4 ms·

I’m reminded of that famous debate between Poincaré and Hilbert at the International Congress of Mathematicians in Paris in 1900. It was then that everyone deci

I’m reminded of that famous debate between Poincaré and Hilbert at the International Congress of Mathematicians in Paris in 1900. It was then that everyone decided to follow Hilbert’s path, and proof came to be valued more than intuition. I think modern math at school and at applied university kind of lost this intuitive part. I try to teach my students that mathematics is, first and foremost, a very precise language of communication. It’s sometimes amusing to ask those who don’t like math to do without it entirely, just to see how much harder it becomes to describe the things around them. Second thing I tell them, formulas are the essence of mechanisms in their purest form. And in this form, they’re much easier to grasp and mentally manipulate. It always amused me, after taking a mechanics course, to imagine that for any formula, you could visualize a mechanism or process that implements it. And third thing, I suppose, the ability to verify one’s own statements as proof. Although, of course, mathematicians would probably tear me apart here for my heresy:sorry, I’m not a mathematician, but an engineer. You can make mistakes by using incorrect assumptions, but at some point, analysis itself will show you that you were mistaken. There’s a wonderful book, How to Prove It by Daniel Velleman, which provides an introduction to proof for the uninitiated like me. I really enjoyed it.

conmod278 15d ago
[flagged]
Geof25 15d ago
People often hate math because it was not explained to them correctly, usually by people who are good mathematicians but know close to nothing about teaching. It was so infuriating to see everyone in the class absolutely fail on a specific subject and the "teacher" assumed that everyone must be stupid then. No self reflection, no questioning himself why he is not getting gaussian distribution in marks, just straight Fs.
- partyficial 15d ago
  a good teacher remembers the journey, not just the destination. socratic method exists. almost none follows it.
  - awesome_dude 15d ago
    >socratic method exists. almost none follows it. I have a hatred for people who think they can use this method. If used incorrectly which it is a great percentage of the time it confuses the student. The person employing the socratic method must actually know the answer and where the student is in their mind. Failure on either account makes it pointless. Ask anyone unfortunate enough to ask for help on IRC
    - Kim_Bruning 15d ago
      I sometimes ask more questions than utter new things when trying to explain something. But that's because I'm trying to focus down and determine exactly where they're at before I just randomly make things worse by accident :-P. I'm not sure if that's the actual socratic method. But people accuse me of using it. Either way, it does seem to work for me.
    - bananaflag 15d ago
      > The person employing the socratic method must actually know the answer and where the student is in their mind. The socratic method also has a much higher chance of revealing where the student is in their mind.
  - moffkalast 15d ago
    That only works if the one you're trying to guide can figure it out mostly on their own and is interested in cooperating. Aka does not work for anything below university level.
- krisoft 15d ago
  > usually by people who are good mathematicians but know close to nothing about teaching. I higly doubt that. Maybe in university level courses. Most people’s only experience with mathematics is an elementary or high school teacher who were probably themselves at best mediocre at the subject. Simply because of selection factors. Those who are good at math are encouraged to go into STEM. There will be of course exceptions everywhere, but that is not what “usually” happens. And thats just about being good at maths the school subject, which is distinct from being “ good mathematicians” the science / research topic. Mathematicians are few and far between, simply because it is a specialist subject. There just aren’t enough of them to go around for them to be the formative experience around math for most people.
  - graemep 15d ago
    I think the problem is partly circular. Most people do not like maths. This includes most primary school teachers - in places I know primary school teachers are not subject specialists so just reflect the population of those with the required level of education in terms of their attitude to maths. If you do not enjoy a subject, any subject, you cannot make it fun for those you teach. In the case of maths specifically its pretty bad: https://worrydream.com/refs/Lockhart_2002_-_A_Mathematician%27s_Lament.pdf https://worrydream.com/refs/Lockhart_2002_-_A_Mathematician%... My daughter hated maths when I took her out of school at the age of nine. A few years later she was very good at it and enjoying maths and STEM subjects. When she went to a sixth form college[1] she liked it well enough to pick it as one of her A levels[2]. [1] https://en.wikipedia.org/wiki/Sixth_form_college https://en.wikipedia.org/wiki/Sixth_form_college [2] https://en.wikipedia.org/wiki/A-level https://en.wikipedia.org/wiki/A-level
