more a grant application, really)
The aim is to create a complete
philosophy of mathematics based directly on applied mathematics, taking the
view that mathematics is not about other-worldly entities like numbers or sets,
nor a mere language of science, but a direct science of structural features of
the real world like symmetry, continuity and ratios.
Applied mathematicians take it for
granted they are studying certain real features of the world - properties like
symmetry and continuity. Modern developments in mathematics such as chaos
theory and computer simulation have confirmed that view, but traditional
philosophy of mathematics has remained fixated instead on complicated formal
results concerning the simplest mathematical entities, numbers and sets. Using
straightforward examples that exhibit the richness of the mathematical study of
complexity, the grant project will develop an Aristotelian realist philosophy
of mathematics that challenges the usual Platonist and other classical options.
In argument readable by an educated philosophical or scientific audience, it
shows how mathematics finds the necessities hidden below the surface of our
For most of the twentieth century, the
philosophy of mathematics was dominated by the competing schools of logicism,
formalism and intuitionism, all of which emphasised the role of human thought
and symbols in creating mathematics. Dating from around 1900, they were
generally regarded as unsatisfactory, especially in explaining applied
mathematics. (Körner 1962) For example logicism, the theory developed by Frege
and Russell that mathematics is just logic, proved untenable on technical
grounds as well as giving no insight into how trivial logical truths could
prove so useful in dealing with the real world.
Those schools shared this problem with
Platonism, the traditional alternative according to which mathematics is about
an abstract or other-worldly realm inhabited by numbers, sets and so on;
Platonism always found it hard to explain the mysterious connection between
that other world and the real objects of our world which are counted and
weighed. Platonism also has significant epistemological problems, being
susceptible to Benacerraf's challenge (1973).
The challenge is to explain how knowledge of mathematics is possible,
given (i) a broadly causal approach to epistemology, and (ii) the view that
mathematical objects are abstract.
Despite this difficulty, many working mathematicians continue to find
Platonism attractive, in part because it seems to be the only realist position
By the time of Eugene Wigner's celebrated
1960 article `The unreasonable effectiveness of mathematics in the natural
sciences', it was clear that new directions in the philosophy of mathematics
were needed. In the last thirty years, there has been a diverse range of
responses to the impasse, but there has been no agreement on what is the
leading direction, or even consensus within particular schools on whether the
problem of the applicability of mathematics is adequately solved. Much of the
best work has been in a Platonist direction. Works such as Colyvan 2001a and
2001b have showed that Platonism has substantial resources and is not easily
dismissed, while Steiner 1998 presented a direct Platonist attack on the
problem of the applicability of mathematics. Nevertheless we believe (for
reasons to be developed more fully in our project) that these authors have not
succeeded in dealing with the argument advanced originally by Aristotle, that
sciences of the real world should be able to deal with real properties
directly, and reference to abstract objects in another world creates
philosophical difficulties without being necessary for explaining the necessary
interconnections between the real properties. In particular we believe an adequate
epistemology for realism has yet to be developed. For this reason we also
disagree with the school led by Resnik (1997) and Shapiro (1997, 2004)
(surveyed in Reck and Price 2000 and Parsons 2004) Although like us they accept
the slogan "mathematics is the science of structure" and they have made many
perceptive observations on the way mathematics looks at structure and patterns,
their theory is in our view vitiated as a complete philosophy of mathematics by
their tendency to regard "structures" as a kind of Platonist entity similar to
numbers and sets.
There have also been nominalist philosophies of mathematics (Field 1980,
Azzouni 1994, Chihara 2004), which we believe are subject to the insurmountable
obstacles that dog nominalism in general. As with the Platonists, they speak as
if Platonism and nominalism are the only alternatives, whereas Aristotelian
realists believe those two schools make the same error, of supposing that
everything that exists is an individual (whether physical or abstract). The nominalists
did however usefully describe some possibilities of discussing mathematical
realities without reference to Platonist abstract entities.
One of the more important developments in
philosophy of mathematics in the last quarter of the twentieth century is the
rise of indispensability arguments for mathematical realism.
According to the Quine-Putnam
indispensability arguments, we must believe in the existence of mathematical
objects if we accept our best physical theories at face value.
Our best physical theories make
indispensable reference to mathematical objects. We agree that indispensability
arguments are important but believe their significance has been misunderstood
because of the Platonism-or-nominalism dichotomy being assumed. That encourages
a fundamentalist attitude to mathematical language, as if numbers must either
exist fully as abstact entities, or not exist in any way at all. Some subtlety
is needed as to what exactly is concluded to be indispensable. (Baker 2003).
Moreover, care must be taken so as to make room in naturalism for the
distinctive methods employed in generating mathematical knowledge (Maddy 1992).
Instead, we will argue, mathematical language is
indeed about some real aspects of the world, but not about abstract objects.
