1.) Gödel's Realism.
Mathematicians are reluctant to refer to the "concepts behind the symbols" as meaningless, or to assert that the entities of mathematics are non-existent, and quite justifiably so, owing to the overt pejorative nature of such terminology. The ontological status of mathematical entities, like the question of paradox, has a long history of philosophical debate, perhaps because of its close relationship to the problem of universals. Gödel has stated his own position forcefully:
It seems to me that the assumption of such objects is quite as legitimate as the assumption of physical bodies and there is quite as much reason to believe in their existence. They are in the same sense necessary to obtain a satisfactory system of mathematics as physical bodies are necessary for a satisfactory theory of our sense perceptions and in both cases it is impossible to interpret the propositions one wants to assert about these entities as propositions about the "data", i.e., in the latter case the actually occurring sense perceptions.[50] These carefully worded paragraphs must be analyzed extensively because they contain subtle but crucial distinctions.
One must admit that there is an important difference between Gödel's statement that "Classes and concepts may, however, also be conceived as real objects" and the more patently realistic "classes and concepts are real objects." For if it were simply a matter of how classes and concepts were conceived, then the fact that one employs "thing" language in mathematics could be regarded as an expected extension of our everyday speech, and thus a convenient, heuristically valuable method of exposition, perhaps devoid of ontological presupposition or committment,
Even if one grants that we have "something like a perception" of mathematical objects, it is difficult to imagine what a phenomenalistic exposition of mathematics would be, or if one is at all possible. Thus the choice of language does not seem justifiable on grounds of epistemological priority, whereas Gödel apparently feels that in the case of sense perceptions, sense data are epistemologically prior to physical objects, but physical objects are a necessary theoretical assumption.[51] As quoted above:
Gödel makes a rather strong comparison between "the question of the objective existence of the objects of mathematical intuition" and the "question of the objective existence of the outer world" which he considers to be "an exact replica."[53] One is inclined to believe that this comparison does not account for the fact that mathematics is not universally understood in the same way that the physical world is "accessible" to virtually everyone. To postulate "mathematical intuition" in this manner tends to establish the mathematician as a "visionary" which is certainly not Gödel's intention, since one would like to believe that "mathematical truths" are as accessible as "physical truths," i.e., of commonplace physical objects. It must be admitted that most people are somewhat mystified by higher mathematics[54] whereas the external world poses no such difficulty, even for phenomenalists. If however Gödel means that "physical intuition" is the intuition of the physicist, the comparison is much more accurate, since the entities of physics are perhaps as equally "abstract" as those of mathematics. Arguments concerning "our knowledge of the external world" are, for the most part, at a "tables and chairs" level.
While it would seem unusual for someone to deny the actual existence of the physical world, the "ontology-free" positions taken by Professors Carnap and Bar-Hillel,[55] as well as Professor A. Robinson's Formalism are founded on the view that it is possible to assume the existence of nathematical objects (to do mathematics as if such objects exist) without believing that such objects actually exist. As one might expect, the source of confusion is 'exist'. For one can maintain that it makes sense to speak of mathematical entities which are not definable or constructible but are describable or characterizable, as in the case of the real number system vis à vis the vicious circle principle. Hence "things existing independently of our definitions and constructions" may be assumed to exist by adopting appropriate axioms such as the power set and infinity, perhaps without committing oneself to the belief in the actual existence of such objects.
