
#REDIRECT Reductio ad absurdum
Reductio ad absurdum (Latin for "reduction to the absurd", traceable back to the Greek language ἡ εις το αδυνατον απαγωγη, "reduction to the impossible", often used by Aristotle) is a type of logical argument where we assume a claim for the sake of argument, arrive at an absurd result, and then conclude the original assumption must have been wrong, since it gave us this absurd result. This is also known as proof by contradiction. It makes use of the law of excluded middle — a statement which cannot be false, must then be true. ==In philosophy and everyday reasoning== A reduction to the absurd can be made to argue many points. Take the following dialogue, for example. :A — ''You should respect C's belief, for all beliefs are of equal validity and cannot be denied.'' :B — ''What about D's belief? (Where D believes something that is considered to be wrong by most people, such as Nazism or the world being flat)'' :A — ''I agree it is right to deny D's belief.'' :B — ''If it is right to deny D's belief, it is not true that no belief can be denied. Therefore, I can deny C's belief if I can give reasons that suggest it too is incorrect.'' A trickier, but even stronger reduction from the philosophical point of view, because it does not rely on A's accepting that D's opinion is wrong, would be the following. : A — ''You should respect C's belief, for all beliefs are of equal validity and cannot be denied.'' : B — :#''I deny that belief of yours and believe it to be invalid.'' :#''According to your statement, this belief of mine (''1'') is valid, like all other beliefs.'' :#''However, your statement also contradicts and invalidates mine, being the exact opposite of it.'' :#''The conclusions of ''2'' and ''3'' are incompatible and contradictory, so your statement is logically absurd.'' In each case, B has used a reduction to the absurd to argue his or her point. Among some people, there is a misconception that ''reductio ad absurdum'' just means "a silly argument". ==In mathematics== Say we wish to prove proposition ''p''. The procedure is to show that assuming "not ''p''" (i.e. that ''p'' is false) leads to a logical contradiction. Thus ''p'' cannot be false, and must, according to the law of the excluded middle, therefore be true. For a simple example, consider the proposition "there is no smallest rational number greater than 0". In a ''reductio ad absurdum'' argument, we would start by assuming the opposite: that there ''is'' a smallest rational number, say, ''r''0. Now let ''x'' = ''r''0/2. Then ''x'' is a rational number, and it's greater than 0; and ''x'' is smaller than ''r''0. But that is absurd — it contradicts our initial assumption that ''r''0 was the ''smallest'' rational number. So we can conclude that the original proposition must be true — "there is no smallest rational number greater than 0". It is not uncommon to use this type of argument with propositions such as the one above, concerning the ''non''-existence of some mathematical object. One assumes that such an object exists, and then proves that this would lead to a contradiction; thus, such an object does not exist. For examples, see Irrational number#Irrationality_of_the_square_root_of_2 and Cantor's diagonal argument. It is important to note that to form a valid proof, it must be demonstrated that given a proposition ''p'', "not ''p''" implies a property that is actually false in the mathematical system being used. The danger here is the logical fallacy of Argument from ignorance, where it is proven that "not ''p''" implies a property "''q''", which ''looks'' false, but is not really ''proven'' to be false. Traditional (but incorrect!) examples of this fallacy include false proofs of Euclid's ''fifth postulate'' (a.k.a. ''the parallel postulate'') from the other postulates. The reason these examples are not really examples of this fallacy is that the notion of proof was different in the 19th century; (Euclidean) geometry was seen as being a 'true' reflection of physical reality, and so deducing a contradiction by concluding something physically implausible (like the angles of a triangle not being 180 degrees) was acceptable. Doubts about the nature of the geometry of the universe led mathematicians such as Bolyai, Carl Friedrich Gauss, Lobachevsky, Riemann, among others, to question and clarify what actually constituted 'geometry'. Out of these men's work, resulted Non-Euclidean geometry. For a further exposition of these misunderstandings see Morris Kline, ''Mathematical Thought: from Ancient to Modern Times''. Although it is quite freely used in mathematical proofs, not every philosophy of mathematics accepts ''reductio ad absurdum'' arguments as universally valid. In schools such as intuitionism, the law of the excluded middle is not taken as true. From this way of thinking, there is a very significant difference between proving that something exists by showing that it would be absurd if it did not; and