Sunday, January 6, 2013

And the future of 'standard' analysis was forged.

"At the time of Brouwer’s death it appeared that your choices were:
(1) accept Brouwer’s theories, give up most of mathematics and give up talking to most mathematicians; or
(2) accept Church’s thesis, give up analysis and give up talking to most mathematicians; or
(3) reject constructive mathematics entirely.
This was not a difficult choice for most mathematicians;" (Constructivity, Computability, and the Continuum, Michael Beeson)

The one moment when you ask: 'Is the constructed real number continuum really necessary?'

(Of course, only considered as enlightenment after you have really studied the continuum).

Intuition 'mismatch'

'To the criticism that the intuition of the continuum in no way contains those logical principles on which we must rely for the exact definition of the concept “real number,” we respond that the conceptual world of mathematics is so foreign to what the intuitive continuum presents to us that the demand for coincidence between the two must be dismissed as absurd. Nevertheless, those abstract schemata which supply us with mathematics must also underlie the exact science of domains of objects in which continua play a role.' (Weyl)

Friday, January 4, 2013