematics-a realist philosophy of geometry but a formalist philosophy of arithmetic and algebra. Geometry has a genuine subject matter, on his view, but arithmetic and algebra can be studied only for the sake of guiding action (cf. Pycior 1987, pp. 277 ff.). His account includes a compartmentalized view of mathematical reasoning, which, I shall argue, fails on the realist side. But the problems Berkeley tried to solve with his philosophy of geometry can be handled using his formalist strategy. This yields a neo-Berkeleian characterization of geometry that is pleasing on its own. And it does not turn the empiricist project in the direction of logical positivism, as might be expected, but in the direction of Wittgenstein. BERKELEY
1. A Lockean Problem, a Berkeleian Solution
I shall begin with Locke, who provided a constant stimulus to Berkeley’s thought. Locke joined the Cartesian revolution with its doctrine that clear and distinct ideas are necessary and sufficient for knowledge. The ‘way of ideas’ promised freedom from interminable scholastic disputes, and both Descartes and Locke agreed that careful conception and comparison of ideas was the only way to avoid error and extend knowledge. This is nicely illustrated by their rejection of formal logic in favor of the natural light (or native rustic reason). Knowledge, according to Locke, consists in the perception of agreement and disagreement of ideas. In intuitive knowledge, the perception is immediate, but in demonstrative knowledge the perception occurs via intermediate ideas. To perceive the agreement between the interior angles of a triangle and two right angles one must discover an intermediate idea which agrees with each of the extremes. In this case (Euclid 1.32) the intermediate idea is gotten by extending CB to D and constructing BE parallel to CA (Fig. 1). Thus if BE is parallel to CA, the theorem that governs two parallels cut by a transversal (1.29) implies that < ACB= < EBD and < CAB= -e EBA; hence *Department of Philosophy, Northern Arizona University, 6011, U.S.A. Received I July 1992; in revisedform 28 October 1992.
Stud. His?. Phil. Sci., Vol. 24, No. 2, pp. 207-225, Printed in Great Britain
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PO Box 6011, Flagstaff,
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Fig. 1.
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Thus when I demonstrate any proposition concerning triangles, it is to be supposed that I have in view the universal idea of a triangle, which ought not to be understood as if I could frame an idea of a triangle which was neither equilateral, nor scalenon, nor equicrural, but only that the particular triangle I consider, whether of this or that sort it matters not, does equally stand for and represent all rectilinear triangles whatsoever, and is in that sense universal. [Principles, $151
For Berkeley, the subject matter of geometry still consists of ideas, indeed, general ideas, but in an entirely new sense: a general idea of a triangle is no longer a concept difficult of attainment, but simply a particular idea used to stand for all triangles. By restricting the general use of particular ideas it is possible to account for the generality of geometric truths. . . . though the idea I have in view whilst I make the demonstration be, for instance, that of an isosceles rectangular triangle whose sides are of a determinate length, I may nevertheless be certain it extends to all other rectilinear triangles, of what sort of bigness soever. And that because neither the right angle, nor the equality, nor determinate length of the sides are at all concerned in the demonstration. [Ibid., $161
The geometer averts the worry that a proof is not generally valid by forsaking particular features of a representative triangle. Using the diagram under this condition does not require an entity which embodies just the characteristics shared by all triangles. Berkeley’s proposal is therefore a new way to understand proof. Traditionally a proof shows one (abstract) general idea predicable of another, whereas he understands a proof as a general procedure for showing some property holds true of all particular ideas of some sort. Thus the proof of I.32 really provides a procedure for demonstrating, of any particular triangle, that its interior angles sum to two right angles. Abstract general ideas are not part of Berkeley’s ontology, but then neither are they unabashedly part of Locke’s: they are ‘fictions and contrivances of the mind . . . something imperfect which cannot exist’ (Essay, 4.7.9). Seeing no other way of achieving generality, Locke begrudged general ideas ontological status, but in so doing he faced a dilemma: he knew general ideas could not be images, but he couldn’t, as an empiricist, avail himself of the rationalist power to grasp ideas which correspond to no perception. Berkeley’s genius consisted in invoking the use of ideas-over and above the ideas themselves-to solve Locke’s dilemma without sacrificing empiricist scruples. Locke’s problem with generality is also implicit in what he says about reasoning. And here too Berkeley’s innovation suggests a way out. Both Locke and Berkeley were content to explicate reasoning as the agreement of ideas without saying much more. But it is worthwhile to see what each can say about the agreement of ideas. Locke’s general ideas are universals, not particular diagrams. Even so, he was impressed by diagrams. They are ‘copies of the
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ideas in the mind’, and by their means mathematicians are able to achieve an enviable certainty (Exsu~, 4.3.19-20)’ I suspect that when Locke described diagrams as copies he was conveniently forgetting his admission that general ideas are fictions and contrivances and taking too literally his famous metaphor that knowledge consists ‘in the view the mind has of its own ideas’ (4.2.1). But with respect to general ideas the visual inspection metaphor takes him no farther than the rationalist who claims to perceive the agreement of ideas by intellectual insight. The only difference is that Locke insists that mathematical ideas are constructed while the rationalist allows that the mind comes stocked with them. Although Locke was correct, in my view, to recognize that ideas agree or disagree in the sense of having affinities and aversions for one another, he was not prepared to explain this insight in a way that avoids the subjective character of rationalism. In contrast, the agreement of geometrical ideas is apparently no problem for Berkeley. Geometer’s diagrams do not stand for hidden or fictitious entities. The diagrams are (in Locke’s phrase) ‘the objects of understanding when a man thinks’, and so their agreement can be explained literally as a visible resemblance: in geometry, demonstration is demonstration ad OCU~OS.~ Thus Berkeley can regard ordinary experience, rather than intellectual insight, as the foundation of geometric truths. And this explains his rejection of infinite divisibility in geometry and infinitesimals in the calculus (cf. Principles, $132, and The Analyst).
2. A Reluctant Formalism There is no demonstration ad oculos in arithmetic and algebra, for, on Berkeley’s account, neither involves ideas which agree with or resemble one another. Already Locke had recognized that the marks of arithmetic and algebra are not copies of general ideas and concluded that ciphers and the like did not so much ‘help the mind.. . to perceive the agreement of any two numbers, their equalities or proportions’ as to help the memory (Essay, 4.3.19).
‘Locke even believed that certainty would be achieved in ethics were someone to invent a moral algebra and so ensure that we could consider our demonstrations at leisure ‘without any danger of the least change in our ideas’ (Essay, 4.3.19). Both mathematics and ethics admit of demonstration because their concepts are archetypes (ideas constructed from materials of sense experience), not ideas abstracted from sense experience (4.4.6-S). Locke never explains how archetypes make demonstration possible. My remarks in Section 6 suggest an explanation of archetypes. % New Theory of Vision Berkeley argues that geometry concerns tangible extension (5159); in the Principles he appeals to visible extension. This sort of image employed makes no difference to my case.
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Thus it is not surprising that Berkeley should conclude that arithmetic and algebra have no subject matter at all. But number being defined a ‘collection of units,’ we may conclude that, if there be no such thing as unity or unit in abstract, there are no ideas of number in abstract denoted by the numeral names and figures. The theories therefore in arithmetic, if they are abstracted from the names and figures, as likewise from all use and practice, as well as from the particular things numbered, can be supposed to have nothing at all for their object. [Principles, $1201
In this passage Berkeley takes for granted that neither signs nor things could constitute a subject matter for arithmetic (and algebra) the way perceptible extension constitutes the subject matter of geometry. Perhaps he reasoned that while there is nothing in a given numeral or thing which necessarily connects it with a number, the diagram of a triangle is necessarily triangular. Berkeley allows that arithmetic and algebra are conversant about signs (AIciphron, VII, §12), but this is not to give a subject matter to arithmetic. In his view, our grasp of number is distinct from our grasp of a particular object or set of objects (Principles, $12; cf. Theory of Vision, $109). He argues that particulars are given different numbers-‘the same extension is one, three or thirty-six, according as the mind considers it with reference to a yard, a foot, or an inch’--depending on what is convenient to the task at hand. Thus, no object of perception, by itself, suffices as an object for arithmetic. Extension, on the other hand, presents itself without any arbitrary combination by the mind; thus extension can be regarded for its own sake, and geometry, over and against arithmetic, qualifies as a science with a subject matter. The lack of a subject matter for arithmetic and algebra does not prevent Berkeley from admiring them; but it does effect a dichotomy in his conception of science and scientific reasoning. Arithmetic and algebra are practical sciences of manipulating symbols to obtain useful conclusions, kinds of logistic. Geometry is a genuine or demonstrative science, which rests on the agreement of ideas and so fits the paradigm of knowledge that Berkeley inherited from Locke. Corresponding to the distinction of genuine from practical science are two senses of reasoning. That distinction turns upon whether or not one can conceive the referents of the terms of a theory. Where such conception is possible, the standard of reasoning consists in agreement between objects of reasoning (referents of the terms). But in arithmetic and algebra there are no objects to agree and disagree except signs and things, and they do not agree with one another in the literal sense in which one can visualize the congruence between the interior angles of a triangle and two right angles. Consequently, the standard of reasoning in arithmetic and algebra is dictated by rules for using symbols rather than the ideas which constitute the subject matter of the discipline. And since there is no ideal backing for the rules, their justification is pragmatic,
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Studies in History and Philosophy of Science 3. A Blindspot and Some Extra-Mathematical
Commitments
Berkeley’s portmanteau philosophy of mathematics has an uncomfortable side effect. In the Philosophical Commentaries he characterizes arithmetic and algebra as ‘sciences purely verbal and entirely useless but for practice in societies of men. No speculative knowledge, no comparing of ideas in them’ (P.C., p. 768). Without a subject matter, arithmetic has no claim to speculative knowledge, over and above practical knowledge; indeed, Berkeley relegates speculative endeavors in arithmetic to d@ciles nugae (Principles, $119). These derogatory comments about questions such as Fermat’s Last Theorem, which are neither subservient to practice nor promote the benefit of life, have no parallel in his remarks on geometry. Yet surely there have been geometrical investigations which promote the benefit of life no more than researches into Fermat’s problem: how should one apply the theorem that the external bisectors of two angles of a triangle are concurrent with the internal bisector of the third? It is unreasonable to expect practical application for every theorem in geometry. Berkeley’s blindness to this sort of objection reveals a philosophical commitment that goes beyond dissatisfaction with rationalist epistemology. Great Britain was much slower than the rest of Europe to replace geometry with algebra in liberal arts curricula because the change had to fight a tradition which conceived geometry as the best training for practical reasoning (Becher 1980; Olson 1971). Although the tradition became most pronounced in the Scottish philosophers, Berkeley accepted it without reservation. It hath been an old remark that geometry is an excellent logic. And it must be owned that when the definitions are clear; when the postulata cannot be refused, nor the axioms denied; when from the distinct contemplation and comparison of figures, their properties are derived by a perpetual well-connected chain of consequences, the objects being still kept in view, and the attention ever fixed upon them; there is acquired an habit of reasoning close and exact and methodical; which habit strengthens and sharpens the mind, and being transferred to other subjects is of general use in the inquiry after truth. [Analyst, $21 Pace the twentieth-century
practice of requiring college algebra, the British tradition saw no liberalizing value in arithmetic and algebra; they were expedients with no tendency, in Dugald Stewart’s phrase ‘to strengthen the power of steady and concatenated thinking’ (Olson 197 1, p. 43).’ The failure of algebra to satisfy the demands of liberal education was due to its proceeding with symbols rather than ideas and so not exercising the mind’s power of conception-of keeping objects in view with the attention ever fixed on them. Berkeley’s rejection of pure arithmetical speculation arises, then, from a ‘Locke himself praises algebra; he was not part of the Hobbesian as ‘a scab of symbols’.