proving that something exists by constructing an actual example of such an object. In mathematical logic, the ''reductio ad absurdum'' is represented as: : if :: : then :: In the above, ''p'' is the proposition we wish to prove; and ''S'' is a set of statements which are given as true — these could be, for example, the axioms of the theory we are working in, or earlier theorems we can build upon. We consider the negation of ''p'' in addition to ''S''; if this leads to a logical contradiction ''F'', then we can conclude that the statements in ''S'' lead to ''p''. Note that the union (set theory), in some contexts closely related to logical disjunction (or), is used here for sets of statements in such a way that it is more related to logical conjunction (and). In the words of G. H. Hardy (A Mathematician's Apology), "Reductio ad absurdum, which Euclid loved so much, is one of a mathematician's finest weapons. It is a far finer gambit than any chess gambit: a chess player may offer the sacrifice of a pawn or even a piece, but a mathematician offers the game." Logic Proofs Mathematical terminology Latin logical phrases
Can someone please explain this: :basically: if :: S union { ¬ t } |-- F ::: then ::: S |-- t I suspect the equivalent point can be made just as well in more widely known notation, perhaps even in plain-old English. --User:Ryguasu : seems that the english version of the above is in the first paragraph. -- User:Tarquin 10:34 Jan 10, 2003 (UTC) : could you at least explain what S and t are? It probably doesn't add clarity that ''A'', used in the English version, does not figure into the symbols. --User:Ryguasu : Ok, thanks for the in-line explanation. I still think the TeX graphic and the textual description should use the same letter to represent "the statement". Changing the text might be easier, but I think, Tarquin, your choice of "t" for the statement in the TeX graphic is unfortunate, as it reminds the untrained eye of "true". Is this a routine convention for this sort of formal symbol representation? --User:Ryguasu ::I would prefer A instead of t, for the sake of consistency within the article. After all, proposition F already has the form of a capital letter. User:FvdP 20:35 Jan 10, 2003 (UTC) F is not a proposition, it represents logical False. Propositions are small letters. small ''a'' could be ok. -- User:Tarquin 20:37 Jan 10, 2003 (UTC) :I should have known (and did realize, but a bit late.) OK for a. I've had the idea to replace F with a more common sign for "false", like , but that would annoy non-mathematically readers more than it would be useful, IMO. User:FvdP :Does it make sense to replace A with a (or pehaps ''a'') in the body of the text as well? This would save us from including an overly philosophical sentence along the lines of "Take a to be the same as A." Also, if we put italic ''a'' in the text, can we make it italic in the graphic as well? --User:Ryguasu ::I (boldly!) jumped in and relaced occurrences of "A", ''a'', and ''t'' with ''p'' (for ''p''roposition), and relaced "B" with ''q''. Apologies if the term "hypothesis" is being used in a technical sense here - ''h'' just seemed less obvious as a choice. :::Ah, ''p'' ! That had become too obvious to be seen (by me). Thanks Chas for the change. User:FvdP ::I agree that seems unneccesarily obtuse here - I'd be just as happy replacing ''F'' with "''false''", i.e.: ::: ::unless that's problematic. Also, the law of the excluded middle is being invoked here - do we need to add a link to intuitionism or the like here? User:Chas zzz brown 01:01 Jan 11, 2003 (UTC) :::I have no definitive opinion on this. The interested reader can get the link through law of the excluded middle. Perhaps we should interest him more by telling a few words on intuitionism ? User:FvdP :::Even in intuitionistic logic, if , one can still conclude that . The real difference is that in classical logic, but not in intuitionistic logic. So I don't think it's a major difference in this case, or particularly worth mentioning. User:Dominus ---- I don't like too much the ''given'' in :''and S is a set of statements which are ''given'' as true'' because it reads (to me) like we're just giving them the status of being true, out of the blue, which can't be. But I don't know what to write instead: *''and S is a set of true statements'' *''and S is a set of statements which are known to be true'' *''and S is a set of statements which have been proven to be true'' *... User:FvdP 01:05 Jan 11, 2003 (UTC) :But that ultimately is all we ''can'' say; that they are given (or is it taken?) as true "out of the blue". For example (loosely speaking), "parallel lines meet at infinity" is not inherently true - but if we ''take'' it as true ''then'' we get euclidean geometry. Alternatively, we could just as easily say that "parallel lines always intersect at exactly two points"; this would then ''imply'' a different geometry. :The statements in ''S'' are either a given set of