THE SYDNEY SCHOOL: MATHEMATICS, THE SCIENCE OF STRUCTURE

MANIFESTO

(OK, more a grant application, really)

PROJECT TITLE

Mathematics, the science of real structure: An Aristotelian realist philosophy of mathematics

AIMS AND BACKGROUND

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 complex world.

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. (KÃ¶rner 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.Â Despite this difficulty, many working mathematicians continue to find Platonism attractive, in part because it seems to be the only realist position available.

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 mathematicians.

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 ideas.

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 oneÂ 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.

SIGNIFICANCE AND INNOVATION

The approach is innovative in starting with applied mathematics, rather than detouring via pure mathematics and wondering how pure mathematics can "apply" to something so removed from it as the physical world. That enables a strong Aristotelian directness to drive our reasoning.

Contemporary philosophy of mathematics has paid insufficient attention to the possible varieties of realism. Attempts to defend realism (Maddy 1992, Steiner 1978) have generally assumed uncritically and unreflectively that realism must be some form of Platonism. Resnik and Shapiro made progress in moving from the idea mathematics is about individual objects to the idea that mathematics is about structural universals.Â But the epistemology of structuralism is far from developed (Hale 1996).Â Our project will fill in this gap, especially by working on theories of abstraction and the cognitive science of pattern recognition.Â We plan to hold a workshop on this topic in July 2005 (before the period of the grant).

Our version of realism is novel, since the possibility of an Aristotelian realism has been largely absent from contemporary debates.Â Our version of realism will be more developed epistemologically than Platonism or the current expositions of structuralism. In particular, we will pay attention to the role of abstraction and proof in generating mathematical knowledge, as well as integrating these with cognitive psychology. At present, there is almost nothing available in the area Â– in our view, because the standard philosophical approaches such as Platonism, logicism and so on draw attention away from such basic facts as that the sense organs are designed to respond to complex patterns.

APPROACH

We will begin by setting out a number of simple mathematical examples with explanations of their philosophical significance. The choice will be different from the well-worn round of examples from set theory, logic and number theory generally examined by philosophers of mathematics, and will concentrate instead on examples form applied mathematics where it is possible to see structural features and the necessities connecting them.

For example, Einstein and most philosophers have thought there cannot be mathematical truths which are at once necessary and about reality. Against this are prima facie examples such as Â“it is impossible to tile my bathroom floor with (equally sized) regular pentagonal tiles.Â” Objections such as those based on the supposed purely logical or hypothetical nature of mathematics have no force against such examples. (Franklin 1989) Another instructive example is EulerÂ’s classic case of the bridges of KÃ¶nigsberg, which created a system where it was impossible to walk over all the bridges without walking over one of them twice. Now recognised as the first result in network topology, the example shows both the possibility of findings necessities (or in this case, impossibilities) in real systems, and the need to see mathematics as about structures that cannot reasonably be called "quantity" Other examples will come from the formal sciences such as operations research and computer science (where proofs of program correctness give examples of mathematical necessities).

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 Â“science of quantityÂ” conception Â– 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 Â– arguing for our view that mathematics studies structural aspects of the real world Â– 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 BenacerrafÂ’s 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.Â 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").Â We anticipate the objection that the natural world does not have the perfect precise structures needed in mathematics.Â 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).Â 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). FregeÂ’s objections are an important reason for the neglect of an Aristotelian approach. However, we will demonstrate that FregeÂ’s criticisms do not touch Aristotelian realism.Â 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.Â 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.Â 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 Â– metaphysical, epistemological, mathematical and quantum-mechanical.Â 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.Â 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:

Â· Year 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.
Â· Year 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.
Â· Year 3: Completion and submission of book containing the full work, probably to Oxford University Press.

NATIONAL BENEFIT

Mathematics and philosophy are the two classical abstract disciplines, and using philosophy to understand mathematics is a project of intrinsic worth. The Australian philosophical tradition has been strong in robust realisms in many fields, including the philosophy of science and mathematics. As explained in Franklin 2003, ch. 12, Australian philosophy has been much more ready than British and especially American philosophy to take a realist view of such scientific entities as forces, universals, natural laws, non-deductive relations between evidence and hypothesis and so on. That has led to a number of leading overseas philosophers who have realist sympathies visiting Australia regularly (most famously the late David Lewis) and consequently to AustraliaÂ’s high profile in the philosophical world. The Australian tradition in the philosophy of science and mathematics is, we believe, as a whole well in advance of most overseas work, which remains mired in the effete Platonisms, nominalisms and idealisms of the Old World. That has not yet become obvious, however, in the philosophy of mathematics. At the same time Australian realist philosophical thought is in closer contact with the thinking of scientists and mathematicians. The present project will build on those strengths so as complete, on a broader scale, an updated Aristotelian philosophy of mathematics that will reveal a characteristically Australian response to a problem that has been of keen interest at the most abstract levels of thought for 2400 years.

