December 30, 2004
Apple Pi
Posted by Jess in
Tech Talk
Notwithstanding the fact that I can't even do basic addition to save my life, math has always been a passion of mine. Though my lack of understanding may be the enticement, however, as I love a good enigma.
Mathematical algorithms are behind every aspect of life and nature, from being able to calculate the largest package that will fit in your mailbox, right down to the reduction of a recipe from four to two people instead.
I have just stumbled upon one answer of which the question had been driving me nuts for ages:
Which came first the mathematical principles themselves, or our counting system, in which 1+1=2, which enables all the mathematical principles to actually work?
The answer I found here, in a paper called The Magnificent Perfect Square, by Roger Logan:
'Over the centuries, mathematicians have expanded the number system four times. Each new expansion was required because the then existing number system was not sufficient to solve certain problems.'
So it would seem the first problem is that my original question was not correct at all
1+1=2 is not complex enough to enable all the principles to work in the first place.
Apparently, mathematics has four number systems in addition to the natural system (1, 2, 3, 4 etc.):
1. Integers (the natural system, but expanded to include 0 and -1, -2, -3 etc.)
The introduction of integers allowed many more algebraic equations to be solved, which would have been impossible to solve previously.
2. Rational (the introduction of fractions)
3. Real (which accounts for infinite decimal point ability)
4. Complex (an ordered pair of real numbers, imaginary)
I was a little disappointed in this learning
my first thought was almost a letdown, that maybe the natural system I love so much wasnt so natural after all. But then I read this paragraph in the same paper:
'The age old question is, "Which came first, the chicken or the egg?" This simplistic summary regarding the expansion of the number system makes it appear that the mathematicians' main concern was to expand the number system so as to solve the then current number problems. However, solving the number problems is what caused the creation, discovery or invention of new numbers, thereby forcing the expansion.'
After my reading, I now have a new question.
Are the expansions considered a discovery, or an invention? And would the answer be fact, or a debatable opinion?
If I tally the facts, I'm sure I'm oversimplifying everything. But I warned you I couldn't do basic addition to save my life.
Permalink
|
TrackBack (0)
Comments
Jess,
Philsophers, scientists and mathematicians a lot smarter than I am have debated this quetison for a long time.
This is one of the better essays on this subject that I've found on the web.
http://www.ex.ac.uk/~PErnest/pome12/article2.htm
It mentions Roger Penrose's book "The Emperor's New Mind". It's a challenging book, but if you enjoy this sort of question, I think you'll like it.
My personal answer to this question, btw, is 42 :-)
-rich
Wow, thanks Rich! Both the article and book look like they may be a bit of a struggle for me, but I'm definately going to do my best. :-)
As for 42, I'm right there with you on that one. My bro, who seems to know me far too well, got me the definitive blankie for Christmas. ;-)
http://www.thinkgeek.com/cubegoodies/blankets/6cf7/
Well, I have been something of a mathematician for most of my life, and I have to say that nearly everything I've come across since, say, grade three or so has been made-up stuff. Some of it, particularly statistics, is closer to numerological mysticism than it is to a fundamental truth. How many hours have I spent finding the most efficient packing of n-dimensional "spheres" into n-dimensional "cubes"? When n is greater than 3, the discussion is the same as discussing angels on pinheads, even though there are well-received rules governing the discussion. You want fabbilism? PBS ran an interesting documentary about the proof for Fermat's "last theorem" on Nova:
http://www.pbs.org/wgbh/nova/proof/
There is no way in the world that Fermat could have approached the problem in the same way, and a lot of what is being thrown around centers on "rules" created more-or-less for this one single purpose. One of the key elements of the "proof" is formally called a Conjecture. (How a "proof" can be founded upon a "conjecture" and still be called a Proof is quite beyond me, and I have been involved in developing a couple of Proofs in my day.)
No, ma'am: if you can't count it, it ain't there. No need to apologize for "mathematical weakness"; just tell people you don't indulge in that sort of fantasy.
Of course, hearing it and reading it like that seems to make such perfect sense... yet, I have to say, I do think it is somewhat of a letdown. My favorite Rhetoric professor in college said once that math exists because someone said it did. I had refused to believe it, then. :-)
Stan, the key to the proof of Fermat was that Wiles proved that the Taniyama-Shimura conjecture was true. It is now, therefore, teh Taniyama-Shimura theorem. It had already been established (in 1985 by Frey) that proving this theorem would prove Fermat.
See this site for details:
http://mathworld.wolfram.com/Taniyama-ShimuraConjecture.html
Also, there's a good book about the proof of Fermat, called Fermat's Enigma
http://www.amazon.com/exec/obidos/tg/detail/-/0385493622/qid=1104540026/sr=8-1/ref=pd_csp_1/102-7020619-9796121?v=glance&s=books&n=507846
-rich
BTW, my personal belief about this:
Mathematical technique is invented. Mathematical facts are discovered. Mathematics simply is what was there all along, waiting for its techniques to be invented and its facts discovered.
-rich
Rich, have you taken a look at Iwasawa? I'll leave the provability to your revue, but IMHO Gödel comes into play heavily here. Remember, Zaphod Beeblebrox survived the Total Perspective Vortex because the Universe in which it existed was created especially for him....
Stan, Iwasawa activates my SEP field :-) Really, I can deal with this level of mathematical thought on a highly simplified conceptual level, but the actual mathematics is way over my head. The way I reconcile Gödel with my view is that it says that mathematics is a tease. It's waiting to be discovered, but it can't ever be fully discovered. The more you discover, the more there is to discover. Mathematics itself is, IMHO, the ultimate infinity. So here's a question back at you: if mathematics itself is infinite, is it aleph-0, aleph-1, aleph-2, aleph-3, an countably higher order infinity than that, or an infinitely higher order of aleph? :-)
-rich
My cat says so. Or at least I think that that which I perceive to be a cat seems to make noises that make me believe that it has said so. Or maybe I am imagining that....
