Category Archives: math

abstractmath.org, math, representations, understanding math

Syntactic and semantic thinkers

2009/10/07 Charles Wells 6 Comments

A paper by Keith Weber

Reidar Mosvold’s math-ed blog recently provided a link to an article by Keith Weber (Reference [2]) about a very good university math student he referred to as a “syntactic reasoner”. He interviewed the student in depth as the student worked on some proofs suitable to his level. The student would “write the proofs out in quantifiers” and reason based on previous steps of the proof in a syntactic way rather than than depending on an intuitive understanding of the problem, as many of us do (the author calls us semantic reasoners). The student didn’t think about specific examples — he always tried to make them as abstract as possible while letting them remain examples (or counterexamples).

I recommend this paper if you are at all interested in math education at the university math major level — it is fascinating. It made all sorts of connections for me with other ideas about how we think about math that I have thought about for years and which appear in the Understanding Math part of abstractmath.org. It also raises lots of new (to me) questions.

Weber’s paper talks mostly about how the student comes up with a proof. I suspect that the distinction between syntactic reasoners and semantic reasoners can be seen in other aspects of mathematical behavior, too, in trying to understand and explain math concepts. Some thoughts:

Other behaviors of syntactic reasoners (maybe)

1) Many mathematicians (and good math students) explain math using conceptual and geometric images and metaphors, as described in Images and metaphors in abstractmath.org. Some people I think of as syntactic reasoners seem to avoid such things. Some of them even deny thinking in images and metaphors, as I discussed in the post Thinking without words. It used to be that even semantic reasoners were embarassed to used images and metaphors when lecturing (see the post How “math is logic” ruined math for a generation).

2) In my experience, syntactic reasoners like to use first order symbolic notation, for example

and will often translate a complicated sentence in ordinary mathematical English into this notation so they can understand it better. (Weber describes the student he interviewed as doing this.) Furthermore they seem to think that putting a formula such as the one above on the board says it all, so they don’t need to draw pictures, wave their hands [Note 1], and so on. When you come up with a picture of a concept or theorem that you claim explains it their first impulse is to say it out in words that generally can be translated very easily into first order symbolism, and say that is what is going on. It is a matter of what is primary.

The semantic reasoners of students and (I think) many mathematicians find the symbolic notation difficult to parse and would rather have it written out in English. I am pretty good at reading such symbolic notation [Note 2] but I still prefer ordinary English.

3) I suspect the syntactic reasoners also prefer to read proofs step by step, as I described in my post Grasshoppers and linear proofs, rather than skipping around like a grasshopper.

And maybe not

Now it may very well be that syntactic thinkers do not all do all those things I mentioned in (1)-(3). Perhaps the group is not cohesive in all those ways. Probably really good mathematicians use both techniques, although Weyl didn’t think so (quoted in Weber’s paper). I think of myself as an image and metaphor person but I do use syntax, and sometimes even find that a certain syntactic explanation feels like a genuinely useful insight, as in the example I discussed under conceptual in the Handbook.

Distinctions among semantic thinkers

Semantic thinkers differ among themselves. One demarcation line is between those who use a lot of visual thinking and those who use conceptual thinking which is not necessarily visual. I have known grad students who couldn’t understand how I could do group theory (that was in a Former Life, before category theory) because how could you “see” what was happening? But the way I think about groups is certainly conceptual, not syntactic. When I think of a group acting on a space I think of it as stirring the space around. But the stirring is something I feel more than I see. On the other hand, when I am thinking about the relationships between certain abstract objects, I “see” the different objects in different parts of an interior visual space. For example, group is on the right, stirring the space-acted-upon on the left, or the group is in one place, a subgroup is in another place while simultaneously being inside the group, and the cosets are grouped (sorry) together in a third place, being (guess what) stirred around by the group acting by conjugation (Note [3]).

This distinction between conceptual and visual, perhaps I should say visual-conceptual and non-visual-conceptual, both opposed to linguistic or syntactic reasoning, may or may not be as fundamental as syntactic vs semantic. But it feels fundamental to me.

Weber’s paper mentions an intriguing sounding book (Reference [1]) by Burton which describes a three-way distinction called conceptual, visual and symbolic, that sounds like it might be the distinction I am discussing here. I have asked for it on ILL.

Notes

Handwaving is now called kinesthetic communication. Just to keep you au courant.
I took Joe Shoenfield’s course in logic when his book Mathematical Logic [3] was still purple.
Clockwise for left action, counterclockwise for right action. Not.

References

Leone L. Burton, Mathematicians as Enquirers: Learning about Learning Mathematics. Springer, 2004.
Keith Weber, How syntactic reasoners can develop understanding, evaluate conjectures, and generate counterexamples in advanced mathematics. Proof copy available from Science Direct.
Joseph Shoenfield, Mathematical logic, Addison-Wesley 1967, reprinted 2001 by the Association for Symbolic Logic.

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category theory, computer science, math, Sketches and forms

Addenda to the 1993 Sketches paper

2009/09/15 Charles Wells Leave a comment

I have uploaded here a version of my 1993 sketches paper with an addendum listing a few relevant papers written since then. I have not kept up with the field well enough to contemplate a complete revision of the 1993 paper.

I recommend that more people update their papers this way. I did it by making a new PDF file with the added references and then using Acrobat to combine it with the old paper into one file. That way I didn’t have to re-TeX the old paper, which is a good thing, since I don’t know where some of the .sty files are.

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math, proof, understanding math

Mastering a proof

2009/09/12 Charles Wells 3 Comments

In response to Grasshoppers and linear proofs, Avery Andrews said:

Maybe a related question is how much time people do/ought spend on really mastering the proofs of theorems in textbooks, ‘mastering’ being, say, able to explain it in any desired amount of detail at least 2 weeks after last looking at it.

There are two different goals:

Mastering the proof of a theorem in a textbook so that you can explain it in any desired amount of detail…
Mastering a proof of the theorem so that you can explain it in any desired amount of detail…

My observation is that most research mathematicians don’t attempt (1); they are satisfied with (2). Trying to understand a written proof in detail can be quite difficult:

The author may use misleading language.
The author may jump over a piece of reasoning that to them is obvious but not to you.
The author may mention a previous step or a theorem that justifies the current step, but get the reference wrong.

And so on.

In my observation the typical mathematician will look at the proof, perhaps getting some idea of the overall strategy of the whole proof or a particular part, and then think about it independently until they come up with a proof or part of it. This may or may not be what the author had in mind. But by thinking through it the reader will solidify their understanding of the proof in a way that reading and rereading step by step is unlikely to do.

When you construct your knowledge like that you are likely to have it in a permanent, well semi-permanent, way.

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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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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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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

Category Archives: math

Syntactic and semantic thinkers

A paper by Keith Weber

Other behaviors of syntactic reasoners (maybe)

And maybe not

Distinctions among semantic thinkers

Notes

References

Addenda to the 1993 Sketches paper

Mastering a proof

Composites of functions

Grasshoppers and linear proofs

References

The revolution in technical exposition II

Introduction to Wikibook on categories

Commonword names for technical concepts

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