Interactive Granular Computations in Networks and Systems Engineering: A Practical Perspective by Andrzej Jankowski

Author:Andrzej Jankowski
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

In his another work [70] (p. 451), Brouwer evokes a highly suggestive metaphor which defines the relationship between mathematics and formal languages expressing it:Formal language accompanies mathematics as the weather-map accompanies the atmospheric processes.

In the context of the aforesaid “FIRST ACT OF INTUITIONISM” by Brouwer, a comment by Martin-Löf from 2007 was quite surprising. More than two decades after his attempts to oppose the intuitions contained in the “FIRST ACT OF INTUITIONISM” with his constructive type theory, he wrote an article entitled “The Hilbert-Brouwer controversy resolved?” [523] in which he says: if you take Brouwers view that the mathematical objects are mental constructions, or thought constructions, then there arises immediately the problem: how do we get to know them? and not much self-reflection is needed to see how we do get to know them. After all, there is no mathematical zoo to which we can be taken in order to have them displayed to us without any linguistic mediation: we go to listen to some lectures or to read some books or articles, which means that we are dependent on language and symbols from the very start. So this view of Brouwers is hardly credible any longer because of the developments in the philosophy of language during the last century, as Dummett has emphasized for the first time in his paper “The philosophical basis of intuitionistic logic” (Dummett 1975). But there is a kind of mediating position, namely that the mathematical objects are not just purely formal sign configurations: they are meaningful sign configurations, and that is what gives them properties which do not come from their combinatorial nature, but come from the meaning with which they are endowed, as beautifully stated by Gödel in the beginning of his “Dialectica” paper (Gödel 1958). It is this view of the mathematical objects, mediating as it is between Hilbert and Brouwer, which has turned out to be the most credible one to my mind: at least it is the one which underlies all of my own work from 1974 until the present day.

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