For mathematicians, = does not mean equality
jeremykun.com
jeremykun.com
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
“The change of subject from “The dog bit the boy” to “the boy was bitten by the dog” is similar to the change of subject in a formula, as for example … In each case, the two sentences state the same relationship, but with different emphasis.” [1]
[1] https://medium.com/q-e-d/that-loser-woman-mathematician-who-...
https://en.wikipedia.org/wiki/Thematic_relation
Sure, algebra was fine, a(x+y)=ax+ay can go either way; but not ratios and other relationships.
What helped me was was geometry, where you can see it's just a relationship. All the components move together; one part isn't priviledged as the result.
e.g. you enlarge a circle. It doesn't make sense to ask whether the radius made the circumference bigger, or the circumference made the radius bigger.
This is simply not true.
Definition 1 ("equation") vs definition 2b ("recipe")
As a theoretical physicist you might find definition 2b uninteresting.
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.
Does Hacker News have built in LaTeX, or is that just very clever use of Unicode?
I also ask "why haven't more symbols been introduced to our keyboards? Just !@#$%&*..."
It's strange, esp. when you realize coming up with a new symbol that everyone uses is easier these days than it was 4 centuries ago!
Take an advanced math test and try to type it out. Being in no sense historically limited by typewriters/keyboards, you might be surprised at the complexity of the symbols.
Because you want to find the value of K, which means that if it was expressed thusly, you'll still need to divide by 2 to get to the desired result.
> 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 the essential point: There is a finite computation process which shows that the denotations are equal.
> [...] if the two things we have were the same then we would never feel the need to state their equality. Concretely we ask a question, 27 x 37, and get an answer, 999. The two expressions have different senses and we must do something (make a proof or a calculation, or at least look in an encyclopedia) to show that these two senses have the same denotation.
Typically, the objects related by equality can be thought to have the same meaning with respect to extension and different meanings with respect to intension. Further, the difference in intension reveals something of the computational content of the extensional object being referred to.
Further topics to explore: the BHK interpretation of intuitionistic proof and the univalence axiom in homotopy type theory. Both of these topics give one some insight on the relationship between the computational content of mathematical objects and how this content pertains to the question of whether two objects are “the same.”
Finally, I did skim the article itself and found it lacking. The author seems to be aware of the fact that there are surprising, highly non-trivial properties of the (seemingly trivial) notion of equality in mathematics. And also to be aware of the fact that the use of ‘=‘ in CS contexts isn’t some sort of abuse of notation. But there seems to be very little of interest here beyond some circumstantial verification of these two general (and well-known) facts about equality in the mathematical and computational contexts.
Thanks! For the interested, Girard's book focuses on the Curry-Howard isomorphism, another great result linking computer programs (actually, the typed lambda-calculus) to mathematical proofs.
f(x) = 2x + 3
For all x in the domain of f, f(x) = 2x + 3
f = { (x, y) in Cartesian product of domain and codomain | y = 2x + 3 }
The articles other point in this example is that we might way “when x = 2, f(x) = 7.” Claiming that x is used both as an indeterminate value and a concrete value. Again, I think the ambiguity is resolved when translate using the correct quantifies, something like “for all x in the domain of f, if x = 2, then f(x) = 7.”
Or perhaps you might claim, “there exists an x in the domain of f such that f(x) = 7.” The important point being that the function f is formally NOT the formula f(x) = 2x + 3, but a particular set of ordered pairs, of which you can make formal statements about in first order logic.
Another example used was
A = {n^2 | n = 1, 2, ... 100}
A = { n^2 | n in {1, 2, ..., 100}}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.
Since assignment is about twice as frequent as equality testing in typical programs, it’s appropriate that the operator be half as long.
Of course, that's wrong. Trying to make it easier to type, a bigger usability problem is created, assigning counterintuitive symbols to functionality that most people would not associate to them.
So the problem is not what = means in maths, it's what it means for people learning a language. Anyway, smadge is right: most of the uses in maths are of the kind "a little imprecission saves tons of explanation". In other words: that's not a question of what = is, but a question of how we use it to get things done.
Oh and then there is the "I'm used to it so it must not be so bad" crowd and the rationalization ensues.
I would like to emphasize the distinction that smadge made, since distinguishing between universal and existential quantification is important to properly discuss different meanings and usage modes of equations.
How do you know? It seems no such problem has been reported, so the burden is on you to show it exists. I don't think people assume that the same token has the same meaning in different languages.
Oh, please. This has been a flamewar since the 80's. Every angle you can think of has been tried before. The real question is why are we still using plain text. A lot of the difference between languages is this kind of absurd minutiae taken too seriously as if we're discussing about the reality fabric instead of mere conventions.
When you're teaching children, this is one of the most common complaints.
However, I think that using this as a justification for programming languages' use of the equals sign for assignment (whether or not it needs any justification at all) I think is a moot point because mathematics is more expository, and isnt always strictly rigorous as long as its clear that everything is well defined and correct in the end (although there is expressiveness in the formalities as well).
> 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 predicate "f(x) = 7" will also hold. It happens there is only a single value that will satisfy "x = 2", but that's not the equality's problem.
So both = signs really are equality here.
This touches on another point that one sees much of in mathematics lectures but little in math lectures. Mathematical notation needs to be unambiguous but also facilitate communication and hence tends to be very terse. Thus when discussing addition on a finite field F_5, one usually starts defining equivalence classes [j] and an addition operator "+" and then says that [3]"+"[3]=[3+3]=[5+1]=[1] followed by a disclaimer like:
"We hence see that this notion is compatible with the previously defined one when identifying ... . Hence, if there is no possible confusion, we will simply write ..."
See also Tao's comments on rigor in mathematics: https://terrytao.wordpress.com/career-advice/theres-more-to-...
All notation can be made rigorous, if one wants to, but discussion is more concise if we let everybody do that by themselves.
