> This isn't just hypothetical either, I've been in enough math classes to see the students of ordinary intelligence struggle with this.
I honestly have not seen that. Maybe some minor confusion between sin and asin maybe, but most of the time it's clear what it's meant.
> I consider using invisible operators to be generally unwise. I wouldn't consider adding ambiguity to save a key or pen stroke a wise trade-off. I'm aware that many mathematicians do, and all I can say to that is that it baffles me. In humility I'm willing to allow that they know something I don't, so perhaps my bafflement is a personal defect. Even so I don't think I want to repair it. In my own work I appreciate the clarity too much.
I have seen tons of times authors omitting notation to make it less cumbersome. It's usually preceded by something like "we omit X for brevity/simplicity in the following". The reason is that symbolic notation exists for density of information and focus. When clarity, details and specifics are required, mathematicians use text.
> LaTeX and other comparable typesetting software adequately solve for this.
Funnily enough, they also solve for sin() and sin if you use \sin (or \mathrm{sin}).
> Reading speed is by chunk and not character count. I challenge the notion that f(x+y) is faster to read than f.(x+y)
The dot is short enough to not change things too much, but compare "xy + yz + zy + xyz" to "xy + yz + zy + xy*z". And this happens a lot, because often you want only symbols for things that matter and remove the redundant things. For example, if you're doing calculus you'll often write down the arguments for the functions, but in differential equations you'll omit them because they're not really important.
> Nevertheless, I continue to agree that learning concepts is the more challenging and interesting part of mathematics. I also welcome improvements in clarifying concepts. Sadly, making a complicated concept easier to understand is a much greater challenge than making an ambiguous and muddled syntax unambiguous and clear. My preference is that we pursue both, because they're complementary.
Yes, but my point is that while notation can sometimes be improved, the relation effort/gains is usually small. For starters, notation is not the hardest things one faces when learning mathematics. Then, you have the issue of improvements in one aspect of notation causing problems in other aspects because the set of symbols we have is limited (for example, dot is used as the dot product in vector spaces too). And of course, the problem of changing notation that is already written. Sometimes the gains are worth the effort, such as the ceiling/floor notation of Iverson (and the bracket, although I don't think it's as standard). But that's reasoning mathematicians use for/against notation changes. It's not because having difficult notation is enjoyable or because it acts as gatekeeping.
I honestly have not seen that. Maybe some minor confusion between sin and asin maybe, but most of the time it's clear what it's meant.
> I consider using invisible operators to be generally unwise. I wouldn't consider adding ambiguity to save a key or pen stroke a wise trade-off. I'm aware that many mathematicians do, and all I can say to that is that it baffles me. In humility I'm willing to allow that they know something I don't, so perhaps my bafflement is a personal defect. Even so I don't think I want to repair it. In my own work I appreciate the clarity too much.
I have seen tons of times authors omitting notation to make it less cumbersome. It's usually preceded by something like "we omit X for brevity/simplicity in the following". The reason is that symbolic notation exists for density of information and focus. When clarity, details and specifics are required, mathematicians use text.
> LaTeX and other comparable typesetting software adequately solve for this.
Funnily enough, they also solve for sin() and sin if you use \sin (or \mathrm{sin}).
> Reading speed is by chunk and not character count. I challenge the notion that f(x+y) is faster to read than f.(x+y)
The dot is short enough to not change things too much, but compare "xy + yz + zy + xyz" to "xy + yz + zy + xy*z". And this happens a lot, because often you want only symbols for things that matter and remove the redundant things. For example, if you're doing calculus you'll often write down the arguments for the functions, but in differential equations you'll omit them because they're not really important.
> Nevertheless, I continue to agree that learning concepts is the more challenging and interesting part of mathematics. I also welcome improvements in clarifying concepts. Sadly, making a complicated concept easier to understand is a much greater challenge than making an ambiguous and muddled syntax unambiguous and clear. My preference is that we pursue both, because they're complementary.
Yes, but my point is that while notation can sometimes be improved, the relation effort/gains is usually small. For starters, notation is not the hardest things one faces when learning mathematics. Then, you have the issue of improvements in one aspect of notation causing problems in other aspects because the set of symbols we have is limited (for example, dot is used as the dot product in vector spaces too). And of course, the problem of changing notation that is already written. Sometimes the gains are worth the effort, such as the ceiling/floor notation of Iverson (and the bracket, although I don't think it's as standard). But that's reasoning mathematicians use for/against notation changes. It's not because having difficult notation is enjoyable or because it acts as gatekeeping.