  - anon48293 15d ago
    [dead]
- Paracompact 15d ago
  Another response to math that makes me sad: "I must be too stupid to understand this," "my brain is too small for this," etc. Different people say it for different reasons, but it's almost always in response to a hand-wavey explanation that doesn't makes sense to anyone not already in the know. Math is so much more about humility and skepticism than it is prodigy.
  - D-Machine 15d ago
    This is tricky, because, in fact, hard math having an intelligence floor is one of the nastier realities of the human condition. Anyone who is even quite intelligent but has really pursued the rigorous stuff, unless they are in fact a prodigy, eventually realizes they have an abstraction ceiling (and this term is a common one thrown around in people studying mathematics, because intelligence denial is so obviously false when you do hit your abstraction ceiling). Most people are correct that they lack the intelligence / mind for a lot of hard math (even epsilon-delta proofs are enough to eliminate the majority of the population, no matter how good a teacher you are, and these are just basic undergrad calc). And yeah, sure, people have different kinds of intelligence and such, but there is still a g-factor, and people of low intelligence almost universally can't do hard math, whereas most people who can do e.g. advanced undergrad math can generally do almost all other advanced undergrad fields reasonably well. The world isn't fair here.
    - x______________ 15d ago
      I would even go as far as saying that mental conditioning and training is also required, on top of mental capabilities.
    - Paracompact 15d ago
      > Anyone who is even quite intelligent but has really pursued the rigorous stuff, unless they are in fact a prodigy, eventually realizes they have an abstraction ceiling Eh. I'm a math PhD who fled academia because it was too much for me. But I have never encountered this term "abstraction ceiling" nor did I succumb to it. I simply ran out of motivation to pursue higher math, especially when following through on learning and research became more and more labor. (It was always labor; but it was a labor I used to love.) I am far from a prodigy. > even epsilon-delta proofs are enough to eliminate the majority of the population, no matter how good a teacher you are Disagree. It's a notoriously hard subject to teach, and with all the demands placed on e-d in so little time in your average curriculum, it doesn't require appeals to IQ to explain its infamy. With enough motivation and practice, the quantifier alternation is comprehensible to any sound mind. What your average mind (and student) lacks is exposure to formalism, abstraction, and how these things tie in with what they are familiar with, which is symbolic manipulation. With the exception of geometric proofs (another educational bugbear), they have little context for what formalism is or why it matters.
- zozbot234 15d ago
  > People often hate math because it was not explained to them correctly Spoiler: this is also why mathematicians hate vibe-math. AIs are outright terrible explainers even when they do have a watertight logical argument—and honestly, this is the load-bearing seam. It goes beyond "proof vs. exposition": the logical derivations AI comes up with fail to even qualify as human-directed proof because of how terrible they are (far below even the most novice mathematician doing their roughest work) at the exposition part.
  - Marha01 15d ago
    > AIs are outright terrible explainers even when they do have a watertight logical argument I think this only applies to cutting edge mathematics (novel proofs of hard problems). I have seen it reported more than once that such AI proofs are cumbersome to follow. But in my experience, when it comes to explaining well-established math that is already in the training data, AIs can be very good teachers (at least with recent models). Especially if you use it along with a textbook and ask it about anything that might not be explained well in the textbook.
  - ogogmad 15d ago
    > AIs are outright terrible explainers Gemini's explanations are very good.
- jacquesm 14d ago
  'From which it is obvious that...'
notarobot123 15d ago
Similarly, programming is also a precise language of communication. Initially, we focused on direct machine behavior but every abstraction above the hardware (including assembly) has been to make that behavior legible to humans. Developed notations and shared procedural abstractions have made thinking about computation more intentionally human and source control has established a protocol for conversing with other humans in the language of a program and changes to that program. The moment just now feels like a neglecting of the idea of communication being central. If the program is a compile target but not sufficiently legible or if the conversation moves too quickly for us to keep up then we retain the effects of computation but loose its meaning as communication. We loose the understanding and the ability to develop and evolve further shared abstractions. Open source programs could be more like motivated explanations of computation. For open source to survive, maybe we should start to make the distinction between free product distribution and programming as communication and community building.
