I agree that $f(z) \circ g(z)$ is not good notation, and so "completely reasonable" was too strong a claim. The point remains, though, that there is nothing wrong with writing $f(x)$ to denote a function, or $\sin(x)/x$ for that matter.

Best wishes,

Matt

I stand by my answer in the case of the particular setup that Anixx proposed, namely a family of functions indexed by an arbitrary set. I completely agree that it is customary, and quite often useful, to write a sequence as a_n, not a:N->R. However, I would point out that it's common, at least in my experience, to use "a" as a name for the sequence (at least when it's not being used for the limit of the sequence), i.e. $a={a_n}_{n\in N}$, which I consider an acceptable middle ground between only ever writing a_n's (and ignoring the sequence as an object itself), and defining "a" to be a function a:N->R.

I will attempt to think of a better way of phrasing my previous post, but I've got to go to sleep for now.

I think that one should be careful making categorical statements of the form "You are too focused on writing the functions with their argments "on display"". Anixx uses a certain system of notation which is completely reasonable, possibly slightly idiosyncratic when considered in one or two of its details, but which has much in common with systems of notation used by many other mathematicians and scientists who use mathematics as a tool. Furthermore, my impression based on Anixx's posting history is that her/his topics of interest are all closely related to classical analysis. I don't see that there is any particular need "to think of functions more categorically" in that context.

Incidentally, how do you denote a sequence? A sequence is a function from the natural numbers to the real numbers (say), and almost everyone I know denotes such a thing $a_n$, not just $a$, i.e. they put the variable $n$ on display in their notation. This is convenient for many purposes, even if it is occasionally bothersome (in that certain expressions and constructions become more convoluted when forced into this notational convention). Furthermore, most people write $a_{n_i}$ for a subsequence, rather than something more functorial such as $a \circ j$ where $j$ is a monotone injection from the natural numbers to itself. The notation $f(x)$ is simiarly traditional in many circles (even if it is not as universal as the sequence notation $a_n$). Like sequence notation, it serves some purposes, and can be bothersome in others.

Finally, I do often denote the identity function from an algebraically closed field (say) to itself by $x$. This is very standard in algebraic geometry, where $x$ stands for both a polynomial variable and the (identity) function on the field of coefficients that it induces. I don't think this reflects a lack of functorial thinking; it is just a particular notation tradition.

Regards,

Matthew

I very much agree with Alexander Woo, in that I think of a function as an object in and of itself, and not as a mere relation between independent and dependent variables.

As to your second comment, it just depends on what I feel like emphasizing. It's obvious that a subscript is just another place to put an argument, i.e. defining the function $f:X->Z$ by $f(x)=B_x(a)$, where {B_x:Y->Z}_{x\in X} is a collection of functions indexed by X and $a\in Y$, is equivalent to defining $f:x->Z$ by $f(x)=B(x,a)$, where $B:X x Y ->Z$ and $a\in Y$, and $B(x,y)=B_x(y)$ where $B_x$ are the functions from before. If for some reason I felt like keeping the setup of an indexed family of functions, it would be perfectly fine to write B_x for a specific x\in X, as that is the name of a specific function. If I converted the indexed family of functions into a big function B (again, they are equivalent), then I would be fine with just talking about B, without requiring myself to write in an x in the first argument of B. However, I expect that people would for the most part understand what I meant if I never explicitly mentioned the change but still wrote B without a subscript.

I am also one of the people who voted in Andrew Stacy's poll, who has no objection to writing $f(x)$ to denote a function.

In the same vein, writing $f(x)/x$ (say) to denote the quotient of one function by another seems perfectly fine to me, in the appropriate context. I can see that there are times when it may be a source of confusion (because of issues of the precise domain), and then one would avoid it, but there are lots of other contexts in which it would be perfectly okay (the most common, for me, being if $f(x)$ was a rational function, or more generally an analytic function, of $x$).

I agree that I wouldn't normally use the composition symbol $\circ$ in the way that Anixx did in his post at the top of this thread, just because I think it looks a little strange, but it's meaning is certainly clear, and I don't see what is so objectionable about it.

Incidentally, I am a pure mathematician.

Let $f,g:\mathbb{C}\rightarrow\mathbb{C}$ be defined by
$$f(x)=\frac{x^3+2x^2+\sin(x)}{x^5-x^3+\cos(x)+5}$$
and
$$g(x)=\frac{x^3-5x+\log(x)}{x^7+3x^2+10}.$$
Consider $f\circ g$.

Giving complicated things names (and using English/your native language to explain them) is a good thing.