tradition that viewed algebra
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A
04
A
A L!!‘ D
B
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D
B
Fig. 2.
conviction that arithmetic and algebra offer at most a mechanical shortcut for problems at hand. And his condescending attitude toward the symbolic approach underscores the literal sense in which he understands genuine reasoning as visual inspection. For without the literal interpretation the algebraic and geometric methods are not so easily distinguished. Complementing this presumption is Berkeley’s failure to make pragmatic demands upon abstruse geometrical speculation (the calculus aside), which results from his traditional view that geometrical speculation is good practice for the faculty of conceiving and inspecting ideas. These observations are important for explaining why Berkeley would have held so tightly to his account of geometry in spite of having envisioned a powerful alternative for understanding mathematical thought. Unfortunately, neither of these observations excuses the most serious flaw in Berkeley’s philosophy of geometry.
4. Theory versus Practice Geometrical reasoning cannot consist in perceiving agreement-in the sense of literal resemblance-between the extremes of a chain of ideas. Geometrical practice shows this. Suppose someone sets out to demonstrate that the bisector of the angle formed by the sides of an isosceles triangle is the perpendicular bisector of the base. The proof begins by drawing a triangle ABC (Fig. 2a) whose sides AB and AC are pretty much the same length and constructing with ruler and compass (or by estimating and drawing) AD the bisector of < CAB. A geometer may take care in drawing the figure, and in attempting to prove a more complex theorem is advised to do so. But in practice, individuals come to see the theorem without an accurate drawing. This does not show-as the logical positivists suggested-that diagrams are irrelevant or that geometrical proofs are really proofs in logic; proofs without diagrams are simply too long to be intelligible (see Section 5 below). Geometers can live with bad drawings
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by using hashmarks on lines and angles to remind themselves that such and such magnitudes are supposed equal (Fig. 2b). But this is quite different from considering visible magnitudes which are literally or discernibly equal. Furthermore a pair of lines that appears equal on first inspection will probably appear unequal when a measuring device is introduced. Thus, Berkeley’s thesis that geometrical truth and reasoning consist in the agreement of sensible ideas-particular figures--cannot be right. In proving the theorem just mentioned one comes to ‘see’ that CD agrees with DB and that < CDA agrees with < BDA even though measurement would probably show that the magnitudes literally disagree. Clearly it is possible to construct an isosceles triangle and the bisector of the angle subtended by its base, but these constructions are generally done freehand, and when instruments are used we find that the sides and base angles agree only more or less. Berkeley cannot seriously maintain that geometrical demonstrations mostly fail for want of an accurate drawing, yet he is committed to this position by his thesis that geometrical diagrams are the very ideas with which geometrical theorems are concerned. There is a subtler but more telling way to show Berkeley’s error. Many geometrical proofs proceed by way of reductio ad absurdum (e.g. Euclid I.6 and 1.19). Like the rest of Euclid’s proofs these demonstrations involve diagrams, but the thrust of RAA proofs is that the situation the diagram, attempts to depict is not possible. Thus the diagram of a proof by RAA represents the hypothesis of the proof only in a weak sense. There is nothing like an isomorphism between the diagram and the conditions of the hypothesis, because the hypothesis turns out to be inconsistent while the diagram is actual. Therefore, Berkeley’s idea that inspection is the modus operandi of geometers has got to be mistaken because proofs by RAA succeed only by showing that there could not have been a suitable object to inspect. Yet surely there is reasoning in proofs by RAA, so another explanation is required.4 Such complaints are not new, of course; the vicissitudes of particular figures led Plato to the forms (Phaedo, 74aac). Empiricism rules out their kin, abstract general ideas, but Plato’s concerns are still relevant: perception cannot reveal ‘Of interest in this connection is the work of Berkeley’s contemporary Girolamo Saccheri, a Jesuit who attempted to use proof by RAA to demonstrate the truth of Euclid’s parallel postulate (Saccheri 1920). Despite the revolutionary character of its hypotheses, Saccheri’s work uses inference techniques that are commonplace. When non-Euclidean geometry began to cause a stir in the nineteenth century, it was the hypotheses, not the methods of reasoning, that came under fire. Saccheri used a figure now called a Saccheri quadrilateral, with two equal sides erected perpendicular to a base. The summit angles of this figure are equal to one another, but equal to, greater than, or less than a right depending on whether the parallel postulate or one of its contraries is assumed. Contrary to his intentions, Saccheri proved many theorems of nonEuclidean geometry in reasoning about his quadrilateral. For our purposes, Saccheri is important for showing explicitly that a figure does not speak for itself, but requires a contribution from the mathematician. What Berkeley needs is an account of geometry which accommodates the contribution of the mathematician.
Berkeley on Mathematical Reasoning
the agreement
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of mathematical consists
geometers
a perceptual sense, as when the duck-rabbit uses do not make a scalene rather
in mistaking
really do, seeing as. Here ‘seeing as’ must not be understood
in
hashmarks a look like an isosceles triangle;
they
relations relations are represented although they are not in the A proof can bring us to ‘see’ that two angles are equal whether or not we literally perceive equality. Paradoxically, perhaps, literally see the inequality of a pair of angles (and so apologize drawing) without that affecting Berkeley came close to recognizing geometers proceed by seeing as figures of a certain sort. Having shown how general use may be made of a right-angled, isosceles triangle Berkeley that ‘ . . . here it must be acknowledged that a man may consider a figure merely as triangular, without attending to the particular qualities of the angles or the relations of the sides. So far he may abstract . . . ’ (Principles, $16). But in geometrical practice, considering or seeing as occurs in a direction opposite from what Berkeley allows. A diagram that depicts accurately a right-angled isosceles triangle may be seen as an equilateral or scalene triangle, and as long as one keeps in view what role each of the elements in the diagram is playing, no confusion results. ‘Seeing as’, in the preceding sense, is independent of the perceptual details of an actual diagram. To do its work a diagram needs to exhibit some of the features of the structure it is seen as; presumably a diagram seen as an equilateral triangle would need three sides. But as long as care is taken to see the sides as equal and to avoid misleading suggestions of the diagram (i.e. to proceed solely in accordance with postulates and axioms), any three-sided figure will do.’ Thus the actual practice of geometers is incompatible with Berkeley’s proposed subject matter for geometry. Berkeley talks as if geometrical diagrams embody particular ideas and so reveal truth by inspection. But in fact diagrams merely function heuristically by representing particular ideas; thus inspection is irrelevant, and Berkeley’s proposed subject matter is as unsatisfactory as abstract general ideas. What’s an empiricist to do?
5Fallacious proofs can be made more plausible by bad drawings (cf. Maxwell 1959, pp. 134, pp. 24-32, for a proof that all triangles are isosceles). Such difficulties arise especially when an additional construction is added to a diagram and there is a question whether that construction (or part of it) lies inside or outside the original figure or between or beyond a pair of points. Here an ‘accurate’ diagram may help, but what is really needed are additional axioms governing outside, inside, between and beyond. Pasch, Hilbert and others ‘filled the gaps’ in Euclid’s geometry with such axioms.
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As I indicated,
extra-mathematical
commitments
prevented
Berkeley
from
divorcing geometry from ideas understood as ‘objects in view with the attention ever fixed upon them’. Yet he could and should have undertaken the split. He could and should have considered geometry, as much as arithmetic and algebra, a system of symbols which is useful for guiding action. Although this move would be consistent with the rejection of rationalist insight, it would have stripped geometry of its subject matter and so helped to undercut the doctrine that knowledge and reasoning consist in the view the mind has of its ideas. Section 6 will argue that this is not too much to ask of an empiricist.
