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Comment by ivansavz | Hacker News Reader

Very nice lecture that connects calculus to the laws of nature. I've always thought applied math has a leg up on abstract math because of the connections to real-world ideas.

I encourage anyone who liked the talk to try out some of the calculations on their own using https://live.sympy.org/ which is an online REPL with build in calculus function like integrate, diff, limit, summation, etc. Here is an example:

   >>> summation((1/2)**(2*n), [n,0,oo])
   4/3

via https://live.sympy.org/?evaluate=summation((1%2F2)**(2*n)%2C... note the answer is an exact rational number (class 'sympy.core.numbers.Rational) and not a float approximation 1.33333...

For a quick tutorial on how to use SymPy, check "Taming math and physics using SymPy" which is avail in printable format https://minireference.com/static/tutorials/sympy_tutorial.pd... or notebook https://nbviewer.jupyter.org/github/minireference/sympytut_n...

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>I've always thought applied math has a leg up on abstract math because of the connections to real-world ideas.

I don't particularly like this dichotomy. Abstract math is connected to the real world because if it wasn't it wouldn't exist. Overall people who say this stuff are just referring to math that is only derived to work from an experimental standpoint and not from an axiomatic standpoint.

Machine learning is one example of a mathematical field with no axiomatic basis and the the pythagorean theorem is an example of one that is derived from the axioms of euclid. Both exist in the real world and are therefore applicable.

I don't think it's worthwhile to engage in applied versus pure "math war" either, but I don't really agree with the point you've made. All applied mathematics has an axiomatic basis. When certain mathematical theories can be "applied" to real world problems, it simply means the relevant axioms and definitions are a robust approximation of reality.

To speak to your example directly: machine learning absolutely has an axiomatic basis. You can conduct legitimate research in implementations and software or hardware optimizations thereof; however, fundamentally every experimental result in machine learning is an application of a variety of theorems in linear algebra, probability theory or calculus.

Imho this is not the right way to think of it. Experimental results aren't applications of any theorem, they are just measurements. And the fact that these measurements may or may not come from machines we know how to "perfectly" measure (eg digital computers) doesn't give them any axiomatic basis. I'm not sure anyone is doing machine learning using symbolic computation, afaik it's mostly about low-precision floats--which do have some foundations themselves, but quite far away from the reals-based vector spaces upon which optimization theory is based. As addition to this point: most convergence results are much better in practice than in theory, ie we do not yet have satisfying predictive theories for experimental optimization.

On this pure vs applied math thing, imho it is indeed a false dichotomy: there are symbolic objects of study and there are empiric or otherwise "preexisting" objects of study. We may use symbolic objects to approximate empiric objects and provide "applied theories" with predictive power, but we may also abstract empiric objects into new symbolic theories (both usually being done together). There is a funny phenomenon in the more abstract domains of math where at some point everyone is routinely talking about "intuition", "seeing things" and "morality" of facts. For me, building up intuition in a domain is about taking an axiomatic theory and transforming your view of it into something of the more empiric kind, one where you believe in an external understanding of how things behave. A symbolic fact being "moral" when it's consistent with the empiric counterpart you developed in your mind. My conclusion: some things are axiomatic/symbolic and some are not, but we mostly treat them the same way, ie by building empiric mental models. How else would we have proof ideas?

> Abstract math is connected to the real world because if it wasn't it wouldn't exist.

I'm not a mathematician by any means, but seeing that we know very few things about the foundations of the real world I think that trying to strongly link mathematics to it (meaning to the real world) also means "dragging" maths down to not really having any clear foundations.

Yes, I do know of the opposite road taken by some very smart people, i.e. tying maths to reality, finding the foundations of maths and how maths work => we now have a pretty good out idea of the foundations of reality (for lack of a better expression) and how reality "works" via its connections with maths.

It's just that even though we've been quite successful up until step 2 (meaning in postulating the maths <-> reality connection and in finding out how maths really works) it is my understanding that until now we have failed quite miserably at step number 3, as we haven't got the slightest idea of what this "reality" is made up of (again, for a lack of a better expression).

seeing that we know very few things about the foundations of the real world

we haven't got the slightest idea of what this "reality" is made up of

What in the world are you talking about? Setting aside unanswerable nonsense philosophical questions, we have a damn good idea what reality is made of.

In math, we actually know and can prove beyond all doubt something to be true.

The same can't be said about the reality as we can never be sure that we've accounted for all variables and filled them correctly. This applies to everything in physics, even something as basic as "how long will this Apple take to fall to the ground"... You can make an estimate and then let reality take it's course. And when you're finally looking at the numbers it might've been correct down to milliseconds... But there is going to be some drift

in math, it's gonna be one exact point, as everything is accounted for. In reality, the Apple is not going to land at exactly the same time, as you forgot to include something like the the current air pressure or a comet vaporizing the planet in flight!

This is true. Not only is it impossible to prove anything in science and therefore reality, but we also view things through a blurry lens. Results arrive with a limited amount of significant figures or in the form of a statistic. Technically, although things cannot be proven in science, things can actually be disproven in science, but due to the blurry lens we are unable to fully do so. Instead science is largely about establishing correlations and to an even harder extent trying to give a blurry statistic to causation.

Do note that for math, you can prove things but two things must be assumed. The first assumption is that all axioms related to the proof are true. The second assumption you make is that logic as we know it is true and always consistent regardless of context.

There is one more interesting thing regarding science. Because nothing can be proven in science and because all science is, is establishing correlations, one thing that we assume is true in the real world is probability. In math the axioms of probability are basically arbitrary ratios assigned to sets with the theorems blossoming outward from different compositions of these sets and ratios. The theory itself has nothing to do with random events. So literally the theory of probability is just about sets and an associated rational number representing a portion of that set.

If we had a 6 sided dice and we rolled it billions of times the reason why the number 5 appears close to a 1/6 portion of all the results is a mystery. We literally assume this is the case and that the axioms of probability which are essentially just sets and ratios actually applies to random events and happenings. All other science is derived from this assumption.

See y4mi’s reply. Also, we cannot account for about 70% or 80% of the material Universe (the one we presume has weight), we even found a fancy name for it, “dark matter”. This is pure mysticism, this is definitely not knowing what reality is made up of.

> I've always thought applied math has a leg up on abstract math because of the connections to real-world ideas.

You may enjoy this conversation from 1972 (https://youtu.be/avSHHi9QCjA) with Cornelius Lanczos and others on his dissatisfaction with the applied/pure math dichotomy.

> I've always thought applied math has a leg up on abstract math because of the connections to real-world ideas.

godel settled this, so it's ultimately moot.

does sympy render latex in the repl...? also haha oo as an alias for float("inf") or whatever is clever.

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