History Main / ProofByExamples

25th Jan '17 7:31:52 PM Game_Fan
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* In mathematics proof by example is no proof at all. One instance of this is a conjecture by Christian Goldbach that "every odd composite number can be written as the sum of a prime and twice a square number" which certainly seems to be true if you try casually testing a few example. It wasn't until much later that a counter example (5777) was found.
** An even more dramatic example is [[https://en.wikipedia.org/wiki/Euler%27s_sum_of_powers_conjecture Euler's sum of power conjecture]] for which the first counter example is 61,917,364,224!

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* In mathematics proof by example is usually no proof at all. One instance of this Many famous examples are used to illustrate this.
** A simple one
is a conjecture by Christian Goldbach that "every odd composite number can be written as the sum of a prime and twice a square number" which certainly seems to be true if you try casually testing a few example. It wasn't until much later that a counter example (5777) was found.
** An even more dramatic example is [[https://en.wikipedia.org/wiki/Euler%27s_sum_of_powers_conjecture Euler's sum of power conjecture]] for which the first counter example counter-example is 61,917,364,224!61,917,364,224!
** There are problem in mathematics that have been tested for trillions upon trillions of examples without finding a counter-example but still lack proof. (The huge numbers of examples are both used to search for counter-examples and in hopes of discovering patterns that might lead to a proof.)
25th Jan '17 7:28:21 PM Game_Fan
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* Usually, in mathematics, [[http://en.wikipedia.org/wiki/Skewe%27s_number no matter how many examples]] [[http://en.wikipedia.org/wiki/Riemann_hypothesis you might have]], proof by example is not a good idea.

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* Usually, in mathematics, [[http://en.In mathematics proof by example is no proof at all. One instance of this is a conjecture by Christian Goldbach that "every odd composite number can be written as the sum of a prime and twice a square number" which certainly seems to be true if you try casually testing a few example. It wasn't until much later that a counter example (5777) was found.
** An even more dramatic example is [[https://en.
wikipedia.org/wiki/Skewe%27s_number no matter how many examples]] [[http://en.wikipedia.org/wiki/Riemann_hypothesis you might have]], proof by org/wiki/Euler%27s_sum_of_powers_conjecture Euler's sum of power conjecture]] for which the first counter example is not a good idea.61,917,364,224!
22nd Jun '15 4:15:56 AM ShorinBJ
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:: Hasty Generalisation

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:: Hasty GeneralisationGeneralization
8th Nov '13 6:09:12 PM RatherRandomRachel
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** A (simple) example of this might be: "A new plant found seems to fit into a particular category with several others. All of the plants within this category need three things to thrive - carbon dioxide, water and a light source. Therefore, it seems likely this plant will also need those three to survive. We should study it to confirm or deny this theory."

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** A (simple) example of this might be: "A new plant found seems to fit into a particular category with several others. All of the plants within this category seem to need three things to thrive - carbon dioxide, water and a light source. Therefore, it seems likely this plant will also need those three to survive. We should study it to confirm or deny this theory."
23rd Sep '13 3:20:52 AM RatherRandomRachel
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** A (simple) example of this might be: A new plant found seems to fit into a particular category with several others. All of the plants within this category need three things to thrive - carbon dioxide, water and a light source. Therefore, it seems likely this plant will also need those three to survive. We should study it to confirm or deny this theory.

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** A (simple) example of this might be: A "A new plant found seems to fit into a particular category with several others. All of the plants within this category need three things to thrive - carbon dioxide, water and a light source. Therefore, it seems likely this plant will also need those three to survive. We should study it to confirm or deny this theory."
23rd Sep '13 3:13:04 AM RatherRandomRachel
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** To clarify. Induction - at its most basic is proving it by proving two things. The Base Case exists (typically for the value of 1 or 0) and that if we assume the theory works at value k (k being any given number) we can prove that it works at k+1. Combine the two and you get the ladder (1 is true, which means 1+1 is true, which means 2+1 is true...)
*** That's mathematical induction, which is not "induction" in the logical sense (''i.e.'', inductive reasoning); rather, it is rigorous deductive reasoning.

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** To clarify. Induction - at its most basic is proving it by proving two things. The Base Case exists (typically for A (simple) example of this might be: A new plant found seems to fit into a particular category with several others. All of the value of 1 or 0) plants within this category need three things to thrive - carbon dioxide, water and that if we assume the theory works at value k (k being any given number) we can prove that a light source. Therefore, it works at k+1. Combine the two and you get the ladder (1 is true, which means 1+1 is true, which means 2+1 is true...)
*** That's mathematical induction, which is not "induction" in the logical sense (''i.e.'', inductive reasoning); rather,
seems likely this plant will also need those three to survive. We should study it is rigorous deductive reasoning.to confirm or deny this theory.
11th Jun '13 2:19:17 PM DCC
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Added DiffLines:

In short, mistaking inductive reasoning for deductive reasoning
19th May '13 10:19:55 PM trumpetmarietta
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** To clarify. Induction - at its most basic is proving it by proving two things. The Base Case exists (typically for the value of 1 or 0) and that if we assume the theory works at value k (k being any given number) we can prove that it works at k+1. Combine the two and you get the ladder (1 is true, which means 1+1 is true, which means 2+1 is true...)

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** To clarify. Induction - at its most basic is proving it by proving two things. The Base Case exists (typically for the value of 1 or 0) and that if we assume the theory works at value k (k being any given number) we can prove that it works at k+1. Combine the two and you get the ladder (1 is true, which means 1+1 is true, which means 2+1 is true...))
*** That's mathematical induction, which is not "induction" in the logical sense (''i.e.'', inductive reasoning); rather, it is rigorous deductive reasoning.
6th Apr '13 7:22:17 PM SenseiLeRoof
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** To clarify. Induction - at it's most basic is proving it by proving two things. THe Base Case exists (typically for the value of 1 or 0) and that if we assume the theory works at value k (k being any given number) we can prove that it works at k+1. Combine the two and you get the ladder (1 is true, which means 1+1 is true, which means 2+1 is true...)

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** To clarify. Induction - at it's its most basic is proving it by proving two things. THe The Base Case exists (typically for the value of 1 or 0) and that if we assume the theory works at value k (k being any given number) we can prove that it works at k+1. Combine the two and you get the ladder (1 is true, which means 1+1 is true, which means 2+1 is true...)
6th Nov '12 3:59:58 PM MasamiPhoenix
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* An attempt at real induction. Inductive logic admits that its conclusions are not ''necessarily'' true, but rather that they are ''probably'' true, and it tends to attempt to be as exhaustive as possible and to eliminate as many alternative explanations as possible, to reduce the possibility that the conclusion is wrong to as close to zero as possible. However, an honest scientist (i.e. practitioner of inductive logic) would freely admit that there is the possibility, however slim, that the entirety of his/her science is entirely wrong.

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* An attempt at real induction. Inductive logic admits that its conclusions are not ''necessarily'' true, but rather that they are ''probably'' true, and it tends to attempt to be as exhaustive as possible and to eliminate as many alternative explanations as possible, to reduce the possibility that the conclusion is wrong to as close to zero as possible. However, an honest scientist (i.e. practitioner of inductive logic) would freely admit that there is the possibility, however slim, that the entirety of his/her science is entirely wrong.wrong.
** To clarify. Induction - at it's most basic is proving it by proving two things. THe Base Case exists (typically for the value of 1 or 0) and that if we assume the theory works at value k (k being any given number) we can prove that it works at k+1. Combine the two and you get the ladder (1 is true, which means 1+1 is true, which means 2+1 is true...)
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