Mathematical "Reality"

What is real? How do you define "real"?

Morpheus, from The Matrix

I was reading an article over at Republic of Math that led me back to an older article about whether mathematics is just a language describing other things. Gary Davis insists that mathematics is not merely a language describing other things, such as physics or chemistry, but it has its own real content.

"My own view," he says, "is that mathematics is a science – it is a science of number, space, pattern and arrangement." I'm inclined to agree with him.

But this question of whether math has "real" content is a curious one. I wonder if the reason mathematicians deal with this question so often is that mathematics is embedded within a scientific community that has, as a rule, embraced materialism as a philosophy. Thus "reality" can only be thought of as particles colliding in space. Every other view of the world is just metaphysical and not "real."

So there's this curious phenomenon of mathematicians throughout not only the 20th century but even today bringing forth a revival of mathematical Platonism, the view that the objects we study really exist, and are in fact even more fundamental than physical reality (Davis seems to agree with this view here).

I can see what's so attractive about this view. Being a young mathematician myself, I most definitely get the feeling often of interacting directly with mathematical concepts. It feels as if the concepts are "out there," as if I can touch them with the hands of my mind. Surely when I prove something like the fact that there are infinitely many primes, I'm actually saying something real, which implies that I'm saying something about real objects, namely numbers. These numbers presumably exist in an eternal realm that is accessible only through the mind.

Christian philosophers throughout the ages have been attracted to this view, as well (e.g. Augustine), which has to be one reason why Christianity has always attached itself to the idea of an immortal soul. The relationship between mathematics and spiritual truth is rather fascinating. In both cases, our intuition seems to tell us that the only way we can know these things is directly through the soul, rather than through anything physical.

I tend to wonder whether this is false on both counts. Does experience really prove that there are any numbers existing in an eternal Platonic heaven? Likewise, does experience really prove that we have direct access to knowledge of the divine through a metaphysical soul?

Yet underlying the question of whether mathematical "reality" exists is the more pedestrian question of whether what we do is important. If all we do is push around symbols all day, then I guess mathematics should be merely subservient to science. That would be a rather hard idea for mathematicians to swallow (although in terms of government funding it's unavoidable).

But I don't see it this way, even without being a Platonist. The job of a mathematician is to create something. Create what? A system of thought. This is profoundly important. Political commentators are responsible for shaping our collective fate as a nation, yet all they really do is produce ideas. Many religious leaders are responsible almost solely for teaching people ideas about life, and yet those ideas shape the way we live. Likewise, mathematicians create ideas that shape the way we do science.

And we don't just come up with systems of thought; we analyze them. We deduce all the important results of that system. Just as Gary Davis points out, the natural numbers are a perfect example. We essentially create a system of natural numbers. It is a system that follows a limited set of rules. We are then able to deduce facts about that system from those few rules. This leads to all of our beautiful theorems about prime numbers and divisibility and so on.

Change one of the rules, and you get something else. There's nothing wrong with that. For instance, you can change the parallel line postulate of Euclidean geometry to get new geometries. That was a huge deal when someone first figured that out, but I wonder if we would've been better prepared for that if we didn't confuse the systems we construct with some Platonic heaven that we can experience with our minds.

Nevertheless, I do think that the systems we construct are certainly real. They exist in our minds! Where else do you want them to exist? Do you think science is any more "real"? Does science describe "reality"? Science tries to describe the "physical world"--but what is that? Is the concept of "physical world" not a system that we ourselves have come up with? It exists within the mind, just like mathematics.

This is not to say science and mathematics don't teach us anything! On the contrary, they teach us precisely because they form ideas within the mind. What else is learning other than to absorb and act on carefully constructed ideas?

Ultimately I believe we can fuse these two questions together. The first is, "What is reality like?" The second is, "What should I believe about reality?" I think it is important to fuse the empirical question together with the normative question, because otherwise we end up chasing phantoms. As in, "Oh, well, that might be a good way of looking at the world... but is that real?" Nothing is ever "real" in the sense being sought after by such a question.

Suppose I ask, "Is the computer in front of me real?" My question is already based on a mental construct called "computer." Our functioning in this world largely consists of building such mental constructs. This is OK! Mental constructs are wonderful! And until something leads me to think that my mental construct has failed me, I will continue to believe that yes, I have a computer sitting in front of me.

Basically what I'm saying is that there are good and bad ways of looking at the world, and nothing more. People try to draw a distinction between how we merely describe reality and what reality actually is, but the only purpose that serves is to gain authority as one who looks at the world in the right way. No, scientists are not allowed to cheat and say science studies real reality. Science is one way among many of looking at the world (it happens to be pretty decent).

So should mathematics be judged solely on how useful it is for scientific progress? I don't think so. I think there are plenty of other considerations. For instance, aesthetic appeal, or perhaps some deeper intuitive concepts. For one, mathematicians always like simple concepts that have big consequences. If the ratio between the number of theorems one can prove about X to the number of axioms it takes to define X is especially high, then so much the better!

(My favorite two definitions in mathematics are the definition of a topology and the definition of a sigma-algebra. Who knew that such simple concepts could tie together so much knowledge?)

My point is that mathematics can be thought of as a subject in its own right not because mathematical objects exist in some ethereal realm, but because they exist in the human mind, and that existence is as real as it gets. We shouldn't think of mental constructions as "mere" mental constructions. The world of the mind is not a "fake" world. So when humans create new concepts and study them, we are doing something quite real--as real as real can be.