How on earth did this happen?
Circa 500,000 years ago, in the Northern Cape region of
South Africa, on an overcast day, with clouds that looked dark and cerebral and
seemed to grumble angrily, a homo heidelbergensis male called Bob
picked up the big chunk of Obsidian lying on the ground in front of him and
bashed it with the smaller rock resting in his other hand. A small, roughly
triangular chunk of obsidian with a sharp point fell off the big chunk. Bob stared
at it in appreciation. Suddenly, he had an epiphany. He picked up the small
chunk with his right hand, manipulated it in his hand – so that its sharp end
was pointing downwards – raised it upwards, and
jammed it into his left wrist. A sharp sear of pain; Bob grunted. He
looked down at his left wrist: a drop of blood had appeared on its surface. So
he had split his skin with the rock! But could it do the same to an animal?
Leaving that question aside for
the moment, Bob decided to communicate the basic discovery that he had made a
rock sharp enough to penetrate skin to the only person in the vicinity: the
female he’d had sex with last night, Marissa, who was crouched on the rocky
platform nearby, drawing crude faces in the sand. “I just made a piece of obsidian
so sharp that it was able to pierce the skin on my wrist,” Bob cried out.
“Like I give a shit about that,
Bob.”
“But, think about it, Marissa.”
At that point, Bob hadn’t figured out what she was actually supposed to think
about. Fortunately, just as the pause was about to linger long enough to expose
him as a fraud, he had another epiphany: “What if, somehow, we could attach
this to a stick and use it to hurt animals?”
Within a month the first
prototype had been developed. And within two, the first kill had been made with
this new tool which we now know, in English,
as
a “spear”.
Then many thousands of years
passed and Bob’s very distant descendant, who was of the species homo sapiens, was
walking up a hill somewhere in Jordan. When he reached the top, he looked down,
in wonder, to see an area in which large swathes of wheat were rippling and swaying
in the wind. Consequently, he ran back to his community’s temporary encampment
about ten kilometres away and ran around telling everyone about it. Within a relatively
short period of time, the entire community had relocated there and, within a relatively short
period of time after that, they had built some shacks. And the longer they
stayed in this same place, the more solid these shacks became.
While in the fields a year after
the discovery of the swathes of wheats, Bob’s distant descendant’s second cousin suddenly
realised that, if they gathered enough wheat seeds, they could plant their own
crops. Within a month, they were sowing seeds for the first time. Within 12,
they were eating the first product of agriculture. As the years passed, the community
turned to a village then a town then to the world’s first city, and other
settlements began to arise near it. Over many thousands of years, the concept
of agriculture spread to most places around the globe, and eventually there
began to exist quite a few sizeable cities with complex societies and complex
water and sewerage systems.
Many thousands of years after the
birth of agriculture, the distant descendant of Bob’s distant descendant (and also
the distant descendant of Bob’s distant descendant’s second cousin) was trying to
buy a cow from another one of Bob’s distant descendant’s distant descendants, but
did not have the requisite grain. However, as he really needed a cow for his
family, he negotiated hard with the vendor and was eventually able to convince
him to relinquish the cow with the corollary that the vendor would carve a notch
into his stall to connote that grain was still owed by the customer. Over the
next few days, the vendor told all his friends about this idea, and gradually
this idea of ‘I owe yous’ caught on, and within a few years the act of carving
into stalls was carried out so frequently that the vendors began to look for
other ways of keeping track of goods owed by customers. Eventually, they discovered
the efficiency of putting a liquid we now know as “ink” on a sharp stick and
using this sharp stick to imprint the important details on a material we would
now call “crude paper”. Gradually, over many decades, this technology increased
in sophistication and began to be used for ever more diverse functions. What we
now call a “complex written language” gradually arose from such use. This was a
truly great leap forward for human civilisation and really accelerated our
progress. Once we could write things down, every human being trying to figure
out a problem would be able to stand on the shoulders of multitudinous others,
thus effectively serving to amplify our individual intelligence a millionfold.
