a) Three Fundamental Theses

In the Foundations, Frege raises and answers two questions:

His answers are:

I have explained that I disagree with both of Frege’s answers. I argued that numbers are not extensions. Nor are they sets. Rather, they belong to a category of their own, the category of quantifier. Furthermore, certain natural numbers are presented to us in perception. We perceive these numbers in the same sense and just as directly as we perceive colors and pitches. In regard to Frege’s second answer, I argued that arithmetic truths are synthetic, as Kant claimed. They are also a priori. But they are a priori, not in that they are “known independently of experience,” but in that they are necessary and universal. We cut the connection with Kant’s transcendental idealism by asserting that Kant’s axiom is false that what is necessary and universal must be independent of experience. Thus, while we agree with Kant that arithmetic truths are necessary and universal, we insist also that they are “known empirically.” This is our empiricism. We shall now go on to defend our answers to Frege’s two questions against some more recent objections and to elaborate on these answers in the full light of contemporary philosophical wisdom.

But before we begin, I shall list some of our most basic assumptions. In philosophy, perhaps more than anywhere else, there is the danger that one loses sight of the woods in favor of the trees. The details of our view may obscure the philosophical framework within which it is developed. Nor could it be amiss to remind the reader that our view, like any other, rests on several metaphysical assumptions. Let me mention three such assumptions which are of special importance.

Firstly, we embrace the thesis of “semantic atomism.” What a sentence represents, the state of affairs it is about, is determined in part by what the words in the sentence represent, and not the other way around. I pointed out earlier that this thesis is not contradicted by Frege’s famous rule never to ask for the meaning of a word in isolation, but only in the context of a sentence. As I explained, Frege’s claim is not that one must know what a sentence says before one can know what the words in it mean, but rather that acquaintance with abstract objects is propositional. It seems obvious to me that the opposing view has things upside down, as upside down as the verificationist view that the meaning of a sentence is its method of verification. It seems obvious to me that one has to know first what a sentence says, before one can think of how it is to be verified.

Secondly, we accept the thesis that truth is univocal: Statements of arithmetic (and of set theory and logic) are true (or false) in precisely the same sense in which statements of physics or botany are true or false. What distinguishes between a true statement of arithmetic and a true statement of botany is the subject matter: The former is about numbers, the latter is about plants.

Thirdly, there is the so-called “principle of acquaintance,” the foundation of our empiricism. We know what there is by perception and introspection, or else by inference from these. The first alternative takes care of chairs, colors, and shapes, but also of (small) numbers and sets. The second accounts for our knowledge of elementary particles, black holes, and spin, but also for our knowledge of the higher infinites and of large sets. In particular, I claim that we know by acquaintance that there are certain numbers, and that we know by inference that there are others.

b) Apeirophobia

While I am at it, I may as well add some further, very general, observations. These ruminations are not meant to strengthen my view, but to put it in the proper perspective.

My view is free from apeirophobia, the horror of the infinite, which colored so much of what was written at the beginning of this century about the foundations of mathematics. I call it a phobia because, as Benacerraf and Putnam point out, there are few if any arguments against the infinite in the relevant literature (see Philosophy of Mathematics, ed. by P. Benacerraf and H. Putnam, p. 5). But it is clear that there are two worries about the infinite. The first is whether or not the notion of the infinite makes any sense; the second, whether or not there is such a thing, given that the notion does make sense.

How do we form the notion of the infinite? How do we come by it? I shall concede immediately that we are not acquainted with an infinity of things (with an infinite set of things). Here lies a fundamental difference between the notion of the number two, for example, and the notion of the denumerable infinite: while we perceive the number two, we never perceive the number aleph zero. But it does not follow that we have no notion of aleph zero, that we have no concept of it. We have notions of many things with which we are not acquainted. Such things we know, as Russell pointed out a long time ago, by description. Even though I am not acquainted with aleph zero, I can nevertheless describe it, as Frege does, as the number of natural numbers (G. Frege, The Foundations of Arithmetic, p. 96). Moreover, we have a notion of infinite numbers in general, that is, of infinite number: An infinite number is distinguished from a finite number in that only an infinite number is such that a set with infinitely many members has subsets which are similar to it. I trust that I do not have to go on. By now it may be conceded by everyone familiar with the technical aspects of set theory that there are descriptions of transfinite numbers.

