User:Double sharp

The zeroes on my Babel are my language wishlist. (And in general, if I speak it or want to speak it, I like the associated national cuisine very much. Though that's usually not the only reason...)

I wish it was standard to transcribe Chinese proper names into Pīnyīn with tone marks (just like how one would not strip off tone marks from Vietnamese names). (They're not that necessary in running text, because the context is usually enough to clarify things. But with proper names, you can't really tell.)

I have edited significantly on inorganic chemistry, Solar System astronomy, geometry, classical music, and chess (including variants; among regional variants, mostly the historical shogi variants). In no particular order.

My favourite star is Spica. (Present company excepted, naturally.) With the analogous caveat, my favourite planets are Mercury and Venus. We have so few rocky planetary bodies to study, and I'd like to know more about Earth's siblings! At least for Luna and Mars we've had many more missions. (I mostly think of "planet" geophysically, so Luna, Io, and Europa are another three rocky planets.) Also, I feel like they are now often unfairly overlooked in the popular imagination in favour of the planets further out, though of course I'd like to know more about those as well. Since I think of "planet" geophysically, for me the lower limit is collapse of most porosity to form a solid, round body, and the upper limit is the onset of hydrogen burning and becoming a red dwarf: for me, brown dwarfs are just high-mass planets (most of them would've stopped fusing deuterium by now, anyway). Of course this has some issues with the lower end, but I'm inclined to think "planetoid" is a good enough fuzzy word for things like Pallas, Vesta, or Hygiea. Maybe also very low-density Tethys and some TNOs like Uni. And maybe also Psyche (probably differentiated, but too small to be round).

This paper by Hamkins convinced me that GCH is Platonically true and should be adopted as an axiom. (That is not his view, as he is a pluralist, and his actual argument is that CH could have been viewed as fundamental; but he did say that if you have a universe view of set theory, it becomes an argument for CH, and I'd add that indeed it turns into an excellent argument for GCH.) The upshot (heavily adapted from him, I just draw a much more universe-view conclusion for it): I consider hyperreals to be outright the natural way to do analysis, as shown by people still saying "now multiply by $dx$ " even though infinitesimals got banished to the shadow realm, and indeed the historical development of the subject where the infinitesimal language came first. In many ways, $\varepsilon$ - $\delta$ formulations make me (I am being somewhat polemic here) think of Renaissance mathematicians inventing many varieties of the cubic to avoid talking about negative numbers (or, if I was being even more polemic, of Vladimir Arnold's joke about the theory of odd numbers, in which obviously even numbers are just ideal objects and the product of terms and the sum of an odd number of terms are always defined): analogously, to me doing a lot of work to bound epsilons to keep them from exploding feels to me like avoiding talking about non-Archimedeanness. I'm not claiming $\varepsilon$ - $\delta$ doesn't work (of course it works beautifully): it simply feels like tying your hands by saying that the objects that you'd naturally like to exist don't exist, and then working with their shadows anyway, which to me is deeply ontologically unnatural. The very fact that $dx$ is useful notation, even for people who are not thinking in infinitesimals, is to me a silent rebuke running through all our notation saying that we have our ontology all wrong and that everything should be formalized on the non-Archimedean side. The fact that $\varepsilon$ - $\delta$ is very nice with the Archimedean property is to me precisely the problem: the very nature of the intuition behind the calculus is non-Archimedean, and what $\varepsilon$ - $\delta$ does is systematically simulate non-Archimedeanness through hierarchies of smaller and smaller Archimedean tolerances. The theory of odd numbers, here we go again. And just look at things like nets. Hello, what are we doing? Inventing larger indexing apparatuses when we could simply answer "there is nothing really wrong with the infinitary construction through sequences, we just need enough saturation so that a sequence indexed by hypernaturals will do what you want"?

