K. Gödel, "What is Cantor's Continuum Problem?" in P, Benacerraf and H. Putnam, eds., Philosophy of Mathematics (Englewood cliffs, 1964), p. 262
K. Gödel, "Russell's Mathematical Logic," in P. Benacerraf and H. Putnam, eds., Philosophy of Mathematics (Englewood Cliffs, 1964), pp. 215-216
Gödel, "What is Cantor's Continuum Problem?" in Benacerraf and Putnam, pp. 262-263.
Ibid., p. 262
I.e., any one of the standard semantical paradoxes such as Grelling's, Richard's., König's, The Liar, etc. Gödel is apparently following F. Ramsey's classification of the paradoxes into "logical or mathematical," and "semantical" or "epistemological." Cf. The Foundations of Mathematics (London, 1931), pp. 20-21
K. Gödel, "On Formally Undecidable Propositions of Principia Mathematica and Related Svstems, I," in M. Davis, ed., The Undecidable (Hewlett, 1965), p. 9, footnote 14
Cf. J. B. Rosser, Logic for Mathematicians (New York, 1953), pp. 197-207
Cf. S. Feferman "Systems of Predicative Analysis, Part I," Journal of Symbolic Logic, 1964, 29:1-13; S. C. Kleene, Introduction to Metamathematics (Princeton, 1952), section 12
As described in Feferman's "Systems"
Ibid., p. 4
Ibid.
Gödel "Russell's Mathematical Logic," in Benacerraf and Putnam, pp. 218-219
E.g. Kleene, Introduction, Section 12; Feferman, Journal of Symbolic Logic, p. 2
Ibid., p. 217. For a careful, precise discussion of (4), and "predicativity" as well, cf. the book of A. Church cited in footnote 39, pp. 346-356.
L. Félix, The Modern Aspect of Mathematics, translated by J. H. Hlavaty and F. H. Hlavaty (New York, 1960), pp. 42-43:
"The mathematician and physicist Boussinesq was astonished to learn in 1875 that there existed continuous functions without derivatives at any point. As Picard reports: 'He said--very seriously, I think--that functions have everything to gain by having derivatives.'"
J. Pierpont, Lectures on The Theory of Functions of Real Variables, Vol. I (Boston, 1905), p. 82
L. M. Graves, The Theory of Functions of Real Variables, 2nd ed. (New York, 1956), p. 14
Journal of Symbolic Logic, especially p. 2
"Russell's Mathematical Logic," p. 219
Kleene, p. 43. This argument is due to F. Ramsey.
Cf. S. Feferman, "Geometric Motivations," The Number Systems (Reading, 1964), pp. 224-226
Ibid., p. 226
Cf. D. Hilbert. The Foundations of Geometry, 2nd ed., trans. by E. J. Townsend (Chicago, 1921), pp. 24-26; K. Borsuk and W. Szmielew, Foundations of Geometry (Amsterdam, 1960). pp. 218-225, 151-154; H. Levi, Foundations of Geometry and Trigonometry (Englewood Cliffs, 1960), pp. 315-320
"Russell's Mathematical Logic," p. 218, footnote 18
A. Tarski., A. Mostowski, and R. M. Robinson, (Amsterdam, 1953), p. 57
R. C. Lyndon, Notes an Logic (Princeton, 1966), pp. 9-10
A. Tarski, Logic, Semantics, Metamathematics (Oxford, 1956), p. 63. An object belongs to the "least class" if it belongs to all classes having certain specifications. But the "least class" is among all those having such specifications.
"What is Cantor's Continuum Problem?" in Benacerraf and Putnam, p. 261
Ibid., p. 262. E.g. Von Neumann--Bernays--Gödel Set Theory.
Cf. Kleene, p. 498
"Zur intuitionistischen Arithmetik und Zahlentheorie," Ergebnisse eines matematischen Kolloquiums, Vol. 4, pp. 34-38
Cf. L. E. J. Brouwer, "Intuitionism and Formalism," in P. Benacerraf and H. Putnam, eds., Philosophy of Mathematics, pp. 66-77
Professor Church has pointed out that Heyting now maintains, in the light of Gödel's result, that intuitionintic arithmetic is superior to classical, not as narrower, but as richer--it makes distinctions that the classical arithmetic fails to make.
We are indebted to Professor Kreisel for this observation.
