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Group: philosophy of mathematics
topics >
Godel's incompleteness theorem
Topic:
mathematical proof
Topic:
mathematical proof as a social process
Topic:
mathematics as a formal system
Topic:
mathematics by proofs and refutations
Topic:
number as a named set of numbers
Topic:
number as a progression for counting and 1-1 relations
Topic:
number and arithmetic as part of language
Topic:
number as the extension of a class of equinumerous classes
Topic:
number representation
Topic:
what is a number
formalism
Group:
mathematics
Group:
philosophy of science
Group:
science
Group:
sets
Topic:
abstraction
Topic:
analytic truth
Topic:
facts as relationships between entities
Topic:
history of mathematics
Topic:
infinity and infinitesimal
Topic:
kinds of numbers
Topic:
logic
Topic:
object and value equivalence
Topic:
problems with analytic truth
Topic:
science as measurement
Topic:
scientific method
Topic:
set definition by extension or intension
Topic:
symmetry
Topic:
topology
Topic:
what is a computer
Summary
There are two main ideas here. One is that mathematics is an objective, ideal reality; a refuge for Platonic existence. The other is about a fluid heuristic, Babylonian mathematics, the freedom to construct and build new theories. (cbb 3/94)
Subtopic: mathematics as primary
Quote: a philosopher can talk about mathematics by dealing with words that have a natural meaning; e.g., "proof", "number" [»wittL_1939] Quote: profound differences in philosophical outlook between mathematicians; e.g., whether every object must have a construction rule [»straC8_1967]
Subtopic: mathematics and science
Quote: philosophy is written in this grand book, the universe, which stands continually open to our gaze Quote: philosophy is written in the language of mathematics, and its characters are triangles, circles, and other geometrical figures Quote: mathematical knowledge is virtually included in the definition of all things Quote: our mathematical knowledge is the same as Divine wisdom but we reason by steps while His is one of simple intuition [»galiG_1632]
Subtopic: mathematics as organized reasoning
Quote: the best scientific theories are fertile sources of mathematical ideas; connects the real world with mathematics [»penrR_1989] Quote: how can mathematics be so admirably appropriate to reality; by not being certain [»einsA_1923]
Subtopic: mathematics as abstraction
Quote: mathematics is organized reasoning that connects statements together; e.g., about the force of gravity [»feynR_1965] Quote: an exercise for the reader. The principle benefit of mathematics is training the mind [»descR_1641a] Quote: through mathematical analysis, there is no problem that cannot be solved [»vietF_1591]
Subtopic: mathematics as ideal reality
Quote: a real proof is fruitful, its conclusion is in a sense more general than its premises Quote: solve algebraic equations by generalization of addition by iteration: get negative numbers and reciprocal powers from subtraction, fractions from division, irrationals from fractional powers, and complex numbers from the square root of -1 [»feynRP_1963] Quote: there is no science but the science of the general Quote: the true, profound analogies arise from the mathematical spirit that disdains matter and embraces form Quote: classical mathematics requires and assumes universals as values of its bound variables; e.g., numbers [»quinWV2_1947]
Subtopic: mathematics as relationship
Quote: mathematical analysis is a language for learning the internal harmony of the world, the only true objective reality Quote: mathematical analysis is primarily the study of temporal and spatial frames Quote: objective reality is that which is common to many thinking beings; it can only be the harmony expressed by mathematical laws [»poinH_1905, OK] Quote: mathematics is an objective, ideal reality that is neither subjective nor physical [»daviPJ_1981] Quote: mathematics is independent of the existence of material objects; mathematical existence means free from contradiction [»poinH_1908, OK] Quote: mathematical concepts have absolute truth and a Platonic existence; e.g., the Mandelbrot set [»penrR_1989] Quote: arithmetic laws are the laws of the laws of nature Quote: although the senses can be confused, corporeal things contain what is clearly understand; at least as the objects of pure mathematics [»descR_1641]
Subtopic: nature as mathematical structure
Quote: mathematicians study the relations between objects, not the objects themselves; i.e., they study form; matter does not engage their attention [»poinH_1902, OK]
Subtopic: expression vs. command
