History UsefulNotes / Mathematics

8th Mar '15 6:56:09 PM 102372
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-> If in other sciences we should arrive at certainty without doubt and truth without error, it behooves us to place the foundations of knowledge in mathematics.
-->-- Roger Bacon

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-> If ->''If in other sciences we should arrive at certainty without doubt and truth without error, it behooves us to place the foundations of knowledge in mathematics.
-->-- Roger Bacon
mathematics.''
-->--'''Roger Bacon'''



[[EverybodyHatesMathematics It is also hated by roughly 99.9% of all people.]] It is the major reason that there aren't more scientists, doctors, engineers, or Wall Street workers. This is probably due to the way it is taught: 6 to 8 years of number crunching, most of which a calculator can do faster and with less chance of error, followed by massive, abstract generalizations (algebra and calculus). To make things worse, students are rarely told ''why'' they need to learn the current topic: science teachers tend to wait until after you've learned the math to show you the uses for it, and a lot of math teachers focus on the techniques rather than the applications. (Or they try to demonstrate the applications by assigning word problems--without realizing that "translate this word problem into math" is ''also'' a skill that needs to be taught, and that assigning said problems to students who don't have that skill won't help.) Worst of all, many elementary school teachers are poorly trained in mathematics, so they don't know good problem-solving techniques, they don't know the particular real-world applications of any given topic, and they don't ''like'' it enough to teach math for its own sake.

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[[EverybodyHatesMathematics It is also hated by roughly 99.9% of all people.]] It people]]. This is the major reason that there aren't more scientists, doctors, engineers, or Wall Street workers. This is probably due to the way it is taught: 6 to 8 years of number crunching, most of which a calculator can do faster and with less chance of error, followed by massive, abstract generalizations (algebra and calculus). To make things worse, students are rarely told ''why'' they need to learn the current topic: science teachers tend to wait until after you've learned the math to show you the uses for it, and a lot of math teachers focus on the techniques rather than the applications. (Or they try to demonstrate the applications by assigning word problems--without realizing that "translate this word problem into math" is ''also'' a skill that needs to be taught, and that assigning said problems to students who don't have that skill won't help.) Worst of all, many elementary school teachers are poorly trained in mathematics, so they don't know good problem-solving techniques, they don't know the particular real-world applications of any given topic, and they don't ''like'' it enough to teach math for its own sake.



* Francis Su's [[http://www.math.hmc.edu/funfacts/ Fun facts]] website. These are short web pages detailing some "fun fact" about mathematics; the maintainer likes to spend the first five minutes of his calculus classes explaining one of these facts.

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* Francis Su's [[http://www.math.hmc.edu/funfacts/ Fun fun facts]] website. These are short web pages detailing some "fun fact" about mathematics; the maintainer likes to spend the first five minutes of his calculus classes explaining one of these facts.



* Martin Gardner's books of [[http://en.wikipedia.org/wiki/Martin_Gardner#Collected_Scientific_American_columns collected Scientific American columns]]. These go into a bit more detail than the fun facts above. Each chapter in these books was originally a magazine article about some topic in recreational mathematics, so each chapter stands alone; reading one (or a few) is not a huge time investment.

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* Martin Gardner's books of [[http://en.wikipedia.org/wiki/Martin_Gardner#Collected_Scientific_American_columns collected Scientific American ''Scientific American'' columns]]. These go into a bit more detail than the fun facts above. Each chapter in these books was originally a magazine article about some topic in recreational mathematics, so each chapter stands alone; reading one (or a few) is not a huge time investment.
1st Aug '13 3:20:34 AM SeptimusHeap
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* Category theory, which describes mathematical structures--that is, objects that attach to other elements in a set--and their relationships between them.[[hottip:*:It's actually a kind of super-abstract algebra, and for all its removedness from practice, it still finds application in abstract computer science -- especially in the theory of calculatability, which tries to determine whether a given problem is solvable, and how many resources solving it would take. One of its main problems is determining whether a given program would stop given a specified input or whether it would work forever. With significantly complex programs and inputs this turns out a surprisingly complex task. It is known (thanks to Turing) that this problem (called the Halting Problem) is not solvable in the general case, yet specific instances can be of interest.]] Category theory is perhaps the most abstract kind of math on this page; it's a bit hard to describe and even mathematicians tend to find those who specialize in it ''off''. Additionally it has given birth to one of the stranger bits of [[http://en.wikipedia.org/wiki/Mathematical_jargon mathematical jargon]]: "[[http://en.wikipedia.org/wiki/Abstract_nonsense abstract nonsense]]".

