At a recent Ethos Institute seminar on ‘Science and the Christian Faith’, a participant asked an important question pertaining to the relationship between mathematical models and reality. Mathematicians and philosophers are still debating this contentious issue, and it looks like the jury will be out for some time yet.
I offer these reflections as a theologian and philosopher, and not as a mathematician.
There can be no doubt that mathematics is held in the highest regard in modern society as many believe in its power to unlock the truths of the universe of which we are a part. Mathematics has been triumphantly described as ‘the language of the universe’ because of its ability to depict physical reality with such precision and elegance.
The veneration of mathematics can be traced to the golden age of Greek philosophy. The great Pythagoras could say that ‘All is number’, and Aristotle who came after him could echo his view approvingly by declaring that ‘The principles of mathematics are the principles of all things’.
Closer to our day, Albert Einstein expressed his amazement at the power of mathematics thus: “How can it be that mathematics, being after all a product of human thought which is independent of experience, is so admirably appropriate to the objects of reality?”
Mathematicians and philosophers are drawn to mathematics because of its sheer beauty. Whether it’s the mathematical constant π or Einstein’s famous E=mc2, the sheer elegance of mathematical models and the way in which they help us to make sense of the physical world is at once stunning and attractive.
Mathematicians and philosophers generally agree that mathematics is in some sense related to the physical world, but how this relationship should be understood is still a matter of considerable debate.
In contrast to the so-called mathematical Platonists who believe that mathematical objects and ideas exist independently from the material world, I hold the view that they are mental abstractions of our perceptions of reality. This means that mathematical concepts are grounded in and therefore dependent on the material world.
The history of mathematics itself bears this out as ‘natural numbers’ emerged very early in human consciousness and systems representing numbers can be traced to very ancient times.
‘The counting of numbers,’ writes Schweitzer, ‘… arose at the dawn of human consciousness, to make it possible to number the oxen in a herd, or the number of coins in a purse, or the number of people in a tribe. Thus numbers are abstracted from concrete reality.’
Sophisticated systems like multiplication tables can be traced to the Sumerian civilisation during the Chalcolithic and Early Bronze Ages. The great Sicilian mathematician, Archimedes, developed a system of numbers that is so sophisticated and precise that it is said that he could calculate the number of grains of sand in the universe! Geometric figures and spaces are also abstractions based on our perceptions and observations of reality concerning spatial relations between objects.
Mathematics, Derek Abbott maintains, is the product of the human imagination that is used to describe or portray reality. Abbott even argues that although the majority of mathematicians claim to hold the Platonist view, they are in fact closet non-Platonists!
But why is the philosophy of mathematics important? It is quite obvious that mathematicians who have very different views about the nature of mathematics could do their work unimpeded.
I think this question is important for at least two reasons.
Firstly, it is important to have a realistic estimate of the power and effectiveness of mathematics. The non-Platonic view, in my opinion, alerts us to the fact that mathematics is a human enterprise and not the ‘miracle’ that some scientists have made it out to be.
Put differently, because perfect mathematical forms do not exist in the physical universe, mathematics is just a mental construct and the models it creates are merely approximations of reality. Seen in this way, mathematics not only has its limits, it is also vulnerable to mistakes and failures.
That said, the precision and effectiveness of mathematics is truly remarkable, prompting Eugene Wigner to write his famous paper entitled, ‘The Unreasonable Effectiveness of Mathematics in the Physical Sciences’ in 1960. ‘The mathematical formulation of the physicist’s often crude experience,’ Wigner writes, ‘leads in an uncanny number of cases to an amazingly accurate description of a large class of phenomena.’
But lest we get carried away with the perceived omnicompetence of mathematics (which the Platonic approach encourages), we should evaluate its successes more closely.
In response to Wigner’s paper, Abbott wrote a piece entitled, ‘The Reasonable Ineffectiveness of Mathematics’ which, following the arguments of Richard W. Hamming, highlights some areas of human inquiry where mathematics has had lesser success.
He notes, for example, that mathematics has less success ‘in describing biological systems, and even less in describing economic and social systems’. One possible reason why this is so, Abbott speculates, could be the way in which these systems are adaptive and mutable. ‘Could it be they are harder to model simply because they adapt and change on human time scales, and so the search for useful invariant properties is more challenging?’, he asks.
But the question of timescale and the limits of human perception should also give us pause when considering the successful mathematical models. Abbott adds: ‘Could it be that the inanimate universe itself is no different, but happens to operate on a timescale so large that in our anthropcentrism we see the illusion of invariance?’
The second reason is related to the first. A realistic estimate of mathematics would prevent us from embracing a naïve epistemological exclusivism (scientism) that dangerously neglects or ignores other kinds of truth.
While mathematical models have a remarkable way of portraying reality, they are also deficient in a number of ways. For example, they present a world of quantities without qualities. As the philosopher and poet Raymond Tallis has brilliantly put it: ‘The energy in Einstein’s equation is not warm or bright or noisy, and the matter is not heavy or sticky or obstructive’.
Mathematics has a very important place in our lives. However, we must never take the hyperboles of Pythagoras or Aristotle too seriously.
Instead we must follow Tallis’ wise counsel and never neglect other kinds of truth, especially truths that are ‘rooted in the actual experience of human beings that lie beyond mathematics: situational truths saturated with qualities and feelings and concerns, and differentiations of space and time (‘here’, ‘now’)’.
Dr Roland Chia is Chew Hock Hin Professor of Christian Doctrine at Trinity Theological College and Theological and Research Advisor for the Ethos Institute for Public Christianity.