> With ML, most of what we are doing is modeling intractable distributions...

I am aware and we specifically don't directly compute those because they are intractable, hence rendering the need for low level ML practitioners to be familiar with their theoretical properties to be mostly unnecessary. MCMC exists for a reason and modern deep learning contains almost zero direct integration. There are lots of sampling but few integrals.

I have seen high schoolers use and implement VAEs without understanding what the reparametrization trick is.

> Value is too abstract and I think you should clarify

The value of LLMs and similar deep learning classifiers/generators is self evident. If your research is only good for publishing papers, you should stay in academia. You are in no position to judge or gatekeep ML research.

> I think this example better clarifies your lack of understanding in areospace engineering rather than your argument.

I am a pilot, software engineer, and a machine learning practitioner with plenty of interdisciplinary training in other scientific fields. I assure you I am more than familiar with the basics of fluid dynamics and flight principles. Granny knows how to suck eggs, no need for the lecture.

> First, I'm not sure I claimed abstract algebra was necessary, but that's a comment about if you were going to argue with me about "math being a language"

You claimed that people needed to know rings, groups and set theory to debate you on understanding ML. ̶I̶ ̶t̶h̶i̶n̶k̶ ̶y̶o̶u̶ ̶a̶r̶e̶ ̶t̶h̶e̶ ̶o̶n̶e̶ ̶w̶h̶o̶ ̶n̶e̶e̶d̶s̶ ̶t̶o̶ ̶g̶o̶ ̶b̶a̶c̶k̶ ̶t̶o̶ ̶s̶c̶h̶o̶o̶l̶ ̶a̶n̶d̶ ̶s̶t̶o̶p̶ ̶g̶a̶t̶e̶ ̶k̶e̶e̶p̶i̶n̶g̶.̶ ̶ ̶Y̶o̶u̶ ̶r̶e̶m̶i̶n̶d̶ ̶m̶e̶ ̶o̶f̶ ̶t̶h̶o̶s̶e̶ ̶f̶u̶n̶c̶t̶i̶o̶n̶a̶l̶ ̶p̶r̶o̶g̶r̶a̶m̶m̶e̶r̶s̶ ̶w̶h̶o̶ ̶w̶o̶u̶l̶d̶ ̶r̶e̶w̶r̶i̶t̶e̶ ̶n̶e̶u̶r̶a̶l̶ ̶n̶e̶t̶w̶o̶r̶k̶ ̶l̶i̶b̶r̶a̶r̶i̶e̶s̶ ̶i̶n̶ ̶H̶a̶s̶k̶e̶l̶l̶ ̶b̶e̶l̶i̶e̶v̶i̶n̶g̶ ̶c̶a̶t̶e̶g̶o̶r̶y̶ ̶t̶h̶e̶o̶r̶y̶ ̶w̶o̶u̶l̶d̶ ̶u̶n̶l̶o̶c̶k̶ ̶s̶o̶m̶e̶ ̶m̶a̶g̶i̶c̶ ̶i̶n̶s̶i̶g̶h̶t̶ ̶t̶h̶a̶t̶ ̶w̶o̶u̶l̶d̶ ̶l̶e̶a̶d̶ ̶t̶h̶e̶m̶ ̶t̶o̶w̶a̶r̶d̶s̶ ̶A̶G̶I̶.̶

̶I̶t̶ ̶m̶u̶s̶t̶ ̶b̶e̶ ̶n̶i̶c̶e̶ ̶u̶p̶ ̶t̶h̶e̶r̶e̶ ̶i̶n̶ ̶t̶h̶e̶ ̶i̶v̶o̶r̶y̶ ̶t̶o̶w̶e̶r̶ ̶o̶f̶ ̶a̶c̶a̶d̶e̶m̶i̶a̶.̶ I pity your students. Those who teach has a duty to encourage value creation and seeking out knowledge for its own sake, not constantly dangling a carrot in front of the student like leading a donkey. Don't gatekeep.

> I don't know how to address this because I'm not sure where I made this claim.

I am referring specifically to:
I'm sure many here recognize the reference[0], but being able to make a model that performs successfully on a test set[1] is not always meaningful. For example, about a year ago I was working a very big tech firm and increased their model's capacity on customer data by over 200% with a model that performed worse on their "test set". No additional data was used, nor did I make any changes to the architecture. Figure that out without math. (note, I was able to predict poor generalization performance PRIOR to my changes and accurately predict my model's significantly higher generalization performance).

