Here is Jason Dyer’s recent attempt to make a proof of the Fundamental Theorem of Arithmetic that is actually nice to read. He’s looking for feedback! Dyer’s proof
This seems to have gone a bit too far in the other direction, I felt like I was wasting my time reading a whole page for an insight that could have been conveyed in one line.
(I wrote the Fundamental Theorem of Algebra post.)
I don’t necessarily disagree; I wrote the original thing as a proof of concept and wasn’t expecting it to be perfect first time out. However, I do have the extra notion—given my design ideas presume things are being delivered electronically, as seems to be the case with most math papers these days—that one could expand or condense different parts of the proof so the amount of detail would be selectable.
Also, however, regarding the fiddly details, some of the insights conveyable in one line are in the original proof are written as the original whereas some of the denser things that DID need expansion were originally mashed together as one line with cryptic variable names. This can happen with math papers quite often due to the demands on rigor. One proof redesign I was working on but never posted (at least not yet, I should dig it out) was that the set-theory definition of ordered pairs (x,y) actually works; it’s head-slappingly obvious but the demands of mathematics require every small detail accounted for.
Also also, a lot of this is dependent on personal preference. One commenter didn’t like extra emphasis the indents-for-suppositions but I’ve found them extremely helpful in reading proofs.
There’s a link to this article that makes a lot of sense. But I get the overall sense that there is a limit to consistency, expressibility, and ease of reading. If you try to go too far in one factor, you’re going to lose some of the other factors. It boils down to a presentation matter, and it depends on what purpose are you making the presentation.
Sure, but the current status quo is largely a result of chance and mathematical fads over the centuries than any concerted effort to find consistent and good symbols and notation (with Leibniz being an exception to this). There’s really no reason to think we can’t do better than the current state of mathematical notation.
Well, I was hinting at this, but I think you should also consider the idea that form follows function. I think the function of mathematical notation for the sake of mathematics proper is for greater and greater abstraction, which involves ignoring any element not considered necessary or relevant to what is being proposed.
Those of us who are, instead, more interested in practical reason and wish to gain some mileage from the achievements of mathematics, are more likely to adopt notation more similar to programming languages, where we want to express relationships that are more grounded and more concrete.
There isn’t any perfect mathematical notation, only notation that is most efficient for your particular usage. Like everything else, finding “good notation” is an economics problem.
Here is Jason Dyer’s recent attempt to make a proof of the Fundamental Theorem of Arithmetic that is actually nice to read. He’s looking for feedback! Dyer’s proof
This seems to have gone a bit too far in the other direction, I felt like I was wasting my time reading a whole page for an insight that could have been conveyed in one line.
(I wrote the Fundamental Theorem of Algebra post.)
I don’t necessarily disagree; I wrote the original thing as a proof of concept and wasn’t expecting it to be perfect first time out. However, I do have the extra notion—given my design ideas presume things are being delivered electronically, as seems to be the case with most math papers these days—that one could expand or condense different parts of the proof so the amount of detail would be selectable.
Also, however, regarding the fiddly details, some of the insights conveyable in one line are in the original proof are written as the original whereas some of the denser things that DID need expansion were originally mashed together as one line with cryptic variable names. This can happen with math papers quite often due to the demands on rigor. One proof redesign I was working on but never posted (at least not yet, I should dig it out) was that the set-theory definition of ordered pairs (x,y) actually works; it’s head-slappingly obvious but the demands of mathematics require every small detail accounted for.
Also also, a lot of this is dependent on personal preference. One commenter didn’t like extra emphasis the indents-for-suppositions but I’ve found them extremely helpful in reading proofs.
There’s a link to this article that makes a lot of sense. But I get the overall sense that there is a limit to consistency, expressibility, and ease of reading. If you try to go too far in one factor, you’re going to lose some of the other factors. It boils down to a presentation matter, and it depends on what purpose are you making the presentation.
Sure, but the current status quo is largely a result of chance and mathematical fads over the centuries than any concerted effort to find consistent and good symbols and notation (with Leibniz being an exception to this). There’s really no reason to think we can’t do better than the current state of mathematical notation.
Well, I was hinting at this, but I think you should also consider the idea that form follows function. I think the function of mathematical notation for the sake of mathematics proper is for greater and greater abstraction, which involves ignoring any element not considered necessary or relevant to what is being proposed.
Those of us who are, instead, more interested in practical reason and wish to gain some mileage from the achievements of mathematics, are more likely to adopt notation more similar to programming languages, where we want to express relationships that are more grounded and more concrete.
There isn’t any perfect mathematical notation, only notation that is most efficient for your particular usage. Like everything else, finding “good notation” is an economics problem.
Quite good.
This is great. I want more.
That’s actually quite nice.