Shapiro’s and Hellman’s Structuralism - Uni · PDF fileShapiro’s structuralism Hellman’s structuralism Von Bu low’s structuralism? Shapiro’s and Hellman’s Structuralism

Shapiro’s structuralismHellman’s structuralism

Von Bulow’s structuralism?

Shapiro’s and Hellman’s Structuralism

Christopher von BulowUniversity of Constance, Department of Philosophy

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www.uni-konstanz.de/FuF/Philo/Philosophie/philosophie/

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Trends in the Philosophy of MathematicsFrankfurt/Main, September 4, 2009

Christopher von Bulow Shapiro’s and Hellman’s Structuralism



(1) Two versions of structuralism

Stewart Shapiro (Philosophy of Mathematics, 1997):

Mathematical objects like the numbers are ‘places instructures’; structures are universals/ones-over-many:properties of systems.

Geoffrey Hellman (Mathematics without Numbers, 1989):

There are neither mathematical objects nor structures(in Shapiro’s sense); mathematical propositions areimplicit generalizations about logically possible systems.




Contents

1 Shapiro’s structuralismSystems and structuresPlaces in structuresAre the office places the mathematical objects?Mathematical objects are unnecessary

2 Hellman’s structuralismGenerality and trivialityPossibility is unnecessary

3 Von Bulow’s structuralism?




Systems and structuresPlaces in structuresAre the office places the mathematical objects?Mathematical objects are unnecessary

(2) The nat.-number structure: a property of systems

Consider the natural numbers: 0, 1, 2, . . .Do they exist??? If so, what are they???

One thing is clear: the natural numbers are supposed to beinterrelated by, e.g., the successor relation in such a waythat the Peano axioms are satisfied.

This way of being interrelated by a two-place relationis a (second-order) property that systems can have,a one-over-many . . .

. . . or a second-order relation between (1) a collection of objectsand (2) a relation on that collection.

This property of systems is the natural-number structure.





(3) Systems having the natural-number structure

The natural-number system (ℕ,Succ) – if it exists –has the natural-number structure.

So do, inter alia, certain set-theoretical systems, for example,

the Zermelo numerals : ∅, {∅},{{∅}}

,{{{∅}}}

, . . .

(where “y is the successor of x” means “y = {x}”)

the finite von Neumann ordinals :

∅, {∅},{∅, {∅}

},{∅, {∅},

{∅, {∅}

}}, . . .

(where “y is the successor of x” means “y = x ∪{x}”)

– assuming that sets exist.





(4) Dubious exemplars, well-understood property

Uncertain:Are there systems having the natural-number structure???(systems consisting of physical objects???)

If such systems exist, is there one special, distinguished systemamong them that contains the numbers???(instead of merely objects which can be used as numbers)

But:We know (reasonably well) what the natural-number structureitself, as a property of systems, is.





(5) Places in structures

The set{{∅}}

‘does’ in the system of Zermelo numerals what{∅, {∅}

}does in the system of finite von Neumann ordinals,

or what 2 does in the system of the natural numbers.They each play the same role in their respective systems.

The set{{∅}}

‘is in’ the system of Zermelo numerals‘where’ 2 is in the system of the natural numbers.They occupy the same place in their different systems.

We call this place the 2-place in the natural-number structure.





(6) Objects of systems occupy places in a structure

For every system having the natural-number structure,its 0-object occupies the 0-place in that system.

Each 0-object’s successor occupies the 1-place in the given system.

And so on.

The natural-number structure has countably many places:the 0-place, the 1-place, the 2-place, . . .

What are places?Do they exist? Can we consider them as objects?





(7) Indiscernibility of places

In the Euclidean plane, for example,since all points have the same geometric properties,every point plays the same role.

So the places in the Euclidean-plane structure, it seems,are all indiscernible, though distinct.

Is this a problem for Shapiro’s structuralism?

It’s not one I worry about.

Reason: Places in structures are like the ‘places’ of relations.

How is that, and how does it help with indiscernibility?





(8) Structures as �-place relations

For example,

“x is less than y”

is a two-place relation; it has an ‘x-place’ and a ‘y -place’.

Now take the natural-number structure: Considered asa property of whole systems, it is a one-place relation.

Whereas if you consider it as a relation between(1) a collection of objects and (2) a relation on that collection,it is a two-place relation.

If, however, you consider the natural-number structureas a relation between countably many objects and one relation,then it is an (!+1)-place relation.

The places of the natural-number structure are likethe argument places of such a relation.





(9) Argument places of symmetric relations

If such argument places are sometimes indiscernible,it is not a problem:

For example, the argument places of the relation

“x is the same age as y”

(or of any other symmetric relation) aren’t discernible either.1

So what? The task of better understanding argument placesis one we are burdened with independently of mathematics.