But because under ZFC hyperreals are not canonical, they are hobbled and not being allowed to compete on the same level playing field as the reals (and yet still people informally use infinitesimal language, which speaks volumes): the reals can be characterised by their canonicity theorem, that they are the unique complete Archimedean ordered field up to isomorphism. Mathematicians would generally consider weaker models, in which the Dedekind reals and Cauchy reals might not be the same, or in which there are multiple algebraic closures of the rationals, to miss something essential about the reals or the algebraic numbers: but all this proves is that canonicity is somewhat dependent on the axiomatic structure you pick. Hyperreals can be canonical, if ZFC+GCH is your theory (or rather, there being a unique hyperreal field up to isomorphism for each desired size); with only ZFC, they are not. The algebraic numbers can be canonical, if ZFC is your theory; but without enough choice, they are not.

I happen to think that the naturalness of infinitesimal language, indeed over the usual epsilon-delta language, means that hyperreals should be canonical; and we know from set theory that they can be canonical. So I'd likewise say that ZFC is too weak and misses an important point about the nature of hyperreal continua: it is not enough to me to say "well, pick any ultrafilter, the important properties will be satisfied", because that still does not legitimize you to use the definite article and there would always be a natural worry that for something, the lack of isomorphism would bite you. It solves the question "can you do nonstandard analysis" (yes) but not at all the foundational question of what you mean when saying "the hyperreals". Given my commitments, the only plausible solution (I trust we do not need to explain why "just pick one ultrafilter and ignore all the others" doesn't solve the problem; it's like saying "fine, we will do everything over the Dedekind reals and ignore the Cauchy reals") becomes saying that all such constructions give the same mathematical object. However, as Hamkins points out, canonicizing hyperreal fields at every desired uncountable regular cardinality and desired level of saturation is outright equivalent to GCH over ZFC: more precisely, the statement I am considering is the statement that there is a unique saturated real-closed field of any uncountable regular cardinality $\kappa$ . This to me is a very strong argument that GCH is true and should be added to the axioms: it is necessary to put the hyperreals on the same ontological foundation as the reals, which I argue they should be. Moreover, speaking abductively, the very simplicity of its statement and independent motivation (GCH is a well-known and well-studied axiom even if it isn't generally promoted to a default one) is striking. The categoricity of hyperreals over ZFC happens to be equivalent to an independently motivated principle about sets, saying that taking the power set on cardinality has the simplest possible behaviour. (Obviously I am not saying that ZFC implies GCH, which is false; I am arguing that ZFC is an incomplete description of mathematical reality.) It also helps that in some sense, GCH and AC are both statements that favour combinatorics and are not nice to measure theory. We already accept AC because of its powerful combinatorial consequences, and accept Vitali sets and Banach-Tarski, so maybe that just means that the Platonic universe likes our combinatorial intuition better than our measure-theoretic intuition: so GCH in that sense is a proposed axiom that to me is philosophically coherent with the flavour of AC (unlike what it would be to adopt Freiling's axiom of symmetry, to me). (You could argue I suppose that this is an argument against AC rather than against GCH, which I'd accept, but I suspect the reason to treat them differently is more sociological than mathematical.)

You could argue instead for going all the way to the surreals, as has been suggested, but that is unsatisfactory for me because while they are a very nice structure, they also lose the quality of being set-sized. Going to $\mathbf {No} (\omega _{1})$ and christening that the canonical "hyperreals" is also to me unsatisfactory without CH, because (as Hamkins notes) the point is that all continuum-size countably saturated real closed fields should be isomorphic. My demand is that all natural specifications satisfying the desired saturation requirements should give you the same structure, because otherwise it is like saying "fine, we shall live with just the Dedekind reals as a default because they are nice for some reason, while accepting that the Cauchy reals might also exist as something else". To me, there is simply a ladder of hyperreal fields $\mathbf {R} _{\aleph _{1}}^{*}\subset \mathbf {R} _{\aleph _{2}}^{*}\subset \mathbf {R} _{\aleph _{3}}^{*}\subset \cdots$ and you choose what saturation level you need for your purposes. (Each level is understood up to isomorphism and the inclusion up to the corresponding compatible embeddings.) For elementary calculus, the first one will suffice: for some other things, you may need to go further up the ladder (to discretize even larger infinities, which is why I would demand GCH instead of merely CH). Or, in short, the hierarchy of filling out the continuum goes $\mathbf {N} \subset \mathbf {Z} \subset \mathbf {Q} \subset \mathbf {R} \subset \mathbf {R} _{\aleph _{1}}^{*}\subset \mathbf {R} _{\aleph _{2}}^{*}\subset \mathbf {R} _{\aleph _{3}}^{*}\subset \cdots$ converging up to $\mathbf {No}$ at proper class size. (The notation is loaded of course, because I want $\mathbf {R} _{\kappa }^{*}$ to be the unique saturated real-closed field of cardinality $\kappa$ when that is uncountable and regular, and there isn't such a unique thing without GCH at $\kappa$ ; with it, I would simply take $\mathbf {No} (\kappa )$ as a nice representative of $\mathbf {R} _{\kappa }^{*}$ . But make no mistake: I want canonicity of the result, not the construction.) This doesn't mean that to me worlds with $\lnot$ GCH are illegitimate objects of mathematical study: on the contrary, they are perfectly legitimate, just as it is legitimate to study a world with so little choice that ${\overline {\mathbf {Q} }}$ does not refer to a single well-defined object. But they are, in my view, pathological and do not describe the true mathematical universe. Yes, I am a hardcore out-of-fashion Platonist, but I think I am being a coherent hardcore out-of-fashion Platonist.

