5 ms·

What Makes for 'Good' Mathematics?

46 points by digital55 3y ago

j2kun 3y 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.
- ducttapecrown 3y 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.
- nerdponx 3y 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 3y 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'
  - BalinKing 3y 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.
- 3y ago
isilofi 3y ago
Good Mathematics doesn't use the axiom of choice.
- j2kun 3y ago
  In undergrad my friends and I made a shirt: Pro-axiom of choice Because every vector space deserves a basis
  - tux3 3y ago
    Something bothers about some of the special cases, but I can't quite describe it.
- IngoBlechschmid 3y 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-choice.pdf https://www.speicherleck.de/iblech/stuff/37c3-axiom-of-choic...
- candlemas 3y ago
  The axiom of choice should be considered supernatural.
paulpauper 3y 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.
- feoren 3y 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 3y ago
I was sad to see how little discussion there was of goodness itself. Seems like a key problem to address for any instrument.
feoren 3y 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.

46 points by digital55 3y 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.

ducttapecrown 3y 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.
nerdponx 3y 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 3y 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'
- BalinKing 3y 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.
3y ago

isilofi 3y ago

Good Mathematics doesn't use the axiom of choice.

j2kun 3y ago
In undergrad my friends and I made a shirt: Pro-axiom of choice Because every vector space deserves a basis
- tux3 3y ago
  Something bothers about some of the special cases, but I can't quite describe it.
IngoBlechschmid 3y 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-choice.pdf https://www.speicherleck.de/iblech/stuff/37c3-axiom-of-choic...
candlemas 3y ago
The axiom of choice should be considered supernatural.

paulpauper 3y 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.

feoren 3y 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 3y ago

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

feoren 3y 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.