I wrote this a few years ago. I came across it again recently so thought I'd lob it on this blog...
Ever since I read Aldous Huxley's ``The Doors of Perception,'' I've been fascinated by the idea that the use of drugs could offer insight into every day life. Huxley's descriptions of the world as it appears under the influence of the hallucinogen Mescalin lead to thoughts on religion, meaning and art. Well I too am an artist - a pure mathematics student - and it struck me that the same process could be applied to that particular branch of learning, if someone were to only try...
My ambition was further strengthened after reading of Carlos Castaneda's experience with drugs, including mescalin, while under the careful guidance of a Yaqui Indian Man of Power, don Juan. In Castaneda's account of his apprenticeship to don Juan, ``The Teachings of don Juan,'' the final experience proves so terrifying that Castaneda decides to leave the apprenticeship for good and steer clear of the mind-bending drugs that are part of Yaqui culture. As it happens he changed his mind some years later and continued the process. However, this aside, it was not the terror of the experience that most impressed me (though it was genuinely fearsome,) but the truly awesome insight that Castaneda gains from his experiences - understanding which survives his return to the conscious plane and which, don Juan assures him, will eventually lead to him becoming a Man of Power.
There was only one thing to be done then. Apprenticeship to Men of Power is restricted to a chosen few and they don't live in England. Likewise, it might have been acceptable for a distinguished thinker like Huxley to indulge in a bit of Mescalin, but your average student Joe might not be looked on so kindly, even if it was possible to get hold of the stuff. What your average student Joe CAN do though, is go to Amsterdam. He won't find Mescalin but for a handful of Euros he can buy a packet of copelandia magic mushrooms, he can chew on them and he can see what happens. This, reader, is what I did and here are some of my thoughts...
Let's start with that archetypal hippy image of the happy day tripper staring fascinatedly at some ordinary object and exclaiming at how cool it is! For Huxley it was his trousers, for me it was the pavement - the same pavement I can look at any day which was now, not just more colourful, but much more highly patterned. Where usually I would see random chaos, now I could see symmetry and relations; an abstract structure underpinning the matrix of blue stone. One aspect is worth a particular mention - I became quite alarmed, at a certain point, about a strange covering that seemed to have been laid down on the ground and which caused my toes to curl! Looking more closely, though, I realised that this covering I was seeing was `the gaps' or the space between the objects strewn on the pavement. I was seeing what a mathematician may call a `complementary image' to my usual vision. The same information was being encoded by my eyes, but my rewired brain was decoding it in an altogether different, if equivalent, manner.
Both this search for symmetry, this perception of abstract structure beneath surface form and this reinterpereting of information to allow it to be analysed in a different way are valuable mathematical priniciples. In fact, in some sense, they completely describe the mathematician's task and method. The Game Theory, for instance which John Nash (subject of the film ``a Beautiful Mind'') dreamt up, and for which he won the Nobel Prize, is an analysis of the abstract structure underlying the interactions of a number of competing or co-operating interests pursuing dependent goals. The significance of his work was that he was able to see these situations in a new, `re-wired' way.
But it's not just in the philosophy of the working mathematician where the tripping hippy may bear a resemblance, the actual activity of doing mathematics can appear very similar: In his biography of the great Hungarian mathematician, Paul Erd\"os, ``The Man who Loved Only Numbers'', Paul Hoffman relates how one day, while trying to solve a particular problem, Erdos and a colleague were sat next to each other in a public place for an hour of cogitating silence. The silence was only brought to an end when one of them said, ``It is not naught. It is one.'' Much rejoicing followed! Who knows what strange mindscape of abstraction, Erd\"os and friend inhabited for that hour?
The case of Erd\"os brings with it, in addition, a somewhat more unusual link to the chemical world; For this most prolific of mathematicians spent his last twenty-five years working nineteen hours a day on the back of a heady cocktail of Benzedrine, Ritalin, strong espresso and caffeine tablets. This is a parallel that will not withstand generalization however!
It is at the point of paradox that the respective paths of mathematics and mushrooms most clearly diverge. Towards the end of my copelandia experience, sometime after I became aware that I was a single human entity who was neither mad nor dead (all facts of which I'd been very unsure), I found myself in a very warm, calm, beautiful state of mind in which the universe was understood and meaning accessed. This occurred through a series of `moments of clarity' in which statements of paradox were the fundamental unit. Time after time I saw truth yet knew that truth lay in its opposite also - in chaos, there was order; in mortal futility, enduring meaning. There is no place for such statements in mathematics.
In this regard, the mathematical process can be though of as the assumption of a collection of axiomatic first principles from which are deduced, using logic, a structure of `therefores': facts which must be true given our assumptions and our previous `therefores'. This process is unambiguous and is verifiable - I have heard a professor of mathematics say ``The great thing about mathematics is that you can convince people!'' Mathematicians give proofs for their arguments. This is in contrast to social scientists or even physical and life scientists who, though they use logic and argument to draw conclusions from data or suppositions, must accept that internally consistent arguments for conflicting positions may be put. No one will ever prove or disprove that drugs prohibition never works or that humans are descended from the apes, but it can and has been proved that there is no projective plane of order 6.
It may not be going too far to say that mathematics could be characterised in this way: as the study of truth that can be proven. But the implications of such truths tend to spill out beyond this boundary of proof. That the truth is beautiful, for instance, is a statement that few mathematicians would dispute but none can prove. The happy fact is that when conscious beings penetrate the abstract thought structures that underpin reality, they find crystalline structures of logic that can move the heart. In the same way that I can't explain why the colours that swirled in front of my eyes that crazy Dutch evening were so lovely yet terrifying, perhaps this much at least will always remain a mystery.