Category Archives: exposition

exposition, Handbook, language, language of math, math, representations

Technical meanings clash with everyday meanings

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Recently (see note [a]) on MathOverflow, Colin Tan asked [1] “What does ‘kernel’ mean in ‘integral kernel’?”  He had noticed the different use of the word in referring to the kernels of morphisms.

I have long thought [2] that the clash between technical meanings and everyday meaning of technical terms (not just in math) causes trouble for learners.  I have recently returned to teaching (discrete math) and my feeling is reinforced — some students early in studying abstract math cannot rid themselves of thinking of a concept in terms of familiar meanings of the word.

One of the worst areas is logic, where “implies” causes well-known bafflement.   “How can ‘If P then Q’ be true if P is false??”  For a large minority of beginning college math students, it is useless to say, “Because the truth table says so!”.  I may write in large purple letters (see [3] for example) on the board and in class notes that The Definition of a Technical Math Concept Determines Everything That Is True About the Concept but it does not take.  Not nearly.

The problem seems to be worse in logic, which changes the meaning of words used in communicating math reasoning as well as those naming math concepts. But it is bad enough elsewhere in math.

Colin’s question about “kernel” is motivated by these feelings, although in this case it is the clash of two different technical meanings given to the same English word — he wondered what the original idea was that resulted in the two meanings.  (This is discussed by those who answered his question.)

Well, when I was a grad student I made a more fundamental mistake when I was faced with two meanings of the word “domain” (in fact there are at least four meanings in math).  I tried to prove that the domain of a continuous function had to be a connected open set.  It didn’t take me all that long to realize that calculus books talked about functions defined on closed intervals, so then I thought maybe it was the interior of the domain that was a, uh, domain, but I pretty soon decided the two meanings had no relation to each other.   If I am not mistaken Colin never thought the two meanings of “kernel” had a common mathematical definition.

It is not wrong to ask about the metaphor behind the use of a particular common word for a technical concept.  It is quite illuminating to get an expert in a subject to tell about metaphors and images they have about something.  Younger mathematicians know this.  Many of the questions on MathOverflow are asking just for that.  My recollection of the Bad Old Days of Abstraction and Only Abstraction (1940-1990?) is that such questions were then strongly discouraged.

Notes

[a] The recent stock market crash has been blamed [4] on the fact that computers make buy and sell decisions so rapidly that their actions cannot be communicated around the world fast enough because of the finiteness of the speed of light.  This has affected academic exposition, too.  At the time of writing, “recently” means yesterday.

References

[1] Colin Tan, “What does ‘kernel’ mean in ‘integral kernel’?

[2] Commonword names for technical concepts (previous blog).

[3] Definitions. (Abstractmath).

[4] John Baez, This weeks finds in mathematical physics, Week 297.

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exposition, language of math, math, representations, understanding math

Thinking about mathematical objects revisited

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How we think about X

It is notable that many questions posted at MathOverflow are like, “How should I think about X?”, where X can be any type of mathematical object (quotient group, scheme, fibration, cohomology and so on).  Some crotchety contributors to that group want the questions to be specific and well-defined, but “how do I think about…” questions  are in my opinion among the most interesting questions on the website.  (See note [a]).

Don’t confuse “How do I think about X” with “What is X really?” (pace Reuben Hersh).  The latter is a philosophical question.  As far as I am concerned, thinking about how to think about X is very important and needs lots of research by mathematicians, educators, and philosophers — for practical reasons: how you think about it helps you do it.   What it really is is no help and anyway no answer may exist.

Inert and eternal

The idea that mathematical objects should be thought of as  “inert” and “eternal”  has been around for awhile.  (Never mind whether they really are inert and eternal.)  I believe, and have said in the past [1], that thinking about them that way clears up a lot of confusion in newbies concerning logical inference.

  • That mathematical objects are “inert” means that the do not cause anything. They have no effect on the real world or on each other.
  • That they are “eternal” means they don’t change over time.

Naturally, a function (a mathematical object) can model change over time, and it can model causation, too, in that it can describe a process that starts in one state and achieves stasis in another state (that is just one way of relation functions to causation).  But when we want to prove something about a type of math object, our metaphorical understanding of them has to lose all its life and color and go dead, like the dry bones before Ezekiel started nagging them.

It’s only mathematical reasoning if it is about dead things

The effect on logical inference can be seen in the fact that “and” is a commutative logical operator. 