5. A Neo-Berkeleian Philosophy of Geometry It is necessary at this point to see how geometry can be understood as a calculus, in analogy with arithmetic and algebra. Consider again the theorem that the bisector of the angle formed by the sides of an isosceles triangle is the perpendicular bisector of the base (Fig. 2b). One begins by drawing and labelling a triangular diagram and ‘letting’ it be an isosceles triangle. The diagram is not a symbol in the sense that it stands for (or ‘copies’) some abstract idea. Rather the diagram is a symbol complex which may represent a concrete situation, in part by a stipulation, but also in part by sharing certain features (such as being triangular) with a situation. The symbol complex can be used as a model of a concrete situation in order to gain additional knowledge about that situation. To prove that AD is a perpendicular bisector, one needs to see two relations, CD= DB and < CDA = c BDA, which are definitive of a perpendicular bisector. Again, seeing that CD = DB is not an instance of perception, but of recognizing that the conditions stipulated in the problem warrant treating CD and DB as having the same magnitude, regardless of how they appear. Clearly, then, the diagram is heuristic. But this does not imply that the diagram is inessential. Geometric inference, as it is usually practiced, involves subsuming some or all of the stipulated characteristics of a diagram under a rule in order to determine a new characteristic of the diagram. Although the rules and stipulations may be formulated in predicate logic and the inference cast as a transformation of logical formulas, this is virtually never done. Not only is the logic proof is logical method tedious and error prone, but the predicate unintelligible without being read in light of the usual geometric proof; one doesn’t know what or when to instantiate without mimicking the structure of
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the informal essential because it makes the judicious application of the rules. Still, the problem that proving the theorem presents is something like a problem in symbolic logic where, given an arrangement of symbols, we use a restricted set of moves to obtain a new arrangement. Only in geometry it is not a matter of obtaining a new symbolic structure but of recognizing that the structure contains additional information in virtue of the originally stated conditions. But just as the inference rules license a new formula, the geometrical rules (definitions, axioms, and previous theorems) license one’s treating parts of the symbol complex in new, presumably helpful, ways. The inference to CD= DB and < CDA = < BDA is licensed by the congruence of triangles CDA and BDA. Once we see the congruence relation, we can then treat CD as equal to DB and indicate that by hashmarks; likewise for the two angles. Here’s how we proceed. The datum that AABC is isosceles allows us to treat AC as equal to AB, and the datum that AD bisects
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proving such theorems (or undertaking purely theoretical constructions) are the same as those used in more practical cases, there is good inductive evidence that the results could have useful physical application.7 It is noteworthy, then, that although Berkeley’s account of demonstration blinds him to a formalist philosophy of geometry, such a maneuver is consistent with and almost demanded by his revolutionary doctrine of meaningfulness as utility in guiding action. In order to complete this neo-Berkeleian approach to geometry it is necessary to say more about his pragmatism. An unwelcome consequence of his philosophy of arithmetic and algebra is that the disciplines are wholly subserviant to practice and so have no business worrying about the questions like Fermat’s Last Theorem. Berkeley seems not to have considered the difficulty of saying which mathematical truths will turn out to be useful in the affairs of life. Tradition has it that the classical Greek problem of doubling the cube was posed by Apollo’s oracle at Delphi. In order to end a plague in Delos the oracle required that Apollo’s altar be doubled in volume (Smart 1988, p. 177). Alas, like Meno’s slave, the priests guessed this would be a cube with double the side of the original and the plague continued. Whether or not the Delians were facing a practical problem that required constructing 3J2-with ruler and compass-is, I suppose, dependent upon Apollo’s standards of mathematical rigor. But the example should make clear that just about any d@iciles nugae could have practical applications. Recently, Fermat’s ‘little theorem’ (if p is prime and a any number not divisible by p, then Ccp-’= 1 (mod p)) was used for developing a secret code that is easy to decode, given a certain piece of information, but almost impossible to decode without it. The technique depends upon the difficulty of factoring large numbers into their prime factors (Stewart 1987, pp. 18-21). Both these examples suggest that Berkeley’s notion of usefulness is too narrow; otherwise he would have not been so quick to deride pure numerical investigations. Had he embraced a more liberal notion of utility, it would have been far more difficult to separate geometry from arithmetic Perhaps
and algebra. Berkeley was thinking
that an arithmetic
or algebraic
result worth
taking seriously must arise from prior concerns with objects of experience (like the optimum shape of a wine cask) and not mathematician’s puzzles. That this ‘Berkeley never considered extreme cases such as Desargues’s (1593-l662) projective geometry that used ‘improper points’ at infinity. No doubt he would have rendered them the same treatment he gave fluxions and infinitesimals. But the pragmatic line I’m adopting could be modified to handle such cases. Improper points at infinity are treated analogously to ordinary points, being subject to the same kinds of rules; thus there is only one point of intersection between two parallel lines. Such a point can even be represented, after a fashion, in a perspectival drawing. Indeed, Desargues was led to improper points in the course of giving a systematic account of perspectival drawing (Kline 1972, pp. 286 ff.). Taking a Hilbertian line, we could say that the justification for the new inference licenses consists in the fact that the ideal theorems do not imply any falsehoods involving garden variety points and magnitudes.
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be short-sighted mathematics.
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is one of the most profound
The theory
of conic
sections
lessons
of the history
of
was developed
in the study
of
famous construction problems such as doubling the cube; subsequently conic sections were pursued for their own sake by Apollonius of Perga (c. 250 B.C.). Almost two millennia later the ellipse (one of the four tonics) finally found practical
application in Kepler’s first law of planetary motion, and Newton’s soon followed. Even for such a long period of maturation, Apollonius’s investment paid big dividends! More importantly, from the mathematicians’ perspective, d@ciles nugae prove useful in mathematics itself by leading to new techniques and questions. Whether these will turn out to be pragmatically valuable is a question for a soothsayer, not a philosopher. Berkeley’s conception of arithmetic and algebra as instruments to direct our practice smacks of a presumption that human practices are independent of mathematical theories that might direct them; that is, he treats arithmetic and algebra as though they are nothing more than time-saving conveniences which are in principle dispensable. This is mistaken for two reasons. First, part of our experience in working with mathematical concepts and this kind of experience suggests new mathematics. Imaginary numbers are suggested by familiar techniques for solving equations; that experience tells us just how the ‘new entities’ should behave and how to make them behave. Berkeley could and should have accommodated this perspective. Second, by failing to see that mathematics can lead to revolutionary ways of conceiving of things, Berkeley also fails to appreciate that pursuing mathematics for its own sake, over the long run, is pragmatically worthwhile. Kepler’s elliptical orbits also illustrate this point, as does Boltzmann’s statistical thermodynamics. We can excuse the Bishop on this score because the history of mathematics and science was hardly developed in the eighteenth century and it would have been very Principia
difficult for him to recognize that often the physicist finds that the mathematician has trod the same ground decades or millenia before (cf. Weinberg 1986, p. 725). Actually, Berkeley comes close to recognizing a more liberal notion of the utility of mathematics. The first concrete applications of imaginary numbers were by Lambert (map construction) and D’Alembert (hydrodynamics) in the 1770s twenty years after Berkeley’s death (Jones 1954, p. 114, p. 262). Yet Berkeley praises J1 for ‘its use in logistic operations’ (Alciphron, VII, $14) and so seems to allow utility with respect to mathematics itself. It is a pity he did not explain further what use in logistic operation amounts to; but it certainly sounds as though he is permitting utility in purely mathematical endeavors. Berkeley may have been ambivalent about innovations in arithmetic and algebra. In more charitable moments he appreciates and encourages symbolic innovations for their intra-theoretical value and recognizes that utility is multi-faceted; in less charitable moments he insists upon a narrower
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conception of use in which arithmetic and algebra are subservient to practice and so denied a life of their own. In the end, the narrower view was predominant, to the detriment of Berkeley’s philosophy of mathematics. Berkeley’s philosophy of mathematics was radical, but not radical enough. Traditionally a science has a subject matter, and Berkeley followed tradition as far as construing geometry as the ‘natural history’ of points, lines and shapes concocted from them. He varied tradition by proposing new candidates for objects of geometrical thought, but the new, empirically correct candidates are plainly not the subject matter of geometry. Based in the doctrine that language need not be referential in order to be meaningful, Berkeley’s philosophy of arithmetic and algebra was genuinely revolutionary in doing away with a subject matter and construing reasoning as the mastery of a calculus. But his account is still infected with the demand that knowledge in the strictest sense is based on the mind’s view of its ideas; thus his shoddy treatment of the speculative regions of arithmetic and algebra. We have seen that there is no basis for such a division between geometry and arithmetic. To be fair to Berkeley, (positional) arithmetic and algebra were, unlike geometry, newcomers on the intellectual scene, and their symbols did not anticipate their practical applications the way geometrical diagrams do. Hence, it was natural for him to view the newcomers as fundamentally different. But there are no grounds for this division besides tradition, and had Berkeley freed himself from the metaphor of visual inspection, he would have approached the mathematical disciplines all in the same way. Before moving to more general considerations, note that the neo-Berkeleian philosophy of geometry is a real possibility for Berkeley and not a mere anachronism. My critique of his philosophy of geometry depends on straightforward observations familiar for centuries. And the notion of ‘seeing as’ is not a twentieth-century invention, but already employed by Berkeley in his philosophy of arithmetic: ‘The same extension is one, three, or thirty-six, according as the mind considers it with reference to a yard, a foot, or an inch.’ An uncritical acceptance of an easily refuted model of reasoning prevented Berkeley from appreciating the power of his own philosophical constructions. In the final section I shall show the considerable extent to which this model is inessential to empiricist epistemology.