But are these descriptions consistent? There used to be much to do about the question of whether or not the notion of an infinite number is consistent. Hilbert and his followers demanded a proof of its consistency. This demand, I submit, is an expression of apeirophobia. Why should one suspect the notion of the infinite to be inconsistent, unless one is unduly afraid of it? I find nothing in or about the notion of aleph zero that smells of inconsistency. Of course, in such matters, intuition is merely the first court of appeal. Who would have thought that there are properties, like the property of being a set (or of being a Fregean extension!), which do not determine sets? I do not wish to give the impression that my intuitions somehow show that the description of aleph zero is consistent. But I am not weary of it, and I am perfectly willing to accept it until a contradiction crops up.

Entirely different is the question of whether or not something answers to the description. Is there such a number as aleph zero? Of course, if the description is inconsistent, then there can be no such number. But the description may be perfectly consistent and yet not describe anything. It is also clear that if there are denumerably many things, then aleph zero exists. But are there? Hilbert argued that there are not, but I am not convinced (see D. Hilbert, “On the Infinite”, pp. 136-37). Nor am I convinced that the world is finite. I simply have no firm opinion on this matter. But there are a few things I am relatively certain of. For example, if it is not the case that there are denumerably infinitely many things, then aleph zero does not exist, for numbers are numbers of things, just as colors are colors of things. If there were no olive green things, then the color olive green would not exist. If there were no five things (of any kind), then the number five would not exist. But even though it may be the case that aleph zero does not exist, we can still speculate about what would be the case if it existed. If there were such a number, for example, then it would be smaller than the number, if it existed, of the subsets of any set with aleph zero many members. The whole theory of transfinite cardinals can be treated in this fashion as being about things which may or may not exist, but which, if they do exist, behave just as described by the theory.

c) Intuitionism and Impredicative Descriptions

I have commented on Hilbert’s concern about the consistency of the notion of the infinite and on what may be called “Russell’s concern” about the truth of the axiom of infinity. In the literature, one finds a third worry, namely, the worry that certain “totalities” are not well-defined. Or perhaps it is more accurate to say that we find the suspicion that these totalities are not well-defined and, based on this suspicion, the view that statements about such totalities do not have a truth-value unless they can be proved or disproved in a certain way. Infinite totalities like the set of all natural numbers are under suspicion. I shall confess immediately that I do not fully understand the intuitionist’s attitude, but I shall make a stab at explaining its source. It seems to me that this source is the belief that certain paradoxes are the result of a vicious circle, and that one can only avoid them by shunning so-called “impredicative definitions.” Let us then briefly refresh our memory of how Russell at one time, and Poincare consistently, diagnosed the trouble caused by certain paradoxes.

In my view, all of the so-called paradoxes—the logical, set-theoretic, and semantic ones—are nothing but proofs that certain things do not exist, (cf. my “Russell’s Paradox and Complex Properties”, and also The Categorial Structure of the World, p. 223-37). What makes these non-existence proofs paradoxical is the fact that they clash with our most basic intuitions. For example, it is fairly obvious that the color olive green is not olive green and that the property of being square is not square. What could be more self-evident, therefore, than that these two properties share the property of not exemplifying themselves? For, olive green does not exemplify itself and neither does the property of being square. But we can easily prove that this self-evident thesis is false, for we can prove that there is no such property as the property of not exemplifying oneself. The assumption that there is such a property, call it “F”, leads to the conclusion that F is F if and only if F is not F. Or consider the property of being a set. That there is such a property cannot be doubted, for if it did not exist, then there would be no sets, and there are sets. Is it not also certain, indubitable, self-evident, that this property, like every other property we can think of, determines a set? Must there not be the set of all of those things which have this property, that is, which are sets? But we can prove that there is no such set. The assumption that this set exists, leads to a contradiction. What the paradoxes teach us, a lesson some refuse to learn, is that we must forever be ready to revise our most cherished intuitions about what there is.

