Without clever innovations in notation, a lot of math (and physics) would be utterly intractable. For instance, without Einstein's notation hack [1], using and manipulating tensors is extremely painful. Arguably, General Relativity would not have been possible without the clarity of the Einstein notation.<p>Also long-established notation like integrals and differentials were once new and innovative, and paved the way for new discoveries.<p>[1] <a href="https://en.wikipedia.org/wiki/Einstein_notation" rel="nofollow">https://en.wikipedia.org/wiki/Einstein_notation</a><p>(edit: capitalize GR, grammar)
This reminds me of a wonderful mathoverflow answer about desirable properties of math notation, authored by none other than Terrance Tao:<p><a href="https://mathoverflow.net/questions/366070/what-are-the-benefits-of-writing-vector-inner-products-as-langle-u-v-rangle/" rel="nofollow">https://mathoverflow.net/questions/366070/what-are-the-benef...</a>
An aside, but it’s so nice that interesting discussions like the one linked in the article are allowed to bloom on smaller Stack Exchange sites like Mathematics.<p>I can’t imagine a similar sort of rumination surviving on Stack Overflow or Server Fault, but the discussions in that thread are really interesting to read.
> I think mathematicians don't yet understand the power of mathematical notation and what it does. We use it, but we don't understand it<p>I think this is where a computer science (but really, "computologist" mindset) differs from a typical mathematical one; us CS (computer-software) people do understand this power very well, at the least it's why I'm interested in CS (computer 'science').<p>> It's almost as if the symbols are doing some of the thinking for you.<p>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.
Actually, not bad idea. This one reminds me about Zhegalkin polynomial ( <a href="https://en.wikipedia.org/wiki/Zhegalkin_polynomial" rel="nofollow">https://en.wikipedia.org/wiki/Zhegalkin_polynomial</a> ), a way to express all boolean functions with minimally sufficient basis: AND and XOR. Such minimal and invariant constructs have some nice properties useful in some class of applications.
One big issue with that notation is that the log of the exponential is not the exponential of the log, so the order really does matter(for any complex valued expressions).<p><a href="https://www.wolframalpha.com/input?i=log%28exp%28x%2Biy%29%29+-+exp%28log%28x%2Biy%29%29" rel="nofollow">https://www.wolframalpha.com/input?i=log%28exp%28x%2Biy%29%2...</a><p>So just having over and underbars loses some information.
> you can solve algebraic equations or calculus problems just by “pushing around the symbols”. But why can you do that? Where is the meaning, and how do the symbols capture the meaning? How does that work? The fact that symbols in general can somehow convey meaning is a deep philosophical mystery<p>The correspondance between symbols and meaning can and has been studied rigorously — it’s a main theme of Gödel Escher Bach and I’d recommend reading it if you find these kind of questions fascinating, even though it doesn’t have much to say about notation. The basic idea is that mathematical notation is working as a formal system whose semantics correspond to those of arithmetic. By pushing around symbols you’re applying inference rules of the formal system that encode axioms of mathematics.
I wonder how much difficulty to "get" maths have to do with difficulty to grasp notation conventions. Probably not much in the big picture. But is there some book or dictionary that lays this out in a novel way that makes the reader feel they can understand it superficially?
In one of Feynman's books he talks about a notation he tried to introduce but couldn't get to stick. He particularly disliked sin^-1 as arcsine, it looked to home like 1/sin. His idea was to use σ (lower case sigma) for sin with the dash at the top extending over the thing to be sined and a similar symbol with the dash extending backwards for arcsine. It sounded like a great idea to me, and I I would be pushing for it if I ever used trigonometric functions in the first place.
This reminds me of being in secondary school and procrastinating/nerd sniping our teacher for A-level Further Maths (so I was roughly 17-18) by arguing that a number system in base e (so 1, 2, 2.1, ..., 2.7, 2.71, ..., 10) would simplify a lot of the maths we were studying.
Related idea which I don't see mentioned yet: William Bricken's iconic arithmetic, with regards to what he calls James Algebra.<p>I don't have my copy of the material handy but it comes down to using different containers to represent logarithms and powers such that<p>(x) ~> #^x<p>[x] ~> log_#(x)<p><x> ~> -x<p>where # is an arbitrary base.<p>Writing expressions next to each other is implied addition. Whole numbers can be written e.g.<p>0 ~> _<p>1 ~> ()<p>2 ~> ()()<p>3 ~> ()()()<p>etc.<p>Operations, like addition, read<p>A+B ~> A B<p>and subtraction<p>A-B ~> A<B><p>where <<x>> ~> x,<p>on to multiplication<p>A*B ~> ([A][B])<p>and division<p>A/B ~> ([A]<[B]>)<p>and exponentiation<p>A^B ~> (([[A]][B])).<p>There are a few axiomatic equations (maybe 3?) that are used to establish the general properties of the system, and from which the rest of it can then be deduced.<p>It also introduces an interesting construction, which he simply calls J ~> [<()>], analogous to the imaginary number i.<p>I'd recommend taking a look at this if TFA tickled your fancy.
A similar idea was put forth on the math stack exchange [1], which I found through this 3blue1brown video [2]. Worth a read/watch!<p>[1] <a href="https://math.stackexchange.com/questions/30046/alternative-notation-for-exponents-logs-and-roots" rel="nofollow">https://math.stackexchange.com/questions/30046/alternative-n...</a><p>[2] <a href="https://www.youtube.com/watch?v=sULa9Lc4pck" rel="nofollow">https://www.youtube.com/watch?v=sULa9Lc4pck</a>
If you find this interesting I'd recommend Florian Cajori's "A History of Mathematical Notations" (1930; I have the 2 volume combined one reprinted in 1993).
The author's example for x^2 + x could be written with the first two symbols swapped. With this it looks fine to me. Putting the 2 first here is like putting the x first in "2x" such that it becomes "x2". I think also maybe if the lines above and below had curved ends so you could see where they start and end clearly then this could be not so bad notation.
Does the author not know that 0 is a number? 0^3 = 0, does that mean that 3 = log_0(0)? The notation differs for a reason. As far as mathematicians not understanding the power of notation... Yes, we do.