Mathematical aristotelianism

Aristotle's philosophy was a reaction to that of his predecessor and teacher Plato. One particular detail that separates the two philosophers is the question of the forms, or universals. While Plato claimed that the forms are eternal, existing independently of their physical actualisations, Aristotle thought that the forms were inherent in the objects, without any possibility of existing independently of the objects. Consequently, mathematical Platonism sees physical objects as instantiations of general forms, while mathematical Aristotelianism views mathematical forms as attributes of particular concrete objects. The question is, do particular objects (such as physical bodies) require the platonic forms for their existence, or do, as Aristotle claimed, the forms depend on particular objects for their existence?
At least when it comes to mathematics, the Aristotelian view runs into trouble because there are mathematical objects and results which cannot possibly have a physical basis. Let's list some examples:

• Infinite sets. Physics tells us that there are only a finite number of (observable) physical entities. (There may exist infinitely many distinct universes, but if you believe this, then you don't need to be convinced of mathematical Platonism.) Aristotle tried to solve this problem by rejecting the existence of 'actual' infinity, and replacing it by 'potential' infinity. It's not entirely clear whether this distinction makes any sense when analysed closer, but in any case modern mathematics could definitely not work without something like actual infinity. Moreover, it is a fact that modern mathematics has tremendous applications to all aspects of science. Hence mathematics is something that uses concepts without physical ground, and at the same time it is something whose consequences deeply affects physical reality.
• The Banach-Tarski paradox. If this result had a physical grounding, it would most likely have to be its standard interpretation involving two physical objects and decompositions of these. However, the mathematical result then implies something which is physically impossible, namely that one solid ball can be decomposed and the pieces rearranged so that to form two balls, each of the same volume as the first. The Banach-Tarski theorem can therefore not possibly have any physical grounding. Thus we have here another example of a mathematical form which cannot possibly be realised physically.

There are more examples of this kind, but it will not change anything to include these. To these examples the Aristotelian can reply that 'real' mathematics is grounded in physical objects, such as a number of pebbles on the ground, and then mathematicians dream up all sorts of abstractions on top of that, which are are just mental constructions with no connection to the real world. Well, the problem with this view is that these 'dreamt-up' abstractions have applications to the physical world which are just as intimate as the relation between three pebbles and the number three. To take a simple example, consider differential and integral calculus discovered by Newton and Leibniz. Operating with infinite sums of infinitely small quantities led to modern calculus and the continuum, techniques essentially incorporating infinite objects, and without which most of physics and its applications in technology, would not have seen the light of day.

Newton's platonism

The following often cited quote by Isaac Newton shows his thoroughly platonist attitude. Since he was primarily a theoretician it is safe to assume that the statement is as much about his mathematical discoveries as they might be about physical facts:

I do not know what I may appear to the world; but to myself I seem to have been only like a boy playing on the seashore, and diverting myself in now and then finding a smoother pebble or a prettier shell than ordinary, whilst the great ocean of truth lay all undiscovered before me.

It is interesting to note that another mathematical genius, Grothendieck, has also compared the mathematical body of truth, unknown to the mind, as an ocean.

Comments on Gardner and Davies

The most recent issue of the EMS Newsletter

http://www.ems-ph.org/journals/newsletter/pdf/2009-06-72.pdf

continues the series on Platonism with contributions by well-known Platonist and puzzle-composer M. Gardner, as well as another article by B. Davies whose first article initiated the debate and has been commented on in a previous post here.

Gardner's article is mainly a personal reply to Hersh and makes the case for 'Aristotelian Platonism' (yes, it sounds paradoxical), that is, the view that mathematical reality subsists in physical reality as forms of material objects. In this view mathematical reality is independent of human minds to the same extent that physical objects are so. I have already pointed out the inadequacy of this view in an earlier post, but will repeat some arguments in relation to Gardner article, because it offers a particularly good illustration.
Gardner says early on:

Consider pebbles. On the assumption that every pebble is a model of the number 1, obviously all the theorems of arithmetic can be proved by manipulating pebbles. You even in principle can prove that any integer, no matter how large, is either a prime or composite.

Already here there is something that is not right. Arithmetic reality cannot subside in physical pebbles because there are only a finite number of pebbles in the universe. If there were infinitely many pebbles, the universe would have infinite mass, and so would collapse into a singularity by the infinite gravitation. Therefore one can never prove anything about all (or infinitely many) integers and claim that this fact is a fact concerning physical entities. One can certainly not prove Euclid's famous result that there is no largest prime number by any manipulation of pebbles (or any other physical objects one happens to fancy playing with).
What Gardner's mathematical Aristotelianism leads to is mathematical finitism, that is, the belief that there are no actual mathematical infinities. Aristotle, and many finitists admit potential infinities, that is, patterns that in principle go on indefinitely, but which, it is claimed, do not actually go on infinitely. Personally I don't think the concept of potential infinity is well-defined, and it is not clear if it has any meaning beyond the muddy word-twisting of philosophy. How can we say that something in principle goes on forever, without it actually doing it? What gives us the right to assert this, and what does 'in principle' even mean here?
On the other hand, infinite sets have a precise mathematical definition which was given by Cantor, namely that to be infinite is to have a proper subset of the same cardinality as oneself.
In any case, modern mathematics could not exist in its present form without actual infinities. It is fair to say that actual infinity is indispensable to mathematics in the same way that mathematics is indispensable in natural science. To doubt the existence of something which is the basis for things one does in every-day life is a very dishonest attitude. This is an analogue of the Quine-Putnam indispensability argument for the reality of mathematical objects.

