Archimedes actual infinities, as revealed in his lost manuscript.
I read in the article 'A Prayer for Archimedes', about the long-lost text by the ancient Greek mathematician which shows that he had begun to discover the principles of calculus.
In the claim put forward by Reviel Netz, an historian of mathematics at Stanford University who transcribed the newly found text, says that the recent discoveries show that Archimedes indeed used the notion of actual infinity.
Infinities, as defined by Aristotle, mentioned here
"The Greek philosopher Aristotle built defenses against infinity's vexing qualities by distinguishing between the "potential infinite" and the "actual infinite." An infinitely long line would be actually infinite, whereas a line that could always be extended would be potentially infinite. Aristotle argued that the actual infinite didn't exist."
An infinitely long line would be actually infinite, hence actual infinity, whereas a line that could always be extended would be potentially infinite, hence potential infinity. Aristotle argued that the actual infinity didn't exist.
I read further
"Archimedes found a relationship between the full area of that slice, which was a section through the plane-sided volume, and the smaller area within it, which was a section through the curved shape. Then he argued that he could use that relationship to calculate the entire volume of the curved shape, because both the curved figure and the straight one contained the same number of slices."
"That number just happened to be infinity—actual infinity."
"The interesting breakthrough is that he is completely willing to operate with actual infinity," Netz says, but he adds that "the argument is definitely not completely valid. He just had a strong intuition that it should work." In this case, it did work, but it remained for Newton and Leibniz to figure out how to make the argument mathematically rigorous."
Archimedes being willing to operate with actual infinity, an infinitely long line, instead of a line that could always be extended. A potentially infinite line, what Aristotle argued that exists, whereas the actual infinity did not exist.
So, there is not infinity as such, whatever name it can be given, actual or potential, since an infinite line can always be extended. There are no boundaries.
But the infinity of decimal numbers, lying between two integer numbers, it has boundaries. The two integer numbers that lies within. Does having boundaries determine the kind of infinity it is? The infinity of decimal numbers between two integers, can not be extended beyond the integer boundaries.
It cannot be potentially infinite, it can only be actual infinity, what the calculus uses. What has driven Archimedes "strong intuition that it should work", and why "In this case, it did work". And it works when dealing with infinities that the boundaries are known or postulated beforehand.
What "it remained for Newton and Leibniz to figure out, how to make the argument mathematically rigorous.
Confusing the issues amidst vague statements, if "the argument is definitely not completely valid." What is valid, what is completely valid, what is not completely valid, and what is definitely not completely valid. How is validity defined, what is required to make an argument valid.
I read further in the same article
"Newton and Leibniz also worked with actual infinity. Leibniz went so far as to say in a letter, "I am so in favor of the actual infinite that instead of admitting that Nature abhors it, as is commonly said, I hold that Nature makes frequent use of it everywhere, in order to show more effectively the perfections of its Author."
Since Newton and Leibniz also worked with actual infinity and produced calculus, would that not validate Archimedes intuition and his willingness to operate with actual infinity?
And what is meant by Leibniz' statement that nature makes frequent use of it, everywhere. Does that not imply the fractality inherent in all nature's objects, chaos driven processes surpassing, traversing fractal dimensions, from the microscales to the macroscales weaving the perfections of its author, chaos. Actual infinities trapped within the delimiting boundaries of any nature's object, all objects.
The statement 'modern calculus no longer makes use of the actual infinite; it sticks with Aristotle's distinction', a matter of taste?