|WikiProject Mathematics||(Rated Start-class, Mid-priority)|
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question from Revolver
I don't understand the last section (Isaac Newton?). All these are very useful AIDS in learning mathematics, but does anyone really think visual pictures or tactile or auditory data are really going to become useful NOTATION for precisely expressing thoughts? Revolver 20:40, 15 Nov 2004 (UTC)
- Mouse clicks on a web page have helped the User interface.
- Venn diagrams help to visualize logic statements
Why shouldn't other sensory records and reactions help in notation? What do you think that the marks on clay tablets were? Although some mathematicians, like Galois and Ulam actually did everything in their heads before committing to paper, other mathematicians found writing, internet, letters, etc. to be useful in propagating their discoveries. Ancheta Wis 21:13, 15 Nov 2004 (UTC)
- Obviously, you didn't read what I wrote. I said they are useful AIDS in discovering and learning math, but there is a difference between AIDS and strict notation itself. "Mouse clicks on a web page are not 'notation'". Venn diagrams help visualise logical statements and statements about sets, but they are NOT notation, unless they are given a precise definition which can be codified. This is true, e.g. in some areas of graph theory and commutative diagrams. Look, I'm not telling you not to use these things, I'm just saying strictly speaking, they're not notation. They're other kinds of aids. So, just keep the distinction clear. Revolver 21:38, 15 Nov 2004 (UTC)
- I would add that informal mathematical discourse (writing, letters), does not preclude the use of formal notation. And not using other nonformalised notation doesn't mean one is left with only "doing everything is one's head" (???) You don't seem to realise that a lot of formal notation is used in INformal discourse. Revolver 21:41, 15 Nov 2004 (UTC)
It looks like an example is in order. I am trying to illustrate a thinking process in the style of a visualizer. The closest example I can come up with is from the Green's theorem article:
- Given P and Q, where
Now we need to imagine functions T and U such that : and
What we need instead of the integrals of T and U or the concrete partial derivatives are 2 sets of mountain ranges - a visualization of the integrals of T and U, which are the summations over a domain which is plane A, with altitudes = the values of the sums of T and U over the area A. Then the straightforward readout of the altitudes along a contour D, summed over the contour projected on A is the value of the integral. It's very concrete this way, and the notation in the Green's theorem article overwhelms the basic idea of a fluid set of T and U (Newton's fluents). Now I admit that the concreteness of the example is probably not in the spirit of a formalism, but Newton did not use the notation that we were trained in, and obviously did not think of things the way we have been trained in. Once we have the conditions
then our imaginations need to find T and U. What we need in a notation is for it to help us transform one thing into a related thing which we can solve. Now we can either work it all out with individual cases, laboriously, where a text-based notation might not help us, until we have translated it into a standard notation, or we might build up a toolbox of models like the mountain ranges (or definite integrals) to help us solve the problems. Our notations could be more visual. The flow fields of a weather map and the colors of a doppler radar map could be used a lot more. What else might we learn with such added notation? Ancheta Wis 23:56, 15 Nov 2004 (UTC)
- You're missing my point. I don't disagree that such things are useful and helpful. I'm not saying don't use them!! All I'm saying is, those things are not notation, they're visualisation of data, doodling, scribbling, imagining, visualising...all very important, but they don't belong here really because they're not notation. By definition, notation is formal. What you're talking about it GREAT, just don't call it "notation". Revolver 10:22, 17 Nov 2004 (UTC)
- Newton did not use our modern notation, true. But, he did mean for the notation he used to have a definite meaning. Of course, if you can develop any of these visualisation processes into formalised language, fine. But I think a clear line should be drawn between formal notation and visualisation processes. They are BOTH useful and complementary; they are NOT the same. Revolver 10:28, 17 Nov 2004 (UTC)
- So the divide we are speaking across is that between Formalism (you) and Constructivism (me). Clearly there needs to be notation which bespeaks the rules and constraints gathered from the centuries of mathematics and the category theory that refers to them. And I bow to all of you who have contributed to this noble subject. Ancheta Wis 15:54, 17 Nov 2004 (UTC) The good thing is that this lays out a program for the improvement of this article.
