The Future of Mathematics in Economics: a Philosophically Grounded Proposal

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The Future of Mathematics in Economics: a

Philosophically Grounded Proposal

Ricardo Crespo - Fernando Tohme

IAE- Universidad Austral and CONICET, [email protected] Acosta 1550 - (1630) Pilar, Argentina

INMABB (CONICET) and UNS, [email protected], Argentina

12 de Octubre y San Juan - (8000) Baha Blanca, Argentina

Abstract

The use of Mathematics in Economics has been widely discussed.In this paper we sustain that Mathematics is a powerful tool enor-mously useful in Economics. However, we also maintain that we needto update the specific way of applying this discipline to the analysis ofeconomic affairs. We present a proposal in this sense, rooted on philo-sophical considerations. After reviewing the traditional philosophicalconceptions of Mathematics which are at the basis of traditional Math-ematical Economics and discussing their shortcomings, we argue why

Economics requires a different mathematical foundation, closer to itsmodeling needs. We introduce an alternative and in accordance withit, suggest some specific mathematical tools that may be applied toEconomics in the near future.

1 Introduction

The use of Mathematics in Economics is a debated topic. Numerous voicesclaim that Economics has become excessively mathematical in its formula-tion. Other, also numerous, voices not only defend but strongly argue for

using and refining the mathematical tools applied to Economics. The firstgroup, say of contrarians, sustains that Economics, due to its mathemat-ical formalisation, has become an abstract discipline detached from reality,or at the very best only able to show an oversimplified picture of facts. Thepartisans maintain, instead, that Mathematics provides the right language

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to express real-world relations and the tools to reason correctly about them.These defenders of the mathematisation of Economics can, in turn, be splitin two. On one hand, we have those who support axiomatical presentationsof the ideas while on the other, we have those who stress the need of a moreempirically based mathematical approach. The former tradition focuses onbuilding logically coherent bodies free from psychological subjective biaseswhile the latter criticises the speculative character ofa prioriapproaches andsupport a two-way interaction between empirical data and formal models.While each side has some valid points, it is interesting to note that both thecontrarians and the empirically leaning partisans concentrate their attackson the economists that build purely abstract models. This latter position

can be seen as being at the core of the Mathematical Economics tradition.Since it has been subservient to specific meta-conceptions of Mathematics,their analysis will help to choose the right mathematical approach on whichto base any future developments in this area.

In this paper we will discuss these problems along two lines of argumen-tation. On one hand we will discuss the shortcomings of the traditionalfoundations of the mathematical tools in Economics. On the other, we willintroduce new approaches and explore their consequences, discussing themeaningfulness of the results and their future perspectives.

The rest of this paper is organized as follows. In section 2 we will sketchthe main traditional philosophical conceptions of Mathematics. Section 3

and 4 extend this discussion to the possible application of Mathematics inthe empirical sciences and to Economics in particular. We will concludethat Economics may require a different mathematical foundation, closer toits modeling needs. Section 5 discusses some of those future lines of devel-opment. Section 6 concludes.

2 Philosophical Views on Mathematics

Philosophers have reflected on the nature of mathematical thinking since itsinception. But the first predominant conceptions of Mathematics were for-mulated by Plato and Aristotle. Afterwards, they remained without major

changes until the 1800s. In this section we will briefly introduce the modernconceptions that arose at the turn of the XXth century. One of these posi-tions isPlatonism, obviously alluding to Plato, but only vaguely inspired inhim. Aristotles ideas, instead, exerted almost no influence on the moderncurrents in philosophy of Mathematics.

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According to Kitcher and Aspray (1988), the German mathematician,logician and philosopher Gottlob Frege posed the problems with whichphilosophers of Mathematics have struggled ever since. Frege maintainedthat the key problem of philosophy of Mathematics was to identify its foun-dations. This idea is still present in different guises. The main ones are thefollowing:

Platonism: In truth, Platonism in Mathematics, as mentioned, isonly loosely inspired in Platos ideas. The label is inspired in Platosthought on the existence of ideas independent of the empirical objectsof which they are as the archetype. More precisely, mathematical Pla-tonism has been defined as the conjunction of three theses. Namely,existence(there are mathematical objects), abstractness(mathemati-cal objects are abstract) and independence(mathematical objects areindependent of intelligent agents and their language, thought, andpractices) (Linnebo 2013). Thus, mathematical entities, according toPlatonists, are really existent beings and are, instead of physical, ab-stract objects. The truth of mathematical statements derives from theobjects to which they refer: the truth is discovered, not constructed.While one of its fathers was Frege, many (and maybe most) mathe-maticians share it. In one way or other, either supporting or criticisingit, this conception crosses all the different visions of Mathematics. Butit still has to answer a relevant question of epistemological and onto-logical nature: how do we have access to mathematical objects (seeBenacerraf 1973)?

