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For mathematicians, = does not mean equality

jeremykun.com

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Re: For mathematicians, = does not mean equality

#81

Post by sykh »

jcranberry wrote: post #67
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.

There is a solution. An equation is really a question.

x+3 = 1

is asking the question, “what value for x makes x+3 the number 1?”

The polynomial x+3 is defined for all values in R, the base ring you are working in. We are trying to find the elements of R for which x+3 is the element 1.

Re: For mathematicians, = does not mean equality

#82

Post by sykh »

dragonwriter wrote: post #66
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 vari…

Let me be more precise. Assume x+3 is an element of R[x] with x an indeterminate and R a ring with characteristic not equal to 3. Then x+3 defines a natural map from R to R. The equation x+3=1 is just a shorthand way of asking for the pre-image of this map.

In two variables we get a map from R^2 to R^2and solutions are ordered pairs. By definition of an element of a polynomial ring over R the variables are ordered.

Re: For mathematicians, = does not mean equality

#83

Post by contravariant »

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.

Could be worse. I've seen things like f(x) = g(x) + O(h(x)) a few times.

Re: For mathematicians, = does not mean equality

#85

Post by [deleted] »

danharaj wrote: post #22
roywiggins wrote: post #15

Earlier quoted context omitted.

CS already abuses equality all the time with big-O notation. Often you see stuff like f(n) = O(N²), when they mean that f ∈ O(N²). It's fine because everyone knows what's going on, but it's not using it in the sense of equality.

Sure it is, O notation denotes equivalence classes and being part of the same equivalence class is a perfectly cromulent notion of equality.

[deleted]

Re: For mathematicians, = does not mean equality

#86

Post by contravariant »

allthenews wrote: post #4

Operators in mathematics are overloaded in a very similar way to operators in computer science (in languages that permit overloading). I think the author hints toward a good point: there is no use arguing over the meaning of "=" in a general sense, because the meaning is contextual. I think this whole discussion is merely indicative of inexperience on the part of computer scientists attempting to navigate mathematics…

While operators are 'overloaded' all the time, equality is a bit of a special case as it is part of logic rather than some algebraic operation.

In model theory you don't require your models to have an equality operator, they have one simply by being logical constructs.

Then again mathematicians use quotient spaces so transparently that you might as well consider:

5 = 1 (mod 4)

as 'overloading the equality operator' even though the technical definition implies that those 5 and 1 are different from the 5 and 1 in the set of natural numbers, and in Z/4Z the symbols 1 and 5 refer to the same object.

Re: For mathematicians, = does not mean equality

#88

Post by cconroy »

Alan Kay had a good answer on quora for this.

Let me confess that I’ve not read every answer. But the ones that I did read were all very concerned with “squaring” etc.

The simplest answer — and I think the reason many people have difficulty with both arithmetic and especially algebra — is that you need to deeply internalize just what the “=” sign symbolizes and asserts: that there is the very same number on each side.

In other words don’t be distracted by the symbols and operations. One way to think about this is that “a number is all the ways you can make it” (i.e. it can be thought of as “processes” (an infinite number of them) as well as a “value”).

This means whatever you can do to any number can be done on both sides of the “=” because there is just the same number underneath the gobblydegook on both sides.

This is what “=” actually means. And it’s why algebra is actually quite easy rather than mysterious or difficult. [0]

[0] http://qr.ae/TU1SxJ

Re: For mathematicians, = does not mean equality

#89

Post by abhishekjha »

contravariant wrote: post #83
jcranberry wrote: post #62

Earlier quoted context omitted.

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.

Could be worse. I've seen things like f(x) = g(x) + O(h(x)) a few times.

A few would be sn understatement as far as my algoritmic anslysis class went.

Re: For mathematicians, = does not mean equality

#90

Post by cconroy »

cconroy wrote: post #88

Alan Kay had a good answer on quora for this. Let me confess that I’ve not read every answer. But the ones that I did read were all very concerned with “squaring” etc. The simplest answer — and I think the reason many people have difficulty with both arithmetic and especially algebra — is that you need to deeply internalize just what the “=” sign symbolizes and asserts: that there is the very same number on each side…

I also think prolog has the most advanced sense of this concept with equalities and unification operators.
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