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Comment by dan-robertson | Hacker News Reader

I recall learning the fundamental theorem of applied maths. The approximate phrasing is “in applied maths, if it looks right then it is” and a more precise phrasing is that “in applied maths, all reasonable series converge, all functions are continuous, differentiable and smooth almost everywhere, all limits (and integrals) exist, all sums or integrals commute, all Taylor series are good approximations, all of the silly well-behaved ness conditions of the theorems you want apply, and every equation has one solution”.

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> every equation has one solution

So when you see the sun rising, cherish the moment, as it has never happened before and will never happen again.

In a philosophical manner of speaking, that's true. You never step into the same river twice.

Indeed. But the scientific approach is different -- it's all about finding patterns that do repeat and making predictions.

The original comment's "theorem" is funny, but unless you are doing a homework problem, you better have a good intuition for when to take a second look at a seemingly simple situation.

> the scientific approach is different -- it's all about finding patterns that do repeat and making predictions

Sure, but if you're scientifically looking to answer "where and when will the sun rise" you're only going to collect enough variables to answer that question, and within an acceptable margin of error, right? If you can measure with greater accuracy and collect more data, then you would probably realize that every sunrise is not strictly identical.

We're splitting hairs at this point but I wonder if the comment I replied to that stated that "it has never happened before and will never happen again" can't be argued to be actually true. In a philosophical sense it's more obvious, but in a scientific sense, the more precision you get in your analysis of a sunrise, the more data you would get that differentiates it from other sunrises, no? After all, our solar system isn't closed and constant and there are minor changes not only in smaller factors like weather on Earth, but also larger factors like the orbit of the earth and the drift of the different planetary bodies.

Well there are 3 fragments related to this, one likely fake. It is a way to wake you up in his crazy paradox.

There are books and articles and may be one can start with the key phrase What is same river

And the hidden issue is you. If you think it is unique every time frame ... what is same river and same you. Can One same you even step into the River

And if step into it and swim in parallel is the molecules mostly the same ...

Back to What is river? If it is keep on changing as any river that is not changing and moving is not a river, the sameness if you accept then you can step into the same river as many time as you like. But that is wrong per the statement. Hence what is “same river”.

The funny thing is, besides this one you picked out, pretty much all the rest of these either are continuity assumptions or follow from continuity. I think that's because physicists tend to dismiss discontinuities and singularities as "unphysical," but it does make good intuitive sense.

I suspect that "all equations have one solution" is a lot easier to satisfy when all functions are continuous, or otherwise nice, as well.

That's mostly true -- however, physicists devote a lot of time to studying phase transitions and characterizing the kinds of discontinuities that are possible etc in that context.

Ah, yes, I forgot about phase transitions. Those are actually interesting discontinuities. It seems to me the uninteresting ones are the ones that have roughly the same behavior all around them (e.g. poles of complex functions are not that interesting).

Would you say that's an accurate characterization?

sorry don't have much more insight here. I think the goal is to generally make any approximation that isn't relevant to the physical behavior one is trying to model. if a pole sitting somewhere isn't related to the phenomenon you're trying to explore, ignore it!

Great point. There are a ton of good models that break down in certain regions (e.g. quantum mechanics and general relativity at large and small physical scales, respectively).

The idea isn’t so much that the sun rises only once, but rather that if you were to solve a differential equation for how the sun’s position in the sky varies over time, there would only be one possible solution (provided that you had sufficient initial conditions). The idea is that your equation is supposed to be a good model for reality and in non-quantum physics, only one thing can happen so either your equation has one solution or it is a bad model for reality.

Thank you, this helps. To my taste, jokes get better when they make more sense. :)

Do you remember where you read it? It's a nice and funny formulation but a quick attempt at googling didn't bring this up for me.

i wonder if it's a folk theorem kind of thing? i heard something similar from one of my physics professors back in undergrad

It’s basically just a joke. I got it from a professor of mine. But like all good jokes, there’s some element of truth in it: most problems in applied maths relate to something physical, and in (non-quantum) physics, something happens and only one thing happens. Therefore your problems are either bad models of reality or they have exactly one solution. It often turns out that even if your methods aren’t technically correct or properly justified, you can still reach that solution.

It is reminiscent of what Euler called “the universality of analysis” (note that over the history of mathematics, analysis has been used to refer to just about everything that isn’t geometry. In this case it means roughly the sort of mathematics a physicist or engineer might know well, so non-abstract algebra with integration and differentiation). He could do crazy things like solve combinatorics problems using infinite expansions of series in ways that didn’t really make sense (I think this example is well known but I can’t think what it is) but get to the right answer. A non-Eulerian example would be Dirac delta functions which aren’t functions at all and can be tricky to formalise but also just magically work within the framework of abstract symbol manipulation that is used in applied maths without significant modification (except when e.g. some fundamental property of your model of the universe depends on the value that the Heaviside step function takes at 0)

> I think this example is well known but I can’t think what it is

Could be the basel problem though he probably used this trick many times in his prolific life:

https://en.wikipedia.org/wiki/Basel_problem#Euler's_approach

> over the history of mathematics, analysis has been used to refer to just about everything that isn’t geometry

Actually, at the start (2500 years ago) the word “analysis” in mathematics was only about geometry.

See e.g. https://www.jstor.org/stable/2250061

> in applied maths ... all functions are continuous

I guess that's related to the fact that all computable functions from real numbers to real numbers are continuous. It seems reasonable that the laws of physics should be computable to a large extent otherwise we'd have been able to build a hypercomputer by now.

A step function is not computable?

If I gave you 0.0000000... as input, you would have to scan forever to find out whether I put in a nonzero digit. Scanning to a partial depth would not result in partial accuracy, because the step function is discontinuous.

In fact, yes! This is counterintuitive if you think of the floats as a model for the reals, but in fact the floats give the wrong intuition, being (a finite subset of) rationals. [1] details this and refers to two related articles (particularly [2], which I find a lighter read). This fact is also remarked on in [3] (by example of the signum function being uncomputable).

For the particular example of a step function, consider the function f which is 1 on the positive reals (>=0) and 0 otherwise. What representation of the real numbers would you choose for the computation of f(0) to terminate in a finite amount of time? It cannot be the binary representation, as those are infinite. Your algorithm, terminating in a finite amount of time, will have checked only a finite number N of binary digits of the argument, and so I can choose x = 0.0{..N..}01 and obtain f(0) = f(x) by this algorithm, which is incorrect. You can choose the Cauchy sequence representation, or the Dedekind cut one, but this problem will persist (and [1] proves this in general).

You can "cheat" by saying that the leading two bits will store the sign of the number: 00 for 0, 01 for positive numbers and 10 for negative ones. But then suddenly arithmetic operations are not computable (see comments on [2])!

[1] https://lukepalmer.wordpress.com/2008/08/11/all-functions-ar...

[2] http://math.andrej.com/2006/03/27/sometimes-all-functions-ar...

[3] Vereshchagin, N., Shen, A. Computable Functions. https://bookstore.ams.org/stml-19/

Sounds like many of my physics teachers.

The more precise statement of it is that physics and much of applied math is done in the topos of synthetic differential geometry, where by removing the law of excluded middle from the logic, you can basically make all of this true.

Well, except the Taylor series part. Numerical analysis is perfectly happy to take you to the cleaners even in ultrafinitist settings.

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