category theory, exposition, language, language of math, math, representations

Composites of functions

2009/09/11 Charles Wells 3 Comments

In my post on automatic spelling reform, I mentioned the various attempts at spelling reform that have resulted in both the old and new systems being used, which only makes things worse. This happens in Christian denominations, too. Someone (Martin Luther, John Wesley) tries to reform things; result: two denominations. But a lot of the time the reform effort simply disappears. The Chicago Tribune tried for years to get us to write “thru” and “tho” — and failed. Nynorsk (really a language reform rather than a spelling reform) is down to 18% of the population and the result of allowing Nynorsk forms to be used in the standard language have mostly been nil. (See Note 1.)

In my early years as a mathematician I wrote a bunch of papers writing functions on the right (including the one mentioned in the last post). I was inspired by some algebraists and particularly by Beck’s Thesis (available online via TAC), which I thought was exceptionally well-written. This makes function composition read left to right and makes the pronunciation of commutative diagrams get along with notation, so when you see the diagram below you naturally write h = fg instead of h = gf. Composite

Sadly, I gave all that up before 1980 (I just looked at some of my old papers to check). People kept complaining. I even completely rewrote one long paper (Reference [3]) changing from right hand to left hand (just like Samoa). I did this in Zürich when I had the gout, and I was happy to do it because it was very complicated and I had a chance to check for errors.

Well, I adapted. I have learned to read the arrows backward (g then f in the diagram above). Some French category theorists write the diagram backward, thus:

CompositeOp

But I was co-authoring books on category theory in those days and didn’t think people would accept it. Not to mention Mike Barr (not that he is not a people, oh, never mind).

Nevertheless, we should have gone the other way. We should have adopted the Dvorak keyboard and Betamax, too.

Notes

[1] A lifelong Norwegian friend of ours said that when her children say “boka” instead of “boken” it sound like hillbilly talk does to Americans. I kind of regretted this, since I grew up in north Georgia and have been a kind of hillbilly-wannabe (mostly because of the music); I don’t share that negative reaction to hillbillies. On the other hand, you can fageddabout “ho” for “hun”.

References

[1] Charles Wells, Automorphisms of group extensions, Trans. Amer. Math. Soc, 155 (1970), 189-194.

[2] John Martino and Stewart Priddy, Group extensions and automorphism group rings. Homology, Homotopy and Applications 5 (2003), 53-70.

[3] Charles Wells, Wreath product decomposition of categories 1, Acta Sci. Math. Szeged 52 (1988), 307 – 319.

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abstractmath.org, Handbook, language of math, math, proof, understanding math

Grasshoppers and linear proofs

2009/09/09 Charles Wells 7 Comments

Below, I give an detailed example of how the context of a proof changes as you read the proof line by line. This example comes from the abstractmath article on context. I mean something like verbal context or context in the computer science sense (see also Reference [1]): the values of all the relevant variables as specified up to the current statement in the proof. For example, if the proof says “Suppose x = 3″, then when you read succeeding statements you know that x has the value 3, as long as it is not changed in some later statement.

Here is the text I will analyze:

Definition: Divides

Let m and n be integers with $m\ne 0$ . The statement “m divides n” means that there is an integer q for which $n=qm$ .

Theorem

Let m, n and p be integers, with m and n nonzero, and suppose m divides n and n divides p . Then m divides p.

Proof

By definition of divides, there are integers q and q’ for which $n=qm$ and $p=q'n$ . We must prove that there is an integer $q''$ for which $p=q''n$ . But $p=q'n=q'qm$ , so let $q''=q'q$ . Then $p=q''n$ .