The rise of empire, the rise of
mass religion. Then the rise of mathematics: the Babylonians and the Egyptians
both developed geometry, multiplication, division, the Pythagorean Theorem (the
theorem was recognised before Pythagoras himself), algebra, linear and
quadratic equations, and the Greeks benefitted from this to make huge advances.
As I am myself no expert on the history of mathematics, the following comes
from the Wikipedia page called the History
of Mathematics: “Greek mathematics was much more sophisticated than the
mathematics that had been developed by earlier cultures. All surviving records
of pre-Greek mathematics show the use of inductive reasoning, that is, repeated
observations used to establish rules of thumb. Greek mathematicians, by
contrast, used deductive reasoning. The Greeks used logic to derive conclusions
from definitions and axioms, and used mathematical rigour to prove them.
Greek mathematics is thought to
have begun with Thales of Miletus (c. 624–c.546 BC) and Pythagoras
of Samos (c. 582–c. 507 BC). Although the extent of the influence is
disputed, they were probably inspired by Egyptian and Babylonian
mathematics. According to legend, Pythagoras travelled to Egypt to learn
mathematics, geometry, and astronomy from Egyptian priests.
Thales used geometry to
solve problems such as calculating the height of pyramids and the
distance of ships from the shore. He is credited with the first use of
deductive reasoning applied to geometry, by deriving four corollaries to Thales'
Theorem. As a result, he has been hailed as the first true mathematician and
the first known individual to whom a mathematical discovery has been
attributed. Pythagoras established the Pythagorean School, whose
doctrine it was that mathematics ruled the universe and whose motto was
"All is number". It was the Pythagoreans who coined the term
"mathematics", and with whom the study of mathematics for its own
sake begins. The Pythagoreans are credited with the first proof of the Pythagorean
theorem, though the statement of the theorem has a long history, and with the
proof of the existence of irrational numbers.
Plato (428/427 BC – 348/347
BC) is important in the history of mathematics for inspiring and guiding
others. His Platonic Academy, in Athens, became the mathematical centre of
the world in the 4th century BC, and it was from this school that the leading
mathematicians of the day, such as Eudoxus of Cnidus, came. Plato
also discussed the foundations of mathematics, clarified some of the
definitions (e.g. that of a line as "breadthless length"), and
reorganized the assumptions. The analytic method is ascribed to
Plato, while a formula for obtaining Pythagorean triples bears his name.
Eudoxus (408–c.355 BC)
developed the method of exhaustion, a precursor of modern integration and
a theory of ratios that avoided the problem of incommensurable magnitudes.
The former allowed the calculations of areas and volumes of curvilinear
figures, while the latter enabled subsequent geometers to make significant
advances in geometry. Though he made no specific technical mathematical
discoveries, Aristotle (384—c.322 BC) contributed significantly to
the development of mathematics by laying the foundations of logic.
In the 3rd century BC, the
premier centre of mathematical education and research was the Musaeum of Alexandria. It
was there that Euclid (c. 300 BC) taught, and wrote the Elements,
widely considered the most successful and influential textbook of all
time. The Elements introduced mathematical rigor through
the axiomatic method and is the earliest example of the format still
used in mathematics today, that of definition, axiom, theorem, and proof.
Although most of the contents of the Elements were already
known, Euclid arranged them into a single, coherent logical framework. The Elements was
known to all educated people in the West until the middle of the 20th century
and its contents are still taught in geometry classes today. In addition
to the familiar theorems of Euclidean geometry, the Elements was
meant as an introductory textbook to all mathematical subjects of the time,
such as number theory, algebra and solid geometry, including
proofs that the square root of two is irrational and that there are infinitely
many prime numbers. Euclid also wrote extensively on other subjects,
such as conic sections, optics, spherical geometry, and
mechanics, but only half of his writings survive.
