Proof, Logic and Formalization

Michael Detlefsen

• Publisher: Routledge
• Year: 2005
• ISBN:
  9781134975273
• Edition: 1
• Topic: Filosofia
• Subtopic: Storia e teoria della filosofia

The mathematical proof is the most important form of justification in mathematics. It is not, however, the only kind of justification for mathematical propositions. The existence of other forms, some of very significant strength, places a question mark over the prominence given to proof within mathematics. This collection of essays, by leading figures working within the philosophy of mathematics, is a response to the challenge of understanding the nature and role of the proof.

1
PROOFS ABOUT PROOFS A DEFENSE OF CLASSICAL LOGIC
Part I: the aims of classical logic

John P.Burgess

Let us remember that we are mathematicians…

Hilbert

SUMMARY

The aims and claims of classical logic are expounded, in preparation for its defense against the counterclaims of non-classical logic.

INTRODUCTION

The question, “Which is the right logic?” invites the immediate counter-question, “Right for what?” Hence a defense of classical logic must consist of two parts: (i) a preparatory exposition of the general aims of classical logic, and (ii) a defense against specific counterclaims of nonclassical logics. The present paper contains the first part of such a defense, expounding and defending a conception of the aims of classical logic.

According to this conception, the object and the method of investigation of classical logic are both mathematical proof. Given its mathematical method, it must be descriptive rather than prescriptive, and its description must be idealized. The appropriateness of its idealization is to be evaluated by the success of its application in the investigation of its object, the scope and limits of mathematics.

The conception just indicated, and expounded below, will be defended as being both historically and scientifically important: historically, because it approximates the conception of the founders of classical logic; scientifically, because it attributes to classical logic an object of investigation of considerable importance, and because its method of investigation of that object has had considerable success. Attacks on classical logic based on the attribution to it of some different aim are misconceived, according to the line of defense to be erected in the present paper.

Detailed consideration of specific logics usually conceived of as alternatives to the classical, such as partial, many-valued, relevance/relevant, intuitionistic, free, and quantum logics, will be deferred to a projected sequel.

I

Classical logic is to be conceived of as having mathematical proof as its object of investigation. Specifically, its object is classical, as opposed to constructivist, mathematics; and deduction as distinguished from axiomatics, how proofs proceed as distinguished from where proofs begin.

Traditional logic was a theory of deduction: in part, as in the Sophistic Refutations, deduction as practiced in philosophical debate; and in part, as in the Prior Analytics, deduction as practiced in mathematical proof. Classical logic was developed by Frege, Peano, Russell, Hilbert, Skolem, Gödel, Tarski, and other founders as an extension of traditional logic mainly, if not solely, about proof procedures in mathematics. This point is so clear from the contents, and indeed the very titles, of their major works that it perhaps need not be argued here.

Nor ought it to be needful to argue the scientific importance of deduction in classical mathematics as an object of investigation. It ought not to be necessary to argue the importance of rigorous deduction in mathematics, or of rigorous deductive mathematics in science. Yet in the present climate of opinion, what ought not to be necessary perhaps is. For much of the recent philosophical literature on mathematics and science, by authors otherwise as diverse as Lakatos (1976), Kitcher (1984), and Tymoczko (1987), has, to put it positively, emphasized factors other than rigorous deduction. As this trend in the recent literature has been partly a reaction against and corrective to an overemphasis on rigorous deduction in earlier literature, any defense of the continued scientific importance of the object of investigation of classical logic must begin by acknowledging that other factors are also important in the investigation of mathematics and the role it plays in science.

First, it must be acknowledged that the requirements of rigor pertain to the context of justification, publication for collective evaluation by a community of colleagues, and not to the context of discovery, private mental processes of individual researchers. No one discovers a theorem by first discovering the first step of the proof, second discovering the second step of the proof, and so on. The role of inductive, analogical, heuristic, intuitive, and even unconscious, thought in the context of discovery has been emphasized by all mathematicians discussing mathematics, and notably in the books of Hadamard (1954) and Polya (1954). Much the same point is at issue in the dicta of Poincaré, “The sole function of logic is to sanction the conquests of intuition,” and Weyl, “Logic is a hygiene we practice to keep our ideas healthy.” As regards the context of discovery, the claim, much repeated in the recent literature, that thought processes in the mathematical sciences resemble those in the empirical sciences, would be entirely acceptable, did it not tend to suggest, what has not been established, that the thought processes of scientific discovery do not resemble those of artistic creation. For it seems, rather, that science is most clearly distinguishable from art in the context of justification, or rather, in having a context of justification, and that there rigorous proof is clearly distinguishable from systematic observation or controlled experiment.

