Structuralism in thePhilosophy of Mathematics.

Porchet Thierrythierry.porchet@epfl.ch

Supervised by Pr. Tony Martin & Pr. Jacques Duparc

June 22, 2011

Section de mathématiques, EPFL

Contents

1 Philosophy of Mathematics in General 3

1.1 Philosophy-First VS. Philosophy-Last-if-at-All . . . . . . . . . . . 31.2 Realism VS. Anti-Realism . . . . . . . . . . . . . . . . . . . . . . 7

1.2.1 Working Realism . . . . . . . . . . . . . . . . . . . . . . . 71.2.2 Philosophical Realism . . . . . . . . . . . . . . . . . . . . 81.2.3 Anti-Realism . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.3 Internal and External Questions . . . . . . . . . . . . . . . . . . 9

2 Structuralism 11

2.1 Structuralism in General . . . . . . . . . . . . . . . . . . . . . . . 122.2 Mathematical Objects in Structuralism . . . . . . . . . . . . . . . 142.3 The Ontological Status of Structures . . . . . . . . . . . . . . . . 182.4 Theory of Structuralism . . . . . . . . . . . . . . . . . . . . . . . 21

3 Epistemology 26

3.1 Bourbaki's Mother Structures . . . . . . . . . . . . . . . . . . . . 263.1.1 Formalization of the "Grouping" . . . . . . . . . . . . . . 293.1.2 The Topological Structure . . . . . . . . . . . . . . . . . . 313.1.3 The Link Between the Firsts Mental Structures and Bour-

baki's Mother Structures . . . . . . . . . . . . . . . . . . 323.2 The Discovery of a Mathematical Structure . . . . . . . . . . . . 34

3.2.1 Toward Pure Mathematics . . . . . . . . . . . . . . . . . . 373.2.2 Back to Ontology . . . . . . . . . . . . . . . . . . . . . . . 39

4 Conclusion 42

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Abstract

Mathematics has been use by virtually every civilizations throughout

history. But mathematics is a peculiar topic, because even though every-

body is capable of using it, to some extent, the question of what exactly

is mathematics isn't an obvious one at all. Other questions, linked to

the latter, are the question of the truth value of mathematics, of how it

is applicable to science, and of how we manage to know anything about

mathematics at the �rst place. The history of the philosophy of mathe-

matics presents various di�erent views on those matters.

We will start this work by brie�y talking about the di�erent variety

of possible philosophical view concerning the philosophy of mathemat-

ics. The �rst point we will discuss is about the extent to which philoso-

phy should in�uence mathematics, or what use philosophy has, regarding

mathematics. We will then look at the di�erent possibilities concerning

the reality of the mathematical objects, i.e. whether mathematical objects

do really exist, and the reality of mathematical truth, i.e. whether mathe-

matical truth really are truth. We will then �nish this introduction to the

philosophy of mathematics by looking more critically at the philosophical

questions concerning mathematics themselves, regarding their legitimacy.

At the end of the 19th century, in great extent due to the work of

Hilbert, the idea that a mathematical theory can be applied to di�erent

systems, as long as the axioms are satis�ed, got more and more impor-

tance, giving birth to modern mathematics. From a philosophical per-

spective, this gave birth to structuralism: the idea that the topic of math-

ematics is the study of the structures in which a mathematical theory is

articulated. This idea gives new perspectives concerning the ontology of

mathematical objects, and we will discuss the di�erent possibilities of in-

terpretation of what mathematical objects are, from a structuralism point

of view. If mathematics is the science of structures, then we need to know

what those structures really are, so we will then discuss the di�erent pos-

sible ontological status of the structures themselves. We will then see how

the theory of structuralism can be rigorously de�ned.

We will end this paper by looking more in details the question of how

knowledge of mathematics is acquired. To do that, we will look at the

work of Jean Piaget concerning the genesis of mathematical concepts, in

particular in the mind of a child. We will see that there is a close link

between the �rsts mental structures that we can observe in the child, and

the Bourbaki's mother structures, which are three very general structures

that can be de�ned somehow as fundamental structures, from which every

mathematical structures is derived. We will then study how, from those

�rsts mental structures, further structure can be constructed, through the

process that Piaget calls re�ecting abstraction, and how this process can

bring us all the way to pure mathematics. In light of these epistemo-

logical considerations, we will go back to the question of the ontology of

mathematical structures.

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1 Philosophy of Mathematics in General

In this �rst chapter, which will serve as an introduction, we will look at phi-losophy of mathematics in general. We will �rst discuss the extent to whichphilosophy is (or should be) in�uencing mathematics. We will then look at thetwo orientations that philosophy of mathematics can take: realism and anti-realism. We will end this chapter by looking a little more critically at thephilosophical questions themselves, in regards to their legitimacy. Most of thecontent of this chapter is based on Philosophy of Mathematics, Structure andOntology [16] from Shapiro. Many of the quotes from other authors are alsotaken from this book.

1.1 Philosophy-First VS. Philosophy-Last-if-at-All

Philosophy of mathematics can be divided in two distinct kinds, that Shapirocalls philosophy-�rst principle and philosophy-last-if-at-all principle. Accordingto the �rst one, philosophy should determine the proper use of mathematics.We must �rst determine what is mathematics, what it subject matter is, etc.,and then work on mathematics itself, in accordance with the philosophical con-ceptions.

As for the second one, Shapiro writes

[T]he thesis [of philosophy-last-if-at-all is] that philosophy is ir-relevant to mathematics. On this perspective, mathematics has a lifeof its own, quite independent of any philosophical considerations. Aview concerning the status of mathematical objects or statements isat best an epiphenomena that has nothing to contribute to math-ematics, and is at worst a meaningless sophistry, the rambling andmeddling of outsiders. If philosophy of mathematics has a job at all,it is to give a coherent account of mathematics as practiced up tothat point. Philosophers must wait on the mathematician [...] andbe prepared to reject their own work, out of hand, if developmentsin mathematics come into con�ict with it. (Shapiro in [16], p.28)

So according to this view, philosophy of mathematics is not important to math-ematics itself, and should only try to make sense of the current practice ofmathematics.

Let's �rst have a look a some examples following the philosophy-�rst princi-ple. For Plato, the subject matter of mathematics was an eternal, unchangingand ideal realm, called the realm of Ideas, or Forms. He was thus very criticalof the geometers of his time, regarding their constructive and dynamic way topractice mathematics. In the Republic, he writes

[The] science [of geometry] is in direct contradiction with thelanguage employed by its adepts. [. . . ] Their language is most lu-dicrous, [. . . ] for they speak as if they were doing something andas if all their words were directed toward action. [. . . ] [They talk]

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of squaring and applying and adding and the like [. . . ] whereas infact the real object of the entire subject is [. . . ] knowledge [. . . ] ofwhat eternally exists, not of anything that comes to be this or thatat some time and ceases to be. (Plato in Book VII of the Republic)

So for Plato, mathematical objects exist prior to anything the mathematiciancan do. The only thing the mathematician can do is discover truth about thosemathematical entities, but he cannot create one. According to his philosophy,the practice of mathematics of that time, consisting of drawing lines and moving�gures, didn't make sense. The Plato's Academy thus tried for some time toresolve the problem of the use of dynamic language, trying to for example to �nda way to introduce the notion of motion into the immovable geometric objects.

At �rst, one might think this problem is just a matter of language. Forexample, if we write, in a Euclidean fashion, that between two points, one candraw a line, it seems equivalent in content to writing, in an Hilbertian fashion,that between to points, there is a straight line. But most problems at that time- squaring of a circle, doubling a cube, trisecting an angle - weren't questions ofexistence, but questions of constructibility. If we take for example the problemof the trisection of the angle, the matter was not to know if there existed anangle of 20◦, but whether we could construct such an angle.

Another more recent example of a philosophical challenge of mathematicsas practiced is the intuitionist philosophy, that took place in the twentiethcentury. According to this view, mathematical objects are mental constructions,and mathematical statements can only refer to those. Intuitionist reject theexistence of any object that has not been constructed.

One of the main consequence of this philosophy on the practice of mathe-matics is the rejection of the law of excluded middle. Let φ be a property ofnumber. According to an intuitionist, contrary to the classical view, the twopropositions ¬∀xφ(x) and ∃x¬φ(x) are not equivalent. The �rst propositionmean that it is false to assert that there exists a construction that shows that φholds of each numbers, where the second one says that it is possible to constructa number x of which φ does not hold, and thus the �rst one does not impliesthe second one.

Another main feature of classical mathematics challenged by the intuition-ists and others is the use of impredicative de�nitions. The de�nition of amathematical entity is impredicative if it refers to a collection of objects inwhich the de�ned entity is. For example, if we de�ne n as being the small-est prime number bigger then 100, the number n would be contained in the set{prime numbers bigger then 100}, then n would be impredicatively de�ned. Im-predicative de�nitions presuppose the existence of all members of the collectionused, e.g. the existence of all prime numbers bigger then 100.

The last example we will take to illustrate the implications that philosophicalconception can have on mathematics is the (in)famous axiom of choice. It canbe stated this way:

for every set A of nonempty sets, there is a function whose domainis A and whose value, for every a ∈ A, is a member of a. (Shapiro

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in [16], p.24)

In other words, there is a function f that ∀a ∈ A, chooses an element of a, i.e.f(a) ∈ a.

This axiom is used widely in today mathematics (often without the math-ematician being aware of it), and thus, rejecting it would mean rejecting mostof contemporary mathematics. But when it was �rst explicitly formulated inZermalo's proof of the well-ordering theorem, many leading mathematicians,like Lebesgue, Baire or Borel, opposed to it. Using Lebesgue's terms, math-ematicians can be divided in two categories: the empirists, who only acceptthe existence of explicitly de�nable function, and the idealists, who accept theexistence of unde�nable functions. According to the idealists perspective, math-ematics is independent of the mind of the mathematician, and thus, de�nabililtywould be irrelevant to matters of existence. For the empirists, some method ofconstruction for the function is required for it to exists.

Historically, the philosophy-�rst principle is not true to the development ofmathematics. Greek mathematicians developed their theories using dynamiclanguages, and the use of impredicative de�nitions, excluded middle, unde�nedfunctions, etc, has been widely used in the development of modern mathematics.

Proponents of the axiom of choice pointed out that this axiom codi�es aprinciple often used in mathematics. Further studies of the axiom of choice andits role in the di�erent branches of mathematics showed that it was essentialin their practice. With some irony, it turned out that the axiom of choice wasa necessity for the work of major opponents against it, due to the implicit useof the choice principle (or a lesser form of it, like the countable choice or thedependent choice) in the development of the central branches of mathematics.In the end, and this is the part the most relevant to this chapter, the axiom ofchoice was not accepted because the philosophical idealist view, or realism, wasaccepted as the right philosophical view, but because the axiom was needed formodern mathematics.

The same goes to the laws of excluded middle or impredicative de�nitions.The studies of logical systems lacking those principles showed that mathemat-ics in general would be notably di�erent, in the practice itself, and more im-portantly, with many lacking important results, without them. In short, theresulting mathematics end up being unattractive. That is why modern math-ematics makes use of all theses tools. The important thing for us here is thatthe reasons for modern mathematics to makes use of impredicative de�nition,excluded middle, axiom of choice, etc, are not related to philosophy.

This brings us to the philosophy-last-if-at-all principle. Like Shapiro puts it,as an argument in favor of philosophy-last,

The fact is that many mathematicians, perhaps most, are notin the least interested in philosophy, much less in speci�c questionsof ontology or semantics; and it is mathematicians, after all, whopractice and further articulate their �eld. For better or worse, thediscipline carries on quite independently of the musings of us philoso-phers. (Shapiro in [16], p.28)

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It is sometime said that mathematicians are Platonists during the week, andformalists on Sunday. In other words, when working, the mathematician willwork as if the mathematical entities were part of a mind-independent, eternalrealm. But when, out of his work, questioned about, for example, how it wouldbe possible for anyone to know anything about entities of an external realm, hewould tend to take a formalist perspective, even if it's not at odds with his work.But when returning to his work, he would continue his practice as a Platonist.The point here being that philosophical considerations would not in�uence thepractice of mathematics.

A view that goes in the direction of the philosophy-last idea, but appliedto science in general, is Quine naturalism. The original goal of philosophyis to understand, or describe, the world, the best way possible. For Quine,science is the only plausible way to aim for that goal. Furthermore, for Quine,mathematics is only relevant in the way that it is part of science. Thus, partsof mathematics that go beyond the parts applicable to science are not acceptedby Quine. For example, for Quine, the class of sets V would be restrained to theclass of Gödel's constructible L, even if many set theorists believe that L ( V.So Quine's view could be put as science-�rst-philosophy-last.

Some philosophers, like Burgess or Maddy, extent the idea of naturalism tomathematics, instead of just to science, thus leaning toward a philosophy-lastpoint of view.

