I consider myself an applied mathematician though technically I’m an economist (macro models, mostly) and operations researcher and differential game theorist (odd combination, but whatever). For context I’m 45 and I’ve spent most of my working life in corporate life.
I’ve always been at a disadvantage academically because I’m rather clumsy with my manipulations, derivations, and I’m a disaster at mental arithmetic. I’m also dyslexic. But starting in the late 1990s when I was in High School I started to become fluent with CAS (Computer Algebra Systems): first Derive, then the Symbolics capabilities of Mathlab, and ultimately Mathematica.
The whole transition is turning out quite well for me: I’m now able to delegate exploring my intuitions to increasingly powerful tools, I no longer have to haul the pyramid blocks up the ramp myself but I can drop them in by helicopter (as if were) and what I bring to the party is intuition and understanding.
I view it a bit like astronomy: in ancient times, before telescopes, keen eyesight was a prerequisite to be an astronomer. Later, after telescopes, anybody with eyesight had enormously enhanced capabilities of observation, and now, with radioastronomy and other forms of remote sensing (neutrino observatories, gravitational interferometers) the whole field has opened up to people who by virtue of being blind would’ve literally been excluded only a few decades ago.
Go forth and multiply: everybody can become a mathematician now. There’s an infinite number of potential universes out there with an infinite number of facts to prove and disprove. And while we’re at it: there’s so much more to mathematics than conjectures, proofs and counterexamples. Solve, model, approximate, fiddle around: as long as you’re not doing trite numerics on arbitrary systems of equations you’ve cooked up for your own amusement you’re good in my books.
Enjoy the tools. Keep your wits about you. Work through the steps that are presented to you. Build your intuition. HAVE FUN.
I think the astronomy and telescope comparison is missing some nuance. You don't delegate actual thought to a telescope, the thinking and acting on the observations you make with it is still done by you. With AI that is different; you get the opportunity to delegate a lot of thinking and acting to it (practically all of it if you really wish), at which point it becomes dubious to call yourself the creator or inventor of some idea.
You can use AI like a telescope, and alleviate your inherent organizational problems or other ailments, which in my opinion is a great use case, as it is empowering. But there will also be plenty of people who are not looking to alleviate any mental/physical quirks they have to do more, but simply want to make a "quick buck" for the least possible effort from the work and thinking of others, without adding much value of their own. Lowering the bar makes things easier for both, the ones who bring value and the ones who merely exploit in some context.
Absolutely true: there’ll be folks who understand no mathematics at all and who will try to elbow their way in. There will also be (extending the metaphor) an ecology of instrument-makers and lens-grinders who somehow become party to the debate. But to be Frank (though I am James) mathematicians will recognise their own.
> I no longer have to haul the pyramid blocks up the ramp myself... what I bring to the party is intuition and understanding.
Hauling the pyramid blocks is what gives people intuition and understanding. It is true that school systems usually have way too much computation -- it is easier to test and grade computation. But the only way that you were able to use computer algebra systems fluently is because you had internalized how algebra worked by hand. If we tell students that they no longer need to learn how to solve equations we are seriously depriving them of a mathematical education.
Also there is some instinctual understanding of the work moving a big heavy rock takes, and how to do it. No such corollary for algebra and hence some civilizations never discovered or understood it.
Maybe. As a parallel, I know C# and F#. They are programming languages. They are built on top of Microsoft products like the JIT and more basically Assembly language.
I have no idea how assembly works - in my years of experience I’ve never had an issue that required I dig that deep into it. Does that mean I can’t be effective with higher-level tooling because I do not know the absolute basics?
The entire point of a programming language design is to give you an abstracted environment with a complete coherent semantics so that you don't need to understand what's going on under the hood.
You can be effective at using C# and F# to solve other problems specifically because they are very well designed at the goal of abstracting over the lower level hardware. But if your goal is to build an intuition of how programming languages themselves are implemented and how computer hardware works, then, using those languages will be counter productive.
If you're trying to use math to solve arithmetical problems, then by all means using AI to help. But if you're trying to advance the math field itself, then actually knowing how math works is probably essential.
Put in more concrete terms: if someone wants to be a working mathematician without learning the fundamentals of math, then how do they even know what prompts to the AI are worth writing? In what way are they adding any value to the process at all?
