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: <a href="https://www.coursera.org/learn/mathematical-thinking" rel="nofollow">https://www.coursera.org/learn/mathematical-thinking</a>
> 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.<p>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!).<p>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):<p><pre><code> * (Existential-Universal) quantification
* (Epsilon-Delta) language
</code></pre>
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.<p>I've seen also a lot of people struggle with <i>definitions</i>.
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).<p>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.
Essentially a lite version of Velleman's _How to Prove It_?<p>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.<p>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.
>>> 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.<p>Ah yes, one of those - also known as a <i>completely useless analogy</i>.
If someone claims to be a "mathematical thinker" then probably they should be able to convey their thoughts in a precise and concise manner.<p>For example by starting with a definition and then expanding/explaining it.<p>Blogspam written by someone who sounds like a teenager with ADHD should be posted in the "iamverysmart" subreddit, not on HN.
So, so much hubris.<p>- "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!"<p>- <i>"You need to identify as an X to be good at X"</i> -- 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.<p>- <i>"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"</i> -- 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 <i>are</i> 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 <i>suck</i>.<p>- <i>"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. "</i> -- the entire paragraph just wastes the readers attention. Teaching <i>is</i> to help learn. What are you saying? Nothing of value.<p>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?