This is wild stuff; I cannot say whether it is mathematically respectable but the man does teach at Rutgers. It is certainly not mainstream. Excerpt:

It is utter nonsense to say that sqrt 2

is*irrational*, because this presupposes that it exists, as a

*number*or

*distance*. The truth is that there is no such number or distance. What does exist is the

*symbol*, which is just shorthand for an

*ideal*object

*x*that satisfies

*x*= 2.

^{2}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 the celebrated theorem of Pythagoras, a right triangle with sides of 1 unit in length will have an hypotenuse with length = the square root of 2. This is an irrational number. But this irrational number measures a quite definite length both in the physical world and in the ideal world. How can this number not exist? It is inept to speak of a symbol as shorthand for an ideal object since, if x is shorthand for y, then both are linguistic items. For example 'POTUS' is shorthand for 'president of the United States.' But 'POTUS' is not shorthand for Obama. 'POTUS' refers to Obama. Zeilberger appears to be falling into use/mention confusion. If the symbol for the sqrt of 2 refers to an ideal object, then said object is a number that does exist. And in that case Zeilberger is contradicting himself.

What's more, it seems that from Zeilberger's own example one can squeeze out an argument for actual infinity. We note first that the decimal expansion of the the sqrt of 2 is nonterminating: 1.4142136 . . . . We note second that the length of the hypotenuse is quite definite and determinate. This seems to suggest that the decimal expansion must be actually infinite. Otherwise, how could the length of the hypotenuse be definite?

As an ultrafinitist, however, Zeilberger denies both actual *and potential* infinity:

. . . the philosophy that I am advocating here is called

*ultrafinitism*. If I understand it correctly, the ultrafinitists deny the existence of

*any*infinite, not [sic] even the potential infinity, but their motivation is `naturalistic', i.e. they believe in a `fade-out' phenomenon when you keep counting. [. . .]

So I deny even the existence of the Peano axiom that every integer has a successor.

As I said, this is wild stuff. He may be competent as a mathematician; I am not competent to pronounce upon that question. But he appears to be an inept philosopher of mathematics. But this is not surprising. It is not unusual for competent scientists and mathematicians to be incapable of talking coherently about what they are doing when they pursue their subjects. Poking around his website, I find more ranting and raving than serious argument.

The ComBox is open if someone can clue us into the mysteries of ultrafinitism. There is also some finitist Russian cat, a Soviet dissident to boot, name of Esenin-Volpin, who Michael Dummett refers to in his essay on Wang's Paradox, but Dummett provides no reference. Is ultrafinitism the same as strict finitism?

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.

Posted by: anonymous | Saturday, August 14, 2010 at 09:04 AM

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)?

Posted by: Alfred Centauri | Saturday, August 14, 2010 at 06:44 PM

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.

Posted by: Tomas | Monday, August 16, 2010 at 04:06 PM

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).

Posted by: Tomas | Monday, August 16, 2010 at 04:23 PM

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.

Posted by: William | Friday, August 20, 2010 at 06:12 AM

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

Posted by: Bill Vallicella | Friday, August 20, 2010 at 11:34 AM