Mathematics does not stand to natural science as a tool stands to a constructed
entity; rather the object of scientific study exemplifies, or instantiates, a
mathematical structure. (What to say of mathematical structures that have no
physical instantiation is an issue that we will also consider carefully.) Thus
a (pure) quantum state is a vector, and a space-time is a
differentiable manifold, and both facts constrain the object in very definite,
mathematically understood, ways.
We will be guided by more hopeful
developments from a number of Australian authors (Armstrong 1988, 1991, Forrest
and Armstrong 1987, Bigelow 1988, Bigelow and Pargetter 1990, Michell 1994, Mortensen
1998), supported by a few overseas writings that are not explicitly in the
philosophy of mathematics (Dennett 1991, Devlin 1994, Mundy 1987) They hark
back to the old theory of medieval and early modern Aristotelians that
mathematics is the "science of quantity" one still visible in some basic
developments of nineteenth-century mathematics (Newstead 2001) but thereafter
ignored.. This work is situated in the Australian realist theory of universals
defended by D.M. Armstrong. Lengths, weights, time intervals and so on are real
properties of things, and so are the relations between those properties. So a
ratio such as 2.71, for example, is conceived to be the (real) relation that
can be shared by pairs of lengths, pairs of weights and pairs of time intervals.
A similar analysis is given of whole numbers like 4, which is a real relation
between a heap of, say, parrots, and the "unit-making" property,
being-a-parrot. This school of thought has unfortunately been little noticed
outside Australia, a situation we hope to remedy. It has also confined itself
to analysing only the most simple and traditional mathematical such as numbers
and sets, thus ignoring the richer mathematical structures like symmetry and
network topology, and the more applied mathematical sciences such as operations
research, where, we believe, the strengths of a structuralist philosophy of
mathematics are both more obvious and better connected with the concerns of
Those concerns have broadened in ways
that demand to be considered philosophically. The last sixty years have seen the creation of a number of new "formal"
or "mathematical" sciences, or "sciences of complexity" - operations research,
theoretical computer science, information theory, descriptive statistics, mathematical
ecology, control theory and others. Theorists of science have almost ignored
them, despite the remarkable fact that (from the way the practitioners speak)
they seem to have come upon the "philosophers' stone" a way of converting
knowledge about the real world into certainty, merely by thinking. (Franklin
1994) In these sciences and more generally in the natural sciences, there has
been a better appreciation of the role of "systems concepts" like "ecosystem", "water cycle",
"energy balance", "feedback" and "equilibrium" are systems concepts. They
provide the language for studying complex interactions. They are generalisable
to other complex systems, such as those in business, and so show the relevance
of scientific systems thinking to the wider world. They unify and give a
perspective on science itself, and on its connections with the science of
complexity, mathematics. (Franklin 2000) The present project will give the
first extended philosophical consideration to the full range of this body of
The part of the project most undeveloped
so far is its epistemology. Once it is established that mathematics deals with
structural aspects of the world, how are those aspects known? Where Platonism
has immense difficulties in explaining how we could know about entities such as
number which it takes to be in "another world", Aristotelian approaches give
promise of a more direct epistemology, since one can sense symmetry (for
example) as well as one can sense colour. Realising that promise is difficult, however,
since one needs to integrate an
Aristotelian theory of abstraction (the cognition of one feature of reality,
say colour, in abstraction from others, such as shape) with what is known from
cognitive psychology on pattern recognition and the comparison of modalities
(for example, how the brain compares felt and seen shape). The well-known role
of proof in establishing mathematical knowledge needs to be integrated as well.
Again, there is little work at present on that topic.
After recalling the
general reasons for accepting an Aristotelian realist position on universals
(these reasons are developed by other writers, but still need collecting and
expounding in a way relevant to the mathematical case), and illustrating them
in the examples just mentioned, we will be in a position to develop the core of
the theory that mathematics is a science of certain real properties. One task
is to distinguish two substantially different kinds of properties that are both
objects of mathematics. An older theory held that mathematics is the "science
of quantity", a newer one that it studies structure or patterns. Both quantity
and structure are real features of the world, but different ones. Both are studied
by mathematics. The division between the two roughly corresponds to the
division between elementary and higher mathematics.
The first component of the project will consist in
an investigation of the indispensability argument and its relation to quantum mechanics.
For while quantum mechanics presents an argument for realism about the complex
number field, it also suggests that this field has primacy. And since this
field subsumes the natural numbers and the reals, it suggests a significant
limitation to the science of quantity conception since that is inextricably
linked with linearly orderable fields. We believe this represents an area of
hitherto untapped connections and arguments that is capable of throwing great
light on the relation between physics and mathematics. Thus one thing we will
be concerned with is the significance of the Montgomery-Olydzko law. This
suggests that the eigenvalues of a random Hermitian matrix (such as might be
found in certain quantum mechanical problems) have the same spacing properties
as the non-trivial zeroes of the Riemann zeta function - which are not spaced
randomly. This is now fairly widely confirmed - but it suggests a connection
between very different areas of science: between the traditional a priori and
traditional a posteriori. (What role quantum mechanics is itself playing in
this connection is still an unsolved question. Professor Barry Mazur of Harvard
has made some interesting comments to us on this problem.)
