On Gödel's Philosophy of Mathematics, Notes and References


On Gödel's Philosophy of Mathematics, Note 1


K. Gödel, "What is Cantor's Continuum Problem?" in P, Benacerraf and H. Putnam, eds., Philosophy of Mathematics (Englewood cliffs, 1964), p. 262

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On Gödel's Philosophy of Mathematics, Note 2


K. Gödel, "Russell's Mathematical Logic," in P. Benacerraf and H. Putnam, eds., Philosophy of Mathematics (Englewood Cliffs, 1964), pp. 215-216

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On Gödel's Philosophy of Mathematics, Note 3


Gödel, "What is Cantor's Continuum Problem?" in Benacerraf and Putnam, pp. 262-263.

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On Gödel's Philosophy of Mathematics, Note 4


Ibid., p. 262

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On Gödel's Philosophy of Mathematics, Note 5


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

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On Gödel's Philosophy of Mathematics, Note 6


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

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On Gödel's Philosophy of Mathematics, Note 7


Cf. J. B. Rosser, Logic for Mathematicians (New York, 1953), pp. 197-207

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On Gödel's Philosophy of Mathematics, Note 8


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

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On Gödel's Philosophy of Mathematics, Note 9


As described in Feferman's "Systems"

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On Gödel's Philosophy of Mathematics, Note 10


Ibid., p. 4

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On Gödel's Philosophy of Mathematics, Note 11


Ibid.

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On Gödel's Philosophy of Mathematics, Note 12


Gödel "Russell's Mathematical Logic," in Benacerraf and Putnam, pp. 218-219

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On Gödel's Philosophy of Mathematics, Note 13


E.g. Kleene, Introduction, Section 12; Feferman, Journal of Symbolic Logic, p. 2

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On Gödel's Philosophy of Mathematics, Note 14


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.

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On Gödel's Philosophy of Mathematics, Note 15


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.'"

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On Gödel's Philosophy of Mathematics, Note 16


J. Pierpont, Lectures on The Theory of Functions of Real Variables, Vol. I (Boston, 1905), p. 82

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On Gödel's Philosophy of Mathematics, Note 17


L. M. Graves, The Theory of Functions of Real Variables, 2nd ed. (New York, 1956), p. 14

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On Gödel's Philosophy of Mathematics, Note 18


Journal of Symbolic Logic, especially p. 2

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On Gödel's Philosophy of Mathematics, Note 19


"Russell's Mathematical Logic," p. 219

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On Gödel's Philosophy of Mathematics, Note 20


Kleene, p. 43. This argument is due to F. Ramsey.

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On Gödel's Philosophy of Mathematics, Note 21


Cf. S. Feferman, "Geometric Motivations," The Number Systems (Reading, 1964), pp. 224-226

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On Gödel's Philosophy of Mathematics, Note 22


Ibid., p. 226

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On Gödel's Philosophy of Mathematics, Note 23


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

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On Gödel's Philosophy of Mathematics, Note 24


"Russell's Mathematical Logic," p. 218, footnote 18

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On Gödel's Philosophy of Mathematics, Note 25


A. Tarski., A. Mostowski, and R. M. Robinson, (Amsterdam, 1953), p. 57

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On Gödel's Philosophy of Mathematics, Note 26


R. C. Lyndon, Notes an Logic (Princeton, 1966), pp. 9-10

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On Gödel's Philosophy of Mathematics, Note 27


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.

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On Gödel's Philosophy of Mathematics, Note 28


"What is Cantor's Continuum Problem?" in Benacerraf and Putnam, p. 261

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On Gödel's Philosophy of Mathematics, Note 29


Ibid., p. 262. E.g. Von Neumann--Bernays--Gödel Set Theory.

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On Gödel's Philosophy of Mathematics, Note 30


Cf. Kleene, p. 498

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On Gödel's Philosophy of Mathematics, Note 31


"Zur intuitionistischen Arithmetik und Zahlentheorie," Ergebnisse eines matematischen Kolloquiums, Vol. 4, pp. 34-38

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On Gödel's Philosophy of Mathematics, Note 32


Cf. L. E. J. Brouwer, "Intuitionism and Formalism," in P. Benacerraf and H. Putnam, eds., Philosophy of Mathematics, pp. 66-77

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On Gödel's Philosophy of Mathematics, Note 33


Professor Church has pointed out that Heyting now maintains, in the light of Gödel's result, that intuitionistic arithmetic is superior to classical, not as narrower, but as richer--it makes distinctions that the classical arithmetic fails to make.

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On Gödel's Philosophy of Mathematics, Note 34


We are indebted to Professor Kreisel for this observation.

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On Gödel's Philosophy of Mathematics, Note 35


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

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On Gödel's Philosophy of Mathematics, Note 36


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)

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On Gödel's Philosophy of Mathematics, Note 37


"What is Cantor's Continuum Problem?" in Benacerraf and Putnam, p. 270

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On Gödel's Philosophy of Mathematics, Note 38


Ibid., p. 271

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On Gödel's Philosophy of Mathematics, Note 39


Cf. A. Church, Introduction to Mathematical Logic, Volume I (Princeton, 1956), p. 56

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On Gödel's Philosophy of Mathematics, Note 40


"What is Cantor's Continuum Problem?" in Benacerraf and Putnam, p. 270. Higher systems may have stronger languages.

