** PHILOSOPHY, MATHEMATICS AND STRUCTURE**

An Interview with James Franklin

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Philosopher1 (2) (Winter, 1995), 31-38)

James Franklin is a senior lecturer in Mathematics at the University of New South Wales. He holds a PhD in algebra from Warwick University, and has written on the history of ideas and on neural nets as well as on the philosophy of mathematics. He is co-author of the textbook

Introduction to Proofs in Mathematicsand the historyThe Science of Conjectureand is completing a book on recent Australian philosophy.

Here he talks to David Shteinman about his views on a new direction for the philosophy of mathematics, based on what mathematics actually tells us about the world.

Jim, as a philosopher and a mathematician, how do you see relations between these two disciplines?

Philosophy and mathematics are the two great armchair disciplines, and it is time there was a rapprochement between the two. Unfortunately, at the moment cold war conditions apply. Philosophers often want to 'use' mathematics somehow, but forget that if you want to pontificate about x, you need to know something about x. On the other side, mathematicians typically talk garbage when they're asked philosophical questions. The New Zealand philosopher, Alan Musgrave, said in this connection, "fish are good at swimming, but poor at hydrodynamics", and that exactly describes mathematicians who dabble a bit in philosophy in their spare time.

So, in a nutshell, what is your position? What is mathematics about?

Mathematics is the science of structure. It studies structural relations. So it can also be described as the science of complexity ('complexity' just means 'a lot of structure'). One reason for the difficult relationship between mathematics and philosophy is that philosophers are trained to a certain blindness to structure. Rae Langton (a philosopher at Monash University) recommends the use of philosophers on Ethics committees because, she says, philosophers are used to complexity. But that's completely wrong.

Philosophers hate complexity.What they want is to see complexityexplained awayin terms of a few simple principles.

In your 1989 paper, 'Mathematics, Necessity and Reality’, you stated that mathematics can be both necessary and about the real world. Is your position Kantian in that the necessary relations require perceptual concepts of space and time - so the subject matter of maths is just relations of perception of space and time?

No. Symmetry is a real property - spatial things have it but so do non-spatial things. Arguments, for example, can be symmetrical by being the same backwards as forwards. Space is easy to think about, but maths studies the relations in anything, not just spatial and temporal things.

A point that runs through several of your papers is that you can have both necessity and reality in mathematics. That's the opposite of Einstein who said, "As far as the propositions of mathematics refer to reality, they are not certain, and so far as they are certain they do not refer to reality"

What's wrong with this idea comes out in the old view of mathematics as the 'science of quantity'. Quantity is a real aspect of things, and mathematics studies it. In the last 200 years, mathematics has expanded to study some other things, like network topology (as in the famous Königsberg bridges problem). These things are not exactly 'quantity', but they're still real aspects of things.

But isn't that about spatial relations?

In that case, it is, but it doesn't have to be. Topological questions can be raised concerning paths in an argument. Or concerning 'slippery slope' arguments, or continuous variation in the 'space' of colours.

So you are against the Kantian position that maths is about relations in space and time only?

Yes. I say mathematics is the science of the real structural properties of things. In this I take an Aristotelian, rather than Platonist, view of the matter. Platonism is a traditional option in the philosophy of mathematics, holding that numbers, sets and so on exist in some independent 'abstract' world, and that that explains the objectivity of mathematical truth. Platonism makes two mistakes about mathematics:

First. Because ancient geometry dealt only with simple shapes, it never dealt with the shapes of real things. This left the impression that geometry could deal only with 'abstract' or 'Idealised' lines or circles. But in fact modem mathematics can deal, via approximation theory, with complex near-circle shapes.

Second. There has always been confusion between the Platonist and Aristotelian view of universals. I hope the revival of Aristotelian realism generally, in the work of Armstrong and others, will make it possible for believers in the objectivity of mathematical truth to avoid rushing to Platonist extremes.

