Thread

1. ForgotMyUUID · 1789802300 · focus · HN ↗

   I’m reminded of that famous debate between Poincaré and Hilbert at the International Congress of Mathematicians in Paris in 1900. It was then that everyone decided to follow Hilbert’s path, and proof came to be valued more than intuition. I think modern math at school and at applied university kind of lost this intuitive part.

   I try to teach my students that mathematics is, first and foremost, a very precise language of communication. It’s sometimes amusing to ask those who don’t like math to do without it entirely, just to see how much harder it becomes to describe the things around them.

   Second thing I tell them, formulas are the essence of mechanisms in their purest form. And in this form, they’re much easier to grasp and mentally manipulate. It always amused me, after taking a mechanics course, to imagine that for any formula, you could visualize a mechanism or process that implements it.

   And third thing, I suppose, the ability to verify one’s own statements as proof. Although, of course, mathematicians would probably tear me apart here for my heresy:sorry, I’m not a mathematician, but an engineer. You can make mistakes by using incorrect assumptions, but at some point, analysis itself will show you that you were mistaken. There’s a wonderful book, How to Prove It by Daniel Velleman, which provides an introduction to proof for the uninitiated like me. I really enjoyed it.

   1. bananaflag · 1789810853 · focus · HN ↗

      As a math prof, I care(d) much more about proof than intuition, not because proof is more important, but exactly because intuition is (I'm a bit Chesterton-ish here haha). You cannot do proof without intuition hence, if you emphasize proof, intuition will take care of itself. Whereas if you emphasize intuition, students won't have any idea of what a rigorous proof should be.

      1. bunderbunder · 1789823597 · focus · HN ↗

         As someone who mostly only applies math, that strikes me as a peculiarly academic take. Intuition is more important for me because it’s what enables me to know what methods are most applicable to whatever practical problem I’m trying to solve. The proof’s purpose is to verify my intuition. It’s just a means to an end. I only take the time to do my own when I can’t confirm what I need from a textbook or paper.

         1. bananaflag · 1789824387 · focus · HN ↗

            > As someone who mostly only applies math, that strikes me as a peculiarly academic take.

            Yeah I was talking strictly about preparing students to become pure mathematicians. No opinion here on other goals.

            1. adastra22 · 1789825552 · focus · HN ↗

               If what you teach is proofs, then wheat you will filter for are students who live proofs.

               1. bunderbunder · 1789826790 · focus · HN ↗

                  And if your job is to train people to become mathematicians, that is absolutely what you should be doing.

                  1. adastra22 · 1789844087 · focus · HN ↗

                     The idea the proofs are the heart and soul of mathematics is an unfortunate unforced error, and will lead to the death of the professions now that machines are better at making proofs.
                  2. Natsu · 1789894880 · focus · HN ↗

                     The explanation of proofs I got in school was terrible. I had no idea that you could actually reduce everything to one of the applications of the axioms until I found metamath's proof explorer.

                     Of course, it's too much work for most normal purposes, and in school they accepted whatever random breakdown people used inconsistently and never explained why.

                     Actually understanding that it wasn't about convincing anyone so much as having a chain of reasoning going all the way back to the axioms was something of a revelation for me.

                     1. analog31 · 1789903401 · focus · HN ↗

                        Fortunately proofs were still part of the curriculum when I was in school. Typically, geometry focused on proofs. When my kids took geometry, proofs took a back seat to problems. This also got us to the point where the college math curriculum has to include a class on how to do proofs.
               2. naijaboiler · 1789863041 · focus · HN ↗

                  or memorize a few

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