I think mathematicians thend to use a lot of "notational slang" or "ad-hoc syntactic sugar", if you will. This will make the notation often look imcomprehensible for people not familiar with the exact domain - however, there is usually a consistent meaning behind it. If one wanted, the notation could be "desugared" into a more rigid (but more verbose) form that expresses the same.
To take big-O-notation as an example, if you write something crazy-looking like
x + 5 = O(x)
The function "f(x) := x + 5" is a member of the set of functions "O(x)".
Where "O(x)" is itself shorthand for a convoluted set expression.
Similarly, if you write
x^3 + O(x)
Contrast that with programming languages, where you often have a rigid syntax but higher-level semantics being quite fuzzy.
E.g., equals() and toString() in java are straight-forward to write but they can "mean" quite a lot of different things depending which kinds of objects you call them.
e.g., the "obvious" meaning of equals (value equality) works only with immutable value types - yet the method is defined on any kind of object. So it might also mean instance equality - or even entity equality if you deal with ORM proxies - or even wilder things...
Computer languages make no exception: they are nothing else than formalisms to express computations. As for every other formalism, the meaning of signs is chosen to be what appears most comfortable in that context by the formalism designer. The statement "x = x+1" has very different interpretations depending on whether you consider it written in C or in standard polynomial equation theory; but in both cases there is a well known meaning for it. In exactly the same way the word "case" has different meaning depending on whether your are reading in English or in Italian.
Oh, and the wonderful notation for integrals, ∫ 2x dx = x² + C
∫ f(x) dx = set of solutions of diff. eq. g'(x) = f(x)
Of course, writing
∫ 2x dx = { x² + C | C ∊ R }
grows old pretty fast, so we drop a couple of characters here and there, but it's pretty consistent with set-builder.
Of course, this only goes to show the article's point that what "=" means here requires a lot of context.
Source that both are in use: https://math.stackexchange.com/questions/196081/the-right-wa...
It bothered me a lot when I took logics lectures that equality wasn't treated as operator or even just anything within a theory (whereas turnstyle was described, at least).
3 * 3 ≡ 1 (mod 4)
3 * 3 = 1
In my current ring theory course, we have indeed written things like 3 * 3 = 1 when working in |F_5 (not sure that notation is going to work as well as I hope, looks alright in the app I use), but it's not the equality symbol is overloaded, but the numbers themselves. Rather than using = to mean numeric equality and equality w.r.t. equivalence classes, we just use the numbers themselves as shorthand for their equivalence classes.
This is not limited to rings of the form Z/nZ.
I should also add that, as generally presented in higher level math, modular arithmetic actually does use literal equality. The abuse is in the numbers.
That is to say, when we write "9" in modular arithmetic, we mean the set of all numbers congruent to 9. In this case in "9=1 mod 4" the equality is literal because the set of numbers congruent to 9 is literally the same set as the set of numbers congruent to 1.
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.
For instance we might say:
sin(x) = x -x^3/6 + x^5/120 + O(x^7)
However, mathematicians do indeed use similar notation in this context, that is, little-o notation. It is in fact true that
sin(x) = x -x^3/6 + x^5/120 + o(x^5), x -> 0.
https://www.wolframalpha.com/input/?i=taylor+series+sin+x
Notice that in your example, you have o(x^5) and an explicit x^5 term. In my example I have O(x^7), but no explicit x^7 term. It is true that I cannot think of a circumstance where you want to do this abuse of notation and would care if you were forced to use little-o or big-O instead of the other.
In my experience, it happens to be more common to use big-O.
The parent comment was right and you are wrong, around zero x^5 absolutely dominates x^7 and the big-O notation is used. See for example here [1]
[1] https://en.wikipedia.org/wiki/Taylor_series#First_example
For instance, we might say x = O(x^2) and x=O(x), but we would not say O(x^2)=O(x).
Interestingly, in my experience, some people will actually say O(x)=O(x^2), but that seems a bit too abusive for my liking.
Before Knuth popularized Big O notation in CS and started the field of analysis of algorithms, already in 1958 N. G. de Bruijn wrote an entire book on Asymptotic Methods in Analysis (not CS): see a few of its leading pages here: https://shreevatsa.wordpress.com/2014/03/13/big-o-notation-a...
And the notation was already being used by Bachmann in 1894 and Landau by 1909 in analytic number theory, well before computers. It was perfectly commonplace to use big-O notation with the equals sign very quickly: see e.g. this paper by Hardy and Littlewood (https://projecteuclid.org/download/pdf_1/euclid.acta/1485887...) from 1914, well before even Turing machines or lambda calculus were formulated, let alone actual computers or analysis of algorithms.
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.
S[ x ↦ V ]
indicates that the new state is equal to old state S, but with variable x now bound to value V.
Math symbols and expressions are inconsistent just like regular languages. But, unlike math, other languages don't claim to be consistent.
It's not surprising that John von Neumann said "in mathematics you don't understand things. You just get used to them." - I've never heard a software developer say this about coding.
For example, I did not enjoy integrals at school because of the 'dx' at the end which means 'with respect to x' but which actually looks like a multiplication (* d * x).
I think that the reason why I never got deep into math is because the language of math is too inconsistent and has too many logical shortcuts and I can't operate in such environment.
(even in my pure math classes, I sometimes imagined that the variables had units, to help find mistakes).
But your point is well taken about the consistency of math notation. Math spent most of its history being scribbled by hand and read by humans. It got the job done. And it was not uncommon to invent a new notation on the fly to replace an abstraction with a single symbol. That's the precursor to the subroutine.
The need for perfect formality of notation is a new thing, brought on by the computer age. This may illustrate the point that programming is not "just math," and math is not a form of programming.
This is sarcasm, right? We say this all the time about legacy spaghetti code or excessively clever magic our languages or (closest to von Neumann's statement, perhaps) the mountain of APIs and libraries and OS we use..
