On Mathematical Errorjwood.faculty.unlv.edu/unlv/Articles/Sherry.pdf · The simplest proof in...

Pergamon Stud. Hist. Phil. Sci.. Vol. 28, No. 3, 393-416, 1997 pp. 0 1997 Elsevier Science Ltd

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On Mathematical Error

David Sherry *

. . . the refuted proposition seems such an elementary mistake that one cannot imagine that great mathematicians could have made it (Lakatos, 1976, p. 87).

That mathematicians are fallible is hardly news. More newsworthy is the thesis

that mathematics itself is fallible (e.g. Quine, 1961; Lakatos, 1976; Kline, 1980;

Kitcher, 1983; Tymoczko, 1986; Crowe, 1988; Maddy, 1990). Fallibilists believe

that long standing communities of mathematicians have been or can be in error

about cherished results. They point to the historical record as evidence of the

‘fallible, corrigible, tentative and evolving’ nature of mathematics (Tymoczko,

1986, p. 21). Prima facie it is difficult to deny propositions like 7+5= 12.’ Even so,

the fallibilist claims there are propositions thought to have been established only to

have been overturned in the progress of mathematics. Frequently mentioned is

Euler’s conjecture that the vertices and faces of a polyhedron outnumber its edges

by 2. Crowe (1988) is typical: ‘Euler’s claim was repeatedly falsified’ (p. 264). But

our epigraph warrants caution, and, in fact, standard historical cases fail to

support the thesis that mathematics is fallible, corrigible or tentative; they serve

only as evidence that mathematics is evolving. Errors implicating an entire com-

munity of mathematicians do not exist in any but a philosophically problematic

sense. This is not an historical accident, but a reflection of mathematics’ nature.

The clearest sense of mathematical error consists in failing to follow a

prescribed procedure. One sort of fallibilist accepts that existing procedures are

trustworthy but finds error in their application (81). A more radical sort faults

the procedures themselves (#2-5).* Both sorts of fallibilist illustrate their views, but analysis will show that these examples cannot establish that mathematics,

like empirical science, is ‘continually faced with the possibility of someone

*Department of Philosophy, Northern Arizona University, P.O. Box 6011, Flagstaff, AZ 8601 l-601 1, U.S.A.

Received 6 February 1996; in revLed form 29 August 1996.

‘Of course, 7+5= 14 in base 8. But this is not a case of giving sense to a denial of 7+ 5= 12, since a sentence and its denial presuppose the same interpretation of the symbols involved.

2Fallibilists also differ in their ontological dispositions. They range from platonic realists (Maddy, Quine), to anti-realist, social constructivists (Bloor, Davis, Hersh). For the realist, error consists in misdescription, though misdescription is recognized only indirectly through application in science. For the anti-realist, error consists in failure to satisfy social norms. Errors of the kind that support fallibilism ought to be phenomena which precede philosophizing about the semantics of mathematics.

PII: 80039-3681(!Mgooo24-6

393

394 Studies in History and Philosophy of Science

discovering a counterexample which will refute even the safest, best confirmed theorem’ (Marchi, 1976, p. 384). Illustrations of fallibilism are plausible only through ignoring crucial features of mathematical practice.

1. Algorithmic Errors

Algorithmic errors thrive in checkbooks, income tax returns and mathematical tables, especially those drawn without the aid of a computer. And faulty mathematical results have been based on them.3 Algorithmic errors occur relative to a set of rules. Davis (1972) uses the potential for algorithmic error to make a case for fallibilism; and Kitcher (1983) uses it to argue for his empiricist philosophy of mathematics (Chap. 2). Davis observes that mathematics involves creating, recognizing, reproducing and concatenating symbols (p. 256). Each of these symbolic operations has a small chance of going wrong, and as the number of operations mounts, the likelihood of error mounts:

As we get away from trivial sums, arithmetic operations are enveloped in a smog of uncertainty (p. 258).

This is evident in light of the following problem, which I have tried to reproduce faithfully (ibid.).

PROBLEM: Given

A=117777777111717171717771711717111111177717177711771177171717171

71777171777171717171777111717111111717777111717171111717177171

B=777771711711117777777111111111177171717111777771717777111717111

117171171717771111111717177777777111717177771111777117177771

Find A+B

The probability of error in finding non-trivial sums like A + B, leads Davis to

remark (tongue in cheek?) that

The sum 12345+54321 is not 66666. It is not a number. It is a probability distribution of possible answers in which 66666 is the odds-on-favorite. (ibid.)

We must be wary of the slippery slope between problems like finding A + B

and trivial sums. The triviality of a sum, or any piece of mathematics, is largely a function of one’s willingness to employ it in an activity, the success of which hangs on doing the mathematics correctly. Under this criterion, finding A+B

might warrant a probabilistic interpretation, but 12345 + 54321=66666 would not. If it were not possible to be certain of the latter sum, there would be no institution of addition.

‘For example, Euler published a list of 64 pairs of amicable numbers (numbers whose proper divisors sum to one another), two of which are mistaken. In one case he failed to recognize that 220,499 was composite, contrary to the hypothesis of his theorem; in the other case he seems to have written 8563 (which is prime) in place of 8567 (which is not). (Lecat, 1935, p. 31; cf. Dickson, 1934, pp. 4246). Lecat’s work, Erreurs des Math&nmaticiens, does not classify the hundreds of errors it catalogues, but refers to them all as ‘logical’ (p. ix).

On Mathematical Error 395

No one avoids taxes by claiming that the auditor’s sum of two five digit figures is enveloped in a smog of uncertainty. Questioning the auditor’s arithmetic leads to a check, and if neither party finds an error, the matter is settled; there is no smog of uncertainty. The auditor admits the possibility of error in a given case; but having checked the case, the possibility disappears. Historically, calculation has played the same role in mathematical proof and, granting errors by individual mathematicians, no more to the chagrin of mathematics than the chagrin of the revenue service. Unless calculation had this kind of finality, there would be no accounting for the effort it takes to inculcate its techniques and no rationale for using it to determine who owes what to whom. Thus the certainty of 12345+54321=66666 is a reflection of the role it plays in our lives. To treat it seriously as a probability is far more involved than adopting a new attitude toward a proposition of pure mathematics; a probabilistic rendering conflicts with our way of life. How could a democracy even consider an income tax or regulated commerce if its inhabitants felt uncertain of the arithmetic propositions necessary for its fair administration?