None of that says that I think ZFC+GCH is the final theory. I don't claim to know everything about the Platonic universe, I just claim that GCH is true in it. So if I eventually had a good argument for something that let me drop GCH from the axioms, it wouldn't be "I suddenly realized it's false", it'd be "I have a good argument for a stronger principle being true, that happens to imply GCH, so that it became a theorem rather than an axiom". (So I guess ultimately my argument is more about "natural mathematical structures Platonically have canonicity properties unless there's some special reason otherwise, the hyperreals don't seem to have such a special reason because somehow the lack of isomorphism over ZFC hardly matters when doing actual nonstandard calculus, and that should inform what axioms we regard as true, just like how we do not in fact argue that maybe there really are multiple distinct algebraic closures of $\mathbf {Q}$ ".)

If you ask me "does infinity exist in the physical Universe", though, I'd probably answer: who knows, but Platonic existence is still some kind of useful existence even if it doesn't equate to existence in the physical Universe. And of course I'm also very self-aware that in a sense this is an argument that nonstandard methods should outright oust $\varepsilon$ - $\delta$ in the curriculum and banish them into the history of mathematics course. Good luck with that. Still I think it would be mathematically justified and you can do a little bit of it yourself anyway by learning the nonstandard way alongside the standard way. Or, even better, before it. Make it your native language. I sure haven't finished it yet, but I've been trying. :P

(This whole business also makes it a bit easier to fit things to intuition, like how the obvious idea that infinitesimal hyperreals correspond to the Cauchy sequences with limit zero is technically not correct without CH, because existence of P-point ultrafilters is independent over ZFC, but is a consequence of ZFC+CH. But that is a bit of a side detail from the main point.)

Also a believer in P=NP. See this Don Knuth interview for why (it's question 17).

Regarding $0^{0}$ , maybe it's a function of what areas of mathematics I tend to think about most often, but $1$ is to me a perfectly serviceable and indeed the only natural value. So I guess $0^{0}=1$ hovers as a blanket convention for my own head. I would rather say that $0^{0}$ is a special kind of indeterminate form, in the sense that $0^{0}$ with constant zeroes is 1, but $0^{0}$ meaning different functions in base and exponent tending to zero separately needs stricter conditions to be 1.

I think it would've been better had we chosen to write numbers by default in octal rather than decimal. Nobody actually cares about 10 being divisible by 5; in practice you double and halve, which is why numbers like "25" (connoting a quarter) and "125" (connoting an eighth) are so common. If that's all you care about, a power of two would've been a better base. But I think it would have been still better to use senary (base six) or duodecimal (base twelve): if you are going to go away from the purity of powers of two, then put in a factor that makes more sense than 5. Thirds are actually worth it. :)

