Okay, suppose someone else simply entered some numbers in a Suduko grid and said, "I wonder whether there is any solution that incorporates these numbers, and, if there is a solution, is it unique?" I concede that the person who did this invented the problem. But if I prove that there is a solution and that it is unique, I haven't invented that fact as that fact was implicit in the original question, but I have discovered that the fact was implicit, have I not?
On 10/7/08 1:37 PM, "glen e. p. ropella" <[EMAIL PROTECTED]> wrote: Thus spake John F. Kennison circa 10/07/2008 10:01 AM: > I would like to respond to Wittgenstein's idea that a mathematical > proof should be called an invention rather than a discovery. When > solving a Suduko puzzle, I often produce a logical deduction that the > solution is unique. It seems clear to me that I discovered that there > is only one solution. I don't see how to make any sense of the idea > that I "invented" the fact that there is only one solution. As cryptic as all the jargon may sound, Wittgenstein's point is very intuitive. Mathematics is built upon a set of _definitions_. We define everything. Hence, all of math is an invention. It's true that you didn't invent the solution to the sudoku puzzle. But the formal system in which sudoku puzzles sit is a human invention. Hence, the solution to the puzzle is also a human invention. The person who invented the game invented the solution. You discovered what s/he invented. Did you discover it first? Unlikely. Can multiple people discover the same thing even after someone previously discovered that thing? Yes. The same would be true for, say, some occult part of an engine being explored by someone ignorant of engines. The novice starts unscrewing things one after another and comes upon a part she didn't know existed. She literally discovered the part. But that doesn't mean that engines aren't human inventions. Some human put it there. Another human discovered it there. Now, fancy pants filosofers will enter at this point and say things like: "but the engine builders designed and installed that engine part purposefully, whereas the solution to a logic puzzle can be an undesigned deductive consequence of the formal system and the way the puzzle was set up". That appeal to intention, purpose, or the magical, metaphysical homunculus in our brains is specious, though. So don't let it get in the way of rational thought. [grin] All mathematics is a human invention ... it's a set of definitions and grammatical manipulations of those definitions. The real questions come when we discuss why our invention mirrors reality so well ... Now _that's_ another issue entirely. The best argument is that we cognitive animals are inventions of reality and, hence, all the thoughts we have (including math) reflect some deep structure of reality. So, since we invented math and reality invented us, then math must be real ... perhaps even a filtered, essential, purified, form of reality. And that's what Wittgenstein was fighting against (or perhaps ultimately for? since he was a big fan of _thought_ in general but not math in particular) using his banal observation that math is a human invention just like Monopoly or Chess. -- glen e. p. ropella, 971-219-3846, http://tempusdictum.com ============================================================ FRIAM Applied Complexity Group listserv Meets Fridays 9a-11:30 at cafe at St. John's College lectures, archives, unsubscribe, maps at http://www.friam.org
============================================================ FRIAM Applied Complexity Group listserv Meets Fridays 9a-11:30 at cafe at St. John's College lectures, archives, unsubscribe, maps at http://www.friam.org
