Post B2IIu7oMftCx4jw9q4 by lemgandi@mastodon.social
(DIR) More posts by lemgandi@mastodon.social
(DIR) Post #B2IC0S5vTXuc2cDlrs by futurebird@sauropods.win
2026-01-15T00:36:25Z
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The nice thing that happened in class today:Grade 5 students solve a puzzle where they put cuneiform numbers in order (there is no guidance, just work with the symbols, how do you order them?)I told them they are like archeologists cracking a code. They did it!"But were is zero?""It wasn't invented yet." I said this seriously. I mean ... it's true. Later that day the same student asked if it was a joke. I got to tell them no! Zero had to be invented. Everything had to be invented!
(DIR) Post #B2IC5YpZrEBsNi0gtc by futurebird@sauropods.win
2026-01-15T00:37:26Z
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This student wants to "invent a new zero" so. Watch out everyone. Math is about to get a lot more... IDK ... but MORE.
(DIR) Post #B2ICDbzPztYkUjS60u by lapis@elekk.xyz
2026-01-15T00:38:49Z
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@futurebird admittedly I'm not the best at math and I do have an anthropology degree (MesoAmerica was one of the places that independently invented Zero but I'm sure you already knew that)But I can't FATHOM counting or math without zero.There must be a way to make sense of it, but I haven't come to that answer
(DIR) Post #B2ICGKa5B8i5N5tkUC by mhoye@mastodon.social
2026-01-15T00:39:20Z
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@futurebird for a while there, no joke, computing was flirting with upper and lowercase zeroes…
(DIR) Post #B2ICJF9Pp9vTOnoVY8 by donlamb_1@mastodon.online
2026-01-15T00:39:49Z
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@futurebird the additive identity 😊
(DIR) Post #B2ICgg9facTboM6jAm by itgrrl@infosec.exchange
2026-01-15T00:44:05Z
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@futurebird …or perhaps less 😜
(DIR) Post #B2ICl9Lczdd9Vepglk by Unixbigot@aus.social
2026-01-15T00:44:52Z
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@futurebird this story has saved my day from being bleh.
(DIR) Post #B2ICnZJLQwvoc0uz2G by silvermoon82@wandering.shop
2026-01-15T00:45:22Z
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@futurebird I've been in the late-capitalist dystopia long enough that "new zero" sounds like "now you need an app and a subscription to do math".
(DIR) Post #B2ICqzs58igVr8nic4 by munin@infosec.exchange
2026-01-15T00:45:58Z
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@futurebird @Unixbigot last time this happened we got Javascript and the infamous "WAT" talk lol
(DIR) Post #B2ID3ss2TmOzuNvSyW by futurebird@sauropods.win
2026-01-15T00:48:20Z
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@SarraceniaWilds We have zero at home!The zero at home: 0⁰
(DIR) Post #B2IDjUMM08323EwamW by jztusk@mastodon.social
2026-01-15T00:55:49Z
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@futurebird Hee-hee, topologist have already got "the line with two origins", but we gotta let this student run free and see what they come up with on their own.
(DIR) Post #B2IE7Qv4RVaKKebGvQ by futurebird@sauropods.win
2026-01-15T01:00:10Z
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@SarraceniaWilds Maybe ϵ would be another "zero at home"
(DIR) Post #B2IEtvaTiuotLZ5mnw by macbraughton@infosec.exchange
2026-01-15T01:08:56Z
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@futurebird that's awesome. I don't share my work very often, but especially because you mention cuneiform, I actually have "invented" a new zero, called zo, in a modern base-60 number system, inspired by the Babylonian system and Wu Xinghttps://github.com/hyxos/hyxos_numerals/blob/main/GRAMMAR.mdThere is a very poorly written and not maintained api to generate the glyphs at https://hyxos.io/docsI'm plodding away in my spare time trying to turn it into something more usable to make it more accessible for everyone... up to this point it's mostly been used by my wife and I to build card game prototypes. I'm hoping to release a much more polished glyph builder this year, I really want to make a typeface, and oh boy, that is a deep, deep rabbithole
(DIR) Post #B2IF0aCJeFtD2Dxoyu by elebertus@mastodon.social
2026-01-15T01:10:05Z
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@futurebird in all seriousness that’s awesome. They were engaged and interacting.Most importantly made an awesome joke lol
(DIR) Post #B2IG41G9aaYdmxxfiS by hypolite@friendica.mrpetovan.com
2026-01-15T00:43:44Z
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@lapis @futurebird If you’re counting physical things, then I believe you don’t need a zero.