̶T̶h̶e̶r̶e̶ ̶a̶r̶e̶ ̶m̶a̶n̶y̶ ̶w̶a̶y̶s̶ ̶t̶o̶ ̶t̶e̶s̶t̶ ̶c̶a̶u̶s̶a̶l̶i̶t̶y̶.̶ ̶T̶h̶e̶ ̶d̶a̶t̶a̶ ̶s̶c̶i̶e̶n̶c̶e̶/̶s̶t̶a̶t̶i̶s̶t̶i̶c̶ ̶w̶a̶y̶s̶ ̶a̶r̶e̶ ̶S̶p̶e̶a̶r̶m̶a̶n̶/̶P̶e̶a̶r̶s̶o̶n̶ ̶r̶a̶n̶k̶s̶ ̶a̶n̶d̶ ̶t̶ ̶t̶e̶s̶t̶s̶.̶ ̶T̶h̶o̶s̶e̶ ̶a̶r̶e̶ ̶g̶e̶n̶e̶r̶a̶l̶l̶y̶ ̶l̶i̶n̶e̶a̶r̶.̶
̶
̶h̶t̶t̶p̶s̶:̶/̶/̶l̶i̶n̶d̶e̶l̶o̶e̶v̶.̶g̶i̶t̶h̶u̶b̶.̶i̶o̶/̶t̶e̶s̶t̶s̶-̶a̶s̶-̶l̶i̶n̶e̶a̶r̶/̶
̶
̶A̶l̶t̶e̶r̶n̶a̶t̶i̶v̶e̶l̶y̶ ̶t̶h̶e̶r̶e̶ ̶a̶r̶e̶ ̶M̶L̶ ̶m̶e̶t̶h̶o̶d̶s̶ ̶l̶i̶k̶e̶ ̶g̶r̶a̶p̶h̶i̶c̶a̶l̶ ̶m̶o̶d̶e̶l̶s̶ ̶b̶u̶t̶ ̶I̶ ̶d̶o̶n̶'̶t̶ ̶t̶h̶i̶n̶k̶ ̶t̶h̶a̶t̶'̶s̶ ̶w̶h̶a̶t̶ ̶y̶o̶u̶ ̶a̶r̶e̶ ̶r̶e̶f̶e̶r̶r̶i̶n̶g̶ ̶t̶o̶ ̶h̶e̶r̶e̶.̶ ̶F̶o̶r̶ ̶d̶e̶e̶p̶ ̶l̶e̶a̶r̶n̶i̶n̶g̶ ̶s̶p̶e̶c̶i̶f̶i̶c̶a̶l̶l̶y̶ ̶t̶h̶e̶r̶e̶ ̶a̶r̶e̶ ̶t̶r̶i̶c̶k̶s̶ ̶w̶i̶t̶h̶ ̶s̶a̶m̶p̶l̶i̶n̶g̶ ̶t̶h̶a̶t̶ ̶y̶o̶u̶ ̶c̶a̶n̶ ̶u̶s̶e̶ ̶t̶o̶ ̶e̶y̶e̶b̶a̶l̶l̶ ̶t̶h̶i̶n̶g̶s̶,̶ ̶g̶u̶i̶d̶e̶d̶ ̶b̶y̶ ̶i̶n̶t̶u̶i̶t̶i̶o̶n̶.̶
̶
̶H̶e̶r̶e̶'̶s̶ ̶a̶ ̶g̶o̶o̶d̶ ̶r̶e̶f̶e̶r̶e̶n̶c̶e̶ ̶o̶f̶ ̶w̶h̶a̶t̶ ̶I̶ ̶m̶e̶a̶n̶:̶ ̶
̶h̶t̶t̶p̶s̶:̶/̶/̶m̶a̶t̶h̶e̶u̶s̶f̶a̶c̶u̶r̶e̶.̶g̶i̶t̶h̶u̶b̶.̶i̶o̶/̶p̶y̶t̶h̶o̶n̶-̶c̶a̶u̶s̶a̶l̶i̶t̶y̶-̶h̶a̶n̶d̶b̶o̶o̶k̶/̶l̶a̶n̶d̶i̶n̶g̶-̶p̶a̶g̶e̶.̶h̶t̶m̶l̶
̶h̶t̶t̶p̶s̶:̶/̶/̶a̶r̶x̶i̶v̶.̶o̶r̶g̶/̶a̶b̶s̶/̶2̶3̶0̶5̶.̶1̶8̶7̶9̶3̶
̶
̶A̶g̶a̶i̶n̶ ̶m̶o̶r̶e̶ ̶o̶f̶ ̶t̶h̶e̶s̶e̶ ̶a̶r̶e̶ ̶e̶m̶p̶i̶r̶i̶c̶a̶l̶ ̶c̶o̶m̶m̶o̶n̶ ̶s̶e̶n̶s̶e̶.̶
No need for mathematical maturity or any grasp of discrete mathematics.

> Maybe I should have been clearer by adding a line that most people do not take calculus in high school. While it is offered there, these are the advance courses, and I'd be wary of being so pejorative. I know a large number of great mathematicians, computer scientists, and physicists who did not take calculus in high school. I don't think we need to discourage anyone or needlessly make them feel dumb. I'd rather encourage more to undertake further math education and I believe the lessons learned from calculus are highly beneficial in real world every day usage, without requiring explicit formula writing (as referenced in my prior post).

Okay fair you have a point. I forgot not all schools offer AP classes and advanced mathematics.

I believe we both share the view that education is important, but disagree on how much mathematical understanding is truly necessary to apply or advance ML. I suppose we will have to agree to disagree.