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| Author | Topic: Question.... (Processes of Logic) | |||||||||||||||||||||||
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John Inactive Member |
quote: But that isn't what he is talking about. In fact, what you've done isn't really proving a negative. It is a round about way of proving a positive, or of proving something about a positive-- that positive being the number you choose to test. Throughout the operation you have values, data, and other such things with which to work. What is meant by 'proving a negative' is different. Consider: Someone tells you that a giant lives in the mountains, but this person offers no proof but instead challenges you to disprove the existence of the giant. If the giant does exist then he will leave evidence and can be proven to exist. This is proving a positive. The giant could also be proven to be something other than a giant-- like a hoax or the result of the consumption of a peculiar fungus. This also is proving a positive. You have proven that something other than a giant can account for the alleged sighting of a giant. But you still haven't proven the non-existnce of the giant, but only that you have no evidence for the giant and do have evidence for other explanations. A believer in the giant could always say 'that was a hoax but the giant is real. Where is the proof that he isn't?' Here is the problem. If the giant does not exist there is no evidence, and there never will be. If the giant does not exist, he leaves no trace. There is no evidence to examine, and no information with which to construct a proof or a disproof. You can't prove, or disprove, a negative, because there isn't any evidence with which to work. So we are stuck with proving, or disproving, positives. ------------------
No webpage found at provided URL: www.hells-handmaiden.com
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John Inactive Member |
quote: Or... there IS a prime number larger than the assumed largest prime, which is exactly what you conclude:
In both cases, there is a number that is prime that we haven't taken account of larger than what we thought was the largest prime. What you have proven is a positive statement, then flipped it, but I can see how this is largely semantic. I don't have a problem with your examples, but they don't address the issue. What crash is refering to is the use of a logical fallacy called Appeal to Ignorance.
Appeal to Ignorance (Proving a Negative): an argument that asserts a claim is true because no one can prove it is wrong; this shifts the burden of proof to the audience or opponent rather than the claimant
No webpage found at provided URL: http://chuma.cas.usf.edu/~pinsky/logicguide.htm quote: Precisely why 'proving a negative' is fallacious. In the real world, objects are always ill-defined, or can be argued to be ill-defined, because we have incomplete knowledge of pretty much everything. Just look at the way creationists argue for a young earth or for God.
quote: Ummm... no it doesn't. I think you are confusing mathematics with logic or with existential propositions-- somewhat in line with statements made by Mr. P. Take, for example, two statements. 1) John exists.2) Not-John exists. ( Not-John taken to mean the negative of John, rather than something like 'a person who is not John' ) Does the first imply the second? Nope. In fact the first and second are mutually exclusive. But take only the second statement. It, in itself, makes no sense. The statement contradicts itself. John can exist, but Not-John cannot. The only real value-- the only value with existence or potential for existence-- is John  . Not-John is just short-hand for the absence of the positive value. It is not a statement of the existence of the negative value. It doesn't work like numbers on a number line with zero in the middle, negatives on the left and positives on the right. Once you get to zero-- nothing, not existence-- that's it. Well, I haven't really said all I'd like but I must leave for work. Take care. ------------------
No webpage found at provided URL: www.hells-handmaiden.com
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John Inactive Member |
quote: So, the negation of the statement...
quote: ... which would be "there is a prime number larger than the assumed largest prime" is somehow a demonstration of proving a negative? You've proven a positive statement and reworded it.
quote: You apparently missed the second part of my sentence-- "because we have incomplete knowledge of pretty much everything." We have incomplete knowledge simply because we do not have absolute knowledge.
quote: You are not making existential statements. 'X is not in the room' is not not equivalent to 'X is not.' Look at the structure of the arguments you make. They all depend upon examining what is present and detectable. You do not examine what is not detectable to prove that it is not there. You prove what is there. The rest is in limbo. You look at what exists, not what doesn't. Try proving that an object doesn't exist without refering to objects that do. You can't do it, because there is no information. The only things for which we have evidence are things that exist. Thus the only things we can prove are things that exist. We cannot prove or disprove things that don't exist.
quote: Usually.
quote: You are not making existential statements. An existential statement is of the form 'X exists' or 'X does not exist.' 'X is not Y' is not equivalent to 'X is not.' It doesn't even matter if X and Y are mutually exclusive, because we have have logic wrong. We may have math wrong. And it appears that there are sections of the universe where both are true.