The so-called semantic paradoxes are in this respect not different from the logical and set-theoretic ones. Consider Grelling’s paradox. We assume that a predicate has a certain property, the property of being heterological, if and only if it does not have the property which it represents. The word ‘long’, for example, is heterological because it is not long. What about the predicate ‘heterological’; is it heterological or not? It can easily be shown that it is heterological if and only if it is not heterological. From this consequence, I conclude that there is no such property as the property of being heterological. By now, I think, we are reminded of Russell’s ramified theory of types and his short-lived attempt to blame the paradoxes on a vicious circle. I shall therefore be very brief; for our goal is not to revisit those ill-conceived attempts to get rid of the paradoxes, but to understand the intuitionist’s notion of a totality that is not “well-defined.”

It is customary to speak of “impredicative definitions.” but it should be obvious that one is dealing with impredicative descriptions. The difference in terminology may appear to be slight and inconsequential, but we must never forget that the word ‘definition’ carries with it an aura of convention, of mere stipulation, which is absent from the term ‘description’. There are true and false descriptions, but most philosophers would hesitate to admit that there are true and false definitions. Now, Russell, as I remarked, at one point blamed a vicious circle for the paradoxes (see B. Russell, “Mathematical Logic as Based on the Theory of Types”). He observed that the paradoxes can be created by using descriptions which contain universal or existential quantifiers that range over the kind of entity to which the described thing belongs. Recall for example the description of the alleged property of not exemplifying itself: The f such that: for all properties g, g is f if and only if g is not g. This description mentions all properties g, and this is obviously a totality to which f itself belongs. Or consider the following description of the set of all sets: The set s such that: for all sets t, t is a member of s. Here the totality of all sets to which s itself belongs is mentioned.

Russell therefore suggested to cure us of paradox by banning all such “impredicative” descriptions. But this cure is almost as bad as the disease. It led Russell to the ramified theory of types and, hence, to the infamous axiom of reducibility. More to the point, it rested on the wrong diagnosis. Impredicative descriptions cannot be blamed for the paradoxes. Cantor’s paradox, for example, cannot be laid at the doorstep of the description just mentioned, for it appears also if we describe the set s in a predicative way: The set s such that: everything which is a set is a member of it. This description is no more impredicative than the description of the set of all olive green things as the set such that everything which is olive green belongs to it. Nor does every impredicative description lead to paradox. Dedekind’s impredicative description of the square root of 2, as far as we know, does not lead to paradox. Furthermore, the following description is impredicative: The person who is as tall or taller than any person in the room right now where I am working. Yet, this description actually describes a person and cannot, therefore, harbor a contradiction.

Our conclusion that impredicative descriptions cannot be blamed for the paradoxes is not surprising in the light of our own analysis of the situation. The paradoxes, I claimed, are straightforward nonexistence proofs which insult our most firmly held convictions. If this is correct, then there exists a proof that there is no such thing as the set of all sets. But, surely, the existence or nonexistence of this set does not depend on how we describe it, predicatively or impredicatively. If there is no such set, then there is no set to which all the things belong which have the property of being a set, nor is there the set such that all sets belong to it.