axioms which we ''postulate'' as true, or theorems following from axioms which essentially comes to the same thing (in that we assume that these theorems really ''do'' follow from some axiom set ''A''; i.e., that ''p'' can be deduced directly from ''A'' as well as from ''S''). From this point of view, we don't so much prove that ''p'' is "True", as that ''p'' is true assuming that (i.e., given that) the statements in ''S'' are true (which generally has been previously proven if needed). :By my phrasing, I was hoping to distinguish between "Truth" and "truth". Philosophically, one can argue that certain axioms are "really" true, e.g. platonism; but it is not required that one hold this philosophical view to use reductio ad absurdum. Cheers User:Chas zzz brown 01:50 Jan 11, 2003 (UTC) ::We can get rid of this "given as true" confusion by adding a second part to the if in the formal notation. It can now read ::: if :::: it is ''not'' the case that S |-- F ::: and :::: S union { ¬ p } |-- F ::: then :::: S |-- t ::Does this deal with both side's objections? --User:Ryguasu :::I have no idea what this is saying. -- User:Zoe ::::Do you understand the formal logic notation used elsewhere on this page? My comment was aimed at those who did. --User:Ryguasu The formal notation is useless and should be relegated to a footnote. Why do I push this heresy? Because anyone who can read it ''already knows'' what a reductio ad absurdum is, and anyone who doesn't know what it is has got Buckley's chance of figuring out how to read ::: or any similar formula. A plain-text example in simple English is badly needed. This should come first, so that the non-expert reader can find it immediately, and be followed by the more formal explanation. User:Tannin 02:15 Jan 11, 2003 (UTC) : Yes, much improved now Chas. It would be better still if it were a non-mathematical example, but (to my shame) I can't think of one that's clear enough! User:Tannin 09:44 Jan 11, 2003 (UTC) It's always nice when an article dramatically improves overnight! :-) The "smallest rational number" example is great. -- User:Tarquin 10:43 Jan 11, 2003 (UTC)~ ---- Text from Reductio Ad Absurdum:
A method of disproving a proposition, by demonstrating that it leads to a contradiction. An example would be the proof that the square root of 2 is irrational. Begin with the proposition that there exist integers M and N with no factors in common, where M*M = 2*N*N. If they exist, then the square root of 2 is the rational number M/N. 2*N*N is even, and therefore M*M is even. This implies that M is even, since an odd number cannot have an even square. Therefore K = M/2 is an integer. Then 2*K*K = N*N, and it follows that N is even. The equation M*M = 2*N*N cannot hold unless both M and N are even. This contradicts the initial assumption that M and N have no factors in common. ---- I'd propose moving this article to Proof by contradiction and redirecting there. ''Reductio ad absurdum'' is the historical name used in philosophy and formal logic, but ''proof by contradiction'' is the more common English name and the one preferred in most modern mathematics articles. --User:Delirium 22:51, Oct 15, 2003 (UTC) ---- I used to be appalled at seeing people (non-mathematicians, mostly) use the term to refer simply to an absurd argument, seemingly translating the Latin to "absurd reduction" instead of "reduction to the absurd". However, I've noticed that Merriam-Webster's lists a second (presumably less formal) meaning of the term as "the carrying of something to an absurd extreme". I'm curious to find out whether this is an instance of a sloppy translation gaining status as an acceptable meaning simply through widespread usage. If anyone can shed some light on the non-specialist use (or abuse) of the term, it could be an interesting aside to this article. User:Rajneeshhegde 04:15, 19 Aug 2004 (UTC) I've never heard it misused like that. Note, however, that the mathematical use is not the primary one. It is mainly used in philosophy. User:Chameleon 09:06, 17 Nov 2004 (UTC) I arrived at this page to find out if there were non-mathematical uses of this term. Specifically, where one tries to kill a project, by "improving" it to the point where it must fail. For instance, take a product that breaks even with costs of $0.90, and a sale price of $1, and improve it so that it now costs $1.05 to make, has to sell for $2, and now is a failure. Perhaps there is another term for that kind of effort. User:Mcr314 02:37, 4 May 2005 (UTC) ==Nazism== I'm out of my depth, so just wondered if anyone else felt that using ''Nazism'' in the article seems a bit odd, especially against the ''flat earth'' thing. --bodnotbod\">User:Bodnotbod_»_.....TALKQuietly)\">User_talk:Bodnotbod">User:Bodnotbod|bodnotbod\">User:Bodnotbod » .....TALKQuietly)\">User talk:Bodnotbod 10:01, Dec 5, 2004 (UTC)
