The source of all great mathematics is the special case, the concrete example. It is frequent in mathematics that every instance of a concept of seemingly generality is, in essence, the same as a small and concrete special case.
… [Kurt Gödel] kept trying to use his ultraprecisionistic logical habits in ordinary human affairs. When at the beginning of World War II he had to answer a bureaucratically and unintelligently designed draft questionnaire, he became confused and sowed even greater confusion. Instead of answering the unanswerable questions with an impatient yes or no, the way most of us did, he would write lengthy and involved essays explaining that if the question meant A, then the answer was X, but if it meant B, then … and so on.
- Paul Halmos, I Want to be a Mathematician: An Automatography, p. 141-142. Springer, 1985.
Mathematics is not a deductive science – that’s a cliché. When you try to prove a theorem, you don’t just list the hypotheses, and then start to reason. What you do is trial and error, experimentation, guesswork.
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Are you ready for Valentine’s Day, gents? Got the reservations? Prepared the gifts? Awesome. Now all you need to think about is your outfit for the day. As much as sneakers and hoodies can be mixed and matched for fashion-forward street style ensembles, there are just certain occasions when it’s better to leave them in the closet; Valentine’s Day is one of those occasions. Here’s a V-Day outfit idea that your date will appreciate:
1. Start with a crisp shirt. Make sure it’s form-fitting, and any wrinkles have been ironed or steamed out. By no means are you limited to a white shirt, but this tends to work well with a variety of color combinations.
2. Add a tie to your ensemble to get extra dapper points. If you’re going on a more casual rendezvous, skip the silk tie. Instead, try a woven or wool tie so you don’t look inappropriately formal for the date.
4. Top it off with handsome outerwear. A sharp blazer that fits perfectly won’t fail you. If your Valentine’s Day weather will be too cold, try something like the Jack Spade hybrid blazer with quilts for extra warmth, or opt for a coat instead.
Don’t just read it; fight it! Ask your own questions, look for your own examples, discover your own proofs. Is the hypothesis necessary? Is the converse true? What happens in the classical special case? What about the degenerate cases? Where does the proof use the hypothesis?