Russell does not explain clearly why he thinks that impredicative descriptions create paradoxes. He says: “In each contradiction something is said about all cases of some kind, and from what is said a new case seems to be generated, which both is and is not of the same kind as the cases of which all were concerned in what was said” (Russell, 1956, p. 61). A little later, he adds: “Thus all our contradictions have in common the assumption of a totality such that, if it were legitimate, it would at once be enlarged by new members defined in terms of itself’ (Russell, 1956, p. 63). This is one place where our insistence that we are dealing with impredicative descriptions rather than definitions is of great importance. As soon as we abandon Russell’s talk about impredicative definitions, we realize that there can be no talk about “generating a new case.” We do not generate a thing by describing it. Nor can we “enlarge a totality” by describing something in terms of it. If we describe the square root of 2 as the least upper bound of the numbers whose square is at most 2, we do not “generate” or “create” that number. The square root of 2 is one of the numbers whose square is at most 2. It is not added to the totality of these numbers by our describing it.

Russell was not the only one who pointed the finger at impredicative descriptions. Poincaré announced that a definition is logically admissible only if it excludes all objects which are dependent on the notion to be defined (see H. Poincaré, “Les mathematiques et la logic,”). This sounds like nothing more than an injunction against circular definitions. But I believe that Poincaré had something else in mind. Before we get to what I take to be his point, we should ask: What does circularity have to do with paradox?

Everyone agrees that circular definitions are worthless. They do not accomplish what they are supposed to. But, granted that they are worthless, do they give rise to paradox? I do not see how they possibly could. It is not even true that impredicative descriptions are circular. The square root of 2 is the same as the number which is the least upper bound of the numbers whose square root is at most 2. One and the same number is described in two different ways. Let us say, for short, that two different features determine the same number. We could say that the two features have the same extension, that they are “equivalent.” All the different descriptions of the same thing are of this sort: they involve equivalent features which determine the same thing. Now, if impredicative descriptions had to be shunned because they are circular, and if their circularity consisted in that they involve equivalent features, then all true identity statements with description expressions would have to be banned. Zermelo thought that this, indeed, is a consequence of Poincaré’s criticism of impredicative descriptions: “Strict observance of Poincaré’s demand [to exclude all impredicative definitions] would make every definition, hence all of science, impossible” (see E. Zermelo, “Neuer Beweis fuer die Moeglichkeit einer Wohlordnung,” p. 191).

But Poincaré, as I hinted a moment ago, is not making the trivial point that circular definitions are worthless. I believe that something more interesting is on his mind, something that leads us back to intuitionism and ill-defined totalities. As I see it, the description of the square root of 2 as the least upper bound of certain numbers is to be banished because it involves the notion of number. The notion of the square root of 2 must be exorcized from the notion of all numbers before the latter can be used to describe the former; for the notion of the square root of 2 is somehow contained in the notion of all numbers. According to this line of reasoning, the notion of all numbers consists in part of the notion of the square root of 2. Therefore, in order to have the notion of all numbers, one must already have the notion of the square root of 2.

The set of all real numbers contains the square root of 2. But we are not talking about this set and its members. Rather, we are talking about certain notions, certain concepts, namely, the notion all numbers and the notion square root of 2. And what Poincaré claims, according to my interpretation, is that the former notion somehow consists in part of the latter. But this seems to me to be a mistake. Taking for granted that ‘all numbers’ is short for ‘all things which are numbers’, what this expression represents consists, among other things, of the property of being a number, but it does not contain the square root of 2 or the property of being a square root of 2. In order to have a notion of all numbers, to put it differently, one needs to have a notion of the property of being a number, but one need not have the concept of the square root of 2. Surely, there are many numbers you have never thought of, and yet you know perfectly well what you mean by ‘all numbers’. Similarly, what such expressions as ‘all sets’, ‘all properties’, and ‘all elephants’ represent involve, among other things, the properties of being a set, of being a property, and of being an elephant. What they represent does not consist, in part, of the set of odd natural numbers, the property of being square, and the elephant Dumbo from the San Diego zoo.

