When I was at school, I enjoyed mathematics a lot more than PE. So when my PE teacher said my mathematics teacher was an idiot for believing that negative numbers existed, I naturally sided with my mathematics teacher. Now while I would still probably side with my mathematics teacher, I wouldn’t be quite so ready to judge my PE teacher as a mathematical ignoramus as I did back then. But during school and when I went on to study mathematics at university, I took it for granted that the mathematical objects we studied existed in some kind of Platonic realm. It was only once I entered the Dominican Order and started reading St Thomas Aquinas that I began to question this assumption.
The Angelic Doctor discusses the philosophy of mathematics in his commentary on Boethius’s De Trinitate. Although mathematics is not directly relevant to Trinitarian theology, Aquinas was concerned that if one’s investigation of the Trinity was to be fruitful, then one should use an appropriate method of investigation. Scientists in our own age would do well to learn from his example, for one often gets the impression that scientists assume that the whole of reality can be understood in mathematical terms, even when this assumption leads to absurd conclusions. It is therefore rather helpful to consider what mathematics is and what it can do so that we can avoid making such a mistake.
Now, according to Aristotle, mathematics is one of the three speculative sciences, the other two being physics and theology. To the modern mind, calling theology a science may seem rather strange, but it is actually very reasonable to do so when one considers what Aristotle meant by a science, namely, a mode of investigation in which things are understood in terms of their principles and causes. So theology is the science that seeks to understand things in terms of the First Cause which is God. As such, theology considers things in so far as they have existence.
Physics, on the other hand, considers things in so far as they are changeable, and so physics seeks to understand the principles of change. Among these principles is matter which is the ultimate subject of change, and so physics is primarily concerned with material things, since only material things are changeable. This is in contrast to theology which considers immaterial things as well as material things, since it is not essential to existing things that they be changeable.
At this point, one might wonder whether there could be room for any other science besides physics and theology, since if physics deals with what is essentially changeable, and theology deals with everything else, then it might look like these two sciences are sufficient to study the whole of reality. But there is another science, and this is because the sciences are not just distinguished from one another according to the kind of things that are investigated, but rather they are also distinguished according to how we think about these things, for science is essentially a human activity. And this is where mathematics comes in. For in the science of mathematics, the mathematician considers the same subject matter as the physicist, namely material/changeable things, but the mathematician considers them without considering any of the changeable aspects of things such as their colour or texture – all that is left for the mathematician to consider is quantity. For instance, instead of considering a triangular piece of paper, the mathematician will just consider the triangle and ignore its papery qualities. Likewise, instead of considering five apples, the mathematician will just consider five without considering the nature of the apples being counted.
Although mathematics is often viewed as a difficult subject, it is a lot easier to learn than physics because the mathematician has less to think about than the physicist, hence the etymology of the word mathematics coming as it does from the Greek word mathein which means “to learn”. Mathematics is nevertheless a very rich subject since there are so many ways quantities can be manipulated and compared to one another. For example, negative numbers are those numbers which decrease a quantity when they are added to it, and so negative numbers are just as real as the operation of decreasing a quantity.
This understanding of mathematics is not uncontroversial – after all, there have been many mathematical developments since Aquinas’s day, and so one might wonder whether his understanding can accommodate all these developments (I think it can). But the great advantage of this understanding compared to more Platonic views is that it is no longer a mystery why mathematics has anything to say about the material world, since the material world is the subject matter of mathematics. Furthermore, with this understanding there is no danger of being tempted to use mathematics to investigate the mystery of the Trinity. For such an investigation, we need the science of theology.
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