What is Mathematics really?

I am irresistibly tempted to follow the common trope of essays like this and start with a dictionary definition. I will however update it to our times and quote Wikipedia:

Mathematics (from Greek μάθημα máthēma, “knowledge, study, learning”) includes the study of such topics as quantity, structure, space, and change.

The difficulty to define mathematics in a single sentence can be seen here. Let’s continue.

Mathematicians seek and use patterns to formulate new conjectures; they resolve the truth or falsity of conjectures by mathematical proof. When mathematical structures are good models of real phenomena, then mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation, measurement, and the systematic study of the shapes and motions of physical objects. Practical mathematics has been a human activity from as far back as written records exist. The research required to solve mathematical problems can take years or even centuries of sustained inquiry.

Rigorous arguments first appeared in Greek mathematics, most notably in Euclid’s Elements. Since the pioneering work of Giuseppe Peano (1858–1932), David Hilbert (1862–1943), and others on axiomatic systems in the late 19th century, it has become customary to view mathematical research as establishing truth by rigorous deduction from appropriately chosen axioms and definitions. Mathematics developed at a relatively slow pace until the Renaissance, when mathematical innovations interacting with new scientific discoveries led to a rapid increase in the rate of mathematical discovery that has continued to the present day.

Ok, so let’s deconstruct this. Many people associate mathematics exclusively with some of the most famous mathematical structures, like numbers, but this is imprecise. Mathematics can be defined by a common methodology, that is, mathematics is defined as work that is made following the mathematical method, just like science can be defined as work that follows the scientific method.

The basic element of the mathematical method is the proof (just like in science we have experiments). Proofs are sequences of deductive arguments leading to a conclusion. Once something is proven, it is accepted as true. Let’s see an example:

We want to prove that there are an infinite quantity of prime numbers (integer numbers that can only be divided by 1 and themselves). Well, assume that there are only a finite amount of prime numbers. Multiply all prime numbers and add one to the result. This number is not divisible by any prime number, because the remainder of the division is always 1. Therefore, it is not divisible by any number between 1 and itself. So it is itself a prime number. It could not have been in our original list of primes however, so we have reached a contradiction. Therefore, there must be an infinite quantity of primes.

Deduction is the process of logically inferring conclusion from premises. When doing deduction, all the information is laid out up front and the conclusions necessarily follow. It is customary to refer to one of Aristotle’s canonical deductive arguments:

Socrates is a man. All man are mortal. Therefore, Socrates is mortal.

In the proof above, the premises are not clearly written, though they are implicitly there. If the proof were written in a didactic book, the book would start with premises, and from then on the reader should assume all previously stated premises. In this case, the set of premises would be all the basic truths about integer numbers, like the fact that you can add one to any number and obtain another.

It is important to note that the mathematical method concerns itself exclusively with deduction. Your choice of axioms or premises is considered “arbitrary”, at least so far as mathematical correctness goes.

Though the vague notion of a deductive argument is kinda clear, reality is quite complicated. People make mistakes in the course of their deductive argumentation. In order to eliminate these mistakes (at least in principle), mathematicians invented axiomatic systems.

Formality

If you study mathematics in university, you will spend a great deal of your time worrying about formality. A formal argument can perhaps be defined as: “a deductive argument that can be expressed as an axiomatic system”.

Axiomatic systems (or formal systems) reduce deduction to a simple mechanical process. Let’s explore a toy example:

Let ab be our only axiom and we’ll have a single inference rule: b -> ab. So we have:

ab

And applying our inference rule we get aab, now we have:

ab

aab

Applying the rule again to all our prepositions we get:

ab

aab

aaab

And so on. Axioms are an arbitrary set of symbols and inference rules specify simple symbolic manipulations.

So what is going on here? The process is rigorous because it is easily repeatable by any human being and was designed to leave no run for reasonable doubt, making its conclusions certain.

Let’s really push it though. What if I really want to disagree with the conclusion “aaab”? Is it not within my rights? Maybe nothing is real, maybe reaching truth is impossible. Why is this “unreasonable”?

Search yourself, you know it to be true.

Yes, you can believe whatever you like, I guess. But you have to admit that it kinda, like, seems true. If you want, change the undoubtable conclusion to the following: “human minds recognise that this is a valid step of deduction”. You can doubt your recognition, but it is there.

Ok, I can follow these steps, but how are they connected to reality? This is the essence of the trick. Formal systems gives us a mechanical procedure we can follow, but they do not justify any sort of conclusion about the world made from these mechanical calculations. There are indeed many formal systems I could specify that generate lists of untruthful propositions, or just complete nonsense.

Formality gives mathematicians a language, and one that guarantees consensus between its speakers (an amazing accomplishment when compared with other disciplines). But to reach truth, we need to justify this language, and this cannot be done mathematically as the argument would be circular.

We need to go deeper

But before we move on, two points should be made here.

First, it’s worth noting that in the day-to-day of a mathematician a more vague notion of formality is used. Depending on the context (specially depending on the listener), arguments can be more or less formal, basically, you can skip steps if you can trust your listeners to fill them out for themselves. In fact, you can actually work your entire career as a mathematician without knowing what an axiomatic system is, but, as I mentioned before, you will definitely think a lot about formality.

Second, if formal arguments are a mechanical process, can we build mechanical machines that generate them? Indeed we can! And do: those are automatic theorem provers. However, they are very slow, so slow that I don’t know of any substantial theorem that was proved by an ATP. Why is that? The reason is that formality is very good at confirming the correctness of deductive arguments, but it does not give us a good way to come up with arguments in the first place, for that we need intuition.