Forbidden quote: It IS the relevant negative! This is what I have been trying to explain to you. You commented on crash's statement that one cannot prove a negative by stating that one can prove that 'X is not Y.' In context, 'God is not a Buick' ( but Jesus built my hotrod ). But the relevant formulation would be 'X is not.' This cannot be proven. A thing's existence can be proven with one case, but its non-existence cannot. The proof could be just around the next corner. You only need one apple to prove that apples exist. You need absolute knowledge to prove that they do not. To prove that purple-people-eaters do not exist you'd have to search not only here but across the entire universe from its beginning to its end and down to its very basic foundations. Even then, all you have is an absence of evidence, not evidence against. There may be other universes... and so on and so on. You can prove that an object is inconsistent with our understanding of the universe, but our understanding could be wrong. This is a relative statement, not an existential one. The orbital path of Mercury is inconsistent with Newtonian orbital mechanics. Does that mean that Mercury doesn't orbit as it does? Nope. It means Newton had a fly in the ointment. Einstein's ideas have replaced Newton's. The same thing could happen to any of our ideas. Logic, math, whatever-- it could all be very wrong. Can you prove that some as yet undiscovered theory for which, obviously, we have no evidence will not replace Einstein's theories? Nope. There is no evidence. That is a major snag when trying to prove a negative. Without absolute knowledge, we can't be sure and we are far from absolute knowledge if such a thing is possible at all.
quote: So, if John exists then John exists? Brilliant. You quite rightly point out that a thing implies the negation of its negation-- a trivial transformation. It does not imply the existence of anything that is not itself. ------------------
No webpage found at provided URL: www.hells-handmaiden.com
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John Inactive Member |
quote: Actually, what you have prove is that it IS a square. You said this yourself-- "I show that it is a square." And then the slight of hand... "X is not a circle." This is a qualitatively different statement. To illustrate, the phrase "X is a square" tells you precisely what X looks like when drawn on a flat sheet of paper. The phrase "X is not a circle" eliminates one of an infinite set of possibilities. The former statement is precise and can be proven, the latter is imprecise and can only be inferred by proving something about what X actually is. This is what is meant by the statement that you can't prove a negative. In other words, you prove a positive-- you prove something about the object in question, then make the assertion that within some some axiomatic system inconsistent properties cannot also be true. You cannot prove the negative other than by proving something about the object. Even indirect proofs work the same way. You prove something about the object-- in the case of an inderct proof, a hypothetical object. You prove that it IS contradictory within some axiomatic system or another. Certainly you are aware that things which are contradictory within one axiomatic system are not necessarily contradictory within other axiomatic systems? Parallel lines never converge or diverge on a flat surface, but curve that surface and they do one or the other. This is complicated by the fact that there is no axiomatic system for the whole of the universe-- at least, there is no such system to which we can currently appeal.
quote: At the subatomic level, we have very little idea of what they "look like."
quote: We have absolute knowledge about nothing at all. We have adequate knowledge ( crossing my fingers ) about quite a few things, but that is not the same thing.
quote: It is a conditional. It is conditional upon the items in the set. I provided a link to the following earlier. I assumed you read it. Guess I was wrong.
A negative existential statement has the form 'N does not exist,' where 'N' is a referring expression.
Forbidden quote: This...
Look at the structure of the arguments you make. They all depend upon examining what is present and detectable. Has everything to do with it. All of your efforts go into proving what is present. None go into proving what is not. Why is that? 1) There are a finite number of 'is presents' while there are an infinite number of 'is not presents.' 2) Proving directly what is not present would require having information about what is not present, and such information does not exist. Information only exists for what actually is present. Within a limited set-- say, a collection of marbles-- where you have knowledge of what lies outside the set you can determine if something is not in the set of marbles. But in an absolute sense, as in the case of the universe, we have no knowledge of what isn't in the set-- only of what is. This is what is meant by "you cannot prove a negative."
quote: This is a textbook case of the fallacy of 'proving a negative.'
Appeal to Ignorance (Proving a Negative): an argument that asserts a claim is true because no one can prove it is wrong; this shifts the burden of proof to the audience or opponent rather than the claimant.
No webpage found at provided URL: http://chuma.cas.usf.edu/~pinsky/logicguide.htm In other words, lack of evidence for is not evidence agaist. Want more?
Argumentum ad Ignorantiam [appeal to ignorance] To recognize the fallacy of appeal to ignorance, look for a conclusion based upon an absence of proof or evidence. Be aware of the two types of cases in which lack of evidence for S is relevant to the truth or falsity of S. But also, be aware that the onus of proof is on the claimant and that no one can prove a negative. Page Not Found | Embry-Riddle Aeronautical University - Faculty quote: You haven't demonstrated that it doesn't exist, only that you have no evidence for its existence. Review the fallacies cited above.
quote: No. It proves that we have no evidence for the thing's existence. Two thousand years ago, no one on Earth had evidence for sub-atomic particles. Does this mean that sub-atomic particles did not exist? Thus, the reason it is a fallacy to conclude that something does not exist because you have no evidence for its existence.