1Yes, in linguistic expressions or other representations of that relation youhave discernible place-holders for argument designations. But the relation itselfand its places aren’t linguistic phenomena.





(10) Are the places the mathematical objects?

I have no scruples about accepting places in structures into anaccount of mathematics.

But I don’t believe that they are the mathematical objects.

Up to now, I have talked about places as somewhat like officesto be occupied, or like roles to be played, by other objects.

Shapiro calls this the places-are-offices perspective.

I suggest that this is a concept of places rather than a perspectiveon them: I call places in this sense the office places of a structure.

Question: Are the office places of mathematical structuresthe mathematical objects?





(11) What are mathematical objects like?

What are numbers like?

Paul Benacerraf, “What Numbers Could Not Be”(Philosophical Review, 1965):

If the numbers exist then their mathematical propertiesconsist in their arithmetical relations to each other:the number 2 is the successor of 1, the predecessor of 3,the first prime, etc. – otherwise it wouldn’t be 2.

And their mathematical properties do not include,e.g., any set-theoretical relations between them:“Is 2 ∈ 4?” is a bad question, it doesn’t make much sense.2

Moral:Mathematical objects (if they exist) have a ‘relational essence’.

2It does make more sense than “Is 2 an element element?” For example,“If 2 ∈ 4 then 2 ∈ 4∪7” seems somewhat plausible.





(12) Essence places (= mathematical objects)

When we talk about mathematical objects like this –not considering them as ‘offices’ but rather as objectsin their own right: e.g., “2+2 = 4”, “5 is prime” –Shapiro considers it as another perspective on places in structures:the places-are-objects perspective.

I suggest that this is another concept of places. I call places inthis sense relational-essence places, or “essence places” for short.

They are ‘places’ which stand in certain mathematical relationsto each other (which are peculiar to the structure they belong to),and only in those mathematical relations.

Essence places are the same as mathematical objects(if they exist).





(13) Office places = essence places?

Shapiro doesn’t distinguish between what I call office places andessence places. Mostly he just presupposes that they are the same.One attempt to motivate the identification:

In contrast to this office orientation, there are contextsin which the places of a given structure are treatedas objects in their own right, at least grammatically.That is, sometimes items that denote places are bonafide singular terms. We say that the vice president ispresident of the Senate, that the chess bishop moveson a diagonal, . . .My perspective . . . presupposes that statements in theplaces-are-objects perspective are to be taken literally,at face value. Bona fide singular terms, like “vicepresident,” “shortstop,” and “2,” denote bona fideobjects. (Philosophy of Mathematics, p. 83)





(14) Offices treated as objects – allegedly

Shapiro: In the sentence

“The chess bishop moves on a diagonal”

– when it does not refer to a particular concrete chess piece –“the chess bishop” denotes a role concrete chess pieces can play.

Analogously, in

“5 is prime”

– when it doesn’t refer to the 5-object of some particular‘concrete’ system with the natural-number structurelike, e.g., the Zermelo numerals –“5” would denote an office place in the natural-number structure.

If so, then this office place would indeed have the primenessproperty that is characteristic of certain numbers,i.e., certain relational-essence places.Thus office places would indeed be essence places.





(15) Hidden generalizations about office occupants

However, this interpretation of so-called ‘places-are-objects’sentences is inadequate:

It is not the role of bishop in chess that moves on a diagonal;it is concrete pieces which move.“The chess bishop moves on a diagonal” is a hidden generalization.

It is not the offices of vice president and president of the Senatewhich coincide; it is their occupants.“The vice president is president of the Senate” is another hiddengeneralization.

Analogously, truths like “5 is prime” do not give us reason to thinkthat it is the 5–office place which is prime.





(16) Offices treated as objects – really

There are actual ‘places-are-objects sentences’:

“The office of president of the USA has a special seal.”

“The 6-place of the natural-number structureis just as occupiable as the 0-place.”

These sentences do treat offices as objects in their own right.But the properties they predicate of them are ones fit onlyfor offices, not for their occupants.





(17) “5 is prime”: ‘numbers-are-objects’ or general

Shapiro’s everyday sentences,

“The vice president is president of the Senate”,“The chess bishop moves on a diagonal”,

are disguised generalizations.

By contrast, mathematical sentences can be read in two ways:

Either “5 is prime” talks about a number, an essence place.Then “prime” expresses a property fit for mathematicalobjects, which has no obvious connection to office places.

Or “5 is prime” is a generalization about arbitrary occupantsof the 5-place. It treats 5 as an office, but it does notpredicate primeness of it.





(18) Identifying office places and essence places?

