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For mathematicians, = does not mean equality

jeremykun.com

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Re: For mathematicians, = does not mean equality

#131

Post by OscarCunningham »

j2kun wrote: post #18
xg15 wrote: post #14

Fully agreed, though to nitpick: > Rather than precisely say, f(2) = 7, we say that for x=2, f(x) = 7. So x is simultaneously an indeterminate input and a concrete value This seems like a perfectly by-the-book piece of second-order logic with two equality predicates. i.e., the statement asserts that if you look at the space of all possible values for x, then for each value where the predicate "x = 2" holds, the other…

Fair point. I'd add that f(x) = 7 can be both equality of functions and equality of evaluations, and binding x=2 suddenly changes the meaning of the equality and the expression.

I think "f(x) = 7" is always equality of the evaluations, and if you want equality of the functions you should write "forall x, f(x) = 7". Admittedly mathematicians are lazy and don't always write that when it's clear from contex, but I don't think the ambiguity is caused by the "=" sign. The expression "f(x) < 7" would be equally (ha) ambiguous.

Re: For mathematicians, = does not mean equality

#133

Post by xiaq »

qxmat wrote: post #129

I have an engineers understanding of higher maths - overly general and very patchy. Short of taking an undergraduate math course, are there any resources to help me parse math notation? For example, while brushing up on endogeneity/exogeneity, E[B'|X] = 0 completely threw me - I had to search Google for the use cases of a bar/pipe aka latex vert/mid. I usually lose interest in a paper if I get stuck trying to decode…

https://en.wikipedia.org/wiki/List_of_mathematical_symbols

Also, if I were you, I would always look up introductory textbooks before trying to read math literature. The example you gave is conditional probability; any textbook on probability theory would cover it.

Re: For mathematicians, = does not mean equality

#134

Post by hexane360 »

skybrian wrote: post #111
ChrisSD wrote: post #19

I agree with the thoughts on the = sign but I'm not so sure about mutations. > If mutation is so great, why do mathematicians use recursion so much? Huh? Huh? > Well, I’ve got two counterpoints. The first is that the goal here is to reason about the sequence, not to describe it in a way that can be efficiently carried out by a computer. Most high level languages try to avoid making the programmer describe the most ef…

The thing that blew my mind was that there are some mathematical programming languages where the point is not to ever actually run the program. Just type-checking it is enough to prove the result. (In the "programs are proofs" sense.) In these languages, it's important not to allow infinite loops because you will never test the code. Even though there's a correspondence, there's always going to be a difference betwee…

https://aphyr.com/posts/342-typing-the-technical-interview

>"You… do realize that the type system is meant to constrain values, right?”

>“No,” you inform him, matter-of-factly. “No, that doesn’t sound right.”

Re: For mathematicians, = does not mean equality

#135

Post by OscarCunningham »

gus_massa wrote: post #40
lvh wrote: post #27

Earlier quoted context omitted.

The usual notation for modular arithmetic uses three dashes, not two, to denote congruence.

At least in my university, in one of the first Algebra courses about integers, factorization and congruence the usual notation is 3 * 3 ≡ 1 (mod 4) but in the next year Algebra course about groups and crazy algebraic structures the notation in the group Z_4 is just 3 * 3 = 1 and everyone understand that you are working in Z_4 (and the * and = symbols are "overloaded" (but no one call them "overloaded")).

I don't think "=" is being overloaded in this example. The symbols "1", "3" and "*" are, since they're working in Z_4 rather than Z, but equality is just equality.

Re: For mathematicians, = does not mean equality

#137

Post by a1369209993 »

kqr wrote: post #126
kps wrote: post #31

(I assume this was inspired by https://news.ycombinator.com/item?id=16803874 ) The use of ‘=’ for assignment in programming languages comes, not directly from mathematics, but indirectly from the use of mathematics in science and engineering. As an example, consider the formula for kinetic energy, commonly written 𝑚𝑣² 𝐾 = ─── 2 Why isn't it written 2 K = m v ², which expresses the same mathematical equality in a s…

While this reasoning may be common, I don't think it's to anyone's benefit. Talking about "the" formula for kinetic energy seems nonsensical, when there are so many ways to state that relationship. Another option is p^2=mK. But there is a good reason to write it as K=mv^2/2 which has nothing to do with specifying a computation. It is the result of symbolic integration of p=mv with respect to v.

And, perhaps more intuitively, its the second integral of m with respect to v. This pattern also shows up in eg A = ∫∫τ∂r² for the area of a circle.

Re: For mathematicians, = does not mean equality

#138

Post by vinchuco »

Humans are compilers of math.

'=', like number, means that concept, regardless of instantiation.

Beyond that note, I don't see the value in listing all the ways you could interpret notation.

Maybe you would also find it entertaining to list all the ways a program can interpret a binary string?

Re: For mathematicians, = does not mean equality

#139

Post by triska »

smadge wrote: post #122

I agree that “=“ as interpreted by people doing math requires context, but in most situations they are able to translate it into a “correct” or formal notion of equality. For example, translating on the fly these ad hoc notions of equality into precise notions of equality in first order logic and/or set theory. For example, f(x) = 2x + 3 Might be translate into something like, For all x in the domain of f, f(x) = 2x…

This comment contains an important key distinction between different usages of "=" that are often casually intermixed in such discussions: There is a major difference in how we quantify the logical variables that occur in formulas.

For example, if we consider the atomic formula x = 5+y, then we may mean the identity ∀x∀y (x = 5+y), where all variables are universally quantified.

Or we may mean ∃x∃y (x = 5+y), where the variables are existentially quantified. To determine whether this holds, we can search for a solution given by a substitution that makes the terms equal modulo some theory E we associate with =. If E is empty, then this corresponds to syntactic unification.

Confusingly, in the literature, sometimes "equation" is used for both, and an entire subthread in this discussion is due to this issue.

When one is asked to "solve for x" etc., then one answers whether there is any solution, thus solving the existentially quantified version. When one means "this identity holds", then one states the universally quantified sentence.

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