Comment by amiga386
Comment by amiga386 a day ago
Comment by amiga386 a day ago
Off: mathematics is kind of moving away from the asymmetric element symbol "∈" to the symmetric type of symbol ":", which, I think is a loss. I'm sad about it.
For example with the element symbol you can do
let x \in R a real number
let U \subset R, U \ni 0 a neighborhood around 0
You must be new to mathematics. There is no standardization. Every paper and book essentially has its own notation system.
I think it’s kinda a weird translation. To be something is to have a relationship with the thing that one is, like an identity relationship, one is one, 1=1 for example. A one way equals sign would communicate to me something like (square)=>(rectangle) to represent that all squares are rectangles, while emphasizing that they do not represent the same thing, as they do not use a standard equals sign, as that would imply a shared identity.
That’s just me spitballing though, I definitely can’t read that character or understand the language. I’m just assuming that your translation is accurate and following the context clues to their logical conclusion.
In most languages, "to be" is used to express at least 3 kinds of relationships, which can be distinguished depending on whether the words connected by "to be" are e.g. pronouns, proper nouns or common nouns:
1. identity: "He is John"
2. membership: "He is engineer"
3. inclusion: "Wolves are carnivores"
For the non-symmetric membership and inclusion relationships, in natural languages the order of the words does not really matter, because a speaker will recognize which of the 2 words connected by "to be" corresponds to a bigger set, of which the other word may be a member or a subset, so "he is engineer" and "engineer is he" will be understood to mean the same, even when one alternative sounds weird (i.e. Yoda speech).
This is why, unlike for the agent and patient of a transitive verb, which need special markers, e.g. the nominative and accusative case markers, in the languages that do not have a fixed word order, for the subject and the nominal predicate that are connected by "to be" no distinct markers are required, they can use the same case (e.g. nominative), because they can always be recognized regardless of their order.
"To be" can also express other relationships, like position in space or time, qualities or quantities and so on, all of which are also distinguished by the kinds of words that are connected by "to be".
In an unambiguous language, like in formal mathematics or in programming, each kind of relationship should use a different notation.
That makes sense. It would really have to be read in context to tell how poorly it might read. I didn’t mean to say it was bad, just saying that I understand the confusion.
Were you around for the thread about if LLMs “know” things? It would have benefited from more precise language.
だ is what linguists call the copula: https://en.wikipedia.org/wiki/Copula_(linguistics)
In English, it's words like "is", "was", "are" or the verb "be". It follows the rules of linguistics, not mathematics, so there shouldn't be pressing need to stress its non-commutativity.
That's helpful context, but I didn't really consider them separate contexts for the purposes of using context clues to try to understand it. I thought you didn't know what that part might mean. What did you mean?
In most Japanese lessons in English, when you are introduced to だ/です/である, it is simply explained as being like "is", or if they've already set a rule that sentences end in verbs, it's introduced as the verb "to be". That conveys, rightly or wrongly, all the things English speakers understand about "is".
This Japanese lesson starts with the assumption that part of the "だ" hiragana looks like "=", which I already don't agree with! Even the dakuten (゛) on the character looks more like an equals than the swooshy bowl they're comparing to "=".
It then says that だ is comparable to equality and mathematical equations, rather than like the lingual is/are/be, and then has to qualify that the equality is non-commutative. They could've saved time by not making that comparison, and instead comparing with is/are/be, which English speakers already understand is not commutative. "John will be early" != "Early will be John" (unless Yoda, you are)
There's a subset of people in the Japanese learning community that claim to have found the "one true way" of understanding Japanese and sometimes this amounts to making very grandiose claims about how to think like a Japanese person (in an almost Neo-Whorfian way). You can tell these people likely never had any sort of formal linguistic training.
Which is whatever - if you want to use some analogies that make sense to you, go for it - just don't pretend that this is what Japanese really is like.
(There is an element of truth to it, namely that Japanese works rather differently in most respects that European languages including English. But also, there are 6-7000 languages in the world and many things that happen in Japanese have analogues in other languages, even if the specifics play out somewhat differently.)
In the case of だ the much more mundane understanding is that it's the (nonpast, plain form) copula. Yes, there are some rules about when you can/should leave it off and when not, but those IMO belong to the realm of pragmatics not semantics.
> I would of course explain that the equality sign is not symmetric with respect to such notations; we have 3=A(5) and 4=A(5) but not 3=4, nor can we say that A(5)=4. We can, however, say that A(0)=0. As de Bruijn points out in [1, 1.2], mathematicians customarily use the = sign as they use the word “is” in English: Aristotle is a man, but a man isn’t necessarily Aristotle.
— Donald Knuth, Teach Calculus via O Notation (https://shreevatsa.wordpress.com/2014/03/13/big-o-notation-a... or http://micromath.wordpress.com/2008/04/14/donald-knuth-calcu...)