special issue on philosophy of mathematics || mathematical rigor--who needs it?

Mathematical Rigor--Who needs it?Author(s): Philip KitcherSource: Noûs, Vol. 15, No. 4, Special Issue on Philosophy of Mathematics (Nov., 1981), pp. 469-493Published by: WileyStable URL: http://www.jstor.org/stable/2214848 .

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Mathematical Rigor-Who Needs It? PHILIP KITCHER

UNiVERSITY OF VERMONT

Historians, philosophers and mathematicians standardly draw a distinction between rigorous and unrigorous mathematics. In drawing the distinction, they usually intend to advance some normative claim. At the very least, rigorous mathematics is supposed to be preferable to unrigorous mathematics. I want to explore the distinction between rigorous mathematics and unrigorous mathematics, in the hope of understanding why rigor is desirable, and when the need for rigor is urgent. My goal is to take my title question at face value and to answer it.

To practice rigorous mathematics is, apparently, to offer rigorous arguments. Logic helps with the characterization of rigorous argument: central to the idea of rigorous reasoning is that it should contain no gaps, that it should proceed by means of elementary steps. (Con- temporary systems of natural deduction attempt to show how inferences can be decomposed into such elementary steps.) Conceive of an argument as a triple, consisting of a set of premises, a conclusion, and a finite sequence of statements. The argument is rigorous if and only -if the sequence of statements has the conclusion as its last member, and every statement which occurs in it is either a premise or a statement obtainable from previous statements by means of an elementary logical inference.'

The demand for rigor in mathematics cannot simply be identified as a request for rigorous argument. To see this, we need only remark that we can devise a rigorous substitute for any unrigorous argument by expanding the set of premises. Obviously, what is needed is a constraint on the premises: rigorous mathematics, we might suggest, consists in providing rigorous arguments from premises which are known to be true. This suggestion will help us to understand why we might want rigor when we do not already know the conclusion of an unrigorous argument. However, mathematicians (and philosophers)

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sometimes ask for rigorous proofs of statements which they already know (and recognize that they know). If they are demanding rigorous arguments from known premises then their demands are easily met. They may simply write down one-line proofs.

Traditional epistemology of mathematics offers two related responses to this point. The first is simply to deny that mathematicians ever know statements which they are unable to prove rigorously. Math- ematicians may sometimes come to accept statements on the basis of plausibility arguments, analogies and so forth, but acceptance on these grounds doesn't constitute knowledge. I think that this response sets the standards for knowledge in mathematics too high. If we can gain scientific knowledge by nondeductive inference then, prima facie, we can obtain mathematical knowledge in the same way.2 However, the idea that rigorous proof is the distinctive feature of mathematical knowledge can be reformulated. A second version of the traditionalist's response is to concede that, on occasion, mathematical truths can be known without rigorous proofs but to insist that such mathematical -knowledge fails to meet the standards we ideally demand of mathematical knowledge. Ideally, mathematical knowledge is obtained by deduction from first principles, statements which are knowable without inference from other statements. What makes such knowledge ideal is the fact that it is less vulnerable to empirical challenge than the knowledge we obtain in other ways. Indeed, it is commonly held that when we know a mathematical truth by following a rigorous deduction from first principles,-in othe-r words, a rigorous proof-we know that truth a priori and for certain. A classic presentation of this ideal can be found in Frege's writings, where it is used to introduce a demand for rigor.

If we ask what gives mathematical knowledge its value, then the answer must be that the value consists less in what is known than how it is known, less in the content of the knowledge than in the degree of the self- evidence and the insight into logical connections. ([11]: 171).3

Without presupposing that the only way to gain mathematical knowledge is by following a rigorous proof, the traditional view of ideal mathematical knowledge can explain why rigor is sometimes demanded: by constructing rigorous proofs of known truths we improve our knowledge of these truths, either by coming to know them a priori and for certain or, at least, making our knowledge objectively more certain than it was before.

I shall call this traditional picture of ideal mathematical knowledge deductivism. It consists of two separate claims: (i) all mathematical knowledge can be obtained by deduction from first principles (statements knowable without inference from other statements); (ii) deduction from first principles is an optimal route to mathematical knowl-



MATHEMATICAL RIGOR-WHO NEEDS IT? 471

edge, providing knowledge which is less vulnerable to empirical challenge than other ways of coming to know the same truths. I think that both claims are incorrect. Although we can sometimes present parts of mathematics in axiomatic form, deducing all the statements we accept from a small number of axioms, the statements taken as axioms usually lack the epistemological features which the deductivist theses attribute to first principles. Our knowledge of the axioms is frequently less certain than our knowledge of the statements we derive from them, and, moreover, it is often implausible to claim that the axioms can be known without inference from other statements. In fact, our knowledge of the axioms is sometimes obtained by nondeductive inference from knowledge of the theorems they are used to systematize. (See [28]: 64-6; I shall enlarge on the point below).

As an illustration, consider the axioms of ZF set theory. The thesis that these statements can be known without inference from others prompts the question of how this knowledge is gained. Some deductivists provide no answer to this question; others wave a hand vaguely in the direction of "intuition."4 If we consider alternatives to deductivism, then we can acheive a more satisfactory explanation of our knowledge of the ZF axioms. These statements are accepted because they systematize and answer questions which arise in other parts of mathematics (for example, in arithmetic and in real function theory). The historical process through which they were articulated and accepted reflects the nature of the evidence which we have for them. (A particularly clear example is the axiom of choice; see [31].)

In this paper, I want to explore the consequences of a nondeductivist view of mathematical knowledge. I don't pretend to have a knockdown argument against deductivism: deductivists can always claim that a satisfactory explanation of our knowledge of those statements they take as first principles will one day be given. However, the considerationsjust indicated motivate exploration of alternatives. This means that we shall have to give up the deductivist explanation of mathematical rigor and its desirability. In what follows I shall try to show three things: firstly, that it is possible to explain mathematical rigor and the legitimacy of demands for rigor without making deductivist as5umptions; secondly, that a nondeductivist account better explains the practice of historical mathematicians than the deductivist account considered in this section; and, finally, that an understanding of mathematical rigor can point the way to articulate further a nondeductivist picture of mathematical knowledge.

II

Even in a nondeductivist theory of mathematical knowledge, deductive inferences are going to play an important role. Frequently, we deduce



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new theorems from statements we already know, and, since the deduction provides our sole basis for believing such theorems, we can agree that the achievement of a rigorous proof would improve our epistemic state. As we saw above, the difficult cases are those in which mathematicians complain that a part of their subject is unrigorous, demanding rigorous proofs of truths they already know. Can we find a construal of these demands other than that offered by deductivism?

One radical hypothesis which we might adopt is that the demands for rigorous proofs of known truths arejust confused. Mathematicians who make these demands have been brainwashed by deductivists into thinking that a rigorous proof recapitulates the correct epistemological order. Now there is some evidence for the influence of deductivism. The prefatory remarks of many mathematicians who claim to provide rigorous proofs of familiar truths suggest that what they have accom- plished is an optimal reconstruction of some part of mathematical knowledge. (See, for example, Cauchy's preface to [8].) However, I think that we can do better than to attribute widespread deductivist confusions which generate unreasonable demands for rigor.

