## On Wildberger’s “Inconvenient Truths”

Dr. Norman Wildberger of the University of New South Wales has a wonderful and prolific YouTube channel in which he discusses a great deal of very interesting mathematics. I have discussed Dr. Wildberger before, regarding a very similar subject, but I wanted to take a moment to discuss a video from his Math Foundations series entitled, “Inconvenient truths about sqrt(2).”

In the video, Dr. Wildberger claims that there are three different ways in which is commonly discussed: the Applied, the Algebraic, and the Analytical. He does a fairly good job of discussing the manner in which the ancient Greeks discovered that there exists no ratio of two whole numbers which can be equal to , which is a topic I have covered here, as well. He then explains what he means by each of the above three categories.

Since we have shown that there is no ratio of two whole numbers which can equal exactly, the Applied path seeks to find ratios which simply come close to equaling that number– approximations with an arbitrarily large or small error. We are not searching for an exact solution, on the Applied path, and indeed we are content to agree that there is no exact solution which can be attained, according to Dr. Wildberger. We can, for example, find that 1.414, when squared, gives a solution quite close to 2, but it is not exactly 2.

For the Algebraic path, we can construct an extension to the rational numbers which contains some exact solution to the question of — Dr. Wildberger gives the example of an arithmetic using pairs of rational numbers and such that . He notes that this can be done in such a way that it conforms to all the usual laws of arithmetic, but objects that the in this scenario “has nothing whatsoever to do with that one-point-four-one-four-et-cetera that we were talking about previously.”

Finally, Dr. Wildberger presents the Analytic path, which he describes as “the square root of 2 is some infinite decimal which starts out 1.414 and goes on in some fashion.” He unequivocally refers to the Analytic path as “wrong thinking,” and unabashedly goes on to claim that such an object “does not exist, my friends.” It is quite clear that Dr. Wildberger has no love for Analysis. Quite the contrary, he is openly hostile to the idea.

While there are minor statements that I could nitpick in Dr. Wildberger’s treatments of the Applied and Algebraic approaches to the square root of 2, it is his handling of the Analytic approach with which I’ll interact in this article. His discussion of the subject is incredibly hyperbolic, highly oversimplified, and entirely uncharitable. Dr. Wildberger doesn’t even pretend to consider the idea that the Analytic approach may have some reasonable underpinnings which he nevertheless finds to be flawed; rather, he simply dismisses the entire field of analysis as being incorrect and accuses it of being the ruin of mathematics. He treats the subject in this manner despite the fact that, as he will well admit, the overwhelmingly vast majority of all the world’s mathematicians from the past hundred years find the Analytic approach to be perfectly good. In fact, Dr. Wildberger rather boldly claims that these other mathematicians “are all wrong. They are seriously wrong.”

The closest which Dr. Wildberger comes to giving an accurate description of the Analytic approach is when he is discussing the number line. According to Dr. Wildberger,

…this Analytic approach to root 2… pretends that, somewhere on the line (which up ’til now only consists of Rational numbers), somewhere there’s a new place, and it’s somewhere between 1 and 2, and there’s a new number called ‘root 2,’ and it has the property that its square is 2, and we can find out what this thing is by making a calculation.

To say that this is a mischaracterization of Analysis is quite an understatement. In truth, Analysis is based upon an assumption regarding the number line, but it does not simply try to plop an object called somewhere between 1 and 2, as Dr. Wildberger claims. Rather, the assumption regarding the number line upon which Analysis is built is a fairly reasonable one– the idea that the number line is continuous. That is to say, Analysis assumes that there are no gaps or holes in the number line. If the number line only consisted of Rational numbers, as Dr. Wildberger claims it did, then there would be a great many holes in it, indeed, as there are a great many mathematical statements which produce values which cannot be expressed as Rational numbers– uncountably infinitely many, in fact.

The idea that the number line is continuous did not originate with Analysis. It had been an openly discussed question in mathematics since at least the ancient Greeks. The Analysts simply decided to explore what it would mean for such a continuum to exist. Quite happily, they found that assuming continuity led to very beautiful developments in mathematics– exactly the opposite of the picture Dr. Wildberger paints.

If one assumes that the number line is continuous, as the Analysts did, then there is no need to try to create a place for to go, despite Dr. Wildberger’s intimations otherwise. It’s already there, occupying a gap between the Rational numbers. Analysis simply asks, “What can we learn about this gap?” It was not arbitrarily placed between 1 and 2, as Dr. Wildberger hints. Analysis helps us to discover that it is there.