Archimedes (c.287–212 BC)
of Syracuse, widely considered the greatest mathematician of
antiquity, used the method of exhaustion to calculate the area under
the arc of a parabola with the summation of an infinite series,
in a manner not too dissimilar from modern calculus. He also showed one
could use the method of exhaustion to calculate the value of π with
as much precision as desired, and obtained the most accurate value of π then
known, 310⁄71 < π < 310⁄70. He
also studied the spiral bearing his name, obtained formulas for
the volumes of surfaces of revolution (paraboloid,
ellipsoid, hyperboloid), and an ingenious system for expressing very large
numbers. While he is also known for his contributions to physics and
several advanced mechanical devices, Archimedes himself placed far greater
value on the products of his thought and general mathematical principles. He
regarded as his greatest achievement his finding of the surface area and volume
of a sphere, which he obtained by proving these are 2/3 the surface area and
volume of a cylinder circumscribing the sphere.
Apollonius of Perga (c.
262-190 BC) made significant advances to the study of conic sections,
showing that one can obtain all three varieties of conic section by varying the
angle of the plane that cuts a double-napped cone. He also coined the
terminology in use today for conic sections, namely parabola ("place
beside" or "comparison"), "ellipse"
("deficiency"), and "hyperbola" ("a throw
beyond"). His work Conics is one of the best known
and preserved mathematical works from antiquity, and in it he derives many
theorems concerning conic sections that would prove invaluable to later
mathematicians and astronomers studying planetary motion, such as Isaac
Newton. While neither Apollonius nor any other Greek mathematicians made
the leap to coordinate geometry, Apollonius' treatment of curves is in some
ways similar to the modern treatment, and some of his work seems to anticipate
the development of analytical geometry by Descartes some 1800 years later.
Around the same time, Eratosthenes
of Cyrene (c. 276-194 BC) devised the Sieve of Eratosthenes for
finding prime numbers. The 3rd century BC is generally regarded as
the "Golden Age" of Greek mathematics, with advances in pure
mathematics henceforth in relative decline. Nevertheless, in the centuries that
followed significant advances were made in applied mathematics, most
notably trigonometry, largely to address the needs of astronomers. Hipparchus
of Nicaea (c. 190-120 BC) is considered the founder of trigonometry for
compiling the first known trigonometric table, and to him is also due the
systematic use of the 360 degree circle. Heron of Alexandria (c. 10–70 AD)
is credited with Heron's formula for finding the area of a scalene
triangle and with being the first to recognize the possibility of negative
numbers possessing square roots. Menelaus of Alexandria (c. 100 AD)
pioneered spherical trigonometry through Menelaus' theorem. The
most complete and influential trigonometric work of antiquity is the Almagest of
Ptolemy (c. AD 90-168), a landmark astronomical treatise whose
trigonometric tables would be used by astronomers for the next thousand years. Ptolemy
is also credited with Ptolemy's theorem for deriving trigonometric
quantities, and the most accurate value of π outside of China until the
medieval period, 3.1416.
Following a period of stagnation
after Ptolemy, the period between 250 and 350 AD is sometimes referred to as
the "Silver Age" of Greek mathematics. During this period, Diophantus made
significant advances in algebra, particularly indeterminate analysis,
which is also known as "Diophantine analysis". The study
of Diophantine equations and Diophantine approximations is a
significant area of research to this day. His main work was the Arithmetica,
a collection of 150 algebraic problems dealing with exact solutions to
determinate and indeterminate equations. The Arithmetica had
a significant influence on later mathematicians, such as Pierre de Fermat,
who arrived at his famous Last Theorem after trying to generalize a
problem he had read in the Arithmetica (that of dividing a
square into two squares). Diophantus also made significant advances in
notation, the Arithmetica being the first instance of
algebraic symbolism and syncopation.”
“Medieval European interest in mathematics was driven by
concerns quite different from those of modern mathematicians. One driving
element was the belief that mathematics provided the key to understanding the
created order of nature, frequently justified by Plato's Timaeus and
the biblical passage (in the Book of Wisdom) that God had ordered
all things in measure, and number, and weight.
Boethius provided a place
for mathematics in the curriculum in the 6th century when he coined the
term quadrivium to describe the study of arithmetic, geometry,
astronomy, and music. He wrote De institutione arithmetica, a free
translation from the Greek of Nicomachus's Introduction to
Arithmetic; De institutione musica, also derived from Greek
sources; and a series of excerpts from Euclid's Elements. His
works were theoretical, rather than practical, and were the basis of mathematical
study until the recovery of Greek and Arabic mathematical works.
