I propose that all numbers besides the counting numbers are completely subjective and do not exist except in human thought and imagination. Most numbers can only describe the universe approximately, and are thus not real. Natural numbers are useful, and rational and complex numbers might still find real world applications, but I can easily construct a whole number that dwarfs the size of the visible universe, like 10^10^10^10^10^10. There also exist surreal numbers and infinite ordinals, which are also numbers despite exhibiting some strange properties. Mathematics is a human endeavor, and though we have used it as a tool to describe the universe and advance technology and engineering, mathematics is only a tool of our own making, nothing more.
Numbers, when applied to the physical world, are simply imperfect labels of things. They are almost always approximations. One can never measure the mass of an object, like an apple, exactly, since there is always some error. Our world is not ideal: if we carefully made a square of side length 1 m and tried to measure the diagonal, it would not be exactly sqrt(2). Our theories for describing nature may not be perfect, either. The laws of Newtonian mechanics provide a very good approximation for how projectiles fly and galaxies twirl, but it is not exact, and it would be a bit arrogant for humans as a whole to presume that theories of quantum mechanics and relativity are the last, best description of the universe. (On the other hand, natural numbers, used for counting, seem to be fundamentally a property of the universe: if I say three electrons, I mean exactly three electrons.)
As it stands, numbers are limited in predicting things. Perhaps it is simple to describe the trajectory of small spherical frictionless cannonballs in a vacuum, but fully-fledged human beings are another thing. We cannot predict the actions of humans, except in a few isolated cases. The universe is effectively non-deterministic – there is much we do not know and cannot know.
Yet we use math everywhere. Even complex numbers, which might initially seem useless, find applications in electrical engineering and other branches of science. Constants like pi or e are ubiquitous. Still, this is only one system of math that happens to work for us. If we were to encounter an advanced alien civilization and compare our knowledge of physics and math with them, we would probably find some similarities and many differences.
Not all numbers are useful for describing the universe. With exponent towers, we can easily make a number so big or so small as to be meaningless, like the aforementioned 10^10^10^10^10^10. Mathematicians have created much larger numbers, like Graham’s number, but that deals with the abstract problem of coloring the edges of n-dimensional hypercubes.
Mathematics, for the most part, is self-consistent – but only because we made it so. We have number systems, like the set of rational numbers, and we can do arithmetic with them. But did we simply make all of this up? Mathematicians have “discovered” the ordinals, some of which are larger than any countable number – effectively infinite. Yet there are infinitely many infinite ordinals, and mathematicians have defined ways to add, subtract, multiply, and divide them. Both the rationals and the ordinals are self-consistent – under the established rules we cannot find a contradiction.
Ultimately, I say that numbers (besides the natural numbers) are only made up by humans. We can create new systems of numbers, as we like; we can play with our existing rules and make “discoveries” and theorems based on those rules. With the numbers we have, we can only approximate the behavior of the universe, and we cannot predict everything. Numbers are only a game – a game of our own making.
No comments:
Post a Comment