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I know that Kronecker claimed it was God's doing, and that even prehistoric humans used some ways of counting. But I am curious where the idea of a sequence of numbers stretching out into infinity appears for the first time explicitly. I suppose another way to phrase it would be to ask who invented infinity. "Ancient cultures had various ideas about the nature of infinity" is what Wikipedia says.

Pythagoreans must have understood it already, and Aristotle even discusses the difference between potential and actual infinities, so it had to be before that. Maybe they borrowed it from Egyptians? Babylonians had a positional system which allows recording arbitrarily large numbers in principle. But is that enough? What gives me pause is the history of zero. Babylonians and Alexandrian astronomers were using it as a placeholder for centuries before the concept of zero as representing nothing was formed, and not by them. And if it happened to zero it could happen to infinity. By the way, how ironic that infinity was discovered before zero.

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    $\begingroup$ Would you prefer an answer based on integers or infinity? $\endgroup$
    – HDE 226868
    Nov 11 '14 at 22:31
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    $\begingroup$ @HDE226868 This particular question is about "infinity of counting". I tacitly assumed that it was the first one conceived, but now that I think about it it's possible that geometric infinity of extension or divisibility preceded it. I was hoping that some early document like Rhind or Plimpton had "and so on" in it or something like that hinting at realization of indefinite continuation of integers, but the answers suggest that probably not. $\endgroup$
    – Conifold
    Nov 12 '14 at 19:17
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Very few (if any) mathematicians before Cantor thought of the SET of integers. Certainly for Euclid it was completely evident that the sequence of integers extends without limit. (He actually has a famous theorem that the sequence of PRIMES extends without limit). Who discovered this we will never know because very few mathematical sources before Euclid survived. Perhaps Pythagoreans, but maybe earlier. (What we know about Pythagoreans comes from much later secondary sources, Pythagoreans themselves were a secret society and did not publish their discoveries).

As I said, most mathematicians imagined infinite sets as POTENTIALLY infinite, that is without limit. To every integer you can add 1 and obtain a larger integer.

However, in theological literature, beginning from early Middle Age, we encounter ACTUAL infinity (infinite things, infinite sets). I think this begins in the Neoplatonic school, but Augustine (of Hyppo) certainly discusses (in V century) that in the City of God, and Cantor mentions him. These discussion continue in the medieval scholastic literature, but there is little mathematics or science in them.

The notion of actual infinity was revived by Cantor in his set theory, and nowadays this is a common language of mathematics.

Remark. Since the Hellenistic times, it is a popular opinion that the first mathematicians (Pythagoras, Thales) "learned something from Egyptians". Some modern authors tend to say that they learned everything from Egyptians. Serious research on the history of mathematics and astronomy does not confirm that. With enormous number of surviving texts, we know pretty much about ancient Egypt. There was NOTHING for people like Thales to learn there. Egyptian astronomy and mathematics was in very primitive state in comparison with contemporary Babylonian and Greek sciences.

Best source: O. Neugebauer. Exact sciences in antiquity.

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    $\begingroup$ I thought it was more or less uncontroversial that Pythagoras studied with priests in Egypt before founding his school. I agree that mathematics in surviving papyri is not overwhelming, but they deal with practical calculations, and priests were secretive, so we might not have the full story. Something in Egypt impressed Greeks enough to single them out as the only non-"barbarians". $\endgroup$
    – Conifold
    Nov 12 '14 at 19:25
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    $\begingroup$ Even that Pythagoras existed is controversial. Both Pythagoras and Thales (according to the much later accounts) indeed traveled to Egypt. However, as I said there was not much in Egypt that they could learn and bring home. $\endgroup$ Nov 12 '14 at 20:14
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    $\begingroup$ Quote from "Precession and the Pyramid Astronomical Knowledge in Ancient Egypt" by Jim Fournier "It follows that the ancient Greeks should be taken at their word when they claim that their knowledge is of great antiquity and was derived from Egyptian sources. Indeed it is nothing if not bizarre that modern scholars of the Greek world should go to great lengths to dismiss such claims on the part of the authors of the primary texts themselves, to instead rely on the advice of modern Egyptologists that the ancient Egyptians had no such knowledge." $\endgroup$ May 17 '15 at 19:50
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    $\begingroup$ The title and contents you quote suggests that this is a pseudo-scientific nonsense, like most of the modern trash written about "astronomical knowledge" in connection with pyramids. $\endgroup$ May 18 '15 at 5:02
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    $\begingroup$ Anyway, if the advanced Egyptian mathematics/astronomy existed but all traces of it are completely lost, there is no subject to talk about. $\endgroup$ May 18 '15 at 19:52
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Actually when we say Integer today, we mean set of all positive whole numbers, negative whole numbers and zero. But this complete set was not discovered/invented in a day. People were working with integers from the very beginning. They might be using different names though(like Whole numbers, Natural numbers, ...). According to Wikipedia

Negative numbers appeared for the first time in history in the "Nine Chapters on the Mathematical Art", which in its present form dates from the period of the Chinese Han Dynasty (202 BC – AD 220), but may well contain much older material.

In an article i found that the word "integer" was first used of whole numbers in 1571 by Thomas Digges (refer this).

The same article further says that,

"The positive and negative numbers did not actually become part of a single "number line" (today's "set of integers") until the 1700's or 1800's."

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    $\begingroup$ Interesting that Digges applied "integer" to positive integers only. But I still wonder if anyone before Pythagoreans thought of them as "neverending". $\endgroup$
    – Conifold
    Nov 11 '14 at 20:42
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    $\begingroup$ @Conifold read this(math.tamu.edu/~dallen/history/infinity.pdf). $\endgroup$
    – Amit Tyagi
    Nov 11 '14 at 21:26
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Initially numbers were used for accounting. Thus, it seems, the numbering systems initially appeared around the time of the Neolithic revolution. Neolithic revolution signifies the change in society, which is characterised by the appearance of organized agriculture, as well as state power. With new technology the food could be produced in excess, stored and redistributed.

This led to the need for counting and accounting.

Linguistics confirms that languages got numerals around this change as well. The Neolithic revolution happened first in Fertile Crescent around 10000 BC. This is roughly when integers were invented.

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