(DIR) Post #B2IG42XCqfnBk8wnw0 by futurebird@sauropods.win
2026-01-15T01:21:39Z
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@hypolite @lapis Zero makes record keeping nicer and when you start using place value (as cuneiform does) to get more out of your limited symbol set ... zero make things less ambiguous. One needs a way to show which place the symbol lives in even if you don't have everything written in neat columns.
(DIR) Post #B2IHjI0yIZxpa56SVk by leon_p_smith@ioc.exchange
2026-01-15T01:40:36Z
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@futurebird Actually I have a (very undeveloped) concept of a lesson with respect to the symmetry group of the square.Basically, after the class has been introduced at least to the intuitive approach to the symmetry group of the square, you give them a problem where they have to "solve" a substitution cipher from {a,b,c,d,e,f,g,h} or whatever to the symmetry group of the square, given the multiplication table of that substitution cipher.The lesson here is that this problem doesn't have a single unambiguous answer: rather you can solve the substitution cipher for a few elements like the identity element and the "rotate by 180 degrees" element, but you can only classify the rest of the substitution cipher up to the symmetry group of the symmetry group of the square, more technically known as the automorphisms of D_4.I was thinking maybe there's an angle to develop as like an alien linguist as part of a Star Trek science team, and perhaps even make it a trick question by making it seem like they are expected to find the one "true" solution.It turns out that the automorphisms of D_4 is isomorphic to D_4, which is definitely a very yo dawg moment, but it turns out this is very much accidental. Groups G that are isomorphic to their own automorphism group include all complete groups, but this is one of a handful of sporadic exceptions of a group that is not complete but also isomorphic to its automorphism group. This includes D_4, D_6, D_∞, and may include a few more unknown examples.It turns out that all the symmetric groups (i.e. groups of permutations of n elements) are complete except for n=2 and n=6. The n=6 exception actually pretty interesting, and @johncarlosbaez likes to talk about it.https://github.com/constructive-symmetry/constructive-symmetry/tree/master/D002_Book_of_Algebrahttps://math.ucr.edu/home/baez/six.html
(DIR) Post #B2IHyMjF3QLIXr00Bc by futurebird@sauropods.win
2026-01-15T01:43:22Z
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@leon_p_smith @johncarlosbaez I wonder if putting it in an addition table format might make it easier?I've been wanting to do some symmetry group stuff. Bookmarking this for summer. I'd need to play around a lot to see if I can find a simple angle.
(DIR) Post #B2IIu7oMftCx4jw9q4 by lemgandi@mastodon.social
2026-01-15T01:53:47Z
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@futurebird So very Cool!
(DIR) Post #B2IJpioVm09QLITCLY by leon_p_smith@ioc.exchange
2026-01-15T02:04:11Z
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@futurebird @johncarlosbaez addition table, multiplication table, it doesn't matter, its an abstract operation. But yeah, I do call it "addition", not multiplication, at least when introducing this stuff.I think I have a reasonably simple angle for introducing the symmetry group of the square, and that's (imperfectly) represented in the repo as it currently exists.I have somewhat developed ideas about how to move from the intuitive approach of my mechanical number line for D_4 to implementing the arithmetic of D_4 using pencil-and-paper calculations. Namely, I think the semidirect product, the 2x2 integer matrix approach, and the permutation-based (i.e. subgroup of S_4) approach are particularly notable.I don't know where I'd place the lesson on automorphisms, as honestly it need not depend on anything other than the intuitive approach. On the other hand, I'd probably want to prioritize at least one or two of the pencil-and-paper approaches to performing addition in D_4.
(DIR) Post #B2IJqhk1FJnpJ1pyJU by catselbow@fosstodon.org
2026-01-15T02:04:16Z
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@futurebird Introduce them to 10-adic numbers, where there's more than one zero.
(DIR) Post #B2IKM9aBmYaydEEeie by babelcarp@social.tchncs.de
2026-01-15T02:10:02Z
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@futurebird If there can be multiple infinities...just sayin'.