In the 12th century, European
scholars traveled to Spain and Sicily seeking scientific Arabic texts,
including al-Khwārizmī's The Compendious Book on Calculation by
Completion and Balancing, translated into Latin by Robert of Chester,
and the complete text of Euclid's Elements, translated in
various versions by Adelard of Bath, Herman of Carinthia, and Gerard
of Cremona.
These new sources sparked a
renewal of mathematics. Fibonacci, writing in the Liber Abaci,
in 1202 and updated in 1254, produced the first significant mathematics in
Europe since the time of Eratosthenes, a gap of more than a thousand
years. The work introduced Hindu-Arabic numerals to Europe, and
discussed many other mathematical problems.
The 14th century saw the
development of new mathematical concepts to investigate a wide range of
problems. One important contribution was development of mathematics of
local motion.
Thomas Bradwardine proposed
that speed (V) increases in arithmetic proportion as the ratio of force (F) to
resistance (R) increases in geometric proportion. Bradwardine expressed this by
a series of specific examples, but although the logarithm had not yet been
conceived, we can express his conclusion anachronistically by writing: V = log
(F/R). Bradwardine's analysis is an example of transferring a mathematical
technique used by al-Kindi and Arnald of Villanova to
quantify the nature of compound medicines to a different physical problem.[119]
One of the
14th-century Oxford Calculators, William Heytesbury,
lacking differential calculus and the concept of limits,
proposed to measure instantaneous speed "by the path that would be
described by [a body] if... it were moved uniformly at the same
degree of speed with which it is moved in that given instant".
Heytesbury and others
mathematically determined the distance covered by a body undergoing uniformly
accelerated motion (today solved by integration), stating that "a
moving body uniformly acquiring or losing that increment [of speed] will
traverse in some given time a [distance] completely equal to that which it
would traverse if it were moving continuously through the same time with the
mean degree [of speed]".
Nicole Oresme at
the University of Paris and the Italian Giovanni di
Casali independently provided graphical demonstrations of this
relationship, asserting that the area under the line depicting the constant
acceleration, represented the total distance traveled. In a later mathematical
commentary on Euclid's Elements, Oresme made a more detailed
general analysis in which he demonstrated that a body will acquire in each
successive increment of time an increment of any quality that increases as the
odd numbers. Since Euclid had demonstrated the sum of the odd numbers are the
square numbers, the total quality acquired by the body increases as the square
of the time.”
“During the Renaissance, the development of mathematics
and of accounting were intertwined. While there is no direct
relationship between algebra and accounting, the teaching of the subjects and
the books published often intended for the children of merchants who were sent
to reckoning schools (in Flanders and Germany) or abacus
schools (known as abbaco in Italy), where they learned
the skills useful for trade and commerce. There is probably no need for algebra
in performing bookkeeping operations, but for complex bartering
operations or the calculation of compound interest, a basic knowledge of
arithmetic was mandatory and knowledge of algebra was very useful.
Luca Pacioli's "Summa
de Arithmetica, Geometria, Proportioni et Proportionalità" (Italian:
"Review of Arithmetic, Geometry, Ratio and Proportion")
was first printed and published in Venice in 1494. It included a
27-page treatise on bookkeeping, "Particularis de
Computis et Scripturis" (Italian: "Details of Calculation
and Recording"). It was written primarily for, and sold mainly to,
merchants who used the book as a reference text, as a source of pleasure from
the mathematical puzzles it contained, and to aid the education of
their sons. In Summa Arithmetica, Pacioli introduced symbols
for plus and minus for the first time in a printed book, symbols that
became standard notation in Italian Renaissance mathematics. Summa
Arithmetica was also the first known book printed in Italy to
contain algebra. It is important to note that Pacioli himself had borrowed
much of the work of Piero Della Francesca whom he plagiarized.
