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Autor: fizyk doświadczalny, pomiarowiec, obecnie już tylko analizujący dane produkowane przez innych, programista (Delphi, JS, VB). Wybrane publikacje. Zdjęcia. Formalne kwalifikacje matematyczne: oceny bdb na studiach, z algebry, analizy, mechaniki teoretycznej itp. Formalne kwalifikacje filozoficzne: wykład na pierwszym roku fizyki: Jan Kurowicki, egzamin z filozofii do doktoratu: Wit Jaworski, oficjalny fotograf Biesiad Filozoficznych Janusza A. Majcherka.

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Modern discussion of the infinite is now regarded as part of set theory and mathematics. /Wikipedia

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Paweł Polak, Rozwój pojęcia nieskończoności. Dialog pomiędzy filozofią a matematyką, Semina Scientiarum 2002

... nieskończoność jest w pewien sposób bliska człowiekowi, ponieważ wpisuje się w wyczuwalną przez niego potrzebę transcendencji

Anaksymander: apeiron — bezkres ... na określenie ilościowej własności pierwotnej materii

... filozofii pitagorejskiej, w której liczbę zaczęto traktować jako zasadę bytu

... Antyfon przyjął, że koło jest wielokątem o nieskończonej ilości boków ... przez Arystotelesa błąd, który polegał na utożsamieniu bardzo wielkiej ilości boków z nieskończoną ich ilością. Rozumienie nieskończoności Antyfona było zbyt ubogie — utoż- samiał on nieskończoność z bardzo dużą liczbą

... Demokryt próbował zbudować matematykę skończoną.

Zenona ... Stagiryta wydzielił więc:
1. nieskończoność podziałów i nieskończoność krańców;
2. nieskończoność aktualną i potencjalną.

Arystoteles odrzucił istnienie nieskończoności aktualnej. ... wszystko w świecie pozostaje skończone.

... wzorcem nieskończoności potencjalnej jest ciąg liczb, który w każdym momencie składa się ze skończonej ilości elementów, ale nie istnieje żadne ograniczenie w jego rozszerzaniu.

Dopiero dzięki pracom Cantora w II poł. XIX wieku udało się wytworzyć odpowiedni grunt dla owocnej współpracy filozofii i matematyki.

Mikołaj z Kuzy. Twierdził on, że nieskończoność nie pozostaje w proporcji do niczego, jest więc jako taka nieznana. Kuzań- czyk podkreślał istotową różnicę pomiędzy tym co skończone, a tym co nieskończone.

... nie da się pominąć aspektu wzajemnych oddziaływań i tworzyć nawet tak abstrakcyjnej nauki jaką jest matematyka, bez uwzględnienia oddziaływań pochodzących z filozofii.

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Galileo, On two New Sciences, 1638
So far as I see we can only infer that the totality of all numbers is infinite, that the number of squares is infinite, and that the number of their roots is infinite; neither is the number of squares less than the totality of all numbers, nor the latter greater than the former; and finally the attributes "equal," "greater," and "less," are not applicable to infinite, but only to finite, quantities.

John Locke, Essay, II. xvii. 7.
Whatever positive ideas we have in our minds of any space, duration, or number, let them be never so great, they are still finite; but when we suppose an inexhaustible remainder, from which we remove all bounds, and wherein we allow the mind an endless progression of thought, without ever completing the idea, there we have our idea of infinity... yet when we would frame in our minds the idea of an infinite space or duration, that idea is very obscure and confused, because it is made up of two parts very different, if not inconsistent. For let a man frame in his mind an idea of any space or number, as great as he will, it is plain the mind rests and terminates in that idea; which is contrary to the idea of infinity, which consists in a supposed endless progression.

Emmanuel Levinas, Philosophy and the Idea of Infinity

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Ludwig Wittgenstein, Tractatuslogico-philosophicus
6.4312
Czasowo pojęta nieśmiertelność duszy ludzkiej — czyli jej wieczne życie po śmierci — nie tylko nie jest niczym zagwarantowana, lecz nade wszystko nie daje wcale tego, co zawsze chciano przez nią osiągnąć. Czy rozwiąże to jakąś zagadkę, że będę żył wiecznie? Czyż takie wieczne życie nie będzie równie zagadkowe jak obecne? Rozwiązanie zagadki życia w czasie i przestrzeni leży poza czasem i przestrzenią. (Nie chodzi tu przecież o rozwiązywanie problemów naukowych.)

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Emmanuel Levinas, Philosophy and the Idea of Infinity

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Internet Encyclopedia of Philosophy. The Infinite

Philosophers want to know whether there is more than one coherent concept of infinity

The density of matter at the center of a black hole is infinitely large. An electron is infinitely small. An hour is infinitely divisible. The integers are infinitely numerous. These four claims are ordered from most to least controversial

Thomas Aquinas ... God is infinitely powerful

Gauss ... scientific theories involve infinities merely as idealizations and merely in order to make for easy applications of those theories, when in fact all physically real entities are finite

Quine ... the first three sizes of Cantor’s infinities are the only ones we have reason to believe in

2,500 years ... actually infinite, potentially infinite, and transcendentally infinite

Aristotle ... “the idea of the actual infinite-of that whose infinitude presents itself all at once-was close to a contradiction in terms…,”

Calculus

Dedekind in 1888

Cantor 1887 ... each potential infinite…presupposes an actual infinite.

