Here I present my brain, it has been learning and evolving for some time, and recently, it noticed that, logically, the math it thought makes sense, actually doesn't.
The source of 'Math'
This goes some time back into the past, when I suddenly felt the urge to see where Math starts, because logically, and this is something I remember was the base of proving stuff, you need to base yourself on something that is true to prove something else. Anybody who knows a little bit about this knows that this directly leads to Axioms, Occam's razor, Goedel and co...
Useless education
Funny we have been thinking we know our very basic math, but we really do not even know that.
Even the pythagorean theorem seems not logical looking at it this way. Looking at proofs, the proofs themselves either use geometric manipulation of squares, triangles making assertions about areas, and some of those proofs came from periods were an area was something intuitive and not really formalized, come to think of it the concept of area itself is pretty much elusive, and looking for the rigorous math definition leads you to Reimann and others, and that's pretty recent in history. What's more annoying, I made it through school and a Bachelor in Engineering and I never once heard of them. What is even more annoying, I felt I knew what an 'area' is although, if I had thought critically and logically, I would have came ot the conclusion that there is something elusive about it, just like I did recently.
All of this post comes after lots of going back to trying to understand the roots of math, using wikipedia and google, some of this are listed in the bottom of the post.
Proof of a proof
One nice idea from this quest is Goedel and his Incompleteness theorem, naively for me right now it means you need to start from something to make any proofs, and that something you started from cannot be proved. I will not go back and read the details, but while taking a shower just now, I became curious as to how Goedel proved this, did he use an axiom as a base, if this axiom was removed, not even this could be proven? This got me to think about what logic is, and about the 'axioms' of logic. Logic seems to be something the brain can very easily accept and use as a base. Again going back to Engineering, much of what is left is the logic. But why? and what is logic, isn't it absurd by itself? what is the logic that logic is based on? Why does the brain readily accept it? (without 'proof').
A group of 'things', excluding 'Neo, the source'
This got me to realize that there is a certain group of things, that all fall into some category for which I don't have a name: Logic (needing logic to make sense), time (continuous/discrete), infinity, zero (1 over infinity!), space and it's size both endless and not being absurd (same for time). All these things feel like one and the same, or belonging to one category. We end up accepting them and even using them, but few of us really grasp them.
Think versus. Grasp
I also remembered vaguely something that I think Einstein said about things a human brain will never grasp, comparing to a table with eyes looking down never being able to see what is above it (I am not sure about the exactness of any of this). But what I recently found interesting, is the fact that we are able to think about these things, even though we might not be able to understand them (by construction?), why this separation? why can't we only think about things we can understand? Does this boundary mean something? and what?
Dump and live on
I wrote this post mainly for one reason: get it off my brain to free it for thinking about more practical stuff.
Feel free to express your opinion about this at
http://forums.aigamedev.com/showthread.php?p=15004#post15004
Some of the references
http://www.mathacademy.com/pr/prime/articles/fta/index.asp?LEV=&TBM=&TAL=&TAN=&TBI=&TCA=&TCS=&TDI=&TEC=&TFO=&TGE=&TGR=&THI=&TNT=&TPH=&TST=&TTO=&TTR=&TAD=
http://www.mathacademy.com/pr/prime/articles/irr2/index.asp
http://www.google.de/search?q=proof+square+root+of+2+is+irrational&ie=utf-8&oe=utf-8&aq=t&rls=org.mozilla:en-US:official&client=firefox-a
http://en.wikipedia.org/wiki/Well-order
http://en.wikipedia.org/wiki/Infinite_descent
http://en.wikipedia.org/wiki/Square_root_of_2
http://en.wikipedia.org/wiki/Rational_number
http://eu.wiley.com/WileyCDA/WileyTitle/productCd-0470211520.html
http://en.wikipedia.org/wiki/Commensurability_(mathematics)
http://www.boost.org/doc/libs/1_37_0/libs/math/doc/sf_and_dist/html/math_toolkit/special/ellint/ellint_intro.html
http://en.wikipedia.org/wiki/Elliptic_integral
http://sci.tech-archive.net/Archive/sci.math/2006-09/msg04719.html
http://books.google.de/books?id=RM1D3mFw2u0C&pg=PA7&lpg=PA7&dq=%22rigorous+definition+of+area%22&source=bl&ots=jiarfVKaP5&sig=OAi9X-H7Hnp92BdfIuiIA911KSc&hl=en&ei=jJdXSomENIed_AahldSdCQ&sa=X&oi=book_result&ct=result&resnum=7
http://www.amazon.co.uk/gp/offer-listing/0133459438/ref=dp_olp_1?ie=UTF8&qid=1247256551&sr=8-1
http://www.amazon.com/gp/product/images/0486439461/ref=dp_image_0?ie=UTF8&n=283155&s=books
http://www.amazon.com/s/ref=nb_ss_b?url=search-alias%3Dstripbooks&field-keywords=Discrete+Mathematics&x=0&y=0
http://www.mathkb.com/Uwe/Forum.aspx/math/16463/Concept-of-measure-in-undergraduate-mathematics
http://www.google.de/search?hl=en&safe=off&client=firefox-a&rls=org.mozilla%3Aen-US%3Aofficial&hs=iW1&num=100&q=%22rigorous+definition+of+area%22&btnG=Search
http://www.youtube.com/results?search_query=The+Fundamental+Theorem+of+Calculus&search_type=&aq=f
http://www.youtube.com/watch?v=MOnnMlMM70Q&feature=PlayList&p=D4E266DF4E3352B1&index=18
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