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Friday, August 13, 2010

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Some mathematicians in the computer era are getting hung up on the digital-analog divide. The square root of 2 is a real quantity, but it cannot be expressed digitally, or in any integer arithmetic base. This only matters to people who will not think of numbers except in digital terms. I'd suggest that the digital fractional numbers are the rarities, and that our universe is full of mostly integers and non-rational quantities.

Bill wrote: "But this irrational number measures a quite definite length both in the physical world and in the ideal world."

Is it true that lengths in the physical world are as definite as that? As I understand it, according to our best understanding of nature to date, physical lengths get kind of "fuzzy" at around the 35th or so decimal place (in meters)?

Dr. Z is not as bad a philosopher as you think. If you look at any process giving us the so called "real" numbers you would be more astonished than reading his arguments. Did you know that in math the real numbers are understood not as some lengths or numbers as we intuitively feel them, but rather as equivalence classes of rational Cauchy sequences? Read it, then read it once again and you will see that there is much more mysticism and confusion than in the "wild stuff" Dr. Z preaches.

Addendum:
"Now what the hell does that mean? A rational number is one that can be expressed as a fraction a/b where both a and b are integers and b is not 0. An irrational number is one that cannot be expressed in this way."
By saying that you presuppose the existence of some irrational numbers. It is very easy to construct rational numbers from the natural ones (Dr. Z would not even agree on that as he views - you can read it in other essays on his webpage - the number axis to be not the real numbers, but rather a necklace hZ_p, where h>0 is some extremely small rational number and Z_p is the set of integers modulo some very large natural number). But for talking about irrational numbers you first have to introduce them (taking all infinite decimal expansions doesn't work and if you doubt it you can read it on Timothy Gowers' webpage (an celebrity in math)). So the argument on the equation x^2=2 only gives us a the information that there is no rational number satisfying it. Yet to talk about some other solution one first has to introduce some other objects. And this is where the real "wild stuff" happens. A plausible answer should be that there cannot exist a right triangle with sides 1 and 1 - there only exist right triangles with rational sides (my own interpretation from now on). Even if we take such drastic position, there would be no problem from the point of an engineer - as one never measures things precisely, there is no difference for him whether the length is rational or irrational (whatever the latter means).

I just noticed this thread (hope it's not too late). Tim Gowers (Fields medal winner) has an elegant and accessible discussion of the root 2 problem on a page here

http://www.dpmms.cam.ac.uk/~wtg10/roottwo.html

There are also links to other discussions on his site. I read these years ago, strongly recommended.

Thanks for that, I'll have to study it.

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