One writer who cannot be accused of deductivist confusions but who emphasizes the need for rigor is Bolzano. (See [4], [5], and, for a reconstruction of Bolzano's position, [19].) Bolzano explains his aims by claiming that there are "objective relations of dependence" among mathematical truths, independent of our cognition of them, and he regards the enterprise of proving theorems as one of disclosing these objective relations of dependence. He stresses that the objective ordering may be quite different from the epistemological ordering. It may be that our knowledge of truths of one kind depends on knowledge of truths of another kind, even though the truths of the latter kind are objectively dependent on those of the former kind. For example, Bolzano contends that our knowledge of properties of continuous functions is dependent on our knowledge of properties of geometrical figures, but he maintains that, because truths of geometry are objectively dependent on truths of function theory, we must not rest content with proofs of theorems about functions which appeal to geometrical principles.

Bolzano's enterprise rests on a distinction between two types of proofs: there are proofs which exhibit a route to knowledge of the theorem proved and there are proofs which give an objective grounding (objective Begriindung) of the theorem. But how are we to interpret the latter notion? Bolzano's answer is given in the following passage:

It is known that Aristotle already ... distinguished two kinds of proofs, namely those which show only the oti, i.e. which show that something is the case, and those which show the 6u5ti, i.e. why something is the case. ([15]: 272)




Given Bolzano's distinction, we can begin to understand how reasonable demands for rigorous proofs of known truths can arise, without making deductivist assumptions. Sometimes we want an explanation of a familiar result. Under these circumstances we canjustifiably demand a proof which shows why the result is true, that is, an "objective" grounding of the theorem. (For further argument that Bolzano's notion of "objective grounding" should be understood in terms of explanation, see [19].)

Let me immediately respond to two objections. The first is that talk of explanation is inappropriate in connection with mathematics. This objection finds support in the recognition that questions of mathematical explanation cannot always sensibly be asked. There are many mathematical truths with respect to which the questions "Has it been explained?," "Does this proof explain it?," "Is this proof more explanatory than that?" seem inappropriate.5 The goal of explanations is to provide understanding, and I suggest that we approach the topic of mathematical explanation via the concept of understanding-or, more exactly, via the concept of afailure in understanding. Bolzano was right to emphasize that there are epistemic contexts in which a person knows that a mathematical statement is true, but does not see why it is true. (There are also, I think, mathematical statements for which there is a permanent danger of this, in that our natural ways of coming to recognize their truth do not show us why they are true.) To take one simple example, we may establish by direct computation that, for n< 100, ,' i r= 1/2 * n(n + 1), but you do not understand the truth until you see it as an instance of a result which holds generally. Hence, I suggest that, although questions of mathematical explanation do not always arise, there are some contexts in which people fail to understand some truths and there are arguments which promote understanding where it was previously lacking.

My talk of understanding may prompt a second criticism, the criticism that such talk is totally subjective. This criticism seems to me to be as mistaken as the parallel complaint about scientific understanding. Relative to epistemic contexts, some scientific statements are objectively puzzling: in such contexts it is rational to ask for an explanation, and rational to appraise some arguments as dissolving the puzzle. I think that the same applies to mathematics.

Unfortunately, even though we succeed in defending Bolzano's distinction between two types of proofs, this distinction cannot be applied directly to our problems of understanding demands for rigor in mathematics. Demands for rigor are not simply identifiable with demands for explanation. To see this, we need only remark that there are cases in which we have a rigorous argument from known premises, but in which we do not understand why the conclusion is true. "Brute



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force" computations provide one kind of case, and, perhaps, the four- color theorem is another. Failure of understanding is not ipso facto a sign of lack of rigor.

There are several ways in which we can fail to understand a mathematical truth, two of which are especially important. The first occurs when, although we can provide an argument for the truth which is composed of individually simple and compelling steps, we can find no way to organize our reasoning so that its global structure is apparent. Some arguments are "unsurveyable," and when our only ground for a mathematical truth is an unsurveyable argument our understanding of the truth is deficient. The second type of failure, with which I am primarily concerned here, occurs when we accept a statement on the basis of an argument which we cannot recast as a gapless argument. Here, our trouble is not with the global features of the argument but with its fine structure. The failures contrast with our successes, cases in which we have an argument for a truth on which we can adopt either of two perspectives, attending at will to its broad lines of inference or to the finer details, (The best presentations of proofs in mathematics books are organized so that the broad lines of the proof are apparent, while enough detail is supplied to enable the intended readers to arrive at the fine structure.)

I therefore propose to view demands for rigorous proofs of known truths as signalling a particular kind of failure of understanding. I will use an example to sharpen the ideas presented so far, examining some features of the most prominent instance of unrigorous mathematics in the history of the subject.

III

Newton and Leibniz introduced the calculus as a technique for answering a number of traditional questions about tangents to curves, maxima and minima, lengths of curved arcs, areas enclosed by curvilinear figures and so forth.6 Some of their predecessors had offered methods for answering some of these questions, at least in special cases. Fermat had a method for finding maxima and minima, Descartes had a method for computing subnormals, and various writers had achieved partial success with the problems of finding lengths of arcs and enclosed areas. However, the technique suggested by Newton and Leib- niz had two distinct advantages over earlier methods. Not only were Newton and Leibniz able to treat a more extensive class of curves, but they also managed to give a unified treatment of problems which had previously been tackled by separate methods. They recognized that the whole class of geometrical questions can be reduced to two fundamen-




anachronistically). To answer a particular question for a particular curve, you represent the curve by an algebraic equation, follow the procedure for differentiating or integrating (depending on the type of question), and then use a specific algorithm to find the answer to the geometrical question. (Their insight thus consists in recognizing the usefulness of calculating new functions from the functions represent- ing curves and showing how these functions provide answers to the old geometrical questions.) I shall concentrate on the simplest example. Let us suppose that you are given the equation of a curve, y= f(x), in a particular coordinate system, and you want to find the slope of the tangent to the curve at a particular point (i.e. you want to find tan a where a is the angle between the x-axis and the line "touching" the curve at the point). You differentiate f(x) with respect to x, evaluate it at the point, and the result is the value of the slope.

Trouble comes with the method for differentiating. Both Newton and Leibniz generalized the method of infinitesimals, which had been popular among their predecessors. Their method for differentiating a function of one variable, f, can be set out (in a way which reflects the historical presentations) as follows:

(1) Construct the difference f(x+d) - f(x), where d is "small," writing it as d(A+B), where B contains all and only those terms containing d as a factor.

(2) Divide the difference by d, obtaining A+B. (3) Ignore terms containing d as a factor, thus obtaining A

as the derivative of f.

Newton and Leibniz had excellent reasons for believing that this method works. Used in conjunction with their specific algorithms for geometrical questions it gave recognizably correct answers in those cases where answers were provided by previous methods. Having extablished the concordance of their results with those obtained by their predecessors, and finding that the method gave no recognizably incorrect results, Newton and Leibniz were amply justified in extending it to give answers to problems which "nobody previously had dared to attempt" (Preface to [25]).

