Here's a possible good faith interpretation: Irrational numbers exist as a limiting process, not as a static atomic unit. That is, irrational numbers are a verb, not a noun.
We use irrational numbers as nouns, when convenient, but this is an abuse, in some sense. When we want some digit of sqrt(2), say, we need to interrogate an algorithm to get it. We talk about how much time it takes to extract the amount of precision we want. At best sqrt(2) can be thought of as an abstract symbol that, when we multiply it by itself, is 2. That is, an algebraic manipulation that we can reduce to an integer under certain circumstances, but it doesn't "exist" the same way that an integer or a rational exists.
> That is, an algebraic manipulation that we can reduce to an integer under certain circumstances, but it doesn't "exist" the same way that an integer or a rational exists.
This depends on your interpretation: some view the reals as completions of that process, in which those “verbs” are “nouns”.
But you can construct a coherent theory in which this is not the case — and nobody is much fussed, because mathematics is full of weird theories and interpretations.
And both integers and rationals are defined by their relations, eg, integers are equivalence classes of pairs of naturals and rationals as equivalence classes of pairs of integers — where the class obeys some algebraic manipulation properties. If you feel there’s some great difference in sequences (and where you find that difference, eg, allowing only constructibles) is a matter of perspective.
Put another way: irrational numbers are functions, not values. You can run the function as long as you want and get more digits but the function won't halt.
He is a sort of strong finitist [0]. Roughly, this means that many usual mathematical objects that rely on notions of infinity, e.g., irrational numbers, do not exist.
I came across his YouTube channel [1] years ago as a undergrad and became really confused about some ideas in logic as a result.
Wildberger has always been this way. Way back in 2007, Marc Chu-Carroll's "Good Math Bad Math" highlighted Wildberger: "This isn’t the typical wankish crackpottery, but rather a deep and interesting bit of crackpottery." In brief, Wildberger is clearly educated, but also clearly rejects axioms that mathematicians accepted a long time ago (infinite sets in this case):
All models are wrong; some models are useful. I'm not sure I'd say I "believe in" the number 7, either. But the number 7 is very useful. It's reasonable to debate how useful irrational numbers are. That is: there are models of math where irrational numbers are "outlawed" in some sense, and it sounds like Prof. Wildberger studies those models. It's a "fringe" model though; most of math is better off simply accepting irrational numbers as they are. Clearly some of the most widely useful math involves exp(x), sin(x), and pi, all of which are either completely outlawed or require laborious workarounds if you're not allowing true irrational numbers. Still, it's a good idea to have some people studying these fringe models to get a better understanding of math as a whole.
In this case, I find the argument "but you can't calculate it!" unconvincing, since every computer will have rational numbers they can't exactly calculate as well. Our computers can't calculate the exact value of 1/3 either; so what? If we're worried about computing things, we should consider whether we can calculate things to arbitrary precision or not within reasonable time. In that sense, pi behaves no worse than 1/3.
> since every computer will have rational numbers they can't exactly calculate as well
It might be better worded as "can't calculate a decimal version of every rational number". One can work quite easily nowadays with exact representations of rational numbers on computer. With Bigint stuff, it is easy to have very large (for human purposes) numerators and denominators. To what extent practical calculations could be done with exact rational arithmetic, I am not sure of though I suspect it is largely not an issue as precision of inputs is presumably a limiting factor.
Wildberger has specific objections to the usual definitions of real numbers and they vary based on the definition. For decimals, it is the idea that doing arithmetic with an infinite decimal is difficult even with a simple example such as 1/9*1/9 which is multiplying .111... times itself, leading to sums of 1s that carryover and create a repeating pattern that is not self-evident from the decimal itself.
For Cauchy sequences, he objects to the absurd lack of uniqueness, particularly that given any finite sequence, one can prepend that sequence to the start of any Cauchy sequence. So a Cauchy sequence for pi could start with a trillion elements of a sequence converging to square root 2. This can be fixed up with tighter notions of a Cauchy sequence though that makes the arithmetic much more cumbersome.
For Dedekind cuts, his issue seems mostly with a lack of explicit examples beyond roots. I think that is the weakest critique.
Inspired by his objections, I came up with a version of real numbers using intervals. Usually such approaches use a family of overlapping, notionally shrinking intervals. I maximized it to include all intervals that include the real number and came up with axioms for it that allow one to skirt around the issue that this is defining the real number. My work on this is hosted on GitHub: https://github.com/jostylr/Reals-as-Oracles
> One can work quite easily nowadays with exact representations of rational numbers on computer
One can also work with exact representations of Pi and sqrt(2). Use a symbolic system like MATLAB or Wolfram Alpha. Yes, if you create dedicated data structures for those exact representations you can work around the limitations of both 1/3 and Pi -- this is my point: the line is not "rational vs. irrational", it's "exact vs. computable to arbitrary precision vs. uncomputable". That is to say: a mathematical model that permits the rationals but outlaws the irrationals is much less likely to be at all useful than a model that permits computable numbers but outlaws/ignores non-computable numbers. I contend most objections to irrational numbers boil down to their general incomputability -- that is, 100% of all irrationals are not computable, and that makes people anxious. There is a coherent computation-focused model that keeps all computable irrationals and disallows the rest that would quell almost everyone's objections to the irrationals. For example, the set of rationals plus computable irrationals is countably infinite. All polynomials have roots.
> For decimals, it is the idea that doing arithmetic with an infinite decimal is difficult even with a simple example such as 1/9*1/9 which is multiplying .111... times itself, leading to sums of 1s that carryover and create a repeating pattern that is not self-evident from the decimal itself.
Right, but this is another example where an objection to irrational numbers can also be levied against 1/9, showing that computability is actually what we care about. And Pi and e and sqrt(2) are all computable, and not in any qualitatively more "difficult" way than the rationals.
> For Dedekind cuts, his issue seems mostly with a lack of explicit examples beyond roots. I think that is the weakest critique.
Yes, that is a weak critique indeed. Any computable real can be turned into a Dedekind cut that you can query in finite time.
> I came up with a version of real numbers using intervals
I haven't dug into your axioms but it seems to follow that if you gave me a Dedkind cut (A, B) then I could produce an Interval Oracle by taking [x, y] => x ∈ A && y ∈ B. Similarly if you gave me an Oracle I could query it to determine inclusion in A and B for any points -- immediately if you allow infinity in the query. That is, Oracle(x, inf) <=> x ∈ A and Oracle(-inf, x) <=> x ∈ B. So at first glance these appear to be equivalent, unless you disallow infinity to the Oracle, in which case I might need O(Log(n)) steps to establish inclusion in the Dedekind steps. So it might be a very slight Is that where the power comes from?
Oh boy, I hope that they missed a joke or misquoted.