The problem of interaction

The main argument against platonism is the problem of how we as human beings can interact with and obtain knowledge of, the platonic realm. In some ways one might say that arguments for platonism, by way of persuasiveness, stand or fall with the way in which they deal with this problem.
As I see it, the problem lies in the way in which we intuitively imagine the meaning of interactions. After all, a philosophical argument, given that it is logically correct, will be persuasive exactly if it is intuitively appealing to a majority.
Despite the modern advances in physics, most people probably still imagine interaction between two entities as two completely separate bodies, and some point in which they touch, or with some ray of energy going between them. This picture of interaction is the psychological source of the problems with dualism and platonism alike, because we can't have any direct physical touch-contact with things that are non-material, and moreover, we don't observe any "rays of energy" or anything similar going from somewhere outside the universe into our brains. So how can humans interact with the platonic realm? This question is very deep and difficult, so we can only begin to sketch an answer, which is by no means complete at this stage.

As stated above, the salient point is the ways in which we can model interaction. We have seen that the Newtonian billiard-ball model of interaction won't do, and neither will the more modern wave-mechanistic paradigm. After all, numbers do not hit us the way balls do, and neither do they induce processes in our brains after being picked up, as waves, by our ears or eyes. What other ways could there possibly be in which to model interaction? The answer may come if we consider analogies with more concrete phenomena. One example is:

How does memory interact with consciousness?

We know that memory is 'dislocated' in the brain, that is, there is no specific parts of the brain that carry memories. Still consciousness clearly interacts with at least some of our memories. We currently do not understand how this works, but whatever the mechanism, it seems very likely that it must be an interaction of a type very different from our everyday life intuitions. It is clear that consciousness and memory are partially separate entities, in the sense that we can be conscious without remembering the names of all the capitals of the world, and conversely we have memories that we are not immediately conscious of. Still these two entities must be considered as somehow intertwined, because consciousness is not physically touching a new point when it digs up old memories. The point is that it is not necessarily a question of an interaction that takes place as a transposition in space. We tend to think about interactions as points in space such that before the interaction there were two separate bodies, and during the interaction there is at least one point in space where the two entities meet. With our present understanding of consciousness and memory, this touching-point model seems to be inadequate.
I do not pretend to be able to explain how consciousness interacts with memory, but I claim that this phenomenon points to a new way of understanding interactions between entities, specifically between 'dislocated' entities.
It is naive and misleading to think about the platonic realm of mathematical entities as somehow 'outside' our physical universe. In the same way it would be wrong to think about memory as spatially located outside of consciousness. Still, not all our memories are in our consciousness, and indeed, at least some of our memories exist independently of our wanting them to be there, and independently of whether we are currently conscious of them or not.

This is an important point which I should follow up another time. Hopefully there will be more to come on this subject.

Reply to Davies: Let Platonism die

It's time to confront the series of articles on platonism published in recent issues of the EMS Newsletter. The first one that was published was E. B. Davies' article entitled Let Platonism die. Inclusion of words derived from death or dying usually indicates some zealous semi-political agitation, but let us instead focus on the arguments Davies puts forward:

Davies starts by expressing indignation over the fact that several Fields medalists have spoken positively about their platonist convictions. After quickly dismissing mystical experience as nonsense, in the best style of scientism or positivism, Davies seems to jump to conclusions when he states:

Although he is a Platonist, Roger Penrose is almost
unique in accepting that his beliefs imply that the math-
ematical brain cannot obey the known laws of physics.

