[FOM] Brouwer on mathematical operations

[josef@us.es]

Dear Lucius,

let me recommend, once again, the work of Mark van Atten on Brouwer (both his introductory book, Wadsworth 2003, and his more sophisticated "Brouwer meets Husserl: On the Phenomenology of Choice Sequences", Springer, 2006).

I'd like to make one remark that I find important. Even if one is inclined to agree that "One cannot inquire into the foundations and nature of mathematics without delving into the question of the operations by which the mathematical activity of the mind is conducted", there is one element in this statement that allows for uncontrolled philosophical hypotheses to enter. And this is, quite simply, the words "activity of the mind".

By "the mind", Brouwer means what has been traditionally described as "the subject" or "the ego"; mental activity is for him a matter of thoughts and perceptions displayed before "the mind's I" (this last expression is an obvious metaphor), completely independent from the physical or material world.

At least some of us today believe that this old understanding (however traditional and sanctioned by philosophers of the past) is heavily biased, and that "the mind" and mental activity are part of concrete human activities -- completely dependent on our physical constitution. Some cognitive scientists use the expression "embodied mind" to refer to this shift.

My remark may not be very helpful, but at least I hope it is a good and reasonable warning. Best wishes to all, 

Jos'e Ferreir'os

Dpto. de Filosofia y Logica, Univ. de Sevilla


> 
> Date: Sat, 27 Dec 2008 09:20:16 -0500
> From: Lucius Schoenbaum <ltsbaum at gmail.com>
> Subject: [FOM] Brouwer on mathematical operations
> 
> "One cannot inquire into the foundations and nature of mathematics  
> without delving into the question of the operations by which the  
> mathematical activity of the mind is conducted. If one failed to take  
> that into account, then one would be left studying only the language  
> in which mathematics is represented rather than the essence of  
> mathematics."  L.E.J. Brouwer
> 
> I am only a student, but I am inclined to agree with Brouwer.  Does  
> anyone else agree also?  If I venture to call intuitionism an  
> "operational" approach in foundations, can anyone think of anything  
> similar existing today in foundations?  What operations do you think 
> Brouwer had in mind?
>