What do we do when we do mathematics?

What Do We Do When We Do Mathematics?

Ernst Snapper

Introduction. First, I mean by "doing mathematics" the usual research activity of the practicing mathematician. Hence philosophizing about mathematics, as is being done in this paper, is not "doing mathematics." On the other hand, much of what is nowadays called "foundations of mathematics," such as mathematical logic and set theory, is not philosophizing about mathematics but is doing mathematics. The title of this paper should be read in this sense.

Second, I am not trying to say what "doing mathematics" is in such a way that a working mathematician will say, "Yes, I recognize that this is what I am doing." On the contrary, mathematicians who are not interested in philosophy will probably not recognize that I am describing their activity. This is in the nature of philosophy. When a philosopher says to a bum on skid row, "Look, Joe, when you are sober and reason accurately, you may be using several of the Aristote- lian principles of logic," the philosopher is right, but the bum will probably not know what the philosopher is talking about.

When it comes to mathematics, I am a conceptualist, meaning that I believe that mathematical objects are creations of the human mind. When we study elemen- tary number theory, we study the natural numbers. What is, say, the number 3? The meadows are full of triples, such as triples of cows or daisies, but no one will ever f ind the number 3 s u n n i n g itself in a meadow. Numbers do not exist in space, and I agree with those who say that they are to be found in our minds. This, however, implies that my 3 is not your 3, since my 3 is in my mind, while your 3 is in your mind. How then are different mathematicians able to communicate? They are able to because the sequences of the natural numbers that occur in the minds of different mathematicians are isomorphic, and isomor-

phism, as opposed to identity, is all that is necessary for communication.

The same holds for all mathematical constructs. When we study the three-dimensional space that sur- rounds us, we replace the straight railroad tracks of the Grand Central Railroad by the straight lines in our minds, and, again, the mental 3-spaces of different mathematicians are isomorphic. In short, mathematicians create mathematical structures in their minds and study those properties of these structures about which they can communicate-- that is, those properties that remain invariant under isomorphisms. Here

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is how Poincar6 put it in [9]: "Mathematicians do not s tudy objects, but the relations be tween objects; to them it is a matter of indifference if these objects are replaced by others, provided that the relations do not change. Matter does not engage their attention, they are interested in form alone."

Hence "doing mathematics" consists of the mental c rea t ion of ma thema t i ca l s t ruc tu res and proving theorems about them. Proving theorems is, of course, a telltale of mathematical activity; if we don' t prove things, we are certainly not doing mathematics.

We can right away be somewhat more precise about "mathematical structures." No matter how complicated such a structure may be, it can always be considered as a set. Say the mathematical structure in question is a group G, meaning that we are given the set G together with a function f: G x G ~ G having the usual properties. This function f, however, is nothing but a subset of the cartesian product (G x G) x G, and the set {G,f}, whose two members are the sets G and f is the group G, considered as a set. In this way every mathematical structure, even the most complicated one, can always be considered as a set. Hence it seems that "doing mathematics" consists of the mental creation of sets and proving theorems about them.

The situation is, however, slightly more complicated. Namely, we also often study general properties of all structures of some fixed kind. Suppose we are studying topology and state that the continuous image of a connec ted space is connected. We are then making a remark about all connected topological spaces, and this collection of spaces is not a set, but is what set theorists call a "proper class." These proper classes are collections that have no cardinal numbers; they are too large for that. I will use the customary terminology of set theory and say "class" to denote either sets or proper classes. A set is a class that has a cardinal number; a proper class is a class that is too vast to have a cardinal number. This gives an efficient criterion by which a non-set-theorist can determine quickly whether a given class is a set or a proper class; a related criterion will be mentioned in Section 4. Set theorists, of course, know that sets can also be defined as those classes that are members of other classes, while proper classes cannot be members of classes.