:How is it odd? Whether you happen to agree with the Nazis or not, it is the typical example used in such reductions. So much so that Godwin's law has been humorously invented as a remark upon the phenomenon. User:Chamaeleon 01:43, 19 Jan 2005 (UTC)
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#REDIRECT Reductio ad absurdum
Reductio ad absurdum
Reductio ad absurdum (Latin for "reduction to the absurd", traceable back to the Greek language ἡ εις το αδυνατον απαγωγη, "reduction to the impossible", often used by Aristotle) is a type of logical argument where we assume a claim for the sake of argument, arrive at an absurd result, and then conclude the original assumption must have been wrong, since it gave us this absurd result. This is also known as proof by contradiction. It makes use of the law of excluded middle — a statement which cannot be false, must then be true. ==In philosophy and everyday reasoning== A reduction to the absurd can be made to argue many points. Take the following dialogue, for example. :A — ''You should respect C's belief, for all beliefs are of equal validity and cannot be denied.'' :B — ''What about D's belief? (Where D believes something that is considered to be wrong by most people, such as Nazism or the world being flat)'' :A — ''I agree it is right to deny D's belief.'' :B — ''If it is right to deny D's belief, it is not true that no belief can be denied. Therefore, I can deny C's belief if I can give reasons that suggest it too is incorrect.'' A trickier, but even stronger reduction from the philosophical point of view, because it does not rely on A's accepting that D's opinion is wrong, would be the following. : A — ''You should respect C's belief, for all beliefs are of equal validity and cannot be denied.'' : B — :#''I deny that belief of yours and believe it to be invalid.'' :#''According to your statement, this belief of mine (''1'') is valid, like all other beliefs.'' :#''However, your statement also contradicts and invalidates mine, being the exact opposite of it.'' :#''The conclusions of ''2'' and ''3'' are incompatible and contradictory, so your statement is logically absurd.'' In each case, B has used a reduction to the absurd to argue his or her point. Among some people, there is a misconception that ''reductio ad absurdum'' just means "a silly argument". ==In mathematics== Say we wish to prove proposition ''p''. The procedure is to show that assuming "not ''p''" (i.e. that ''p'' is false) leads to a logical contradiction. Thus ''p'' cannot be false, and must, according to the law of the excluded middle, therefore be true. For a simple example, consider the proposition "there is no smallest rational number greater than 0". In a ''reductio ad absurdum'' argument, we would start by assuming the opposite: that there ''is'' a smallest rational number, say, ''r''0. Now let ''x'' = ''r''0/2. Then ''x'' is a rational number, and it's greater than 0; and ''x'' is smaller than ''r''0. But that is absurd — it contradicts our initial assumption that ''r''0 was the ''smallest'' rational number. So we can conclude that the original proposition must be true — "there is no smallest rational number greater than 0". It is not uncommon to use this type of argument with propositions such as the one above, concerning the ''non''-existence of some mathematical object. One assumes that such an object exists, and then proves that this would lead to a contradiction; thus, such an object does not exist. For examples, see Irrational number#Irrationality_of_the_square_root_of_2 and Cantor's diagonal argument. It is important to note that to form a valid proof, it must be demonstrated that given a proposition ''p'', "not ''p''" implies a property that is actually false in the mathematical system being used. The danger here is the logical fallacy of Argument from ignorance, where it is proven that "not ''p''" implies a property "''q''", which ''looks'' false, but is not really ''proven'' to be false. Traditional (but incorrect!) examples of this fallacy include false proofs of Euclid's ''fifth postulate'' (a.k.a. ''the parallel postulate'') from the other postulates. The reason these examples are not really examples of this fallacy is that the notion of proof was different in the 19th century; (Euclidean) geometry was seen as being a 'true' reflection of physical reality, and so deducing a contradiction by concluding something physically implausible (like the angles of a triangle not being 180 degrees) was acceptable. Doubts about the nature of the geometry of the universe led mathematicians such as Bolyai, Carl Friedrich Gauss, Lobachevsky, Riemann, among others, to question and clarify what actually constituted 'geometry'. Out of these men's work, resulted Non-Euclidean geometry. For a further exposition of these misunderstandings see Morris Kline, ''Mathematical Thought: from Ancient to Modern Times''. Although it is quite freely used in mathematical proofs, not every philosophy of mathematics accepts ''reductio ad absurdum'' arguments as universally valid. In schools such as intuitionism, the law of the