Logicism: Another strand in the Philosophy of Mathematics initiatedby Frege is based on the idea that mathematical concepts are definedin logical terms, derived from logical laws and, in sum, reduced tologic (see Carnap 1964): Arithmetic would be only a further devel-oped Logic, every arithmetic theorem a logical law, albeit a derivedone (Frege 1964, p. 107). This program was continued by BertrandRussell and Alfred Whitehead in Principia Mathematica. The verynature of the logicist approach is summarized by the following claim:Philosophy asks of Mathematics: What does it mean? Mathematicsin the past was unable to answer, and Philosophy answered by intro-ducing the totally irrelevant notion of mind. But now Mathematicsis able to answer, so far at least as to reduce the whole of its propo-sitions to certain fundamental notions of Logic (Russell 2010). This

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reveals that in a certain way, Russell was a Platonist realist; for him,as for his contemporary colleague George E. Moore, logical entitiesare real objects. Other logicist positions admitted, instead, a conven-tional foundation. Many criticisms of logicism have been advanced,based on the idea that not all Mathematics can be reduced to Logic(Benacerraf and Putnam 1964). Nevertheless, the deepest criticismsgo against its Platonism. More precisely, since Logic deals with signs-second intentions, which are not more than signs-, reducing Mathe-matics to Logic either implies a Platonist position (thinking that thesesecond intentions are real abstract objects) or a non-realist stance (ifsecond intentions are seen as having a lower ontological status). As

noted by W.V.O. Quine (1964), old medieval claims on the nature ofuniversals reappeared in the XXth century at the root of these dif-ferent views. So, for instance, the Platonist logicist camp conceivesuniversals as abstract real objects.

Intuitionism: this foundational program was initially developed bythe Dutch mathematician Luitzen E. J. Brouwer, inspired in Kantsideas. For Brouwer (1999), in Kantian spirit,man always and every-where creates order in nature and in particular mathematical judg-ments are synthetic a priori. The idea is that humans construct theobjects of Mathematics and that our knowledge of their fundamentalproperties is based on an a priori intuition. Through this intuition,we are able to recognize a potential infinity of mathematical entities(the natural numbers), and these can form the basis for further con-structions (Kitcher and Aspray 1988). In this way, [t]he truth ofa mathematical statement can only be conceived via a mental con-struction that proves it to be true (Iemhoff 2013; see also Heyting1964). Mathematics, for intuitionists, consists in building mental con-structions which are inductive and effective. Furthermore, in thisview classical procedures cannot be used, like proving claims by con-tradiction or applying the axiom of choice. But this leads to the re-

jection of some universally accepted theorems and, in consequence

the mathematical community has almost universally rejected intu-itionism (Snapper 1979). With respect to the intuitionist positionon the nature of mathematical objects, we have that Intuitionistsaffirm the existence of mathematical objects but maintain that theseobjects depend on or are constituted by mathematicians and their ac-tivities (Linnebo 2013). In other words, they are conceived as being

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mental objects. This fits well with the medieval conception of univer-sals called conceptualism, which holds that there are universals butthey are mind-made (Quine 1964). Mathematical objects are thusinvented, not discovered. Or as Brouwer (1999) claims, mathematicalexactness exists in the human intellect.

