Reply #84: Yes, you're right about the square/circle statement [View All]

[Guaranteed]

being even more difficult to disprove, because they are more clearly analytic concepts. But, not quite so obviously (but still clearly), so are "2," "+," "=," and "4." Given the definitions of all those concepts, put together in such an example as I gave, the statement is universally and logically true. Now, they ARE abstract concepts in that they may or may not truly exist in the world (although I would argue that the concept of "2" exists more in reality than a TRUE square or circle), but once their definitions are accepted the statement is undeniable.

I don't believe that concepts such as numbers come to the point of knowledge empirically. I seem to remember that it was John Stuart Mill who argued that that was the case (and we've discussed him before) as he was a pretty hardcore empiricist. But we don't "witness" numbers in the classic empirical sense- instead, there is an a priori tendency for our mind to group things together abstractly (as Kant probably would have argued). One can't point out the thing that one would call "2"- you have to understand the concept of grouping first.

I'd also have to say that if quantum theory has begun to suggest that statements such as "2 + 2 = 4" are not necessarily true, then things have gone dreadfully wrong in its study, especially considering that it is a purely theoretical one and is in fact built upon knowledge a priori.