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This isn't entirely true. Numerous mathematicians have publicly bemoaned the difficulty of communication between mathematicians in different specialties. So it'
by john_b 14y ago
This isn't entirely true. Numerous mathematicians have publicly bemoaned the difficulty of communication between mathematicians in different specialties. So it's definitely not something that is ignored.
The question is what to do about it. We're not just talking about confusion arising from different notation between mathematical specialties (resolving that would be as easy as defining your notation in an appendix), but different, equally valid ways of mathematical thinking.
In some ways, it may be better to think of different mathematical specialties as different programming languages. Proficiency in one will help you, but won't guarantee that you can interpret another. Except that in mathematics, the differences are more extreme. If you have two Turing-complete programming languages, then you have two different tools that can solve the same class of problems. But different fields in mathematics deal with entirely different mathematical objects which require a conceptual instead of notational leap on the part of the reader. It's not simply a matter of figuring out how to write for loops or manipulate strings in the new language. You actually have entirely different ideas in each, and trying to impose some common notational standard among them is fraught with problems.
- Mgccl 14y agoI second this. In programming languages, different paradigms differs as much as how different field in math. So a more concrete analogy would be one jumping from different paradigms of programming languages. Like someone using a imperative language like Java for their entire life and must learn to read a Haskell program.
- skybrian 14y agoFair enough. But even though there's no hope of coming up with a single programming language for everything, we do have quite good tools for popular areas. It seems like being able to mechanize the error-checking of proofs in certain of the more useful and popular subfields of math might be just what's needed by non-mathematicians?
- tikhonj 14y agoWell, we do have proof assistants. I think people are starting to use those in more diverse fields of math now--I remember seeing an article about an algebraic proof in Coq, but I don't remember any of the details.
- bcoates 14y agoI agree with you and the OP that creating a standard notation for all math is problematic. And you're right that there are conceptual difficulties behind the math that are not going to be automatically resolved by clearer communication. But a lot of the difficulties are accidents of culture. To continue the programming language/mathematical notation allegory, you can use whatever notation you want to describe your program in a Lisp with macros. But you have to actually define what your notation means. Neglecting to provide the definition of the macros would be the equivalent of a logical argument that leaves off its premises or the state of math publications today from the perspective of the poor sap that has to read them (and the mind of the person who wrote it).