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What Is Math About? Is It Discovered or Invented?

Philosopher Edward Feser suggests that the velociraptor, an extinct birdlike dinosaur, might illustrate the problem

Pasadena City College philosopher Edward Feser (pictured) offers some thoughts that may be relevant to the current war on math. Pointing to a recently published article by mathematician James Franklin, he writes,

What is mathematics about? The Platonist says that it is about a realm of abstract objects distinct from both the world of concrete material things and the human mind. The nominalist says that it is not really about anything, since mathematical entities are in no way real. The Aristotelian approach rejects nominalism and agrees with Platonism that mathematical entities are real. But it disagrees with the Platonist about the location of these entities. They are, for the Aristotelian, properties of concrete particular things themselves, rather than denizens of a Platonic “third realm.”

Edward Feser, “What is mathematics about?” at Edward Feser (April 9, 2021)

So mathematics is deeply bound up with questions about ultimate reality. For example, do 2 plus 2 = 4 because we have always found that to be true or do the math symbols reflect a reality about the world of numbers? Is the world of numbers real apart from our perceptions of it?

Dr. Feser suggests that the velociraptor, an extinct birdlike dinosaur, might be a handy illustration of the nature of the problem:

After they died out, the universal velociraptor was no longer instantiated, and before human beings discovered its remains, no human mind entertained that universal. So, during the long interval between extinction and discovery, the universal could be found neither in the world of concrete material things nor in human minds. But what was true about the essence and properties of velociraptors did not change during that time. What could have grounded that fact?

Edward Feser, “What is mathematics about?” at Edward Feser (April 9, 2021)

We sense that the velociraptor “existed,” in some way, during that intervening period. But how?

The view Feser espouses, scholastic realism, is that all things are present in the mind of God, including things we could not know about, or not yet anyway. That includes dinosaurs before we dig up the fossils but also truths about mathematics that we don’t know yet. That’s why mathematician Gregory Chaitin and others would say that the truths of math are discovered rather than invented.

Note 1: Feser discusses this stuff in more detail in Five Proofs of the Existence of God (2017), Chapter 2.

Note 2: The image of Velociraptor mongoliensis (2017) is courtesy Fred Wierum CC BY-SA 4.0.

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What Is Math About? Is It Discovered or Invented?