Proof and Knowledge in Mathematics

Michael Detlefsen

• Publisher: Routledge
• Year: 2005
• ISBN:
  9781134916757
• Edition: 1
• Topic: Philosophy
• Subtopic: Philosophy History & Theory

These questions arise from any attempt to discover an epistemology for mathematics. This collection of essays considers various questions concerning the nature of justification in mathematics and possible sources of that justification. Among these are the question of whether mathematical justification is a priori or a posteriori in character, whether logical and mathematical differ, and if formalization plays a significant role in mathematical justification,

1
PROOF AS A SOURCE OF TRUTH

Michael D.Resnik

SUMMARY

For mathematical realists an especially pressing question is that of how a proof, for example of Euclid’s theorem, can establish a conclusion about mathematical objects, for example that there are infinitely many prime numbers. In trying to give a realist answer to this question, I interpret it as the question of how a proof of, say, Euclid’s theorem could (i) induce us to believe that there are infinitely many prime numbers and (ii) give us good reasons for so believing. My answer to the last question is that our proof of Euclid’s theorem induces us to believe that there are infinitely many primes, because we have been prepared through our mathematical education to understand its component statements and to follow its reasoning. We have also been prepared, through learning an appropriate background theory, to see the proof as presenting, arranging and transforming information about numbers, and hence have also been prepared to draw from it the conclusion that there are infinitely many prime numbers. That takes care of the belief-inducing part of my answer. The good reasons part is trickier. We want to know why the proof of Euclid’s theorem gives us good reasons for believing that there are infinitely many primes. Now viewed from our own standards of reasoning it is clear that Euclid’s proof gives us good reasons for believing its conclusions. We need only check the proof to see that it conforms to our standards of rigor. But for our present purposes this is not enough. Everything is based upon accepting an “appropriate background theory.” We need assurance that such a theory is not just an elaborate mythology passed from generation to generation by the priesthood of mathematics. Towards this end I turn to speculative history about the development of ancient mathematics, hoping thereby to give a rough idea of the sort of “background theory” that connects proofs with mathematical objects and through comparing its genesis with developments in science to dispel the mythological clouds that some might see clinging to it.

I.
INTRODUCTION

Taken literally and seriously, mathematics affirms truths about numbers, functions, sets, spaces and other entities, which are as real as rocks and yet inhabit neither space-time nor our minds. There are good reasons, which I shall not consider here, for philosophers of mathematics to take mathematics seriously and literally.¹ Opposing those reasons is the ensuing mystery such realism makes of mathematical knowledge. If numbers and their cousins are outside of space-time, then they cannot transmit information to our sensory detectors. If they are also neither individual nor collective mental constructions, then we are not free to imagine or stipulate or otherwise “dream up” their properties. How then can we acquire knowledge about them? That question drives mathematical realists to despair and makes reluctant nominalists of many sensible people.

A special case of this question is the truth/proof problem. It may be formulated as follows: when mathematicians prove mathematical statements they construct diagrams, write formulas, and produce arguments. On the face of it, none of these spatiotemporal or mental transactions could provide them with information about the abstract world of mathematical objects. Thus how can proving a mathematical statement show that it is true? For that matter, how can anything mathematicians do produce information about mathematical objects?²

To assess the magnitude of this problem compare proving in mathematics with observing in, say, biology. To find out whether a certain species of fish carries its eggs internally until its young hatch and swim free, biologists might capture members of the species, provide them with a suitable breeding ground and observe whether they bear their young live. Although our biologists might encounter serious problems arranging for this experiment and the results might be inconclusive, there is no serious question that observing the fish provides information about them. One reason why there is no serious question about this is that physiologists and psychologists have developed (rudimentary) theories of how information about events in fish tanks is transmitted to our brains. On the other hand, our methods for acquiring mathematical knowledge seem to contrast sharply with perception, and we do not have even rudimentary scientific theories concerning the mechanisms whereby we learn about the mathematical realm.

This essay attempts to solve the truth/proof problem. Section II formulates a more tractable version of the problem. Sections III and IV are concerned with proofs, how they work and what they achieve. Section V begins with a brief and incomplete summary of my view of mathematical objects as positions in patterns. This opens the way for a historically grounded (but still hypothetical) account of how the study of numerical patterns may have linked the subject-matter of mathematics with proofs.

II.
REFORMULATING THE TRUTH/PROOF PROBLEM

Formalists and intuitionists do not have a truth/proof problem. For them a mathematical statement is true just in case it is provable, and proofs are syntactic or mental constructions of our own making. Their example suggests that we might solve the problem by closing the gap between ourselves and mathematical reality, but it also shows the danger of sacrificing mathematical realism in the process. Thus my first task will be to formulate a version of mathematical realism which still produces a genuine truth/proof problem.

The version I have in mind is immanent realism about mathematical objects.³ With other realist views, it holds that mathematical objects are abstract entities existing independently of us and our constructions and theories. It also holds that (most of) the claims of contemporary mathematics are true, and that they are true independently of our holding them to be true. Thus immanent realism about mathematical objects is incompatible with formalism, intuitionism, deductivism, and the other familiar antirealist views in the philosophy of mathematics. Despite its title, immanent realism affirms that mathematical reality transcends our own existence, beliefs, and experience. Immanent realism derives its title from its immanent conception of truth. It uses a conception of truth that applies only to sentences within its own language, whereas transcendent versions of realism employ conceptions of truth that transcend their home language through applying to sentences in a variety of languages.

