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 :
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 .) 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.
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.
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.
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!"
 18 Unconventional Essays on the Nature of Mathematics, by Reuben Hersh. Springer, 2005. ISBN 978-0387257174
 Attitude, in abstractmath.org.