I propose a dictionary of mathematical terms to be defined SYNTACTICALLY.Typically, in a dictionary, the RELATION BETWEEN DEFIENDUM (term DEFINED) and DEFINIENS (term DEFINING) is SEMANTIC: between TERM and REFERENCE (MEANING). As I note elsewhere, this is a "semantic dictionary", whereas a TWO-LANGUAGE DICTIONARY (say, English-German or English-Latin) is a "syntactic dictionary".
That is, A TWO-LANGUAGE DICTIONARY IS (sometimes or mostly)
- A SYNTACTIC DICTIONARY in which A TERM IN LANGUAGE 1 IS REPLACED BY A TERM IN LANGUAGE 2 -- A RELATION ONLY BETWEEN SIGNS: THE SYNTACTIC RELATION
;But A ONE-LANGUAGE DICTIONARY IS
- A SEMANTIC DICTIONARY in which A TERM IS REPLACED BY A REFERENT: THE SEMANTIC RELATION
.That's WHAT WE NEED TO UNDERSTAND MATHEMATICS (AND SCIENCE)! MATHEMATICS IS A SPECIAL LANGUAGE, WHICH CAN RELATE TO ANY UNIVERSAL LANGUAGE. But do we have this courtesy? this TOOL? NO!!!
Our textbooks present ENGLISH TERMINOLOGY and MATHEMATICAL TERMINOLOGY -- BUT GUIDANCE IN THE REPLACEMENT STEP! (THE CRITICAL TEACHING STEP) IS OMITTED! (See, Ophelia, why "Mathematical Teaching" is so IATROGENIC? A FAULTY LEARNING PROCESS WHICH CREATES LEARNING DIFFICULTIES!)
But a MATHEMATICAL SYNTACTIC DICTIONARY would correct this egregious omission.
What I propose will do this more effectively than in any other way, since it makes use of a POWERFUL DEVICE ("BNF") OF COMPUTER SCIENCE WHICH HAS WITHSTOOD 40 YEARS OF ANALYSIS AND CRITICISM.
BNF: BACKUS NORMAL FORM -> BACK NAUR FORM In 1955, an IBM team, headed by computer scientist, John Backus, announced completion of FORTRAN, the second (after COBOL) "high-level programming language". To explain it, Backus developed a SYNTACTIC PROCEDURE for DEFINING TERMS. This became known as "BNF" ("Backus Normal Form"). Later, in 1960, an IBM team in Vienna, Austria, led by Peter Naur, announced the development of another high-level programming language, ALGOL. Naur modified the Backus Normal Form and used it to define terms in ALGOl. Dutch computer scientist, E. W. Dijkistra, suggested keeping the initials "BNF", and simply change the "N" from "Normal" to "Naur".
The basic form was as follows: <language-term-1>::= <language-term-2>. You REPLACE TERM-ON-LEFT by TERM-ON-RIGHT.
Can anything be simpler? You don't have to UNDERSTAND EITHER TERM. The TERM-ON-LEFT could be in Russian, spelled by the Cyrillic alphabet, and the TERM-ON-THE-RIGHT COULD BE IN THE ADVANCED MATH OF LIE GROUP THEORY. You simply REPLACE ONE BY THE OTHER! Purely SYNTACTIC -- RELATING SIGNS, WITHOUT REGARD FOR MEANING OR REFERENCE.
Backus said he used the "::=" CONNECTOR between terms because it wouldn't be confused with other notation. Many now drop one of the colons, and so shall we.
<language-term-1>::= <language-term-2>. You REPLACE TERM-ON-LEFT by TERM-ON-RIGHT.
I got the idea for this dictionary from a complaint of a former student. I'll use his problem to illustrate BNF.
"Hey, Prof! I took your 'Introductory" course and did OK. But the Chem Prof's giving me grief. What's he mean, 'ratio of pressure-to-
volume of a gas'? Sounds like maybe math. But I don't remember it."I'd gone over this. So, I adapted my explanation to his present case. "You have a measurement of the gas pressure and of the gas volume. To get their ratio, you DIVIDE THE PRESSURE MEASUREMENT BY THE VOLUME MEASUREMENT. Listen to me! When you hear 'ratio', think 'divide'!"
"Gotcha -- 'ratio' means 'divide'. Thanks!"
In BNF:
<ratio-of-pressure-to-volume>:= <pressure>/<volume>.Actually, this is applied math (in physics). And I don't want to get flypapered by dimensional discusssions, so I'll write, in BNF (but without colors), a purely mathematical ratio.
<ratio-of-three-to-four>:= 3/4
(Note that there are no brackets on the RIGHT. The brackets on the right in the "chemistry" example are necessary because we're referring to PRESSURE and VOLUME by the English language of the LEFT. If we put numbers in, we would drop the brackets, as we do in this last case.) Now, some simple examples.
<etsettery>:= ...
<numerals>;= 0, 1, 2, 3, ...
<natural numbers (denoted by numerals)>;= 0, 1, 2, 3, ...
<positive integers>:= +1, +2, +3, ...
<negative integers>:= -1, -2, -3, ...
<rational numbers (denoted as ratios of integers)>:= 1/2, 3/7, -5/11, ...
I CHALLENGE you to contribute to this effort to better the lives of teachers and students.