You've heard of Bourbaki. Of course, you have. The nice folks who inspired "The New Math"! Yes, Hamlet, I said "folks", because Bourbaki is man-y. Hype this.

In the 50's, some mathematicians at the University of Nancy, in Nancy, France, worried because many advances in mathematics were not appearing in textbooks. So they decided to do something about this. But, deciding it was a big job, requiring several mathematicians, but should look like the work of only one, they thought about using a pseudonym. What pseudonym?

A short time before this, students had pulled off an elaborate joke. Faculty and students of U. of Nancy were invited to hear the eminent physicist, Nicholas Bourbaki, speaking at one of the auditoriums. But the speaker appeared to have had a little too much dinner wine. After babbling incoherently, he collapsed into a snoring nap, so the audience tiptoed out. Next day, the joke was revealed and the reason for the name of the speaker, assumed by a student in disguise. The name "Bourbaki" was taken from a stupid General Bourbaki of World War I who suffered a disastrous defeat.

This suggested to the Nancy mathematicians the pseudonym for their collective writings: "Nicholas Bourbaki". The next few years, several volumes issued forth, under this authorship. One amusing item was the appearance of French road signs on the edge of the text, warning about "difficulties of travel" through the text.

Bourbaki advocated basing all of Mathematics on Set Theory. So, a few years later, some American mathematicians, working on a Grant from The National Science Foundation, developed what became known as "The New Math", basing it on Set Theory.

But I got involved with this the year before "The New Math" appeared in 1958. In 1957, I initiated and co-organized the first National Foundation Seminar in Puerto Rico -- for high school math teachers. I gave a series of "Lectures on The Foundations of Mathematics". Not knowing about "The New Math" program, to be announced the next year, yet much of my "program" was equivalent to what became known as "The New Math", but I vehemently objected to The Platonist view of set theory, since it did not jive with what needs to be taught in Arithmetic.

Briefly, standard sets ignore multiplicities of elements. Thus {a, a, a, b, b, c} = {a, b, c}, ignoring multiplicity of the first two elements. But this means that YOU CANNOT USE SETS TO DESCRIBE THE FACTORS OF COMPOSITE NUMBERS SUCH AS 120 = 2³·3²·5, since {2,2,2,3,3,5} = {2, 3, 5}, making -- SET-WISE -- THE FACTOR-SET of 120 LOOK THE SAME AS THE FACTOR-SET of 30! And, in general, AN INFINITY OF DISTINCT RESULTS COLLAPSE INTO A SINGLE SET! Thus, what became known as "The New Math" DID NOT SUPPORT "The Old Math".

Still not knowing what the NSF would do the next year, I advocated labeling the standard set as "the t-set", since it considered only TYPE ("kind"), and advocated an "o-set", which also considered ORDER ("degree"), say, writing for 120, [2,2,2,3,3,5]. Remember this was in the summer of 1957. In the 1960's, THE MULTISET appeared in the literature, as a HYBRID: MAPPING A SET INTO INTEGERS: {2,3,5} → 3,2,1. My form "kept it in the family". And when my o-sets are given indicator ("truth"-)tables, this leads from t-Logic to o-Logic, my 1957 anticipation of "Fuzzy Logic" in the 70's.

What was the objection to such an extension? Ah! The two most used NONCONSTRUCTIVE PROOFS became KAPUT! One proof invovled "The Axiom of Choice". (You can use it to prove that the moon can be subdivided into 5 parts, put back together and put in your pocket!) The other was "proof by contradiction" (a.k.a., tertium non dature. If you can't prove it directly, you assume it's false and get a contradiction -- as in double negation is positive.

Now, the Dutch mathematician, Jan Egbert Brouwer had objected to this, giving as an argument: "If a criminal, in committing a crime, manages to destroy the only evidence that can be used to convict him, must I declare him 'innocent'? No, I want a third verdict, 'Not proven'." (My o-Logic allows just that. And many years later a mathematical fable of mine was published, "The Battle of the Frog and the Mouse", taking its title from a comment from Albert Einstein about this argument. And I mention this and other arguments of Brouwer ("The Frog" against the Platonist arguments of David Hilbert ("The Mouse")).

Actually, as I learned after sketching my o-Logic, Scottish Jurisprudence allows this third verdict, "Not Proven". On late night television, you may see a 1950 movie, Madeleine, about the famous 19th century trial of heiress Madeleine Smith, accused poisoning her false lover by arsenic in a cup of chocolate. Tried in Scotland, the Crown could not prove that Madeleine put the arsenic in the chocolate but that she had it in her possession. The Jury did not wish to declare her "innocent", and returned a verdict of "Not Proven". So Madeleine Smith went "free" under a lifetime-cloud of suspicion. (The movie was directed by the great David Lean, who created it as a star-vehicle for Ann Todd, then Lean's wife.) Yes, Ophelia, you've heard of such a verdict recently. Senator Arlen Specter of Pennsylvania, in 1998, wished to have a verdict issued in the Senate of "Not Proven" regarding the charge of Clinton lying, if a "Guilty" charge was defeated.

Back to 1957. Soon after the end of the Seminar, I received a furious letter from an official of The National Science Foundation about my lectures. I could not understand why until "The New Math" came out next year, espousing the Bourbaki (and Platonist) position of set-theoretic-based Mathematics.

Over the years, I've submitted many proposals to NSF:

• Methodocopoeia (like the Pharmocopoeia of physicians) for teachers to explain different methods for teaching a concept or procedure.
• Handbook for Mathematics Teachers (like the Handbook for English Teachers), using icons and abbreviations in correcting papers so students can see the correct form in the Manual and participate in their learning process.

• Syntactic Dictionary of Mathematical and Scientific Terms, using BNF (Backus-Naur Form) for definitions, freeing teachers from having to act liking walking-dictionaries or tape-recorders.

• Program for Dividing Math Class Days into Days for the Strategy of Problem-Solving and Days for the Tactics of Problem-Solving.

• Study-Pert, allowing capable students to monitor their progress.

• Axiomatics vs. Generatics, let eahc student judge which presentation of Arithmetic and NumAlgebra "works" for her/him.

• Arithmetic-in-History: Show origin of Counting in tallies, tally-sticks, knots in cords, etc. and calculation in Roman Numerals and in Decimals, so students will appreciate the "short-cuts", but also see there are methods to fall back on.

• Mathematics Appreciation: Just as you can Appreciate Art with drawing or painting or scuplting, or Appreciate Music without singing or playing an instrument, you can Appreciate Mathematics by seeing its Strategies, without getting lost in its Tactics.

• Library of Science Kits, so students can prepare for lab courses, and afterward review what was covered via equipment.

• SKOGA: Science Kits for Growth Areas: more affluent children learn science by preparing kits and manuals for needy children.

• Taming the Scary Word-Problems or "Math-Lib" (emulating "Ad-Lib") to allow students to formulate their own word-problems, since the math is the abstract form, not the language.

None of these is available, after all these years. Yet, I NEVER RECEIVED ONE WORD OF ACKNOWLEDGEMENT FROM NSF REGARDING THESE PROPOSALS. By innocently challenging their 1958 program in 1957, I became PERSONA NON GRATA for a life-time!

So I agree with Hestenes about "the Bourbaki virus".