"Doing mathematics" consists of the mental creation of classes and proving theorems about them. It follows that the two fundamental questions in the philosophy of mathemat ics are " H o w do we create classes?" and " H o w do we prove theorems?" Each question has several different answers, and my main contention is:

Thesis. Any sound method to create classes may be com- bined with any sound method to prove theorems, and such a combination is a mode of mathematics.

I do not believe that mathematics is a unique doc-

trine, but that there are several kinds of mathematics, depending on how one chooses to create classes and chooses to prove theorems. At present, several different methods to create classes are known, namely the one practiced in classical mathematics (Section 4) and those practiced in intuitionism and related construc- tivist programs (Section 5). With regard to constructi- vist programs, I will restrict the discussion to intuitionism. We also possess several different procedures to prove theorems, the reason being that philosophy gives us various theories of truth. Each theory of truth gives rise to its own proof method (Section 1), and one's selection of a proof method is determined by the theory of truth one adopts. The proof method of classical mathematics is based on the correspondence theory of truth (Sections 2, 4), while the proof method of intuitionism is based on the pragmatic theory of truth (Sections 3, 5).

We should become accustomed to the fact that there are many modes of mathematics, probably many more than are known at present. We certainly don't have the slightest reason to believe that all methods to create classes and prove theorems have been discov- ered by now. Furthermore, the known methods could be paired in different ways and thereby give rise to new kinds of mathematics.

Section 1. H o w Do We Prove Theorems?

Proof and truth are related notions. After all, we prove that something is true. There are several theories of truth in philosophy, and each carries with it its own indiv idual logic. This logic de te rmines the proof method associated with the given truth theory: Truth determines proof and not vice versa. The practice of the logic of a given truth theory--equivalently, the practice of the proof method associated with that truth theory- - i s the proof method the mathematician uses. Clearly, then, there is no unique proof method for mathematics, but this method depends on the choice of truth theory. Once this theory has been chosen, the mathematician is committed to prove theorems ac- cording to the logic of that theory. For example, classical mathematics uses the correspondence theory of truth (Section 2), while intuitionism uses the pragmatic theory of truth (Section 3).

If the logic of the chosen truth theory has been given mathematical treatment, meaning that it has been expressed as a formal logic in a formal language, proving in that truth theory can be formulated either in that formal language or in the natural language of the mathematician. If the logic in question has not been mathemat ized, proving in the adopted truth theory can, of course, be formulated only in a natural language. So far, only the logic of the correspondence theory and the pragmatic theory have been mathema- tized, and I will restrict myself to these two theories.

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However, every viable truth theory gives rise to a method of proof that is there for mathematicians to use.

Section 2. The Correspondence Theory of Truth

In this theory, the objects that are true or false are the declarative sentences of the language used. Suppose this language is English, whence an example of a declarative sentence is, "Snow is white." Every declarative sentence refers to a fact, the present sentence to the fact that snow is white. The fact to which a declarative sentence refers can be something in the real world outside of us, as in "Snow is white," or it can be something mental inside of us, as in "'Seven plus five is twelve." We should realize that it does not make sense to ask whether a fact is true or false. Facts can occur, can be hoped for or feared, but truth and falsity do not apply to them. In the remainder of this paper, "sentence" always means "'declarative sentence."

Whether a sentence is true or false depends on how it "corresponds" to the fact to which it refers. The sentence "'Snow is white" is true because snow is indeed white. The sentence "Seven plus five is twelve" is true because seven plus five is indeed twelve. In general, a sentence S is true if it corresponds in such a way to the fact F to which it refers that we can say that it describes F correctly, in short, that S concurs with F. If S does not concur with F, we say that S is false, equivalently, that 7 S is true (-1 stands for "not").

The above philosophical definition of truth cannot satisfy a mathematician who is not trained in philosophy. Mathematicians want their definitions to be clear and unambiguous, not vague and uncertain. The main reason for this dissatisfaction is that "philosophical definition" is actually a misnomer, as Russell has pointed out in [10]. One should say, instead, "philosophical analysis" because, in contrast to a mathematical definition, a philosophical definition is simply an analysis of an idea into its constituents. The above definition analyzes the idea of truth into its constituents of sentence, fact, and correspondence. I will not say "philosophical definition" again.

