
:''See also the table of mathematical symbols.'' ---- Mathematical notation is used in mathematics, and throughout the physical sciences, engineering, and economics. The complexity of such notation ranges from relatively simple symbol representations, such as 1 (number) and 2 (number); to conceptual symbols, such as addition and derivative; to equations, function (mathematics)s, and variables. ==Definition== *A mathematical notation is a writing system used for recording concepts in mathematics. **The notation uses symbols or symbolic expressions which are intended to have a #Precise semantic meaning. **In the history of mathematics, these symbols have denoted #Counting, #Geometry becomes analytic, #Ideographic notation, and #Counting is mechanized. The notation can also include symbols for parts of the conventional discourse between mathematicians, when viewing mathematics as a language. The media used for writing is recounted below, but common materials currently include paper and pencil, or perhaps computer screen and keyboard, as well as board and chalk. One key point behind mathematical notation is the ''systematic adherence to mathematical concepts'' as recounted below. (But see also some related concepts: Topic (linguistics), Logical argument, Cogency, Mathematical logic#Technical reference Model theory#Definition Mathematics#Major themes in mathematics) ==Expressions== A Expression (mathematics) is a sequence of symbols which can be evaluation. For example, if the symbols represent numbers, the expressions are evaluated according to a conventional order of operations which provides for calculation, if possible, of any expressions within parentheses, followed by any multiplications and divisions done from left to right, finally any additions or subtractions done from left to right. In a computer language, these rules are implemented by the compilers. For more on expression evaluation, see for example, the computer science topics: eager evaluation, lazy evaluation, evaluation operator. #Definition ==Precise semantic meaning== :''See model (abstract)#Abstract models vs models in mathematics'' Precision is necessary so that we can know ''what'' we are investigating. Suppose that we have statements, denotation by some formal sequence of symbols, about some objects (for example, numbers, shapes, patterns). Until the statements can be shown to be valid, their meaning is not yet resolved. While reasoning, we might let the denoted symbols refer to those objects, perhaps in a model (abstract). The semantics of that object has a heuristic side and a deductive side. In either case, we might want to know the properties of that object, which we might then list. Those properties might then be expressed by some well-known and agreed-upon symbols from a table of mathematical symbols. This mathematical notation might include annotation such as *"All x", "No x", "There is an x" (or its equivalent, "Some x"), "A set", "A function" *"A mapping from the real numbers to the complex numbers" *#Definition ==Counting== It is believed that a mathematical notation was first developed at least 50,000 years ago in order to assist with counting. Early mathematical ideas for counting were represented by collections of Rock (geology)s, sticks, bone, clay, Rock (geology), wood carvings, and knotted ropes. The tally stick is a timeless way of counting. Perhaps the oldest known mathematical texts are those of ancient history Sumer. The census quipu of the Andes and the Ishango Bone from Africa both used the tally mark method of accounting for numerical concepts. ==Geometry becomes analytic== The mathematical viewpoints in geometry did not lend themselves well to counting. The natural numbers, their relationship to fraction (mathematics)s, and the identification of continuous quantities actually took millennia to take form, much less allow for the development of notation. It was not until the invention of analytic geometry by René Descartes that geometry became more subject to a numerical notation. However, some symbolic shortcuts for mathematical concepts came to be used in the publication of geometric proofs, for example. The power and authority of the custom of geometrical style of Theorem and Proof was even followed by Isaac Newton's Philosophiae Naturalis Principia Mathematica, though he did not use geometry to invent his concepts, but instead blazed a new trail through the invention of calculus to understand the Philosophiae Naturalis Principia Mathematica. ==Counting is mechanized== After the rise of Boolean algebra and the development of positional notation, it became possible to mechanize simple circuits for counting, first by mechanical means, such as gears and rods, using rotation and translation to represent changes of state (computer science), then by electrical means, using changes in voltage and current (electricity) to represent the analogs of quantity. Today, of course, computers use standard circuits to both store and change quantities, which represent not only numbers, but pictures, sound, motion, and control. ==Computerized notation== The rise of expression evaluators such as calculators and slide rules were only part of what was required to mathematicize civilization. Today, keyboard-based notations are used for the e-mail of mathematical expressions, the Internet shorthand notation. The wide use of programming languages, which teach their users the need for rigor in the statement of a mathematical expression (or else the compiler will not accept the