December 17, 2022
Dear Nikola,
a) Thank you very much for your visit.
b) Please feel free to come back whenever you wish.
c) I admit to being a mathematical imbecile who failed his first midterm calculus exam at Harvard.
d) For 76 years, I have tried over and over to become familiar with and proficient in mathematics. Einstein's definition of insanity, doing the same thing over and over and expecting different results, applies to to my efforts to learn. I can't even learn to stop trying to learn.
e) Please feel no obligation to read or to reply to this letter.
f) Kleene reports Galileo as remarking in 1638 that the squares of the positive integers can be placed in a 1-1 correspondence with the positive integers themselves, despite the ancient axiom that the whole is greater than any of its parts.
g) https://se-scholar.com/se-blog/2017/6/23/who-said-the-whole-is-greater-than-the-sum-of-the-parts suggests that whatever the mathematical merits of Kleene's report, it is philologically unreliable. Aristotle is reported as having said

Metaphysics Book 8: 1045α ... περὶ δὲ τῆς ἀπορίας τῆς εἰρημένης περί τε τοὺς ὁρισμοὺς καὶ περὶ τοὺς ἀριθμούς, τί αἴτιον τοῦ ἓν εἶναι; πάντων γὰρ ὅσα πλείω μέρη ἔχει καὶ μὴ ἔστιν οἷον σωρὸς τὸ πᾶν [10] ἀλλ᾽ ἔστι τι τὸ ὅλον παρὰ τὰ μόρια, ἔστι τι αἴτιον, ἐπεὶ καὶ ἐν τοῖς σώμασι τοῖς μὲν ἁφὴ αἰτία τοῦ ἓν εἶναι τοῖς δὲ γλισχρότης ἤ τι πάθος ἕτερον τοιοῦτον. ὁ δ᾽ ὁρισμὸς λόγος ἐστὶν εἷς οὐ συνδέσμῳ καθάπερ ἡ Ἰλιὰς ἀλλὰ τῷ ἑνὸς εἶναι.

As translated by Hugh Trendennick: With regard to the difficulty which we have described in connection with definitions and numbers, what is the cause of the unification? In all things which have a plurality of parts, and which are not a total aggregate but a whole of some sort distinct from the parts, there is some cause ; inasmuch as even in bodies sometimes contact is the cause of their unity, and sometimes viscosity or some other such quality.But a definition is one account, not by connection, like the Iliad , but because it is a definition of one thing.

In Euclid Book 1, Common Notion No. 8, I find: "καὶ τὸ ὅλον τοῦ μέρους μεῖζον ἐστιν." which I translate: "and the whole is greater than its part." Note that τοῦ μέρους is genitive singular.  Euclid has nothing to say about the value of the sum of the parts of the whole, especially where one or more parts of the whole, or even the whole in its entirety might be negative.

In the fog of my ignorance, I find myself unable to explain how infinity being beyond numbers, can be countable. In my feeble unmathematical intuition, as the indeterminacy of the wave function collapses into definition when one measures, so the infinity function collapses into fineteness or number when one counts, when one adds to infinity or subtracts from it, as does the infinity of Hilberts Hotel when though filled to infinity, it admits yet another guest.  In searching for what is meant by infinity and infinitesimality I stumbled an account of hyper-real numbers.  https://en.wikipedia.org/wiki/Hyperreal_number Perhaps they point to a realm of meta-meta-mathematics, which secures the validity of metamathematics and hence of all mathematics.  Rhetoric is a discipline which is often overlooked. 

My ruminations suggest to me yet another science: The study of reason as insanity. Goethe warned me that if I started on that path, the devil would unavoidably intercept me. EJM

 Mephistopheles (in Fausts langem Kleide):

Verachte nur Vernunft und Wissenschaft,
Des Menschen allerhöchste Kraft,
Laß nur in Blend- und Zauberwerken
Dich von dem Lügengeist bestärken,
So hab’ ich dich schon unbedingt – 1855
Ihm hat das Schicksal einen Geist gegeben,
Der ungebändigt immer vorwärts dringt,
Und dessen übereiltes Streben
Der Erde Freuden überspringt.
Den schlepp’ ich durch das wilde Leben, 1860
Durch flache Unbedeutenheit,
Er soll mir zappeln, starren, kleben,
Und seiner Unersättlichkeit
Soll Speis’ und Trank vor gier’gen Lippen schweben;
Er wird Erquickung sich umsonst erflehn, 1865
Und hätt’ er sich auch nicht dem Teufel übergeben,
Er müßte doch zu Grunde gehn!

Translated by A. S. Kline
 Mephistopheles (In Faust’s long gown.)

Reason and Science you despise,
Man’s highest powers: now the lies
Of the deceiving spirit must bind you
With those magic arts that blind you,
And I’ll have you, totally – 1855
Fate gave him such a spirit
It urges him ever onwards, wildly,
And, in his hasty striving, he has leapt
Beyond all earth’s ecstasies.
I’ll drag him through raw life, 1860
Through the meaningless and shallow,
I’ll freeze him: stick to him: keep him ripe,
Frustrate his insatiable greed, allow
Food and drink to drift before his eyes:
In vain he’ll beg for consummation, 1865
And if he weren’t the devil’s, why
He’d still go to his ruination! 

PS: don't blame me for the translation