The notation "x = x + 1" is awful because at lhs x denotes a reference to an integer while at rhs x denotes the value hold by the reference. If you know C, it is similar to the difference between an integer pointer *x and an integer x. As an illustration, here are two programs that are doing the same thing, one in C and one in Haskell.
#include <stdio.h>
int main() {
int x = 0;
x = x + 1;
printf("%d\n", x);
return 0;
}
import Data.IORef
main :: IO ()
main = do
xref <- newIORef 0
x1 <- readIORef xref
writeIORef xref (x1 + 1)
x2 <- readIORef xref
print x2You don't say "angles are equal" - you say they are congruent. It's their measures that are "equal".
Although congruency implies measure equality, it doesn't really mean that, but that the shapes are the same (can be rotated/translated to coincide).
Two triangles on a page are congruent, because they have different position. The three corners of an eauikao trisne are congruent, because their corner+angles are equal. If you take angle to mean corner, you say congruent. If you aren't also talking so about their position, you say equal.
In math you can slice things every which way, so it's impossible to use the different words for every different concept, so you have to establish a context
Equal vs congruent (and equal vs equivalent, which are also synonymous in math) is a crutch for beginners who are over reliant on their informal intuition.
> 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 efficient way to handle variables. The idea is to describe your algorithms and how they connect and allow the compiler (or interpreter) to figure out how to use registers etc to implement it. Of course that ideal breaks down sometimes but most high level programmers don't normally need to stress the low level details too much.
> My second point is that mathematical notation is so flexible and adaptable that it doesn’t need mutation the same way programming languages need it. In mathematics we have no stack overflows, no register limits or page swaps, no limitations on variable names or memory allocation, our brains do the continuation passing for us, and we can rewrite history ad hoc and pile on abstractions as needed to achieve a particular goal.
It's true that there's a limit to abstractions even the highest level languages can make if they want to remain general purpose. However I think languages can handle immutable variables as a default.
That's not to say I agree that programming should always follow mathematical notation. But I also don't think it's a bad ideal in many cases.
> Huh? Huh?
The deal here is surely that induction and other recursive approaches are conducive to being reasoned about in traditional mathematical contexts (e.g. taking a walk). Mutation is impossible to keep track of, mentally. Though others' mileage will vary on that.
There is no karma to be had from complaining about HN's software, and yet it is so very inadequate.
I'm loathe to infer that you think that the problem is "me" because I think the underlying question is whether the comments should be a friendly place for people who do not have time or inclination to read the article. You may see that as an appalling, lazy, degeneration in discourse; the reality is that reading the comments without wading through a blog post is a valid tactic. If clearer methods of quoting were available the two or three of us involved here would have wasted less time.
The "engineer-forward" alternative in which everyone strives to speak from a totalizing position of authority is just, frankly (as a technical person myself) unattainable.
Even though there's a correspondence, there's always going to be a difference between writing a program so that you can actually run it and writing proofs, where you don't, and ridiculously inefficient algorithms don't matter at all, so long as they don't diverge.
>"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.”
It's true, you can't rely on tail call optimizations on every language. But practically all modern compilers for imperative languages transform your code into SSA form[1] in one of their intermediate stages, so the code:
int x = 1
x = x + 1
int x0 = 1
int x1 = x0 + 1
This means that mutating a variable, at least when dealing with local primitive values, is equivalent to assigning the value to a new variable.
Even in other cases, in-place mutation is not necessarily more efficient than copying and modifying values. For multi-threaded code, mutation often requires relying on synchronization which can be more expensive than copying around data.
> Simply stated, the goals of mathematics and programming are quite differently aligned. The former is about understanding a thing, and the latter is more often about describing a concrete process under threat of limited resources.
In the current day and age, if a programmer would wants to write the most efficient code, they need to understand a lot about their multi-stage optimizing compiler, Out-of-order CPUs, OS threads and so on - and they would still need to benchmark their implementation against others. It is not clear anymore that mutation is always faster.
I think that in 99% of the cases, clear, safe and maintainable code trump the micro-optimizations that may (and often may not) be gained from using mutable data. Immutable data is generally easier to reason about, always safer from data races and other bugs, and - in most languages languages - more maintainable.
The age of limited resources and straightforward compilers is long over, but it left mutable variables as its heritage. I'd argue they're still common not because they are necessary to deal with hardware limitations, but just because generations of programmers have gotten so used to them, it's hard to give up on them.
[1] https://en.wikipedia.org/wiki/Static_single_assignment_form
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]
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.
x y + 2 = 0
has infinitely many solutions. One of them is (1, -2).
First, (1, 2) only makes sense if you assume that x is first and y is second, or in other words that they correspond to X_1 and X_2 for some vector X. This is probably a reasonable assumption for x and y, but what if you have some other arbitrary choice of variables?
a ρ + 2 = 0
Then the only unambiguous way to write a solution is with explicit labels: a = 1, ρ = -2.
Second, another way to “solve” the equation “x y + 2 = 0” is: “y = -2/x, for any x ∈ ℝ \ {0}”. You could argue that this is also a different kind of “equation” than the one you started with: the reason it can be seen as a solution is not (just) that it’s simpler than the original equation, but that it provides an algorithm to enumerate the set of individual solutions, as well as the set of solutions given some proposed x value. (That is, a set with one or zero members depending on whether x = 0.) However, even if it is a different kind of object, there’s no way to represent it in standard notation that doesn’t conflate ‘questions’ with ‘answers’. If you really want to use tuples, you could go for “{(x, -2/x) | x ∈ ℝ \ {0}}”, which avoids the equals sign – but the ∈ is playing a similar role, an algorithm (enumerate all members of this set) disguised as a test (is this value a member of the set?).
edit: Upon further reflection, I might actually just be expressing violent agreement with the point you were trying to make. shrug
If we start with x y + 2 = 0 we can apply the function
f(x) = x - 2
to both sides of the equation. This gives us the equation
x y = -2
This is a different equation than the one we started with but these two equations have the same solution set. Can go a step further and transform this equation to
y = -2/x
This equation has the same solution set as the first equation. All three equations are equivalent. Solutions are ordered pairs of numbers.
