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Comment by xanderlewis | Hacker News Reader

Friendly warning to those who might not be aware: the vast majority of comments below posts like the above will be left by (otherwise intelligent) programmers who think mathematics is a closed system where one attempts to solve endless Olympiad-type problems. I wouldn’t take any of it seriously at all. Better to listen to what those who actually know what the subject is about have to say.

Unfortunately, mathematics (especially pure mathematics) is by its very nature very, very poorly understood by those who haven’t worked as a mathematician. Even worse, those who don’t understand are seemingly not at all aware of their misunderstanding and are entirely confident in their (very wrong) characterisation of the subject.

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I did systems administration for a university math department for several years. I came to the conclusion that mathematics (and perhaps philosophy) were both topics where it was likely that no staff members in that department could describe "what goes on here" and that possibly even within the department, one professor may not be able to describe what another professor's actually doing.

The closest I could come to describing math is "some abstract process where imagined structures are characterized and extended; the most critical part of the process is identifying where seemingly independent structures are found to actually be fungible in some previously undiscovered way".

A simple example is

  "hey, did you know that x^i is the unit circle?" 

  "what's i?" 

  "i is defined as if you square it the result is -1" 

  "what does that have to do with circles?"

The clue is burried in the text of OP's article.

> Mathematicians can also consider wholly redirecting their skill sets to work on real world problems. I’ve actually been encouraging mathematicians to consider thinking about working on government or other large-scale societal issues.

The fact that this is a radical departure from the norm is part of why mathematics (and philosophy) is often seen as some intangible or ungrokable science to many outsiders, as they're generally approaching it from a perspective of "Okay, but why, what is this useful for?", and the answer "For the science of it" doesn't tend to land with people that aren't already passionate about said science/discipline and are just trying to figure out what it even is or involves.

Doesn't help that there is a pervasive sentiment in American Academia (not sure about elsewhere) about Math being *the* hard science, and I mean hard as in difficulty, so a lot of people get intimidated by it before they ever give it a chance very early on in their academic life and carry that through the rest of their education.

So there ends up being a rather small pool of people that are in(to) the field, and rather high friction for stimualting interest in it from outsiders from the way that it's taught, and a massive difference in the perspective of it's use between it's diaspora and the unmathed masses.

Mathematicians who do work on "real world" problems are mostly doing so with theoretical physicists, genomicists, and cryptographers. None of these count as what people consider valuable other than because they are hard.

That said, few 50 (or even 40) years ago would have predicted that completely abstract number theoretical computations about primes, discrete logarithms, and elliptic curves would be the foundation of our monetary system.

Never said they didn't.

> I did systems administration for a university math department for several years. I came to the conclusion that mathematics (and perhaps philosophy) were both topics where it was likely that no staff members in that department could describe "what goes on here" and that possibly even within the department, one professor may not be able to describe what another professor's actually doing.

And this is indeed why it is not going to be taken seriously as an academic or (more importantly) an economic endeavour done by humans anymore.

That won't stop the career mathematicians from protesting and having a cry here trying to justify themselves.

Could you clue us in? What is mathematics actually about?

Good question, and maybe I should have provided an alternative description rather than just criticising.

Unfortunately, it is actually surprisingly hard to pin down, and I think mathematicians (and, as a student, I count myself as one to some degree at least) now have the task of making this a lot clearer. If we want to justify our existence in the face of new machines that can seemingly ‘do our work for us’ (so far in a restricted context), we should give a robust defence of our practice. If we can’t do this, we simply don’t deserve the funding (which, by the way, again contrary to some misguided statements here, isn’t very much anyway!). I think all of this will become clearer to outsiders as time passes, but for now it’s not easy to give a quick answer — though I can try.

Mathematics is about understanding things. Isn’t that what every subject is about? Well, I suppose so, but mathematics more specifically does something like the following:

(1) observe some phenomenon in ‘reality’.

(2) attempt to formalise that phenomenon in such a way that it can be manipulated purely symbolically.

(3) use this (perhaps fairly arbitrary; remember that we can invent as many formal systems as we like) system to deduce from our initial assumptions new facts that would otherwise have been very non-obvious.

