On the Nature of Mathematics

Language, Science, or Art?

One of the oldest of all fields of study is that now known as mathematics. Often referred to, used, praised, and disparaged, it has long been one of the most central components of human thought, yet how many of us know what mathematics really is? Many have likened it to a language, while others claim it is a science, and a few more an art. In truth, it stands in a category all by itself, yet it incorporates aspects of those three and more.

At the simplest level, mathematics is a vast symbolic logic system possessing a few simple postulates from which an unimaginable, though finite, number of statements can be proven or disproven. These postulates were all arbitrarily selected by various men and women ranging from the long-forgotten past to the very present. The symbols commonly used to express the entities of mathematics and the rules for manipulating them have also been arbitrarily selected. These symbols are truly abstract; they represent nothing. A "2" is not two sheep, two random objects, or anything we can describe; it is simply a "2". Likewise, the rules we use with these symbols mean nothing--how do we "+" things? What does "=" mean? Even in combination, as in "2+2=4", we are not referring to anything in our world, simply to constructs of the human mind. It is one of the commonest errors to confuse these abstractions with things in our reality. However, these abstractions can be used to help talk about our reality. Through common agreement, we all know that saying, "Two sheep plus two sheep equal four sheep," is shorthand for "Taking a quantity of sheep that is the same as the number of hands a person has and placing it with a like amount of sheep yields an amount of sheep that is the same as the number of fingers (excluding the thumb) that a person has on his or her left hand." Certainly, through the blending of mathematics with a more conventional language, we have greatly simplified the task of describing many things in our world, although we must never make the mistake to think that we are dealing with pure numbers, and not numbered quantities, outside the world of pure mathematics.

Most of the symbols used in mathematics have been inspired by the real world. Numbering probably arose when someone decide that this business about hands and fingers was a mess, and that naming these quantities would be very useful. At some later point, someone thought of using these names for quantities without anything being quantified, and became the first true mathematician. Other concepts, such as Euclidean lines and perfect circles, also came from reality, but did not, do not, and cannot really exist. Any object made of matter cannot be perfectly circular; it will have ridiculously small imperfections, and nothing we can encounter is one-dimensional. Even imagining a one-dimensional line, not to mention a zero-dimensional point, is arguably beyond that capability of our minds. No matter what we dream up, it's still too thick.

Most of us tend to assume that the rules we apply to these symbols are some sort of natural laws, but they too have been chosen arbitrarily. We could just as easily define that "2+2=0", and that through any point outside of a line, there pass an infinite number of lines parallel to the first line, and mathematicians sometimes do. Indeed, there are an infinite number of possible mathematical systems that are completely self-consistent. The one we use all the time has the advantage of describing the way our world works, when properly related to it by means of a normal language. However, science over the past century has shown that perhaps we have chosen the wrong system, and that lines which seem to be parallel actually meet because the very space in which we live in is curved. This is handled by a model known as Riemann's or spherical geometry, as opposed to standard Euclidean geometry. The fact that alternate mathematics may also have their uses in describing our world, coupled with the fascination they hold for mathematicians tired of delving into the much-studied mathematics of tradition, is certainly justification enough to study them.

This brings up an important point. Mathematics, after its sheep-counting inception, has been studied primarily not for its utility but for its own intrinsic worth. The answer to the perennial, "When are we ever going to use this stuff?" is often a shocking (to the lay public), "Never!" Many branches of mathematics were developed simply because some mathematician thought they looked interesting. True, many times some later physicist or other such scientist has come along and found an amazingly practical use for this mathematics (sometimes to the consternation of the abstraction-loving mathematician), but this was not why that branch of the field was originally mapped out. This abstract quest for knowledge is something that the public has never understood. While they often accept that art and sometimes even science may be pursued merely for interest or pleasure, the concept of math that is not pragmatically useful has simply evaded them.

In many ways, mathematics is like an art. People picture mathematics as solving immensely complicated problems by means of massive number crunching, but few things could be further from the truth. Such tasks are generally found among accountants and engineers, two fields that make great use of mathematics, but not among true mathematicians. The mathematician's job is to find out new things that can be proven or disproven from the postulates. He works out the way for doing this once, and then never bothers with it again. Mathematics is discovery, not rote. To this end, rather than the ability to handle such massive calculations (which real mathematicians, no longer caring to actually work out nasty, real-world problems, often leave to computers), mathematics requires a blend of creativity, powerful reasoning ability, and a tinge of insanity (to dare to try ways that no one else has ever used, simply because they "obviously wouldn't work"). Vision is needed, as many proofs rely on completely different fields that have underlying fundamental similarities dues to the abstract and logical nature of math. Lastly, the mathematician, like a good artist, must love his work, for the artist that works only for a living and not because he loves his creations will do inferior work.

Mathematics, rather than being a science in its own right, is the foundation of all other sciences. Mathematics itself does not rely on the natural world for its information but on constructs of the human mind instead. (Although much of math was inspired by nature, it is still artificial.) Unlike science, there is no true knowledge to be gained from math--when the postulates were set, that determined what theorems would be true, false, or unprovable within that system. All that has ever remained is a process of identification. Also unlike science, mathematical knowledge is absolute. Mathematicians have no need of hypotheses, for the postulates tell that something is either right or wrong. This absoluteness is good for the other sciences, as they are based on mathematics. Mathematics is the language in which their scientific models are written, be it the equations of quantum physics and chemical kinetics or the statistics of population ecology.

As mentioned, mathematics is also a kind of language. Very unlike our everyday languages, it can unambiguously describe situations involving its numbers, figures, and operations, yet it is limited. Not in the sense that it cannot talk about our world, only its abstractions, but limited in that it cannot talk about itself. In English, I can refer to the very sentence I am writing, and even discuss its grammatical structure, but mathematics cannot talk about itself, it can only express mathematical concepts. In technical terms, it lacks the capacity of a metalanguage. Mathematics is of particular value when used in combination with normal languages to provide a sort of shorthand, allowing us to relate mathematical abstractions to certain physical concepts, letting us to count sheep easily, as well as build bridges that don't fall down.

So, while its practitioners would call it an art (and its detractors a black one), logicians would refer to it as a language, and most believe it to be a simple science, mathematics is really a blend of all these, forming a sort of foundation for modern human thought. Yet, we must also remember that mathematics has sprung entirely from the human mind, and if we find any errors within it, they must be due entirely to our own mistakes or faulty assumptions. Whether it be that the postulates we have chosen do not match how our physical world operates, or the more chilling prospect that our mathematics is actually not self-consistent, causing the work of millennia to come crashing down like a house of cards without supports, we can blame no instruments or complex quirks of the universe, but only ourselves.

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© Andrés Santiago Pérez-Bergquist, All rights reserved. The reproduction of this work, by any means electronic, physical, or otherwise, in whole or in part, except for the purposes of review or criticism, without the express written consent of the author, is strictly prohibited. All references to copyrighted and/or trademarked names and ideas held by other individuals and/or corporations should not be considered a challenge to said copyrights and trademarks.

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