  • “x > 1 and x < 3″ means exactly the same thing as “x < 3 and x > 1″
  • “He picked up his umbrella and went outside” does not mean the same thing as “He went outside and picked up his umbrella”.

The most profound effect concerns logical implication.  “If  x > 1 then x > 0″ says nothing to suggest that x > 1 causes it to be the case that x > 0.  It is purely a statement about the inert truth sets of two predicates lying around the mathematical boneyard of objects:  The second set includes the first one.  This makes vacuous implication perfectly obvious.  (The number -1 lies in neither truth set and is irrelevant to the fact of inclusion).

Inert and eternal rethought

There are better metaphors than these.  The point about the number 3 is that you think about it as outside time. In the world where you think about 3 or any other mathematical object, all questions about time are meaningless.

  • In the sentence “3 is a prime”, we need a new tense in English like the tenses ancient (very ancient) Greek and Hebrew were supposed to have (perfect with gnomic meaning), where a fact is asserted without reference to time.
  • Since causation involves this happens, then this happens, all questions about causation are meaningless, too.  It is not true that 3 causes 6 to be composite, while being irrelevant to the fact that 35 is composite.

This single metaphor “outside time” thus can replace the two metaphors “inert” and “eternal” and (I think) shows that the latter two are really two aspects of the same thing.

Caveat

Thinking of math objects as outside time is a Good Thing when you are being rigorous, for example doing a proof.  The colorful, changing, full-of-life way of thinking of math that occurs when you say things like the statements below is vitally necessary for inspiring proofs and for understanding how to apply the mathematics.

  • The harmonic series goes to infinity in a very leisurely fashion.
  • A function is a machine — when you dump in a number it grinds away and spits out another number.
  • At zero, this function vanishes.

Acknowledgment

Thanks to Jody Azzouni for the italics (see [3]).

Notes

a.  Another interesting type of question  “in what setting does such and such a question (or proof) make sense?” .  An example is my question in [2].

References

1.  Proofs without dry bones

2. Where does the generic triangle live?

3. The revolution in technical exposition II.

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abstractmath.org, exposition, Handbook, language of math

Learning by osmosis

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In the Handbook, I said:

The osmosis theory of teaching is this attitude: We should not have to teach students to understand the way mathematics is written, or the finer points of logic (for example how quantifiers are negated). They should be able to figure these things on their own —“learn it by osmosis”. If they cannot do that they are not qualified to major in mathematics.

We learned our native language(s) as children by osmosis.  That does not imply that college students can or should learn mathematical reasoning that way. It does not even mean that college students should learn a foreign language that way.

I have been meaning to write a section of Understanding Mathematics that describes the osmosis theory and gives lots of examples.  There are already three links from other places in abstractmath.org that point to it.  Too bad it doesn’t exist…

Lately I have been teaching the Gauss-Jordan method using elementary row operations and found a good example.   The textbook uses the notation [m] +a[n] to mean “add a times row n to row m”.  In particular, [m] +[n] means “add row n to row m”, not “add row m to row n”. So in this notation ” [m] +[n] ” is not an expression, but a command, and in that command the plus sign is not commutative.   Similarly, “3[2]” (for example) does not mean “3 times row 2”, it means “change row 2 to 3 times row 2”.

The explanation is given in parentheses in the middle of an example:

…we add three times the first equation to the second equation.  (Abbreviation: [2] + 3[1].  The [2] means we are changing equation [2].  The expression [2] + 3[1] means that we are replacing equation 2 by the original equation plus three times equation 1.)

This explanation, in my opinion, would be incomprehensible to many students, who would understand the meaning only once it was demonstrated at the board using a couple of examples.  The phrase “The [2] means we are changing equation [2]” should have said something like “the left number, [2] in this case, denotes the equation we are changing.”  The last sentence refers to “the original equation”, meaning equation [2].  How many readers would guess that is what they mean?

In any case, better notation would be something like “[2] ← 3[1]”. I have found several websites that use this notation, sometimes written in the opposite direction. It is familiar to computer science students, which most of the students in my classes are.

Putting the definition of the notation in a parenthetical remark is also undesirable.  It should be in a separate paragraph marked “Notation”.

