History UsefulNotes / Mathematics

6th Aug '17 5:07:42 PM Berrenta
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* The [[{{Trivia/Math}} math trivia page]] on this very wiki.
24th Jul '17 8:06:37 PM FordPrefect
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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.[[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]]".

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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.[[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 to be 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 describe, and even mathematicians tend to find those who specialize in it ''off''. Additionally 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]]".
24th Jul '17 8:03:49 PM FordPrefect
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** Note that fluid dynamics is actually a field of ''physics'', but it uses such complex and sophisticated models that the whole branches of mathematical analysis were developed purely to support these models and solve these problems. That's actually how mathematics and physics are usually related: physicists find some problems or processes and develop a mathematical model of it, while mathematicians (who are quite often the same people, these groups tends to intersect a lot) work them out.

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** Note that fluid dynamics is actually a field of ''physics'', but it uses such complex and sophisticated models that the whole branches of mathematical analysis were developed purely to support these models and solve these problems. That's actually how mathematics and physics are usually related: physicists find some problems or processes and develop a mathematical model of it, while mathematicians (who are quite often the same people, these groups tends to intersect a lot) work them out.
24th Jul '17 8:01:17 PM FordPrefect
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* 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 Martin Gardner's books]] of collected ''Scientific American'' columns]].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.
28th Dec '16 8:33:38 PM Kayube
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Most math researchers work in a subfield of the areas listed above, and have a special type of problem that they work on. The biggest field is probably algebraic geometry, as there are many ways to get at it, and there are many theories that can be useful in solving problems there.
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Most math researchers work in a subfield of the areas listed above, and have a special type of problem that they work on. The biggest field is probably algebraic geometry, as there are many ways to get at it, and there are many theories that can be useful in solving problems there.
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there.
21st Sep '16 10:17:46 AM Morgenthaler
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* {{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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* {{Vihart}} Creator/{{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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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.

to:

* 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]]".

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 [[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.
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