Commonly in basic courses like calculus we tell students that the last form is preferable and we write solutions as (x, -2/x). We call this set the graph of the equation but really it’s the solution set of the equation.
In algebraic geometry x y + 2 = 0 is preferable. The solution set is called an algebraic variety. The solutions are ordered pairs in affine space.
The rules of algebra, as taught in low level courses, are rules that allow one to transform a given equation into simpler equation. In one variable the goal is to end up with something like x = 3 because such an equation is easy to solve. The solution is the 1-tuple 3.
Not all sets are recursively enumerate so using enumerate as you did can cause problems.
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.
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.
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.
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.
A solution necessitates a question, and questions associated with equations involving variables aren't restricted to: what is set of possible values which satisfy those equations?
No it can’t. x in this case is a bound variable. It’s not that it just so happens to take on -2, but that it is already bounded to -2 to make the statement true.
It really says: There exists some x in Z such that x + 3 = 1. What is that x, or what is a proof of the statement? The answer is in the form of a logical implication.
x is actually just, in the language of computer science, syntactic sugar. In reality x+3 is really the infinite tuple
(3, 1, 0, 0, .....)
I think this is incorrect.
Lets continue to assume we are working over Q. Without further context I would take "x+3=-1" to mean that x,3, and -1 are all elements of Q. 3 and -1 being the obvious elements; and x being an a-priori unknown elements which we can easility derive to be 2.
Notably, x+3 is not a polynomial in the technical sense. If we wanted to consider x+3 a polynomial, we would be asking for the t value such that (x+3)[t] = (-1)[t]. Where (-1) is also a polynomial, and (g)[t] is the map Q[x] X Q -> Q given by standard polynomial evaluation.
Sure, this question is equivalent, but I see nothing in the original equation "x+3=-1" to suggest any involvement of formal polynomials.
(3, 1, 0, 0, ....)
A polynomial ring in one variable is an infinite direct sum of the base ring with addition component wise and multiplication defined in a certain way. The expression x+3 meets the definition of a polynomial.
What I am questioning is the necessity to interpret "x+3" as an element of Q[x].
>The expression x+3 meets the definition of a polynomial.
Only if you take x=(0,1,0,...) and adopt the convention that any member q of the base field Q is assumed to represent qx^0 = (q,0,0,...).
That is to say, x+3 only meets the definition of a polynomial because you insist on interpenetrating it as such.
However, we can also handle "x+3=-1" without ever defining the notion of a polynomial.
Eg, we can say, suppose x \in Q such that "x+3=-1". From this premise, we can derive presisly what specific element of Q x must be.
In a more general setting, we might only be able to derive a set of potential values that x could have, or derive that x cannot possibly exist.
As I mentioned in my prior comment, I see no reason to intererperet the "x+3" in "x+3=-1" as a polynomial. If we were to do so, the question would be asking: find t \in Q such that (x+3)[t]=(-1)[t]. Where (g)[t] is polynomial evaluation.
Applying the definition of polynomial evaluation, we would get that the above equation implies: t+3=-1.
Are you now going to insist that "t+3" is a polynomial. Bearing in mind that we have defined t to be an element of Q, which was necessary to apply it as the second argument of polynomial evaluation; and we only got "t+3" as the output of polynomial evaluation, which is defined to result in an element of the base field.
We could modify are notion of polynomial evaltuation to instead be of the form R[x] X R[x] -> R[x], which also gives us (for free) the ability to apply polynomials to other polynomials. But if we were to do this, then when we say that the solution to "x+3=-1", is -4, we are taking "-4" itself to be a polynomial.
In practice this is fine (we identify the base field with the subring of degree 0 polynomials all the time). However, this entire approach breaks down when you start working with functions that do not fit within the framework of polynomial rings.
For instance, suppose I said that "(x+3)! = 120". Are you still going to insist that "x+3" is a polynomial?
What if I define a function id: Q -> Q. In the equation "id(x+3) = 2, are you still going to insist that "x+3" is a polynomial?
That’s what we mathematicians do. In the context of the original post it is absolutely clear that x+3 is a polynomial. There is no other reasonable interpretation.
When you write things like:
Notably, x+3 is not a polynomial in the technical sense. If we wanted to consider x+3 a polynomial, we would be asking for the t value such that (x+3)[t] = (-1)[t]. Where (-1) is also a polynomial, and (g)[t] is the map Q[x] X Q -> Q given by standard polynomial evaluation.
it gives the impression that you don’t know what a polynomial is. The second sentence I quoted is not true. (EDIT: see note below, my interpretation of what was written was wrong.)
Of course if you change context then different interpretations arise. Which of course is the whole point of my original post. Like all spoken languages mathematical language is nuanced. Things must be interpreted in context.
When presented with the equation x+3=-1 x+3 is a polynomial. -1 is a polynomial.
I gather you do not think x^2 - x + 1 = 0 is a polynomial equation. Is x^3+4x a polynomial? Is there any other reasonable interpretation using accepted mathematical conventions? Perhaps you don’t think 2/(x+3) is a member of R(x). What is it a member of then?
Edit:
x+3 is a polynomial that defines a natural map from R to R. To solve the equation x+3=-1 is asking for the pre-image of -1 of this map. This is what it means to solve this equation. It’s solution set is an algebraic variety. I see no other reasonable interpretation. The whole branch of algebraic geometry is about precisely this. Studying zero sets of polynomial equations.