I never said one should not know the fundamentals of mathematics, I was trying to express it’s a damned lot easier when once understood you can delegate the lesser aspects to machinery. Abacus, calculator, computer, CAS, now AI: the mathematician’s understanding and intuition are still fundamental, but the instruments they can rely upon are levelling the playing field (and indeed, it is a great field in which to play).
First of all, programming isn't necessarily a very skilled task as being a mathematician.
Secondly, I've encountered compiler bugs many times, so it is very helpful.
That's the difference between a senior and a junior: the junior goes to the senior and the senior figures it out. You can remain a junior at any age.
While true that doing things the hard way is critical to learn a discipline, the OP’s point is about what is possible with AI once you’ve built those skills and intuition. I think in the future it will be common for those passionate about a discipline to strategically avoid AI until they’ve “built the muscles.”
Exactly. I learnt to haul the blocks myself, but it has always been an uncertain and effortful exercise; now I can move them around as effortlessly as if I were playing Tetris.
This might be clumsy but I will say it anyway. If music is math, how do you explain people who have innate musical abilities without formal academic training in notation, music theory, circle of fifths, etc.
Well i can tell you what i think: music isn't math and whoever told you that is wrong.
To further underline the point: entire cultures' musical traditions don't follow the circle of fifths, or notation as you're likely familiar with it, or even notes that follow the kinds of ratios you might use to describe western musical scales of any kind. The mathematical representations of western music were applied in retrospect, math never played any role in the development of western music anyway.
So considering math was never a factor in the development of western music and the math used to describe western music doesn't even apply universally, i just can't really consider "music is math" to be a meaningful statement at all
2,000 year old Chinese music uses sanfen sunyi. You take a pipe length, cut it by a third, then extend the new one by a third, alternating. Hey look.....a petatonic scale.
Arabic music pitches even more. The maqam system uses intervals that fall between Western half steps, often called "quarter tones," Some used "commas" which divide the whole tone into 9 parts. This gives you octave with about 53 steps.
Just to make sure I follow, do you mean the "music is math" idea fails, or that my question does? I'm not saying math has nothing to do with music. Frequency ratios and rhythmic divisions are clearly there. My point is that you can have deep musical ability without ever thinking in those terms, which suggests music is more than the math that describes it.
I’m saying that though argument by analogy is very persuasive, “math is music” is a metaphor that is fundamentally unsound as the grounds for reasoning from. I’m not talking about the mathematical aspects of music (which, as you correctly point out, can be found to exist) but that music is performative and aesthetic, whereas mathematics is all about internal consistency.
Sheet music is no more music than mathematical notation is mathematics. (Indeed both notations have roots in the same kind of two-dimensional seventeenth century calligraphic tradition.) However, as you point out, mathematics is indeed the study of patterns, structures, and relationships, and music is not. It is a structure and has relationships, but it is not a “metaphor of metaphors” as mathematics is. Case in point: modern visual art post-photography is much more affine to music than mathematics is, because after depiction became not quite trivial but literally photographic from the daguerreotype onwards, the role of the visual arts became the solicitation of emotion and the conveyance of narrative. Music has syntax, indeed it has very many, but it is not the study of syntaxes in the manner in which mathematics is.
After looking at the title for a day, the idea occured that, without mathematicians doing mathematics, those people would end up in banking, so seeing the top post is most amusing.
One of my local contacts is a semi retired bankster, who I am going to prank by pretending that theres something stuck in her fangs, looks like the still quivering flesh of a young accounts executive, which seems to be the thing that happens to mathematicians, money that is, triggers some sort of primitive predetory instinct better left dormant.
qubex · · focus · HN ↗
I’ve always been at a disadvantage academically because I’m rather clumsy with my manipulations, derivations, and I’m a disaster at mental arithmetic. I’m also dyslexic. But starting in the late 1990s when I was in High School I started to become fluent with CAS (Computer Algebra Systems): first Derive, then the Symbolics capabilities of Mathlab, and ultimately Mathematica.
The whole transition is turning out quite well for me: I’m now able to delegate exploring my intuitions to increasingly powerful tools, I no longer have to haul the pyramid blocks up the ramp myself but I can drop them in by helicopter (as if were) and what I bring to the party is intuition and understanding.
