Topics Mathematics

Question As a teacher of high school mathematics and a former student of philosophy, I try to merge the two to engage my students in meaningful conversations about the significance of some mathematical properties. Recently, however, I could not adequately defend the statement "a=a" as being necessary for our study of geometry when one student challenged "When is a never NOT equal to a?" What would you tell them? (One student did offer the defense that "Well, if we said a=2 and a=5 then a=a would be false, causing problems.")

Accepted:
October 12, 2005

Comments

Identity is an important notion in mathematics. There certainly areexamples of geometrical theorems that demonstrate identities, some ofthem very important. Consider, for example, the (Euclidean) theoremthat the three lines from the vertices of a triangle bisecting theopposite sides meet in a single point. Any time one uses a word like"single", identity is in play. Frege gives a similar example in section8 of Begriffsschrift to illustrate the same point:Mathematical identities can have substantial content. Elementaryarithmetic may be a better illustration of how important identitiesare, though, since basic arithmetical facts are all equalities, thatis, identities.

That said, the question arises how identity is to be characterized.There are various ways to proceed. But any characterization is going tohave to deliver four basic properties of identity: reflexivity (thatis, a=a), symmetry (that is, a=b → b=a), transitivity (a=b & b=c →a=c), and substitutivity, which says (roughly) that if you have …a…and a=b, then you have …b…. There are redundancies here, though.Reflexivity follows from transitivity and symmetry; transitivity andsymmetry can be proven from reflexivity and substitutivity. And, inso-called higher-order logics, one can define identity in terms ofsubstitutity and prove reflexivity (and therefore transitivity andsymmetry, too).

It'd be hard, I think, to find a natural mathematical context inwhich reflexivity was important in itself. But it's easy to find caseswhere transitivity is used, and it's not too much harder with symmetry,though one has to pay very careful attention to the language beingused. (We tend, informally, to say things like, "So a and b are thesame" or even "So they are the same", and the order of the argumentsgets a bit lost.) So I'd tell the students that it is really identitythat is critical, and reflexivity is an important property of identity.

I'm not sure whether you're asking (1) What role does the reflexivityof identity (i.e., every object is equal to itself) play in geometry?,or (2) What justification can be offered for the reflexivity ofidentity? As regards (1), I assume that in an explicit axiomatizationof geometry, there would be axioms dealing with identity. As Richardpoints out above, in such an axiomatic system we will want to derivetheorems in which "=" figures; so we had better have some axioms thattell us under what conditions identity statements can be proved. Asregards (2), I'm not sure what to advise you if a student is unwillingto grant that an object is identical to itself. I would infer that s/hedoesn't understand what "identical to" means and would treat the matter as a case of miscommunication.