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Multivalued Functions

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Multivalued functions

I am reconstructing the abstractmath website and am currently working on the part on functions. This has generated some bloggable blustering.

The phrase multivalued function refers to an object that is like a function {f:S\rightarrow T} except that for {s\in S}, {f(s)} may denote more than one value. Multivalued functions arose in considering complex functions such as {\sqrt{z}}. Another example: the indefinite integral is a multivalued operator.

It is useful to think of a multivalued function as a function although it violates one of the requirements of being a function (being single-valued).

A multivalued function {f:S\rightarrow T} can be modeled as a function with domain {S} and codomain the set of all subsets of {T}. The two meanings are equivalent in a strong sense (naturally equivalent). Even so, it seems to me that they represent two different ways of thinking about multivalued functions.: “The value may be any of these things…” as opposed to “The value is this whole set of things.”) The “value may be any of these…” idea has a perfectly good mathematical model: a relation (set of ordered pairs) from {S} to {T} which is the inverse of a surjective function.

Phrases such as “multivalued function” and “partial function” upset some uptight types who say things like, “But a multivalued function is not a function!”. A stepmother is not a mother, either.

I fulminated at length about this in the Handbook article on radial category. (This is conceptual category in the sense of Lakoff, Women, fire and dangerous things, University of Chicago, 1986.). The Handbook is on line, but it downloads very slowly, so I have extracted the particular page on radial categories here.

Functions generate a radial category of concepts in mathematics. There are lots of other concepts in math that have generated radial categories. Think of “incomplete proof” or “left identity”. Radial categories are a basic mechanism of the way we think and function in the world. They should not be banished from mathematics.

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language of math, math, representations, understanding math

Different names for the same thing

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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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