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Tuesday, March 21, 2006

Mathematical miracles

Kenny Easwaran has a post on the essences of mathematical concepts. As I commented there, the next step is to think about natural kinds and laws in mathematics. A further step would be to think about miracles. What could be meant by a mathematical miracle? Well, the term has already been used. Frank Morley's result that
The three points of intersection of the adjacent trisectors of the angles of any triangle form an equilateral triangle
was apparently deemed so surprising that it was called Morley's miracle. (See Connes' proof of it on p.6 of this.)

When this topic was raised during my talk at the IMA n-categories workshop. André Joyal suggested that the fact that the complex numbers are algebraically closed after merely adjoining the roots of one equation (x2 + 1 = 0) to the reals is a miracle. However, someone else pointed out that there are more general results concerning algebraic closure of fields. Perhaps the physical parallel to this may be surprising consequences of physical laws, such as that helium balloons in planes move forward on take off. Indeed, it is very common for people to present seeming miracles, and then explain them away. Speaking about a state sum model for 3-manifold invariants, using quantum 6j symbols to label edges of a triangulation, John Baez points out that the symbols compose just right for the resulting partition function to be independent of the triangulation, saying "See week38 for an explanation of this seeming miracle: it's actually no miracle at all." Perhaps Joyal's intuition, however, is that it is still surprising that the reals just need an extension of degree 2 to achieve closure. This is not an instance contradicting a mathematical law, but then in our secular age we've also given up on controventions of physical laws, unless we mean to overthrow them.

A seemingly inexplicable surprise seems to be the modern, non-theological, sense of miracle. By any sensible measure, most of the truths expressible in a formal mathematical language are true for no good reason. But we wouldn't want to call them miracles, just as there are unmiraculous brute physical facts - 'a uranium atom within a mile of me decays within a picosecond of a solar neutrino passing through my big toe' or 'the distance between the sun and Pluto now is x metres'. To be surprising there must be theoretical considerations which tend to count against the occurrence of the result. Once again, mathematics and physics don't seem to be so very dissimilar.


Akira Bergman said...

I like to think of the closure of complex numbers as a miracle. All it takes is two equations to reach the bottom of the complex field.
x^2-1=0 --> x={+1,-1}
x^2+1=0 --> x={+i,-i}
It it opens up in a binary tree with square-root taking;
reminding the modular group which has a 2*2 format and maps onto a unit circle.

March 22, 2006 1:01 AM  
Fabien Besnard said...

One can consult this page about Morley's theorem. Conway's proof is particularly simple and also of an unusual kind.

March 22, 2006 4:24 PM  
John Baez said...

In mathematics every "miracle" is a call to understanding... we can't help wanting to demystify it! It's an irritant, like the grain of sand that makes an oyster form a pearl. In this case, we hope the result is a pearl of wisdom.

But, even after a miracle has been "understood" in some way, we can still try to confront it afresh and be awed by it yet again... which may lead us to some newer, deeper understanding.

So, ultimately, it's not a matter of whether certain facts are or are not miracles in some "objective" sense. We move between two poles: the desire to be awed by mathematical facts, and the desire to understand them and have them seem obvious. We need both.

March 23, 2006 3:55 AM  
david said...

Are there then any cases where something is now completely obvious, and there is no prospect of it ever being anything else but obvious? In Minneapolis you were showing us how even 2 + 3 = 5 isn't obvious.

March 23, 2006 9:45 AM  
John Baez said...

David writes:

Are there then any cases where something is now completely obvious, and there is no prospect of it ever being anything else but obvious? In Minneapolis you were showing us how even 2 + 3 = 5 isn't obvious.

I think you almost answered your own question! If we can find new ways to look at even such simple facts as 2+3 = 5, there's probably nothing too obvious to be worth pondering. The trick is finding ways to see things afresh.

As Grothendieck said:

In our acquisition of knowledge of the Universe (whether mathematical or otherwise) that which renovates the quest is nothing more nor less than complete innocence. It is in this state of complete innocence that we receive everything from the moment of our birth. Although so often the object of our contempt and of our private fears, it is always in us. It alone can unite humility with boldness so as to allow us to penetrate to the heart of things, or allow things to enter us and taken possession of us.

This unique power is in no way a privilege given to "exceptional talents" - persons of incredible brain power (for example), who are better able to manipulate, with dexterity and ease, an enormous mass of data, ideas and specialized skills. Such gifts are undeniably valuable, and certainly worthy of envy from those who (like myself) were not so "endowed at birth, far beyond the ordinary".

Yet it is not these gifts, nor the most determined ambition combined with irresistible will-power, that enables one to surmount the "invisible yet formidable boundaries" that encircle our universe. Only innocence can surmount them, which mere knowledge doesn't even take into account, in those moments when we find ourselves able to listen to things, totally and intensely absorbed in child's play.

March 24, 2006 9:44 PM  

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