Let’s take a simple example, “i”. Frege decided that to represent what he wanted to represent, he should use a kind of graphical notation. They mostly used base 60—not base 10—which is actually presumably where our hours, minutes, seconds scheme comes from. My opinion: Because most mathematics (the maths you would generally refer to as "maths") is for indicating relations or comparisons or existences. Select[Table[Cyclotomic[i,x],{i,20}],Exponent[#,x]<6&], In[7]:= And at the end of the 1800s and the beginning of the 1900s, when abstract algebra and mathematical logic were really getting going, there was another burst of interest and activity. Most people who actually use math notation have some feeling for that. They mostly look too fragile and not blobby enough to be binary operators. Look at examples, too; every time you look at an example and think "oh, what a good idea! If there is any new notation, what should it be, for example? Kashi also had an algorithm for calculating nth roots. And I think there weren’t too many other possible choices. In Mathematica 3 we went to a lot of trouble to develop fonts for over 1100 characters of relevance to mathematical and technical notation. notation | Proofs | Character Feature Preview: New Review Suspensions Mod UX, Review queue Help Center draft: Triage queue. So that means that there are various obsolete Greek letters left in their number system: like koppa for 90 and sampi for 900. The "gold standard" for rigorous mathematical descriptions of algorithms is (arguably) the book "Introduction to Algorithms," sometimes known as "CLRS" after the initials of each of the four authors, though on occasion I've heard it referred to as just "Cormen" (the primary author.). And you end up with a completely well defined syntax. And even though the field of mathematical logic scaled back a bit from Russell and Whitehead, it’s still the field that has the most complicated notation of any, and the least standardization. (Notably, all ordinary human languages seem to have single words for And or Or, but none have a single word for Nand.) So alpha was 1, beta was 2, and so on. Here’s what we ended up with for the Greek letters. paper showing the notation he made up for his original universal Turing machine, But here’s the surprising fact—that certainly surprised me a lot. So then double-struck “i” is the specific object that we call ImaginaryI. But it is not clear to what extent the kinds of linguistic structure associated with parts of speech in ordinary language are mirrored in mathematical notation. fac 0 = 1 fac n = n * fac (n - 1) is equivalent to the mathematical statements But it does. One can’t generate too much text that actually means very much automatically. Physiologically, I think it works by using nerve impulses that end up not in the ordinary visual cortex, but directly in the brain stem where eye motion is controlled. In the 15th century, Ghiyath al-Kashi computed the value of π to the 16th decimal place. Given any expression, I can always convert it to TraditionalForm. We don’t have any contemporary versions of Euclid. I mean, we might not have found any artifacts from earlier representations. More often than not notation is clutter free as opposed to words, but as J.M mentioned, sometimes words are just better. Well, the idea we had—actually I think I was in the end responsible for it—was to use double-struck characters. Boole had shown around 1850 that one could represent basic propositional logic in mathematical terms. An example is Grimm’s Law, which describes general historical shifts in consonants in Indo-European languages. And as so far as I know, no serious linguistic study of mathematical notation has been made at all in the past. There’s a little book called Mathematics into Type put out by the AMS, but it’s mostly about things like how putting scripts on top of each other requires cutting pieces of paper or film. But for TraditionalForm, it would be good to have some principles. Why does Ukranian "c" correspond English "h"? Here we had a key idea: you make another character, that’s also a lowercase “i” but it’s not an ordinary lowercase “i” and you make that be the “i” that’s the square root of -1. s[x_][y_][z_]:= x[z][y[z]]. But if they get complicated Typical uses include: Math and computation Algorithm development Modeling, simulation, and prototyping Thanks for contributing an answer to Mathematics Stack Exchange! in math notation, and things very quickly started looking quite modern. But computer languages have historically been different. But they do provide relevant reminders. And in traditional science and mathematics there are certainly graphical notations that work just fine, though typically for fairly static constructs. Arithmetic comes from Babylonian times, geometry perhaps from then but certainly from Egyptian times, and logic from Greek times. Here’s an example of that. And the most basic thing for arithmetic is numbers. 