Going against the idea that philosophy could induce revisions of mathemat-ics, David Lewis writes

I laugh to think how presumptuous it would be to reject mathe-matics for philosophical reasons. How would you like to go and tellthe mathematicians that they must change their ways [. . . ]? Willyou tell them, with a straight face, to follow philosophical argumentwherever it leads? If they challenge your credentials, will you boastof philosophy's other great discoveries: That motion is impossible,[. . . ], that it is unthinkable that anything exists outside the mind,that time is unreal, that no theory has ever been made at all probableby evidence, [. . . ], that it is a wide-open scienti�c question whetheranyone has ever believed anything, [. . . ]? Not me! (David Lewis in[8])

Of course, mathematics is not impermeable to any kind of mistakes, andit's perfectly plausible that in the future, parts of mathematics will have to berevised to correct some mistakes unknown at the moment. But purely philo-sophical considerations seems to be insu�cient to generate such revisions.

So to my sense, the purpose of philosophy of mathematics is to give aninterpretation of mathematics. What is the subject matter of mathematics?What are mathematical entities, i.e. what is their ontological status? Howcan we learn anything about mathematical entities? How can mathematics beapplied to science? But as Shapiro writes

[I]t is mathematics that is to be interpreted, and not what thephilosopher hopes mathematics can be or should be, and not what a

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prior (or a priori) philosophical theory says mathematics should be.(Shapiro in [16], p. 32)

1.2 Realism VS. Anti-Realism

The di�erent theories in philosophy of mathematics can be categorized in re-gard of their views concerning the ontology of mathematical objects, and thetruth-value of mathematical statements. Concerning the ontology, the philoso-phers that believes that mathematical objects exist independently of the math-ematician are called by Shapiro realists in ontology1. The opposite view isthe anti-realism in ontology, for which mathematical objects are products ofthe mind, and do not exists independently of the mathematician. In a sim-ilar fashion, philosophers who believe that mathematical statements have anobjective truth-value, independent of the mathematicians, are called realists intruth-value, as opposed to the anti-realists in truth-value. So combining thoseclassi�cations, we have four categories of philosophical view.

We will �rst develop a view called working realism, which would describepeople who accept the implications of a traditional realist philosophy aboutmathematics. This view is more aimed at methodology then a full philosophy.We will then talk a little about the di�culties underlying a realism program.And �nish by talking about some anti-realist views.

1.2.1 Working Realism

As said above, the working realism is more about methodology then philosoph-ical matters. It is concerned about how mathematics is practiced, about thedi�erent tools that are used, like the ones we talked about earlier (impredica-tive de�nitions, axiom of choice, excluded middle, etc), and about how thoseare to be described. Any mathematician who isn't trying to revise mathematicsis most likely to be a working realism.

Shapiro divides working realism in di�erent levels, each one including theone below, and without real sharp boundaries. As he puts it

The �rst, and weakest level applies to those mathematicianswhose practice can be characterized as conforming to the [. . . ] prin-ciples and inferences [like impredicative de�nitions, axiom of choice,etc]. This working realism is purely descriptive and, moreover, suchmathematicians themselves may or may not accept the description,were it o�ered to them. One is a working realist in this sense if oneseems to use excluded middle, the axiom of choice, and the like, un-critically, even if one does not acknowledge or otherwise admit thatone's practice adheres to these principles. (Shapiro in [16], p. 38)

Examples of this level of working realism are Lebesgue, Borel and Baire,who, as mentioned before, opposed Zermalo's axiom of choice, even though it

1They are sometimes called Platonists, in the sense that the mathematical objects areviewed as members of an external, ideal, realm.

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was implicitly used in their work. Shapiro calls it third-person working realism,because "characterization of practice may be disputed by those who engage inthe practice ([16], p. 38)".

The second level, that Shapiro calls �rst-person working realism, applies tothe mathematicians who acknowledge the use of the items in question in theirwork, but without taking any commitment on the question. They admit usingexcluded middle, axiom of choice, etc, and that so far, no problem has occurred.But they would admit the possibility that in the future, those principles wouldbe refuted, and that parts of their work could be discarded on that ground.

The third level is no longer descriptive, but takes a normative perspective.

The principles and inferences in question are accepted as templatesto guide further research and as grounds for criticizing others. Thesemathematicians hold, for whatever reason, that mathematics shouldconform to classical logic, impredicative de�nition, choice, and soforth. (Shapiro in [16], p. 39)

Of course, this division in levels is arbitrary, and some points of view canbe put somewhere in between two levels, but it gives a good idea of the generalideas.

Like we said earlier, working realism is not a full philosophy. It doesn't sayanything about ontology, epistemology, semantics, etc. It only talks about howmathematics is (or must be) practiced. The idea is that mathematics is practicedas if mathematical objects were part of a Platonist realm, but there is no claimthat they actually are part of such a realm. So if an anti-realist theory does notimply any revision in mathematics, it is compatible with working realism. Butworking realism can be a starting point for more complete philosophical viewson mathematics.

1.2.2 Philosophical Realism

When going further then working realism in philosophical questions, one has toconsider the ontology of mathematical objects, and the truth-value of mathe-matical statements. What Shapiro calls a philosophical realist is someone whobelieve that mathematics have a real subject matter, and that mathematics isobjective. The problem that rises from such a point of view is in regard tothe epistemology. If mathematical objects are part of an external and atempo-ral realm, how can we know anything about them. To be able to know aboutsomething, there must be some sort of causal link between the knower and theobjects in question. But if mathematical object are presupposed to live in suchan external and atemporal world, they cannot, by de�nition, have any kind ofinteraction with our world.

One other problem that rises when taking a realist point a view is the ques-tion of the applicability of mathematics to science, and to everyday life in gen-eral. The fact that mathematics is used in science is a double edged sword.On one side, application is a great motivation for realism, since mathematicsis a central feature of science. Like we have mentioned before, Quine's view is

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that mathematics exists because it is indispensable to science, and that scienceworks. But the question that rises then is how exactly are mathematics ap-plied to science? What is the link between science, and acausal, atemporal andexternal objects?

1.2.3 Anti-Realism

Opposed to the realism view is the anti-realism. Its aim is to give an accountof mathematics and its uses, without postulating that mathematics is about amind-independent, objective realm, and that mathematical truth is objectiveand non-vacuous. One of the main reasons for anti-realism is the di�cultyrealism face in regards to epistemology, that we mentioned before.

One possibility, developed by Hellman and Chihara, is the introduction ofmodality, i.e. of logical notions of possibility and necessity. So this kind ofprogram reduces mathematical entities to logical possibilities. Another way,developed by Field, is to take mathematical knowledge in a similar sense thenlogical knowledge. Field's idea is that science can be done without mathematics,but that mathematics is a good tool to facilitate science. His idea is that math-ematics is conservative over science, in a similar way that logic is conservativeover everyday life. But mathematical entities are, according to him, �ctionalentities. Lastly, another idea, developed by Tennent and Dummett, is to replacethe concept of truth, with a concept of warranted assertability.

At least at �rst sight, there seems to be less epistemological problems withnotions of logical knowledge, or assertability. So that would be an advantageover realists views.

1.3 Internal and External Questions

One thing that is, I think, worthy of looking into is the matter of the legitimacyof the questions posed by philosophers regarding mathematics. Some writershave put a distinction between the internal and external questions.

For Carnap, for example, before asking questions of ontological nature, be itin philosophy, physics, or mathematics, a linguistic framework must be properlyformulated, with explicit and rigorous syntax and rules. An example of sucha framework can be Peano's �rst-order arithmetic, as a framework for talkingabout arithmetic. Once such a framework has been set, there can be two sortsof ontological questions: internal, or external. Let us take the question "donumbers exists?". The internal version of it would be to ask whether "∃x ∈ N"is true or not (which has a quite trivial answer). The external version is thetraditional question of asking, without reference to a particular framework, "donumber really exists?", i.e. do number exists independently of the mind, or ofany linguistic framework. For Carnap, only the internal questions are legitimate,and he calls the external questions "pseudo-questions".

As said above, this does not only apply to mathematics, but also to phi-losophy or physics. In physics for example, Stephen Hawking, has a view thathe calls Model-dependent realism, for which "there is no picture - or theory -

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independent concept of reality ([5])". The only relevant way to look at reality isthrough a physical model2, or framework. Concerning the question of whetheror not quarks really exist, the matter has only a truth-value within a model,but none independent of it. Within the framework of modern quantum physics,the quarks do exist. Without laking sense of humor, he writes

It is certainly possible that some alien beings with seventeen arms,infrared eyes and a habit of blowing cotted cream out their earswould make the same experimental observation that we do, butdescribe them without quarks. Nevertheless, according to model-dependent realism, quarks exist in a model that agrees with ourobservations of how sub nuclear particles behave. (Hawking in [5])

So according to this view one could say that for us, quarks exist (in our currenttheoretical state), but for the aliens described here, they don't.

When considering a framework in general, according to these views, the onlylegitimate question is whether or not to accept it. But the acceptance of such aframework doesn't carry a belief that the entities in the framework really exist- are part of the Reality with a capital R. Carnap writes

If someone decides to accept the thing language, there is no ob-jection against saying that he has accepted the world of things. Butthis must not be interpreted as if it meant his acceptance of a be-lief in reality of the thing world; there is no such belief or assertionor assumption, because it is not a theoretical question. To acceptthe thing world means nothing more than to accept a certain formof language, in other words, to accept rules for forming statementsand for testing, accepting, or rejecting them [. . . ] But the thesis ofthe reality of the thing world cannot be among those statements,because it cannot be formulated in the thing language or, it seems,in any other theoretical language. (Carnap in [3]))

So for Carnap, the only questions worthy of attention, or "theoretical" ques-tions, must be internal to a linguistic framework. Traditional ontological ques-tions, like "do the number really exists?", are not among them. Regarding thiskind of questions,

[one] might try to explain [. . . ] that it is a question of the ontologicalstatus of numbers; the question of whether or not numbers have acertain metaphysical characterization called reality [. . . ] or statusof "independent entities". Unfortunately, these philosophers have sofar not given a formulation of their question in terms of the commonscienti�c language. Therefore [. . . ] they have not succeeded in givingto the external question [. . . ] any cognitive content. Unless and untilthey supply a clear cognitive interpretation, we are justi�ed in oursuspicion that their question is a pseudo-question. (Carnap in [3])

2By that I of course mean a model from the science of physics, not a model made of materialobjects.

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So until there is (if such a thing is possible), a complete universal explicit frame-work describing the reality as a whole, thus making the actual external questionsinternal, those are not worthy of consideration.

Another similar distinction is the one made by Hilary Putnam, between whatare called "internal realism", or "internalism", and "metaphysical realism". Ac-cording to Putnam, the latter is the view according to which

the world consists of some �xed totality of mind-independent objects.There is exactly one true and complete description of "the way theworld is". (Putnam in [11])

And the aim of science and philosophy is to come up with this unique andcomplete description of the universe, this theory of everything. It presupposesthat there is

a God's Eye view of truth, or more accurately, a No Eye view of truth- truth as independent of observers altogether (Putnam in [11])

Internalism rejects such a view. We have di�erent "conceptual schemes" todescribe reality, and Putnam's view is realism within those conceptual scheme.This implies an ontological relativity.

[I]t is characteristic of [internal realism] to hold that what objectsdoes the world consist of? is a question that it only makes sense toask within a theory or description. (Putnam in [11])

Having taken mostly geometric courses during my studies, I like to look atthis as if reality is a manifold, and conceptual schemes are maps on it. There isno reason for there to be a map that is de�ned over the whole manifold, but itmakes perfect sense to locally look at reality through a map.

So this view is a little less extreme then Carnap's one. Indeed, a conceptualscheme is a more permissive notion then the rigorously de�ned framework re-quired by Carnap to talk about anything. But I think the most important thingin the context of this work is the rejection of the unique "God's Eye" descrip-tion of reality. Which applied to mathematics means that there can be di�erentway to describe mathematical objects, that would all be legitimate, even if theycan seem to be incompatible3. This goes into the direction of Structuralism, ofwhich we will now talk about.

2 Structuralism

We are now getting to the heart of the matter that interests us in this work:Structuralism. We will start by giving an intuitive, sometime maybe naive,description of the general idea behind structuralism. We will then discuss thematter of the ontological status of mathematical objects, and then, the onto-logical status of the structures themselves. We will then end by giving a formal

3Or, in the words we'll de�ne in the next chapter, there can be di�erent systems exempli-fying a single structure, even if the di�erent systems in themselves seems incompatible.

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and precise de�nition of structures, and the rules inherent to them. As for theprevious chapter, most of the content of this chapter is based on [16].

