Mathematicians Transcend Geometric Theory of Motion | Hacker News Reader

Mathematicians Transcend Geometric Theory of Motion

quantamagazine.org

119 pointstheafh57 comments

Well, this is how I learned Manolescu left UCLA and is now at Stanford

Also, yet another article that makes me rethink my choice to take Homological Algebra with no real algebraic topology coursework. Merkurjev was teaching so at least I have all his notes. It kind of forced me to give up any concrete basis and just handle abstract tools.

If you already understand homological algebra, Peter May’s AT book might get you where you want to be quickly. It’s free on his website.

good advice

I urge you to read Hatcher's book (or better, tom Dieck's book published by EMS) immediately. I can't even imagine learning homological algebra without a bunch of concrete topological examples to compute with. That sounds confusing.

Homological algebra without algebraic topology or geometry sounds like the driest possible exercise in pointlessness.

> The planet’s position and momentum can be described by six numbers, three for each property. If you represent each of the different configurations of the planet’s position and momentum as a point with six coordinates, you’ll create the phase space of the system. In this case, it has the shape of flat six-dimensional space. The motion of a single planet can be represented as a line weaving through this space.

Sounds like ephemeris?

https://en.m.wikipedia.org/wiki/Ephemeris

There are a decent amount of symmetries between old ideas and the newest accepted theories. The good ones were typically drawing from the same well, just not able to get to the source.

In this case it's a description of the 6-dimensional coordinates used in Lagrangian mechanics, which dates to the late 1700s.

https://en.wikipedia.org/wiki/Lagrangian_mechanics https://en.wikipedia.org/wiki/Position_and_momentum_space

The symplectic geometry from the article is an abstract version of ideas from Hamiltonian mechanics, which is from the early 1800s.

https://en.wikipedia.org/wiki/Hamiltonian_mechanics

F=ma and you need initial conditions on position and velocity to pin down a trajectory… you can always apply a change of coordinates (e.g. ephemeris) to make computation easier in some specific case.

I don't know much about entanglement, but does this now explain why it happens? If we think of the universe eventually returning to some kind of steady state (i'm not sure precisely what i mean by steady state), and if there are certain limited ways for two 'entangled' particles to 'reunify' (again, i don't know much but i'm assuming in some final state everything would cancel out), then wouldn't it make sense that we can only see these particles in specific restricted states?

This has nothing whatsoever to do with entanglement.

Ah, Hm. So you did get what i was suggesting, though? The idea was that maybe you could think of particles exactly the same way as you do orbiting planets that eventually return to an original configuration. I am saying you could (possibly) think of entangled particles as kind of like billiard balls. You wouldn't think about bouncing balls, though, you would think about particles and the guaranteed force between them. The thought is that if the universe is just a wave system, then if you take all nondeterminism out of it, it's just gonna loop forever right? I mean if we think about gravity and basic bodies, like a universe of just two bodies, it's just gonna loop i think. The particles will eventually end up where they started, just like a pool table.

So maybe if two particles are entangled, they phased together in some way at one time and must phase together in that way until the end of time, unless you bump it wrong again.

I dunno.

Yeah I'm pretty sure I understand what you're talking about and as far as I can tell it doesn't really have anything to do with quantum mechanics at all.

Entanglement is "just" the result of the fact that the space of possible states of a combined quantum system isn't the Cartesian product of the state-spaces of the subsystems that make it up.

If I have two classical systems, one of which has state-space {A, B} (i.e. the first system is either in state A or state B) and the other has state-space {0,1,2} (i.e. the second system is either in state 0, 1 or 2) then the system I get from combining them has 6 possible states {(A,0), (A,1), (A,2),(B,0), (B,1), (B,2)}.

Thats not how quantum state-spaces combine, they combine with the tensor product, rather than the Cartesian product, so the state-space of the combined system is much richer than what you'd get if you try to use the classical "Cartesian product" rule to combine them.

I think the gulf between research math and teaching math is so wide that it may as well be a different subject altogether. The vast majority of mathematicians are teachers, not researchers. If all mathematicians have at the very minimum PHDs, then is the difference in ability so great?

I wouldn't consider someone a mathematician unless they were a researcher. Otherwise what does a mathematician do? Many people have undergraduate or advanced degrees in math, but most of them don't "do math" for a living (i.e. research).