Nor is it true that Analysis claims “we can find out what this thing is by making a calculation.” We already know what this thing is: it is the square root of two. Dr. Wildberger is conflating “what this thing is” with the manner by which we symbolize this thing when using a particular notation. That is to say, Dr. Wildberger is attempting to claim that the number **is** its decimal representation. This is why he takes such offense at the ellipsis which is used to show that the decimal representation is incomplete. For Dr. Wildberger, the decimal representation **is** the number.

This is, of course, a silly notion. The symbols which we use to represent an idea are not equivalent to that idea. Nobody thinks, for example, that the color blue necessarily consists of the letters “b,” “l,” “u,” and “e.” Nor would anyone claim that is a more proper symbol for the number it represents than is “two” or “два” or “二” or || …or even . Similarly, it seems more than a little misguided that Dr. Wildberger is so inordinately attached to the decimal representation of the square root of 2. The fact of the matter is that, so long as it is clear that we are talking about the square root of 2, then it doesn’t matter if we represent that notion with or with or with or with “the ratio of the magnitude of the diagonal of a square to that of one of its sides.”

So when Dr. Wildberger writes…

…and asks, “Is this a correct and meaningful statement?” the answer to both is, “Yes.” None of the displayed digits is incorrect and the ellipsis acknowledges that the display is incomplete. This statement gives us a good bit of information about , and that alone makes it meaningful. When Dr. Wildberger asks about moving the ellipsis to display fewer and fewer digits, the expression remains correct and meaningful, but becomes less useful as we omit more information. The simple fact of the matter is that a mathematical statement can most certainly be “meaningful” without carrying perfectly complete information.

Even when Dr. Wildberger presents the question of in an attempt to show that the ellipsis is absurd, he is misguided, as this statement actually does have meaning– it tells us that is equal to a number. Now, Dr. Wildberger is correct to point out that one is not likely to get any credit for such an answer on homework or an exam, but his reasoning is incorrect. As Dr. Wildberger well knows, good math homework and exams care less about the completeness of the answer than they do about how the student arrived at that answer. After all, which should receive more credit on a test: a correct answer with incorrect work shown or an incorrect answer with the correct work shown? So, while may be a *technically* correct response, it does nothing to show that the student has any understanding of whatever mathematical concept is actually being tested.

This idea that the decimal expansion of contains an infinite number of non-repeating digits seems to be the only real objection which Dr. Wildberger presents in this video, but his opposition to it seems misplaced, at best. In the description to the video, Dr. Wildberger notes that he will further discuss the logical problems which he purports to exist in the treatment of irrational numbers in his videos on Cauchy sequences and Dedekind cuts, so I will be sure to watch these as well; however, his bold pronouncement that “none of them work” seems more than a little arrogant. We’re not talking about some fringe development in a little known field which is sparking controversy and debate. On the contrary, Dr. Wildberger is overtly stating that hundreds of years worth of the world’s greatest mathematical discoveries are completely wrong.

I believe I understand why Dr. Wildberger makes such outlandish claims. In some of his other work, I have seen him explicitly reject the axioms of infinity and of choice utilized in modern set theoretic frameworks. Certainly, without these axioms, our understanding of the irrationals becomes far less rigorous. However, Dr. Wildberger’s aversion to these axioms has led him to caricature his opposition rather than to treat the opposing viewpoint with even the remotest sense of charity. As such, it seems fairly difficult to take his claims on the subject seriously.

Norman Wildberger’s video on the square root of 2 does not contain the “inconvenient truths” which it purports to show. Worse, it contains rather convenient falsehoods which Dr. Wildberger has utilized in his attempt to denigrate Analysis.

Without suggesting that Dr Wildberger’s argument isn’t flawed, discussing it does make me realize there’s things about the structure of irrational numbers I don’t understand. That is, I’ve never felt doubts about irrational numbers as things, but I’m aware that very many intelligent people, many of them mathematicians, were skeptical of them for various reasons. And I’ve not understood their reasons, and why their reasons for skepticism were mistaken or answered by later reasoning. It feels like a gap in my mathematics education.

