A
mathematician is a person whose primary area of
study and/or research is the field of
mathematics. Mathematicians are concerned with
particular problems related to
logic,
space,
transformations,
numbers and more general ideas which
encompass these concepts.
Some notable mathematicians include
Sir
Isaac Newton,
Johann
Carl Friedrich Gauss,
Archimedes of Syracuse,
Leonhard Paul Euler,
Georg Friedrich Bernhard
Riemann,
Euclid of
Alexandria,
Jules Henri
Poincaré,
David Hilbert,
Joseph-Louis Lagrange, and
Pierre de Fermat.
Some scientists who research other fields are also considered
mathematicians if their research provides insights into
mathematics—one notable example is
Edward
Witten. Conversely, some mathematicians may provide insights
into other fields of research—these people are known as
applied mathematicians.
Education
Mathematicians usually cover a breadth of topics within mathematics
in their undergraduate education, and then proceed to specialize in
topics of their choice at the graduate level. In some universities,
a qualifying exam serves to test both the breadth and depth of a
person's understanding in mathematics; should s/he pass, s/he is
permitted to work on a doctoral dissertation.
There are notable cases where mathematicians have failed to reflect
their ability in their university education, but have nevertheless
become remarkable mathematicians. Fermat, for example, is known for
having been "Prince of Amateurs", because he never did research in
university and took Mathematics as a hobby. A majority of these
cases were those of
child
prodigies.
Motivation
Mathematicians do research in fields such as
logic,
set theory,
category theory,
modern algebra,
number theory,
analysis,
geometry,
topology,
dynamical systems,
combinatorics,
game
theory,
information theory,
numerical analysis,
optimization,
computation,
probability and
statistics. These fields comprise both
pure mathematics and
applied mathematics, as well as
establish links between the two. Some fields, such as the theory of
dynamical systems, or game theory, are classified as applied
mathematics due to the relationships they possess with physics,
economics and the other sciences. Whether probability theory and
statistics are of theoretical nature, applied nature, or both, is
quite controversial among mathematicians. Other branches of
mathematics, however, such as logic, number theory, category theory
or set theory are accepted to be a part of pure mathematics,
although they do indeed find applications in other sciences
(predominantly
computer science and
physics). Likewise, analysis, geometry and
topology, although considered pure mathematics, do find
applications in theoretical physics -
string theory, for instance.
Although it is true that mathematics finds diverse applications in
many areas of research, a mathematician does not determine the
value of an idea by the diversity of its applications. Mathematics
is interesting in its own right, and a majority of mathematicians
investigate the diversity of structures studied in
mathematics
itself. Furthermore, a mathematician is not someone who merely
manipulates formulas, numbers or equations - the diversity of
mathematics permits for researchers in other areas too. In fact,
the theory of equations and numbers (formulas to a lesser extent in
theoretical mathematics, but to some extent in applied
mathematics), can lead to deep questions. For instance, if one
graphs a set of solutions of an equation in some higher dimensional
space, he may ask of the geometric properties of the graph. Thus
one can understand equations by a pure understanding of abstract
topology or
geometry - this idea is of importance in
algebraic geometry. Similarly, a
mathematician does not restrict his study of numbers to the
integers; rather he considers more abstract
structures such as
rings, and in
particular
number rings in the context
of
algebraic number theory.
This exemplifies the abstract nature of mathematics and how it is
not restricted to questions one may ask in daily life.
In a different direction, mathematicians ask questions about space
and transformations, but which are not restricted to geometric
figures such as squares and circles. For instance, an active area
of research within the field of
differential topology concerns itself
with the ways in which one can "smoothen" higher dimensional
figures. In fact, whether one can smoothen certain higher
dimensional spheres remains open - it is known as the
smooth Poincaré conjecture. Another
aspect of mathematics,
set-theoretic topology and
point-set topology, concerns objects of a
different nature to those in our universe, or in a higher
dimensional analogue of our universe. These objects behave in a
rather strange manner under deformations, and the properties they
possess are completely different to those objects in our universe.
For instance, the "distance" between one point on such an object,
and another point, may depend on the order in which you consider
the pair of points. This is quite different to ordinary life, in
which it is accepted that the straight line distance from person A
to person B is the same (and not different to!) that between person
B and person A.
Another aspect of mathematics, often referred to as "foundational
mathematics", consists of the fields of
logic
and
set theory. Here, various ideas
regarding the ways in which one can prove certain claims are
explored. This theory is far more complex than it seems, in that
the truth of a claim depends on the context in which the claim is
made, unlike basic ideas in daily life where truth is absolute. In
fact, although some claims may be true, it is impossible to prove
or disprove them in rather natural contexts!
Category theory, another field within "foundational mathematics",
is rooted on the abstract axiomatization of the definition of a
"class of mathematical structures", referred to as a "category". A
category intuitively consists of a collection of objects, and
defined relationships between them. While these objects may be
anything (such as "tables" or "chairs"), mathematicians are usually
interested in particular, more abstract, classes of such objects.
In any case, it is the
relationships between these
objects, and
not the actual objects which are
predominantly studied.
The Nobel Prize is never awarded for work in the field of
theoretical mathematics. Instead, the most prestigious award in
mathematics is the
Fields Medal,
sometimes referred to as the "Nobel Prize of Mathematics". The
Fields Medal is considered more of a prestige than a mere reward in
that it is only awarded every four years, and the amount of money
awarded is small in comparison to that of the Nobel Prize.
