Some fundamental formulas from metrical algebraic geometry (seminar)

This coming Tuesday in the Pure Maths Seminar at UNSW I will be giving a talk: Here are the details, in case you are in Sydney and are interested. The talk is at the School of Mathematics and Statistics, UNSW, Kensington campus, building the Red Centre, which is up the main walkway from Anzac Parade.

________________________________________________________________
Speaker:   Norman Wildberger (UNSW)
Title:         Some fundamental formulas from metrical algebraic geometry
When:       12:00 Tuesday, 9 June 2015
Where:      RC-4082, Red Centre, UNSW

Abstract:  At the heart of metrical algebraic geometry there are hierarchies of beautiful algebraic formulas, starting in one dimension and then working their way up both in dimension and complexity. These  can be viewed as generalizations of formulas of Archimedes, Ptolemy, Brahmagupta, Bretschneider, von Staudt and others, as well as classical results from Euclidean geometry.

One of the interesting aspects of our approach to these formulas, based on Rational Trigonometry, is the pursuit of generality: using general fields and arbitrary quadratic forms. Interesting connections emerge with the theory of special functions, differential geometry and operator analysis.

Most of these formulas would be challenging to find without the use of a computer: this is one reason why they have laid hidden for so long. We’ll see that there are many unanswered questions that invite exploration.

The talk will be accessible to undergraduates.

16 thoughts on “Some fundamental formulas from metrical algebraic geometry (seminar)”

1. Jim G.

This is an amazing integration of these various theorems. I doubt many would have suspected they could be so interconnected, nor that when this result was found that it would be so symmetrical.

It’s kind of sad then that this field of pure mathematics has got so hung up on this wishful real number thinking. In books of math proofs I read I feel there’s this kneejerk attitude that our primary resort is analysis and taking limits, in some arguments just about every proof involves it. Not to knock the efforts of such competent and intelligent people at all, but I get this image of training wheels on a child’s bicycle. Whereas Newton and Euler seemed to look for and find lots of elegant and symmetrical and beautiful things, there seems to be a resignation from the beginning that we are just going to have to use ugly inelegant methods to find the answer, because after all that answer’s bound to be vague and messy. It’s almost like the difference between optimism and pessimism to my mind.

I watched this old video on YouTube where W.V. Quine discusses his philosophy of science. At one point he is asked about mathematical concepts, and he talks about the “reality” of sets. He defended these sets as incontrovertible, as though they are just plainly obvious once you’ve done the math through.

This was darkly comical, because I had just seen your most recent seminar, in which you closed with a slide showing the standard definition of sets, in which, if I’m correct, they are not even proven, but simply *presumed* via axioms! It seemed a little embarrassing when such smart and accomplished people bungle fundamentals so badly. And I guess it would take a bit of courage on the part of many who’ve spent their lives on this to admit this garden path of infinities that people have put so much effort into traveling down might actually be a dead end.
And that stubbornness may well take another generation to relinquish. Perhaps it will simply take another generation of people, collectively making significant progress using a less pessimistic approach to demonstrate the useful difference between these ways.

1. Lito P. Cruz

I am beginning to see why you do not believe in real numbers. Is it because they are not computable? You pointed out in your debate YouTube, the necessity of being able to write things down. I accept your point. If you can not write it down you can not write an algorithm to process it. Am I in the right track in understanding you?

LPC

2. njwildberger: tangential thoughts Post author

Yes that is quite right. It is easy to talk, but when one has to write something down for all to see, it’s a different game. An arithmetic which is not supported by explicit calculations that everyone can verify is not an arithmetic at all — it is perhaps only a dream, or a belief system. Such is the current story with “real numbers”. I have in vain tried to get my fellow mathematicians to show me even one simple but explicit example of “real number arithmetic”, which is not trivial, and does not essentially reduce to rational number arithmetic.

I am still waiting..

1. L P Cruz

Admitting that we are wrong does not come easy specially for academics who have large egos. My current position now is that real numbers and infinite sets are metaphysical concepts. I am ok in so far as talking about them metaphysically. This then invites questions on what we mean by “real” and “reality”. I am not so confident that mathematicians will recognise these ideas as for what they are — metaphysical. What do you think?

LPC

2. Jim G

Last week I discovered for the first time that there are some logicians (apparently prominent ones among them) who study “para-consistent” logic. From what I gather, rather than abandon infinite sets due to their logical problems, the idea is that we should instead revise logic itself, to accommodate certain kinds of contradictions. My first reaction is that I would be hard pressed to invent an idea any more perfectly backwards than that one.

Admittedly, I know very little about their work, and perhaps there is some worthwhile area in which this could be profitably explored, but I hope the goal of such a project is not to somehow try to re-build all mathematics with different logic. My intuition is the result would be a lot of sophistry leading nowhere. Meanwhile, those same old principles of logic tracing back millennia explain all sorts of theorems with all sorts of practical import in the world of science. The old logic works just fine and is not begging to be replaced outright with something less consistent.

3. extranosky

Not all is rosy in the land of logic. It is still an interesting field for me but you won’t get hired to do research in that field (as a general rule, at least it is very hard in Australia). You have good points to make in your debate but some of these have been ignored or missed because I think your opposer missed the idea of computability. Perhaps that is where you might work on articulating your arguments better cause now I get ya. I mentioned you in https://unpocologico.wordpress.com/2015/10/05/why-real-numbers-are-not/ in defence of your thoughts.

LPC

4. Alan Gibbs

I have just come across your work on rational trigonometry and waiting for delivery of your book. I have a question on rational triangles which you define as a set of three rational points. Considering two of these points you define the quadrance between/of these rational points as the sum of the squares of the difference between the coordinates. No problem there. Now the quadrance is the sum of two rational squares, which is a rational number. I understand that there are number theoretic restrictions on such a rational number, so every rational number/integer is not the quadrance of the side of a rational triangle. Have I got this correct?
If yes, then the integer 3 cannot be the quadrance of the side of a rational triangle and I recon that there are no equilateral rational triangles and the spread 3/4 does not exist. What do you think?
Regards……….AJ

1. njwildberger: tangential thoughts Post author

Hi, In the rational plane, there is a restriction on the kinds of numbers that can appear as spreads between lines. There is no equilateral triangle in such a plane, and so no spread of 3/4. That is all quite correct. Of course there are rational approximations to that. Also if we go to three dimensions, then equilateral triangles are found easily.

1. Alan Gibbs

Thanks for your prompt reply. Good to know I was on the correct track. I have found all your papers and videos very stimulating and informative. Another question if I may: some times you carry surds through your computations, even nesting them. Is there any way to extend the rationals to take this into account?
Regards……..AJ

5. Alan Gibbs

I have just noticed that you didn’t comment that a quadrance of 3 was impossible. Do you agree?

Also a quadrance of 2 is only possible if one of the squares in the sum is not zero, but 1. Thus I think it is impossible to have a rational right triangle of quadrances of 1 and 2 and hypotenuse 3. Do you agree.

I have just noticed that you have used this in your discussion of A4 paper. Where am I going wrong?!

1. njwildberger: tangential thoughts Post author

The fact is that which quadrances and spreads are realisable depends on the dimension. Roughly speaking, the higher the dimension, the more possibilities for relationships between lines. So there is no 1,2,3 quadrances triangle in two dimensions, but there is in three dimensions. Actually the story also depends very much over which field we are assuming our numbers come from. If we extend our field, we have more possibilities for quadrances and spreads.