would be up to such things. I should have made it clearer that the area I have found graphical notations difficult to handle is in representing traditional mathematical actions and operations. That’s very unusual. In fact, Euclid used it in his geometry. A fair amount of work has been done on presenting
So these guys pretty much used plain text, or sometimes things structured like Euclid. In fact, I’m told—in a typical tale often heard of authors being ahead of their publishers—that Russell ended up having to get fonts made specially for some of the notation they used. It works like this: It turns out that there are actually very few tweaks that one has to make to the core of mathematical notation to make it unambiguous. write Sin–2x because
Let’s do an example. But on Friday October 29, 1675 he wrote the following on a piece of paper. Well, then one has something like APL—or parts of
We can edit just fine. For example, whereas he used letters to stand for variables, he used astronomical signs to stand for complete expressions: kind of an interesting idea, actually. How to break the cycle of taking on more debt to pay the rates for debt I already have? So here’s how a table of numbers look in Greek notation. Often such fonts end up having letters that are so different in form from ordinary English letters that they become completely unreadable. But in physics it is still often considered excessively abstract, and explicit subscripts are used instead. But not really about what this meant for the notation for the expressions. Because any letter one might use for that symbolic thing could be confused with a piece of the number. But I decided to actually take a look at it. All these traditions are quite old. I think Whitehead and Russell probably win the prize for the most notation-intensive non-machine-generated piece of work that’s ever been done. But here is a Babylonian tablet that relates to the square root of two that uses Babylonian letters to label things. Other challenges included script and Gothic (Fraktur) fonts. much consistency to that across all the different pieces of math that people
He does make use of a few conventions, but if you skip over all prefaces and notes on conventions and go straight to the actual algorithms themselves you will find them quite readable. More often than not notation is clutter free as opposed to words, but as J.M mentioned, sometimes words are just better. What's the verdicts on hub-less circle bicycle wheels? Well, basically one needs a completely rigorous and unambiguous syntax for math. fraction of texts having math in them than I think you’d find on the web today,
First, on screen one can routinely use color. selection | Frequency distribution of symbols |
But anyway, what Leibniz really brought to things was an interest also in mathematics. How to manage a remote team member who appears to not be working their full hours? For integrals, he had been using “omn.”, presumably standing for omnium. And I think there weren’t too many other possible choices. Actually, he said he didn’t think it was a terribly good notation, and he hoped he could think of a better one soon. Well, this fine abstract Babylonian scheme for doing things was almost forgotten for nearly 3000 years. We tried all sorts of other graphical forms. Babbage wrote one of his rather ponderous polemics on the subject in 1821. And it works quite well when formulas are very simple. In the early 1900s mathematical logicians talked quite a bit about different layers in well-formed mathematical expressions: variables inside functions inside predicates inside functions inside connectives inside quantifiers. But actually, there was a more serious conceptual problem with the letters-as-numbers idea: it made it very difficult to invent the concept of symbolic variables—of having some symbolic thing that stands for a number. And I’ve long thought that it would be very nice to be able to use actual special characters for these, rather than combinations of ordinary ASCII characters. We thought very hard about that. You may notice those jaws on the right-hand side of the cell. rev 2020.11.12.37996, Sorry, we no longer support Internet Explorer, Stack Overflow works best with JavaScript enabled, Where developers & technologists share private knowledge with coworkers, Programming & related technical career opportunities, Recruit tech talent & build your employer brand, Reach developers & technologists worldwide. What should the special one look like? I will discuss the extent to which mathematical notation is like ordinary human
Even things like punctuation marks have barely been looked at. Mathematical notation in algorithms. MATLAB is a high-performance language for technical computing. the Nand operation): {f[f[a,a],f[a,a]]==a,f[a,f[b,f[b,b]]]==f[a,a], f[f[a,f[b,c]],f[a,f[b,c]]]==f[f[f[b,b],a],f[f[c,c],a]]}, Note: (a b) is equivalent to Nand[a,b]. OK. I’ve talked a bit about notation that is somehow possible to use in math. One thing that’s very curious is that, almost without exception, only Latin and Greek characters are ever used. And of course those of us who’ve spent some part of our lives designing computer languages definitely care about this phenomenon. This notation is used to define the lower bound of an algorithm. But basically modern stuff was being used. mean? And most mere mortals couldn’t remember them. And the point is that this expression is completely understandable to Mathematica, so you can evaluate it. It has all sorts of other fun features though. Ask a mathematician if they understand. big directions that one can take. But that’s what we wanted to do. The good ones tend to get invented pretty much all at once, normally by just one person. There are about half a million of these
Actually, very strangely “a” is the second most common. might make use of more general structures, and whether human cognitive abilities
A specific challenge that I had recently was to come up with a good symbol for the logic operations Nand, Nor, and Xor. were writing. I mean, I certainly know that if I think in Mathematica, there are concepts that are easy for me to understand in that language, and I’m quite sure they wouldn’t be if I wasn’t operating in that language structure. The Egyptians did have some notation for operations—they used a pair of legs walking forwards for plus, and walking backwards for minus—in a fine hieroglyphic tradition that perhaps we’ll even come back to a bit in future math notation. In fact, Donald Knuth is his multi-volume opus "The Art of Computer Programming," doesn't use any special notation at all for his descriptions of algorithms. And if one traces things back, there seem to be three basic traditions from which essentially all of mathematics as we know it emerged: arithmetic, geometry, and logic. understanding something very close to standard mathematical notation. There are all kinds of problems with this scheme for numbers. And, of course, in those days we’re not talking about TrueType or Type 1 fonts; we’re talking about pieces of lead. What is the best algorithm for overriding GetHashCode? have a special notation. Because languages like Chinese and Japanese have thousands of ideograms. (We’re currently also developing a sans serif font.) Boole had shown around 1850 that one could represent basic propositional logic in mathematical terms. Our buffer memory of five chunks, or whatever, seems to do well at allowing us to parse them. But what I’ve found, at least in many cases, is that there are pictorial or graphical representations that really work much better than any ordinary language-like notation. And not only have there been grammars for language; in the last centuries or so, there have been endless scholarly works on proper language usage and so on. After all, we already know English so we wouldn’t have to learn anything new to talk to Mathematica.