Let's see. An op-ed in the New York Times entitled “Doubleday and Darwin”, with the following opening paragraph:
As I sat in my high school math class one day, my teacher asked a question that I doubt will find a consensus opinion in my lifetime: “Was math invented or was it discovered?” To this day, I still scratch my head.
Yeah, I think I can be persuaded to read the whole thing.
The writer is baseball player Douglas Glanville. He finds some interesting parallels between the rules of baseball and the evolutionary process. He writes:
But humility reminded me that there are rules by which this game is played. Rules that, when changed, can quickly turn a natural into an unnatural. And if you change just one -- the length from the plate to the mound, the number of strikes and balls, or even something much less significant -- the landscape of the game changes. And with it changes that group of people who could be deemed "standout" performers in the game.
The adaptive landscape as applied to baseball. Cool!
Change in the game is inevitable. Some of it comes from amendments to the rule book by the powers that be, some is evolutionary. Either way, some players have the ability to fit right into whatever the current system may be, others require just a little more work to remain high achievers . . . and others just get phased out.
And later:
There are quantifiable skills that can make someone naturally compatible with the rules of the game, but it's almost more important to be adaptable. Baseball can update pretty dramatically for a National Pastime. It has the ability to stay both classic and current, without contradiction. On the table of baseball rule changes for late this season is the instant replay, intended to help umpires on difficult home run calls. In the end, if it's added to the rule book, the game will go on; players -- and umpires -- who don't adapt won't.
As the game changes, what is deemed “talent” changes right along with it. A player is discovered only in the shadow of these rules -- rules that were invented and that have matured over time.
And now we have the idea that notions of fitness depend on the environment. Glanville knows his evolution well.
I recommend reading the whole thing. I am reminded of a piece of dialogue from an episode of Columbo:
MURDERER: (Angrily) What's your point, Columbo?
COLUMBO: Point? No point. It's just interesting, that's all.
Oh, and here's Glanville's answer to the “invented or discovered” question:
If I were to make my baseball experience the basis of answering the question about discovery vs. invention posed by my high school math teacher, I'd say that math was discovered . . . through the lens of our invention.
Sounds good.
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Baseball is not popular like Football
http://www.lidasatis.org -lida
The writer seems to defy my prejudiced notions about the thoughtfulness of jocks. But how serious is the debate, really, about the invention v. discovery of mathematics? I have some difficulty understanding how mathematics could be anything but invented. I'm open to enlightenment.
Comstock:
For more "enlightenment" on this question than you probably really want, see:
http://en.wikipedia.org/wiki/Philosophy_of_mathematics
While I lean toward your view (specifically the "embodied mind" sort), my impression is that that is the minority view. Perhaps JR (or others) can confirm or refute that impression.
- Charles
I would say that the *terms* are invented, but the *concepts* were discovered. Triangles obeyed the Pythagorean Theorem (Shouldn't this be a Law by now?) long before Pythagoras - or the Egyptians for that matter.
Without getting in any formal philosophy, I think KeithB hits the nail on the head. An "invention" is surely something which exists by contingency. We can imagine an alternative history in which the Internet was never invented. A "discovery" is, well, discovering something that was pre-existing. In any history, electrons (which could be used to carry information in the form of something like the Internet) must exist (but might not be discovered).
As KeithB says, surely a right-angled triangle always has, and always will, obey the Pythagorean Theorem. Where things get tricky, I think, is by what we mean by a "triangle"? Is this just an abstract idea which humans "invented"? After all, a _perfect_ triangle does not exist in nature...
Forget math--I like his baseball ideas. Baseball really does seem to have evolved like a gentic algoritm where the fitness function optimized a balance between pitching and hitting. The distance to the mound, the height of the mound, the distance to the bases. Also, its huge rulebook is like complex genome:
Q: Is a foul ball a strike?
A: A foul ball is a strike. Unless you have two strikes, then it isn't. Oh, unless you are bunting in which case it is. Or even if you are not bunting but you just barely tip the ball it is a strike, as long as the catcher catches it. If the catcher doesn't, it isn't...
