The Language of
Mathematics: Utilizing Math in Practice
by Robert L. Baber, Bob@RLBaber.de
last modified 2008 October 24
All rights to the material on this web
page
are reserved by the author, Robert L.
Baber.
What is mathematics? Someone once answered that question with "What
mathematicians do". That, of course, begs the question, "What do
mathematicians do?" They reason logically about things – artificial,
abstract things.
Many of those things, although artificial and abstract, are useful in
modelling actual things in the real world, for example
- structures of buildings, dams, bridges, roads, etc.
- mechanical devices and equipment
- machines, engines and all kinds of energy conversion devices and
systems
- vehicles of all types – land, underwater, water surface, air,
space
- electrical circuits, devices and equipment for many different
applications
- communication systems – wired and wireless – and their components
- systems for cryptography
- molecules, atoms, nuclei, subatomic particles
- chemical reactions and chemical reactors
- systems for generating and distributing electrical power
- nuclear decay and interaction processes, nuclear reactors
- heating and cooling systems
- computer software
- economies (production,
consumption, imports, exports, trade, income, employment, prices, etc.)
- financial and economic markets (prices, trading volumes, options,
derivative instruments, risk assessment, etc.)
- business systems and processes (sales, inventories, various
assets and liabilities, etc.)
- astronomical systems, processes and phenomena
- atmospheric processes and phenomena
- ecological systems and processes
- geological processes and phenomena
- biological systems and processes
- structural aspects of languages, natural and artificial
- etc., etc.
Such models enable
us to understand, describe and predict things in the real world better,
to our considerable benefit.
In order to reason logically about things, mathematicians have
developed a particular language with particular characteristics. That
language – the language of mathematics – and other languages developed
by human societies are similar in some respects and different in some
ways.
My experience learning, using and teaching others mathematics and how
to use it in practice has convinced me that looking at mathematics as a
language can facilitate the learning process, understanding and the
ability to apply mathematics in practice. I
believe that it can even enable some people to learn how to use
mathematics effectively who would otherwise be completely turned off
mathematics by their early exposure to it – and there are many such
people in today's world.
To help others to learn mathematics or to improve their ability to
apply it beneficially in their own work, I am currently writing a book
on
mathematics as a language. This book will be of interest to the
following groups of readers:
- persons with a general or an intellectual interest in
mathematics, science or language
- engineers, technicians, managers, consultants and others who
could benefit vocationally
and professionally by a greater ability to use and apply mathematics in
their work
- students in tertiary educational institutions
- students in secondary schools especially interested in
mathematics, science or languages
- teachers of mathematics, science or languages in tertiary
educational institutions (universities, polytechnics and vocational and
technical schools)
- teachers of mathematics, science or languages in secondary schools
- teachers in primary schools who introduce pupils to mathematics
and especially to word problems
The following links
will lead you to earlier material I wrote for undergraduate software
engineering students in the Department of Computing and Software at
McMaster University in Hamilton, Ontario, Canada. My new book on the
language of mathematics will expand considerably on this material.
If you have any comments or questions regarding the above material or
viewing mathematics as a language, please feel free to contact me by
email at Bob@RLBaber.de.
The
Language of Mathematics: Utilizing Math in Practice
Draft
table of contents
The following is the table of contents of the current draft of The Language of Mathematics: Utilizing
Math in Practice as of
2008 October 24. The book is not yet finished, so both the table of
contents and the title are subject to change.
Preface
1.
Introduction
What
is language? What is mathematics? Why use mathematics? Mathematics and
its
language. Goals and intended readership. Guidelines for the reader.
2.
Preview: Some Statements in English and the Language of Mathematics
Selected
examples: An ancient problem: planning the digging of a canal. A
numerical thought puzzle. A nursery rhyme. The energy in Earth's
reflected sunlight vs.
in produced crude oil.
3.
Elements of the Language of Mathematics
Values.
Variables. Functions. Expressions. Composing functions in standard
functional
notation. Infix notation. Tree notation. Prefix and postfix notation.
Tabular
notation. Graphical notation. Specialized notational forms for
expressions.
Advantages and disadvantages of the different notational forms.
Evaluating
variables, functions and expressions. Complete (total) evaluation.
Partial evaluation. Undefined values of functions and expressions.
4. Important structures and concepts in the
Language of Mathematics
Common
structures of values. Sets. Arrays (indexed variables) and subscripted
variables. Sequences. The equivalence of array variables, functions,
sequences and variables. Direct correspondence. Relations. Finite state
machines. Infinity.
Series and
quantification. Convergence and limits. Calculus. Probability theory.
Mathematical model of a probabilistic process. Mean, median, deviation
and variance. Independent probabilistic processes. Dependent probabilistic processes. Theorems.
Symbols and
notation.
5.
Solving problems mathematically
Manipulating
expressions. Proving theorems. Solving equations and other Boolean
expressions.
Solving optimization problems.
6.
Linguistic Characteristics of English and the Language of Mathematics
Universe
of discourse. Linguistic elements in the Language of Mathematics and in
English.
Grammatical agreement in the Language of Mathematics. Verbs: tense,
mood,
voice, action vs. state or being. Ambiguity. Limitations and
extendability of
the Language of Mathematics. The languages used in mathematical text.
Evaluating statements in English and expressions in the Language of
Mathematics. Meanings of Boolean expressions in an English language
context.
Mathematical
models and their interpretation. Dimensions of numerical variables.
Example:
Mixing components.
7.
Translating English to Mathematics
General
considerations. Sentences of the form “… is (a) …” (Singular forms).
Sentences
of the form “…s are …s” (Plural forms). Describing dynamic processes in
the
Language of Mathematics. Accuracy, errors and discrepancies in
mathematical models. Errors translating the actual problem into
English. Errors translating the English text
into
a mathematical model. Errors transforming
the mathematical model into a mathematical solution. Errors transforming the mathematical solution into an
implemented solution. Examples: Criterion for
searching an array,
Students
with the same birthday, A logical puzzle, Specifying the initial state
of a
board game, Model of a simple card game, Controlling the water level in
a
reservoir, Shopping mall door controller.
8.
Conclusions
Appendix
A. Representing numbers
Appendix
B. Symbols in the Language of Mathematics
Appendix
C. Sets of numbers
Appendix
D. Special structures in mathematics
Appendix
E. Mathematical logic
Appendix
F. Waves
and the wave Equation
Appendix
G. Glossary:
English to the Language of Mathematics
Appendix H. Programming languages and
the Language of Mathematics
Appendix
I. Other literature