Let S be a sentence and F the fact to which it refers. If we have, somehow, settled conclusively that S concurs with F (i.e., that S is true), I will say that S has been verified. The correspondence theory does not re- quire that a sentence or its negation has been verified before one is allowed to say that the sentence must be either true or false. On the contrary, every sentence is considered to be either true or false, but not both. This holds even for sentences whose truth value we shall never know, such as, "There was human life before the big bang." This shows that the correspondence theory bestows a platonic existence on the notion of truth. I will express the fact that in the correspondence theory every sentence is either true or false, even if no verification has been or can be carried out, by saying

that the correspondence theory of truth is verification- free.

I now turn to the logic of the correspondence theory. By restricting the language to a first-order predicate language L, Tarski was able to give a mathematical definition, as opposed to a philosophical analysis, of truth in the metalanguage of L [11]. The re- sulting mathematical theory of truth is the basis for the customary first-order logic with its customary formal proofs which we all teach in our courses in mathematical logic. If one adopts the correspondence theory of truth, one is thereby committed to use classical logic when proving theorems. See [5, Vols. 1, 2] for the correspondence theory and its mathematization.

Section 3. The Pragmatic Theory of Truth

The pragmatic theory of truth, by William James, Charles Peirce, John Dewey, and other pragmatists, bloomed in the first quarter of this century [5, Vols. 5,6]. The expositions of this theory seem to vary with the pragmatist who represents them, and I will stick with Dewey's exposition.

The objects that are true or false are again the (declarative) sentences of the language used, say English. A sentence S again refers to a fact F, and to verify S means again to settle conclusively that S concurs with F. The great difference wi th the correspondence theory is that truth is not given any kind of platonic existence, and hence we have no reason to say that every sentence is either true or false. On the contrary, a sentence S is said to have no truth value whatsoever until such time as an investigation has been completed that

In contrast to a mathematical definition, a philosophical definition is s imply an analysis of an idea into its constituents.

verified either S or ~ S. Hence the pragmatic theory of truth is just the opposite of verification-free; pragmatic truth is verification. We should look at an example.

A slight variation of one of Dewey's own examples goes as follows [3]. A man is lost in the woods and says to himself, "If I walk due north, I will find my way out of these confounded woods." Is this sentence true or false? When spoken in the middle of the woods, the sentence is without truth value because it has not as yet been investigated. If the man does walk north and finally steps out of the woods, then, at that moment, the sentence has become true. If, after walking many hours north, the man concludes that he is get- ting deeper and deeper into the woods, then, when he reaches that conclusion, the sentence has become false.

The question, of course, is what Dewey meant pre- cisely by an "investigation." Here we are again in a philosophical situation where no sharp, mathematical definition can be given; there are too many different

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types of sentences for that. In all cases, however, an investigation must satisfy the following two conditions in order to be acceptable by pragmatists. In the first place, the investigation must be as concrete and practical as the sentence in question allows. For example, in the case of the man lost in the woods, a per- missible investigation necessarily consists of the man walking north. Second, it must always be possible to complete the investigation in a finite length of time. If one specializes the sentences to those that occur in mathematics, the requirements for an investigation to be acceptable by pragmatists become the conditions intuitionists impose on proofs in order that they be acceptable by them. Such proofs are of course called "constructive," and I will borrow this term and say, in general, that an investigation acceptable to a pragmatist is a constructive investigation. In the same vein, I will call a verification of a sentence based on a constructive investigation a constructive verification. We can now sharpen the assertion that pragmatic truth is verification to: pragmatic truth is constructive verification.

What about the logic of the pragmatic theory of truth? Clearly, classical logic won't do. The sentence "There are infinitely many twin primes" is currently without truth value, and there goes the law of the excluded middle. What about negation?