Arithmetic calculation and similar procedures can be enveloped in a smog of uncertainty, though only well beyond five figure sums. Historically, mathematicians have not been concerned with problems like finding A+ B, but the problem of finding A +B is emblematic of a 20th century trend toward extremely long proofs. The Appel-Haken proof of the four colour theorem, for example, involved so many calculations that an IBM 370-16OA was required to perform them. Computers can err, and here it is reasonable to attribute uncertainty to the result; the situation is worse for human computers. Recent proofs, described by Paul Erdiis as ‘at the limit of the amount of information the human mind can handle’, do raise issues of confidence and have led mathematicians to consider probabilistic methods of proof (Kolata, 1976; Zeilberger, 1993). But traditionally computations have been checkable by a single mathematician at a single sitting. Some have suggested that mathematics has nearly run out of statements with short proofs (Kolata, 1976, p. 990; but c$ Horgan, 1993, p. 94); yet, even if mathematics is developing a new, fallible branch, this does not justify skepticism about the whole institution, particularly the portion embedded in the life of non-mathematicians. To indict the whole institution, skeptics need to locate traditional computational results which, having generated sufficient interest to invite peer scrutiny, survived that scrutiny and then turned out to be wrong.4

“In a defence of the empirical status of mathematics Detlefsen and Luker (1980) observe Kempe published a mistaken proof of the 4CT that remained uncriticized for 11 years. Cauchy, Lame and Kummer all believed that they had proved Fermat’s Last Theorem at one time. And Rademacher claimed in 1945 that he had solved the Riemann hypothesis (p. 816). From examples like this they conclude that certitude is limited ‘even in more traditional mathematics’. Such observations show nothing more than the fallibility of individual mathematicians and certainly will not support their suggestion that Gauss’s boyhood proof that 1+2+ . ..+ 100=5050 might be mistaken (pp. 807 ff.)!


2. Errors in proof

Davis extends his skepticism to proofs by likening them to calculations. Proving is the process of passing from an axiom string to a theorem string by

a finite sequence of allowable elementary transformations (p. 255) and thus,

. . . the authenticity of a mathematical proof is not absolute, but only probabilistic (p. 260).

If this logician’s notion of proof were not an idealization, but a reflection of actual practice, then one would not have to go far before having to accept theorems with a probabilistic character. The simplest proof in geometry, reduced to elementary transformations, runs to more than 100 lines (Mueller, 1981, p. 4). Certifying, by allowable transformations, every proposition in a proof of advanced mathematics would undoubtedly invite errors. Thus mathematicians continue to rely upon informal proof.

According to Davis, informality is a matter of leaving out most the elementary allowable transformations by relying upon theorems previously proved or simply skipping parts of an argument (p. 259). This leads him to comment, poetically but cynically.

Thus, far from being an exercise in reason, a convincing certification of truth, or a device for enhancing our understanding, a proof in a textbook on advanced topics is often a stylized minuet which the author dances with his readers to achieve certain social ends. What begins as reason soon becomes aesthetics and winds up as anesthetics. (ibid.)

Undoubtedly, some proofs in advanced textbooks and journal articles contain mistakes. And perhaps some proofs serve social ends more than truth. The question is whether the existence of some shoddy informal proofs casts doubt on all informally proved results, particularly those which have been sufficiently circulated to be part of the ‘tried and tested’ core of mathematics (Davis and Hersh, 198’7, p. 63). The plausibility of Davis’s analysis rests on conceiving of an informal proof as shorthand for a lengthy series of elementary transformations. But one should be suspicious about an account of proof in terms of transformations some of which were not made explicit until the late 19th century. A different view of informal proof, one which is more faithful to actual

practice, will make Davis’s analysis seem less compelling, In this spirit, we turn to informally proved propositions which have been alleged to constitute mistakes in mathematics (ibid.).

It is well known that proofs in Euclid’s Elements do not satisfy modern demands of rigor because the theorems are not formal consequences of the axioms. Here is a typical account of the logical situation in the Elements.

. . . In many places [Euclid’s] conclusions do not follow from his stated premises by formal logic alone, yet the reader finds the reasoning highly convincing because


Euclid’s book contains a diagram depicting the geometrical situation under discussion, a diagram which enables the reader to feel he sees that Euclid’s conclusions must hold... . mhese logical gaps nevertheless are very much to be deplored, even if they do not definitely mean that Euclid’s reasoning was invalid. For these logical gaps are unintentional: it was simply because he was not aware of them that Euclid failed to close them... . Euclid’s goal when he set out to systematize geometry surely was to try to produce proofs that would be valid on account of their logical form alone (though he probably would not have described his goal in this way). (Barker, 1964, pp. 3940; cf. Crowe, 1988, p. 262).

This passage reeks of Whig history, judgment of the past on the basis of current criteria. There is no evidence, of which I am aware, that Euclid was attempting to produce proofs valid on account of their logical forrn5 The range of logical forms available to Euclid is inadequate for representing the reasoning of the Elements (Mueller, 1974). And we cannot impute to Euclid an implicit grasp of logical form given the unflagging manner in which he ignores formal validity (Heath, 1956, Chap. I, p. 256). Euclidean proof must devolve upon something else. Barker intimates that the diagrams and meaning played a crucial role, though to Euclid’s detriment. I shall take this suggestion seriously in giving a general account of informal proof ($3) after dwelling further upon the ‘error’ in Elements I. 1.

The usual take on the first proposition of the Elements is that it requires an additional axiom of continuity (Heath, 1956, Chap. I, pp. 242 ff.). But it is unclear whether this or similar omissions constitute errors. It is simply false that the formal invalidity of Euclid’s arguments took a toll on his results. Euclid’s approach worked in the very strong sense that none of his theorems drops out as a falsehood under the new regime; the Euclidean corpus, which underwrote most mathematics and physics before the 19th century, was preserved in the rigorous axiomatization of geometry.6 This suggests the errors are matters of style rather than substance.

Klein and others have called attention to

. . . a real danger that a pupil of Euclid may, because of a falsely drawn figure, come to a false conclusion. It is in this way that the numerous so-called geometric sophisms arise (Klein, 1939, p. 201).

But, as Mueller (1981) puts it,

. . . in the history of Euclidean geometry no such fallacious arguments are to be found. There are indeed many instances of tacit assumptions being made, but these assumptions were always true. In Euclidean geometry, conceived as the description of intuitively grasped truth, precautions to avoid falsehood are really unnecessary (p. 5).

‘Proclus’s commentary on the Elements says nothing about the formality of sound mathematical reasoning (cf. Pro&s, 1970).

6Not every proposition enjoys the same status, though. For instance, EZemenfs, Chap. I, p. 4, is elevated to an axiom in Hilbert’s geometry.


