Etymology
The word mathematics comes from Ancient Greek máthēma (μάθημα), meaning "that which is
learnt,"[36] "what one gets to know," hence also "study" and "science". The word for "mathematics"
came to have the narrower and more technical meaning "mathematical study" even in Classical
times.[37] Its adjective is mathēmatikós (μαθηματικός), meaning "related to learning" or "studious,"
which likewise further came to mean "mathematical." In particular, mathēmatikḗ
tékhnē (μαθηματικὴ τέχνη; Latin: ars mathematica) meant "the mathematical art."
Similarly, one of the two main schools of thought in Pythagoreanism was known as
the mathēmatikoi (μαθηματικοί)—which at the time meant "learners" rather than
"mathematicians" in the modern sense.[38]
In Latin, and in English until around 1700, the term mathematics more commonly meant
"astrology" (or sometimes "astronomy") rather than "mathematics"; the meaning gradually
changed to its present one from about 1500 to 1800. This has resulted in several mistranslations.
For example, Saint Augustine's warning that Christians should beware of mathematici, meaning
astrologers, is sometimes mistranslated as a condemnation of mathematicians. [39]
The apparent plural form in English, like the French plural form les mathématiques (and the less
commonly used singular derivative la mathématique), goes back to the
Latin neuter plural mathematica (Cicero), based on the Greek plural ta mathēmatiká (τὰ
μαθηματικά), used by Aristotle (384–322 BC), and meaning roughly "all things mathematical",
although it is plausible that English borrowed only the adjective mathematic(al) and formed the
noun mathematics anew, after the pattern of physics and metaphysics, which were inherited from
Greek.[40] In English, the noun mathematics takes a singular verb. It is often shortened
to maths or, in North America, math.[41]
Definitions of mathematics
Leonardo Fibonacci, the Italian mathematician who introduced the Hindu–Arabic numeral system invented
between the 1st and 4th centuries by Indian mathematicians, to the Western World.
Main article: Definitions of mathematics
Mathematics has no generally accepted definition. [6][7] Aristotle defined mathematics as "the
science of quantity" and this definition prevailed until the 18th century. However, Aristotle also
noted a focus on quantity alone may not distinguish mathematics from sciences like physics; in
his view, abstraction and studying quantity as a property "separable in thought" from real
instances set mathematics apart.[42]
In the 19th century, when the study of mathematics increased in rigor and began to address
abstract topics such as group theory and projective geometry, which have no clear-cut relation to
quantity and measurement, mathematicians and philosophers began to propose a variety of new
definitions.[43]
A great many professional mathematicians take no interest in a definition of mathematics, or
consider it undefinable.[6] There is not even consensus on whether mathematics is an art or a
science.[7] Some just say, "Mathematics is what mathematicians do."[6]
Three leading types
Three leading types of definition of mathematics today are called logicist, intuitionist,
and formalist, each reflecting a different philosophical school of thought. [44] All have severe flaws,
none has widespread acceptance, and no reconciliation seems possible. [44]
Logicist definitions
An early definition of mathematics in terms of logic was that of Benjamin Peirce (1870): "the
science that draws necessary conclusions."[45] In the Principia Mathematica, Bertrand
Russell and Alfred North Whitehead advanced the philosophical program known as logicism, and
attempted to prove that all mathematical concepts, statements, and principles can be defined
and proved entirely in terms of symbolic logic. A logicist definition of mathematics is Russell's
(1903) "All Mathematics is Symbolic Logic."[46]
Intuitionist definitions
Intuitionist definitions, developing from the philosophy of mathematician L. E. J. Brouwer, identify
mathematics with certain mental phenomena. An example of an intuitionist definition is
"Mathematics is the mental activity which consists in carrying out constructs one after the
other."[44] A peculiarity of intuitionism is that it rejects some mathematical ideas considered valid
according to other definitions. In particular, while other philosophies of mathematics allow objects
that can be proved to exist even though they cannot be constructed, intuitionism allows only
mathematical objects that one can actually construct. Intuitionists also reject the law of excluded
middle (i.e., ). While this stance does force them to reject one common version of proof by
contradiction as a viable proof method, namely the inference of from , they are still able to
infer from . For them, is a strictly weaker statement than . [47]
Formalist definitions
Formalist definitions identify mathematics with its symbols and the rules for operating on
them. Haskell Curry defined mathematics simply as "the science of formal systems". [48] A formal
system is a set of symbols, or tokens, and some rules on how the tokens are to be combined
into formulas. In formal systems, the word axiom has a special meaning different from the
ordinary meaning of "a self-evident truth", and is used to refer to a combination of tokens that is
included in a given formal system without needing to be derived using the rules of the system.
