9 ms·
Floor, Ceiling, Bracket
- lordleft 6y agoSomething I think makes mathematics both incredibly powerful but difficult to break into is the notation
- beagle3 6y agoI don't think notation is what's making things hard. The concepts behind are the hard thing. If you can't figure out the concept, the notation doesn't make much sense - and it is the notation that you meet first, so it seems to be the culprit. Which is not to say that all notation is equally good - some is exceptionally bad, and other just confusing. The article mentions [x] which was i use for almost 200 years despite being confusing - but was then fixed into something non-confusing. Similarly, for most uses, Leibniz's differential notation (dy/dx) is superior to Newton's (y with a dot on top) - and is now universally used in those uses - but for a long time Newton's dominated, mostly for political reasons (yes, over 250 years later). But these are the exceptions: Usually, like in the floor/ceil case, when a better notation comes along, it is quicky (30-40 years...) adopted.
- ncmncm 6y agoConcepts can be explained and demonstrated. You can read the demonstration, and eventually understand. If you can find it. But there is no way to look up any given dumb-ass notation, or even to know how to pronounce it. Most notations are used to mean a dozen different things, so you can't even be sure, if you read about it, you are reading about the right one, if you do find it. And if you find the right one, it still won't say how to pronounce it, so you can't even ask without sounding like a dummy. It is all very insular. Or, maybe, was, before Youtube.
- casselc 6y agoIt doesn't help with stuff found in the wild, but if you're working from a text most math books will either list all symbols at the beginning of the index with references to where the notation is introduced and defined, or they have a separate symbol index or glossary of notation.
- _Microft 6y agoLook at this abomination that someone posted in the comment section there: ⌈e⌉ = ⌊π⌋ This is … evil.
- deleted 6y ago[deleted]
- sigjuice 6y agoHow is it evil? To me, it is a mildly amusing coincidence.
- imglorp 6y agoWhich is greater? e^pi or pi^e? https://www.youtube.com/watch?v=SPHD7zmLVa8 https://www.youtube.com/watch?v=SPHD7zmLVa8
- UI_at_80x24 6y ago⌈⌉ Characters like these remind me of the C64 and how they could be strung together to make graphics [ANSI/ASCII/NCURSES are close].
- rzzzt 6y agoThere is a Unicode block dedicated to these symbols: https://en.wikipedia.org/wiki/Box_Drawing_(Unicode_block) https://en.wikipedia.org/wiki/Box_Drawing_(Unicode_block)
- banana_giraffe 6y agoI'm always a little sad how poorly these can work: https://imgur.com/a/mCLZVwp https://imgur.com/a/mCLZVwp I'm sure there's some amount of font twiddling and changing to a different terminal that'll fix this, but using these characters out of the box is problematic.
- UnquietTinkerer 6y ago
- wperron 6y agoWhenever someone tells me "it's just syntax sugar, it doesn't matter" I usually point them to mathematics. This is both beautiful and horrifying in equal measure and _I love it_
- JadeNB 6y agoI knew Iverson introduced his bracket because I've always heard it called "the Iverson bracket"—and I absolutely love it—but I had no idea he invented the floor and ceiling functions, too. At the risk of flippancy, who would think that the creator of APL was also the source of such intuitive notations? Perhaps you have to be willing to explore crazy out-there notation to be able to find these occasional gems.
- johndcook 6y agoAgreed. Say he introduced 100 bits of notation, 3 good and 97 bad. The net result is that the world absorbed 3 good ideas, and the rest have vanished from memory. Not a bad track record. I hope to give the world three good ideas. Besides notation, APL introduced some ideas about vector computing that have been adopted in languages like R and Python (NumPy).
- remcob 6y agoI use brackets with a subscript as notation for the modulo operation. This way expressions with multiple / nested moduli become much more readable. I'm particularly fond of how concisely the Chinese Remainder Theorem can be stated: https://2π.com/17/chinese-remainder-theorem https://xn--2-umb.com/17/chinese-remainder-theorem
- User23 6y agoI find these kind of discussions of notation really interesting. They're not at all the bike-shedding they might first appear to be. One of the last EWDs[1] is an interesting meditation on the topic. [1] https://www.cs.utexas.edu/users/EWD/transcriptions/EWD13xx/EWD1300.html https://www.cs.utexas.edu/users/EWD/transcriptions/EWD13xx/E...
- defrex 6y agoWhilst it's obviously powerful, I often find myself wishing math used syntax even half as easy to understand as any decent programming language. I suppose it's a result of being developed on a chalk board, but math seems be value _terseness_ above all else. Rather than a handful of primitives and simple named functions, it's single greek characters and invented symbols. Those kind of shenanigans would never pass a code review, but somehow when we're talking about math they're "elegant" and "powerful". I call bullshit. Math syntax is bad.
- phailhaus 6y agoOooh don't even get me started on how they name things after people instead of anything remotely descriptive or helpful. Imagine if you named functions after yourself.
- taeric 6y agoShell's sort would like a word. Or Bloom's filter, for that matter...
