It is fortunate that metaphysics requires little mathematics. What it does require would be crucial, indispensable, but the boundary between the two lies deep in the foundations of mathematics where the main issues can usually be stated in elementary arithmetical terms.
Many decades ago the author defied all expectation and passed his mathematics ‘O’ level exam, and it remains his highest qualification in this arcane art. By a fabulous piece of good fortune he sat it in the year that the examination Board introduced simple set theory into the curriculum. The examiners were not quite sure how it would go down so played it safe, and the exam was a doddle. A few Venn diagrams to sketch and hardly any calculations. And yet, this turns out to be exactly what is required for metaphysics.
If we cannot understand the basic principles of set theory then we cannot understand the basic problems of metaphysics. They would be different ways of thinking about the same logical issues. When a metaphysician looks at a logical problem it comes with external referents attached: Something-Nothing, Mind-Matter and so on, When a mathematician looks at such problems the logic is freed from any contingent meaning and the problem is reduced to its simplest and most general logical form: Zero- One, One-Two and so on. In either language we can ask: Which came first? What comes before number and quantity? Can Zero exist without One, or One without Zero?
Two books, one recent and one long famous, usefully illustrate the relationship between metaphysics and mathematics. The first would be The World According to Quantum Mechanics by the physicist Ulrich Mohrhoff. Almost the whole of this book is incomprehensible to me. It is a text book covering the mathematics of quantum mechanics explaining how all the parts fit together with occasional fun tests to which I cannot even understand the explanations of the answers let alone the questions. Yet there is the odd remark here and there, and one chapter in particular, where the language becomes more simple. These are the thoughts of the author most relevant to metaphysics, and they can be understood with little reference to anything but the most basic arithmetic. Such thoughts must nearly always focus on the basic principles of set theory for these form the foundation of mathematics, the structure of thought that we uncover when we reduce mathematics to its most basic logical and numerical operations. Here we are almost in metaphysics and are already in psychology.
The second would be Das Kontinuum by the physicist, philosopher and mathematician Hermann Weyl. Again, most of this book is incomprehensible to me and would be to most people. Yet the pattern is the same. There are passages, and one chapter in particular, where things become, if not simpler, much more general, and that therefore require little more of the reader that than the simplest set theory. This is where the discussions gets down to basics and address the metaphysical implications of the mathematics. These two books appear to be in complete agreement as to the nature and meaning of what these implications are, and thus about the nature of reality.
In metaphysics set theory could be called ‘category theory’ and it would be all about the categories of thought. Suppose we were to take all the ideas in our mind, every single one of them without exception, and put them into one category called the ‘set of all ideas’. No problem, one might think. In fact it would be impossible. The ‘set of all ideas’ is itself an idea. In mathematics this problem is called ‘Russell’s Paradox’ since it stopped in its tracks Russell’s ten year attempt to ‘axiomatise’ set theory.
Such a simple problem, and yet one that causes important and difficult problems in mathematics, psychology, theism, consciousness studies, metaphysics, logic and, as some physicists would see it, theoretical physics. Paul Davies’ Mind of God is all about this problem. It is a problem of self-reference. The most general set has to contain itself, in which case it is not a set but a more complicated idea. Simple stuff but not trivial. If the set of all ideas cannot be an idea, then would this be a proof that the origin of the intellect cannot ever be an idea? Kant thought so, and the Buddha insists on it.
All this is not even up to the level of my long-forgotten set-theoretical examination questions. Yet it is profound and important. Kant’s solution to this problem was to adopt as an axiom for psychology a phenomenon that is not an instance of a category. This would be a phenomenon that we cannot think or imagine, but, rather, what would be required in order for us to think or imagine in the first place. He proposes that this is the proper subject for any rational psychology, thus for any theory of the mind.
In mathematics Russell’s problem is solved by George Spencer Brown, at one time a colleague of Russell, and in just the same way. In his Laws of Form he describes a calculus (a formal system of sets) which is axiomatised on a phenomenon that cannot be categorised.
In metaphysics this would be same solution as is given by the Buddhist philosopher-sage Noble Nagarjuna in his Fundamental Wisdom of the Middle Way, where he logically refutes all other metaphysical theories. In the Tao Te Ching we find Lao Tsu proposing that in no case can we categorise the world as a whole as this or that. Hence Russell’s Paradox would not arise for their view.
In this way a little set theory can reveal that an effective solution for Russell’s problem as it appears in metaphysics, psychology and mathematics would be to assume that the origin of everything is a phenomenon that is not available to our minds as an idea. It could never be an idea since it would be their origin and environment. This approach would also be very much required by physics if it is ever to explain why it cannot describe the world completely.
The whole debate between religion and science, to the extent that it is about reason and logic, may come down to how we go about solving this simple problem of set theory. It is this problem that creates the gap that some people would put God in. It appears in many guises. The Christian doctrine of the Holy Trinity will seem incoherent unless we solve it. The doctrine of Divine Simplicity would be a solution. The transcendence of dualism in philosophy would be impossible without a solution, since the only solution would be to abandon dualism. Unless it is solved metaphysics is a dead end, and then consciousness studies would have to see consciousness as some sort of conjuring trick. Set-theory would have no fundamental axiom. Cosmology would be the search for a theory to describe a clearly impossible event. The problem could hardly be more important or interesting.
Yet simple. This is typical. All the most important questions are simple. Unfortunately it can take a lot of time and effort to realise this, which makes them not so simple after all. We can, if the Buddha is to be believed, verify his solution for the ‘set of all ideas’ in our own experience. But this would not usually be an easy way to do it. In set theory it is not so hard, and Russell was happy with Brown’s simple solution. He failed to see that is also the solution given by the perennial philosophy, or ‘mysticism’ in my terminology, but he did see that the solution works in logic.
It works everywhere. If this is not strong evidence for the truth of the Upanishadic view of Reality that can be accepted by doubters then I doubt if there is any, unless it is a miracle on the road to Damascus.