Bad but interesting mathematical notation idea

Do you think mathematical innovation is dominantly enabled by clever design?

>using and manipulating tensors [without summing over repeated indices] is extremely painful

It's a good notation, but writing a few sigmas in front of every equation is at worst a constant-factor overhead. I know of a text that does it that way in a (misguided?) attempt to be more approachable to nonspecialists.

I often see Einstein notation as APL vs C :)

I'm sorry to nitpick, but, for some reason, Tao has one of those names that just begs for inadvertent misspellings, and it seems like a shame. His first name is "Terence".

Very interesting read. Nowhere on his desiderata does "uniqueness" come in. The rest of his examples show that it is indeed not a desirable quality, as the same abstract object benefits from different representations to make apparent its relation to other abstract concepts in each specific context. Or simply to reduce cognitive overhead.

> The desiderata listed above are primarily concerned with lowering the "recurring costs", but the "one-time costs" are also a significant consideration if one is only using the mathematics from the given field X on a casual basis rather than a full-time one.

I think this is important and applies well to programming languages. Many programming techniques and styles can be extremely powerful when continuously working in the area whereas just jumping in they can be totally maddening. Sometimes this sort of thing builds up a cognitive wall to newcomers. It has its uses, but it's frustrating when you're one of those casual users having to deal with the clever ideas in the codebase.

> (Unambiguity) Every well-formed expression in the notation should have a unique mathematical interpretation in 𝑋

This is never the case in physics because the equality sign "=" is loaded and can mean equality, definition, proportionality, identity and equivalence. Physicists read the equality sign case by case, as the case may be, with one of these meanings. In scholasticism this used to be called casuistry.

My initial reaction was akin to one of the top level comments: These aren't the same, what a silly question!

But as I read through the answers with people simply entertaining it (in a very literal manner), I found myself really questioning it as well. It's odd how different disciplines approach seriousness in their ways.

Alas, different stack exchange communities have different cultural mores. The Japanese language stack exchange. for example, tends towards a certain ideological purity that the Latin stack exchange does not. Some communities tend to have the early members treat the communities as their own personal fiefdoms.

“That doesn’t work … but I like that one part.” Seems to be the history of invention.

It's a bit like trying to prove something: Maybe your approach fails, but you can still gain a lot of insight by contemplating why it failed.

> It's quite literally the symbols doing some of the thinking for you, specifically the symbols are doing the computing or calculating part of thinking.

I was feeling bad earlier this week when thinking about taking the derivative of x^3. I, of course, didn't derive the answer geometrically. I did what most everyone does, I imagined the "3" symbol floating down and being replaced by n-1.

But there is no reason to feel bad! I was outsourcing the computation to the symbols. The notation is good enough that some easily-remembered symbolic rules are just as good as (or actually equivalent to) the real computation.

Note: XOR and AND are the obviously correct operations to use here, since they are the field operations over Z mod 2 :)

pretty sure that's because log is not well defined over the complex numbers, it's just defined over branch cuts.

similar to how

    sqrt(x^2) != (sqrt((x)^2)

when x is Real but not Positive Real, because sqrt is not well defined over the full Reals.

It's not a flaw of notation, it's a flaw in the attempted math.

Bad/Confusing/Complex notation can absolutely ruin a learner's attempt to grok a piece of math, it's not a trivial issue at all, and it comes up in all levels of math. But good notation doesn't necessarily mean easy understanding. It's just like language. If I speak a different language, forget about communication in any deep way. But even if I'm speaking the same language, it's still not guaranteed we can communicate well or at all. Absence of shared language guarantees the impossibility of communication, but it's presence guarantees nothing.

The author is correct about one thing : Notation is incredibly under-appreciated and under-discussed by mathematicians, although it has immense power to shape thinking. Some visionaries like Charles Babbage, Kenneth Iverson or Stephan Wolfram might talk about it every 50 year or so, but that's it. It's mind boggling how the primary tool of communication is just developed ad-hoc on a whim with only minimal explanation and formalization.

>But is there some book or dictionary that lays this out in a novel way

What do you mean by "this"? is it specifically the overbars-and-underbars notation used in the article? as far as I know, that's the first time I encountered it. But the idea that exponentials, logarithms and roots all basically say the same thing and that notations should reflect that is discussed a lot, 3blue1brown has a very well known video that explains it in a visual way.

https://youtu.be/sULa9Lc4pck

Math notation, and there can be several is like a foreign language. You can only to learn it by using it a lot in practice. You can't just read a book that tells you how to speak a foreign language and that would be it.

You have to learn and understand the symbols, and then you have to learn and understand what is being said using those symbols. You need to learn the letters and the words made with them and then sentences made out of words. It's a lot to learn. But practice makes perfect.

Don't think it has much to do with the notation. Take the integral sign for example. What are we talking about Riemann, Riemann–Stieltjes, Lebesgue, some weird stochastic integral? It's the same symbol. You won't "understand" it better by using a different symbol. Even the Riemann integral, to rigorously "understand" it requires lots of concepts from analysis like least upper bounds (infimum, supremum), partitions, etc. The notation is hardly the stumbling block here, there's no way of getting around understanding the details if you really truly want to "get" it. But again, in practise, when you're doing integration by hand, you've learned a few tricks that work and you hardly ever think about it at the level of analysis.

It's nice to be reminded that not everyone has good ideas all the time, not even Feynman.

I think that's in "Surely You're Joking, Mr Feynman". As I recall, he gave up on it because no-one else could understand him - notation is both a tool of thought and of communication with others. A similar idea is Abelson's (I think) remark about writing code for people and only incidentally for machines.

> a number system in base e (so 1, 2, 2.1, ..., 2.7, 2.71, ..., 10)

I don't think I understand the significance of those dots. How would one write, say, 100 in this number system?

With that said, the usual "base" notation works perfectly well for non-integer radices, though it can behave in unexpected fashion. Thus, I might write 12012 for e^4 + 2e^3 + e + 2, which is approximately 99.49, and so can be regarded as the "e-adically integer" part of 100.

Except that's not how bases work. The digits would be of the form e^n. See [1]. Since e is not an integer, its number system uses 3 symbols (0, 1, 2) as 3 is bigger than e. Thus, 20 = 2e^1 + 0e^0 = 5.43656...

I don't know about simplify. Subtracting 1 from the above would yield an irrational number. So certainly not useful for arithmetic. How about for mathematical proofs? Well, any integer above 3 is also irrational. Makes Taylor series annoying. Even mundane things like the factorial would not have a nice representation.

[1] - https://en.wikipedia.org/wiki/Non-integer_base_of_numeration