What did Poincaré mean when he said "Mathematicians do not deal in objects, but in relations among objects"?

Question

The mathematician Henri Poincaré had said, "Mathematicians do not deal in objects, but in relations among objects; they are free to replace some object by others so long as the relations remain unchanged. Content to them is irrelevant; they are interested in form only."

I am not quite sure if this is true. For example, number theorists are interested in prime numbers, which are mathematical objects, and they are interested in prime numbers themselves, not just the relations between prime numbers and other objects.

So what did Poincaré really mean?

Answer 1

One position in the philosophy of maths is Structuralism (as has been pointed out in the comments). According to Structuralism, mathematics is not about numbers, sets, functions, etc., but about structures. It identifies common mathematical objects with places and relations in structures.

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Some time ago, people thought that numbers are really sets. For example, 0 is the null set, 1 is the set of the null set, 2 is the set of the set of the null set, and so on.

But this way of identifying numbers with sets is just one way to do it. Here is another. 0 is the null set, 1 is the powerset of 0, 2 is the powerset of 1, and so on. (which gives us the following sequence: {}; {{}}; {{}, {{}}};...

(This is a very short and probably bad summary of Benacerrafs 'What Numbers Could Not Be' (1965))

This insight motivates the Structuralist's position, that any sequence of stuff will do.

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But what are these structures? According to some, they are abstract objects in their own right. Here we have the problem of supplying a satisfying epistemology of abstracta. According to others, they are concrete. Here we have the problem that mathematics is thought to be non-contingent, while what is concrete, and how much concrete stuff there is, is contingent. And according to a third position, they are merely logically possible, which again dodges some problems and runs into others (do we know that there are possible structures? etc)

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Literature: The relevant SEP link is here

Resnik (1981) - Mathematics as a Science of Patterns

Shapiro (1997) - Philosophy of Mathematics - Structure and Ontology

Hellman (1989) - Mathematics Without Numbers - Towards a Modal-Structural Interpretation

Answer 2

I believe Poincare is referring to the notion of isomorphism. Many mathematical statements are qualified by the phrase "up to isomorphism," so that the statements apply not just to one object, but to an "isomorphism class" of "equivalent" objects. Roughly, an isomorphism is a bijection between two sets which preserves all the relevant structure one has associated with the sets. Put another way, an isomorphism tracks the relations between the elements of the two sets, and shows them to be the same under the identifications of the bijection. More abstractly, there is a category theoretic definition of isomorphism which avoids talk of sets, elements, and bijections.

Answer 3

Basically, what mathematics is all about is abstraction. Abstraction means that you try to remove as many of the specific properties as possible, and see what you can still prove.

Note that already the natural numbers are an abstraction in that sense: Three dogs are something very different than three humans, and both are very different from three houses, three apples or three stars. But they have all in common the "three-ness". That is, if you speak about the number three, you don't care about whether you are speaking about three dogs, three humans or three houses. All you care about is the "three-ness". The objects you might count with it don't matter.

And that is exactly where the power of abstraction comes from. As soon as you know that three is not divisible by two, we know that this applies no matter what objects you are speaking about. It means you cannot equally distribute three dogs to two people, but also that you cannot equally distribute three apples in two rooms. It tells you that if two people own three houses (and don't have shared ownership), they don't both have the same number of houses. And so on.

And as such, the divisibility (and thus also the primeness) is not about the objects (dogs, humans, houses, etc.) but about their relations (two of X cannot be equally distributed over three of Y; two is not a divisor of three).

Note that you don't even need "one" to refer to one single thing; you can use "one" to refer to one dozen, and you'll still find that if you distribute three dozen objects equally to two dozen destinations, it's not possible for those destinations to get a whole number of dozens; if you have boxes of one dozen objects each, you'll have to open one of the boxes.

Answer 4

If you look at it from the perspective of an axiomatic system, then the relational view becomes much more obvious. For instance, take geometric axioms. You have statements like "2 points define a line", "3 points define a plane" and so on. Yet when you get down to the basic terms like "points", they remain undefined. If objects remain undefined, then they are clearly place-holders for the relation. In fact...

Hilbert once remarked that instead of points, lines and planes one might    
just as well talk of tables, chairs and beer mugs.[3] His point being that 
the primitive terms are just empty shells, place holders if you will, and 
have no intrinsic properties.

This is from here.

If the relations are all that matter, and the objects are simply useful as place-holders for the relationships, then can the objects be dispensed with entirely? YES! And this is where you get Combinatory Logic.

Answer 5

Poincaré wrote this paragraph in chapter 2 of Science and Hypothesis (1905). It was in the context of discussing geometry. Geometry, not in the classical Euclidean sense, but in the modern sense of mathematical analysis, which starts by defining the real numbers - the real continuum. Anyway this distinction between geometries does not seem important for the example that Poincaré had in mind. The example was about lines and points. In a sense a geometrical line is made of points. But not as if the points were self standing objects, antecedent to the lines. The points depend upon the lines, just as the lines depend upon the points. It is the relations between them that define space, and geometry.

Before going any further, let me make a preliminary remark. The continuum thus conceived is no longer a collection of individuals arranged in a certain order, infinite in number, it is true, but external the one to the other. This is not the ordinary conception in which it is supposed that between the elements of the continuum exists an intimate connection making of it one whole, in which the point has no existence previous to the line, but the line does exist previous to the point.

Poincaré contrasts this character of geometry (and mathematics in general) with sense-based intuition. The concepts of space cannot be based on visual intuitions alone, because these intuitions are not fine enough, and they actually lead to paradoxes (Zeno's paradoxes, and others). Mathematicians define space in ways that overcome the paradoxes.

We are next led to ask if the idea of the mathematical continuum is not simply drawn from experiment . . . The rough results of the experiments may be expressed by the following relations: A = B, B = C, A < C, which may be regarded as the formula of the physical continuum. But here is an intolerable disagreement with the law of contradiction, and the necessity of banishing this disagreement has compelled us to invent the mathematical continuum.

Answer 6

Poincare was one of the originators or of topology; and a thinker as Weil was of the continuum; as Tobolski points out a line is more than its points, one must also include its topology.

If one identifies points with objects, topology as relations than one can begin to make some sense of this thought of Poincare.

It's worth pointing out that some contemporary treatments dispense with points altogether and have just the topology - use pointless topology.

In a different direction; ala Kinsellas answer, category theory deliberately obfuscates or makes opaque the notion of an object and solely concentrates on the notion of a relation; or in its language - a morphism; one of the roots of this idea is the structuralism, not of Levi-Strauss; but of the Bourbaki school.

Answer 7

As an example, in the field of arithmetic geometry, one replaces the notion of "prime number" with the notion of "closed point of the scheme Spec Z". One also replaces the notion of "integer" with "regular function on Spec Z". The factorization of an integer? Oh, you mean the roots (with multiplicity) of a function!

In this way, we can turn problems about number theory into problems about geometry, and use all of our tools and intuitions about modern geometry to attack hard number theory problems. Or even to improve our understanding of easy ones.

In fact, depending on your point of view, you can even view this as one-upping Poincaré's quote: the relations have changed too, but preserving the relations between the relations.