quote: Yeah, sort-of. Via the rule of inference called addition ( in the symbolic system set out by Irwin Copi ) if you have a proposition A, you can add anything to the right of it seperated by 'or' -- k, l, x, whatever. Then, if you can infer the negation of one side, you can infer the other. For example: p v q~p Therefore, q Now, look what happens when you use A v ~A. A v ~AA Therefore.... nothing. This isn't a valid operation. A v ~A~A Therefore, ~A. This one is pointless. We already had ~A. It is tautalogical. What you are looking for is an inference from lack of evidence, and such cannot be done. Also, if that is an exclusive 'or' which I assume you meant it to be, then yes, we assume it to be true. But this isn't logic, this is the foundation of logic. This is the stone upon which logic is built. It cannot be proven. Sub-atomic particles appear to violate this rule quite frequently, in fact. Logic is an artificial system. It deals with propositions irregardless of their actual existence.
quote: Because I can't get you to understand that proving what IS does not prove what ISn't. Proving what is, only proves what is; or proves something about what is. That you have a circle, may imply within our system of mathematics that this circle is not a square. It does not prove that squares do not exist or even that circles and squares cannot exist, in reality, in the same object; but proves only that our mathematics forbid it. A property of the circle-- a positive thing-- is that it doesn't have the properties of a square. You haven't proven the non-existence of squares.
quote: Something like that. I am trying to get you to prove a negative without reliance on an axiomatic system. Prove an apple. Point to it. Pick it up. Now prove that apples do not exist. You can't point to not-apple. You can't pick up not-apple. In fact, there is no information about not-apple. There is only information about apple. Why can't you rely on an axiomatic system? Because no such system can be proven. All are built on assumptions and it is possible to build self-consistent systems that are mutually contradictory-- plane geometry vs. various curved-surface geometries. Proving something within plane geometry only proves it within plane geometry, not within curved surface geometries.
quote: Different use of the term... I defined my meaning long ago. You should have figured that out by now, but in case you haven't:
The person making a negative claim cannot logically prove nonexistence. And here's why: to know that a X does not exist would require a perfect knowledge of all things (omniscience). To attain this knowledge would require simultaneous access to all parts of the world and beyond (omnipresence). Therefore, to be certain of the claim that X does not exist one would have to possess abilities that are non-existent. Obviously, mankind's limited nature precludes these special abilities. The claim that X does not exist is therefore unjustifiable. As logician Mortimer Adler has pointed out, the attempt to prove a universal negative is a self- defeating proposition. These claims are "worldwide existential negatives." They are only a small class of all possible negatives. They cannot be established by direct observation because no single human observer can cover the whole earth at one time in order to declare by personal authority that any X doesn't exist.
{Replaced long display of URL, with "Link", to restore page width to normal, also fixed another quote box below - Adminnemooseus}
Link quote: But the relevant formulation would be that there is no case where X exists. Period. Can you prove that no right triangles have angles adding to more than 180 degrees? Yep. Can you prove that no right triangles have angles adding to 180 degrees? Yup. You can prove that too. You just have to switch geometries. But these are not proofs or disproofs of real world objects. These are abstractions.
quote: You should consider what happens at a singularity, or inside the event horizon of a black hole, or at sub-atomic scales. Illogical things, as logic is defined, happen all the time. Nature glibly ignores your convictions.
quote: LOL.... then it is OK to divide by zero. That is what happens at a singularity when some values go to zero and others go to infinity. Seriously, you are convinced of this why? We've invented math over the past few thousand years. There is no guarantee that what we have invented is an accurate representation of the world.
quote: Which is shorthand for "For every number that is prime, there is a larger number that is also prime." Simply rephrasing it doesn't change the meaning.
quote: I think you misread the article, but it doesn't matter. The point was only to show you what a negative existential statement looks like.
quote: Ummm... you jumped in, rather arrogantly, to explain to crash that he was wrong to claim that you cannot prove a negative. So, no, you don't get to be arbiter. If you are arguing a different negative, then you are not arguing the POINT. Remember the first thing I said to you? "That isn't what he is talking about."
quote: ... that you are stubbornly arguing the wrong point-- wrong in reference to the meaning of the phrase that started this exchange.
quote: You are confused. This is equivalent to "X is not Y." Have you ever heard anyone say "Prove that God is not a piano, a redhead, a martian?" Nope. People say "Prove that God does not exist." This should illustrate to you the inherent differences of the claims you are making and what is intended by the phrase "you cannot prove a negative." It is about existence and non-existence, not about qualities.
quote: This is kinda silly. If Y is all that can be --> Everything that exists is Y.Then if X is not Y --> X is something that doesn't exist. Then X is not --> Then X is something that doesn't exist. The conclusion is assumed in the second premise. What you want to do is prove Y, or define Y. That means observing and recording everything that is, was and will be across all of space and any other spaces there might be. Once you have done that, you still cannot infer that X doesn't exist, or that other things do not exist. All you have is a lack of evidence. Inferring based on lack of evidence is a fallacy.