Philosophers of mathematics often concentrate almost exclusively on mathematicians as theorem generators: the mathematician has been regarded simply as a person who proves theorems. However, mnuch mathematical work (including much of Leibniz' and Newton's work) is directed towards what I shall call problem-solving. To solve a problem is (unsurprisingly) to produce a correct answer to a question. Correct answers might, of course, be produced on the basis of a hunch or a lucky guess, but I shall restrict the notion of problem-solving, applying



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it only in those cases in which the mathematician produces an argument for the correctness of the solution.

The argument produced to support a solution might be a rigorous argument from known premises. However, I am chiefly interested in those cases in which it is not. Newton and Leibniz advanced problem- solutions of this type. In extending their methods to new cases, they used the recognized successes of the methods as inductive grounds for believing that they would yield correct answers to previously un- answered questions. (The success of the methods could be verified in some cases by performing computations or using approximations.)

One trouble with this situation is that, although one has reason to believe that some newly generated solution is correct, one does not see why it is correct. We have here a failure of understanding. It might appear that this is easily correctable. Doesn't the success of the method of infinitesimals provide evidence for the principles which underlie it, so that we could reasonably systematize our knowledge and explain our conclusions by adopting those principles and deducing our solutions from them? This suggestion is along the right lines: the success of the method and its incorporation in a unified treatment of traditionally unrelated questions indicates its potential for systematization and explanation. However, a search for "the principles which underlie the method" divulges the real source of the failure of understanding. What are these principles? Attempts to replace the formal procedure (1)-(3) with a set of premises for explanatory arguments quickly runs into trouble.

Use of the calculus to answer questions about tangents presup- poses the assumption:

[A] Following the procedure (1)-(3) and evaluating the result at x=a gives the slope of the tangent to the curve y=f(x) at (a, f(a)).

From [A] and standard principles of algebra, one can deduce many statements of the form rThe slope of the tangent to the curve y=f(x) at (a, f(a)) is z,r statements which are recognizably correct. To understand why these statements are true we appear to need an interpretation of the formal procedure (1)-(3) which will enable us to replace [A] by acceptable statements from which we can deduce the conclusions generated by [A]. Consider the obvious approach.

(4) The slope of the tangent to the curve y=f(x) at (x, f(x))=[-f(x+d)-f(x)]/d where d is "small."




(5) If f(x+d)-(x)=d(A+B), where B contains all and only terms which occur with d as a factor, and if d is "small," then [f(x+d)-f(x)]/d=A.

(4) and (5) together enable us to reach the conclusions we want, and they appear to bring out principles underlying [A]. However, embar- rassment arises once we ask what we mean in saying that d is "small." If d is non-zero than both (4) and (5) are false: (4) is false because [f(x+d)-f(x)]/d is the slope of a chord joining neighboring points on the curve, and, in general, this will be different from the slope of the tangent; (5) is false because if its antecedent is true then [f(x+d)- f(x)]/d=A+B, and, in general, B will be non-zero. If d is zero then both (4) and (5) are illformed. Hence, there appears to be no way of inter- preting (4) and (5) so they will be acceptable.7

Here we find a situation in which a demand for rigor may legiti- mately arise. [A] tells us that a certain type of reasoning will lead to true conclusions, but when we try to find a rigorous argument which shows us why that reasoning leads to true conclusions, we fail. Throughout the eighteenth century, mathematicians agreed that there must be an explanation for [A], and even Berkeley, the most acute critic of the calculus, concurred with this. Indeed, Berkeley himself proposed an explanation. Although (4) and (5) are, strictly speaking, false, the errors compensate. Unfortunately, Berkeley's work only announces a program, it does not complete it. (Berkeley described it as a "hint.")8 Equally programmatic were the efforts of some Leibnizians to reinterpret (4) and (5) in a way which would make them both true. The Marquis de l'Hopital and Jean Bernoulli suggested that it is possible to regard (4) and (5) as holding for non-zero infinitesimal values of d. Infinitesimals are quantities that can be neglected in some equations, so that (5) is true. Moreover, we can analyze the notion of a tangent's "touching" a curve by supposing that the tangentjoins points which are infinitely close. Thus (4) can be defended. However, the l'Ho^pital- Bernoulli program faces the problem of avoiding the obvious inconsistencies generated by applying standard algebraic laws to infinitesimals. To deduce the desired conclusions from their versions of (4) and (5), l'Hopital and Bernoulli needed principles about infinitesimals. Unable to specify these principles, they proceeded in practice by selectively applying standard algebraic laws, so that their treatment did not amount to a satisfactory explanation.9

Newton proposed a different strategy for explaining [A] and other similar statements which assert that the method of infinitesimals gives correct answers to geometrical questions. Instead of viewing (4) and (5) as the principles underlying the use of the calculus in solving the problem of tangents, he introduced the notion of limit and suggested



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that the tangent to a curve at a point should be interpreted as the limit of a sequence of chords, connecting the point to successively closer points. (The situation is more complicated than this because Newton adopts a kinematic view of geometry: curves are regarded as swept out by the motions of points and the tangent is regarded as the line in which a generating point would travel if the instantaneous velocity were held constant. The instantaneous velocity is then viewed as a limit. I shall ignore these complications in what follows.) The slope of the tangent is thus identified as the limit of [f(x+d)-f(x)]/d as d goes to zero, and Newton proposes that the procedure (1)-(3) is an easy way of computing the limit. More explicitly, Newton's systematization of the calculus uses the principles

(7) The slope of the tangent to the curve y-f(x) at

(x,f(x) = lim [f(x+ d)-f(x))]/d. d--> 0

(8) If [f(x+d)-f(x)]/d=(A+B) where B contains all and only terms in which d occurs as a factor, then the value of B can be made as small as one pleases, in comparison with the value of A, by taking d to be sufficiently small.

Newton uses (7) and (8) to explain [A]. (See [16]: 46-9, [27]: 141-3). There are two problems with Newton's own presentation of this

approach. In the first place, his defense of (8) is extremely casual- especially so, since he wants (8) to apply to cases where B contains an infinite series of terms. Secondly, and more importantly, although (7) and (8) expose the underlying structure of Newton's reasoning, they distort the fundamentally geometrical character of his approach. For Newton the concept of limit is primarily geometrical, that is, it applies to sequences of geometrical entities, tangents are limits of sequences of chords, curves are limits of sequences of linear figues. (Thus my use of 'lim' in (7) should not be taken to suggest that Newton's concept of limit is ours.) Similarly, Newton's notion of a real number (or a "quantity") is that of a ratio of line segments. Hence, when he presents his explanations of [A], he formulates (7) and (8) in geometrical terms, regarding his achievement as one of integrating his new techniques within the framework of traditional mathematics.

This last point is important not only for comprehending Newton's proposal but also for understanding the ideas of his disciples and the attitudes of his opponents. Newton regards traditional mathematics as consisting of two disciplines, arithmetic and geometry. The development of algebra by Viete and Descartes provides a convenient way of formulating and solving problems about the entities studied in these disciplines. Algebra provides methods which enable us to shortcut




traditional arithmetical or geometrical arguments, but Newton would insist that those methods must be interpretable as arithmetical or geometrical results, and the task of making the calculus rigorous consists entirely in explaining why the reasoning leads to correct geometrical results. For Newton, this means that one must find ways of replac- ing the algebraic reasoning with geometrical arguments, so as to explain the truth of [A] and similar statements which assert the success of the method of infinitesimals in coping with other geometrical questions.