1.e4 is best by test, Fischer was right about that. And the best answer is 1...e5. But hey, if you're not a GM, just about anything sensible will be fine. (Alas, I appear to have mellowed out from a King's Gambit player to a Queen's Gambit player.) The fact that computers play much better chess than we do does not stop us from having fun. We wouldn't have victories and defeats without mistakes. Antichess survives as a game despite being weakly solved, and let's face it: practically chess is already weakly solved by Stockfish NNUE. Though, if normal chess is not exciting enough for you, why not try the Capablanca-family variants? :)

The correct version of the periodic table, insofar as there is one for a model (so let's say: the consistent version), has helium in group 2. You can have it as Charles Janet's form (below), or keep the s-block at the left end just like usual (because quantum effects lower s orbital energies and so the big energy gap happens before them), but either way 1s² overrides chemical properties. :) As Eric Scerri has pointed out, the periodic table classifies abstract elements (atoms with their electronic structure) that are preserved across chemical conditions, not elements as simple substances that are not: salt contains sodium and chlorine the atoms, and their overlapping orbitals, but it doesn't contain sodium the reactive metal and chlorine the toxic gas. Otherwise, it would be difficult to understand why nitrogen and bismuth are in the same group. Actually this reassignment is starting to get more and more serious consideration these days, but it goes without saying that I do not support changing our default periodic table format on Wikipedia just yet. For me, an element is philosophically a type of atom (as distinguished by Z); you place an element on the PT by considering its characteristic set of valence electrons and orbitals when engaging in bonding interactions with other kinds of atom. Mendeleev was kind of doing this by proxy by considering valences as primary for group assignments, and once the quantum revolution happened, we understood why that worked. Likewise I'm completely in favour of Sc-Y-Lu-Lr as group 3. (There are a few papers that suggest some 4f valence usage for Lu, but most seem to reject it. I am convinced there cannot be much, because it is so hard to get it for late lanthanoids already; but even if there does happen to be a bit in this starting-to-be-more-relativistic area, La 4f is clearly much greater. And in that case I'd still say Madelung is better to follow for simplicity, since it is a model anyway, and we don't split apart groups for such inhomogeneous elements as N and Bi.)

Yes, the classification into blocks ignores relativity, but it doesn't matter too much. It will probably matter for period 8, but that is still theoretical and more calculations in that area would be helpful. And honestly, putting elements past 118 or so on the normal periodic table is inherently difficult, since they will be inescapably relativistic and cannot be expected to follow the old pattern. Doing so is inherently a simplification to an even greater degree than it would be for the elements we already know: that is not to say that we shouldn't do it, but we should go into this with our eyes open. The destruction of the Madelung rule in period 8 (because of intruder levels) is also important, as does the fact that it is really one end on a continuum ranging from neutral atoms to hydrogen-like atoms. (Once you remove two electrons, (n-1)d and (n-2)f fall below ns, e.g. Ca [Ar]4s² vs Ti²⁺ [Ar]3d².) It is what makes me sceptical of group-theoretic approaches to justifying Madelung: what, are they going to happily continue past 118 and "prove" that probably tin-like element 168 is a noble gas? What about the "wrong" position of 9s? The Madelung rule is rather something that we need to study experimentally, with justifications like Demkov-Ostrovsky being a better way to look at it from QM principles, choosing the potential that seems to approximate things best just like the nuclear shell model. No one complains about that there; indeed, a Nobel Prize got awarded for it. :)

With analogous caveats about what's actually necessary for life, I vote for mercury as a favourite elemental metal, and fluorine as a favourite elemental nonmetal. I am defining this in the Mott sense of whether the stable phase at absolute zero conducts or not. Periodicity makes polonium, astatine, and radon fairly interesting, but I hesitate to call them "favourites" because they are not known well and are unhealthier to be around than mercury. Nonetheless it is really a shame that they are unstable, because they would finally put paid to the school myth that groups show homogeneous behaviour (false) and that astatine must therefore be a black solid (calculated to be probably false). Metallicity appears at some point when we go down groups 13 to 16, so why shouldn't it eventually happen in groups 17 and 18? I suspect oganesson would be a metal. :)

