### NI - Pg. 11 - Names as God > Archbishop Nikon of Vologda accompanied the marines, a representative of the highest authorities in the Russian Orthodox Church, he lectured the assembled monks on the details of their "Name Worshipping heresy" in a voice trembling with fear and emotion: > > "You mistakenly believe that names are the same as God. But I tell you that names, even of divine beings, are not God themselves. The name of Jesus is not God. And the Son is less than the Father. Even Jesus said, 'the Father is greater than me.' But you believe you have both Christ and God." Interesting that both parties are wrong, for while names are not God, the Son is also not less than the Father. They are co-equal. I also think this verse is interesting regarding the name of God: > [!bible] [Exodus 23:20-21 - ESV](https://bolls.life/ESV/2/23/) > 20. “Behold, I send an angel before you to guard you on the way and to bring you to the place that I have prepared. > 21. Pay careful attention to him and obey his voice; do not rebel against him, for he will not pardon your transgression, for my name is in him. This verse is often associated with Christophony, or the pre-incarnate Christ. ### NI - Pg. 17-18 - Egerov, Florensky, and Essence > One of the members of a Name Worshipping circle was the mathematician Dmitri Egorov, who actually met with the patriarch of the Church and begged him to forgive the Name Worshippers. > Some of the other Name Worshipping groups had already given up on gaining the support of the Church hierarchy, and perhaps even took pride in their "heretical" status. Thus there were differences in emphasis among the Name Worship. (Archimandrite) David, still reaching out to the top leaders of the Church, went to considerable lengths to try to prove that the charge that the Name Worshippers were cultists believing in magic or polytheism was incorrect. He agreed that if one took literally the assertion "The Name of God *is* God," the charge of polytheism had some plausibility, since the name of God is different in many different languages. He tried to avoid this conclusion by saying that the name of God should not be understood in terms of "letters" or "specific words", but instead as an "essence" that stands behind the name. David criticized some of his fellow Name Worshippers, including the priest Pavel Florensky, for believing in the "magic" of the word "God" ("Bog"). > Florensky refuted the charge that he believed literally in the divinity of the letters making up the word "God." He enlisted the help of philosophers and writers like Aleksei Losev and Sergei Bulgakov in giving his Name Worshipper circle a Neoplatonistic orientation that tried to reconcile his ideas with Christianity. He and his supporters were not interested in trying to win over the official Church; rather, they were devoted to exploring the meaning of symbolism, linguistics, and "signifiers," of which they saw the word "God" as the most important. > Florensky was particularly devoted to tge relevance of Name Worshipping in mathematics, the field in which he was trained at Moscow University by Dmitri Egerov. Florensky saw a relationship between the naming of "God" and the naming of sets in set theory: > Both God and sets were made real by their naming. In fact, the "set of all sets" might be God Himself. I have found myself pondering the fact that God contains all things apart from evil without being contained Himself. In that way, Florensky is interesting to me, but this is where we stray: "Both God and sets were made real by their naming." What a foolish statement. God is real. That is all. All else results of Himself. I also think God could be thought of more as the Set of All Uncontained Sets as well as the Set of All Sets. I think about this a bit in [[The Recursive Actuality, Possible to Possible, The Sifted Reality]]. [[NI - Pg. 17-18 - Egerov, Florensky, and Essence]] ### NI - Pg. 19-23 - A Set's Definition is the Set Itself and the Definition of Infinity is Ineffible and the Definition of Infinity is Ineffible > A "set" is a collection of objects sharing some property and given a "name." For example, the set of all giraffes in South Carolina could be named "South Carolina Giraffes." This set obviously has a finite number of elements. By its description this set is different from the set of all flowers in your garden or all inhabitants of Cyprus, but in each case, the number of elements in these sets is finite. More interesting are sets with an infinite number of elements, such as the set of all integers(1,2,3,4,5,6...) later denoted **N**, where the ellipses mean that one thinks of the entire series of integers as potentially never ending. The set of all the points on a line segment is also infinite, but of a different sort. These