
What Makes the Hardest Equations in Physics So Difficult? (2018) - c89X
https://www.quantamagazine.org/what-makes-the-hardest-equations-in-physics-so-difficult-20180116/
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bnjmn
One of my favorite takes on this topic:

"From... elementary theories we build up descriptions of more and more complex
systems. But in all these efforts we take for granted that we may use any
language we wish and as many [languages] as necessary. That is, we choose
whatever mathematical formalism is most useful and then interpret the symbols
and measurement operations in very highly developed natural language. To a
large degree, the simplicity of natural laws arises through the complexities
of the languages we use for their expression."

– H. H. Pattee

~~~
mr_overalls
I did an undergrad in physics decades ago, and it pleases me to no end to see
that someone else has made this observation. Especially when dealing with
quantum weirdness, the advice of professors to focus "follow the math" always
seem to gloss over the extensive interpretation (Copenhagen and otherwise)
that gave context and meaning to the math & experimental results.

~~~
guygurari
In my experience, the reason to “follow the math” is that it is the best and
perhaps only way to truly understand a physical theory, especially one as
strange as quantum mechanics. One can understand the math first, and then
develop an understanding of the context and meaning. But this abstract
understanding is anchored in math. This is important because when the meaning
gets too obscure, one can return to the math to resolve any confusion. The
other way around does not work.

~~~
hutzlibu
"This is important because when the meaning gets too obscure, one can return
to the math to resolve any confusion. The other way around does not work."

Note: my math is not very advanced, at least not good enough to understand
quantum mechanics

But why does it not work the other way? When the math tells me something very
wrong, can't the conctext and meaning show where the math modell is wrong?

As far as I understood, every physical modell is only a limited model of
reality, so they all have flaws. Meaning the math can be wrong when applied to
reality, which one could spot, with the understanding of reality?

~~~
aliceryhl
> When the math tells me something very wrong, can't the context and meaning
> show where the math model is wrong?

The point is that when learning a new part of physics, it is far more likely
that your intuition was wrong as the math was right, even if it's surprising.

Sure, the mathematical model is not a perfect model, but it can still be
_very_ good, so if you disagree with it, you're very likely to be wrong.

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RhysU
Big whorls have little whorls Which feed on their velocity, And little whorls
have lesser whorls And so on to viscosity.

\- Lewis Fry Richardson, 1922

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ocfnash
Note that finite-time singularities for the good old Newtonian n-body problem
have been known to exist for quite a while. (See for example [1].)

It's curious to think that a mathematical phenomenon like this can hint at new
physics.

[1] Saari, D. and Xia, Z., "Off to Infinity in Finite Time",
[https://www.ams.org/notices/199505/saari-2.pdf](https://www.ams.org/notices/199505/saari-2.pdf)

~~~
xelxebar
That paper is from 1993. Hard to believe that it took such a long time to find
a concrete pathology in a Newtonian system. Now the question is whether these
blowups only occur on initial condition sets of zero measure.

John Baez also has a nice, accessible series of articles called "Stuggles with
the Continuum" [0]. As a side product, it gives a nice perspective on the
development of modern physics as a series of attempts to fix these infinities
(only to create more subtle one).

[0]:[https://johncarlosbaez.wordpress.com/2016/09/08/struggles-
wi...](https://johncarlosbaez.wordpress.com/2016/09/08/struggles-with-the-
continuum-part-1/)

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davidw
Since these equations are something I've never studied and certainly don't
hear about on a day to day basis, the thing they bring to mind is the scene in
Cryptonomicon when Waterhouse doesn't get that a simple question on an army
test is actually a simple question and starts in on some complex math. It's an
entertaining read.

~~~
waterhouse
[The Navy] gave [Lawrence Waterhouse] an intelligence test. The first question
on the math part had to do with boats on a river: Port Smith is 100 miles
upstream of Port Jones. The river flows at 5 miles per hour. The boat goes
through water at 10 miles per hour. How long does it take to go from Port
Smith to Port Jones? How long to come back?