formula) are all contributing toward a more mathematical viewpoint across all walks of life. There is a part of mathematics which is not algebraic, but which seems to use a mathematics#Change of the mind. For those people with such minds and imaginations, like Isaac Newton's, if they are to benefit from the wide availability of mathematical devices, then they will need to be served by more graphical, visual, aural, tactile, and temporal modalities in notation, as a first step. ==Ideographic notation== In the history of writing, ideographic symbols arose first, as more-or-less direct renderings of some concrete item. This has come full circle with the rise of the computer visualization systems, which can be applied to abstract visualizations as well, such as for rendering some projections of a Calabi-Yau manifold. Examples of Information visualization which properly belong to the mathematics#Spatial relations, can be found, for example in computer graphics. The need for such models abounds, for example, when the measures for the subject of study are actually random variables and not really ordinary mathematical functions. ==See also== *Table of mathematical symbols ==Notes== *Florian Cajori, ''A History of Mathematical Notations'' (1929), 2 volumes. ISBN 0486677664 ==External links== * [http://www.cut-the-knot.org/language/index.shtml Mathematics as a Language] * [http://members.aol.com/jeff570/mathsym.html Earliest Uses of Various Mathematical Symbols] ==See also== * Abuse of notation *Begriffsschrift Mathematical notation
Mathematics
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? User: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 etc 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. User: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. User: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. User: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 : and 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? User: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". User: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. User: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. User: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. User: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. User: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. User:Revolver 22:45, 19 Nov 2004 (UTC) == Definition == 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. User:BrianjdUser talk:BrianjdSpecial:Contributions/Brianjd | Talk:HTML#Restricted 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? User:Ancheta Wis 08:22, 23 Apr 2005 (UTC)
See other meanings of words starting from letter:
Words begining with Mathematical_notation:
Mathematical_notation
Mathematical_notation
Mathematical_notation
Mathematical_notation/to_do
Mathematical notation
:''See also the table of mathematical symbols.'' ---- Mathematical notation is used in mathematics, and throughout the physical sciences, engineering, and economics. The complexity of such notation ranges from relatively simple symbol representations, such as 1 (number) and 2 (number); to conceptual symbols, such as addition and derivative; to equations, function (mathematics)s, and variables. ==Definition== *A mathematical notation is a writing system used for recording concepts in mathematics. **The notation uses symbols or symbolic expressions which are intended to have a #Precise semantic meaning. **In the history of mathematics, these symbols have denoted #Counting, #Geometry becomes analytic, #Ideographic notation, and #Counting is mechanized. The notation can also include symbols for parts of the conventional discourse between mathematicians, when viewing mathematics as a language. The media used for writing is recounted below, but common materials currently include paper and pencil, or perhaps computer screen and keyboard, as well as board and chalk. One key point behind mathematical notation is the ''systematic adherence to mathematical concepts'' as recounted below. (But see also some related concepts: Topic (linguistics), Logical argument, Cogency, Mathematical logic#Technical reference Model theory#Definition Mathematics#Major themes in mathematics) ==Expressions== A Expression (mathematics) is a sequence of symbols which can be evaluation. For example, if the symbols represent numbers, the expressions are evaluated according to a conventional order of operations which provides for calculation, if possible, of any expressions within parentheses, followed by any multiplications and divisions done from left to right, finally any additions or subtractions done from left to right. In a computer language, these rules are implemented by the compilers. For more on expression evaluation, see for example, the computer science topics: eager evaluation, lazy evaluation, evaluation operator. #Definition ==Precise semantic meaning== :''See model (abstract)#Abstract models vs models in mathematics'' Precision is necessary so that we can know ''what'' we are investigating. Suppose that we have statements, denotation by some formal sequence of symbols, about some objects (for example, numbers, shapes, patterns). Until the statements can be shown to be valid, their meaning is not yet resolved. While reasoning, we might let the denoted symbols refer to those objects, perhaps in a model (abstract). The semantics of that object has a heuristic side and a deductive side. In either case, we might want to know the properties of that object, which we might then list. Those properties