So there is no reason to believe that office places areessence places/mathematical objects.

Can’t we just postulate that they are the same?

No.

For one thing, office places do not have what it takesto be essence places:





(19) Office places are not essence places

On the numbers, there is one special successor relation:the successor relation.

On the office places of the natural-number structure, there isno such distinguished, privileged successor relation.

The ‘canonical’ successor relation on the office placesis merely particularly easy to specify :

Office place q is a place-successor of office place p iffin any system S with the natural-number structure,q’s occupant is the S-successor of p’s occupant.

Thus you get: 0-place ↷ 1-place ↷ 2-place ↷ 3-place ↷ . . .

This relation is only pragmatically, not metaphysically, special:1-place ↷ 0-place ↷ 3-place ↷ 2-place ↷ . . . works just as well.





(20) Essence places are not office places

Symmetrically, the numbers/essence places do not havewhat it takes to be the number–office places:

Office places are special in being occupiable by arbitrary objects.

The way for a system’s objects to occupy the numbersconsists in there being an isomorphism from that systemto the system of the numbers.

‘Occupying’ the number 0 means being mapped onto 0by that isomorphism.

If that were all it takes to be an office place in the ℕ-structure,then any system having the ℕ-structure could do the job just as well.

So the numbers/essence places are not especially good candidatesfor being the ℕ-structure’s office places.





(21) Mathematical objects: by stipulation only

The office places and the essence places of a structuredo not seem to be the same; the office places of structuresare not plausible candidates for the corresponding essence places,the mathematical objects.

So Shapiro’s structuralism leaves mathematical objectsas mysterious as before:

If we want to believe in them we have to stipulate them,like in traditional platonism.





(22) Are mathematical objects needed for semantics?

Do we really need mathematical objects?

Paul Benacerraf, “Mathematical Truth” (Journal of Philosophy,1973):

We would like to have a Tarski semanticsfor mathematical discourse, and for that we needmathematical objects as denotations of singular termslike “2”.

But Tarski semantics is just a model of semantics,not a satisfying account of semantics.





(23) The numbers vs. . . .

Furthermore realism in ontology does not give a very faithfulaccount of how mathematical discourse acquires its content:

How do we find things out about the numbers?

We talk as if we had one particular system of objects in our hands,about which we know certain things, e.g., the Peano axioms.

Certain questions we just don’t ask:

“What color does 5 have?”“Does 5 have any elements?”





(24) . . . vs. a group’s elements

How do we find things out about, say, groups?

We say,

“let G be a group, let ∘ be its multiplication,and let e be its unit element”,

and then we go on talking as if we had one particular systemof objects in our hands: the group G, about which we knowcertain things, e.g., the group axioms.

Certain questions we just don’t ask:

“What color is the unit element?”“Do any of the objects of G have elements?”“How many objects are there in G ?”





(25) No relevant difference in practice

In the case of ‘the group G ’, nobody would conclude thatafter saying

“Let G be a group”,

we are indeed talking about one particular system:“the arbitrary group” perhaps, whose objects have onlythose properties which follow from the group axioms.

What we are doing is, proving theorems aboutall groups whatsoever.

But there is no relevant difference between the casesof group theory and arithmetic.

This indicates that arithmetic, too, is really aboutall systems whatsoever which have the right structure.




Generality and trivialityPossibility is unnecessary

(26) Another semantics of mathematical discourse?

If there are no numbers, if mathematics is actually generalwhere it seems to be about particular objects,then what is the meaning of mathematical propositionslike “2+2 = 4”?

One structuralist answer comes from Geoffrey Hellman:modal-structuralism.





(27) “∀”: the danger of triviality

If arithmetic isn’t about ‘the numbers’ but rather aboutarbitrary systems having the natural-number structure,then, it seems, what “2+2 = 4” really means is

“∀ systems S with the ℕ-structure: 2S +S 2S = 4S .”

But then along comes a philosopher and says:

“What if there are no systems that have the ℕ-structure?Then ‘∀ ℕ-systems S : 2S +S 2S = 4S ’would be trivially true,just like ‘∀ ℕ-systems S : 2S +S 2S = 5S ’ !”





(28) “□∀”: triviality vanquished (?)

Hellman’s solution:Arithmetic isn’t just about all ℕ-systems which actually exist,but rather about whichever ℕ-systems might possibly exist.

So, what “2+2 = 4” really means is

“□∀ ℕ-systems S : 2S +S 2S = 4S .”

Since it seems possible that there be infinitely many objects –

“♢∃S : S is an ℕ-system”

– this explication of “2+2 = 4” is not in danger of being trivial.