0) Definition: Divides	Changes the status of the word “divides” so that it becomes the definiendum. The scope is the following paragraph.
1) Let m and n be integers	m and n are new symbols in this discourse, constrained to be integers
2) with $m\neq 0$	another constraint on m
3) The statement “m divides n” means that	This sentence fragment gives the rest of the sentence (in the box below it) a special status.
4) there is an integer q for which n = qm.	This clause introduces q, another new symbol constrained to be an integer. The clause imposes a restraint on m, n and q, that they satisfy the equation n = qm. But we know this only in the scope of the word Definition, which ends at the end of the sentence. Once we read the word Theorem we no longer know that q exists, much less that it satisfies the constraint. Indeed, the statement of the definition means that one way to prove the theorem is to find an integer q for which n = qm. This is not stated explicitly, and indeed the reader would be wrong to draw the conclusion that in what follows the theorem will be proved in this way. (In fact it will in this example, but the author could have done some other kind of proof. )
5) Theorem	The placement of the word “Theorem” here announces that the next paragraph is a mathematical statement and that the statement has been proved. In real time the statement was proved long before this discourse was written, but in terms of reading the text in order, it has not yet been proved.
6) Let m, n and p be integers,	We are starting a new context, in which we know that m, n and p are all integers. This changes that status of m and n, which were variables used in the preceding paragraph, but now all previous constraints are discarded. We are starting over with m, n, and p. We are also starting what the reader must recognize as the hypotheses of a conditional sentence, since that affects the context in a very precise way.
7) with m and n nonzero.	Now m and n are nonzero. Note that in the previous paragraph n was not constrained to be nonzero. Between the words “Let” and “with” in the current sentence, neither were constrained to be nonzero.
8 ) and n divides p.	More new constraints: m divides n and n divides p.
9) Then m divides p.	The word “then” signals that we are starting the conclusion of the conditional sentence. It makes a claim that m divides p whenever the conditions in the hypothesis are true. Because it is the conclusion, it has a different status from the assumptions that m divides n and n divides p. We can’t treat m as if it divides p even though this sentence says it does. All we know is that the author is claiming that m divides p if the hypotheses are true, and we expect (because the next word is “Proof”) that this claim will shortly be proved.
10) Proof	This starts a new paragraph. It does not necessarily wipe out the context. If the proof is going to be by the direct method (assume hypothesis, prove conclusion) — as it does — then it will still be true that m and n are nonzero integers, m divides n and n divides p.
11) By definition of divides, there are integers q and q′ for which n = qm and p = q’n .	Since this proof starts by stating the hypothesis of the definition of “divides”, we now know that we are using the direct method, and that q and q’ are new symbols that we are to assume satisfy the equations n = qm and p = q’n. The phrase “by definition of divides” tells us (because the definition was given previously) that there *are* such integers, so in effect this sentence chooses q and q′ so that n = qm and p = q’n. The reader probably knows that there is only one choice for each of q and q′ but in fact that claim is not being made here. Note that m, n and p are not new symbols – they still fall within the scope of the previous paragraph, so we still know that m divides n and n divides p. If the proof were by contradiction, we would not know that.
12) We must prove that there is an integer q” for which p = q”n	q’’ is introduced by this sentence and is constrained by the equation. The scope of this sentence is just this sentence. The existence of q’’ and the constraint on it do not exist in the context after the sentence is finished. However, the constraints previously imposed on m, n, p, q and q’ do continue.
13) But p = q’n = q’qm	This is a claim about p, q, q′, m and n. The equations are justified by certain preceding sentences but this justification is not made explicit.
14) so let q” = q’q	We are establishing a new variable q″ in the context. Now we put another constraint on it, namely q” = q’q. It is significant that a variable named q″ was introduced once before, in the reference to the definition of divides. A convention of mathematical discourse tells you to expect the author to establish that it fits the requirement of the definition. This condition is triggered by using the same symbol q″ both here and in the definition.
15) Then p = q”n	This is an assertion about p, q″ and n, justified (but not explicitly) by the claim that p = q’n = q’qm.
16)	The proof is now complete, although no statement asserts that.

