An overview of
the history of mathematics
Mathematics
starts with counting. It is not reasonable, however, to suggest that early
counting was mathematics. Only when some record of the counting was kept and,
therefore, some representation of numbers occurred can mathematics be said to
have started.
In Babylonia mathematics developed from 2000 BC. Earlier a
place value notation number system had evolved over a lengthy period with a
number base of 60. It allowed arbitrarily large numbers and fractions to be
represented and so proved to be the foundation of more high powered
mathematical development.
Number
problems such as that of the Pythagorean triples (a,b,c)
with a2+b2 = c2 were
studied from at least 1700 BC. Systems of linear equations were studied in the
context of solving number problems. Quadratic equations were also studied and
these examples led to a type of numerical algebra.
Geometric
problems relating to similar figures, area and volume were also studied and
values obtained for π.
The Babylonian
basis of mathematics was inherited by the Greeks and independent development by
the Greeks began from around 450 BC. Zeno
of Elea's paradoxes led to the atomic theory of Democritus.
A more precise formulation of concepts led to the realization that the rational
numbers did not suffice to measure all lengths. A geometric formulation of
irrational numbers arose. Studies of area led to a form of integration.
The theory of
conic sections shows a high point
in pure mathematical study by Apollonius.
Further mathematical discoveries were driven by the astronomy, for example the
study of trigonometry.
The major
Greek progress in mathematics was from 300 BC to 200 AD. After this time
progress continued in Islamic countries. Mathematics flourished in particular
in Iran,
Syria
and India.
This work did not match the progress made by the Greeks but in addition to the
Islamic progress, it did preserve Greek mathematics. From about the 11th
Century Adelard
of Bath, then
later Fibonacci,
brought this Islamic mathematics and its knowledge of Greek mathematics back
into Europe.
Major progress
in mathematics in Europe began again at the beginning of the 16th
Century with Pacioli,
then Cardan,
Tartaglia
and Ferrari
with the algebraic solution of cubic and quartic equations. Copernicus
and Galileo
revolutionised the applications of mathematics to the study of the universe.
The progress
in algebra had a major psychological effect and enthusiasm for mathematical
research, in particular research in algebra, spread from Italy to Stevin
in Belgium
and Viète
in France.
The 17th
Century saw Napier,
Briggs
and others greatly extend the power of mathematics as a calculatory science
with his discovery of logarithms. Cavalieri
made progress towards the calculus with his infinitesimal methods and Descartes
added the power of algebraic methods to geometry.
Progress
towards the calculus continued with Fermat,
who, together with Pascal,
began the mathematical study of probability. However the calculus was to be the
topic of most significance to evolve in the 17th Century.
Newton,
building on the work of many earlier mathematicians such as his teacher Barrow,
developed the calculus into a tool to push forward the study of nature. His
work contained a wealth of new discoveries showing the interaction between
mathematics, physics and astronomy. Newton's
theory of gravitation and his theory of light take us into the 18th
Century.
However we
must also mention Leibniz,
whose much more rigorous approach to the calculus (although still
unsatisfactory) was to set the scene for the mathematical work of the 18th
Century rather than that of Newton.
Leibniz's
influence on the various members of the Bernoulli family was important in
seeing the calculus grow in power and variety of application.
The most
important mathematician of the 18th Century was Euler
who, in addition to work in a wide range of mathematical areas, was to invent
two new branches, namely the calculus of variations and differential geometry. Euler
was also important in pushing forward with research in number theory begun so
effectively by Fermat.
Toward the end
of the 18th Century, Lagrange
was to begin a rigorous theory of functions and of mechanics. The period around
the turn of the century saw Laplace's
great work on celestial mechanics as well as major progress in synthetic
geometry by Monge
and Carnot.
The 19th
Century saw rapid progress. Fourier's
work on heat was of fundamental importance. In geometry Plücker
produced fundamental work on analytic geometry and Steiner
in synthetic geometry.
Non-euclidean
geometry developed by Lobachevsky
and Bolyai
led to characterisation of geometry by Riemann.
Gauss,
thought by some to be the greatest mathematician of all time, studied quadratic
reciprocity and integer congruences. His work in differential geometry was to
revolutionise the topic. He also contributed in a major way to astronomy and
magnetism.
The 19th
Century saw the work of Galois
on equations and his insight into the path that mathematics would follow in
studying fundamental operations. Galois'
introduction of the group concept was to herald in a new direction for
mathematical research which has continued through the 20th Century.
Cauchy,
building on the work of Lagrange
on functions, began rigorous analysis and began the study of the theory of
functions of a complex variable. This work would continue through Weierstrass
and Riemann.
Algebraic
geometry was carried forward by Cayley
whose work on matrices and linear algebra complemented that by Hamilton
and Grassmann.
The end of the 19th Century saw Cantor
invent set theory almost single handedly while his analysis of the concept of
number added to the major work of Dedekind
and Weierstrass
on irrational numbers
Analysis was
driven by the requirements of mathematical physics and astronomy. Lie's
work on differential equations led to the study of topological groups and
differential topology. Maxwell
was to revolutionise the application of analysis to mathematical physics.