Leibniz' attitude is very different. One of his principal themes is that the new algebra has outstripped the methods of traditional mathematics. It is a science of "quantity in general" which has particular applications to arithmetic and geometry, (natural numbers and geometrical magnitudes are particular quantitites). Because Leibniz con- ceives of algebra as having a wider scope than traditional methematics, he rejects the demand that algebraic reasonings are only shortcuts for the solution of arithmetical or geometrical problems. Algebraic equations are not just equations of curves, and they can be manipulated using the procedures of the calculus even when the manipulation receives no geometrical interpretation. Leibniz constantly exhorts his followers to extend the algebraic techniques, and, even though he admits that his methods are imperfectly understood, he postpones the task of explaining their success.'0

I have tried to reinterpret the initial phase in the history of the calculus in a way which avoids commitment to deductivist assumptions. Leibniz and Newton developed a problem-solving technique they could use to produce new mathematical knowledge. The technique was rationally adopted because the results it generated were well- confirmed, and because it promised systematization of its results. But a demand for rigor was reasonable because of difficulties in understanding why the technique was successful. Straightforward attempts to replace the technique with rigorous reasoning produced arguments whose premises appeared to be false.

Of course, many of the writers to whom I have referred accepted deductivism. Their demands for rigor, (or their appraisals of their own achievements), were often couched in terms of self-evidence and certainty, as well as in terms of explanation and understanding." l The goal is frequently characterized as one of deducing the results of the calculus from self-evident first principles. I suggest that these char- acterizations stem from faulty philosophical assumptions. The writers in question fail to make Bolzano's distinction between two kinds of proof. Responding to genuine failures of understanding, they are led to misdiagnose the problem as a deficiency in. the certainty (self- evidence, etc.) of their knowledge.



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IV

I shall use the foregoing discussion of the method of infinitesimals as a model for understarfding mathematical rigor. We might naturally ex- pect to explain how demands for rigor can rationally arise in mathematical practice by giving a definition of 'rigorous mathematics' and showing, on the basis of this definition, that it is sometimes reasonable to want rigorous mathematics. But I think that this approach is faulty. 'Rigorous' and 'unrigorous,' as used by mathematicians (historians, philosophers) in the contexts I am considering, are correlative terms. To put the point another way, when a demand for rigor arises, what is being demanded is something which stands in a particular relation to current practice. For want of a better word, I shall call the relation the rigorization relation.

Let me begin by identifying a mathematical practice. A mathematical practice is the set of reasonings accepted by a community of mathematicians; a reasoning is a sequence of statements. Intuitively, the practice of a community consists of all the sequences members of the commu- -nity would be prepared to write down in support of the statements they would be prepared to assert. It will standardly include instances of proof-techniques and problem-solving techniques.

A rigorization of a mathematical practice is a set of rigorous replacements for all the reasonings in the practice. A rigorous replacement for a reasoning is an argument meeting the following conditions:

(i) it is a rigorous argument (in the sense of ?I); (ii) its premises are true;

(iii) its conclusion is the conclusion of the reasoning it replaces;

(iv) on the basis of the replacement, we can explain how the original reasoning led to its conclusion and thus see why it led to a true conclusion (or, more briefly, we can use the replacement to explain the success of the original reasoning).

There is a natural way to construe clause (iv) if the reasonin'g replaced was an argument with clearly stated premises and an explicit commentary: a rigorous replacement should have the same structure as the original argument but it should "fill in the gaps." I have not adopted this as a general requirement because I think it is inapplicable to the interesting cases. The reasoning employed in the method of infinitesimals (for example) was not presented as an argument with clearly stated premises and an explicit commentary. Furthermore, the proposals for rigorization advanced by Newton and Berkeley were not in




principle wrong, even though Newton intended to change the vocabu- lary used in the original reasoning and Berkeley intended to explain the success of the reasoning in terms of cancelling errors in the underlying premises.

In the remainder of this section I want to consider some issues which are raised by the analysis just given. I shall pursue these issues insofar as they are relevant to my central purpose, namely the understanding of demands for rigor without appeal to deductivist assumptions.

Let us begin with a point which I set on one side above (footnote 1). I have been supposing that there are absolutely elementary logical inferences and that the inferences permitted by the basic rules of standard natural deduction systems are absolutely elementary. This double assumption is necessary if we take the view that what we currently regard as rigorizations of past mathematical practices are indeed rigorizations of those practices. But the assumption can be criticized. A limited criticism would concede the claim that there are absolutely elementary inferences, but deny our right to assume that our current efforts to delineate them succeed. From this perspective, our proposals for rigorizing past practices are the best available approximations to an ideal at which we aim, and it is possible that future work will supersede our proposals as they have overtaken those of our predecessors.

There is a more radical challenge. One might deny that there are any absolutely elementary inferences, claiming that the idea of a simple logical inference only makes sense within the context of some logical theory. It would be consistent to suppose that logical theorizing progresses, that we can sensibly speak of successive refinements in the decomposition of inferences.'2 What is denied is that there is a limit to the process of refinement, a point at which inferences are resolved into absolutely elementary parts.

This radical challenge can be accommodated within the framework I have developed. We would simply have to allow the notion of rigorization to admit of further relativization. A rigorization would be a rigorization of a prior practice in the light of a conception of elementary logical inference, and condition (i) in the definition of 'rigorous replacement' would be understood as bearing an implicit relativization to a conception of elementary inference and a corresponding idea of rigorous reasoning. A rigorous replacement for a reasoning would be a reconstruction of it which breaks it down into elementary steps from true premises, the notion of an elementary step varying with context.

The questions of whether there are absolutely elementary inferences and whether, if there are, contemporary logical theory succeeds in delineating them, are fascinating, but I shall not try to answer them here. (I suspect that our current acceptance of some inferences as



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elementary in just the latest stage in the refinement of logical theory. This suspicion rests simply on induction from the history of logic.) As I have just indicated, my analysis is neutral on these issues, and it can be used to investigate the rationality of demands for rigor without settling them. A demand for rigor reasonably arises when there are successful reasonings which cannot be reconstructed as- arguments involving elementary steps from true premises, given prevalent ideas about what steps are elementary and what statements are acceptable as true. The example of the method of infinitesimals shows clearly that concern for rigor is rationally motivated when mathematicians are unable to find a way to reconstruct successful reasonings in terms of inferences they take as elementary and premises they can accept as true. We can understand the concern independently of deciding whether the idea of absolutely elementary inferences makes sense.