It is true that Penrose is a platonist. However, it does not follow from platonism that the "mathematical brain" cannot obey the known laws of physics. Why not? Well, even though platonism postulates a type of perception with the capacity to reach beyond the physical world as currently understood, this does not imply that the mechanisms of the brain must themselves go beyond the laws of physics as currently understood. This only follows if one also assumes that the universe is causally closed, that is, that something beyond the physical universe cannot interact with the physical universe. This is something which platonism does not assert.
There are many aspects of the human mind that are currently not understood. Indeed, nobody would claim that we understand for example how human consciousness works. Platonism implies that there is a brain function which is not covered by today's explanations of the workings of the mind. This is not a very radical claim, because there are many brain functions which are currently not understood. If you claim that consciousness exists as a brain function, then you are claiming that the human brain works in ways which are not explained by current science. Does this make you an unscientific mystic? Obviously not. Does this imply that consciousness must be due to some physical laws not currently known? No. If one reads Penrose one learns that his original motivation to develop his ideas about consciousness as a product of quantum effects was to construct an alternative to the predominant view of the mind as a computer. This view of the brain as a computer does not fit well with Gödel's incompleteness theorem, and this, rather than any necessary need to defend platonism by postulating new physical laws, has been Penrose's goal.

Given the failure of Davies' claim that platonism requires some extra-physical brain mechanism, the remainder of his argument looses its relevance. Davies mentions how modern brain science has revealed mechanisms of sense perception and the perception of number. This is all very well, because of course we understand mathematics with our brains, and of course the way in which we understand certain things can be described in terms of neuro-cognitive models. Platonists would not deny such obvious facts. The point is that it is fully possible that our brains are able to produce mental states of awareness of objects whose existence is not dependent on the spacio-temporal world, but which nevertheless interact with it because these objects provide the forms in which the landscape of the universe takes shape.

Two Fields medalists on platonism

The Fields medal is the most prestigious prize that is awarded to mathematicians. One of the most recent Fields medalists, Andrei Okounkov was interviewed in the EMS Newsletter. Among other things, he didn't hesitate to express the following in reply to a question:

Much of your work has deep connections to physics. Does
that mean that you find it essential that mathematics is
related to the natural word, or that you would even think
of it as subservient to the other natural sciences?

O: When I said “our world” earlier I didn’t mean just the
tangible objects of our everyday experience. Primes are
as real as planets. Or, in the present context, should I say
that celestial bodies are as real as primes?

What I hope will slowly emerge from various posts on this blog is that a large majority of the best mathematical minds see mathematical platonism as a natural and convincing position. The skepticism often found among philosophers and empirical scientists does not take this fact seriously. Either mathematical platonism is true, or mathematics is the only area of human intellectual endeavour where its most prominent practitioners have less understanding about the nature of their subject than people from outside that area.
Here is an excerpt from the same article; an interview with Terrence Tao, who won a Fields medal the same year as Okounkov:

Many mathematicians are Platonists, although many may
not be aware of it, and others would be reluctant to admit
it. A more “sophisticated” approach is to claim that it is
just a formal game. Where do you stand on this issue?

T: I suppose I am both a formalist and a Platonist. On the
one hand, mathematics is one of the best ways we know
to try to formalise thinking and understanding of con-
cepts and phenomena. Ideally we want to deal with these
concepts and phenomena directly, but this takes a lot of
insight and mental training. The purpose of formalism in
mathematics, I think, is to discipline one’s mind (and fil-
ter out bad or unreliable intuition) to the point where
one can approach this ideal. On the other hand, I feel the
formalist approach is a good way to reach the Platonic
ideal. Of course, other ways of discovering mathematics,
such as heuristic or intuitive reasoning, are also impor-
tant, though without the rigour of formalism they are too
unreliable to be useful by themselves.

Two remarks are in order here. First of all, the interviewer obviously didn't find a better word to use than "sophisticated". One must not be led to believe that formalism is somehow deeper, better developed in its details, or somehow the preferred view of sophisticated minds. What was probably meant above was that platonism is the "first" position one naturally develops as a mathematician, and that with later reflection, one might be led to formalism. Obviously this doesn't say anything about the relative merits of truths of the two positions.
Secondly, Tao says he's both a formalist and a platonist. The common meaning of these terms, as used in the philosophical debate, is such that one excludes the other. Mathematical formalism as a philosophical position usually means that mathematics is ontologically seen as formal symbols. Platonism on the other hand posits the independent existence of mathematical objects, beyond the mere symbols, which are modelled by the formal systems used in mathematics. The latter seems to be the position of Tao, and it's a fair bet that most working mathematicians (but not all, of course) would agree.