C

GD A B

Fig. 1

The tacit assumptions include principles like x+y=y+x and if x>y, then

x+z>y +z. It could hardly have occurred to Euclid that the order in which two magnitudes were added could affect their total. Propositions like these were articulated only after mathematics reached a level of abstraction in which it could make sense to deny them; thus, principles of commutativity were not articulated until the idea of algebraic structure emerged, in the 1840s (Eves, 1990, p. 16). It was not accidental that only in 1844 did Hamilton publish his paper on quaternions, the first structure for which commutativity (of multipli- cation) does not hold. Neither Euclid nor anyone else who worked before the 19th century had occasion to look for such systems. The interpretation which ‘undercuts’ the reasoning in I.1 shows similarly that Euclid was in no position to grasp his ‘error’.

Having constructed circles of radius AB about A and B (Fig. Z), Euclid asserts the existence of a point of intersection C, which the axioms of the Elements and logic alone do not guarantee. Consider a ‘space’ S which consists of the Cartesian plane with all points having an irrational coordinate removed. Define a straight line as a set of points in S satisfying an equation of the form y =mx+ b and a circle as a set of points in S satisfying an equation of the form (x+a)2+Cy+ b)2=k. Such a structure satisfies modern formulations of Euclid’s axioms.’ Thus it seems we can use Euclid’s theory to reason about S. Let AB be the segment [0, l] of the line y =O and consider the circles x2+ y2= 1 and (x - 1)2+ y2= 1, centered at (0,O and 1 ,O), respectively. Through algebra we determine that these equations (circles) have common solutions (intersect) at (l/2, 432) and (l/2, -d/2), neither of which is in S. So in S these two circles do not intersect, and the inference which I.1 sanctions is invalid. To rule out this counterexample we require an assumption that in crossing the boundary between two regions of space one intersects the boundary. S allows a boundary which may be crossed without being intersected, a concept which would have seemed peculiar, if not absurd, to Euclid. Without being in a position to give sense to such a concept, Euclid has no grounds for doubting that the two circles in I.1 intersect.

7The modern approach paraphrases a postulate like ‘let it be postulated to produce a straight line connecting two points’ as ‘given two points, there exists a line on which both of them lie’. The departure from Euclid is considerable because production in the sense of a task that can be accomplished is replaced by an antecedently given domain of points and lines.


If these sorts of cotmterexamples indicate errors at all, let us refer to them

as ex post facto errors, since they are recognizable only from a perspective adopted subsequent to adopting the system in which the alleged error occurs. Ex post facto errors are, as far as I can determine, the only kinds of errors in the core of mathematics.8 I put forward the claim that mathematical errors (other than algorithmic errors) are ex post facto as an historical observation, but the observation requires philosophical elucidation. We look again to Euclid. Even if Euclid’s inferences are formally invalid, there is still a sense in which they are correct. Mueller defended Euclid’s tacit assumptions by claiming that they were all true, but this is awkward in light of structures for which the tacit assumptions do not hold. Hence, ‘true’ cannot mean the assumptions may be used with impunity.9 ‘True’ must mean that Euclid’s tacit assumptions work for the range of phenomena-mathematical or otherwise-to which he intended to apply the theory. This observation suggests a way to explicate the validity of Euclid’s inferences. Euclid takes for granted a family of characteristic uses for his terms; and these uses constitute part of the meaning of those terms. Thus, it does not occur to him that ‘circle’ denotes a set of ordered pairs satisfying an equation. The validity of his inferences presupposes the family of characteristic uses, which rules out counterexamples like the discontinuous space S. Euclid had no idea of the diversity (or, depending on your point of view, perversity (cf. Poincare, 1918, pp. 32-33) of uses to which ancestors of his concepts would be put. For him, geometry was the study of shapes and sixes of material objects. His ‘failure’ to notice a counterexample is on all fours with his ‘failure’ to hold to the standards of modern logical theory. Neither affect Euclid’s geometry even though they may affect attempts to use it in contexts for which it was never

intended. None of this implies that a particular employment of Euclidean geometry is

immune from error. An individual can misapply the techniques of the Elements or employ a technique that is not part of the Elements. The sophisms mentioned by Klein involve such transgressions. The ‘proof that all triangles are isosceles depends upon a particular point of intersection falling within rather than outside a given triangle (Maxwell, 1959, p. 3 ff.). But nothing in Euclid dictates where the point is to fall; so the sophism constitutes an error with respect to the

*This claim is supported by a recent series of essays on revolutions in mathematics (Gillies, 1992). They debate whether there have been revolutions in mathematics akin to political and scientific revolutions; despite their differences, all hands agree that the frequent mark of a revolution-an entity (king, theory) being ‘overthrown and irrevocably discarded’does not apply to mathematics. Instead, the debate concerns the tiner point whether a displacement of previous mathematics counts as a revolution (ibid., p. 4). Euclidean geometry has, perhaps, been displaced, but not irrevocably discarded.

‘Even Euclid’s least controversial assumptions, e.g. commutativity and associativity of addition, have their limits. For instance, in the addition of transfinite ordinals o+2#2+w.


Elements. But the Euclidean corpus, and the techniques it embodies, are, with respect to their intended applications, above error.

What is the source of this certainty? Geometry originated as a body of techniques for solving practical problems, for instance, constructing an altar with twice the area of a given altar (Seidenberg, 1962). Euclid’s first postulate also began life as an practical technique, stretching a rope between two pegs, to determine a straight line. When there is a spontaneous and universal agreement on the results of applying such a procedure, it becomes a conceptual technique. The agreement is not a convention that might easily be otherwise, like driving on the right-hand side of the road; it is a natural response of an organism, though, to be sure, an organism living a particular kind of life. Once the procedure assumes the mantle of a conceptual move, it serves as a standard or criterion, and, provided it is applied correctly, its denial lacks sense. It is certain. If an architect and an engineer dispute the straightness of a foundation, say, the rope is stretched and the matter is settled, just like the case of the tax payer and auditor. This is not to deny that a situation could arise in which the stretched rope would be inconclusive; the distance involved might be so great that there was no consensus on whether the rope was stretched. And no doubt there is a point at which elementary geometry would suffer the same fate as finding A+ B. But Euclid’s Elements and other classics of the geometrical tradition, indeed, mathematics through the 19th century fall short of this point. In general, problematic cases lie outside the domain for which a given technique was originally intended, and therefore cast no doubt upon applications of that technique within its original domain.

Between rope stretching and Euclid’s letting it ‘be postulated to draw a straight line from any point to any point’, abstraction takes place. Rope stretching is replaced by a postulate with ideal points and lines but no mention of the means of construction. But the objective pull of the postulate resides in the role in daily life played by the conceptual move of determining a straight line. The objective pull continues even when mathematical techniques are applied within mathematics itself. lo For many philosophers, this pragmatic approach is unsatisfactory (e.g. Folina, 1994, pp. 204206). They worry that counterexamples to existing mathematics or, worse, contradictions within existing mathematics may be found in the future. This position is difficult to attack because the skeptic has the future as an ally: no one knows how mathematics will develop. But I shall urge that this kind of skepticism misunderstands informal proof. Informal proof is not simply heuristic, as the skeptic suggests. It does not merely give hint of a mathematical result but is constitutive of that result.

