is because those choices work.
Almost all of those choices are made in our subconscious mind in the context of deciding what to do in survival circumstances.
Long before our species evolved, the brains of animals, which I presume contained some kind of subconscious mind, or a pre-cursor to it, was able to calculate trajectories of moving objects, etc. in order to catch things or avoid things etc.
These calculations, of course, did not use any kind of mathematical or language symbols, and there was no conscious awareness of what was going on; only the useful results.
Then along came Euclid, et. al. who found that you can duplicate some of the rudimentary examples of some of these calculations and capture them in language and symbols. For the things he couldn't quite capture, like why logical expressions like 'and' and 'or' and 'if' make sense, he just assumed that there was some inexplicable 'truth' available to our minds that we could apply. Similarly, the axioms were taken to be obvious truths which needed no proof.
In reality, these were simply concepts used by our subconscious minds that made it possible to predict enough of what might go on in the world in order to help us survive on a day to day basis.
The reason Euclidean geometry 'worked' was because for the purposes of survival, the world was flat enough. The calculations based on a flat geometry yielded answers that were close enough. Much later, we routinely used the Pythagorean theorem to survey land because the curvature of the earth was so slight over the small distances of the surveys that it didn't introduce a noticible error.
When we got to surveying larger portions of the earth, we did need to correct for that error and switch to spherical geometry, in which, as has been pointed out earlier, the Pythagorean theorem does not hold.
Now, with our high-tech surveying instruments, we still consider the path of light to be a straight line, even though we recognize that it is really curved a little because of the gravity of the earth. Again, the error is so slight that we can ignore it.
Mathematics has tried all kinds of postulate combinations and derived all kinds of theoretical structures from them. Most of them bear no relationship to anything real that is known.
But, for some of those mathematical systems, the system of numbers in particular, we have discovered a remarkable correspondence between the world we perceive and the theorems of that system.
(I forgot to mention that mathematicians, since the grudging acceptance of Lobachevsky's non-Euclidean geometry, have given up on the notion that any 'axiom' represents any kind of truth. Axioms and postulates are now considered to be the same thing: simply an arbitrary choice of a starting condition for developing a particular mathematical theory, with no connection to any truth whatsoever.)
As we discover that our world doesn't exactly match the mathematical theory, such as discovering that the world is not flat after all, we simply abandon the old math theory and adopt a new one that seems to fit better with our experiences.
I think we are, or should be, on the threshhold of realizing that the old 'smooth' continuous math of Newton's and Leibniz's calculus does not match the 'graininess' we find in our world at the smallest scales. I think it is high time we developed a new 'grainy' math to better deal with this situation.
I think another threshhold we should be crossing is to acknowledge that macro behavior of the world can only be expressed as statistical results of micro, or more fundamental, behavior. Here, I think Dick's work can show the way. He has developed a theorem of statistics, or probability theory, that says that any statistical observations of sets of numbers must obey his fundamental differential equation. He then goes on to show that solutions to his differential equation are simply the familiar Maxwell, Schroedinger, and Einsteinean equations.
It is exciting for me to speculate on what some other, yet to be discovered, solutions to Dick's equation might suggest about the world we perceive, and maybe even some portions of reality which we don't perceive.
In summary, Math IS a human invention, and we keep reinventing it as we discover that world doesn't quite match up. The marvelous mystery is that the world matches up to any degree at all.
Warm regards,
Paul |