Abstract
Proclus spoke for antiquity in branding the infinite as incomprehensible and unknowable, remarking that “the imagination recognizes the infinite by not understanding it.” Philosophers long maintained the incomprehensibility of the infinite — according to Descartes, “incomprehensibility is comprised within the. formal concept of the infinite” — though they later tried to explain it by reference to the limitations of the human mind. Hume still spoke for many when he said that “the capacity of the human mind is limited, and can never attain a full and adequate conception of infinity”, where by ‘adequate conception’ he meant a clear and distinct image in the mind. Hobbes had already elaborated this thesis as follows: Whatsoever we imagine is fiinite. Therefore there is no idea or conception of anything we call infiinite. No man can have in his mind an image of infinite magnitude, nor conceive infinite swiftness, infinite time, infinite force, or infinite power. When we say anything is infinite, we signify only that we are not able to conceive the ends and bounds of the thing named, having no conception of the thing but of our own inability ([131], p. 36).
In science nothing capable of proof ought to be accepted without proof.
R. Dedekind
Our ideas are not always proofs of the existence of things.
J. Locke
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Notes
Hence, for Locke, our ideas need not prove the existence of things even if they are clear and distinct.
One could perhaps invoke unconscious mental processes which continue, so to speak, ‘on their own’, but infinite repetition hardly seems plausible as an unconscious desire of mankind, unless perhaps, we view the desire for immortality as such.
[169], p. 292.
See Cantor [44], p. 409. See Lodge [171] for modern scepticism about continuity.
See Kustaanheimo [160] for an introduction to this program and Kanitscheider [141], for a discussion of how it approximates differential equations of classical physics in finite geometries by recursions. In fact, there are several research programs underway seeking to solve fundamental problems in physics by introducing discrete space and time.
These geometries still contain the parallel postulate, but no continuity axioms, and modified order and congruence axioms. See Kustaanheimo [159].
These attempts are discussed at the end of Chapter III below. Becker [6] formulated a certain principle of ‘transcendental idealism’ (which is very close to a basic assumption made by Kaufmann) according to which questions undecidable in principle could not exist. Carnap [46], after arguing for the finite decidability of all questions on the basis of his pheonomenalistic ‘construction theory’, goes on to express his agreement with both positivism and idealism on such questions In the thesis of the decidability of all questions, we agree with positivism as well as idealism; cf. Becker: .. . “According to the principle of transcendental idealism, a question which is in principle (in essence) undecidable does not have any meaning at all . . . For there are no states of affairs which are in principle inaccessible to consciousness” ([46], p. 292).
Gödel accepted Turing’s analysis as providing a definitive clarification of a ‘formal system’, but not for ‘effectiveness’ in general. His reservations are discussed in Chapter IV. I have followed the formulation in Kreisel [155] of Gödel’s argument.
See Wang [258], pp. 224ff. for a discussion of the lack of any correlation between those domains known to be theoretically decidable or undecidable and those, respectively, for which we have had success in programs for mechanical theorem-proving or those for which we have not. Indeed, he emphasizes that the best such programs have so far been for subdomains of theoretically undecidable domains.
See Smith [236], Vol. I, p. 294.
See Tarski [245], p. 54.
See Frege [87], pp. 8ff.
See Buck [39] for a detailed discussion and proof of this example. The example below of an inconsistent recursion is also taken from Buck.
See Brouwer [36], [38].
I am following the elegant formulation of the Dedekind-Peano theory of numbers given by Henkin [113].
This is equivalent to Dedekind’s requirement that N should be the ‘chain’ of 0, i.e. the intersection of all sets including 0 and closed under’. This is his basis for (D5).
See Steiner [239] for a discussion of the role of isomorphism in the Hilbert-Frege controversy. We discuss it in Section 2 of Chapter III.
See Henkin [113], pp. 337ff.
Veronese [257], p. 45, complained of Dedekind’s use of the “elegant theorem” (2) to reduce meaningful acts of calculation to purely symbolic manipulations.
See Buck [39], pp. 133ff., for a discussion of the difficulty of even devising a notation for an explicit description of such functions.
