Day 138 - Summary Dispatch

Back in Oxford today, after waking up at 7am, last minute checking, then a two hour drive to Oxford, nice brunch, new shirt, some food, unpacking, Frege, maths, iplayer, Pizza Hut, maths discussions, outside games, consequences.

Day 137 - Dorothea Hamilton Fyfe

Really, this is just too much. The jaw juts in an unbecoming fashion, and the Provost is never welcome where he treads too lightly. Let no man be your measure, while striking down that which shall measure too soon. For in these things we find the light, the laughter, and the wisdom which guides us over hillock and under dale.

Can four sights pierce the fog? Or must obfuscation ever cripple us.

Day 136 - The Ragman's House

My Easter vacation is nearly over - five weeks has gone past in an amount of time that appears to defy science as we understand it, yet still the physicists ignore my letters.
On Sunday I return to Oxford, and there are only five weeks until my exams start. I believe so, anyway, since the Examination Schools still have not released the exam timetables for any of the mathematics department. They appear to be mocking us cruelly, in a show of such brazen scorn that there can only be two possible explanations:

1) One hideously unfortunate afternoon, everybody involved in such things arrived home from work and then XXXXXXXXXX XXXX XXXXX XXXXXXXX XXXXXX XXXXX beyond space and XXXXXX rending XXXXX XXXX XXXXXXXX XXXXXX horror.
2) One unfortunately hideous afternoon, everybody involved in such things was kidnapped and replaced by agents from the Anti-Mathematical Team, who have set about quietly dismantling all university mathematics departments, and are trying to deny they ever existed.

Continue to fight the good fight, my brothers and sisters. Hide that textbook under the floorboards, and let the No-Maths Police come knocking. Whisper axioms and laws of inference to your children when all written records have been burnt, so that the knowledge will not die. Shelter the Fields medallists in your attics, and supply them with paper and pencils. Though our freedom may be eroded, our morals corrupted, our children sold for air conditioning research, pure mathematics will live on.

Harry visited a couple of days ago - we went to Whipsnade with Julia. This was pretty cool, and deserves a proper blog post (Harry/Julia, I'm looking at you). He should also be very excited that he's been mentioned in my blog post.

I have not forgotten his request for a blog on what I was saying about philosophy and free will. I will, I promise, when I have time to think. Things are so hectic round here, I am literally rushed off my feet.

Ouch.

Day 133 - Ave Maria

I have been reading Michael Dummett's "Frege: Philosophy of Language" for the past...hour and a half, it seems, which has resulted in a sore neck and a contemplative mood. Part of my recent revision efforts has been a resolution to tackle Dummett's two mammoth works on Frege (the other is called Frege: Philosophy of Mathematics).

Considering Frege is only 2/3 of one of my six papers, this may be a poor balance of my priorities. I maintain, however, that thinking deeply about such issues makes me a better philosopher generally - indeed, a better thinker generally.

If anyone wants to think properly about Frege's work, I strongly recommend Dummett - though not, perhaps, as an introduction level book. Seriously, though, Dummett is a very clever and insightful chap. According to Wikipedia, he is also an emiment scholar on the history of Tarot. Cooooool.

I will now quote a bit from his Preface:

"There is also a quite different reason why I have taken so long [to write this book]...I conceived it my duty to involve myself actively in opposition to the racism which was becoming more and more manifest in English life...I make no apology for this decision, nor do I regret it. Bertrand Russell, in a television interview given shortly before his death, was asked whether he thought that the political work on which he was engaged at the end of his life was of more importance than the philosophical and mathematical work he had done earlier. He replied, 'It depends how successful the political work is: if it succeeds, it is of much more importance than the other; but, if it does not, it is just silly.'...It was only at the stage at which, outwitted by those who could, after all, draw on a long tradition of the tactics of handling subjugated populations, I felt that I no longer had any very significant contribution to make, that I thought myself justified in returning to writing about more abstract matters of much less importance to anyone's happiness or future."

