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[faust]

It's tough, and I don't think there is a consistent way of saying what is or isn't a number across all of mathematics. You could say something like "any structure that allows arithmetic in a way such that it contains the natural numbers", and that would cover a lot, but it would also make you call matrix rings or rational functions numbers, and arguably even weirder things like isomorphism classes of surfaces.

I think the process of calling something a number is more inductive - like, you start with the natural numbers, wich are the "original" numbers. And whenever you encounter a problem that you cannot solve using your existing classes of numbers (what is 2-5? What is 2/9? What is sqrt(-1)? How many elements does the set of natural numbers contain?), mathematicians are prone to call the structure they come up with to solve that issue a class of "numbers" again.

Which would make a number "any element of a structure that solves a general problem for which the natural numbers provide solutions in special circumstances".

[pacovf]

That’s how I understand it as well.

[Watno]

f ∈ O(n^2)

If you write this, it's a shorthand for f ∈ O(g) where g : n ↦ n^2. But f(x) ∈ O(g(x)) is nonsensical because f(x) isn't a thing.  f is a function; f(5) is a number, "f(x) = 3x" is a shorthand for f = {...  (-1, -3), (0,0), (1,3), ...}. But f(x) itself is literally not anything.

What makes it acceptable to use n^2 as a shorthand for n -> n^2, but not f(x) as a shorthand for x -> f(x)?

[silverspawn]

I don't think I understand what you mean. How is f(x) used a shorthand for x -> f(x) in the expression f(x) = O(g(x)), and what does x -> f(x) even mean?

If you just mean x -> f(x) as a description of f, then you're using f(x) as a shorthand for... f. ?

[heron]

Idk this f(x) business seems fine.

f(x) \in O(g(x)) means that there exists m,M such that if x > M then |f(x)| ≤ Mg(x).

f(x) is the number that you get by applying f to x. I guess you might complain that this is a statement about f, not a particular number f(x).
But from the sounds of it you don't have a problem with saying x \in O(x^2), which would have the same problem, so this argument does not hold up for me.

tl;dr: how can x be a thing but f(x) not be a thing.

[silverspawn]

f(x) is the number that you get by applying f to x.

But what is x?

[silverspawn]

Furthermore, if f(x) does denote a number as you say, then f(x) \in O({whatever}) says that a number is in a complexity class.

[Cuzz]

Pedantically f is a function (an ordered triple (X,Y,S) where S is a subset of XxY satisfying the function axioms, or really better an equivalence class of such ordered triples where (X,Y,S) ~ (X',Y', S') iff S = S') and f(x) is an element of the codomain Y.

But using f(x) in place of f is one of the more ubiquitous and mild abuses of notation in mathematics. And it's useful because while the definition of a function doesn't require any particular formula or pattern for specifying what the elements of S are, often in practice functions do arise from formulas and writing something like "the function f(x)" is less clunky than "the function f defined by f(x) = ..." or "the function f: x -> f(x)" or whatever. 

[silverspawn]

And it's useful because while the definition of a function doesn't require any particular formula or pattern for specifying what the elements of S are, often in practice functions do arise from formulas and writing something like "the function f(x)" is less clunky than "the function f defined by f(x) = ..." or "the function f: x -> f(x)" or whatever. 

How does this apply to f(x) \in O(g(x))? I *don't* object to f(x) = x^2 to define a function, but that's not what's happening here.

[Cuzz]

And it's useful because while the definition of a function doesn't require any particular formula or pattern for specifying what the elements of S are, often in practice functions do arise from formulas and writing something like "the function f(x)" is less clunky than "the function f defined by f(x) = ..." or "the function f: x -> f(x)" or whatever. 

How does this apply to f(x) \in O(g(x))? I *don't* object to f(x) = x^2 to define a function, but that's not what's happening here.

Well for one it keeps track of what the independent variable is in cases where it might be ambiguous.

[silverspawn]

what the independent variable is

This is not a thing! You've just written down the formalism for a function in the previous post. There is no independent variable that is part of what a function is. There is no difference between f(x) = x^2 and f(y) = y^2. Both are short hands for the same object.

[Cuzz]

Are you objecting to the literal symbol “f(x)” or an actual spelled out expression like (3x^3-1)/x? Can you quote an instance of what you find so objectionable?

[Cuzz]

what the independent variable is

This is not a thing! You've just written down the formalism for a function in the previous post. There is no independent variable that is part of what a function is. There is no difference between f(x) = x^2 and f(y) = y^2. Both are short hands for the same object.