That we teach people rules they can apply to find the answer does not detract that what is really going is as I’ve described and as you did describe with the second quoted text.
I assume you are refering to the sentence:
> If we wanted to consider x+3 a polynomial, we would be asking for the t value such that (x+3)[t] = (-1)[t]
Bearing in mind that the example I have in mind is the equation "x+3=-1" with the solution of "2", in what sense in the above sentence not true?
>When presented with the equation x+3=-1 x+3 is a polynomial. -1 is a polynomial.
Fair enough. In that case, I assume you would consider the equation "x+3=-1" to be false, as it is clear that (3,1,0,0,...) != (-1,0,0,...). Unless of course you are asking, as I had suggested, that you are looking for the particular element at which polynomial evalutation yields an equal result on both sides. If this is the case then, as far as I can tell, you are introducing the machinery of formal polynomials for the sole purposes of overloading the "=" symbol in a confusing way.
>I gather you do not think x^2 - x + 1 = 0 is a polynomial equation.
Define "polynomial equation" If you are asking if I would consider that an equation taking place in R[x], then (absent some other context) the answer is no. Even with other context I would say that "x^2 - x + 1 = 0" is false as a polynomial equation. You might be able to get me to call equations done in the quotient ring R[x]/<x^2-x+1> polynomial equations, in which case "x^2 - x + 1 = 0" would be both a polynomial equation and true at the same time.
If you are asking if I would call "x^2 - x + 1 = 0" in an informal setting, then the answer is yes. However, I do not see how this is relevant, as the whole point of this comment chain was the formal notion of polynomials.
>x^3+4x a polynomial?
Informally, yes. Formally, it depends on context. However, absent some context, I would not consider "x^3 +4x" to be a formal polynomial.
>Is there any other reasonable interpretation using accepted mathematical conventions?
Yes, x^3 +4x is the member of the base ring corresponding to "(x * x * x) + (4 * x)", where x is some other member of the ring.
>Perhaps you don’t think 2/(x+3) is a member of R(x). What is it a member of then?
I am glad you asked. I believe my above answer regarding x^3+4x still applies. However, let me ask you: Is x^3+4x a member of R(x)?
>What is it a member of then?
Again, depending on context. Without context, I would consider 2/(x+3) to be a member of R.
I know of no mathematician who thinks x^2-x+1=0 is anything other than a polynomial equation. Specifically it’s shorthand notation for the variety of the ideal generated by x^2-x+1. And in general you don’t look associate this variety with the quotient ring of the ideal generated by the polynomial. You look at the quotient of the radical of that ideal.
Without any further information the only reasonable interpretation of x+3 is that it is a polynomial. Without any further context in an algebraic equation x is a variable and is not assumed to be an element of the base ring.
In the context of function spaces like C(R) it’s a different matter. And viewing x+3 as an element of C(R) the only reasonable interpretation of x+3=1 is that we are finding the pre-image of 1. And to do this for a complicated function means solving an equation. And solving an equation by hand, the context of my original comment, means reducing the equation to a simpler one. In the case I gave this means reducing x+3=1 to the simpler equation x=-2 whose solution is obtained by inspection. That’s the goal of all the algebraic manipulations we bore beginning algebra students with. Reduce complicated equation to simpler equation. One whose solution is obtained by inspection.
Your view of how to interpret x^3+4x is too simplistic because the only to way to algebraically manipulate that object is by considering it as an R[x] or R(x). You have to view the x as an indeterminate in some larger ring than the base ring.
For this to be the case, there would need to be a statement in the first place. And that statement would involve the =, so you necessarily still have the = symbol representing something other than questioness. I would further say that the equation itself is still just a statement, and any "question" interpretation is based entirely on the context where the equation is presented.
Further, lets take seriously the notion that "x+1" is a polynomial in the formal sense. What does it mean to find the set of values for which x+1=2 is true? Normally, I would say that we are looking for the set of x values which makes that equation true. However, we are insisting that the LHS is a polynomial (again, in the formal sense). This means that there is no variable. The "x" in the LHS is literally a value. It makes no sense to ask what values of (0,1,0,0,...) make that equation true. The fact that we give (0,1,0,...) a standard name of x does not suddenly make the question sensical. Nor does the fact x is often used to represent variables.
>I know of no mathematician who thinks x^2-x+1=0 is anything other than a polynomial equation.
To be clear, outside of very particular contexts I would still call x^2-x+1=0 a polynomial equation, because it is extremly useful to talk about polynomials without invoking all of the machinery of formal polynomials.
>And viewing x+3 as an element of C(R) the only reasonable interpretation of x+3=1 is that we are finding the pre-image of 1.
I disagree. Viewing x+3 as an element of C(R), the only reasonable interpretation of x+3=1 is the statement (x↦x+3)=(x↦1).
Viewing x+3 as an element of R would allow us to treat the equation x+3=1 in the "obvious" way. We can prove that the statement x+3=1 implies that x=-2.
Further, I would agree with you that, absent other context, when given an equation which contains an "x" in it, there is some implication that we are supposed to solve for x.
>Your view of how to interpret x^3+4x is too simplistic because the only to way to algebraically manipulate that object is by considering it as an R[x] or R(x).
Why? Suppose I don't know what R[x] or R(x) is. We certainly don't teach highschoolers what either of those are, and they seem to be able to do "algebra" just fine.
Here is a simple approach to dealing with x^3+4x without considering it a member of R[x]. For concreteness, I want to solve x^3+4x=0.
Suppose x \in R such that x^3 +4x = 0.
By the distributive property, this equation is true iff x(x^2 +4x)=0.
By direct calculation, we can verify that x=0 is consistent with this equation, and therefore consistent with the original equation.
Consider the case where x != 0.
Note that the function f(n) = n/x is a bijection. Therefore, we have x(x^2 + 4)=0 iff f(x(x^2+4)) = f(0).
