What’s the nature of mathematical objects?
The question is made difficult by a centuries-old assumption of Western philosophy: “There are two kinds of things in the world. What isn’t physical is mental; what isn’t mental is physical.”
Mental is individual consciousness. It includes private thoughts—mathematical and philosophical, for example—before they’re communicated to the world and become social—and also perception, fear, desire, despair, hope, and so on.
Physical is taking up space—having weight or energy. It’s flesh and bones, sound waves, X-rays, galaxies.
Frege showed that mathematical objects are neither physical nor mental. He labeled them “abstract objects.” What did he tell us about abstract objects? Only this: They’re neither physical nor mental.
Are there other things besides numbers that aren’t mental or physical?
Yes! Sonatas. Prices. Eviction notices. Declarations of war.
Not mental or physical, but not abstract either!
The U.S. Supreme Court exists. It can condemn you to death!
Is the Court physical? If the Court building were blown up and the justices moved to the Pentagon, the Court would go on. Is it mental? If all nine justices expired in a suicide cult, they’d be replaced.
The Court would go on. The Court isn’t the stones of its building, nor is it anyone’s minds and bodies. Physical and mental embodiment are necessary to it, but they’re not it. It’s a social institution. Mental and physical categories are insufficient to understand it. It’s comprehensible only in the context of American society.
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Mathematics consists of concepts. Not pencil or chalk marks, not physical triangles or physical sets, but concepts, which may be suggested or represented by physical objects.
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In reviewing The Mathematical Experience, the mathematical expositor and journalist Martin Gardner made this objection: When two dinosaurs wandered to the water hole in the Jurassic era and met another pair of dinosaurs happily sloshing, there were four dinosaurs at the water hole, even though no human was present to think, “2 + 2 = 4.” This shows, says Gardner, that 2+ 2 really is 4 in reality, not just in some cultural consciousness. 2 + 2 = 4 is a law of nature, he says, independent of human thought.
To untangle this knot, we must see that “2” plays two linguistic roles. Sometimes it’s an adjective; sometimes it’s a noun.
In “two dinosaurs,” “two” is a collective adjective. “Two dinosaurs plus two dinosaurs equals four dinosaurs” is telling about dinosaurs. If I say “Two discrete, reasonably permanent, non-interacting objects collected with two others makes four such objects,” I’m telling part of what’s meant by discrete, reasonably permanent non-interacting objects. That is a statement in elementary physics.
John Stuart Mill pointed out that with regard to discrete, reasonably permanent non-interacting objects, experience tells us
2 + 2 = 4.
In contrast, “Two is prime but four is composite” is a statement about the pure numbers of elementary arithmetic. Now “two” and “four” are nouns, not adjectives. They stand for pure numbers, which are concepts and objects. They are conceptual objects, shared by everyone who knows elementary arithmetic, described by familiar axioms and theorems.
The collective adjectives or “counting numbers” are finite. There’s a limit to how high anyone will ever count. Yet there isn’t any last counting number. If you counted up to, say, a billion, then you could count to a billion and one. In pure arithmetic, these two properties—finiteness, and not having a last—are contradictory. This shows that the counting numbers aren’t the pure numbers.
Consider the pure number 10^(1010). We easily ascertain some of its properties, such as: “The only prime factors of 10^(1010) are 2 and 5.” But we can’t count that high. In that sense, there’s no counting number equal to 10^(1010).
Körner made the same distinction, using uppercase for Counting Numbers (adjectives) and lowercase for “pure” natural numbers (nouns). Jacob Klein wrote that a related distinction was made by the Greeks, using their words “arithmos” and “logistiké.”
So “two” and “four” have double meanings: as Counting Numbers or as pure numbers. The formula
2 + 2 = 4
has a double meaning. It’s about counting—about how discrete, reasonably permanent, non-interacting objects behave. And it’s a theorem in pure arithmetic (Peano arithmetic if you like). This linguistic ambiguity blurs the difference between Counting Numbers and pure natural numbers. But it’s convenient. It’s comparable to the ambiguity of non-mathematical words, such as “art” or “America.”
The pure numbers rise out of the Counting Numbers. In a process related to Aristotle’s abstraction, they disconnect from “real” objects, to exist as shared concepts in the mind/brains of people who know elementary arithmetic. In that realm of shared concepts, 2 + 2 = 4 is a different fact, with a different meaning. And we can now show that it follows logically from other shared concepts, which we usually call axioms.
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Once created and communicated, mathematical objects are there. They detach from their originator and become part of human culture. We learn of them as external objects, with known properties and unknown properties. Of the unknown properties, there are some we are able to discover.
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Why do these objects, our own creations, so often become useful in describing nature? To answer this in detail is a major task for the history of mathematics, and for a psychology of mathematical cognition that may be coming to birth in Piaget and Vygotsky. To answer it in general, however, is easy. Mathematics is part of human culture and history, which are rooted in our biological nature and our physical and biological surroundings. Our mathematical ideas in general match our world for the same reason that our lungs match earth’s atmosphere.
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Psychological and historical studies won’t make mathematical truth indubitable. But why expect mathematical truth to be indubitable? Correcting errors by confronting them with experience is the essence of science. What’s needed is explication of what mathematicians do—as part of general human culture, as well as in mathematical terms. The result will be a description of mathematics that mathematicians recognize—the kind of truth that’s obvious once said.
Certain kinds of ideas (concepts, notions, conceptions, and so forth) have science-like quality. They have the rigidity, the reproducibility, of physical science. They yield reproducible results, independent of particular investigators. Such kinds of ideas are important enough to have a name. Study of the lawful, predictable parts of the physical world has a name: “physics.”
Study of the lawful, predictable, parts of the social-conceptual world also has a name: “mathematics.” A world of ideas exists, created by human beings, existing in their shared consciousness. These ideas have objective properties, in the same sense that material objects have objective properties. The construction of proof and counterexample is the method of discovering the properties of these ideas. This branch of knowledge is called mathematics.
