Thursday, 2 May 2013

The impact of Godels theories on modern science and maths

The greatest most intelligent minds that walked this earth in terms of understanding the machinery of the universe were likely Albert Einstein and Godel, I still think to this day. We would be wise to take a historical note of what Einsteins later years were spent doing and what conclusions he came to, as the scientific community ran off and made mainstream with a set of his theories he himself was deeply uneasy with. Einstein and Godel seemed to reach similar conclusions, but seemingly one of them coped better with the implications of this than the other in the end.

The solid mathematical foundations that a lot of these modern day Nobels are awarded for in linearized maths, have a slightly darker and mirkier past in a historical context. And in fact, the whole materialist myopic view of linear maths as revealing deeper and deeper truths to us about the universe is fatally flawed from the get go.

This is the true story of how some of our most intellectually stimulated minds untied the previously cosy relationship the universe seemed to have with the certainties of mathematics, and how these facts have been acknowledged but largely ignored. It's a story of how such deep questions being asked back then of such high importance resulted in the fact that when some of the greatest minds of the time engaged their mind with such questions their brain dare not look away from the evidence that perplexed them so much, and how pursuit of meaningful answers to these issues pushed them first to the brink of insanity, then over to madness and suicide.

But for all the human tragedy of great minds lost due to seeking meaning from life from maths and logic, what they saw is still true - the intellectuals at the time that took over the consensus opinion, assigning Einsteins work greater credibility than the original creator himself did, whilst in the case of Godels work largely ignoring it; so to this date we have yet to inherit at large the conclusions they themselves made.

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George Cantor (1845-1918) was a religious professor of maths who started a paradigm shift in the world of established maths and science, that maybe he did not appreciate at the time. The profundity of a brand new question, not based on previous knowledge or even a similar school of thought in maths at the time; he asked himself "how big is infinity"?

It’s just an incredible feat of imagination. It’s, to me, the equivalent of taking mind enhancing drugs for that era (1800-1900) Others before him, going back to the ancient Greeks at least, had asked the question but it was Cantor who made the journey no one else ever had, and found the answer. But he paid a price for his discovery. He died utterly alone in an insane asylum.

The question is what could the greatest mathematician of his century have seen that could drive him insane?

Cantor had auditary hallucinations from a little boy that he attributed to god as calling him to maths. So for Cantor, his mathematics of the nature of infinity had to be true, because God had revealed it to him.

Cantor soon discovered he could add and subtract infinities conceptually, and in fact discovered there was a vast new mathematics opening up infront of him - maths of the infinite. This out of the boxing thinking had revealed something special, and he could feel it as a sort of profound insight into the nature of maths he was previously blind to.

By 1884 Cantor has been working solidly on the Continuum Hypothesis for over 2 years. At the same time the personal and professional attacks on him for his heretical "maths of the infinites" had become more and more extreme. Due to this, the following may of that year he had a mental breakdown. His daughter describes how his whole personality is transformed. He would rant and rave and then fall completely and uncommunicatively silent. Eventually he is brought here to the NervenKlinik in Halle, which is an asylum.

Even after concerted further effort he could still not solve the Continuum Hypothesis, he came to describe the infinite as an abyss. A chasm perhaps between what he had seen and what he knew must be there but could never reach. He realized that there’s a way in which in order to understand something you have to look very hard at it but you also have to be able to sort of move away from it and kind of see it in a kind of holistic context, and the person who stares too hard can often can lose that sense of context.

After the death of a close relative, Cantor went on to say that he "could no longer" even remember why he himself had left music in order to go into maths. That secret 'voice' which had once called him on to mathematics and given meaning to his life and work. The voice he identified with God. That voice too had left him.

Here I divert from Cantor, because if we treat Cantor’s story in isolation it does little to bridge the gap in the idea that Cantor had dislodged something was part of a much broader feeling of that time. That things once felt to be solid were slipping. A feeling seen more clearly in the story of his great contemporary- a man called Ludwig Boltzmann.

The physics of Boltzmann’s time was still the physics of certainty, of an ordered universe, determined from above by predictable and timeless God-given laws. Boltzmann suggested that the order of the world was not imposed from above by God, but emerged from below, from the random bumping of atoms. A radical idea, at odds with its times, but the foundation of ours. Ernest Marc one of the most influential er philosopher of science at that time stated: 'I cannot see, I don’t need it, they do not exist so why we should bring them in the game.'