In Italy, during the first half
of the 16th century, Scipione del Ferro and Niccolò Fontana
Tartaglia discovered solutions for cubic equations. Gerolamo
Cardano published them in his 1545 book Ars Magna, together
with a solution for the quartic equations, discovered by his
student Lodovico Ferrari. In 1572 Rafael Bombelli published
his L'Algebra in which he showed how to deal with
the imaginary quantities that could appear in Cardano's formula for
solving cubic equations.
Simon Stevin's book De
Thiende ('the art of tenths'), first published in Dutch in 1585,
contained the first systematic treatment of decimal notation, which
influenced all later work on the real number system.
Driven by the demands of
navigation and the growing need for accurate maps of large
areas, trigonometry grew to be a major branch of
mathematics. Bartholomaeus Pitiscuswas the first to use the word,
publishing his Trigonometria in 1595. Regiomontanus's table of
sines and cosines was published in 1533.
During the Renaissance the desire
of artists to represent the natural world realistically, together with the rediscovered
philosophy of the Greeks, led artists to study mathematics. They were also the
engineers and architects of that time, and so had need of mathematics in any
case. The art of painting in perspective, and the developments in geometry that
involved, were studied intensely.”
“The 17th century saw an unprecedented explosion of
mathematical and scientific ideas across Europe. Galileo observed the
moons of Jupiter in orbit about that planet, using a telescope based on a toy
imported from Holland. Tycho Brahe had gathered an enormous quantity
of mathematical data describing the positions of the planets in the sky. By his
position as Brahe's assistant, Johannes Kepler was first exposed to
and seriously interacted with the topic of planetary motion. Kepler's
calculations were made simpler by the contemporaneous invention
of logarithms by John Napier and Jost Bürgi. Kepler
succeeded in formulating mathematical laws of planetary motion. The analytic
geometry developed by René Descartes (1596–1650) allowed those
orbits to be plotted on a graph, in Cartesian coordinates. Simon
Stevin (1585) created the basis for modern decimal notation capable of
describing all numbers, whether rational or irrational.
Building on earlier work by many
predecessors, Isaac Newton discovered the laws of physics
explaining Kepler's Laws, and brought together the concepts now known
as calculus. Independently, Gottfried Wilhelm Leibniz, who is
arguably one of the most important mathematicians of the 17th century, developed
calculus and much of the calculus notation still in use today. Science and
mathematics had become an international endeavor, which would soon spread over
the entire world.
In addition to the application of
mathematics to the studies of the heavens, applied mathematics began
to expand into new areas, with the correspondence of Pierre de
Fermat and Blaise Pascal. Pascal and Fermat set the groundwork for
the investigations of probability theory and the corresponding rules
of combinatorics in their discussions over a game of gambling.
Pascal, with his wager, attempted to use the newly developing probability
theory to argue for a life devoted to religion, on the grounds that even if the
probability of success was small, the rewards were infinite. In some sense,
this foreshadowed the development of utility theory in the 18th–19th
century.”
“The most influential mathematician of the 18th century was
arguably Leonhard Euler. His contributions range from founding the study
of graph theory with the Seven Bridges of Königsberg problem to
standardizing many modern mathematical terms and notations. For example, he
named the square root of minus 1 with the symbol i, and he
popularized the use of the Greek letter
to stand for the ratio of a circle's circumference to
its diameter. He made numerous contributions to the study of topology, graph
theory, calculus, combinatorics, and complex analysis, as evidenced by the
multitude of theorems and notations named for him.
Other important European
mathematicians of the 18th century included Joseph Louis Lagrange, who did
pioneering work in number theory, algebra, differential calculus, and the
calculus of variations, and Laplace who, in the age of Napoleon,
did important work on the foundations of celestial mechanics and
on statistics.”
“Throughout the 19th century mathematics became increasingly
abstract. In the 19th century lived Carl Friedrich Gauss (1777–1855).
Leaving aside his many contributions to science, in pure
mathematics he did revolutionary work on functions of complex
variables, in geometry, and on the convergence of series. He gave the
first satisfactory proofs of the fundamental theorem of algebra and
of the quadratic reciprocity law.
