Pretty much the most interesting blog on the Internet.— Prof. Steven Landsburg

Once you get past the title, and the subtitle, and the equations, and the foreign quotes, and the computer code, and the various hapax legomena, a solid 50% English content!—The Proprietor

Tuesday, September 22, 2015

Ultrafinitism Is Interesting

Norman Wildberger

Lately one has been trawling YouTube for lectures on subjects in higher mathematics, physics, computer science, and economics. These lectures, like the ones from MIT OpenCourseWare (generally excellent, but limited), can be run in the corner of one's screen to refresh one's recollection, or perhaps even learn new ideas not covered in one's student days, while performing other tasks which do not require total concentration. In this course, one ran across the productions of one Norman J. Wildberger.

And what extraordinary productions they are! For Wildberger, as a (the?) representative of the Ultrafinitist school, abjures most of the content and toolkit of modern mathematics. Infinity, infinite sets, the real numbers, the transcendental numbers, the transfinite numbers, functions, limits, uncountable sets, axiomatics, Cauchy sequences, Dedekind cuts—all of these are to Wildberger mere pretenses and frauds unworthy of the attentions of a true mathematician.

To Wildberger, the only valid mathematical objects are individual integers (like \(5\)), rationals (like \(\frac{5}{3}\)), and objects directly constructed from them, like specific polynomials with integer or rational coefficients (like \(x^2+1\)) and the ratios of such polynomials (like \(\frac{x^2+1}{x}\)). About algebraic numbers (i.e., the roots of rational polynomials, like \(\sqrt{2}\) or \(i\)) he, at the very least, has grave doubts which demand that they be handled with much greater care than mathematicians ordinarily employ. Transcendental numbers (like \(\pi\) or \(e\)) are right out. So are the sets of any of the objects he accepts, like the integers, because they are infinite.

Ordinarily, a sensible person confronted with such claims denouncing most of the established and useful content of a genuine science would merely back away, while smiling and nodding politely and perhaps glancing about desperately for some object to defend oneself with should this obvious madman decide to charge.

However, in the case of Wildberger such a reaction, while understandable, may be a mistake. For Wildberger is no ordinary mathematical madman, such the ones that heatedly insist that \(\pi\) exactly equals \(\frac{22}{7}\). Rather, he exhibits the full conversancy with the subjects of higher mathematics one would expect from a professor of mathematics at a major Western university (Prof. Wildberger teaches at the University of New South Wales). When such a person makes such extraodinary claims, it behooves the philosopher to give them more careful consideration:

Some of Wildberger's claims have merit. His explanation that educators of higher mathematics perform a sort of shell game—perhaps conceding to neophytes that there are some foundational issues with some concepts, but deferring them to later courses, and then, in those later courses, waving them away as something surely resolved in the neophytes' courses—would often be correct. And Wilderberger's skill at deriving—using only his idiosyncratic methods and terms, abjuring all the concepts he deems haram—to reach results one previously would have thought to require these forbidden concepts, is both surprising and respectable.

That said, Wildberger's claims are not necessarily true. Using his method, he derives nothing unknown to conventional mathematics nor does he disprove anything shown using conventional mathematics. Hence he offers no reason for a conventionally-trained mathematician to give up his tools and instead relearn the subject in Wildberger's terms.

Wildberger's oft-promised proof that conventional mathematics is wrong also invariably fails. All Wildberger does is point to some conclusion of conventional mathematics—such as the Peano SquareOne shudders to contemplate what must be Wildberger's feelings on the Banach Tarski Paradox—which are perhaps surprising to the uninitiated and then almost-literally jumping up and down, while exclaiming Do you not see how very, very wrong this is?! Wildberger is surely entitled to his discomfort, but the rest of us are as surely entitled to our indifference or even delight at such propositions. Neither sentiment proves anything.

Wildberger's history of mathematics too is flawed. He tells a story of all great mathematicians being virtuous Finitists until Cantor—surely under the influence of the Devil himself—whispered his honeyed poison into the ears of such great mathematicians as Hilbert and succeeded in corrupting them, even as noble Finitist like Kronecker warned of the danger.

But this is not right. For example, long before Cantor, Euler—of whom Wildberger speaks with the reverence he is due—would count as his greatest, among many great, legacy the formula \(e^{i \pi}+1=0\), perhaps the greatest formula in mathematics. Wildberger would not object to the symbols \(=\), \(+\), \(1\), and \(0\), for all of them are to him well-defined and meaningful. Not even \(i\), the frequent source of disbelief in higher mathematics, would necessarily raise his ire, for \(i\) is the root of a rational polynomial, a concept the validity of which he admits. But \(e\) and \(\pi\)? They are transcendental numbers and therefore, to Wildberger, frauds. Hence, so must be any formula purporting to establish a relationship between these meaningless concepts.

A future post or posts may deal with the author's attempts to disprove—or perhaps establish—some of Wildberger's other claims.