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## Tuesday, December 27, 2011

### What is the origin of mathematics ?

If radio astronomers would discover a signal containing a repeating sequence of prime numbers then they would claim to have found extra-terrestrial intelligent life. Why? Because they consider mathematics as universal throughout the entire universe to which only intelligent life forms have access.

Mathematics is universal. That basically means that mathematics is discovered and not created. We use, for example, $\pi$ as the ratio between circumference and diameter of a circle, that ratio is the same everywhere in the universe. Not the symbol $pi$, the decimal number system and so forth.

We take the existence of mathematics for granted, we don't question when and how it was created. Where does that vast body of mathematics come from? Was it created with the Big Bang? If so, than the ( mathematical ) models of physicists that explain their Big Bang theory look naively simple.

It's rather vague to discuss if there was mathematics before the Big Bang. But -if- there was a point from which everything was created than that creation must have included -all- of mathematics. If we don't accept that than we are saying that we are the most intelligent life form in the universe, because mathematics is created by us and not discovered by us.

What -is- the origin of mathematics?

1. Not sure I agree I think maths is a way in which humans impose structure on the world around us.
Hence it comes from us. A lot of maths has no connection with the real world. You seem to be dangerously close to Platonism here, which would claim that every mathematical concept corresponds to an element of reality. That turns mathematics into some form of mysticism. I accept that some of your hero's such as Hardy are Platonists but I beg to differ.

2. Imagine a technologically advanced humanoid civilization somewhere in the galaxy. They live on a planet, so they must experience the force we call gravity. They must have invented calculus if they use machines that can acccelerate and so on. You could argue that calculus is so inter-related with physics that it basically -is- physics. Riemann laid the foundation for differential geometry. At the time it was mostly ignored because there were no real applications. Until Einstein used it for his relativity theory. Groups, in all sorts, are part of the real world as symmetries from the elementary particles to the spirals of galaxies. Groups have a -deep- connection with prime numbers which by themselves are still a mystery in many ways.

Lastly. Think of music. Intelligent life forms will have art and thus music. Their way of communicating -about- music will probably be completely different. As will be their instruments. But hen they start playing they are generating soundwaves we will most likely find enjoyable to hear. What will probably surprise us most is that the music doesn't sound alien at all. Because, as mathematics, music is universal to intelligent life.

It is not a joke that radio-astronomers are looking for prime sequences to detect intelligent life.

3. But it's only a small subset of maths that is directly relevant to the real world. In particle physics it's only SU(3), SU(2) and U(1) groups that are relevant. In larger groups this is probably but not definitely proved that E8 is probably the most relevant.

As for your argument vis a vis calculus and physics, Whilst Calculus plays a fundamental part in physics when considered mathematically as your experience of M208 would have shown you in order to make it rigorous you have to use such counterintuitive concepts as the epsilon delta definition of continuity. Hardly natural and quite ugly, indeed most physicists hardly encounter it at all. I cannot believe that such a convoluted definition isn't a human construct or if you like a humanoid construct.

As for your final point vis a vis music, if aliens do exist then they will probably have a completely different aural structure to us and will respond to different frequencies. It is highly unlikely (unless they are human clones) that their music will be the same as ours as they will have a completely different response to frequencies than we have.

4. Limit mathematics to the knowledge of the structure of the set of prime numbers. If they ( ... ) understand physics, then they will have knowledge of Lie Groups ( as you say ) which are built upon the theory of Finite Groups. And finite groups are intimately linked to the primes. - That is what troubles me, what if the primes are not 'our creation'?

5. I accept prime numbers are a problem for my position in that there is obviously some fundamental nature to them that any civilisation will have to use if it is to advance beyond mere counting. On the other hand, as there are an infinite number of primes then sooner or later no matter how you finely you subdivide the universe, you will encounter a prime number that does not correspond to anything in the universe.

Just as you can never see a real triangle because of the irrationality of the length of the hypotenuse (Which pythagoras tried to suppress as it would undermine his position).

I believe it was Kronecker who said God made the integers, the rest is mans creation. He would have been more accurate to say the primes.