As I understand it, the problem with irrationals– both historically and in modern objections– has to do with the difficulty in defining them rigorously and precisely. It really wasn’t until Cauchy sequences and Dedekind cuts that we had means for a general definition of irrationals, but both of these rely on the use of controversial infinite sets.

If one rejects the idea of an infinite set, as Dr. Wildberger does, then there does not seem to be any good way to rigorously define what we mean by irrational numbers.

So, using the sqrt(2) example, Wildberger would say that we are only justified in saying that there exists no Rational number equal to sqrt(2); and he argues that we are NOT justified in saying that because of this, there must be some number equal to sqrt(2) which is not Rational.

“Analysis assumes that there are no gaps or holes in the number line”

so, other than intuition, what is the basis for this assumption? Dr Wildberger rejects analysis simply because the basis of it seems weak.

“Quite happily, they found that assuming continuity led to very beautiful developments in mathematics”

that still is no proof of the basic assumptions. surely we could get beutiful results from some “mathematics” based on some crazy axioms. that is no measurement of the quality of a mathematical theory.

“That is to say, Dr. Wildberger is attempting to claim that the number is its decimal representation”

No. the problem is that it is not completed represented in a way that allows us to do basic arithmetic operations, decimal or not.

“The simple fact of the matter is that a mathematical statement can most certainly be “meaningful” without carrying perfectly complete information”

As an aproximation, yes. Now realize you can make such approximations with truncated decimals or rationals, there is no need to use “real” numbers to do so.

“however, his bold pronouncement that “none of them work” seems more than a little arrogant. We’re not talking about some fringe development in a little known field which is sparking controversy and debate.”

it sparked controversy and debate in the 19th century. In the 20th it became widely accepted because methematicians accepted some assumptions not because they were clearly and carefully proved.

“I believe I understand why Dr. Wildberger makes such outlandish claims. In some of his other work, I have seen him explicitly reject the axioms of infinity and of choice utilized in modern set theoretic frameworks. Certainly, without these axioms, our understanding of the irrationals becomes far less rigorous. However, Dr. Wildberger’s aversion to these axioms has led him to caricature his opposition rather than to treat the opposing viewpoint with even the remotest sense of charity.”

Yes, because in his opinion the assumptions/axioms are outlandish too. How do you prove the existence of the continuum? How do do operations on sets that you cant be sure which elements it holds (infinite sets)? Accepting axioms just because we want to believe in them or seems intuitively true or because they give beautiful results or because everyone else accept them.. THAT is not more rigorous. It is the exact opposite.

Thank you for reading and taking the time to reply! I will try to touch on all your points, but please forgive me if I’ve missed anything or misunderstood you at any point.

I see no reason to think that “the basis of it seems weak.” It seems no less weak, to me, than any other area of mathematics.

Of course not. Axioms are axioms because they are assumptions which themselves are not– and cannot be– proven.

Sure it is. There are no basic arithmetic operations which I can’t do on √2, and that is a symbol which represents that number completely.

Nor is that even a very useful method of determining the value of a symbol. For example, I’m sure you would agree that 0 is a perfectly valid symbol for representing the number zero, and yet there are a number of basic arithmetic operations which we cannot do on 0. For example, x/0 and 0^0.

Sure, truncated decimals and other rationals are perfectly fine for approximations. That has nothing to do with the point I was addressing in my statement. Dr. Wildberger had made the claim that symbolic representations of a number which are suffixed by an ellipsis are meaningless. I contend that they are not meaningless, in the least, and quite the contrary that they convey quite a bit of meaning very clearly.

It is wholly disingenuous to pretend that the wide acceptance which these methods gained in the 20th century was solely due to the fact that “[mathematicians] accepted some assumptions.” Mathematicians in the 20th century were able to develop a foundation which provided a rigorous basis for these methods and which was perfectly internally consistent and which led to the development of highly useful fields of mathematics which would not have been otherwise possible. The controversy fell to the fringes because those philosophers and mathematicians who opposed the infinite were not nearly as successful in producing new, useful results; nor is their axiomatic rejection of the infinite any more provable than the axiomatic acceptance of it.

I’m curious now– do you think that ANY axioms of mathematics are provable? Should we reject ALL axioms of mathematics? That would seem to leave us with nothing at all. And if we shouldn’t reject ALL axioms of mathematics, what standards do you propose for differentiating a good axiom from a bad one? Is it even possible to make any such set of standards without basing the standards themselves on some axiomatic assumptions?