Consider again the man lost in the woods and denote the sentence he spoke to himself by S. Suppose that after walking north for many hours and seeing the woods around him thicken all the time, he concludes that 7S is true. Then, the sole ground for his conclusion is that he has completed an investigation which shows that S is contradictory. The negation of the pragmatic theory of truth is the intuitionistic negation.

If one investigates along these lines the interpreta- tion of the connectives and quantifiers in pragmatism, one comes to the conclusion that the logic of the pragmatic theory of truth is intuitionistic logic. Here I mean by

The sentence "'There are infinitely many twin primes" is currently without truth value, and there goes the law of the excluded middle.

"intuitionistic logic" not any formal logic, but simply the modes of reasoning accepted by intuitionists. When Heyting formalized these modes of reasoning [8, Chap. VIII, he formalized the logic of pragmatism.

A complete justification that intuitionistic logic is the logic of the pragmatic theory of truth would take an- other paper, but the basic reason is that intuitionistic truth is nothing but pragmatic truth restricted to those sentences that occur in intuitionism. Both kinds of truth are identified with constructive verification.

The question arises, of course, whether it is correct to identify truth and verification, as the pragmatists and intuitionists do. Carnap has warned against con-

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fusing truth and verification [2]. Truth is atemporal, while verification is temporal in the sense that a sentence ma y not be verified today but verified to- morrow. The logic of a theory of verification depends on how strong a verification is being used. If there is no verification at all, as in the case of the correspondence theory of truth, classical logic emerges. The stronger the verification, the more the logic shifts from classical logic to intuitionistic logic. The strongest possible verification is constructive verification, and its logic is intuitionistic logic. Since I agree with Carnap that truth and verification should not be confused, I will not speak of pragmatic truth or intuitionistic truth again, but only of constructive verification.

Section 4. Classical Mathematics

A mode of mathematics is determined by stipulating how classes should be created and theorems should be proved (Thesis). When we search for the philosophical basis of a mode of mathemat ics , we should ask, "Which philosophy underlies its creation of classes?" and "Which philosophy underlies its proof method?"

In classical mathematics, one uses classical logic to prove theorems, whether the proving is being done in a natural language or in a formal language. Hence the philosophical basis for its proof method is the correspondence theory of truth (Section 2). The situation with regard to class creation is, however, not nearly as satisfactory.

No one has been able to discover the philosophy on which the class creation of classical mathematics is based. This is why one has to admit that, from the point of view of philosophy, classical mathematics is not in good shape. In this sad situation, let us mo- mentarily forget about philosophy and simply review the familiar ways classical mathematicians create classes. The quest ion is not how the formation of classes can be axiomatized, but how the working mathematician forms them. There are two ways.

I. Formation of classes in terms of a given class. Such a class formation is always relative, in the sense that one is given a class and forms a new class in terms of the given one. Examples are the formation of the set of the rational numbers in terms of the set of the integers, of the set of the real numbers in terms of the set of the rational numbers, etc.

II. Formation of classes in terms of a given property. Whenever we are given a property of things, we may always form the class of all things that have that property. As everyone knows, reckless handling of this class causes the set theoretic paradoxes, such as those of Russell and Burali-Forti. "Reckless" does not mean that the class may not be formed, as many mathematicians erroneously believe. "Reckless" means that the created class may be a proper class and hence should not be treated as a set. Let us look at this more closely.

Properties are of two kinds. A property of the first kind is such that, if X is a set all of whose members have that property, there is always an object that has that property but does not belong to X. Examples of such properties are: being a nonempty set of fixed car- dinality; being a set that is not a member of itself; being an ordinal; being a connected topological space.