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* Category theory, which describes mathematical structures--that is, objects that attach to other elements in a set--and their relationships between them.[[hottip:*:It's [[note]]It's actually a kind of super-abstract algebra, and for all its removedness from practice, it still finds application in abstract computer science -- especially in the theory of calculatability, which tries to determine whether a given problem is solvable, and how many resources solving it would take. One of its main problems is determining whether a given program would stop given a specified input or whether it would work forever. With significantly complex programs and inputs this turns out a surprisingly complex task. It is known (thanks to Turing) that this problem (called the Halting Problem) is not solvable in the general case, yet specific instances can be of interest.]] [[/note]] Category theory is perhaps the most abstract kind of math on this page; it's a bit hard to describe and even mathematicians tend to find those who specialize in it ''off''. Additionally it has given birth to one of the stranger bits of [[http://en.wikipedia.org/wiki/Mathematical_jargon mathematical jargon]]: "[[http://en.wikipedia.org/wiki/Abstract_nonsense abstract nonsense]]".
16th Mar '13 9:51:40 AM MystyrNile
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Give UsefulNotesOnMathematics here.
22nd Feb '13 6:52:10 AM smspain
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* Graph theory studies graphs--not in the sense of "the graph of ''y=x[[superscript:2]]''", but in the sense of "a bunch of objects, called vertices, some of which are connected to each other''. A family tree is a graph (people are vertices, and parents are connected to their children). The pictures on the TriangRelations page are directed multi-graphs. Graph theory has direct applications in computer networking, and multimedia compression algorithms we use every day such as MP3 or JPEG use graphs to calculate the way they will encode the file in order to reduce its size.

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* Graph theory studies graphs--not in the sense of "the graph of ''y=x[[superscript:2]]''", but in the sense of "a bunch of objects, called vertices, some of which are connected to each other''. A family tree is a graph (people are vertices, and parents are connected to their children). The pictures on the TriangRelations page are directed multi-graphs. Graph theory has direct applications in computer networking, and multimedia compression algorithms we use every day such as MP3 or JPEG use graphs to calculate the way they will encode the file in order to reduce its size. Graph theory also forms the theoretical basis of social network analysis and the discipline of "graphical modeling"--representing the relations between variables as a graph.
26th Dec '12 12:20:22 PM jate88
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* [[http://www.youtube.com/user/vihart Vihart]] and [[http://www.youtube.com/user/numberphile?feature=fvstc Numberphile]] are youtubers who show awesome math stuff for people who aren't scientists, engineers, or mathematicians.

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* [[http://www.youtube.com/user/vihart Vihart]] {{Vihart}} and [[http://www.youtube.com/user/numberphile?feature=fvstc Numberphile]] are youtubers who show awesome math stuff for people who aren't scientists, engineers, or mathematicians.
25th Dec '12 10:13:37 PM jate88
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Added DiffLines:

*[[http://www.youtube.com/user/vihart Vihart]] and [[http://www.youtube.com/user/numberphile?feature=fvstc Numberphile]] are youtubers who show awesome math stuff for people who aren't scientists, engineers, or mathematicians.
27th Oct '12 3:17:34 PM SapphireCrow
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* Set theory is the study of collections (sets) of objects, and ways that these collections can interact. So you can have the set of all integers, a subset (the set of all positive integers, or the set of all even integers); you can have the union of two sets (the set of all positive integers which are either even or positive or both) or the intersection of two sets (the set of all integers which are both positive and even). Set theory also includes a notion of a bigger set: for example, {1,2,3,4} is a bigger set than {7,8,9}.

to:

* Set theory is the study of collections (sets) of objects, and ways that these collections can interact. So you can have the set of all integers, a subset (the set of all positive integers, or the set of all even integers); you can have the union of two sets (the set of all positive integers which are either even or positive or both) or the intersection of two sets (the set of all integers which are both positive and even). Set theory also includes a notion of a bigger set: for example, {1,2,3,4} is a bigger set than {7,8,9}.{2,3,4}.
29th Jun '12 6:55:06 PM slvstrChung
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There are some fields of mathematics that act as a basis, or foundation, of the other fields. Mathematicians like to base their work on as few assumptions as possible; this is both an aesthetic ideal of mathematics, and means that they are less likely to discover that two of their assumptions contradict each other. Thus, the basis of mathematics involves brutally simple things and things that we really do need for everything.