“‘Hence they were relied upon for sorting through puzzling proofs about circle squaring (Mueller, 1969, pp. 293).


3. Informal Proof

Usually a mathematical theory is represented as a formal deductive system. While this device simplifies discussion, it can be misleading. Formalization is guided by informal theory, and the fact that a theory has been formalized is no guarantee that its subsequent, informal development will not make that formalization obsolete (cf. Lakatos, 1978, pp. 6169). Geometry, even in the widest sense, has never been pursued by applying formal logic to axioms; that is, its various branches have been developed informally in the manner of the Elements. Thus, correctness in geometry is neither logical nor empirical truth, puce usual discussions (e.g. Sklar, 1974, pp. 88-146). Even if there is a single geometry which describes physical space, the alternatives have an integrity that goes beyond formal theoretical development and so require an account which is neither logical nor empirical.

Of course the various geometries are reducible to set theory, but the situation is the same there, contrary to the view that there is a fact of the matter concerning the correct description of the universe of sets (Maddy, 1990, p. 28).” The gathering of extrinsic evidence which, according to Maddy, determines correct descriptions of the set theoretic universe requires that one develop theories with competing axioms and compare results. One evaluates proposed axioms in a Quinean way, with respect to fruitfulness for mathematics as a whole (ibid., pp. 145-146). Does the proposed axiom, for instance, answer open questions in a way that is consistent with existing mathematics and aesthetically pleasing? Maddy’s approach-a tempting version of fallibilism-is plausible on the assumption that testing is a non- committal matter of drawing out formal logical consequences. In a formal proof content plays no role, and so sense can be made of trying out one or another formulation of a concept. But in informal reasoning-and reasoning in set theory is informal--concepts function essentially, and there is nothing tentative about trying out one or another formulation of a concept: To try

out a concept is to create new mathematics. This explains why conflicting set theories seem real to the people who study them. We need an account of correctness in an idiom other than extrinsic evidence. That account is found in an analysis of informal proof.

Informal proofs have been aptly characterized as thought experiments (Lakatos, 1976, p. 7; Mueller, 1969; Brown, 1991, Chap. 3). Thought experiments aim to derive insights from applications of the grammar of various concepts in easily imaginable, if not commonplace contexts. Einstein applied the grammar of basic concepts of physics to a weight suspended from a spring hanging in an elevator in order to make a profound point about the equivalence

“Riskin (1994) gives examples of set theoretic work outside ZF that show explicitly that Maddy’s view is at odds with mathematical practice (pp. 118 ff.).


Fig. 2

of accelerated motion and gravitation (cf. Reichenbach, 1980, p. 85 ff.). The grammar of a particular concept consists of a network of conceptual connections that are learned in learning that concept. For instance, in learning the concept ‘two’ one learns that whatever is two is also greater than one, less than three, etc. The novelty of a thought experiment lies not in the concepts used or the object to which they are applied, but rather in these particular concepts’ not having been applied in this situation before. In a strong sense, then, a mathematical thought experiment-i.e. an informal proof-amounts to upply-

ing mathematics to a mathematical object’* and so taking advantage of two sets of conceptual resources.

The logical force of a thought experiment derives from the status of the conceptual connections which constitute it. In most cases, it is clear that the moves that constitute a thought experiment apply in the context in which they are used; thus we accept an informal proof for the same reason that we accept other applications of mathematics. Sometimes, however, it is unclear whether a particular technique applies in a new context. And sometimes the melding of two sets of conceptual resources involves a conflict and so it becomes impossible to adhere completely to the grammars of the concepts involved. We consider such cases in $4, after an illustration.

The illustration is from Plato’s Meno (Plato, 1961, pp. 80-87), where Socrates helps an uneducated slave grasp that the side of a square double a given square is the diagonal of the given square. Just before beginning his string of questions Socrates inquires: ‘He is a Greek and speaks our language? (82B). An affirmative answer frees Socrates to apply mathematical techniques, including counting, adding and multiplying, to the figures which he draws in the sand (Fig. 2). The slaveboy could not grasp the theorem if he had not already

‘*A mathematical object is just an object treated mathematically. For instance, a triangle drawn on the board is a mathematical object when one treats it as having straight sides etc. regardless of its appearance.


mastered the grammar of the terms Socrates employs.i3 There is more to the proof than these techniques, of course. There is the diagram which provides the

context for their application, and Socrates surely needs to know how to represent the double square (DBEH) so that the slaveboy can see its special relation to the original square (ABCD).

Even if he has never seen quite such a diagram before, the slaveboy has sufficient understanding of its parts and their relations on the basis of prior conceptual mastery. Moreover, he has sufficiently mastered the conceptual techniques Socrates employs to recognize that the figure drawn is susceptible to treatment by means of those concepts. The slaveboy then grasps the connection between a square and its double by tracing out conceptual connections necessitated by the grammar of concepts he has already mastered.

It is worth noting that the squareness of BDHE is never established (though this is easily done), for it is jumps like this which can lead to error. Socrates probably takes it for granted that the symmetry of the drawing enables the slaveboy to see that BDHE is a square, and likewise that the diagonal bisects the square.14 This is standard practice in informal mathematics; it invites error but does not constitute error. One need only be prepared to justify an inference with respect to the techniques that one brings to the case.

Meno, 8tL87, also contains a lesson on mathematical error. The slaveboy errs when he is guessing rather than following a line of thought, where a line of thought consists in applying correctly techniques he has learned. Socrates demonstrates the error of the guesses by showing that they conflict with procedures the slaveboy already embraces. On this reading, correctness of reasoning and conclusion is entirely a function of prior practice. Of course, mathematical reasoning is not always so transparent. Socrates’ is uncontroversial because the context in which it is applied is familiar fare, i.e. precisely the kind of case for which the techniques were originally learned. This is not always so; sometimes it is unclear whether a grammatical technique is applicable in the context of a given thought experiment. Such cases constitute heuristic reasoning, and seem tailor made for fallibilism.

4. Heuristic Reasoning

From the perspective of scientific and technological achievement, no other era in the history of mathematics is as important as the development of the calculus in the 17th and 18th centuries. And no other era creates as many

“These are part of everyday language, and for a time, no doubt, Greek mathematics was accessible to anyone with a grasp of the language and normal powers of concentration. By the time of Euclid, mathematics had reached a level of complexity that demands vocabulary and techniques mastered only through protracted study; though, to be sure, the study is grounded in everyday language.

