How far can non-specialists get: How far can a non-specialist get in A Brief History of Time, Stephen Hawking’s famous first book of “popular science” directed at non-specialists?
As we noted last week, we returned to that ancient question after a recent Science Times piece. But first, consider the piece which appeared in yesterday’s Sunday Review.
Professor Frenkel had himself all tangled up over the mysterious nature of the Pythagorean theorem. Needless to say, the New York Times thought this was good solid stuff:
FRENKEL (2/16/14): If Pythagoras had not lived, or if his work had been destroyed, someone else eventually would have discovered the same Pythagorean theorem. Moreover, this theorem means the same thing to everyone today as it meant 2,500 years ago, and will mean the same thing to everyone a thousand years from now—no matter what advances occur in technology or what new evidence emerges. Mathematical knowledge is unlike any other knowledge. Its truths are objective, necessary and timeless.Professor Frenkel was taking it hard.
What kinds of things are mathematical entities and theorems, that they are knowable in this way? Do they exist somewhere, a set of immaterial objects in the enchanted gardens of the Platonic world, waiting to be discovered? Or are they mere creations of the human mind?
This question has divided thinkers for centuries. It seems spooky to suggest that mathematical entities actually exist in and of themselves. But if math is only a product of the human imagination, how do we all end up agreeing on exactly the same math?
Might we start with a basic point? We assume that Frenkel is a brilliant mathematician. That doesn’t make him a good baseball player, or a good “philosopher.”
In these ruminations, he has moved outside his field. We say it pretty much shows.
We’re always amazed when brilliant mathematicians get tangled up in such matters. A similar problem comes into play when brilliants physicists try to explain modern physics to people who aren’t brilliant physicists.
They simply may not have the tools. Beyond that, they may not realize that they don’t have the tools.
Frenkel continued his rumination. The analysts continued to writhe:
FRENKEL (continuing directly): Some might argue that mathematical entities are like chess pieces, elaborate fictions in a game invented by humans. But unlike chess, mathematics is indispensable to scientific theories describing our universe. And yet there are many mathematical concepts—from esoteric numerical systems to infinite-dimensional spaces—that we don’t currently find in the world around us. In what sense do they exist?“How do humans manage to access this hidden reality?” “How do we all end up agreeing on exactly the same math?”
Many mathematicians, when pressed, admit to being Platonists. The great logician Kurt Gödel argued that mathematical concepts and ideas “form an objective reality of their own, which we cannot create or change, but only perceive and describe.” But if this is true, how do humans manage to access this hidden reality?
We’d be inclined to start our answer with this: Humans figure it out!
Frenkel started his piece with the famous old Pythagorean theorem. Hoping to break the professor’s fever, we’ll ponder that “mathematical entity” tomorrow.
(Is the P-the an “immaterial object?” Funny someone should ask!)