It's refreshing to read a good summary of the theory of evolution in a different light. In terms of teaching, I was always one to think that it was good to teach multiple methods of knowledge acquisition. The baseball analogy is a good one in another sense: what's the creationist approach? "God created baseball in its present form." True, and IDiot would say that it is obvious that there was an "intelligent designer" that made baseball. However, that IDiot would be missing the point of the analogy, since the analogy discusses the method of evolution, and (as far as I know) there isn't any designed process guiding the way the rules and strategies changed over time.
KeithB and Doormat:
I don't think the argument that any sufficiently intelligent entity would of necessity come up with essentially the same mathematical construct (the essence of the Pythagorean Theorem example) is considered to be definitive for the "discovered" contention. If we call that argument "inevitable discovery", the analogous "inevitable invention" argument should work just as well.
Not to argue for or against either answer, just to suggest that the issue may not be all that straightforward. Consider that it apparently remains open to debate even among those whose mathematical/philosophical sophistication perhaps exceeds yours and without question greatly exceeds mine.
- Charles
Invention and discovery of both a measurement and simulation system can amount to descriptions of the same process. So, as has been more often the case here than not, the either/or limitation does not apply.
Actually baseball does not evolve by means analogous to natural selection or adaptation.
Any evolution is the result of outside forces designing the changes to adapt to their tastes in entertainment. The rules of nature's games are, as far as we have determined, fixed. Not so with rules designed by humans to simulate reality in a game scenario.
The Pythagorean theorem is actually contingent; it is contingent on space being flat (Euclidean). In non flat space (and no "real" space is perfectly flat), a2 + b2 can be greater or less than c2 or even zero. I'm not sure what this implies for the discovery verses invention question, but since all mathematics derives from the laws of logic, the appropriate question is more likely to be wether the laws of logic are invented or discovered. Its a little difficult to think of the laws of logic as being contingent.
Someone once said that god invented the integers, and everything else is man's work.. The Italian mathematician, Peano, would probably have said that god invented the number "1", and everything else is man's imagination.
All mathematics do not derive from the laws of logic. Mathematics can equate objects by number regardless of exact similarity. But no two apples are exactly alike according to the logic that first defined them.
"and no "real" space is perfectly flat"
Oops, I had forgotten about non-euclidean geometries, but I am not sure what that does to my argument - just gives us more things to discover, I think. 8^)
And once you talk about "real space", you get into physics, not math.
I'm surprised no one's mentioned Stephen Jay Gould's book-length treatment of the mathematical parallels between baseball and the evolutionary process, Full House.
Doug Glanville is writing a regular column for the New York Times this year during the baseball season. He seems to be a pretty intelligent guy, to judge by the columns I've read. According to his Wikipedia page, he majored in systems engineering at Penn, and is now the president of a consulting company for start-up businesses. I remember him well from his days with the Chicago Cubs.
Having been subjected to some classes that covered philosophy of math, I remember that the discovered/invented debate has some roots in Platonism. According to hard-core Platonists, all objects and all concepts exist as a perfect form in some mysterious "realm of the forms." Whenever we experience an object or think of a concept, we are experiencing an imperfect reflection of one of the Forms from that mysterious realm. Thus, if Platonism is true, then we do indeed discover things like math rather than invent them. Soft-core, modern Platonists might say something less extreme or mysterious (and similar to previous comments in this thread). They believe that given the basic properties of our universe, some things are just true. It's true, for example, that electrons exist whether we are aware of them or not, and so our noticing them counts as a discovery rather than an invention. Also, certain logical relations exist whether we are aware of them or not, and so when we notice one of those (like the pythagoean theorem) it also counts as a discovery rather than an invention.
Also, btw, Russell and Whitehead (around the turn of the 20th century) made a famous attempt to ground all of mathematics in the very basic laws of logic. As I understand it, they were pretty much successful, at least for regular math up to, like, calculus or something.
Aside from wikipedia, I would recommend the Stanford online (free) encyclopedia of philosophy entry for anyone who is interested (in this or any other topic).
http://plato.stanford.edu/entries/philosophy-mathematics/
thanks.
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