There is another point here:  No verbal definition of this notation, however well written, can be understood as well as seeing it carried out in an example.  This is also true of matrix multiplication, whose definition in terms of symbols such as a_ib_j is difficult to understand (if a student can figure out how you do it from this definition they should be encouraged to be a math major), whereas the process becomes immediately clear when you see someone pointing with one hand at successive entries in a row of one matrix while pointing with the other hand at successive entries in the other matrix’s columns.  This is an example of the superiority (in many cases) of pattern recognition over definitions in terms of strings of symbols to be interpreted.  I did write about pattern recognition, here.

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astounding math, computer science, exposition, math

Logarithms mod an integer

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Recently on MathOverflow the following question was asked: “9=2^x\ \text{mod}\ 11.   What is x and how do you find x?”  This question was soon closed as being inappropriate for MO.  In fact it is a very interesting question.

The general question is:  Solve a^x=b\ \text{mod}\ m (a, b, x, m all integers).  This is the discrete logarithm problem.  All known general algorithms for solving it run in exponential time.  It is used in cryptography, for example in the RSA algorithm and the Diffie-Hellman algorithm.

For example, the RSA algorithm is implement in this way:  Find two very large primes p and q.  This can be done in polynomial time.  Let m=pq.  Find integers d and e relatively prime to (p-1)(q-1) such that de = 1 \ \text{mod}\ m; this can also be done efficiently.  Note that for any integer a, a^{de}=a\ \text{mod}\ m.  You make m and e public but keep d, p and q private.  Then someone else can encode a message as an integer a and transmit a^e\ \text{mod}\ m (which can be calculated pretty fast)  to you.  Since you know the logarithm d you can decode it by calculating a^{de}=a\ \text{mod}\ m.   (What if the message is too long?  Details, details…)

The fact that you can find large primes fast but you can’t find discrete logarithms fast means that this method is safe.  The fact that no one can prove you can’t find discrete logarithms fast means cryptographers worry about this a lot.

 

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

Composites of functions

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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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exposition, language, math

The revolution in technical exposition II

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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

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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 Isaac 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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Joseph Priestley

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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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exposition, language, language of math, math, representations

Commonword names for technical concepts

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

Abstractmath.org after four years

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I have been working on the abstractmath website for about four years now (with time off for three major operations). Much has been written, but there are still lots of stubs that need to be filled in. Also much of it needs editing for stylistic uniformity, and for filling in details and providing more examples in some hastily written sections that read like outlines. Not to mention correcting errors, which seem to multiply when I am not looking. The website consists of four main parts and some ancillary chapters. I will go into more detail about some of the parts in later articles.

The languages of math. This is a description of mathematical English and the symbolic language of math (which are two different languages!) with an emphasis on the problems they cause people new to abstract math (roughly, math after calculus). At this point, I have completed a fairly thorough edit of the whole chapter that makes it almost presentable. Start with the Introduction.

Proofs. Mathematical proofs are a central problem for abstract math newbies. People interested in abstract math must learn to read and understand proofs. A proof is narrated in mathematical English. A proof has a logical structure. The reader must extract the logical structure from the narrative form. The chapter on proofs gives examples of proofs and discusses the logical structure and its relationship with the narration. The introduction to the chapter on proofs tells more about it.

Understanding math. There are certain barriers to understanding math that are difficult to get over. Mathematicians, math educators and philosophers work on various aspects of these problems and this chapter draws on their work and my own observations as a mathematician and a teacher.

All true statements about a math object must follow from the definition. That sounds clear enough. But in fact there are subtleties about definitions teachers may not tell students about because they are not aware of them themselves. For example, a definition can really mislead you about how to think about a math object.

The section on math objects breaks new ground (in my opinion) about how to think about them. I also discuss representations and models and images and metaphors (which I think is especially important), and in shorter articles about other topics such as abstraction and pattern recognition.

Doing math. This chapter points out useful behaviors and dysfunctional behaviors in doing math, with concrete examples. Beginners need to be told that when proving an elementary theorem they need to rewrite what is to be proved according to the definitions. Were you ever told that? (If you went to a Jesuit high school, you probably were.) Beginners need to be told that they should not try the same computational trick over and over even though it doesn’t work. That they need to look at examples. That they need to zoom in and out, looking at a detail and then the big picture. We need someone to make movies illustrating these things.

These other articles are outside the main organization:
Topic articles. Sets, real numbers, functions, and so on. In each case I talk just a bit about the topic to get the newbie over the initial hump.
Diagnostic examples. Examples chosen to evoke a misunderstanding, with a link to where it is explained. This needs to be greatly expanded.
Attitudes. This explains my point of view in doing abstractmath.org. I expect to rewrite it.

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