Topic: infinity and infinitesimal

topics > Group: mathematics

philosophy of mathematics

continuum in mathematics
infinite sequences
kinds of numbers
Liar's paradox and Russell's paradox
Turing machine
unbounded precision
what is a number

Subtopic: continuous up

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 up

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 up

Quote: space should be discrete because continuous space yields infinities; so geometry does not extend down into infinitely small space [»feynR_1965]

Subtopic: differential up

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 up

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 up

Quote: Zermelo's axioms for set theory: extensionality, elementary sets, separation, union, choice, infinity [»zermE19_1908]

Subtopic: inductive proof up

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 up

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 up

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 up

Quote: SANE supports extended precision, NaNs, Infinities, unordered comparisons, rounding, and floating point exceptions; no signaling NaNs

Subtopic: contingent truth up

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 up

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 up

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 up

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 up

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 up

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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