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SIGNIFICANCE OF MATHEMATICS FOR ECONOMICS
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Abstract
Mathematics plays a very important role in economics. This role has been
increasing in importance in last years. Mathematics is thus increasingly important
in terms of the expression and communication of ideas in economics. This in itself is
a matter of interest, particularly with respect to the public understanding of
economics. Mathematics is increasingly significant for economics, namely its role in
the economy itself. Increasingly activity in financial markets (particularly in
derivatives trading) is governed by mathematical models. We will focus here, rather,
on the effect of mathematisation on the content of economics. This is preceded by a
brief account of the history of the role of mathematics in economics.
INTRODUCTION
The role of matematics in economics has been significant for almost a century,
and has been increasing in importance particularly in recent years. A comparison of
academic journals now with, say, fifty years ago reveals a tremendous increase in
mathematical expression; Backhouse (1998) [2] reports an increase in the incidence of
algebra in articles in the two leading economics journals from 10% in 1940 to 80% in
1990. The same is true also of textbooks at all levels.
Grubel and Boland (1986) [9] report a concern, identified through a
questionnaire survey of leading economists, with the increasing use of mathematics in
published research and graduate teaching. However the survey results also implied that
emphasis on pure mathematics was a rational response to incentives within the
profession. Concern with the extent of mathematical expression may derive from the
view that it crowds out other modes of expression, i.e. the issue is over how ideas are
communicated. But there is a more fundamental issue concerning the role of
mathematics in economics, namely its potential effect on content. One interpretation is
that economics has been undergoing technical change, employing more mathematics
and more sophisticated statistical techniques, which has improved the productivity of
the discipline; the change in content is thus one of undoubted improvement. But
concerns have been raised that mathematisation has proceeded at the cost of attention to
matters which cannot be expressed mathematically, ie the alternative modes of
communication can actually allow analysis in areas closed to mathematics. The issue is
thus the fundamental one of what we understand by the discipline of economics and
what it can achieve. This issue too feeds back into the issue of the public understanding
of economics as a discipline.
HISTORY OF MATHEMATICS IN ECONOMICS
Mathematics first took on a significant role in economics in the last century in
the build-up to what is commonly referred to as the Marginalist Revolution. This was a
period in which Classical concerns with production, growth, and the distribution of the
fruits of growth among social classes, were being replaced by concern with market
exchange. The focus thus shifted from the level of the economy and social classes to the
level of the individual. Leon Walras, in particular, set out to establish the conditions for
a successful co-ordination of market exchange, and he did so mathematically. Along
with Augustin Cournot, he is responsible for the introduction of the systematic
application of mathematics to economics.
At the same time, there was a concern that economics should be seen as a
discipline on a par with the physical sciences (Drakopoulos, 1991) [7]. Walras’s father,
like many other economists of the time, saw mathematics as the vehicle for achieving
this goal. Further, just as the physical sciences were being built up in axiomatic fashion
on the basis of units of energy, etc, economics was being built up axiomatically on the
basis of units of utility. The motivation of individuals in the economy engaging in
market exchange is understood as the maximisation of utility, a human motivation
which clearly lends itself to mathematical treatment. Walras (1965) [17] went further:
“It is only with the aid of mathematics that we can understand what is meant by the
condition of maximum utility”. And indeed, the term “Marginalist Revolution” refers to
the mathematical result of the marginal conditions for market equilibrium, as derived by
calculus.
Mathematics and Formalism
The role of mathematics in economics can usefully be discussed in relation to
the role of formalism in economics. While the two terms are often used interchangeably
(Krugman, 1998) [14], an argument need not be mathematical to be formal (see Chick,
1998) [4]. Further, it has been argued recently (by Backhouse, 1998) that, unlike
mathematics, formalism entails the tighter condition of fixity of meaning. Weintraub
(1998) has demonstrated the changing meaning of terms in mathematics. In particular,
formalism also includes the notion of rigour; but scientific rigour may itself be subject
to different meanings. Thus, while, at the turn of the century, scientific rigour referred to
testing against empirical evidence, it is now associated more with mathematical
axiomatisation.
We have noted above the greater extent of mathematical formalism in pure than
in applied economics. But formalism also has consequences for empirical testing; it
requires the notion of fixity of meaning applied to data too. This allows for reference to
‘the facts’ as objectively measured phenomena with fixed meaning independent of
theory. Indeed Mirowski (1991) [16] argues that the very act of measurement imposes a
mathematical structure. For example, the conventional market diagram presumes
homogeneity of commodity space which is not in fact fixed in nature; Mirowski argues
that the degree of homogeneity will vary depending on the changing social perception
of market activities. Insofar as economics embraces formalism, therefore, it embraces a
particular general approach to mathematics which derives from logical positivism and
has implications at both the pure and applied levels. Within this general approach, there
are then different uses made of mathematics depending on whether the research is pure
or applied.
Mathematics and logic
Keynes’s first work (Keynes, 1973) [13] addressed the problem of induction, in
reaction against Russell and Whitehead’s attempts to construct mathematical logic on
rationalist grounds. He was concerned with how we establish reasonable grounds for
belief in the absence of the conditions for certainty. Certainty for Keynes was the
special case, only possible within a closed, atomic structure (in mathematics or in
reality), ie those to which classical logic apply. Use of mathematics (based on classical
logic) therefore requires justification in terms of the degree to which the case
approximates to a closed, atomic structure. Keynes supported the use of mathematics as
an aid to thought but argued that the onus was on the economist to demonstrate that its
application was appropriate to the subject matter.
Axiomatisation is a type of formalism which relies particularly on classical
logic, and which characterises the formalist approach to pure economic theory.
Correspondence between theory and reality occurs only at the level of the axioms and at
the level of the propositions which emerge from the application of deductive logic.
There has been much discussion of the realism or otherwise of the axioms (see for
example Hausman, 1992) [12]; the issue of testing we will address in the next
subsection.