might then be expressed by some well-known and agreed-upon symbols from a table of mathematical symbols. This mathematical notation might include annotation such as *"All x", "No x", "There is an x" (or its equivalent, "Some x"), "A set", "A function" *"A mapping from the real numbers to the complex numbers" *#Definition ==Counting== It is believed that a mathematical notation was first developed at least 50,000 years ago in order to assist with counting. Early mathematical ideas for counting were represented by collections of Rock (geology)s, sticks, bone, clay, Rock (geology), wood carvings, and knotted ropes. The tally stick is a timeless way of counting. Perhaps the oldest known mathematical texts are those of ancient history Sumer. The census quipu of the Andes and the Ishango Bone from Africa both used the tally mark method of accounting for numerical concepts. ==Geometry becomes analytic== The mathematical viewpoints in geometry did not lend themselves well to counting. The natural numbers, their relationship to fraction (mathematics)s, and the identification of continuous quantities actually took millennia to take form, much less allow for the development of notation. It was not until the invention of analytic geometry by René Descartes that geometry became more subject to a numerical notation. However, some symbolic shortcuts for mathematical concepts came to be used in the publication of geometric proofs, for example. The power and authority of the custom of geometrical style of Theorem and Proof was even followed by Isaac Newton's Philosophiae Naturalis Principia Mathematica, though he did not use geometry to invent his concepts, but instead blazed a new trail through the invention of calculus to understand the Philosophiae Naturalis Principia Mathematica. ==Counting is mechanized== After the rise of Boolean algebra and the development of positional notation, it became possible to mechanize simple circuits for counting, first by mechanical means, such as gears and rods, using rotation and translation to represent changes of state (computer science), then by electrical means, using changes in voltage and current (electricity) to represent the analogs of quantity. Today, of course, computers use standard circuits to both store and change quantities, which represent not only numbers, but pictures, sound, motion, and control. ==Computerized notation== The rise of expression evaluators such as calculators and slide rules were only part of what was required to mathematicize civilization. Today, keyboard-based notations are used for the e-mail of mathematical expressions, the Internet shorthand notation. The wide use of programming languages, which teach their users the need for rigor in the statement of a mathematical expression (or else the compiler will not accept the formula) are all contributing toward a more mathematical viewpoint across all walks of life. There is a part of mathematics which is not algebraic, but which seems to use a mathematics#Change of the mind. For those people with such minds and imaginations, like Isaac Newton's, if they are to benefit from the wide availability of mathematical devices, then they will need to be served by more graphical, visual, aural, tactile, and temporal modalities in notation, as a first step. ==Ideographic notation== In the history of writing, ideographic symbols arose first, as more-or-less direct renderings of some concrete item. This has come full circle with the rise of the computer visualization systems, which can be applied to abstract visualizations as well, such as for rendering some projections of a Calabi-Yau manifold. Examples of Information visualization which properly belong to the mathematics#Spatial relations, can be found, for example in computer graphics. The need for such models abounds, for example, when the measures for the subject of study are actually random variables and not really ordinary mathematical functions. ==See also== *Table of mathematical symbols ==Notes== *Florian Cajori, ''A History of Mathematical Notations'' (1929), 2 volumes. ISBN 0486677664 ==External links== * [http://www.cut-the-knot.org/language/index.shtml Mathematics as a Language] * [http://members.aol.com/jeff570/mathsym.html Earliest Uses of Various Mathematical Symbols] ==See also== * Abuse of notation *Begriffsschrift Mathematical notation
Mathematical notation
Mathematics
Mathematical notation
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? User: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 etc 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. User: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. User: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. User: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 : and 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? User: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". User: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. User: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. User: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. User: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. User: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. User:Revolver 22:45, 19 Nov 2004 (UTC) == Definition == 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. User:BrianjdUser talk:BrianjdSpecial:Contributions/Brianjd | Talk:HTML#Restricted 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? User:Ancheta Wis 08:22, 23 Apr 2005 (UTC)
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