Worse than insisting on the reality of something people could not see, to base physics on atoms meant to base it on things whose behavior was too complex to predict. Which meant an entirely new kind of physics – one based on probabilities not certainties. Boltzman worked tirelessly at his idea irrespective, and as Boltzmann got older and more exhausted from the struggle, he'd get mood swings, mood swings that became more and more severe. More and more of Boltzmann’s energy was absorbed in trying to convince his opponents that his theory was correct. He wrote, “No sacrifice is too high for this goal, which represents the whole meaning of my life.”

The last year of Boltzmann he didn’t do any research at all, I’m talking about the last 10 years. He was fully immersed in a dispute, philosophical dispute, tried to make his point – writing books which were most of the time the same repeating the same concept and so on. So you can see he was in a loop that didn’t go ahead. By the beginning of the 1900’s the struggle was getting too hard him.

Boltzmann had discovered one of the fundamental equations, which makes the universe work and he had dedicated his life to it. The philosopher Bertrand Russell said that for any great thinker, “This discovery that everything flows from these fundamental laws… comes”, as he described it, “with the overwhelming force of a revelation: like a palace emerging from the autumn mist, as the traveler ascends an Italian hillside,”

And so it was for Boltzmann. But for him, that palace was at Duino in Italy, where he hung himself.

A new generation of mathematicians and philosophers, were convinced if only they could solve the problem of the nature of infinity Maths could be made perfect again. Kurt Godel was born the year Boltzmann died 1906. He was an insatiably questioning boy, growing up in unstable times. His family called him Mr Why.

What Godel later showed in his Incompleteness Theorem is that no matter how large you make your basis of reasoning, your set of axioms in arithmetic there would always be statements that are true but cannot be proved. No matter how much data you have to build on, you will never prove all true statements.  

Godel's incompleteness proof involves constructing statements that are well-formed within the system in question, and cannot be proven true within the system, but can be proven true via analysis outside the system.

Mere undecidable statements (that cannot be proven true or false at all) are far easier to construct and do not render a formal system incomplete.

There are no holes in Godels argument. It is, in a way, a perfect argument. Thus the present tense of this paragraph, it stands unimpeachably strong to this day. The argument is so crystal clear, and obvious.

Yet still to this day, very few want to face the consequences of Godel. People want to go ahead with formal systems, and Godel explodes that formalist view of mathematics that you can just mechanically grind away on a fixed set of concepts. There’s a very ambivalent attitude to Godel even now a century after his birth. On the one hand he’s the greatest logician of all time so logicians will claim him but on the other hand they don’t want people who are not logicians to talk about the consequences of Godel’s work because the obvious conclusion from Godel’s work is that logic is a failure - let’s move onto something else, as this will destroy the field.

Godel too felt the effects of his conclusion. As he worked out the true extent of what he had done, Incompleteness began to eat away at his own beliefs about the nature of Mathematics. His health began to deteriorate and he began to worry about the state of his mind. In 1934 he had his first breakdown. But it was after he recovered however, that his real troubles began, when he made a fateful decision.

Almost as soon as Godel has finished the Incompleteness Theorem, he decides to work on the great unsolved problem of modern mathematics, Cantor’s Continuum Hypothesis. Godel, like Cantor before him, could neither solve the problem nor put it down - even as it made him unwell. Again, the mind so engaged the brain dare not look away from the evidence that perplexed the mind so much. He calls this the worst year of his life. He has a massive nervous breakdown and ends up in a sanatoria, just like Cantor himself.

Alan Turing is the next person to enter this brief history. Turing was most well known for breaking the Enigma code; but he is also the man who made Gödel’s already devastating Incompleteness Theorem even more devastating.

Computers being logic machines was Turings predominant world view, and he showed that since they are logic machines incompleteness meant there would always be some problems they would never solve. A machine fed one of those problems, would never stop. And worse, Turing proved there was no way of telling beforehand which these problems were.

With Gödels work there was the hope that you could distinguish between the provable and the unprovable and simply leave the unprovable to one side. What Turing does, is prove that, in fact, there is no way of telling which will be the unprovable problems. So how do you know when to stop? You will never know whether the problem you’re working on is simply fundamentally unprovable or extraordinarily difficult. And that is Turing’s Halting Problem.