Finally Lie Groups as they occur in physics only approximately apply unless you assume all fundamental particles have the same mass which they don't. One of the big mysteries to me is why the top quark has so much more mass than the other ones. The idea that the third generation of quarks obey a symmetry similar to the others seems a bit of wishful thinking. On the other hand it seems to be the best we can do given our limited knowledge of physics beyond the energy scale of the Large Hadron Collider. Unlike some of the hype I think that the Standard model is best seen as a relatively concise summary of the current state of particle physics but certainly not the final answer.

So it still seems to me that mathematics is a really useful abstraction which in some cases works really well when applied to real world problems but there will always be a gap between the level of fit and the abstraction.

6. I always used to think that physicists crammed the laws of physics ( as they perceived them ) into beautiful streamlined mathematical equations. It is easy to hide the things you don't understand yet in a constant. I don't know if this is true but if it is and as you say "Lie Groups as they occur in physics only approximately apply unless you assume all fundamental particles have the same mass which they don't." then my arguments fail and the prime numbers could be nothing more than our own constructions.

It would however imply that the we are nowhere near understanding the 'laws of physics', that we are only cramming observations into formulas. I don't know what I want to believe.

7. I hope I haven't destroyed your faith in mathematics consider some of the finest works of Art or music, these are undoubtedly human creations but they still inspire awe and wonder. I would argue that mathematics and physics are the same. It doesn't matter that they are human constructions they are still wonderful and the theory of Lie groups is beautiful (unlike the epsilon delta definition of continuity).

Again when one applies mathematics to the real world once one has certain empirical information one can get amazing correlations between the predictions and what one can measure in the lab.
However this usually occurs after quite a complicated calculation. In particle physics for example to calculate a typical scattering cross section (the probability that a certain reaction will occur after a number of events)or decay rate will usually involve the caclulation of the contributions of hundreds of Feynman diagrams to the scattering amplitude. It is these quantities which are measured at CERN and also the fact there is no way to calculate these quantities from superstring theory casts doubt on whether superstring theory will ever be testable.

Still once this has been done and these days it's usually done by computer then in most cases the scattering cross section can be calculated to almost arbitrary precision. But this involves the process of renormalisation whereby the predicted masses and charges are absorbed (crammed in as you would say) into the actual meausured values.

So the situation is that one doesn't have a way as yet of calculating fundamental empirical information masses of particles, their charges etc but once one has measured these values one can make accurate predictions based on the laws of physics.

So don't despair, yes on the one hand we still seem to be groping in the dark and as you've pointed out vis a vis Godel's theorem there do seem to be fundamental limits to what mathematics can achieve. Enjoy the aesthetic pleasure from Lie Group theory and all the other branches of maths that you find appealing. Also celebrate the fact that some of this maths does have some application to the real world but don't get carried away with the idea that there will always be a one to one correspondence between maths and the real world.

Have a happy mathematical new year. Are you doing any OU courses this year or are you going to concentrate on your own interests which you seem to be pursuing in a more productive manner than the OU is. I'll be doing Topology in it's last incarnation and MS324 waves as I missed not doing any serious calculus last year. In October it will be MST326 and M381. Then I will have done enough maths to get me onto the MSc but I will take that slowly and I'm not sure I want to do all the courses. Definitely calculus of variations, Applied Complex Analysis, Approximation theory and Functional analysis. But I'm not sure about anything else just yet.

Anyway all the best and I hope our fruitful dialogue will continue
Chris

## Welcome to The Bridge

Mathematics: is it the fabric of MEST?
This is my voyage
My continuous mission
To uncover hidden structures
To create new theorems and proofs
To boldly go where no man has gone before

(Raumpatrouille – Die phantastischen Abenteuer des Raumschiffes Orion, colloquially aka Raumpatrouille Orion was the first German science fiction television series. Its seven episodes were broadcast by ARD beginning September 17, 1966. The series has since acquired cult status in Germany. Broadcast six years before Star Trek first aired in West Germany (in 1972), it became a huge success.)