2.1 Structuralism in General

In every day life, we tend to organize, or describe, what we perceive, by lookingat it as something in a pattern, or in a structures. Shapiro talks about a capacitythe human have of pattern recognition, i.e. he is capable of recognizing commonstructures that govern di�erent things. The slogan of structuralism is thatmathematics is the science of structures. But let's start by a more precisede�nition. Shapiro writes

I de�ne a system to be a collection of objects with certain rela-tions. An extended family is a system of people with blood and mar-ital relationships, a chess con�guration is a system of pieces underspacial and "possible-move" relationships, a symphony is a systemof tones under temporal and harmonic relationships, and a baseballdefense is a collection of people with on-�eld spacial and "defensive-role" relations. A structure is the abstract form of a system, high-lighting the interrelationships among the objects, and ignoring anyfeatures of them that do not a�ect how they relate to other objectsin the system. (Shapiro in [16], p. 73-74)

We will go deeper in the epistemological aspect in the next chapter, butlet's have a brief look, maybe a naive one, at it. At �rst, we see systemsof things. For example, di�erent chess boards with di�erent kinds of pieces,some in wood, other in plastic, etc... Every chess board with its pieces, andtheir possible moves, is what Shapiro calls a system. By noticing the commonfeatures of all systems (the 64 squares grid, the allowed moves for each pieces,etc) and abstracting the other (the material in which the board and the pieces aremade, the size, the shape of the pieces, etc), we build the abstract structure ofchess games. Another more mathematical example of this is the natural numberstructure. We �rst have collections of physical objects. In a Fregean fashion,we notice a property of equinumerosity between the di�erent collections, andabstract from this the concept of number. By abstraction from the process ofadding one object to a collection, we obtain the concept of the successor function,or "+1". By noticing that this function can be applied to any number, i.e. byrecurrence, we can de�ne this function over all the natural number. The naturalnumbers so constructed, with their relations (e.g. the di�erence between twonumber, being the number of time the successor function must be applied to oneof them, in order to obtain the other), is then a system of the natural numbersstructure. After noticing that there are other systems for which the relationsbetween their objects are the same as the ones between the objects of the systemwe just constructed (e.g. a graduated line, each graduation corresponding toone number), one might want to further abstract on the objects of a system, andonly keep the relations between the objects, taking the objects in question only

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as places4 between which the relations apply, we obtain the natural numbersstructure 5.

An important, maybe even central, feature of structuralism is resumed bywhat Hilbert wrote in a letter to Frege in 19006.

[A] concept can be �xed logically only by its relations to otherconcepts. These relations, formulated in certain statements I callaxioms, thus arriving at the view that axioms [. . . ] are the de�nitionsof the concepts. I did not think up this view because I had nothingbetter to do, but I found myself forced into it by the requirementsof strictness in logical inference and in the logical construction ofa theory. I have become convinced that the more subtle parts ofmathematics [. . . ] can be treated with certainty only in this way;otherwise one is only going around in a circle. (Hilbert, in a Letterto Frege, 09.22.1900, quoted by Shapiro in [16], p. 164)

In the course of history of mathematics, people have often tried to giveintrinsic de�nitions of the mathematical objects. For example, for Euclid, apoint is de�ned as "that which has no part". From the perspective that Resnikcalls "ontological Platonism", numbers are "on a par" with physical objects.This implies that one can state the essence of a number, on itself, withouthaving to refer to any other numbers. This would be like a de�nition of thenumber in question. Hilbert, and structuralists in general, reject this idea,which would, according to Hilbert, only be "going around in circle". Accordingto structuralism, a mathematical object is de�ned by its relations with otherobjects. These relations form a structure, and studying the structure is therelevant thing to do.

The example of chess games we gave above is of a structure abstractedfrom physical systems. But it is possible to grasp a structure without havinga physical exempli�cation of it. The study of structures, independently of thefact that they have a physical exempli�cation or not, is what we would call puremathematics. As Resnik puts it:

In mathematics, I claim, we do not have objects with an "inter-nal" composition arranged in structures, we have only structures.The objects of mathematics, that is, the entities which our math-ematical constants and quanti�ers denote, are structureless pointsor positions in structures. As positions in structures, they have noidentity or features outside a structure. (Resnik in [14])

And also:

4We'll come back to the question of the ontological status of these places later.5Again, this is a naive way to look at thing, from the epistemological perspective. But the

point here is to give the intuitive notion of a structure. See chapter 3 for a better account ofthe epistemological aspect.

6I did a semester project on the subject of the views of Frege and Hilbert, so I will not gotoo deep in the matter. See [18].

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Take the case of linguistics. Let us imagine that by using theabstractive process [. . . ] a grammarian arrives at a complex struc-ture which he calls English. Now suppose that it later turns out thatthe English corpus fails in signi�cant ways to instantiate this pat-tern, so that many of the claims which our linguist made concerninghis structure will be falsi�ed. Derisively, linguists rename the struc-ture Tenglish. Nonetheless, much of our linguist's knowledge aboutTenglish qua pattern stands; for he has managed to describe somepattern and to discuss some of its properties. Similarly, I claimthat we know much about Euclidean space despite its failure to beinstantiated physically. (Resnik in [15])

A mathematician does not require to see an entire system exemplifying astructure to be able to understand it. It is even most likely that most mathe-matical structures have no physical system exemplifying them. Indeed, for allwe know, the physical world is composed only by a �nite number of objects,atoms, particles, or whatever. And even if it wasn't, a human being only livesfor a �nite time (and furthermore, for all we know, life forms in general haveexisted only for a �nite time), and thus can only perceive a �nite number ofthings. But most mathematical structure have an in�nite number of places inthem. The set theoretic hierarchy structure has a proper class of positions.

2.2 Mathematical Objects in Structuralism

We will now consider more precisely the ontological status of mathematicalobjects, in a structuralist perspective. The basic idea is that mathematicalobjects7 are places in a structure. In the natural number structure, the number2 is the second place of the structure. This structure can be exempli�ed bydi�erent systems, with di�erent objects holding the di�erent places, but thestructure remains the same. If we take the example of a chess game again, thereis a distinction to be made between the concept of the white king, and the pieceof wood that exempli�es the role of the king in a particular chess board. Thesame goes for any mathematical structure, and the objects that compose it.

This perspective solves one problem against which some realists in ontologyhave come across, and that had been used sometimes as an argument towardanti-realism. Frege considered numbers to be objects, and tried to ground arith-metic with only logical consideration8. In his �rst attempt, even if he was ableto give a good account of sentences like "the number of φ is y", where φ is apredicate like for example "moons of Jupiter", or "cards in a deck", he facedthe problem of lacking a criterion to determine the truth-value of sentences like"Julius Caesar= 2". The truth value of this equality is of course obvious tocommon sense, but Frege's goal being to ground arithmetic only through logic,he needed a way to determine the truth value of such a sentence in order fornumbers to be objects. The heart of the matter being that it is required to have

7This is a little oversimplifying, but more subtle consideration will be treated further.8See [18] for a more developed account of Frege's logicist program

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a criterion to individuate the items in our ontology in order to have bona �deobjects. This problem is known under the name of the Caesar problem. Fregethen tried to solve this problem introducing the notion of the extension of a con-cept, which could be described in todays terminology as the class of all objectsfalling under the concept. With this concept, the number 2 would be de�nedas the collection of all pairs, i.e. the extension of the concept "composed of twoelements". Julius Caesar not being an extension, this would have solved theCaesar problem. But the concept of extension happened to be an inconsistentone, due to Russel's paradox.

The discovery that every �eld of mathematics can be modeled in set theorybrought the idea that the set-theoretic hierarchy could be the ontology for allof mathematics. The only needed objects for an ontological realism are thensets. The problem is that, in for example the case of arithmetic, there isn't aunique way to instantiate the structure with sets. Von Neumann for examplede�nes the successor function, for φ a set, as being S(φ) := φ∪{φ}, and so with0 := ∅, 1 = S(0) = {∅}, 2 = S(S(0)) = {∅, {∅}}, etc. . . , where Zermelo de�nesthe successor function as being S(φ) := {φ}, and thus with 0 := ∅, 1 = {∅}, 2 ={{∅}}, etc. . . So according to von Neumann's de�nition, we have 1 ∈ 4, whereaccording to Zermelo, 1 6∈ 4. There being no criterion to take one de�nitionover the other, both being de�ned in set theory, how are we to decide if "1 is amember of 4" or not? Benacerraf and Kitcher concluded that number are notobjects. But we believe that this conclusion is base on an ill conception of whatit is of being an object.

As Shapiro writes:

I would think that a good philosophy of mathematics need notanswer questions like "Is Julius Caesar = 2?" and "Is 1 ∈ 4?" Rather,a philosophy of mathematics should show why these questions needno answers, even if the questions are intelligible. It is not that wejust do not care about the answers; we want to see why there is noanswer to be discovered - even for a realist in ontology. (Shapiro in[16], p. 79)

The idea here is, I think, close to Carnap's idea of pseudo-questions. Anumber is a place in the natural number structure. In some sense, the naturalnumber structure is the framework we're in when discussing matters of numbers.There are, in this framework, rules to determine matters of identity, like 1 = 1,1 6= 4. It also makes sense to look into identity between numbers de�ned bya description in the language of arithmetic. 7 is equal to the largest primenumber smaller then 10. It also makes sense to identify numbers with physicalconceptions, in a Fregean fashion, and determine that the number of cards ina deck is 52. But it doesn't make sense to try to de�ne the notion of identitybetween places in the natural number structure, and other objects, like JuliusCaesar, because Julius Caesar cannot be described in the language of arithmetic.

It also makes sense to ask about relations de�nable in the language or arith-metic, and expect a determinate answer. For example, we have that 1 < 4, andthat 2 divides 4. Those matters are internal to the framework of arithmetic, to

15

the natural number structure. But the relation ∈ is not de�ned in this structure.So it makes no sense to ask whether "1 ∈ 4?". "It is similar to asking whether1 is braver than 4, or funnier. (Shapiro in [16], p. 79)" Questions about matteras "being an element of", "being funny", "begin equal to Julius Caesar", areoutside of the framework of arithmetic, and thus are pseudo-questions.

Following these considerations, one might conclude that the concept of iden-tity only makes sense between objects of the same structure. But this would begoing to far. Indeed, there are numerous cases where it makes perfect sense, andeven gives insight, to identify positions in di�erent structures. For example, wecan identify the real numbers with the points in a Euclidean line or the complexnumbers with the Euclidean plane. It would be unwise to object to the identi-�cation of the positions of the natural number structure with their counterpartin the integer-, rational-, real- and complex-number structures. As Parson putsit:

[O]ne should be cautious in making such assertions as that iden-tity statements involving objects or di�erent structures are meaning-less or indeterminate. There is an obvious sense in which identityof natural numbers and sets is indeterminate, in that di�erent in-terpretations of number theory and set theory are possible whichgive di�erent answers about the truth of identities of numbers andsets. In a lot of ordinary, mathematical discourse, where di�erentstructures are involved, the question of identity or non-identity ofelements of one with elements of another just does not arise (evento be rejected). But of course some discourse about numbers andsets makes identity statements between them meaningful, and someof that [. . . ] makes commitments as to the truth value of such iden-tities. Thus it would be quite out of order to say (without referenceto context) that identities of numbers and sets are meaningless orthat they lack truth-values. (Parson in [9])

One slogan of structuralism is that mathematical objects are places in struc-tures. Intuitively though, there is a di�erence between objects and places in astructure. Depending on the context, there are two ways to look at things, thatShapiro calls places-are-o�ces and places-are-objects.

One might want to compares di�erent instantiations of a place in a structure.For example, one might say that the von Neumann 2 has more elements thanthe Zermelo 2, or that the king of one chess board is better carved than theone from another chess board. This perspective is the places-are-o�ces. Inthis perspective, a background ontology is presupposed, to supply objects that�lls the places in the structure. In the case of a chess game, the backgroundontology is a collection of di�erent movable objects with spacial positions ona 64 squares grid, possible moves, etc. In the case of arithmetic, one possiblebackground ontology (but others could do the trick) is sets.

On the other hand, in some contexts, places in a structure can be treated asbona �de objects. In the context of a chess game for example, we can say that thebishop only moves in diagonal, without having to refer to any exempli�cation

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of the chess game structure, i.e. without having to take one particular chessboard with its pieces. In this perspective, arithmetic for example, is about thenatural numbers structure, and its subject matter is the places in this structure.The same seems to be applicable to any other �eld of mathematic. The obviousadvantage of this view is to give a clear de�nition of what mathematical objectsare.

In general mathematical talk however, the distinction between o�ces ando�ce holders, i.e. o�ces or objects, can be a relative one. Sometimes, places ofone structure can be said to occupy the places of another structure. For example,one can say that the Euclidean line exempli�es the real numbers structure. Inthat case, The Euclidean line is considered from the places-are-objects perspec-tive, where the real numbers structure is considered from the places-are-o�cesperspective. The places of one structure can even sometimes be �led with theplaces of the very structure considered. For example, one can say that the evennumber exemplify the natural number structure.

The legitimacy of this distinction can however be disputed. The places-are-object perspective can be viewed simply as a convenient way to generalizestatements that applies to all systems instantiating a particular structure. I liketo look at this argument in the sense that in the places-are-object perspective,the considered objects could just be equivalence classes, under the equivalencerelations of occupying particular o�ces, in a places-are-o�ce perspective. Such aview would eliminate the places-are-object perspective, by using the equivalenceclasses as o�ce holders, that is to say, the collection of these equivalence classeswould be seen as a system exemplifying the structure in question. Some cautionis however needed, because one would have to be sure that every relation de�ningthe structure can be well de�ned for a system of equivalence classes. Anotherthing to be careful about in such consideration is the size of such equivalenceclasses, that could very well be proper classes.

