Tag Archives: azzouni

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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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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