There are analogous difficulties in the problem of mathematical objects and the problem of physical objects. In both cases, our everyday speech is a "thing" language. Although the question of our perception of physical objects has centered around the sense data debate, it is difficult to construe an analogous entity for mathematics. Gödel, in his analysis of Russell's comparisons between "the axioms of logic and mathematics with the laws of nature and logical evidence with sense perception," indicated that arithmetic is "the domain of the kind of elementary indisputable evidence that may be most fittingly compared with sense perception."[56] This still does not answer the question of how we "perceive" arithmetical truths. In the case of the manipulation of symbols, this in a sense reduces to the perception of physical objects, viz. the symbols themselves. "Logical evidence" or "mathematical data" may be regarded as the examination of numerical calculations, and reasoning from tables and diagrams as in the following:
(i) Theorems of number theory are often generalizations of the "observation" of calculations. Statements about primes and perfect numbers, for example, can be regarded as generalizations (laws) obtained from noticing that 7 is a prime, 6 is a perfect number, etc. Euclid's Theorem, asserting the infinity of primes, can be "verified" by calculation as was indicated above.[57] The axioms of arithmetic enable us to prove this assertion in general, to "make it possible for these 'sense perceptions' to be deduced."[58] For any given prime, we have always been able to calculate a greater one.
(ii) Tables and diagrams have proved to be heuristically valuable in algebra and geometry. Measurements made on carefully drawn diagrams "verify" various theorems. Historically, geometry is the science of the measurement of physical objects.[59] Tables listing the elements of a group and the results of a particular composition can be used to "verify" theorems of group theory, e.g. by "counting" the number of its subgroups. Such tables are an algebraic aspect of "experiments with physical objects."
(iii) The question of computability has been reduced to the possibility of performing specified operations on theoretical calculating machines, i.e. of performing tasks with "physical" objects:
(iv) Many problems of "large" cardinality have their origin in the observation that certain familiar infinite cardinals have particular properties, and the question is posed as to the existence of others (usually uncountable). This is clearly analogous to the case of number theory mentioned above:
If the view that we "perceive" or "experience" mathematical objects by examining calculations and constructions is plausible, one is in a position to offer an interpretation of Gödel's usage of 'real'. Gödel stated that he "considers mathematical objects to exist independently of our constructions and of our having an intuition of them individually...."[62] If mathematical objects were finite in number, "an intuition of them individually" might be possible. But this is not even possible for the integers. For example, there is a finite, effective procedure by which we may test an integer to see if it is a prime. While it is possible in principle to test any given integer (disregarding the amount of time or material required by the present state of technology), it is impossible to test all integers. Hence one who accepts Euclid's Theorem, must further accept the existence of a number which is prime by virtue of Euclid's Theorem, not because it has been tested. Thus one must acknowledge the existence of mathematical objects which we are unable to examine (experience), even if one restricts the domain of objects to the potential infinite. The "mathematical phenomenalist" then, one who admits mathematical truths insofar as they are verifiable by examining calculations and constructions, is forced to place a bound on the size of the finite structures he allows, relative to the technological possibility of scrutinizing such objects. Nevertheless, it may be objected that "mathematical phenomenalism" could accept those truths which are theoretically" verifiable by examination, and need not reject the potential infinite. It would then appear that "theoretically verifiable" would inevitably reduce to some form of "lawlike" behavior of mathematical objects, and the problem of truth just reappears in a different form. Note that Gödel distinguishes 'existence' from 'knowable': In discussing the third form of the vicious circle principle, he indicated that if mathematical objects are assumed to exist independently of our constructions, this form of the principle would not be violated "if 'presuppose' means 'presuppose for the existence' not 'for the knowability.'" This tends to corroborate our identification of "knowledge by experience" with "the examination of constructions and calculations."[63] Hence even if "mathematical phenomenalism" is extended to allow the potential infinite, it falls far short of allowing classical analysis, because it is unreasonable to maintain that an uncountable totality is "in principle subject to examination." Taking classical analysis as our criterion for "a satisfactory system of mathematics," we see that we are forced to acknowledge the existence of mathematical objects which we have no way, even in principle, of examining. Thus, mathematical objects exist independently of experience, as opposed to being phenomenal or apparent, and are therefore real.[64] I.e., replacing "independently of our constructions and of our having an intuition of them individually" by "independently of experience" yields the dictionary definition.
