Infinity is a fascinating and seductive topic, but it is also a contentious one and, throughout intellectual history, has been attacked on religious, philosophical, and practical grounds. Today, we turn away from the vastness of infinity and toward those who deny its existence.

Consider the Pythagoreans, followers of the famous 6th century BC Greek philosopher. According to the Pythagoreans, numbers are the first of all beings the “dominant and self-produced bond of the eternal continuance of things.” Pythagoreans saw numbers in everything, from the motions of the planets to musical harmony. But they saw only rational numbers, i.e. ratios of finite integers; according to legend, the Pythagorean Hippasus was drowned at sea after proving that $\sqrt{2}$ is irrational.

As we have mentioned in earlier posts, actual infinity did not establish a real foothold in mathematics until the late 19th century and the work of Dedekind and Cantor, which naturally engendered fierce opposition. Perhaps the most prominent opponent of Cantorian set theory was Leopold Kronecker, who is widely known for having said, “God made the integers, all else is the work of man” and thought that irrational numbers do not exist. I have given this exact quote before, but I will do so again now:

I don’t know what predominates in Cantor’s theory – philosophy or theology, but I am sure that there is no mathematics there.

Finitism, the philosophy of mathematics that denies the existence of infinite objects, is today a minority view but is nonetheless very much alive, and a number of mathematicians are doing work to remove infinities (and, hence, irrationalities) from mathematics. Among these is N.J. Wildberger, a professor at the University of New South Wales who is developing what he calls Rational Trigonometry, a reworking of trigonometry replacing the notions of ‘distance’ and ‘angle,’ which readily produce irrational numbers even when applied only to points with rational coordinates, with ‘quadrance’ and ‘spread’ to measure the amount of separation between two points and two lines, respectively.

(A related but separate movement, which we will return to in a later post, is the attempt by prominent scientists such as Max Tegmark and Raphael Bousso, to remove infinities from physics. The assertion that our physical universe is entirely finite is a weaker (and, in my opinion, much more plausible) assertion than the assertion that infinity does not exist mathematically, and we will devote attention to it in the future.)

There are some, though, for whom the finitists do not go far enough. These people, known as ‘ultra-finitists,’ not only deny the existence of mathematical infinity but even refrain from accepting the existence of very large finite integers. For example, let $N$ be the largest integer less than $e^{e^{e^{79}}}$, a number known as Skewes’ number and that has appeared in proofs in number theory. An ultra-finitist will likely refrain from accepting the existence of $N$ on the basis that this natural number has not actually been calculated and may in fact be too large to be physically calculated at all.

Ultra-finitism in its modern guise was initiated by Alexander Yessenin-Volpin, a mathematician who was a prominent human-rights activist in the Soviet Union, for which he was imprisoned in 1968. There is a wonderful anecdote about him from Harvey Friedman, who had occasion to confront him about his extreme ultra-finitist views.

I have seen some ultrafinitists go so far as to challenge the existence of 2100 as a natural number, in the sense of there being a series of “points” of that length. There is the obvious “draw the line” objection, asking where in 21, 22, 23, … , 2100 do we stop having “Platonistic reality”? Here this … is totally innocent, in that it can be easily be replaced by 100 items (names) separated by commas. I raised just this objection with the (extreme) ultrafinitist Yessenin-Volpin during a lecture of his. He asked me to be more specific. I then proceeded to start with 21 and asked him whether this is “real” or something to that effect. He virtually immediately said yes. Then I asked about 22, and he again said yes, but with a perceptible delay. Then 23, and yes, but with more delay. This continued for a couple of more times, till it was obvious how he was handling this objection. Sure, he was prepared to always answer yes, but he was going to take 2100 times as long to answer yes to 2100 then he would to answering 21. There is no way that I could get very far with this.

Harvey M. Friedman “Philosophical Problems in Logic”

Perhaps the most prominent ultra-finitist working today is Doron Zeilberger, a mathematician at Rutgers. Zeilberger, like many ultra-finitists, believe there is a largest natural number. When asked the inevitable question about what happens when you add 1 to it, he replies that, in a very elegant circularity, you go back to 0. Zeilberger also has a fascinating web page of his Opinions. I present to you in full Opinion 146: Why the “fact” that 0.99999999…(ad infinitium)=1 is NOT EVEN WRONG.

The statement of the title, is, in fact, meaningless, because it tacitly assumes that we can add-up “infinitely” many numbers, and good old Zenon already told us that this is absurd.

The true statement is that the sequence, a(n), defined by the recurrence

a(n)=a(n-1)+9/10n   a(0)=0   ,

has the finitistic property that there exists an algorithm that inputs a (symbolic!) positive rational number ε and outputs a (symbolic!) positive integer N=N(ε) such that

|a(n)-1|<ε for (symbolic!) n>N   .

Note that nowhere did I use the quantifier “for every”, that is completely meaningless if it is applied to an “infinite” set. There are no infinite sets! Everything can be reduced to manipulations with a (finite!) set of symbols.

We end with the classic instructional video, “Look Around You (Maths),” which addresses ultra-finitism in its first segment. Enjoy!