Same for most science - physics, chemistry, etc… I have a Physics PhD and I still wouldn’t consider myself a physicist as applying the physics skills is not what I do day to day.

with a bachelor's in Biochem, in my head I say I'm formally trained in biochemistry. But iut loud I just say "my bachelor's was in biochem"

often as large as a draft prospect and nba player with years of experience

I was ABD in math at a top 25 program when I went to a talk given by a graduate student from Berkeley. He was in my area and we had the same length of them spent on the subject. As close to equals in terms of experience and area of study as one can get. He was far better than me and I knew that I’d never understand the subject as well as him. Shortly after I quit the Ph.D. program. I realized I would never do anything worthwhile in the field. There is a large variation in talent within the community of professional mathematicians. Outliers amongst outliers.

In trained in MMA for a number of years and sparred against some local fighters. They were better than me but I could get some hits in. I could cause them to expend some effort. Once I sparred with a low level UFC fighter. He thoroughly destroyed me. It was like I was 5 years old. Outliers amongst outliers.

It's not so hard to get a PhD in math, in the sense that if you're willing to attend a low-ranked program and have at least a moderate affinity for mathematics, you could probably do it. It takes a lot of time to learn all the prerequisites, but that's why the undergrad degree is 4 years and the PhD typically 5+. Then you just find a suitable advisor, ask to be handed a dissertation problem and some ideas for the solution, write down that solution, and graduate. Anyone who's been doing math research for a few years has a collection of problems they know how to solve but haven't written up for various reasons, which they can give for this purpose. (Usually: the question is too boring or simple, no one cares, and there are bigger impact things to do instead).

Doing research that meaningfully advances mathematics, as opposed to being make-work in service of getting a degree? Much harder.

The higher up you go the tower of abstractions the more you notice difference in skill. A person with even just a slightly worse foundation will soon find that their tower is unstable and it is hard to add more on top. I'd say that mathematics is currently the field with the tallest such towers, it is just abstractions on top of abstractions over and over and you have to understand every step thoroughly, so difference in skill should be about as high as it can get in any mental field.

The math most STE (not M) people learn in college was mostly well understood 200 years ago. A lot of things happened in those 200 years just adding more and more on top, getting to the forefront today requires you to build your understanding extremely high.

What if mathematical theorems are described by executable code instead of just symbols then ? Currently it seems impossible for reader to verify all theorems in a math paper.

There are efforts in that direction: https://leanprover.github.io/theorem_proving_in_lean4/

A major advancement happened this year where a recent paper from a fields medalist on an incredibly abstract topic was formally proved in Lean. The theorem was that Scholze’s new condensed mathematics was logically consistent with real functional analysis.

https://www.quantamagazine.org/lean-computer-program-confirm...

Currently transforming a new piece of maths from its "standard" form (i.e. what working mathematicians actually use) to something a computer can understand is a big task that usually takes a team of experts something of the order of years.

Maths is really hard and proofs require a tonne of steps. For this reason mathematicians have to be comfortable jumping over the standard pedestrian intermediate steps in proofs and just focusing on the important stuff. This is necessary because including all the details would obscure the important stuff (imagine directions for driving somewhere with steps like "now walk up to the car", "now click the opener", "now open the car door", "now sit down in the drivers seat").

Computers (currently) are way too dumb to skip these steps so you have to walk them through it.

And yet… the gold standard for demonstrating that a proof has no “bugs” is to execute it within a proof assistant. The late Fields medalist Vladimir Voevodsky gave a memorable talk on this subject, as part of his motivation for developing Homotopy type theory as a foundational language for mathematics that is very close to the syntax used in proof assistants.

Is it a consensus among mathematicians that is the gold standard? Voevodsky for sure was an advocate but I’m not aware of there being a consensus on the subject.

From my experience as a mathematician, I would say that only a very small minority of mathematicians would accept computer verification of proofs, let alone view it as a gold standard.

Thats a bit like saying the gold standard for programming is formal verification. Sure in an ideal world maybe every piece of code would be formally verified, but its not realistic and (for the vast majority of use cases) wouldn't be overly useful. A much more realistic "gold standard" for maths is to be peer reviewed and read and understood by a sufficient number of working mathematicians.

This question seems to always come up on Hacker News - figuring out how to even formulate the statement of theorems in theorem provers such as Lean is a massive research undertaking in itself, requiring much creativity and novel work. Let alone figuring out how to formulate the proofs. I’d recommend you have a look at some of the talks that Buzzard has given on this to understand the complications (both technical and social) and progress that has been made so far.

See this. It uses first-order logic.

http://www.mizar.org/library/

Logicians study this I think in computability theory. It's a hard problem.

https://en.m.wikipedia.org/wiki/Computably_enumerable.

Tell them what you're going to tell them.

Tell them.

