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Comment by dsacco | Hacker News Reader

> It's often said that one of the benefits of studying math is that it improves critical thinking skills. I believe this to be true, but I also believe that if our primary concern is improving our day to day critical thinking skills, then we would be better served studying natural language philosphy than mathematics, because on a day to day basis we tend to think and reason and debate and engage with one another in natural language.

I agree. For this reason, my favorite philosopher has always been Wittgenstein. I also have an affection for the existentialists, but language games demonstrate that philosophy can be eminently practical and relevant despite the pejorative connotations of the term, "philosophizing."

That said, I really don't think philosophy should be its own major in most schools. It would be better, in my opinion, to teach philosophy as we teach arithmetic and language, rather than a focused degree.

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And what if someone wants to become a philosopher? That's like saying we shouldn't teach match as a major because every kid gets taught arithmetic. Well, some people want to do more with math than arithmetic.

Frankly, I disagree that mathematics should be its own degree at the undergraduate level as well.

Let me clarify: I think people should be able to become philosophers, just like they can become mathematicians. But I’d advocate for changing the undergraduate curriculum - move proof-based mathematics into a targeted graduate degree, and emphasize computational mathematics like calculus and (linear) algebra, but make them core components of a much more generalized degree. Add in language and logic studies that borrow significantly from philosophy.

With very few exceptions, you’re not doing research mathematics (i.e. being a mathematician) without significant graduate study. Similarly, for majors like philosophy that are “watered down” until graduate or postgraduate work, I don’t think they should be eligible for targeted degrees at the undergraduate level. This would have the added benefit of 1) delaying time to major decision for college-aged students and 2) keeping their options open for eligibility to targeted graduate degrees later on, by limiting the number of prerequisites. You don’t need undergraduate number theory to do well in an early graduate number theory course. The 300 and 400 level philosophy courses you take in an undergrad setting will mostly not have value on paper, so in my opinion we should shift them to the degree that has value on paper.

But at this point I’m pontificating about the education system, not philosophy in particular :)

If you don't teach proof-based mathematics at the undergraduate level then aspiring mathematicians would have to take those courses at the graduate level. This would cause advanced degrees to take considerably longer to obtain.

Like programming, writing proofs is a skill that takes practice to get good at. If you were to suggest that we shouldn't teach programming until graduate-level CS courses, you would be laughed out of the room. I think the best mathematicians in history started learning how to prove things as children. How about we start there, instead?

> If you don't teach proof-based mathematics at the undergraduate level then aspiring mathematicians would have to take those courses at the graduate level. This would cause advanced degrees to take considerably longer to obtain.

No it wouldn't. Most math PhDs require six or so courses before the research - you can begin with analysis or abstract algebra for proofs. The current system does have unnecessary redundancy.

> Like programming, writing proofs is a skill that takes practice to get good at. If you were to suggest that we shouldn't teach programming until graduate-level CS courses, you would be laughed out of the room.

Programming is more comparable to the computational mathematics that I've already mentioned should be in undergrad; more to the point, a lot of computer science theory need not be taught in undergrad either. Computer science theory is much closer to proof-base mathematics.

> I think the best mathematicians in history started learning how to prove things as children.

Approximately no one is one of the best mathematicians in history. That's an attractive idea, but I think it's more realistic to assume they were extraordinarily precocious than that proofs are easier to digest at a younger age. I could point to people who mastered computational mathematics at a young age too (like Einstein, at 14), but that doesn't say anything about which is easier to learn. I'm talking about which is more widely applicable for practical work.

Most of the math majors I know didn't have much trouble with proofs as 19-22 year olds. You're assuming a pretty low ability.

The parent mentioned children, and I mentioned a (very talented) 14 year old. You're talking about 19-22 year olds; I have no reservations about that age bracket tackling proofs, but that's not really the point.

Your philosophy towards mathematics education is interesting. If you don't mind me asking a direct question: what's the point?

Here's my take on the purpose of math courses in college: any college graduate should be able to teach themselves something "like" calculus or linear algebra upon graduation. So you should teach calculus and especially linear algebra in such a way that a student who has never seen e.g. graph theory can pick that up a bit of graph theory their own using the thinking skills they acquired by studying linear algebra or calculus. Ditto for dynamic programming or combinatorics or basic probability or...

So if you're not teaching proofs, wth are you teaching all semester? A few conceptual underpinnings that take about a week to explain, and then a whole bunch of crap Mathematica can do for you anyways.

> computational mathematics like calculus and (linear) algebra.. move proof-based mathematics into a targeted graduate degree

Memorizing symbolic calculations and/or understanding (how to use) numerical algorithms do not endow students with the skills required to learn new mathematics. Proof-based calculus and algebra courses do teach those skills.

The recent post on HN about PID control comes to mind. I'd expect someone with a bit of practice at writing proofs to be able to teach themselves why PID controllers work and avoid pitfalls. But I would not expect your average human-meat-based-derivative-and-integral-calculator to be able to understand the same.

To put it in terms of a folk saying about fishing: if you teach a man to calculate, he can perform a specific calculation. If you teach a man how to write proofs, he can learn any calculation he might need throughout his lifetime.

I think you and I mean different things by computational mathematics. Calculus isn't analysis, but it still has substantial rigor to it. You might not be able to develop calculus from scratch after studying it, like you could by working through real analysis, but that doesn't mean studying calculus has to be hollow and rote. For example, I find Spivak's Calculus to be a good compromise between full on analysis and the kind of formula memorization you're talking about.

Ideally computational mathematics teaches more than just the simple "what" - it should address at least one of the "how" or "why", but it doesn't necessarily need to address all three. I also disagree that the point of the education should be for students to be capable of teaching themselves novel mathematics (and to be honest, I'm pessimistic most graduating math undergrads could do that simply because they made it through something like Rudin).

The difference between mathematics and philosophy here is that an undergraduate math degree is considered as the most appropriate training for middle school math educators, whereas there is no obvious career path an undergraduate degree in philosophy leads to.

That's a good point. But if that's the case, undergraduate mathematics does not need to introduce abstract algebra, topology, linear algebra, analysis, or any of the other higher level courses. In fact, they could finish out at calculus.

I think there are overlooked advantages for middle and high school teachers to be conversant with higher levels of math than they’ll ever directly teach to young people

Econ should move that way too. Most Econ undergrad degrees are so watered down than Econ grad schools prefer Math and Physics majors who can actually do the work.

I was taught proof-based mathematics in high school…

I concur with your last statement. Similar to mathematics, it make sure a very good second degree as well. Econ majors are a dime a dozen, but Econ/philosophy double majors are much more likely to Ben precise thinkers. Same benefit to precision when pairing it with CS.

Wittgenstein is one of my favorites as well for the same reason. The Tractatus should be required reading for everyone, not just philosophy students.

If one reads Tractatus, then a reading of Philosophical Investigations is likely needed shortly thereafter.

I can't resist:

https://www.youtube.com/watch?v=uSdHoNJu5fU

Philosophical Investigations is better.

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