<i>old proofs of theorems may become false proofs. The old proofs no longer cover the newly defined things. The miracle is that almost always the theorems are still true; it is merely a matter of fixing up the proofs. It is claimed that an ex-editor of Mathematical Reviews once said that over half of the new theorems published these days are essentially true though the published proofs are false.</i><p>I'm only around 55% of the way through (according to my scrollbar --- and thank you Readability!) but it's an incredibly interesting read. It's very lengthy, but I really advise taking the time to look it over. It really tickles those brain cells. Thus far, I'm not seeing anything "new" in this article, but seeing all these incredible things expressed and summarized up close is amazing.
I think our sense of geometry comes from hunting (not body decoration).<p>It's a little bit like mathematicians invent little "chains of reasoning" rather than "mathematics", and that these chains are interesting and useful; even if their original assumptions turns out to be incorrect, the reasoning is still valid. In the marketplace/ecosystem of mathematics, people then choose the ones that they find most useful and/or interesting.<p>I love the thought that when we meet aliens, they have utterly different mathematics from us, so it reveals how parochial our particular toolbox is. This has actually happened, in a sense, with Chinese mathematics. Apparently, their approach to "proof" was algorithmic rather than declarative - not just a different toolbox, but a different kind of toolbox.
The discussion of beauty in mathematics resonates deeply with me and I seem to always relate such discussions with music. As a musician turned math enthusiast, I think the thought processes involved in the creation of music parallel those involved in the creation of proofs quite nicely. My obsession with music seems to complement my obsession with mathematics on some level that I cannot quite define. As far as I know, there is not a distinctly important practical relationship between math and music (please correct me if I am wrong), but I still feel as though there is some connection. I do not know if this is a personal thing or not, but I cannot stop relating the two!
Mathematics is a product of the mind; the mind is a product of evolution; evolution is a product of natural laws. If you want answers to these questions, it helps to look at what we are, exactly, and how we came to be.<p>In fact "simple" mathematics are not simple at all by any objective measure. Starting with any truly formal system, you need a stupendous number of deductions to get to things like elementary laws of arithmetic, or basic plane geometry. Mathematical proofs are not formal proofs--they are instructions for our brains. Evolution made the relevant parts of our brains the same, so same instructions lead to same results. That's why there's never any argument over whether a proof is correct, once a few people got to study it in detail. This also explains Hamming's observation that when proofs turn out to be "wrong" after math has evolved a bit, theorems are still usually correct. We find a new, better route to the same place in our brain, and recognize the hazards of the old route, now deprecated.<p>Okay, here is the key bit: if evolution made the relevant parts of our brains the same, that means it has arrived at a maximum, or at least a local maximum. What is the nature of this maximum? Physiologically, there are constraints on the amount of brain circuity our body can maintain. Brains consume a lot of energy, take up space, etc. So naturally, evolution ended up with a design where the same circuity can serve the greatest possible number of functions.<p>Of course, evolution only concerns itself with those functions relevant to our survival and reproduction. But there is nothing niche about those goals. If some general pattern occurs often in our quest for survival, then it likely occurs often in other quests that evolution never knew about--like building airplanes.
The original article by Wigner: <a href="http://www.dartmouth.edu/~matc/MathDrama/reading/Wigner.html" rel="nofollow">http://www.dartmouth.edu/~matc/MathDrama/reading/Wigner.html</a>
There is also, more recent, the unreasonable effectiveness of data: <a href="http://www.computer.org/portal/web/csdl/doi/10.1109/MIS.2009.36" rel="nofollow">http://www.computer.org/portal/web/csdl/doi/10.1109/MIS.2009...</a>
So after all that verbiage, he concludes:<p>Some math was designed to be useful.<p>Science is by definition those practical problems to which math can be applied.<p>He still doesn't know.<p>Very disappointing.
" Is it not remarkable that 6 sheep plus 7 sheep make 13 sheep; that 6 stones plus 7 stones make 13 stones? Is it not a miracle that the universe is so constructed that such a simple abstraction as a number is possible? To me this is one of the strongest examples of the unreasonable effectiveness of mathematics. Indeed, l find it both strange and unexplainable."<p>this is confusing because what he is talking about is -counting- not mathematics -mathematics- is an academic field that may include -counting- as one of its areas of study -but- it is confusing to reduce mathematics to counting