January 23, 2022 Dear Nikola, Thank you for speaking with me on the phone last Friday, January 21. Meanwhile I have been enjoying myself reading vicariously about topics in mathematics and about mathematicians, exercising my freedom to explore the most egregiously unconventional thoughts, such as the hypothesis that mathematics is a stylized kind of fictitious prose, or that it is a species of poetry; or that it is a slippery slope to intellectual (and moral) chaos; that mathematicians are like haruspices exploring and exploiting the entrails of the human mind, or like diviners of old with purported insight into the secrets of existence, or like frustrated latter-day poets, vying for recognition, publicity and professorships. Perhaps my view is distorted by the fact that I have spent my life practicing a profession that presumes to understand illness as the connector between life and death. Perhaps when on my virtual deathbed I presume to search mathematics for truths that been unable to find elsewhere, I do nothing more than gaze into a mirror where I see the failures of my own intellect. Perhaps because I have spent so much time and energy writing, consideration of the role of language in mathematics and the meaning of words by mathematicians are the portals through which I try to gain access to the mathematics citadel. The key to these portals is the specificity of meaning of symbols and/or words and/or the absence of specificity of meaning of symbols and/or words. Both the logician and the mathematician have an unconditonal need that symbols and words should be consistent precise and unequivocal. The poet asserts the opposite. He trades on the words' undefined spectra of meanings. Goethe wrote: Und doch haben sie recht, die ich schelte: Denn, daß ein Wort nicht einfach gelte, Das müßte sich wohl von selbst verstehn. Das Wort ist ein Fächer! Zwischen den Stäben Blicken ein paar schöne Augen hervor, Der Fächer ist nur ein lieblicher Flor, Er verdeckt mir zwar das Gesicht, Aber das Mädchen verbirgt er nicht, Weil das Schönste, was sie besitzt, Das Auge, mir ins Auge blitzt. Und doch haben sie recht, die ich schelte: And yet they whom I scold, have reason, Denn, daß ein Wort nicht einfach gelte, For, that a word should have more than a simple meaning Das müßte sich wohl von selbst verstehn. is a clearly self-evident, manifest. Das Wort ist ein Fächer! Zwischen den Stäben The word is a fan, and between its staves Blicken ein paar schöne Augen hervor, the gaze of a pair of beautiful eyes, Der Fächer ist nur ein lieblicher Flor, The fan is merely a lovely veil, Er verdeckt mir zwar das Gesicht, although it hides her face from me Aber das Mädchen verbirgt er nicht, it does not conceal the girl behind, Weil das Schönste, was sie besitzt, because her most beautiful possession, Das Auge, mir ins Auge blitzt. her eye, sends its lightning gleam into mine. The meaning of a word, the meaning of language is never fixed. It is unavoidably a social phenomenon. Words and sentences are links that connect and bonds that bind the individual minds of each of us to the other. It is only from the dynamic, fluid, continuing interaction among us that words, sentences, and mathematical symbols derive their meanings. The various disciplines of education may be construed as exercises to engender the relative reliability of symbolic meanings which make possible the fusion of individual intentions into a common language, which make possible the fusion of individual effort into the common enterprises of technology, and not least, the performances of "symphonies" of "sounding together" of musical orchestras. I blame my (relative) inability to assimilate mathematical propositions on a peculiar intellectual if not indeed spiritual intransigence of mine, on my stubbornness, on my need and insistence to fashion my world, the world of my representations, (die Welt meiner Vorstellungen), by and for myself. I very much hope and wish that you are and stay well. EJM Appendix: Russells paradox: the set of all sets that are not members of themselves. Hilberts proposed axiomatic foundation of mathematics. Brouwers intuitionist dispensation with the rule of the excluded middle. The unrestricted axiom of comprehension in set theory states that to every condition there corresponds a set of things meeting the condition: (∃y) (y={x : Fx}). The axiom needs restriction, since Russell's paradox shows that in this form it will lead to contradiction. According to the unrestricted comprehension principle, for any sufficiently well-defined property, there is the set of all and only the objects that have that property. Let R be the set of all sets that are not members of themselves. If R is not a member of itself, then its definition entails that it is a member of itself; if it is a member of itself, then it is not a member of itself, since it is the set of all sets that are not members of themselves. Quine's paradox is a paradox concerning truth values, stated by Willard Van Orman Quine.[1] It is related to the liar paradox as a problem, and it purports to show that a sentence can be paradoxical even if it is not self-referring and does not use demonstratives or indexicals (i.e. it does not explicitly refer to itself). The paradox can be expressed as follows: "yields falsehood when preceded by its quotation" yields falsehood when preceded by its quotation. If the paradox is not clear, consider each part of the above description of the paradox incrementally: it = yields falsehood when preceded by its quotation its quotation = "yields falsehood when preceded by its quotation" it preceded by its quotation = "yields falsehood when preceded by its quotation" yields falsehood when preceded by its quotation. With these tools, the description of the paradox may now be reconsidered; it can be seen to assert the following: The statement "'yields falsehood when preceded by its quotation' yields falsehood when preceded by its quotation" is false. In other words, the sentence implies that it is false, which is paradoxical—for if it is false, what it states is in fact true.