If a property is of the first kind, the sets all of whose members have that property are always growing, and their growth never stops. Clearly, there cannot be a set that consists of all objects having that property. These objects form a proper class, and there is no logical ob- jection whatsoever against using this proper class. Proper classes have no cardinal number and cannot be members of classes. However, they can serve as do- mains of functions, and many set theoretic operations apply to them. If one treats a proper class as a set, however, one is in danger of committing what Russell called "vicious circularity," and a set theoretic paradox results [7]. The set theoretic paradoxes are not caused by forming proper classes, but by treating a proper class as a set.

This t reatment of the set theoretic paradoxes is somewhat controversial. Russell viewed the formation of the proper class itself as committing a vicious circularity and hence as being logically objectionable. I simply do not agree with him.

A property is of the second kind if it is not of the first kind, i.e., if the objects that have the property form a set. This splitting up of properties into those of first and second kind is convenient for the working mathematician who is not a set theorist. Namely, there is never any doubt to which kind a given property belongs and hence whether the objects that have the property form a proper class or a set. This criterion for sethood is not needed by set theorists, because they determine sethood by the membership relation.

S e c t i o n 5. I n t u i t i o n i s m

One cannot doubt that, from the point of view of philosophy, intuitionism is in much better shape than classical mathematics. The reason is that intuitionists give good answers to both questions "Which philosophy under l ies the intuit ionistic way of creating classes?" and "Which philosophy underlies the intuitionistic way of proving theorems?" Let us review both answers.

An intuitionistic proof is a constructive verification, and hence the ph i lo sophy under ly ing the proof method of intuitionism is the theory of constructive verification with its intuitionistic logic (Section 3).

With regard to the creation of classes, intuitionists start with the creation of the natural numbers 1, 2, 3, �9 . . (0 is omitted from these numbers). They accept the Kantian thesis that we become aware of these numbers through our intuition of time and temporal succession. A careful study of this thesis along intu-

itionistic lines gives insight in the mental processes that give us the natural numbers.

The thesis that is the foundation for the intuitionistic way of creating classes says that doing mathematics consists entirely of carrying out, one after the other, only those mental processes that give us the natural numbers [6, Chap. IV]. In a creation of a class, only finitely many of these mental processes may be

From the p o i n t o f v i e w o f ph i losophy , in tu- i t ion i sm is in much bet ter shape than clas- s ical mathemat ics .

used, which causes these class creations to be constructive. A further analysis of this thesis gives rise again to two types of class formations.

I. Spreads. This class formation is the analog of the "formation of classes in terms of a given class" of classical mathematics. A spread is defined in terms of two given data, the spread law and the complementary law [8, Chap. III].

II. Species. This class formation is the analog of the "formation of classes in terms of a given property" of classical mathematics. In fact, a species is a property which a mathematical object can be supposed to possess [8, Chap. III]. In order for a mathematical object to be a member of a given species, the object must satisfy two conditions: (1) It must possess the property in question, and (2) it must be definable independently of the definition of the species.

This second condition shows that no species can be a member of itself and, in fact, removes all traces of circularity, vicious or nonvicious, from the intuitionistic membership relation. It thereby protects intuitionism from the set theoretic paradoxes and makes distinguishing between species of the first and second kind unnecessary.

Differences of opinion can arise, however, as to whether a given class construction really consists of only those mental processes that give us the natural numbers. This has led to constructive programs, related to but divergent from intuitionism. For example, choice sequences are accepted by intuitionists, but n o t

in the constructive program of Errett Bishop.

S e c t i o n 6. E p i l o g u e

In this paper, two questions have been considered: "What is the philosophical foundation for the way we create classes?" and "What is the philosophical foundation for the way we prove theorems?" I have based the answer to the second question on the philosophies of truth and verification. The first question has a good answer in in tui t ionism--namely, the intuitionistic way to create classes is based on the Kantian philosophy of temporal succession. I have not been able to

answer this first question in the case of classical math-

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ematics. The discovery of the philosophy underlying the classical way to create classes constitutes a major research problem for the philosophy of mathematics. As soon as this problem is solved, we may say that mathematics is, not only technically but also philo- sophically, in good shape. But not until then.