By direct computation, we get that this is true iff x^2+4 = 0.
We know that x^2 >=0, and 4>0.
Therefore, x^2+4 > 0.
This is a contradiction, which means that the case where x!=0 is impossible.
This means that we have proven that x=0.
Now, suppose we were working over C.
Continuing from x^2 + 4 = 0, we can show that:
x^2+4 = 0 iff
(x+2i)(x-2i) = 0 iff
x+2i = 0 OR x-2i = 0 iff
x=2i OR x=-2i
Since the cases x=0 and x!=0 are exhaustive, we have proven the statement x \in {0, 2i, -2i}.
I solved this using the method we teach school children and without invoking any notion of polynomials.
We don’t teach high schoolers what is really going on. We mask what is really going on because making it all precise is not effective or helpful at this stage of development. We teach rules to manipulate equations. We don’t use the language of algebraic geometry because that is too complicated. We don’t say to them the variety of the ideal generated by x+2 is the same as the ideal generated 4x+8 are the same so that x=-2 is equivalent to 3x+1=-x-7.
However, we are insisting that the LHS is a polynomial (again, in the formal sense). This means that there is no variable. The "x" in the LHS is literally a value. It makes no sense to ask what values of (0,1,0,0,...) make that equation true. The fact that we give (0,1,0,...) a standard name of x does not suddenly make the question sensical. Nor does the fact x is often used to represent variables.
You are clearly not an algebraist. The polynomial
(3, 1, 0, 0, 0, ....)
Induces a natural map from R to R that is generally called “evaluation”. It maps the number 5 to 8 for instance. We abuse notation and say to begininnng students, “replace x with 5”. We dumb things down. Instead of asking for the pre-image of 8 under this natural map we ask for what values of x do we get a value of 8. The shorthand way of writing this is to say solve the equation:
x+3=8
That you don’t know this is disconcerting since you’ve obscuoulsy had more than an elementary mathematical education. You are confusing the simplistic view of what is taught in basic courses with what is really going on.
Ask a million mathematicians, “is x^2+3x a polynomial” and without hesitation they’ll say yes. Because in standard uasage of that expression it is a polynomial. That’s the default interpretation.
Read the first chapter of any beginning algebraic geometry textbook. x^2+3x-1=0 is an algebraic variety. This is the standard interpretation of that equation.
I am well aware of this. If you look at my comments within this very chain, you will see that I made reference to polynomial evaluation, which I will call Ev(f, x). I Am not denying that the partial application given by f' = x -> f(f,x) is naturally induced by the polynomial f. Nor that this is so natural that it often makes sense to identify f with f' so that we would consider f=f', even though they are different types of objects.
>Ask a million mathematicians, “is x^2+3x a polynomial” and without hesitation they’ll say yes.
As will I, because the distinction between formal polynomials and expressions which can be naturally modeled as polynomials is so unimportant that it is almost never worth thinking about.
Put another way, how would you compute the following sets:
{ x \in C | x^3 - x = 8 }
{ x \in C | x^3 = x + 8 }
{ x \in C | log(x) = x^x }
{ x \in C | sin(x) = .7 }
{ x \in N | exists y \in N such that 5x + 3y = 1 }
Are you really claiming that these questions are ill-posed without stating them in terms of algebraic geometry?
You don’t understand the underlying algebraic theory. This is evidenced by your claim
The "x" in the LHS is literally a value. It makes no sense to ask what values of (0,1,0,0,...) make that equation true.
You can’t write such a statement if you really understand that “what values make x+3 the number 5” is a nicer way of conveying the question “under the natural evil map induced by x+3 what is the pre-image of 5”. The text of your I quoted is wrong.
Then you are not reading my comments.
> To be clear, outside of very particular contexts I would still call x^2-x+1=0 a polynomial equation, because it is extremely useful to talk about polynomials without invoking all of the machinery of formal polynomials.
>You don’t understand the underlying algebraic theory.
My claim is that the underlying algebraic theory is not necessary to rigorously state or solve the problems under discussion.
>You can’t write such a statement if you really understand that “what values make x+3 the number 5” is a nicer way of conveying the question “under the natural evil map induced by x+3 what is the pre-image of 5”.
My claim is that "what values make x+3 the number 5" translates directly into { x \in F | x+3=5}. I further claim that the "x+3" in this interpenetration is not a polynomial in the formal sense
I am not disputing that this is set is the same set as the pre-image of 5 under the natural map induced by x+3.
..x=0 is consistent..
You don't see the problem with this? Saying x=0 and that x is an element base ring means that x is the element 0. You can't later in your problem write "...x=2i..." if you are going to persist in your view that x is an element of the base ring. What you have shown is that the variety of the ideal generated by x^3+4x is the same as the variety of the ideal generated by x, x-2i, x+2i if we are talking about C as the base ring. If the base ring is Q then a different thing is shown.
You can't logically say, in a consistent manner, that x is in R and x^3+4x = 0 and that x is 3 different values. An element of a ring is not three different elements. An element of a ring is itself. If you want to vary the object x then you need to enlarge your ring to an algebraic structure that admits x so that it behaves the way you wish to view it. Your view of what is really happening when solving an equation is not rigorously sound. The proper way to view this is in the context of algebraic geometry.
I never said that x=0. I said that x could be 0. More specifically, I stated that the statement x^3+4x=0 AND x=0 is not inconsistent.
All I have shown in is that (assuming we are working in C), the statement x^3 + x=0 implies that x \in {0, 2i, -2i}.
I suppose you could complain that I have not defined a sense in which {0, 2i, -2i} is correct while {0, 2i, -2i, 7} is incorrect, as it is still a true statement that x^3+4x=0 implies x \in {0, 2i, -2i, 7}.