Australian philosophy will thus be strengthened in general, giving it a greater attractiveness to the many overseas visitors and students who already appreciate its unique strengths. The nature of the project will also lead to closer ties and interdisciplinary work between philosophers and mathematicians Â— traditionally, mathematicians have seen philosophers of mathematics as having little to say of relevance to mathematics, a criticism in our view largely justified. Our closer attention to what mathematicians, especially applied mathematicians, are actually doing should help overcome that problem.

If our plans for communicating our results at a more popular level (see below) are successful, that would raise the general community perception of mathematics, and we believe make the mathematical sciences more attractive to a broad range of potential students who are currently put off by the rather forbidding image mathematics sometimes has of an esoteric and technical art of little relevance to basic human concerns.

COMMUNICATION OF RESULTS

The results will be published in an academic book, for which a leading overseas university press will be sought in order to maximise its impact. There will be some preceding academic papers in the areas on which we have not so far written, especially epistemology. We will also give conference presentations on preliminary work, both in Australia and overseas. Some material is available on CI FranklinÂ’s personal website, http://www.maths.unsw.edu.au/~jim ; as the project progresses, this will be reorganised into a website on the whole project, where an overall view of our "line" will be given and our published and preliminary work will be available for download. We would expect the website to attract sympathetic researchers around the world who would disseminate our ideas in their communities. Unlike most philosophers of mathematics, we will also speak at conferences on mathematics such as the annual conference of the Australian Mathematical Society.

It is in the nature of the theory that it can be communicated without reference to complicated technical results in mathematics or logic, so we expect to promote it also in works aimed at the wider intellectual audience that reads such popular books on mathematics as those by Ian Stewart, Keith Devlin and Reuben Hersh. Structural features particularly lend themselves to diagrams (Franklin 2000), so there is an opportunity to increase the bookÂ’s impact through generous use of illustrations.

DESCRIPTION OF PERSONNEL

James Franklin and Adrian Heathcote as CIs will plan and direct the project. They have some of the central parts of the project already written, but certain parts still need writing, notably the theory of quantity as an abstract universal (Franklin) and the mathematical interpretation of quantum mechanics (Heathcote). The planning and writing of the final book will require considerable work, as the relations between disparate parts are problematic in detail, even if reasonably clear in principle. Many philosophies of mathematics have fallen foul of the difficulties of reconciling their metaphysics and epistemology in this area, and in addition we plan to integrate the essentials of the cognitive psychology of pattern recognition. The task is substantial, but in view of the advanced state of preparatory work, we estimate it should be achievable in 0.1 FTE work over three years. That comparatively relaxed pace will allow the research assistantÂ’s more intensive work to finish at the same time as that of the CIs.

The Research Associate to be employed on the grant will work in consultation with the CIs but to a degree independently. Some of the work to be done by the RA will be on general tasks and co-authorship with the CIs, some, at a later stage, will be independent research in the same direction but with a special concentration on epistemology, the area in which the work of the CIs is undeveloped. The RA will need to be qualified at doctoral level, either in philosophy of mathematics or in metaphysics and epistemology with an interest in philosophy of mathematics Â– that level will ensure an immediate start on the project is possible.Â The philosophy background of the RA will complement the backgrounds in mathematics and philosophy of physics of the CIs. The expected background of the RA in epistemology, especially, will be essential to making progress on that currently undeveloped part of the project. We anticipate no problem in finding a suitably qualified person.
James Franklin
March 2005

