Structure
Many mathematical objects, such as
sets of numbers and
functions, exhibit internal structure. The structural properties of these
objects are investigated in the study of
groups,
rings,
fields and other abstract systems, which are themselves such objects. This
is the field of
abstract algebra.
An important concept here is that of
vectors, generalized to
vector
spaces, and studied in
linear algebra. The study of vectors combines three of the fundamental areas
of mathematics: quantity, structure, and space.
Vector calculus expands the field into a fourth fundamental area, that of
change.
Number theory |
Abstract algebra |
Group theory |
Order theory |
Space
The study of space originates with
geometry -
in particular,
Euclidean geometry.
Trigonometry combines space and numbers, and encompasses the well-known
Pythagorean theorem. The modern study of space generalizes these ideas to
include higher-dimensional geometry,
non-Euclidean geometries (which play a central role in
general relativity) and
topology.
Quantity and space both play a role in
analytic geometry,
differential geometry, and
algebraic geometry. Within differential geometry are the concepts of
fiber bundles and calculus on
manifolds.
Within algebraic geometry is the description of geometric objects as solution
sets of
polynomial equations, combining the concepts of quantity and space, and also
the study of
topological groups, which combine structure and space.
Lie groups
are used to study space, structure, and change.
Topology in
all its many ramifications may have been the greatest growth area in 20th
century mathematics, and includes the long-standing
Poincar� conjecture and the controversial
four color theorem, whose only proof, by computer, has never been verified
by a human.
Geometry |
Trigonometry |
Differential
geometry |
Topology |
Fractal geometry |
Change
Understanding and describing change is a common theme in the
natural sciences, and
calculus
was developed as a powerful tool to investigate it.
Functions arise here, as a central concept describing a changing quantity.
The rigorous study of real numbers and real-valued functions is known as
real
analysis, with
complex analysis the equivalent field for the complex numbers. The
Riemann hypothesis, one of the most fundamental open questions in
mathematics, is drawn from complex analysis.
Functional analysis focuses attention on (typically infinite-dimensional)
spaces of functions. One of many applications of functional analysis is
quantum mechanics. Many problems lead naturally to relationships between a
quantity and its rate of change, and these are studied as
differential equations. Many phenomena in nature can be described by
dynamical systems;
chaos
theory makes precise the ways in which many of these systems exhibit
unpredictable yet still
deterministic behavior.
Calculus |
Vector calculus |
Differential equations |
Dynamical systems |
Chaos theory |
Foundations and philosophy
In order to clarify the
foundations of mathematics, the fields of
mathematical logic and
set theory
were developed, as well as
category theory which is still in development.
Mathematical logic is concerned with setting mathematics on a rigid
axiomatic
framework, and studying the results of such a framework. As such, it is home to
G�del's second incompleteness theorem, perhaps the most widely celebrated
result in logic, which (informally) implies that any
formal system that contains basic arithmetic, if sound (meaning that
all theorems that can be proven are true), is necessarily incomplete
(meaning that there are true theorems which cannot be proved in that system).
G�del showed how to construct, whatever the given collection of
number-theoretical axioms, a formal statement in the logic that is a true
number-theoretical fact, but which does not follow from those axioms. Therefore
no formal system is a true axiomatization of full number theory. Modern logic is
divided into
recursion theory,
model
theory, and
proof
theory, and is closely linked to
theoretical
computer science.
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Mathematical logic |
Set theory |
Category theory |
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Discrete mathematics
Discrete mathematics is the common name for the fields of mathematics most
generally useful in
theoretical computer science. This includes
computability theory,
computational complexity theory, and
information theory. Computability theory examines the limitations of various
theoretical models of the computer, including the most powerful known model -
the
Turing machine. Complexity theory is the study of tractability by computer;
some problems, although theoretically solvable by computer, are so expensive in
terms of time or space that solving them is likely to remain practically
unfeasible, even with rapid advance of computer hardware. Finally, information
theory is concerned with the amount of data that can be stored on a given
medium, and hence concepts such as
compression and
entropy.
As a relatively new field, discrete mathematics has a number of fundamental
open problems. The most famous of these is the "P=NP?"
problem, one of the
Millennium Prize Problems.
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Combinatorics |
Theory of computation |
Cryptography |
Graph theory |
Applied mathematics
Applied mathematics considers the use of abstract mathematical tools in
solving concrete problems in the
sciences,
business,
and other areas. An important field in applied mathematics is
statistics,
which uses
probability theory as a tool and allows the description, analysis, and
prediction of phenomena where chance plays a role. Most experiments, surveys and
observational studies require the informed use of statistics. (Many
statisticians, however, do not consider themselves to be mathematicians, but
rather part of an allied group.)
Numerical analysis investigates computational methods for efficiently
solving a broad range of mathematical problems that are typically too large for
human numerical capacity; it includes the study of
rounding errors or other sources of error in computation.
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Mathematical fluid dynamics
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Common misconceptions
Mathematics is not a closed intellectual system, in which everything has
already been worked out. There is no shortage of open problems. Mathematicians
publish many thousands of papers embodying new discoveries in mathematics every
month.
Mathematics is not
numerology,
nor is it
accountancy; nor is it restricted to
arithmetic.
Pseudomathematics is a form of mathematics-like activity undertaken outside
academia,
and occasionally by mathematicians themselves. It often consists of determined
attacks on famous questions, consisting of proof-attempts made in an isolated
way (that is, long papers not supported by previously published theory). The
relationship to generally-accepted mathematics is similar to that between
pseudoscience and real science. The misconceptions involved are normally
based on:
- misunderstanding of the implications of
mathematical rigor;
- attempts to circumvent the usual criteria for publication of
mathematical papers in a
learned journal after
peer
review, often in the belief that the journal is biased against the
author;
- lack of familiarity with, and therefore underestimation of, the existing
literature.
The case of
Kurt
Heegner's work shows that the mathematical establishment is neither
infallible, nor unwilling to admit error in assessing 'amateur' work. And like
astronomy,
mathematics owes much to amateur contributors such as
Fermat and
Mersenne.
Mathematics and physical reality
Mathematical concepts and theorems need not correspond to anything in the
physical world. Insofar as a correspondence does exist, while mathematicians and
physicists may select axioms and postulates that seem reasonable and intuitive,
it is not necessary for the basic assumptions within an axiomatic system to be
true in an empirical or physical sense. Thus, while many
axiom systems are derived from our perceptions and experiments, they are not
dependent on them.
For example, we could say that the physical concept of two apples may be
accurately
modeled by the
natural number 2. On the other hand, we could also say that the natural
numbers are not an accurate model because there is no standard "unit"
apple and no two apples are exactly alike. The modeling idea is further
complicated by the possibility of
fractional or partial apples. So while it may be instructive to visualize
the axiomatic definition of the natural numbers as collections of apples, the
definition itself is not dependent upon nor derived from any actual physical
entities.
Nevertheless, mathematics remains extremely useful for solving real-world
problems. This fact led physicist
Eugene Wigner to write an article titled "The
Unreasonable Effectiveness of Mathematics in the Natural Sciences".
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