Apr 1, 2010
Mathematics describes the real world of atoms and acorns, stars and stairs. Simple abstract equations define complex physical things beautifully, elegantly. Why should this be? The more I think about it, the more astonished I get.
Albert Einstein said: “The most incomprehensible thing about the universe is that it is comprehensible.” Physicist Eugene Wigner wrote of “the unreasonable effectiveness of mathematics” in science. So is mathematics invented by humans, like chisels and hammers, cars and computers, music and art? Or is mathematics discovered, always out there, somewhere, like mysterious islands waiting to be found? The question probes the deepest secrets of existence.
Roger Penrose, one of the world’s most distinguished mathematicians, says that “people often find it puzzling that something abstract like mathematics could really describe reality.” But you cannot understand atomic particles and structures, such as gluons and electrons, he says, except with mathematics. “These equations are fantastically accurate,” he remarks. “Newton’s theory has a precision of something like one part in 10 million. Einstein’s theory something like one part in 100 billion.”
Penrose famously believes that mathematics has an independent existence, a “Platonic existence” (following Plato’s “forms”) that is radically distinct from physical space and time. “Certainly,” he says, “mathematicians view mathematics as something out there, which seems to have a reality independent of the ordinary kind of reality of things like chairs, which we normally think of as real. It’s sometimes referred to as a ‘Platonic world,’ a Platonic reality. … I like to think of mathematics as a bit like geology or archeology, where you’re really exploring beautiful things out there in the world, which have been out there, in fact, for ages and ages and ages, and you’re revealing them for the first time.”
Penrose describes three kinds of “worlds”— physical, mental, and mathematical (Platonic)—and he wonders whether “the mathematical reality of the Platonic world gives reality to these worlds.”
But not everyone agrees. Philosopher Mark Belaguer notes that “there are tons and tons of mathematical structures that are of no use at all in studying the physical world” and the reason that “mathematicians started studying those structures that turned out to be useful was because they lived in the physical world.”
But do abstract mathematical objects really exist in the world? Belaguer cites four opposing views: the “mentalistic” view that mathematical objects are all in our head; the “physicalistic” view that mathematical objects exist in the physical world; the Platonic view that the mathematical objects are nonphysical and nonmental abstract objects; and the “anti-realist” view that there aren’t mathematical objects at all.
(Belaguer defines an “abstract object,” such as a number, as “not physical, not mental, and not entering into causal relations. So an abstract object is not like any object we ever encounter in our ordinary lives. And belief in those kinds of abstract objects is called Platonism because it was famously Plato’s view that there were such things.”)
Belaguer asserts: “The right kind of Platonism is the strongest kind of realism you can have. And ‘fictionalism’ is the strongest kind of anti-realism you can have because the mathematics is literally false.” Interestingly, he defends both of these diametrically opposed views because, he says, “the only thing they disagree on is: Do the abstract objects exist or not?” And, he concludes, “it doesn’t look like we have any way of knowing whether they exist. So we can’t discover whether Platonism or fictionalism is the right view.”
But is one or the other correct? No, Belaguer says, “there’s no fact of the matter about whether the abstract objects exist. … There’s no right answer to it.”
The effectiveness of mathematics is clear. Why, then, is the essence of mathematics so foggy? Is math mental, physical, Platonic, or just not real?
I ask mathematician Gregory Chaitin. “When you’re a mathematician and you find something that feels really fundamental,” he says, “you may think that if you hadn’t found it, somebody else would have because in some sense it’s got to be there. But some mathematics feels much more contrived. … If you look into the inner recesses of many mathematicians, and I include myself, you find that we have this theological medieval belief in this Platonic world of perfect ideas of mathematical concepts.” But, he muses, “is it all a game that we just invent as we go along?”
Chaitin offers his “final conclusion after a lifetime obsessed with mathematics.” Mathematicians, he says, “should behave a little bit more like experimental scientists do.” He argues that “if they do computer experiments and see that something seems to be the case, but they can’t prove it, and yet this something is a very useful truth if it were true, then maybe they should add that as a new axiom. Mathematicians will reel back in horror, but I think my work pushes in this direction. I’ve been forced against my will toward saying that mathematics is empirical—or, to put it in other words, we invent it as we go.” Concluding, Chaitin laments, “I’m not quite sure where we are.”
Invented or discovered? Chaitin’s ideas about math have changed, from certainty to uncertainty about whether math has always existed. If, after a lifetime in math, Chaitin is “not quite sure,” where does that leave me? And, yes, it does matter. Math is fundamental to existence!