This view, of course, is very common among various kinds of ‘constructivists’.
See Henkin [113] for a discussion of this point.
See below for discussion of this difficult point. See also Helm [112].
Bolzano’s assumption that propositions with different subjects must themselves be different is also made by Dedekind in defending (P2)’; see below.
Frege’s formulation runs as follows: So the sense of the word ‘true’ is such that it does not make any essential contribution to the thought. If I assert that ‘it is true that sea-water is salty’, I assert the same thing as if I assert ‘sea-water is salty’. This enables us to recognize that the assertion is not to be found in the word ‘true’, but in the assertoric force with which the sentence is uttered ([90], p. 251).
Essentially this criticism of Bolzano’s argument was made by König, [152], p. 83.
A lucid and direct formulation of this shift can be found in the introduction of Lyndon [178]. He begins by noting that: Logic is often said to deal with the laws of thought. Here is meant not the historical or psychological principles governing the processes of thought, but rather those formal structural properties of thought which appear to reflect properties of the real world ([178], p. 1).
But for a theoretical development we need suitable idealizations for thought, reality, and their relation: For thought we substitute language, or, more precisely, a formalized version of parts of everyday language . . . For reality we substitute something called a structure, which is hardly more than a collection of things suitable for being correlated, as meanings, to various expressions in the language. For the connection between thought and reality we substitute an interpretation, that is, a function assigning to certain expressions in the language, as their meanings under the interpretation, certain objects in the stucture (ibid., p. 2).
See Moss [189] for the evolution of Russell’s thinking on infinity.
Strangely enough many writers willing to bracket the parallel postulate still cling to the incommeasurabilities that depend on it. In elliptic and hyperbolic geometry the diagonal of a square can be commeasurable with its side; only in Euclid’s must they be incommeasurable, for here their ratio must be the same in all squares. Many still imagine that incommeasurability in turn implies infinite divisibility, but Berkeley indicated long ago the correct line in a note to himself, “to inquire most diligently concerning the incommeasurability of diagonal and side — whether it does not go on the supposition of units being divisible ad infiinitum . . . and so the infinite divisibility deduced therefrom is a petitio principi” ([15a], p. 79). He saw clearly that “in geometry it is not prov’d that an inch is divisible ad infiinitum” (ibid., p. 77). This allowed him to entertain a “diagonal of a particular square commeasurable with its side, they both containing a certain number of m. v.” (ibid.), thereby harmonizing geometry with his theory of vision as comprising a finite number of “minimum visibles” which he would identify with its points. Berkeley’s critics have not appreciated the resources of his finitism, some even believing that incommeasurability and the pythagorean theorem refute it. But he met this objection squarely: “One square cannot be the double of another. Hence the Pythagoric theorem is false” (ibid., p. 19). Berkeley was wrestling with what a contemporary geometer still calls the “Pythagorean paradox”: although “it was intuitively evident to the Pythagoreans, and as I write it is intuitively evident to me, that a common measure can be found for any pair of straight lines” ([251], p. 15), their theorem together with arithmetic reasoning implies that none can exist for the diagonal and side of a square. Arithmetic logic denies what geometric intuition affirms. Non-euclidean geometry lessens the conflict somewhat, but only at the cost of intuitively appealing features of Euclid’s. Berkeley’s finitism cries out rather for the finite geometries of Järnefelt and Kustaanheimo, constructed from the residue classes of integral numbers modulo a prime number, which have euclidean geometry as their limit. Indeed, In the Euclidean geometry certain properties of the finite geometry are smoothed out. But then the finer details, i.e. the fine structure of the plane, space, a.s.o. depending on the repartition of the prime numbers, have disappeared. One could also say that the Euclidean geometry draws a veil over the finer details of the plane, space, a.s.o. This fine structure comes into view only in the finite geometrical models ([138], p. 168).
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Webb, J.C. (1980). Mind, Number, and the Infinite. In: Mechanism, Mentalism and Metamathematics. Synthese Library, vol 137. Springer, Dordrecht. https://doi.org/10.1007/978-94-015-7653-6_2
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