and

"There is some irony for me in the fact that the man about whose philosophical views I have devoted, over years, a great deal of time to thinking, was, at least at the end of his life, a virulent racist, specifically an anti-semite. This fact is revealed by a fragment of a diary which survives among Frege's Nachlass...shows Frege to been a man of extreme right-wing political opinions, bitterly opposed to the parliamentary system, democrats, liberals, Catholics, the French and, above all, Jews, who he thought ought to be deprived of political rights and, preferably, expelled from Germany. When I first read that diary, many years ago, I was deeply shocked, because I had revered Frege as an absolutely rational man, if, perhaps, not a very likeable one. I regret that the editors of Frege's Nachlass chose to suppress that particular item. From it I learned something about human beings which I should be sorry not to know; perhaps something about Europe, also."

I agree. I too revere Frege as an absolutely rational man, and I was deeply shocked when I read that. I learnt something that I am sorry to be true; but I should be sorry not to know it.

Back to 'more abstract matters' now.

Day 129 - Towers and Blocks and Towers

A lot of rumours get bandied around about the time I wrestled a bear. Tempers get heated, minds get bored, and tongues begin to wag. Some of the more outrageous lies that I've heard about the night shock even me, Tom "the Unshockable" (my WWF alias). Let me just set the record straight, mohitos:

1) I was not drunk. Neither was the bear.
2) It was not started out of malice or a desire for random violence. It was an honest, honourable duel between two men of different species.
3) No weapons were involved.
4) While knuckling my forehead, the bear did not start expounding the work of the 19th century German philosophy and mathematician Gottlob Frege. He chose instead for a cheeky series of limericks, parodying the psychologistic approach of Husserl.
5) At no point were bodily gases used as a flotation aid.
6) I was technically dead for seven hours.
7) It was a draw, as all parties agreed.
8) The bear is currently alive and well, and running a small crepe stall in Berlin. We exchange letters on a regular basis, and now count each other as friends.

Day 126 - Fit as a Fiddler

Did you get it right? That's right, the numbers were the Fighting Fantasy books which I don't own in the series. Pretty cunning, no? Still, if you tried hard enough, you would have figured it out. Any failure to do so reflects a sheer lack of integrity and dedication on your part. Frankly, I don't want somebody with such low moral fibre and backbone reading this blog. It's worth more than that, goshdarnit! Pay these jewelled words some respect, and go back to whatever stinking hole you crawled from, with your platitudes and your apologies. I will have none of it! Dry those tears with the harsh iron fist of my cruelty, then leave this sacred server, and take the stench of despair and misery along with you, you dry-mouthed tall-headed back-muchest cetacean. I want none of your lunacy; don't try to drag me down, back in the hellish miasma that you managed to haul your flabby bulk out of with one twisted claw, digging deep into the good clean earth of my dignity.

Now they're gone, we're left together, us people of intelligence and wit. Kudos!

Day 125 - A Puzzling Outcome

Script writing continues, now with a partner. We have a plot. We have characters. We have a scene structure. We have five pages of solid gold. We have EPIC.

Now, a numerical puzzle:

21 22 26 30 34 36 43 44 47 50 52 53 55

Spot the pattern!

Day 124 - Sequence marches on

Hogshine:

Read your blog? Only when drunk. I have it on good authority (an Oxford professor of theoretical physics) that string theory has fallen from its pedestal within physics circles as well - although, if I recall correctly, it hasn't been replaced by another theory, but rather by a trend towards more throw-things-together-and-see-what-happens kind of stuff (hence the big Higgs thing).
248 dimensions? Pah! I will not rest until we've modelled the universe in 196883 dimensions, so we can use the Monster group.

Also, hello Mr Gilbert! Always a pleasure to receive greetings/book recommendations/book plugs. I can't tell whether that's actually you, or some clever advertising automata (amazing what they can crank out in the labs, these days). Either way, your book sounds interesting, and I will check it out. Probably through some library, however, unless you care to send me a free publisher's copy...go on...

Day 123 - That poor horse

My last EPIC post attracted some EPIC comments, so without further ado, let's get to it.

Haniff:

Thanks for your interesting comment! I don't think we've ever met, but I appreciate your readership. It warms the cockles of my heart to know that an actual internet person reads my blog.