However, you can easily make this intuition rigourous by saying that the question is to compute the set {x | x^3+4x=0}. Sure, this is invoking machinery not explicitly present in the statement x^3+4x=0. I will even concede that we do not make this machinery explicit when teaching highschool students. However, it is far less machinery than your approach.
I am not claiming that the algebraic approach is not rigourously sound; merely that it is not the only rigourously sound approach.
As far as I can tell, you are claiming that it is the only rigourously sound way of stateing the question.
>You can't logically say, in a consistent manner, that x is in R and x^3+4x = 0 and that x is 3 different values.
I believe I have made this point clear, but I never claimed x is 3 different values. The claim I made was that x is a member of the set {0, 2i, -2i}
This means that we have proven that x=0.
Later you have x=2i or x=-2i. Persisting with the idea that you can view x as an element in the base ring and at the same time allowing its value to change or vary indicates you don’t really understand these issues. If x is in R it is a single value. If you want to vary it you need to expand R to include x and add the approroate algebraic structure.
It’s shocking that you think x^3+4x is not a polynomial. The whole discussion I started originally was that math language, like all other human languages, is nuanced and there are lots of abuse of notations. This is ok because math is written for humans by humans. The standard interpretation of x^3+4x is that it’s a polynomial. This is not disputable.
To be clear, by the time I got to this statement, I had changed the question (from a base of Q to C)
Further, the complete statement I was making at that point was:
If ((x^3+4x=0) AND x != 0) then ((x=2i) OR (x=-2i))
I am not allowing x to vary at all here. Suppose, for the sake of arguement, we had x=2i. It would still be true that ((x=2i) OR (x=-2i)).
> Persisting with the idea that you can view x as an element in the base ring and at the same time allowing its value to change or vary indicates you don’t really understand these issues.
Persisting with the idea that I am doing this indicates that your are not reading what I am writing.
That is not overloading "=" at all. It is overloading the word "solution".
What your teacher probably told you is:
"x+3=1 is equivalent to x=-2 by .... Hence, the solution of x+3=1 is the same the solution of x=-2 (which is -2 in both cases). Since the solution of x=-2 is so obvious, we also say the solution is x=-2 to mean that it is the same solution as of the equation x=-2."
Same difference as in "Q:Would you like a large coffee or a small coffee? A: Large " The correct answers should only be either phrase "A large coffee" or "a small coffee", but the answer was an adjective.
I’m a mathematician so it’s something my teacher said as such but something that I noticed consciously when I started doing some programming.
=
eq
eql
equal
equalp
string=
...
But think going deeper into this is quite fun
Your post goes to the point at the heart of philsophical number theory.
What does equality mean ?
Yup – you got functinal equivalence, isomorphism, and temporary assignment of values.
But I think you could prove – that all these types of equality – are “instances” of “different implementations” of “equivalence.
They are no more equivalent than 1 = 1 is equivalent.
I.e. 1 = 1 means I think we can define a bijective “counting function” that proves there’s the “same number” of “elemetns” in the “sets”
I think (not sure) – if you define – counting fucntion / same number / elements / sets differently – you get the differing definitions of equivalence you enumerate.
The interesting thing for me is that 1 = 1 is defined clear in 4 of peano’s axioms
https://en.wikipedia.org/wiki/Peano_axioms#Formulation
And you could mentally – try to develop different (and potentially) – more powerful notions of “equivalence” – with differing axioms
A final point… the prevalence of several “similar” concepts of equivalence in computer science – may point to an underlying “platonic idea” of equivalence – that either exists dormant in the world awaiting for us to discover it; or is a useful “technologocial” construct – that has accelerated “progress”
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.
- John Baez ( http://math.ucr.edu/home/baez/week147.html )
--
\
/
--
0 < i < n
n
--
\
/
--
i = 1 ---
\
/
---
i∈[1,n]'=', 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?
i_1 =/= i_2.
As xg15 noted, it's perfectly fine to say (x=2) => (x + 3 = 5). The problem the first example really addresses is that in mathematics, the namespaces are loosely defined, but in programming they aren't. 'i' can mean several things at once, and it doesn't really matter because those things never really interact in the same context. In programming, you need to specify the name 'i' every time you want to reference it, so it's important that you have a stricter namespace rule.
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 symbols that were available back when High level languages were first written?
https://stackoverflow.com/questions/1741820/what-are-the-dif...
https://en.wikipedia.org/wiki/Assignment_(computer_science)#...
Edit: Misunderstood, thought you were talking about math, not programming languages.
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.
> A = { n^2 : n = 1, 2, …, 100 }
> In Python, or interpreting the expression literally, the value of n would be a tuple, producing a type error. (In Javascript, it produces 2.[link] How could it be Javascript if it didn’t?)
That linked JS code uses ^, which is xor, not pow. Math.pow([], 2) = NaN. Or maybe that was the joke and it flew completely over my head.
See, for example, https://www.destroyallsoftware.com/talks/wat
(n + O(√n))(n + O(log n))^2 = (n + O(√n)(n^2 + O(n)) = n^3 + O(n^(5/2))
without dealing with a mess of sets and quantifiers. Please take a look at some works where asymptotic expressions are dealt with proficiently; you'll understand. (de Bruijn's book https://news.ycombinator.com/item?id=16834297 for example, or at least Chapter 9 of Concrete Mathematics.) I recently worked out an example here: https://cs.stackexchange.com/a/88562/891 — replacing all O() equations with “∈” and “⊆”, though it can be done ( https://math.stackexchange.com/a/86096/205 ), is just cumbersome and only distracts from what's going on.
In any case your notation works perfectly fine as an equality of sets. It's not that unusual to add and multiply sets together, there's really only one sensible definition, and the slight abuse of notation to use x instead of {x} is perfectly acceptable if you're careful.
