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Is Mathematics Discovered Or Invented?
An anonymous reader points out an article up at Science News on a question that, remarkably, is still being debated after a few thousand years: is mathematics discovered, or is it invented? Those who answer "discovered" are the intellectual descendants of Plato; their number includes Roger Penrose. The article notes that one difficulty with the Platonic view: if mathematical ideas exist in some way independent of humans or minds, then human minds engaged in doing mathematics must somehow be able to connect with this non-physical state. The European Mathematical Society recently devoted space to the debate. One of the papers, Let Platonism die, can be found on page 24 of this PDF. The author believes that Platonism "has more in common with mystical religions than with modern science."
The language we use to describe mathematics is not the math itself. The math exists regardless of the symbolism used to describe it. Hence, you are incorrect. It is all discovered, but the means to describe it and put it to use is invented.
Yes, it's also amazing that the equation isn't 2.14332544988e=2.14332544988mc^2.
Yes, sorry, I'm being a smart-ass and it's not polite. But c^2 is just a constant.
What's purple and commutes? An Abelian grape.
The late mathematician Paul Erds used to say, perhaps metaphorically, that the most elegant proof of every mathematical theorem was written in a great book in God's library. When he came up with a beautiful proof, he would say it was one from the book.
Feynman also felt like coming up with a proof was more discovery than invention. He said that the proof felt like it was already there all along, raising the question of where "there" is.
Check it out. cool stuff.
RS
Shoes for Industry. Shoes for the Dead.
Your reasoning is subtly but fundamentally flawed. Yet as with all subtlties, pinpointing the exact nature of the flaw is difficult without having a back-and-forth conversation.
You are right on target with respect to Ohm's law and Hooke's law -- but quite off base with your general assertion. The deep laws of physics *are* eerily symmetric, independent of our need to describe them so.
For example, the inverse-square law of gravity or electromagnetism can be derived as a consequence of living in a 3-dimensional universe. (Integrate your favorite conserved quantity over concentric spherical surfaces and you get something that must "fan out" as 1/r^2). Nothing very suprising there. Nevertheless the deeper into exploration of physical laws you get, the more surprising interconnections pop up independent of our need to observe them.
Your assertion that "momentum" is simply a convenient and observed quantity is both false and misleading. "Momentum" is a fundamental quantity that relates directly and ... well, fundamentally to the nature of energy, space, time, et cetera. It is particularly noteworthy that the nature of space and momentum should relate to our perception of time -- a property/dimension/quality which is quite distinct from all others in its one-way observable nature. The laws of "physics" seem to be time-invariant, yet the laws of "thermodynamics" which are equally fundamental seem to recognize that time is somehow special.
Thus, it is misleading to imply that our physical laws are simple and elegant because we have simple and elegant requirements to describe the universe. An accurate description of the universe need not be simple -- and often it is not. For instance, I understand (although lack the mathematical sophistication to prove) that the electron spin g-factor has a theoretical value of exactly 2. Yet it is observed to be approximately 2.00232 and is one of the most precisely measured physical constants. So it is not always simple truth and beauty. Which makes it all the more surprising when the simplicity is there nevertheless.
And while it is true that the inverse-square law breaks down at relativistic energies, even that corrective factor of "gamma" remains mathematically simple, and in fact geometrically constructable via a pythagorean triangle analysis of a certain thought-experiment.
My point is that the easy examples are easily explained away by laymen, yet the surprisingly simple nature of the fundamental laws of the universe continue to pop up where you wouldn't expect. That is why expert scientists, true geniuses, of the sort that don't come along every day, routinely make comments about the "beauty" of physics. They have a deep understanding and "feeling" about the way the universe fits together that isn't captured by your example about momentum.