Translating from English into Mathematics

Translatingfrom Englishinto Mathematics : Translating from English into Mathematics SFWR ENG 4G03 2002 Robert L. Baber

English and Mathematics as Languages : English and Mathematics as Languages English is a language. So is Mathematics. Both have rules of grammar (syntax) semantics When writing in any language, pay attention to grammar and semantics. Get both right.

English and Mathematics: A Difference : English and Mathematics: A Difference In English and other natural languages ambiguity desired, intentionally possible unambiguous statements almost impossible In Mathematics ambiguity not desired, intentionally prevented ambiguous statements almost impossible (even in probability theory, fuzzy logic)

Mathematics and Engineering : Mathematics and Engineering Therefore, mathematics is the language of engineering.

Different World Views : Different World Views English and other natural languages express both static and dynamic views states and actions (verbs of being and action) Programming languages primarily dynamic world view (changes) Mathematics static world view only Fundamental conceptual differences

Static vs. Dynamic Views : Static vs. Dynamic Views These very different world views pose a conceptual hurdle for the translator. The translator must bridge the gap between dynamic and static view of problem statement, dynamic world view of programming and purely static world view of mathematics. Not hard, but requires conscious attention.

Translating between Languages : Translating between Languages Translating a statement from one language to another is a multistep (not single) process. 1. statement in source language to a mental understanding of the meaning of the statement 2. reformulate mental understanding into target language view, concepts, culture 3. mental understanding of the meaning of the statement to a statement in the target language The first and last statements must mean the same.

Translators : Translators Knowing two languages: not enough to translate A good translator knows well the two languages AND the subject being translated AND how to translate These three things are different.

Organization and Style : Organization and Style When writing in English or any other natural language, one pays careful attention to organization of the essay, report, etc. style of expression When writing in Mathematics, to do the same: clear, complete, concise — KISSS understandable interesting

Strategies : Strategies Understand the meaning of the original Obtain all needed information Close the gap between the English text and mathematics Divide and conquer (complexity)

Strategy: Understand the original : Strategy: Understand the original describe specific instance of general problem distinguish essentials from background draw a diagram express in intermediate or mixed language identify objects referred to identify implicit (but false) "information" identify missing information identify relationships between essential objects identify special cases

Strategy: Obtain all needed information : Strategy: Obtain all needed information ask the author of the task description identify gaps in the description of the task identify implicit "information" ask if implicit "information" may be assumed identify data present and ask about related details ask if missing information is really needed read carefully, thoroughly, precisely

Strategy: Close gap English – math : Strategy: Close gap English – math express implicit information explicitly reduce vagueness and ambiguity reword English text to be closer to mathematics (express in intermediate, mixed language)

Strategy: Divide and conquer : Strategy: Divide and conquer construct a table distinguish between specific cases introduce an auxiliary mathematical function modularize

A Small Translator’s Glossary : A Small Translator’s Glossary English: Mathematics and, but:  (and) or:  (or) for all, each, every, any: ,  (and) series, universal quantification for no, none: ,  (and) series, universal quantification with a negated assertion there is (are), there exist(s), for some, at least one: ,  (or) series, existential quantification

A Small Translator’s Glossary : A Small Translator’s Glossary English: Mathematics integer: ...Z sorted: i=1n-1 A(i)A(i+1), (A i : iZ  1in-1 : A(i)A(i+1)) if (when, whenever) … then … : …  … search, find, equal, present: = exchange, rearrange, different order, different sequence, merge, copy, sort: permutation

Your Translator’s Glossary : Your Translator’s Glossary A professional translator compiles his/her own translation glossary over time based on own accumulated experience You should, too.

Exercise : Exercise Consider an array D with index values ranging from 1 to n. The subject of this example is part of a specification for a subprogram that will count how many times a particular given value occurs in the array D. The goal of this exercise is to write a postcondition for the subprogram, relating the various relevant variables’ values when the search is complete.

Exercise : Exercise Understand the task in the original language identify objects referred to (look for nouns in the original English text): array D, index value, times (count), particular given value, relevant variables' values identify missing information: names of variables for: index, times (count), particular given value. Are there any other relevant variables?

Exercise : Exercise Identify missing information: names of variables for index: assume "i" times (count): Ask the author of the task. assume "count" particular given value: Ask the author of the task. assume "key" Are there any other relevant variables? (no?)

Exercise : Exercise Close the gap between the English text and mathematics reword the English text to be closer to mathematics: the English verb count

Exercise : Exercise The English verb count means, in programming language and in terms closer to mathematics, add 1. But this is a dynamic (action) concept. The corresponding static (state, relational) concept in mathematics is the function addition with 1, i.e. +1.

Exercise : Exercise The occurrence of the particular given value in an array element in D, i.e. D(i)=key is a condition for the addition with 1. The repetition over a variable number of index values suggests quantification with the function addition and with the argument 1, i.e. (+ i : …  D(i)=key : 1)

Exercise : Exercise Identify relationships between essential objects: array D, index value, particular given value: D(i)=key above and count (+ conditionally with 1): count = (+ i : iZ  …i…  D(i)=key : 1) range of i missing. Refer to original English text: 1 to n. Then, count = (+ i : iZ  1in  D(i)=key : 1)

Exercise: New glossary entry : Exercise: New glossary entry Now we have a new entry for our glossary: count: (+ i : iZ  …i…  … : 1), where the … define the range of the quantified variable and the condition for counting

Summary : Summary Knowledge of English and Mathematics necessary but not sufficient to translate into Mathematics knowledge of subject area also needed translating skills needed The three are different.

Summary : Summary Compile your own glossary Make intermediate steps, expressions, languages conscious Modularize Organize systematically KISSS

Reference : Reference Baber, Robert L., Translating English to Mathematics, 2002, http://www.cas.mcmaster.ca/~baber/Courses/General/EnglToMath.pdf