This century saw the development
of the two forms of non-Euclidean geometry, where the parallel
postulate of Euclidean geometry no longer holds. The Russian
mathematician Nikolai Ivanovich Lobachevsky and his rival, the
Hungarian mathematician János Bolyai, independently defined and studied hyperbolic
geometry, where uniqueness of parallels no longer holds. In this geometry the
sum of angles in a triangle add up to less than 180°. Elliptic
geometry was developed later in the 19th century by the German
mathematician Bernhard Riemann; here no parallel can be found and the
angles in a triangle add up to more than 180°. Riemann also
developed Riemannian geometry, which unifies and vastly generalizes the
three types of geometry, and he defined the concept of a manifold, which
generalizes the ideas of curves and surfaces.
The 19th century saw the
beginning of a great deal of abstract algebra. Hermann
Grassmann in Germany gave a first version of vector
spaces, William Rowan Hamilton in Ireland
developed noncommutative algebra. The British mathematician George
Boole devised an algebra that soon evolved into what is now called Boolean
algebra, in which the only numbers were 0 and 1. Boolean algebra is the
starting point of mathematical logic and has important applications
in computer science.
Augustin-Louis
Cauchy, Bernhard Riemann, and Karl Weierstrass reformulated the
calculus in a more rigorous fashion.
Also, for the first time, the
limits of mathematics were explored. Niels Henrik Abel, a Norwegian,
and Évariste Galois, a Frenchman, proved that there is no general
algebraic method for solving polynomial equations of degree greater than four
(Abel–Ruffini theorem). Other 19th-century mathematicians utilized this in
their proofs that straightedge and compass alone are not sufficient
to trisect an arbitrary angle, to construct the side of a cube twice the
volume of a given cube, nor to construct a square equal in area to a given
circle. Mathematicians had vainly attempted to solve all of these problems
since the time of the ancient Greeks. On the other hand, the limitation of three dimensions in
geometry was surpassed in the 19th century through considerations
of parameter space and hypercomplex numbers.
Abel and Galois's investigations
into the solutions of various polynomial equations laid the groundwork for
further developments of group theory, and the associated fields of abstract
algebra. In the 20th century physicists and other scientists have seen group
theory as the ideal way to study symmetry.
In the later 19th
century, Georg Cantor established the first foundations of set
theory, which enabled the rigorous treatment of the notion of infinity and has
become the common language of nearly all mathematics. Cantor's set theory, and
the rise of mathematical logic in the hands of Peano, L. E.
J. Brouwer, David Hilbert, Bertrand Russell, and A.N. Whitehead,
initiated a long running debate on the foundations of mathematics.
The 19th century saw the founding
of a number of national mathematical societies: the London Mathematical
Society in 1865, the Société Mathématique de France in 1872,
theCircolo Matematico di Palermo in 1884, the Edinburgh Mathematical
Society in 1883, and the American Mathematical Society in 1888.
The first international, special-interest society, the Quaternion Society,
was formed in 1899, in the context of a vector controversy.
In 1897, Hensel
introduced p-adic numbers.”
“The 20th century saw mathematics become a major profession.
Every year, thousands of new Ph.D.s in mathematics were awarded, and jobs were
available in both teaching and industry. An effort to catalogue the areas and
applications of mathematics was undertaken inKlein's encyclopedia.
In a 1900 speech to
the International Congress of Mathematicians, David Hilbert set
out a list of 23 unsolved problems in mathematics. These problems,
spanning many areas of mathematics, formed a central focus for much of
20th-century mathematics. Today, 10 have been solved, 7 are partially solved,
and 2 are still open. The remaining 4 are too loosely formulated to be stated
as solved or not.
Notable historical conjectures
were finally proven. In 1976, Wolfgang Haken and Kenneth
Appel used a computer to prove the four color theorem. Andrew
Wiles, building on the work of others, proved Fermat's Last
Theorem in 1995. Paul Cohen and Kurt Gödel proved that
the continuum hypothesis is independent of (could neither
be proved nor disproved from) the standard axioms of set theory. In 1998 Thomas
Callister Hales proved the Kepler conjecture.
