Reply to Hersh: On Platonism

Reuben Hersh is a well-known contributor to the debate on the philosophy of mathematics. He wrote a follow-up to Davies' article, also published in the EMS Newsletter (June 08). He seems to be in general agreement with Davies, but focuses on a different aspect of criticism of platonism, namely the social-construction view of mathematics.
Already in the first paragraph, Hersh declares that most practicing mathematicians have a "rough-and-ready, naive" view of the philosophical underpinnings of mathematics. This is a very typical comment which mathematicians have heard from philosophers so many times that they finally stopped listening. It is a rather arrogant attitude to claim that people who spend a lot of time reflecting on various arguments for or against ideas, have a more developed and mature view of things than people who deal with the actual objects in question on a day to day basis. Since Hersh has written much about the importance of founding philosophy on the experience of mathematics, one might have hoped that he would pay specific attention to the experience of mathematicians who de facto experience mathematics closer and deeper than anybody else. Unfortunately, Hersh has chosen the view that most mathematicians do not understand what it is they are working with, and are philosophically naive and illusioned.
This type of attitude of philosophers towards mathematicians has been very damaging for the relations between philosophy and mathematics in the past, and one may hope that the future of the philosophy of mathematics will be more interested in a dialogue with mathematicians, rather than continuing the tradition of dismissing mathematicians as philosophically naive.

Hersh chooses to reject Platonism on the grounds that it is incompatible with the materialist outlook of reality, a view whose correctness he takes as self-evident and unquestionable. This type of view is a relic from the positivist era, and as such has lost much of its power since the days when it was at its most popular. Nowadays there are many rational scientifically-minded people who are not materialists, and we now know that materialism/mechanism/reductionism and so forth are not proved by science, but in fact just assumptions that one may or may not hold in connection with a scientific view of the world. One simply can't base an argument against platonism on a premise that lacks any proof or self-evident qualities.
One problem with Hersh's social-construction view of mathematical reality is that it could just as well be applied to, for example, physical science. If mathematical objects are social constructions, then why are not the objects of physics, such as planets, forces, energy, also social constructions? Despite the social aspects of science, its 'corporeal ground', its postmodern critique, deconstruction, and whatever, almost no sane person will deny that planets are really 'out there'. In the same way, a social-construction view of mathematics cannot necessarily imply that the platonist 'out there' view must be rejected. Of course there are cultural aspects to any human activity, but most people still hold that there are matters of fact beyond the cultural sphere. These facts are relations between objects existing independently of human culture. What this comes down to is the much repeated observation that a social-construction view of mathematics can't be used as an argument against platonism, it can only be used as an alternative after one has chosen to reject platonism.
The article also mentions the problem of mathematical 'intuition' and how, if the platonic realm exists, humans can obtain knowledge of it. Here Hersh mentions the curiously irrelevant observation that blood flow through various parts of the brain can be correlated to mathematical thought. As I have written before in the post on Davies' article, this is completely irrelevant as to the existence of a platonic realm, but again, it can be used to fill the vacuum after one has chosen to reject platonism. Hersh completely agrees with this when he writes:

From this point of view, the facts presented by Davies are relevant and interesting. They do not refute Platonism, they are part of the scientific program that one focuses on after rejecting Platonism.

In spite of Hersh being (rightfully) sceptical of Davies' line of argument in rejecting platonism, Hersh agrees with Davies in dismissing the possibility of a mathematical intuition interacting with a platonic realm:

It seems that Davies regards the evidence that thinking takes place in the brain as proof that there is no such “intuition”, in the sense of a special mental faculty for connecting to “out there.” But with or without neurophysiological evidence, it is pretty clear that the posited “intuition” is an ad hoc artifact, lacking any specificity or clear description, let alone empirical evidence.

The lack of specificity or clear description of the interaction between the mind and the abstract objects of platonism is no argument for or against the existence of such a function of the brain.
As I described in the post on Davies' article, we have no specific or clear description of how memory is stored, or how consciousness interacts with it or with other faculties of the mind. Does this provide a basis for rejecting the existence of stored memories in the mind? Obviously not. It is probably a risky and possibly not very useful strategy to limit one's philosophical positions to what science can tell us at the moment. Philosophy is at its most fruitful when it is not pretending to be science.
Hersh formulates his materialistically based conclusion despite admitting the lack of the proper empirical evidence asked for above:

The mental, social and cultural, including the mathematical, are grounded in the
physical – the flesh and blood of past and present humans, especially mathematicians. We can recognize this, even while the detailed nature of this grounding – just how our thoughts are carried out by our brains – may
never be completely understood.

