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Galileo's paradox is the observation that the natural numbers can be put into one to one correspondence with the square numbers, showing that an infinite set can be bijected to a proper subset of itself.
I remember reading that this was already known to Arab mathematicians in the 12th or 13th century, but now I can't find any references via Google.
Are there online references to the origin of this discovery?
The answer depends on what "this" means. According to Mancosu's Measuring the Size of Infinite Collections of Natural Numbers (reprinted in his book Abstraction and Infinity):
It is actually unclear when the paradox, in the numerical form I just gave, appears. In the Greek tradition we have paradoxes that are related, but are not identical, to it; in this tradition what is claimed to be paradoxical is the existence of different sizes of infinity."
An early geometric example appears in Proclus's commentary on the Book I of Elements:"But if from one diameter two semicircles are produced, and if an indefinite number of diameters can be drawn through the center, it will follow that the number of semicircles is twice infinity". Proclus expresses the Aristotelian view that infinities have no "comparative sizes" (because actual infinities do not even exist), which, ironically, is what "Galileo's paradox" confirms. The "Arab mathematicians" is most likely Thābit ibn Qurra al-Ḥarrānī (826-901), best known for his work on the parallel postulate (anticipating elements of hyperbolic geometry), and amicable numbers. Thābit was apparently the first known source to challenge the Aristotelian view and discuss "comparative sizes" of infinities more in the spirit of Cantor. In Questions asked of Thābit ibn Qurra (narrated by his student) we read:
"We questioned him also regarding a proposition put into service by many revered commentators, namely that an infinite cannot be greater than an infinite. – He pointed out to us the falsity of this (proposition) also by reference to numbers. For (the totality of) numbers itself is infinite, and the even numbers alone are infinite, and so are the odd numbers, and these two classes are equal, and each is half the totality of numbers. That they are equal is manifest from the fact that in every two consecutive numbers one will be even and the other odd; that the (totality of) numbers is twice each of the two [other classes] is due to their equality and the fact that they (together) exhaust (that totality), leaving out no other division in it, and therefore each of them is half (the totality) of numbers."
This is the view that the paradox challenged. According to Murdoch's entry in Cambridge History of Later Medieval Philosophy, there was a lively discussion of the issue in the medieval Middle East (but not with numbers as examples), with Al-Shahrastani, Averroes and Maimonides participating. All of them are from 12th century, none is a mathematician, and only Averroes is an Arab. Murdoch further remarks:
"Although the paradox was treated in other ancient sources they were either not available in the Latin Middle Ages or never cited; the same must be said of the Islamic discussions of the paradox. On the basis of presently available evidence, then, philosophers and theologians of the Latin West appear to have realized the importance of the paradox on their own”.
In the West Bradwardine (1300 – 1349) was apparently first to apply one-to-one correspondence to show that an infinity can be equal to one of which it is a proper part. But his argument in Geometria Speculativa is part of a reductio to argue against eternity of the world, a side of a contradiction, not a valid conclusion. Oresme (c. 1325 – 1382) comes closest to phrasing the paradox as it is phrased today in his commentary on Aristotle's Physics:
"All infinite multitudes are simply infinite, therefore there is none that is greater than or less than another... consider the multitude of odd numbers and [perfect] squares; then, when there is the first, the second, the third and fourth, and thus, according to the order, without end all the numbers, it follows that there is no less than the multitude of odd numbers which lasted for the multitude of all numbers, and thus arguably the multitude of the parts of the even and the odd numbers is not less than the whole, or the whole multitude is not more than its parts."
By the way, Bradwardine and Oresme also anticipated Galileo's work on uniformly accelerated motion, including a proof of the mean speed theorem, although they did not apply it to falling bodies (or anything else physical), see Was Galileo a plagiarizer?