About the primes thing - you're absolutely right, and it is by no means intended as any sort of convincing argument or proof, more as heuristic ramblings on why primes might crop up (though they do so disguised as the zeta function). I thought it was interesting about the taxis, etc - obviously, in this case, my rough remarks wouldn't apply to the taxis themselves, but perhaps to some factor determining their distribution (i.e. the preference of the drivers, or something). The 'objects' I spoke about can be anything, or any property, with some suitable 'force' and 'group' relation defined appropriately.
Or, if you think I'm chatting baloney, chuck it out with the rest of the processed meats. They contain unhealthy amounts of salt, you know.

Dimension thing - thanks for mentioning Charles Hinton. I've never heard of him, and after a brief dose of Wikipedia, I need to read his books. Again, my ramblings were primarily heuristic (goshdarnit, I love that word). My point was really that we can't exactly visualise a 4D cube, but we can quite easily imagine such a thing, taking into account non-visual factors.
Although, since I'm finding it hard to picture infinitely many coloured cubes, I need to modify that 'quite easily'. I'll get back to you after I read some Charles Hinton.

String theory sounds crazy, and I know almost nothing about it. I shy away from physics and applied mathematics, mostly because it scares me, and I can't do it. Apparently it's no longer the hot new theory among theoretical physicists, however. I can't remember what is.

I agree that mathematics can't answer everything, nor should it want to. Any discipline thinking it can answer everything is just asking for trouble. That's why, in mathematics, they generally pick a smaller problem they think they can solve. Then a smaller one, that they actually can solve. And so on.

Pointing to a creator...I'm not so sure, at least not how the mysteries of mathematics come into it. Then you have to go into the metaphysics of mathematics, and what kind of status the propositions really have, and so on. One big medieval mess, as Merlin would say.

Can of infinite worms in infinite universes? Great metaphor, if a bit demanding on the clean up afterwards. Yeah, I agree we'll never fully understand them. Some famous mathematician said something similar, but much better than I could.

Julia:

You missed out, but I still love you.

Harry:

Sadly, I have met you in real life, and was already aware that you read my blog, so the thrill of seeing your comment was dampened slightly. Next time, please adopt some cunning pseudonym, so that my life can abound with artificial mysteries once more.

Yes, I am a genius.

Thanks for the interesting occurrences of primes. I'd heard about the cicada thing before, and in that case, the primality is pretty important since, as you said, it reduces the risk of coinciding with predator cycles. I think other animals with similar sorts of cycles also tend to it in prime numbers...some sort of Australian frog, if I recall.
Discrete phenomena wise, primes themselves probably don't occur much more often, except in odd cases like that one, where primality is important. They do, however, occur surprisingly often in quantum physics (at least, in their zeta function disguise) - at least, so I have been led to believe, and why would they try to mislead me?
Primes in biology, I share your scepticism, but I would also love to be proved wrong. If everything ran on primes, my gut feeling tells me things would work out better. This bears further research, when I am less tired.

The dimension thing - I wasn't really thinking about physics use of higher dimensions at the time, pretty much just higher dimensional polyhedra. It might let me down when it comes to paths, etc - and certainly, intuition there is a lot harder than intuition in three dimensions, but I still suspect it to be achievable.
Even if there is space of four dimensions, or however many you feel like, it's not the space that we perceive visually, and hence we can't hope to visualise it...visually. Our intuition/imagination techniques, are more powerful than simple visual visualisation, so we're not so restricted as this would suggest.

Your question put to Mr Haniff, I am also interested in, and I eagerly await his comment reply.

Yes, I do see primes in my cereal. They scare the living bejeesus out of me.

Day 121 - Unexpected Retrievals

There were three things I intended to blog about today - here's what I wrote down to remind myself:

Primes occuring in nature
Visualising dimensions
Trivial when expressed properly

Now let's see how they turn out when I actually write them. Behind-the-scenes blogging, that is. Straight from my brainstorm to your brainwave (why does the brain have such weather-themed imagery?).