6. Agreement Revisited Earlier we considered how an empiricist could explain reasoning in mathematics. Locke seems to bar himself from an illuminating account by appealing to (abstract) general ideas, which are fundamentally mysterious. Berkeley’s explanation of geometrical reasoning seems more successful because it avoids
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mysterious entities and explains agreement of ideas in terms of visible resemblance. Ultimately, though, this strategy fails because it is unfaithful to actual practice. Berkeley conflates seeing two magnitudes as equal (similar, etc.) with seeing the equality of two magnitudes simpliciter; the former is essential to geometrical practice while the latter, if it does occur, is incidental. The attempts of Locke and Berkeley to provide philosophical insight into geometry fail equally by building a case around an alleged subject matter: both propose to make the same use of the subject matter, viz. explaining (some portion of) mathematics in terms of objects surveyed by the mind’s eye, but in each case the subject matter is philosophically unilluminating. Nonetheless, there must be agreement of some sort in mathematics unless Locke was completely out of touch with the phenomena. Take a less obvious sum, 827+466= 1293. Locke maintains that we perceive that 1293 and the sum of 827 and 466 agree, but this is unenlightening when understood as a claim about abstract general ideas. A better account is possible in terms of the rules and techniques that constitute addition: having mastered the techniques of addition and having applied them to ‘827 +466’ to obtain ‘1293’, we are then justified in using ‘1293’ in the place of the sum, and that is one sense of perceiving that two ideas agree. More generally, we perceive that two ideas agree when we can get from the one to the other by a recognized procedure, but the agreement itself consists in one of the ideas permitting our application of the other idea. This account of agreement entails that a mathematical idea (archetype!) is a symbol embedded within a system of symbols that serves practical ends. The procedural rules of the system, such as the addition tables, are themselves justified by their practical success.* I offer the preceding as a model for agreement of mathematical ideas. In the arithmetic example agreement consists in seeing one symbol complex as the result of applying rule-governed transformations to another complex. To be sure, this is different from the geometric case in which, as a result of sanctioned moves, one portion of a symbol complex is seen as equal (less, similar) to another portion of the complex. But the difference lies in the level of abstraction with which the symbols are used. Arithmetic symbols are generally applicable, while geometry is applicable only to objects that are structurally similar to the symbols of geometry.9 And in both cases perceiving agreement can be understood as mastering a set of techniques which permit one to recognize that given symbols stand in certain relations.
nOne can of course replace the addition tables by a more compact set of rules such as the Peano axioms and a separate deductive apparatus. But these starting points would have to be justified inductively as well. It is curious that Frege, who promoted this strategy, never considered the question of justifying the rules of logic. 9Note, however, that the sequence of numerals ‘I’, ‘ 2 ’ , ., ‘n’ is structurally similar to a group of n objects to which it is applied. SHIPS 24:2-E
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To elaborate this line of thought, it is helpful to consider the following objection:‘0 are not the very structures which diagrams or physical situations are seen as the subject matter of geometry, and does not talk of such structures commit us to their existence? ‘Seeing as’ has many senses. To see a diagram as an isosceles triangle, say, is not to see that the diagram could become something that at present it is not, as when we shuddered at seeing Dan Quayle as the President of the U.S.A.; it is more like seeing an upcoming curricular change as an opportunity. To see a curricular change as an opportunity is to focus one’s attention on certain features of the change rather than others, to see that it creates some desirable possibilities, even as it abolishes some of the status quo. This is not like inserting (in thought) one thing in the place of another. Likewise, to see a diagram as an isosceles triangle is to accord it a certain treatment, to sanction and disallow various inferences about parts of the diagram. It is not a matter of seeing the diagram as another kind of thing with which we are familiar. So construed diagrams (and numerals) do not stand for transcendent, mathematical objects; they do not stand for anything. Rather, they are part of a linguistic apparatus-mathematics-which may be used to model concrete situations. Thus the agreement of ideas, at least mathematical ideas, rests on linguistic mastery. Having gone this far, what is left of empiricism? The problem which generated Berkeley’s philosophy of geometry was that of finding an empirically suitable subject matter for the discipline. Berkeley’s ‘solution’ could not explain geometrical reasoning, so, taking my cue from Berkeley, I proposed that the best avenue for one who rejects rationalist intuition is to forego subject matter for a symbolic system. A strong suggestion of the neoBerkeleian view of geometry is that reasoning be understood generally as a linguistic activity rather than a mental activity of comparing ideas. This will be difficult for Locke and Berkeley to accept, for it makes superfluous the privileged view of the contents of our minds, which figures prominently in their philosophies. But, in fact, a privileged view of the contents of our minds is a feature of both rationalist and empiricist philosophies of mathematics. That is, both philosophies attempt to explain (the possibility of) mathematical knowledge in terms of inspection of ideas that are themselves sufficient to tell us what to make of them and what inferences to draw from them. Our discussion has shown two things about ideas understood as objects for mental inspection. Not only are they unnecessary as regards the actual practice of mathematics, but they are incapable of doing the job. Either as in the case of Descartes and Locke, the process is mysterious and so not objective (cf. Essay, 4.4. l), or, as in the case of Berkeley, the objects admit a variety of uses which would be incompatible if the objects (ideas) inspected determined their own logical significance. ‘This
objection
was raised by Adrian
Riskin.
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Thus if there is any hope for the way of ideas, it lies in understanding
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the
agreement of ideas through means other than the metaphor of the mind as ideal spectator. The way of ideas, 1 submit, is not finished, for as we have seen, it can be given sense independent of the visual metaphor by appeal to linguistic mastery. Its true value lies in its antiformalist methodology, that is, in its rejection of formal logic as a valuable epistemological tool (Sherry 1992).” Nor is empiricism finished, for the linguistic approach makes no appeal to intellectual insight and still bases knowledge upon experience; only now the relevant experience is sucessful managing of practical affairs rather than viewing the contents of the mind. What is finished is the distinction between genuine reasoning, on the one hand, and practical or formal reasoning on the other. That distinction depends essentially upon the model of reasoning as quasivisual inspection; we have seen that the model is useless for mathematics and I suspect the same is true elsewhere.12 Locke was within striking distance of a linguistic account of the agreement of ideas. In a late addition to the Essay he tells how it gradually dawned upon him that words are ‘scarce separable from our general knowledge’ (3.9.21). This remark suggests he may have suspected the visual inspection metaphor was just that. And in the final chapter of the Essay he supplements the traditional division of human knowledge into the theoretical and the practical with a third science, the doctrine of signs or logic (4.21). Yet it seems never to have occurred to him that language offered a way out of the puzzles arising from the conception of ideas as entities viewed by the mind. For Berkeley, the metaphor was a methodological principle that led to brilliant insights and dead ends. By taking the metaphor seriously, Berkeley was pushed to discover that the use of language could make arithmetic intelligible even though it lacked ideal content. He fell short by not considering that language and its uses might be the very stuff of ideas. Even though Berkeley’s formalism contained the details of such an approach, his commitment to ideas as entities for inspection prevented him from an unmitigated linguistic approach. We have seen that, in part, a popular view about the role of geometry in liberal education sheltered this commitment from Berkeley’s penetrating philosophical scrutiny. One hundred years would have to pass until algebra became ascendant and philosophers would feel comfortable attacking more traditional views of
“It is important
to realize that rejecting formal logic does not amount to rejecting formalism approach to geometry is formalist but with no concern for the logical
simpliciter. The symbolic forms of its statements.
‘Contemporary formalist philosophy of mathematics, such as Hilbert’s embraces a distinction between the real (or contentual) and the ideal (or formal) portions of mathematics (Hilbert 1925). Our investigation of Berkeley suggests that either that distinction, too, is unwarranted, or the mathematical situation changed sufficiently to warrant such a distinction. I hope to write on both questions in the future.
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geometry. Our investigation shows that the raw materials for this revolution were present; only custom and habit stood in the way. Alas, no one escapes his age, and we still admire Berkeley for proceeding so far.
7. An Ironic Conclusion It is interesting that the points where Berkeley’s philosophy of mathematics disparages mathematical practice-his famous criticisms of infinite divisibility and infinitesimals in the higher geometry (calculus) as well as the less notorious remarks about d@iciles nugae-all result from his inability to overcome the
deep inclination to tie geometry to a subject matter which the mind inspects. But the criticisms are innocuous as soon as one adopts the attitude that the significance of those regions of mathematics lies in the use of symbols rather than some ideal content. A full-fledged linguistic approach to mathematics would have prevented Berkeley from developing precisely the theses in the
philosophy of mathematics for which he is best known.
Acknowledgements - Michael Malone and George Rudebusch provided useful criticism of an earlier draft of this essay, and I benefited from conversation with Ian Dove and John Moulton.
References Aristotle 1941, ‘Posterior Analytics’, in The Basic Works of Aristotle, ed. McKeon (New York: Random House). Becher, H. 1980, ‘Woodhouse, Babbage, Peacock, and Modern Algebra’, Historia Mathematics
7, 389400.
Berkeley, G. 1948-1957, The Works of George Berkeley Bishop of Cioyne, eds Lute and Jessop (London: Thomas Nelson). Hilbert, D. 1902, The Foundations of Geometry, trans. Townsend (Chicago: Open Court). Hilbert, D. 1925 ‘On the Infinite’, reprinted in P. Benacerraf and H. Putnam (eds), Readings in Philosophy of Mathematics, 2nd edition (Cambridge: Cambridge University Press, 1983). Jones, P. 1954, ‘Complex Numbers: An Example of Recurring Themes in the Development of Mathematics’, The Mathematics Teacher 47, 106-114, 257-263, 340-345. Kline, M. 1972, Mathematical Thought from Ancient to Modern Times (Oxford: Oxford University Press). Locke, J. 1975, An Essay Concerning Human Understanding, ed. Nidditch (Oxford: Oxford University Press). Maxwell, E. 1959, Fallacies in Mathematics (Cambridge: Cambridge University Press). Mueller, I. 1981, Philosophy of Mathematics and Deductive Structure in Euclid’s Elements (Cambridge, Mass: MIT Press).
