[HN Gopher] What is mathematical thinking? (2012)
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What is mathematical thinking? (2012)
Author : mjreacher
Score : 74 points
Date : 2022-03-21 09:45 UTC (13 hours ago)
(HTM) web link (devlinsangle.blogspot.com)
(TXT) w3m dump (devlinsangle.blogspot.com)
| holyyikes wrote:
| [deleted]
| rq1 wrote:
| 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_.
| ifbutthen wrote:
| 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.
| ronald_raygun wrote:
| yeah there are like 10 million ways of defining cont. or
| limits in R^N that turn out to be the same
| ifbutthen wrote:
| 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.
| rq1 wrote:
| 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.
| Oron7 wrote:
| 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.
| ggm wrote:
| 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.
| injb wrote:
| >>> 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_.
| Jensson wrote:
| 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.
| 2bitlobster wrote:
| 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
| lordgrenville wrote:
| Don't Coursera classes always start $TODAY?
| sidpatil wrote:
| Some do. Others have fixed starting dates.
| diegoperini wrote:
| Some classes are browsable everyday but the classroom
| experience had a schedule last time I checked.
| harshreality 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.
| kaba0 wrote:
| > 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
| hans1729 wrote:
| 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?
| lesam wrote:
| Also: "my work was extremely valuable to a group of academics,
| but not worth publishing."
|
| What?
| alberto_ol wrote:
| 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."
| devnonymous wrote:
| 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'
| hans1729 wrote:
| >..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.
| devnonymous wrote:
| > 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 ?
| hans1729 wrote:
| 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.
| mejutoco wrote:
| 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
| devnonymous wrote:
| 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.
| karmakaze wrote:
| > 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?
| harry8 wrote:
| 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?
| hdjjhhvvhga wrote:
| 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.
| jonnybgood wrote:
| A great exposition on how mathematical thinking is performed is
| Imre Lakatos' Proofs and Refutations.
| rvba wrote:
| 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.
| devnonymous wrote:
| Ehe ...you do realise you're speaking about this guy here
| right? http://profkeithdevlin.com/
| new_guy wrote:
| Does it magically stop being blogspam if we know who that is
| supposed to be?
| devnonymous wrote:
| 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.
| rvba wrote:
| Does he have a Putnam?
| zd123 wrote:
| What does that have to do with anything?
|
| Is there a point to your question?
| devnonymous wrote:
| 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.
| rvba wrote:
| So he doesnt have a Putnam?
| devnonymous wrote:
| So you haven't actually read the post?
| carapace wrote:
| It's an old HN in-joke:
| https://news.ycombinator.com/item?id=35079
| Jensson wrote:
| > 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.
| jcranberry wrote:
| 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.
| ouid wrote:
| 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.
| [deleted]
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