Is geometry purely mathematical or does it rely on spatiality which is beyond mathematics?

How does Godel's incompleteness theorem affect the way that mathematicians understand and see mathematics as well as the world (if at all)? I'm not even close to a mathematician, but even a slight dose of the idea and theorem were enough to affect me so I suppose that I'm curious.

Do imaginary numbers exist?

I've just read about Cantor'd diagonal argument, and I have some questions about it... Let's say we want to map every real number between 0 and 1 to natural numbers. If I'm not mistaken, that can be done this way: If we have number of form 0.abcdef... (letters stand for decimals, but only some are shown since there is infinite amount of them), then we can produce number N which equals 2^a * 3^b * 5^c * 7^d * 11^e * 13^f * (next prime)^(next decimal). For example, number 0.12 equals to 2^1 * 3^2 (* 5^0 * 7^0 * ...) = 18. Given any natural number N, we can easily determine which real number it represents (if any). My first question is: is all this consistent with Cantor's diagonal argument? (Can both be true at the same time?) Cantor proved there is no one-to-one mapping (not just any mapping), is that important for his result? If yes, it somehow seems intuitive to me, at least at the first sight, that one-to-one mapping can be achieved by simply removing natural numbers that don't represent any real...

I have been reading discussions on this site about the Principia and about Godel's incompleteness theorem. I would really like to understand what you guys are talking about; it seems endlessly fascinating. I was an English/history major, though, and avoided math whenever I could. Consequently I have never even taken a semester of calculus. The good news (from my perspective) is that I have nothing to do for the rest of my life except for working toward the fulfillment of this one goal I have: to plow through the literature of the Frankfurt School and make sense of it all. Understanding the methods and arguments of logicians would seem to provide a strong context for the worldview that inspired Horkheimer, Fromm, et al. So yeah, where should I start? Do I need to get a book on the fundamentals of arithmetic? Algebra? Geometry? Or do books on elementary logic do a good job explaining the mathematics necessary for understanding the material? As I said, I'm not looking for a quick solution. I...

Are physical and logical truths distinct and, if so, how are they related? Is one more fundamental than the other? By ‘physical truth’ I mean something true in virtue of the laws of physics, such as ‘masses attract other masses’ (gravity) and by ‘logical truth’ I mean something true in virtue of logical or mathematical principles, like ‘2 + 2 = 4’. Could there be a world where some of the physical truths of our world were false but all of the logical truths of our world were true? That is, a world where masses always repelled other masses but 2 + 2 = 4? Conversely, could there be a world where some of the logical truths of our world were false but all of the physical truths of our world remained true? That is, a world where 2 + 2 = 5 but where, as in our world, masses attract other masses? [We’ve been discussing this hours and feel in desperate need of professional guidance - please help!]

Is there a difference between a number as an abstract object and as a metric unit used to measure things?

My mathematics teacher says that a line is an infinite sum of points. I disagree and I think that she must not have thought it through very deeply. I argue that instead that though a line can be theoretically be described as a sum of smaller lines that in no way can a line be said to be described as a continuity of points because a point is not in any way extended. If a line has an atomic unit then that unit must have the same properties as the line itself and a point has an altogether different property than a line. (That you can fit a point inside a line only shows their common property of spaciality, it does not demonstrate that a line is in any way composed of points) I hope you understand what I am saying. Do you think I am right?

Is there a correct formulation of set theory? For example, it's been proven by Gödel and Paul Cohen that the continuum hypothesis can neither be proven nor disproven in ZFC. Should we take from this that there exists a hitherto undiscovered formulation of set theory that can conclusively establish whether the continuum hypothesis is true or false?

Suppose there is an infinitely long ladder in front of me. I do not know that this ladder is infinitely long, only that it is either a very long (but finitely long) ladder, or an infinitely long ladder. What kind of evidence would I need to give me reasonable assurance (I don't need absolute certainty) that this ladder is indeed infinitely long? I could walk a mile along the ladder and see that it still shows no signs of stopping soon. But the finitely long ladder would still be a better hypothesis in this case, because it explains the same data with a more conservative hypothesis. If I walk two miles, the finitely long hypothesis is still better for the same reasons. No matter what test I perform, the finitely long hypothesis will still better explain the results. Does this mean that, even if infinite objects exist, empirical evidence will never provide reasonable assurance that they exist?