- skydhash 15d ago
  > Open source programs could be more like motivated explanations of computation. It is already that. Every time a method/function is created, a structure is defined, a variable is added, a file is created or renamed,… It’s all for the purpose of human communication. The computer only need binary in a single file. But people feels like they should be able to jumpninto curl code without any understanding of networking, or linux code with no knowlede of computer architecture. Few code are meant for total beginners.
- Xirdus 15d ago
  Programming was never about communication. It was always about making the machine do the thing we want. Back in the day, a good game programmer knew which time intervals had writable video memory and which CPU cycles drew which scanlines, and spent more time rearranging the code to hit these timings than to write the actual algorithm. Later programmers (I hesitate to call them good) learned everything there is to learn about OS internals and wrote theoretically nonsensical and invalid code that still worked thanks to those internals, to save CPU cycles and especially memory use. And the next generation of programmers took the principles of late binding and abstraction to the logical extreme and created architectures that cannot be described in words anymore, only in diagrams - but are crazy good for code reuse, traceability and A/B testing.
bananaflag 15d ago
As a math prof, I care(d) much more about proof than intuition, not because proof is more important, but exactly because intuition is (I'm a bit Chesterton-ish here haha). You cannot do proof without intuition hence, if you emphasize proof, intuition will take care of itself. Whereas if you emphasize intuition, students won't have any idea of what a rigorous proof should be.
- contubernio 15d ago
  As a math professor, I care much more about the key idea, heuristics, and motivation than the proof. With the others in place the proof is clear, something an AI or a student can do.
  - sigbottle 15d ago
    Well, it's knowing when to push and when to not. You probably have an intuition for, I don't know, abstract algebra objects (I don't know your field of specialty :P), without needing to symbolically manipulate all of it, but you developed a deep intuition for them through many proofs and attempts at proofs with them.
    - CrazyStat 15d ago
      Im glad you brought up abstract algebra—that was the one class in my math undergrad that I never developed an intuition for. I learned to do the proofs by pushing symbols around and putting bars on top of them but I never felt like I understood what was happening.
  - lupire 15d ago
    That's leaning into engineering, away from math. Heuristics aren't always accurate. Math history before proof is the history of delusion. Idea, heuristics, and motivation aren't nearly enough for correctness outside of a sandbox.
- fidotron 15d ago
  Surely this implies these LLM generated proofs require the LLMs to have mathematical intuition . . . and honestly I don't think many people believe that, and rightly so, certainly not in the way Poincaré was on about. Maybe it's been done, but I'd like to see an LLM recreate Euclid from questioning without having seen it during training.
derangedHorse 15d ago
Intuition happens naturally and people are prone to inducting the wrong conclusions. Proofs provide a framework for rigorously analyzing drawn conclusions such that it can be used to build intuition in others. If math is about sharing the insights gained in a particular class of problems, proofs are the means to getting there.
philipov 15d ago
"How To Prove It" is used to initiate people. It was required reading for an introductory class on formal mathematics at university.
boredatoms 15d ago
> a very precise language of communication I find that difficult to match to my own experience, in that there is seemingly endless domain specific notation that heavily obscures communication
- mejutoco 14d ago
  Just because you do not know a language it does not mean it is not precise, once you learn it.
- HappMacDonald 14d ago
  So, like learning Chinese then
a-dub 15d ago
> There’s a wonderful book, How to Prove It by Daniel Velleman the name sounds familiar but i don't think i have read that one, i did enjoy "introduction to mathematical reasoning" by eccles. personally my relationship with mathematical proofs has been complicated. it took some work to understand basic proofs (dedekind cuts, ideas vs. instructions with mathematical notation), but all of the theory of computation proofs, which supposedly are difficult for many, were completely intuitively easy for me. i think mathematicians are facing a similar confusion as computer programmers. the medium used to require precise thinking and the simple act of reading, writing and composing it was a mechanism for thinking and learning. in the llm era, the question is: should there be a new mechanism and if so, what should it look like?
deleted 14d ago
[deleted]

