Mathsemantics: Making Numbers Talk Sense.

Edward MacNeal, a market research consultant specializing in air travel analysis, has served as executive secretary of the International Society for General Semantics. This puts him in a ringside seat from which to observe our many foibles and failings with elementary math. It all started when he and his wife devised a test for job applicants "good with numbers" and found that only about one in four could add 2 apples to 5 oranges and get 7 fruit. The whole test appears in an appendix for those who want a challenge. The discussion is, by design, not "rigorous" in the mathematical sense; instead, its commonsensical and conversational approach makes the most serious material highly readable.

Other books that overlap with this one are John Paulos' recent Innumeracy, and Darrell Huff's classic, How to Lie with Statistics. Mathsemantics shows why we have the problems described in the other two books, and to an extent how we might start avoiding them. MacNeal concludes that our problem is one of adult education; I agree, and further recommend that every teacher of elementary mathematics should be beaten over the head with this book until its contents are fully absorbed.

I found the discussion of environmental problems "by the numbers" in the last chapter unconvincing, for lack of good empirical grounding; and there is a silly confusion on page 83, taken from a quote from Gina Kolata, between the use of a slashed zero in "new math" to denote the empty set and the use of a slash in computer typefaces to avoid confusing 0 with O. My mathematical bogometers were also triggered by a casual reference to 19th century mathematicians as having "ruled out" division by zero (0/0 is a topic for a small book of its own, actually!) as if there were some nice way to define it. These little problems do not much diminish the work, and I would love to see someone properly equipped build on Mac-Neal's observations to patch some of the cracks in the philosophy of mathematics. Maybe this would make a good retirement project for the Mathsemantic Monitor?

Response by Edward MacNeal

I thank Craig Presson for reviewing my book, Mathsemantics, which I gather he liked. I just wish, of course, that he hadn't felt quite so much need to balance his praise with the three "little problems" he mentions, two of which concern nothing, uh, that is, zero.

My wife has asked me to report that she refuses to accept any "credit" for devising the recruitment quiz (and indeed I confess my responsibility on page 2 of the book). She will, however, admit to administering the quiz, as our personnel director, if pressed. I'm reminded to thank her once again, this time by a reader's suggestion that any new edition carry the subtitle: The true story of how Ed MacNeal found out the literal truth of "Good help really is hard to find."

While on exonerations, let me hasten to assure Presson and all readers that I didn't mean to imply that Gina Kolata or any computer-typeface manufacturer confuses the two meanings (empty set and zero) of a slashed zero. I was only trying to say that we're probably stuck with the symbol-confusion. I might hive added that because international-signage-convention uses slashes to mean "not," a slashed-zero should mean not-zero; right?

Presson is surely correct that a separate book could be written on the subject of division by zero (although not by me). My brief remark about 19th century mathematicians' "ruling out" division by zero was prompted by James Newman's footnote (on page 293 of The World of Mathematics) that although excluded by Martin Ohm as an inadmissible procedure as early as 1828, it wasn't until 1881 that the necessity of excluding division by zero was explained in elementary school books on algebra. I found the historical aspect interesting.

I'd love to learn from Presson's difficulties with my environmental remarks, potentially more consequential than my zero asides. The possible problems, however, are so numerous that -- to reduce waste -- I'll withhold comment until he gives me more to go on.

Finally, Presson flatters me outrageously by suggesting that "the Mathsemantic Monitor" might be able to "patch some of the cracks in the philosophy of mathematics." I really appreciate the warmth of the expressed thought, it's terribly nice, but math as such isn't my field, nor is the philosophy of mathematics. I visit those exalted levels of abstraction only to further a more worldly mathsemantic aim, that of making numbers talk sense. As Paulos said (Innumeracy, page 89), "Mathematics is too important to be left to the mathematicians."