(DIR) Post #B2IO9EWwTXCy1poMfw by bassthang@mastodon.au
2026-01-15T02:52:31Z
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@futurebird Dedekind showed that any two models of Peano arithmetic are isomorphic. In laymen's terms, if there is something that works like we expect arithmetic to, it will have just the one zero. This is not obvious, and your student is to be commended for trying things out!
(DIR) Post #B2IWBxPzLzMZSTMvFQ by kristinHenry@vis.social
2026-01-15T04:22:37Z
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@futurebird In a more perfect world, I would have had you as a teacher when I was a kid. Even for a few months.I'm so thankful that there are kids out there, right now, with you as their teacher.
(DIR) Post #B2Idi8CtTZXrrgX54y by joriki@infosec.exchange
2026-01-15T05:46:55Z
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@futurebird Summer reading https://en.wikipedia.org/wiki/Zero:_The_Biography_of_a_Dangerous_Idea
(DIR) Post #B2IjKAb5XMAW3Lo5ZI by dahukanna@mastodon.social
2026-01-15T06:49:47Z
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@futurebird I wish I had that “math” class that inspired “zero indignation” - 😉😀😁😆🤣
(DIR) Post #B2IwctJLKVJ9LZlQRs by david_chisnall@infosec.exchange
2026-01-15T09:18:51Z
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@futurebird C has at least four kinds of zero. I’m sure there’s space for at least one more.
(DIR) Post #B2IwquhUxZyTi8kUFM by quaithe@mastodon.social
2026-01-15T09:21:21Z
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@futurebird The "invention" of zero is really the invention of the number line - that's where we got the idea that "nothing" is a number. From realising that numbers are a continuum. It's where we go beyond just counting. Ironically, zero is actually a countable number. But we had to go beyond to figure that out.
(DIR) Post #B2JuRulnYkgPw6fic4 by encthenet@flyovercountry.social
2026-01-15T20:29:01Z
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@futurebirdA good video on the history of 0:https://youtu.be/ndmwB8F2kxA
(DIR) Post #B2MZKMFbzH3JHq3HpQ by leon_p_smith@ioc.exchange
2026-01-17T03:14:50Z
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@futurebird @johncarlosbaez Anyway, my point is that "decoding alien hieroglyphs" is actually a pretty good way to start to get an intuition for what an automorphism is.For example, if you were given all the alien hieroglyphs that correspond to their integers, and given access only to their addition table, then you'd be able to figure out what 0 is, as any identity must be unique, if it exists. Similarly you'll be able to figure out which two hieroglyphs correspond to ±1, in the sense that you could call one of them "1 up" and "1 down", and from there figure out what "2 up" and "3 down" are, but you'll never be able to decide if "up" correponds to positive and "down" corresponds to negative, or vice-versa.This intuition is captured by the fact that the group of integers under addition has exactly one non-trivial automorphism: we can negate everything everywhere and things will still work out. (And in fact, this is the only such change that is guaranteed to work perfectly in all cases.)Of course, if you then gain access to the alien's multiplication table, you can multiply "1 up" by "1 up", and that answer will correspond to the positive direction. Thus we can fully decode the alien's integers, which corresponds to the fact that the ring of integers exhibits only the identity automorphism: when multiplication is involved, we can't just flip everything's sign and expect things to work out.This intuition is a bit hard to operationalize, though, as the addition tables are infinitely large. In reality, if the alien heiroglyphs are truly capable of expressing arbitrarily large members of an infinite set, such as the rational numbers, the notation must involve some regularity. That regularity can provide insight into the alien's interpretation of their rationals in ways that don't correspond to what could be learned from their operation tables alone.The automorphisms of the group of rationals under addition correspond to multiplying by a non-zero rational number, capturing the intuition that you'll never be able to definitively decode the scale of the alien's unit of measurement from the addition table alone. But if you see something like 1/10000, you can guess it's probably not the unit, versus something much simpler like 1/1.However, the field of rationals exhibit only the trivial automorphism, meaning that you could fully decode alien rationals from their addition and multiplication table.Switching to a finite system avoids these complications, and also is capable of providing much more interesting examples of automorphism groups than your more widely-appreicated number systems can.