The common perception of mathematics

I have from time to time brought up the question of the nature of mathematics in discussions with people from other disciplines. The common perception of mathematics by non-mathematicians is that mathematics is simply the language of the quantitative, a formalism developed to solve real-life problems. It follows from this view that mathematical objects have about the same ontological status as the words of normal language - at best they point to something that's real, namely, physical objects, but they are not themselves real. It is difficult to explain to non-mathematicians why this view is an inadequate description of mathematics. In the June 2000 issue of the EMS Newsletter, Jean-Pierre Bourguignon writes:

Some fallacies about mathematics. For
some people, mathematics is just the language
of the quantitative. They base their judge-
ment on the fact that mathematics enter-
tains a special relationship with language;
this opinion is shared by some of our fellow
scientists. We mathematicians know how
wrong this is, and how much effort goes
into building concepts, establishing facts,
and following avenues we once thought
plausible but turn out to be dead ends.
This widespread belief forces us to consid-
er more carefully how mathematics inter-
acts with other disciplines and what is the
exact nature of the mathematical models
that appear ever more frequently.

The challenge of explaining to non-mathematicians the perception of mathematics obtained from years of deep study of the subject is made difficult for several reasons. There seems to be a general mistrust, in academic or philosophical discussions, of the judgement of mathematicians regarding their own subject. After all, wouldn't a long and intense study of the characters of the Lord of the Rings eventually engender the view that these characters have an existence of their own?
There is a cognitive difficulty in understanding mathematics the way working mathematicians understand it. Not only does one need a tremendous effort of abstraction, it is also not enough to just read about the results of mathematics in reviews or books, because these presentations are usually mathematics after it has crystallized into formalism. Mathematics as a living development is a completely different thing (as mentioned in the above quote), and this can only be understood by actually engaging in research, or by listening to what mathematics have to say about their work. Here is Bourguignon again:

A special link with truth: Confronted with
a mathematical statement, a mathemati-
cian's goal is to prove that it is "true". What
does this mean? Of course, now that math-
ematicians agree to work in the context of
a potentially completely formalised theory,
ts meaning can be none other than 'the
statement can be deduced from the basic
axioms agreed upon'. In fact, if we are to
address this question in the context of the
relations of mathematics to society, we are
forced to take it in a broader, more philo-
sophical, perspective because we have to
confront it with reality. This amounts to
deciding whether there is a mathematical
reality of which this statement is a part,
and of the status of this reality in relation
with ordinary 'sensible' reality. On this
broader issue, mathematicians have differ-
ent views: from the 'Platonists' at one end
of the spectrum who believe that they 'dis-
cover' new territories and new facts of the
mathematical world, to the 'intuitionists' at
the other end who view mathematical con-
structions as purely human scaffoldings
based on consensus opinions among a
restricted community - in other words,
that mathematics is 'invented'. I have no
doubt that the vast majority of mathemati-
cians are closer to the Platonist viewpoint
than to the other one, or at least that they
spend a good portion of their professional
time behaving as if they were, as André
Weil once put it.

One of the symptoms of non-mathematicians perception of mathematics is the view that mathematics is not a science. If maths is a formal language that deals with logical relations between fictional entities or mental constructions, then how can it be a science? Many scientists view science as an empirical activity, first and foremost. For most working mathematicians this is an unfamiliar view which does not resonate with their everyday experience of mathematics as an activity of uncovering synthetic and deeply meaningful truths. In his conclusion Bourguignon writes:

From all the above challenges concerning
the image of mathematics, the most serious
seems to me the need to make it evident that
mathematics is a science and alive. The rest
should follow.