Serving the Quantitative Finance Community
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- katastrofa
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Re: Philosophy of Mathematics
I prefer to attribute it to Cybenko ;-P Don't we all live in the world of gospels/models? MLers would tell you that indeed the theory doesn't give any useful hints about the size of the network or the learning algorithm. But at least it deals with deterministic functions. It's not half as wrong as claiming that models are capable of describing a phenomena involving aleatory or epistemic uncertainty, as e.g. epidemiologic or climate models do.
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- Cuchulainn
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Re: Philosophy of Mathematics
One thing worse about talking about Cybenko and that's not talking about Cybenko. Imagine explaoing Measure Theory to a data scientist.
Trick1 deus ex machina, Trick2 Antioch granada
Trick1 deus ex machina, Trick2 Antioch granada
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- Cuchulainn
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Re: Philosophy of Mathematics
But at least it deals with deterministic functions.
They are non-constructive. Everett Bishop would turn in his grave..
They are non-constructive. Everett Bishop would turn in his grave..
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- katastrofa
- Posts: 7929
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- Location: Event Horizon
Re: Philosophy of Mathematics
"The essence of the present theory is that no probability, direct, prior, or posterior, is simply a frequency". Kolmogorov's measure theory applied to probability calculus is just a veil covering a much deeper problem. You folk de Finetti and Keynes wrestled with it.
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- Cuchulainn
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Re: Philosophy of Mathematics
I have just made the decision to end my mission as a high school mathematics teacher, a month and a half after starting it following a hasty recruitment process.
When I accepted this position, it was with enthusiasm, commitment and a certain amount of idealism, as I have expressed on this network. I arrived without any preconceptions, driven by the desire to pass on knowledge that I consider essential and to contribute, at my level, to the education of the younger generations. Aware of the challenges of teaching today, I had not anticipated the reality I would face once I was in office.
I felt helpless in the face of the students' attention deficit. A significant proportion of teenagers struggle to concentrate for more than a few minutes, to make a long-term effort, and to let themselves be caught up in the slightest distraction. In this atmosphere of constant dispersion, capturing their interest and maintaining a constructive class dynamic becomes a trying and sometimes impossible daily exercise.
I also discovered that a large part of my students had not acquired the basics necessary to understand high school lessons. Fundamental prerequisites are missing. How can we construct mathematical reasoning when the foundations themselves – such as fractions – are not mastered? This shortcoming considerably slows down the progress of the students and creates a feeling of powerlessness, both for them and for the teacher.
I realized that math in high school is approached in a utilitarian way. Rather than trying to understand, we are content to learn "recipes" dictated by the program, with the sole objective of reproducing them identically on the day of the evaluation, and getting a good grade. This approach empties learning of its substance and hinders the acquisition of a true mathematical culture. A real loss of meaning...
With these observations, I very quickly felt that I did not have the keys to solve the equation I had been entrusted with and that I probably would not find them.
I had a hard time with it.
So I made the decision to leave my post without further delay. Not because of a lack of willpower, but because I don't recognize myself in this educational system whose evolution I had not measured before discovering it from the inside.
This is the first time in my life that I feel from the beginning of the mission that I will probably not be able to complete it. My departure is therefore an acknowledgement of failure – mine. The students have little to do with it: they are as we have educated them. As for my fellow professors, they carry out their mission with passion and professionalism, and try to keep the faith in spite of everything: I admire them.
I understood that it is our entire society, and not just the National Education "machine", that must rethink the education of children in general, and education in particular.
When I accepted this position, it was with enthusiasm, commitment and a certain amount of idealism, as I have expressed on this network. I arrived without any preconceptions, driven by the desire to pass on knowledge that I consider essential and to contribute, at my level, to the education of the younger generations. Aware of the challenges of teaching today, I had not anticipated the reality I would face once I was in office.
I felt helpless in the face of the students' attention deficit. A significant proportion of teenagers struggle to concentrate for more than a few minutes, to make a long-term effort, and to let themselves be caught up in the slightest distraction. In this atmosphere of constant dispersion, capturing their interest and maintaining a constructive class dynamic becomes a trying and sometimes impossible daily exercise.