Startling as the Halting problem was, the really profound part of Incompleteness, for Turing, was not what it said about logic or computers, but what it said about us and our minds. Were we or weren’t we computers? It was the question that went to the heart of who Turing was.

This tension between the human and the computational was central to Turing’s life – and he lived with it until, the events which led to his death. After the war Turing increasingly found himself drawing the attention of the security services. In the cold war, homosexuality was seen as not only illegal and immoral, but also a security risk. So when in March 1952 he was arrested, charged and found guilty of engaging in a homosexual act, the authorities decided he was a problem that needed to be fixed.

They would chemically castrate him by injecting him with the female hormone, Oestrogen. Turing was being treated as no more than a machine (which in a sadly ironic way is what he was trying to prove himself). Chemically re-programmed to eliminate the uncertainty of his sexuality and the risk they felt it posed to security and order. To his horror he found the treatment affected his mind and his body .He grew breasts, his moods altered and he worried about his mind. For a man who had always been authentic and at one with himself, it was as if he had been injected with hypocrisy.

On the 7th June 1954, Turing was found dead. At his bedside an apple from which he had taken several bites. Turing had poisoned the apple with cyanide. Turing had passed, but his question remained. Whether the mind was a computer and so limited by logic, or somehow able to transcend logic, was now the question that came to trouble the mind of Kurt Godel.

Having recovered from his time in the mentally unstable sanctum, by the time he got here to the Insititute for Advanced Study in America he was a very peculiar man. One of the stories they tell about him is if he was caught in the commons with a crowd of other people he so hated physical contact, that he would stand very still, so as to plot the perfect course out so as not to have to actually touch anyone. He also felt he was being poisoned by what he called bad air, from heating systems and air conditioners. And most of all he thought his food was being poisoned.

Peculiar as Gödel was his genius was undimmed. Unlike Turing, Godel could not believe we were like computers. He wanted to show how the mind had a way of reaching truth outside logic. And what it would mean if it couldn’t.

So, why so convinced was Godel that humans had
this spark of creativity? The key to his belief
comes from a deep conviction he shared with one of the few close friends he ever had, that other Austrian genius who had settled at the Institute, Albert Einstein.

Einstein used to say that he came here to the Institute for Advanced Studies simply for the privilege of walking home with Kurt Godel. And what was it that held this most unlikely of couples together. On the one hand you’ve got the warm and avuncular Einstein and on the other the rather cold, wizened and withdrawn Kurt Godel. The answer for this strange companionship comes I think from something else that Einstein said.. He said that "God may be subtle but he’s not malicious." And what does that mean? Well, it means for Einstein is that however complicated the universe might be there will always be beautiful rules by which it works. Godel believed the same idea from his point of view to mean, that God would never have put us into a creation that we could not then understand.

The question is, how is it that Kurt Gödel can believe that God is not malicious? That it’s all understandable? Because Gödel is the man who has proved that some things cannot be proven logically and rationally. So surely God must be malicious? The way he gets out of it is that Gödel, like Einstein, believes deeply in Intuition - That we can know things outside of logic, maths and computation; because we just intuit them. And they both believed this, because they both felt it. They have both had their moments of intuition, moments of sudden conceptual realisation that were by far more than just chance.

Einstein talked about new principles that the mathematician should adopt closing their eyes, tuning out the real world you can try to perceive directly by your mathematical intuition, the platonic world of ideas and come up with new principles which you can then use to extend the current set of principles in mathematics. And he viewed this as a way of getting around the limitations of his own theorem. He no longer thought that there was a limit to the mathematics that human beings were capable of. But how could he prove such subjectives?

The interpretation that Gödel himself drew was that computers are limited. He certainly tried again and again to work out that the human mind transcends the computer. In the sense that he can’t understand things to be true that cannot be proved by a computer programme. Gödel also was wrestling with some finding means of knowledge which are not based on experience and on mathematical reasoning but on some sort of intuition. The frustration for Gödel was getting anyone to understand him.

Gödel was trying to show what one might call mathematical intuition of the kind we see in the brains of Synesthesia Savants such as Daniel Tammet in current times, and he was demonstrating that this is outside just following formal rules. What he had shown was that for any system that you adopt, which in a sense the mind has been removed from it because it's you that's used to lay down the system, but from there on mind takes over and you ask what’s it’s scope? And what Gödel showed is that it’s scope is always limited and that the mind can always go beyond it.

