Topics Mathematics

Question For a long time I have been very concerned with clarifying mathematics, primarily for myself but also because I plan to teach. After decades of reading and questioning and thinking, it seems to me that the philosophical views of mathematics are nonsensical. What does it MEAN to question whether mathematical objects exist outside of our minds? It sounds absurd. It seems clear to me that mathematics is a science like all the others except that verification (confirmation) is different. It is the science of QUANTITY and its amazing developments and offshoots (like set theory). And all sciences are products of our minds. They are our constructions, as are most of the physical objects in our immediate worlds. Shoes, sinks, forks, radios, computers, computer programs, eyeglasses, cars, planes, airports, buildings, roads, and on ad nauseam, are ALL our constructions. Nature didn't produce any of them. We did. What does it MEAN to speak of a "PHYSICAL" circle? A circle is OUR IDEA of a plane locus equidistant from a point. A transistor is no less real because it is OUR invention. How can anyone MAKE such a distinction? Who cares what Plato thought about mathematics? He didn't know what an algebraic number is. He didn't know what a p-adic number is. Hardly any mathematics had been invented yet twenty five hundred years ago. Why do people respect in his speculations, his fictions? And the same is true of the other contenders that presume to account for mathematics. We are surrounded by our inventions and their properties. My father used to have to get his car greased. No one does that anymore. Now we have much better bearings. Can you please explain to me why there is so much bizarre speculation about the nature of mathematics? I hope you answer. I am truly perplexed. (I started as a philosophy major but switched to electrical engineering.) Thank you, George F.

Accepted:
September 9, 2011

Comments

"...[A]ll sciences are products of our minds. They are our constructions, as are most of the physical objects in our immediate worlds."

That is no doubt true, but it misses a crucial point: Scientific theories are of course human creations. But what those theories are about are (generally) not human creations. People do not make quarks, atoms, or molecules, fields, stars or galaxies, bacteria, birds, or insects, etc, etc, etc. Nor is it up to us whether the theories we invent are true. And even when physicists discuss colliding billiard balls, the fact that the balls were made by us is neither here nor there. They are external objects, and it is not up to us how they will behave.

Much the same is true of mathematical objects. I see no reason whatsoever to believe that numbers are a human creation, any more than tectonic plates are. And it is just a confusion to think that a circle is an idea. Surely whatever ideas I have are in my mind. Is mathematics supposed to be about the contents of my mind? Should we stop proving things and instead do MRIs to find out if the circle can be squared? What is, perhaps and in some sense, our creation is the concept of a circle. But in that sense, whatever it is, the concept of a molecule is also our creation. That does not make DNA a product of our minds, and the corresponding facts about mathematics do not make set products of our minds, either.

So what precisely are circles? What exactly is mathematics about? If it's true that every even number is the sum of two primes, even though we can't prove it, what makes it true? These are the questions that motivate philosophers, and it does nothing to clarify or help answer them simply to say that mathematics is "the science of quantity". The notion of quantity itself is badly in need of clarification.

Nothing I have just said is at all original. It's largely borrowed from Gottlob Frege, and I can do little more than recommend reading his Foundations of Arithmetic.