In article <Dsyrpn.5I1@mv.mv.com> Alberto C Moreira <email@example.com> writes: >firstname.lastname@example.org (john baez) wrote:
>Also, you're talking at a level far beyond that which I was addressing. >When doing mathematical research, one probably has to use whatever tools >one has at his/her disposal, but learning it is another story. And when >we're talking about teaching students, do we want to emphasize informal >thought?
I suspect we should emphasize formal thought and informal thought and the road between the two: formal thought without the backing of informal thought tends to be lacking in zest, while informal thought without the backing of formal thought tends to bog down in vagueness.
>Logic itself is your frame, or isn't it ? I'm not at that level, but I >have difficulty seeing any kind of mathematics that doesn't anchor >itself in a logical frame. I know that category theory people, just >like lambda calculus people, like to see category theory as a formal >system par with set theory and others. But all three of them are still >based on a logic system, or at least I do so far believe.
Certainly once one starts proving theorems in category theory one can, just like in set theory, formalize the reasoning with the help of an axiomatic system, and it's good to know how to do this.
>>Think of how Ed Witten won the Fields Medal, the biggest >>math prize of all! His reasoning was not formal, and in many cases we >>still don't know how to make it formal --- but his discoveries were
>I'm >not familiar with the work of Ed Witten, are his proofs "informal", >in the sense that they don't fit within a logic system ?
Yes, they are informal. He is primarily a physicist, yet he won the highest prize in mathematics, the Fields Medal. Many of his arguments and computations have been formalized by other mathematicians, and many of them have not yet been fully formalized. They have revolutionized many subjects.