quote: No, I believe I am talking about something quite different from either.
quote: Just for the record, do any numbers exist? Does 3 exist, or is it a abstraction? Part of a system we've made up to keep track of things? Saying a number exists is not like saying a rock exists. Even your proof of 'no largest prime' works by proving what DOES exist within the rules of the system. Remember?
quote: Only if you include within that definition "within human knowledge" ( or some equivalent ) will this work. In other words, you are no longer making a claim of existence or non-existence, but a claim of this-case or that-case doesn't exist. The relevant claim is that NO CASE EXISTS.
quote: There is always a larger prime.
quote: That's right, BY DEFINITION. This may be the case on a plane as we define it, but does such actually exist anywhere? That is the element you seem to be missing.
quote: Now perhaps you are getting closer to what I am trying to say. How does one, in the real world, get a solid definition of something that does not exist? There is no evidence of PPE's, thus there is nothing upon which to base a definition. There is no definition, there can be no test.
quote: With things that do not exist you can't really know what must happen if they did exist. You have no evidence, no way to investigate the thing that doesn't exist. How can you know what must happen? You can guess, sure. Fine. A lot of science works this way. If your test works out you have positive evidence for a thing's existence, hitherto unknown. If your test fails, you have lack of evidence for, not evidence against a thing's existence. For example, lets test for ghosts. If ghosts exist we ought to be able to film them? Well... why? We don't know anything about ghosts. We have no evidence. Perhaps we've proven that ghosts aren't caught on film, rather than that there are no ghosts. If ghosts do not exists we will never have evidence concerning them, if ghosts do exist we might never have evidence. That depends upon the nature of ghosts, but it is only if they do exist that we will ever have direct evidence with which to work.
quote: When making absolute claims one must include every case imaginable. This is not fallacious. And certainly isn't this fallacy.
quote: Do you assert that we have infalible knowledge? Unless you do, you agree with my statement. We could be very very wrong. ummm... you might brush up on your informal logic, because you've mis-applied this one as well.
quote: Newton's mechanics are a description, not a thing. Strike this one up as "still just not getting it."
quote: You apparently missed Gdel. Ever hear of the Russell paradox? Math and logic are riddled with contradiction and based upon assumption. Why exactly can these not be wrong?
Page Not Found quote: No. I am not claiming that one ought to be able to prove anything. However, it is this sort of negative that is relevant to the statement crash made.
quote: A sufficient definition would require absolute knowledge, even to determine whether or not it is sufficient. We 'solve' this by introducing assumptions-- things like 'within the confines of plane geometry' or 'visible things inside this room.' We limit our scope. But it is assumption.
quote: What you prove in this paragraph is that either the definition is wrong -- ie. it does refer to a real thing but that are mistaken about what that thing is-- or that Santa doesn't exist. In the absence of Santa's existence, there is no way to sort out the definition. The only way to sort out the definition is to have a Santa to investigate, which of course, would mean you were investigating and proving a positive. ------------------
No webpage found at provided URL: www.hells-handmaiden.com [This message has been edited by Adminnemooseus, 05-10-2003]
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John Inactive Member |
quote: And you've missed the point is several different ways.
quote: This is true only within a defined ruleset-- a defined ruleset which, by the way, is itself self-contradictory. There is no guarantee that it applies to the real world. Sorry. Your incredulity is irrelevant. The second way you missed it, and more directly the intent of my statements, is that the functional component of a proof-- any proof-- is the proving of something about the relationships between your premises-- proving a true statement about that/those relationships. Any proof, if it is correct, is a true statement-- true not false, positive, not negative. Constructing a proof, by default means constructing a true statement. One can construct a hundred false arguments-- the conclusion does not follow from the premises-- and it simply does not matter. The arguments are irrelevant. They don't matter. The only ones that do matter are the true ones.
quote: And you have to think more carefully about the statements being made. If I say 'square' you know what I mean. You can draw it. If I say 'not a circle' it conveys virtually no information. It eliminates one of an infinite set of shapes. They are quite different statements. If you can draw 'not a circle' in the same way that you can draw 'square' -- with a compass and strait edge-- then lets see it. Until then, you cannot claim they are the same type of statement. The fact is that the only things you can produce are things-- circles, squares, etc. And you could draw shapes to infinity and beyond and not run out of shapes. But you wish to draw upon the limited set of shapes, generalize to all shapes-- that is what geometry is-- and conclude that nothing violating your sample set of shapes can exist. This is absurd. It is generalization from specific to universal. You'll find that that is fallacious. All we can do is accept the information we have, draw conclusions, and run with it, all the while knowing the conclusions must be tentative and the next turn may change everything.
