The corresponding row vector is denoted by x^T when we need to distinguish them. We can also ignore the transpose for readability, if the shape is clear from context.
I am tilting at windmills, but I am continually annoyed at the sloppiness of mathematicians in writing. Fine, you don’t like verbosity, but for didactic purposes, please do not assume the reader is equipped to know that variable x actually implies variable y.
All that being said, the writing style from the first chapter is very encouraging at how approachable this will be.
I am tilting at windmills, but I am continually annoyed at the sloppiness of mathematicians in writing. Fine, you don’t like verbosity, but for didactic purposes, please do not assume the reader is equipped to know that variable x actually implies variable y.
I am a practicing mathematician who felt the same way you did when I started, and who still writes their papers in a way that many of my colleagues feel is gallingly pedantic. With that as my credentials, I hope I may say that it can be much worse as a reader to read something where every detail is spelled out, because a bit of syntactic sugar begins to seem as important as the heart of an argument. Where the dividing line is between precision and obfuscation depends on the reader, and so inevitably will leave some readers on the wrong side, but a trade-off does have to be made somewhere.
I wish the formality would be included in an appendix — as someone who has had to implement a lot of things (and more than once, found errors).
But I agree with your general point: understanding the recipe and general thrust of the approach is often more important, because even if the exact proof misses some technical detail, that can often be patched.
It is weird to be honest. I first learned Coq and then started taking upper level maths classes. My group theory proofs were panned by my TAs as overly verbose, very precise, and I was specializing on H_1 and H_2s everywhere and having IHns flying around like crazy because I could not fathom how one proves things without formally connecting things up.
Then my profs told me I was not “wrong”, but proofs or expositions are to most mathematicians not programs (ha! How did I not know. You teach me natural deduction and expect me not to program?), more like convincing arguments/prose. At some point one abstracts.
Humans, even talented mathematicians, have limited context. A big part of any mathematics text is abstraction for the sake of understanding. It can be confusing at first, but once you learn how to read mathematics with these abstractions in place, reading everything spelled out with great verbocity and pedantic accuracy is frustrating and tiring. You're forcing your eyes to parse and interpret a whole bunch of symbols that your brain doesn't need.
Of course, some mathematicians take it too far and use these abstractions to obfuscate and prove how smart they are. Like everything, it's a balance.
I personally wasn't a fan of this particular shorthand when I read this book but I got used to it quickly.
Comments
All that being said, the writing style from the first chapter is very encouraging at how approachable this will be.
I am a practicing mathematician who felt the same way you did when I started, and who still writes their papers in a way that many of my colleagues feel is gallingly pedantic. With that as my credentials, I hope I may say that it can be much worse as a reader to read something where every detail is spelled out, because a bit of syntactic sugar begins to seem as important as the heart of an argument. Where the dividing line is between precision and obfuscation depends on the reader, and so inevitably will leave some readers on the wrong side, but a trade-off does have to be made somewhere.
I wish the formality would be included in an appendix — as someone who has had to implement a lot of things (and more than once, found errors).
But I agree with your general point: understanding the recipe and general thrust of the approach is often more important, because even if the exact proof misses some technical detail, that can often be patched.
Indeed. Lamport says that this was part of what inspired his interest in formal proofs: https://mathoverflow.net/questions/35727/community-experienc....
Could there be a compromise where the verbosity is kept but the key points are highlighted, grouped or presented in a different color.
I would certainly appreciate if math papers were more explicit and "hand-holding" but understand why trained mathematicians would find that tedious.
There's no reason except inertia why there couldn't be. Lamport actually proposed a system for this: https://lamport.azurewebsites.net/pubs/lamport-how-to-write.....
It is weird to be honest. I first learned Coq and then started taking upper level maths classes. My group theory proofs were panned by my TAs as overly verbose, very precise, and I was specializing on H_1 and H_2s everywhere and having IHns flying around like crazy because I could not fathom how one proves things without formally connecting things up.
Then my profs told me I was not “wrong”, but proofs or expositions are to most mathematicians not programs (ha! How did I not know. You teach me natural deduction and expect me not to program?), more like convincing arguments/prose. At some point one abstracts.
Humans, even talented mathematicians, have limited context. A big part of any mathematics text is abstraction for the sake of understanding. It can be confusing at first, but once you learn how to read mathematics with these abstractions in place, reading everything spelled out with great verbocity and pedantic accuracy is frustrating and tiring. You're forcing your eyes to parse and interpret a whole bunch of symbols that your brain doesn't need.
Of course, some mathematicians take it too far and use these abstractions to obfuscate and prove how smart they are. Like everything, it's a balance.
I personally wasn't a fan of this particular shorthand when I read this book but I got used to it quickly.