excluded middle is not taken as true. From this way of thinking, there is a very significant difference between proving that something exists by showing that it would be absurd if it did not; and proving that something exists by constructing an actual example of such an object. In mathematical logic, the ''reductio ad absurdum'' is represented as: : if :: : then :: In the above, ''p'' is the proposition we wish to prove; and ''S'' is a set of statements which are given as true — these could be, for example, the axioms of the theory we are working in, or earlier theorems we can build upon. We consider the negation of ''p'' in addition to ''S''; if this leads to a logical contradiction ''F'', then we can conclude that the statements in ''S'' lead to ''p''. Note that the union (set theory), in some contexts closely related to logical disjunction (or), is used here for sets of statements in such a way that it is more related to logical conjunction (and). In the words of G. H. Hardy (A Mathematician's Apology), "Reductio ad absurdum, which Euclid loved so much, is one of a mathematician's finest weapons. It is a far finer gambit than any chess gambit: a chess player may offer the sacrifice of a pawn or even a piece, but a mathematician offers the game." Logic Proofs Mathematical terminology Latin logical phrases
Reductio ad absurdum
Can someone please explain this: :basically: if :: S union { ¬ t } |-- F ::: then ::: S |-- t I suspect the equivalent point can be made just as well in more widely known notation, perhaps even in plain-old English. --User:Ryguasu : seems that the english version of the above is in the first paragraph. -- User:Tarquin 10:34 Jan 10, 2003 (UTC) : could you at least explain what S and t are? It probably doesn't add clarity that ''A'', used in the English version, does not figure into the symbols. --User:Ryguasu : Ok, thanks for the in-line explanation. I still think the TeX graphic and the textual description should use the same letter to represent "the statement". Changing the text might be easier, but I think, Tarquin, your choice of "t" for the statement in the TeX graphic is unfortunate, as it reminds the untrained eye of "true". Is this a routine convention for this sort of formal symbol representation? --User:Ryguasu ::I would prefer A instead of t, for the sake of consistency within the article. After all, proposition F already has the form of a capital letter. User:FvdP 20:35 Jan 10, 2003 (UTC) F is not a proposition, it represents logical False. Propositions are small letters. small ''a'' could be ok. -- User:Tarquin 20:37 Jan 10, 2003 (UTC) :I should have known (and did realize, but a bit late.) OK for a. I've had the idea to replace F with a more common sign for "false", like , but that would annoy non-mathematically readers more than it would be useful, IMO. User:FvdP :Does it make sense to replace A with a (or pehaps ''a'') in the body of the text as well? This would save us from including an overly philosophical sentence along the lines of "Take a to be the same as A." Also, if we put italic ''a'' in the text, can we make it italic in the graphic as well? --User:Ryguasu ::I (boldly!) jumped in and relaced occurrences of "A", ''a'', and ''t'' with ''p'' (for ''p''roposition), and relaced "B" with ''q''. Apologies if the term "hypothesis" is being used in a technical sense here - ''h'' just seemed less obvious as a choice. :::Ah, ''p'' ! That had become too obvious to be seen (by me). Thanks Chas for the change. User:FvdP ::I agree that seems unneccesarily obtuse here - I'd be just as happy replacing ''F'' with "''false''", i.e.: ::: ::unless that's problematic. Also, the law of the excluded middle is being invoked here - do we need to add a link to intuitionism or the like here? User:Chas zzz brown 01:01 Jan 11, 2003 (UTC) :::I have no definitive opinion on this. The interested reader can get the link through law of the excluded middle. Perhaps we should interest him more by telling a few words on intuitionism ? User:FvdP :::Even in intuitionistic logic, if , one can still conclude that . The real difference is that in classical logic, but not in intuitionistic logic. So I don't think it's a major difference in this case, or particularly worth mentioning. User:Dominus ---- I don't like too much the ''given'' in :''and S is a set of statements which are ''given'' as true'' because it reads (to me) like we're just giving them the status of being true, out of the blue, which can't be. But I don't know what to write instead: *''and S is a set of true statements'' *''and S is a set of statements which are known to be true'' *''and S is a set of statements which have been proven to be true'' *... User:FvdP 01:05 Jan 11, 2003 (UTC) :But that ultimately is all we ''can'' say; that they are given (or is it taken?) as true "out of the blue". For example (loosely speaking), "parallel lines meet at infinity" is not inherently true - but if we ''take'' it as true ''then'' we get euclidean