If I understand you, what you have in mind for the article is a discussion of the evolution of formal systems, the formal grammar, etc. with a set of requirements for a well-formed set of expressions etc, and its impact on notation. Ancheta Wis 20:00, 17 Nov 2004 (UTC)
- I don't think I mean to be that restrictive. And I don't think our difference of opinion is based on a distinction between formalism and constructivism. Informal mathematical discourse (which is basically all mathematical discourse) incorporates formal notation...very little actual mathematics is 'formal' in the strict sense of the word. By "notation" I just mean, a symbol or symbolic expression intended to have a precise semantic meaning. (I say, "intended", because there are always philosophical questions about certain things.) In this sense, I don't consider, say, Venn diagrams to be notation, because they aren't intended to have a precise meaning, there's simply visualisation of certain things about sets. The same goes for the various physical ways of getting a hold on Green's theorem (or Stoke's theorem, generally) through visualising flux and vector fields. These are more pictures intended to evoke a concept, idea, or thought process. But drawing a picture of a ball with flux arrows and drawing it chopped up into pieces as a way to illustrate Stoke's theorem is not "notation" to me, because it's not intended to have a precise meaning. Confusing the two can be deadly, e.g. when the picture leads you to a false conclusion. I certainly think these things are useful, I think any math person is lying to say they don't use tools like this all the time, but they're not what I would call notation. Maybe this is just a disagreement over terms. Revolver 22:41, 19 Nov 2004 (UTC)
- Just to be doubly clear — by formal, I do not mean formal in the sense of the formalist school of philosophy, I mean formal in the sense of symbolic expressions intended to have precise meaning, regardless of whether this occurs in the context of a strict formal system. Revolver 22:45, 19 Nov 2004 (UTC)
- Maybe we just need a better example of a kind of visualization which is well-defined enough to count as notation? I'm thinking of something like a knot diagram, which is constructed so it identifies one mathematical object unambiguously, and can be transformed according to precise rules (Reidemeister moves). EricDBurgess (talk) 16:59, 5 September 2011 (UTC)
I would have thought that "mathematical notation", by definition, is notation used in mathematics. The discussion at Talk:Mathematics#Tautologous definition? indicates that this may not be so. The article doesn't help here. Brianjd | Why restrict HTML? | 04:03, 2005 Apr 23 (UTC)
- The key idea is consistency of notation - the rules of the grammar, and the parser which detects conformance with the rules. It takes a mathematical POV to utilize the notation consistently. Do you think a sentence should be crafted for the article? Ancheta Wis 08:22, 23 Apr 2005 (UTC)
There should be a mention of TeX because it led to the standardization of mathematics typography.
About evaluating expressions and computers
The current claim under the section Expressions is simply wrong:
- A mathematical expression is a sequence of symbols which can be evaluated. ... In a computer language, these rules are implemented by the compilers.
First of all, what on Earth is a computer language? Without delving further in that, let us assume the author of the piece of text meant programming languages.
The rules of evaluating the value of expressions is contained in the semantics of a programming language. What is then implemented on the computer is these rules, the semantics, using the built-in semantics of, usually, a register machine. What the compiler actually does is translate a program written in programming language A to a program written in programming language B, and often A is a high-level language, such as C or Java, and B is machine language (either the native one or one of a virtual machine). The semantics of the machine language depend on how the computer has been put together and since this is mostly electronics, it's not relevant here.
In any case, saying that a compiler implements the rules of evaluating expressions, i.e. that a compiler implements the semantics is false. The compiler only transforms the program or expression from one form to another, hopefully in a way that is executable by the machine. The machine can then be used to execute the program, and this is where evaluation happens.
(In certain special cases, the compiler can implement some of the rules, for example when you use macros in Lisp or Scheme to do computation at compile time, but this is certainly not the whole picture.) 220.127.116.11 —Preceding unsigned comment added by 18.104.22.168 (talk) 09:36, August 29, 2007 (UTC)
Placement of exponents
Example of differing notation
I'd like to insert an example of how the same notation can mean different things in different contexts. Here are a few possibilities:
- |x| could be an absolute value, a norm, or the cardinality of x
- f 2(x) could be (f(x))2 or f(f(x)).
- sin2(x) and sin-1(x) are very different but have parallel notation.
Is this all?
I found this site looking for a description of some various things like a bold letter, a vector etc. and found nothing? Perhaps this page needs a list of what a differential looks like what the times sign looks like etc... —Preceding unsigned comment added by 22.214.171.124 (talk) 02:31, 6 November 2008 (UTC)