In general, a practice is unrigorous when it contains reasonings for which there is no rigorous replacement in the practice (call these the unrigorous reasonings of the practice). If these reasonings are replaced with arguments that contain only elementary steps (by the standards of those whose practice it is) and which proceed from premises which do not appear to coflflict with accepted statements, then the resulting practice will usually be called 'rigorous.' Of course, if standards of elementary inference change, or difficulties are discovered in some of the premises adduced, what was hailed as a rigorous practice will come to seem unrigorous. Shifts in standards of rigor, unsuspected troubles with newly proposed premises and renewed demands for rigor, typify the nineteenth century endeavors to solve the problems posed by the calculus. Cauchy's contemporaries hailed his work as making the calculus rigorous; subsequent research uncovered the need for further rigorization. (See [13] for an account of the puzzles generated by Cauchy's application of his concepts of convergence and continuity to Fourier series expansions.) Weierstrass was credited with the achievement of a rigorous practice; once again, subsequent developments prompted a demand for rigor. When the set-theoretic presuppositions of the Weierstrassian enterprise were articulated in a natural way, inconsistencies appeared. Today we claim that current set-theoretic treatments provide a rigorous practice. So far, our claims have sur- vived.'

To sum up, an unrigorous practice is one which needs rigorization, and a rigorous practice is one which rigorizes a prior unrigorous practice. We should note explicitly that a rigorous practice is not simply a set of reasonings, for each of which there is a rigorous replacement in the practice. That is always easily attainable by deleting any unrigorous reasonings.




So far, I have discussed rigor in connection with reasoning, but I have said nothing about the need for rigor in definition. It would be tempting to think that the concept of a rigorous definition can be captured in terms quite different from those which I have used to understand the rigorization of a practice: to give a rigorous definition of a term is to formulate explicitly the conditions for the application of the term which were implicit in prior usage. But I think that a clear understanding of what is required if one is to formulate successfully the implicit conditions of prior mathematical usage, involves an appeal to the notion of rigor which I have been developing. The implicit conditions of application of mathematical terms are intimately linked to the accepted reasonings involving that term. To see this in detail, consider the typical way in which demands for rigorous definitions arise and how these demands are satisfied. Frequently, a mathematical term will be introduced by reference to an example, or a family of examples, without any explicit statement of the limits of applicability of the term. Subsequently, the mathematical community may come to accept successful but unrigorous reasonings involving the term, and a proposal for rigorization may include an explicit statement of the conditions of application of the term, a statement which is regarded as giving a rigorous definition of the term, articulating the conditions implicit in prior usage. What standardly occurs is that a demand for rigor in definition arises against the background of a set of unrigorous reasonings, whose rigorization seems to call for an explicit equivalence involving a particular term. The demand is satisfied by producing an equivalence which enables reconstruction of the problematic reasonings.13

The example of the method of infinitesimals illustrates this point. L'Hopital and Bernoulli diagnosed (part of) the problem as stemming from the lack of an explicit characterization of the concept of a tangent. They proposed that a tangent to a curve should be identified as a line joining points separated by an infinitesimal distance. The diagnosis shows how a demand for rigor in definition can arise from a problem of an unrigorous reasoning; the proposal for solution reflects the fact that what is demanded of a rigorous defintion is an equivalence which helps with the reconstruction of prior unrigorous reasonings. Later episodes in the history of the calculus supply further examples. The rigorous definitions of 'function,' 'continuous function,' 'derivative' and 'inte- gral' were motivated by the recognition of unrigorous reasonings and assessed in terms of their ability to reconstruct such reasonings. (An exceptionally clear example is the rigorous analysis of 'continuity' provided by Dedekind. As is apparent from Dedekind's discussion, acceptance of the analysis is based on its ability to complete previous problematic proofs of fundamental results about real numbers. See [9]

?VII.)



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In general, a rigorization of a practice should not only enable us to understand the unrigorous reasonings of the practice but it should also advance our understanding of the conclusions of those reasonings. (This is especially obvious in those cases in which rigorization involves the definition of old terms.) If an unrigorous problem-solving technique promises to systematize its conclusions-as the method of infinitesimals did-then the replacement should enable us to see why the original reasonings led to true conclusions and why those conclusions are true. A rigorization should cash in the explanatory promise of the old technique.

It is time to introduce an obvious complication into my account. I have dealt with what might appropriately be called explicative rigorization. Explicative rigorization shows that the accepted reasonings were fundamentally correct. However, it is possible that the set of accepted reasonings should contain mistaken reasonings, so that, although the demand for rigor was originally posed as a request for replacements of all the unrigorous reasonings of the practice, it is answered by correcting

-the practice. A corrective rigorization of a practice consists of a set of rigorous replacements for some of the unrigorous reasonings of the practice and the deletion of the rest of the unrigorous reasonings of the practice; on the basis of the replacements given, we must be able to explain what was wrong with the deleted reasonings. Intuitively, an explicative rigorization accounts for prior success, a corrective rigorization shows why we were successful to the extent that we were. As the amount of correction increases, we may become less happy to talk of rigorization at all. In extreme cases, if all the unrigorous arguments of the old practice are deleted, we will be inclined to say that the practice has not been rigorized but discarded. (There is an obvious analogy here with the reduction/replacement dichotomy in discussions of natural science.)

Further distinctions can easily be introduced. We can obviously talk about partial rigorization of a practice, that is, of rigorization of a subset of the practice. Moreover, we can identifyprogressive rigorization as cumulative partial rigorization. These refinements allow us to construe many of the remarks of mathematicians, philosophers and historians. For example, we can understand Cauchy's achievement, retro- spectively, as one of partially rigorizing the calculus, and we can see Weierstrass' work as extending the partial rigorization begun by Cauchy.

The final point that I want to discuss in this section concerns the

confirmation of new principles via rigorization. My remarks about rigor in definition indicate the way in which proposed equivalences can

gain support. More generally, a statement may win acceptance because, on the hypothesis that it is true, we can rigorize a previous unrigorous




practice. It is easy to give a schematic account of this kind of argument for new principles:

(a) Reasonings in class C have been successful, but there is no apparent way to reconstruct them as sequences of elementary inferences from currently accepted premises.

(b) We can find rigorous replacements for the reasonings in C if we add p to our set of accepted premises.

(c) There is no evidence against p. ... p

Arguments of this general type are very common in the history of mathematics, and I think that they play an important role in the growth of mathematical knowledge. Usually, however, the basic form is over- laid with complications. The introduction of a new principle may help us to provide rigorous replacements for some, but not all, of the current unrigorous reasonings; or there may be alternative incompat- ible principles which help with some subset of the current unrigorous reasonings (or, worse still, which help with different proper subsets of the current unrigorous reasonings); or the proposed principle may have difficulties of its own. As an example of the last type, consider the l'Hopital-Bernoulli principles about infinitesimals. Although these could be used to reconstruct infinitesimalist reasoning, they presented the problem of how to limit the algebra of infinitesimals so that con- tradictory results would not be forthcoming.

It is worth noting explicitly that,just as rigorization can lead to the extension and modification of mathematics, it can also generate changes in logical theory. Historically, requests for rigor have rarely been made against the background of a completely formulated logical theory. Instead, the kinds of inferences which are regarded as acceptable steps for the decomposition of arguments are tacitly recognized by those who make the demands. The formulation of the demands may lead to an explicit presentation of those inferential transitions which are taken as elementary, and even to the revision of prior inferential practices. Obviously, the most striking example of this development is the emergence of modern formal logic.

