Axioms and Illusions: The Limits of Knowledge and Proof

Introduction

The worlds of mathematics, science and philosophy all share a central pursuit: knowledge. In mathematics, a statement must be proved in order to be accepted as an undeniable fact. However, a fundamental reality of mathematics is that there will always exist truths which cannot be proven. This raises the question: should we adhere to the principle that knowledge requires proof, given that certain unproven statements appear no less valid than proven ones?

The method in which we formulate a valid proof is very specific. While its roots date all the way back to Thales of Miletus in the 6th century BC, the first specific categorisation and definition of proof is attributed to Aristotle in his book Posterior Analytics. In this work, and indeed in modern logic, it is said that a proof is established on the basis of axioms or postulates (assumed truths), single-step deductions and ultimately the conclusions to which they lead. Most notably, however, conclusions are sometimes used to disprove initial assumptions rather than to confirm it (commonly known as proof by contradiction).

Set Theory: The Axioms of Mathematics

One might expect that, applying the principle of axioms, all mathematical problems could be resolved without any issue. However, this basic assumption became the subject of significant debate following Georg Cantor’s development of set theory (1874-84). The resulting crisis split mathematicians into two groups: the intuitionists and the formalists. The intuitionists believed that mathematics exists solely within the human mind and that statements do not have objective truth or falsity until a human directly proves them. Thus, they rejected proof by contradiction and consequently declared that language was only a flawed means by which we describe our pure thoughts. However, the formalists opposed this and argued that mathematics is a strict game of symbol manipulation governed by a set of specific rules. Most importantly, they contended that symbols themselves are mathematics, rather than merely the things which present it.

An interesting illustration of dispute between them is Russell’s Paradox. This is a contradiction of set theory, one could say, because the laws of it were poorly defined at first (where any well-defined property could generate a set); there must exist a set ‘R’ which contains all the sets that do not contain themselves. If R contains itself, then by definition it must not; yet if R does not contain itself, it satisfies the condition and therefore must contain itself, thus a paradox is formed. A better way of explaining this is as follows, there is a village in which every man must either shave his own beard or have it shaven by the barber, but not both. Examining the barber’s circumstance brings around the exact same issue as before. However, this paradox was resolved by establishing precise axioms in set theory that restrict how sets are constructed, thus preventing self-referential conditions such as 'does not contain itself' from creating a valid set.

Hilbert's Three Principles

Set theory links to the idea of proof because it defines the qualities of mathematics with axioms and facts, which a proof requires. Formalism was embodied by David Hilbert, who is famously quoted to state “We must know. We will know.” Hilbert believed in three principles of mathematics which related to our knowledge and proof, each of which he thought to be true; completeness, consistency and decidability. The first says that all true statements in a given part of mathematics may be proven using the axioms of that section. The second is that no statements may defy or contradict one another. Thirdly, there exists an algorithm which can always look at any statement and tell whether it is true or false; the better way to put this is that it can look at any mathematical or computational system and tell what will happen.

Gödel’s Incompleteness Theorem

In response to the Hilbert's first claim, the logician Kurt Gödel published the Incompleteness Theorem in 1931. At the time, maths and logic were thought to be separate and different things. However, Gödel decided against this and made a system called the Gödel Numbering, where basic symbols and axioms were each assigned basic integers. Complex mathematical statements could then be created by assigning them a unique single number by taking the consecutive prime numbers and raising each to the exponent corresponding to each symbol's code, then taking the product of all the symbols. For example, the statement 1+1=2 would be

The direct successor (letter S, Gödel number 7) of 0 (letter 0, Gödel number 6) plus (letter +, Gödel number 11) is equal to (letter =, Gödel number 5) the direct successor (letter S, Gödel number 7) of the direct successor (letter S, Gödel number 7) of 0 (letter 0, Gödel number 6)

and thus would have the Gödel number

Furthermore, he made the statement P(x, y) mean ‘x is the valid proof for the statement with Gödel number y’. He then created a formula with a blank spot for a variable ‘x’ which says, 'If you take the formula with code number x and enter x into its own blank spot, the resulting statement, call it d, cannot be proven.' If you then plug ‘d’ into the blank spot, you make a new statement, g: 'The statement formed by plugging statement d into template d cannot be proved.' Thus it is saying, 'I cannot be proven,' which is evidently contradictory. If the final statement is true, there is no proof for it, and if it isn’t, there shouldn’t be, because it is false. This disproved Hilbert’s first principle. As for the second, we can’t know completely because of Gödel’s incompleteness theorem.