Could you elaborate on the Aristotelian position. Is it about abstracting 'from' or 'out of' the world?

To 'abstract' in Aristotle does not mean that the abstractions live in some other world. To say biology is the science of life, and hence

considers just the 'living' aspect of thingsdoes not mean that 'life' is other-worldly, or that biology does not study the life that actual living things have. It just means biology is the study of living things. It is the same with mathematics. It studies certainaspects of the real world - the structural aspects.

I recall your example of the actual impossibility of tiling a bathroom floor with (normal sized, close-fitting) pentagon-shaped tiles.

Yes, your bathroom floor is a real thing with a structure of a near-flat Euclidean plane. And you can prove of that structure that it can't be tiled with pentagons.

So you have proved something necessary about reality.

But coming back to Kant's position: we cannot talk about, say, a 15-dimensional sphere because we can't form the a priori concept of a 15-dimensional sphere.

That seems to be a fact about our psychology, like most of Kant's philosophy. In fact if there could be 15 dimensions then we could talk about aspects of the structure of 15dimensional 'space', and that talk would be true. What we can conceive or imagine is just not relevant - that's confusing epistemology and ontology. Kant is just psychologising. He was proven wrong by the existence of 4-dimensional geometry (as in special relativity) and non-Euclidean geometry (in general relativity). Maths correctly describes the structure of these alternative geometries.

Do you agree, though, that the propositions of mathematics aresynthetic a priori?

1 don't like the language. Both words are too psychological, so that both 'yes' and 'no' are misleading answers. The Aristotelian position is to ask what there is in the real world. You don't ask first what conceptions you have of the real world - that's a second-order epistemological question, to be considered later.

So you start with experience, not conceptions.

Yes. And for maths you start with the different relations (between the things) you find out there.

So the material world and its structure are the sources of the relations that maths studies?

Yes. And that's what you see in any child learning numbers and shapes. It's true that there is more in mathematics than has been experienced. A 5-year-old will have trouble, initially, in going beyond small numbers. But after some more abstraction one can abstract number, and the process of always adding one to a number that one already has, so that one understands numbers beyond the scale one has seen. It is the same with dimensions - we understand the difference between 1, 2 and 3 dimensions, and we can abstract the notion of dimension and then consider many dimensions.

What about Cantor's grading of infinities? That's well beyond experience, surely?

True. But Cantor only came to them by a process of abstracting the notion of cardinality of pairing off sets -itself. Once you do that, there seems no reason why there can't be infinities of different cardinality, as Cantor suggests. The fact that 1 can't "clearly conceive them" is no reason for not believing in them. To say otherwise is to confuse epistemology and ontology again.

This confusion of epistemology and ontology could you elaborate?

You see it again and again. Hume, for example, argues that it is impossible for space to be infinitely divisible. He says: we can't conceive of it, so it can't be. It's not a good argument. In fact, it's closely related to the argument that David Stove awarded the prize in his competition to find the worst argument in the world: "We can know things only as they are related to us/ under our forms of perception and understanding/ in so far as they fall under our conceptual schemes, etc.; therefore, we cannot know things as they are in themselves". In all these, the trouble arises from asserting a necessary connection between what we do or can know, and what is.

So I suppose General Relativity shows that there are things out there which we do not find in our immediate sensory experience.

Yes, the physics shows that you just have to put up with curved 4-dimensional space-time (or embrace it joyfully, depending on your appetite for paradox).

But physicists, like Einstein, don't see it that way…

Physicists and engineers have a funny view of mathematics. They think of it as a bag of tricks, methods, formulas and what not to get from one experimental result to another - a kind of 'theoretical juice extractor' to get predictions from theories. It drives mathematicians crazy. It's true that mathematics does have that aspect to it, but it isn't central. Mathematics in, say, special relativity, acts as a way of describing the structure space-time has. Which is not to say that mathematics is 'just a language'. The structure is the thing, and mathematics is the science that describes the structure.