REFERENCES

Armstrong, D.M., 1988, `Are quantities relations? A reply to Bigelow and PargetterÂ’ Philosophical Studies 54, 305-16.
Armstrong, D.M., 1991, `Classes are states of affairs', Mind 100, 189-200.
Azzouni, J., 1994, Metaphysical Myths, Mathematical Practice, Cambridge, Cambridge University Press.
Baker, A., `The indispensability argument and multiple foundations of mathematicsÂ’, Philosophical Quarterly 53, 49-67.
Benacerraf, P., 1965, `What numbers could not beÂ’, Philosophical Review 74, 495-512.
Benacerraf, P. 1973. Â‘Mathematical TruthÂ’, Journal of Philosophy 70, 661-79.
Bigelow, J., 1988, The Reality of Numbers: A Physicalist's Philosophy of Mathematics, Clarendon, Oxford.
Bigelow J. and R. Pargetter, 1990, Science and Necessity, Cambridge University Press, Cambridge.
Chihara, C.S., 2004, A Structural Account of Mathematics, Clarendon, Oxford.
Colyvan, M., 2001a, The Indispensability of Mathematics, Oxford University Press, New York.
Colyvan, M., 2001b, `The miracle of applied mathematicsÂ’, Synthese 127, 265-77.
Dennett, D., 1991, `Real patternsÂ’, Journal of Philosophy 88, 27-51.
Devlin, K.J., 1994, Mathematics: The Science of Patterns, Scientific American Library, New York.
Field, H., 1980, Science Without Numbers: A Defence of Nominalism, Princeton University Press, Princeton.
Fine, K., 2001. Limits of Abstraction. Â Oxford University Press,Â Oxford.
Forrest P. and D.M. Armstrong, 1987, `The nature of number', Philosophical Papers 16, 165-186.
Frege, G., 1884/1950. Foundations of Arithmetic. Blackwell, Oxford.
Franklin, J., 1989, `Mathematical necessity and reality', Â Australasian J. of Philosophy Â 67, 286-294.
Franklin, J., 2000, `Diagrammatic reasoning and modelling in the imagination, the secret weapons of the Scientific Revolution', inÂ 1543 and All That, Image and Word, Change and Continuity in the Proto-Scientific Revolution, Â ed. G. Freeland & A. Corones (Kluwer, Dordrecht), pp. 53-115
Franklin, J., 1994, `The formal sciences discover the philosophers' stone', Studies in History and Philosophy of Science Â 25, 13-33
Franklin, J., 2000, `Complexity theory, mathematics and the unity of science', History, Philosophy and New South Wales Science Teaching Â Third Annual Conference, ed. M. Matthews, pp. 91-4.
Franklin, J., 2003, Corrupting the Youth: A History of Philosophy in Australia, Macleay Press, Sydney.
Hale, B., 1996. Â‘StructuralismÂ’s Unpaid Epistemological DebtsÂ’, Philosophia Mathematica, 4, 124-47.
Heathcote, A., 1990, `Unbounded operators and the incompleteness of quantum mechanicsÂ’, Philosophy ofÂ Science S90, 523-34.
KÃ¶rner, S., 1962, The Philosophy of Mathematics: An Introduction, Harper, New York.
Mac Lane, S., 1986, Mathematics: Form and Function, Springer, New York.
Maddy, P., 1990. Realism in Mathematics, Clarendon Press, Oxford.
Maddy, P., 1992. Â‘Indispensability and Mathematical PracticeÂ’, Journal of Philosophy 89, 275-289.
Maddy, P., 1997. Naturalism in Mathematics, Clarendon Press, Oxford.
Michell, J., 1994, `Numbers as quantitative relations and the traditional theory of measurementÂ’, British Journal for the Philosophy of Science 45, 389-406.
Mortensen, C., 1998, `On the possibility of science without numbersÂ’, Australasian J. of Philosophy 76, 182-97.
Mundy, B., 1987, `The metaphysics of quantityÂ’, Philosophical Studies 51, 29-54.
Newstead, A.G.J., 2001, `Aristotle and modern mathematical theories of the continuumÂ’, in D. Sfendoni-Mentzou, ed, Aristotle and Contemporary Science, vol. 2, Lang, New York.
Parsons, C., 2004, `Structuralism and metaphysicsÂ’, Philosophical Quarterly 54, 57-77.
Quine, W.v.O., 1951/1980. Â‘Two dogmas of empiricismÂ’, in From a Logical Point of View, Harvard University Press, Harvard.
Reck, E., and M. Price, 2000, `Structures and structuralism in contemporary philosophy of mathematicsÂ’, Synthese 125, 341-383.
Resnik, M.D., 1997, Mathematics as a Science of Patterns, Clarendon, Oxford.
Shapiro, S., 1997, Philosophy of Mathematics: Structure and Ontology, Oxford University Press, New York.
Shapiro, S., 2004, `Foundations of mathematics: metaphysics, epistemology, structureÂ’, Philosophical Quarterly 54, 16-37.
Steiner, M., 1975. Mathematical Knowledge, Cornell University Press, Ithaca.
Steiner, M., 1998, The Applicability of Mathematics as a Philosophical Problem, Harvard University Press, Harvard.
Weyl, H., 1952, Symmetry, Princeton University Press, Princeton.
Wigner, E., 1960, `The unreasonable effectiveness of mathematics in the natural sciencesÂ’, Communications in Pure and Applied Mathematics 13, 1-14.
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