The problem is that f = h + O(g) is generally used in a way that's false as an equality of sets, it's usually used to mean f ∈ h + O(g), which just makes using '=' needlessly sloppy notation (especially since you can make it correct by just changing a single character, no mess or quantifiers required).
Still, that's nothing compared to the suggestions on stack overflow which suggest using = to mean ⊆. Which is a terrible idea, equivalent to using = instead of ≤. In fact you point out one of the reasons it's terrible yourself, when you note that all equality signs only work one way, going against all mathematical convention.
Even stronger, ⊆ has some very nice properties. It's a total order on the big O sets, which is one of the nicer properties you can have (it's merely a partial order on the f + O(g) sets, but you can't have everything). Why on earth would you sacrifice all that just so you can avoid a scary math symbol?
The purpose of notation is to communicate effectively, and ideally notation should match the thoughts that the writer has and wishes the reader to have. That is, notation should match thought, rather than humans change their thinking to match notation.
When a mathematician (who works often with asymptotics, say in analytic number theory) writes something like O(n), they are not thinking of a set; they are thinking of “some unspecified quantity that is at most a constant times n”. (As de Bruijn illustrates with his L() example: https://shreevatsa.wordpress.com/2014/03/13/big-o-notation-a...) So for example, the sentence
(n + O(log n))^2 = n^2 + O(n log n)
is thought of (by the person writing it) as something like
“when you take n and add a quantity that is at most a constant times log n, and square it, you get n^2 plus at most a constant times n log n”
and not as something like
“the set of functions obtainable by adding the function n ↦ n to a member of the set of functions that map n to at most a constant times log n, and squaring the sum, is a subset of the set of functions obtainable by adding the function n ↦ n^2 to a member of the set of the functions that map n to at most a constant times n log n”,
even if the latter is the fully formal and precise way of articulating it. (Consider “the sky is blue” versus “the sky is a member of the set of all blue things”, where “the set of all blue things” was never a part of the original speaker's thoughts.)
That's one reason for preferring the equals sign.
I think what's happening is that we lack a good theory or notation for talking about unspecified things the way we think of them, and set theory (and associated notation) is the closest thing that anyone's bother to develop. (An unspecified thing is “just” a member of some set. And when someone wants to be fully formal, that works fine enough and in fact the differences disappear. As Thurston says (https://shreevatsa.wordpress.com/2016/03/26/multiple-ways-of...): “Unless great efforts are made to maintain the tone and flavor of the original human insights, the differences start to evaporate as soon as the mental concepts are translated into precise, formal and explicit definitions.”) Using the equals sign here captures the original thought better than ∈ or ⊆.
What's so terrible about a one-way = sign anyway? The problem can't be that it goes “against all mathematical convention”, because using it that way is the mathematical convention for over a century, ever since soon after Bachmann introduced it in 1894. It seems that as people learn more mathematics, they learn to accept a great many things and extensions to notation, but the thought of an asymmetric = sign, used like “is”, is just too much to bear for some people. But everyone who works with asymptotics enough does get used to it and comes to appreciate it, so it can't be that either...
If you seriously think you're not missing out by removing the distinction between ⊆ and =, consider that O and subset relations are enough to denote all relations that you'd normally use the different big Os for that nobody ever bothers to remember.
So f being of order o(g) is equivalent to O(f) being a proper subset of O(g) (O(f)⊊O(g)), f being of order Theta(g) is simply equivalent to O(f)=O(g), and f being of order Omega(g) is simply O(f)⊋(g). You might need to restrict yourself to nonnegative monotonically increasing functions, but that's pretty much done in practice anyway. And this might differ from some of the existing definitions that are being used, but those do vary a bit across sources, and these definitions are as sensible as any, and are compatible with the natural partial order on sets of the form O(f) given by inclusion.
Seriously though there's nothing you'll gain from throwing away a nice inequality relation just because it saves you from writing ⊆.
I'm also not sure what you have against thinking of something as sets, but you can't be precise in mathematics and insist you're not using sets, at least not without going through a lot more trouble then you'd ever avoid that way. In particular it's fine to think of:
>“when you take n and add a quantity that is at most a constant times log n, and square it, you get n^2 plus at most a constant times n log n”
and
>“the set of functions obtainable by adding the function n ↦ n to a member of the set of functions that map n to at most a constant times log n, and squaring the sum, is a subset of the set of functions obtainable by adding the function n ↦ n^2 to a member of the set of the functions that map n to at most a constant times n log n”,
as being the same thing. One is just written in slightly more obnoxious math speak. However if you insist that the meanings are different then something is wrong, because that would imply that the most straightforward mathematical interpretation is apparently different from what you meant.
I have two points:
Firstly, although the article does a good job making the point that there's no reason to get on programming languages' case for using formalized, ad hoc syntax, it does a terrible job of backing up it's actual title. The bottom line is the equals sign is very closely associated with the concept of RST equivalence in mathematics, to the point where a specificity such as the symbol for the isomorphism (which, ironically, he used as a counterexample) is abstracted away, because isomorphism is an RST relation. If two things are isomorphic, they are, in some sense, the same (or equal), so use of an equals sign is natural.
Secondly, if you had read the first source you posted, maybe you wouldn't have claimed programmers/CS people were the ones pedantically insisting on calling it an abuse of notation since, on page 6, Bruijin writes, "The trouble is, of course, due to abusing the equality sign =."[1] Furthermore, after defining the parameters around the use of the O-symbol, he also writes, on page 7, "It is obvious that the sign = is really the wrong sign for such relations, because it suggests symmetry, and there is no such symmetry."[1]
1. https://shreevatsa.wordpress.com/2014/03/13/big-o-notation-a...
console.log([1,2,3,4,5,6,7] ^ 2);
This produces 2 because ^ is the bitwise XOR operator in Javascript. Arrays are not numeric types, so they appear to be coerced to 0 for this comparison. In short, they are effectively logging "0 XOR 2", which is 2.