Are you suggesting that maths discovers actual structural relations?

Well, yes, but it discovers more than are actually there - in the same sense that biologists can consider more species than are actually there. It discovers what you can have, as well as what possibilities are actually instantiated.

Could you elaborate on this notion of structure? In your 1989 paper you gave the example of Newton deriving the elliptical orbit of the planets from his laws.

He started with the universal law of gravitation. That gives you the local structure of what happens in a small area of space and time how the situation at one moment determines the situation at the next moment. It's the same as the way a bank calculates your bank balance for tomorrow from the balance today by adding interest according to a formula. The

mathematicalwork is in finding the global structure that results - the ellipse in the Newtonian case, or the exponential growth in the bank balance case. The global structure results from the gradual build-up of the effects of the local structure. Now the mathematical work doesn't tell us what the structure is, just the relation between the local and the global structure.

But by structure you don't just mean 3-dimensional physical structure?No; for example group theory studies the structure of symmetry, and symmetry can be realised in both physical and non-physical things (in plots of narratives, for example, as the literary structuralists said). Continuity is another structure that can exist in time as well as space, and also in the meanings of words, where the vagueness of the meaning of most words means that there can be borderline cases, with varying degrees of being borderline.

What about number theory? In your 1994 paper you mentioned that philosophers of mathematics have always been stuck on the question, "What are numbers?", but this is not very revealing from a structural point of view.

That's right. Numbers and sets, which philosophers traditionally concentrate on, are a sort of degenerate structure. They arise just from the fact that things are different from other things. They supervene, that is, on the relation 'is not equal to', which is blind to most interesting structures. The more exciting structures are the complex ones like topological structures, continuity and symmetry.

Philosophers would be better off concentrating on the rich structures of Operations Research and computer simulation; they are much more revealing about what mathematics can tell us. But with philosophers' trained to reject anything that isn't simple, what can you do?

You would include queueing theory?

Yes. A queue has a subtle (probabilistic) structure which mathematics can reason about in the sense that it can look at the interaction between the local structure driving it - the people arriving randomly to join the queue - and the global structure - the fluctuating length of the queue.

Yes, having done queueing theory myself I must admit it was very interesting but I couldn't see its philosophical relevance at the time!

Because you were trained to think of things like that as mere technicalities.

Do you think the formal sciences, as you call them, can fuse the necessity and reality aspects of mathematics - as the structures it examines definitely exist in the real world?

Yes. The more structure you have, the more necessity there is to find in the relations between the structures.

What are the areas of maths where this is happening?

Group Theory and Topology on the pure mathematics side. There's also algebraic topology, which tells you about the different kinds of surfaces Dieudonné said it was the central topic in pure mathematics this century. In applied maths there is Operations Research, and there is mathematical modelling or computer simulation. Simulating by computer is just cloning a structure. We throw away the irrelevant physical bits of a system and study its structure alone - directly.

But computer simulation - does it have the intellectual content that deriving equations analytically does?

The mental operation in writing equations lies in identifying

the structure that the equation describes- not in the string of symbols that is the actual equation. What leads to different equations is the different structures. It's quite difficult to describe structures, and it's unfortunate that the technique of doing it is rarely taught in mathematics and computer science courses.

But surely that's what physics does? Aren't you just really talking about mathematics doing physics?

First, a lot of physics is experimental measurement and data interpretation. In theoretical physics, it's understood that the equations like Newton's laws describe what's truly going on, but, as the Einstein quote showed, physicists too can take a surface view of the mathematical symbols, as if equations are primarily a linguistic entity that one manipulates according to rules. And it's worse for those who view the matter from a greater distance, like philosophers.

The philosophers just don't understand equations and what's going on!

Philosophers have been misled by Frege, Russell and Wittgenstein into thinking that mathematics is primarily a manipulation of symbols. Perhaps if Wittgenstein had persevered with his engineering a little longer, he would have got beyond the bag-of-tricks view of mathematics that engineering students are fed.

