The earliest known documented knowledge of infinity is presented in the Veda- Yajur Veda which states that "if you remove a part from infinity or add a part to infinity, still what remains is infinity". The Indian Jaina mathematical text Surya Prajinapti (ca. 400 BC) classifies all numbers into three sets: enumerable, innumerable and infinite.
It recognizes five different types of infinity:
The concept of different orders of infinity would remain unknown in Europe until the late 19th century. In Europe, the traditional view derives from Aristotle: "...it is always possible to think of a larger number: for the number of times a magnitude can be bisected is infinite. Hence the infinite is potential, never actual; the number of parts that can be taken always surpasses any assigned number." This is often called potential infinity; however there are two ideas for this:
1. It is always possible to find a number of things that surpasses any given number, even if there are not actually such things and
2. We may quantify over-infinite sets without restriction.
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The second view is found written here (in Latin) by medieval writers such as William of Ockham:
"Sed omne continuum est actualiter existens. Igitur quaelibet pars sua est vere existens in rerum natura. Sed partes continui sunt infinitae quia non tot quin plures, igitur partes infinitae sunt actualiter existentes." which translates to: "But every continuum is actually existent. Therefore any of its parts is really existent in nature. But the parts of the continuum are infinite because there are not so many that there are not more, and therefore the infinite parts are actually existent."
The parts are actually there, in some sense. However, on this view, no infinite magnitude can have a number. For whatever number we can imagine, there is always a larger one: (RE: Aquinas saying "There are not so many (in number) that there are no more". Aquinas also argued against the idea that infinity could be in any sense complete, or a totality.
The parts are actually there, in some sense. However, on this view, no infinite magnitude can have a number. For whatever number we can imagine, there is always a larger one: "There are not so many (in number) that there are no more".Aquinas also argued against the idea that infinity could be in any sense complete, or a totality.
Galileo (during his long house arrest in Siena after his condemnation by the Inquisition) was the first to notice that we can place an infinite set into one-to-one correspondence with one of its proper subsets (any part of the set, that is not the whole). For example, we can match up the "set" of even numbers {2, 4 6, 8...} with the natural numbers {1, 2, 3, 4...}.
It appeared, by this reasoning, as though a set which is naturally smaller than the set of which it is a part is in some sense the same size. He thought this was one of the difficulties which arise when we try, "with our finite minds", to comprehend the infinite.
"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."[On two New Sciences, 1638]
The idea that size can be measured by one-to-one correspondence is today known as Hume's principle, although Hume, like Galileo, believed the principle could not be applied to infinite sets. Locke, in common with most of the empiricist philosophers, also believed that we can have no proper idea of the infinite. They believed all our ideas were derived from sense data or "impressions", and since all sensory impressions are inherently finite, so too are our thoughts and ideas.
(continued from above) Modern discussion of the infinite is now regarded as part of set theory and mathematics, 
and generally avoided by philosophers. An exception was Wittgenstein, who made an impassioned attack upon axiomatic set theory, and upon the idea of the actual infinite, during his "middle period". He said that the relation m = 2n does not correlate the class of all numbers with one of its subclasses.
"It correlates any arbitrary number with another, and in that way we arrive at infinitely many pairs of classes, of which one is correlated with the other, but are never related as class and subclass. Neither is this infinite process itself in some sense or other such a pair of classes ...In the superstition that m = 2n correlates a class with its subclass, we merely have yet another case of ambiguous grammar."(Philosophical Remarks § 141, cf Philosophical Grammar p. 465) Unlike the traditional empiricists, he thought that the infinite was in some way given to sense experience. "... I can see in space the possibility of any finite experience ...we recognize [the] essential infinity of space in its smallest part."
INFINITY SYMBOL
The infinity symbol is derived from the device known as a mobius (named after a nineteenth century mathematician) strip.
It is suggested that by the name it is sometimes called - the infinity symbol that looks like a number 8 turned on it’s side; the lemniscate, derives from the Latin lemniscuses; meaning "ribbon". You could walk forever along a simple loop formed from a ribbon. A mobius strip (pictured) is a strip of paper which is twisted and attached at the ends, forming a two dimensional surface. The symbol had been in use to represent infinity for over two hundred years before August Ferdinand Mobius and Johann Benedict Listing discovered the Möbius strip in 1858.
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