Mathematician solves algebra's oldest problem using intriguing number sequences
phys.org
phys.org
This is about solving polynomial equations using Lagrange inversion. This method, as one might have guessed, is due to... Lagrange. See https://www.numdam.org/item/RHM_1998__4_1_73_0.pdf for a historical survey. What Wildberger is suggesting is a new(?) formula for the coefficients of the resulting power series. Whether it is new I am not sure about -- Wildberger has been working in isolation from others in the field, which is already full of rediscoveries. Note that the method does not compete with solutions in radicals (as in the quadratic formula, Tartaglia, Cardano, del Ferro, Galois) because it produces infinite sums even when applied to quadratic equations.
Phys.org has gotten no part of the story correct.
I’ll get my coat.
The proof they give that the hyper-Catalan series solves the Lagrange inversion problem is very good from a pedagogical point of view - I don't think I'll ever be able to forget it now that I've seen it. The only thing this paper is missing is a direct, self-contained combinatorial proof of the factorial-ratio formula they gave for the hyper-Catalan numbers - digging though the chain of equivalences proved in the references eventually got too annoying for me and I had to sit down and find a proof myself (there is a simple variation of the usual argument for counting Dyck paths [1] that does the trick).
Another thing to note is that the power series solution isn't just "a power series" - it's a hypergeometric series. There are lots of computation techniques that apply to hypergeometric series which don't apply to power series in general (see [2]).
[1] https://jlmartin.ku.edu/courses/math724-F13/count-dyck.pdf (for instance) [2] https://sites.math.rutgers.edu/~zeilberg/AeqB.pdf
Thank you for acknowledging this. Every time Norm's work comes up on HN there is a subcurrent of comments about how his philosophy of math is wrong or dumb whose are arguments can be summed up as "Lol no infinity wtf".
Do I personally agree with his philosophy? No. But I still watched all his videos because they are entertaining, thoughtful, and his is rigorous in his definitions and examples.
Journal article here: https://www.tandfonline.com/doi/full/10.1080/00029890.2025.2...
So setting aside the new method's practical implications, replacing an infinitely accurate approximation with a different infinitely accurate approximation doesn't feel any different.
Maybe it's a gut reaction because power series can seem so "nice" to them in their experience.
Maybe if someone explained Computable Topology to them, then they could be more accepting? But if their judgement comes from the gut, instead of intellectual integrity or reason, then I'm not sure it would be worth trying it.
What Wildberger is suggesting is that, rather than taking an nth root (solution to x^n = A where A is a fraction) as a "fundamental" operation, what if we took power series with "hyper-Catalan" coefficients as fundamental operators? (This is where I get a bit fuzzy because I haven't read and understood his work.)
Galois proved that you can't have a general algorithm for solving polynomials of degree >= 5 if all you can use are +,-,*,/, and nth roots. But what if you can use a different operation besides nth roots? That's what Wildberger is proposing and apparently it works for higher degrees.
Stepping back a bit, this is very much in line with Kronecker's notion that God made the natural numbers and all else is man's handiwork. There's no avoiding infinite series for computing non-rational roots of equations, but it is possible to choose series that are easier to work with.
> This is why, Prof. Wildberger says he "doesn't believe in irrational numbers."
OK. I’m with you.
> His new method to solve polynomials also avoids radicals and irrational numbers, relying instead on special extensions of polynomials called "power series," which can have an infinite number of terms with the powers of x.
So, he believes this other infinite, never ending thing exists? Isn’t that a tad inconsistent?
They seem to make some very nuanced distinctions that don't make sense to me, but which satisfy them.
Wildberger believes you can always get more precise calculations with more terms, but there is never a “completed” infinite object.
He had extensive videos on YouTube but something happened. Not sure if he ever got them all back.
(Emphasis added)
As a mathetician, he doesn’t have a problem with the concept of infinity, it’s about the structure of the definition and how you are able to reason about it. Power series are heavily studied and understood.
The different sizes of infinity are also heavily studied and understood.
And cube roots aren’t? Worse, the Taylor series are power series, and the cube root function can be described as a power series.
Oh boy, I hope that they missed a joke or misquoted.
I actually quite liked "Divine Proportions". As far as I know Wildberger is eccentric, but not exactly a crackpot.
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.
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.
I came across his YouTube channel [1] years ago as a undergrad and became really confused about some ideas in logic as a result.
"Dirty Rotten Infinite Sets and the Foundations of Math" http://www.goodmath.org/blog/2007/10/15/dirty-rotten-infinit...
Wildberger also wrote a book on geometry with nothing allowed but rationals. (Or something like that.)
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.
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 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?
As others have noted (the author apparently "doesn't believe in irrational numbers"), this press release is laughable and utterly absurd. Wildberger did not "solve algebra's oldest problem", or anything remotely close to that.
I checked out phys.org -- I assumed this would be the webpage of some prominent national society or something -- but it turns out to be some randos that have a publishing outfit.
I did, however, look up the original paper. Unfortunately it seems to be paywalled, although I have access through my university.
The actual paper seems to for the most part be sober, legitimate, and potentially interesting (albeit on the same scale that many many other published math papers are interesting). Except for a bit of hyperbole in the introduction, it doesn't traffic in exaggerated claims. Seems to be a legitimate effort, somewhat off the beaten track.