It seems like outsiders have a decent grasp of (3) and the application of AI to it, but have very little idea about the other two steps. It seems to be widely assumed among non-mathematicians that problems are essentially god given and that the job of a mathematician is therefore to chug away on these problems, manipulating symbols and trying out tools, in the hope of learning a yes/no answer to each one.

The first two steps are by far the hardest and most important, and they’re also the parts that AI seems currently unable to help with.

NOTE: this is not a deeply insightful description of what the subject is about, and there are many better characterisations out there. I think Tao and various others have written recently about why complicated and inscrutable AI-generated proofs aren’t nearly as valuable as one might imagine. (That’s not to say there’s no value to such proofs; perhaps in time, as technology improves, mathematicians will come to accept AI as part of the process.)

If you want to understand all of this issues better, reading the recent slew of guest posts on Tao’s blog would be a very good start.

Despite loving Mathematics and having considered being a Mathematician myself, that wasn't a very robust defense. Also, even though I'm a big Terence Tao fan, I'm not sure he has sorted that out this defense in full himself.

This post which he forwarded was quite poor in my opinion. Confusing, all over the place with AI criticisms and promotion of the AI hazing being done by mathematicians.

X thousand mathematicians who want to protect their livelihoods signed a bunch of letters against AI. Duh. We've seen similar movements from every profession that has been displaced ever.

Terence Tao uses AI and has made a few good points on how to use it. But defensiveness leaks into almost every defense of the role of humans in Mathematics that I've read, even his own at times.

To be clear, I actually believe that Mathematicians aren't going away, but I dont have enough knowledge about the life of a professional mathematician to articulate a path forward.

This "path forward" is what I'd like to see. We need a top mathematician with enough intellectual honesty (Terence Tao qualifies, I think) to start this questioning with "there's actually no role for human Mathematicians" as one of the options on the table and go from there.

Though your points above may all be valid, you're asking us to just accept your view that the parent's wasn't a very robust defense.

Can you provide an explanation for why their claim that (1) and (2) are the hardest parts is false?

Or perhaps provide an alternative definition of Mathematics that is more explanatory than the one they've provided?

For me, mathematicians seem to be still having an inner discussion rather a making these essays for the more general public. In any case, I think that statements of the type "there's actually no role for human Mathematicians" are completely non-serious, so it'd sad that the discussion concentrates on that.

I agree, but it's still something we shouldn't eliminate a priori. I think Mathematics as a profession will be greatly enhanced by AI, but I can't prove that.

There’s a simple point to be made: if no one in the world understands an AI proof, what’s the point?

Maybe another AI could use the proof.

How would we know it was correct?

If you feed an AI nonsense in its training data, it will generate nonsense

My understanding when I was practicing is that the trend in modern mathematics is to focus on spaces with a certain kind of structure, and maps between those spaces that preserve it, and then what are invariants are preserved by those maps. Structure-preserving maps between categories of such spaces - "functors" in the language of category theory - are especially neat.

That's certainly different from Olympiad-style problems.

Wondering about the world, using the simplified language of "mathematical logic"

Pure mathematics reduces to abstract "relationships" and their implications largely.

To know what something is about, a natural way is to do it yourself.

Wow, I'm glad AI is going to start humbling people.

> To know what something is about, a natural way is to do it yourself.

literally what software engineers were doing for decades though

software is mostly just simple math, for the most part, until you need to do something more complex for some hairy algos lol

Actually, if you were to convert most software into mathematical notation it would look incomprehensible. We are not interested in gatekeeping as much by making things look more convoluted than they need to.

This is a great point. When I was learning machine learning in the early 00s, there were many papers where I would struggle to understand exactly what the mathematics was trying to communicate, but where I would look at the matlab source code and say to myself "that's all?"

A lot of programmers consider themselves to be applied mathematicians.

Maybe more in years past when Comp Sci was a subset of Math Departments.

In my observation, most programmers and engineers lose their math chops over time. The math they need is mostly baked into their tools, such as CAD. If a problem requires more advanced math, it's given to a "math person" in the department. Often, the "math person" is also not allowed to touch the production code.

the author right in that post pitches his ideas as generalizable way beyond math, so why the snobbery?

i'd be on board with this concept but the author seems to believe his point is generalisable to all under industries, hence committing the same fallacy you're talking about at a large scale.

but you're still right. i disregarded his take, as you would with mine re. math.

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