Second and third, it must be acknowledged that as one moves from pure to more and more concretely applied mathematics, and from present-day to more and more remotely past mathematics, one eventually reaches a point where rigorous proof is less and less insistently demanded. The cases of applied and of past mathematics are related, inasmuch as past mathematics tended to be closer to applications in science. As regards rigor in proofs, the standards of Euler were far lower than those of Weierstrass. And it must be acknowledged that this is not because Euler was some mere obscurantist. Applications often cannot wait for rigorous proof. Indeed, it is a commonplace that the calculus, essential to modern science, would never have been developed if Newton and Leibniz had held themselves to a standard as high as that of Weierstrass.

What perhaps needs to be emphasized in the present climate of opinion is that, inversely, Weierstrass was not some mere pedant, and that applications often must wait for rigorous proof—or rather, new applications often emerge from the attempt to supply rigorous proof to the mathematics of old applications. Indeed, it is also a commonplace that the differential geometry and functional analysis applied in relativistic and quantum physics would never have been developed if Reimann and Hilbert had held themselves to a standard no higher than that of Euler. The period of about a century during which the standards of present-day pure mathematics have been in force is a small fraction of the history of civilization, but it is a largish fraction of the history of science. It is a period during which progress in mathematics has been cumulative, by addition of new theorems, while that of empirical science has been often radically revisionary, by amendment of old theories. This is an important difference, and one that will be overlooked by any approach that attends too exclusively to factors other than rigorous proof, to the most concrete applications, or to the most remote past.

Fourth, however, it must be acknowledged that, even in the context of publication in present-day pure mathematics, theorems and proofs are not everything. Principles resisting rigorous statement, conjectures resisting rigorous proof, with inductive, analogical, or heuristic arguments for them, and increasingly often computer verifications, simulations, or explorations, can also be found. In this regard the column of Wagon (1985) and its sequels must especially be cited. Enforcement of a high standard of rigor consists less of excluding such material from the literature than of assigning it a subordinate place, and especially of distinguishing it clearly from proofs of theorems. Wagon’s column holds far more human interest than the typical theorem-proof-theorem-proof paper in a technical journal. So, too, do the studies, so common in the recent literature, emphasizing discovery, or applications, or the past. But scientifically, investigation of such factors belongs inevitably within the domain of the so-called human or soft studies, disciplines like psychology, sociology, history. To attend too exclusively to factors other than rigorous proof is to overlook the one factor that is susceptible to being investigated mathematically.

II

Classical logic is to be conceived of as having mathematical proof as its method of investigation. Its founders were all by education, and all but Russell by profession, mathematicians, and they never forgot it. This is so clear from their major works that it perhaps need not be argued here.

What do need clarification are the consequences of a conception of classical logic as a mathematical logic, a discipline similar in method to mathematical physics or mathematical economics (differing from them only in that its object is the very process of mathematical proof of which it and they are instances, giving it a self-referential character). One consequence is that, like mathematical physics, it deals with an idealization of reality. Its artificial languages are conspicuously simpler in grammar than natural languages, for example. No more than mathematical physics is mathematical logic to be condemned just for being unrealistic or idealized. Rather, like mathematical physics, it is to be evaluated by the success of its applications.

Another consequence is that, like mathematical economics, it provides descriptions rather than prescriptions, although, again like mathematical economics, its results can serve as minor premises in arguments with prescriptive major premises leading to prescriptive conclusions. Since the distinction between descriptive and prescriptive as applied to logic may be unfamiliar, and since the claim is often repeated in the recent literature that logic is a normative, not a factual, discipline, some elaboration on this point is perhaps desirable.

Whenever a community has a practice, the project of developing a theory of it suggests itself. When the practice is one of evaluation, a distinction must be made between descriptive and prescriptive theories thereof. The former aims to describe explicitly what the community’s implicit standards have been: the theory is itself evaluated by how well it agrees with the facts of the community’s practice. The latter presumes to prescribe what the community’s standards ought to be: the community’s practice is evaluated by how well it agrees with the norms of the theory. Logic, according to almost any conception, is a theory dealing with standards of evaluation of deduction, much as linguistics deals with standards of evaluation of utterances. The distinction between descriptive and prescriptive is familiar in the case of linguistics: no one could confuse Chomsky with Fowler. It is not less important in the case of logic.