Intuitionists will, of course, not agree that there are several sound methods to create classes and prove theorems. They recognize only one method to create classes and one method to prove theorems, namely their own. This, however, reduces much of classical mathematics to meaningless combinations of words and, by Descartes's cogito, this implies that I do not exist. I simply cannot accept that. On the other hand, I also disagree with classical mathematicians who claim that intuit ionism is not mathematics. Intuit ionism combines a perfectly valid method of class formation with a perfectly valid proof method and hence is a perfectly valid mode of mathematics.

Dummett, in his penetrating paper [4], also dis- cusses the question "'How do we prove theorems?" He asks whether we are compelled to use intuitionistic logic rather than classical logic for this purpose and bases his answer on the theory of meaning. Since the theory of meaning is dependent on the theories of truth and verification and I used only these latter two theories and not the first, my discussion clearly ran on a lower philosophical level than Dummett's. I was able to run on this lower level because I never asked the question whether one of the two logics under discussion should be favored over the other. I don' t believe so. If one studies truth, one must use classical logic; if

one studies constructive verification, one must use intuitionistic logic; and in mathematics, as in all other intellectual pursuits, one is free to choose whether to pursue truth or verification.

In fact, Dummett 's conclusions do not refute that one has this free choice in mathematics. On the one hand, he argues that a conceptualist attitude toward the natural numbers, such as I have taken in this paper, does not in itself force one to favor intuitionistic logic over classical logic unless one is hardheaded enough to "deny that there exists any proposition which is now true about what the result of a computa- tion which has not yet been performed would be if it were to be performed." On the other hand, he also gives a powerful argument whose purpose is to show that, independent of whether one is a conceptualist or a platonist with regard to mathematical objects, one must favor intuitionistic logic over classical logic. This argument leans heavily on Wittgenstein's ideas about language; after having completed it, Dummett says, "I shall not take the time here to attempt an evaluation of the argument . . . . " If such an evaluation would unde- niably show that the argument must be accepted, my reaction would be that there must be someth ing wrong with Wittgenstein's theory of language. As of now, I don' t see any clash between Dummett's paper and my own.

Anyone who writes a paper nurtures great hopes for it. My hope is that this paper will be a step toward regaining for the philosophy of mathematics the re- spect it once enjoyed.

References

1. P. Benacerraf and H. Putnam, Philosophy of Mathematics, 2d Ed., Cambridge University Press, 1983. (paperback)

2. R. Carnap, "Truth and Confirmation," in H. Feigl and W. Sellars, eds., Readings in Philosophical Analysis, New York, 1949.

3. J. Dewey, Essays in Experimental Logic, Chicago, 1916. 4. M. Dummett, "The Philosophical Basis of Intuitionistic

Logic," reprinted in [1]. 5. The Encyclopedia of Philosophy, Macmillan and The Free

Press, New York and London, 1967. (This beautiful work can be bought for $20 through the Book of the Month Club.)

6. A. A. Fraenkel, Y. Bar-Hillel, and A. Levy, Foundations of Set Theory, North-Holland, Amsterdam, 1973.

7. K. G6del, "Russell's Mathematical Logic," reprinted in [1].

8. A. Heyting, Intuitionism, an Introduction, 2d Ed., North- Holland, Amsterdam, 1966.

9. H. Poincar6, Science and Hypothesis, Dover, New York, 1952. (paperback)

10. B. Russell, Principles of Mathematics, 1st Ed. W. W. Norton, New York, 1903. (paperback)

11. A. Tarski, "The Concept of Truth in Formalized Lan- guages," reprinted in Tarski, Logic, Semantics, Metamath- ematics, Oxford, 1956.

Department of Mathematics and Computer Science Dartmouth College Hanover, NH 03755 USA

5 8 THE MATHEMATICAL INTELUGENCER VOL. 10, NO. 4, 1988

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What do we do when we do mathematics?