Considering last week I made a tech article, I figured this one could be more abstract. I wish to talk about whether mathematics exists objectively in the universe. I’ll be looking at the three most popular views, and how they compare to each other.
Mathematics as a system is elegant, exhaustive and frequently even beautiful. In the words of Eugene Wigner it is “Unreasonably effective when put to use in all other sciences.” But where does math come from? Whereas biology is the study of living organisms, Physics the study of the universe and its forces, chemistry the study of chemicals; Math, is the study of math. Unlike all the other sciences, math lacks an empirical component; math cannot be observed in nature. This has caused some mathematicians and philosophers to doubt that math, or numbers, have an objective existence. Math and all of its elements, such as calculus, parabolas, and subtraction do not actually exist.
Mathematical Platonism is the belief that numbers are real abstract objects, outside space and time and it is the most popular view among mathematicians. Platonists say that math is real, and that it has an objective existence. They think that beyond the mathematics humans already know, there is more math. This argument seems logical at first, however there are several examples against this, the most prominent being Andrew Wiles proof of Fermat’s last theorem. Wiles spent years in his house working to create a proof, not to find it. Another big problem with Platonism is the question, how are these mathematicians so reliably accessing this world of abstract mathematics? As well, seeing as how math is only provable with more math, a tautology arises. The only math which is known by humans is the math knowable, by humans. This seems obvious at first, but it is really important. Platonists believe in an objective mathematical universe that contains concepts human’s might one day discover, or might never discover; and in the absence of a direct observation of mathematics, in the same way a biologist can directly observe animals, Platonism boils down to a kind of faith. A faith in a mathematical entity, or set of entities, which is out there in the universe waiting to be discovered; and would be there regardless if there are humans or not. This causes the view of Platonism to have a few, quite frayed loose ends.
Moving away from Platonism, Mathematical Nominalism is the view that man’s mathematical claims are true, but should best be understood through everyday objects. Children are not born with the inherent knowledge of numbers; they are taught it using things like pencils and blocks. Whereas the Platonist would say six times seven is forty-two, the nominalist agrees, but says that it only has a purpose in the sense that if there are six groups of seven objects, there are 42 objects. The nominalists say that is all there is to mathematics. Although this is the most common view for regular people, the argument breaks down when applied to irrational and imaginary numbers. What kind of objects do these numbers represent? For the Platonist it is easy; it is just another number. However for the nominalist it is quite contradictory to say the square-root of negative one can represent anything physical.
Finally, there is the unintuitive view of Mathematical Fictionalism. The Fictionalist says that although mathematics is useful, it has no real value outside the set of rules that people have designed for it. They say that, similar to a game or a story, the mathematical statement ten divided by two equals five is about as true as Batman sped down the street in his Batmobile. Within the confines of the story it makes perfect sense; but outside the story, ten, five, Batman and his Batmobile are all equally fictional. Much like an atheist can agree that the Ten Commandments set a good groundwork for the laws of society, a Fictionalist can recognize the benefits of using mathematics. However, simply because something works does not make it true. A Fictionalist sees math merely as a tool humans developed to model regularities in the universe, which are not inherently mathematic. From this it is clear that Fictionalism, unlike Platonism or Nominalism, is nearly flawless from a logical standpoint.
Philosopher Alain Badiou says “Mathematics is thus, a rigorous aesthetic; it tells us nothing of real being, but forges a fiction of intelligible consistency.” Since humans want to describe, discover, and probe; human’s are great at creating systems to do so. Numbers, operations, even measurements, are systems people created and designed to assist in mankinds overall understanding of the universe. If there were something better than math at describing the universe, people would use that. However, just because it is not inherently embedded in the universe, does not mean math is worthless. It is still beautiful, elegant and works remarkably well to approximate reality.
So what do you guys think? Is math a part of the universe, just a tool, a language, or maybe something else? let me know in the comments down below!
Until next time