Formalism: A fourth foundational program developed in the twenti-eth century was started by David Hilbert, who advocated for the fullaxiomatisation of mathematical theories in order to ensure that theyare free of contradictions. In this view the most important task ofMathematics is to provide consistency proofs of mathematical theo-ries. On the other hand, the meaning of the formal constructions isirrelevant: for the strict formalist to do Mathematics is to manipulatethe meaningless symbols of a first order language according to explicit,syntactic rules (Snapper 1979). But in 1931 Kurt Godel proved thatmathematical theories constructed under this approach are either in-complete or cannot be proven to be free of contradictions.However, ithas been clearly shown that Hilbert conceived formalism only in re-lation to the foundations instead of the whole body of Mathematics(Corry 1997, 2002, 2004). Other mathematicians took the extreme po-sition of applying formalism to the whole corpus of Mathematics. Thisis the case of the group of French mathematicians known under thepseudonym of Nicolas Bourbaki. Mathematics, for Jean Dieudonne,one of the Bourbakians, is not more than a play of chess: On foun-dations we believe in the reality of Mathematics, but of course whenphilosophers attack us with their paradoxes, we run to hide behindformalism and say: Mathematics is just a combination of meaninglesssymbols. (cited in Corry 2004).It is fair to say that Hilbert did not goas far as the extreme Bourbakian position, being also concerned withthe application of Mathematics to the real world (Weintraub 2002,Giocoli 2009). However, the general flavor and emphasis of Hilbertsideas was on axiomatic consistency. While for mathematicians of theXIXth century (for instance Vito Volterra) mathematical rigor meant

correspondence to real facts, for Hilbert it was a consistent outcomeof a deductive process applied to the assumptions, rather than basedupon the observation of reality (Giocoli 2009). In effect, at the turnof the XXth century the meaning of rigor in Mathematics shifted fromempirical correspondence to axiomatic consistency (Weintraub 2002).With respect to the nature of universals in Formalism, it is clear that

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it holds the nominalist position (Quine 1964). According to this view,universals are no more thanflatus vocis, empty or mere names, words,or sounds without corresponding objective realities.

For different reasons, Platonism, Logicism, Intuitionism and Formalismhave failed to provide Mathematics with the philosophical foundations Fregelooked for. Modern Mathematics has in fact abandoned this quest despitethat some authors think that the Philosophy of Mathematics holds the key toa firm foundation for the discipline (Snapper 1979). In this sense, it is inter-esting to note that from the medieval views on universals, one has not beenadopted by the aforementioned alternative foundations of Mathematics. Itis the one based on Aristotles idea that universals are present in singular

concrete objects as their essence (universal in essendo), in concepts express-ing this essence (universal in repraesentando), and in words referring to theplurality of things that share the same feature (universal in significando).Thus, while Platonic realism refers to abstract objects, conceptualism toobjects only existing in the mind and nominalism denies the existence ofuniversal categories, the Aristotelian view refers to real world things. Thelatter position has been neglected in the discussion on the foundations ofMathematics. More precisely, there has been no empiricist tradition in thePhilosophy of Mathematics in the early XXth century (Kitcher and Aspray1988). But then, this could have been the right foundations for a Mathe-matics of the empirical sciences like Economics.

3 Mathematics for the Real World

A reasonable take on this matter should be based on the idea that Mathe-matics (be it Formalisms consistency, Intuitionisms insights or Logicismslaws) should be in some correspondence with real facts. To verify this, empir-ical tests may be needed. Interestingly enough, as noted by Eugene Wigner(1960), such tests may have been passed successfully: mathematical lan-guage has more to commend it than being the only language which we canspeak; it shows that it is, in a very real sense, the correct language. The ap-

propriateness of the language of Mathematics for the formulation of the lawsof Physics is a wonderful gift that is hard to understand. Kline (1985) eventalks about a magical power or an inner mysterious strength. But theexplanation should come from a relation between Mathematics and the ac-tual world. Recent views maintain that mathematical objects should be seenas fictional entities grounded on physical objects (Mendell 2004). Other

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authors see Mathematics as the study of quantitative and structural aspectsof things, like ratios, patterns, or relations: [mathematicians] are scientistswho study patterns or forms that arise in nature (Franklin 2009). Theidea is that mathematical symbols are instrumental signs of mathemati-cal concepts which are either formal or natural signs of real structuresor properties (Franklin 2009, pp. 101-102):

Aristotelian realism unifies Mathematics and the other naturalsciences. It explains in a straightforward way how babies cometo mathematical knowledge through perceiving regularities, howmathematical universals like ratios, symmetries and continuitiescan be real and perceivable properties of physical and other ob-

jects, how new applied mathematical sciences like operations re-search and chaos theory have expanded the range of what Math-ematics studies, and how experimental evidence in Mathematicsleads to new knowledge.