I also discovered that a large part of my students had not acquired the basics necessary to understand high school lessons. Fundamental prerequisites are missing. How can we construct mathematical reasoning when the foundations themselves – such as fractions – are not mastered? This shortcoming considerably slows down the progress of the students and creates a feeling of powerlessness, both for them and for the teacher.
I realized that math in high school is approached in a utilitarian way. Rather than trying to understand, we are content to learn "recipes" dictated by the program, with the sole objective of reproducing them identically on the day of the evaluation, and getting a good grade. This approach empties learning of its substance and hinders the acquisition of a true mathematical culture. A real loss of meaning...
With these observations, I very quickly felt that I did not have the keys to solve the equation I had been entrusted with and that I probably would not find them.
I had a hard time with it.
So I made the decision to leave my post without further delay. Not because of a lack of willpower, but because I don't recognize myself in this educational system whose evolution I had not measured before discovering it from the inside.
This is the first time in my life that I feel from the beginning of the mission that I will probably not be able to complete it. My departure is therefore an acknowledgement of failure – mine. The students have little to do with it: they are as we have educated them. As for my fellow professors, they carry out their mission with passion and professionalism, and try to keep the faith in spite of everything: I admire them.
I understood that it is our entire society, and not just the National Education "machine", that must rethink the education of children in general, and education in particular.
Re: Philosophy of Mathematics
That’s such a confusing ramble without some context for when and where (we could throw in by whom, but I know you like to protect your sources). It’s hard to map the combination of “high school”, “professors” and “National Education” to any English speaking secondary school system that I can think of.
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- Cuchulainn
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Re: Philosophy of Mathematics
It's a random link from LinkedIn from a French maths teacher.That’s such a confusing ramble without some context for when and where (we could throw in by whom, but I know you like to protect your sources). It’s hard to map the combination of “high school”, “professors” and “National Education” to any English speaking secondary school system that I can think of.
In many European countries a match teacher is addressed as "Professor". In Italy, dottore <==> BA/BSc degree.
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- Cuchulainn
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Re: Philosophy of Mathematics
You're welcome.Thanks, that helps.
I take students for UCB etc. with Calc III and get them to ODE/PDE level (including 1,2 Black Scholes PDE) in 2-3 months. Calc on its own is like a dead end.
Most other courses reflect the originator's background (applied maths, physics) which is not appealing to MFE students. We are holistic/eclectic. // pm..
https://www.datasim.nl/onlinecourses/97 ... -equations
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- Cuchulainn
- Posts: 22926
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- Cuchulainn
- Posts: 22926
- Joined:
Re: Philosophy of Mathematics
"I stared at the differential equation for many, many months. I made hundreds of silly mistakes that led me down blind alleys. Nothing worked..." (Black, 1988).
“In order to solve this differential equation you look at it till a solution occurs to you.”
― George Pólya
“In order to solve this differential equation you look at it till a solution occurs to you.”
― George Pólya
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- Cuchulainn
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Re: Philosophy of Mathematics
ChatGPT-5 has become less of a toy and more of a partner in math research for me.
While many of my colleagues still prefer to laugh at its mistakes and strange answers, my own experience this past month proved to be quite different (and surprisingly meaningful).
How exactly can the model help a mathematician?
Proof assistance
I’ve been working on a long paper about automorphisms of twisted Chevalley groups, which are, in a sense, generalizations of the usual Chevalley groups. Those, in turn, can be seen as generalizations of classical linear groups like SLₙ(𝖱) or SOₙ(𝖱).
Describing automorphisms of these groups, I found the technical work much harder than anything I’d done with ordinary Chevalley groups. Sometimes, things got complicated.
My usual way of using ChatGPT was simple: I’d paste in a page or two and ask it to fix typos. But right after the release of ChatGPT-5, I gave it a short lemma to polish — and instead of just making small language corrections, the model told me: “Your proof looks incomplete.”
I replied, “Well, then complete it.” The model did, and quite nicely.
Next, I gave ChatGPT another lemma that I hadn’t fully worked out myself, asking it to check the draft and suggest possible endings. The model proposed three different ideas, all based on properties of root systems that I couldn’t bring to mind immediately.