Another problem that can arise with this view is that one has to be sure thatthe equivalence classes are not empty. If we take for example the natural numberstructure. This structure is in�nite, but as we noted above, we have reasonsto believe that there are only a �nite number of objects in the universe9. Sothere seems to be no physical system �lling all the places in the natural numberstructure. This could mean that once reaching the upper bound of number ofphysical objects, the equivalence classes would be empty. And how about higherin�nite system? How about the set theory hierarchy? But more about that inthe next section.

In any case, the ontological status of the places of structures is directly linkedto the ontological status of the structure. Commitment to one view or the otherconcerning the ontology of structures, as we will see, eliminates the relativitybetween the o�ces and the o�ce-holders that we mentioned before.

9The number of atoms in the universe would be something in the order of 1080, which isof course enormous, but still �nite.

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2.3 The Ontological Status of Structures

Let's now turn ourselves to the question of the ontological status of the struc-tures themselves. As we have seen, structures can be exempli�ed by numeroussystems. They are similar to the concept of Form, or Idea introduced by Plato.In contemporary philosophy, one would speak of token and type. Several tokencan be of on particular type. In a similar sense, several systems can exemplifya certain structure.

The ontological status of types, or universals has preoccupied philosophersover the time, with numerous views. Two traditional views basically stand out.The �rst one, due to Plato, sometimes called "ante rem realism", is the viewthat universals exist prior to any instantiation, those universals can be called"ante rem universals". The concept of Redness exists prior to the existenceof any red object. The second view, due to Aristotle, sometimes called "in rerealism", with its "in re objects", is the view that universals are ontologicallydependent of their instantiations. Without any red object, there is no form ofRedness. But according to this view, the universals, once instantiated, do exists,in some sense. Some variations of this view are the conceptualist perspective,that says that universals are mental constructions, or the nominalist view thatsays that they are linguistic constructions (which can mean their non-existence).

In the context of structuralism, the equivalent of the �rst view is ante remstructuralism. Structures exists prior and independently to any system exem-plifying them. Structures can exists even if no system exempli�es them at all.This perspective takes the places-are-objects perspective at face value. Places instructures are bona �de objects. In some sense, every structure exempli�es itself.That is to say, the system of the structure's places exempli�es the structure.

The Platonist perspective having been used to introduce this idea, it couldbe somehow misleading. As Shapiro puts it:

In the history of philosophy, ante rem universals are sometimesgiven an explanatory primacy. It might be said, for example, thatthe reason the White House is white is that it participates in theForm of Whiteness. Or what makes a basketball round is that itparticipates in the Form of Roundness. No such explanatory claimis contemplated here on behalf of ante rem structures. I do not hold,for example, that a given system is a model of the natural numbersbecause it exempli�es the natural-number structure. If anything,it is the other way around. What makes the system exemplify thenatural-number structure is that it has a one-to-one successor func-tion with an initial object, and the system satis�es the inductionprinciple. That is, what makes a system exemplify the natural-number structure is that it is a model of arithmetic. (Shapiro in[16], p. 89-90)

This view however poses a problem of the epistemic access to the realm ofthose ante rem structures. How are we to know anything about these structures.

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One who tends toward the in re structuralism might be attracted by the con-sideration at the end of the last section. Like we said, in the places-are-objectsperspective, statements over the places themselves can be viewed as general-izations over all systems, in the places-are-o�ces perspective. Statements like"2 + 3 = 5" would thus mean something like "in every system S of the natural-number structure, the S-addition of the item at the second position with theitem at the third position is S-equal to the item at the �fth position". With thiskind of translation, or paraphrasing, statement about ontology, like "3 exists"becomes "in every system of the natural number structure, there is an object atthe third position", "number exists" becomes "in every system of the naturalnumber structure, there are objects �lling all places". So such matter becometrivial, they get a status like analytic truth.

So in this perspective, mathematical objects are just shorthands for talk-ing about any system exemplifying a particular structure. But they are notconsidered as bona �de objects. This elimination of any reference to abstractmathematical objects gave this view the name of eliminative structuralism. AsParson presents it:

It [. . . ] avoids singling out any one [. . . ] system as the naturalnumbers.[. . . ] [Eliminative structuralism] exempli�es a very naturalresponse to the considerations on which a structuralist view is based,to see statements about a kind of mathematical objects as generalstatements about structures of a certain type and to look for a way ofeliminating reference to mathematical objects of the kind in questionby means of this idea. (Parson in [9])

So according to this view, only the places-are-o�ces is perspicuous. So aswe said before, a background ontology is required. For a structure to exists,there has to be at least one system exemplifying it. This is one of the maindi�culty upon which eliminative structuralism is confronted. The di�cultyresides in the size the background ontology must have in order to account forsome (maybe most) part of mathematics. Let's take once again the simpleexample of the natural number structure. Let Φ be a sentence in the languageof arithmetic. For S a system of the natural number structure, let's denoteby Φ[S] the interpretation, or translation, of Φ in the language of S. In asimilar fashion as above, stating Φ for the natural numbers (seen as places inthe structure) accounts for something like

(Φ′) ∀S a system of the natural numbers structure,Φ[S]. (1)

Now if like we said before, there is no in�nite system of objects, there is nosystem exemplifying the natural numbers structure. So we have Φ′ ∧ (¬Φ)′,and this, for any proposition Φ of arithmetic, rendering arithmetic vacuous.Following the same idea, real analysis or Euclidean geometry requires a back-ground ontology the size of the continuum, and set theory requires a backgroundontology the size of a proper class.

So even if we make the hypothesis (against current physical theories) thatthere are an in�nite number of objects in the universe, we would be stuck with

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a countable number of objects. Or if as Field, we hold that every space-timepoint is a physical object, we would still "just" have something the size of thecontinuum. So by sticking to the requirement of a physical system instantiatingthe structure, all branches mathematics requiring an ontology larger then thecontinuum would be rendered vacuous. As Shapiro puts it: "Even with arith-metic, it is counterintuitive for an account of mathematics to be held hostageto the size of the physical universe.([16], p. 86)".

One possibility is to postulate that enough abstract objects do exists for allstructures to be exemplifyiable. We call this view the ontological eliminativestructuralism. One possible way would be to postulate the existence of the setsfrom the set theoretic hierarchy. So the background ontology would be V10.

The problem with this perspective is that set theory cannot be understoodas the study of its structure. And if we take the slogan that mathematics isthe science of structure, it would then mean that set theory is not a branch ofmathematic. Shapiro remarks:

Perhaps from a di�erent point of view, set theory can be thoughtof as the study of a particular structure U , but this would requireanother background ontology to �ll the places of U . This new back-ground ontology is not to be understood as the places of anotherstructure, or, if it is, we need yet another background ontology forits places. On the ontological option, we have to stop the regress ofsystem and structure somewhere. The �nal ontology is not under-stood in terms of structures, even if everything else in mathematicsis. (Shapiro in [16], p. 87)

I personally think it unintuitive to consider set theory as not being a branchof mathematics, or at least, as a branch of mathematics in essence di�erentfrom the others. Since it is possible to ground mathematics via category theory,set theory can be seen as a the study of a structure. But then, category theorywould have the status of the essentially special part of mathematics. So thechoice between set theory and category theory as the background ontology isan arbitrary one. And if set theory and category theory are both to be taken asthe studies of particular structure, a kind of ultimate background ontology hasto be postulated. But what would this background ontology be? And in whatsense would it be di�erent then the rest of the mathematics? In what sense,other then the fact that we need it to serve as a background ontology for themathematics, would it not be studyable as a bona �de structure?

Another possible way to answer to the problem of the lack of physical systemis through modality. For example, instead of looking at arithmetic as beingabout all systems exemplifying the natural numbers structure, the idea is tolook at it as being about all possible systems exemplifying the structure. Wecall this view the modal eliminative structuralism. Following this perspective,

10It is important to note that it is also possible to get a background ontology by usingcategory theory or topos theory. But not being familiar with these �elds, I will not developfurther.

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equation (1) becomes

(Φ′) 2∀S a system of the natural-number structure,Φ[S], (2)

where "2" means "it is necessary that".For a structure to exist then, according to this view, it is only required that

it is logically possible for it to have a system exemplifying it. So in the end,what is needed is for a background ontology to be logically possible, but thereis no need for it to exists, in a strict sense, anymore.

Hellman summarizes this view:

Mathematics is the free exploration of structural possibilities, pur-sued by (more or less) rigorous deductive means. (Hellman in [6])

This also gives an information about the epistemological question, and he writes:

The price of course is taking a logical modality as primitive, raisingquestions of evidence and epistemic access not unlike those raisedby platonist ontologies [. . . ]. (Hellman in [6])

So there remains an epistemic problem similar to the one we had with the anterem structuralism.

This perspective of modal eliminative structuralism is the one that appealsto me the most, for reasons of ontological economy. The ante rem structuralismpresupposes the existence of structures in a platonic sense, where the ontologicaleliminative structuralism needs for there to be a background ontology the sizeof a proper class, and I fail to see in what sense such a background ontologycould exists external to any mathematical theory. On the other hand, I don'tsee any conceptual problem11 with notion of logical possibilities.

In any way, I think it is important for consideration about the ontology ofstructure to look at the way we learn about them, or construct them, i.e itis important to look at the epistemology of the matter, that we will treat inchapter 3.

Nevertheless, the three possibilities exposed here are conceptually close toeach other, with only subtle di�erences. They are in some sense equivalent.

2.4 Theory of Structuralism

Let us now consider structuralism from a more mathematical point of view.Mathematics being at aim here, it is now time to put philosophical considera-tions on the side, and focus on a more formal perspective. It is important forus to have a well de�ned framework, in Carnapian sense, to be able to reallygrasp structuralism.

One central notion around which structuralism revolves is the idea thattwo systems (the structure itself being a system at the same time or not) canexemplify the same structure.

11except for the matter of the epistemic access to the knowledge of modal logic

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Several di�erent relations can be de�ned for this purpose, and we will proposetwo of them. The �rst one is of course isomorphism. Two systems are isomorphicif there is a one-to-one correspondence between the objects and the relations.Formally:

De�nition 2.4.1. Two systems S and S ′ are isomorphic if ∃f : S → S ′ abijection such that, ∀R a relation in S there is a relation R′ in S ′ such that

∀m,n ∈ S,mRn⇒ f(m)R′f(n).

But for the present purpose, isomorphism is a to strong notion. Indeed,one would like to say that the systems S = 〈N,+, ·〉 and S ′ = 〈N,+, ·, <〉 bothexemplify the natural number structure. But there is no isomorphism betweenthe two preserving the "<" ("less then") relation, for the trivial reason thatthere is no binary relation that could correspond with it in S. Notice that itis possible to de�ne the "<" relation in the system S by means of addition:x < y ⇔ ∃z(z 6= 0 ∧ x + z = y), so the structure S ′ is not "richer" then S ′.Similarly, the Euclidean plane can be de�ned by means of di�erent primitives(e.g. points or lines), but it is natural to require that they all exemplify thesame structure.

Resnik has formulated a more appropriate equivalence relation between sys-tems.

De�nition 2.4.2. LetR by a system and P a subsystem12. P is a full subsystemof R if ∀x ∈ R, x ∈ P (i.e. they have the same object), and if every relation inR can be de�ned in terms of relation in P.

So the only di�erence between R and P is that some de�nable relationspresent in R are omitted in P. So our previous system S ′ is a full subsystem ofS.

De�nition 2.4.3. Let M and N be two systems. M and N are structure-equivalent if there is a system R such thatM and N are each isomorphic to afull subsystem of R.

Notice that the de�nition of a full subsystem makes use of the notion of de-�nability, which is a linguistic notion. So the notion of full subsystem (and thusof structure-equivalence) is dependent on the metalanguage. For example, in�rst-order logic, the systems 〈N, S〉 (with S the successor function) and 〈N,+, ·〉are not structure-equivalent, because addition cannot be de�ned just by meansof the successor function. But they are structure-equivalent in second-orderlogic.

To continue rigorously developing the theory of structuralism, di�erent thingsare required, depending on the view considered regarding the ontology of struc-tures.

As we have said above, the ontological in re structuralism requires a back-ground ontology. A natural (but again, not unique) choice is to take the set

12i.e. a system with some, but not necessary all, objects and relations of R

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theoretic hierarchy V, with ZF theory, as the background ontology. Shapirowrites

[W]e need an account of systems in the background ontology. This[. . . ] has been done already, via standard model theory. An n-ary"relation" is a set of n-tuple and an n-place function is a many-oneset of (n+ 1)-tuples. A system is an ordered pair that consists of adomain and a set of relations and functions on it. [. . . ] In set the-ory, isomorphism and structure equivalence are also easily de�ned,thus completing the requisite eliminative theory of structuralism.(Shapiro in [16], p. 92)

And concerning modal structuralism

With admirable rigor and attention to detail, Hellman [. . . ] developsthe modal option. Modal operators are added to standard formallanguage, and the aforementioned notions of "system" and isomor-phism are invoked. A sentence of arithmetic, say, is rendered as astatement about all possible systems that satisfy the (second-order)Peano axioms. (Shapiro in [16], p. 92)

The ante rem option requires a little more work. The structure being theultimate ground on which mathematics would stand, a theory of structure isneeded. Where for the ontological option, set theory can be used, the ante remstructuralism needs to be axiomatized "from nothing". This formalization canbe done in a similar fashion as the way category theory is formalized, or in asimilar way as set theory. We will here account for the formalization Shapirogives in [16], which follow the latter option.