If one has chosen ontological materialism, then Hersh's arguments are a natural and rational way to proceed from there. However, this position has some major disadvantages. Firstly, as we have already observed, it demands that one ignores the overwhelming consensus and experience of practicing mathematicians, and therefore alienates a whole group whose opinion is most relevant for the matter. Secondly, one looses the possibility of a good explanation of how mathematics can be so universal, stable in time, and above all, utterly useful for understanding the physical universe. Hersh dismissively calls this a puzzle (no doubt, he chose this word consciously, rather than the word 'problem'), while in fact it is a major problem and a great mystery (to use Wigner's words), a mystery whose only satisfactory explanation to date is platonism. It is nowhere nearly enough to say that all human cultures can observe and count pebbles or fingers, and that therefore they develop the same mathematics, and as long as we can see pebbles and count our fingers, mathematics will be universal. The thing is that the universality and timelessness of mathematics is manifest on a much higher level which cannot simply be reduced to counting pebbles. Examples of this include the remarkably congruent and simultaneous but independent discoveries of infinitesimal calculus (Leibniz and Newton), and non-Euclidean geometry (Lobachevski, Gauss, Bolyai), respectively. An example of the 'mysterious' effectiveness of mathematics is the application of group theory (which started as an area of pure mathematics) to physics.
If one adheres to platonism one must explain how our minds interact with the platonic realm. If one rejects platonism one must explain why mathematics is culturally and geographically universal, tolerant to the test of time, and why it is that mathematics is so efficient in describing our physical universe, even when it was not originally conceived for this purpose! These two are both very difficult problems, but given that we go beyond scientism and materialism, we have a chance of finding good solutions to them. I claim that reality does not simply consist of things that are empirical on the one hand, and things which are socially constructed on the other. This type of ontology (empirical objects + social constructions) is of course what remains after the two big trends in Western thought, exemplified by positivism and social constructivism, respectively. The problem is that both of these two trends ignored, or did not provide an adequate place for, a fundamental domain of intellectual pursuit: mathematics.

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.

Evolution and human nature

The Templeton Foundation has asked several distinguished researchers the question

"Does evolution explain human nature?"

The responses vary, and are in the form of essays available here:

http://www.templeton.org/evolution/

Of particular interest is Martin Nowak's reply which mirrors the ethos of this blog in a sharp and succinct way. Nowak is professor of biology and mathematics at Harvard University, and has this to say:

"Music is part of human nature. There is also something very intuitive about numbers and geometric objects, and the ability to do some basic math seems to be part of human nature.

Yet the great theorems of mathematics are statements of an eternal truth that comes from another world, a world that seems to be entirely independent of the particular trajectory that biological evolution has taken on earth. The great symphonies of Beethoven and Mahler capture glimpses of a beauty that is absolute and everlasting. Beyond the temporal, materialistic world there is an unchanging reality.

My position is very simple. Evolution has led to a human brain that can gain access to a Platonic world of forms and ideas. This world is eternal and not the product of evolution, but it does affect human nature deeply. Therefore evolution cannot possibly explain all aspects of human nature."

Grothendieck's platonism

Alexandre Grothendieck is one of the last century's greatest mathematicians. There is no controversy in describing him as the mathematician who has pursued the most abstract type of mathematics ever. In his mathematics there are usually no numbers directly involved. Most of the time there are even no letters denoting numbers involved. The level of abstraction is breathtaking and, to some, deeply fascinating. Nevertheless, his work does have more concrete implications for relations between numbers. There is a huge collection of articles by and about him on the internet. A brief recent introduction can be found here:

http://www.sciencenews.org/view/generic/id/31898/titl/Math_Trek__Sensitivity_to_the_harmony_of_things

The following passage from Grothendieck's Recoltes et Semailles expresses, in a somewhat poetical way, his scientific realism, in particular mathematical realism. It seems like he is saying that what we call inventions and imagination are in fact very close to what platonists refer to as intuition or 'the mind's eye':

"What makes the quality of a researcher’s inventiveness and imagination is the quality of his attention to hearing the voices of things."

Perhaps Grothendieck is also saying that not only is the platonic realm a static life-less structure which we can tap for information, but he seems to suggest that physical and metaphysical entities are actively revealing themselves in various ways, ways which we of course may or may not pick up.