On the bus to the university, I thought through one of the ideas and found the proof. Once I got to my office, I described to ChatGPT, very roughly and in Russian, what I wanted to do — and it produced the completed proof, perhaps the hardest one in that section.
Honestly, if a human collaborator had helped me this much with 2–3 lemmas, I would have added them as a coauthor.
Applying ideas by analogy
Another time, I needed to generalize one specific result from Chevalley groups over local rings to Chevalley groups over arbitrary rings. Over local rings, the result had three major steps. I had figured out how to generalize the first one, but I hadn’t started thinking about the other two.
I gave ChatGPT an old paper (for local rings), plus my notes on the first step, and asked it to continue. And it did.
The result, once again, was good enough to count as coauthor-level help.
Search
In another project, I had promised my coauthor to obtain a certain result about automorphism groups of free groups. It wasn’t that easy, and although I had a rough idea of the proof, I kept postponing the work.
Eventually, I opened ChatGPT, explained what I wanted — and within five minutes it found the result, published 25 years ago.
Beyond the jokes, these moments suggest there’s real depth in human–LLM collaborations.
While many of my colleagues still prefer to laugh at its mistakes and strange answers, my own experience this past month proved to be quite different (and surprisingly meaningful).
How exactly can the model help a mathematician?
Proof assistanceI’ve been working on a long paper about automorphisms of twisted Chevalley groups, which are, in a sense, generalizations of the usual Chevalley groups. Those, in turn, can be seen as generalizations of classical linear groups like SLₙ(𝖱) or SOₙ(𝖱).
Describing automorphisms of these groups, I found the technical work much harder than anything I’d done with ordinary Chevalley groups. Sometimes, things got complicated.
My usual way of using ChatGPT was simple: I’d paste in a page or two and ask it to fix typos. But right after the release of ChatGPT-5, I gave it a short lemma to polish — and instead of just making small language corrections, the model told me: “Your proof looks incomplete.”
I replied, “Well, then complete it.” The model did, and quite nicely.
Next, I gave ChatGPT another lemma that I hadn’t fully worked out myself, asking it to check the draft and suggest possible endings. The model proposed three different ideas, all based on properties of root systems that I couldn’t bring to mind immediately.
On the bus to the university, I thought through one of the ideas and found the proof. Once I got to my office, I described to ChatGPT, very roughly and in Russian, what I wanted to do — and it produced the completed proof, perhaps the hardest one in that section.
Honestly, if a human collaborator had helped me this much with 2–3 lemmas, I would have added them as a coauthor.
Applying ideas by analogyAnother time, I needed to generalize one specific result from Chevalley groups over local rings to Chevalley groups over arbitrary rings. Over local rings, the result had three major steps. I had figured out how to generalize the first one, but I hadn’t started thinking about the other two.
I gave ChatGPT an old paper (for local rings), plus my notes on the first step, and asked it to continue. And it did.
The result, once again, was good enough to count as coauthor-level help.
SearchIn another project, I had promised my coauthor to obtain a certain result about automorphism groups of free groups. It wasn’t that easy, and although I had a rough idea of the proof, I kept postponing the work.
Eventually, I opened ChatGPT, explained what I wanted — and within five minutes it found the result, published 25 years ago.
Beyond the jokes, these moments suggest there’s real depth in human–LLM collaborations.
Re: Philosophy of Mathematics
Can one reverse engineer AI to find out what someone, Cuch, say, “said” to ChatGPT to write so eloquently?
Re: Philosophy of Mathematics
He probably used Claude instead.Can one reverse engineer AI to find out what someone, Cuch, say, “said” to ChatGPT to write so eloquently?
if Cuch had used ChatGPT everything would have started with something being "rapidly evolving", like the "rapidly evolving Greenlandic number names..."
Reverse engineer, I'd love to say you can but only to a certain degree and it's way after word2vec has taken place.
Linkedin: https://www.linkedin.com/in/jasonbelldata/
Author of Machine Learning: Hands on for Developers and Technical Professionals (Wiley).
Contributor: Machine Learning in the City (Wiley).
Author of Machine Learning: Hands on for Developers and Technical Professionals (Wiley).
Contributor: Machine Learning in the City (Wiley).