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.
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...
> 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.
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.
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.
knappa•11h ago
Oh boy, I hope that they missed a joke or misquoted.
rssoconnor•11h ago
I actually quite liked "Divine Proportions". As far as I know Wildberger is eccentric, but not exactly a crackpot.
superidiot1932•11h ago
GTP•10h ago
greesil•10h ago
abetusk•10h ago
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.
DietaryNonsense•10h ago
tromp•10h ago
zmgsabst•9h ago
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.
wannabebarista•10h ago
I came across his YouTube channel [1] years ago as a undergrad and became really confused about some ideas in logic as a result.
[0] https://en.wikipedia.org/wiki/Ultrafinitism
[1] https://www.youtube.com/@njwildberger/playlists
zitterbewegung•10h ago
BeetleB•10h ago
jrdres•9h ago
"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.)
feoren•9h ago
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.