[FOM] Choice of new axioms 1

praatika@mappi.helsinki.fi praatika at mappi.helsinki.fi
Fri Feb 17 08:37:44 EST 2006

Quoting Andrej Bauer <Andrej.Bauer at andrej.com>:

> I put this forth as a general strategy for mathematics, 
> not just ZFC. 

Your view seems then amount to what has been called "if-then-ism" (Putnam) 
or "deductivism" (Resnik) in the philosophy of mathematics. Such a view has 
been advanced for example by Russell, Hilbert and Putnam at some point of 
their career (as well as certain logical positivists, as a way to try to 
salvage logicism (fails)). It is a widely held opinion that this view faces 
deep problems, but unfortunately I can't go to them now (I am away from my 
office, in London, and terribly busy at the moment) - maybe little later.  
(Maybe somebody else here can continue?)

> And since I am not advocating total mathematical anarchy,
> yet, there should be some principle for selecting certain 
> axioms (and logics) over others.

Yes, that's one of the problems...

> A visible consequence of what I am suggesting is this: young
> mathematicians should _not_ be taught a single "standard" kind 
> of logic and axiom system. Instead, students should be taught 
> how to think with or without classical logic, axiom of choice,
> large cardinals, powersets, etc. In effect I am proposing that 
> f.o.m.ers make themselves indispensable for the forseeable 
> future. Surely you must agree :-)

Yes, I think I agree with you on that, even if I think myself as a kind of 

All the Best, Panu

Panu Raatikainen
Ph.D., Academy Research Fellow,
Docent in Theoretical Philosophy
Department of Philosophy, 
University of Helsinki

Visiting Fellow, 
Institute of Philosophy,
School of Advanced Studies, 
University of London

E-mail: panu.raatikainen at helsinki.fi

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