[QED Stepfiles]

Originally Posted by Cavernio

"It is definitely possible to think logically, and purely logically, and it's something that mathematicians have been doing for a long time."

Prove it. I'm not sure you can. Of course, that would be because our brain has developed from our senses, and any proof you'd offer would involve a general acceptance as a language being purely logical, which I say would not be possible because we're not purely logical beings. So I suppose I've asked you to do the impossible in proving that, but you might be able to convince me anyways.

The rest of that post is moot IMO because you have not proven the above statement. If we take math as representing our world around us, 1*1=1 does not need to be proven logically.

"Rather, mathematicians work more predominantly in "First Order Logic," a mathematical language that uses completely unambiguous language to a satisfactory level of precision."

And these are the same mathematicians who've been using logic 'for along time' as you say? My knowledge (which is obviously incomplete, but not necessarily wrong), is that 'formal' math in fact started as geometry, which is clearly an offshoot of our visual sense. Also, why is our language of math displayed visually?

Yes, I understand that my arguments are far from proof, however, that does not mean they're not valid to consider.

Actually... the whole formal math starting from geometry thing may be accurate to some degree, but today mathematics is definitely not built up on geometry. Geometry is built up off of logic. Actually... euclidean geometry is a bit dated... there are a hell of a lot of assumptions in euclidean geometry (for example, why do we have to use the standard metric -i.e. measure distance and length the standard way- for the space?). Geometry is a result of logic.

First order logic was not developed in Ancient Greece or anything... the whole concept of mathematical logic and consistency is a relatively recent phenomenon. I'm not really up to writing a huge long post about first order logic, but suffice to say that it's precise and unambiguous. And, yes, language in general is not logical, but it is possible to fix all the ambiguities in language, and make it absolutely precise (thus, the creation of F.O.L). Humans may not be logical beings inherently, but we do have the capacity to think logically and unambiguously if we really try. And really trying in that capacity is essential to understanding mathematics.