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Reasoning
Olson, R. 1971, ‘Scottish Philosophy
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and Mathematics
175&1830’, Journal of the
History of Ideas 32, 2944.
Passmore, J. 1953, ‘Descartes, the British Empiricists, and Formal Logic’, Philosophical Review 62, 545-553.
Pycior, H. 1987, ‘Mathematics and Philosophy: Wallis, Hobbes, Barrow, and Berkeley’, Journal of the History of Ideas 48, 265-286.
Saccheri, G. 1920, Eucfides Vindicatus, ed. and trans. G. Halsted (Chicago: Open Court). Sherry, D. 1987, ‘The Wake of Berkeley’s Analyst’, Studies in History and Philosophy of Science 18, 455480.
Sherry, D. 1992, ‘Locke and Leibniz on the Utility of Formal Logic’, unpublished typescript. Smart, J. 1988, Modern Geometries (Pacific Grove, California: Brooks/Cole). Stewart, I. 1987, The Problems of Mathematics (Oxford: Oxford University Press). Weinberg, S. 1986, Notices of the American Mathematical Society 33, 725
1. A Lockean Problem, a Berkeleian Solution
I shall begin with Locke, who provided a constant stimulus to Berkeley’s thought. Locke joined the Cartesian revolution with its doctrine that clear and distinct ideas are necessary and sufficient for knowledge. The ‘way of ideas’ promised freedom from interminable scholastic disputes, and both Descartes and Locke agreed that careful conception and comparison of ideas was the only way to avoid error and extend knowledge. This is nicely illustrated by their rejection of formal logic in favor of the natural light (or native rustic reason). Knowledge, according to Locke, consists in the perception of agreement and disagreement of ideas. In intuitive knowledge, the perception is immediate, but in demonstrative knowledge the perception occurs via intermediate ideas. To perceive the agreement between the interior angles of a triangle and two right angles one must discover an intermediate idea which agrees with each of the extremes. In this case (Euclid 1.32) the intermediate idea is gotten by extending CB to D and constructing BE parallel to CA (Fig. 1). Thus if BE is parallel to CA, the theorem that governs two parallels cut by a transversal (1.29) implies that < ACB= < EBD and < CAB= -e EBA; hence *Department of Philosophy, Northern Arizona University, 6011, U.S.A. Received I July 1992; in revisedform 28 October 1992.
Stud. His?. Phil. Sci., Vol. 24, No. 2, pp. 207-225, Printed in Great Britain
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PO Box 6011, Flagstaff,
1993. 0
AZ 8601 I-
0039-3681/93 $6.00+0.00 1993. Pergamon Press Ltd
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Thus when I demonstrate any proposition concerning triangles, it is to be supposed that I have in view the universal idea of a triangle, which ought not to be understood as if I could frame an idea of a triangle which was neither equilateral, nor scalenon, nor equicrural, but only that the particular triangle I consider, whether of this or that sort it matters not, does equally stand for and represent all rectilinear triangles whatsoever, and is in that sense universal. [Principles, $151
For Berkeley, the subject matter of geometry still consists of ideas, indeed, general ideas, but in an entirely new sense: a general idea of a triangle is no longer a concept difficult of attainment, but simply a particular idea used to stand for all triangles. By restricting the general use of particular ideas it is possible to account for the generality of geometric truths. . . . though the idea I have in view whilst I make the demonstration be, for instance, that of an isosceles rectangular triangle whose sides are of a determinate length, I may nevertheless be certain it extends to all other rectilinear triangles, of what sort of bigness soever. And that because neither the right angle, nor the equality, nor determinate length of the sides are at all concerned in the demonstration. [Ibid., $161
The geometer averts the worry that a proof is not generally valid by forsaking particular features of a representative triangle. Using the diagram under this condition does not require an entity which embodies just the characteristics shared by all triangles. Berkeley’s proposal is therefore a new way to understand proof. Traditionally a proof shows one (abstract) general idea predicable of another, whereas he understands a proof as a general procedure for showing some property holds true of all particular ideas of some sort. Thus the proof of I.32 really provides a procedure for demonstrating, of any particular triangle, that its interior angles sum to two right angles. Abstract general ideas are not part of Berkeley’s ontology, but then neither are they unabashedly part of Locke’s: they are ‘fictions and contrivances of the mind . . . something imperfect which cannot exist’ (Essay, 4.7.9). Seeing no other way of achieving generality, Locke begrudged general ideas ontological status, but in so doing he faced a dilemma: he knew general ideas could not be images, but he couldn’t, as an empiricist, avail himself of the rationalist power to grasp ideas which correspond to no perception. Berkeley’s genius consisted in invoking the use of ideas-over and above the ideas themselves-to solve Locke’s dilemma without sacrificing empiricist scruples. Locke’s problem with generality is also implicit in what he says about reasoning. And here too Berkeley’s innovation suggests a way out. Both Locke and Berkeley were content to explicate reasoning as the agreement of ideas without saying much more. But it is worthwhile to see what each can say about the agreement of ideas. Locke’s general ideas are universals, not particular diagrams. Even so, he was impressed by diagrams. They are ‘copies of the
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ideas in the mind’, and by their means mathematicians are able to achieve an enviable certainty (Exsu~, 4.3.19-20)’ I suspect that when Locke described diagrams as copies he was conveniently forgetting his admission that general ideas are fictions and contrivances and taking too literally his famous metaphor that knowledge consists ‘in the view the mind has of its own ideas’ (4.2.1). But with respect to general ideas the visual inspection metaphor takes him no farther than the rationalist who claims to perceive the agreement of ideas by intellectual insight. The only difference is that Locke insists that mathematical ideas are constructed while the rationalist allows that the mind comes stocked with them. Although Locke was correct, in my view, to recognize that ideas agree or disagree in the sense of having affinities and aversions for one another, he was not prepared to explain this insight in a way that avoids the subjective character of rationalism. In contrast, the agreement of geometrical ideas is apparently no problem for Berkeley. Geometer’s diagrams do not stand for hidden or fictitious entities. The diagrams are (in Locke’s phrase) ‘the objects of understanding when a man thinks’, and so their agreement can be explained literally as a visible resemblance: in geometry, demonstration is demonstration ad OCU~OS.~ Thus Berkeley can regard ordinary experience, rather than intellectual insight, as the foundation of geometric truths. And this explains his rejection of infinite divisibility in geometry and infinitesimals in the calculus (cf. Principles, $132, and The Analyst).
2. A Reluctant Formalism There is no demonstration ad oculos in arithmetic and algebra, for, on Berkeley’s account, neither involves ideas which agree with or resemble one another. Already Locke had recognized that the marks of arithmetic and algebra are not copies of general ideas and concluded that ciphers and the like did not so much ‘help the mind.. . to perceive the agreement of any two numbers, their equalities or proportions’ as to help the memory (Essay, 4.3.19).
‘Locke even believed that certainty would be achieved in ethics were someone to invent a moral algebra and so ensure that we could consider our demonstrations at leisure ‘without any danger of the least change in our ideas’ (Essay, 4.3.19). Both mathematics and ethics admit of demonstration because their concepts are archetypes (ideas constructed from materials of sense experience), not ideas abstracted from sense experience (4.4.6-S). Locke never explains how archetypes make demonstration possible. My remarks in Section 6 suggest an explanation of archetypes. % New Theory of Vision Berkeley argues that geometry concerns tangible extension (5159); in the Principles he appeals to visible extension. This sort of image employed makes no difference to my case.
