Abstract
An attempt was made to show how we can plausibly commit to mathematical realism. For the purpose of illustration, a defence of natural realism for arithmetic was developed that draws upon the American pragmatist’s, Hillary Putnam’s, early and later writings. Natural realism is the idea that truth is recognition-transcendent and knowable. It was suggested that the natural realist should embrace, globally, what N. Tennant has identified as M-realism (Tennant 1997, 160). M-realism is the idea that one rejects bivalence and assents to the recognition-transcendent requirement. It was argued that over-all—for all domains—the natural realist should be a M-realist, with the aim of clarifying the realist debate for arithmetic.
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Notes
Wright says that various regions of discourse may be more successful for realism than others (Wright 1993, 2).
The term natural realism is used by Putnam in his Dewey Lectures (Putnam 1999, xii), but the usage here is completely independent of that and must not be confused with what he intends.
Brouwer writes, “Classical algebra of logic, founded by Boole, developed by De Morgan, Jevons, Peirce, and perfected by Schröder, furnishes a formal image of the laws of common-sensical thought. This common-sensical thought is based on the following, conscious or subconscious, threefold belief: First, in a truth existing independently of human thought and expressible by means of sentences called ‘true assertions’, mainly assigning certain properties to certain objects or stating that objects possessing certain properties exist or that certain phenomena behave according to certain laws. Furthermore in the possibility of extending one’s knowledge of truth by the mental process of thinking, in particular thinking accompanied by linguistic operations independent of experience called ‘logical reasoning’, which to a limited stock of ‘evidently’ true assertions mainly founded on experience and sometimes called axioms, contrives to add an abundance of further truths. Finally, using the term ‘false’ for the converse of true, in the so-called ‘principle of the excluded third’ saying that each assertion, in particular each existence assertion and each assignment of a property to an object or of a behaviour to phenomenon, is either true or false, independently of human beings knowing about this falsehood or truth...” (Brouwer 1975, 551).
Brouwer writes, “Only after intuitionism had recognized mathematics as a autonomic, interior constructional mental activity which although it has found extremely useful linguistic expression and can be applied to an exterior world, nevertheless neither in its origin nor in the essence of its method has anything to do with language or an exterior world, on the one hand, axioms become illusory, on the other hand the criterion of truth and falsehood of a mathematical assertion was confined to mathematical activity itself, without appeal to logic or to a hypothetical omniscient being” (Brouwer 1975, 551). Dummett notes that intuitionists, however, are wary of extending ideas about meaning beyond mathematics (Dummett 1977, 138).
Wright 1993, 85.
As Wright concludes, the anti-realist shows a failure of nerve by separating “in principle” from “actual” (Wright 1993, 32).
Griffin writes, “The correspondence theory of truth states that a proposition is true if and only if it ‘corresponds’ to the way the world is. A proposition whose truth actually transcends our ability to recognize it, may still correspond to the world, but, of course, we will not, in this case, be able to recognize how the world is in this respect” (Private Communication: 18 September 1999).
Personal Communication from N. Griffin, 18 September 1999: “Realism is consistent with scepticism, but does not entail it. Scepticism (about subject S) asserts that we know nothing about S, i.e., that the truth-values of all propositions about S actually transcend our ability to recognize them. Realism merely asserts that these truth-values may transcend our ability to recognize them…Most realists would claim that we do know the truth-values of some propositions about S, but not the truth-values of others. But two limiting cases are possible: (i) the sceptical realist who says that in fact we don’t know the truth-values of any of them; (ii) the ‘optimistic’ optimistic realist, however believes that truth is recognition-transcendent; that the fact that we do know the truth-values is not what makes the propositions true; that it is possible that we didn’t know them and this would not affect what the values were—we’re just incredibly fortunate in knowing all of them. Neither of these extreme positions is very plausible. Anti-realism, by contrast, is incompatible with scepticism. This is its one big attraction.”
According to a natural realist decidable in principle only requires an abidance of the law of non-contradiction: ¬ (∃x)(Px & ¬Px); and LEM: (∀x)(Px v ¬Px), with an exclusive disjunction. Natural realism is thus more liberal (more realist) than Dummett’s notion of “decidable in principle” (Dummett 1977, v, 19, 24; 1978, xxvii, 6, 248).
Wright 1993, 240.
Wright 1992, 109, 148.
Tennant provides a usefully summary of the incompleteness results: [N]o acceptable formal system can capture all the truths of classical arithmetic (Tennant 1997, 189): ¬ (∃S)(∀ϕ) (ϕ is true → S | ϕ). The incompleteness results can be construed as a challenge to the anti-realist, i.e., they provide the possibility of a true but undecidable statement (Gardiner 2000, 97). Dummett’s solution is to reject that there are true but undecidable statements. Gödel’s solution is is to accept that there is some undecidable statements (those which are undecidable now) have truth-values and are meaningful (Gödel 1990, 268). Suffice it to say that Gödel’s incompleteness theorem does not offer a knock-down argument against anti-realism (or realism).
See:(Butterworth 1999, 63–4).
One could be an orthodox realist about the Peano axioms, but retain anti-realism for the axiom of choice. Definition(Every set has a choice function). Let S be a set and F be a local function. Then by definition f is a choice function for S: iff: [f: P(S) - {0} → S and ∀x , P (S) - {0} [f’x , x]]. Thus a choice function f for a non-empty set S “chooses” an element f’T from every non-empty subset T ⊆ S. Of course every such T has such an element. The problem is that, in the general case, we have no way of uniformly describing a particular element in each such set. This is why we must posit a choice function, which makes the selection for us, so to speak. Of course the empty set is a choice function for the empty set (Mayberry 2000, 163).
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Gupta, A. The Truth about Realism: Natural Realism, Many Worlds, and Global M-Realism. Philosophia 47, 1487–1499 (2019). https://doi.org/10.1007/s11406-019-00060-0
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DOI: https://doi.org/10.1007/s11406-019-00060-0