As I have written before, the Aristotelian view of mathematics is enough if one is only considering mathematics at the level of pebble counting or simple geometry. A similar mistake is made by conceptualists like Nunez who thinks that by explaining the axioms of mathematics in terms of cognitive structures, one has thus reduced all of mathematics to a mental construction. Gardner has no problem fitting the primality of 17 or Klein bottles into this picture, but already at the level of complex numbers and derivatives he begins to struggle, and has to resort to saying that these entities are probably somehow embedded into the physical universe, even though they do not have concrete material models. Already here we can notice a drift away from Aristotle who said that mathematical forms have to be carried by physical objects, to something more Platonistic, namely that the forms are somewhat more autonomous and that forms are instantiated by physical objects.
One can go even further. Try to explain how algebraic schemes or infinite dimensional representations of Lie groups depend upon the physical structure of the universe (and not the other way around), and how their existence depends on the existence of physical objects (or human minds!). Then explain how these abstract entities can play important roles in mathematical physics, which is a science describing physical objects. It is pointless to try to see these abstract entities as something residing in for example elementary particles. However, we know that certain abstract mathematical entities have a direct relation with these very same elementary particles, because we can describe the latter using the former.
In many ways I agree with Gardner's views, and it is commendable to try to defend mathematical realism in public philosophical debates using the common sense principle that existence is first and foremost material existence (a principle I don't agree with). Some people are using a similar approach in an effort to reconcile science with religion. The problem is that it doesn't quite work all the way. I have tried to explain above why it doesn't work for mathematics. Perhaps I will some day write something about why I do not think it can work adequately in the science-religion bridge building.

Regarding Davies' article, it mainly adds some details to his first one, and gives replies to the other contributions that appeared in this debate, noticeably the Platonistic one by Mumford. Davies mentions the non-Platonism of P. Cohen. This is a well-known theme basically dealing with a very special form of Platonism, namely set-theoretical Platonism. Platonists do not necessarily believe in the existence of sets, but a more reasonable view is the one mentioned in Mumford's article mentioned above, namely that set theory offers one possible model for mathematics. Mathematical facts can be grasped either through Russell-Whitehead's Principia Mathematica, or equivalently, though Quine's New Foundations. These are just two different formalisations of the same mathematical objects. If one is a formalist like Cohen professed to be in Davies' quotation, then one would regard the question of different formalisations describing the same objects as absurd and nonsensical. Still it is obviously the case that different models exist for the same mathematical entity. Set-theoretical Platonism may be wrong, but it is not the only type of mathematical Platonism.

Davies admires P. Davis' history of negative numbers and contrasts this with the discovery of the moons of Jupiter or America. The argument is that if it took such a long time for mathematicians to accept negative numbers, then it must mean that these are a social construction rather than a discovery of some objective existence. The point is however that there are many instances in the history of science where facts and discoveries have been accepted only after a prolonged debate. It took a while and heated debates before the existence of the vacuum was accepted in physics, and the debate continues with the latest findings of quantum physics. Today we are debating whether dark matter exists, or whether the anomalies in galaxy dynamics are simply due to inaccuracies in our physical formulas. How long did it not take until Darwin's theory was accepted. Is it even accepted now? Deep and complicated facts take time to digest and understand. Mathematical objects like negative numbers are abstract, so there is no surprise that it takes a while for them to be generally accepted. Of course mathematicians do not have a 'direct perception' of the platonic realm. We have to constantly push ourselves to our limits in order to get mathematical understanding and insights. Historically, only people who pushed their knowledge, perception and abstraction sufficiently, could grasp the negative numbers, but once the negative number were put into a mathematical and pedagogical framework it became much easier for subsequent generations to grasp them.

To compare abstract entities with very concrete ones like continents is like comparing apples and pears (even worse!). A future P. Davis appearing in 500 years could quote many thinkers from the ancient Greeks to young-earth creationists who lived around the year 2000, and based on this make the claim that the old-earth theory was a social construction. Would people then be right in believing this future hypothetical social constructivist? The answer is obviously no.

In another part of his article, Davies offers an interesting passage:

Platonism is relatively harmless, but no form does anything to explain why the orbits of the planets correspond so closely to the solutions of Newton’s law of
gravitation. Saying that the equations control the motion is vacuous unless one can at least begin to explain how this might happen, and I do not know of any significant
attempt to do this. My own approach is to admit that we do not know why the world exhibits so much regularity, but to regard this as a problem about the world rather
than about mathematics. Mathematics is simply our way of describing the regularity.

Contrary to Davies' statement, Platonism does indeed offer (the only available) explanation for the quantifiable (meaning it can be expressed quantitatively) regularity of the universe. It is however not an physicalist-positivist explanation of the kind that would satisfy Davies, and so he prefers to leave the whole question as a mystery. If our goal is to explain the universe, then it is better to take existing explanations seriously rather than adhering to some preconceived ontological commitments (i.e., physicalism) and hence ignore the explanations offered.
Davies quotes one of the most prominent living mathematicians, Michael Atiyah, who squarely puts himself in the conceptualist camp in saying that "Mathematics is part of the human mind". Of course there are several mathematicians who are Platonists, and several who are not (although they are in minority), but those who are not, like Atiyah and Davis, must leave the regularity of the universe and the unreasonable effectiveness of mathematics, as a mystical unexplained problem.