In this sense, a mathematical proposition would just express a well-founded empirical generalization or law about the properties and behaviorof objects (Gaskin 1964). The consistency of mathematical developmentsmight express the rationality of the real world. As Pierre Duhem concludes,it is impossible for us to believe that this order and this organization pro-duced by theory are not the reflected image of a real order and organization

(quoted in Kline 1985). Consequently, though we can imagine mathematicaldevelopments without previous induction, they could reflect a (maybe yetunknown) piece of reality. That is, Mathematics would then be the scienceof possible consistent objects.

In concrete applications it becomes fundamental to check whether theseconstructions can be really found in the world. However, we must considerthat it is usually easier to use data to check out a theory in the natural thanin social sciences. In the latter there is a kind of complexity, aggravatedby human freedom, that makes patterns irregular and predictions very dif-ficult. Experiments are not repeatable in exactly the same way, and exactlaws may be lacking. Economics tries to avoid the inscrutability of sub-

jectivity by normatively seeking the allocation of means in order to satisfygiven ends. But in this way it only indicates how things should be accordingto first principles. Predicting real world behaviors is more involved and canbe achieved only if a) statistical regularities are known; b) there are sys-tematical ways in which people schedule and coordinate their social actions;and c) there are known causal regularities between natural and social life

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(MacIntyre 1984). Furthermore, in many cases the knowledge is sought inorder to solve practical problems. In this case, the rigour required in theirsolution is quite different from the rigour necessary in axiomatic systems(Boumans 2005). More precisely, the range of validity of conclusions mustbe contextualised, yielding onlylocalknowledge.

We can summarize saying that the validity of a social or economic knowl-edge is grounded in local theories confirmed by data. Any generalisation oflocal theories requires a context that may involve natural laws, institutions,social routines or structures. The mathematical tools needed to build suchtheories should be able to somehow express the fluctuating and inexact na-ture of social reality.

4 From Debreus to von Neumanns View

Authors like Weintraub (2002) and Giocoli (2009) have examined the historyof the relations between Mathematics and Economics in the last century,finding that the abstract character of the different currents in the philosophyof Mathematics -Platonism, Formalism, Logicism, and Intuitionism- had agreat influence on the way in which economic theories were built. Theseauthors considered two different kinds of questions in the discipline. Thefirst kind involves its body of knowledge, i.e. the theories, facts and methodsused in Economics. The other questions refer to the image of knowledge:

the problems that deserve to be tackled in Economics and the perspectivesadopted. Thus the increasing demand for mathematical rigor in Economicswould be a result of the change of image of the discipline, from a mereinstrument for other disciplines to a self-containing full-fledged science. Arelated change of image led Economics from being conceived as the studyofsystems of forces to be seen as the examination of systems of relations(Giocoli 2009):

According to system-of-forces (SOF) view, Economics is a dis-cipline whose main subject is the analysis of the economic pro-cesses generated by market and non-market forces, including -

but by no means exclusively- the processes leading the systemto an equilibrium. According instead to the system-of-relations(SOR) view, Economics is a discipline whose main subject is theinvestigation of the existence and properties of economic equi-libria in terms of the validation and mutual consistency of givenformal conditions, but which has little if anything to say about

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the meaningfulness of these equilibria for the analysis of realeconomic systems.

There is a parallelism between this shift of image and Mathematics shiftof image, including the concept of rigour. This shift does not mean that theinterest on economic applications has been lost, but that the formal tendencyhas prevailed. As Giocoli asserts, what really drove the transformation ofmodern neoclassical Economics in the direction of formalism and Mathemat-ics was the economists desire to achieve the greatest possible generality andconceptual integrity of their analysis. This scientific goal was naturallyencouraged by the increase in rigour of mathematical formalisms. Thus,according to this author, Economics became a branch of logic concealed

behind an empiricist faade.The formalist tendency is clear in Gerard Debreus case. His connec-

tions with the Bourbaki group, the circumstances of his incorporation to theCowles Commission, and his contacts with the American Bourbakian math-ematicians have been widely discussed by Weintraub, Giocoli and Duppe(2011). In The Mathematization of Economic Theory, Debreu (1991) as-serts: Being denied a sufficiently secure experimental base, economic the-ory has to adhere to the rules of logical structure and must renounce thefacility of internal inconsistency. He also states that in proving existenceone is not trying to make a statement about the real world, one is trying toevaluate the model (quoted in Duppe 2011). It goes without saying that

Debreus work had an enormous impact on Economic Theory. Samuelsoneven speaks about an Age of Debreu in the discipline.