Much mythology has been read into Gödel's widely quoted remark that:
Under the interpretation of 'real' given above, a reality is nothing more than a system of real objects. By a system of real objects, we mean that the objects of mathematics are governed by regularities, viz. the axioms are to the mathematical objects as the laws of physics are to physical objects. One argument in favor of this view is the fact that no mythological allusions are to be found in any of Gödel's writings. Gödel seems to be saying that mathematical objects are as concrete, as stable, and as well-behaved as physical objects, and that the axioms do indeed govern their behavior. In this respect, mathematical objects are neither illusory, nor ephemeral; neither figment, nor allegory. They are real.
2.) Gödel's Interpretation of 'exist'.
We have attempted to explain Gödel's mathematical realism in terms of his view of the independent existence of mathematical objects. It remains then to analyze Gödel's usage of 'exist' in this context. Our problem is what Professor Sierpinski called "the great and ancient problem of existence," when he asked our question: "But what does it mean: 'to exist'."[66] One cannot overemphasize the difficulty caused by 'exist'. Its tortured history in philosophical discourse, at present hopelessly inconclusive, as well as the seemingly endless and inapplicable ways in which it occurs in everydav speech, severely limit one's confidence in the correctness of any interpretation of Gödel's usage.
One can however observe that an aspect of Gödel's usage of 'exist', its close relationship to the criterion of clarity, allows a comparison with other philosophers' usage, in particular Hume and Descartes. Gödel stated that he "requires only that the general mathematical concepts must be sufficiently clear for us to be able to recognize their soundness and the truth of the axioms concerning them...."[67] Here we find a subtle but nonetheless important distinction between intuition and clarity. What we have an intuition of may be said to be clear. Concepts, for example of infinite totalities, may be clear and yet our intuition of these entities may be rather weak. For example our intuition of elementary arithmetic and logic enables us to formalize such theories with rather strong convictions and general acceptance. Our intuition of the arithmetic of large cardinals, and the logic concerning them is much weaker. These reasonings cannot be said to be immediate. Nevertheless, most mathematicians would agree that these concepts are clear, although not basic or primitive.[68] That we are capable of distinguishing two different powers of infinite is evidence for the belief that the concept of infinite Dower is clear.
Descartes, in the Fifth Meditation, analyzes the concept of a triangle in a manner remarkably similar to Gödel's position that the entities of mathematics are not mind-dependent:
Descartes' remark that in the case of a triangle, "there may perhaps be no such figure anywhere in the world outside of my thought" raises an issue which some critics of Gödel's view seem to misunderstand. Gödel is indicating that there is an analogy between the existence of mathematical objects and the existence of physical objects. He did not assert that they were physical objects, or existed in space[70] and certainly not in any mythological "heaven." In what sense then can mathematical objects be said to exist? Perhaps in the sense of Hume's Maxim:
It is not our intention to claim that Gödel is a Humean; rather we wish to point out that Hume, whose conception of the problem of the existence of the external world is similar to Gödel's, held that the reality of mathematical objects is a product of our clear conception of them.[72]
One desirable feature of a constructivist viewpoint is the possibility of actually "building" mathematical objects, e.g. in terms of computing machinery, or physical interpretations of mathematics. Gödel is primarily concerned with the clarity and force[73] of our conception of mathematical objects. Computability or physical interpretations are of secondary importance. Hence if one understands 'actual' in the sense of factual physical existence (in space and time), mathematical objects are not actual. Since 'real' and 'factual' are often considered synonymous., this may be one source of confusion.[74]
Another aspect of 'exist' in mathematical contexts which should not be confused with "existence in terms of clarity," is existence in the sense of consistency. Often mathematicians will ask if certain objects having specified properties exist. By the context of the discussion, it is clear that the question being asked is whether or not the assumption of the existence of such objects is consistent with other axioms being assumed, and unfortunately, these are not always specified.