Furthermore, the recipient of the Fields Medal must be (roughly)
under 40 years of age at the time the medal is awarded. Other
prominent prizes in mathematics include the
Abel Prize, the
Nemmers Prize, the
Wolf Prize, the
Schock
Prize, and the
Nevanlinna
Prize.
Differences with scientists
Mathematics differs from natural
sciences in
that physical theories in the sciences are tested by experiments,
while mathematical statements are supported by proofs which may be
verified objectively by mathematicians. If a certain statement is
believed to be true by mathematicians (typically because special
cases have been confirmed to some degree) but has neither been
proven nor dis-proven, it is called a
conjecture, as opposed to the ultimate goal:
a
theorem that is proven true. Physical theories may be
expected to change whenever new information about our physical
world is discovered. Mathematics changes in a different way: new
ideas don't falsify old ones but rather are used to
generalize what was known before to capture a broader
range of phenomena. For instance,
calculus
(in one variable) generalizes to
multivariable calculus, which
generalizes to analysis on
manifolds. The
development of
algebraic geometry
from its classical to modern forms is a particularly striking
example of the way an area of mathematics can change radically in
its viewpoint without making what was proved before in any way
incorrect. While a theorem, once proved, is true forever, our
understanding of what the theorem
really means gains in
profundity as the mathematics around the theorem grows. A
mathematician feels that a theorem is better understood when it can
be extended to apply in a broader setting than previously known.
For instance,
Fermat's little
theorem for the nonzero integers modulo a prime generalizes to
Euler's theorem for the invertible
numbers modulo any nonzero integer, which generalizes to
Lagrange's theorem for
finite groups.
Doctoral degree statistics for mathematicians in the United
States
The number
of Doctoral degrees in mathematics awarded each year in the
United
States has ranged from 750 to 1230 over the past 35
years. In the early seventies, degree awards were at their
peak, followed by a decline throughout the seventies, a rise
through the eighties, and another peak through the nineties.
Unemployment for new doctoral recipients peaked at 10.7% in 1994
but was as low as 3.3% by 2000. The percentage of female doctoral
recipients increased from 15% in 1980 to 30% in 2000.
As of 2000, there are approximately 21,000 full-time faculty
positions in mathematics at colleges and universities in the United
States. Of these positions about 36% are at institutions whose
highest degree granted in mathematics is a bachelor's degree, 23%
at institutions that offer a master's degree and 41% at
institutions offering a doctoral degree.
The median age for doctoral recipients in 1999-2000 was 30, and the
mean age was 31.7.
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Women in mathematics
While the majority of mathematicians are male, there have been some
demographic changes since
World War II.
Some prominent female mathematicians are
Hypatia of Alexandria (ca. 400 AD),
Labana of
Cordoba (ca. 1000),
Ada Lovelace
(1815–1852),
Maria Gaetana
Agnesi (1718–1799),
Emmy Noether
(1882–1935),
Sophie Germain
(1776–1831),
Sofia Kovalevskaya
(1850–1891),
Rózsa Péter
(1905–1977),
Julia Robinson
(1919–1985),
Olga Taussky-Todd
(1906–1995),
Émilie du
Châtelet (1706–1749), and
Mary
Cartwright (1900–1998).
The
Association for
Women in Mathematics is a professional society whose purpose is
"to encourage women and girls to study and to have active careers
in the mathematical sciences, and to promote equal opportunity and
the equal treatment of women and girls in the mathematical
sciences."The
American
Mathematical Society and other mathematical societies offer
several prizes aimed at increasing the representation of women and
minorities in the future of mathematics.
Quotations about mathematicians
The following are quotations about mathematicians, or by
mathematicians.
- A mathematician is a device for turning coffee into
theorems.
- :—Attributed to both Alfréd
Rényi and Paul Erdős
- Die Mathematiker sind eine Art Franzosen; redet man mit
ihnen, so übersetzen sie es in ihre Sprache, und dann ist es
alsobald ganz etwas anderes. (Mathematicians are [like] a sort
of Frenchmen; if you talk to them, they translate it into their own
language, and then it is immediately something quite
different.)
- :—Johann Wolfgang von
Goethe
- Each generation has its few great mathematicians...and [the
others'] research harms no one.
- :—Alfred W. Adler (1930- ), "Mathematics and
Creativity"
- In short, I never yet encountered the mere mathematician
who could be trusted out of equal roots, or one who did not
clandestinely hold it as a point of his faith that x squared + px
was absolutely and unconditionally equal to q. Say to one
of these gentlemen, by way of experiment, if you please, that you
believe occasions may occur where x squared + px is not altogether
equal to q, and, having made him understand what you mean, get out
of his reach as speedily as convenient, for, beyond doubt, he will
endeavor to knock you down.
- :—Edgar Allan Poe, The
purloined letter
- A mathematician, like a painter or poet, is a maker of
patterns. If his patterns are more permanent than theirs,
it is because they are made with ideas.
- :—G. H.
Hardy, A Mathematician's
Apology
- Some of you may have met mathematicians and wondered how
they got that way.
- :—Tom Lehrer
- It is impossible to be a mathematician without being a poet
in soul.
- :—Sofia Kovalevskaya
See also
Notes
References
External links