I have several comments to make about this kind of analysis that are (mostly) not included in the abstractmath article.

a) This is supposed to be what goes through an experienced mathematician’s head while they are reading the proof. Mostly subconsciously. Linguists (as in Reference [1]) seem to think something like this takes place in your mind when you read any text, but it gets much denser in mathematical text. Computer scientists analyze the operation of subprograms in this way, too.

b) Comment (a) is probably off the mark. With a short proof like that, I get a global picture of the proof as my eyes dart back and forth over the various statements in the proof. Now, I am a grasshopper: I read math stuff by jumping back and forth trying to understand the structure of the argument. I do this both locally in a short proof and also globally when reading a long article or book: I page through to find the topic I want and then jump back and forth finding the meanings of words and phrases I don’t understand.

c) I think most mathematicians are either grasshoppers or they are not good readers and they simply do not learn math by reading text. I would like feedback on this.

d) If (a) is incorrect, should I omit this example from abstractmath? I don’t think so. My experience in teaching tells me that

some students think this is perfectly obvious and why would I spend time constructing the example?,
others are not aware that this is going on in their head and they are amazed to realize that it is really happening,
and still others do not understand how to read proofs and when you tell them this sort of thing goes on in your head they are terminally intimidated. (“Terminally” in the sense that they dye their hair black and become sociology majors. They really do.) Is that bad? Well, I don’t think so. I would like to hear arguments on the other side.

e) Can you figure out why item 8 of the analysis is labeled as “8 )” instead of “8)”?

Time is running out. I have other comments to make which must wait for a later post.

References

G. Chierchia and S. McConnell-Ginet (1990), Meaning and Grammar. The MIT Press.

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exposition, language, math

The revolution in technical exposition II

2009/08/18 Charles Wells 7 Comments

In the last post I talked about the bad side of much technical exposition and the better aspects of popular science writing (exemplified by Priestley). These two streams have continued to the present. Stuffy, formal, impersonal technical exposition has continued to be the norm for works intended for academic credit. Math and science expositions written for the public have been much looser and some have been remarkably good. I described two of them in a previous post.

The revolution mentioned in the title of this post is that some aspects of the style of popular science writing have begun infiltrating writing in academic journals. Consider these sentences from Jody Azzouni's essay in [1]:

It's widely observed that, unlike other cases of conformity, and where social practices really are the source of that conformity, one finds in mathematical practice nothing like the variability found cuisine, clothing, or metaphysical doctrine. (p. 202).

Add two numbers fifteen times, and you do something different each time — you do fifteen different things that (if you don't blunder) are the same in the respect needed; the sum you write down at the end of each process is the same (right) one. (p. 210).

Written material gives the reader many fewer clues as to the author's meaning in comparison with a lecture. Azzouni increases the comprehensibility of his message by doing things that would have been unheard of in a scholarly book on the philosophy of math thirty years ago.

He uses italics to emphasis the thrust of his message.
He uses abbreviations such as "it's".
He says "you" instead of "one": He does not say "If one adds two numbers fifteen times, one does something different each time…" This phrase would probably have been nominalized to incomprehensibility thirty years ago: "A computation with fifteen repetitions of the process of numerical addition of a fixed pair of integers involves fifteen distinct actions."

In abstractmath.org I deliberately adopt a style that is similar to Azzouni's, including "you" instead of "one", "it's" instead of "it is" (and the like), and many other tricks, including bulleted prose, setting off proclamations in purple prose, and so on. (See [2].) One difference is that I too use italics a lot (actually bold italics), but with a difference of purpose: I use it for phrases that I think a student should mark with a highlighter.

My discussion of modus ponens from the section Conditional Assertions illustrates some of these ideas:

Method of deduction: Modus ponens

The truth table for conditional assertions may be summed up by saying: The conditional assertion “If P, then Q” is true unless P is true and Q is false.

This fits with the major use of conditional assertions in reasoning:

Method of deduction

If you know that a conditional assertion is true and

you know that its hypothesis is true,

then you know its conclusion is true.

In symbols:

When “If P then Q” and P are both true,

______________________________________

then Q must be true as well.

This notation means that if the statements above the line are true, the statement below the line has to be true too.

This fact is called modus ponens and is the most used method of deduction of all.

Remark

That modus ponens is valid is a consequence of the truth table:

If P is true that means that one of the first two lines of the truth table holds.

If the assertion “If P then Q” is true, then one of lines 1, 3 or 4 must hold.

The only possibility, then, is that Q is true.