At this point, I have assembled the materials needed to answer my title question. In the final section of this paper, I shall try to explain how demands for mathematical rigor can be more or less urgent.

v

I pointed out in ?111 that Newton and Leibniz adopted very different attitudes to the question of finding a rigorous version of the method of



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infinitesimals. Newton bequeathed to his successors the view that the task of making the calculus rigorous is important and should be under- taken by showing how the algebraic procedures abbreviate rigorous geometrical argument. Leibniz' successors also followed the master's directive. They freely manipulated algebraic expressions, with few qualms about the arithmetical or geometrical significance of what they were doing. How are we to understand this difference in attitude? Is the Leibnizian neglect of questions of rigor irresponsible and careless? Or is the Newtonian concern for rigor misguided fussiness?

Let us begin by noting that successful rigorization pays two kinds of epistemic dividends. Directly, it brings us understanding where there was prior failure to understand. Indirectly, it may enable us to extend our reasonings into new areas of research, to solve problems which we would not have been able to tackle without a clearer view of a particular technique. These indirect benefits may be negligible or they may be extremely large. Moreover, as we shall see below, they may increase or decrease because of shifts in the focus of problem-solving -research. Clearly, as the indirect benefits of rigorizing a practice appear larger, the problem of rigorization will seem more urgent to the community.

Decisions to undertake a program of rigorization will not only be affected by estimates of the epistemic benefits to be obtained. Consid- erations of the probability of success are also relevant. Now, one's estimate of the probability of success may depend on appraisal of the state of mathematical practice and what is needed for its rigorization. For example, one may hold that current practice could be rigorized by a conservative rigorization, a set of replacements using only premises accepted by the community prior to the rigorization. Or one may believe that any successful rigorization will be revolutionary, introducing previously unaccepted premises, including, perhaps, premises which violate prior metamathematical views about the ontology or organiza- tion of mathematics.14 Somebody who thinks that a revolutionary rigorization is required is likely to be less sanguine about the chances of success than someone who thinks that all that is needed is to apply familiar principles.

These considerations enable us to understand the divergence of Newtonians and Leibnizians on the issue of making the calculus rigorous. Newton and his disciples (and also Berkeley) believed that it would be possible to provide a conservative rigorization of the calculus. Leib- niz and his followers contended that a successful rigorization of the calculus would be revolutionary. Indeed, Leibniz suggested that the algebraic methods used in the calculus could not be accommodated within traditional views of the nature of mathematics. Proceeding on the grounds that the indirect benefits of immediate rigorization were



MAIHEMA'l ICAL RIGOR-WHO NEEDS IT? 487

slight (in other words, that problem-solving could continue without rigorization) and that the probability of successful immediate rigorization was small (that, initially, it was unclear how to frame new principles), Leibniz was led to advocate postponement of work on the problem of rigorization.

Early estimates of probability and utility may be too rough to indicate a preferred course of action. Indeed, I think it would be foolish to look for a "decision" between Newton and Leibniz in the context of the late seventeenth century situation. However, by the middle of the eighteenth century, matters were different. The success of continental mathematics and the stagnation of British research made further work in the Newtonian tradition irrational.

There are two related epistemological questions to be asked about someone who attends to (or ignores) an issue of mathematical rigor. Firstly, is it rational, in the circumstances, to take the problem of rigor seriously (or to ignore it)? Secondly, is it rational, in the circumstances, to adopt a (possibly programmatic) proposal for rigorization? These questions arise naturally when one looks at the history of the calculus, and my discussion indicates answers to them. Newton proceeded rationally in attending to issues of rigor in the calculus because, given the state of mathematics at the time, it was reasonable to worry that con- tinuation of the unrigorous practice might break down and reasonable to believe that a geometrical rigorization of existing practices could be given. Nor was Leibniz unreasonable in investigating the extent to which one could stretch the use of the new methods without rigorizing them. Leibniz' successors produced so many apparently successful reasonings, which could not readily be reinterpreted using the methods favored by eighteenth century Newtonians, that, by 1750, the Newtonian claim that all the unrigorous reasonings of the calculus could be reconstructed according to Newtonian proposals was no longer defensible. Similarly, the achievements of the Bernoullis, Euler, and other continental mathematicians, undercut the thesis that the question of rigorizing the calculus was an urgent one.

Two accidental features elevated the importance of completing the Newtonian program of rigorization despite the growing evidence against, it. In the first place, Newton's disciples wished to vindicate the master's approach to the calculus by showing that it, unlike its continental counterpart, involved no departure from "the received methods of the Ancients" ([27]: 142-3; see also [26]: i-iv, 1-3, 33-4, 575-6). Sec- ondly, Berkeley's challenge to the calculus prompted a flurry of writings on the issue of rigor, and reinforced the belief in a geometrical solution. In his presentation of the calculus, Colin Maclaurin, the most talented of Newton's successors, set out the fundamentals of the calculus, as well as new applications of it, using intricate geometrical



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arguments. Explicit references make it clear that this cumbersome style was at least partly motivated by desire to alleviate Berkeley's worry that algebraic procedure's can deploy "empty" symbols to gloss over defects in reasoning. Maclaurin's efforts at reconstruction cover only a frag- ment of the analysis developed by his Leibnizian contemporaries.

I claim that the approach to mathematical rigor which I have adopted provides a satisfactory explanation of the different attitudes to the calculus adopted in the late seventeenth and early eighteenth centuries. The explanation brings out a general moral. The strategies of attending to issues of rigor and of ignoring such issues are beset with opposite dangers. If one addresses the question of rigorization too early, one may simply draw a blank.15 Or, by trying to carry out a program which promises initially to reconstruct most of the unrigorous reasonings, one may find oneself able to accommodate an ever smaller fraction of the (successful) techniques developed by one's less fastidi- ous contemporaries. But those who postpone work on rigorization may find that the enterprise of problem-solving breaks down. Unclarity about some reasonings may lead to a predicament in which mathematicians neither know what they are talking about nor whether what they are saying is true, so that they are unable to resolve the questions which most interest them.

As a last piece of evidence in favor of my approach to rigor, I want to look briefly at the way in which the issue of rigorizing the calculus finally became urgent. Contrary to popular belief, the impetus for making the calculus rigorous did not come from worries about infinitesimals, but from a complex of mathematical problems, whose most important element was the difficulty of using infinite series representations of functions. Leibniz and Newton had deployed the method of infinite series to apply the method of infinitesimals to recalcitrant functions, and, during the eighteenth century, the question of finding the values of infinite series of numbers became of interest in its own right. From the beginning, it was clear that apparently straightforward techniques for series summation could produce peculiar results. By the mid-century, Euler, whose methods for computing infinite sums were as imaginative as anybody's, had recognized explicitly that, while judi- cious substitutions in standard infinite series representations of functions lead to unproblematic conclusions, other substitutions lead to unacceptable statements. (See [10] Series 1, Vol. 14, pp. 589-91).

Cauchy and Abel, writing in the 1820's, declared this situation to be scandalous. ([8]: ii-iii, [1] Vol. II pp. 256-7, 263). Cauchy proposed that the vague analogy with finite sums (which had been in use since Newton and Leibniz) does not suffice to give meaning to infinite sum expressions and that the value of an infinite sum expression should be defined as the limit of the sequence of partial sums (when the series is




convergent) and should be left undefined when the series is divergent. Over half a century earlier, Euler had made the underlying concept of convergence explicit and had rejected the solution to the problem of rigor which Cauchy was to propose. ([10] Series 1, Vol. 14 pp. 592-5, Vol. 15 pp. 70-75.) This raises an interesting question: why was the problem of rigorizing the use of infinite series urgent for Cauchy (and Abel) but not for Euler, and why did Cauchy adopt and Euler reject the same proposed rigorization?