Turing and the Halting Problem

Hilbert's third principle of decidability was disproven just five years later, in 1936 by Alan Turing. Firstly, he imagined the Turing-machine, a computer which takes an input of 1s and 0s. It can perform one of four options: either move left, move right, halt, or write over the code (these depend on the specific computer but don’t matter as they don’t change the outcome of the experiment). The question which he asked himself (which he found had a likeness to Hilbert’s principle) was whether, if given the code and software, an algorithm could always find the outcome of the machine and whether it will halt in a finite amount of time, or without playing it. This is known as the Halting Problem. It must be said that the Turing-machine can run any algorithm in the world, if given enough time, even ones for modern software such as Instagram or TikTok, and therefore are not limited in computing power.

We may assume that there exists a machine called ‘h’ which, given the program and input (P, I) of a machine, can tell whether the Turing-machine will halt or not, in a finite amount of time. We do not know why or how it works, we just assume for contradiction that it does. Therefore, if given input I, program P halts, ‘h’ outputs ‘HALTS’; if given input I, program P loops forever, ‘h’ outputs ‘LOOPS’. Like any computer, components may be added onto it, they are as follows, if ‘h’ outputs ‘LOOPS’, ‘h’ halts immediately, and if it outputs ‘HALTS’, ‘h’ forces itself into an infinite loop (it hard to explain how it does this, but it certainly can).

We may now call this whole machine with components ‘H’. If we were to enter (P,I) of ‘H’ into itself, there would be a contradiction. For if ‘H’ concludes that it halts, it forces itself into a loop, and therefore was wrong; or it said that it halted, in which case it is in an infinite loop and is also wrong. Furthermore, machine ‘h’ cannot exist and Hilbert's views were invalidated. Mathematics is neither decidable nor complete, and could even be contradictory.

Practical Certainty and Moorean Truths

Although we may not know everything that is to be proven with certainty, we may still know unprovable things. In philosophy, there are people who claim statements such as, ‘you are in a computer simulation, and you can’t know you aren’t!’ and ‘you don’t know you have two hands!’. Although these statements may sound ridiculous, there are certainly some elements of truth in them, for it is true that they cannot be proven wrong. However, they are wrong in the sense that we may not know these things, as there is absolutely zero doubt about their falsity. Philosopher G E Moore invented the idea of ‘Moorean Truths’ which state that we know something if we do not have any reasonable doubt about it. His famous example of this is ‘Here is one hand, and here is another,’ demonstrating that we know everyday empirical facts with much greater certainty than abstract and skeptical premises.

This principle may be applied similarly to mathematics, as there are many things which we know with certainty but cannot prove such as the Collatz, Goldbach and Twin Prime Conjecture. These conjectures are most certainly true, and mathematicians have almost no doubt about this, so why should we not take them to be true? One would think that there would be no logical reasons to do so, however some examples in the past of similar circumstances have shown otherwise.

The Pólya Conjecture was a theorem which stated that for all n > 1, at least 50% of the numbers up to n must have an odd number of prime factors. This was taken for granted, from its founding in 1919 up until 1958, when it was disproved upon testing numbers up to 900 million. Because of this warning, mathematics takes a precaution not to assume that things are correct just because they appear to be. For all we know, these three conjectures could be disproven in the future, as was Pólya’s. In short, we can’t assume things if we wish to be completely certain.

Conclusion

We cannot know everything for certain, and we should be weary about what we take for granted. As per Gödel’s and Turing’s conclusions, we cannot prove all true statements and there exists nothing which can decide the outcome of a system without any doubt. Although this may sound rather disappointing, it nuances the study of mathematics by introducing a philosophical element. This prevents mathematics from falling into the same trap as physics, a field with many certainties and unknowns existing in conjunction with each other as well as many immense difficulties in the making of new theorems. Despite these limitations, mathematics strikes a good balance between knowing everything (such as biology, being quite unexciting) and having extraordinarily challenging problems, as in quantum physics. Additionally, it saves us from having to search endlessly for true proofs; as did Whitehead and Russell in their Principia Mathematica in which it took him over 360 pages to prove that 1 + 1 = 2, before stating that it was not proper and had to be revisited in their second volume; whilst also being able to have interesting conversations surrounding the pursuit of truth.

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