Remark

Modus ponens is not a method of proving conditional assertions. It is a method of using a conditional assertion in the proof of another assertion. Methods for proving conditional assertions are found in the chapter on forms of proof.

This section also includes a sidebar (common in magazines) that says: "The first statement of modus ponens does not require pattern recognition. The second one (in purple) does require it."

Informality, bulleted lists, italics for emphasis, highlighted text, sidebars, and so on all belong in academic prose, not just in popular articles and high school textbooks. There are plenty of other features about popular science articles that could be used in academic prose, too, and I will talk about them in later posts.

Note: Some features of popular science should not be used in academic prose, of course, such as renaming technical concepts as I discussed in the post of that name. An example is referring to simple groups as "atoms of symmetry", since many laymen would not be able to divorce their understanding of the words "simple" and "group" from the everyday meanings: "HOW can you say the Monster Group is SIMPLE??? You must be a GENIUS!"

References

[1] 18 Unconventional Essays on the Nature of Mathematics, by Reuben Hersh. Springer, 2005. ISBN 978-0387257174

[2] Attitude, in abstractmath.org.

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exposition, language

The revolution in technical exposition

2009/08/18 Charles Wells 1 Comment

Most of the posts on G&G are in the streams math or language. Many articles are also in various subcategories. The articles in each stream can be found by looking to the column to the left of this post and scrolling down to "categories". (That word has too many meanings…) I have added a new stream, exposition, and have put four earlier articles in the stream. They concern expository prose in the sciences.

Old fashioned mathematical and scientific exposition appears to be designed to put as many barriers as possible in the way of the reader. Some of its properties:

Highly formal
Full of pronouncements worded in an impersonal way (noun phrases, everything objectified)
All traces obliterated of how the results came to be discovered
No intuitive explanations

References [2] and [3] go into detail about some of these characteristics.

Steven Johnson, in the Invention of Air [1] describes the classical expository style of Isaac Newton as having these properties. (But see I saac buys him a prism ). He also says that Priestley's book [4] on electricity is in some sense the first popular science book. It is narrative, not didactic; it uses "I" a lot; it goes into great detail about how the experiments were conducted (read his account of Benjamin Franklin's experiments starting on page 222), including what were in his opinion the many mistakes of other researchers, and occasionally attempts intuitive descriptions of electricity.

I see that I accidentally published this post, so I will stop here and continue in another post.

References

[1] Steven Johnson, The Invention of Air. Riverhead Books, 2008. ISBN 9781594488528. Reviewed in my post on Priestley.

[2] O’Halloran, K. L. (2005), Mathematical Discourse: Language, Symbolism And Visual Images. Continuum International Publishing Group. ISBN 978-0826468574.

[3] Halliday, M. A. K. and J. R. Martin (1993), Writing Science: Literacy and Discursive Power. University
of Pittsburgh Press. ISBN 978-0822961031

[4] Joseph Priestley, The History and Present State of Electricity, with Original Experiments (1775).

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exposition

Joseph Priestley

2009/08/17 Charles Wells 2 Comments

The Invention of Air, by Steven Johnson. Riverhead Books, 2008. 978-1-59448-852-8. This is a biography of Joseph Priestly:

He discovered that, although animals put in a closed box with no source of air died pretty quickly, plants put in a similar box did not die. This led him to conceive a primitive form of the idea of the cycle of nature. (Note 1.)
He discovered oxygen (apparently not really based on the previous discovery above), but did not understand what he discovered. He continued to believe in phlogiston to the end of his life.
He invented soda water because he lived near a brewery.
He cofounded the first Unitarian Church in England and wrote extensively about the corruptions of Christianity such as the Trinity.
He supported America’s independence and the French Revolution. Concerning the latter, he exhibited considerable naiveté.
Because of the last two things listed, a mob burned down his house and laboratory, his church and the house of one of his supporters. In consequence he moved to America.
He engaged in much correspondence with Thomas Jefferson with the result that Jefferson was relieved to find that he could still consider himself a Christian, of the Unitarian variety, of course. (Nowadays Unitarians don’t consider themselves Christian but then they did.)
He wrote a bunch of sharp attacks on John Adams, in particular accusing him of dastardly behavior in signing the Alien and Sedition Act, and of opposing further advances of science. Guess which attack made Adams the most furious. (The latter.)
Thomas Jefferson and John Adams were bitter enemies for many years, but engaged in an extensive and reasonably polite correspondence during the last years of their lives. Much of the correspondence involved Adams defending himself against Priestley’s criticisms.