It might be asked whether Gödel believes that mathematical truths are "eternal" since he does believe that mathematical objects exist "independently" of experience. We see no incompatibility with his view of mathematical existence and the view that mathematical truths are tenseless. We are again resisting any attempt to read "mythology" into his view of the existence of mathematical objects. Gödel's realism, then, may be viewed as a form of scientific realism, without mythological or ontological overtones. Thus his views are in accord with those of Professor Carnap, and indeed indicative of Gödel's membership in the Vienna Circle.
3.) Some Criticisms of Gödel's Realism.
The ontological questions raised by considering mathematical objects as real objects, by "believing in their existence," represent an area of disagreement among many philosophers who, for the most part, are in agreement as to the methods and content of mathematics. These critics do not dispute the clarity or the necessity of an adequate development of mathematics, as understood by Gödel, but cannot agree on the "ontological status" of mathematical objects.
One point of view holds that it is the content and methods of mathematics which are crucial; the ontology is of secondary importance or to be ignored entirely. Hence those disagreements which do not effect any change in the content or methods of mathematics are regarded as linguistic or verbal disputes, arguments about words and not things (The Fallacy of the Market place), and often the relevance of such arguments is attacked as well.[75]
One may thus question the meaningfulness of statements such as "Classes exist." Hume rejected statements of this sort in the Treatise, Book I, Section VI, "Of the idea of existence, and of external existence." Hume wrote:
As was pointed out above, Gödel is primarily concerned with the clarity and force of our conception (ideas) of mathematical objects. One can follow Hume and simply refuse to consider questions involving 'existence' in this context (called "external" by Professor Carnap).[77]
Professor Bar-Hillel, calling his position "ontology-free," has followed this line of thought. He has presented a forceful "trivialization" of ontological disputes which leave classical mathematics untouched. First he compares the Formalism of H. B. Curry to that of Carnap:
Then, Bar-Hillel, in the spirit of William James, asks for the difference that makes a difference:
Nevertheless, two other philosophers, Abraham Robinson, and W. V. O. Quine, have maintained that there are important distinctions to be made. Both accept classical mathematics, but are troubled by indiscreet usage of 'real' and 'exist'. Professor Quine, whose joint paper with Nelson Goodman on nominalism began with the solemn benediction "We don't believe in the existence of abstract entities," is often regarded as a nominalist[80] and a logicist. These labels are misleading because one might think that Quine has rejected classical mathematics. Such is not the case:
Quine's concern with quantification over classes represents his preference for a nominalistic system whenever possible:
Note Quine's usage of 'real' in this context. Quine is nevertheless reluctant to admit that classes are real. He is forced into this ontological commitment by his own rule regarding the values of bound variables, and in this respect Quine's reason for regarding classes as real differs considerably from Gödel's reasons for regarding classes (and other mathematical objects) as real.
Professor A. Robinson considers himself a formalist. In "Formalism 64"[83] he rejects the real or actual infinite as meaningless but counsels us to continue to do mathematics in a "business-as-usual" manner. This view is important because it stands in opposition to those of Gödel and Quine. Gödel believes in the existence of infinite classes (e.g. the set of all reals) and holds that they are real. Quine maintains that even those who reluctantly quantify over classes are nevertheless ontologically culpable; committed to accepting classes as real objects. Professor Robinson maintains that while infinite classes are not real, it is still possible to do classical mathematics in the usual way, i.e. quantify over infinite classes, but not be committed to the real existence of infinite classes. It hardly seems possible to reconcile the views of Gödel, Quine, and A. Robinson. But Bar-Hillel and Carnap hold that no such reconciliation is necessary because ontological disputes are fruitless and irrelevant, especially when the disputants are in agreement mathematically.
4.) Gödel's Realism vis à vis Gödel's Methodology of Mathematics.