Gödel, "The Consistency of the Axiom of Choice and the Generalized Continuum Hypothesis," Proc. Nat. Academy of Science, 1938, 24:556-557; "What is Cantorts Continuum Problem?" in Benacerraf and Putnam, p. 261, footnote 11; Hermann Weyl, Das Kontinuum, no date given, reprinted by Chelsea Publishing Company (New York), Section 6, pp. 19-29
By 'decide' we do not mean that there is an effective procedure, but merely that there is a proof, in the mathematical sense. Gödel lists these possibilities as "demonstrable, disprovable, or undecidable." (See Benacerraf and Putnam, p. 262)
"What is Cantor's Continuum Problem?" in Benacerraf and Putnam, p. 270
Ibid., p. 271
Cf. A. Church, Introduction to Mathematical Logic, Volume I (Princeton, 1956), p. 56
"What is Cantor's Continuum Problem?" in Benacerraf and Putnam, p. 270. Higher systems may have stronger languages.
Cf. The Modern Aspect of Mathematics, Chapter 1. E.g. Professor Félix quotes Hermite as saying: "'I turn aside in horror from this lamentable plague of functions which do not have derivatives.'" (p. 10)
"What is Cantor's Continuum Problem?" in Benacerraf and Putnam, p. 272
The axiom of choice provides an excellent example of the kind of debate which can take place once an axiom has "implausible" consequences. Examples of this are cited in Appendix A.
Nevertheless, there are applications of the theory of Alephs e.g. "The Problem of Infinite Matter in Steady-State Cosmology," by R. Schlegel in Philosophy of Science, 1965, 32:2l-31.
See Appendix B for further comments on Schlegel's article.
I.e. Absolute Geometry. Cf. Borsuk and Szmielow, loc. cit.
"What is Cantor's Continuum Problem?" in Benacerraf and Putnam, pp. 263-264
Philosophy of Science, 1965, 32:22
Ibid., pp. 22-24
Benacerraf and Putnam, p. 271
"Russell's Mathematical Logic," in Benacerraf and Putnam, p. 220
Cf. W. V. Quine's claim that "among the various conceptual schemes...the phenomenalistic--claims epistemological priority." ("On What There Is," in Benacerraf and Putnam, p. 196) Nominalists are not in agreement on this issue. Goodman argues against the plausibility of a sharp distinction in Chapter IV of The Structure of Appearance, 2nd ed. (New York, 1966), especially strongly in Chapter IV, Section 4.
"Russell's Mathematical Logic," in Benacerraf and Putnam, p. 220
"What is Cantor's Continuum Problem?" in Benacerraf and Putnam, p. 272
We are indebted to Professor Capaldi for this observation. Indeed most accomplished mathematicians would admit that Gödel's mathematical eyesight" is exceptionally acute.
This position is crystallized by Professor Bar-Hillel in "On a neglected ontology-free philosophy of mathematics," Problems in the Philosophy of Mathematics, T. Lakatos, ed. (Amsterdam, 1997), p. 136. Professor Robinson states his position in "Formalism 64," of Logic, Methodology, and Philosophy of Science, T. Bar-Hillel, ed, (Amsterdam, 1965), pp. 228-246. (Bar-Hillel discusses Carnap's Position.)
"Russell's Mathematial Logic," in Benacerraf and Putnam, p. 213
See Chapter I, Section 5, for a discussion of Gödel's concept of verification by calculation.
"Russell's Mathematical Logic," in Benacerraf and Putnam, p. 213
Experiments with physical objects are discussed in Foundations of Geometry and Trigonometry, by H. Levi.
H. Hermes, Enumerability, Decidability, Computability, translated by G. T. Herman and 0. Plassmann (New York, 1965, p. 3, footnote 1
This aspect of Gödel's methodology, as well as the relationship between generalizations of intuitively desirable theorems and higher axioms is indicated in Gödel's The Consistency of the Continuum Hypothesis, Seventh Printing (Princeton, 1966), p. 70, Note 12. In addition to the references cited we should like to add: H. Gaifman, "Infinite Boolean Polynomials, I," Fundamenta Mathematicae, 1964, 54:229-250. Cf. also P. Suppes, Axiomatic Set Theory (Princeton, 1960), Section 8.3, pp. 251-252
"What is Cantor's Continuum Problem?" in Benacerraf and Putnam, p. 262
E.g. Borel, whose vehement objections to the axiom of choice were cited above, defines a known number to be a calculable number. Borel disagrees with Sierpinski over the notion of effectively denumerable. (Cf. W. Sierpinski, Cardinal and Ordinal Numbers, Chapter II, especially pp. 34-36, and E. Borel Eléments de la Théorie des Ensembles (Paris, 1949), Section 66, pp. 229-232.)