Mathematical collaborations of
unprecedented size and scope took place. An example is the classification
of finite simple groups (also called the "enormous theorem"),
whose proof between 1955 and 1983 required 500-odd journal articles by about
100 authors, and filling tens of thousands of pages. A group of French mathematicians,
including Jean Dieudonné and André Weil, publishing under the pseudonym "Nicolas
Bourbaki", attempted to exposit all of known mathematics as a coherent
rigorous whole. The resulting several dozen volumes has had a controversial
influence on mathematical education.
Differential geometry came
into its own when Einstein used it in general relativity. Entire
new areas of mathematics such as mathematical logic, topology,
and John von Neumann's game theory changed the kinds of questions
that could be answered by mathematical methods. All kinds
of structures were abstracted using axioms and given names
like metric spaces, topological spaces etc. As mathematicians
do, the concept of an abstract structure was itself abstracted and led
to category theory. Grothendieck and Serre recast algebraic
geometry using sheaf theory. Large advances were made in the qualitative
study of dynamical systems that Poincaré had begun in the
1890s. Measure theory was developed in the late 19th and early 20th
centuries. Applications of measures include the Lebesgue integral, Kolmogorov's
axiomatisation of probability theory, and ergodic theory. Knot
theory greatly expanded. Quantum mechanics led to the
development of functional analysis. Other new areas include Laurent
Schwartz's distribution theory, fixed point theory, singularity
theory and René Thom's catastrophe theory, model theory,
and Mandelbrot's fractals. Lie theory with its Lie
groups and Lie algebras became one of the major areas of study.
Non-standard analysis, introduced
by Abraham Robinson, rehabilitated the infinitesimal approach to
calculus, which had fallen into disrepute in favour of the theory
of limits, by extending the field of real numbers to the Hyperreal
numbers which include infinitesimal and infinite quantities. An even
larger number system, the surreal numbers were discovered
by John Horton Conway in connection with combinatorial games.
The development and continual
improvement of computers, at first mechanical analog machines and then
digital electronic machines, allowed industry to deal with larger and
larger amounts of data to facilitate mass production and distribution and
communication, and new areas of mathematics were developed to deal with
this: Alan Turing's computability theory; complexity
theory; Derrick Henry Lehmer's use of ENIAC to further number
theory and the Lucas-Lehmer test; Claude Shannon's information
theory; signal processing; data analysis; optimization and
other areas of operations research. In the preceding centuries much
mathematical focus was on calculus and continuous functions, but the
rise of computing and communication networks led to an increasing importance
of discrete concepts and the expansion
of combinatorics including graph theory. The speed and data
processing abilities of computers also enabled the handling of mathematical
problems that were too time-consuming to deal with by pencil and paper
calculations, leading to areas such as numerical
analysis and symbolic computation. Some of the most important methods
and algorithms of the 20th century are: the simplex algorithm,
the Fast Fourier Transform, error-correcting codes, the Kalman
filter from control theory and the RSA
algorithm of public-key cryptography.
At the same time, deep insights
were made about the limitations to mathematics. In 1929 and 1930, it was proved
the truth or falsity of all statements formulated about the natural
numbers plus one of addition and multiplication, was decidable, i.e.
could be determined by some algorithm. In 1931, Kurt Gödel found that
this was not the case for the natural numbers plus both addition and
multiplication; this system, known as Peano arithmetic, was in
fact incompletable. (Peano arithmetic is adequate for a good deal
of number theory, including the notion of prime number.) A consequence
of Gödel's two incompleteness theorems is that in any mathematical
system that includes Peano arithmetic (including all
of analysis and geometry), truth necessarily outruns proof, i.e.
there are true statements that cannot be proved within the system.
Hence mathematics cannot be reduced to mathematical logic, and David
Hilbert's dream of making all of mathematics complete and consistent needed to
be reformulated.
One of the more colorful figures
in 20th-century mathematics was Srinivasa Aiyangar
Ramanujan (1887–1920), an Indian autodidact who conjectured or
proved over 3000 theorems, including properties of highly composite
numbers, the partition function and its asymptotics,
and mock theta functions. He also made major investigations in the areas
of gamma functions, modular forms, divergent series, hypergeometric
series and prime number theory.
