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paradoxe de Galilée.md Normal file
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sources/0 - cours/LOGOS S2/le savoir en mathématiques/(Brancato) Cantor Leibniz Spinoza.md
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> [!PDF|yellow] [[sources/0 - cours/LOGOS S2/le savoir en mathématiques/(Brancato) Cantor Leibniz Spinoza.pdf#page=3&selection=25,0,29,47&color=yellow|(Brancato) Cantor Leibniz Spinoza, p.3]]
> The main reason why Spinoza and Leibniz cannot be regarded, in Cantor’s mind, as true anticipators of his notion of infinity, is their rejection of the concept of an infinite number.
^rejection-of-infinite-number
> [!PDF|yellow] [[sources/0 - cours/LOGOS S2/le savoir en mathématiques/(Brancato) Cantor Leibniz Spinoza.pdf#page=3&selection=34,3,80,48&color=yellow|(Brancato) Cantor Leibniz Spinoza, p.3]]
> Starting from Galileo Galilei, mathematicians and philosophers debated on whether or not it was possible to conceive a number capable of expressing infinity and, if that was the case, which properties such number would have. Galilei concluded that this concept would inevitably lead to a contradiction, because admitting the possibility of an infinity expressed in numerical form means also admitting that infinity is somehow homogeneous to finite numbers, sharing then with them properties like « being greater than », « being equal to » and « being less than » something else. Intuitively, such properties do not seem to apply very well to the notion of infinity, because for instance the series of natural numbers and the series of their squares, if analysed in the same way of finite sets, don’t seem to contain as many elements in the same interval, while at the same time they are both supposed to represent infinite sets, meaning that in infinity they should instead be equal. More precisely, it seems that in the same interval a one-to-one correspondence between the two sets cannot be established. Similar reflections lead Galilei to a position that will be very influential in the early modern period: pertaining infinity, qualifications such as less or greater have no place, meaning that an infinite number homogeneous to its finite numerical counterparts would represent a contradictory concept.
^galileo-paradox