What is mathematical thinking? (2012) | Hacker News Reader

What is mathematical thinking? (2012)

devlinsangle.blogspot.com

85 pointsmjreacher46 comments

Wow, so, so much venom on this thread. For those of you who enjoyed the article (like myself), you'll see that his class ("Introduction to Mathematical Thinking") starts today as a Coursera Stanford MOOC here: https://www.coursera.org/learn/mathematical-thinking

Don't Coursera classes always start $TODAY?

Some do. Others have fixed starting dates.

Some classes are browsable everyday but the classroom experience had a schedule last time I checked.

> The opening paragraph is a parody of such writing. This comment was added a day after initial publication, when a letter from a reader indicated that he missed the fact that the opening was a parody, and complained that he found it difficult to read. That difficulty was, of course, the whole point of the opening, but that point is lost if readers don't recognize what is going on. So I added this remark.

Read this ^ before reading the article

This particular 10-year old post does a disservice to Prof. Devlin's work. I actually took his course on Mathematical Thinking and found it quite interesting, as was the book withe the same title I read several years ago. He is able to make certain concepts very clear, and I wish more profs had a similar attitude (many already do!).

That said, I believe most people here already took a course on discrete mathematics including logic and so on, so I don't think it would be particularly interesting to HN audience.

Two foundational mathematical thinking constructing constructs (if I may say):

  * (Existential-Universal) quantification

  * (Epsilon-Delta) language

Once people start to not only understand it, but actually think that way, they start to be scientists and mathematicians. It opens the door of refutable and constructive thinking.

I've seen also a lot of people struggle with definitions.

One problem with epsilon-delta definitions is that they don't scale. In many topological spaces, there's no concept of distance and so epsilon-delta stuff has to be replaced with more general open sets. About the same reason why sequences have to be replaced with nets and filters.

yeah there are like 10 million ways of defining cont. or limits in R^N that turn out to be the same

R^n is Euclidean and so epsilon-delta defns work just fine. I am talking about topologies like cofinite topology or cocountable topology where the picture doesn't look like a grid to which you can apply distance related notions of epsilon- room here and delta-long there.

Yes of course, I didn't mean that we should stick to metrizable spaces.

Though it's intuitive for humans and its limits are quite broad, if I may say :)

Eg. we usually don't start defining the derivative with distributions.

I didn't mention geometry neither btw though it's foundational too imo.

Mathematical thinking is actually pretty easy to characterize as thinking that is equipped with a proof checker. Mathematics itself is then just the study of proof checkers. Interestingly, this places much of it firmly as a subset of the natural sciences, as proof checkers are physically realizable as compilers (well, certain compilers anyway).

Anything else you bring to the study of mathematics is a heuristic, made admissible by composition with the proof checker. Many of these heuristics seem universal, but they aren't, and it's harmful to assume so.

> Interestingly, this places much of it firmly as a subset of the natural sciences, as proof checkers are physically realizable as compilers (well, certain compilers anyway).

That's a bold assertion. I think entire books have been written as to what kind of "science" mathematics is, if it even is one, and what the ontological status of mathematical truths is.

Regardless of what mathematics truly "is", though, the process of doing mathematics is substantially different from that of doing research in the natural sciences. The arbiter of truth in the latter is experiment, and in particular, inductive reasoning. But experiment and inductive reasoning have no place in mathematical truth (although they may be used to generate hypotheses).

And that's leaving aside all of the fundamentally non-scientific questions of mathematical philosophy, such as "is mathematics true/real" or, somewhat more practically, "should we assume the axiom of choice, or the law of excluded middle, or the existence of different kinds of infinities".

Entering a proof into a physical proof checker is mathematics. That's what mathematicians are, after all. That's what publication is, or aspires to be, anyway.

That is at most a very small part of doing mathematics.

You're not getting my point here. People are physical proof checkers. Proof is the language of mathematics. The language is not the content.

How does that relate to what I originally wrote?

Essentially a lite version of Velleman's _How to Prove It_?

I don't think you learn mathematical thinking very well by hearing it described. I think you learn it by immersing yourself in mathematical logic, and then proofs (basic proofs, any proofs). Formal math usually appears in HS geometry starting with truth tables and logical operators, not as a long-winded explanation of how you should think about geometry from a math perspective.