Lawrence immediately saw that it was a trick question. You would have to be
some kind of idiot to make the facile assumption that the current would add or
subtract 5 miles per hour to or from the speed of the boat. Clearly, 5 miles
per hour was nothing more than the _average_ speed. The current would be
faster in the middle of the river and slower at the banks. More complicated
variations could be expected at bends in the river. Basically it was a
question of hydrodynamics, which could be tackled using certain well-known
systems of differential equations. Lawrence dove into the problem, rapidly (or
so he thought) covering both sides of ten sheets of paper with calculations.
Along the way, he realized that one of his assumptions, in combination with
the simplified Navier-Stokes equations, had led him into an exploration of a
particularly interesting family of partial differential equations. Before he
knew it, he had proved a new theorem. If that didn’t prove his intelligence,
what would?

Then the time bell rang and the papers were collected. Lawrence managed to
hang onto his scratch paper. He took it back to his dorm, typed it up, and
mailed it to one of the more approachable math professors at Princeton, who
promptly arranged for it to be published in a Parisian mathematics journal.

Lawrence received two free, freshly printed copies of the journal a few months
later, in San Diego, California, during mail call on board a large ship called
the U.S.S. _Nevada_. The ship had a band, and the Navy had given Lawrence the
job of playing the glockenspiel in it, because their testing procedures had
proven that he was not intelligent enough to do anything else.

~~~
umvi
Reminds me of my (old) college professor who once gave a test where the first
question was about a sailboat with an on-board fan blowing into the sail.

It was meant to be an easy question to see if students understood Newton's
third law, but one student filled in the entire test with momentum
calculations showing that the boat would actually move forward at X velocity
because the sail would essentially redirect some % of the air backwards like a
reverse thruster (conservation of momentum). He left the rest of the test
blank because he blew the whole time limit on the first question.

The professor was perplexed when grading this student's exam and built a
"sailboat" out of a pinewood derby car with a dowel rod mast and aluminum foil
sail. He taped a handheld fan to the car, pointed into the sail, and indeed,
the car moved forward (this part he demoed to the class as he was telling the
story and just before he did it, he took a poll to see how many people thought
it would move forward, backward, or stay still - "stay still" won the poll)

The student reportedly got 100% on the test and the professor threw out that
question on future exams.

~~~
mkl
This was tested (and confirmed) on an episode of Mythbusters:
[https://mythresults.com/blow-your-own-sail](https://mythresults.com/blow-
your-own-sail)

Their boat went about 10% the speed of just pointing the fan backwards.

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mannykannot
Given that the equations involve continuous functions of some bulk properties
of a fluid, while the fluid itself is made up of discrete particles, is there
anything fundamentally or metaphysically problematical if these blow-ups are
unavoidable? Or is it about finding practical ways to avoid them when modeling
real-world scenarios?

~~~
lgeorget
I guess that it's just that it would be super convenient if the equation that
work for a fluid would work for any part of the fluid, however small it is.

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farisjarrah
As a non-mathematician, I found the Numberphile video on YouTube very helpful
for breaking down the basics of Navier-Stokes equations in a very easy to
comprehend way:
[https://www.youtube.com/watch?v=ERBVFcutl3M](https://www.youtube.com/watch?v=ERBVFcutl3M)

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quantum_state
From the perspective of trying to get to the actual physical behavior of such
systems, the exercise is purely theoretical since we know there are other
mechanisms at work to prevent the “ blow up” ...

~~~
tom-thistime
(I'm not a fluid flow person and this is probably not a very good comment.)

Yeah, that seems likely. But sometimes it's difficult to draw the line between
issues that are "purely theoretical" and issues that make a practical
difference. Conceivably at some scale the Navier-Stokes equations match up
with molecular dynamics simulations in a way that is illuminating. Then in
that situation maybe the problems with the N-S equations become nice
indicators of what's going on.

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Causality1
Because we're drilling down into finer and finer descriptions. If there was a
black hole in earth orbit where it could be closely observed I'm sure we'd
find small details of its behavior and the behavior of dust crossing its event
horizon that require difficult math to explain.