Quote: sodium and potassium exist because there is a corresponding mathematical structure guaranteeing their stability [»thomR_1975] Quote: there are geometric objects in biology that prescribe the only possible forms capable of self-reproduction Quote: laws of nature should be valid in any 4-d system of coordinates that preserves neighboring events; like Gauss's theory of surfaces [»einsA_1956, OK] Quote: a law found by measurement is necessarily mathematical, with mathematical consequences [»drakS_1980] Quote: Galileo saw mathematics as essential to physics and reading the great book of nature
Subtopic: mathematics as transformation
Quote: assembly codes are strings of commands, but the rest of mathematics uses expressions instead of commands; we say 'x + y' instead of 'add x to y' [»straC8_1967]
Subtopic: mathematical objects
Quote: measure the angle between curves by the angle between straight lines that are perpendicular to the curves at their point of intersection [»descR_1641a] Quote: bit-vector encoding of the n-queen problem; natural, simple, and efficient [»zongQ2_2002] Quote: mathematics is independent of the existence of material objects; mathematical existence means free from contradiction [»poinH_1908, OK]
Subtopic: Babylonian mathematics
Quote: in the realm of ideas, the reproducible ideas are mathematical objects; similar to science being about reproducible, physical results [»daviPJ_1981] Quote: the most important mathematical problems are clean and simple to state [»demiRA_1977] Quote: the essence of mathematics is freedom to construct and make assumption; then these constructions impose their order on us [»daviPJ_1981] Quote: the logics and mathematics in the brain must be structurally very different from known languages; because of imprecision [»vonnJ_1958] Quote: the sole natural object of mathematical thought is the whole number [»poinH_1905, OK]
Subtopic: mathematics as random
Quote: unlike the Euclidean method, the Babylonian idea is to remember enough to work something out or reconstruct what is needed [»feynR_1965] Quote: physicists do Babylonian mathematics because different formulations of the same law give different clues for related laws [»feynR_1965] Quote: in a formalized theory, the tools are completely prescribed by its syntax; in proof-analysis, tools are unconstrained [»lakaI_1976] Quote: mathematicians use a situational logic that is neither formal, mechanistic, nor irrational blind guessing [»lakaI_1976]
Subtopic: mathematics vs programming
Quote: maybe pure mathematics is random; e.g., distribution of the primes; there's nothing there to prove [»chaiGJ_2001]
Subtopic: evolution of mathematics
Quote: mathematics is what programming becomes when freed of all considerations of efficiency [»schwJT_1972]
Subtopic: algebra
Quote: new theories replace the dominant, mathematical theories; e.g., arithmetic to geometry to algebra to arithmetic, probability to measure theory [»lakaI_1976]
Subtopic: logic
Quote: algebra is the natural expression of man's interest in the numerical thought
Subtopic: imaginary numbers
Quote: unrestricted quantification is only useful for truth functions; e.g., mathematics [»quinWV_1947]
Subtopic: problems with mathematics and philosophy
Quote: the imaginary number i is the solution to the self-referential equation x=-1/x [»browGS_1972, OK]
Quote: mathematical methods in philosophy can only build car-castles while philosophical methods in mathematics result in mere verbiage [»kantI_1781, OK] Quote: mathematical entities have no independent being; they are neither ideas nor sensible primary beings [»aris_322a] Quote: mathematical entities are not separate from sensible things; not first principles; no coherency
[»aris_322a]
Topic: Godel's incompleteness theorem (19 items)
Topic: mathematical proof (23 items)
Topic: mathematical proof as a social process (14 items)
Topic: mathematics as a formal system (30 items)
Topic: mathematics by proofs and refutations (31 items)
Topic: number as a named set of numbers (15 items)
Topic: number as a progression for counting and 1-1 relations (22 items)
Topic: number and arithmetic as part of language (30 items)
Topic: number as the extension of a class of equinumerous classes (23 items)
Topic: number representation (16 items)
Topic: what is a number
(55 items)
Group: formalism (9 topics, 478 quotes)
Group: mathematics (23 topics, 560 quotes)
Group: philosophy of science (10 topics, 406 quotes)
Group: science (45 topics, 1960 quotes)
Group: sets (7 topics, 148 quotes)Topic: abstraction (62 items)
Topic: analytic truth (51 items)
Topic: facts as relationships between entities (22 items)
Topic: history of mathematics (57 items)
Topic: infinity and infinitesimal (37 items)
Topic: kinds of numbers (24 items)
Topic: logic (84 items)
Topic: object and value equivalence (60 items)
Topic: problems with analytic truth (20 items)
Topic: science as measurement (36 items)
Topic: scientific method (42 items)
Topic: set definition by extension or intension (18 items)
Topic: symmetry (11 items)
Topic: topology (29 items)
Topic: what is a computer
(62 items)