Mathematics is a science for weirdos. It is the basis for all the other branches of science, either directly or indirectly, it’s full of misconceptions, and often closer to philosophy than the scientific endeavour of describing nature. Mathematicians work in the abstract, in symbols, and set up their own rules and conditions whenever they want. They love thought experiments, and impossible situations that open up for interesting problems. Some say maths is anything that’s got numbers, or classical shapes, but if I’ve learned anything in my formal teaching, is that mathematics, above everything, is about relationships, connections, and bare logic, quite a lot like philosophy.

As a student, I was fascinated by the story around Euclid’s Elements, and how we discovered spherical and hyperbolic geometry. Ancient Greeks, famous for their interest in this subject, constructed their mathematics exclusively on geometrical facts; no numbers, no algebra, no x to be found. Euclid was one of those Ancient Greek mathematicians who wrote the most reedited book in history, the Elements, which is quite like a textbook of geometry and some primitive number theory. Everything he states on those lines is proven basing himself on a collection of definitions, and five postulates. Those postulates, also known as axioms, are unprovable truths that we take for granted, really basic statements such as “you can draw a circle of any radius centered in any point of the plane” or “all right angles are the same”. Everything that follows is sustained only and exclusively on those lines.

Ever since his book was published two and half millenia ago, there was a very large debate on whether one of his axioms was unnecessary. Many believed that it could be proven from the other four, even Euclid himself, though he kept it in place. The “fifth postulate” stated that “given a [straight] line and a point outside of it, you can draw one, and only one, parallel line to the former one that contains that point”. Euclid tried to make use of it as rarely as he could, and many mathematicians have given “proofs” that it wasn’t necessary.

It was a Hungarian mathematician I the 19th century, Bolyai János, who finally decided to see what would happen if you rewrote Euclid’s Elements removing the fifth axiom altogether. After all, not taking it for granted doesn’t imply the conclusions aren’t true. His results were, not only unexpected, but significant: not only such a geometrical surface could exist without the axiom, but he obtained two of them, the hyperbolic surface, and the spherical one. Spherical geometry is the one in which parallel lines can’t exist, as any diameter line along the surface of a sphere will intersect an other in exactly two opposed points. The hyperbolic case is the one in which every line has infinitely many parallel lines going through any given point, and the hyperbolic plane can be pictured as a trumpet-like cone, which is infinitely long, becoming ever narrower and narrower.

Bolyai’s work suggests that modifying one’s starting point, it’s axioms, can radically modify the outcome of one’s deductions, without those being less true, and it’s not only in mathematics where we see those kind of results, Albert Einstein’s Relativity starts in a very similar manner too. In Einstein’s case, several experiments in the late 19th Century seemed to indicate that light travelled at a finite speed, which given the current physical theory, such a thing went against the concepts of movement, and time. Einstein, in a similar way to what Max Planck had done in it’s time, threw away the existing theory, and started reasoning from a handful of postulates, in fact just three of them. He assumed light travelled at a finite speed, as experiments indicated, he assumed that there was no preferred frame of reference, as Galileo had done back in his day, and he assumed as well that you could swap frames of reference with it all making sense.

The result of those postulates was the proof that the Lorentz Transformations, developed by Dutch physicist Hendrik Lorentz, did explain the changes between frames of reference, as well as the undermining reality of our universe by which space and time aren’t absolute, which is one of the deepest results in physics we’ve ever developed.

All of those cases indicate us that the axioms we choose, the facts we take as starting points for our reasoning, are fundamental for our results. In fact, I’d say more than that: it’s all about that choice, because everything we develop out of them is done logically, thus we can all agree on the reasoning. Logic is common to all, we all agree upon it, and we all make use of it. Logic gates are at the heart of how we reason, regardless if it’s an exclusively human perception or universally common. Therefore, we can’t argue upon a logical conclusion, as it is inequivocal if properly obtained.

In the same way that the criterion in mathematics is rigorous proofs, and the criterion in physics and experimental sciences is rigorous experimentation, the criterion in philosophy is no other than rigorous arguments. The most rigorous kind of argument to philosophers has traditionally been the deductive one, as it is the one that obtains the strongest conclusions, though it depends on previously verified truths. Back on our discussion, if we established a set of postulates and definitions to start with, and we proceeded exclusively deductively, all of our conclusions should be true if, and only if, our preestablished axioms are as well.

The origin of those postulates, ironically, can be no other than inductive reasoning, or blind assumptions. Just like in Einstein’s case, his postulates were inspired in experimental evidence, and in Bolyai and Euclid’s case, their axioms were solely based on inductive intuition. They can’t be developed in a deductive way, as there is nothing prior, yet in fact, it doesn’t really matter what their source is. The whole idea of an axiom is assuming a truth that we can’t proof, quite like making an act of faith.

Nevertheless, as everything developed from our original assumptions is done deductively, thus of common agreement, this suggests us that arguments in philosophy aren’t about the conclusions, nor how we obtain them, but really they’re all about our initial convictions, and of mistaken deductions, which we will not take into account, as they are not part of the subject of this essay. Putting that aside, most physicists will agree that, if known all the specific initial conditions, we could predict how a system would evolve at any point in time. Perhaps same happens here. Known the initial assumptions, we can deduce what conclusions will that individual achieve.

Therefore, our ideology and opinion is essentially composed of some original convictions that we take for granted, either because we don’t question them at all, or because they can’t be questioned, as in the case of definitions. In essence, our diversity of opinion seems to only depend on our choice of what to use as our foundation, on what truths we understand as fundamental to start our reasoning. It’s all about the axioms, not only the ones you choose to believe, but also the ones you don’t choose matter, as in the case of Bolyai.

This doesn’t mean that there is a single undermining philosophical truth of the Universe, but quite the opposite. The fact that you can obtain equally valid conclusions, beginning from different starting points, despite them being completely separate results, implies that there is no absolute truth but different points of view, and the only common universal thing, as far as we have been able to experience, is bare logic. This also makes many arguments impossible to settle, as much as absurd. Political ideologies’ differences radicate in a very simple question: what the point of a social order is. Fascism and similar ideologies will argue that the aim is the good of the nations, of the group, of the abstract union of the individuals. Socialist tendencies will argue that the aim is the good of the members of such society, the citizenship, of as many of them possible, regardless of their individuality. Liberals will argue that the aim is the good of individuals as such, who can’t be nulled of such condition. The good of the group, of the majority, and of the individuals. Everything after that are the implications of assuming one of those postulates.

We fundament all of our knowledge in a series of postulates, which might be related to observations or not, but the simple fact that in doing so, depending on what choice we make, we can deeply change the logical conclusions of them, thus the outcoming opinions, indicates us both the lack and existence of absolute truth in the world we live in. We can say there is no absolute axioms, no absolute point to start, because if you start from anything that is true at some extent, you can only obtain true conclusions reasoning deductively from there, thus there is no central main truth of the universe, but many different things that are, depending on our frame of reference. However, there is an absolute, seemingly undeniable fact, which is the persistence of logic as an undeniable form or reasoning, yet it’s validity radicates only on the validity of the arguments of start, which we already mentioned.

In the end, truth might not be about absolutes, but about what’s real, about what we perceive as naturally part of the world we live in. Perhaps our universe is absolutely relative from it’s starting point.