Mathematics has many decisions to make. The bifurcation of axiom systems in geometry and in set theory, the discovery of nonstandard models of arithmetic and analysis, the general unsolvability of the Decision Problem, and many other recent advances, as well as the ever-present question of consistency, have forced mathematicians into evaluating the various aspects of their discipline. Mathematics is not really accustomed to leading such an examined life; indeed most people feel that mathematics would be the last place one would find such difficulties. Gödel however has always been sensitive to the methods of resolving open questions, and the necessity of evaluating axiom systems. His methodology offers a partial but nonetheless satisfactory solution. That his solution is partial is due to his own Incompleteness Theorems. That his solution is satisfactory can be seen from its acceptance by the majority of mathematicians with otherwise widely divergent viewpoints. Those who accept classical analysis as a criterion for adequacy are in substantial agreement with Gödel. The differences in the views of Bar-Hillel, Carnap, Curry, Gödel, Quine, and A. Robinson are important, but are not as crucial as the fact that their mathematical agreement far outweighs their ontological differences. They are certainly of the same mathematical kind, though not of the same ontological kin. That such ontological ecumenism is possible in mathematics tends to devalue the relevance or need for ontology in mathematics. Yet one ought not disrespect the "ontic consciences" of Goodman, Quine, and A. Robinson.
All of this points to the above-mentioned possibility of separating Gödel's methodology from his realism. Gödel correctly predicted the independence of the continuum hypothesis. He further predicted the search, presently taking place, for higher axioms which would decide the issue and be generally accepted. This and numerous other examples confirm Gödel's methodology as reflecting actual mathematical practice; not because he tells mathematics what to do, but because he tells what mathematics does. Clearly one can assent to Gödel's methodology completely and yet reject his realism completely. Rather than enter a dispute which has remained unresolved for centuries, we prefer to rest our case on the above distinction.
Another British Empiricist who employed 'real' in this context was John Locke. In Book Four, Chapter IV, "Of the Reality of Knowledge," Locke had two relevant sections. In Section 6, "Hence the reality of Mathematical Knowledge," he held that:
Note that Locke distinguished between objects existing as physical objects and mathematical objects in Section 8, "Existence not required to make it real." He states that:
Gödel's rejection of Russell's "logical fictions" may be seen as a refusal to regard nathematical objects as "insignificant chimeras of the brain."
Professor Church pointed out that Leibniz argued for the "real existence of mathematical objects" in Dialogus de connexione inter res et verba. Translated as "Dialogue on the Connection between Things and Words" by Leroy E. Loemker, we find marked similarities to the views of Locke quoted above:
Gödel's realism, although similar to that of Locke and Leibniz, places emphasis on the fact that the "axioms force themselves upon us as being true." This answers a question, untouched by Locke and Leibniz, why we choose one system, or set of axioms, and not another; that the choice of a mathematical system is not arbitrary.
Classes and concepts may, however, also be conceived as real objects, namely classes as "pluralities of things" or as structures consisting of a plurality of things and concepts as the properties and relations of things existing independently of our definitions and constructions.
...as physical bodies are necessary for a satisfactory theory of our sense perceptions and...it is impossible to interpret the propositions one wants to assert about these entities as propositions about the "data", i.e.,...the actually occurring sense perceptions.[52]
Algorithms...are frequently called calculi. This name originates from the calculi (small pieces of limestone) which the Romans used for calculations.[60]
Propositions which, if true, are extremely strong axioms of infinity.... In contradistinction to Mahlo's axioms the truth (or consistency) of these axioms does not immediately follow from 'the basic intuitions underlying abstract set theory'...nor can it, as of now, be derived from them. However, the new axioms are supported by rather strong arguments from analogy, such as the fact that they are implied by the existence of Stone's representation theorem to Boolean algebras with operations on infinitely many elements.[61]
...if the meanings of the primitive terms of set theory...are accepted as sound, it follows that the set-theoretical axioms describe some well-determined reality....[65]