conmod278 15d ago

[flagged]

Geof25 15d ago

People often hate math because it was not explained to them correctly, usually by people who are good mathematicians but know close to nothing about teaching. It was so infuriating to see everyone in the class absolutely fail on a specific subject and the "teacher" assumed that everyone must be stupid then. No self reflection, no questioning himself why he is not getting gaussian distribution in marks, just straight Fs.

partyficial 15d ago
a good teacher remembers the journey, not just the destination. socratic method exists. almost none follows it.
- awesome_dude 15d ago
  >socratic method exists. almost none follows it. I have a hatred for people who think they can use this method. If used incorrectly which it is a great percentage of the time it confuses the student. The person employing the socratic method must actually know the answer and where the student is in their mind. Failure on either account makes it pointless. Ask anyone unfortunate enough to ask for help on IRC
  - Kim_Bruning 15d ago
    I sometimes ask more questions than utter new things when trying to explain something. But that's because I'm trying to focus down and determine exactly where they're at before I just randomly make things worse by accident :-P. I'm not sure if that's the actual socratic method. But people accuse me of using it. Either way, it does seem to work for me.
  - bananaflag 15d ago
    > The person employing the socratic method must actually know the answer and where the student is in their mind. The socratic method also has a much higher chance of revealing where the student is in their mind.
- moffkalast 15d ago
  That only works if the one you're trying to guide can figure it out mostly on their own and is interested in cooperating. Aka does not work for anything below university level.
krisoft 15d ago
> usually by people who are good mathematicians but know close to nothing about teaching. I higly doubt that. Maybe in university level courses. Most people’s only experience with mathematics is an elementary or high school teacher who were probably themselves at best mediocre at the subject. Simply because of selection factors. Those who are good at math are encouraged to go into STEM. There will be of course exceptions everywhere, but that is not what “usually” happens. And thats just about being good at maths the school subject, which is distinct from being “ good mathematicians” the science / research topic. Mathematicians are few and far between, simply because it is a specialist subject. There just aren’t enough of them to go around for them to be the formative experience around math for most people.
- graemep 15d ago
  I think the problem is partly circular. Most people do not like maths. This includes most primary school teachers - in places I know primary school teachers are not subject specialists so just reflect the population of those with the required level of education in terms of their attitude to maths. If you do not enjoy a subject, any subject, you cannot make it fun for those you teach. In the case of maths specifically its pretty bad: https://worrydream.com/refs/Lockhart_2002_-_A_Mathematician%27s_Lament.pdf https://worrydream.com/refs/Lockhart_2002_-_A_Mathematician%... My daughter hated maths when I took her out of school at the age of nine. A few years later she was very good at it and enjoying maths and STEM subjects. When she went to a sixth form college[1] she liked it well enough to pick it as one of her A levels[2]. [1] https://en.wikipedia.org/wiki/Sixth_form_college https://en.wikipedia.org/wiki/Sixth_form_college [2] https://en.wikipedia.org/wiki/A-level https://en.wikipedia.org/wiki/A-level
- anon48293 15d ago
  [dead]
Paracompact 15d ago
Another response to math that makes me sad: "I must be too stupid to understand this," "my brain is too small for this," etc. Different people say it for different reasons, but it's almost always in response to a hand-wavey explanation that doesn't makes sense to anyone not already in the know. Math is so much more about humility and skepticism than it is prodigy.
- D-Machine 15d ago