‘@fIris is a very different matter from the slaveboy’s seeing a square, since the thought experiment need not fail on account of a sloppy drawing.


suspicions of overly casual, if not faulty reasoning, suspicions that seem to be justified by the rigorization of the calculus in the 19th century (e.g. Kline, 1980, Chaps 67; Grabiner, 1974). The moral Grabiner draws from reflections upon these two periods is clearly fallibilist:

Unless there were the prior appearance of major errors, standards could never improve in any important way... Only if we accept the possibility of present error can we hope that the future will bring a fundamental improvement in our knowledge

(P. 364).

The 19th century was pivotal in the development of mathematics, but, in my view, it is a mistake to think of it as correcting errors of the past.

Much of the suspicious reasoning in analysis involved infinite series, and Euler is notorious in this regard. Jacques Bernoulli tried, unsuccessfully, to find the sum of the reciprocals of the squares.

1 +;+;+A+ . . . .

Euler managed to sum this series by setting familiar infinite series equal to 0 and treating these configurations as equations of infinite degree. He then applied to these equations existing rules governing the relations between roots and coefficients of$nite equations and this led him to the sum n2/6 (Euler, 1925, pp. 73-86; cf. Polya, 1954, pp. 8-21; Putnam, 1975). Euler’s procedure fulfills the preceding description of informal proof except for the fact that it is unclear whether algebraic techniques apply to infinite equations; thus it is heuristic. Euler’s reasoning is plausible, though not demonstrative. Its plausibility depends upon several things: (1) he had earlier calculated 1+ l/4+ l/9+ l/16+... to seven places and his computation of 7~~16 agreed to the last place; (2) the result was useful for summing other interesting series, the sums of which agreed with calculation to several places; and (3) the method permitted the derivation of sums of important series already known, e.g. Leibniz’s series, 1 - l/3 + l/ 5 - l/7 + . . . = n/4. Euler’s approach is clearly inductive, for its warrant lies not in a demonstration, but in successful consequences.

Polya (1954) discusses heuristic reasoning extensively, and his analysis is heralded as an anticipation of the fallibilist view that mathematics is tentative in the same way as empirical science (Tymoczko, 1986, p. 97). In Tymoczko’s eyes, though, Polya falls short by allowing that mathematics also includes demonstrative reasoning, that is, reasoning that is beyond controversy and final. For the fallibilist, mathematical reasoning consists entirely in heuristic reasoning, which supports conjectures without establishing them beyond a doubt. Euler himself was no fallibilist. In 1743 he published a paper that begins by recognizing the unsatisfactory nature of his summation of the reciprocals of the squares by means of the roots of an infinite equation; it then presents two


methods that use uncontroversial techniques for summing infinite series (Euler, 1925, pp. 77-186).

Both he and Polya recognize the distinction of demonstrative from heuristic reasoning and require demonstrative reasoning for a finished piece of mathematics. But how is ‘demonstrative reasoning’ to be understood here? In his preface, Polya ties demonstrative reasoning to formal logic: ‘Demonstrative reasoning has rigid standards, codified and clarified by logic (formal or demonstrative logic), which is the theory of demonstrative reasoning’ (ibid., p. v). But this seems not to be his real meaning. For instance, his discussion of a generalization of the Pythagorean theorem (Elements, VI.31), gives no indication that this reasoning, which is informal, is merely heuristic. Further- more, a formally acceptable solution of Bernoulli’s problem was not possible at a time in which formal logic involved only monadic predicates and the theory of real numbers was not axiomatized. Thus, when Polya attributes to Euler a completely rigourous proof of 1 + l/4 + l/9 + l/l 6 + . . . + 7r2/6, he seems to take for granted that informal mathematics admits of demonstrative reasoning (p. 21). In other words, Euler’s appeal to more usual considerations amounts to applying existing techniques in familiar contexts, as in the proof from Meno. Mathematicians have generally been clear that heuristic and demonstrative methods are complementary but distinct techniques (Putnam, 1975, p. 76). Thus fallibilists must make their case by showing that reasoning regarded as demonstrative by a mathematical community has been mistaken, and this is not easy to do as the history of infinite series shows.

Grabiner points out that while ‘there are surprisingly few mistakes in 18th century mathematics’15 errors started to emerge

. . . near the end of the 18th century, when there was increasing interest among mathematicians in complex functions, in functions of several variables, and in trigonometric series (pp. 358-359).

A famous example is Cauchy’s theorem (1821) that a series of functions, each continuous on an interval, is itself continuous on that interval. Abel criticized this theorem (1826) with the function

sin2x + sin3x sinx-- --

2 3 . ..)

which is discontinuous at x=(2n+l)q n integral (Lakatos, 1976, pp. 27-31; Grattan-Guinness, 1970, p. 78). At first glance this counterexample seems to be different from the discontinuous space that refutes Elements I.1 because it involves mathematics that was available to Cauchy; thus Abel’s criticism could be part of the scrutiny that a piece of mathematics undergoes before it enters the

“This is more surprising to a committed fallibilist.


archives. But this counterexample is not a simple matter of carelessness on Cauchy’s part. In the early 19th century various uses of ‘continuity’ and ‘convergence’ were current. This variety was occasioned by functions like Fourier’s series, to which 18th century techniques did not apply unambigu- ously. Thus Fourier, in whose work the ‘counterexamples’ to Cauchy’s theorem first appeared, counted as continuous functions like sinx - sin2x/2 + sin3x/3 - . . .

(Fourier, 1955, p. 44); of course, Abel’s criticism requires that such functions be discontinuous. Furthermore, Abel’s criticism rests on the claim that sinx - sin2x/2 + sin3x13 - , . . converges to a discontinuous limit function. The concept of convergence which Cauchy employed in 1821 excluded such functions; indeed, Cauchy proved his theorem specifically to discredit Fourier’s work by showing that trigonometric series representations of discontinuous functions do not converge to those functions (Grattan-Guinness, 1970, p. 78). Cauchy’s error, then, is ex post facto. l6 Clear recognition of Abel’s counterexample presupposes subsequent work by Dirichlet (on the convergence of trigonometric series to discontinuous functions) and Weierstrass (on the distinction of uniform from non-uniform convergence).

The revolution in analysis was prompted not by a general malaise, but by a specific group of problems whose solution required the adoption of additional techniques. Interestingly, Grabiner points out that Cauchy’s e -6 techniques- the starting point for the rigourous calculus-were gleaned from approximation techniques developed by Lagrange and others (pp. 361-362). Cauchy saw that heuristic reasoning with infinite series could be made demonstrative by limiting oneself to the sequence of partial sums and then applying Lagrange’s approximation techniques. This accords with the account of proof offered in 63.