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Thus it is not surprising that Berkeley should conclude that arithmetic and algebra have no subject matter at all. But number being defined a ‘collection of units,’ we may conclude that, if there be no such thing as unity or unit in abstract, there are no ideas of number in abstract denoted by the numeral names and figures. The theories therefore in arithmetic, if they are abstracted from the names and figures, as likewise from all use and practice, as well as from the particular things numbered, can be supposed to have nothing at all for their object. [Principles, $1201
In this passage Berkeley takes for granted that neither signs nor things could constitute a subject matter for arithmetic (and algebra) the way perceptible extension constitutes the subject matter of geometry. Perhaps he reasoned that while there is nothing in a given numeral or thing which necessarily connects it with a number, the diagram of a triangle is necessarily triangular. Berkeley allows that arithmetic and algebra are conversant about signs (AIciphron, VII, §12), but this is not to give a subject matter to arithmetic. In his view, our grasp of number is distinct from our grasp of a particular object or set of objects (Principles, $12; cf. Theory of Vision, $109). He argues that particulars are given different numbers-‘the same extension is one, three or thirty-six, according as the mind considers it with reference to a yard, a foot, or an inch’--depending on what is convenient to the task at hand. Thus, no object of perception, by itself, suffices as an object for arithmetic. Extension, on the other hand, presents itself without any arbitrary combination by the mind; thus extension can be regarded for its own sake, and geometry, over and against arithmetic, qualifies as a science with a subject matter. The lack of a subject matter for arithmetic and algebra does not prevent Berkeley from admiring them; but it does effect a dichotomy in his conception of science and scientific reasoning. Arithmetic and algebra are practical sciences of manipulating symbols to obtain useful conclusions, kinds of logistic. Geometry is a genuine or demonstrative science, which rests on the agreement of ideas and so fits the paradigm of knowledge that Berkeley inherited from Locke. Corresponding to the distinction of genuine from practical science are two senses of reasoning. That distinction turns upon whether or not one can conceive the referents of the terms of a theory. Where such conception is possible, the standard of reasoning consists in agreement between objects of reasoning (referents of the terms). But in arithmetic and algebra there are no objects to agree and disagree except signs and things, and they do not agree with one another in the literal sense in which one can visualize the congruence between the interior angles of a triangle and two right angles. Consequently, the standard of reasoning in arithmetic and algebra is dictated by rules for using symbols rather than the ideas which constitute the subject matter of the discipline. And since there is no ideal backing for the rules, their justification is pragmatic,
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Commitments
Berkeley’s portmanteau philosophy of mathematics has an uncomfortable side effect. In the Philosophical Commentaries he characterizes arithmetic and algebra as ‘sciences purely verbal and entirely useless but for practice in societies of men. No speculative knowledge, no comparing of ideas in them’ (P.C., p. 768). Without a subject matter, arithmetic has no claim to speculative knowledge, over and above practical knowledge; indeed, Berkeley relegates speculative endeavors in arithmetic to d@ciles nugae (Principles, $119). These derogatory comments about questions such as Fermat’s Last Theorem, which are neither subservient to practice nor promote the benefit of life, have no parallel in his remarks on geometry. Yet surely there have been geometrical investigations which promote the benefit of life no more than researches into Fermat’s problem: how should one apply the theorem that the external bisectors of two angles of a triangle are concurrent with the internal bisector of the third? It is unreasonable to expect practical application for every theorem in geometry. Berkeley’s blindness to this sort of objection reveals a philosophical commitment that goes beyond dissatisfaction with rationalist epistemology. Great Britain was much slower than the rest of Europe to replace geometry with algebra in liberal arts curricula because the change had to fight a tradition which conceived geometry as the best training for practical reasoning (Becher 1980; Olson 1971). Although the tradition became most pronounced in the Scottish philosophers, Berkeley accepted it without reservation. It hath been an old remark that geometry is an excellent logic. And it must be owned that when the definitions are clear; when the postulata cannot be refused, nor the axioms denied; when from the distinct contemplation and comparison of figures, their properties are derived by a perpetual well-connected chain of consequences, the objects being still kept in view, and the attention ever fixed upon them; there is acquired an habit of reasoning close and exact and methodical; which habit strengthens and sharpens the mind, and being transferred to other subjects is of general use in the inquiry after truth. [Analyst, $21 Pace the twentieth-century
practice of requiring college algebra, the British tradition saw no liberalizing value in arithmetic and algebra; they were expedients with no tendency, in Dugald Stewart’s phrase ‘to strengthen the power of steady and concatenated thinking’ (Olson 197 1, p. 43).’ The failure of algebra to satisfy the demands of liberal education was due to its proceeding with symbols rather than ideas and so not exercising the mind’s power of conception-of keeping objects in view with the attention ever fixed on them. Berkeley’s rejection of pure arithmetical speculation arises, then, from a ‘Locke himself praises algebra; he was not part of the Hobbesian as ‘a scab of symbols’.
tradition that viewed algebra
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A
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A
A L!!‘ D
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Fig. 2.
conviction that arithmetic and algebra offer at most a mechanical shortcut for problems at hand. And his condescending attitude toward the symbolic approach underscores the literal sense in which he understands genuine reasoning as visual inspection. For without the literal interpretation the algebraic and geometric methods are not so easily distinguished. Complementing this presumption is Berkeley’s failure to make pragmatic demands upon abstruse geometrical speculation (the calculus aside), which results from his traditional view that geometrical speculation is good practice for the faculty of conceiving and inspecting ideas. These observations are important for explaining why Berkeley would have held so tightly to his account of geometry in spite of having envisioned a powerful alternative for understanding mathematical thought. Unfortunately, neither of these observations excuses the most serious flaw in Berkeley’s philosophy of geometry.
4. Theory versus Practice Geometrical reasoning cannot consist in perceiving agreement-in the sense of literal resemblance-between the extremes of a chain of ideas. Geometrical practice shows this. Suppose someone sets out to demonstrate that the bisector of the angle formed by the sides of an isosceles triangle is the perpendicular bisector of the base. The proof begins by drawing a triangle ABC (Fig. 2a) whose sides AB and AC are pretty much the same length and constructing with ruler and compass (or by estimating and drawing) AD the bisector of < CAB. A geometer may take care in drawing the figure, and in attempting to prove a more complex theorem is advised to do so. But in practice, individuals come to see the theorem without an accurate drawing. This does not show-as the logical positivists suggested-that diagrams are irrelevant or that geometrical proofs are really proofs in logic; proofs without diagrams are simply too long to be intelligible (see Section 5 below). Geometers can live with bad drawings
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by using hashmarks on lines and angles to remind themselves that such and such magnitudes are supposed equal (Fig. 2b). But this is quite different from considering visible magnitudes which are literally or discernibly equal. Furthermore a pair of lines that appears equal on first inspection will probably appear unequal when a measuring device is introduced. Thus, Berkeley’s thesis that geometrical truth and reasoning consist in the agreement of sensible ideas-particular figures--cannot be right. In proving the theorem just mentioned one comes to ‘see’ that CD agrees with DB and that < CDA agrees with < BDA even though measurement would probably show that the magnitudes literally disagree. Clearly it is possible to construct an isosceles triangle and the bisector of the angle subtended by its base, but these constructions are generally done freehand, and when instruments are used we find that the sides and base angles agree only more or less. Berkeley cannot seriously maintain that geometrical demonstrations mostly fail for want of an accurate drawing, yet he is committed to this position by his thesis that geometrical diagrams are the very ideas with which geometrical theorems are concerned. There is a subtler but more telling way to show Berkeley’s error. Many geometrical proofs proceed by way of reductio ad absurdum (e.g. Euclid I.6 and 1.19). Like the rest of Euclid’s proofs these demonstrations involve diagrams, but the thrust of RAA proofs is that the situation the diagram, attempts to depict is not possible. Thus the diagram of a proof by RAA represents the hypothesis of the proof only in a weak sense. There is nothing like an isomorphism between the diagram and the conditions of the hypothesis, because the hypothesis turns out to be inconsistent while the diagram is actual. Therefore, Berkeley’s idea that inspection is the modus operandi of geometers has got to be mistaken because proofs by RAA succeed only by showing that there could not have been a suitable object to inspect. Yet surely there is reasoning in proofs by RAA, so another explanation is required.4 Such complaints are not new, of course; the vicissitudes of particular figures led Plato to the forms (Phaedo, 74aac). Empiricism rules out their kin, abstract general ideas, but Plato’s concerns are still relevant: perception cannot reveal ‘Of interest in this connection is the work of Berkeley’s contemporary Girolamo Saccheri, a Jesuit who attempted to use proof by RAA to demonstrate the truth of Euclid’s parallel postulate (Saccheri 1920). Despite the revolutionary character of its hypotheses, Saccheri’s work uses inference techniques that are commonplace. When non-Euclidean geometry began to cause a stir in the nineteenth century, it was the hypotheses, not the methods of reasoning, that came under fire. Saccheri used a figure now called a Saccheri quadrilateral, with two equal sides erected perpendicular to a base. The summit angles of this figure are equal to one another, but equal to, greater than, or less than a right depending on whether the parallel postulate or one of its contraries is assumed. Contrary to his intentions, Saccheri proved many theorems of nonEuclidean geometry in reasoning about his quadrilateral. For our purposes, Saccheri is important for showing explicitly that a figure does not speak for itself, but requires a contribution from the mathematician. What Berkeley needs is an account of geometry which accommodates the contribution of the mathematician.
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the agreement
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of mathematical consists
geometers
a perceptual sense, as when the duck-rabbit uses do not make a scalene rather
in mistaking
really do, seeing as. Here ‘seeing as’ must not be understood
in
hashmarks a look like an isosceles triangle;
they
relations relations are represented although they are not in the A proof can bring us to ‘see’ that two angles are equal whether or not we literally perceive equality. Paradoxically, perhaps, literally see the inequality of a pair of angles (and so apologize drawing) without that affecting Berkeley came close to recognizing geometers proceed by seeing as figures of a certain sort. Having shown how general use may be made of a right-angled, isosceles triangle Berkeley that ‘ . . . here it must be acknowledged that a man may consider a figure merely as triangular, without attending to the particular qualities of the angles or the relations of the sides. So far he may abstract . . . ’ (Principles, $16). But in geometrical practice, considering or seeing as occurs in a direction opposite from what Berkeley allows. A diagram that depicts accurately a right-angled isosceles triangle may be seen as an equilateral or scalene triangle, and as long as one keeps in view what role each of the elements in the diagram is playing, no confusion results. ‘Seeing as’, in the preceding sense, is independent of the perceptual details of an actual diagram. To do its work a diagram needs to exhibit some of the features of the structure it is seen as; presumably a diagram seen as an equilateral triangle would need three sides. But as long as care is taken to see the sides as equal and to avoid misleading suggestions of the diagram (i.e. to proceed solely in accordance with postulates and axioms), any three-sided figure will do.’ Thus the actual practice of geometers is incompatible with Berkeley’s proposed subject matter for geometry. Berkeley talks as if geometrical diagrams embody particular ideas and so reveal truth by inspection. But in fact diagrams merely function heuristically by representing particular ideas; thus inspection is irrelevant, and Berkeley’s proposed subject matter is as unsatisfactory as abstract general ideas. What’s an empiricist to do?