Recall that the ante rem view considers the places-are-object perspective.So a structure has a collection of places, and a �nite collection of functions andrelations on those places. Two structures are identi�ed if they are isomorphic.

The �rst axiom Shapiro introduces is an axiom of existence

Axiom 1 (In�nity). There exists a structure with an in�nite number of places.

Shapiro then de�nes the notion of a system:

[A] system is de�ned to be a collection of places from one or morestructures, together with some relations and functions on those places.For example, the even-number places of the natural-number struc-ture constitute a system, and on this system, a "successor" functionscould be de�ned that would make the system exemplify the natural-number structure. The "successor" of n would be n + 2. Similarly,the �nite von Neumann ordinals are a system that consists of placesin the set-theoretic hierarchy structure, and this system also exem-pli�es the natural-number structure, once the requisite relations andfunctions are added. (Shapiro in [16], p. 93-94)

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But I have a problem with this de�nition. In the places-are-object perspective,places in a structure can constitute a system of the structure, and I have nothingto say against that. But it seems that Shapiro restricts the notion of systemto systems of places of structures. As we have seen, some (simple) structurescan be exempli�ed by physical objects and the like. I fail to see why Shapirowould restrict the notion of system in such a way. Systems of places constitutea particular kind of systems, but I believe that the notion of system in generalmust not be restricted to them. But for reasons of convenience, future referencesto "systems" will by to systems of this kind.

The next axioms Shapiro poses concern the notion of "substructures".

Axiom 2 (Subtraction). Let S be a system.

� If R is a relation of S, there exists a structure S ′ isomorphic to the systemconsisting of the places, functions, and relations of S, except R.

� If f is a function of S, there exists a structure S ′′ isomorphic to the systemconsisting of places, functions, and relations of S, except f .

Axiom 3 (Subclass). Let S be a system and c a subclass of the places of S.There exists a structure S ′ isomorphic to the system that consists of c, but withno relations and functions.

Axiom 4 (Addition). Let S be a structure.

� If R is a relation on the places of S, there exists a structure S ′ isomor-phic to the system consisting of the places, functions, and relations of Stogether with R.

� If f is a function from the places of S to the places of S, there exists astructure S ′′ isomorphic to the system consisting of the places, functions,and relations of S together with f .

These axioms account for the possibility to add or remove functions andrelation from a structure.

In a similar fashion as the powerset axiom of set theory, we have:

Axiom 5 (Powerstructure). Let S be a structure, and s be the collection of itsplaces. There exists a structure T and a binary relation R such that

∀s′ ⊆ s∃x ∈ T ,∀z (z ∈ s′ ↔ Rxz) .

That is to say, every subset of the places of S is related to a place in T . Sothe collection of the places of T is at least the size of the powerset of the placesof S.

Axiom 6 (Replacement). Let S be a structure and f a function such that forall places x of S, f(x) is a place of a structure Sx. There exists a structure Tand a function g such that for all places z in all Sx, there is a place of T y suchthat g(y) = z.

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That is to say, there exists a structure T that is at least the size of the unionof the places of the structures Sx. In other words, there is a structure T atleast as large as the result of replacing every places x of S with the collectionof all the places of Sx. This axiom is essentially equivalent to the axiom ofreplacement of set theory.

We now need an axiom to account to the fact that any mathematical theorycan be viewed as the study of a particular structure.

Axiom 7 (Coherence). Let Φ be a coherent second order formula. There existsa structure that satis�es Φ

This axiom makes use of the notion of "coherence", which needs to be de-�ned. Consistency is not enough because the theorem of completeness is falsefor second-order logic. There are consistent second-order theories that have nomodel. Coherence here accounts more for something like "satis�ability". Thisnotion is usually formulated in terms of set-theory, but we want to avoid this.

Not having an axiom of foundation, Shapiro poses the equivalent of there�ection principle in set theory as the following axiom:

Axiom 8 (Re�ection). Let Φ be a �rst or second order sentence in the languageof structure theory. If Φ is true, then there is a structure S that satis�es axioms1-7 and Φ.

Now for someone familiar with the axiomatic of set theory, it is clear thatstructure theory is little more then a variant of set theory. But as Shapiro putsit:

[F]or the present purposes, structure theory is a more perspicuousand less arti�cial framework than set theory. If nothing else, struc-ture theory regards set theory (and perhaps even structure theoryitself) as one branch of mathematics among many, whereas the on-tological option makes set theory (or another designated theory) thespecial foundation. (Shapiro in [16], p. 96).

As for the modal option, without going into the details, this program consistbasically in adding "2" and "3" in the usual formulas, to account for "necessity"and "possibility".

Shapiro writes:

In short, on any structuralist program, some background theoryis needed. The present options are set theory, modal model theory,and ante rem structure theory. The fact that any of a number ofbackground theories will do is a reason to adopt the program of anterem structuralism. Ante rem structuralism is more perspicuous inthat the background is, in a sense, minimal. On this option, we neednot assume any more about the background ontology of mathematicsthan is required by structuralism itself. But when all is said anddone, the di�erent accounts are equivalent. (Shapiro in [16], p. 96)

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I agree with Shapiro in the sense that ante rem structuralism is more eco-nomic regarding the background ontology, in regards to the formalization. Butfrom the ontological perspective, I think the modal option is the more economic.I tend to reject the idea of some sort of "mystical" realm of mathematical ob-jects, or structure, so I fail so see in what sense other then being a logicalpossibility, a mathematical concept, like a set or a structure (except of coursethe one that can be physically exempli�ed, but they are a minority) can exists13.But then again "the di�erent accounts are equivalent".

3 Epistemology

A too often neglected way to apprehend philosophy of mathematics is throughpsychology. The problem with psychological matters is that it's a topic aboutwhich everybody thinks he has a great knowledge, through the process of in-trospection. But introspection isn't an appropriate tool for several reason, towhich we will go back later. I believe that this is a mistake made in manyepistemological theories, and especially in the domain of mathematics, since it'sa subject studied in the great part through a purely mental activity. That iswhy I believe it is important to look at the psychological aspect of the matter.Piaget studied this through what is called the genetic method, which consists instudying the genesis of the conceptions used by fully grown up minds, by lookingat how they progressively construct themselves in the minds of children.

This chapter focuses on the discoveries made through the genetic method,and to what extent they are in accordance with the concepts of structuralism.Most of the content in this chapter is based on the second part of ÉpistémologieMathématique et Psychologie [1], written by Jean Piaget.

3.1 Bourbaki's Mother Structures

Under the name of an imaginary mathematician Nicholas Bourbaki, a group ofmathematicians developed a theory about the architecture of mathematics ([2]).Following the idea introduced by the work of Hilbert, they focused their workon the axiomatic, with the aim of studying the relations between the di�erentbranches of mathematics.

In their work, they use the word structure, to describe a form characterizedby a group of axioms. They write

It can now be made clear what is to be understood, in general, bya mathematical structure. The common character of the di�erentconcepts designated by this generic name, is that they can be appliedto sets of elements whose nature has not been speci�ed; to de�ne astructure, one takes as given one or several relations, into which theseelements enter [. . . ] then one postulates that the given relation,

13I sadly lack the time to study deeper this modal option, the ontology of possibilia andthe like, in the context of this paper.

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or relations, satisfy certain conditions (which are explicitly statedand which are the axioms of the structure under consideration).(Bourbaki in [2])

So as we can see, this notion of structure is close to the notion of structure wepreviously presented.

Their idea is that at the base of mathematics can be found fundamentalstructures, which they call "mother structures", in an a priori non-deduciblenumber. According to this view, every mathematical structure can be derivedfrom those mother structures by combining them and/or enriching them withnew axioms.

From a psychological point of view, this perspective presents great interest.As Piaget puts it

One can immediately see the interest of such a endeavor in regardto the psychological problems that rise from the existence of math-ematics, and this in regard to three points of view: (1) that of theuse of the notion of "structure" that raises the question of a possi-ble comparison with the mental structures; (2) that of the notion ofa mathematical �liation of structures that raises the question of apossible comparison with genetic �liations; (3) that of the methodemployed to discover the structures (before axiomatically justifyingthem), method of which the analysis can provide some indicationsor at least suggestions about the type of existence of structures inregard to the relations between the subject and the object.14

According to Bourbaki, there are three type of mother structures, from whichevery mathematical structure known to this day can be derived15 :

� the algebraic structures, such as groups, rings and �elds,

� ordering structures, like partial order, linear order or well-order,

� topological structures, de�ning the notions of limits, continuity and neigh-borhood.

From these three structures (or type of structures) can be derived every otherstructures by di�erentiation, which consist in adding axioms to the structure,and combinations, which consist in using, in an ad hoc fashion, principles ofother mother structures, like it is the case for example for algebraic topology

14On voit d'emblée l'intérêt d'une telle tentative quant aux problèmes psychologiques quesoulève l'exitence des mathématiques, et cela à trois points de vue: (1) celui du recours à lanotion de �structure� qui soulève la question d'une comparaison possible avec les structuresmentales; (2) celui de la notion d'une �liation mathématique des structures, qui soulève laquestion d'une comparaison possible avec les �liations génétiques; (3) celui de la méthodeemployée pour découvrir les structures (avant de les justi�er axiomatiquement), méthodedont l'analyse peut fournir quelques indications ou tout au moins suggestions sur le typed'existence des structures eu égard aux relations entre le sujet et l'objet (Piaget in [1], p. 177)

15But again, they admit that a new mother structure could be necessary in the future, toaccount for potential new branches of mathematics

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(use of algebraic tools in the study of topological spaces), or topological algebra(use of topological notions in the study of algebraic spaces).

Prior to his knowledge of the conceptions of Bourbaki, Piaget tried to classifythe di�erent operatory structure empirically observed in the development of thechild's intelligence. He found three starting structures type:

the structures of which the form of reversibility is the inversion, orthe cancellation (A−A = 0) and that one can describe by referringto algebraic or group model; the structures of which the form ofreversibility is the reciprocity and that one must describe in termsof relation and order; and the structures based on continuum, inparticular the spacial structures of which it is remarkable that theelementary form are of a topological character before metric andprojective constructions!16

In 1952, a symposium on the mental and mathematical structures took place.It was opened by two talks, one about the Bourbaki mathematical structures,and the other about Piaget's mental structures, and the convergence betweenthe two talk, totally independent from one another, was stunning.

However, where the Bourbaki's mother structures are found with the aimof having the most possible generality, the genetic method aims at �nding thestructures that are the most elementary in their development. Piaget writes

Observation and experimentation have [. . . ] shown us that, if wecall "operations" the actions interiorized, reversible (in the sense: be-ing executable in both ways) and coordinated in structure of wholes,and if we call "concrete" the operations that take place in the ma-nipulation of objects or in their representation accompanied by lan-guage but not only about propositions of verbal statements (theoperations about those, independently of any manipulation, beingcalled "hypothetico-deductive"), all structures at the level of con-crete operation reduce themselves to a single model, that can benamed "grouping".17

This system of "grouping" has no real interest from a logical perspective, dueto the numerous limitation that gives it no generality. But since it seems to be

16les structures dont la forme de réversibilité est l'inversion ou annulation (A − A = 0)et que l'on peut décrire en se référant à des models algébriques ou de groupe; les structuresdont la forme de réversibilité est la réciprocité et que l'on doit décrire en termes de relationsd'ordre; et les structures à base de continu, en particulier les structures spatiales dont il estremarquable que les formes élémentaires sont de caractère topologique avant les constructionsmétriques et projectives! (Piaget in [1], p. 182)

17L'observation et l'expérience nous ont [. . . ] montré que, si l'on appelle �opération� lesactions intériorisées, réversibles (au sens de: pouvant être exécutées dans les deux sens) etcoordonnées en structures d'ensemble, et si l'on appelle �concrète� les opérations intervenantdans la manipulation des objects ou dans leur représentation accompagnée de language maisne portant pas sur de seules propositions ou énoncés verbaux (les opérations portant sur ceux-ci, indépendament de toute manipulation, étant appelées �hypotético-déductives�), toutes lesstructures du niveau des opérations concrètes se réduisent à un seul modèle, que l'on peutdésigner sous le nom de �groupement�.(Piaget in [1], p. 185)

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a starting point for other structures, it presents an interest in the psychologicalperspective.