Although the history of mathematics is filled with heuristic reasoning,

mathematicians have generally agreed when a rigourous demonstration has been given and when results were only heuristically obtained.17 The path from an heuristically obtained result to its rigourous demonstration can take different forms. Sometimes one simply jettisons the heuristic proof. In other cases one sees how to transform an heuristic proof into a more acceptable idiom. In yet other cases an heuristic technique, owing to its fruitfulness, can be elevated to a conceptual one. The latter move requires the judgment of experience, but history suggests that mathematicians have been capable in this regard. Of course, as in the paradoxes of set theory, untoward results have been obtained by applying techniques of rigourous demonstration to cases markedly

‘6The case is similar for alleged errors like Lagrange’s claim that continuous functions have derivatives everywhere (Kline, 1980, p. 61; Crowe, 1988, p. 264). Weierstrass’s continuous everywhere but differentiable nowhere function indicates an ex post facto error, for the example, is not, for Lagrange, a continuous function. See Youschkevitch (1976) and Grattan-Guinness (1970), p. 7. - -

~ ,

“This is not to say philosophers agreed with the mathematical community. Sherry (1987) defends Newton’s calculus against Berkeley’s charge of merely heuristic reasoning.


different from those for which the technique was developed. But this does not mean that mathematicians were mistaken in their judgment because embracing a conceptual technique does not commit one to the pretentious claim that the technique is guaranteed to work in cases that have not yet emerged over the mathematical horizon. As the discussion of the logical gaps in Euclid’s EZements

showed, untoward consequences of applying a familiar technique in a new case neither brings mathematics to a halt nor leads mathematicians to abandon results in the core. This casual response is possible because the troublesome cases involve creating new mathematics rather than uncovering a flaw in existing mathematics. Mathematicians make stipulations which cover the troublesome case (e.g. the axiom of continuity or Zermelo-Frankel set theory) and move on. Thus, even in the face of paradox, results whose proofs have been sanctioned by peer scrutiny can stand as correct. This is not to say that all such theorems are correct upon substitution of current concepts (though, not surprisingly, most of them are). Nor is it to say that the concepts on which these theorems depend for their correctness are useful for current mathematical research. The point is rather that theorems such as Cauchy’s stand with respect to the concepts from which they are constructed. And though these conceptual structures are subject to displacement, displacement does not entail fallibility. To elaborate this point, we turn to Lakatos’s classic presentation of fallibilism (Lakatos, 1976).

5. Lakatos

Lakatos investigates the history of Euler’s conjecture that for polyhedra the number of vertices minus the number of faces plus the number of edges is two. This example is supposed to illustrate the general thesis that the growth of mathematical knowledge cannot be understood without fallibilism (pp. 86-87, 140). But this is not the best explanation of the historical data.

In order to prove Euler’s conjecture, Cauchy imagines a polyhedral surface made of thin rubber, removes a face, stretches the remaining surface into a plane and proceeds to reason about the network of polygons comprising that plane.18 The network can be triangulated and the triangles systematically removed so that: (1) there is no net change in Y- E+F, and (2) a triangle, for which V- E+F= 1, results. Adding the face initially removed yields the required V- E+F=2 (pp. 7-8). The technique of deforming a polyhedral surface into a polygonal network was novel, as was the idea of treating a polyhedron as a surface rather than a solid. But clearly these techniques are not invented from whole cloth; there is precedent for each in the practical sphere. One does not need to be a carpenter to know that a box may start as a network

“Cauchy did not actually think of the polyhedron with one face removed being stretched into a flat polygonal network. Rather, he mapped the one into the other, again a familiar technique (89, note).


Fig. 3

of flat pieces. Virtually everyone has the linguistic resources to imagine and reason with the deformation from which Cauchy begins. Once a flat polygonal network is obtained, triangulation of a polygon, a standard technique, is brought to bear. Decomposition is standard technique too. That the network can be created, triangulated and then decomposed without altering V- E+ F is warranted by carrying out these operations in a paradigm case (cube, say) and recognizing that the procedure is applicable to any polyhedron. As Lakatos allows, the mathematical community accepted Cauchy’s demonstration as sound (8, note).

But counterexamples are put forward-hollow cubes, twin tetrahedra, star polyhedra, picture frames, and even the cylinder (pp. 13-23). A hollow cube (i.e. a cube with a cubical cavity in its interior) is not Eulerian, since I/--E+ F=2 for both the inner and the outer cube, and so V-E+ F=4; (Fig. 3) even with a face removed, a hollow cube cannot be stretched into a plane. But the hollow cube and company are counterexamples only if they are bonajide

polyhedra, and it certainly occurred to mathematicians that they were not (14 ff.) Thus, Lakatos’s character Delta:

Why should the theorem give way, when it has been proved? It is the ‘criticism’ that should retreat. It is fake criticism. This pair of nested cubes is not a polyhedron at all. It is a monster, a pathological case, not a counterexample. (ibid.)

Even though hollow cubes are easily conceived, they are controversial precisely

because they were not, in Cauchy’s day, part of the existing mathematics of polyhedra. Euclid had defined and proved theorems about particular polyhedra and even showed how to inscribe a polyhedron within the space between two concentric spheres without touching the inner sphere (XII, p. 17).19 All the polyhedra which Euclid considered are Eulerian, and it is only charitable to infer that mathematicians who accepted Cauchy’s thought experiment had in mind this use of ‘polyhedron’. Indeed, Euler’s conjecture would have garnered no attention had ‘polyhedron’ denoted the pathological cases. Accepting the hollow cube as a counterexample requires stretching the concept polyhedron beyond its intended range of application. Unlike Socrates’ criticism of the slaveboy’s guess that doubling the side doubles the area, this criticism is not

“Euclid did not define ‘polyhedron’.


based in conceptual techniques to which Cauchy is already committed.20 The hollow cube is a refutation only in the ex post facto sense. Once hollow cubes and the like are taken to be relevant to the study of polyhedra, new mathematics is in the offing. But is the earlier mathematics incorrect?

Fallibilists rarely mention that Lakatos himself worries that polyhedral monsters refute Cauchy’s proof only if one changes the rules of the game: yet he recognizes the distinction ‘between refutations which onZy reveal a silly mistake and refutations which are major events in the growth of knowledge’ and so mirrors our distinction of algorithmic from ex post facto error with the distinction between logical and heuristic counterexamples. Lakatos seems to agree with the line we have taken when he writes, ‘This argument is perfectly correct within Cauchy’s narrow conceptual framework’ (note 8); however, he continues, ‘... but [it is] incorrect in a wider one, in which “polyhedron” refers to, say, picture-frames’. The argument is so obviously incorrect in the wider framework that Cauchy could not have accepted it. So why introduce ‘incorrectness’ here?