  This is tricky, because, in fact, hard math having an intelligence floor is one of the nastier realities of the human condition. Anyone who is even quite intelligent but has really pursued the rigorous stuff, unless they are in fact a prodigy, eventually realizes they have an abstraction ceiling (and this term is a common one thrown around in people studying mathematics, because intelligence denial is so obviously false when you do hit your abstraction ceiling). Most people are correct that they lack the intelligence / mind for a lot of hard math (even epsilon-delta proofs are enough to eliminate the majority of the population, no matter how good a teacher you are, and these are just basic undergrad calc). And yeah, sure, people have different kinds of intelligence and such, but there is still a g-factor, and people of low intelligence almost universally can't do hard math, whereas most people who can do e.g. advanced undergrad math can generally do almost all other advanced undergrad fields reasonably well. The world isn't fair here.
  - x______________ 15d ago
    I would even go as far as saying that mental conditioning and training is also required, on top of mental capabilities.
  - Paracompact 15d ago
    > Anyone who is even quite intelligent but has really pursued the rigorous stuff, unless they are in fact a prodigy, eventually realizes they have an abstraction ceiling Eh. I'm a math PhD who fled academia because it was too much for me. But I have never encountered this term "abstraction ceiling" nor did I succumb to it. I simply ran out of motivation to pursue higher math, especially when following through on learning and research became more and more labor. (It was always labor; but it was a labor I used to love.) I am far from a prodigy. > even epsilon-delta proofs are enough to eliminate the majority of the population, no matter how good a teacher you are Disagree. It's a notoriously hard subject to teach, and with all the demands placed on e-d in so little time in your average curriculum, it doesn't require appeals to IQ to explain its infamy. With enough motivation and practice, the quantifier alternation is comprehensible to any sound mind. What your average mind (and student) lacks is exposure to formalism, abstraction, and how these things tie in with what they are familiar with, which is symbolic manipulation. With the exception of geometric proofs (another educational bugbear), they have little context for what formalism is or why it matters.
zozbot234 15d ago
> People often hate math because it was not explained to them correctly Spoiler: this is also why mathematicians hate vibe-math. AIs are outright terrible explainers even when they do have a watertight logical argument—and honestly, this is the load-bearing seam. It goes beyond "proof vs. exposition": the logical derivations AI comes up with fail to even qualify as human-directed proof because of how terrible they are (far below even the most novice mathematician doing their roughest work) at the exposition part.
- Marha01 15d ago
  > AIs are outright terrible explainers even when they do have a watertight logical argument I think this only applies to cutting edge mathematics (novel proofs of hard problems). I have seen it reported more than once that such AI proofs are cumbersome to follow. But in my experience, when it comes to explaining well-established math that is already in the training data, AIs can be very good teachers (at least with recent models). Especially if you use it along with a textbook and ask it about anything that might not be explained well in the textbook.
- ogogmad 15d ago
  > AIs are outright terrible explainers Gemini's explanations are very good.
jacquesm 14d ago
'From which it is obvious that...'

notarobot123 15d ago

Similarly, programming is also a precise language of communication. Initially, we focused on direct machine behavior but every abstraction above the hardware (including assembly) has been to make that behavior legible to humans. Developed notations and shared procedural abstractions have made thinking about computation more intentionally human and source control has established a protocol for conversing with other humans in the language of a program and changes to that program. The moment just now feels like a neglecting of the idea of communication being central. If the program is a compile target but not sufficiently legible or if the conversation moves too quickly for us to keep up then we retain the effects of computation but loose its meaning as communication. We loose the understanding and the ability to develop and evolve further shared abstractions. Open source programs could be more like motivated explanations of computation. For open source to survive, maybe we should start to make the distinction between free product distribution and programming as communication and community building.