I think the superheavies exist, but not quite in the same way that tungsten or even plutonium exists. Their existence is mostly potential rather than actual, with the exception that we can turn it into reality briefly in the relevant facilities. As far as existence of Og vs existence of 119 goes, it's really a matter of human knowledge as far as I'm concerned: we're sure the former can be made, and so it exists in that sense, whereas we are not yet sure about the latter (though of course everyone expects that it will exist). Of course there is a continuum as half-lives decrease, not to mention other factors: I think francium exists more than dubnium does. Once we get far enough, and reach Z values where every possible nuclide would not survive long enough to get an electron cloud, then I'll agree that the element does not exist in this world. I suspect the continent of stability is likely: just as covalent bonding gives way to metallic bonding, so should individual baryons give way to quark matter.

There are no singularities in the real world, only gaps in our current picture of physics. But that's just me spewing opinions. :)

I have a lot of favourite composers in classical music (yeah, my username is for the musical accidental because it looks cool), but if you ask what period I love the best: the Classical period and the first Romantic generation. (Well, Franz Schubert will forever stand between them, and his music has a special place in my heart.) I'd also mention Charles-Valentin Alkan as a perhaps not-too-well-known name who is also on my list of favourites. Also, considering all the chemistry edits I do, it would be odd not to mention Alexander Borodin explicitly, though he's later than what I'm most keen on. :) I wish the standard range of the piano was F0-D8 at least, adding some notes to each end (it is silly that the standard repertoire is not fully playable: D8 is needed for Scriabin's sixth sonata, G0 and A♭0 would be helpful for Liszt's Harmonies du soir and Ravel's Scarbo, G0 is marked in the third piano concerto of Bartók and it and A♭0 are also in the Busoni transcription of the St Anne Fugue, F0 in the Bartók piano sonata; it would be nice to have eight octaves up to F8, but maybe that would make things difficult to add notes to every upright). And also that the standard piano was 7/8-sized. Seriously, for me ninths hurt and tenths are impossible except in slow passages on the edge (e.g. the end of Schumann's Fantaisie, 1st movement). It is truly aggravating to find the chromatic scales in thirds in the Don Juan Fantasy easier than the leaping tenths in the left hand in the ensuing variations. That's not how it's supposed to work! For similar reasons I actually find the Chopin Op 10/2 etude easier than 10/1. :( But octaves are okay; I can play the Erlkönig accompaniment without strain. (A tip: whenever the other hand is not playing, you can split the octaves between hands, 3-2-1 in each, to give yourself a break.)

Fixed do is pointless in English (and German, and other languages that use letter names). We already have an absolute system for note names: they're just letters. "Do" should always be the local tonic, whether major, minor, or whatever other mode. So the minor scale is do-re-me-fa-so-le-ti-do. The major scale is fundamental, and the parallel minor scale comes as an alteration of it: for me, the closest minor key to C major is not A minor (the relative) but C minor (the parallel). Also, the really "natural" minor scale is the harmonic minor. (The tonic must be minor; the dominant must be major to be functional; so we need a minor subdominant to keep the minor chords in the majority in the most important three.) The variable degrees (6th, 7th, Neapolitan 2nd) arise as chromatic alterations to avoid awkwardness in the circle of fifths, because here the diminished fifth is so much closer to the tonic. Well, in major or minor the circle of fifths is I-IV-VII-III-VI-II-V-I; but in major the d5 is IV-VII, whereas in minor it is either VI-II or N-V (N meaning Neapolitan ♭II). For this reason, movable do with la-based minor is the one thing I would admit is worse than fixed do, because it doesn't make sense: it is not consistent about making "do" the tonic, which was the entire point of movable do. And I say this while having perfect pitch (albeit with the ability to switch to thinking in functions, read transposed scores, and reset A to 415 Hz if needed; nonetheless, A = 392 Hz is too much for me to accept). And yes, I am very much a Stufentheorist rather than a Funktionstheorist...

For languages that already use the sol-fa note names as the absolute names of the notes, I guess scale degree numbers are the best option I can think of, though syllable count might be an issue. For atonal music, singing the German note names isn't a bad option: as long as you stay in single-sharp or single-flat territory (which atonal music really should anyway), they are all monosyllabic.

