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1. **WHAT IS MATHEMATICS?**

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**Mathematics** includes the study of such topics as quantity (number theory), structure (algebra), space (geometry), and change (mathematical analysis). It has no generally accepted definition.^{}

Mathematicians seek and use patterns^{} to formulate new conjectures; they resolve the truth or falsity of conjectures by mathematical proof. When mathematical structures are good models of real phenomena, mathematical reasoning can be used to provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation, measurement, and the systematic study of the shapes and motions of physical objects. Practical mathematics has been a human activity from as far back as written records exist. The research required to solve mathematical problems can take years or even centuries of sustained inquiry.

Rigorous arguments first appeared in Greek mathematics, most notably in Euclid’s *Elements*. Since the pioneering work of Giuseppe Peano (1858–1932), David Hilbert (1862–1943), and others on axiomatic systems in the late 19th century, it has become customary to view mathematical research as establishing truth by rigorous deduction from appropriately chosen axioms and definitions. Mathematics developed at a relatively slow pace until the Renaissance, when mathematical innovations interacting with new scientific discoveries led to a rapid increase in the rate of mathematical discovery that has continued to the present day.^{}

Mathematics is essential in many fields, including natural science, engineering, medicine, finance, and the social sciences. Applied mathematics has led to entirely new mathematical disciplines, such as statistics and game theory. Mathematicians engage in pure mathematics (mathematics for its own sake) without having any application in mind, but practical applications for what began as pure mathematics are often discovered later.^{}

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.^{}

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.^{}

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 τὰ μαθηματικά (*ta mathēmatiká*), 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. In English, the noun *mathematics* takes a singular verb. It is often shortened to *maths* or, in North America, *math*.^{}

Mathematics has no generally accepted definition. Aristotle defined mathematics as “the science of quantity” and this definition prevailed until the 18th century. 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. Three leading types of definition of mathematics today are called logicist, intuitionist, and formalist, each reflecting a different philosophical school of thought. All have severe flaws, none has widespread acceptance, and no reconciliation seems possible.^{}

An early definition of mathematics in terms of logic was Benjamin Peirce’s “the science that draws necessary conclusions” (1870). 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 “All Mathematics is Symbolic Logic” (1903).^{}

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.” 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 — a stance which forces them to reject proof by contradiction as a viable proof method as well. ^{}

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”.^{} 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.

A great many professional mathematicians take no interest in a definition of mathematics, or consider it undefinable. There is not even consensus on whether mathematics is an art or a science. Some just say, “Mathematics is what mathematicians do.”^{}

2. **WHAT IS THE ESSENTIAL DIALECTIC OF MATHEMATICS?
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*The Essential Dialectic of Mathematics is:
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**{Number-Theory ⇆ Geometry ⇅ Algebra} ↻ Mathematical-Analysis**

3. **WHAT IS THE INTERMEDIARY** DIALECTIC **OF MATHEMATICS?
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*The Intermediary Dialectic of Mathematics is:
*

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4. **WHAT IS THE COMPLETE**** **DIALECTIC **OF ****MATHEMATICS?**

*The Complete Dialectic of Mathematics is:*

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