In a new way of thinking, physicist Stephen Wolfram offers the shocking idea that simple rules, not complex mathematics, construct reality. “For a long time, I’ve been interested in the essence of mathematics,” Wolfram says. “Is today’s mathematics the only possible mathematics, or is it a mathematics that is sort of a great artifact of our civilization?” His “resounding” conclusion is that “the mathematics that we have today is in fact a historical artifact. Now throughout history, that’s not what mathematicians have tended to conclude. They’ve [generally] thought that mathematics is the most general possible formal abstract system.”
He continues: “If you look at the history of mathematics, that’s certainly not how it originally started out. In ancient Babylon, arithmetic was for commerce, geometry for land surveying.” Then came “the progressive generalization of arithmetic and geometry, plus one key methodological idea: that one can make theorems and abstract proofs of those theorems,” he says. “One can ask the question: If one arbitrarily looks at formal systems, will they tend to have the character of mathematics as we know it today? Will they tend to have the feature that one can successfully prove theorems? I think in both cases the answer is no, not really.”
Wolfram suggests that we “deconstruct mathematics” by recognizing that all our mathematics are based on a certain set of axioms, which are quite simple. “But there’s a whole universe of possible mathematics out there,” he states. “What are they like? And where does our particular mathematics lie in this universe of possible mathematics? Is it possible mathematics number one? Is it possible mathematics number 10? Is it possible mathematics number quintillion? … The answer depends on exactly how you enumerate the space. But roughly, our mathematics is about the 50,000th possible axiom system. So right there in the universe of possible axiom systems, the universe of possible mathematics, there’s logic.”
He goes on: “I suspect that if we were to just sort of ask mathematical questions arbitrarily, the vast majority of them would end up turning out to be unsolvable. We just don’t see it because the particular way that our mathematics has progressed historically has tended to avoid it. Now, you might say, ‘But mathematics is a good model of the natural world.’ I think there’s kind of a circular argument here because what’s happened is that those things which have been successfully addressed in science when studying the natural world are just those things that mathematical methods have successfully allowed us to address.”
Wolfram appreciates that human mathematics is “one of the wonderful things that has been produced by a huge amount of human effort.” Nonetheless, he concludes, it is an artifact.
Thus, Wolfram rejects the idea that our mathematics has deep significance. Rather, he looks to the vastly large “space of all possible mathematics.” But math as mere artifact still troubles me.
Nobel laureate Frank Wilczek tells me that mathematics is both invented and discovered,” but he thinks “it’s mostly discovered.” Mathematics, he says, “is the process of taking axioms, definite sets of assumptions, and drawing out the consequences. So, devising axioms is invention, and drawing out the consequences is discovery.”
He explains that, “Occasionally, you have to introduce new sets of axioms like the passage from Euclidean geometry to non-Euclidean geometry. These are epical events in mathematics, which, in a sense, are inventions.”
But isn’t the world constructed with non-Euclidean geometry, such as Einstein’s theory of relativity, so that it was somehow always there?
“Inventions have to come from somewhere,” Wilczek responds. “So they could be inspired by natural phenomena. … You can invent [all kinds of] axioms, but most of them won’t be interesting. And the ones that are interesting are discoveries, so even the inventions have some element of discovery. So as I said, mathematics is more discovered than invented, and this only makes it more so.”
So, is mathematics invented or discovered? Here’s what we know. Mathematics describes the physical world with remarkable precision. Why? There are two possibilities.
First, math somehow underlies the physical world, generates it. Or second, math is a human description of how we describe certain regularities in nature, and because there is so much possible mathematics, some equations are bound to fit.
As for the essence of mathematics, there are four possibilities. Only one is really true. Math could be: physical, in the real world, actually existing; mental, in the mind, only a human construct; Platonic, nonphysical, nonmental abstract objects; or fictional, anti-realist, utterly made up. Math is physical or mental or Platonic or fictional. Choose only one.
In peering down the dark well of deep reality, mathematics brings us closer to truth.
Robert Lawrence Kuhn speaks with Roger Penrose, Mark Belaguer, Gregory Chaitin, Stephen Wolfram, and Frank Wilczek in “Is Mathematics Invented or Discovered?”—the ninth episode in the new season of the Closer To Truth: Cosmos, Consciousness, God TV series (48th in total).
The series airs on PBS World (often on Thursdays, twice) and many other PBS and noncommercial stations. Every Thursday, participants will discuss the current episode.
P.S. Click here to visit our Closer to Truth archive.