What Makes the Hardest Equations in Physics So Difficult? • Hacker News Board

Re: What Makes the Hardest Equations in Physics So Difficult?

#71

Post by carterehsmith » Thu, Jan 18, 2018, 3:53 AM UTC

kazinator wrote: post #69

carterehsmith wrote: post #68

Earlier quoted context omitted.

>> The pages you're referencing are all crackpottery. So... I can trust Ramanujan and Abel, that published results on these things, and Terrence Tao that has a nice writeup, and a bunch of others, or I can trust HN user "Kazinator", who... published some middle-school algebra "proof" on HN. Guess who is crackpot here. >> https://en.wikipedia.org/wiki/Ramanujan_summation

I would say, it is he who cannot separate "summation" from "Ramanjuan summation" due to the overlapping notation.

There are different definitions of "summation". Also different definitions of "integration".

I can see how overlapping notation can be confusing for laymen (me included).

But that does not mean that people that came up with those are crackpots, just because they reused the "=" symbol.

Re: What Makes the Hardest Equations in Physics So Difficult?

#72

Post by improbable22 » Thu, Jan 18, 2018, 1:35 PM UTC

oldandtired wrote: post #64

improbable22 wrote: post #57

Earlier quoted context omitted.

"much smaller today" meaning literally smaller: Navier didn't know about atoms, 10^-10 m, but now we know about things at 10^-18 m or so. Sure, our awareness of how much we don't know has grown over time. The point I was trying to make is that theoretical models (like N-S) not only don't have to be perfect to be useful, but more, are useful precisely because they are not complete. By ignoring irrelevant details we ge…

I don't disagree with the idea that a model doesn't have to be perfect to be useful. But I do disagree with the prevalent idea that such theories are the "truth" when they are not. Understanding that a model or theory is useful even when we ignore certain aspects of reality is quite different to the often displayed belief that a specific theory is "gospel" even in the face of anomalies and discrepancies of the real w…

It's certainly possible to find such views in theoretical physics. But typically not from very serious people.

In my understanding the big shift was the understanding of renormalisation, Kadanoff and Wilson, around 1970. This took airy ideas about useful approximation and turned them into serious tools, which are both useful for everyday things and illuminating about why any of it works.

Re: What Makes the Hardest Equations in Physics So Difficult?

#73

Post by [deleted] » Thu, Jan 18, 2018, 6:51 PM UTC

carterehsmith wrote: post #71

kazinator wrote: post #69

Earlier quoted context omitted.

I would say, it is he who cannot separate "summation" from "Ramanjuan summation" due to the overlapping notation.

There are different definitions of "summation". Also different definitions of "integration". I can see how overlapping notation can be confusing for laymen (me included). But that does not mean that people that came up with those are crackpots, just because they reused the "=" symbol.

[deleted]

Re: What Makes the Hardest Equations in Physics So Difficult?

#74

Post by kazinator » Thu, Jan 18, 2018, 8:23 PM UTC

carterehsmith wrote: post #71

kazinator wrote: post #69

Earlier quoted context omitted.

I would say, it is he who cannot separate "summation" from "Ramanjuan summation" due to the overlapping notation.

There are different definitions of "summation". Also different definitions of "integration". I can see how overlapping notation can be confusing for laymen (me included). But that does not mean that people that came up with those are crackpots, just because they reused the "=" symbol.

Indeed, and "summation" is not the same as "addition".

Unfortunately, the syntax chosen for representing the sequence is that of an additive series, where a binary + operator is interposed between terms.

The semantics being shown does not seem to follow from a redefinition of that operator per se as stand-alone binary operator.

You really want to show this as, say

  fun([1 2 3 4 ... ])

a function applied to a vector. Why the algebraic rules seem to work is because fun is a linear operator; i.e.

  n fun([x0 x1 ... ]) = fun([nx0 nx1 ...])

and

  fun(v0 + v1) = fun(v0) + fun(v1)

We can ply these rules back to the original 1 + 2 + 3 ... notation and then they look like algebra.

Basically, none of this means that the natural numbers add to -1/12; only that the sequence of natural numbers can be fed into some decimating calculation which ends up with -1/12.

Well, no kidding; the sequence of natural numbers can be fed into a decimating calculation which converges on any value you want, if you can freely choose the decimating calculation, and that calculation can be chosen to be linear operator.

Note that we don't have:

   fun([x0 x1 x3 ... x42 ...]) = fun([x3 x42 x0 ... x1 ...])

which would be required to hold if this were addition. We can't change the positions of the terms. Why? Because they correspond to different powers in a power series.

Re: What Makes the Hardest Equations in Physics So Difficult?

#75

Post by andrewflnr » Sat, Jan 20, 2018, 12:21 AM UTC

adrianratnapala wrote: post #43

andrewflnr wrote: post #32

Earlier quoted context omitted.

I guess if you're looking for pure mathematical abstraction, Navier-Stokes doesn't seem like the most interesting without the physics connection. But that's just me.

I suppose what is interesting is the very difficulties we have deciding if there are smooth solutions. It would be interesting if there are whole class of PDEs which are difficult in this way. And if there isn't , then what is special about the NS? That too, is interesting.

Fair point, there.