Tell them what you told them.

The first and last parts are missing from this article. You have to take a gestalt approach and scan the whole thing to get an idea of what it's about.

It's an article and not a power-point slide. There's nothing wrong with the format.

The outline you gave is more appropriate for slide-deck talks where the audience is captive and they're apt to be unconscious during the middle.

It's also the standard for scientific papers.

That depends on the field, especially in areas of pure maths I see this "slide-deck" style way less.

there isn’t a universal standard for scientific papers.

more’s the pity.

I think science has the abstract (tell em what you'll tell em) then the body and a discussion which is more "OK, now that I got you're attention, here is what I actually think but can't prove".

Right, but it's also not a scientific paper.

Science is a very broad term, and writing style varies between research groups, let alone between fields.

The reason I'd like part 1 (tell them what you're going to tell them) is it helps me answer the question, "Is this relevant and interesting enough to me to spend the time reading it?"

In a sense, I need part 1 because I'm not a captive audience. If this were (say) a lecture in a college class, then it's a foregone conclusion that I'm using the time, so I might as well pay attention.

But since it's a web article, I have the choice to keep reading or close the browser tab. I'd prefer to be able to make an informed choice.

This is what I was getting at. I had to scan the whole thing to find out if it was something I wanted to read about. Turns out, it wasn’t.

Sometimes the journey is its own destination.

I like the way quanta writes their math articles and I hope they never go towards a stilted formulaic approach to writing. :shrug:

I despise this structure and the kind of writing and presentations it tends to produce. The absolute worst, which I see all too often in presentations, is its fractal form:

    1. Tell them that you are going to:
        1. Tell them what you are going to tell them.
        2. Tell them.
        3. Tell you what you told them.
    1. Tell them what you are going to tell them.
    2. Tell them that now you will tell them what you will tell them.
    3. Tell them.
    4. Tell them that you are done telling them what you will tell them.
    5. Tell them that you will tell them what you told them.
    6. Tell them what you told them.
    7. Tell them that you are done telling them what you told them.

A structure I like much better which permits the above structure but allows other variations is:

1. Explain how to tell if this is worth their time.

2. Tell them.

3. If there was a lot, suggest what's worth remembering.

The focus here is on how it benefits the audience and not on some arbitrary structural form.

It seems you're suggesting the same strategy, and just don't like when it's done poorly.

Not at all. There are many ways to let an audience decide whether the rest of the material is worth their time. Summarizing to give them a preview is only one (and often the least interesting one).

Other ways:

* Describe a problem that the audience also has, so that they understand that you are aligned with them.

* Tell an engaging anecdote so that they expect it will be a rewarding experience. (The idea that a piece of writing should entertain, inform, or persuade and that those are mutually exclusive is another canard that I find to be completely toxic and antithetical to good writing. Good writing should entertain, inform, and persuade.)

* Telegraph that the time investment will be smaller by getting started and making the overall thing shorter.

* Describe previous failures to solve a problem.

* Give them an interesting insight right off the bat, which implies there may be more to come.

* Tell them something personal which conveys whether you are likely to be a person with interesting things to say.

* Throw out a detailed, hard to acquire fact, which implies that you have other hard-won knowledge.

Note that what all of these have in common is that the intro material is unique and is not simply a pre-statement of information they will encounter lately.

Also, the fact that I made step 3 optional is significant. Most writing and presentations don't need a summary and a summary will often detract. If you want to stick in the audience's memory, what you really need is a climax, and "here's what I just said, said again" is about the most anti-climactic ending you can imagine.

It's also a trade-off between length of presentation or article in that case and amount of interesting content. Also I think Dynamical Systems are quite a mixture of theory and practical applications, so it makes sense to mix that.

The opening is a value proposition at its core. Attention is a commodity to be earned. Recognizing this is the key to good openings.

It's unfortunate that they omit Floer's suicide. Nobody wants to talk about the negative effects of a career in mathematics.

You're saying that an article about "Mathematicians Transcend Geometric Theory of Motion" should also be about "the negative effects of a career in mathematics"?

Genius and dead at 34 - if you read between the lines, it said it all.

Are mathematicians statistically more likely to commit suicide?

I don’t think the arrow of causation flows from career to cause of death. More likely both have a similar root cause.

>Arnold predicted that every phase space of a certain type contains a minimum number of configurations in which the system it describes returns to where it started.

is this article talking about Ergodicity without mentioning Ergodicity? https://en.wikipedia.org/wiki/Ergodicity

All dynamical systems have a phase space and ones with the return property don’t necessarily have to be ergodic.

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