Today I watched, with Julia (under some duress), a one-off Horizon programme called Alan and Marcus Go Forth and Multiply. Basically, the comedian Alan Davies gets taken on a whirlwind tour of higher mathematics by Oxford professor Marcus Du Sautoy. The usual things come up - Monty Hall problem, dimensions, prime numbers, and it does prevent a couple of things (such as a potted description of what Perelman achieved) in interesting and original ways, helped along by Alan's charm and Marcus' obvious enthusiasm and intelligence. I highly recommend viewing it - on with the point, as they say in the pencil factory.
While discussing the Riemann zeta function and its relation to the prime numbers, they also demonstrated an interesting parallel in nature - the frequency of the vibrations of a quartz crystal correspond closely to (I presume) the modulus of a certain section of the zeta function.
This got me thinking about why prime numbers are so important in nature, and in particular, quantum physics, etc.
It's not that surprising.

Let's suppose we have a large group of objects of some kind, held together somehow. Further assume that these bonds are relatively weak - in other words, it won't take much to break up this big group into several smaller groups. Now, also suppose that the forces that act upon this group to break it up act pretty much symmetrically over the whole group. In other words, this breaking-up force will be distributed evenly. This suggests that the parts it is broken up into will be evenly sized as well.
For example, a group with 6 objects in it, broken up in a symmetric fashion, won't break up into a group of 5 objects and a group of 1 object. It will be broken up into groups of the same size, e.g. 2 groups of 3 objects.

See where I'm going with this? The size of groups of objects will tend towards factorisation, which in turns entails that they will tend towards sizes which are prime numbers, which can't be factored in a non-trivial way.

Now, this is a very general heuristic sketch - billiard balls, for example, don't naturally group into 2s or 3s. If, however, the bonds between objects in questions were sufficiently weak, and the breaking-up forces were sufficiently symmetrical, such a tendency towards primes would occur.
The objects may not be physical - they may be waves, or frequencies of waves, or properties of frequencies of waves, and so on. In any physical situation in which have some kinds of objects grouped together, and forces acting on them symmetrically, it would naturally tend towards primes.

Moreover, such weak bonds and symmetrical forces would be more likely to occur at very small scales, where there is simply less room for forces to act non-symmetrically. So, primes would crop up frequently in quantum physics. Since the zeta function generates the primes, in some fashion, it's also expected that it would generate, in some fashion, these quantum phenomena.

Alriiiiiiight, enough heuristics.

Another problem mentioned in this program was how to visualise in more than three dimensions. This is usually seen as some in built limitation of our brains, in that we can only visualise spacially in up to three dimensions - which is only natural, since we perceive things in three dimensions.
I think, however, that it's a lot easier than people think, and we're being held back by what think a dimension should be.

Take the paradigmatic cases: for 0-dimensions a point, 1 dimensions a line, 2 dimensions a square, 3 dimensions a cube, and 4 dimensions a hypercube, and so on.
Up to 3 dimensions, we can visualise these easily. We can build them from the previous one, by extending it in a different direction, in some sense.
We stretch out a point along a length, to get a line. We stretch out a line along a width, to get a square. We stretch out a square along a height, to get a cube. Then our minds hit a brick wall - where is the fourth direction in which to stretch our cube?

Take your pick! The fourth direction does not need to be spatial. Perhaps it can't be, since all the space we know and love only has three dimensions. A dimension is just another co-ordinate; another piece of information telling us where to look for a point.

Example. A point is 0 dimensions, because no co-ordinates need to be given, we don't need any information to locate any point within the point. Because there's only one point. We know where it is, we don't need to look.
Now, for a line, we need 1 number, to know how far along the line our point is, and then we're done. A line is 1-dimensional.
In a square, we need 1 number to tell us which line the point lies on - which cross-section of the square we need. Then another number to tell us how far along that line.
Similarly the cube - the first co-ordinate tells us the square, the second the line, the third which point on that line.

So! To make a fourth dimensional cube, a hypercube, watch carefully (I have nothing up my sleeves). Take time to be the fourth dimension. Suppose a cube suddenly springs into existence at time 0, and winks out at time 1. Now, if we searching through time as well as space, we have a 4-dimensional cube. The first co-ordinate tells us at which time we're looking - which cube we're searching in. The second which square within that cube, the third which line within that square, the fourth which point within that line.