They never taught me all that in school! By the way, I probably got all sorts of things wrong in the summary above. So you’d better read the book from cover to cover.

Scientists should read this book, too; it gives them a new sense of how important they were regarded by the politicians in England, America and France, in comparison to these days. Politicians should read this book as well, but they won’t.

Popular science

The author claims (pp. 34-35) that Priestley’s work (Note 2) explaining the wonderful new discoveries about electricity constitute the first popular science book (at least of the narrative kind.)

Note

1. See Priestley’s Experiments and Observations on Different Kinds of Air, Volume III, Book 9, Part 1. (1790).

2. Joseph Priestley, The History and Present State of Electricity, with Original Experiments (1775).

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category theory, math, understanding math

Introduction to Wikibook on categories

2009/08/12 Charles Wells Leave a comment

Below is my newly rewritten introduction to the Wikibook on categories. I am posting it here because of course the Wikibook version is likely to change at any time.

== Introduction ==

This Wikibook is an introduction to category theory. It is written for those who have some understanding of one or more branches of abstract mathematics, such as group theory, analysis or topology. The book contains many examples drawn from various branches of math. If you are not familiar with some of the kinds of math mentioned, don’t worry. If practically all the examples are unfamiliar, this book may be too advanced for you.

===What is a category?===

A category is a mathematical structure, like a group or a vector space, abstractly defined by axioms. Groups were defined in this way in order to study symmetries (of physical objects and equations, among other things). Vector spaces are an abstraction of vector calculus.

What makes category theory different from the study of other structures is that in a sense the concept of category is an abstraction of a kind of mathematics. (This cannot be made into a precise mathematical definition!) This makes category theory unusually self-referential and capable of treating many of the same questions that mathematical logic treats. In particular, it provides a language that unifies many concepts in different parts of math.

In more detail, a category has objects and morphisms or arrows. (It is best to think of the morphisms as arrows: the word “morphism” makes you think they are set maps, and they are not always set maps. The formal definition of category is given in the chapter on categories.)

The category of groups has groups as objects and homomorphisms as arrows.
The category of vector spaces has vector spaces as objects and linear maps as arrows.

The maps between categories that preserve structure are called functors.

The underlying set of a group determines a functor from the category of groups to the category of sets.
The fundamental group of a pointed space determines a functor from the category of pointed topological spaces to the category of groups. The fact that it is a functor means that a continuous point-preserving map from a pointed space S to a pointed space T induces a group homomorphism from the fundamental group of S to the fundamental group of T.

Categories form a category as well, with functors as arrows. Most fundamentally, functors between specific categories form a category: its morphisms are called natural transformations. The fact that category theory has natural transformations is arguably the single feature that makes category theory so important.

===History===

Category theory was invented by Samuel Eilenberg and Saunders Mac Lane in the 1940’s as a way of expressing certain constructions in algebraic topology. Category theory was developed rapidly in the subsequent decades. It has become an autonomous part of mathematics, studied for its own sake as well as being widely used as a unified language for the expression of mathematical ideas relating different fields.

For example, algebraic topology relates domains of interest in geometry to domains of interest in algebra. Algebraic geometry, on the other hand, goes in the opposite direction, associating, for example, with each commutative ring its spectrum of prime ideals. These fields were among the earliest to be studied using tools of category theory. Later applications came to abstract algebra, logic, computing science and physics, among others.

===Aspects of category theory===

Because the concept of a category is so general, it is to be expected that theorems provable for all categories will not usually be very deep. Consequently, many theorems of category theory are stated and proved for particular classes of categories.

Homological algebra is concerned with Abelian categories, which exhibit features suggested by the category of Abelian groups.
Logic is studied using topos theory: a topos is a category with certain properties in common with the category of sets but which allows the logic of the topos to be weaker than classical logic. It is characteristic of the malleability of category theory that toposes were originally developed to study algebraic geometry.

