Editing 1724: Proofs
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==Explanation== | ==Explanation== | ||
− | [[Miss Lenhart]] is teaching a math class. She begins a proof when one of her students ([[Cueball]]) interrupts her asking if this is one of those dark-magic (unclear, incomprehensible) proofs. She | + | [[Miss Lenhart]] is back teaching a math class. She begins a proof when one of her students ([[Cueball]]) interrupts her asking if this is one of those dark-magic (unclear, incomprehensible) proofs. She says no, but it soon turns out that it is; Cueball exclaims that he just knew it would be. |
− | The proof she starts setting up resembles a {{w|proof by contradiction}}. | + | The proof she starts setting up resembles a {{w|proof by contradiction}}. This kind of proof assumes that a particular theorem is true, and shows that this assumption leads to a contradiction, which disproves the initial assumption. For example assumption that √2 is a {{w|rational number}} means that, for some natural ''a'' and ''b'', √2=''a/b'', where ''a/b'' is an {{w|irreducible fraction}}. Yet, multiplying this equation by itself, we get 2=''a²/b²'' which in turn rearranges to 2''b²''=''a²'', therefor ''a²'' is even (as any integer multiplied by 2 is even), which means that ''a'' is an even number, as an even number squared is always even and an odd number squared is always odd. This means, that ''a=2k'' and ''2b²=(2k)²=4k²'', meaning ''b²''=2''k²'', so ''b'' must be even too. But if both ''a'' and ''b'' are even, ''a/b'' cannot be irreducible. Contradiction means that the initial assumption is false, and √2 cannot be a rational number. |
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Alternatively, instead of a proof by contradiction the setup could be for a one way function. For example, it is relatively easy to test that a solution to a differential equation is valid but choosing the correct solution to test can seem like black magic to students. | Alternatively, instead of a proof by contradiction the setup could be for a one way function. For example, it is relatively easy to test that a solution to a differential equation is valid but choosing the correct solution to test can seem like black magic to students. | ||
− | The way | + | The way, Ms Lenhart's proof refers to the act of doing math itself, is characteristic to metamathematical proofs, for example {{w|Gödel's incompleteness theorems}}, which, at first sight, may indeed look like black magic, even if in the end they must be a "perfectly sensible chain of reasoning" like the rest of good mathematics. While standard mathematical theorems and their proofs deal with standard mathematical objects, like numbers, functions, points or lines, the metamathematical theorems treat other theorems as objects of interest. In this way you can propose and prove theorems about possibility of proving other theorems. For example, in 1931 {{w|Kurt Gödel}} was able to prove that any mathematical system based on arithmetics (that is using numbers) has statements that are true, but can be neither proved nor disproved. This kind of metamathematical reasoning is especially useful in the {{w|set theory}}, where many statements become impossible to prove and disprove if the {{w|axiom of choice}} is not taken as a part of the axiomatic system. |
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− | + | In the title text the decision of whether to take the axiom of choice is made by a deterministic process, that is a process which future states can be developed with no randomness involved. The axiom of choice itself is a non-constructive axiom in mathematics, it asserts the existence of objects, without providing a method of constructing them, in particular there is no deterministic process by which we can define objects whose existence can only be proved using the axiom of choice. The axiom of choice was mentioned earlier in [[804: Pumpkin Carving]]. | |
− | Although Miss Lenhart did retire a year ago after [[1519: Venus]], she seems to have returned here for a math course at university level, but continues the trend she finished with in her prior class | + | Although Miss Lenhart did retire a year ago after [[1519: Venus]], she seems to have returned here for a math course at university level, but continues the trend she finished with in her prior class. |
==Transcript== | ==Transcript== |