3.1.1 Formalization of the "Grouping"

With that aim, it was formalized by J.B. Grize in [4], and we will describe thisformalization here. Let 〈M,⊂,+,−〉 be a system, withM a non-empty set18,⊂ a relation, + and − binary operations, and let x, y, z ∈M. Let us then de�netwo additional symbols:

x ≈ y ⇔ (x ⊂ y) ∧ (y ⊂ x) (3)

andx ⊂1 y ⇔ (x ⊂ y)∧

(¬x ≈ y)∧(∀z(x ⊂ z ∧ y ⊂ z)⇒ (x ≈ z ∨ y ≈ z)).

(4)

The relation ⊂ can be understood as "is in", and is a partial order relation, ≈ isan equivalence relation that can be understood as something close to equality,and ⊂1 would be "is directly in"19.

De�nition 3.1.1. The system 〈M,⊂,+,−〉 is a grouping if it satis�es thefollowing rules

� (Re�exivity) x ⊂ x,

� (Transitivity) x ⊂ y ∧ y ⊂ z ⇔ x ⊂ z,

� (G0)

1. y ∈M∧ x ⊂ y ⇒ x ∈M,

2. x ∈M∧ x ⊂1 y ⇒ (y − x) ∈M,

3. x ∈M∧ x ⊂1 y ⇒ (x+ (y − x)) ∈M,

� (G1) x+ (y + z) ≈ (x+ y) + z,

� (G2) x+ y ≈ y + x,

� (G3) x ⊂ y ⇒ x+ z ⊂ y + z,

� (G4) x ⊂ y ⇔ x+ y ≈ y,

� (G5) y ⊂ x+ z ⇒ y − x ⊂ z,

� (G6) y ⊂ x+ (y − x),

18Things aren't really clear in [1], and I lacked the time to go deeper into this matter. Butit seems the general domain are the concrete operations, and M is a subset, or subclass, ofthose.

19In [1], the symbol→ is used instead of ⊂, and↔ instead of ≈, but I think it is more clearwhat is meant with the use of these symbols.

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� (G7) x ⊂1 y ⇒ x ⊂ y − (y − x),

� (G8) ∃o ∈M, o ⊂ x.

Due to the numerous limitations, this structure cannot be reduced to astructure of group or to a structure of ordering (and of course, neither to atopological structure). The access the child has to the elements ofM is limitedby (G0), and this is a feature really di�erent from what we are used to inmathematics. For example, as Piaget writes:

[I]n a group, any two elements x and y of the system generate by theircomposition x Op y (where Op is the direct operation of the group'sinverse) a third element z of the system without going through theintermediary elements between x and y, and this with complete mo-bility. In a grouping such as A+A′ = B;B +B′ = C; etc., one canon the contrary only compose in a contiguous way, little by little (forexample A + C = D − B′ − A′), the mobility of the system beingrestrained.20

As one more concrete example of this fact, until some point in his development,given three sticks A, B and C, if the child, by comparing them, sees that A < Band B < C, he will not be able to conclude that A < C.

This is all very technical, and hard to understand, especially for someone notused to this kind of theoretical mix between logic and psychology. But goingdeeper into the understanding of this grouping structure would go out of thesubject of this paper. For a full development of this formalization, see [4]. Theimportant thing to us here is to know that there is a mental structure, thatcan be formalized in logico-mathematical terms, and that can be observed inthe child's behavior. By giving a somehow detailed formalization, my aim is toshow that the �rst observable mental structure isn't, at this point, close to afully developed mind's logic, and to the usual mathematical structures.

Piaget categorizes 8 di�erent kind of occurrences of this structure, dependingon whether they apply to classes or to relations, and on the type of operation,namely asymmetric addition, symmetric addition, and two sorts of multipli-cation that Piaget calls "co-univoques" and "bi-univoque"21. For example, thestructure applying to classes, with asymmetric addition, would be used in simpleembedding, like "cats"⊂"mammals"⊂"living beings". Piaget adds:

Note [. . . ] that if the natural number system, obtained under anapproximative form during the same period of concrete operation,seems far away from this elementary structure of grouping, it is on

20[E]n un groupe deux éléments quelconques x et y du système engendrent par leur compo-sition x Op y (où Op est l'opération directe ou inverse du groupe) un troisième élément z dusystème sans passer par les intermédiaires entre x et y, et cela avec une mobilité complète. Enun groupement tel que A+A′ = B; B+B′ = C; etc., on ne peut faire au contraire les compo-sitions que de façon contiguë, donc de proche en proche (par exemple A+C = D−B′ −A′),la mobilité du système étant ainsi restreinte. (Piaget in [1], p. 187)

21See [10] for details, or [17] for short de�nitions of those operations and the cases in whichthey apply.

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the contrary possible to show that: (1) genetically the constructionof the natural numbers is done by progressive "synthesis" of [the �rstkind of grouping]; (2) axiomatically, [. . . ] it is possible to assign acoherent status to this "synthesis" [of those grouping] by gatheringtheir operations, which ipso facto brings to remove the compositionlimitations and allows to deduce the �ve Peano's axioms, includingrecurrence.22

An important distinction between the di�erent kinds of grouping, that isof importance to us in our aim to make a correspondence between elementarystructures and the Bourbaki's mother structures is the following. The di�erentkinds of grouping structures can be separated into two categories, depending onthe form of their "reversibility". Let us call an inversion the correspondencebetween an operation T and its inverse T−1, in the sense that the compositionof T and its inverse correspond to the null operation, for example T = +a andT−1 = −a.

The second kind of reversibility is reciprocity. In the case of the relations,reciprocity consists in inverting either the items, or the relation itself, or both.So the 3 forms of reciprocity are something like

R(a < b) = b < a,

R′(a < b) = a > b,

R′′(a < b) = b > a.

The reciprocity in more speci�c cases is fairly technical. See [1], p. 191-193, forthe detailed explanation. But what is of interest to us is to know that there areelementary mental structures that somehow can relate to the Bourbaki's motherstructures.

3.1.2 The Topological Structure

With the two kinds of structures delimited by the form of reversibility, a thirdelementary structure is of interest to us. This concerns the operations relativeto space and time.

Piaget writes

In considering �rst only the school's teaching of geometry, onemight think that, possessing a certain logic and a certain arithmetic,the child applies them without more consideration to perceptive �g-ures, geometry being thus just an applied mathematics in the sense

22Notons [. . . ] que si le système des nombres naturels, acquis sous forme approximative aucours de cette même période des opérations concrètes, paraît fort éloigné de cette structureélémentaire de groupement, il est au contraire possible de montrer que: (1) génétiquement laconstruction des nombres naturels s'e�ectue par �synthèse� progressive [des premiers types degroupements]; (2) axiomatiquement, [. . . ] il est possible d'assigner un statut cohérent à cette�synthèse� [de ces groupements] en réunissant leurs opérations, ce qui conduit ipso facto àlever les limitations de composition et permet de déduire les cinq axiomes de Peano y comprisla récurrence. (Piaget in [1], p. 189)

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that it has been made over a long time in the �eld of science itself.But the child possesses or develops a geometry in actions way beforebeing given any teaching and the �rst question that need discussionis to know whether this spontaneous construction shows an ordercloser to the historical order of acquisitions (Euclidean geometryat �rst, then projective and �nally topology) or to the theoreticalconstruction order[. . . ].23

Observation and experimentation have shown several things, going in the di-rection of the primacy of topology in the mind of the child. Before the notionsof invariant regarding movements (e.g. moving length) and those of the projec-tive transformations, invariant concerning neighborhoods, notions of open andclosed, interior and exterior, continuity and discontinuity, etc, are observed. In-variants of the Euclidean geometry are only constructed by the age of 7-8 years,and �nally, natural systems of coordinates only appears at the age of 9-10 years.

There is thus reason to consider, with the two sorts of mental structuresde�ned by the notions of reversibility, that there is a third kind of mental struc-tures, characterized by essentially topological conceptions.

3.1.3 The Link Between the Firsts Mental Structures and Bour-

baki's Mother Structures

Supposing that there are no other mother structure structures (every observedstructures corresponding to one of those), Piaget proposes that the structureswith the inversion form of reversibility correspond to the algebraic structures(the neutral element being constructed by composition of the operation andits inverse), the ones with the reciprocity form of inversion corresponds to theordering structures, and of course, the topological considerations correspondsto the topological structures.

Let's have a deeper look at how exactly those two kind of structures, math-ematical and mental, are related. Let's call "M -structures" the mathematicalstructures, and "G-structures" the structures studied from a genetic perspec-tive. There are some important di�erences between them. First of all, theM -structures are constructed through a process of re�ection from the mathe-matician developing the theory about it, where for the G-structure, the subjectisn't conscious of them, and they manifest themselves only through the behaviorand reasoning from the subject. Secondly, as Piaget puts it

23A ne considérer d'abord que l'enseignement scolaire de la géométrie, on pourrait penserque, en possession d'une certains logique et d'une certaine arithmétique, l'enfant les appliquesans plus à des �gures perceptives, la géométrie ne constituant ainsi qu'une mathématiqueappliquée au sens où elle a été longtemps conçue sur le terrain de la science elle-même. Maisl'enfant possède ou élabore une géométrie dans l'action bien avant d'être soumis à un enseigne-ment et la première question à discuter est de savoir si cette construction spontanée présenteun ordre plus proche de l'ordre historique des acquisitions (géométrie euclidienne au départ,puis projective et en�n topologique) ou de l'ordre de construction théorique[. . . ].(Piaget in[1], p. 198).

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The conditions that the relations inherent to the M -structure sat-isfy are the axioms of those structures, where in a G-structure, theconditions remain immanent to its functioning and the subject getsno axiomatization out of it.24

And lastly, those conditions are the starting points from which the rest of anM -structure is deduced, where for a G-structure, they form the rules that theprinciples of deduction obey.

Nevertheless, there are also some important similarities between these twokinds of structures. First of all, both kinds apply to di�erent variety of ob-jects, of very di�erent natures (even if the G-structure present a lower degreeof generality). In the terminology of the last chapter, both kinds of structurescan be exempli�ed by di�erent systems. Secondly the relations that constitutethe M -structure corresponds to the operations of the G-structure. And lastly,the characterization of the G-structures in term of the type of reversibility cor-responds to the conditions, or axioms, of the M -structures, as we mentionedbefore.

Those di�erences and similarities leads us to believe that there is some kindof "genetic �liation" between the M -structures and the G-structures. WhatPiaget proposes is the following:

Under the hypothesis that the three G-structures alone cover thetotality of the natural structures, and under the hypothesis that themathematician [. . . ] only constructs the mathematical beings by us-ing the "natural" thinking, simply re�ned by an uninterrupted serieof progressive abstraction, and of abstractions proceeding not fromthe empirical objects (perceptions, etc.) but from the actions andoperations he exerts on those objects [. . . ], it is necessary to admitthat this construction of mathematical beings will be conditioned bythe characters of the three elementary G-structures.25

If afterward, one studies the common features of the mathematical structures,in a similar way as Bourbaki, the mathematician will �nd certain general "re-lations" and their "conditions", which will be at odds with the previously men-tioned similarities between G- and M -structures. But by nature, this kind oftheoretic studies, brings structures with a maximum degree of generality fromthe start. We will come back more deeply to this concept, but this presents animportant feature of this theory, which Piaget calls re�ecting abstraction26. This

24Les conditions auxquelles satisfont les relations propres aux structures M sont les axiomesde ces structures, tandis qu'en une structure G les conditions demeurent immanentes à sonfonctionnement et le sujet n'en tire aucune axiomatique. (Piaget in [1], 201)

25Dans l'hypothèse où les trois seules structures élémentaires G couvrent l'ensemble desstructures naturelles, et dans l'hypothèse où le mathématicien [. . . ] ne construit les êtremathématiques qu'en utilisant la pensée �naturelle�, simplement a�née par une série initer-rompue d'abstractions progressives, et d'abstractions procédant non pas à partir des objetsempiriques (perceptions, etc.) mais à partir des actions et des opérations qu'il exerce sur cesobjets [. . . ], il est nécessaire d'admettre que cette construction des êtres mathématiques seraconditionnée par les caractères des trois structures élémentaires G (Piaget in [1], p. 202)

26"abstraction ré�echissante"

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process of abstraction is distinguished with the abstraction simply regarding theobjects (generalizing principles that apply to di�erent objects in a similar way),in the sense that it is constructive. Piaget writes

[The] re�ecting abstraction consists in getting from a system of ac-tions or operations at an inferior level certain characters for whichit assures the re�ection (in a quasi physical sense of the word) onactions or operations of a superior level. [. . . ] In a word the re-�ecting abstraction proceeds by reconstructions that go beyond, byintegrating them, the anterior constructions.27

And he conclude by writing

It results from this that the construction of mathematical beingsconstitute an enlargement of the elements of the natural thinkingand the construction of M -structures an enlargement of particularmathematical beings.28

Note how this echoes with the structuralist conception of mathematics.

3.2 The Discovery of a Mathematical Structure

Let us now get more in detail into the question of the discovery (or invention,but we'll go back to this distinction later) of a mathematical structures. Twoquestions can be raised about this matter. The �rst being the question of themental process creating a new idea, and the second one being the question ofthe ontology of these newly created idea.