An important use purpose of categorical reasoning is to identify within a given argument that part which is trivial and separate it from the part which is deep and proper to the particular context. For example, in the study of the theory of the GCD, the fact that it is essentially unique simply follows from the uniqueness of the product in any category and is thus really trivial. On the other hand, the fact that the GCD of the integers A and B can be expressed as a linear combination of A and B with integer coefficients—GCD(a, b) = ma + nb, for some integers m and n —is a much deeper fact that is special to a much more restricted situation.

===Note on terminology===

Most variations in terminology are discussed in the place where the terminology is defined. Here it is important to point out one annoying terminological problem: The adjective corresponding to “category” is “categorical”. Since “categorical” in logic means having only one model up to isomorphism, this can cause cognitive dissonance; in any case, the use of “categorical” in this book has nothing to do with the idea of having only one model.

Some authors use “categorial” instead. Unfortunately, this means something else in linguistics. This book follows majority usage with “categorical”.

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language

Must, have to, gotta

2009/08/06 Charles Wells 2 Comments

A volunteer helping in an intermediate-level ESL class reports that one day the teacher introduced “must” and “have to”, in contexts such as

You must renew your visa = You have to renew your visa.
You must have a ticket to get into the show = You have to have a ticket to get into the show.

In the volunteer’s discussion group this provoked two phenomena:

1. A heated discussion about “have to have”. Many students thought that was crazy and couldn’t figure out what it meant. They didn’t think “have to renew” was crazy, but the usage was unfamiliar to many of them.

2. Partway through the discussion in the subgroup moderated by the volunteer, a student suddenly Saw The Light: “They’re talking about GOTTA!” (You gotta renew your visa. You gotta have a ticket to get into the show.)

“Must” “have to” and “gotta” occur in three different registers of English. In America, in my experience, “must” is uncommon in speech and occurs mostly in formal writing. “Have to” (or “hafta”) is informal and widely used in both speech and writing. In street-conversation, “gotta” is the usual usage. It is uncommon in writing. “You gotta” would be spelled “You’ve got to”. (You do hear “you’ve gotta” as well as “you gotta”.)

New immigrants are exposed to English in the work place and on the street, not in the home and not usually in formal circumstances. The teacher should have given “gotta” as a third alternative way of saying “must” right from the start, since clearly that is the term most familiar to most of them. She should probably have also pointed out the pronunciation “hafta”, which is not obvious from the “have to” spelling unless you are a Sophisticated Amateur Linguist like me.

I should add that negating these expressions introduces complications. “Must not” does not mean the same thing as “don’t have to”. “Don’t got” is considered wrong, and plenty of people who say “gotta” in conversation, including me, don’t say “don’t gotta”; I would say “don’t have to” or “don’t need to” instead.

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exposition, language, language of math, math, representations

Commonword names for technical concepts

2009/06/30 Charles Wells 4 Comments

In a previous post I talked about the use of commonword names for technical concepts, for example, “simple group” for a group with no proper normal subgroups. This makes the monster group a simple group! Lay readers on the subject might very well feel terminally put-down by such usage. (If he calls that “simple” he must be a genius. How could I ever understand that? See note 1.) Mark Ronan used of “atom of symmetry” instead of “simple group” in his book Symmetry and the Monster, probably for some such reason.

Recently I had what used to be called a CAT scan and (perhaps) what used to be called a PET scan on the same day. The medically community now refers to CT scan or nuclear imaging. This may be because too many clients were thinking of doing sadistic testing on cats or other pets. But I have not been able to confirm that.

The nurse called the CT scan an x-ray. Well, of course, it is an x-ray, but it is an x-ray with tomography. She explicitly said that calling CT scans x-rays was common usage in their lab. In the past, other medical people have said to me, “It used to be called CAT scan but now it is CT scan.” But no one said why.