I do not think the assimilation of . to ³ is historically justified from at least the Classical period proper onward (i.e. from mature Haydn), and would consider playing such passages as written to be the only correct approach in Beethoven, Schubert, and Chopin. (I suppose there is a sort of example of the colliding rhythms in the Act II finale from Mozart's Entführung, but there Mozart puts Blonde's part in ¹²
₈ instead; regardless, I also think Belmont should not assimilate the dotted rhythm in his part to her rhythm.) Quantz in 1752 already spoke against it, and J. S. Bach in the E minor Fugue from WTC II already showed he knew both notations; although C. P. E. had a different opinion about assimilation, this clearly shows his father disagreed, as there would be no sense of writing both notations in the same piece unless some difference were intended. And in the case of Schubert and Chopin, it's absurd to say that composers who were willing to write nested tuplets (the Adagio D 178, and the Nocturnes Op 15/1 and Op 32/2 as well as the Allegro de concert Op 46, respectively), would have been unwilling to write ³ if that was what they wanted. They did in fact write it (Geisternähe D 100, Das Geheimnis D 793, the Adagio D 178; the Etude Op 25/10), and wrote ³ as well (Wanderer Fantasy D 760; Andante spianato and Grande Polonaise Brillante Op 22; also, going further back in time, the slow movement of Mozart's 29th symphony – BTW, this isn't relevant, but the minuet and trio has a fun case of seeing written F3 in a violin part.). Schubert went to the trouble of changing the metre in the G major sonata (D 894) to compound and writing out all the ties to get ¹²
₈ correct, so of course he cared deeply about the difference. As for Beethoven, one need only look at Op 111/ii, which is chock-full of ³ and ³ ; who shall doubt his seriousness? (Besides, in the first song of An die ferne Geliebte there's a triplet in the piano RH with the first two notes tied, creating the exact same rhythm. If he really wanted alignment with triplets, he had ways to write it clearly instead of using the dot to mean something that it did not mean then.) I have assimilated in Chopin's Polonaise-Fantaisie before, and am now sorry I did it. :) Later we of course have all the occurrences of this figure in Berlioz (e.g. the Chant de bonheur from Lélio) and Schumann (e.g. the third of the oboe romances Op. 94). (The work of David Montgomery is an indispensable source here, especially with his refutation of the alignment argument: the fact of the matter is that in that time period, aligning different note values did not indicate simultaneity, so arguments from that are all founded on sand.)

I wish my heroes George Gamow (nice interview with him) and Lev Landau had gotten elements, like Einstein and Fermi did. Okay, actually I have many other heroes as well across fields, e.g. Li Shanlan, Yuen-ren Chao (Jaw Yuanrenn – normally I'd give Pīnyīn for Chinese names but in this case I feel like I should give GR instead), the Marquis de Condorcet, David Bronstein, E. T. A. Hoffmann, ... But Gamow and Landau are my physicist heroes. :)

I like reading Tolkien for the astonishing worldbuilding, even if I tend to find myself always on the side of characters the narrative doesn't like (Fëanorians, I am looking at you). What does "canonical" mean, when the author left the work unfinished, other than (1) a fan's personal taste or (2) just everything he wrote on the subject of his fictional world? In the context of (1), I believe the Sun and Moon existed from the beginning of Arda (and yes, Venus too; it existed before Eärendil). Tolkien never completed the revisions, but it's not like he completed many other things either; I see no problem with him not having fully explained how Eärendil got associated with Venus in this conception, when he never fully wrote Eärendil's adventures anyway. Had JRRT lived longer, I assume he would have rewritten a few more narratives in some more ways contradicting the earlier drafts, and another few hundred pages doing the same for the languages and worldbuilding. What a wonderful world that would be!

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This form of periodic table is congruent with the order in which electron shells are ideally filled according to the Madelung rule, as shown in the accompanying sequence in the left margin (read from top to bottom, left to right). The experimentally determined ground-state electron configurations of the elements differ from the configurations predicted by the Madelung rule in twenty instances, but the Madelung-predicted configurations are always at least close to the ground state. The last two elements shown, elements 119 and 120, have not yet been synthesized.