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Comment by mnemonicsloth | Hacker News Reader

Math.

They tell you it's useful (it is) and beautiful (in a way, for a slightly stretched definition of beauty). What they don't tell you right away is that there's no end to the stuff. Sooner or later you have to say "enough." And since a significant fraction of what I learned before I said that was useless [1], I wish I had said it sooner.

[1] I know you're not supposed to say math is useless. It's the kind of thing a disaffected sixth grader would do. But there are a lot of unnecessary proofs in math: things that seem perfectly obvious and are in fact true, but that require a long and counterintuitive argument to prove. Time and again I read the argument, pondered it, more than half-memorized it when the books could just as easily have said "A complex argument is necessary to establish what is obviously the case. Find it if you want to in Appendix J."

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> things that seem perfectly obvious and are in fact true,

The problem is there are lot of things that seem perfectly obvious and are, in fact, false. It's not easy to tell them apart.

I think it depends on what you mean by false. I've always thought of falsity as a continuum. At one end are obvious non sequiturs like 1=3. A little further along are the novice blunders everybody makes while learning the theory. At the far end are propositions where you have to cook up a really abstruse counterexample to show they aren't true. But that's tantamount to saying that they almost are true. And in fact a lot of the time you can treat them as true and not get into trouble.

Physicists, to take a notorious example, spend a lot more time on the real line than mathematicians do, but in their day to day work they ignore most of what we know about the real number system.

> At the far end are propositions where you have to cook up a really abstruse counterexample to show they aren't true. But that's tantamount to saying that they almost are true. And in fact a lot of the time you can treat them as true and not get into trouble.

It seems as if you totally didn't get the point of proof based mathematics. Falsity is not a continuum, either something is false or its true. If you can produce a counterexample the proposition is false.

What are you, some kind of Bourbakiste?

Let a little intuition into your life.

This is not entirely true and depends on the logic system that you choose. Something that helped open my eyes to the different possibilities is this article, which talks about producing a formal logical system for statements in Buddhism:

https://aeon.co/essays/the-logic-of-buddhist-philosophy-goes...

This is written by a professor that I think has produced some really interesting results. Anyway, he discusses something called a plurivalent logic, which is a kind of paraconsistent logic:

https://en.wikipedia.org/wiki/Paraconsistent_logic

These logical systems allow for more than true or false. For example, they allow for neither true nor false as well as both true and false. Outside of their general theoretical interest, there are direct applications to systems with contradictory information. I like the paper "A Useful Four-Valued Logic" by Nuel Belnap:

https://link.springer.com/chapter/10.1007/978-94-010-1161-7_...

which discusses a 4-value logic system and its application to databases.

Math really opens you up to new ideas and changes your way of thinking, but you're right it's a serious life long investment, not just a little tool you can pick up on the side.

> What they don't tell you right away is that there's no end to the stuff

Is there an end to any field worth studying? I would argue the allure of studying such fields lies in the endless horizons. The possibilities of expanding human knowledge by pushing beyond what is known.

It seems what they really forget to tell you was "if you don't like Math - you don't have to study it."

Hi, there. Professional mathematician here. Sorry to hear about your experience. In truth, I believe you. That said, I believe there are some historical and pedagogical reasons for your experience.

1. Often, math is used to teach skills outside of their mathematical usefulness and I wish we'd be more honest about it. For example, take the controversial topic of long division. As a professional, I never use. Ever. In fact, we use a different algorithm computationally on a computer. That said, why did we teach it? It's one of the first algorithms that students learn. As such, it teaches organization and a methodical following of steps. Now, is this the only way to teach this? Of course not. In years past, we taught square roots. Candidly, a similar sort of precision can be taught with computer programming. However, long division can also fulfill this role even though, again, it's practically not that useful.

2. As far as proofs, you're right. It's just a complex argument that something is true or not. In fact, it's an incredibly flawed process as well since spoken and written language tends to lack the precision to be absolutely sure. Now, there are very formal ways to prove things using techniques from, for example, the mathematical logic community, which can be computationally realized in proof based systems like Coq or Isabelle. However, this is hard, so no one really does it. As such, why do we stick with a possibly flawed proof system that's a pain for most people?

Well, I'm sure there are lots of reasons, but the big one for me is that there many circumstances where intuition breaks down. In my studies, the first big breakdown in intuition occurred during calculus and the first experience with the infinite. Another big breakdown occurred in the transition between real analysis (calculus) and functional analysis. For example, the unit sphere is compact in finite dimensions, but not in infinite. It just works differently.

Now, does that mean that intuition isn't good or used? Of course not. However, the proofs help bring forth the brittleness, or robustness, of a situation by systematically breaking down where things work or do not work. For me, it's a way to add structure to a problem that I'm working with in order to ensure I know what's going on.

OK, so I'm a mathematician and my interests may be different. However, think about it from an engineering discipline. We can model things like fluid flow or electromagnetics and achieve really good, useful results. Most of the time. All of these equations have assumptions behind them and when these assumptions are violated, everything breaks. For example, do the governing equations and algorithms work when the domain has a reentrant corner?

As another example, we use optimization solvers in many domains from engineering design to machine learning. The equations that these solvers use for constrained optimization depend on something called constraint qualifications. When these qualifications don't hold, everything breaks and we don't find solutions. For me, the constraint qualifications such as when the tangent cone coincides with the linearized cone (Abadie CQ) aren't particularly intuitive. It's an artificial construction that results from the proof of equivalence between two formulations. However, it's also essential for the algorithms to work.

Anyway, none of this is meant to refute your experience. I completely believe you. Really, it's a way to provide some clarity as to why some of these things are taught in this way and why they may be valuable for other reasons. I do believe that math can be taught better. I also believe that there's not a universal way to teach math and that different approaches resonate with different people.

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