quote: If you want an adequate definition, you need to include all components in the definition.
quote: You are not using a relevant definition and are stubbornly refusing to use, or to even attempt to understand, the relevant definition. Skipping the Santa bits as you've misread the article and once again missed the reason I posted the article in the first place, which was to demonstrate a negative existential statement of the type at issue.
quote: What you have proven is that if there is a largest prime our system of mathematics is contradictory. Ignoring that our system of mathematics may be contradictory-- and in fact is-- lets look at how your proof works. You assume a largest prime and prove a contradiction, and set this up so that it applies to any prime you fill into the blank. What you've actually got is an infinite series of assumptions and an infinite series of proofs that there is a larger prime than the assumed one, or for any prime there must be a larger one. You can translate this into English as a negative, but that isn't how the proof works. The proof is an infinite series of positives.
quote: So... do this in the real world. ''
quote: Holmes was not really much of a logician.
quote: Because talking about things for which there is no evidence is just making things up. Why is this so hard to understand? A idea remains a phantasm, a thought experiment, until there is some positive proof of its existence. You can bitch and moan about this or that being impossible, but educated people throughout history have done the same; and been wrong about it. Any statement about what is imposible, or possible even, is conditional upon our understanding of the universe. If a relevant portion of that understanding is flawed, then the impossible suddenly becomes possible. Only when there is evidence for a thing does it step out of that limbo and into actuality.
quote: Can you? How? How do you know the properties of a square? Initially, someone measured one, then another, and generalized. Same with circles, same with triangles. And we end up with something we call geometry. It is the generalization from specific cases to the universal and as such will always be questionable. Hate to break it to you. Without a square-circle to measure, you can't know with certainty, its properties. You can't know if you've named all of them, or if you've got all the properties correct. You can't check your assumptions, in other words. What you've got is thought experiment. What you've got is "if the assumptions of plane geometry are true, then square circles can't exist." Notice how the whole structure is conditional, and that there is no guarantee that it applies to the real world?
quote: hmmm... if space were to be contracted to a point as it is a a singularity, would not the dimensions of a square and a circle match? The radius of a circle under such circumstances would be zero, as would any measurement you made of a square. Thus, they would appear to be the same.
quote: Why are you equating definition with existence?
quote: It is getting damned frustrating to attemp a debate with you when you choose to remain ignorant of the meaning of the phrase which started the debate. You can't win by arguing against the wrong definition. Additionally, you really ought to brush up on what is meant by 'ad hoc.' There is some question as to whether such hypothesis are technically fallacious.
While "ad hoc hypotheses" are not usually called "fallacies", "ad hoc" is definitely a term of criticism, that is, it is a bad thing for a hypothesis to be ad hoc.
No webpage found at provided URL: http://gncurtis.home.texas.net/2003_01_01_archive.html Strictly, 'ad hoc' means 'for this case only.' No one is making that claim. The claim being made is we don't know what is outside our experience. Since our experience is limited, we have to conclude that our claims knowledge are tentative. There may be things we don't know. To do otherwise is to commit the most-definitely-a-fallacy of generalizing from the specific to the universal. This ought to be common sense, though such seems to be none too common.
quote: Frequently.
quote: Could be. Of two things which appear to us to be the same, or make no difference, one or the other may well be true and the other false. Whether we are ever able to sort this out is another thing. This is a rule of thumb, not hard logic.
quote: Here you go agian, refusing to understand what is being stated.
quote: What would those cases be? The author doesn't say, and I can't think of any. Nor can I find a similar statement elsewhere, as here:
Page not found - Intrepid Software Surely you must know what these two cases are, as you know that your case falls into one of them. Or perhaps this is just what it looks like-- avoidance.
quote: ... that you are proving only within a defined rule set. And that you are translating positive mathematics into negative English and calling it proof of a negative.
quote: More like 'assume.' Unless you have absolute and infallible knowledge you are arguing from ignorance.
quote: How is it not a false dilemma? For every real thingie-- say, cat-- there are three possibilities. 1) Evidence we have that supports the proposition. One case of 'cat' and its over. 2) Evidence we have which does not support 'cat.' Finding a dog does not prove or disprove 'cat.' Finding that dog is not 'cat' does not prove or disprove 'cat' thought you seem to think it ought to. 3) Then there is evidence we don't have which might go either way. Until we have the evidence, we won't know which way it goes. Until we have absolute and infallible knowledge, there will always be this option. What you want to do is deny the third possibility. You can't do so without making the idiotic claim that we have absolute and infallible knowledge.
quote: No kidding?
quote: You really aren't getting this are you? You can't elucidate all possibilities-- ever, not in the real world.
