What Is a Number?
It's not easy to say whether numbers were invented, discovered, or generated
Felix, my youngest grandson, has aged out of Sesame Street. Since leaving, though, he calls himself “The Count.” Of late he’s busy counting his Legos. Trump, keen to remain POTUS, wanted more votes to count. No luck. Of late he’s counting the legal actions he faces.
For better or worse, though, we can’t count without numbers. But inquiring minds — and little Felix has such a mind — sometimes pause to ask just what a number is. Alert: this post will now touch upon matters philosophical.
Once upon a time, I directed an Honors Program. Often I’d ask candidates whether numbers were invented or discovered. Even the brightest students were puzzled. I’m still working on the question.
Suppose we invented numbers. Questions arise. Who invented them and when? Could we forget our invention? If so, what would happen to numbers?
Alternatively, suppose we discovered numbers. Where were they until we found them? How had they come to be there? Are there more to find?
Maybe it’s best to say that we generate numbers. But what do we generate them from? Surely not from a batch of chemicals. And it won’t do to say we generate them from other numbers. We’d have to ask, wouldn’t we, where those numbers came from.
Numbers are peculiar. They are not like physical objects. Consider a classroom full of chairs. There is a certain number of them, but the number isn’t in the room with the chairs. Nor is it outside the room.
Now just maybe someone will say that numbers must be physical, since we can erase the numbers on the whiteboard in that room full of chairs. But erasing a numeral isn’t erasing a number. Erasing all the numerals in the world only makes it harder for us to keep track of numbers.
Wait! What does Wikipedia say? “A number is a mathematical object used to count, measure, and label.” This answer just kicks the philosopher’s question down the road. After all, what’s a mathematical object? If we can’t answer this question, we can’t answer the question of what a mathematical truth is.
Doing philosophy, Bertrand Russell says, calls for a robust sense of reality. So why not consider mathematical realism? On this view, the truths of mathematics are true independently of what we make of them. They describe mathematical facts. Platonism is even more specific: Mathematics is about independently existing mathematical objects. Think numbers!
Given a moderate realism, how do we learn of such objects? Thomists do not treat numbers as Platonic forms. Nor are they mere inventions. Jacques Maritain contends that abstraction plays a key role. We grasp entities drawn from quantifiable sensible data. We then construct or reconstruct them at this level of abstraction. (Note: some see abstraction as a highlighting or illuminating.) As such, numbers are beings of reason. They correspond to the accidents and properties of bodies in the physical world, including chairs and the diminishing number of hairs on my head!
With regard to diminishing numbers, dare I end on a desultory note? Our national life expectancy continues to go down. The timing of when one’s “number is up” is, as they say, of existential significance. So significant as to incite some of us to explore philosophy as a preparation for death! Plato, after all, reports that Socrates taught that “those who practice philosophy in the right way are in training for dying, and they fear death least of all men.” Proverbs goes further: “Fear of the Lord is the beginning of wisdom.”
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