It is interesting to contrast the case of Debreu to John von Neumanns.The latter was a disciple of David Hilbert who, according to Giocoli, in Eco-nomics held a view closer to the system-of-forces image than to the system-of-relations one. von Neumann and Oskar Morgensterns The Theory ofGames and Economic Behavior(1944) presented his view on the discipline.

Despite his Formalist origins, von Neumann (1960) was aware of the needof empirical sources in the applications of Mathematics:

As a mathematical discipline travels far from its empirical source,

or still more, if it is a second and third generation only indirectlyinspired by ideas coming from reality, it is beset with gravedangers [] In other words, at a great distance from its empiri-cal source, or after much abstract inbreeding, a mathematicalsubject is in danger of degeneration.

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For von Neumann, the solution was clear: Whenever this stage isreached, the only remedy seems to me to be the rejuvenating return to thesource: the reinjection of more or less directly empirical ideas. Accordingto Giocoli, Economics was not ready to adopt these ideas and this explainswhy von Neumann left after a while the field.

Another interesting case to consider is that of Wassily Leontief, whowas a fervent advocate of the use mathematical models in conjunction withempirical data. In his presidential address to the American Economic Asso-ciation (1970), he referred to the lack of empirical backing for many theories,stating, for example, that [t]he weak and all too slowly growing empiricalfoundation clearly cannot support the proliferating superstructure of pure

or, should I say, speculative economic theory. Indeed, between 1970 and1980, for example, more than half of the articles published in the AmericanEconomic Review used mathematical models without data. This tendencyhas since been progressively reverted. It is true that it is difficult to obtainreliable, homogenous and repeatable data to undertake real experiments.But data has been collected relentlessly in recent years. Still, in the pre-sentation of the hundredth volume of the Journal of Econometrics (2001),Nobel prize-winning economist James Heckman points to the persistenceof a threefold division: Economic Theory (including Game Theory), Mathe-matical Statistics (favored by theoretical econometricians) and the collectionof statistical observations. The future of empirical analysis depends on the

combination of these three elements -without one of the three, it becomesuseless. Impediments to this convergence stem from the sophistication ofthese branches, each one with its own approaches and communities. Buton the other hand, the impetus towards a unification of sub-disciplines arestrong, as reflected in the following cite (Backhouse 1988, pp. 1856-1857):

What is the solution? It is emphatically not to abandon formaltechniques altogether - that would be to court disaster. What isrequired is, as von Neumann points out, that theory be kept closeto its empirical roots. This is turn requires: (1) that economistsput sufficient effort into empirical work [. . .]; (2) that they takeempirical evidence seriously; (3) that they be aware of the con-

ceptual gap between theories and reality [. . .]; and (4) that theybe aware that mathematical imperatives may result in changesin the questions that are being addressed.

The next section will present new mathematical visions and devices thatwe think can contribute to achieve these goals.

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5 The futureThe previous section presented some of the more influential views the role ofMathematics in Economics put in the context of the foundations of Mathe-matics, personified by the positions of Debreu and von Neumann. But it isinteresting to note that some foundational inquiries may lead to further re-consider them. Bourbakis approach, for instance, has lost its preeminence,mostly because of its, hard to satisfy, requirement of expressing every rel-evant idea in the language of set theory. The developments of the last fewdecades have shown that there are alternative and more convenient ways ofrepresenting mathematical ideas. Central to these advances has been therole ofcategory theory. So, for instance Caramello (2010), Goldblatt (1984),Lambek (1994) and Lawvere (2009), among many, have argued in differentways in favor of the unification of Mathematics by means of the category-theoretic language. Furthermore, Lambek has even proposed the free toposas the suitable candidate for the world of Mathematics, noting that the freetopos is an ob ject of itself. Related to this, the recent strong push in favor ofHomotopy Type Theoryis based (besides of the need for computer-checks onproofs) on a semantics grounded in higher order categories (The UnivalentFoundations Program 2013).