It may be objected that we cannot acquire the concept of number without first acquiring the different concepts of all individual numbers. But this objection is so obviously misguided in regard to other notions that there is no reason to believe that it works for numbers. For example, we acquire the notion of olive green without having become acquainted with all of the olive green things in the world. Nor do we acquire the notion of an elephant by having thought of all individual elephants. How do we acquire these concepts? The answer is obvious. Some things are olive green, and on some occasion we notice that one of these things has this particular color. Similarly, some things are numbers; they have the property of being a number. The number two has this property, and so does the square root of 2. We acquire the concept of number by noticing, for example, that the number two is a number. A single act of perception is sufficient, just as in the case of olive green. Of course, we may have been in the presence of olive green things before, without perceiving that they were olive green or that olive green is a color. And we may have been in the presence of numbered things before, without perceiving how many of them there are or that their number has the property of being a number. What this shows is that it may take time before some property is perceived, not that we cannot perceive it. It is also clear that it may take a higher mental development to perceive such “higher” features as those of being a color, of being a number, or of being a set, than what is required for the perception of olive green, of the number two, and of the set of three pencils on my desk. But, again, this does not diminish the fact that such “abstract” properties can be perceived, and that they can be perceived in single acts of perception.

My reply to this last objection gives us a clue why an intuitionist may hold that impredicative descriptions are flawed and that ill-defined totalities are illegitimate. My reply assumes that there are such properties as the color olive green, being a color, and being a number. If we reject this ontological premise, then our reply to Poincaré will not do. If the word ‘number’ represents, not a property which many individual numbers share, but instead (as one says: commonly or indifferently) every individual number, then we could not possibly perceive what the word represents by perceiving an individual number. The view that ‘number’ is a common name of individual numbers rather than the proper name of the property of being a number may easily be taken to imply that we do not fully understand the word until we are familiar with all numbers. It may be taken to imply, to get back to our example, that we cannot fully understand ‘number’ in the description expression for the square root of 2, unless we are already familiar with the square root of 2. We are faced with the following dilemma: Either one knows the square root of 2, in which case one need not define it; or else one does not know it, in which case one cannot define it in terms of the notion of number, for one does not have this notion. In this manner, I suggest, the common name doctrine of “general terms” may lead to a rejection of impredicative descriptions. But if this is the source of the intuitionist’s worry, then we need not share it, for the common name doctrine is quite obviously false. If it were true, then it would follow that we do not know what ordinary property words mean (represent). But we do know what they mean. The fact that we talk intelligibly about elephants, colors, and numbers, without being acquainted (in whatever fashion) with all elephants, all colors, and all numbers, reduces the common name doctrine to absurdity.

d) “Mengenfurcht” and “Mengenliebe”

The development of set theory is not only a monumental intellectual achievement, but also of great philosophical significance; for it amounts to nothing less than the discovery of an ontological category and its laws. This point needs emphasis. All too often, set theory is thought of as a branch of mathematics. In reality, it reaches far beyond mathematics. Of course, the most fascinating sets are infinite sets and, among these, sets of points and sets of numbers. But we must keep in mind that there are sets of peas and sets of pitches as well. The notion of a set, in short, is not a mathematical but an ontological notion, even though the most interesting sets are sets of mathematical things. (The situation is similar for relations: the notion of a relation is not a mathematical notion, even though some of the most interesting relations are relations among numbers.)

Set theory, to say it again, is nothing less than the theory of an ontological category. No wonder that its discovery and development met, on the one hand, with utmost suspicion and outright hostility and, on the other, with boundless enthusiasm. While philosophically-minded mathematicians often went so far as to suspect the very notion of a set, mathematically-minded philosophers soon elevated sets to the only category worthy of their attention. While the former blamed the set-theoretic paradoxes on the very notion of a set, the latter replaced properties by sets, relations by ordered sets, and, ultimately, philosophy by set theory. Both extremes are equally foolish. We must not succumb to “Mengenfurcht.” But we must also beware of the laughable view that philosophy is nothing but applied set theory.