5Fallacious proofs can be made more plausible by bad drawings (cf. Maxwell 1959, pp. 134, pp. 24-32, for a proof that all triangles are isosceles). Such difficulties arise especially when an additional construction is added to a diagram and there is a question whether that construction (or part of it) lies inside or outside the original figure or between or beyond a pair of points. Here an ‘accurate’ diagram may help, but what is really needed are additional axioms governing outside, inside, between and beyond. Pasch, Hilbert and others ‘filled the gaps’ in Euclid’s geometry with such axioms.
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As I indicated,
extra-mathematical
commitments
prevented
Berkeley
from
divorcing geometry from ideas understood as ‘objects in view with the attention ever fixed upon them’. Yet he could and should have undertaken the split. He could and should have considered geometry, as much as arithmetic and algebra, a system of symbols which is useful for guiding action. Although this move would be consistent with the rejection of rationalist insight, it would have stripped geometry of its subject matter and so helped to undercut the doctrine that knowledge and reasoning consist in the view the mind has of its ideas. Section 6 will argue that this is not too much to ask of an empiricist.
5. A Neo-Berkeleian Philosophy of Geometry It is necessary at this point to see how geometry can be understood as a calculus, in analogy with arithmetic and algebra. Consider again the theorem that the bisector of the angle formed by the sides of an isosceles triangle is the perpendicular bisector of the base (Fig. 2b). One begins by drawing and labelling a triangular diagram and ‘letting’ it be an isosceles triangle. The diagram is not a symbol in the sense that it stands for (or ‘copies’) some abstract idea. Rather the diagram is a symbol complex which may represent a concrete situation, in part by a stipulation, but also in part by sharing certain features (such as being triangular) with a situation. The symbol complex can be used as a model of a concrete situation in order to gain additional knowledge about that situation. To prove that AD is a perpendicular bisector, one needs to see two relations, CD= DB and < CDA = c BDA, which are definitive of a perpendicular bisector. Again, seeing that CD = DB is not an instance of perception, but of recognizing that the conditions stipulated in the problem warrant treating CD and DB as having the same magnitude, regardless of how they appear. Clearly, then, the diagram is heuristic. But this does not imply that the diagram is inessential. Geometric inference, as it is usually practiced, involves subsuming some or all of the stipulated characteristics of a diagram under a rule in order to determine a new characteristic of the diagram. Although the rules and stipulations may be formulated in predicate logic and the inference cast as a transformation of logical formulas, this is virtually never done. Not only is the logic proof is logical method tedious and error prone, but the predicate unintelligible without being read in light of the usual geometric proof; one doesn’t know what or when to instantiate without mimicking the structure of
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the informal essential because it makes the judicious application of the rules. Still, the problem that proving the theorem presents is something like a problem in symbolic logic where, given an arrangement of symbols, we use a restricted set of moves to obtain a new arrangement. Only in geometry it is not a matter of obtaining a new symbolic structure but of recognizing that the structure contains additional information in virtue of the originally stated conditions. But just as the inference rules license a new formula, the geometrical rules (definitions, axioms, and previous theorems) license one’s treating parts of the symbol complex in new, presumably helpful, ways. The inference to CD= DB and < CDA = < BDA is licensed by the congruence of triangles CDA and BDA. Once we see the congruence relation, we can then treat CD as equal to DB and indicate that by hashmarks; likewise for the two angles. Here’s how we proceed. The datum that AABC is isosceles allows us to treat AC as equal to AB, and the datum that AD bisects
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proving such theorems (or undertaking purely theoretical constructions) are the same as those used in more practical cases, there is good inductive evidence that the results could have useful physical application.7 It is noteworthy, then, that although Berkeley’s account of demonstration blinds him to a formalist philosophy of geometry, such a maneuver is consistent with and almost demanded by his revolutionary doctrine of meaningfulness as utility in guiding action. In order to complete this neo-Berkeleian approach to geometry it is necessary to say more about his pragmatism. An unwelcome consequence of his philosophy of arithmetic and algebra is that the disciplines are wholly subserviant to practice and so have no business worrying about the questions like Fermat’s Last Theorem. Berkeley seems not to have considered the difficulty of saying which mathematical truths will turn out to be useful in the affairs of life. Tradition has it that the classical Greek problem of doubling the cube was posed by Apollo’s oracle at Delphi. In order to end a plague in Delos the oracle required that Apollo’s altar be doubled in volume (Smart 1988, p. 177). Alas, like Meno’s slave, the priests guessed this would be a cube with double the side of the original and the plague continued. Whether or not the Delians were facing a practical problem that required constructing 3J2-with ruler and compass-is, I suppose, dependent upon Apollo’s standards of mathematical rigor. But the example should make clear that just about any d@iciles nugae could have practical applications. Recently, Fermat’s ‘little theorem’ (if p is prime and a any number not divisible by p, then Ccp-’= 1 (mod p)) was used for developing a secret code that is easy to decode, given a certain piece of information, but almost impossible to decode without it. The technique depends upon the difficulty of factoring large numbers into their prime factors (Stewart 1987, pp. 18-21). Both these examples suggest that Berkeley’s notion of usefulness is too narrow; otherwise he would have not been so quick to deride pure numerical investigations. Had he embraced a more liberal notion of utility, it would have been far more difficult to separate geometry from arithmetic Perhaps
and algebra. Berkeley was thinking
that an arithmetic
or algebraic
result worth
taking seriously must arise from prior concerns with objects of experience (like the optimum shape of a wine cask) and not mathematician’s puzzles. That this ‘Berkeley never considered extreme cases such as Desargues’s (1593-l662) projective geometry that used ‘improper points’ at infinity. No doubt he would have rendered them the same treatment he gave fluxions and infinitesimals. But the pragmatic line I’m adopting could be modified to handle such cases. Improper points at infinity are treated analogously to ordinary points, being subject to the same kinds of rules; thus there is only one point of intersection between two parallel lines. Such a point can even be represented, after a fashion, in a perspectival drawing. Indeed, Desargues was led to improper points in the course of giving a systematic account of perspectival drawing (Kline 1972, pp. 286 ff.). Taking a Hilbertian line, we could say that the justification for the new inference licenses consists in the fact that the ideal theorems do not imply any falsehoods involving garden variety points and magnitudes.
Berkeley on Mathematical Reasoning
be short-sighted mathematics.
219
is one of the most profound
The theory
of conic
sections
lessons
of the history
of
was developed
in the study
of
famous construction problems such as doubling the cube; subsequently conic sections were pursued for their own sake by Apollonius of Perga (c. 250 B.C.). Almost two millennia later the ellipse (one of the four tonics) finally found practical
application in Kepler’s first law of planetary motion, and Newton’s soon followed. Even for such a long period of maturation, Apollonius’s investment paid big dividends! More importantly, from the mathematicians’ perspective, d@ciles nugae prove useful in mathematics itself by leading to new techniques and questions. Whether these will turn out to be pragmatically valuable is a question for a soothsayer, not a philosopher. Berkeley’s conception of arithmetic and algebra as instruments to direct our practice smacks of a presumption that human practices are independent of mathematical theories that might direct them; that is, he treats arithmetic and algebra as though they are nothing more than time-saving conveniences which are in principle dispensable. This is mistaken for two reasons. First, part of our experience in working with mathematical concepts and this kind of experience suggests new mathematics. Imaginary numbers are suggested by familiar techniques for solving equations; that experience tells us just how the ‘new entities’ should behave and how to make them behave. Berkeley could and should have accommodated this perspective. Second, by failing to see that mathematics can lead to revolutionary ways of conceiving of things, Berkeley also fails to appreciate that pursuing mathematics for its own sake, over the long run, is pragmatically worthwhile. Kepler’s elliptical orbits also illustrate this point, as does Boltzmann’s statistical thermodynamics. We can excuse the Bishop on this score because the history of mathematics and science was hardly developed in the eighteenth century and it would have been very Principia
difficult for him to recognize that often the physicist finds that the mathematician has trod the same ground decades or millenia before (cf. Weinberg 1986, p. 725). Actually, Berkeley comes close to recognizing a more liberal notion of the utility of mathematics. The first concrete applications of imaginary numbers were by Lambert (map construction) and D’Alembert (hydrodynamics) in the 1770s twenty years after Berkeley’s death (Jones 1954, p. 114, p. 262). Yet Berkeley praises J1 for ‘its use in logistic operations’ (Alciphron, VII, $14) and so seems to allow utility with respect to mathematics itself. It is a pity he did not explain further what use in logistic operation amounts to; but it certainly sounds as though he is permitting utility in purely mathematical endeavors. Berkeley may have been ambivalent about innovations in arithmetic and algebra. In more charitable moments he appreciates and encourages symbolic innovations for their intra-theoretical value and recognizes that utility is multi-faceted; in less charitable moments he insists upon a narrower
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conception of use in which arithmetic and algebra are subservient to practice and so denied a life of their own. In the end, the narrower view was predominant, to the detriment of Berkeley’s philosophy of mathematics. Berkeley’s philosophy of mathematics was radical, but not radical enough. Traditionally a science has a subject matter, and Berkeley followed tradition as far as construing geometry as the ‘natural history’ of points, lines and shapes concocted from them. He varied tradition by proposing new candidates for objects of geometrical thought, but the new, empirically correct candidates are plainly not the subject matter of geometry. Based in the doctrine that language need not be referential in order to be meaningful, Berkeley’s philosophy of arithmetic and algebra was genuinely revolutionary in doing away with a subject matter and construing reasoning as the mastery of a calculus. But his account is still infected with the demand that knowledge in the strictest sense is based on the mind’s view of its ideas; thus his shoddy treatment of the speculative regions of arithmetic and algebra. We have seen that there is no basis for such a division between geometry and arithmetic. To be fair to Berkeley, (positional) arithmetic and algebra were, unlike geometry, newcomers on the intellectual scene, and their symbols did not anticipate their practical applications the way geometrical diagrams do. Hence, it was natural for him to view the newcomers as fundamentally different. But there are no grounds for this division besides tradition, and had Berkeley freed himself from the metaphor of visual inspection, he would have approached the mathematical disciplines all in the same way. Before moving to more general considerations, note that the neo-Berkeleian philosophy of geometry is a real possibility for Berkeley and not a mere anachronism. My critique of his philosophy of geometry depends on straightforward observations familiar for centuries. And the notion of ‘seeing as’ is not a twentieth-century invention, but already employed by Berkeley in his philosophy of arithmetic: ‘The same extension is one, three, or thirty-six, according as the mind considers it with reference to a yard, a foot, or an inch.’ An uncritical acceptance of an easily refuted model of reasoning prevented Berkeley from appreciating the power of his own philosophical constructions. In the final section I shall show the considerable extent to which this model is inessential to empiricist epistemology.
