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.
cevi · 9h ago
The (actual) article has a fairly detailed literature review in the introduction, and makes it pretty clear that the main idea was sort-of known already if you squint - but it looks like nobody had put the whole theory together elegantly and advertised it properly. The fact that they couldn't find some natural slices of the hyper-Catalan numbers on OEIS supports that.
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]).
Do I understand correctly that this gives a formula to solve polynomic equations, which needs no square roots, but it also doesn‘t need complex numbers?
Qem · 1d ago
Is this something with potential to change mathematics as taught in schoolbooks, or just some sort of galactic algorithm[1] or research niche that will never spread beyond the expertise domain of half a dozen research groups across the whole world?
This will not be taught in textbooks. Wildberger works in a math world where "infinity" or even "really really big numbers" don't exist. There is nothing mathematically wrong about that world, but others (like me) do not find this problem context to be a useful framework to work in.
Qem · 22h ago
Interesting. So sort of a polar opposite to hyperreal numbers[1]. Instead of postulating de existence of numbers to the infinite and beyond, postulating there is no infinite.
> There is nothing mathematically wrong about that world
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.
ogogmad · 22h ago
His channel is sometimes the best way of learning certain topics in classical geometry.
michelpp · 22h ago
Agreed, his videos on Tropical Algebra are also very good!
ogogmad · 23h ago
Note that their solution is a formal power series solution, which will be divergent for some polynomial equations. Divergence is unavoidable because of the multiplicity of solutions.
So the mathematicians have found an algorithm for producing the roots of quintic and higher order polynomials in terms of power series? Normally one would find these roots using an iterative algorithm so this seems like a big deal.
ducttapecrown · 22h ago
The mathematician in question doesn't believe in infinity as an axiom, so he has a different understanding of what finding irrational roots means than others. From what I've gleaned from the other comments, it sounds like he does use an iterative algorithm to produce a power series solution.
Someone · 22h ago
> So, when we assume ∛7 "exists" in a formula, we're assuming that this infinite, never-ending decimal is somehow a complete object.
> 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?
meroes · 22h ago
It could be the reporting?
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.
comte7092 · 22h ago
> Irrational numbers, he says, rely on an imprecise concept of infinity and lead to logical problems in mathematics.
(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.
Someone · 18h ago
> Power series are 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.
mcphage · 22h ago
> Power series are heavily studied and understood.
The different sizes of infinity are also heavily studied and understood.
CoastalCoder · 22h ago
Personally, I find it hard to track how mathematicians reason about infinities.
They seem to make some very nuanced distinctions that don't make sense to me, but which satisfy them.
dhosek · 22h ago
A lot of it comes down to set theory and the idea of mappings between sets. So countably infinite sets are those for which there exists a bijective (one-to-one and onto) mapping f:ℤ→X which is an extension of the idea of counting a finite set by showing that there’s a mapping of the set {1, 2, …, n}→X for a set X of size n. From this, we’re able to find larger infinities like |ℝ| where there is no bijective map from ℤ to ℝ. David Foster Wallace (yes, that David Foster Wallace) wrote a reasonably readable book about infinity that could be a good starting point in exploring the subject.
I don't get this. If they are using power series, aren't they in practice still getting an approximation of the result anyway? Why would this be a better way of approximating the solution of a polynomial equation?
6gvONxR4sf7o · 1d ago
I think it’s using the series as a generating function, not as a calculation.
zmgsabst · 23h ago
It’s only an approximation after finitely many terms — the full series limit is exact.
seplox · 23h ago
I think the incongruity that the original commenter was pointing out is that Wildberger critiqued radicals by saying that they're imprecise approximations that rely on the problematic concept of infinity.
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.
GTP · 2h ago
Indeed, this is what confuses me. But also, could you please elaborate on the practical implications? Why does this work better in practice?
ogogmad · 23h ago
It seems that the authors are skeptical of real numbers (even computable ones???) while being perfectly comfortable with power series. I don't see how one of these can be acceptable and the other not. Sadly, their point of view seems incoherent.
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.
brendanyounger · 21h ago
Yes, you're still calculating an infinite decimal, no matter how you approach the problem.
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.
jfengel · 18h ago
I feel like physics is tending in the opposite direction: God made the complex numbers as an algebraically closed field, and provided a few groups to operate on. The rest we made up -- including the integers.
FilosofumRex · 20h ago
The Greeks fought √2 to the last Pythagoreans standing but √2 won, so best of luck to prof Wildberger.
knappa · 1d ago
> This is why, Prof. Wildberger says he "doesn't believe in irrational numbers."
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.
No comments yet
abetusk · 1d ago
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.
DietaryNonsense · 1d ago
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.
tromp · 1d ago
That would make more sense if you use the word computation instead of function.
zmgsabst · 23h ago
> 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.
wannabebarista · 1d ago
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):
Wildberger also wrote a book on geometry with nothing allowed but rationals. (Or something like that.)
zitterbewegung · 1d ago
In mathematics there are fringe philosophies but you still just need proofs that are kosher.
superidiot1932 · 1d ago
No, he really doesn't believe in them.
GTP · 1d ago
Never show him a square whose side is 1, or he wouldn't believe in its diagonal :D
greesil · 1d ago
If you watch it closely, you can see the hypotenuse converging.
BeetleB · 23h ago
I suspect that in reality it's transcendental numbers be doesn't believe in. Usually such folks believe in irrational but algebraic numbers.
feoren · 22h ago
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.
jostylr · 2h ago
> 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
impendia · 22h ago
I'm a mathematics professor, specializing in number theory.
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.
gweinberg · 22h ago
Ok, but given that the phys.org writeup is crap, we poor readers who have been kicked out of our universities with nothing but a degree to show for it have no clue what the paper actually says, and no way to find out. Except perhaps by spending money.
jostylr · 2h ago
The current top comment has a link to the paper and the pdf is freely available there. It says it is an open access article.
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.
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
I’ll get my coat.
[1] https://en.wikipedia.org/wiki/Galactic_algorithm
[1] https://en.m.wikipedia.org/wiki/Hyperreal_number
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?
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.
And cube roots aren’t? Worse, the Taylor series are power series, and the cube root function can be described as a power series.
The different sizes of infinity are also heavily studied and understood.
They seem to make some very nuanced distinctions that don't make sense to me, but which satisfy them.
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.
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.
No comments yet
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.
[0] https://en.wikipedia.org/wiki/Ultrafinitism
[1] https://www.youtube.com/@njwildberger/playlists
"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
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.