There are plenty of contexts where x^2 and y^2 are definitely not the same thing, like say in a polynomial ring R[x,y].


Eventually you want to actually do some math and communicate what you’re trying to do to an audience who you reasonably expect to understand what you mean when you write things without getting bogged down in formalism at every turn.

Like xt^2 is O(x) but not O(t). The label on the variable matters here. I could spend a lot of energy writing down the formalism that makes this make sense but everyone knows what I mean.

[silverspawn]

I'm saying that the symbol f(x) by itself is not an object of any kind. f is an object, and if you take a specific number, then f([that number]) is another object.

Despite this, the statement "f(x) = x^2" does make sense. It's not implying that f(x) is a thing, rather it's a shorthand for something like f = (R, R, {(x,x^2) | x \in R}), so here the x is a properly bound variable. It's saying that 'if we apply f to some number x, we get this thing, and that's true for every x in R'. All is well here.

But then people say things like 'the graph of f(x)' or 'f(x) has a root at x = 5' or 'f(x) \in O(g(x))', and none of the replies in this thread have convinced me that this makes even an atom of sense. These are the things I'm objecting to.

It would be one thing if people just wrote f(x) intending to mean the same as f (although why if it's longer??) but I'm worried that most people actually do think there is a logic behind this, like you when you just said that it keeps track of what the independent variable is. A textbook I was reading just a few weeks ago said something like 'suppose f is a function of x and g a function of y' which DOESN'T MEAN ANYTHING. But I think people have this image of several variables that can depend on each other in their head, so that somehow if y=f(x), then y and x are two things and y depends on the other thing -- even though this image is completely incompatible with every formal definition of a function (at least every definition I've ever read)

There was a stack exchange post about this recently, but I don't think anyone found a solution.

[silverspawn]

There are plenty of contexts where x^2 and y^2 are definitely not the same thing, like say in a polynomial ring R[x,y].


Eventually you want to actually do some math and communicate what you’re trying to do to an audience who you reasonably expect to understand what you mean when you write things without getting bogged down in formalism at every turn.

Like xt^2 is O(x) but not O(t). The label on the variable matters here. I could spend a lot of energy writing down the formalism that makes this make sense but everyone knows what I mean.

I think you think that I'm against various elegant ways of writing things down, but I'm not familiar with any example where what I'm saying forces you to be less elegant. f \in O(g) is shorter than f(x) \in O(g(x)) and doesn't convey any less information. And I have no issue with f \in O(n) or even xt^2 \in O(x).

[Cuzz]

I don't think anyone is claiming there is a full rigorous logic behind this. It's about whether you abuse terminology and notation in ways that aid the communication of whatever it is you're trying to communicate or obfuscate it.

I agree that "the graph of f(x)" is awkward.

"f(x) has a root..." might be defensible sometimes. For example, if it seems perfectly fine to say "f(x) = x^2-25" and "x^2-25 has a root at x=5," then I don't see why "f(x) has a root at x=5" should be automatically rejected.

Similarly, if you can say  "f(x) = x^2-25" and "x^2 - 25 \in O(x^2)" then why not "f(x) \in O(x^2)?" "f(x)" becomes a symbolic replacement for the expression "x^2-25" and it seems awkward to make that replacement illegal in some specific context.

I don't really think I'm gonna satisfy you, because I can tell that you just have a different attitude about math. But at the end of the day you're not really doing math if you can't clearly communicate your ideas, and that clarity of communication depends on the culture of how modern math is written, not all of which perfectly aligns with formalism or everything would be written in ZFC machine code and be completely worthless to any real audience. This blog post informs how I think about a lot of this https://terrytao.wordpress.com/career-advice/theres-more-to-mathematics-than-rigour-and-proofs/

[Cuzz]

There are plenty of contexts where x^2 and y^2 are definitely not the same thing, like say in a polynomial ring R[x,y].


Eventually you want to actually do some math and communicate what you’re trying to do to an audience who you reasonably expect to understand what you mean when you write things without getting bogged down in formalism at every turn.

Like xt^2 is O(x) but not O(t). The label on the variable matters here. I could spend a lot of energy writing down the formalism that makes this make sense but everyone knows what I mean.

I think you think that I'm against various elegant ways of writing things down, but I'm not familiar with any example where what I'm saying forces you to be less elegant. f \in O(g) is shorter than f(x) \in O(g(x)) and doesn't convey any less information. And I have no issue with f \in O(n) or even xt^2 \in O(x).

Is there like a quote from a book or a paper or something that actually demonstrates what you dislike and how you would rewrite it? I admit I'm still confused about whether you're taking issues with the appearance of the actual character "x" or the entire concept of representing a function by a symbolic formula.

