I recommend reading the discussion (to which I contributed) of the post “Why aren’t all functions well-defined?” on Gower’s Weblog. It resulted in an insight I should have had a long time ago.
I have been preaching the importance of different ways of thinking about a math object (different images, metaphors, mental representations — there are too many names for this in the math ed literature). Well, mathematicians at least occasionally use different names for a type of math object to indicate how they are thinking about it.
Examples
We talk about a relation and we talk about multivalued functions. Those are two different ways of talking about the same thing (they are the same by an adjunction). A relation is a predicate. A multivalued function is a function except that it can have more than one output for a given input. But they are the same thing.
We talk about an equivalence relation and we talk about a partition of a set (or a quotient set). The category of equivalence relations and the category of partitions of sets are naturally isomorphic, not merely equivalent. But one is a special kind of relation and the other is a grouping.
Let’s be open about what we do
We should be explicit about the way we think about and do math. We have several different ways to think about any interesting type of math object and we should push this practice to students as being absolutely vital. In particular we (some of us) use different names sometimes for the same object and we refuse to give them up, muttering about “reductionism” and “nothing buttery”.
Some students arrive in class already as (pedantic?)(geeky?) as many mathematicians (I am a recovering pedant myself). We need to be up front about this phenomenon and explain the value of thinking and talking about the same thing in different ways, even using different words.
It used to be different but now it’s the same
A kind of opposite phenomenon occurs with some students and mathematicians of a certain personality type. Consider the name “multivalued function”. Of course a multivalued function is not (necessarily) a function. Your mother-in-law is not your mother, either. I go on about this (using ideas from Lakoff) in the Handbook under “radial concept”. Pedantic types can’t stand this kind of usage. “A multivalued function can’t be a function”. “Equivalence relations and partitions are not the same thing because one is a relation and the other is a set of sets.” “The image of a homomorphism and the quotient by its kernel are not the same thing because…”
This attitude makes me tired. Put your hands on the tv screen and think like a category theorist.
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You say
right after calling out people for saying
I put it to you that a category theorist would be exactly the person to say this. The quotient group and the image are isomorphic, but not “the same thing”.
You are right; I should not have said “the same thing” because “thing” is ambiguous. The isomorphism is a natural isomorphism, so the quotient and the image are the same mathematical concept. To ask whether they are the same mathematical object is not worth asking. And indeed, “thing” sounds more like an object than a concept.
Two comments. First, unless you allow multi-valued functions to take on less than one as well as more than one value, they are not the same as relations. A minor point, but must be made. Second a category theorist is interested in concepts and is perfectly happy to give different names to different things depending on how he conceptualizes them. What John Armstrong said above is just wrong.
Incidentally, I have no patience for people who say that a multi-valued function isn’t allowed to be a function, that a line is no parallel to itself, and that a square is not a rectangle nor a circle an ellipse.
Michael, I think you missed my point. I never said that category theorists wouldn’t give different names to different things. What I said was that a category theorist is more likely than many other mathematicians to recognize that two concepts that are casually called “the same thing” are actually two distinct, yet isomorphic, things.
The multivalued function example is not in the same class as the others you mentioned. Saying a square is not a rectangle excludes a special case from the definition of rectangle, and I agree with you that that is usually a mistake. To say a function is not a multivalued function would be to make that kind of mistake, too.
But saying a multivalued function is not a function is different. A multivalued function is technically not a function, but it is like a function in some respects and the adjective points out the way it differs. A mother in law is not technically a mother but is like a mother and the “in law” modifier points to the connection. Purists object to constructions like this. Wrongly, in my opinion. That kind of modification of a concept using an adjective that gives a concept that is strictly not an example of the original concept but is related to it in a specific way is a very common technique in ordinary English speech and native speakers understand how it works (usually subconsciously) without any bother at all.