Statistical mechanics was developed by Maxwell,
Boltzmann
and Gibbs.
It led to ergodic theory.
The study of
integral equations was driven by the study of electrostatics and potential
theory. Fredholm's
work led to Hilbert
and the development of functional analysis.
Notation and
communication
There are many
major mathematical discoveries but only those which can be understood by others
lead to progress. However, the easy use and understanding of mathematical
concepts depends on their notation.
For example,
work with numbers is clearly hindered by poor notation. Try multiplying two
numbers together in Roman numerals. What is MLXXXIV times MMLLLXIX? Addition of
course is a different matter and in this case Roman numerals come into their
own, merchants who did most of their arithmetic adding figures were reluctant
to give up using Roman numerals.
What are other
examples of notational problems. The best known is probably the notation for
the calculus used by Leibniz
and Newton.
Leibniz's
notation lead more easily to extending the ideas of the calculus, while Newton's
notation although good to describe velocity and acceleration had much less
potential when functions of two variables were considered. British
mathematicians who patriotically used Newton's
notation put themselves at a disadvantage compared with the continental
mathematicians who followed Leibniz.
Let us think
for a moment how dependent we all are on mathematical notation and convention.
Ask any mathematician to solve ax = b and you will be given the
answer x = b/a. I would be very surprised if you were
given the answer a = b/x, but why not. We are, often
without realising it, using a convention that letters near the end of the
alphabet represent unknowns while those near the beginning represent known
quantities.
It was not
always like this: Harriot
used a as his unknown as did others at this time. The convention we use
(letters near the end of the alphabet representing unknowns) was introduced by Descartes
in 1637. Other conventions have fallen out of favour, such as that due to Viète
who used vowels for unknowns and consonants for knowns.
Of course ax
= b contains other conventions of notation which we use without noticing
them. For example the sign "=" was introduced by Recorde
in 1557. Also ax is used to denote the product of a and x,
the most efficient notation of all since nothing has to be written!
Brilliant
discoveries?
It is quite
hard to understand the brilliance of major mathematical discoveries. On the one
hand they often appear as isolated flashes of brilliance although in fact they
are the culmination of work by many, often less able, mathematicians over a
long period.
For example
the controversy over whether Newton
or Leibniz
discovered the calculus first can easily be answered. Neither did since Newton
certainly learnt the calculus from his teacher Barrow.
Of course I am not suggesting that Barrow
should receive the credit for discovering the calculus, I'm merely pointing out
that the calculus comes out of a long period of progress starting with Greek
mathematics.
Now we are in
danger of reducing major mathematical discoveries as no more than the luck of
who was working on a topic at "the right time". This too would be
completely unfair (although it does go some why to explain why two or more
people often discovered something independently around the same time). There is
still the flash of genius in the discoveries, often coming from a deeper
understanding or seeing the importance of certain ideas more clearly.
How we view
history
We view the
history of mathematics from our own position of understanding and
sophistication. There can be no other way but nevertheless we have to try to
appreciate the difference between our viewpoint and that of mathematicians
centuries ago. Often the way mathematics is taught today makes it harder to
understand the difficulties of the past.
There is no
reason why anyone should introduce negative numbers just to be solutions of
equations such as x + 3 = 0. In fact there is no real reason why
negative numbers should be introduced at all. Nobody owned -2 books. We can
think of 2 as being some abstract property which every set of 2 objects
possesses. This in itself is a deep idea. Adding 2 apples to 3 apples is one
matter. Realizing that there are abstract properties 2 and 3 which apply to
every sets with 2 and 3 elements and that 2 + 3 = 5 is a general theorem which
applies whether they are sets of apples, books or trees moves from counting
into the realm of mathematics.
Negative
numbers do not have this type of concrete representation on which to build the
abstraction. It is not surprising that their introduction came only after a
long struggle. An understanding of these difficulties would benefit any teacher
trying to teach primary school children. Even the integers, which we take as
the most basic concept, have a sophistication which can only be properly
understood by examining the historical setting.
A challenge
If you think
that mathematical discovery is easy then here is a challenge to make you think.
Napier,
Briggs
and others introduced the world to logarithms nearly 400 years ago. These were
used for 350 years as the main tool in arithmetical calculations. An amazing
amount of effort was saved using logarithms, how could the heavy calculations
necessary in the sciences ever have taken place without logs.
Then the world
changed. The pocket calculator appeared. The logarithm remains an important
mathematical function but its use in calculating has gone for ever.
Here is the
challenge. What will replace the calculator? You might say that this is an
unfair question. However let me remind you that Napier
invented the basic concepts of a mechanical computer at the same time as logs.
The basic ideas that will lead to the replacement of the pocket calculator are
almost certainly around us.
We can think
of faster calculators, smaller calculators, better calculators but I'm asking
for something as different from the calculator as the calculator itself is from
log tables. I have an answer to my own question but it would spoil the point of
my challenge to say what it is. Think about it and realize how difficult it was
to invent non-Euclidean geometries, groups, general relativity, set theory,
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