This post won’t be as melancholy as the title may suggest; it forms part of my quest to acquire the basic ingredients of human existence. Mathematics is the unspoken language of nature, in which we must strive to achieve fluency if we are to bask in the richness of the universe in which we are marooned. When we speak in our mother tongue we tend not to concern ourselves with proof. We may perceive a distinction between a “lie” and the “truth” but we allow for expression of opinion, without which the world would be much more honest but considerably less interesting. Take the statement I can’t go to drinks this evening, I am really busy. How taken aback you would be if the response to your statement was; prove it? Conversely, if I came to you with a statement for finding the nth prime number in the language of mathematics, how taken aback would I be if the response didn’t come; prove it? Is it that we just don’t require rigorous proof in the spoken language, or are we just accepting a set of truths deeming them not to require any derivation? There are parallels between the mathematical language and spoken word; the most important being underlying conventions or axioms (rules taken as true), which in our own human way we take for granted. Remove these rules – chaos reigns.
Kurt Friedrich Gödel was a mathematician, logician and philosopher and widely regarded as one of the most important figures in mathematical logic (and certainly the most important contemporary). At the tender age of 25, Gödel was able to prove his incompleteness theorems, the results of which have far reaching impact. I really should try and pack a little more into my years. His theorems can be summarized as;
- Within any system where enough arithmetic can be carried out there are statements written in the language of this system that can’t be proved or disproved.
- A system cannot prove that the system itself is consistent, assuming it is in the first place (which we could never prove!).
Roughly speaking a “system” means a set of rules within which we can create new theorems. We generally call these rules axioms; they are the truths of the system if you like. We then have operations we may perform within our system, which coupled with the axioms allow us to determine for any formula (in the language of the system) if it is a proper derivation or proof within the system. The natural numbers are a very basic example, coupled with the arithmetic operators we know so well. If the above definition didn’t come naturally it might be worth a reread; it is quite important. Now a system can be described as complete if for any statement we generate within the system we can prove/derive the statement (within the system). A system is consistent if, for every provable statement I cannot also correctly prove the contra. To use an example; if I can prove I am a man, for the system to be consistent I better not be able to also prove I am a woman; which, to my knowledge I cannot.
So what does all of this mean? It is really important to understand that a statement being false is not analogous with a disprovable statement. Although not a perfect example, consider the idea that no number of white swans may prove the statement that all swans are white – all swans might be white, but even with a large army of white swans I am nothing more than a man with lots of swans. In order to get your head around this we think about computer programming; if you read about Gödel’s theorems this is what you will often see cited.
We use a “perfect” computer where we don’t worry about petty things like processors, electricity and alike. Just one big super computer that exists (in your mind). The computer can perform the usual arithmetic operations of addition, subtraction, division and multiplication. You can program your computer to perform tasks, you really can pick anything you like it’s your computer. You might be interested in knowing if a value is larger than 100 – well tell the computer to subtract 100 from the value and if the result is 0 or less display false. We can get more sophisticated and program our wonderful computer to determine primes (a sequence of divisions), Fibonacci numbers or square numbers. We can do whatever we like with our supercomputer, right? Wrong.
As a direct consequence of Gödel’s theorems, there are statements (only involving arithmetic, we aren’t asking the meaning of life), correctly inputted into our not so super computer which it cannot prove or disprove. The computer genuinely cannot decide if the outcome is true or false; nothing worse than an indecisive computer. To make things spookier, say you introduce a new axiom to the computer, a cheap work around to say that the bothersome statement is either true or false (it may well be intuitive) things don’t get better. You would either generate more statements the computer cannot decide between (1) or you would create results that contradict the pre-improved computer which are valid in the new post-improved computer – inconsistent (2). When this whole math-bomb dropped people hoped it was just a quirk of the system as defined rather than the reality of systems we know and love. Alan Turing, among others showed that this is reality. An incomplete or inconsistent mathematical meltdown keeping logicians up at night pondering all they once knew.
The theory has been used in a huge number of areas; including as an attempt to prove the existence of God I know some of my readers will be delighted to know! But alas, that is not the direction I intend to take for the swansong; instead I delve inside your mind which is much more interesting. I exaggerate a little, we will all need to journey deep inside our own minds. If we consider our brains as essentially machines (your choice), then do Gödel’s theorems apply? We hope – dearly – that our brains are consistent (although I do sometimes find myself questioning this watching political commentary today). So we arrive at the conclusion, based on the above, that there are basic statements with arithmetic operations we cannot prove or disprove within our own brain – which would be the strongest line of evidence that we cannot ever construct or program a human mind. Rodger Penrose is a firm believer of this, who controversially states that a human mind is capable of knowing the truth about these Gödel-disprovable statements; this form of intelligence is never able to be computed. More on this in the future, but I should tell you this has been met with some severe criticism; centrally that we cannot know a human brain is consistent without much more understanding of the inner workings, especially due to the number of mistakes we make we could very plausibly be neurologically inconsistent, (and perhaps some more than others!).
That aside, Gödel’s theorems raise some profound questions around the foundations of mathematics and the nature of our brains. The mathematical proof is quite dense and I do not intend to outline it; if you are interested I actually find Wikipedia the most usable but please let me know if you have more user-friendly sources. The language of logic is a difficult beast to tame.