Group:
philosophy of mathematics
Group:
sets
Topic:
continuum in mathematics
Topic:
elements
Topic:
infinite sequences
Topic:
kinds of numbers
Topic:
Liar's paradox and Russell's paradox
Topic:
Turing machine
Topic:
unbounded precision
Topic:
what is a number
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Subtopic: continuous 
| Quote: when instances approach each other continually and eventually merge, the consequences or outcomes must do so also; e.g., parabola is an ellipse
| | Quote: rule of continuity -- no change takes place by leaps and bounds
|
Subtopic: continuum
| Quote: every body has an infinite number of parts; i.e., the continuum is infinitely divisible
| | Quote: there are no atoms; in every particle there is a world of innumerable creatures [»leibGW_1679]
| | Quote: in every parcel of matter there is an infinity of created things
|
Subtopic: discrete vs. continuous
| Quote: space should be discrete because continuous space yields infinities; so geometry does not extend down into infinitely small space [»feynR_1965]
|
Subtopic: differential
| Quote: Leibniz's infinitesimal calculus was independent of what differentials actually are; avoids metaphysical disputes [»kramS2_1996]
| | Quote: in Zeno's argument there is an infinite number of steps but not an infinite amount of time; instead, a derivative concerns the ratio of infinitesimals [»feynRP_1963]
| | Quote: the scientific definition of speed is the ratio of infinitesimal distance over infinitesimal time as the time gets smaller and smaller; not known to the Greeks
| | Quote: there are infinitely many instants which correspond to infinitely many degrees of diminished speed
|
Subtopic: in the limit
| Quote: the circumference of an infinite circle and a straight line are the same thing [»galiG_1632]
| | Quote: if moveables descend rays from a point, they form a circle of ever widening diameter; a wonderful feature of nature [»galiG_1638]
|
Subtopic: set theory
| Quote: Zermelo's axioms for set theory: extensionality, elementary sets, separation, union, choice, infinity [»zermE19_1908]
|
Subtopic: inductive proof
| Quote: proof by induction allows us to generalize, to pass from the finite to the infinite [»poinH_1902, OK]
| | Quote: we define the natural numbers as those to which proofs by mathematical induction can be applied
| | Quote: mathematical induction is the essential characteristic that distinguishes the finite from the infinite [»russB_1919, OK]
|
Subtopic: infinite from finite
| Quote: number proceeds from unity to ten to 30 to 100 and on to infinity
| | Quote: Frege sketches the proof that every natural number has a successor and therefore forms an infinite sequence [»fregG_1884]
| | Quote: infinite things can be compounded out of the combination of a few; e.g., numbers from digits [»leibGW_1679]
| | Quote: finite-state grammars are the simplest that generate an infinite number of sentences from a finite apparatus [»chomN9_1956]
|
Subtopic: unbounded entities
| Quote: zero, one, and infinity are the only reasonable numbers in program language design [»maclBJ_1987]
| | Quote: a Petri net can be an infinite net because its parts can not communicate with arbitrarily far away parts [»petrCA1_1966]
| | Quote: Petri constructed a Petri net representation of a Turing machine; used automatic tape constructors; equivalent to an infinite net [»petrCA1_1966, OK]
|
Subtopic: floating point infinity
| Quote: SANE supports extended precision, NaNs, Infinities, unordered comparisons, rounding, and floating point exceptions; no signaling NaNs
|
Subtopic: contingent truth
| Quote: possible truths are those that do not lead to contradiction; contingent truths lack resolution even if continued to infinity [»leibGW_1686]
|
Subtopic: division by zero
| Quote: Bhaskara II (1150): in a/0 there is no alteration, as no change takes place in the infinite and immutable God [»dattB_1935]
|
Subtopic: infinite being
| Quote: free and contingent things furnish an infinite series of reasons, which God alone can see through; e.g., of space and time
| | Quote: the certitude and perfect reason of contingent truths is known only to God, who grasps the infinite with one intuition
| | Quote: God exists because only an infinite, perfect being can be the ultimate cause of a thing that thinks and has the idea of God [»descR_1641]
| | Quote: God constitutes duration and space; God is eternal and infinite, omnipotent and omniscient [»newtI_1685, OK]
|
Subtopic: infinite regress
| Quote: psychology must posit internal representations, but something is a representation only for or to someone; i.e., a homunculus that leads to an infinite regress [»dennDC_1978a]
| | Quote: the process of lemma-incorporation yields an infinite regress unless proof is a game or there are trivially true lemmas [»lakaI_1976]
|
Subtopic: no infinite regress
| Quote: we know when we know how to explain adequately; adding factors infinitely should take endless time
| | Quote: the end of every movement must be one of the divine bodies moving in the sky; no other ultimate source for movement; no infinite regress [»aris_322a]
| | Quote: there is a definite beginning and the reasons for things are not infinite [»aris_322a]
| | Quote: there can be no infinite regress in the production of things
| Quote: where there is no first term, there is no explanation at all; no infinite regress
|
Related Topics
Group: philosophy of mathematics (11 topics, 330 quotes)
Group: sets (7 topics, 148 quotes)
Topic: continuum in mathematics (7 items)
Topic: elements (22 items)
Topic: infinite sequences (7 items)
Topic: kinds of numbers (24 items)
Topic: Liar's paradox and Russell's paradox (25 items)
Topic: Turing machine (30 items)
Topic: unbounded precision (9 items)
Topic: what is a number
(55 items)
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