(Philosopher 1 (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 Mathematics and the
history The Science of Conjecture
and 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 complexity
explained away in 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 things does 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 certain aspects 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 are
synthetic 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 mathematical
work 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?
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.
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
equations describe 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 within sets. 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
arguments are 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 are symmetrical with
respect to the information you have, your sensory evidence. Not just that you
can't be certain which one holds, but that there is no reason to 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
Gleick's book Chaos is good. Even better is M. Mitchell Waldrop's
Complexity - a more recent book about the study of
self-organising phenomena, which are like chaotic ones, but more organised. For
something more philosophical, Bigelow and Pargetter's Science and Necessity
is a very interesting Realist treatment of some aspects of science
and mathematics.
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 second billion 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?
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.
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.
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