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2014 | OriginalPaper | Buchkapitel

3. Intuitionism

verfasst von : Andreas Kapsner

Erschienen in: Logics and Falsifications

Verlag: Springer International Publishing

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Abstract

As promised, this chapter should supply enough information about intuitionistic mathematics and its logic to make it possible to follow the rest of the book. I will begin by saying more about Brouwer’s philosophy and the practical consequences for mathematics he and his followers drew from it. Next, I will present the logic and some semantical theories for it, with an emphasis on the BHK interpretation and the Kripke semantics.

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Vorheriges Kapitel Constructivism

Nächstes Kapitel Gaps, Gluts and Paraconsistency

One should, for full effect, turn to his collected works. Especially, references to Indian mysticism have been purged in the reprints one finds in important anthologies such as Benaceraff and Putnam’s The Philosophy of Mathematics.

An excellent resource for comparisons between Brouwer’s, Heyting’s and Dummett’s views is Placek (Placek 1999). A shorter overview is in Chap. 7 of Shapiro (2000).

Even though this example is repeated over and over again, it is a very recent trend to add the solution to the intuitionist’s challenge: \(\sqrt{2}^{\sqrt{2}}\) is irrational. The fact of the matter is that this has only recently been established, and the proof is said to be rather complicated (cf. Burgess (2009), p. 121).

Warning: In Brouwer’s writing, this is called “Law of Excluded Third”; I follow Dummett in attaching this label to the double negation of LEM.

See Dummett (2000).

See also TOE, p. 293, and Dummett (1998), p. 126.

This seems to be one of the points at which a certain amount of idealization of our capacities is required. It would not seem right to attribute understanding of Fermat’s Last Theorem only to those few who could recognize Wiles’s proof to be correct. Even worse is the proof of the Four Color Theorem, which has been proven, but only with the aid of computers that surveyed an extensive range of cases too large for any single human being to check. Rather than to say that we really do not understand these mathematical theorems, N. Tennant proposes the following refinement of Dummett’s requirement (picking up an idea he attributes to J. Cogburn):

“For a speaker S to be credited with a grasp of the meaning of a sentence \(\phi \) we should have good grounds for believing that, if presented with some finite piece of discourse \(\pi \), S would be able to deliver a correct verdict on any aspect of \(\pi \) that is relevant to arriving at a correct judgement of the form ‘\(\pi \) is a proof of \(\phi \)’ or of the form ‘\(\pi \) is a disproof of \(\phi \)’ or of the form ‘\(\pi \) is neither a proof nor a disproof of \(\phi \)’, that is, for any such aspect \(\alpha \), S would, after some time, be able to judge whether \(\alpha \) was as it ought to be, in order for \(\pi \) to have the status in question (Tennant (2002), p. 154).”

In other words, one has to be able to recognize each step in the proof as correct or incorrect, even if one cannot survey the whole proof.

See Williamson (1994), who adds insult to injury by arguing that the intuitionist cannot even say that, by way of historical coincidence, there is a chance that we might not actually decide whether Goldbach’s conjecture or its negation holds.

Cf. van Dalen (2004).

Cf. Wansing (1993), p. 21.

“I’ll give a million dollars to the local orphanage... when pigs fly.”

Introduced in Johansson (1936).

Also, expect to see \(t\ge s\) as a notational variation of \(s\le t\) occasionally.

As far as intuitionistic logic is considered, it does not hurt to restrict one’s attention to trees. The consequence relation that arises from tree models is not different from the consequence relation one gets from the more inclusive set of partially ordered models. This changes when co-implications are added to the vocabulary of intuitionistic logic, cf. Restall (1997). But I shall leave co-implications for a later point in the story.

That the BHK interpretation is alluded to in this paragraph is not accidental to the exposition. I will argue presently (in Sect. 3.7.3) that BHK and Kripke semantics need each other to satisfy Dummett’s needs.

Happily, this has no effect on the logic.

Dummett comes to a similar conclusion, although he is concentrating in his discussion on Beth trees (Dummett (2000), p. 287). The difference between Kripke models and Beth trees is not too great, at least in those respects that interest me here. I choose to discuss Kripke models because it is much easier to compare them to the semantics for the logics that follow (at least for those logics that have been characterized and given semantics in the literature).

One may not wish to think of this option as anything distinct from option 1. If we know of a method of obtaining a proof, then we have a proof, or else we would not know that we would obtain a proof. This depends a bit on how closely individuated proofs are taken to be, and on how strict we are with the term “proof.” For example, if an informal but valid argument is a proof, then there might be not much point in drawing the distinction between (1) and (2). Nothing much in the following argument will change if they are collapsed.

Prawitz (1987), p. 157.

See Martino and Usberti (1994) for an extended argument against Prawitz’s proposal and untensed constructive truth in general.

Dummett notes that there is another property usually ascribed to truth that we would have to do without:

It is worth noting that beside the tenselessness, another feature that is commonly assumed to hold for truth seems to fail for the notion of truth at hand: This notion of truth does not satisfy the disquotational scheme, at least if negation is understood in the intuitionistic way (LBM p. 166).

From

“\(p\)” is true iff \(p\)

it follows intuitionistically that

“\(p\)” is not true iff \({\sim }p\)

but the “only if” part is wrong: For “\(p\)” to be not true, it is enough that we are in a situation where \(p\) is undecidable. More doubts about the unrestricted validity of the disquotational scheme can be found in Dummett (2004), p. 14 ff.

How easy will depend on how much of our temporal and intellectual restrictions we are willing to idealize away.

Note that both options would agree on the fact that the statement is decidable. The example looks deceptively similar to the one with the seventy 7s I used before to illustrate the phenomenon of undecidability. However, the important difference is that in the present case, there is a method known to us that will decide the issue, and we know that this method will decide the issue. In the former example, the method of simply going on to compute more and more digits was far from certain to decide the question about the string of 7s.

TOE, p. 297.

In the case of mathematics, this has us assigning value \(1\) to statements that are mathematically impossible, which is slightly counterintuitive. Again, this sense of implausibility is assuaged by pointing out that the possibility that is reflected in the accessibility relation is an epistemic one.

Titel: Intuitionism
verfasst von: Andreas Kapsner
Verlag: Springer International Publishing
Buch: Logics and Falsifications
Print ISBN: 978-3-319-05205-2

Electronic ISBN: 978-3-319-05206-9

Copyright-Jahr: 2014
DOI: https://doi.org/10.1007/978-3-319-05206-9_3

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