## Fields of mathematics

As noted above, the major disciplines within mathematics first arose out of the need to do calculations in commerce, to understand the relationships between numbers, to measure land, and to predict astronomical events. These four needs can be roughly related to the broad subdivision of mathematics into the study of quantity, structure, space, and change (i.e., arithmetic, algebra, geometry, and analysis). In addition to these main concerns, there are also subdivisions dedicated to exploring links from the heart of mathematics to other fields: to logic, to set theory (foundations), to the empirical mathematics of the various sciences (applied mathematics), and more recently to the rigorous study of uncertainty.

### Quantity

The study of quantity starts with numbers, first the familiar natural numbers and integers ("whole numbers") and arithmetical operations on them, which are characterized in arithmetic. The deeper properties of integers are studied in number theory, whence such popular results as Fermat's last theorem. Two famous unsolved problems in number theory are the twin prime conjecture and Goldbach's conjecture.

As the number system is further developed, the integers are recognised as a subset of the rational numbers ("fractions"). These, in turn, are contained within the real numbers, which are used to represent continuous quantities. Real numbers are generalised to complex numbers. These are the first steps of a hierarchy of number systems that include the quaternions and octonions. Consideration of numbers larger than all finite natural numbers leads to the concept of transfinite numbers. In this formalism, infinite cardinal numbers, the aleph numbers, allow meaningful comparison of the size of infinitely large sets.

 $1, 2, 3\,\!$ $-2, -1, 0, 1, 2\,\!$ $-2, \frac{2}{3}, 1.21\,\!$ $-e, \sqrt{2}, 3, \pi\,\!$ $2, i, -2+3i, 2e^{i\frac{4\pi}{3}}\,\!$ Natural numbers Integers Rational numbers Real numbers Complex numbers

### 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.

### Space

The study of space originates with geometry - in particular, Euclidean geometry. Trigonometry combines space and number, 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 Poincaré conjecture and the controversial four color theorem, whose only proof, by computer, has never been verified by a human.

### 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.

 <--96px--> $\frac{d^2}{dx^2} y = \frac{d}{dx} y + c$ 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.

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 Gödel's second incompleteness theorem, perhaps the most widely celebrated result in logic, which (informally) implies that there are always true theorems which cannot be proven. Modern logic is divided into recursion theory, model theory, and proof theory, and is closely linked to theoretical computer science.

 $P \Rightarrow Q \,$ Mathematical logic Set theory Category theory

### 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 soluble 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. It is widely believed that the answer to this problem is no.

 $\begin{matrix} (1,2,3) & (1,3,2) \\ (2,1,3) & (2,3,1) \\ (3,1,2) & (3,2,1) \end{matrix}$ 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.

Mathematical physicsAnalytical mechanicsMathematical fluid dynamicsNumerical analysisOptimizationProbabilityStatisticsMathematical economicsFinancial mathematicsGame theoryMathematical biologyCryptographyOperations research

## Common misconceptions

Mathematics is not a closed intellectual system, in which everything has already been worked out. There is no shortage of open problems.

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.

### Relationship between 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 most systems of axioms are derived from our perceptions and experiments, they are not dependent on them. Nevertheless, mathematics remains extremely useful for solving real-world problems. This fact led Eugene Wigner to write an essay, The Unreasonable Effectiveness of Mathematics in the Natural Sciences.

### What mathematics is not

Mathematics is not about unrestricted theorem proving, any more than literature is about the construction of grammatically correct sentences.

Mathematics is not accountancy. Although arithmetic computation is crucial to accountants, their main concern is to verify that computations are correct through a system of doublechecks. Advances in abstract mathematics are mostly irrelevant to the efficiency of bookkeeping, but the use of computers clearly does matter.

Mathematics is not numerology. Numerology uses modular arithmetic to reduce names and dates down to numbers, but assigns emotions or traits to these numbers intuitively or on the basis of traditions.

## References

1. Philip E. B. Jourdain, The Nature of Mathematics, in The World of Mathematics, James R. Newman, editor, Dover, 2003, ISBN 0486432688.
2. Howard Eves, An Introduction to the History of Mathematics, Sixth Edition, Saunders, 1990, ISBN 0030295580.
3. Ivars Peterson, Mathematical Tourist, New and Updated Snapshots of Modern Mathematics, Owl Books, 2001, ISBN 0805071598.
4. The Oxford Dictionary of English Etymology, 1983 reprint. ISBN 0-19-861112-9.
5. Oxford English Dictionary, second edition, ed. John Simpson and Edmund Weiner, Clarendon Press, 1989, ISBN 0-19-861186-2.
6. Mikhail B. Sevryuk (January 2006). "Book Reviews" (PDF). Bulletin of the American Mathematical Society 43 (1): 101-109. Retrieved on 2006-06-24.
7. Earliest Uses of Various Mathematical Symbols (Contains many further references)
8. Waltershausen, Wolfgang Sartorius von: Gauss zum Gedächtniss, 1856. (Gauss zum Gedächtnis 1965 reprint by Sändig Reprint Verlag H. R. Wohlwend: ISBN 3-253-01702-8, ASIN: B0000BN5SQ).
9. Einstein, Albert (1923). "Sidelights on Relativity (Geometry and Experience)".
10. Ziman, J.M., F.R.S. (1968). Public Knowledge:An essay concerning the social dimension of science.
11. "The Fields Medal is now indisputably the best known and most influential award in mathematics."Monastyrsky, Michael (2001). Some Trends in Modern Mathematics and the Fields Medal. Canadian Mathematical Society. Retrieved on 2006-07-28.