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There are some fields of mathematics that act as a basis, or foundation, of the other fields. Mathematicians like to base their work on [[OccamsRazor as few assumptions as possible; possible]]; this is both an aesthetic ideal of mathematics, and means that they are less likely to discover that minimizes the chances of two of their those assumptions contradict contradicting each other. Thus, the basis of mathematics involves brutally simple things and things that we really do need for everything.
4th Dec '11 4:40:19 PM Tominator2
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* Category theory, which describes mathematical structures--that is, objects that attach to other elements in a set--and their relationships between them.[[hottip:*:It's actually a kind of super-abstract algebra, and for all its removedness from practice, it still finds application in abstract computer science -- especially in the theory of calculatability, which tries to determine whether a given problem is solvable, and how many resources solving it would take. One of its main problems is a problem of stopping. For example, whether a given program would stop given specified input, or whether it would work forever. With significantly complex programs and inputs this turns out a surprisingly complex task, and there's the general agreement (unproven, though) that it's essentially unsolvable in general terms.]] Category theory is perhaps the most abstract kind of math on this page; it's a bit hard to describe and even mathematicians tend to find those who specialize in it ''off''. Additionally it has given birth to one of the stranger bits of [[http://en.wikipedia.org/wiki/Mathematical_jargon mathematical jargon]]: "[[http://en.wikipedia.org/wiki/Abstract_nonsense abstract nonsense]]".

to:

* Category theory, which describes mathematical structures--that is, objects that attach to other elements in a set--and their relationships between them.[[hottip:*:It's actually a kind of super-abstract algebra, and for all its removedness from practice, it still finds application in abstract computer science -- especially in the theory of calculatability, which tries to determine whether a given problem is solvable, and how many resources solving it would take. One of its main problems is a problem of stopping. For example, determining whether a given program would stop given a specified input, input or whether it would work forever. With significantly complex programs and inputs this turns out a surprisingly complex task, and there's task. It is known (thanks to Turing) that this problem (called the Halting Problem) is not solvable in the general agreement (unproven, though) that it's essentially unsolvable in general terms.case, yet specific instances can be of interest.]] Category theory is perhaps the most abstract kind of math on this page; it's a bit hard to describe and even mathematicians tend to find those who specialize in it ''off''. Additionally it has given birth to one of the stranger bits of [[http://en.wikipedia.org/wiki/Mathematical_jargon mathematical jargon]]: "[[http://en.wikipedia.org/wiki/Abstract_nonsense abstract nonsense]]".
3rd Dec '11 4:23:31 PM Infophreak
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* Category theory, which describes mathematical structures--that is, objects that attach to other elements in a set--and their relationships between them.[[hottip:*:It's actually a kind of super-abstract algebra, and for all its removedness from practice, it still finds application in abstract computer science -- especially in the theory of calculatability, which tries to determine whether a given problem is solvable, and how many resources solving it would take. One of its main problems is a problem of stopping. For example, whether a given program would stop given specified input, or whether it would work forever. With significantly complex programs and inputs this turns out a surprisingly complex task, and there's the general agreement (unproven, though) that it's essentially unsolvable in general terms.]] Category theory is perhaps the most abstract kind of math on this page; it's a bit hard to describe and even mathematicians tend to find those who specialize in it ''off''.

to:

* Category theory, which describes mathematical structures--that is, objects that attach to other elements in a set--and their relationships between them.[[hottip:*:It's actually a kind of super-abstract algebra, and for all its removedness from practice, it still finds application in abstract computer science -- especially in the theory of calculatability, which tries to determine whether a given problem is solvable, and how many resources solving it would take. One of its main problems is a problem of stopping. For example, whether a given program would stop given specified input, or whether it would work forever. With significantly complex programs and inputs this turns out a surprisingly complex task, and there's the general agreement (unproven, though) that it's essentially unsolvable in general terms.]] Category theory is perhaps the most abstract kind of math on this page; it's a bit hard to describe and even mathematicians tend to find those who specialize in it ''off''.
''off''. Additionally it has given birth to one of the stranger bits of [[http://en.wikipedia.org/wiki/Mathematical_jargon mathematical jargon]]: "[[http://en.wikipedia.org/wiki/Abstract_nonsense abstract nonsense]]".
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