Once we understand the difference in the mathematical problems which concerned Cauchy and Euler, the answer is evident. Euler took questions of computing the sums of infinite series to comprise an important family of mathematical questions. Since his arsenal of techniques for attacking these questions included the device of represent- ing a convergent series as the difference of two divergent series, and since, in some cases, the use of this device was his only method for finding the sum of the convergent series, he was reluctant to jettison divergent series. Moreover, Euler's inability to find rigorous replacements for his reasonings with infinite series expressions did not hinder his problem-solving; in the cases which interested him, he could readily check to see if his reasonings were successful. (Of course, with the advantage of hindsight and the theory of convergence which Cauchy bequeathed to us, we can judge that Euler did not clearly understand the results he set down. That should not lead us to deny that his terms refer and that some of his claims are true.) By contrast, Cauchy's research into the representation of functions by infinite series was crippled by the lack of a reliable procedure for using infinite series expressions. Central questions in the theory of real functions were unresolved, pending a rigorization of infinite series techniques. Al- though Cauchy's proposed rigorization could not accommodate some peripheral parts of the prior practice-the results, cherished by Euler, which had been obtained through essential use of divergent series-it was extremely fruitful in addressing the problems which were at the focus of early nineteenth contury research. To put the point simply, Euler's problems could be tackled without rigorization and the Cauchy proposal would have diminished his range of successful techniques, while Cauchy's problems demanded rigorization of series techniques and Cauchy's proposal filled the bill. (The success of some Euler summations could be left as a minor curiosity, to be explained later.)16

The episode I have just sketched shows how the urgency of a problem of rigor (and, correspondingly, the attractiveness of a proposal which solves the problem at a certain cost) can vary as the focus of mathematical research changes. I do not mean to suggest that mathematicians' appraisals of the importance and interest of questions are whimsical or arbitrary. Indeed, I think that such appraisals are subject



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to standards of rationality, whose description is central to understanding the growth of mathematical knowledge. My point is that changes in the assessment of what counts as an important problem can increase the need for rigorization. The search for rigor begins with the recognition that the current state of understanding with respect to particular reasonings is inadequate to the solution of the problems of current interest.

Not only does my approach to mathematical rigor make sense of historical demands for rigor, it makes better sense of those demands than rival deductivist interpretations. In the first place, it does not attribute to those who seek rigor exalted epistemological interests. Although the prefatory remarks of reformers often announce that their goal is to deduce a body of statements from self-evident first principles, their mathematical work usually exhibits specific local concerns. If we take seriously the passages in which Cauchy and Abel declare their dedication to deductivist standards, we shall be puzzled to find that their mathematics reveals a quick fall from grace. Instead of -regarding them as incompetents who aspired to do a deductivist job and botched it, my interpretation views them as responding to particular mathematical difficulties.'7 The epistemological commentaries can be seen as misguided window-dressing. Secondly, we can begin to understand the piecemeal character of much foundational research. Frequently, a mathematician (Cauchy, for example) will labor to reconstruct some reasonings while accepting others which, from the perspective of later research, are equally unrigorous. This should not surprise us. Mathematical concepts are clarified and reasonings reconstructed to the extent required for solving problems of immediate interest. Thirdly, the account I have offered begins to show how the demand for rigor is intertwined with the growth of mathematical knowledge. New principles (like those advanced by Cauchy) are confirmed by their ability to solve problems of rigorization which have become urgent. We are no longer driven to suppose that, by sharpen- ing their powers of "intuition," mathematicians came to apprehend a new first principle.

Who needs mathematical rigor? Some mathematicians at some times, but by no means all mathematicians at all times. I have tried to show how we can recognize demands for rigor and evaluate their urgency, without committing ourselves to deductivism. Yet, in answering my original questions, I have been led to further issues on which, at most, I have barely touched. What are the criteria for accepting a proposed rigorization? What features determine the importance of a particular mathematical problem? Under what circumstances is it reasonable to believe that accepted views of the nature of mathematics (or of logic) require revision? These questions are parallel to issues which




arise in connection with scientific change. Traditional epistemology of mathematics ignores them. I hope to have shown that the questions, and the parallel, should be taken seriously.18

REFERENCES

[1] N. Abel, Oeuvres Completes, (2 Volumes), New York, 1965. [2] G. Berkeley, Works, Volume 4 (eds. A. Luce and T.Jessop), London: Nelson, 1950. [3] J. Bernoulli, Die Differentialrechnung, Leipzig: Ostwald, 1924. [4] B. Bolzano, Rein analytischer Beweis, Leibzig: Ostwald, 1905. [5] , Theory of Science (translation by R. Georg of parts of Wissenschaftslehre),

California, 1972. [6] C. Boyer, History of the Calculus and its Conceptual Development, New York: Dover,

1949. [7] F. Cajori. History of the Conceptions of Limits and Fluxions in GreatBritainfrom Newton to

Woodhouse, Chicago: Open Court, 1919. [8] A. Cauchy, Cours d'Analyse, Paris, 1821 (reprinted as Volume 3 of Series 2 of

Oeuvres Completes, Paris, 1881-1932). [9j R. Dedekind, Continuity and Irrational Numbers (in Essays on the Theory of Num-

bers, edited by W. Beman, New York: Dover, 1901). [10] L. Euler, Opera Omnia, Teubner, 1911-36. [ 1] G. Frege, Nachgelassene Schriften, (eds. H. Hermes, F. Kambartel, F. Kaulbach),

Hamburg: F. Meiner, 1969. [12] K. Godel, "What is Cantor's Continuum Problem?", (in P. Benacerraf & H. Put-

nam, eds., Readings in the Philosophy of Mathematics, Prentice-Hall, 1964). [13] Ivor Grattan-Guinness, The Development of the Foundations of Analysis from Euler to

Riemann, MIT, 1970. [14] Ivor Grattan-Guinness, "Berkeley's Criticism of the Calculus as a Study in the

Theory of Limits," Janus, 56, 1970, 213-27. [15] I. Kant, Critique of Pure Reason (Translated by N. Kemp Smith), New York: St.