So you're not a logicist or a formalist?

Definitely not.

Could we discuss chaos theory and how it fits into your view of mathematics.

Chaos theory is very nice from my point of view, because it's about how a simple local structure can determine a very complex global structure. With the older mathematics of linear systems, it was easier to ignore the difference between the two, the local and the global, but chaos theory rubs your nose in it. Chaos theory is about systems like turbulent fluids that develop in

time,but the fact that one axis represents time is irrelevant from a strictly mathematical point of view. It is just that the differential equationsdescribe a simple local structure, and what develops out of it is globally complicated.

So the mathematical equivalent of "Go West, Young Man" is "Get Into Chaos Theory''?

Yes. This and other mathematical developments of the last fifty years, like Operations Research, systems engineering, genetic algorithms - are exciting in allowing the use of computers, of pictorial thinking, of direct applicability. And they tell you a lot more philosophically than some of the older branches of mathematics, like logic.

But in OR - Operations Research- say in modelling a queue, the gap between model and reality is narrow, and one doesn't feel one is doing anything so fundamental.

But that's an advantage. You're just stating true facts about the queue, and seeing where they lead. You are not diverted, philosophically speaking, by temptations to introduce Platonist entities, like you are in number theory and set theory. Who would want to talk about the Queue In Itself, when you have real queues before you, which you're reasoning about successfully?

But in terms of breakthroughs, you're not suggesting OR is going to make mathematics forge ahead? Isn't there a cost in loss of breadth in a topic like OR that's close to reality? Would topology, say, be broader?

True. But there's no difference in principle. The structure in topology may be more exciting than in OR, but that is not important philosophically.

And do number theory and set theory not reveal anything about the structure of the world?

They do reveal something, about the simplest type of structure. But philosophers have fixated on the fact that you can construct all of maths in set theory, as if that makes set theory more basic, and - here is the real subtext, surely excuses them from studying the rest. 1 don't see that fact as an exciting one. It just diverts attention from the fact that you are constructing structure

withinsets. You have to know what structure you're looking for before you can construct it out of sets. In the same way, if it turned out that all mathematical structures could be made out of wood, it wouldn't follow that mathematics was a branch of carpentry!

Let's now talk on a more general level about mathematics and philosophy in general. Does mathematics help with the big philosophical questions?

It's been said that western thought has been blind to the reality of relations. Well, doing mathematics the right way - OR and chaos theory before number and set theory - is the ideal therapy. It gives a good feel for the reality of relations: symmetry, ratios and so on. That is a help for some other philosophical questions. Take Leibniz's Best of All Possible Worlds theory. It's a kind of 'bump in the carpet' theory - if you fiddle around here to improve things, it makes things worse somewhere else. Now this may appear crazy, because it's easy to imagine fixing up one bit without affecting distant parts of the world fabric. But a feel for the qualitative theory of differential equations - of which chaos theory is a part - will, I think, make you a lot more sympathetic to Leibniz. Chaos theory shows exactly how interconnectedness can quickly lead to unexpected interactions, and strange effects of what you do here or there.

That could affect ethics too?

Yes. For example, a standard objection to Utilitarianism is that you just can't predict the consequences of an action, so you can't choose the action that leads to the most happiness, because you don't know which action that is. But we do have a way of predicting consequences, and it's pretty good (though of course it's fallible). It involves running a mental simulation, which we visualise in the imagination. The word "imagination" has unfortunately been hijacked by the literary people, but it used to mean a literal mental visualisation facility. How good it is is shown by the fact that we use it every day to drive a car - to predict the state of the traffic ahead of time. And we mostly stay alive after using it. This ability is very important, and is essential for any consequentialist ethical theory. But imagination is limited in the complexity of the situations it can handle; it's not much good for whole worlds.

Are there any other mathematical structures of use in philosophy?

