What Makes for 'Good' Mathematics?

https://www.quantamagazine.org/what-makes-for-good-mathematics-20240201/

by digital55 • 2 years ago

46 59 2 years ago

j2kun4• 2 years ago

I have sort of a contrarian take on this, at least by traditional academic standards. I think applicability is extremely undervalued in modern mathematics. Tao references algebraic geometry in the interview, but the people who do algebraic geometry in applied settings ("applied" is also being quite generous here, in my opinion) are doing math that is basically unrelated to algebraic geometry as practiced by theoretical mathematicians. Even in connections to computer science, the people studying algebraic circuit complexity-which a priori should benefit a lot from algebraic geometry-basically can't use any modern algebraic geometry theory because (to my understanding) the theoreiticians choose to work in a setting that is so elegant it rules out interesting algebraic circuits. And circuit complexity is still quite far removed from applications the way I think of it.

I think the best and most beautiful mathematics is driven directly by applications, and made better for the pressure to apply to a real problem (not a problem that just mathematicians care about).

A quote from Milind Tambe's 2018 AAAI keynote talk sums up why:

Deploying the algorithms in the field often gives us new insights into what might be wrong with our models and provides us new research directions.

If you've never heard of Tambe's work, you should really check it out. He literally sends his grad students to do field work to study how their algorithms are performing.

Too many mathematicians consider "applications" as "can be used in other, purely theoretical academic papers," and based on my research for my next book (pmfpbook.org) I find that large swaths of what is claimed to be applied math is actually not useful for solving the real world problems they claim to be useful for. A math thing will be tried in production, to much acclaim and fanfare, and then it will be quietly discarded in favor of simpler, more holistic, or less complicated solutions that end up performing better. But the academic publications praising it persist for decades, and nobody seems to get the message that it's not used.

One thing Tao gets right is that the best math is interdisciplinary. Connecting ideas across different theories is one great way to demonstrate the power of a theory. I would add that, to be great a theory needs to _grounded_ in a practical setting, where it's not just mathematicians judging the math for its beauty but an external reality pushing back. For my aesthetic, a pretty theory is just not enough.

nerdponx• 2 years ago

An effect similar to this is what ultimately drove me away from pursuing an advanced degree in economics.

I kept seeing what were supposedly great innovations in our understanding of economic theory, which were mathematically elegant and completely unrealistic with respect to how people actually behave. What's the value in a closed-form expression that doesn't describe any real-world phenomenon?

I came to realize later that there is plenty of "good" economics research and modeling out there, but I had no desire to enter a field where models were not expected to be useful, in the sense of Box's aphorism.

ysofunny• 2 years ago

What's the value in a closed-form expression that doesn't describe any real-world phenomenon?

it keeps smart, driven, and potentially disruptive people busy chasing around closed forms while "real" power politics are kept firmly and rigorously under the control of incumbent powers.

this may be a perspective too cynical, nonetheless I offer it as an answer to your question; still working on making this funny somehow as it hits close to why I couldn't make it through the academic-industrialized pipeline of institutional corporations (big universities) 'manufacturing' and 'selling' graduated professionals in the labor 'marketplace'

BalinKing2• 2 years ago

This example feels categorically different to me, since economics is fundamentally about modeling the real world, whereas many would argue that the mathematics' ability to do so is more of a convenient byproduct. This reads to me like an indictment of economics in particular, rather than a general statement about the value of the real-world applicability of research.

ink_13• 2 years ago

The idea that academic work should be grounded in "applicability" is a lazy canard. Nobody was much convinced that decades of work in number theory was worth much of anything until applications in cryptography were discovered. We just don't know what we'll find.

Academic research in every field is undertaken for its own sake. If you want to work on "applicability", go get a job in industry. Finance is still hiring quants, for example.

j2kun4• 2 years ago

People are welcome to do whatever they like. I'm expressing an opinion, my value judgement, that the math done for its own sake is just not as beautiful to me as math that has applications.

You call it laziness, and I call it a refined view built up over decades of study and repeatedly hearing mathematicians bullshit and be misinformed about how applicable their work is.

chongli6• 2 years ago

decades of work in number theory

Centuries of work. To Galois and at least all the way back to the Chinese Remainder Theorem, and to Euclid as well! Mathematicians have been fascinated by this stuff for a very, very long time.

umutisik4• 2 years ago

This assumes that the parts of number theory that ended up being useful could not have been developed after people realized you could do public key cryptography with primes.

If the work is undertaken for its own sake, there should not be a need to argue about how it will be useful in the future.

PH95VuimJjqBqy3• 2 years ago

Nobody was much convinced that decades of work in number theory was worth much of anything until applications in cryptography were discovered. We just don't know what we'll find.

There are plenty of examples of this, Bezier Curves is one such.

sfpotter7• 2 years ago

This is simply wrong. There is loads of academic research born out of industrial collaborations which is absolutely focused on how useful it is. Applied research exists in academia.

On the flip side, if you think the only place you can do something "applicable" is by working as a quant... again, wrong!

If anything, the idea that research is free to disregard applicability (read: "relevance to anyone outside academia") is what's a lazy canard. :-) People who are wrestling with whether their research is applicable are struggling to figure out how to connect it to the rest of society, which is valuable. It's responsible of them to do so.

eachro2• 2 years ago

Every year I see tons of CS students captivated by the beauty of algorithms in theoretical CS. I lose track of all the bright eyed undergraduates saying they love thinking about algorithms and would ideally like to spend their summer doing research on these sorts of problems. More often than not, I end up telling them they should focus their efforts on the systems side of things and chat with an OS or DB professor rather than a prof in TCS, but very few of them actually take this advice.

gcanko• 2 years ago

I’m curious why would it better to focus on the system side of things instead of algorithms?

umutisik4• 2 years ago

As a former mathematician, I completely agree. Academic math is good stuff but you end up making too many assumptions that end up being hard to reconcile with a working system.

