An ancient question: how many numbers are there?

In the sixth century B.C.E. Anaximander of Miletus gave a name to the infinite, calling the indeterminate, or “something without bound, form, or quality,” apeiron. But limitlessness, and non-rationality, and ineffability were all descriptions of what infinity was not. The closest anyone came for centuries to a positive definition was “potentiality” as opposed to “actuality,” in the influential terms of Aristotle. But this formulation did little to help define the indefinable. Even Galileo, nearly two thousand years later, bowed his weighty head before the limitless. Contemplating the series of infinite integers (1,2,3,4...) and the series of infinite even numbers (2,4,6,8...), he gave up: clearly both could continue without limit, and yet wasn’t one precisely one half as large as the other?
And thus we crack open the shell of one of the hardest problems in Metaphysics. Now, I must admit that I love the thesis of this particular article: that mysticism, and not pure reason, is necessary to apprehend the truth. That is exactly what I would like to believe to be true, here as elsewhere.

For that reason, let us turn aside from it, and explore something else. Dr. Anthony Kenny talks about the problems of 'potentiality and actuality' as expressed by the famous Islamic philosopher Avicenna. (This is from pp. 193-5 of Kenny's Medieval Philosophy.)
If we take 'essence' in the generic sense, then the distinction between existence and essence corresponds to the distinction between the question 'Are there Xs?' and 'What are Xs?' That there are quarks is not at all the same thing as what quarks are.... But if we take the distinction to be one about individual essences, then it seems to entail the possibility of individual essences not united to any existence; individual essences of possible, but non-existent individuals. The essence of Adam, say, is there from all eternity; when God creates Adam, he confers actuality on this already present possibility.
Dr. Kenny does not want us to accept this idea.
Let us ask how an individual humanity -- say the humanity of Abraham -- is itself individuated. It is not individuated qua humanity: that is something shared by all humans. It is not individuated by belonging to Abraham: ex hypothesi, it could exist, and be the same individual, even if Abraham had never been created but remained a perpetual possibility. It can only be identified, as Avicenna says, by the properties and accidents that accompany it -- that is to say, by everything that was true of the actual Abraham -- that he migrated from Ur of the Chaldees, obeyed a divine command to sacrifice his son... Of course, since Abraham's essence was there before Abraham existed, it could not be individuated by the actuality of these things, but only by their possibility.
This natually looks like Saul Kripke's assertion that Aristotle could have been Aristotle even if he'd gone into shoemaking instead of philosophy. Names are, Kripke said, a 'rigid designator' for a given thing; what that thing does, or might have done, is still captured by the designation across various possible worlds. He did this here; he did that there; but it's still the same thing. Joe in this world lost his Mustang to me in a poker game; in another world, the same Joe decided to spend the night reading philosophy, and therefore kept his car. (Wise Joe! Even if it gave him a headache!)

Well, all that takes us right back around to the article: and the mysticism.
Rocking in the belly of the Imperial Russian Navy ship as it sailed, in June 1913, through sparkling Aegean waters toward the Monastery of St. Pantaleimon on Mount Athos, the Archbishop Nikon of Vologda braced himself. He was determined. Even before hermits in the deserts of Palestine practiced the “Prayer of the Heart” in the fourth century, Christianity had known mystical sects. Later called hesychast monks from the Greekhesychia, or stillness, such mystics had believed in the power of glossalia, or “praying without ceasing,” with control of breathing and the heartbeat, to reach union with God. Already in the fourteenth century Gregory Palamas, a Constantine monk, had settled on Mount Athos preaching hesychasm as a true alternative to the staid rationalism of Byzantine Christianity. Now, in modern times, to the great consternation of leaders of the Russian Orthodox Church, a Russian monk named Ilarion had instituted the “Jesus Prayer” among his followers (“Lord Jesus Christ, Son of God, have mercy on me, a sinner”—sometimes shortened to “Lord Jesus Christ,” or just “Jesus”—repeated over and over again), a prayer considered heretical for harking back to mystical times. Ilarion admitted that when reciting the prayer worshippers needed to be careful. There were three “stages of immersion”—the oral, the mental, and finally the “Prayer of the Heart”: if one jumped between them prematurely, warm blood could descend to the lower parts of the body and lead to sexual arousal. Archbishop Nikon of Vologda clenched his fists.

The last thing Nicholas II wanted was for bickering monks to invite an invasion of the Greek army into the monastery; the czar didn’t care much about the theological dispute, but he was not about to lose a Russian protectorate in the Aegean. Later, after the gunboatDonets had lowered its anchor and Russian marines stormed the monastery with clubs, water hoses, and bayonets, each side would claim a different story. Whether monks were brutally murdered, soldiers were beaten, or only a small number of fanatics were rather quietly subdued didn’t in the end really matter: after all, nearly a thousand monks were hauled back on the ships to Russia, where their leadership was thrown in jail, and the rest were defrocked and banished to far-off provinces. The Name Worshippers of Mount Athos had been shut down. What mattered most were the defiant interruptions to the angry sermon of Archbishop Nikon of Vologda, who had marched into the monastery courtyard behind the troops. “You mistakenly believe that names are the same as God,” his voice trembled. “But I tell you that names, even of divine beings, are not God themselves.” Corralled, water-drenched, their arms twisted violently behind their backs, the monks would not be silenced. “Imia Bozhie est’ sam Bog!” some of them were clearly heard shouting, their eyes alight. “The Name of God is God!”....

Throwing himself into set theory back in Moscow, Luzin maintained strong ties with Florensky, and here is where the escapades of the monks of the Aegean return to our story. It is not clear precisely when both men first learned of Name Worshipping, but already in 1906 they enjoyed calling each other by names other than their own. When news of the rebellion on Mount Athos reached Russia in 1913, Florensky spoke up publicly in its favor, and befriended monks who had endured firsthand the navy’s brutal attack on St. Pantaleimon. Soon two worlds were becoming entwined. Lebesgue had asked whether a mathematical object could exist without defining (meaning naming) it, and now the answer was becoming clear. Just as naming God via glossolalian repetition was a religious act that brought the deity into existence, so naming sets via increasingly recursive definitions was a mathematical act that conferred a reality in the world of numbers. Cantor and before him the ancient Neoplatonists had shown the way, but this was only the beginning. Infused with mysticism, Florensky believed, new forms of mathematics and religion were being born, ones that by rejecting determinism would rescue mankind from catastrophe. In both cases—God and infinity—the key to bringing abstractions into reality was bestowing upon them a name.
What is the power of a name? And, as Kripke warns us to consider, just what are we naming? There is a truth lying there as deep and as dangerous as the sea.

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