This process has been in many ways a response to the need of bet-ter notations, allowing a more abstract and detached (from any concretedetails) characterisation of mathematical objects and relations. Startingwith Grothendiecks successful reconstruction of Algebraic Geometry (seeGrothendieck 1986) and followed by the large advances in the Langlandsprogram(Frenkel 2005), the trend towards the unification through abstrac-tion of large swats of Mathematics has made the usefulness of a flexibleyet rigorous language more evident than ever to mathematicians. Withoutdelving much into Wigners question, it has also become clear in the Sci-ences that mathematical tools of representation and analysis are necessaryat the very least as a language ensuring precision and clarity (Spivak 2014).Furthermore, given the increasing influence of Computer Science models inthe representation of real world phenomena, the use of category-theory inthe expression of scientific theories has only grown in the last decades. Eco-nomics, nevertheless, has been slow in adopting this new framework. Thisis even more surprising considering the widespread influence of the claim ofPaul Samuelson (1952) that Mathematics is for Economics just a language.If so, at the very least for its pure notational value, category theory shouldhave been more widely applied in the discipline.

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In any case, this is relevant when we consider the slow drift from eco-nomic models based on linear algebra and functional analysis to the moreabstract tools used in, for instance, the study of mechanism design and epis-temic game theory. Some of the core aspects in these investigations can bebetter seen in the light of categorical representations:

The understanding of back and forth relation between individualandaggregatebehavior.

The need for, on one hand, the a priorimodeling process and, on theother, the examination ofreal world data, with the hope that thesetwo approaches will, in the end, converge.

The context of decision-making which should, in turn, emerge fromthe decisions made.

There are many interesting consequences of addressing these problems.On the theoretical side these concerns make Economics closer to some ar-eas of Computer Science and Physics. In fact, a large body of research inthe former discipline involve transactions in distributed virtual societies(Shoham and Leyton-Brown 2009). In the latter, the role of exchanges inquantum contexts are being avidly studied, noting its similarities with in-tentional settings (Khrennikov 2010).

On the practical side, the aforementioned studies open the door to thedesign of better tools for decision-making and economic interactions. Par-ticularly important is the possibility of profiting from the abundance ofinformation (Big Data) to make inferences in complex and noisy environ-ments.

Let us discuss these and related issues in turn.

5.1 Circularity

Economics has always recognised the important role of beliefs, expectations,assumptions, etc. in decision-making. In particular, when these mentalconstructs refer to the behavior of other agents. But maybe since Keynesintroduced the beauty contest analogy it is clear that the main issue ofanalysis must be the beliefs about the others beliefs. That is, any agentanalysing a situation has to figure what the other agents think, even abouthis own beliefs.

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This circularity has been shown hard to tame. Very involved arguments,mostly based on theorems on stochastic processes over different spaces, haveensured the existence of fixed point in the process (Mertens and Zamir1984) (Brandenburger and Dekel 1993). But only with the aid of category-theoretical tools more general results have been obtained, less dependent onthe structure of the space of states of nature (i.e. non-intentional aspectsaffecting results) (Vassilakis 1991a).

An analogous problem appears in the foundations of the mechanismdesign literature. If the choice of rules is what matters, then under whichrules should this decision be made? An under which ones is thisdecisionmade, etc., etc. That is, the goal is to find a fixed point in this process of

rules-changing rules of changing rules-. . . (Lipman 1991)(Vassilakis 1991b).Again, tools drawn from category theory provide positive answers.The pervasiveness of circularities poses a question that is closely related

to that of the existence of fixed points. Namely that of finding ways ofgrounding the circularities. This, of course could be impossible if theprocess involved an infinite regress. Here the mathematical distinctions arecrucial. Even if a fixed point exists, does it obtain at the first infinite ordinal() or is the process transfinite?

These questions are certainly important for the study of supertasks,which albeit being only theoretical constructions may work well as ideal-isations of process in which a hard to come by result may be observed to

pursue a new round of activities (Svozil 2003). Situations like these oneslurk in close variants of the circular processes discussed above, particularlywhen they are transfinite.1

If no solutions can be ensured to obtain at the level, the only alternativeis to accept that the world in the model is open and therefore prone tonew and unexpected outside intervention. This is of course the realm ofmany scenarios studied in Theoretical Computer Science. Categorical toolsallow to see them ascoalgebraicconstructs (studied as functors in categories)and its general properties assessed as co-limits in the appropriate diagram(Adamek 2005).

In formal terms all the circularity questions amount to find a solution to

the following equation, = s,,B()

wheres involves the objective elements in the problem, including the outsidecontingencies (due to the open world assumption) while is obtained by the

1In the latter case it is called a hypertask(Clark and Read 1984).

