I've used this title before in Sep. 2007, even though it was really a discussion of George Lakoff's and Rafael E. Nunez's book, Where mathematics comes from. In fact, it was just my 6th post on this blog. This essay predates that post by 5 years (2002) and I found it by accident after someone returned a USB to me that I had lost. Though there is some repetition, this essay is written in the context of an overall epistemology, whilst the previous one is an argument against a specifically defined philosophical position. To avoid confusion, I will rename the Sep.07 post after the title of Lakoff's and Nunez's book.
I would argue that it is a mixture of both, in the same way that our scientific investigations are a combination of inventiveness and discovery. The difference is, that in science, the roles of creativity and discovery are more clearly delineated. We create theories, hypotheses and paradigms, and we perform experiments to observe results, and we also, sometimes, simply perform observations without a hypothesis and make discoveries, though this wouldn’t necessarily be considered scientific.
But there is a link between science and mathematics, because as our knowledge and investigations go deeper into uncovering nature’s secrets, we become more dependent on mathematics. In fact I would contend that the limit of our knowledge in science is determined by the limits of our mathematical abilities. It is only our ability to uncover complex and esoteric mathematical laws that has allowed us to uncover the most esoteric (some would say spooky) aspects of the natural universe. To the physicist there appears to be a link between mathematical laws and natural laws. Roger Penrose made the comment in a BBC programme, Lords of Time, to paraphrase him, that mathematics exists in nature. It is a sentiment that I would concur with. But to many philosophers, this link is an illusion of our own making.
Stanilas Debaene, in his book, The Number Sense, describes the cognitive aspect of our numeracy skills which can be found in pre-language infants as well as many animals. He argues a case, that numbers, the basic building blocks of all mathematics, are created in our minds, and that there is no such thing as natural numbers. The logical consequence of this argument is that if numbers are a product of the mind then so must be the whole edifice of mathematics. This is in agreement with both Russell and Wittgenstein, who are the most dominant figures in 20th Century philosophy. I have no problem with the notion that numbers exist only as a concept in the human mind, and that they even exist within the minds of some animals up to about 5 (if one reads Debaene’s book) though of course the animals aren’t aware that they have concepts – it’s just that they can count to a rudimentary level.
But mathematics, as we practice it, is not so much about numbers as the relationships that exist between numbers, which follow very precise rules and laws. In fact, the great beauty of algebra is that it strips mathematics of its numbers so that we can merely see the relationships. I have always maintained that mathematical rules are, by and large, not man made, and in fact are universal. From this perspective, Mathematics is a universal language, and it is the ideal tool for uncovering nature’s secrets because nature also obeys mathematical rules and laws. The modern philosopher argues that mathematics is merely logic, created by the human mind, albeit a very complex logic, from which we create models to approximate nature. This is a very persuasive argument, but do we bend mathematics to approximate nature, or is mathematics an inherent aspect of nature that allows an intelligence like ours to comprehend it?
I would argue that relationships like π and Pythagoras’s triangle, and the differential and integral calculus are discovered, not invented. We simply invent the symbols and the means to present them in a comprehensible form for our minds. If you have a problem and you cannot find the solution, does that mean the solution does not exist? Does the solution only exist when someone has unravelled it, like Fermat’s theorem? This is a bit like Schrodinger’s cat; it’s only dead or alive when someone has made an observation. So mathematical theorems and laws only exist when a cognitive mind somewhere reveals them. But do they also exist in nature like Bernoulli’s spiral found in the structure of a shell or a spider web, or Einstein’s equations describing the curvature of space? The modern philosopher would say Einstein’s equations are only an approximation, and he or she may be right, because nature has this habit of changing its laws depending on what scale we observe it at (see Addendum below), which leads paradoxically to the apparent incompatibility of Einstein’s equations with quantum mechanics. This is not unlike the mathematical conundrum of a circle, ellipse, parabola and hyperbola describing different aspects of a curve.
So what we have is this connection between the human mind and the natural world bridged by mathematics. Is mathematics an invention of the mind, a phenomenon of the natural world, or a confluence of both? I would argue that it is the last. Mathematics allows us to render nature’s laws in a coherent and accurate structure – it has the same infinite flexibility while maintaining a rigid consistency. This reads like a contradiction until you take into account two things. One is that nature is comprised of worlds within worlds, each one self-consistent but producing different entities at different levels. The best example is the biological cells that comprise the human body compared to the molecules that makes up the cells, and then in comparison with an individual human, the innumerable social entities that a number of humans can create. Secondly, that this level of complexity appears to be never ending so that our discoveries have infinite potential. This is despite the fact that in every age of technological discovery and invention, we have always believed that we almost know everything that there is to know. The current age is no different in this respect.
The philosophical viewpoint that I prescribe to does not require a belief in the Platonic realm. From my point of view, I consider it to be more Pythagorean than Platonic, because my understanding is that Pythagoras saw mathematics in nature in much the same way that Penrose expresses it. I assume this view, even though we have little direct knowledge of Pythagoras’s teachings. Plato, on the other hand, prescribed an idealised world of forms. He believed that because we’ve had previous incarnations (an idea he picked up from Pythagoras, who was a religious teacher first, mathematician second), we come into this world with preconceived ideas, which are his ‘forms’. These ‘forms’ are an ideal perfect semblance from ‘heaven’, as opposed to the less perfect real objects in nature. This has led to the idea that anyone who prescribes to the notion that mathematical laws and relationships are discovered, must therefore believe in a Platonic realm where they already exist.
This aligns with the idea of God as mathematician. Herbet Westron Turnbull in his short tome, Great Mathematicians, rather poetically states it thus: ‘Mathematics transfigures the fortuitous concourse of atoms into the tracery of the finger of God.’ But mathematics does not have to be a religious connection for its laws to pre-exist. To me, they simply lie dormant awaiting an intelligence like ours to uncover them. The natural world already obeys them in ways that we are finding out, and no doubt, in ways that we are yet to comprehend.
Part of the whole philosophical mystery of our being and the whole extraordinary journey to our arrival on this planet at this time, is contained in this one idea. The universe, whether by accident or anthropic predestination, contains the ability to comprehend itself, and without mathematics that comprehension would be severely limited. Indeed, to return to my earliest point, which converges on Kant and Eco’s treatise in particular, Kant and the Platypus, our ability to comprehend the universe with any degree of certainty, is entirely dependent on our ability to uncover the secrets and details of mathematics. And consequently the limits of our knowledge of the natural world is largely dependent on the limits of our mathematical knowledge.
Addendum: This post has become popular, so I'm tempted to augment it, plus I've written a number of posts on the topic since. When studying physics, one is struck by the significance of scale in the emergence of nature's laws. In other words, scale determines what forces dominate and to what extent. This demonstrable fact, all by itself, signifies how mathematics is intrinsically bound into reality. Without a knowledge of mathematics (often at its most complex) we wouldn't know this, and without mathematics being bound into the Universe at a fundamental level, the significance of scale would not be a factor.