We will develop arguments that the Aristotelian
realist view has the greatest chance of explaining this connection; just as it
has the best chance of explaining what we call inverse indispensability in
general. On this argument there is also an "unreasonable dependence" of
mathematics on physics. The discovery of the infinite number of exotic
differential structures on four dimensional manifolds (making four dimensional
manifolds unique in differential geometry) offers a very striking example of
this phenomenon - since the exotic structures arose out of mathematical
physics. This inverse indispensability can only really be explained, we argue,
on the Aristotelian view.
After establishing our
metaphysical case arguing for our view that mathematics studies structural
aspects of the real world we will move to epistemological issues. Theory on
how mathematics can be known is an underdeveloped part of structural
philosophies of mathematics, and is well recognised as a major difficulty for
realist philosophies of mathematics in general. In this second component of the
project, we will show that Benacerrafs challenge can be overcome by our brand
of realism. The fundamental dilemma for realists was identified (by Benacerraf
1973) as the problem of providing a naturalistic (or broadly causal)
epistemology for mathematics, if mathematics indeed refers to something real.
How can those objects affect us, so that we can know about them? That is very
difficult to explain on a Platonist view, since Platonic objects do not have
causal power. Aristotelian views such as ours permit us to develop a much more
plausible and direct answer, since structural features of real things, such as
symmetry, can affect us in the same way as, for example, their colour, and so
can be directly perceived. On our Aristotelian view, the objects of mathematics
do not exist outside of space and time, but are immanent in space and
time. Consequently, we hold that some
simple mathematical ideas are indeed acquired in a causal manner.
It is true that some of
the more complicated entities spoken of in mathematics, such as the Hilbert
spaces of quantum mechanics, do not seem to be directly perceivable. In order
to move from simple perception of patterns to sophisticated mathematical
theorising, it is necessary to form abstract ideas of structures and
quantities. Therefore, we (and especially the research assistant employed by
the grant) will pay special attention to the role of abstraction in generating
mathematical knowledge. Aristotelians hold that mathematicians abstract or
"separate in thought" features of objects that they perceive in the real world.
We will survey various interpretations of abstraction, and present a theory on
which abstraction draws attention to mathematical features of existing physical
objects (but does not bring into existence any kind of Platonist "abstract
objects"). We anticipate the objection
that the natural world does not have the perfect precise structures needed in
mathematics. We will therefore consider
the role of idealisation in abstraction, and compare it to the uses of
idealisation in physics (e.g. massless points and frictionless planes). In neither case should idealisation
undermine the reality of the phenomena studied.
We will rebut various
objections that have been raised against the meaningfulness of possibility of
abstraction, notably by Frege (1884/1950). Freges objections are an important
reason for the neglect of an Aristotelian approach. However, we will
demonstrate that Freges criticisms do not touch Aristotelian realism. In particular, objections having to do with
how the individuality of mathematical objects is preserved if they are obtained
by abstraction do not apply to our theory.
Since mathematical objects are universals, they are not individual particulars
and not subject to this objection.
The several components
of the project cohere very well, since a proper understanding of the
indispensability of mathematics and physics to one another yields rich results
in metaphysics and epistemology.
Finally, the theoretical work of the project is complemented throughout
by the extensive knowledge of a working mathematician.
The main lines along
which our argument should proceed are clear, but there is much detailed work to
be done to consider and reinterpret existing material, and to ensure coherence
between the various parts of the project metaphysical, epistemological,
mathematical and quantum-mechanical. We
anticipate finding that the understanding of the relation of mathematics and
physics produced by consideration of the indispensability argument in the first
part of the project will shape our epistemology in the second part of the
project. Throughout our findings will
be grounded by the examples of a working mathematician.
In the light of this
plan, we would anticipate the three years of the work on the grant being
structured as follows:
1: CI Franklin to complete current writing on "quantity" as an object of
mathematics, CI Heathcote to research and write on issues relating to quantum
mechanics, both CIs to work with research assistant on initial research on
epistemological issues of abstraction, pattern recognition and proof.
2: Research assistant to work intensively on epistemology, with input from CIs;
research assistant or CI Heathcote to visit Cambridge and St Andrews for
conferences; submission of several academic papers to journals; planning of
book and negotiation with possible publishers.
3: Completion and submission of book containing the full work, probably to
Oxford University Press.
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Armstrong, D.M., 1991,
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of Arithmetic. Blackwell, Oxford.
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Journal for the Philosophy of Science 45, 389-406.
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