Here is one response. By stretching concepts mathematicians improve knowledge, moving from naive2r conjectures (‘all polyhedra are Eulerian’) to sophisticated ones (‘all polyhedra in which circuits and bounding circuits coincide are Eulerian’). We can say, as Euclid’s reformers did, that the sophisticated case involves making explicit what Cauchy and his supporters had in mind when they accepted the original thought experiment. But here too presuppositions are made explicit only against a background different from the one in which Cauchy worked, a background in which ‘polyhedron’ is no longer understood with respect to a few simple paradigms but with respect to an abstract definition in terms of polytopes and incident matrices (pp. 108-l 16).22 Clearly the principles of Poincare’s proof encompass more than the homogeneous family of shapes Cauchy considered. PoincarC’s proof even explains why the alternating, rather than the direct sum is invariant (Steiner, 1983, p. 520). Yet let us be clear on what improvement here entitles us to say about the fallibility of ancestors of Poincare’s proof. What we find in Poincart’s proof is a more expansive network of conceptual connections than Cauchy’s proof offers. The network includes connections with portions of mathematics (like matrix algebra), of which Cauchy had no inkling. But does a less expansive network constitute an error? The fact that a subsequent theory is better connected may count against the convenience of its application; a fully

“Lakatos characterizes Cauchy’s proof as a guess (p. 5), but it is unlike the slaveboy’s guess, which is not couched as a proof.

“Feferman (1978) observes that ‘naive’ is misleading; far from being naive, Cauchy’s proof idea is ‘already well-advanced; it has significant structure and steps’ (p. 317). In my own terms, an heuristic insight has been worked out by means of existing, rigourous techniques.

“Adopting a definition in these terms has awkward consequences, like counting a polygon with a missing edge as a polyhedron (p. 108).


formalized version of Euclidean geometry is less useful to an engineer than its naive ancestor. Wider framework or no, Cauchy’s proof stands as a criterion that can be applied (inside or outside mathematics) the way any piece of mathematics is applied: having represented something by means of Cauchy’s concept of polyhedron, one may then infer properties of that thing on the basis of the conceptual connections established by the proof. The connections involve no more and no less than the conceptual resources which Cauchy employed in the proof; thus there is no pretense of applying the result to cases that fall outside the domain of Cauchy’s concepts. There is a family resemblance between Cauchy’s theorem and Poincart’s theorem, and one can understand how the latter evolved from the former. But there are no grounds for saying that both mathematicians were trying to solve the same problem and one was more

successful. To clarify this point further, compare Cauchy’s ‘mistake’ to a mistake in

physical theory. More accurate observations showed that Newtonian mechanics was mistaken in its determination of the movement of the perihelion of Mercury; a subsequent theory succeeded where Newton’s failed. Movement in the perihelion of Mercury constitutes (in Kuhn’s phrase) an anomaly for the theory because the movement of the perihelion of Mercury lies in the domain of Newtonian mechanics; it is the kind of case for which the theory was created, and not a phenomenon unknown in Newton’s day. Logical counterexamples in mathematics are the closest analogues to anomalies in physical science. The significance of the latter is a function of their being phenomena for which the paradigm was created; and analoguously, logical counterexamples lie within the domain of the concepts which constitute a proof. Peer reviewers look for and use them in evaluating proposed additions to the mathematical archives. But whereas empirical science coexists with anomalies, a piece of mathematics is simply discarded in the face of an anomaly.

Taking seriously the standard examples of the fallibility of mathematics presumes that mathematics coexists with anomalies. Not surprisingly, Lakatos (1978) refers to counterexamples to Cauchy’s proofs as ‘anomalies’ along with the approximately elliptic character of the planetary orbits (p. 98). But this is difficult to reconcile with Delta’s speech quoted above. Physicists never referred to perturbations as fake criticisms or pathological cases. The presumption that mathematics coexists with anomalies can survive only by supposing that mathematicians characteristically employ concepts which are unclear. Thus Cauchy’s proof of Euler’s conjecture would be merely heuristic if it did not employ a definite concept of polyhedron. But informal proofs are not merely heuristic aids to discovery; they are not comparable to the dream of a snake biting its tail which suggested to Kekult that the benzene molecule is a ring (Brown, 1991, p. 89). Granted, both Kekule and Meno’s slave came to an insight, but only the latter’s insight was a function of grammatical connections


constituting an informal proof. KekulC would not have had to ignore the workings of his language in order to see that his insight was fallacious; experiment could have taught him that. Meno’s slave, on the other hand, could not have been dissuaded from his insight without surreptitiously stretching one of the concepts in the proof. Unless the linguistic practices underlying a proof are clear in the mind of those trying to follow it, there is no mathematical insight!23 This is not to say that mathematical insight is infallible, that the ‘Aha!’ of mathematical insight cannot accompany an error. Such errors are not revealed by stretching concepts. Rather, they are revealed by showing that one failed to adhere to techniques which were brought to the proof to begin with.

It is certainly permissible to stretch the domain of ‘polyhedron’ and so develop a new concept. A departure from prior conceptual techniques will leave one with an unclear concept, that is, until new techniques are in place. In the context of groping for a new concept, reasoning will be heuristic in so far as mathematicians are unsure whether their techniques apply to new kinds of cases. And here the method of proofs and refutations is defensible, but as a method of coming to understand the commitments of developing a concept one way rather than another, not as a means to overthrow an existing piece of mathematics. Eventually, as with Poincare’s proof, the mathematical community settles upon a new concept, displacing but not overthrowing its ancestor.

Philosophers will no doubt continue to raise skeptical questions about the concepts of mathematics, but there are no historical grounds for such skepticism. At least prior to uncheckable proofs, mathematicians as a community have proved capable of surveying the range of possible logical counterexamples and so avoided gaffes. Hence, fallibilists have had to resort to heuristic counterexamples to make their case.

Lakatos might ask how mathematics escapes the discomfort of coexisting with anomalies? The answer lies in a fundamental distinction between mathematical and empirical concepts. We can get some idea of what is involved by another comparison. Critics of Copernicanism used a thought experiment to refute the hypothesis that the earth rotates (cf. Feyerabend, 1978, Chaps 67). They argued that if the earth rotated from west to east, then a stone dropped from a tower should land hundreds of yards to the west; since the stone would land at the foot of the tower, there is no diurnal rotation. Galileo defused (rather than refuted) this line of reasoning by distinguishing the apparent motion of the stone from its real motion. The real motion of the stone consists of downward and circular components, and so describes a long arc which intersects the circular path of the tower at its foot. The tower argument and Galileo’s response run on distinct concepts of motion, which Feyerabend

*30f course an informal proof can be clarified and honed, but this takes place within the context of the linguistic practices which make it possible in the iirst place.