quote: Boy, do you love the trivial! The 'item is not P', or 'we do not have P', is not the same as 'P is not.'
quote: Sorry, but it is. A thing is a thing is a thing is a thing. There is no logic involved. It is saying the same thing with extra words. A tautology is true no matter what. Its truth value is built in-- assumed a priori. It can be a fallacy, but isn't always. Inductive systems are based on such assumptions. Doesn't make them true in the real world. They are just assumed to be.
quote: There are no processes sufficiently defined. Assuming certain limitations is necessary but you don't get absolute proofs, only conditional ones, ones conditional upon your assumptions.
quote: Then try doing it the other way around. There is a rat on my desk who keeps walking over my keyboard. Prove what he isn't, to prove what he is. He isn't a cat. He isn't a bird. He isn't a she. He isn't a martian. He isn't.... You can never complete the task. That is a large part of the point. That should tell you there is something quite different about the two statements-- "He is a rat" and "He is not a rat." You can prove the first without bothering with the second. But trying to prove the second without bothering with the first leads you into an impossible task.
quote: Mathematics is a made up system. It is based upon assumption and is internally inconsistent. How exactly can it be SUFFICIENT?
quote: Indeed. You've been constructing and arguing against such straw men since this discussion began. And now appear to be getting a bit testy that we are't taking you scarecrows seriously.
quote: You've got the analogy backwards. You've given a positive proof-- that of some principles of aerodynamics. You've given proof that such objects can fly. The difficulty would be if I tried to prove it impossible for heavier that air objects to fly. A few hundred years ago I might have been able to do so, based on all the best human knowledge; but we all know that that proof would have been flawed. To prove that something can happen, you just have to find a case demonstrating it. You don't have to understand all the reasons why. You can stumble upon an example and figure out what is going on later. To prove something can't happen, you have to assume that you know all the relevant information. It is this assumption I am asking you to drop.
quote: Then you are not arguing the fundamental point.
quote: You are funny guy. You stubbornly refuse to have the statement explained to you. You argue a different point. And complain that we don't argue this different point. Did you mean this to as funny as it is? Or should I be laughing at you, rather than with you?
quote: As illustration of the limitations of axiomatic systems it is quite appropriate. You assume one system and insist that it describes everything everywhere. But there are multiple such systems and none of them, necessarily, apply to the universe.
quote: Because it is quite easy to prove that something does exist without relying on a system of axioms. One just have to find an example of the object. If the object doesn't fit into some pre-existing logical construct, then tough. The object exists. There it is. It can contradict everything we think we know. Doesn't matter. There it is. It does not work the other way around.
quote: Then there is not much debate, and you have admitted to arguing a straw man.
quote: BS. There is absolutely no reason real things have to be consistent. And at very fundamental levels things appear to be radically at odds with what we'd call consistency.
quote: You... YOU... jumped into a discussion that was already in progress. You used definitions not like those in use ( though not directly stated ). You refused to accept definitions appropriate to the discussion, and instead insist that we use yours. Amazing ... !!! It is you who is arguing a straw man.
quote: You have got to be joking?
quote: Actually, this is pretty obviously your error.
quote: Question: Do you know that Platonism isn't a mathematical concept?
quote: This is just about the most pitiful thumbnail version of Platonism I have ever seen.
quote: So... there is a largest Real number then?
quote: And you know this how? Because you say so? Because you think so? I don't know if there is an internal logic or not, but that logic, if it exists, is damn sure not the logic we know and love. I'm not sure what to call this... equivocation? Arguing from ignorance? Just plain 'it is cause I say it is'?
quote: So a system of definitions has existence outside the minds of the people who made made up the definitions? You're joking.
quote: Sorry, but you've been implying it all along. Every time you insist that something is mathematically or logically impossible and thus it IS IN FACT impossible, you are implying that math and/or logic is in fact an accurate representation of the world.
quote: This is the root of the most common problem with attempting to prove a negative and probably the impetus for most chat-room utterances of the phrase. You do end up being asked to disprove an infinite series of 'nots.' And there is no way to settle on a definition if there is nothing to define, if there are no dimensions that can be measured-- whatever. This is the colloquial meaning of the phrase, if you will.
quote: But to make some claim of actual existence or non-existence, you do. Otherwise what you have is a generaliation from the specific to the universal. I can't be more plain than that. This is induction's dirty little secret. If you want to make the conclusions conditional upon some set of, possibly, arbitrary definitions that is another thing altogether. And I don't think this is the case. Your statements about mathmatics make my think that you consider math to be far from arbitrary.
quote: Why must you confuse definition with reality?
quote: Your whole point is that you manipulate definitions? You rest your case upon the manipulation of definitions? Defining doesn't make a thing, just describes it-- in a very limited way, usually.
quote: I made a statement about existentials. And defined the usage at great length. I believe it was a logician named Adler(?) who first useged the phrase as I have. You've thus far ignored this and continue to harp on another usage. This is textbook equivocation on your part, though I doubt it is intentional. You must pay attention to meaning and not get stuck on the language. Browse through the dictionaries at Onelook. There are more ways than yours to use a word.