I view it a bit like astronomy: in ancient times, before telescopes, keen eyesight was a prerequisite to be an astronomer. Later, after telescopes, anybody with eyesight had enormously enhanced capabilities of observation, and now, with radioastronomy and other forms of remote sensing (neutrino observatories, gravitational interferometers) the whole field has opened up to people who by virtue of being blind would’ve literally been excluded only a few decades ago.
Go forth and multiply: everybody can become a mathematician now. There’s an infinite number of potential universes out there with an infinite number of facts to prove and disprove. And while we’re at it: there’s so much more to mathematics than conjectures, proofs and counterexamples. Solve, model, approximate, fiddle around: as long as you’re not doing trite numerics on arbitrary systems of equations you’ve cooked up for your own amusement you’re good in my books.
Enjoy the tools. Keep your wits about you. Work through the steps that are presented to you. Build your intuition. HAVE FUN.
Archer6621 · · focus · HN ↗
You can use AI like a telescope, and alleviate your inherent organizational problems or other ailments, which in my opinion is a great use case, as it is empowering. But there will also be plenty of people who are not looking to alleviate any mental/physical quirks they have to do more, but simply want to make a "quick buck" for the least possible effort from the work and thinking of others, without adding much value of their own. Lowering the bar makes things easier for both, the ones who bring value and the ones who merely exploit in some context.
qubex · · focus · HN ↗
tylerhou · · focus · HN ↗
Hauling the pyramid blocks is what gives people intuition and understanding. It is true that school systems usually have way too much computation -- it is easier to test and grade computation. But the only way that you were able to use computer algebra systems fluently is because you had internalized how algebra worked by hand. If we tell students that they no longer need to learn how to solve equations we are seriously depriving them of a mathematical education.
piker · · focus · HN ↗
forshaper · · focus · HN ↗
qubex · · focus · HN ↗
SamuelAdams · · focus · HN ↗
I have no idea how assembly works - in my years of experience I’ve never had an issue that required I dig that deep into it. Does that mean I can’t be effective with higher-level tooling because I do not know the absolute basics?
munificent · · focus · HN ↗
You can be effective at using C# and F# to solve other problems specifically because they are very well designed at the goal of abstracting over the lower level hardware. But if your goal is to build an intuition of how programming languages themselves are implemented and how computer hardware works, then, using those languages will be counter productive.
If you're trying to use math to solve arithmetical problems, then by all means using AI to help. But if you're trying to advance the math field itself, then actually knowing how math works is probably essential.
Put in more concrete terms: if someone wants to be a working mathematician without learning the fundamentals of math, then how do they even know what prompts to the AI are worth writing? In what way are they adding any value to the process at all?
qubex · · focus · HN ↗
LtWorf · · focus · HN ↗
Secondly, I've encountered compiler bugs many times, so it is very helpful.
That's the difference between a senior and a junior: the junior goes to the senior and the senior figures it out. You can remain a junior at any age.
keeda · · focus · HN ↗
qubex · · focus · HN ↗
jasondigitized · · focus · HN ↗
queenkjuul · · focus · HN ↗
To further underline the point: entire cultures' musical traditions don't follow the circle of fifths, or notation as you're likely familiar with it, or even notes that follow the kinds of ratios you might use to describe western musical scales of any kind. The mathematical representations of western music were applied in retrospect, math never played any role in the development of western music anyway.
So considering math was never a factor in the development of western music and the math used to describe western music doesn't even apply universally, i just can't really consider "music is math" to be a meaningful statement at all
jasondigitized · · focus · HN ↗
2,000 year old Chinese music uses sanfen sunyi. You take a pipe length, cut it by a third, then extend the new one by a third, alternating. Hey look.....a petatonic scale.
Arabic music pitches even more. The maqam system uses intervals that fall between Western half steps, often called "quarter tones," Some used "commas" which divide the whole tone into 9 parts. This gives you octave with about 53 steps.
qubex · · focus · HN ↗
jasondigitized · · focus · HN ↗
qubex · · focus · HN ↗
qubex · · focus · HN ↗
metalman · · focus · HN ↗
qubex · · focus · HN ↗