geometry. Alternatively, we could just as easily say that "parallel lines always intersect at exactly two points"; this would then ''imply'' a different geometry. :The statements in ''S'' are either a given set of axioms which we ''postulate'' as true, or theorems following from axioms which essentially comes to the same thing (in that we assume that these theorems really ''do'' follow from some axiom set ''A''; i.e., that ''p'' can be deduced directly from ''A'' as well as from ''S''). From this point of view, we don't so much prove that ''p'' is "True", as that ''p'' is true assuming that (i.e., given that) the statements in ''S'' are true (which generally has been previously proven if needed). :By my phrasing, I was hoping to distinguish between "Truth" and "truth". Philosophically, one can argue that certain axioms are "really" true, e.g. platonism; but it is not required that one hold this philosophical view to use reductio ad absurdum. Cheers User:Chas zzz brown 01:50 Jan 11, 2003 (UTC) ::We can get rid of this "given as true" confusion by adding a second part to the if in the formal notation. It can now read ::: if :::: it is ''not'' the case that S |-- F ::: and :::: S union { ¬ p } |-- F ::: then :::: S |-- t ::Does this deal with both side's objections? --User:Ryguasu :::I have no idea what this is saying. -- User:Zoe ::::Do you understand the formal logic notation used elsewhere on this page? My comment was aimed at those who did. --User:Ryguasu The formal notation is useless and should be relegated to a footnote. Why do I push this heresy? Because anyone who can read it ''already knows'' what a reductio ad absurdum is, and anyone who doesn't know what it is has got Buckley's chance of figuring out how to read ::: or any similar formula. A plain-text example in simple English is badly needed. This should come first, so that the non-expert reader can find it immediately, and be followed by the more formal explanation. User:Tannin 02:15 Jan 11, 2003 (UTC) : Yes, much improved now Chas. It would be better still if it were a non-mathematical example, but (to my shame) I can't think of one that's clear enough! User:Tannin 09:44 Jan 11, 2003 (UTC) It's always nice when an article dramatically improves overnight! :-) The "smallest rational number" example is great. -- User:Tarquin 10:43 Jan 11, 2003 (UTC)~ ---- Text from Reductio Ad Absurdum:
A method of disproving a proposition, by demonstrating that it leads to a contradiction. An example would be the proof that the square root of 2 is irrational. Begin with the proposition that there exist integers M and N with no factors in common, where M*M = 2*N*N. If they exist, then the square root of 2 is the rational number M/N. 2*N*N is even, and therefore M*M is even. This implies that M is even, since an odd number cannot have an even square. Therefore K = M/2 is an integer. Then 2*K*K = N*N, and it follows that N is even. The equation M*M = 2*N*N cannot hold unless both M and N are even. This contradicts the initial assumption that M and N have no factors in common. ---- I'd propose moving this article to Proof by contradiction and redirecting there. ''Reductio ad absurdum'' is the historical name used in philosophy and formal logic, but ''proof by contradiction'' is the more common English name and the one preferred in most modern mathematics articles. --User:Delirium 22:51, Oct 15, 2003 (UTC) ---- I used to be appalled at seeing people (non-mathematicians, mostly) use the term to refer simply to an absurd argument, seemingly translating the Latin to "absurd reduction" instead of "reduction to the absurd". However, I've noticed that Merriam-Webster's lists a second (presumably less formal) meaning of the term as "the carrying of something to an absurd extreme". I'm curious to find out whether this is an instance of a sloppy translation gaining status as an acceptable meaning simply through widespread usage. If anyone can shed some light on the non-specialist use (or abuse) of the term, it could be an interesting aside to this article. User:Rajneeshhegde 04:15, 19 Aug 2004 (UTC) I've never heard it misused like that. Note, however, that the mathematical use is not the primary one. It is mainly used in philosophy. User:Chameleon 09:06, 17 Nov 2004 (UTC) I arrived at this page to find out if there were non-mathematical uses of this term. Specifically, where one tries to kill a project, by "improving" it to the point where it must fail. For instance, take a product that breaks even with costs of $0.90, and a sale price of $1, and improve it so that it now costs $1.05 to make, has to sell for $2, and now is a failure. Perhaps there is another term for that kind of effort. User:Mcr314 02:37, 4 May 2005 (UTC) ==Nazism== I'm out of my depth, so just wondered if anyone else felt that using ''Nazism'' in the article seems a bit odd, especially against the ''flat earth'' thing. --bodnotbod\">User:Bodnotbod_»_
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