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I know personally made notations are generally a bad thing, but I've not seen any reason to stop using the notation I've made, and it feels more natural to use. Now, this my seem like a biased question to begin with, but my hoping was that there would be a natural reasoning why we don't already use notation like this to begin with.

Now let me get to the point: angled brackets for parameter: $\langle*\rangle$. I can't seem to understand the point of using $(*)$ to denote a parameter, when this can too easily be misunderstood with multiplication. To show an example of how this notation makes things clearer, let me demonstrate with the classical notation of $\ln(x+y)^2$. With classical notation, it's hard to distinguish this from either $\ln((x+y)^2)$ and $\ln(x+y)\times\ln(x+y)$. Using $\langle*\rangle$ for the parameter, would make this simple to distinguish, as the latter would be written $\ln\langle x+y\rangle^2$. The first would be written as $\ln\langle(x+y)^2\rangle$, or simply $\ln(x+y)^2$, as the angled brackets would be redundant. Some other examples:

$$ \begin{align} \text{Normal notation:} \\ \sin(x)^2 &\stackrel{?}{=} \sin x^2 \textbf{ or } \sin x\times\sin x = \sin^2x\\[1em] \text{My notation:} \\ \sin(x)^2 &= \sin x^2 \text{, while }\\ \sin\langle x\rangle^2 &= \sin x \times\sin x = \sin^2 x \\[2em] \text{Normal notation:} \\ f(a+b) &\stackrel{?}{=} f\times a+f\times b\textbf{ or } \text{a function of }a+b\\[1em] \text{My notation:} \\ f(a+b) &= f\times a+f\times b \text{, while }\\ f\langle a+b\rangle &= \text{a function of }a+b \end{align} $$

This notation also allows for one to postfix the function, as $\langle *\rangle$ would always be the parameter of some function $f$.

$$ \begin{align} f\big\langle g \langle x \rangle \big\rangle = \big\langle\langle x \rangle g\big\rangle f \end{align} $$

This allows $n!$ to be written as $!\langle n\rangle$, or $|x|$ as $||\langle x\rangle$, so functions and parameters can easily be distinguished. Now I figure that if this notation was useful, we'd use it already. Yet I haven't found any drawbacks by using this notation, as the angled bracked $\langle \rangle$ are somewhat uncommon in mathematics, yet are really easy to draw and distinguish from normal brackets $( )$.

So the question really is; why do we use the same brackets for multiplication as we do for parameters, and would it be beneficial to change this notation?

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    $\begingroup$ If for no other reason: tradition. Would it be beneficial to change? No, it would cause too much upheaval. The current notation just isn't bad enough to justify the disruption a change like that would cause, even if it would otherwise have some minor benefits. You need a huge benefit to justify that sort of change. $\endgroup$ Commented Oct 10, 2014 at 12:29
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    $\begingroup$ This new notation reminds me of some mathematics-oriented programming languages, namely Wolfram language (the one used in Wolfram Mathematica). There f(x) is equivalent to f*x while f[x] is equivalent to f@x, i.e. f is a function applied to x. $\endgroup$ Commented Oct 10, 2014 at 12:34
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    $\begingroup$ Or inner products. Notation is overloaded all the time. $\endgroup$ Commented Oct 10, 2014 at 12:39
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    $\begingroup$ Actually one should not write $\ln(x+y)^2$ to mean $\ln(x+y)\cdot\ln(x+y)$. One should write $\ln^2(x+y)$ or $(\ln(x+y))^2$ and all the problems are solved. $\endgroup$ Commented Oct 10, 2014 at 12:42
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    $\begingroup$ No, $\ln^2(x)=\ln(x)\cdot\ln(x)$, while $\ln(\ln(x))$ means $\ln(\ln(x))$ $\endgroup$ Commented Oct 10, 2014 at 12:48

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Okay I think I've got enough responses to answer this question myself. I might add more to this list if more things come up, or modify it if what is written is incorrect. I've tried not to be too biased in this list, although I must admit that I'll start using this notation.

Pros

  • Already in use

    The Wolfram language already uses notation like this to distinguish multiplication $f(x)$ from applying a function $f[x]$, although this uses square brackets $[]$ instead of angled brackets $\langle\rangle$. If programming languages finds it sensible to use similar notation, it could work in mathematics too.

  • Information is added, not removed

    If you've already written a complicated expression like $\exp(h(f(a+b)^2-f(a-c)))$ where $h$ is a constant and $f$ is a function and $a,b,c$ are parameters, this can be difficult to read. Writing $\exp\langle h(f\langle a+b\rangle^2-f\langle a-c\rangle)\rangle$ might make it clear what's parameters, and what's to be multiplied.

  • Easy to learn

    This would be a lot easier to learn than notations like $\binom{n}{k}$ or $n!$, because the notation is already similar to the one in use.

Cons

  • Existing functions:

    There exist functions using $\langle \rangle$ notation, such as $\langle u, \phi\rangle$, and these could be difficult to incorporate in this new system, because there would be no symbol for the function. Although the lack of a symbol could be seen as a symbol.

  • Speed of typing

    It would be slightly more inconvenient to use $\langle \rangle$ instead of $()$ as it would require one to write more symbols, since "\langle" is longer than "(", although this problem would only occur when writing $\LaTeX$, not on paper, and could be incorporated as a keyboard shortcut.

  • Traditions

    It would require some learning to be accustomed to this new notation, and would not be worth the hassle to teach people. Its benefits wouldn't outweigh its disadvantages.

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I never ever write $\sin x^2$ when I want to write $\sin x \cdot \sin x$ - absolutely don't do this - as it will definitely cause massive confusion. Normally angular brackets are used to define the component form of a vector, for example $$\vec{F}(x,y,z)=\langle P,Q,R \rangle \equiv P\vec{i}+Q\vec{j}+R\vec{k}$$ But as long as you state what you mean in the text, it should be ok.

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  • $\begingroup$ You realize that this question is 4 years old? I don't think OP is still waiting for a second answer. $\endgroup$ Commented Mar 7, 2019 at 13:12
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    $\begingroup$ A late answer is better than no answer. QED. $\endgroup$ Commented Mar 7, 2019 at 14:20

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