Apparently, much of the disagreement over terminology, as well as acceptability of axioms, stems from confusing the meaning of a term in everyday speech or technology, with its meaning in mathematical contexts which should be recognized as metaphorical, but nevertheless syncategorematic. For example, it has been argued against the axiom of choice that it is impossible to make an "infinity" of choices, as though it were a physical process. Similar objections have been voiced against the power set axiom in the case of non-denumerable sets. Lusin calls the power set of the set of all reals a "totalité illégitime." (Cf. Sierpinski, Cardinal and Ordinal Numbers, p. 84, and other references cited there.) In each case, objections invariably involve what may or may not be done in the case of infinite sets. It is our contention that arguments about the various axioms inevitably reduce to a question of the meaningfulness of the concept of infinity, since each of the various "paradoxical" results tends to be a product of holomerism rather than the other axioms.
Mathematical knowledge is not generally divided into categories, especially where a classical existence proof is considered sufficient without further comment. Sierpinski's Elementary Theory of Numbers, however, employs classical logic throughout but is particularly sensitive to the distinction between beinig able to prove that a method yields desired results, and actually carrying out the process to obtain a numerical solution. The following example is indicative of the various mathematical senses of 'know':
As has been proved above, for every natural number n > 1 we are able to find the factorization into primes effectively provided we are not daunted by long calculations, which may possibly occur.In some cases these are too long to be carried out even with the aid of the newest technical equipment. For instance this happens in the case of the number 2101 - 1 , which has 31 digits. (We know that this number is composite.) We do not know any of its prime divisors, although we do know that the least of them has at least 8 digits. We do not know any of the prime divisors of the number F19 = 2219 - 1 , either. It is not known whether this number is a prime or not. We know a prime divisor of the number F1945 , namely 5 * 21947 + 1 , although we do not know any other of its prime divisors, which, as we know, do exist.
--W. Sierpinski, Elementary Theory of Numbers, translated by A. Hulanicki (Warsaw, 1964), p. 112. (The Corrigendum adds that 2101 - 1 has been factored.)
It is to be observed that Gödel's view of the existence of mathe,matical objects glosses over these distinctions, and, as we have seen, the existence of a mathematical object in no way depends upon an actual calculation, or even the possibility of one:
...it is possible that the existence of an actual calculation is contradictory to the laws of nature (e.g. because there is not enough material in the world for writing down the result in decimal notation, or because mankind does not exist long enough for the effective carrying out of such a calculation).--Hermes, Enumerability, p. 5
The American College Dictionary (New York, 1960), p. 1008: 'real', Definition 7 "Philosophy", part c--"independent of experience as opposed to phenomenal or apparent."
"What is Cantor's Continuum Problem?" in Benacerraf and Putnam, p. 263
Cardinal and Ordinal numbers, p. 91
"What is Cantor's Continuum Problem?" in Benacerraf and Putnam, p. 262
Cf. Gödel's remark about arguments from analogy in Note 2, p. 70 of The Consistency of the Continuum Hypothesis, Seventh Printing, discussed above. (Chapter II.1, p. 38)
R. Descartes, Meditations on First Philosophy, 2nd ed., translated by Laurence J. Lafleur (New York, 1960), p. 61
See Appendix B of Chapter I
D. Hume, A Treatise of Human Nature, L. A. Selby-Bigge, ed. (Oxford, 1964), p. 235
Note that in the Index, prepared by Selby-Bigge we find under "Existence" the remark that: "reality of objects of mathematics prove by our possession of a clear idea of them." (Hume, Treatise, p. 659) Cf. the various textual references given there and under "Mathematics."
The literature, in making Gödel a Platonist, a Kantian, a Cantorian, and so forth, seems to have overlooked this possibility. We choose to view Gödel as a unique philosopher and not try to classify him.
A. Robinson, who held in "Formalism 64" (loc. cit.) that infinite structures do not exist and are meaningless, did not however deny that the concept of infinity was unclear. It is necessary to observe in what sense Professor Robinson has employed what we consider pejoratives; in his sense they are not.