Davis and Davis/Hirsch wrote the book(s) on this. The mathematical experience (1981) is a great read.

What is mathematics really? (1997) also fine.

Mathematical thinking is logical, relational, recursive, quantitative and analytical thinking. All these types of thinking are expressed through various mathematical techniques. A good book on the subject is "Essentials of Discrete Mathematics" by David Hunter.

A great exposition on how mathematical thinking is performed is Imre Lakatos' Proofs and Refutations.

>>> It’s one of those analogies that is brilliant if you are sufficiently familiar with all four components, but hopeless as a way to explain one in terms of the other three.

Ah yes, one of those - also known as a completely useless analogy.

That is his point, it was an useless analogy, that is why he said it in that way and why he tried to find another way of explaining it.

He doesn't have to spell out that those conditions makes the analogy irrelevant, it is an obvious conclusion, anyone who read to that part would realize his analogy was worthless so he doesn't have to spell it out, in a math proof what he did would be the same as spelling it out. But to you it looks like he wasn't himself aware the analogy was irrelevant, because he didn't' say it straight out. But he crafted the text in precisely such a manner that you and basically anyone else reading this text would realize the analogy was worthless without him stating so, that is a part of mathematical thinking: how to communicate things precisely.

Of course regular people like redundancy and repetition, which is why I repeat my self a lot when I write on forums like this, it looks like this mathematician hasn't fully realized that he can't state things just once when writing things for non-mathematicians to read.

If someone claims to be a "mathematical thinker" then probably they should be able to convey their thoughts in a precise and concise manner.

For example by starting with a definition and then expanding/explaining it.

Blogspam written by someone who sounds like a teenager with ADHD should be posted in the "iamverysmart" subreddit, not on HN.

Ehe ...you do realise you're speaking about this guy here right? http://profkeithdevlin.com/

Does it magically stop being blogspam if we know who that is supposed to be?

Yes, A blog post from someone authoritative about the content certainly warrants higher consideration before being dismissed as blogspam by a random passing comment by someone who isn't, imho.

Does he have a Putnam?

So I take it you didn't read the post you commented on, else you'd know that the post describes the distinction between mathematical thinking and doing mathematics.

So he doesnt have a Putnam?

So you haven't actually read the post?

It's an old HN in-joke: https://news.ycombinator.com/item?id=35079

What does that have to do with anything?

Is there a point to your question?

> they should be able to convey their thoughts in a precise and concise manner.

But that is exactly the thing, most people aren't able to convey their thoughts in a precise manner, it requires a lot of words. Programmers knows this as well, it is hard to describe things precisely, human written language is very imprecise. The difference is that mathematicians has to explain things precisely to other humans, while programmers just has to explain things precisely to a computer, never to other humans, it is fine if the other humans just gets a cursory understanding.

So when it comes to explaining things precisely to other humans mathematicians are the experts, not programmers, programmers still thinks that imprecise explanations are good enough for human consumption.

I don't think mathematicians are always precise. I think what mathematicians are good at is mapping intuition to and from precision. So when speaking to each other for the sake of brevity they'll speak intuitively, but rely on an assumed intuition-precision mapping in order to have common understanding.

So, so much hubris.

- "My analogies are brilliant, but you need to already understand the field for them to be brilliant, which I learned after tons of people told me that the analogy is actually not brilliant. But they are brilliant!"

- "You need to identify as an X to be good at X" -- No. Polar opposite. The best people in any area don't identify as anything. If anything, the opposite is the case - once you think you're a great mathematician, you stop becoming exactly that.

- "If you had any difficulty following that first paragraph (only two sentences, each of pretty average length), then you are not a good mathematical thinker" -- or your semantics just suck and you're used to dealing with reading shitty semantics. That doesn't make you a good mathematical thinker, it just means that you opted for a bigger buffer than 99% of people need in their day to day jobs, even those which are able to either come up with or understand foreign omplex models on the fly. Also, the length of the sentence doesn't matter, its the density and arrangement of information in it - and, like I said, your semantics suck.

