For mathematicians, = does not mean equality • Hacker News Board

Re: For mathematicians, = does not mean equality

#61

Post by sykh » Fri, Apr 13, 2018, 11:11 PM UTC

Think about the equation

x + 3 = 1

Typically we write that the solution is "x = –2". This to me is the most abusive form of usage for "=" in mathematics. The solution to the equation is –2. The solution to the equation x = –2 is also –2.

Solving the equation x = –2 is very easy. We can solve it just by looking at the equation. What we are really doing when solving an equation is transforming the original equation into a simpler equation with the same solution set. Tt gets tedious to write this all out so we just say things like "the solution is x = -2" when we've transformed the original equation to x = –2. This is weird because x is not the number -2. x is a variable that can assume a myriad of values. The only value of x that solves the equation is –2.

As the article states the abuse of the = sign in mathematics is rampant. We do it mostly without realizing it. In this sense mathematical language mimics human languages. All human languages are prone to abuse of rules and to shifting with the times.

The notation in mathematics, while much more precise than spoken human languages, is abused frequently and the purpose is to make things cognitively easier. The ancient Greeks didn't have symbols for numbers and in their mathematics they wrote everything out in Greek. This makes it very hard to do tedious calculations. Using symbols in lieu of writing out all the minutia makes doing math easier provided you learn the contextual meaning of the symbols. Over the centuries symbols have been introduced as a shorthand for complex ideas/objects/operations. If you want everything precisely stated then reading Principia Mathematica ought to cure you of this desire. Mathematics is written by humans for humans.

Code is written by humans for computers and hence the notation needs to be rigorously defined in the language you are using and why your code needs to be commented.

Re: For mathematicians, = does not mean equality

#62

Post by jcranberry » Fri, Apr 13, 2018, 11:16 PM UTC

abhishekjha wrote: post #21

Also the asymptotic notation where f(n) = O(n^2) means a set membership operation.

IMO one of the most irritating abuses of notation that I've come across given that it requires no additional effort to use the 'is an element of' symbol instead.

Re: For mathematicians, = does not mean equality

#63

Post by dragonwriter » Fri, Apr 13, 2018, 11:16 PM UTC

theoh wrote: post #59

ChrisSD wrote: post #38

Earlier quoted context omitted.

For the record the "Huh? Huh?" is a quote from the article.

OK. Downvotes happily accepted; but I think we all operate in "peephole" mode now; the cost of tracing sources back to the original is significant. There is no karma to be had from complaining about HN's software, and yet it is so very inadequate.

That it was a quote from the article seemed clear in the post in which the quote was presented, so while I agree that it can be difficult to clearly present quotes in some situations on HN and a good blockquote formatting facility would be preferable, I don't think that's really a problem here.

Re: For mathematicians, = does not mean equality

#64

Post by abhishekjha » Fri, Apr 13, 2018, 11:19 PM UTC

jcranberry wrote: post #62

abhishekjha wrote: post #21

Also the asymptotic notation where f(n) = O(n^2) means a set membership operation.

IMO one of the most irritating abuses of notation that I've come across given that it requires no additional effort to use the 'is an element of' symbol instead.

I agree. And given it is always introduced in an standard CS course I don’t see why they couldn’t have gone ahead with membership notation.

Re: For mathematicians, = does not mean equality

#65

Post by CJefferson » Fri, Apr 13, 2018, 11:21 PM UTC

sykh wrote: post #61

Think about the equation x + 3 = 1 Typically we write that the solution is "x = –2". This to me is the most abusive form of usage for "=" in mathematics. The solution to the equation is –2. The solution to the equation x = –2 is also –2. Solving the equation x = –2 is very easy. We can solve it just by looking at the equation. What we are really doing when solving an equation is transforming the original equation int…

I disagree. If you are saying 'the solution's is -2, you have to be clear what the problem is. This becomes clearer when you have a problem with multiple variables. Then saying 'x=-2, y=3', makes clear the value each variable is taking in the solution.

Re: For mathematicians, = does not mean equality

#66

Post by dragonwriter » Fri, Apr 13, 2018, 11:22 PM UTC

sykh wrote: post #61

Think about the equation x + 3 = 1 Typically we write that the solution is "x = –2". This to me is the most abusive form of usage for "=" in mathematics. The solution to the equation is –2. The solution to the equation x = –2 is also –2. Solving the equation x = –2 is very easy. We can solve it just by looking at the equation. What we are really doing when solving an equation is transforming the original equation int…

> Typically we write that the solution is "x = –2". This to me is the most abusive form of usage for "=" in mathematics. The solution to the equation is –2.

I disagree, though I think it's fine to think of a bare “-2” as a solution when you have a single variable, when you deal with equations or systems of multiple variables it breaks down. Sure, you can think of the solution in terms of untagged tuples when the variables have conventional orderings, but that just highlights that the lack of tagging the variable with the value is a shorthand, not the “true” form of the solution. And it's as much a shorthand in the one-variable case.

Re: For mathematicians, = does not mean equality

#67

Post by jcranberry » Fri, Apr 13, 2018, 11:35 PM UTC

sykh wrote: post #61

Think about the equation x + 3 = 1 Typically we write that the solution is "x = –2". This to me is the most abusive form of usage for "=" in mathematics. The solution to the equation is –2. The solution to the equation x = –2 is also –2. Solving the equation x = –2 is very easy. We can solve it just by looking at the equation. What we are really doing when solving an equation is transforming the original equation int…

Suppose x is an integer such that x + 3 = 1. Then, x is -2. There's no solutions here, just implications and an alternative way of defining the value of x.

I think variables in equations are not meant to express the existence of variance within an equation, but a sense of context-dependency of the value of x. At least, IMO.

Re: For mathematicians, = does not mean equality

#68

Post by pron » Fri, Apr 13, 2018, 11:36 PM UTC

lou1306 wrote: post #56

Some more food for thought on the meaning of =, from Girard's "Proofs and Types" [0]: > There is a standard procedure for multiplication, which yields for the inputs 27 and 37 the result 999. What can we say about that? A first attempt is to say that we have an equality "27 x 37 = 999". This equality makes sense in the mainstream of mathematics by saying that the two sides denote the same integer [...] but it misses…

Just a bit of background: Girard is paraphrasing Frege's famous paper On Sense and Reference[1] which is an investigation into the meaning of equality. As a result of that investigation, Frege shows that terms in a language have at least two kinds of meanings (sense and reference or denotation), which Girard presents in a programming context.

[1]: http://www.scu.edu.tw/philos/98class/Peng/05.pdf

Re: For mathematicians, = does not mean equality

#69

Post by Zarel » Fri, Apr 13, 2018, 11:36 PM UTC

thetruthseeker1 wrote: post #46

When I learnt programming, i was confused by x=x+1; After I understood what it really meant, I wondered why they didn’t use some other symbol to capture this semantic. Say something like x <- x+1 ; Which implies assignment rather than equality - That way this would be unambiguous and I feel is more clear. I now guess the choice of using ‘=‘ was probably an attempt at making a (compromised) choice given the limited sy…

Heh, I just read an explanation of this the other day:

https://www.hillelwayne.com/post/equals-as-assignment/

The summary is that C used it because assignment was a lot more common than equality checking, and back then, typing fewer characters was a big deal.

Re: For mathematicians, = does not mean equality

#70

Post by lvh » Fri, Apr 13, 2018, 11:50 PM UTC

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")).

Sure; I think we’re in violent agreement here, it’s absolutely the case that people write the simpler version when the meaning is clear from context. I’ve definitely done that a bunch.