On still looking into Hawking’s Brief History!


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.

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?
Professor Frenkel was taking it hard.

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?

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?
“How do humans manage to access this hidden reality?” “How do we all end up agreeing on exactly the same math?”

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!)


  1. Over time people have not agreed on exactly the same math.

    Math may be "objective" but it is only experienced in our minds, just as the rest of the world is only experienced through our minds, despite existing separate from us. I don't really see how math is different. I am hoping you will explain this tomorrow.

    1. Math is objective in the way that Frenkel describes it: the humans -- and probably extraterrestrial non-humans -- would always converge on the Pythagorean theorem. Daniel Dennett makes this point in Darwin's Dangerous Idea. He asks if you had a choice of one great manuscript being destroyed, would you select Newton's Principia Mathematica or Melville's Moby Dick. The answer he suggest is the Newton's work, as it would have been written in some form sooner or later. And, of course, Charles Leibniz invented calculus around the same time as Newton.

      I don't see how one can say that math exists only in our minds if so many minds converge on the same understanding; as well as mathematical models being so accurate in predicting the physical world around us.

      I am not sure what Bob's problem was with that piece in the Sunday Times. I read it. I will read it again today. Sure, it requires some leaps of imagination, but the newspaper space is limited.

    2. Thank you for your explanation. I said math exists only in our minds because we have no direct experience of anything outside of our minds because we can only experience external reality through our sense and through our cognitive processes. These shape our experience regardless of the nature of an external reality. It is likely that so many minds converge on the same understanding because they are all human minds with fundamentally the same brains and cognition. Also, don't forget that both Leibnitz and Newton read Descartes. That makes them less independent than they seem.

    3. Again, Fenkel's point is that mathematical truth exists objectively, outside our minds. We can only discover it as well as the language to explain it.

      Thus, the truth of the Pythagorean theorem existed before Pythagoras. And had Pythagoras not existed, someone else would have discovered it.

    4. Wouldn't you need to give some importance to triangles to discover it? What if you lived in a culture where buildings were round, as in some parts of Eastern Europe or Africa? What if your religion placed no emphasis on harmony or proportion, as the Greeks did? Would you still explore such relationships? Might you develop a math of randomness and chaos instead of regularity? Would there be facts discovered that we don't know about yet today? Is that the same math, even though we haven't discovered it ourselves?

    5. Whether or not you put any importance at all on triangles, it does not change the objective truth that the sum of the squares of two sides of a right triangle equals the square of the hypotenuse.

      For all but the last few hundred years of human existence, human beings were convinced that the earth was the center of the universe and the sun and stars revolved around it.

      That didn't make it true. Nor did the earth suddenly start revolving around the sun the instant humans discovered that it did.

    6. So, was the math that those humans believed in the same as current math? I don't think so. And if there are humans with different access to knowledge, they will develop different maths and those may or may not match "objective" reality, to the extent that human guesses about that reality are accurate. Was Zeno right in his speculations about infinite series? If you are going to suggest that math is different because it rests on logic, that doesn't work because logic depends on its premises. I don't see how this solves the problem.

    7. Modern math has changed the Pythagorean theorem? In what way?

      I am merely saying that Pythagoras described a truth that existed before he did, and continues to exist.

    8. How many right triangles do you find in the world that are not man made? If the triangles did not exist, would the truth about them?

    9. That depends on whether you think triangles exist even if you live in a world with no actual closed figures composed of three line segments.

    10. Man-made or not, the sum of the squares of the sides of a right triangle is equal to the square of the hypotenuse.

      That truth has always existed, and it will continue to exist.

      The only thing man added was the language to express it in terms man can understand. And that language neither adds or subtracts to the objective truth that the sum of the squares of the sides of a right triangle is equal to the square of the hypotenuse.

      It would be true on earth, on Mars, on Jupiter, or whereever, regardless of whether right triangles exist there or not.

    11. I always thought Jacob Bronowski demonstrated the Pythagorean Theorem most elegantly in the Ascent of Man series. The right triangle is necessarily a result of how we experience the real world.


  2. The problem with the NYTimes essay, is the failure to distinguish between philosophy and physics and to explain how math could be applied to each or used to explain each.

    Nice job, Bob.

  3. As one moves into more advanced mathematics, one finds areas that look less and less like something "real". Some fields of mathematics amount to choosing a series of definitions and axioms and then figuring out what theorems can be proved. As the set of axioms becomes more arcane and less related to the real world, that field of mathematics more comes to resemble figuring out the winning strategies in a game with arbitrary rule, like chess or go. It's reasonable to believe that any advanced culture would figure out that 1 + 1 = 2, but there's no reason to believe that some other society would define a game with rules identical to the rules of chess.

  4. Something "real"? Is that an inside math joke? Or is it too "complex" and "irrational"?

    All fields of mathematics amount to choosing definitions and axioms and proving theorems that follow from the choices.

    Your analogy to chess or go is exact, but only in the sense that it's completely inapt. There are no opponents in math on which to base a "strategy."

    What happens over and over again is that the "arcane" and "less related to the real world" areas of mathematics turn out to have remarkable power in describing that real world. The quaternions extend the complex numbers, and when Hamilton discovered (invented?) them in the 1840s, few people beside him thought they'd be useful. They don't commute, i.e., the order of operations matter unlike our familiar mathematical objects, in which 3+7 is the same as 7+3. But quaternions are very useful in areas that deal with things that rotate, areas like computer graphics and spacecraft control.

    This happens so often that they physicist Eugene Wigner has called it the "unreasonable effectiveness of mathematics in the natural sciences." No one says this about either chess or go.

  5. deadrat, here's the analogy spelled out. In chess, the equivalent of a mathematical theorem might say, e.g., from a certain position of pieces on the board, white can mate in 4 moves. One can prove this "theorem", using logic similar to a mathematical proof.

    It's true that some fields of mathematics unexpectedly turned out to have a real-world application, but many fields do not. E.g., the Banach-Tarski paradox. Using group theory, real analysis, the axiom of choice, and non-measurable sets, one can subdivide a sphere into a finite number of pieces and re-arrange these pieces to form a larger sphere. This theorem has no real-world meaning or application. Non-measurable sets don't correspond to anything in the natural world.

    1. Ah! Chess problem, not chess game. Yeah, that's a valid analogy. Here's another "Chess problem is to mathematics as single-octave kinderklavier with painted-on black keys is to a symphony."

      If the Banach-Tarski paradox has no real-world meaning, then the axiom of choice doesn't obtain there. But group theory and real analysis do.

  6. I would be interested in where Bob is headed with Hawkings and his book on "Time". I read once that Hawking had the notion that the universe, then believed to be expanding universe would expand to a point and start collapsing...At which time, Hawking believed clocks would start to run backwards...The story was that the more brilliant Richard Feynman convinced him otherwise.

    1. It's pretty clear where Bob is headed with Hawking and his book: Campaign 2000 and the War on Gore.

  7. The Question, it would seem, is: Do I really have to sit through those silly Matrix movies to get all the hip references professors et al. feel they must make if they want to get their points across to the younger set? Attempts so far have been futile. I think I'd rather sit though "Hot Tub Time Machine."

    Such discussions remind me of the scene in "Animal House" where Tom Hulce's freshman mind is blown by Professor Donald Sutherland's comparison of atomic structure to the solar system.