In 1908 is published Enquête de �l'Enseignement Mathématique� sur laméthode de travail des mathématiciens, by H. Fehr, with the cooperation ofTh. Flournoy and Ed. Claparède, which was some kind of methodologicalstudy of the work of mathematicians. This paper is said to have motivated H.Poincaré to hold a conference about L'invention mathématique, and the ideaswere then developed by J. Hadamard.

According to them, the mathematical work is separated in di�erent phases,that can be illustrated by Beth's description of a common experience one canhave when studying a su�ciently di�cult mathematical problem:

After a series of failed attempts one gets tired and is �nally forcedto give up in his research. Then, after a certain resting period thesolution presents itself suddenly, without any conscious e�ort and

27[L]'abstraction ré�échissante consiste à tirer d'un système d'action ou d'opérations deniveau inférieur certains caractères dont elle assure la ré�exion (au sens quasi physique duterme) sur des actions ou opérations de niveau supérieur. [. . . ] En un mot l'abstractionré�échissante procède par reconstructions qui dépassent, en les intégrant, les constructionsantérieures. (Piaget in [1], p. 203)

28Il en résulte que la construction des êtres mathématiques constitue un élargissement deséléments de la pensée naturelle et la construction des structures M un élargissement des êtresmathématiques particuliers. (Piaget in [1], p. 203)

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with a surprising clarity and certainty. The second phase of theconscious work consists then in verifying what has just been found.29

Hadamard describes this process in four phases: preparation, incubation,illumination, and veri�cation. Poincaré suggests that the incubation phase,phase of rest in appearance only, is in fact a phase of unconscious work. Hedoes not suggest that there is some sort of subliminal self, equal (or superior)to the conscious self, but that there is some sort of automatic process, in whichthe mind tries numerous possibilities, with only some of them going to theconscious mind. Beth suggests some sort of Freudian repression of unacceptablepossibilities.

So according to this view, the �rst phase consists in setting the importantnotions in places, and the unconscious part of the mind would do the rest. Bethinsists however on the fact that the �rst phase must be done in a su�cientlye�cient way for the unconscious process to be fruitful. This pushes Beth toreject the idea of a purely automatic process, since it is somehow directed bythe preliminary work. He also argues that some sort of random process of com-bination of idea, with only the worthy ones making their way to the consciousmind, would require incredible calculation capacities for the brain that seemsunimaginable.

Piaget agrees with Beth on this. However, as he puts it:

[N]othing is more relative in the domain of the thoughts, evenat the highest level [. . . ], then the distinction between consciousnessand the unconscious. The unconscious is only the expression of thehelplessness of our introspection. There are not two distinct domainsof the mind separated by a border, but one and only work of themind, thus, even in the most lucid states, we only perceive a smallpart (centered on the results obtained and not on the process itself),and that eludes us almost totally when we don't control it closely.30

The topologist Leray even suggests that the original idea that seems to riseout of nowhere at the phase of illumination appears this way only because weforgot having the idea prior to this. According to him, most of the work isdone in the preparation part. This part consists in considering many di�erentpossibilities, to which one would attribute di�erent levels of importance andof probabilities to succeed. The right idea can appear during this preliminary

29Après une série de tentatives infructueuses on se fatigue et en�n on est forcé d'abandonnerla rechercher. Ensuite, après une certaine période de repos, la solution se présente tout a coup,sans aucun e�ort conscient et avec une clarté et une certitude surprenantes. Alors la deuxièmephase de travail conscient ne consiste qu'à véri�er et a formuler ce qu'on vient de trouver.(Beth in [1], p. 97)

30[R]ien n'est plus relatif dans le domaine de la pensée, même au niveau le plus élevé [. . . ],que la distinction entre la conscience et l'inconscient. L'inconscient n'est que l'expression del'impuissance de notre introspection. Il n'existe pas deux domaines mentaux séparés par unefrontière, mais bien un seul et même travail de l'esprit, donc, même dans les états les pluslucides, nous n'apercevons qu'une très faible partie (centrée sur les résutats obtenus et nonpas sur le processus comme tel), et qui nous échappe à peu près totalement lorsque nous nele contrôlons plus de près. (Piaget in [1], p. 214)

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phase, but be given no credit. This phase goes like this until the mind is moreand more saturated. The second phase, that Poincaré supposed to be an un-conscious automatism creating ideas, can start. But according to Leray, theunconscious process consists instead of getting rid of every ideas and develop-ments that are not useful. The conscious research can then continue, but with amind less saturated by di�erent directions of research. The right direction, ne-glected before, gets then more credibility, and the solution can be found quickly.But it only seems new because we forgot having considered it before. Note thatthis conception however still suppose a purely unconscious part of the mind,which could get rid of the bad ideas.

Let us now go back to the matter of the ontology of the mathematical struc-tures, but this time apprehending it from a psychological perspective, ratherthen from a purely philosophical one. According to Piaget (and we tend toagree with him), the matter of knowing if mathematics is discovered (a pri-ori mathematics), or invented (a posteriori mathematics), cannot be properlytreated through introspection. Like we said earlier, introspection isn't an ob-jective tool. As Piaget puts it:

[C]oncerning the intellectual aspect of the behavior, introspec-tion presents the [following] �aws: (1) it is incomplete, because themechanism of the research in itself eludes the consciousness, in op-position to its direction (question), to its results, partial (emergenceof hypothesis) or total, and to the retroactive veri�cation (demon-stration); (2) it is tendentious, because it is impossible to introspectones own thoughts without taking side more or less unconsciously infavor of the beliefs to which we are attached. Those belief are thenas persistent as they are fundamental (Platonism, idealism, nomi-nalism, etc.)[. . . ].31

He also adds that to try to solve this problem psychologically, a great numberof data, that we don't have at this day, would be required.

Piaget proposes a solution in between the a priori and the a posterioripoints of view. He poses the hypothesis that the re�ecting abstraction, whichwe mentioned before, consists in the following. Having a structure at a certainlevel, the re�ecting abstraction reconstructs the structure at a superior level,integrating it in a larger structure.

First we have to make something clear. The �rst instances of structurespresent a low degree of generality, and are somehow linked to the objects thatare concerned. But even if the term abstraction can mean to generalize the

31[E]n ce qui concerne l'aspect intellectuel de la conduite, l'introspection présente les [. . . ]défauts [suivants]: (1) elle est lacunaire, car le mécanisme comme tel de la recherche échappeà la conscience, par opposition à sa direction (question), à ses résultats partiels (émergencedes hypothèses) ou totaux et à la véri�cation rétroactive (démonstration); (2) elle est ten-dancieuse, car il est impossible d'introspecter sa propre pensée sans prendre parti plus oumoins inconsciemment en faveur des croyances auxquelles on est attaché. Ces croyances sontalors d'autant plus tenaces qu'elles sont fondamentales (platonisme, idéalisme, nominalisme,etc.)[. . . ].(Piaget in [1], p. 216-217)

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structure to objects of di�erent natures, the re�ecting abstraction concerns theactions. This is at odd with our structuralism conception, in the sense that theobjects are not the heart of the matter, but the relations (in a large sense ofthe word) between them are.

He adds something that I think is very interesting to the matter at hand:

In that case, there is a possibility of an endless regression, regardingmental life, and the starting point of the structures might have tobe looked for in the nervous structures, at �rst, and �nally, in theorganic structures in general.32

This perspective would give a whole new sense to the notion of natural mathe-matics. Piaget adds that numerous structures can be found in living organisms,that can be looked as mathematizable structures. Some remarkable geometricaltransformations can be for example observed in the evolutions of the forms of�shes, mollusks, and the like.

From this perspective, the epistemological starting point of the knowledgeof mathematical structures is to be found in organic structures, and Piagetproposes that even the sensory-motor structures (where some form of groupstructure can be found), could come from those organic structures. And all thedevelopment from those natural structures, to the elementary structures thatgovern the actions and the re�ection, to the mathematical structures of whichwe are consciously aware, is made through the re�ecting abstraction. This couldsuggests some form of holistic presence of mathematics in everything concerningthe mind. But maybe this would be going to far, and we will avoid going deeperinto this matter.

3.2.1 Toward Pure Mathematics

Let us now turn more speci�cally to the question of how pure mathematics, thatis to say, mathematics that is totally detached from any physical experiment,and studied only through an hypothetico-deductive method, comes to birth inthe mind, and if there is a clear di�erence between applied mathematics andpure mathematics.

As a starting point to this question, it seems appropriate to look at themore elementary cases of logico-mathematical experience. Initially, mathemati-cal truth are accepted only through an experimental observations. For example,at an early stage, the child will not accept without testing it that 2 + 3 = 3 + 2or that a collection of 5 objects, counted from left to right, or from right to left,would give the same result. Another example: if a collection of n elements canbe divided in two collection of equal number of elements, the child will not de-duce, until a certain stage of his development, that the collection of n+1 elementcould not be divided the same way. But later in his development (around 7-8

32En ce cas, il y a possibilité d'une régression sans �n, en ce qui concerne la vie mentale, etle point de départ des structures sera donc peut-être à chercher dans les structures nerveuses,tout d'abord, et �nalement dans les structures organiques en général [. . . ]. (Piaget in [1], p.217)

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years, according to Piaget), those cases would be treated only through a reason-ing, with no need for experimentation. Some other examples can be found, withdi�erent degree of complexity, in which the child will have to verify empirically(even if he might have some capacity to deduce them beforehand).

But even though empirical experiment have, until some point, an central rolein the logico-mathematical experience, it is important to insist that there are,and this from the start, crucial di�erences from the physical experience. Thisdi�erence is important to understand how the experimentation is so quicklyreplaced by proper deduction, and how it implies from the start, the possibilityof pure mathematics.

The central di�erence is that the physical experience applies to the objects,where the logico-mathematical experience applies to the actions of the subjecton the objects. The subject then gains mathematical knowledge by abstractionon these actions. For example, if the child discovers that a large rock is heavierthen a small one, the experience is physical, because the weight is a propertyof rocks, independently of anything the child does to them. But when countingrocks in a line, from left to right, and from right to left, and discovering thatthe result is the same, the experience is logico-mathematical, because it is notabout the rocks, but about the actions of ordering and counting objects.

An important feature of this kind of logico-mathematical experience is thatthe action of the subject, is part of a mental structure. The subject has no directknowledge of those mental structures, and the place of his actions in those.But the action cannot be conceptualized outside of such a mental structure.By logico-mathematical experiment, and processes of re�ecting abstraction, abetter suited mental structure is constructed, from the previous one. By thisprocess, the action becomes an operation in a mental structure. It can thus beconceptually composed with other operations, in a process of deduction, withouthaving to physically instantiate it in an action.

It is noteworthy to remark that, even though concrete experimentation is atan early stage inherent to the logico-mathematical development of the mind, itdoes not imply that mathematical knowledge is empirical. However, physicalexperimentation can be a motivation to rethink the conceptual schemes (ormental structures), and further develop the mathematical knowledge throughre�ecting abstraction. This is a odd with the fact that physical experimentations,can motivate development even in pure mathematics, like it was for example thecase with the development of the calculus.

The transition from empirical veri�cation to pure deduction is however aprogressive one, and isn't an abrupt one. There are a certain number of inter-mediary stages, that in the end can lead to a deduction from pure hypothesis.We will not go into the details of those intermediary stages, but let us just saythat their development is in accordance with the idea of the re�ecting abstractiongoverning their evolution.

Piaget summarize the general mental process of constructing a structure (inthe general sense) from a previous one:

Expressed in its schematic form, the process characterizing such

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a passage reduces itself to the following: (a) to construct a moreabstract and more general structure from a more concrete and par-ticular structure, it is �rst necessary to abstract certain operatorylinks of the previous structure in order to generalize them in the nextstructure; (b) but as much this abstraction then this generalizationsuppose that the links thus abstracted be "re�ected" (in the propersense) on the new though plane, in order to constitute a generalizedreplica; (c) this "re�ection" consists in new operations being aboutthe previous operations while at the same time extending them. It isthen those new operations, necessary to abstract the previous links,that will constitute the novelty of the derived system, where the ab-straction from the previous operation insure the continuity betweenthe two systems. Finally (d) those new operations allow to unite ina new totality some systems until then separated.33

This is the case at the elementary stages, and there is no reason to thinkthat such a process could end.

So let us summarize the whole construction of mathematics. By starting atthe sensory-motor stage34 some mental structure govern the behavior, and theactions in particular. By a process of re�ecting abstraction, motivated by physi-cal experimentation, the actions are progressively converted into operations, andthe mental structure progressively developed toward a detachment from physicalinstantiation. This allows the use of pure deduction to gain knowledge. Furtherdevelopment and detachment to the objects �nally leads to pure mathematics.The process of re�ecting abstraction giving more and more general structures,structures with di�erent genetic starting point can �nally merge into commonstructures.

