In previous posts (here and here), I argued that mathematics was a product of the human imagination, and that the test of mathematical creations was not how real they were but how useful or valuable they were.
Recently, Russian mathematician Edward Frenkel, in an interview in the Economist magazine, argued the contrary case. According to Frenkel,
[M]athematical concepts and ideas exist objectively, outside of the physical world and outside of the world of consciousness. We mathematicians discover them and are able to connect to this hidden reality through our consciousness. If Leo Tolstoy had not lived we would never had known Anna Karenina. There is no reason to believe that another author would have written that same novel. However, if Pythagoras had not lived, someone else would have discovered exactly the same Pythagoras theorem.
Dr. Frenkel goes on to note that mathematical concepts don’t always match to physical reality — Euclidean geometry represents an idealized three-dimensional flat space, whereas our actual universe has curved space. Nevertheless, mathematical concepts must have an objective reality because “these concepts transcend any specific individual.”
One problem with this argument is the implicit assumption that the human imagination is wholly individualistic and arbitrary, and that if multiple people come up with the same idea, this must demonstrate that the idea exists objectively outside the human mind. I don’t think this assumption is valid. It’s perfectly possible for the same idea to be invented by multiple people independently. Surely if Thomas Edison never lived, someone else would have invented the light bulb. Does that mean that the light bulb is not a true creation of the imagination, that it was not invented but always existed “objectively” before Edison came along and “discovered” it? I don’t think so. Likewise with modern modes of ground transportation, air transportation, manufacturing technology, etc. They’re all apt to be imagined and invented by multiple people working independently; it’s just that laws on copyright and patent only recognize the first person to file.
It’s true that in other fields of human knowledge, such as literature, one is more likely to find creations that are truly unique. Yes, Anna Karenina is not likely to be written by someone else in the absence of Tolstoy. However, even in literature, there are themes that are universal; character names and specific plot developments may vary, but many stories are variations on the same theme. Consider the following story: two characters from different social groups meet and fall in love; the two social groups are antagonistic toward each other and would disapprove of the love; the two lovers meet secretly, but are eventually discovered; one or both lovers die tragically. Is this not the basic plot of multiple stories, plays, operas, and musicals going back two thousand years?
Dr. Frenkel does admit that not all mathematical concepts correspond to physical reality. But if there is not a correspondence to something in physical reality, what does it mean to say that a mathematical concept exists objectively? How do we prove something exists objectively if it is not in physical reality?
If one looks at the history of mathematics, there is an intriguing pattern in which the earliest mathematical symbols do indeed seem to point to or correspond to objects in physical reality; but as time went on and mathematics advanced, mathematical concepts became more and more creative and distant from physical reality. These later mathematical concepts were controversial among mathematicians at first, but later became widely adopted, not because someone proved they existed, but because the concepts seemed to be useful in solving problems that could not be solved any other way.
The earliest mathematical concepts were the “natural numbers,” the numbers we use for counting (1, 2, 3 . . .). Simple operations were derived from these natural numbers. If I have two apples and add three apples, I end up with five apples. However, the number zero was initially controversial — how can nothing be represented by something? The ancient Greeks and Romans, for all of their impressive accomplishments, did not use zero, and the number zero was not adopted in Europe until the Middle Ages.
Negative numbers were also controversial at first. How can one have “negative two apples” or a negative quantity of anything? However, it became clear that negative numbers were indeed useful conceptually. If I have zero apples and borrow two apples from a neighbor, according to my mental accounting book, I do indeed have “negative two apples,” because I owe two apples to my neighbor. It is an accounting fiction, but it is a useful and valuable fiction. Negative numbers were invented in ancient China and India, but were rejected by Western mathematicians and were not widely accepted in the West until the eighteenth century.
The set of numbers known explicitly as “imaginary numbers” was even more controversial, since it involved a quantity which, when squared, results in a negative number. Since there is no known number that allows such an operation, the imaginary numbers were initially derided. However, imaginary numbers proved to be such a useful conceptual tool in solving certain problems, they gradually became accepted. Imaginary numbers have been used to solve problems in electric current, quantum physics, and envisioning rotations in three dimensions.
Professor Stephen Hawking has used imaginary numbers in his own work on understanding the origins of the universe, employing “imaginary time” in order to explore what it might be like for the universe to be finite in time and yet have no real boundary or “beginning.” The potential value of such a theory in explaining the origins of the universe leads Professor Hawking to state the following:
This might suggest that the so-called imaginary time is really the real time, and that what we call real time is just a figment of our imaginations. In real time, the universe has a beginning and an end at singularities that form a boundary to space-time and at which the laws of science break down. But in imaginary time, there are no singularities or boundaries. So maybe what we call imaginary time is really more basic, and what we call real is just an idea that we invent to help us describe what we think the universe is like. But according to the approach I described in Chapter 1, a scientific theory is just a mathematical model we make to describe our observations: it exists only in our minds. So it is meaningless to ask: which is real, “real” or “imaginary” time? It is simply a matter of which is the more useful description. (A Brief History of Time, p. 144.)
If you have trouble understanding this passage, you are not alone. I have a hard enough time understanding imaginary numbers, let alone imaginary time. The main point that I wish to underline is that even the best theoretical physicists don’t bother trying to prove that their conceptual tools are objectively real; the only test of a conceptual tool is if it is useful.
As a final example, let us consider one of the most intriguing of imaginary mathematical objects, the “hypercube.” A hypercube is a cube that extends into additional dimensions, beyond the three spatial dimensions of an ordinary cube. (Time is usually referred to as the “fourth dimension,” but in this case we are dealing strictly with spatial dimensions.) A hypercube can be imagined in four dimensions, five dimensions, eight dimensions, twelve dimensions — in fact, there is no limit to the number of dimensions a hypercube can have, though the hypercube gets increasingly complex and eventually impossible to visualize as the number of dimensions increases.
Does a hypercube correspond to anything in physical reality? Probably not. While there are theories in physics that posit five, eight, ten, or even twenty-six spatial dimensions, these theories also posit that the additional spatial dimensions beyond our third dimension are curved up in very, very small spaces. How small? A million million million million millionth of an inch, according to Stephen Hawking (A Brief History of Time, p. 179). So as a practical matter, hypercubes could exist only on the most minute scale. And that’s probably a good thing, as Stephen Hawking points out, because in a universe with four fully-sized spatial dimensions, gravitational forces would become so sensitive to minor disturbances that planetary systems, stars, and even atoms would fly apart or collapse (pp. 180-81).
Dr. Frenkel would admit that hypercubes may not correspond to anything in physical reality. So how do hypercubes exist? Note that there is no limit to how many dimensions a hypercube can have. Does it make sense to say that the hypercube consisting of exactly 32,458 dimensions exists objectively out there somewhere, waiting for someone to discover it? Or does it make more sense to argue that the hypercube is an invention of the human imagination, and can have as many dimensions as can be imagined? I’m inclined to the latter view.
Many scientists insist that mathematical objects must exist out there somewhere because they’ve been taught that a good scientist must be objective and dedicate him or herself to the discovery of things that exist independently of the human mind. But there’re too many mathematical ideas that are clearly products of the human mind, and they’re too useful to abandon merely because they are products of the mind.