Personally, I stopped caring much about beauty because doing work guided by some beauty heuristics didn't make me happy. Doing work that is useful to many people does make me happy; and there ends up being beauty in it somehow.

somethingsome• 2 years ago

I think the best and most beautiful mathematics is driven directly by applications, and made better for the pressure to apply to a real problem (not a problem that just mathematicians care about).

That's the thing, in my opinion, when you study math, you start with applications, you work your way to solve complex applicative problems, and then, those problems start to be less interesting, in part because they become easy for you, and the other part is that most of the problem is just 'mechanical and technical' work. Not really interesting.

But what you like in math, it's not the maths themselves, it's to be challenged by problems. So you start to work on abstracts problems, until they become easy, then you go one more step in the abstraction, to find challenging problems, etc..

So basically, I find it somewhat natural to be driven always to more abstract theory, with maybe the hope that one day they will be applied. It's far from you to solve too 'technical' problems in the sense that 'it's just an application of XYZ problem to solve, now, I don't want to implement it, as it will take me a lot of time, while I can find pleasure again in discovering new hard things'

I'm also on the applicative side, but I understand the attraction of the abstract side, and sometimes, I allow myself to work with just a bit of higher abstraction.

So with enough luck, some day, a mathematician, with the right level of abstraction at that moment, will discover an applicative problem that is of your interest, and dedicate to it some time, maybe a lot of time, and maybe even, his/her whole carreer, making it more abstract, elegant and practical.

ducttapecrown2• 2 years ago

"The people who do algebraic geometry in applied settings are doing math that is basically unrelated to algebraic geometry as practiced by theoretical mathematicians" seems like it is just a gatekeeping statement. It is basically related---it is algebraic geometry.

isilofi• 2 years ago

Good Mathematics doesn't use the axiom of choice.

j2kun4• 2 years ago

In undergrad my friends and I made a shirt:

   Pro-axiom of choice
   Because every vector space deserves a basis

tux3• 2 years ago

Something bothers about some of the special cases, but I can't quite describe it.

IngoBlechschmid• 2 years ago

I agree, the axiom of choice disincentives us from striving for more elegant solutions.

That said, the axiom of choice is always available in Gödel's sandbox of "constructible sets", and by "Shoenfield absoluteness", some results can escape the sandbox.

For instance, the result that every vector space has a basis is equivalent to the axiom of choice. We have it in Gödel's sandbox and we don't have it outside, if we prefer to not use the axiom of choice in our meta theory. But every purely number-theoretic consequence of that result flows from the sandbox to the ambient universe.

In this sense, the axiom of choice can be regarded as a useful fiction. Not necessarily true in a literal sense, but true enough for many purposes.

I gave a 37c3 talk on this topic: https://www.speicherleck.de/iblech/stuff/37c3-axiom-of-choic...

candlemas• 2 years ago

The axiom of choice should be considered supernatural.

paulpauper• 2 years ago

Good or interesting mathematics imho is about finding the exceptions. Like, why something works for a particular case when otherwise it does not, or the converse. An example is why a general quintic cannot be solved with radicals, or finding non-trivial quintics that can be solved with radicals.

feoren4• 2 years ago

An example is why a general quintic cannot be solved with radicals

A general quintic can be solved if you allow the "Bring radical", which is a function f(x) that gives the (unique) real root of x^5 + x + 1. When I first heard this, it sounded like a total cop-out, almost like saying "you can find the answer if you already know the answer". But what convinced me was the argument that the Bring radical is not more difficult to compute than the quintic root x^(1/5). So if you care about computability, then "solving" a quintic using roots + the Bring radical is equally as useful as solving it using roots alone.

h0p3• 2 years ago

I was sad to see how little discussion there was of goodness itself. Seems like a key problem to address for any instrument.

feoren4• 2 years ago

There are two different things people mean when they say "mathematics". (Probably lots more, but I'll focus on two.) There's mathematics that is invented, and then there's mathematics that is discovered. The "math that is discovered" was already there. It must be true. It was true before our universe existed, and would be true in every other conceivable universe. It is very difficult to believe that any of this math that we discover could possibly not be useful. It's what reality is made out of. How could a deeper understanding of reality not be useful? And I assert that it is simply not possible to know whether a given discovery would be useful before you discover it; how could you possibly evaluate a discovery's usefulness before you even know what it is?

The "math that is invented" is attempting to describe this more fundamental math. These are the terms we come up with and write down in papers, the symbols we use, etc. Sometimes we discover that The X Theorem and The Y Theorem are actually describing the same fundamental math in different ways. Clearly this descriptive math can vary in usefulness. How simply and clearly does it describe the underlying "ideal" math? Does thinking of it this way lend itself to immediate application to make human lives better? Does it help us discover even more math?

Almost everyone who criticizes math is missing this distinction: they're criticizing the writing, the models, even the institution of math. Some of those criticisms have merit; most don't. Either way, they tend to overlook the fundamental math that was already true, already there long before our universe. You simply cannot identify whether or not a given mathematician is "wasting their time", because you cannot know what underlying fundamental math they might be about to discover, and what its uses might be.

If I were in charge of giving out grants for general-purpose math research, I would not attempt to quantify "usefulness" at all. The main metric I would try to optimize for would be breadth. I'd prioritize the math that seems to be exploring areas that haven't already been thoroughly explored, or making connections between areas of "described math" that currently seem only distantly related. In our current institutions of math, there are forces that push toward increased breadth and forces that push toward conformity (decreased breadth). It might be enough to simply work against those conformity forces -- demanding "usefulness" is a conformity force.

What Makes for 'Good' Mathematics?

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