Definitions of existential - OneLook Dictionary Search quote: Numbers exist? Show me one.
quote: 'cause there is always a larger one. What is your point?
quote: ??????????
quote: Definitions don't make the thing. We make definitions to fit the thing, not the thing to fit the definition. You appear to be equating our ideas of things with the things themselves.
quote: You actually believe we have infallible knowledge of something?
quote: I know you are a mathematician. It shows. You reek of it. Credentialing? Seems you are the first one to get to that. I haven't said anything about what formal training I have had or haven't had, not to you anyway. So... bullseye on that argument from authority, bud.
quote: Deductive logic is basically subtraction, hence the name. Inductive logic is an inference from specific cases to the universal. Math is essentially the same. We count five rocks, then split the pile and count again. hmmm... two in one pile, three in another. And so on. Eventually we-- our ancestors, really-- recognized a pattern and formalized that pattern into a basic arithmetic. This is a generalization from the specific to the universal and it is a violation of one of the rules of deductive logic. Now, take i. This is a number which when multiplied by itself equals negative 1. hmmm..... but a positive times a positive is a positive. And a negative times a negative is a positive. So i quite blatantly implies a violation of the rules of multiplication. How is that for consistency? # added by edit
quote: Ya sure about this?
And for any question A, for any affirmation A, you can ask if it's possible to settle the matter by either proving A or the opposite of A, not A. ... That's called completeness. ... Another interesting question is if you can prove an assertion (A) and you can also prove the contrary assertion (A). That's called inconsistency, and if that happens it's very bad!
Page Not Found Inconsistent does not look synonymous with incomplete. You are saying that there is a transformation that allows you to equate 'can't answer one way or the other' with 'can get two answers that contradict one another'? ------------------
No webpage found at provided URL: www.hells-handmaiden.com [This message has been edited by John, 05-15-2003] [This message has been edited by John, 05-15-2003]
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John Inactive Member |
I know I took a long time to respond, perhaps he's just taking his time. I hope so. I was having fun.
Don't suppose you'd like to play devil's advocate? ------------------
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John Inactive Member |
... but I was having so much fun!!!
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John Inactive Member |
quote: Well, no, not really. Just explain how i does not violate the basic rules of multiplication.
quote: Now, why should I be careful? Godel's proof may not apply to all systems but it damn sure does apply to those systems in question-- those very systems you rely upon. So why should I be careful?
quote: Godel's proof works for systems that are almost trivially simple, but you know this. Why are you pretending that something significant can be constructed that does not violate that proof? Sounds to me like you are just being diversionary, and effectively just skipping around the issue-- which, if you've still missed it, is that logic and mathematics which you have made out to be absolute and infallible are the very things which get screwed by Godel's proof.
quote: What are you talking about? Seriously, I don't know where this came from.
quote: Let me put it this way, when you attempt to prove a conclusion, or formulate an argument, do you attempt to create true argument, or a false one? If you attempt to create a true argument, you are by default proving a positive. Reductio ad absurdam works similarly. You have an argument you hope to be true. You reverse the conclusion, thereby creating a false argument. If indeed it does prove to be contradictory, you can infer that the original arguement was true. And you are in the same position-- that of proving a positive. But even when you have reversed the conclusion and found a contradiction, you still have a true argument-- false premises, false conclusion, true argument.
quote: Yes. Godel again. Thank you for pointing that out. But why? I am not claiming that math, or logic, is an unquestionable arbiter of truth. This doesn't hurt me.
quote: Prove any statement, about your hair, that is not true-- the statement as a whole, not just 'the conclusion has a ~ in it."
quote: Why, if something is true, is the ONLY way to prove it true by proving something else false? Doesn't make sense. The only practical way, perhaps. It can certainly be easier, but it is all conditional upon whatever system you used to make the inference. There is no guarantee that that system is correct.
quote: Different from your point, perhaps, but germaine to the issue that started this in the first place.
quote: ummmm.... I believe that I did. And, in fact, you brought up drawing with a compass and straight edge in an earlier post. It was just up in post #48.
quote: And do you understand what is sufficient support for a universal statement such as X can or cannot exist? You need have absolute knowledge of the universe. Or have a guaranteed system for working out answers, which would be the equivalent of absolute knowledge. You have neither, we have neither, but you still insist on making absolute claims about existence.