And what I believe to be more important here is that I find in myself an infinity of ideas of certain things which cannot be assumed to be pure nothingness, even though they may perhaps have no existence outside of my thought. These things are not figments of my imagination, even though it is within my power to think of them or not to think of them; on the contrary, they have their own true and immutable natures. Thus, for example, when I imagine a triangle, even though there may perhaps be no such figure anywhere in the world outside of my thought, nor ever have been nevertheless the figure cannot help having a certain determinate nature of form, or essence, which is immutable and eternal, which I have not invented and which does not in any wav depend upon my mind. This is evidenced by the fact that we can demonstrate various properties of this triangle, namely, that its three angles are equal to two right angles, that the greatest angle subtends the longest side, and other similar properties. Whether I wish it or not, I recognize clearly that these are properties of the triangle, even though I had never previously thought of them in any way when I first imagined one. And therefore it cannot be said that I have invented them.[69]
'Twill not be surprizing after this, if I deliver a maxim, which is condemn'd by several metaphysicians, and is esteem'd contrary to the most certain principles of human reason. This maxim is that an object may exist, and yet be no where: and I assert, that this is not only possible, but that the greatest part of beings do and must exist after this manner.[71]
The idea of existence, then, is the very same with the idea of what we conceive to be existent. To reflect on any thing simply, and to reflect on it as existent, are nothing different from each other. That idea, when conjoin'd with the idea of any object, makes no addition to it. Whatever we conceive, we conceive to be existent.[76]
In spite of many divergencies, Curry's philosophy of mathematics is most closely related to that of Carnap. Like him, he rejects any ontological commitments and stresses acceptability as the criterion by which mathematical theories should be judged. He calls his view empirical formalism to distinguish it from Hilbert's version of formalism, from which it diverges indeed considerably; pragmatical formalism would probably be a better label. Curry's insistence that the formalist definition of mathematics (as he gives it) requires no philosophical presuppositions and that philosophical differences should be transferred rather to the level of acceptability squares well with Carnap's present views, and his distinction between discussions around the truth of some given mathematical statement within a given system and that of the acceptability of the system as a whole is probably equivalent to that between internal and external existence questions of Carnap.[78]
We are not convinced that the distance between the pragmatic Platonism of Gödel and the pragmatic formalism of Carnap and Curry is as great as the customary formulations would make one think. Believing in the existence of sets because they are necessary for obtaining some satisfactory system and accepting some set theory because it is helpful for obtaining some satisfactory system--is the abyss between these views really so steep?[79]
For a constructivistic set theory, with its economy of means, there is both philosophical and aesthetic motivation. And there is a methodological motive as well, knowing as we do the threat of paradox. There are all these counsels for keeping things down as best we can without stifling classical mathematics. And on the other hand there are motives for stepping things up. That there is again an aesthetic one can scarcely be doubted. Also a practical motive for generosity in the ontology of set theory is coming to be felt from the side of abstract algebra and topology, as MacLane has stressed (p. 25), because of the big totalities involved (all groups, all spaces, all...).[81]
Once we admit classes and relations irreducibly as values of variables of quantification, and only then, we are committed to recognizing them as real objects. The range of values of the variables of quantification of a theory is the theory's universe.[82]
I doubt not but it will be easily granted that the knowledge we have of mathematical truths is not only certain, but real knowledge, and not the bare empty vision of vain, insignificant chimeras of the brain; and yet, if we will consider, we shall find that it is only of our ideas.[84]
All the discourses of the mathematicians about the squaring of a circle, conic sections, or any other part of mathematics, concern not the existence of any of those figures; but their demonstrations, which depend on their ideas, are the same, whether there be any square or circle existing in the world or no.[85]
...I notice that, if characters can be applied to ratiocination, there is in them a kind of complex mutual relation (situs) or order which fits the things; if not in the single words at least in their combination and inflection, although it is even better if found in the single words themselves. Though it varies, this order somehow corresponds to all languages...For although the characters are arbitrary, their use and connection have something, which is not arbitrary, namely a definite analogy between characters and things, and the relations which different characters expressing the same thing have to each other. This analogy or relation is the basis of truth. For the result is that, whether we apply one set of characters or another, the products will be the same, or equivalent or corresponding analogously.[86]
Copyright (c) 1968, 1998 Harold Ravitch, Ph.D. All Rights Reserved