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Koshkin
I wonder what a good textbook on theoretical physics would look like if it
assumed all the math involved as a prerequisite. (Say, a 10-volume set
"compressed" into, what, 300 pages still containing a _complete_ treatment of
the subject?)

~~~
yummypaint
Jackson's electrodynamics is probably the closest thing to this I've seen. He
often mentions offhand that it's trivial to show one thing or another, but
when you actually do the derivation it's many pages.

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peteradio
Doesn't this point to some renormalization strategy being necessary? Or at
least some correction to how renormalization is being applied if its leading
to subspace infinities. Not sure how mathematicians view renormalization when
you are trying to ask questions about indefinitely far distances into the
future but it seems like there needs to be some necessary coarseness which
suppresses the infinities, not sure why those terms wouldn't naturally arise.

------
HenryKissinger
"When I meet God, I am going to ask him two questions: Why relativity? And why
turbulence? I really believe he will have an answer for the first."

~~~
amelius
Meaning that the latter is an emergent property, while the former is not?

~~~
pretzell
Possibly, but I either interpreted, or heard this somewhere before, that
turbulence is way harder to make sense of and predict (?). Not a physicist,
but essentially that both are nuts, but that turbulence is even nuttier

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aSplash0fDerp
From a laymens POV, it seems like half of the equation would represent the
energy used in the bonds of each molecule representing smoothness (almost
moving as a solid), with the other half laying out a dynamic list of finite
variables that disrupt or exceed the energy forming the bonds.

Its almost like they would have to define a theoretic limit similar to
9.808175174 m/s^2 to limit the finite possibilities of outside forces acting
on the molecules before the molecules themselves become the outside force,
with those that exceed the limits be classified with a custom equation, and
those below the threshold fitting nicely in a bow wrapped package.

And just to add more SWAG, is it going to be easier to produce these
finding/understandings in space, where theoretic limits are more well defined
and we'll be forced to use custom equations within our atmosphere looking for
an answer from a perspective that is environmentally more complex?

Its all greek to me...

[https://en.m.wikipedia.org/wiki/Navier%E2%80%93Stokes_equati...](https://en.m.wikipedia.org/wiki/Navier%E2%80%93Stokes_equations)

~~~
aSplash0fDerp
OK. I was thinking more of a Newtons apple moment in space, where someone with
a syringe and fluid may make an elegant observation on blow-up or collapse.

Eddys in a river with more eddys creating eddys is a dizzying theory though.

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dang
Discussed at the time:
[https://news.ycombinator.com/item?id=16159744](https://news.ycombinator.com/item?id=16159744)

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ncmncm
If you think the Navier-Stokes equations are hard, try the version that plasma
physicists have to use, instead.

Rather than confront those, cosmologists have chosen to pretend that, while
every single thing they can see is plasma (excepting, uniquely, planets), none
of it _does_ anything plasma-ish.

Since plasma physics is scale-invariant, freaky phenomena seen in labs should
be playing out at stellar, galactic, and super-cluster scale. If they don't,
it needs explanation why not.

Huge props to solar physicists, who confront plasma physics, face-to-face,
daily.

~~~
xelxebar
I picked up a copy of Chandrasekar's "Hydrodynamic and Magnetodynamic
Stability" a while back and started working through it. The basics are
relatively approachable, but dang the differential equations can get gnarly.
It feels like a large portion of the art is in asking the right question that
can aviod toppling into a mess of non-linearities.

Really good book, as far as I got through it. The beginning has a nice
derivation of Navier-Stokes, which was mew to me.

~~~
ncmncm
The mess of non-linearities is where the action is, at least in nature.
Designing systems to stay out of that mess is, as you say, art.

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mkagenius
Is it like purely random event - akin to butterfly effect and its mostly
indeterminate ahead of time? (I am not an expert)

~~~
mrob
"Butterfly effect" refers to chaotic systems, i.e. those where the outcome is
highly sensitive to the starting position. They are deterministic, not random,
but they are hard to predict because this sensitivity means any small errors
in simulation can give completely wrong results.

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bawana
Physicists who like to preserve their jobs. If everyone taught like grant
Sanderson there would be too much understanding.

