Topic: mathematics as a formal system

topics > philosophy > Group: philosophy of mathematics

meaning and truth

abstraction by name
analytic truth
Godel's incompleteness theorem
limitations of formalism
mathematical proof
number representation
problems with analytic truth
Turing machine
what is a computer
what is a number


Formalism is the use of symbols and rules to build a system. One starts with some collection of symbols, a syntax for combining the symbols into statements, some collection of axioms or true statements, and a set of rules for transforming statements into new statements. Given a finite description one can then generate a infinite number of true statements. With a semantic model, one can generate true statements about the world.

Euclid's geometry was a model for formal systems until the discovery of hyperbolic and spherical geometries by Bolyai (1823), Lobachevski (1826) and Riemann (1854). These geometries demonstrated the inadequacies of proofs in a natural language. Leibnitz (1646-1716) attempted to develop a "characteristica universalis" in which one could express and develop thoughts exactly. It was modeled on Lull's ars universalis (1274). Boole developed a symbolic logic in 1847, Frege formalized logic in 1884, and Hilbert formalized geometry in 1899. These lead to Whitehead and Russell's formalization of mathematics in 1910. Many others have extended these formalizations and have developed formalizations of their own. The Turing machine is perhaps the simplest.

The search for truth is a motivation behind these systems. Since one starts with symbols and follows rules that manipulate symbols there appears to be no room for doubt. Complex statements are built out of simpler statements, and the truth of a statement rests on the primitives of the system. A hint of trouble is the meta-language that is required to interpret the symbols. Godel proved the impossibility of Hilbert's goal for a self-contained formalization of mathematics. Common sense holds that mathematics must be more than just symbols. (cbb 4/94)

Subtopic: Frege's concept writing up

Quote: to demonstrate that arithmetic is analytic need to prove its fundamental propositions with utmost rigour; makes clear the primitive truths [»fregG_1884]
Quote: a proof must be a step-by-step procedure; devised a concept writing to reduce their length; conforms with rules [»fregG_1884]
Quote: concept writing has a format for representing a derivation; e.g., (B->A and B) implies A [»fregG_1879]
Note: logicism--Frege showed that arithmetic rests on logic; he developed logic and the notion of proof; defined 'generality', e.g., all A are B [»fregG_1892, OK]
Quote: Frege's ideography will unify the formula languages and extend them to new fields [»fregG_1879]
Quote: need a meta language for an ideography in order to define its basis [»fregG_1879]
Quote: the signs of arithmetic, geometry, and chemistry realized Leibniz's idea of a universal characteristic; the original idea was too gigantic

Subtopic: constructivisim up

Quote: profound differences in philosophical outlook between mathematicians; e.g., whether every object must have a construction rule [»straC8_1967]
Quote: construction rules for mathematics leads to rigour and exact mathematical reasoning [»straC8_1967]
Quote: construction rules for mathematics leads to intense concern about syntax, the way in which things are written

Subtopic: role of definitions up

Quote: use definitions to simplify a derivation; does not change the content of a proposition [»fregG_1879]
Quote: in logical and mathematical systems want both concise notations and primitive notations that are economical in vocabulary and grammar
Quote: combine primitive and concise notations by rules of translation which define primitive equivalents for concise terms [»quinWV1_1951]
Quote: a definition in philosophy describes a concept while a definition in mathematics constructs a concept
Quote: mathematical definitions are names or abbreviations of speech to remove tedious drudgery [»galiG_1638]

Subtopic: algebraic geometry up

Quote: combine geometrical analysis with algebra by reducing proportions to simple symbols and relations [»descR_1637]

Subtopic: formal reasoning up

Quote: proofs can be formalized as a finite sequence of formulas of a formal theory; a part of why proofs are convincing to mathematicians [»tymoT2_1979]
Quote: a formal proof does not hinge on the meaning of specific terms; the meaning is carried by the non-specific, arithmetic/logical terms [»lakaI_1976]
Quote: in a formalized theory, the tools are completely prescribed by its syntax; in proof-analysis, tools are unconstrained [»lakaI_1976]
Quote: logic and mathematics are only concerned with the truth value of forms; i.e., that which remains unchanged when every constituent is changed [»russB_1919, OK]
Quote: formal reasoning concerns statements that do not mention actual things; e.g., if all alphas are betas and x is an alpha than x is a beta [»russB_1919, OK]
Quote: mathematical logic is seldom used in proofs; it should be a calculational alternative to human reasoning [»dijkEW12_1989]

Subtopic: formal reasoning as computation up

Quote: formal systems or formalisms are the same as Turing machines [»godeK_1931]
Quote: the Hilbert functional calculus can be modified so that an automatic machine will find all provable formulae [»turiAM11_1936]
Quote: mathematics does not rest on logic; instead arithmetic and logic are the same [»wittL_1939]
Quote: a proof is like a program: the axioms are input, finding a proof is writing a program, verification is running the program [»daviPJ3_1972]

Subtopic: programming as mathematics up

Quote: computing is a new field of mathematics; its central concepts are ill defined, like calculus when it was the Method of Fluxions [»straC8_1967]

Subtopic: math is not logic up

Quote: even though Russell has translated mathematical procedures into logic does not mean that this explains mathematics [»wittL_1939]
Quote: if math is about numerals instead of definitions, math is about scratches on the blackboard; absurd; math is not about how symbols are used [»wittL_1939]
Quote: calculation is not aggregative mechanical thought; only possible if mathematical notation has been thoroughly developed

Related Topics up

Group: formalism   (9 topics, 478 quotes)
Group: meaning and truth   (18 topics, 634 quotes)

Topic: abstraction by name (29 items)
Topic: analytic truth (51 items)
Topic: Godel's incompleteness theorem (19 items)
Topic: limitations of formalism (93 items)
Topic: logic (84 items)
Topic: mathematical proof (23 items)
Topic: number representation (16 items)
Topic: problems with analytic truth (20 items)
Topic: reductionism (51 items)
Topic: Turing machine (30 items)
Topic: what is a computer (62 items)
Topic: what is a number
(55 items)

Updated barberCB 8/04
Copyright © 2002-2008 by C. Bradford Barber. All rights reserved.
Thesa is a trademark of C. Bradford Barber.