[FOM] intuitionists on source of mathematics

Mark van Atten Mark.vanAtten at univ-paris1.fr
Sun Jan 29 09:45:33 EST 2006

This is a belated reply to the question Richard Haney asked on Tue, 17 
Jan 2006 17:53:35:

 > Can anyone direct me to specific resources (including sections and page
 > numbers, if available) where intuitionists discuss their philosophical
 > views "that the unique source of mathematics is the intuition, and the
 > criterion of acceptability of mathematical concepts, constructions, and
 > inferences is intuitive clarity"?  (The quote is from N. A. Shanin,
 > *Constructive Real Numbers and Function Spaces*, Amer. Math. Soc.
 > Transl. of Math'l. Monographs, Vol. 21 (1968), p. 7, section "0.3".)

Brouwer's dissertation remains a useful source in these matters. For 
example, on p.52 (in Collected Works vol.I), where one reads:

`In the third chapter it will be explained why no mathematics can exist 
which has not been intuitively built up in this way, why consequently 
the only possible foundation of mathematics must be sought in this 
construction under the obligation carefully to watch which constructions 
intuition allows and which not, and why any other attempt at such a 
foundation is doomed to failure.'

On p.97, he says that notions such as `continuous, entity, once more, 
and so on' are irreducible; in a handwritten note to that passage, 
Brouwer explains that these notions are just so many `polarizations' of 
the basic intuition (of the passage of time (from which all content has 
been abstracted)). This note has been published in D. van Dalen, L.E.J. 
Brouwer en de grondslagen van de wiskunde, Utrecht:Epsilon 2001 (2nd 
ed.2005), p.136.

Mark van Atten.

13 rue du Four, F-75006 Paris, France
tel ++ 33 (0)1 43 54 94 60
fax ++ 33 (0)1 43 25 29 48

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