6. Agreement Revisited Earlier we considered how an empiricist could explain reasoning in mathematics. Locke seems to bar himself from an illuminating account by appealing to (abstract) general ideas, which are fundamentally mysterious. Berkeley’s explanation of geometrical reasoning seems more successful because it avoids
Berkeley on Mathematical
Reasoning
221
mysterious entities and explains agreement of ideas in terms of visible resemblance. Ultimately, though, this strategy fails because it is unfaithful to actual practice. Berkeley conflates seeing two magnitudes as equal (similar, etc.) with seeing the equality of two magnitudes simpliciter; the former is essential to geometrical practice while the latter, if it does occur, is incidental. The attempts of Locke and Berkeley to provide philosophical insight into geometry fail equally by building a case around an alleged subject matter: both propose to make the same use of the subject matter, viz. explaining (some portion of) mathematics in terms of objects surveyed by the mind’s eye, but in each case the subject matter is philosophically unilluminating. Nonetheless, there must be agreement of some sort in mathematics unless Locke was completely out of touch with the phenomena. Take a less obvious sum, 827+466= 1293. Locke maintains that we perceive that 1293 and the sum of 827 and 466 agree, but this is unenlightening when understood as a claim about abstract general ideas. A better account is possible in terms of the rules and techniques that constitute addition: having mastered the techniques of addition and having applied them to ‘827 +466’ to obtain ‘1293’, we are then justified in using ‘1293’ in the place of the sum, and that is one sense of perceiving that two ideas agree. More generally, we perceive that two ideas agree when we can get from the one to the other by a recognized procedure, but the agreement itself consists in one of the ideas permitting our application of the other idea. This account of agreement entails that a mathematical idea (archetype!) is a symbol embedded within a system of symbols that serves practical ends. The procedural rules of the system, such as the addition tables, are themselves justified by their practical success.* I offer the preceding as a model for agreement of mathematical ideas. In the arithmetic example agreement consists in seeing one symbol complex as the result of applying rule-governed transformations to another complex. To be sure, this is different from the geometric case in which, as a result of sanctioned moves, one portion of a symbol complex is seen as equal (less, similar) to another portion of the complex. But the difference lies in the level of abstraction with which the symbols are used. Arithmetic symbols are generally applicable, while geometry is applicable only to objects that are structurally similar to the symbols of geometry.9 And in both cases perceiving agreement can be understood as mastering a set of techniques which permit one to recognize that given symbols stand in certain relations.
nOne can of course replace the addition tables by a more compact set of rules such as the Peano axioms and a separate deductive apparatus. But these starting points would have to be justified inductively as well. It is curious that Frege, who promoted this strategy, never considered the question of justifying the rules of logic. 9Note, however, that the sequence of numerals ‘I’, ‘ 2 ’ , ., ‘n’ is structurally similar to a group of n objects to which it is applied. SHIPS 24:2-E
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To elaborate this line of thought, it is helpful to consider the following objection:‘0 are not the very structures which diagrams or physical situations are seen as the subject matter of geometry, and does not talk of such structures commit us to their existence? ‘Seeing as’ has many senses. To see a diagram as an isosceles triangle, say, is not to see that the diagram could become something that at present it is not, as when we shuddered at seeing Dan Quayle as the President of the U.S.A.; it is more like seeing an upcoming curricular change as an opportunity. To see a curricular change as an opportunity is to focus one’s attention on certain features of the change rather than others, to see that it creates some desirable possibilities, even as it abolishes some of the status quo. This is not like inserting (in thought) one thing in the place of another. Likewise, to see a diagram as an isosceles triangle is to accord it a certain treatment, to sanction and disallow various inferences about parts of the diagram. It is not a matter of seeing the diagram as another kind of thing with which we are familiar. So construed diagrams (and numerals) do not stand for transcendent, mathematical objects; they do not stand for anything. Rather, they are part of a linguistic apparatus-mathematics-which may be used to model concrete situations. Thus the agreement of ideas, at least mathematical ideas, rests on linguistic mastery. Having gone this far, what is left of empiricism? The problem which generated Berkeley’s philosophy of geometry was that of finding an empirically suitable subject matter for the discipline. Berkeley’s ‘solution’ could not explain geometrical reasoning, so, taking my cue from Berkeley, I proposed that the best avenue for one who rejects rationalist intuition is to forego subject matter for a symbolic system. A strong suggestion of the neoBerkeleian view of geometry is that reasoning be understood generally as a linguistic activity rather than a mental activity of comparing ideas. This will be difficult for Locke and Berkeley to accept, for it makes superfluous the privileged view of the contents of our minds, which figures prominently in their philosophies. But, in fact, a privileged view of the contents of our minds is a feature of both rationalist and empiricist philosophies of mathematics. That is, both philosophies attempt to explain (the possibility of) mathematical knowledge in terms of inspection of ideas that are themselves sufficient to tell us what to make of them and what inferences to draw from them. Our discussion has shown two things about ideas understood as objects for mental inspection. Not only are they unnecessary as regards the actual practice of mathematics, but they are incapable of doing the job. Either as in the case of Descartes and Locke, the process is mysterious and so not objective (cf. Essay, 4.4. l), or, as in the case of Berkeley, the objects admit a variety of uses which would be incompatible if the objects (ideas) inspected determined their own logical significance. ‘This
objection
was raised by Adrian
Riskin.
Berkeley on Mathematical
Reasoning
Thus if there is any hope for the way of ideas, it lies in understanding
223
the
agreement of ideas through means other than the metaphor of the mind as ideal spectator. The way of ideas, 1 submit, is not finished, for as we have seen, it can be given sense independent of the visual metaphor by appeal to linguistic mastery. Its true value lies in its antiformalist methodology, that is, in its rejection of formal logic as a valuable epistemological tool (Sherry 1992).” Nor is empiricism finished, for the linguistic approach makes no appeal to intellectual insight and still bases knowledge upon experience; only now the relevant experience is sucessful managing of practical affairs rather than viewing the contents of the mind. What is finished is the distinction between genuine reasoning, on the one hand, and practical or formal reasoning on the other. That distinction depends essentially upon the model of reasoning as quasivisual inspection; we have seen that the model is useless for mathematics and I suspect the same is true elsewhere.12 Locke was within striking distance of a linguistic account of the agreement of ideas. In a late addition to the Essay he tells how it gradually dawned upon him that words are ‘scarce separable from our general knowledge’ (3.9.21). This remark suggests he may have suspected the visual inspection metaphor was just that. And in the final chapter of the Essay he supplements the traditional division of human knowledge into the theoretical and the practical with a third science, the doctrine of signs or logic (4.21). Yet it seems never to have occurred to him that language offered a way out of the puzzles arising from the conception of ideas as entities viewed by the mind. For Berkeley, the metaphor was a methodological principle that led to brilliant insights and dead ends. By taking the metaphor seriously, Berkeley was pushed to discover that the use of language could make arithmetic intelligible even though it lacked ideal content. He fell short by not considering that language and its uses might be the very stuff of ideas. Even though Berkeley’s formalism contained the details of such an approach, his commitment to ideas as entities for inspection prevented him from an unmitigated linguistic approach. We have seen that, in part, a popular view about the role of geometry in liberal education sheltered this commitment from Berkeley’s penetrating philosophical scrutiny. One hundred years would have to pass until algebra became ascendant and philosophers would feel comfortable attacking more traditional views of
“It is important
to realize that rejecting formal logic does not amount to rejecting formalism approach to geometry is formalist but with no concern for the logical
simpliciter. The symbolic forms of its statements.
‘Contemporary formalist philosophy of mathematics, such as Hilbert’s embraces a distinction between the real (or contentual) and the ideal (or formal) portions of mathematics (Hilbert 1925). Our investigation of Berkeley suggests that either that distinction, too, is unwarranted, or the mathematical situation changed sufficiently to warrant such a distinction. I hope to write on both questions in the future.
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geometry. Our investigation shows that the raw materials for this revolution were present; only custom and habit stood in the way. Alas, no one escapes his age, and we still admire Berkeley for proceeding so far.
7. An Ironic Conclusion It is interesting that the points where Berkeley’s philosophy of mathematics disparages mathematical practice-his famous criticisms of infinite divisibility and infinitesimals in the higher geometry (calculus) as well as the less notorious remarks about d@iciles nugae-all result from his inability to overcome the
deep inclination to tie geometry to a subject matter which the mind inspects. But the criticisms are innocuous as soon as one adopts the attitude that the significance of those regions of mathematics lies in the use of symbols rather than some ideal content. A full-fledged linguistic approach to mathematics would have prevented Berkeley from developing precisely the theses in the
philosophy of mathematics for which he is best known.
Acknowledgements - Michael Malone and George Rudebusch provided useful criticism of an earlier draft of this essay, and I benefited from conversation with Ian Dove and John Moulton.
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