And now that we’ve officially embarked on Summer, it’s time for something completely different, as Monty Python famously remarked.
Mathematics is a thing apart, like nothing else in our world. It’s a human invention (I think), but not like other inventions. It exists in our minds and nowhere else – although its effects are felt everywhere since science, engineering, and technology have adopted it as their language. Yet, despite the fact that most people lump math and science together, they are very different.
Science is the observation of natural phenomena combined with the attempt to understand, explain, and predict what will happen next. It is rooted in the real world (and I’m going to avoid the whole metaphysical question about what is “really real.”) But although science uses mathematics, mathematics doesn’t use science. Indeed, it has been argued, and I’ve argued myself, that math is more akin to art or philosophy than it is to science, because at its center is an act of creation, often from nothing except the mind itself. Indeed, mathematics could not exist unless it were created by the human mind. I know from my own experience, and the descriptions of others, that arriving at a mathematical insight brings a sense of harmony, beauty, and wonder, which mathematicians call “elegance,” that is like the sense of creative rightness that artists experience. Indeed, one description of this creative sense, based on the observations of the 19th Century French mathematician, Henri Poincaré runs as follows:
Mathematical solutions are selected … on the basis of “mathematical beauty,” of the harmony of numbers and forms, of geometric elegance. “This is a true esthetic feeling which all mathematicians know,” Poincaré said, “but of which the profane are so ignorant as often to be tempted to smile.” But it is this harmony, this beauty, that is at the center of [mathematics].
Why mathematics is different
And again, mathematics is unlike art or philosophy, because the creations of mathematics can be proven (within the assumptions that bound them) to be correct or incorrect. This is what it means to prove a theorem: you demonstrate its correctness. And there is no doubt involved; once proven, no one can later disprove such a theorem without altering the assumptions of the underlying mathematical system. No piece of art can be so demonstrated to be correct, and those works of philosophy that can be proven are, themselves, effectively sub-sets of mathematics based on logic.
I was struck by this conclusion in part because I recently read an article written in 1960 by physicist Eugene Wigner, called “The Unreasonable Effectiveness of Mathematics in the Natural Sciences.” This classic essay describes the uncanny way in which natural phenomena can be described by mathematics, to the point where many natural phenomena, before they were understood, were correctly predicted by their mathematical descriptions. Wigner asks a central question: Why should this be? Why is it that an invention of the human mind can be capable of being so good at describing nature, particularly in cases when the mathematics involved was invented long before it was ever applied to science, as has happened in many instances in the history of science.
Wigner’s comments have been discussed and debated by many people since then. One commentator, physicist Max Tegner, suggested in 2007 that the reason that mathematics is so good at describing reality is that the physical world is completely mathematical. Others have commented that it is our own myopia that causes us to be able to explain the world in mathematical terms: we are only capable of observing those aspects of reality that are capable of being described by mathematics. Both statements are unprovable, and are regarded as a philosophical in nature, rather than mathematical, yet Tegner’s comment raises an interesting point that stimulated two additional thoughts.
Perhaps mathematics is reality
One is the slightly whimsical comment by nuclear physicists that subatomic particles are really just mathematical constructs. I suspect this is just a means of saying that subatomic particles are physicists’ way of balancing the books of quantum mechanics. But as someone trained as a mathematician, the thought struck me that perhaps this observation is literally true, that reality is nothing more than mathematics, and that explains why mathematics can describe reality, for they are the same thing.
The capstone of my conjecture comes from the cult-classic book that has been repeatedly ridiculed because it was a cult-classic, Zen and the Art of Motorcycle Maintenance. In this book, the author, Robert Pirsig, pursues the concept of Quality. To understand why this is important, you will have to read the book, which I recommend in any case. Pirsig is attempting to prove that the philosophic proposition that Quality is at the center of our perception of the universe, that, indeed, it creates the universe we perceive. In this, he is running counter to the large body of Western philosophic thought. Indeed, he had been challenged to prove that Quality existed, in part because he felt that it was vital to human existence, and that it was beyond our ability to define.
To prove the existence of Quality, he used a logical technique from the philosophical school of realism. His argument ran like this:
A thing exists if the world can’t function normally without it. If we can show that the world without Quality functions abnormally, then we have shown that it exists whether or not we can define it.
He then considered what our world would be like if we subtracted Quality from it. Art would cease to exist, because if you can’t distinguish good art from bad, then there is no point in art: a bare wall is as valuable as a painting (and let’s please agree to ignore so-called artists who produce bare walls and call it art). Likewise, poetry and literature would disappear, as would sports and acting and all kinds of performance. In the marketplace, only one kind of clothing would exist, and only plain foods. Indeed, virtually everything in our world would change, and dramatically for the worse, with one exception. In this way, he argued, it is demonstrated that Quality exists. Now let me return to my own thoughts by drawing on his observation.
The one exception in Pirsig’s world-without-Quality was that mathematics, science, logic, and philosophy would be unchanged. He didn’t venture to guess why this should be so. I don’t know, either, but I would like to speculate.
Or perhaps mathematics contains reality
Perhaps mathematics is a thing apart from all the other things we create because it truly is the underlying reality of existence, the frame that contains existence. Perhaps it is mathematics that creates us, rather than the other way around, and it is our discovery of this underlying reality that leads to the mathematician’s sense of elegance and rightness.
I once speculated, in a presentation to a group of computer scientists, that once we learned enough about reality, we would find that reality was a computer (or rather, an information processor), that the elements of reality, from subatomic particles all the way up to ourselves and our minds, are packets of information, and that existence is the processing of these packets.
Or, to put it more poetically, perhaps we are merely mathematics in the mind of God.
It’s a thought.
 Pirsig, Robert M., Zen and the Art of Motorcycle Maintenance, Bantam New Age Books, New York, copyright 1974.
 See, for instance, the following link for a copy of the original paper: http://www.physik.uni-wuerzburg.de/fileadmin/tp3/QM/wigner.pdf
 Pirsig, pp.193-4.