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On Gödel's Philosophy of Mathematics, Note 41


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)

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On Gödel's Philosophy of Mathematics, Note 42


"What is Cantor's Continuum Problem?" in Benacerraf and Putnam, p. 272

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On Gödel's Philosophy of Mathematics, Note 43


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.

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On Gödel's Philosophy of Mathematics, Note 44


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.

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On Gödel's Philosophy of Mathematics, Note 45


I.e. Absolute Geometry. Cf. Borsuk and Szmielow, loc. cit.

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On Gödel's Philosophy of Mathematics, Note 46


"What is Cantor's Continuum Problem?" in Benacerraf and Putnam, pp. 263-264

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On Gödel's Philosophy of Mathematics, Note 47


Philosophy of Science, 1965, 32:22

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On Gödel's Philosophy of Mathematics, Note 48


Ibid., pp. 22-24

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On Gödel's Philosophy of Mathematics, Note 49


Benacerraf and Putnam, p. 271

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On Gödel's Philosophy of Mathematics, Note 50


"Russell's Mathematical Logic," in Benacerraf and Putnam, p. 220

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On Gödel's Philosophy of Mathematics, Note 51


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.

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On Gödel's Philosophy of Mathematics, Note 52


"Russell's Mathematical Logic," in Benacerraf and Putnam, p. 220

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On Gödel's Philosophy of Mathematics, Note 53


"What is Cantor's Continuum Problem?" in Benacerraf and Putnam, p. 272

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On Gödel's Philosophy of Mathematics, Note 54


We are indebted to Professor Capaldi for this observation. Indeed most accomplished mathematicians would admit that Gödel's mathematical eyesight" is exceptionally acute.

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On Gödel's Philosophy of Mathematics, Note 55


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.)

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On Gödel's Philosophy of Mathematics, Note 56


"Russell's Mathematical Logic," in Benacerraf and Putnam, p. 213

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On Gödel's Philosophy of Mathematics, Note 57


See Chapter I, Section 5, for a discussion of Gödel's concept of verification by calculation.

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On Gödel's Philosophy of Mathematics, Note 58


"Russell's Mathematical Logic," in Benacerraf and Putnam, p. 213

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On Gödel's Philosophy of Mathematics, Note 59


Experiments with physical objects are discussed in Foundations of Geometry and Trigonometry, by H. Levi.

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On Gödel's Philosophy of Mathematics, Note 60


H. Hermes, Enumerability, Decidability, Computability, translated by G. T. Herman and 0. Plassmann (New York, 1965, p. 3, footnote 1

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On Gödel's Philosophy of Mathematics, Note 61


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

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On Gödel's Philosophy of Mathematics, Note 62


"What is Cantor's Continuum Problem?" in Benacerraf and Putnam, p. 262

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On Gödel's Philosophy of Mathematics, Note 63


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 being 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 mathematical 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

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On Gödel's Philosophy of Mathematics, Note 64


The American College Dictionary (New York, 1960), p. 1008:  'real', Definition 7 "Philosophy", part c--"independent of experience as opposed to phenomenal or apparent."

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On Gödel's Philosophy of Mathematics, Note 65


"What is Cantor's Continuum Problem?" in Benacerraf and Putnam, p. 263

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On Gödel's Philosophy of Mathematics, Note 66


Cardinal and Ordinal numbers, p. 91

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On Gödel's Philosophy of Mathematics, Note 67


"What is Cantor's Continuum Problem?" in Benacerraf and Putnam, p. 262

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On Gödel's Philosophy of Mathematics, Note 68


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)

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On Gödel's Philosophy of Mathematics, Note 69


R. Descartes, Meditations on First Philosophy, 2nd ed., translated by Laurence J. Lafleur (New York, 1960), p. 61

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On Gödel's Philosophy of Mathematics, Note 70


See Appendix B of Chapter I

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On Gödel's Philosophy of Mathematics, Note 71


D. Hume, A Treatise of Human Nature, L. A. Selby-Bigge, ed. (Oxford, 1964), p. 235

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On Gödel's Philosophy of Mathematics, Note 72


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

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On Gödel's Philosophy of Mathematics, Note 73


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.

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On Gödel's Philosophy of Mathematics, Note 74


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

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On Gödel's Philosophy of Mathematics, Note 75


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

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On Gödel's Philosophy of Mathematics, Note 76


Loc. cit., pp. 63-67

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On Gödel's Philosophy of Mathematics, Note 77


"Empiricism, Semantics, and Ontology," Meaning and Necessity, Section 2:  'Linguistic Frameworks,' pp. 206-213

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On Gödel's Philosophy of Mathematics, Note 78


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.]

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On Gödel's Philosophy of Mathematics, Note 79


Ibid., p. 346

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On Gödel's Philosophy of Mathematics, Note 80


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.

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On Gödel's Philosophy of Mathematics, Note 81


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.

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On Gödel's Philosophy of Mathematics, Note 82


Ibid., Quine, p. 28

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On Gödel's Philosophy of Mathematics, Note 83


Loc. cit.

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On Gödel's Philosophy of Mathematics, Note 84


John Locke: An Essay Concerning Human Understanding, A. D. Woozley, ed. (New York, 1964), Section 6, p. 349

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On Gödel's Philosophy of Mathematics, Note 85


Ibid., Section 8, p. 350

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On Gödel's Philosophy of Mathematics, Note 86


Gottfried Wilhelm Leibniz:  Philosophical Papers and Letters, Vol. I (Chicago, 1956), pp. 278-283

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Copyright (c) 1968, 1998 Harold Ravitch, Ph.D. All Rights Reserved