Intuition

The true essence of mathematics lies in intuition.

Mathematicians reason about mathematics using their intuition of the mathematical objects they are discussing. Intuition is a controversial subject, maybe because a lot of the time it lives under the realm of language. Some argue that it is not worth sharing intuitions because they vary from person to person. But however nebulous it’s inner workings are, it is a fact that a mathematician’s main tool is his intuition.

It is because of this intuition that we are able to prove theorems. Our intuition considers mathematical objects differently from their pure formal existence, and so allows us to make some inferences very easily (and others with great difficulty). It is also intuition that allows us to take a mathematical theory and apply it elsewhere, only then obtaining propositions about the real world.

So by now you have a pretty good picture of how mathematics works. But little progress was made to the central question: does mathematics reach truth? Can we trust the mathematical method? Why does it work?

The rest of this essay will be an attempt to argue this point. We must however leave the realm of precise definitions and empirical facts, to enter the dark depths of philosophical discourse.

Many people argue that we can trust mathematics because it works: when we use it to build things, these things don’t fall apart. But this is not satisfactory: we have an intuitive sense of mathematical truths before we put them to the test. And consequently when we have proven something, we don’t test it, we just accept it with complete certainty. Like, try to imagine a world where 2 + 2 = 5. This isn’t just some random fact we have acquired from observing 2s and 4s in the world, it is incomprehensible to imagine any other possibility. Our brain rejects any attempt to deny mathematics.

Could this be the essence of the matter? Our brain? What if our brain is not a blank slate, but a machine built with certain specifications and capabilities? This is precisely the basis of Kant’s argument in the Critique of Pure Reason. If this is true, perhaps one of these innate capabilities is mathematical reasoning. Formal mathematics would then be an attempt to codify an innate capability of our brain to do reasoning, and by denying mathematics, we deny something built into our very brains.

But if mathematics is inside our head, why should it be “the language of reality”? This “brain theory” explains why we feel mathematics is true, but it doesn’t explain why it is indeed true.

Pull up your sleeves and let’s get philosophical. Assuming the brain is not a blank slate, we can continue with this line of reasoning, and claim that when we observe things, we observe them under the rules of our brain, and so things we perceive would follow the rules of mathematics exactly because our brains imbue them with mathematical properties in the very action of perceiving.

Hang on, that sounds like a pretty argument, but isn’t it just a bunch of bullshit?

Well, maybe, I mean, to actually argue this with certainty, you would have to have a deep understanding of how the brain works, something humanity hasn’t achieved yet. But it is meaningful to see if it makes sense, that is, if we can follow this reasoning maintaining coherence with our experience of reality. At the very least, it is how things could work, and gives us a theory that could guide scientific experiment and inquiry.

So if things get mathematical inside our brains, what would happen if there was something in reality that did not follow mathematical rules? That is, it seems that our observation of the world is pretty complete, in the sense that there isn’t anything obviously missing, because after observing things, we create mathematical rules to predict their behaviour and those seem to work pretty well, isn’t this then an argument in favour of mathematics existing in nature? If so, no progress was made in the end, and reality does indeed follow mathematical patterns.

Well, yes, but I think the question is misguided. The question of whether objects follow mathematical patterns assumes the notion of an object. I would argue that the very division of the world in objects is a contribution of our brain, and this division of objects (and other things like concepts) is made in a way which is coherent with mathematical reasoning.

Let’s think about how we come to identify the world as divided in a group of objects in the first place.

Let’s start with the basics: why do we need object? Why don’t we just comprehend the entire world as a single unity which cannot be split into many component parts. According to Douglas Hofstadter (in his excellent book Gödel, Escher, Bach), this is exactly what Zen Buddhists try to do: attempt to deny any division of the world into parts, including any honest attempt at language, as this division is a prerequisite of language.

Well, for starters, it would be pretty difficult to get anything done without objects. Like, how are we supposed to get food into our mouths if we can’t admit that food is an object. I guess that the whole concept of survival depends on our notion of objects: to claim that I am alive (or claim anything about myself for that matter), I need to separate myself from the rest of the universe.

Well, this seems like a pointless intellectual exercise.

But I think there is something to it. It gives me a feel, so far as that is possible, of imagining the world from outside my own mind. The world doesn’t need to understand itself, it just needs to exist. And so maybe understanding really is something that only makes sense inside the minds of beings that inhabit this world, a naturally selected phenomenon that helps these beings survive in nature.

So, this is how it might work: the world is. Just that, it just is. Then beings pop up with certain innate intellectual capabilities. These beings use these intellectual capabilities to try to understand the world. In doing so, they create a sort of model of the world, which is not identical with it, but is the product of their intellectual capabilities exposed to the natural environment. They then use these models to guide their actions and nature rewards that with bountiful harvests and warm baths. Mathematics codifies some of the most important human intellectual capabilities, and so mathematical truths can be applied to any model of the world created by a human mind. These models of the world are not unique, in the sense that there might be many models to explain a certain phenomenon, and sometimes we struggle to create any sort of model at all: the apparent paradox of the beginning of the universe is an example.

I’ll end the argument here. More could be said about the mind’s capabilities of creating models of the world, but it wouldn’t really be about mathematics anymore, and I think I have kept you long enough.

Feedback and appreciation are always welcome. Till next time!