You may, however, get an analogous problem if you apply thismethod to set theory, at least if you want your systems to consistof physical objects:

Is it really logically possible that there be as many physical objectsas there are sets?





(29) “♢∃”: asserted or presupposed?

What is it I don’t like about this proposal?

Look at the ‘categorical component’ of arithmetic:the assertion that systems with the ℕ-structureare logically possible:

♢∃S : S is an ℕ-system.

Mathematicians usually don’t assert things like that;rather, they tacitly presuppose them.

Is “♢∃ℕ-system” really part of the content of arithmetic?

I think it is rather like a working hypothesis. Someone whobelieves this hypothesis false shouldn’t bother doing arithmetic,at least not for its own sake, because it would be futile.





(30) Three kinds of premisses for proofs

The possibility presupposition isn’t essential, however:

Consider three kinds of mathematical proof:

(a) The proof starts with

“Let S be a system with structure so-and-so”,

where one is certain that systems like these (might) exist(e.g., finite systems or ℕ-systems).

(b) The proof starts with

“Suppose conjecture so-and-so is true”,

where one is uncertain whether so-and-so is true.

(c) The proof starts with

“Suppose so-and-so were the case”,

where one is more or less certain that so-and-so islogically false.





(31) In all cases: substantial results

All three of these cases are ubiquitous and deliver resultswith real mathematical content:

In case (a) (certain truth) one obtains a categorical assertion:

“In systems like that, such-and-such is the case.”

In case (b) (uncertain truth) one obtains a conditional:

“If conjecture so-and-so is true then such-and-such holds.”

Case (c) (certain falsehood) arises, e.g.,in proofs of P by reductio ad absurdum of ¬P,or in proofs of P→Q by reductio ad absurdum of P∧¬Q.





(32) Different kinds of content?

In all these three kinds of proof, the same thing goes on,epistemically:

You ‘assume’ P – never mind whether P is logically possible –and derive consequences.

Assuming in this manner the impossible works just the same wayas assuming the possible or starting from a virtual certainty.

For Hellman, however, different things are going on in case (a)from cases (b) and (c):

In case (a), where you talk about a kind of system you believe in,you start by asserting,

“There might be systems like this!”

By contrast, in cases (b) and (c), you don’t start like that.





(33) Modal-structuralism is inadequate w. r. t. practice

But cases (a)–(c) are all analogous with respect to mathematicalpractice.

For me, this implies that the sorts of content these kinds of proofdeliver are the same.

Since Hellman’s characterization treats them completelydifferently, it must be inadequate.

Mathematical content comprises more than what is logicallypossible (e.g., “If 2+2 = 5 then (2+2)3 = 53”).




(34) Maths as the science of conceivable structure

What is my constructive counterproposal?

I haven’t got more than very rudimentary and vague suggestionsyet:

Mathematics is nowhere about particular systems,like ‘the numbers’ or ‘the Euclidean plane’.

Perhaps it is about particular structures or types of structure,about particular ways systems might conceivably be(where not everything that’s ‘conceivable’ is also logically possible).




(35) Maths, the universe and everything

Mathematics is so general that there is no circumscribed realmof systems or things it is about.3

(That’s a point Hellman often makes, rightly.)

There is not even a however-vaguely-circumscribedspectrum of realms of systems mathematics is about,e.g., the spectrum of the logically possible, as in Hellman’s account:

Whenever you circumscribe such a ‘realm’ or ‘spectrum of realms’,mathematics will transcend it.

E.g., Hellman’s prohibition of quantifying over all possible systems,or of functions and relations between different possible ‘universes’,will not forever stop mathematicians.

3Though probably the real problem lies in attempting to circumscribe,or in supposing as circumscribed, ‘everything there is’.




(36) Mathematical content

Perhaps mathematical theorems are not propositionswith a well-defined, exhaustible content,but rather tools with no pre-determined domain of applicability.

They are applicable to whatever someone might specify some day.

Perhaps mathematics is not about arbitrary conceivable systemsbut about arbitrary descriptions or representationsof (structural) kinds of systems.

Some of these descriptions are of impossible systems,but you can still reason mathematically aboutwhatever coherent parts those descriptions have.




(37) How to philosophize about mathematics

How to make this precise? How to formalize it?

A formalism that explicates a philosophy of mathematicsmay be a nice model, but it is still just more mathematics;and you can’t fully understand mathematics just by doingmore mathematics.

Instead, look at how physical agents – machines with goals –can arrive at, and apply, mathematical knowledge(or mathematical information, or capabilities).


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Shapiro’s and Hellman’s Structuralism - Uni · PDF fileShapiro’s structuralism Hellman’s structuralism Von Bu low’s structuralism? Shapiro’s and Hellman’s Structuralism