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iteration of a process represented by B().In an alternative set-theoretical framework, an entity like is called anon-well foundedset (Barwise and Moss 1996). But in more general settings,it can be seen as a (categorical) fixed-point (Moss and Viglizzo 2004).

In economic matters can be a profile of beliefs, actions, rules, expecta-tions, etc. It acts both as the contextand thecondition for decision making.And it can arise from the aggregatedecisions of the agents, through a B()that can be very complex (involving a large number of steps of computation)or it can be an emergent property, i.e. an unexpected consequence of theinteractions.

This general and abstract problem captures most of the main issues in

theoretical Economics, from partial to general equilibrium questions, as wellas all the problems involving strategic behavior and asymmetric information.The mathematical tools applied to its study yield different solutions underdifferent settings and parameter structures. But it is clear that only a verygeneral mathematical framework can yield the appropriate responses.

5.2 Expectations

Albeit being related with the problem of the circularities in decision-making,there exists a particular view, proper of macroeconomic models. In this set-ting a standing agent is represented as making decisions in time on the basisofrational expectationsabout the actual law of motionof the economy, seen

in the light of the perceived law of motion(Sargent 2008). More precisely, inthe analysis of rational expectations the key is the relation between actual(ALM) and perceived (PLM) laws of motions. Since both have the samerepresentation we can think of:

ALM

= T(

PLM

)

The question is, under which conditions, the actual and the perceivedlaws of motion are identical? This is far from being a trivial issue: the actual

law of motion results from the decisions made by agents that only haveexpectations based on the perceived one. Positive results involve here theability of the analyst to process (and therefore know) as much informationas the standing agent. In this case, we can see that T is such that thereexists a such that:

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=T()

But in some studies it is assumed that an analyst knows less than anactual agent. An application of Chaitins theorem (1982) shows that no

can exist.

5.3 Data and a priori models

Last but not least, let us consider how Mathematics can provide the righttools to connect data with theoretical models. The idea is that any such ex-ercise should have access to a class ofrough data, which has to be translated

into the theoretical language. The relations and objects of study must beclearly distinguished (Tohme and Crespo 2013).

On the other hand, a theoretical model of the domain, must be estab-lished, in such a way that a category-theoretical adjunctioncan be estab-lished between it and the structured data model. The idea is that a sort ofdata/theory duality must be specified (Ellerman 2007).

Finally, and here is where the real gist of the mathematical frameworkcomes at play, in the toposin which the potential adjunctions between dataand theoretical models operate, an optimal one must be found. Particularlythe one with the highest valuation. This means that the best theoreticalmodel is obtained, one that is closer to the real world data (Doring and

Isham 2011) (Naufel do Amaral and Haeusler 2007).

5.4 Discussion

The integration of the aforementioned components in the analysis will yield aoverarching view of economic phenomena, in which close relations with dataare accompanied by the adequate treatment of the ever present circularitiesof social life. It is for sure that Mathematics will provide the right tools forthis task. Particularly important, as we tried to make clear, is the languageused in the last decades to unify and make rigorous mathematical thought.That is, category theory and its related outgrows.

6 Conclusions

Human reason is a weak instrument with a limited capacity for intuition.It proceeds step by step to achieve knowledge. Mathematics is a powerful

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tool enormously useful in assisting human reasoning. It correctly expressesrelations and structures and embodies the necessary rules for a correct think-ing. Sciences, including Economics, benefit a lot from using Mathematics.However, not all mathematical developments are adequate for any kind ofreality. In the second section of this paper, we have sketchily presented theXXth century predominant conceptions of Mathematics. We consider thatQuines assimilation of these conceptions to medieval accounts of universalsis sharp and enlightening. When trying to apply Mathematics to real phe-nomena, conceptions of mathematical entities as abstract real objects (Logi-cism), mind constructions (Intuitionism), or empty names (Formalism), donot seem to be actually suitable. In the paper we maintained that a XXth

centurys neglected conception, namely, mathematical entities as universalspresent in singular objects, is what is needed. After reviewing in section 4the connection of Mathematical Economics with its contemporary mathe-matical ideas, in section 5 we proposed a new mathematical approaches toEconomics in accordance with the lastly mentioned definition of universals.By introducing three possible applications, we showed how this conceptionadapts to the Economics subject circularity, context-dependence, and inex-actness, calling for a tight relation with empirical data.

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The Future of Mathematics in Economics: a Philosophically Grounded Proposal

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