Martin's, 1965. [16] Philip Kitcher, "Fluxions, Limits and Infinite Littlenesse," Isis, 64, (1973), 33-49. [17] "Kant and the Foundations of Mathematics," Philosophical Review,

84(1975), 23-50. [18] "Theories, Theorists and Theoretical Change," Philosophical Review,

87(1978), 519-47. [19] , "Bolzano's Ideal of Algebraic Analysis," Studies in the History and Philosophy

of Science, 6(1975), 229-69. [20] "Frege's Epistemology," Philosophical Review, 88(1979), 235-62. [21] "'On the Uses of Rigorous Proof," (Review of [22]), Science 196(1977),

782-3. [22] Imre Lakatos, Proofs and Refutations, Cambridge, 1976. [23] , "Cauchy and the Continuum," (in Mathematics, Science and Epistemology,

Cambridge, 1978). [24] G. Leibniz, Mathematische Schriften (ed. C. Gerhardt), Halle, 1849-63. [25] G. de L'H6pital, Analyse des Infiniments Pelits, Paris, 1696. [26] C: Maclaurin, Treatis of Fluxions, Edinburgh, 1742. [27] I. Newton, The Mathematical Works of Isaac Newton, Volume 1, (ed. D. T. Whiteside),

New York: Johnson, 1964. [28] Hilary Putnam, "What is Mathematical Truth?", (in Mathematics, Matter and Method,

Cambridge, 1975). [29] A. Robinson, Non-standard Analysis, North-Holland, 1966. [30] Mark Steiner, "Mathematical Explanation," Philosophical Studies, 1978, 135-51. [31] E. Zermelo, "A new proof of the possibility of a well-ordering," (in J. van

He"jenoort (ed.), From Frege to Godel, Harvard, 1967).



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NOTES

'There are two importantly different versions of this approach. If we assume that there are absolutely elementary logical steps, then we can take an argument to be absolutely rigorous if it is decomposed into such steps. But it is possible to deny the assumption, claiming that there are no inferences which are absolutely simple. We would thus obtain a relativized notion of rigor: an argument would be rigorous, relative to a set of inferences taken as elementary, if it were decomposed into steps which are identified as elementary in the set. On this latter conception, a demand for rigor makes tacit reference to an accepted interpretation of elementary inference. For the time being, I shall ignore this possible relativization of rigor, supposing that there are absolutely elementary logical inferences. I shall also suppose that the notion of an absolutely elementary inference is adequately captured in standard systems of natural deduction.

2Moreover, if we suppose that rigorous proofs (by our lights) are required for mathematical knowledge, then we shall have to conclude that there was little, if any, such knowledge before the twentieth century. I take it that the picture of our ancestors ignorantly amassing a body of statements which we, at last, know has little appeal.

3This passage is typical of many in Frege's writings. I have tried to make clear the epistemological character of Frege's intentions in [20].

4For an explicit formulation of the appeal to intuition in the context of a realist view of set theory, see [12]. "Intuition" is also a favorite term of constructivists who usually employ it in equally vague ways. One exception to this charge is Kant, [ 15], who does give a relatively precise account of how intuition gives us basic mathematical knowledge (not, of course, knowledge of the ZF axioms!). But, as I try to show in [17], the specificity of Kant's account makes it vulnerable to decisive objections.

5Here I diverge both from the position taken by Mark Steiner in [30] and the view I adopted in [ 19]. Although I believe that the reasons put forward in [30] and [ 19] motivate some talk of mathematical explanation, I now think that both Steiner and I were too hasty in assuming that questions about mathematical explanation always make sense. The fact that we sometimes distinguish between explanatory and non-explanatory proofs of a theorem doesn't imply that any given proof of any given theorem is either explantory or nonexplanatory.

6For articulation and defense of the historical claims about Newton made in this section, see [16]. A standard historical account of Newton, Leibniz and the early calculus can be found in [6]. Unfortunately, this account is permeated by crude philosophical views which often lead to extremely distorted interpretations. Nevertheless, it contains some useful information and organizes the primary sources.

7This point was already made lucidly by Berkeley in The Analyst. See [2]. 8For a clear discussion of Berkeley's program and its difficulties, see [14]. 9For the l'H6pital-Bernoulli apporach, see [3] and [25]. Since Robinson's develop-

ment of non-standard analysis in [29], several writers have suggested that non-standard analysis can be used to reconstruct the views of eighteenth and nineteenth century mathematicians who used the term 'infinitesimal.' (See, for example, Lakatos' discussion of Cauchy in [23].) As I have suggested in [18], hypotheses about the referents of the tokens of past scientists are tested by their ability to explain why the scientists made the remarks they did, to account for their responses to observation, their inferences and so forth. It seems to me (at best) unclear that Robinsonian treatments of classical infinites- imalists provide a satisfactory reconstruction of the reasoning of the mathematicians in question.

,1For some typical passages, in which Leibniz claims that the new mathematics has outstripped the "methods of the Ancients," and that what is important is to continue the extension of problem-solving-techniques, see [14] Vol, II p. 219, Vol. V pp. 258-9, 322. Especially interesting are Leibniz' attempt to convert Huyghens by showing that he can solve problems which Huyghens is unable to tackle ([24] Vol. II pp. 43-165) and his frank admission that the extent of the new techniques may make them difficult to understand: "Thus one must not be surprised if our new calculus of differences and sums, which includes consideration of the infinite, and thus outruns the powers of imagination, has not immediately come to perfection" ([241 Vol. V p. 307). Leibniz' recommendation is to work at applying and extending the calculus! (ibid.)




"IThese themes are intertwined in Berkeley'sAnalyst (in [2]). See especially ??1, 2, 8, 10, 32.

12It would also be consistent to deny that logical theorizing progresses, but I think that that position is extremely unattractive.

13Full discussion of this process would lead into interesting issues about conceptual change in mathematics. I believe that the general view of conceptual change in science, which I have described in [18], applies, in particular to mathematics, and that the framework presented in that paper can be used to understand the evolution of such concepts as those of function, derivative, continuity and so forth. Lakatos' discussion in [22] can also be viewed as an illustration of the ways in which the demand for rigorous definition can arise and how proposed equivalences can be confirmed. (Lakatos, of course, would not see his discussion in this way. I have tried to indicate in [21] how the Popperian overstatements of his essay can be excised.)

14Examples of such "mathematical revolutions" are the shift from the view of mathematics as consisting of arithmetic and geometry to the view of mathematics as algebra (which occurs in the eighteenth century continental tradition begun by Leibniz) and the shift from the view of analysis as arithmetic to the acceptance of set theory as fundamental.

151t is tempting to credit Leibniz with the insight that the needed new principles will emerge as the techniques are applied and extended. Anomalous results, which mark out the limits of application of a problem-solving-technique, will point the way to formulate a rigorous replacement for it. (Again, this point is similar to a central theme of Lakatos' [22].)

16Traditional accounts of the history of the calculus (such as [6], [7]) suppose that the problem of infinitesimals justifiably vexed mathematicians throughout the eighteenth and early nineteenth centuries. This interpretation faces severe difficulties which I shall not try to detail here: it is sufficient to say that there were no mathematical problems in using the method of infinitesimals until the early nineteenth century and mathematicians happily used infinitesimalist reasonings well into the 1840's. The problems with series were far more acute. For an illuminating account of the period which does much more justice to the concerns of the mathematicians involved than the traditional histories, see [14].

17This does not mean that they were always successful in solving the problems they set themselves. They weren't. But their mistakes were not silly and outrageous lapses, but subtle and fruitful mathematical errors.

18Many people have helped me in preparing this paper. Patricia Kitcher and Timothy McCarthy offered extremely valuable suggestions about an early draft, and John Corcoran's extensive written comments on that draft prompted significant changes. After the paper was read at the symposium I benefitted from the comments of several members of the audience, especially David Auerbach, Paul Benacerraf, Ivor Grattan-Guinness, Michael Resnik and Paul Ziff.



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