Why is one plus one two? Why is anything into zero zero? Why is two to the power zero one? Any mathematician would categorically reply to you, sighing deeply, that these are axioms in math; that they’re obvious. But are they really that obvious? And what really are axioms? Most significantly, what even is math, using certainties in a world where only uncertainty is certain? Is the universe really just math, or are we being presumptuous of our talents? Was math invented, or discovered?
What are axioms?
Merriam-Webster defines an axiom as ‘a statement accepted as true as the basis for argument or inference/an established rule or principle or a self-evident truth/a maxim widely accepted on its intrinsic merit’. A common feature of any mathematics textbook, they are those statements which we commit to our memory without question and utilise for the rest of our lives. If you are reading this, it’s almost guaranteed that you have studied some level of mathematics (as you are literate). Here are the most basic principles (incomplete, since math is built on a multilayered system of axioms), the building blocks of all algebra and geometry, really:
A number is equal to itself
Equations are symmetric around equality signs
Things equal to the same thing are equal to each other
Two equal quantities when added, subtracted, multiplied or divided by equal quantities remain equal
The whole of a quantity is greater than a part.
A line can be drawn from a point to any other point
A finite line can be extended infinitely
Any circle can be drawn given a centre and a radius
All right angles are ninety degrees
If a line intersects two other lines such that the sum of the interior angles on one side of the intersecting line is less than the sum of two right angles, then the lines meet on that side and not on the other side
Is math obvious?
All of these seem quite obvious or self-evident to us. For example, the ninth point lists that ‘all right angles are ninety degrees’, which cannot possibly be wrong, since right angles are by definition 90°. However, that would imply that mathematics is defined… by mathematics? Since we ourselves define that inside the realm of mathematics all right angles are 90°, there is no possible outcome in mathematics where a right angle is not 90°. Yet where is this right angle in reality? There are no right angles outside mathematics. One could argue that we ‘discovered’ the existence of right angles. But from what? In accordance to the quantum uncertainty principle, there is no such thing as a right angle, so where did we ‘discover’ a right angle?
Similar qualms apply to quantities such as numbers (of all kinds – real, imaginary, complex and whatever new ones they make up in the future). Mathematical realists, who believe that we have discovered and not invented mathematics, believe that they are most secure in this argument. Counting is something intrinsic to man – from our paleolithic hunter-gatherer predecessors to our digital successors – we have measured our resources via counting. If there are two sheep, logically speaking, the number of sheep are two, and therefore can be denoted by an equivalent of our numeral two (2). However, is the twoness really a quality of the sheep? Can we really say that just because there are two sheep, the sheep have the quality of twoness? Suppose we added a third sheep. Now, the number of sheep are three, or 3. Does that mean that without, by any physical means, affecting the sheep we have converted their quality of twoness to the quality of threeness? That doesn't add up. The numeral value of an item is not a defining characteristic of its physical object. Therefore, numbers are not linked to the object, rather, they are linked to what we perceive and imagine the object to be.
These questions are often raised by a sect of philosophers who contrast the mathematical realists – the mathematical fictionalists. As the name suggests, these philosophers believe that math is the application of human imagination as a means of identifying patterns in and understanding the universe. In a nutshell, they believe that math is invented. Keeping in mind, we can proceed to hypothesize what is math.
What is math?
Complicated.
To elaborate, mathematics is the ‘science’ (put liberally, in very abstract terms) of numbers, quantities and space abstractly or as applied to other disciplines. Of course, in my opinion, if Math is science, Macbeth is history. Mathematics, in its primary function, is meant to unlock the fundamental quandaries of the universe – how is anything and everything as it is? This is often taken by realists to mean that math is the language of the cosmos. They believe that since math was discovered by humanity, math has existed throughout the entirety of space and time, defining its every move: that math is the universe.
On the other hand, if math was the universe, why is there no physical quantity like the number one? Where in the universe do we find an example of the simplest two-dimensional shape – the triangle? You may differ, stating that numbers and geometric objects have an objective and not a tangible existence. That the universe communicates via math, that while there may be no physical quantity in mathematics, it explains our universe in terms we can grasp and furthers our understanding.
To put this in brazen phrases, this means that humans utilise math, an imaginary non-empirical concept created with a measure of perfection, to explain science, a real empirical concept in which perfection is impossible and exceptions always exist. This would be akin to acknowledging that dreams define the world, and not the other way around. Certainly, we’d all love to be who we dream we are, but that clearly isn’t the case. Perhaps, then, mathematics was not discovered, ergo a part of the cosmos, but rather was invented.
Take a history textbook. A history textbook does not create history – it explains it. History textbooks are not written before history is made. History textbooks are written after history is made. Sure, we try to predict the future, but we always correct where we went wrong when we realise how history actually proceeds. Similarly, math did not create the universe. The universe isn’t mathematical (I mean: everytime a scientist decides to apply a theory to the universe, the universe vengefully throws in an exception [E.g. Universal Theory of Gravity, General Theory of Relativity]). However, we cannot dispute that mathematics tries to explain the universe (key word: tries). Just as history textbooks come after history, math tries to keep up with the pace of the universe. Just like historians and scholars try to predict the future based on the past, we try to predict the universe based on the math we know.
On the contrary, we cannot and should not construe this to mean that math is the universe. The universe is not mathematical, and math is not the language of the cosmos. If the universe was speaking to us, math is the language in which we’ve written down the transcript. Therefore, math does not factually in any possible way exist: it is imagined and invented by the fascinating machinations of the human mind. There may still be some of you who doubt that mathematics is imaginary. On what other principle would the universe work? How else can we describe the utterly confusing workings of the universe? With logic. To our brains, the most intuitive mechanism to explain the cosmos is mathematics, up until now. Without conflating mathematics with logic, we have to acknowledge that pure logic and other logic-based systems can exist, and may be a potential alternative to mathematics itself.
Conclusion?
Mathematics is fiction. Just as beautifully written, and just as true, as the Lord of the Rings trilogy. Despite the pantheistic panic this causes amongst mathematical realists, who have a nearly-religious belief that abstract mathematical concepts inhabit the depths of our universe, we should understand that mathematics is not based on any empirical data. Yet, this does not imply that it should be discarded as worthless. For now, and the foreseeable future, mathematics may as well remain the sole logic-based system available to humankind. It is not the best (I mean with every passing day, math withdraws further and further from reality, with equations over equations amounting to scientifically produced gibberish). We must not confine ourselves to the limitations of mathematics. As Ron Garan once said, ‘We are limited by our imagination alone.’
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