3.2.2 Back to Ontology

With this at hand, we can go back to the question of the ontological status ofstructures, of which we talk in the previous chapter. We'll start to go deeperinto the question of whether mathematics are invented or discovered. Piagetwrites:

33Exprimé sous sa forme schématique, le processus caractérisant un tel passage se réduit[. . . ] à ceci: (a) pour construire une structure plus abstraite et plus générale à partir d'unestructure plus concrète et plus particulière, il est d'abord nécessaire d'abstraire certainestliaisons opératoires de la structure antérieure de manière à les généraliser dans la structureultérieure; (b) mais tant cette abstraction que cette généralisation supposent que les liaisonsainsi abstraites soient �réféchies� (au sens propre) sur un nouveau plan de pensée, de manièreà en constituer la réplique généralisée; (c) or cette �ré�exion� consiste en nouvelles opérationsportant sur les opérations antérieures tout en les prolongeant. Ce sont alors ces nouvellesopérations, nécessaires pour abstraire les liaisons antérieures, qui constituent la nouveauté dusystème dérivé, tandis que l'abstraction à partir des opérations antérieures assure la continuitéentre les deux systèmes, En�n (d) ces nouvelles opérations premettent de réunir en nouvellestotalités des systèmes jusque là séparés. (Piaget in [1], p. 259)

34with a possible regression to organic structures, and maybe even to chemical structuresand the like

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An invention is the creation of a new and free combination, untilthen realized neither in nature nor in the mind of the subjects, even ifthe newly combined elements where known previously: for example,the creation of the steam engine is an invention, in the sense of acombination between steam and an vehicle, both known before it.[. . . ]

A discovery is the meeting between a subject and an object, untilthen unknown by him but that existed before such a meeting: forexample the discovery of America.35

Piaget's point is that re�ecting abstraction can neither be reduce to a purediscovery, nor to a pure invention. It is not a discovery because the relations ofthe newly constructed structure are not of the same nature then the previousones, since it's on a di�erent level. When the child discovers, empirically, theresults of an action, for example that the results of a sum is independent fromthe order in which it is made, the re�ecting abstraction consists in translatinga system of material actions in a system of interiorized operations. The newlyconstructed operation is based on something previously existing, but is enrichedby the discovery. So this could not be reduced to a pure discovery. The sameapplies to the construction of new structures, if the process used is also aninstance of the re�ecting abstraction.

As for the perspective of an invention of mathematics, Piaget writes:

[A] construction that proceeds through "re�ecting abstraction" [. . . ]cannot be reduced to an "invention" either, in the sense de�nedabove of a free and new combination, because the new elements thattakes then place on top of those discovered are never "free", in thesense that they could have been di�erent. [. . . ] The nature of amathematical construction, on the contrary [. . . ], is that its degreeof liberty is dependent to the mode of demonstration and of formal-ization, as for the fundamental theorems, they impose themselves bynecessity.36

He summarize by writing:

35Une invention est la création d'une combinaison nouvelle et libre, non réalisée jusque-là nidans la nature ni dans l'esprit des sujets, même si les éléments combinés de façon nouvelle sontconnus antérieurement: par exemple, la création de la locomotive à vapeur est une invention,au sens d'une combinaison entre la vapeur et le véhicule, tous deux connus avant elle. [. . . ]Une découverte est la rencontre entre un sujet et un objet, jusque-là inconnu de lui mais

qui existait tel quel avant cette rencontre : par exemple la découverte de l'Amérique. (Piagetin [1], p. 218-219)

36[U]ne construction procédant par �abstraction ré�échissante� [. . . ] ne se réduit pas nonplus à une �invention� au sens dé�ni plus haut d'une combinaison nouvelle et libre, car leséléments nouveaux qui interviennent alors en plus de ceux qui sont découverts ne sont jamais�libres�, au sens où ils auraient pû être di�érents. [. . . ] Le propre d'une construction mathé-matique, au contraire [. . . ], est que son degré de liberté ne tient qu'au mode de démonstrationet de formalisation, tandis que les théorèmes fondamentaux s'imposent avec nécéssité. (Piagetin [1], p. 221-222)

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From a psychological point of view, the nature of the mathematicalconstruction and creation, seems to reduce itself neither to discov-eries, nor to inventions, but to a unde�ned sequence of combinationthat are at the same time new, and however interior to a system ofpossibility well determined.37

So given a structure, or group of structures, there are numerous possiblestructures that one can construct. And the more developed the mathematicalknowledge, the more there are possible structures. So the development of a newmathematical structure consists in picking one of the possible structures. Con-cerning the nature of the possible structures, Piaget does not develop too deeplythe question. He however remarks that knowing anything about possible struc-tures, before they are constructed, is impossible. Since the structures changetheir ontological status when we have access to them, i.e. when they stop beingpossibilities, and start being actual mathematical structures, they change theirnature. So the only structures we have access to are the ones that are alreadyconstructed. There is, according to Piaget, no way to objectively know anythingabout the possible structures that have not been constructed yet. I personallydon't want to go this far, and think that the modal logic can be a way to studythe possible structures. However, this perspective poses a problem of epistemicnature, as to how we are to know this modal logic.

This gives some perspective regarding the di�erent ontological perspectivesthat where treated in the previous chapter. I would like to present my pointof view concerning the ontological matter, that would somehow be in betweenthe ante rem structuralism and the eliminative modal structuralism. Recallthat the ante rem structuralism supposes the existence of all the mathematicalstructures, in some sort of Platonic sense, prior to anything the mathematiciandoes. In this perspective, places in a structure are bona �de objects. As forthe eliminative modal structuralism, the idea is that the structures as them-selves are only generalization over the places-are-o�ces perspective, but withconsideration of all the logically possible systems.

I believe that, if the hypothesis of the re�ecting abstraction is accurate,since it applies to the relations, and not to the objects themselves, progressivelygetting rid of any physical exempli�cation of them, the places-are-object per-spective presents itself naturally. But instead of taking a Platonist point of viewconcerning the structures, I think the use of modality is more perspicuous. Myposition is that the ante rem structures exists prior to the mathematician only inthe form of logical possibilities. But once constructed by the mind, they becomeactual, in a sense changing their ontological status. Notice that this change inthe ontological status of the structure is a relative one, since it depends on themind of every individual. But since the notion of mathematical structures is

37D'un point de vue psychologique, le propre de la contruction et de la création mathéma-tiques, semble donc être de ne se réduire ni à des découvertes ni à des inventions, mais à unesuite indé�nie de combinaisons à la fois nouvelles et cependant intérieures à un système depossibilités bien déterminées.(Piaget in [1], p. 222)

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directly linked to the mind38, I think it is natural to have such a relativity.Notice that if we suppose the existence of an ideal constructor, some sort

of superior mind, capable of constructing every possible structures instantly,we have the ante rem structuralism. And if we consider that a structure fromthe places-are-object perspective can be view as system exemplifying the verystructure in question, then the status of the structure, not yet constructed, asbeing logical possibilities, gives them the status of a possible system exemplifyingthe structure, required by the eliminative modal structuralism. That is why Isay that this perspective is somehow in between the ante rem structuralism andthe eliminative modal structuralism.

4 Conclusion

We have started to look at the di�erent possible points of view concerning thephilosophy of mathematics. First, we discussed the two points of view concern-ing the role of philosophy regarding mathematics: the philosophy-�rst principle,according to which a clear de�nition of what mathematics is, and how it shouldbe practiced, should come �rst, in order to do mathematics right, and the otherpoint of view, philosophy-last-if-at-all, according to which mathematics can sus-tain itself without philosophy, and that the role of philosophy is to give aninterpretation of mathematics as practiced. We then discussed the di�erentpoints of view concerning realism and anti-realism in ontology and truth-value.We ended up looking more critically at the internal/external dichotomy con-cerning the philosophical questions, and to what extent question external to aparticular framework have any legitimacy.

We then focused on the structuralism point of view, �rst giving a generalidea of the linked notions, and then taking a more rigorous approach. We haveseen that there are two perspectives (without a sharp border between them)concerning the places in a structure: the places-are-objects, according to whichplaces in a structure can be studied as bona �de mathematical objects, with-out having to take any particular system exemplifying the structure, and theplaces-are-o�ces perspective, according to which the places in a structure areplaces to be �lled with objects of particular systems, exemplifying the structure.We then had studied three di�erent possible ontological status of the structuresthemselves. The ante rem structuralism, taking the places-are-objects point ofview at face value, asserts that all structures exists, prior to the mathematician.The two other views take the eliminative structuralism perspective, accordingto which places in structures have no reality in themselves, but are only conve-nient way to generalize talks about every systems exemplifying a structure. Aswe have seen, this perspective requires a background ontology. The ontological

38We have however supposed the existence of some structure prior to the mind, that wouldappear in nature. So it might be necessary to express some reservation as to the extent towhich structures are to be linked to the mind, or as to which structures are. But this issomehow out of the subject of this paper. I thus ask the reader some indulgence regardingthis particular point.

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eliminative structuralism asserts that there somehow exists a background on-tology, for example, set theory. The other possibility is the modal eliminativestructuralism, making use of the notion of possible systems, in order to havesystems exemplifying any structures. We then studied more precisely how theante rem structuralism theory can be de�ned, in a similar fashion as set theory,and took a very quick glance at how the modal eliminative structuralism canbe formalized. But it would be interesting to study this matter deeper. Thematter of what exactly are logical possibilities, if this perspective is perspicuousalso would require a deeper study.

We then studied the work of Jean Piaget concerning the epistemology ofmathematics, through the genetic approach. We have seen that in the develop-ment of his mind, the child constructs mental structures, that can be classi�edas members of the structure type of grouping, and of the structure that dealswith time and space. We have seen that the structures of grouping can be di-vided in two sorts of structures, depending on the form of reversibility used,and that those structures can be linked to the Bourbaki's mother structuresthat are the algebraic structures and the order structures. We also saw that thelast structure makes uses of principle very close to the mathematical principleused in topology, and thus can be linked with the last Bourbaki's mother struc-ture. We then studied the matter of the re�ecting abstraction, which, taking astructure, rethink the relation and operation of it, and reconstruct them in astructure at a superior level. And we have seen how this principle of re�ect-ing abstraction can bring us all the way to pure mathematics. This principlebringing a possible endless regression in the mental construction of structures,it brought the idea that the origin of mathematical structures could be found inorganic structures, and even maybe in physical structures. In light of those con-siderations, we tried to give an ontological interpretation of structuralism thatis somehow in between the ante rem and the modal eliminative structuralism.The idea being that structure exists prior to the mathematician, but as possiblestructures. But once those structure are mentally built, it can be studied initself with the places-are-object perspective. Here again, a deeper study of thequestion of realm of possibilia could be interesting.

For further studies, the matter of (modal) logic itself could be an interest-ing one to study. Indeed, as we have seen, the development of mathematicalstructures is bound by the laws of logic, so the matter is of great importanceregarding structuralism. But where do these laws come from? Maybe a studyof the question using the genetic approach could bring some light into this ques-tion. Further studies of anti-realists points of view, like Field's nominalism,could also be interesting to bring some perspective.

References

[1] Evert W. Beth & Jean Piaget: Épistémologie Mathématique et Psy-chologie, Presses Universitaires de France, 1961.

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[2] Nicolas Bourbaki: The architecture of mathematics, American Mathe-matical Monthly 57, 1950.

[3] Rudolf Carnap: Empiricism, semantics, and ontology, Revue Interna-tionale de Philosophie, 1950.

[4] Jean-Blaise Grize: Du groupement au nombre, esssai de formalisation,Etude d'Epistemologie génétique, Vol XI, 1960.

[5] Stephen Hawking & Leonard Mlodinow: The Grand Design, BantamPress, 2010.

[6] Geoffrey Hellman: Structuralism without structures, PhilosophiaMathematica (3) 4, http://www.philosophy.ru/library/philmath/eng/

hellman_stucturalism_wihout_structures.pdf, 1996.

[7] Goeffrey Hellman: Towards a Modal-Structural Interpretation of SetTheory, Springer, http://www.jstor.org/stable/20116825, 1990.

[8] David Lewis: Mathematics is megethology, Philosophia Mathematica, 1993.

[9] Charles Parson: The structuralist view of mathematical objects, Synthese84, 1990.

[10] Jean Piaget: Traité de logique, Essai de logistique opératoire, Colin, 1950.

[11] Hilary Putnam: Reason, truth and history, Cambridge University Press,1981.

[12] Willard V. Quine: From a Logical Point of View, Harvard UniversityPress, 1961.

[13] Willard V. Quine: Ontological Relativity and Other Essays, ColombiaUniversity Press, 1969.

[14] Michael Resnik: Mathematics as a science of patterns: Ontology andreference, Nous 15, 1981.

[15] Michael Resnik: Mathematics as a science of patterns: Epistemoloty,Nous 16, 1982.

[16] Stewart Shapiro: Philosophy of Mathematics, Structure and Ontology,Oxford University Press, 1997.

[17] http://www.fondationjeanpiaget.ch

[18] Thierry Porchet: La question de l'existence en philosophie des math-ématiques : un aperçu des points de vue de Frege et de Hilbert, section ofmathematics, EPFL, 2009.

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