By Dirk van Dalen (auth.), Mark van Atten, Pascal Boldini, Michel Bourdeau, Gerhard Heinzmann (eds.)
With logicism and formalism, intuitionism is without doubt one of the major foundations for arithmetic proposed within the 20th century; and because the seventies, particularly its perspectives on common sense became vital additionally outdoors foundational stories, with the improvement of theoretical laptop science.
The target of the booklet is threefold: to check and entire the old account of intuitionism; to give fresh philosophical paintings on intuitionism; and to offer examples of recent technical advances and purposes of intuitionism. This quantity brings jointly 21 contributions via cutting-edge best authors on those themes, and surveys the philosophical, logical and mathematical implications of the strategy initiated in 1907 in L.E.J. Brouwer's dissertation.
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Extra info for One Hundred Years of Intuitionism (1907–2007): The Cerisy Conference
See, for instance, (Posy 2000). I point out in (Posy 2000) that this reasoning is actually a version of the Hangman paradox. I show there that there is a fuller theory of forced indeterminacy at work here. In particular it is worth noting that the problem of forced indeterminacy occurs even in ﬁnite contexts. See (van Atten and van Dalen 2002). 108–109). Van Atten has informed me that though his reading of this passage in Brouwer does indeed involve a second-order provisional restriction, it does not quite coincide with the second reading of the passage that I am suggesting here.
Nowadays we call this a mild assertabilism: In order to assert ∃xP (x) it suﬃces to have a guarantee that we can eventually ﬁnd an object c such that P (c) holds. There is a subtle point here: r is indeterminate in many ways: Most prominently we don’t know whether it is smaller than or equal to 1/2; we don’t even know whether we will ever ﬁnd that out. But still, r counts as a legitimate object. And I should add that this weaker requirement in no way diminishes the ontological status of the things whose existence is thus established.
The striking mathematical change is that Brouwer now adopts the reductive conception of the continuum: He is prepared to allow that the continuum is indeed composed of independently generated real numbers. And I emphasize that he makes this change without abandoning his commitment to the ‘viscous’ topology of the continuum. This is a mathematical achievement of the ﬁrst rank, involving a radical new way constructively to construe inﬁnite sequences and inﬁnite sets of these. Brouwer’s main mathematical moves in achieving this coup — choice sequences, spreads and the uniform continuity theorem — are by now familiar notions, well studied in the literature on intuitionism: The theory of choice sequences allows inﬁnite sequences whose elements may be randomly generated or even freely chosen; so long as we are guaranteed that the sequence of elements can be continued without stop.