What about symmetry, the subject of group theory? Symmetry

argumentsare a staple of philosophy. The best argument for global scepticism is a symmetry argument. A deceitful demon world and the real world are scenarios between which one must choose, on the basis of perceptual experience. The sceptic will argue that the two aresymmetricalwith respect to the information you have, your sensory evidence. Not just that you can't be certain which one holds, but that there isno reasonto prefer one to the other.

Another place where symmetry might help is with Nietzsche. He says that different people have very different rights (to use an unNietzschean idiom) without, others would say, his having demonstrated any morally relevant difference between them. The difference he describes between 'slaves' and 'masters' does not seem to be a morally relevant one. It is normally presumed that people are all relevantly the same in important moral respects, and there is a strong onus of proof against anyone who asserts an asymmetry, who says that people have very different rights.

Are you taking an area of mathematics and seeing what it can tell us for philosophy?

No, it's not that mathematics strictly

tells you anything.It just attunes you to structural things, like symmetry, that turn out to be relevant elsewhere. You can easily forget such things and be misled by surface dissimilarities, like skin colour.

Do we run into Hume's famous Is/Ought problem deriving prescriptive 'oughts' from the structures out there, the 'is'?

Yes, the problem is there. But in the moral rights cases, the symmetry between persons is evident to everyone, and it takes training to suppress it. We clearly get by on a presumption of equality of rights, and that can take us a long way, whether we do or do not agree on the origin of those rights.

Yes, we have seen what happens in asymmetrical societies. Are there any other philosophical questions where mathematics helps?

There's the problem of induction. When you look at it first, you tend to presume that any solution to it must involve laws of nature or some principle of the Uniformity of Nature. But that can't be right, because inductive arguments occur in mathematics, which is true in all worlds, uniform or not. For example, it is a fact that in the first billion digits of π each digit appears about 1/10 of the time. Now that is a good reason to believe that the same will be true in the

secondbillion digits of π. But that is an inductive argument. And π is the same in all possible worlds.

So then induction can be applied everywhere.

Yes. Whatever the justification for induction is, it's got nothing to do with contingent facts of nature. So, induction is based on a purely logical principle, as David Stove argued. Of course, laws of nature might make inductions even more reliable than they would otherwise be.

What would you recommend for further reading?

Gleick's book

Chaosis good. Even better is M. Mitchell Waldrop'sComplexity - amore recent book about the study of self-organising phenomena, which are like chaotic ones, but more organised. For something more philosophical, Bigelow and Pargetter'sScience and Necessityisa very interesting Realist treatment of some aspects of science and mathematics.

PAPERS WRITTEN BY JAMES FRANKLIN ON THE PHILOSOPHY OF MATHEMATICS:

'Philosophy and mathematical modelling',

Teaching Mathematics and its Applications,2 (1983): 118-9.'Is statistical inference non-deductive logic?'

Cogito(U.N.S.W.) 1(5) (1983):26-38.'Non-deductive logic in mathematics',

British Journal for the Philosophy of Science,38 (1987): 1-18.'Mathematics, the computer revolution and the real world',

Philosophica,42 (1988): 79-92.'Mathematical necessity and reality',

Australasian Journal of Philosophy,67 (1989): 286-294.On the uses of symmetry arguments: 'Healthy scepticism',

Philosophy,66 (1991): 305-324.'Achievements and fallacies in Hume's account of infinite divisibility',

Hume Studies,29 (1994): 85-101.'The formal sciences discover the philosophers' stone',

Studies in History and Philosophy of Science,25 (1994): 191-209.'Artifice and the natural world: Mathematics, logic, technology' in

Cambridge History of Eighteenth Century Philosophy,ed. K. Haakonssen, to appear.Topological structure in logical spaces', to appear.

'Logical probability rises from the dead',

Erkenntnis55 (2001): 277-305.

Followup: see the site of the Sydney School in the philosophy of mathematics.