[silverspawn]

For example, if it seems perfectly fine to say "f(x) = x^2-25" and "x^2-25 has a root at x=5," then I don't see why "f(x) has a root at x=5" should be automatically rejected.

I think this quote hits our disagreement perfectly.

The first two of these statements are logically coherent. The first makes a set of statements that defines a function ... f(0) = -25, f(1) = -24 etc. ... and the second uses 'x^2 - 25' as defining and then referencing a function. In that statement, x^2 - 25 denotes a precise object, that object is a function, and so it may have a root. But in the third case, you already have the function f, and then you're writing (x) behind it. It's difficult for me to even critique why this is wrong because I still don't think it means anything. In particular, if x were a number, then f(x) would be a number rather than a function, so saying f(x) has a root would be wrong since a number doesn't have a root.

Since you think it's defensible, can you tell me what f(x) means here? Just, literally, what is it?

Is there like a quote from a book or a paper or something that actually demonstrates what you dislike and how you would rewrite it? I admit I'm still confused about whether you're taking issues with the appearance of the actual character "x" or the entire concept of representing a function by a symbolic formula.

Doesn't this post answer this? In every specific case I've complained about so far, the solution is just to write f rather than f(x).

[Cuzz]

Since you think it's defensible, can you tell me what f(x) means here? Just, literally, what is it?

A macro for "x^2-25"

[silverspawn]

So the exact same thing as f?

[Cuzz]

So the exact same thing as f?

No, because "f(y)" would be a macro for "y^2-25," and those are two different strings, so they can't both be the exact same thing as f.

[pacovf]

Well, if you want to be super rigorous with notation, then f(x) = x^2-25 is not a function definition either, it’s an equation for variable x, which you might be able to solve if f was defined earlier. The notation for defining functions is different (and clunkier). f(x) for f is a frequent abuse of notation where it avoids more cumbersome notation, although it is true that it is used unnecessarily in some cases.

It’s been a while since I’ve had to do super formal maths, but maybe part of the confusion comes from thinking that “x” necessarily refers to some specific number/object. That’s only sometimes true, for example in a set of equations. Maybe the concept of “dependent variable”, as Cuzz puts it, is the right term when “x” is not meant to represent a specific object, but rather to let you speak about rules that convert other from a given category. I do not know how formalised that notation approach is, but in effect is what people do in most of the cases you’re complaining about.

[silverspawn]

So the exact same thing as f?

No, because "f(y)" would be a macro for "y^2-25," and those are two different strings, so they can't both be the exact same thing as f.

But math is formalized in terms of sets*, not strings. And y^2-25 refers to the same function -- and hence the same set -- as x^2-25.

I feel like you want to maintain that there are cases where writing y and x is different. This is true, but this is not one of those cases. And in complexity theory (where I see people using the f(x) =O(g(x)) thing), those cases basically never come up.

* or operators if you like lambda calculus, but the point stands.

[silverspawn]

It’s been a while since I’ve had to do super formal maths, but maybe part of the confusion comes from thinking that “x” necessarily refers to some specific number/object. That’s only sometimes true, for example in a set of equations. Maybe the concept of “dependent variable”, as Cuzz puts it, is the right term when “x” is not meant to represent a specific object, but rather to let you speak about rules that convert other from a given category. I do not know how formalised that notation approach is, but in effect is what people do in most of the cases you’re complaining about.

I don't think that's it. I differentiate 'specific-value' from 'bound value' in my thinking, so if f(x) = ax + c and f(2) = 4 and f(5) = 42, then a and c are specific but unkown, and x is bound, i.e., it's true for all x in R or whatever set we're working in. That doesn't make 'the graph of f(x)' any better.

[pacovf]

I don’t like “the graph of f(x)” either, but in most cases it’s clear what they mean. It’s the same as when someone says “the graph of x^2”, for example, which is also iffy notation but saves a lot of words / formulas.

I find the other example you bring up much more clear cut though. While ideally we would write f=O(g), the truth is that very often g ends up being something like x^2 or n*ln(n) or whatever. I find it nicer to write f(x) = O(x^2) than f = O(x^2), although in both cases the intent of the writer is completely clear. Specifying the variable is even better when you have functions of two or more variables, as Cuzz brought up earlier.

Ultimately, when enough people use a given shorthand, then you just end up having to learn it. This is not the worst case of notation abuse by a long shot. However, I would expect the writer to become more rigorous for contexts in which the distinction between the function and its output is not obvious, such as if you have functions of functions or whatnot.