More precisely, the field of application of the immanent realists’ truth-predicate is restricted to their (our) own language. They take truth and reference as merely disquotational: Using a classical metalanguage, they accept instances of the usual Tarski disquotational schemata,

“p” is true if and only if p,

“N” denotes x if and only if x=N,

x satisfies “P” if and only if Px,

and the laws relating the denotation, satisfaction, and truth-conditions of compound expressions to those of their components. Immanent realists rest content with a Tarskian definition of truth based upon list-like specifications for the references of primitive terms.⁴ These specify references for primitive terms of the object language by using the same terms within the metalanguage. For example, in specifying the references for the primitive terms of first-order number theory immanent realists might use the following definition:

t refers 1 to y if and only if t=“0” & y=0;

t refers 2 to ‹x, y› if and only if t=“S” and Sxy;

t refers 3 to ‹x, y, z› if and only if t=“+” and x+y=z or t=“X” and xXy=z.

More importantly, unlike their transcendent kin, immanent realists seek no additional word-world theory, such as the causal theory of reference, stated in terms applicable to all languages and by means of which one might determine the reference of a foreigner’s term or even that of an unknown term of one’s own language.

Within the immanent realists’ framework one can formulate the familiar doctrines that have traditionally divided mathematical realists from antirealists. For example, in arguing about whether there are unprovable mathematical truths, realists and antirealists can find material enough in our own language without needing to turn to sentences in arbitrary languages. On the other hand, for-going a universal theory of truth forces immanent realists to abandon some beliefs often identified with realism. These include, for instance, the belief that mathematicians, no matter what language they speak, aim to construct true theories, and the belief that if there is a Martian mathematics then it affirms many of the truths that we do. Despite this, immanent realism with respect to mathematical objects is still saddled with all the traditional objections to mathematical objects, for it still maintains that they exist outside of space-time and are independent of us and our theorizing. In particular, immanent realism still faces the truth/ proof problem.

For both immanent and transcendent realists this problem comes to the worry of how proving something, say Euclid’s prime number theorem, establishes that it is true. But to the immanent realist, to establish that Euclid’s prime number theorem is true is just to establish that there are infinitely many primes. More generally, to establish that p is true is just to establish p. So for the immanent realist, the truth/proof problem amounts to the problem of explaining why p is provable only if p.

But this generates a new problem—the problem of giving the truth/proof problem a nontrivial and philosophical interpretation. One could read the truth/proof problem as the mathematical task of proving the soundness of our rules of proof. But this cannot be the correct way to understand the problem, because even formalists could recognize such a soundness proof as a bona fide mathematical result and construe it in their own terms. On the other hand, turning from mathematical technicalities could lead us to philosophical trivialities. It is tempting to say, “Well, it’s just part of the meaning of proof that one cannot prove p unless p.”

The way out of this dilemma is to remember where we began. Our original problem was to show how activities involving reasoning, paper and pencil calculation, drawing diagrams, and the like could establish anything about the mathematical realm. Immanent realists still face this problem. But they do not have to deal with an additional problem of explaining how some nontrivial, truth-making relation manages to obtain between provable mathematical sentences and mathematical reality.

Proving has an ontic dimension for most constructivists; to them proving p establishes its truth in the sense of making p true. But for us it does not; proving p establishes, shows, or demonstrates p only in the epistemic sense of providing us with good reasons for believing p. This suggests that we interpret the truth/proof problems as this pair of questions:

1. In view of the gap between us and mathematical reality why does proving p induce us to believe p?
2. Why are the reasons a proof provides good reasons?

For many of us a glib answer to both questions may apply. This is the response that in our mathematics courses we are trained to accept proofs as giving good reasons, and we have been conditioned to believe things which we think we have good reasons to believe. Some, influenced perhaps by Wittgenstein, would interject that the glib answer applies to all of us—practicing mathematicians and uninspired mathematics students alike. Mathematics, on their view, is not a science, and there are no mathematical facts. There is nothing but a certain social practice in which proving plays a major role. Thus, they would continue, the question we should ask is: how did our practice of proving mathematical statements evolve?

That is a good question too, and answering it will take us a large way towards answering the two questions I posed above. However, from my realist perspective, no answer to those two questions can be fully satisfactory unless ultimately we find some connection between proving and mathematical objects. But first let us take a closer look at proving.

III.
HOW PROOFS ARE USED AND WHERE THE TRUTH/PROOF PROBLEM ARISES

Consider some of the ways in which mathematicians use proofs. They are used, of course, for demonstrating new results, but also for giving alternative demonstrations of previous results. Moreover, under that heading we can distinguish, on the one hand, proofs which show that a previous result can be given a weaker or more economical demonstration (as in replacing a nonconstructive proof by a constru...

Table of contents

• COVER PAGE
• TITLE PAGE
• COPYRIGHT PAGE
• NOTES ON CONTRIBUTORS
• PREFACE
• 1. PROOF AS A SOURCE OF TRUTH
• 2. REFLECTIONS ON THE CONCEPT OF A PRIORI TRUTH AND ITS CORRUPTION BY KANT
• 3. LOGICISM
• 4. EMPIRICAL INQUIRY AND PROOF
• 5. ON THE CONCEPT OF PROOF IN ELEMENTARY GEOMETRY
• 6. MATHEMATICAL RIGOR IN PHYSICS
• 7. FOUNDATIONALISM AND FOUNDATIONS OF MATHEMATICS
• 8. BROUWERIAN INTUITIONISM