Syntax is about the least of people's worries when they worry about 'mathematical rigor'.
Math is made for people to read and write, and they have different 'domain specific languages' for different parts of math.
Now what would be useful is a tool, perhaps something like a large language model, to automatically translate from the notation used in one area of math into another.
That can't be a fully mechanical procedure (hence the need for something flexible like an LLM), because it's part and parcel of human mathematics to abuse notation here and there in the name of ergonomics.
My personal pet peeve is defining a function like f(x) := x + 3, and then treating f(x) as the name of the function, instead of just f. But really, it's just a harmless abuse of notation when done by some humans to other humans.
While I agree with the point you make, I think that syntax has a huge impact. Maybe not on mathematicians themselves, but I am certain that mathematical syntax has done more to scare people away from the field than the actual mathematical problems themselves.
For example the convention to use the greek alphabet for certain things. This is totally arbitrary and you could have also used emoticons instead (had they existed). But what this means is that the pupil, before tackling the meat of the mathematical problem has to accept that weird looking letter they have never seen for no real reason whatsoever.
And I say that as someone who can fluently read the greek alphabet.
I do not think mixing up 'f(x)' and 'f' is harmless, given mathematics is all about clarity. In any quality text, 'f(x)' denotes the value of 'f' at 'x', and 'f' denotes the function 'f' itself. Also, speaking of notation, I wonder why you used ':=' instead of '=' to define 'f'. There is no computation going on, right?
eru|2 years ago
Math is made for people to read and write, and they have different 'domain specific languages' for different parts of math.
Now what would be useful is a tool, perhaps something like a large language model, to automatically translate from the notation used in one area of math into another.
That can't be a fully mechanical procedure (hence the need for something flexible like an LLM), because it's part and parcel of human mathematics to abuse notation here and there in the name of ergonomics.
My personal pet peeve is defining a function like f(x) := x + 3, and then treating f(x) as the name of the function, instead of just f. But really, it's just a harmless abuse of notation when done by some humans to other humans.
atoav|2 years ago
For example the convention to use the greek alphabet for certain things. This is totally arbitrary and you could have also used emoticons instead (had they existed). But what this means is that the pupil, before tackling the meat of the mathematical problem has to accept that weird looking letter they have never seen for no real reason whatsoever.
And I say that as someone who can fluently read the greek alphabet.
salutis|2 years ago
eigenket|2 years ago
Judging maths on its syntax is like judging a poem or work of literature on its font. It really isn't a central thing.