Cardinal numbers ... ℵ ... ℵ0, ℵ1; continuum problem

Cantor ... is not invention but rather is discovery about a mind-independent reality.

actual infinities are indispensable in mathematics and science

whether the set of all cardinal numbers has a cardinal number ... if it does, then it doesn’t

Russell’s Paradox of 1901 ... the set of all sets that are not members of themselves

Zermelo-Fraenkel’s set theory (ZF) was the best way or the least radical way

the concept of "infinite set" within ZF was claimed by many philosophers to be the paradigm example of how to provide a precise and fruitful definition of a philosophically significant concept.

we can never use the word “infinity” coherently because infinity is ineffable or inherently paradoxical

Infinity and the Mind

the infinite is beyond the grasp of the human mind

contemporary philosophers of psychology believe mental pictures are not essential to having any concept

whether we can coherently think about infinity to the extent of being said to have the concept

If we understand negation and have the concept of finite ...

might be thought of by a powerful enough mind.

Infinity in Metaphysics

person’s brain contains approximately 1027 atoms

some version of transcendental infinity that makes infinity be somehow beyond human comprehension

Levinas says the infinite is another name for the Other ... facing a practically incomprehensible and unlimited number of possibilities upon encountering another conscious being ... should say instead that there are too many possibilities to be faced

Cantor claimed his work was revealing God’s existence and that these mathematical objects were in the mind of God ... God gave humans the concept of the infinite so that they could reflect on His perfection

The connection between infinity and God exists in nearly all of the world’s religions.

The multiverse theories of cosmology in the early 21st century allow there to be an uncountable infinity of universes ... Augustine had this worry when considering infinite universes, and he responded that "Christ died once for sinners...."

Infinity in Physical Science

... examples where infinity occurs within physical science
(1) Standard cosmology based on Einstein’s GTR implies the density of the mass at the center of a simple black hole is infinitely large
(2) The Standard Model of particle physics implies the size of an electron is infinitely small.
(3) General relativity implies that every path in space is infinity divisible.
(4) Classical quantum theory implies the values of kinetic energy of an accelerating, free electron are infinitely numerous.
are not something that could be measured directly

George Berkeley and David Hume denied the physical reality of even potential infinities on the empiricist grounds that such infinities are not detectable by our sense organs. ... instrumentalists also ...

reality looks “as if” there are physical infinities ... useful mathematical fiction

theoretical terms that refer to infinities, then infinities must be accepted

Standard Model ... time is a continuum, and space is a continuum ... mass is a continuum as well as energy

space consists of discrete units called loops

Brian Greene ... the notion of being able to divide distances or durations into ever smaller units likely comes to an end at around the Planck length (10-35 m) and Planck time (10-43 s).

Roger Penrose ... The continuum still features in an essential way ... we need to take the use of the infinite seriously

Singularities ... A theory that involves singularities...carries within itself the seeds of its own destruction.

Strings have an infinite number of possible vibrational patterns each corresponding to a particle that should exist if we take the theory literally.

Big Bang ... stopped shrinking ... 10-35 meters

Gauss ... scientific theories involve infinities merely as approximations or idealizations

Penrose ... To my mind, a physical theory which depends fundamentally upon some absurdly enormous...number would be a far more complicated (and improbable) theory than one that is able to depend upon a simple notion of infinity

Erwin Schrödinger remarks, “The idea of a continuous range, so familiar to mathematicians in our days, is something quite exorbitant, an enormous extrapolation of what is accessible to us.”

Infinity in Cosmology

Immanuel Kant (1724–1804) declared that space and time are both potentially infinite in extent because this is imposed by our own minds. ... geometry of space must be Euclidean

The volume of spacetime is finite at present if we can trust the classical Big Bang theory.

Multiverse, then both space and time are actually infinite

Infinity in Mathematics

Bertrand Russell ... thinking in an unfamiliar way

The series s1 + s2 + s3 + … converges to S if, and only if, for every positive number ε there exists a number δ such that |sn+h + sn| < ε for all integers n > δ and all integers h > 0. In this way, reference to an actual infinity has been eliminated.

infinitesimal object is as small as you please but not quite nothing ... an infinite number of infinitesimal steps

Robinson: h is infinitesimal if and only if 0 < |h| < 1/n, for every positive integer n ... the hyperreal line

A constructivist, unlike a realist, ... an unknowable mathematical object is impossible. ... potential infinites can be constructed, actual infinities cannot be

Brouwer ... intuitionist school ... Numbers are human creations.

pi is intuitionistically legitimate because we have an algorithm ... number g is not legitimate ... n consecutive 7s in the decimal expansion of pi

there is no evidence supporting the belief in the existential character of the totality of all natural numbers ... is not a closed realm of things existing in themselves

Finitists, .. the actually infinite set of natural numbers does not exist.

ultrafinitist .. numbers such as 2100 and 21000 can never be accessed by a human mind

Quine .. some actually infinite sets are indispensable to all these scientific theories .. All this success is a good reason to believe

Quine .. only the first three alephs: ℵ0 for the integers, ℵ1 for the set of point places in space, and ℵ2 for the number of possible lines in space (including lines that are not continuous)

Zermelo-Fraenkel Set Theory

Using the axiom of choice, .. set is infinite .. for every natural number n, there is some subset whose size is n.

The power set axiom (which says every set has a power set, namely a set of all its subsets) then generates many more infinite sets of larger cardinality, a surprising result that Cantor first discovered in 1874

The Axiom of Choice and the Continuum Hypothesis

Platonists tend to like the axiom

mathematics’ most unintuitive theorem, the Banach-Tarski Theorem, requires the axiom of choice

A set is always smaller than its power set.

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