Paul Erdős published more
papers than any other mathematician in history, working with hundreds of
collaborators. Mathematicians have a game equivalent to the Kevin Bacon
Game, which leads to the Erdős number of a mathematician. This
describes the "collaborative distance" between a person and Paul
Erdős, as measured by joint authorship of mathematical papers.
Emmy Noether has been
described by many as the most important woman in the history of mathematics. She
revolutionized the theories of rings, fields, and algebras.
As in most areas of study, the
explosion of knowledge in the scientific age has led to specialization: by the
end of the century there were hundreds of specialized areas in mathematics and
the Mathematics Subject Classification was dozens of pages long. More
and more mathematical journals were published and, by the end of the
century, the development of the world wide web led to online
publishing.”
So, of course, some of the other
significant 19th and 20th Century developments (of all
types) for mankind not covered in this article on mathematics were the
industrial revolution (this was an absolutely momentous one, obviously), the
discovery of penicillin, the creation of the atom bomb, the rise of America as
a global superpower, the invention of the television, the creation of
spacecraft (and the subsequent trip to the moon), the civil rights and feminist
movements, the ‘postmodern’ intellectual movement, the rise of commercialism,
“globalisation”, the invention of the video game, the invention of the personal
computer, 9/11 and the invention of
smartphones.
And so now I’m here, wearing very
complex polyester clothes to cover my animal body, lying on the very complex
structure that is my bed, in my bedroom, which is itself part of the extremely
complex structure that is my house, made out of brick (which is itself a very
complex item) and wood cut to precise lengths and plaster and tiles (which are
also very complex items) and glass (which is also a very complex item), put together
many years ago based on precise plans by
multiple men whose job it is to put together houses, using very complex
electrical tools like drills as well as complex, mechanical ones like hammers
and nails, and in my bedroom are lots of truly remarkable creations of human
civilisation, like books about all sorts of things, both non-fiction and
fiction, designed for kids, teens and adults, and CDs (which, it goes without
saying ,are also truly remarkable things), and toys, and sports trophies and a
backpack and hats and a couple of soccer shinpads and the electrical guitar that I never really
used and pencils and pens and folders and coins and a clock and a roll-on
deodorant container and a computer mouse and an old newspaper and a plastic bag
and a corkboard covered in all these ribbons I won for various things, both
academic and athletic, through primary and high school, and a white, chipboard
IKEA set of drawers one level of which contains lots of old phones and phone
chargers, and I am typing this up on a pretty modern laptop, which is a truly
remarkable object, in lots of ways, and which allows me, with the internet
added, to access basically the aggregate knowledge of all of humanity (and, in
particular, I can access the wonderful database of information that is
Wikipedia, as I did for this document) and to listen to most of the music ever
recorded and to watch footage of people doing things all over the world and to play
computer games which immerse me in what is effectively another virtual universe
of significant complexity and to both write and then publish my writings on a
range of highly complex topics using a highly complex language system and my 12
years of state-mandated education in literacy and my general education in
thought. And none of this makes me in the least bit unusual. In fact, I live in
a city of millions of people in a country of millions of people in a world of 7
billion people and billions of these 7 billion have available to them similar
things to me, and these billions like me all live in extremely complex
communities with clean drinking water accessible at a tap in every home and very good sewerage
systems and roads and organisations and shops at which you can buy food,
clothes or various items and technologies of all possible kinds which either make one’s life more convenient or
entertaining, and schools and universities and businesses and parks and ovals
and sports clubs and restaurants and bars and nightclubs and national parks and
stadiums and public transport and buildings with truly incredible architecture and
art galleries and museums and theatres and concert halls and libraries and centres
of government where lots of people wearing suits that we all chose in a remarkable process which occurs on a truly massive scale make decisions concerning us
all.
In his wildest dreams Bob
couldn’t have imagined that this would be the eventual result of his simple
discovery that rocks can be very sharp that one overcast afternoon.