Benacerraf and Putnam remark that today “very few philosophers and mathematicians of any school would maintain that the notion of, say, an arbitrary set of real numbers is a completely clear one, or that all the mathematical statements one can write down in terms of this notion have a truth-value which is well-defined in the sense of being fixed by a rule—even a nonconstructive rule—which does not assume that the notion of an “arbitrary set” has already been made clear” (Benacerraf and Putnam, p. 15). I doubt that this is really true for mathematicians, as Benacerraf and Putnam claim. But be that as it may, it seems to me that the notion of an arbitrary set is no less clear than most notions in arithmetic, geometry, and physics. Ordinarily, we think of sets as being determined by properties, for example, by the property of being a planet that orbits the sun. But the existence of a set, we must constantly remind ourselves, does not depend on the existence of a “determining” property. The set whose only members are the color olive green, the moon, and Napoleon, exists just as truly as the set of planets, although it may well be the case that it has no “determining” property. And the axiom of choice, which I acknowledge without a moment’s hesitation to be a truism, explicitly states that there are “arbitrary” sets (cf. Goedel’s comment on the axiom of choice in his “What is Cantor’s Continuum Problem,” p. 259, footnote 2). No, neither the notion of a set nor the notion of an arbitrary set contains a confusion or ambiguity.

Nor do I share the peculiar view of those who hold, in Benacerraf’s and Putnam’s words, that “if we can show that a proposition is undecidable from the assumptions we currently accept, the question of its ‘truth’ or ‘falsity’ vanishes in a puff of metaphysical smoke” (Benacerraf and Putnam, p. 15). I call this view “peculiar” because it is hard for me to imagine that even the most “formalistically” inclined philosopher can fail to acknowledge after Goedel that truth is one thing, derivability quite another. Of course, if a proposition is undecidable from our current assumptions (axioms), then we do not know whether it is true or false. But it is silly to conclude from our ignorance that “since nothing else is relevant, the question of truth does not arise” (Benacerraf and Putnam, p. 15). Quite to the contrary, the question of truth becomes all the more pressing. As long as we do not merely play the game of trying to find out what follows from what, but are interested in truth at all, we must face up to the question of whether or not the proposition in question is true; for, if it is true, we must add it to our axioms, and if it is false, we must not. The situation is no different in set theory from what it is in geometry and physics. What would we think of a physicist who claims that the question of truth does not arise for what looks like a basic thesis of physics because he has just discovered that this thesis does not follow from the accepted assumptions of physics? Or what would we make of a geometer who declares that the question of truth does not arise for the parallel axiom since it is independent of his other axioms? Axiomatization is the servant of truth, not its master!

The situation in set theory (and arithmetic) is in this respect not different from the situation in physics or botany. In particular, there is no reason in the world why the continuum hypothesis should be treated differently from hypotheses in other fields of inquiry. Since it is independent of the standard axioms of set theory, the question arises naturally whether or not it is true and, hence, if true, should be added to the axioms. Considerations of the sort listed by Goedel suggest that the continuum hypothesis is false (Goedel, 1964b, pp. 266-68). If so, then it should of course be rejected. It would be comforting, needless to say, if we could discover a set-theoretical truth which is (a) independent of the standard axioms, and (b) together with these axioms implies the falsehood of the continuum hypothesis. We shall have to hope for such a fortuitous event.

It must have occurred to other philosophers that there is a certain irony to the fact that while philosophically-minded mathematicians often are leery of set theory, mathematically-minded philosophers substitute set theory for philosophy. If the notion of set is that unclear, that suspect, then it would surely be prudent not to treat set theory as if it contained the solutions to all philosophical problems. But if you look at the philosophical journals, you may conclude that a surprising number of philosophers cannot deal with a philosophical problem unless they have first transformed it into a problem of constructing some kind of set-theoretical model. Perhaps it is not too far from the truth that while some mathematicians live in dread of set theory because of its affinity to philosophers, some so-called philosophers worship it because of its distance from philosophy.