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. And one of the surprising things is that so far as I know, there’s almost never been any kind of introspective study done on the structure of mathematical notation. So the things that can be represented that way are the things we tend to study. Well, one has to qualify that a bit. MATLAB is a high-performance language for technical computing. Well, this fine abstract Babylonian scheme for doing things was almost forgotten for nearly 3000 years. There were other people who had thought about similar kinds of things, mostly from the point of view of ordinary human language and of logic. And if you try to give them more, particularly all at once, they just become confused and put off. Well, I was curious what that distribution was like for letters in math. But I at least have never found it at all useful for anything analogous to notation. Traditional mathematical notation represents mathematical objects but not mathematical Because as soon as you try to type fast—or do editing—you just keep on finding that your computer is beeping at you, and refusing to let you do things that seem like you should obviously be able to do. In early mathematical notation—say the 1600s—quite a few ordinary words were mixed in with symbols. This is a fragment of Newton’s manuscript for the Principia With functions, there have also been some trends to reduce the mention of explicit parameters. Part of the issue is that with math one can expect to generate things automatically. In the early to mid-1600s there was kind of revolution But,a little like Leibniz, he wasn’t satisfied with just inventing a universal notation for math. new web site functions.wolfram.com, though as of right now this has not powers of things instead of one over things and so on. And I’d really like to be able to come up with a kind of language representation of networks. But the double struck idea was the best. actually been implemented there. discuss some of the key ideas that made this possible, as well as some features Well, that particular problem I think we solved in a particularly neat way. By using our site, you acknowledge that you have read and understand our Cookie Policy, Privacy Policy, and our Terms of Service. In the study of ordinary natural language there are various empirical historical laws that have been discovered. I don't know where I should look for to learn how to describe my algorithms with a mathematical notation. And actually one of the biggest issues with numbers, I suspect, had to do with a theme that’ll show up many more times: just how much correspondence should there really be between ordinary natural language and mathematical language? Various people asked about searching mathematical formulas. But anyway, he was serious about notation. All these traditions are quite old. When introducing the elements of ring and eld theory, algorithms o er concrete tools, constructive proofs, and a crisp environment where the bene ts of rigour and abstraction become tangible. One could have no special notation. It ends up seeming rather mystical. I mean, why do we have to have operators that are just prefix, or infix, or overfix, or whatever? You see, what’s mostly been studied in linguistics has actually been spoken languages. For a while, at the beginning of the 1900s, there was almost no effect from what had been done in mathematical logic. Mathematical notation: f(n) = Ω(g(n)). Well, this is a tricky idea, and it took thousands of years for people generally to really grock it. I have tried to find clear ways to represent them and their evaluation. Well, I kind of suspected that there wasn’t In summary, I think you should differentiate between conveying an algorithm to your readers, and analyzing an algorithm in some formal sense. Some people asked about differences between it’s not clear what that means. An example is the Theorema project. And But when one looks at what happened beyond that, there’s quite a bit of diversity. But if one wants to be fully compatible with traditional textbooks one needs something different. However, as early as the 1920s, it was pointed out that one could use so-called combinators to specify such data flow, without ever explicitly having to name parameters. They are for the most part described in plain English. If one asks about all possible equivalences things become impossibly difficult, for basic mathematical reasons. By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy. People often have this view that mathematics is somehow the way it is because that’s the only conceivable way it could be. At least to us now, Diophantus’ notation for polynomials looks extremely hard to understand. Well, the idea that operations are even something that has to represented probably took a long time to arrive. I think it’s very much the same kind of issue that comes up with things like short command names. Well, the first representations for numbers that we recognize are probably notches in bones made perhaps 25,000 years ago. So I went through Well, apart from these things, and with a few exceptions like the “square intersection” sign he used for equal, Leibniz pretty much settled on the notation that still gets used today.

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