examples raise the question of a definition of "infinity," something mathematicians had not managed to produce in two thousand years. Yet, as Weyl observed, "mathematics is the science of infinity." Set theory tries to provide a framework in which all of mathematics can be fitted, and a definition of infinity was a crucial part of its elaboration. > > Most non-mathematicians think they have some idea of what "infinity" means; they would probably say something without end or limit" For mathematicians, however, the problem of infinity has been deeply puzzling. Discussions about infinity will play an important role in our story about French and Russian mathematicians in the early twentieth century. > How does one define infinity? Does it really exist, or is it only an abstraction? Is there only one "infinity," or are there several, perhaps many? Can some infinities be "larger" than others? Georg Cantor, a German mathematician, created set theory from his deep inquiry into these questions. Cantor gave infinity a mathematical definition after 2500 years of unsuccessful efforts, and the ultimate result of his labors was to make set theory the lingua franca of mathematics. The evolution of cultural conceptions of infinity before and after Cantor reveals the significance of his achievement. > The first glimmer of a conception of infinity probably came at the birth of civilization. Is it possible to fathom the first non-trivial thoughts of our ancestors millenia ago, watching the unbounded horizon, feeling time passing continuously from the past to an unknown and frightening future? When did they begin to conceive of unlimited space and time? And did they, from the start, combine this idea with a concept of the unlimited power of a divine or non-human being they thought was above them? Divine perfection eventually became synonymous with almightiness, that is, infinite might. Was infinity a divine prerogative from the beginning? In all probability we will never know. But we have some early clues in an ancient Greek word that combines all of these concepts: áréipov, or apeiron. > This word is found in the first philosophical text of the Greek tra-dition, attributed to Anaximander of Miletus, who probably lived from 610 to 540 B.C.B. He described the ultimate material principle as apeiron, "the Infinite" or indeterminate; "something without bound, form, or quality." So from the start the word contains a contradiction: it attempts to express what is not expressible (the ineffable). > Hermann Weyl, one of the leading mathematicians of the first part of the twentieth century, was inspired by the history of the school of Pythagoras (c. 569-500 B.C.E.) and its fascination with infinity. > He wrote: > Aside from the fact that mathematics is the necessary instrument of natural science, purely mathematical inquiry in itself, according to the conviction of many great thinkers, by its special character, its certainty and stringency, lifts the human mind into closer proximity with the divine than is attainable through any other medium. Mathematics is the science of the infinite, its goal the symbolic comprehension of the infinite with hu-man, that is finite, means. It is the great achievement of the Greeks to have made the contrast between the finite and the infinite fruitful for the cognition of reality. Coming from the Orient, the religious intuition of the infinite, the apeiron, takes hold of the Greek soul. This tension between the finite and the infinite and its conciliation now becomes the driving motive of Greek investigation. > The Greek word apeiron contained three main ideas that persisted in later centuries: > > - the limitlessness of space and time > - a non-rational, religious, or mystic aspect to infinity > - the indefinability and impossibility of description (ineffability) of infinity > > All three of these characteristics are negative, defining what infinity is not (not limited, not rational, not definable) rather than what it is, Aristotle (384-322 B.C.E.) introduced a distinction that was also usually accepted in later times: infinity is a potentiality, not an actuality. He noted that if one takes a line segment (one-dimensional space) it is possible to cut that segment in half, and then cut the resulting half in half again, and so on endlessly. As he observed, "It is always possible to think of a larger number: for the number of times a magnitude can be bisected is infinite. Hence the infinite is potential, never actual; the number of parts that can be taken away surpasses any assigned number." Aristotle's approach remained the dominant one for centuries. It lies at the heart of calculus and most often mathematical treatments of infinity until the time of Cantor Even today, the idea of infinity as only a potential is