- dhosek 6y agoOr currying
- Jtsummers 6y agoDijkstra's algorithm, Turing machines, von Neumann architecture, Hoare triples, Hamming distance, Levenshtein distance, Fisher-Yates (or Knute) shuffle, ... https://en.wikipedia.org/wiki/List_of_algorithms https://en.wikipedia.org/wiki/List_of_algorithms - many of the items in this list are named after people.
- beagle3 6y agoKrebs' cycle, HeLa cells and Lugol's formula in medicine as well; And also an Allen key, a Phillips screwdriver, many of the physical units (Ampere, Volt, Tesla, Weber, Newton, ...). It happens in all fields. For that matter, the Hebrew word for masturbation is named after Onan, who was described in the bible as having done so.
- RodgerTheGreat 6y agoThe reverse operators in APL are very visually suggestive if you imagine a matrix and mirror it about the centerlines- Reverse around last axis: ⌽ Reverse around first axis: ⊖ The grade operators (indices by which one could index to sort ascending or descending) are likewise easy to remember- Grade up: ⍋ Grade down: ⍒ I suspect that the reason only a handful of APL's notational ideas made it into mainstream mathematics is because few mathematicians felt the need to describe algorithmic processes, and those who did were willing to settle for big sigma/pi, set builder notation, piecewise function notation, or a handwave at ALGOL, Pascal, or whatever else was in vogue at the time. Their loss, I'm afraid.
- taeric 6y agoThis somewhat begs the question that those symbols are the best they can be, though. That is, yes, if you know one of those set of "grade" symbols, and you know that you can grade "up" or "down", then it somewhat follows. If you just show me those symbols flat out, though, I have no idea what they do. (Granted, I'm not entirely clear what it means to "grade" in this sense, either...) Of course, that also begs the question of "do we need characters for all operations?" But, I'm not entirely sure we do. What is the advantage? There are plenty of operations that are just fine with their symbol being a word. (As evidenced by programming, in general, right?)
- beagle3 6y ago"grade" (as opposed to "sort") means "find a permutation that puts this into order". So the sorted version of the list "name" is in APL, name[⍋name] (read "name indexed by the permutation that puts name into ascending order), but if you have two corresponding lists, "name" and "age", then you can get names sorted by age using name[⍋age]. It is, in that sense, a more fundamental operation than sorting. Before Iverson added the celing and floor symbols, those weren't known either, even though you had a better chance of guessing them. And w.r.t to "characters for all operators" - this is, of course, subjective. But I've never met a programmer who prefers COBOL's "ADD 1 TO X GIVING X" to C's "++x"; Anything that's common enough deserves a symbol, both because it's shorter and easier to recognize visually, and because it reduces the language barrier (in the same way that "3+4" is easy for a 6-year old Thai in the way that "3 plus 4" in English letters isn't) In my opinion Arthur Whitney's refinements of Iverson's ideas (called "K", an APL-related-language) is the right way to go. He mostly converged on ~50 operations that deserve symbols. Many algorithms end up as orthogonal combinations of those, e.g. |/0(0|+)\ which is a complete, efficient O(n) solution of the maximum-subarray-sum problem; or ",//:" which is a complete, though not very efficient, implementation of "flatten".
- tlb 6y agoKnuth [0] discusses in more detail some advantages of indicator notation, in particular that, when you eliminate the limits on summations, it makes it much easier to do algebra on summations. [0] https://arxiv.org/abs/math/9205211 https://arxiv.org/abs/math/9205211
- deleted 6y ago[deleted]
- emmelaich 6y agoRandom notes .. 1. The small Perl special vars $/ $. etc ... are mostly quite mnemonic. 2. Amiga called the left and right brackets 'bra' and 'ket' which I always thought was clever.
- bradrn 6y ago> Amiga called the left and right brackets 'bra' and 'ket' which I always thought was clever. Wouldn’t know about Amiga, but this is standard notation in quantum mechanics (though you may be aware of this already). <φ| is a ‘bra’, and |ψ> is a ket — the two are adjoints of each other. Naturally, <φ|ψ> is a ‘braket’. (As my QM lecturer is fond of saying, ‘Dirac invented both the notation, and the pun’.)
- hrishi 6y agoEchoing a lot of the sentiment here, I wish math would simply use english (or natural language). It clearly seems too much to ask for a programming language, especially when it comes to pure math. At an undergrad math course, we were allowed to state our proofs in english without any notation. This was the only course that did this. I found it easier to write and reason about my proofs, it was incredibly easy to read my peers', and the professor had no issues with my work while he did have some trouble with some of the notation. When I first started, the question I had was if there existed an encyclopedia or an OED for mathematical notation. I was told there wasn't. I then enquired how I would go about deciphering something I didn't understand, and was told that I should ask my professors and peers. I brought one of the papers I had found online to him, and he asked me to write to the author because he couldn't understand it either. Meanwhile I can download any old piece of code, and given enough time with the compiler and myself, I can understand what it means to express. I shudder to think 9f the sheer body of work that relies on a vague understanding of the notations in the underlying proofs. For a science that believes in precision of logic, I ask: how is this still a thing?