- "That then, is mathematical thinking. How do you teach it? Well, you can’t teach it; in fact there is very little anyone can teach anyone. People have to learn things for themselves; the best a “teacher” can do is help them to learn. " -- the entire paragraph just wastes the readers attention. Teaching is to help learn. What are you saying? Nothing of value.

Posts like these are why I can't take academics serious. This is high-school levels of ignorance, crossing into five different fields that the author is not even close to being competent in. What the hell?

Also: “my work was extremely valuable to a group of academics, but not worth publishing.”

What?

I'm not sure you either read the entire post or if you did, you grasped the point of the post. Prof. Delvin [1], the author, has been pushing to make math literacy and mathematical thinking more accessible for a long time now. This post is a guide to what that means.

[1] http://profkeithdevlin.com/

..so much for the irony of calling out hubris and stating 'the author is not even close to being competent'

>..so much for the irony of calling out hubris and stating 'the author is not even close to being competent'

I don't see how your answer relates to my criticism at all. What I'm pointing out is that the author touches on a variety of topics where he clearly exceeds his field of expertise. I'm not saying that he is incompetent as a mathematician at all, nor that he is ill-intended in writing the post.

> I don't see how your answer relates to my criticism at all. What I'm pointing out is that the author touches on a variety of topics where he clearly exceeds his field of expertise. I'm not saying that he is incompetent as a mathematician at all, nor that he is ill-intended in writing the post.

Forgive me, I'm confused. You did claim hubris. Which topics, precisely, does the good professor touch upon that is not within his field of expertise as a mathematician and an educator ? ... within this post that you claimed is full of hubris ?

From the post:

>If you want to be good at activity X, you have to start to see yourself as an X-er – to act like an X-er.

This is what struck me the most. It's a terrible generalization (beyond the limits of a mathematicians competence) that doesn't hold up - you can be good at any given topic without actually identifying as anything. Since I have no degree in psychology, I'm in no position to actually claim to know better. That being said: the notion that identifying with anything somehow enables you to reach deeper ends of a field is simply off. Afaik, identifying with anything is more a blockage than anything else. Also, the mere fact that he is an educator is meaningless to me in this context, as some of the most ignorant/filled with hubris people I've come across can call themselves educators, so if anything I take that term as counter-indicative for general competence.

That being said, I may have judged too harshly given that I missed the clarification in the last paragraph.

Isn't skipping the preceding statement:

> Unless you get inside the activity and identify with it, you are not going to be good at it.

..taking the statement you quoted out of context ?

I don't quite agree it is a terrible generalization. I do not think you can become good at something without actually identifying as a 'do-er' of the said thing; without actually getting so involved with the thing that it becomes part of your identity.

Also, on the thing you say about educators -- who else, if not educators, would you listen to if the person telling you that math education needs to be impress upon students, mathematical thinking (as opposed to than bland mathematical procedures )? Wouldn't they have the most context ?

PS: also, unless you are a mathematician and/or an educator, I hope you do see the irony of your comments dismissing as 'exceeds his field of expertise', the opinions of someone who is both.

> If you want to be good at activity X, you have to start to see yourself as an X-er – to act like an X-er.

This fails the mathematical thinking bar. Where's the causality? Evidence, applicability across domains?

I don‘t have a strong opinion about it, but there is a similar point made in Atomic habits about identifying as an athlete to do more exercise. I thought it was an unexpected connection

I remember seeing a talk from him (at least I think so or my memory is going to mush) saying he decided to get into giving MOOCS because Khan Academy was so popular and so absolutely terrible. Seemed like hubris at first blush but I gave him the benefit of the doubt and enrolled.

I've ended up doing quite a bit with Sal Khan & Co and have no hesitation is recommending it. (& Co includes Grant Sanderson who has branched off from multivariable calculus at Khan to the whole 3b1b youtube phenom, which is great and also I believe Ben Eater who is also terrific). I didn't get very far with Devlin's coursera MOOC or see the value myself but others mileage may well vary. If you got something useful out of it, please tell us what so those of us who didn't have some chance of seeing the point we missed? But maybe a lot of these kinds of things come down to style and taste?

At the end of the post there is a note

"The opening paragraph is a parody of such writing. This comment was added a day after initial publication, when a letter from a reader indicated that he missed the fact that the opening was a parody, and complained that he found it difficult to read."

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