Indeed Borel conceded:
La théorie des ensembles s'est développée encore bien davantage au cours des vingt dernières années. Ce développement est caractérisé par une tendance de plus en plus grande vers l'abstrait.... Ces déductions de plus en plus abstraites supposent, comme nous allons le voir, des postulats tels que l'axiome de Zermelo ou la notion des nombres transfinis de Cantor ; mais on ne peut contester le droit des mathématicians à utiliser ainsi des postulats, à condition qu'ils précisent ces postulats, de manière que l'on connaisse avec précision les hypothèses sous lesquelles sont valables les résultats obtenus.--Théorie des Ensembles, p. 225
Compare Gödel's usage of 'force' with Hume's usage of 'vivacity'. Hume uses 'force' and 'vivacity' synonymously:
An idea assented to feels different from a fictitious idea, that the fancy alone presents to us: And this different feeling I endeavor to explain by calling it a superior force, or vivacity, or solidity, or firmness, or steadiness. This variety of terms, which may seem so unphilosophical, is intended only to express that act of the mind, which renders realities more present to us than fictions, causes them to weigh more in the thought, and gives them a superior influence on the passions and imagination.--Hume, Treatise, p. 629
(Perhaps Gödel rejected Russell's notion of "logical fiction" on these grounds.) See Appendix A, Chapter II for views of Locke and Leibniz. Both rejected a "fictional' view of mathematical entities.
In contradistinction to A. Robinson's view of what may actually be done, Paul J. Cohen remarked:
There is no reason to believe that in the real world this process cannot be done countably many times and yield finally a countable standard model for ZF.--Set Theory and the Continuum Hypothesis (New York, 1966), p. 79
We wish to leave the question of whether a denumerably infinite (unendlich) process can "finally yield" a model open at present. Gödel as well employs 'construction' in the traditional mathematical sense:
Paul J. Cohen...invented a powerful method for constructing denumerable models.--Continuum Hypothesis, 7th printing, p. 70
This may be seen as an outgrowth of logical positivism, although we hesitate to place any labels. Cf. Professor Carnap's The Logical Structure of the World/Pseudoproblems in Philosophy (Los Angeles, 1967), Part V, Section C, pp. 273-28O of The Logical Structure of the World: "The Constructional or Empirical Problem of Reality," and Section D, pp. 281-287: "The Metaphysical Problem of Reality;" also Pseudoproblems in Philosophy, Section II, Part B, pp. 332-343: "Application to the Realism Controversy."
Cf. also "Empiricism, Semantics, and Ontology" reprinted in Meaning and Necessity, 4th ed. (Chicago, 1964). Here Carnap defines 'real':
To recognize something as a real thing or event means to succeed in incorporating it into the system of things at a particular space-time position so that it fits together with the other things recognized as real, according to the rules of the framework.--Meaning and Necessity, p. 207
Loc. cit., pp. 63-67
"Empiricism, Semantics, and Ontology," Meaning and Necessity, Section 2: 'Linguistic Frameworks,' pp. 206-213
A. A. Fraenkel and T. Bar-Hillel, Foundations of Set Theory (Amsterdam, 1958), p. 342. [The Preface (p. x) states that Bar-Hillel was the principal author of this section although both authors assume the responsibility for the views expressed.]
Ibid., p. 346
N. Goodman and W. V. 0. Quine, "Steps toward a Constructive Nominalism," Journal of Symbolic Logic, 1947, 12:105-122. Neither Goodman nor Quine holds this view at present but the remark has caused quite a stir in the literature.
W. V. 0. Quine, Set Theory and its Logic (Cambridge, 1963), p. 329. Both Quine and Gödel ("What is Cantor's Continuum Problem?" in Benacerraf and Putnam, p. 262, footnote 12) refer to Saunders MacLane's "Locally small categories and the foundations of set theory," in Infinitistic Methods, proceedings of a 1959 symposium (Warsaw, 1961), pp. 25-43.
Ibid., Quine, p. 28
Loc. cit.
John Locke: An Essay Concerning Human Understanding, A. D. Woozley, ed. (New York, 1964), Section 6, p. 349
Ibid., Section 8, p. 350
Gottfried Wilhelm Leibniz: Philosophical Papers and Letters, Vol. I (Chicago, 1956), pp. 278-283