the intuitive concept of the layperson, who knows very well that any specific number he or she mentions can always be exceeded. > Take, for example, the unusual answer proposed by Alexander lessenin-Volpin (Aleksandr Esenin-Volpin), a Russian logician of the ultra-finitist school who was imprisoned in a mental institution in Soviet Russia. Yessenin- Volpin was once asked how far one can take the geometric series of powers of 2, say (2¹, 2² 2³,..., 2¹⁰⁰). He replied that the question "should be made more specific." He was then asked if he considered 2' to be "real," and he immediately answered yes. He was then asked if 2² was "real." Again he replied yes, but with a barely perceptible delay. Then he was asked about 2³, and yes, but with more delay. These questions continued until it became clear how Yessenin-Volpin was going to handle them. He would always answer yes, but he would take 2¹⁰⁰ times as long to answer yes to 2¹⁰⁰ than he would to answering to 2'. Yessenin-Volpin had developed his own way of handling a paradox of infinity. What a wealth of definition. Isn't it interesting that the great minds of Greece occurred during the period of prophetic silence? [[NI - Pg. 19-23 - A Set's Definition is the Set Itself and the Definition of Infinity is Ineffible and the Definition of Infinity is Ineffible]] ### NI - Pg. 94-95 - The Vision of God, the Noetic Sight > Luzin would have immediately noticed that James, in his discussion of mysticism, referred to Plotinus, quoting him as saying: "In the vision of God what sees is not our reason, but something prior and superior to our reason." I think this is axiomatic and obvious to those who have spent time contemplating faith. > James described two specificities of mysticism that must have resonated with Luzin's proper mathematical interests. First, James was concerned with the capacity of humans to have direct access to knowledge, and he spoke of certain "states of insight into dephts of truth unplumbed by the discursive intellect. They are illuminations, revelations...all inarticulate though they remain." James observed that these insights have a "noetic quality," meaning that those mystics who possess them believe they are "states of knowledge." The word "noetic" comes from the Greek *nous*(a philosophical term for mind or intellect) and would have reminded Luzin of Plato's view of mathematics, where "one seems to dream of essesnce." Moreover, James characterized a mystical state as being "ineffable," that is, it "defies expression, that no adequate report of its content can be given words." The Eastern Orthodox Church describes the noetic vision as necessary in the carving of their icons. [[NI - Pg. 94-95 - The Vision of God, the Noetic Sight]] ### NI - Pg. 103 - Strong Names > For a while Egorov's and Luzin's chief assistants in managing Lusitania were three students, each with his own function: Pavel Alexandrov was the "Creator," Pavel Uryson the "Keeper," and Viacheslav Stepanov the "Herald" of the mysteries of Lusitania. Some great names for use in my book. ### NI - Pg. 116 - The Set of Students > All students on entering Lusitania were assigned a name taken from set theory. Recruits were called ℵ₀. Each time a student had some sort of success, such as a first publication, first lecture delivered to the Mathematics Society, graduation from the university, or passing the master's examination, that person's ℵ number would be raised. Alexandrov and Uryson soon achieved the high rank of ℵ₅. Luzin himself was given the name ℵ₁₇. Egerov was ℵ_ω, the "omega" subscript indicating that his status was higher than Luzin's but still not as high as the status of the Continuum. Papers that circulated among the members of Lusitania - what would now be called "pre-prints" -- were often ornamented with the coat of arms of the author, an elaborate rendition of their ℵ number. A fun system to measure progression. ### NI - Pg. 190-191 - Plato's Cratylus, Functions > The idea that a "name" has more in itself than mere word assigned is very old and goes back to at least Plato's Cratylus; the concept has reappeared many times in succeeding centuries. After all, *logos* is a central concept in western culture. I'll give this a read. > The modern theory of functions initiated by Baire and Lebesgue after the introduction of set theory led Lebesgue to inquire into a precise extension of the notion of functions, extending the explicit analytic expressions (polynomial, trigonometric) of earlier mathematics, but ones that could still be described or named. In doing so, Lebesgue and the French school were asking questions that would find a satisfactory framework only twenty years later with the theory of recursivity. Time to look into functions. [[NI - Pg. 190-191 - Plato's Cratylus, Functions]]