Here’s the man who has said, certain things cannot be proved within any rational and logical system. But he says that doesn’t matter, because the human mind isn’t limited that way. We have Intuition. But then of course, the one thing he really must prove to other people, is the existence of intuition. The one thing you'll never be able to prove. It would be synonymous in many regards to trying to prove the strong version of the gaia hypothesis.

Because he couldn’t prove a theorem about creativity or intuition it was just a gut feeling that he had and he wasn’t satisfied with that. And so Gödel had finally found a problem he desperately wanted to solve but could not. He was now caught in a loop, a logical paradox from which his mind could not escape. And at the same time he slowly starved himself to death.

Using mathematics to show the limits of mathematics is…is….is psychologically very contradictory. It’s clear in Gödel’s case that he appreciated this - his own life has this. What Gödel is, is the mind thinking about itself and what it can achieve at the deepest level.

It's a paradox of self-reflection. The kind of madness that you find associated with modernism is a kind of madness that’s’ bound up with not only rationality but with all the paradoxes that arise from self-consciousness from the consciousness contemplating it’s own being as consciousness or from logic contemplating it’s own being as logic.

Even though he’d shown that logic has certain limitations he was still so drawn to the significance of the rational and the logical. That he desperately wants to prove whatever is most important logically even if it’s an alternative to logic. How strange and what a testimony to his inability to separate himself - to detach himself from the need for logical proof; Gödel all of all people.

Cantor originally had hoped that at its deepest level mathematics would rest on certainties, which, for him, were the mind of God. But instead, he had uncovered uncertainties. Which Turing and Godel then proved would never go away; they were an inescapable part of the very foundations of maths and logic. The almost religious belief that there was a perfect logic, which governed a world of certainties had unsurprisingly unravelled itself.

Logic had revealed the limitations of logic. The search for certainty had revealed uncertainty.

The notion of absolute certainty, is, there is no absolute certainty, in human life, in maths, in logic neither in science. The only certainty that has withstood the test of time to date is; that what we think is certain and true has a limited axiomatic scope, and the conscious mind is the only force in the universe that can transcend proclamations of truth by virtue of conceptualizing and defining its limited scope, thus transcending certainties to higher values of truth it itself previously set the scope of. In this regard, focused right by powerful minds, it's self transcendental in the fact that it's forever able to define the scope, lay it down, then re-analyse it and go beyond it.

Such realizations he said are only from becoming
shut off from the outside world and looking fully
internally, a sort of mathematical medication practice. It's the ability to see the full axiomatic scope of an internally self evolving mathematical framework; the message seeming to be that its always going to be expanded better by the internal mind, as long as no external influences of a social culture not open to questioning the scope and truth of the axioms it was predicated on, which was his current outside world over 50 years back. And, I feel, this culture largely remains so today, though certainly not to the extent it did 50 years back. The fact Godels Incompleteness Theorems still have a lot of applicability to many theories I think has been largely overlooked, or even rejected, by certain disciplines predicated on mathematical grounds and potentially spurious axioms, all of which can likely be viewed as a more wholistic viewpoint and expanded on with the power of the minds ability to always see the limits of the system.

But if consciousness in its normal form is indeed non computational, non algorithmic and not based on logic (incompleteness theorem) associated with turing machines then how are we ever going to try to understand it in terms of them without just tying ourselves up in knots made of the same paradoxes that drove the aforementioned geniuses mad?

To finish, applying Godels theorem more vigorously to current dominant paradigms could have such a catalyzing effect in developing new, mathematically sound theories based on the more creative functions of human inspiration; whilst also pointing out certain unprovable assumptions that underlay some materialist sciences.

The problem is that today, some knowledge still feels too dangerous.

Because our times are not so different to Cantor or Boltzmann or Gödel’s time.

We too feel things we thought were solid, being challenged, feel our certainties slipping away.

And so, as then, we still desperately want to cling to belief in certainty.
It makes us feel safe.

At the end of this journey the question, I think we are left with, is actually the same as it was in Cantor and Boltzmann’s time.

Are we grown up enough to live with uncertainties?

Or will we repeat the mistakes of the twentieth century and pledge blind allegiance to yet another certainty?

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