The Academy: Axiom 1

September 3, 2011

The rule

This post is not trying to do anything clever. It is making a statement that seems self-evident:

There are three ways to gain understanding of the world:

  • Personal experience
  • Systems of rules
  • Stories

All are equally important, and each has its strengths and weaknesses.

The important point is not the content of the statement but the stating of it. This is not just something that feels correct (to me) but something that feels fundamental. This mirrors one of the quests of mathematics to find the simplest statements on which to build the whole subject. I have my suspicions that the same thing would not work completely here, though writing the “Elements of the Academy” with this as one of the axioms might make a curious exercise!

This axiom maps onto the world of academia. The Sciences are primarily concerned with the use of rules to understand the world; the Arts centred on the creation of objects that attempt to transfer personal experience; and the Humanities write, dissect and try to understand the stories of the world.

All three areas, of course, do and should take advantage of the strengths of the other two methods as well as their primary concern.

The story

As a mathematician I obviously come from the grand tradition of finding rules to understand the world. For much of human history this was known to be rather limited in its scope. It was applicable to commerce, certainly; but also to questions of measurement, and to the study of the stars and music. Then, with the acceptance of arguments based on infinitesimals and the genius of Newton and Liebniz, the models of calculus opened up a vast array of phenomena to understanding through rules. It was so successful that many started to believe that it would eventually explain everything.

I do not believe this to be the case. Chaos theory shows that even perfect models can be severely limited by small, unavoidable, measurement errors. The work of Gödel and Turing shows that even in the purely theoretical world, there are unanswerable questions. Some even believe that as fundamental a system as arithmetic might contain contradictions. Before we even get to these hard limits we must deal with the soft limits imposed by the great ideas that we have yet to have.

Unfortunately, or fortunately depending on situation and personal preference,  the world offers many questions that we cannot answer with a systematic, rules based approach. Questions we cannot ignore. I wanted to define for myself the other options, and place them in some imagined framework.

The personal experience

I don’t believe I have said much here. It is, as I stated, self-evident. I also think it is important. It has been useful and practical to me. So, if you have managed to read this far, I thank you, but ask one further thing. Think about it yourself and see if it is a useful for you too.

Acknowledgements

This post grew out of a string of tweets, out of which grew very valuable discussion with  Colin Wright (@ColinTheMathmo) and Daniel Colquitt (@danielcolquitt), on twitter and elsewhere.

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Posted by gelada

Future Fantasy

June 17, 2009

Lets suppose we achieve the impossible, a world in which only the human endeavour necessary goes into the essentials. Food, shelter and so on. Where the remainder of our time and energy as a species is available to do what we will. What should it be spent on. I thought I would allow myself a little fantasy time and write some things down. Its not an ordered list, or an exclusive one, there are many overlaps. Its also a list for the whole of humanity, not everyone will do everything, but these are some of the things I hope people would be doing more of:

1. 10,000 hour skills, and not just useful ones. The lost ones. Going round the science museum recently I was struck by all the lost expertises. How many people these days knows how to make a mechanical integrator for a tape based differential equation solver? Lets have more people who can thatch a roof to perfection or create ornate plasterwork.

2. Learning. For its own sake, just for the joy of knowing. Learning both in the sense of being taught established knowledge and studying to make new knowledge, to increase our knowledge of the world. Past, present and future…

3. Lazing around. Just because there are workaholics like me who can often only get satisfaction from doing something does not mean everyone is like that. Some people really do get honest and simple pleasure from lying on the beach, that should be celebrated.

4. Family time. Not “quality time with the kids” but doing whatever it is you want to do together. Dinner, walk in the park, football match, being there not because you should, but because you want to.

5. Drinking great beer and wine, eating good food, which of course requires that people make more of it! Down with processed crap and McDonalds!

6. Random monkey noises, any world would be better with more random monkey noises.

7. Journalism. Yes, journalism. The thing that is meant to be being killed by twitter and blogs. Not the journalism of oracles speaking from hidden sources, but people who dip into the flow of information and process it, comprehend it and pass it on. Who do the hard job of working out what is happening right now.

8. Sport. Both to watch and to play.

9. Computer games, writing but also playing. If someone can be honest with themselves and get fulfillment in a virtual world good on them.

10. Getting excited and making things.

And to finish the most important one of them all:

11. Extending this list, finding that collection of things that fulfills you honestly and completely. To sum it up finding what has meaning to you personally not just some value decided by society.

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Posted by gelada

Working with constraints

November 2, 2008

There is a theory that one can have too much freedom, at least in art.  With constraints the imagination is forced to work harder, and might achieve an elegance and beauty unobtainable when a simple answer can be used.  Rules and constraints can also be broken, at the right time, giving an aesthetic that is unobtainable for a completely free setting.  In particular a set of rules gives a possibility of some idea of perfection, and thus adding a deliberate flaw can, for many, increase the aesthetic appeal of a work of art.  It is not possible to have a flaw in a work which is completely free.  

 These ideas have been developed many times, especially in the modern era as freedom became more fashionable, and thus setting rules became more rebellious than breaking them (as an aside I would be interested if anyone has examples of such ideas from before the middle of the 19th century).  I would like to mention two of my heroes who also wrote theory on this subject.  The first is Max Bill, who gave a vision of concrete art, stating in his essay The Mathematical Approach in Contemporary Art (no prizes for guessing my interest):

…despite the fact the basis of this Mathematical Approach to Art is in reason, its dynamic content is able to launch us on astral flights which soar into unknown and still uncharted regions of the imagination.