skydhash 15d ago
> Open source programs could be more like motivated explanations of computation. It is already that. Every time a method/function is created, a structure is defined, a variable is added, a file is created or renamed,… It’s all for the purpose of human communication. The computer only need binary in a single file. But people feels like they should be able to jumpninto curl code without any understanding of networking, or linux code with no knowlede of computer architecture. Few code are meant for total beginners.
Xirdus 15d ago
Programming was never about communication. It was always about making the machine do the thing we want. Back in the day, a good game programmer knew which time intervals had writable video memory and which CPU cycles drew which scanlines, and spent more time rearranging the code to hit these timings than to write the actual algorithm. Later programmers (I hesitate to call them good) learned everything there is to learn about OS internals and wrote theoretically nonsensical and invalid code that still worked thanks to those internals, to save CPU cycles and especially memory use. And the next generation of programmers took the principles of late binding and abstraction to the logical extreme and created architectures that cannot be described in words anymore, only in diagrams - but are crazy good for code reuse, traceability and A/B testing.

bananaflag 15d ago

As a math prof, I care(d) much more about proof than intuition, not because proof is more important, but exactly because intuition is (I'm a bit Chesterton-ish here haha). You cannot do proof without intuition hence, if you emphasize proof, intuition will take care of itself. Whereas if you emphasize intuition, students won't have any idea of what a rigorous proof should be.

contubernio 15d ago
As a math professor, I care much more about the key idea, heuristics, and motivation than the proof. With the others in place the proof is clear, something an AI or a student can do.
- sigbottle 15d ago
  Well, it's knowing when to push and when to not. You probably have an intuition for, I don't know, abstract algebra objects (I don't know your field of specialty :P), without needing to symbolically manipulate all of it, but you developed a deep intuition for them through many proofs and attempts at proofs with them.
  - CrazyStat 15d ago
    Im glad you brought up abstract algebra—that was the one class in my math undergrad that I never developed an intuition for. I learned to do the proofs by pushing symbols around and putting bars on top of them but I never felt like I understood what was happening.
- lupire 15d ago
  That's leaning into engineering, away from math. Heuristics aren't always accurate. Math history before proof is the history of delusion. Idea, heuristics, and motivation aren't nearly enough for correctness outside of a sandbox.
fidotron 15d ago
Surely this implies these LLM generated proofs require the LLMs to have mathematical intuition . . . and honestly I don't think many people believe that, and rightly so, certainly not in the way Poincaré was on about. Maybe it's been done, but I'd like to see an LLM recreate Euclid from questioning without having seen it during training.

derangedHorse 15d ago

Intuition happens naturally and people are prone to inducting the wrong conclusions. Proofs provide a framework for rigorously analyzing drawn conclusions such that it can be used to build intuition in others. If math is about sharing the insights gained in a particular class of problems, proofs are the means to getting there.

philipov 15d ago

"How To Prove It" is used to initiate people. It was required reading for an introductory class on formal mathematics at university.

boredatoms 15d ago

> a very precise language of communication I find that difficult to match to my own experience, in that there is seemingly endless domain specific notation that heavily obscures communication

mejutoco 14d ago
Just because you do not know a language it does not mean it is not precise, once you learn it.
HappMacDonald 14d ago
So, like learning Chinese then

a-dub 15d ago

> There’s a wonderful book, How to Prove It by Daniel Velleman the name sounds familiar but i don't think i have read that one, i did enjoy "introduction to mathematical reasoning" by eccles. personally my relationship with mathematical proofs has been complicated. it took some work to understand basic proofs (dedekind cuts, ideas vs. instructions with mathematical notation), but all of the theory of computation proofs, which supposedly are difficult for many, were completely intuitively easy for me. i think mathematicians are facing a similar confusion as computer programmers. the medium used to require precise thinking and the simple act of reading, writing and composing it was a mechanism for thinking and learning. in the llm era, the question is: should there be a new mechanism and if so, what should it look like?

deleted 14d ago

[deleted]