Blam. A 4-dimensional cube, plain and simple. What's wrong with that? We can't hold it in our minds eye like the others, since our eyes can only see cross-sections of time. We are still, however, perfectly aware of time, and use duration in our visualisations, so we can easily visualise this magically appearing and disappearing cube. In a way, we already do when we visualise any cube. Now we're just taking notice of the time it's there for. Nothing mysterious or esoteric about the fourth dimension.

We can do something else, however. Instead of time, why not use colour as the fourth dimension? Let 0 be the colour white, 1 the colour black, and a continuous sliding scale of grey in between. Now let's have an infinite collection of cubes, one for each possible shade. Note we need infinite cubes, just as there are infinitely many squares in a cube, and infinitely many lines in a square, we need infinitely many cubes in a hypercube.

Now, to find any point, we specify the first co-ordinate - precisely which shade of grey we're looking at, then pick up that cube. The second gives us the cube, and...you've gotten the idea by now.

In fact, any continuous variable that can be applied to cubes could be used. How much that particular cube is adored by the media, on a scale from 0 to 1, taken continuously. The probability that cube will turn into a flying pig, ideally. They're not spatial things, but we can think about them, we can visualise them. We can visualise time, and colour, and billion other non-spatial things. Use them in your search for higher dimensions.

This can be carried out higher dimensions. Want a 5-dimensional hyperhypercube? No problem! Just take your infinitely many different shades of grey cubes, and have them existing over a duration as well (not hard - most things exist over some duration). 6 dimensions? Take our temporary coloured cubes and just factor the height that the mould on them grows to. Let your mind run free!

Phew, how exciting. The third thing: "trivial when expressed properly." Here's what I meant:

I believe that any mathematical proposition, when expressed in the right way, becomes obviously and trivially true. The difficulty in mathematics lies in working out exactly how to say what you want to prove; translate it into the right language, look at it in the right way, and there's nothing to it.
A quick example, till I think of some more: factorisation of integers into primes. Look at it one way, and it's pretty obvious - primes are, by definition, things which can't be divided anymore. Take any integer, and divide it into smaller bits, then those bits again, and so on, and eventually you'll have to stop, and all you'll have left are primes. Easy.
Now, however, think about the integers like this: you start with 0. Then add 1 to make 1. Then add 1 to make 2, and so on. The integers are everything you get proceeding in this way, adding 1 each time. Now work out analysis, then complex analysis, then the theory of the Riemann zeta function, then let the primes be defined by the sequence which satisfies a certain functional equation involving the zeros of the zeta function. Now try and prove factorisation into primes. A lot harder.

Better examples to follow!

Day 120 - Happiness runs in a circular motion

Happy April Fool's Day! I love my little japes, so in this blog post, there is one false fact. All the others are, as far as I know, true.

Start of Script Frenzy today! I stayed up until midnight last night, and on the bridge from the old day into the new, I paused to write a quick scene. I don't know what my script is about, or what scenes will be written, or who the characters are. I'll be thinking, however. Until then, don't ask about my script. I will write about somebody I just found out about today: Alexander Grothendieck.

Grothendieck (born 1928) is one of the leading mathematicians of the 20th century - described by one mathematician as the "Einstein of mathematics". He brought in not only new theorems, but entirely new ways of thinking about and doing mathematics. In the 50s, he pretty much reinvented algebraic geometry from the ground up, and attempted similarly ambitious schemes in many other areas of mathematics.
Since reality loves a stereotype, this brilliant mathematician is also a bit...fractured. A radical left-wing anarchist and ecological activist, he withdrew from mathematics in the 1970s to focus on his ecological and conservation interests. In 1991, he withdrew from life altogether, disappearing from his home one night, and his current whereabouts are unknown. He receives no visitors, and has withdrawn from not only the mathematical community, but the community in general, it seems.
He has written an incredible amount - many of his opuses stretch to the thousands of pages. He's written long philosophical and biographical accounts of a similar length. Seriously, look him up on Google/Wikipedia. Fascinating.