The situation about PET scan is more complicated. I didn’t raise the question with the nurse, and Wikipedia has separate articles about PET scans and nuclear imaging, even though they both use positrons and tomography. The chemicals mentioned for PET are isotopes of low-atomic-number elements, whereas the nuclear medicine article mentions technetium99 as the most commonly used isotope. Nowhere does it explain the difference. I wrote a querulous note in the comments section of the NM article requesting clarification.

Note 1. “If he calls that ‘simple’ he must be a genius. How could I ever understand that?” Do not dismiss this as the reaction of a stupid person. This kind of ready-to-be-intimidated attitude is very common among intelligent, educated, but non-technically-oriented people. If mathematicians dismiss people like that we will continue to find mathematics anathema among educated people. We need people to feel that they understand something about what mathematicians do (I use that wording advisedly). Even if you are an elitist who doesn’t give a damn about ordinary people, remember who funds the NSF. See co-intimidator.

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language of math, math, representations, understanding math

Mathematical concepts

2009/06/10 Charles Wells 5 Comments

This post was triggered by John Armstrong’s comment on my last post.

We need to distinguish two ideas: representations of a mathematical concept and the total concept. (I will say more about terminology later.)

Example: We can construct the quotient of the kernel of a group homomorphism by taking its cosets and defining a multiplication on them. We can construct the image of the homomorphism by take the set of values of the homomorphism and using the multiplication induced by the codomain group. The quotient group and the image are the same mathematical structure in the sense that anything useful you can say about one is true of the other. For example, it may be useful to know the cardinality of the quotient (image) but it is not useful to know what its elements are.

But hold on, as the Australians say, if we knew that the codomain was an Abelian group then we would know that the quotient group was abelian because the elements of the image form a subgroup of the codomain. (But the Australians I know wouldn’t say that.)

Now that kind of thinking is based on the idea that the elements of the image are “really” elements of the codomain whereas elements of the quotients are “really” subsets of the domain. That is outmoded thinking. The image and the quotient are the same in all important aspects because they are naturally isomorphic. We should think of the quotient as just as much as subgroup of the codomain as the image is. John Baez (I think) would say that to ask whether the subgroup embedding is the identity on elements or not is an evil question.

Let’s step back and look at what is going on here. The definition of the quotient group is a construction using cosets. The definition of the image is a construction using values of the homomorphism. Those are two different specific representations of the same concept.

But what is the concept, as distinct from its representations? Intuitively, it is

All the constructions made possible by the definition of the concept.
All the statements that are true about the concept.

(That is not precise.)

The total concept is like the clone plus the equational theory of a specific type of algebra in the sense of universal algebra. The clone is all the operations you can construct knowing the given signature and equations and the equational theory is the set of all equations that follow from them. That is one way of describing it. Another is the monad in Set that gives the type of algebra — the operations are the arrows and the equations are the commutative diagrams.

Note: The preceding description of the monad is not quite right. Also the whole discussion omits mention of the fact that we are in the world (doctrine) of universal algebra. In the world of first order logic, for example, we need to refer to the classifying topos of the category of algebras of that type (or to its first order theory).

Terminology

We need better terminology for all this. I am not going to propose better terminology, so this is a shaggy dog story.

Math ed people talk about a particular concept image of a concept as well as the total schema of the concept.

In categorical logic, we talk about the sketch or presentation of the concept vs. the theory. The theory is a category (of the kind appropriate to the doctrine) that contains all the possible constructions and commutative diagrams that follow from the presentation.

In this post I have used “total concept” to refer to the schema or theory. I have referred the particular things as “representations” (for example construct the image of a homomorphism by cosets or by values of the homomorphism).

“Representation” does not have the same connotations as “presentation”. Indeed a presentation of a group and a representation of a group are mathematically two different things. But I suspect they are two different aspects of the same idea.

All this needs to be untangled. Maybe we should come up with two completely arbitrary words, like “dostak” and “dosh”.

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Gyre&Gimble

Composites of functions

Grasshoppers and linear proofs

References

The revolution in technical exposition II

The revolution in technical exposition

Joseph Priestley

Introduction to Wikibook on categories

Must, have to, gotta

Commonword names for technical concepts

math, language and other things that may show up in the wabe