The second is the Oulipo, a literary group, described by Raymond Queneau as:

 

Rats qui ont à construire le labyrinthe dont ils se proposent de sortir.

(Rats who construct the labyrinth from which they propose to escape)

 

The ideas of the Oulipo are perhaps more explicitly similar to the points I mention at the beginning, for the simple reason that my thinking has been heavily influenced by them.  An archetypal example of their work and possible my favourite novel is George Perec’s Life: A Users Manual.  This work had an incredibly complex set of rules for its structure, yet wears it lightly.  One can read, and enjoy, the book without considering, or even being aware, of the fiendish construction process.  In addition the story of the book, (grand, complex artistic schemes) describes the book itself.  The schemes in the book eventually fail, and the book itself only has 99 chapters rather than the 100 that the rules declare it should have.  One of its many themes, therefore, is a beautiful study on many levels of the aesthetics of flawed perfection.

One final idea that working with constraints introduces is the element of puzzle.  Can one find the constraints or rules given the finished piece.  This is an idea that I want to develop more fully, so for the moment I will just leave a forward reference.

The ideas of the Oulipo and Max Bill seem so close that it is amazing to find so few links between the two traditions, for example using google.  While I was writing this there were only 93 hits for “Max Bill” and Oulipo, ”concrete art” and oulipo did even worse gaining just 23 and “constructive art” and oulipo only 2!  I think that it is therefore safe to assume that the ideas were developed separately in Literature and visual art.  Can one use this convergent evolution to justify the ideas, though simple are deep, profound and interesting?

Non-philosophic constraints

I did not mention above that one can also take on constraints for purely utilitarian reasons.  Either one is forced to because of the materials one is using, or certain constraints can solve a practical problem.  As a personal example I have recently been playing with images that can be created on squared paper.  For no other reason than I have been using a pad of squared paper as a notebook (and that I am not happy with my drawing skills).  A simple piece made on the computer from some of this thinking is below (guesses on the construction are again welcome, but no prize this time!):

 

Root 5 grids

Root 5 grids

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Posted by gelada

Mathematics, Computers and Zeilberger

October 25, 2008

This piece is a ramble through a collection of thoughts linked to and influenced by the opinions of Doron Zeilberger. It starts with the uncertainty of proof and discusses the importance of computers before concluding with the future of mathematics in the large but finite.

I am never sure how many mathematicians feel this, but I often doubt my own proofs. I find myself certain of some fact, with a chain of reasoning to support it, that, at a later date, fails on some small detail. Only when all the details are worked out do I start to feel a sense of comfort, but even then I fear something will leap out at me and the ideas I thought were quite brilliant fall apart completely.

I therefore always take comfort from Zeilberger’s opinions. He discusses this problem, and actually takes it further:

Mathematics is arguably the most certain body of knowledge, but of course, nothing is certain in this world, and it is a distinct possibility that the Pythagorean Theorem, and even 2+2=4, are wrong, and it just so happened that Nature and/or God programmed the human mind so that it will overlook the gaps in their alleged proofs. Complete certainty (even of death and taxes, and certainly of mathematical facts) is an unreachable ideal, but one can at least try to improve the reliability…

Opinion 91

However he also provides a potential solution:

…we need computers. Computers abhor ambiguity, and trying to teach computers mathematics is also good for us humans, since it forces us to discover hidden ambiguities and resolve them.

Opinion 85

I agree with this entirely. It is so much easier to check if I understand something if I can pull out mathematica or python and actually program it. If my understanding is gibberish I will not see what I expect, but get gibberish back from the computer. It is therefore easy to check if something has failed. Furthemore, I can follow through what happened, and find where it failed (assuming it was not a bug in the code) and thus identify where my intuition was wrong. On the other hand if the output of the code agrees with my expectation, I know that something has gone right. I can now take this further, instead of slowly working through examples for a conjecture by hand I can immediately check them. I even learn mathematics better this way. I probably learnt more about Hyperbolic geometry in constructing the image below than in all the lectures I have attended on the subject (though maybe not in discussions).

A tiling of the hyperbolic plane with 7 equilateral triangles round each point.

A tiling of the hyperbolic plane with 7 equilateral triangles round each point.

To me, therefore, there is an obvious, strong case for using computers for both research and teaching, and encouraging all mathematicians and mathematics students to use SAGE or Mathematica (even though I use it, I do feel that it is a little too expensive). However one can take things further. Maybe computers are not just useful, but essential to the future of mathematics, returning to Zeilberger:

According to this criterion [level of abstraction], most of human mathematics is completely useless. It was developed by humans for human consumption. In order for humans to understand it, it had to proceed in tiny steps, each comprehensible to a human. But if we take the “mesh size” of each step, dA, to be larger, one can do potentially much bigger and better things, and the computer’s dA is much larger, so we can (potentially) reach a mountain-top much faster, and conquer new mountain-tops where no humans will ever tread with their naked brains.

Opinion 72

To me this vision of mathematics is particularly exciting. Mathematics started by studying the behaviour of the small; low numbers and simple equations. The next step was a mathematics of infinity, resting on the observation that in many cases infinity could be used to approximate large numbers, and the continuum to approximate large collections of small things. The problem is that this only holds in “many cases”, it is not true in general. In fact in the growing quantitative study of biology and sociology, made possible by DNA sequences and the data in the internet, it regularly does not hold. We therefore require a mathematics of the large but finite. Computers, with mathematician trainers, provide a way of achieving this. It should be an exciting adventure, after all we have much to learn even in the simplest cases, to quote Tim Gowers:

…there is more to say about the whole question of multiplying large number than you might think…

Joint LMS/Gresham College lecture

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Posted by gelada

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