412 Studies in History and Phiiusuphy of Science

terms ‘operative’ and ‘non-operative’ (pp. 83 ff.).24 According to the operative concept, real and apparent motion are the same. According to the non-operative concept, motion is relative to stable surroundings; thus the Aristotelians are guilty of not taking into account stable surroundings (fixed stars) with respect to which the stone follows a long, curving path.

Clearly there are parallels between this case and the disagreement between Cauchy and his critics over whether hollow cubes, etc. count as polyhedra. Both involve a disagreement over what is the proper concept to use. But there are important disanalogies. First, Galileo’s approach defuses without refuting the tower argument; what refutes the argument, ultimately are data, like the phases of Venus, which constitute anomalies for the Aristotelian world view, with its operant concept of motion. Second, Aristotelians and Copernicans agree on the phenomenon (the motion of the stone), but disagree on the analysis of the phenomenon. Parties disagreeing over the extension of ‘polyhedron’, on the other hand, disagree from the outset, and this is what rules out any straight- forward (logical) refutations. Finally, the correct analysis of the motion of the stone is an empirical problem in the deep sense that it requires fitting that motion into the entire causal nexus. There are anomalies in physical science because the causal nexus is both exceedingly complicated and unavoidable. Nothing like the causal nexus is involved in the dispute over what to count as a polyhedron. Mathematicians are responsible to fit their theories to some phenomena, namely the domains of the concepts they employ. But these domains are far more manageable than the causal nexus, and whereas the causal nexus is more or less forced upon us, the polyhedral character of the hollow cube is not.

Mathematicians are therefore much freer to develop networks of conceptual connections than physical scientists. As emphasized earlier, mathematical concepts have their start in basic practices, but at some point mathematicians synthesize practical techniques into conceptual constructions for which no application is intended, and this is the case in the history of the Euler conjecture.25 With this creative freedom, mathematicians give up their right to refutation beyond the grammar of the concepts which constitute a proof. They invite a plurality of theories, the likes of which is not found in physical science.26

Lakatos’s inclination to treat heuristic counterexamples as showing the incorrectness of a mathematical theory stems, I suspect, from antipathy toward the pluralism to which our account of informal proof leads us. While Lakatos

“‘The genius of Galileo’s criticism lay in getting his Aristotelian opponents to see that the distinction of real and apparent motion was not a neologism-as classifying a cylinder as a polyhedron surely is-but that the linguistic resources for making and reasoning with the distinction already existed (Galileo, 1953, p. 71).

%Saccheri’s anticipation of non-Euclidean geometry is also an excellent example of mathematics for its own sake emerging from the synthesis of basic, practical techniques.

Z6Philosophers may talk as though there are countless alternatives to a given theory that are consistent with the data, but in point of fact scientists are concerned with only a handful of rivals.


rejects the notion that mathematics is a monolithic body which grows by the

accumulation of truths, he suggests instead that mathematics grows as conjectures are made and criticized. Mathematics is still a body of truths, but none of them has a permanent place. In this conception mathematics is still monolithic in the sense that conflicting theories are not allowed to co-exist. The antipathy toward pluralism is evident in the perspective which Lakatos and his followers adopt toward the arrival of non-Euclidean geometry: ‘[it] shattered infallibilist conceit’ (p. 139).27 Such pronouncements are misguided. Non-Euclidean geometry had a shattering effect. Along with imaginary numbers, it helped to shatter the idea that mathematics was closely tied to physical reality and so bring forth pure mathematics as we know it today (Nagel, 1979, Chaps 8-9; Sherry, 1991). The acceptance of non-Euclidean geometry did refute various philosophical conceptions, but Euclidean geometry remained intact.28

The success of non-Euclidean geometry was an inducement to stretch concepts-an uncommon but not unheard of practice before the 19th century (again, imaginary numbers did as much). And, as it turned out, interesting mathematics is often made possible by stretching concepts. But the interest of a piece of mathematics is a different issue from its correctness. Interest is primarily a function of the purposes of professional mathematicians, only indirectly is it a function of the conceptual techniques which make demonstration possible. With respect to interest, I agree with social constructivists who maintain that the dynamics of a community influence the development of mathematics.29 Sociological dynamics can lead communities to adopt new practices, but, as we have argued, this does nothing more than displace earlier practices.

6. Final Observations

Fallibilism subsists on an equivocal use of ‘refutation’ and ‘counterexample’. The doctrine seems newsworthy in so far as ‘refutation’ and ‘counterexample’ suggest that a problem has been in our midst, only we failed to notice it. This is the usual sense of these terms, according to which the proponent of a proposition or proof is chagrined by an oversight. But when we look at examples which illustrate fallibilism, we find that ‘refutation’ and ‘counterexample’ are being used heuristically. A hollow cube is a counterexample to

“‘mhe existence of geometrical objects where there is more than one parallel to a line through a given point refutes Euclid’s geometry’ (Berkson, 1978, p. 299).

**A mathematical colleague underscores this point by observing that the distance formula

d=+-xJ2+(y, -J#+...

which runs throughout calculus texts presupposes the Euclidean framework. “Bloor (1982) proposes a sociological explanation of the history of Euler’s conjecture (pp. 207

IT.).


Euler’s conjecture only if we disregard the standards of correctness which established it as a theorem in the first place. A heuristic counterexample requires that a particular proposition or proof employ concepts other than the ones it actually employs. There is no question of someone being chagrined. The same point can be put differently. Fallibilism is supposed to be the key to understanding mathematical progress, and certainly one makes progress in recognizing an error. Thus Meno agrees readily that his slave is better off having recognized that his guess is incorrect, even when he cannot correctly answer the main question (84b). But fallibilists cannot subscribe to progress on this model since the ‘errors’ of Cauchy and others are not recognizable from the perspective of techniques they already embrace. The errors of the past are indeed heuristic, and we have every right to be skeptical of philosophical conclusions derived from them.

New conceptual techniques can lead to progress, but this is not to be judged with respect to past errors. Rather, we should ask whether we are now able to do things-particularly in the practical sphere-which were previously denied us. Cardano stretched the concept of number by introducing roots of negative numbers, and in so doing he produced heuristic counterexamples. For instance, they ‘refute’ the theorem & I&=&&. But these were not the signs of progress. Only in the fullness of time (e.g. in D’Alembert’s use of imaginary numbers in fluid mechanics), did it become clear that Cardano’s innovation was not, as he saw it, ‘subtle but useless’. Fallibilism is a misnomer for the freedom mathematicians enjoy in developing new conceptual structures. Rather than supporting fallibilism, the history of mathematics provides incentive not to restrict that freedom.30

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