4 ms·

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

by GTP 1y ago

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 1y ago
I think it’s using the series as a generating function, not as a calculation.
zmgsabst 1y ago
It’s only an approximation after finitely many terms — the full series limit is exact.
- seplox 1y 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.
  - ogogmad 1y 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.
  - GTP 1y ago
    Indeed, this is what confuses me. But also, could you please elaborate on the practical implications? Why does this work better in practice?
brendanyounger 1y 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 1y 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.

by GTP 1y ago

6gvONxR4sf7o 1y ago

I think it’s using the series as a generating function, not as a calculation.

zmgsabst 1y ago

It’s only an approximation after finitely many terms — the full series limit is exact.

seplox 1y 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.
- ogogmad 1y 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.
- GTP 1y ago
  Indeed, this is what confuses me. But also, could you please elaborate on the practical implications? Why does this work better in practice?

brendanyounger 1y 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 1y 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.