[FOM] Alternative Foundations/philosophical

[Timothy Y. Chow]

Harvey Friedman wrote:

> With regard to the "liberation Movement", if one is concerned with fully 
> complete rigorous presentations, then has history shown that generally 
> speaking one either doesn't have this at all, or one has it done 
> incorrectly, replete with inconsistencies?
>
> Isn't an example of this kind of thing, the idea of using general 
> category theory as an alternative foundation, with the "liberated" use 
> of things like the category of all categories? Hasn't that been recently 
> shown to lead to convincing inconsistencies within the usual mindset of 
> general category theory?

I'm not sure what you're driving at.  Alternative foundations, like all of 
math, must of course be correct.  If there's something incorrect, then of 
course there's a problem, but at the same time it should be possible to 
identify exactly what is incorrect (or insufficiently justified), and then 
engage in the usual mathematical dialogue to straighten it out.  Unless 
someone digs in his heels and refuses to acknowledge specific objections 
and engage in dialogue (as in the case of Hsiang's purported proof of the 
Kepler conjecture), the issue of correctness is something we know how to 
deal with.  So I don't understand what you're driving at when you cast 
general, vague aspersions like those above.  If you have a specific error 
in mind that you're worried is unfixable, let's hear what it is. 
Otherwise it just sounds like you're saying that because people have made 
mistakes before in this subject area, we should be skeptical about the 
subject as a whole.  If that's the objection then we might as well just 
give up on all of mathematics.  Let him who has never made a mathematical 
error be the first to cast a stone.

Tim