The familiar case of linguistics can help clarify a point about intuition important for logic. The data for descriptive theorizing consist of evaluations of members of the community whose evaluative practices are under investigation (e.g. “That’s not good English”). These evaluations are of particular examples, not general rules. For even if it is supposed that members of the community, in making their evaluations, are implicitly following rules (e.g. even if it is supposed that grammatical rules are “psychologically real”), it cannot be supposed that they are capable of bringing these rules to explicit consciousness (e.g. how many lay native speakers of English can correctly state the phonetic rules for forming plurals?). Indeed, they are seldom acquainted with the technical jargon (e.g. “gerund”) in which such rules are stated. Moreover, even the evaluations of particular examples are usually not stated in technical jargon: a negative evaluation will be primarily one of overall infelicity (e.g. “That isn’t said”), and only secondarily if at all an indication of the dimension of infelicity involved (e.g. “ungrammaticality” as understood by professional linguists, as distinguished from overcomplexity, too obvious untruth or truth, etc.). While there is the least danger of bias when the data consist of spontaneous evaluations of spontaneous examples, in order to gather enough data elicited evaluations of contrived examples must be used. Indeed, theorists who are themselves members of the community (e.g. native English speakers investigating the grammar of English) often use their own impressions of the felicity or infelicity of particular examples as their main source of data. Such impressions are intuitions in an everyday sense, impressions of whose source and grounds one is unconscious. Intuitions in this sense are notably fallible and corrigible, especially in the case of a theorist out to establish a pet theory. If they conflict with the intuitions of other theorists, and especially if they conflict with the evaluations of unbiased lay members of the community, they must be retracted or at least restricted (e.g. to a dialect, rather than the whole language). To insist on them even given conflicting data (e.g. “People talk that way all the time, but it’s wrong”) would be to engage in prescriptive theorizing.

To apply this point to logic: descriptive logic is a branch of what has been called naturalized epistemology, epistemology so conceived that the epistemologist becomes a citizen of the scientific community, appealing to the results of the (other) sciences, and perhaps his or her own “intuitions” as a member of the community, in attempting to explain its practices from within; prescriptive logic is a branch of what may be called alienated epistemology, epistemology so conceived that the epistemologist remains a foreigner to the scientific community, evaluating its practices from without, attempting to provide a foundation for them or a critique of them, appealing perhaps to “intuitions” conceived of as fundamental critical insights from some extra-, supra-, or preter-scientific source of wisdom.

III

A conception of classical logic as a descriptive logic may be implicit in the conception of classical logic as a mathematical logic, as a branch of mathematics and hence of science; but no clear, detailed, explicit account of the distinction between descriptive and prescriptive logic, such as has just been attempted, is present in the major works of the founders. Nonetheless, in defense of the historical importance of such a conception, it can be argued that the work of the earlier founders is at least not explicitly prescriptive, and that that of the later founders is implicitly descriptive.

Such a claim may seem most doubtful in the case of Frege, whose work may seem so much a part of the process of increasing the standards of rigor in mathematics from Eulerian to Weierstrassian levels that was under way in his day. Yet even in his case half a dozen points can be cited in opposition to any interpretation according to which he was mainly concerned to prescribe to mathematicians a new notion of what constitutes full rigor.

First, the process of increasing the standards of rigor was itself less a matter of a new notion of what constitutes full rigor than of a new policy about how much rigor should be demanded. Mathematicians in Euler’s day were less unaware of the fact that they were not demanding of themselves the highest standard of rigor, than unworried about it. Many quotations in the pertinent chapters of the standard history by Kline (1972), such as the following from d’Alembert (p. 619), illustrate this point: “More concern has been given to enlarging the building than to illuminating its entrance, to raising it higher than to giving proper strength to the foundations.”

Second, in the process of increasing the standards of rigor, logicians, few in number and marginal in status, had little influence; and Frege seems to have been painfully aware of this fact. Hilbert is here the exception who proves the rule, since his great influence derived entirely from his great eminence in branches of mathematics other than logic.

Third, Frege himself, in the opening sections of the Grundlagen (1953), cites increasing the standards of rigor not as something that ought to be done but rather as something that is being done. He attempts to motivate his own project by appealing to an already accepted adage to the effect that in mathematics nothing capable of proof should be accepted without it.

Fourth, his own project was less concerned with securing the rigor of the steps that lead from one theorem to the next than with pushing back the starting point for theorem proving. Work of Dedekind and others was already achieving a reducti...

Table of contents

• COVER PAGE
• TITLE PAGE
• COPYRIGHT PAGE
• NOTES ON CONTRIBUTORS
• PREFACE
• 1. PROOFS ABOUT PROOFS A DEFENSE OF CLASSICAL LOGIC: PART I: THE AIMS OF CLASSICAL LOGIC
• 2. PROOFS AND EPISTEMIC STRUCTURE
• 3. WHAT IS A PROOF?
• 4. HOW TO SAY THINGS WITH FORMALISMS
• 5. SOME CONSIDERATIONS ON ARITHMETICAL TRUTH AND THE -RULE
• 6. THE IMPREDICATIVITY OF INDUCTION
• 7. THREE INSUFFICIENTLY ATTENDED TO ASPECTS OF MOST MATHEMATICAL PROOFS: PHENOMENOLOGICAL STUDIES
• 8. ON AN ALLEGED REFUTATION OF HILBERT’S PROGRAM USING GÖDEL’S FIRST INCOMPLETENESS THEOREM