quote: I never said we were using a existential mathematical operator. You assumed that and have been tripping over it ever since. I, as I have explained repeatedly, used another meaning of the term 'existential.' It is a meaning introduced to this discussion by crash or maybe Mr. P. and, I believe, first used in this way by a logician named Adler. It is comical that you can't grasp this.
quote: So you agree that if there is a largest prime then our system of mathematics is contradictory? That was the question. Now, how is it you are sure that our system of mathematics is NOT contradictory? Assumption? Ya just can't imagine it being otherwise? You have a modus tollens. P = largest prime. M = math is contradictory. So: if P, then M.~M therefore ~P. The trick is getting that not-M. You have valiantly avoided the fact that it simply isn't true. Math is contradictory. Lots and lots of people have been struggling with the consequences for a nubmer of years.
quote: The proof depends upon the fact that you COULD do so. That is what variables are all about. No amount of semantic BS will change that. The first conclusion drawn in your proof is that there is a number larger than the assumed prime, whatever that assumed prime may be. That is an infinite series of positives, which you then translate into English as a negative.
quote: All that math has given you a false sense of precision. In the real world, you'll never pin down 'me' or 'not-me.' There are infinite elements to the definition. Of course, you will object that not all elements need to be included. For practical purposes, this is true, and you end up with grossly truncated definition. But for statements that are universal absolutes, it is not. Every time you claim something can or cannot be because math says so, you are making a statement about a universal absolute.
quote: Geez!!!! I've said so more times than I remember. There is no reason to believe that math is anything more than a symbolic system we made up, and of course we made it match the world we observe but there is no guarantee that we got it right or that it applies to the whole universe.
quote: Nor does it mean contrary or variant theories are false, as you appear to be arguing.
quote: Science is the attempt to reduce the complex world to formulas simple enough that we can comprehend them. We adopt the simple an elegant as 'the truth' but the absurdly complex might actually be true.
quote: Admin(s), not long before you arrived here, requested we keep the size of messages down. Drive space costs money. Sorry. I know it can be hard to follow.
quote: Do you actually believe that Euclid invented geometry? Euclid died in 265 BC. The Egyptians had knowledge of geometry 2500 years earlier. So did the Babylonians, so did the Sumerians, so did Greeks long before Euclid. Do you actualy believe the Egyptians, who made damn near square bases for there pyramids, had no knowledge of geometry? Come on! This statement is staggering in its ignorance.
quote: Lol... Where do you think Euclid, and the mathemmaticians from whom he draws, first got these definitions? Pulled them out of thin air? Got drunk and made up stuff? Nope. The definitions were matched to what they observed around them. That is why euclidian geometry is that of a plane. The world Euclid could observe was planar. That is also why no one questioned Euclid until fairly recently. That is also why euclidian geometry ain't universally applicable the real world-- the real world, on a large scale, isn't what Euclid observed. He didn't have the technology to observe anything different, so he created an abstraction of his tabletop, essentially. And it is a generalization from the specific to the universal-- assuming, of course, that you consider it to have a relationship with the real world. Euclid drew from, say, 20 percent of the world's mathematical knowledge-- or from a small chunk of one planet-- and created a geometry that supposedly applies everywhere. How is this not generalization from the specific to the universal? Now, you want to call Euclids postulates 'definitions.' How does this help? It only makes the whole structure one more step removed from reality. And you want to argue that one can determine truths in the real world via systems such as this? If geometry is only definition, why should anyone care? Definitions can be made to prove anything. There are no rules. Invoke observation and you are right back to over-generalization.
quote: Bud, you are making my case. And you are contradicting yourself. You know there are unproven and/or unprovable assumptions and yet still insist that you can get ABSOLUTE answers about the real live observable universe out of axiomatic system.
quote: Perhaps not. Assuming Einstein's conception of gravity, there might not be any true plane anywhere. But, no, that isn't what I was getting at. Geometry, any axiomatic system, is conditional upon its assumptions. If ( assumptions are true ), then ( system is true ). You know this, which is why it is so hard for me to understand why you feel that you can make absolute determinations based on an axiomatic system. No matter how well the system works, the fact remains that it rests on assumption. What you appear to be doing is arguing that since something is contradictory, or impossible, within an axiomatic system, it is absolutely impossible. This amounts to 'proving' one of your assumptions-- non-contradiction--, which is circular, which is invalid.
quote: hmmm... then back up from that singularity until you have four points. Is that a circle or a square?
quote: I think it is fair to say that you have frequently argued otherwise.
quote: Do you think you might have it in you to stop whining about things of which you are also guilty? Just a thought.
quote: Do you actually know what this means? The way you use it makes me think that you don't. Ad hoc is a statement used to brush aside facts that refute an argument. Producing additional evidence or stating previously unstated evidence is not ad hoc. If it were, all of science would be ad hoc. Nor is attempting to explain a previous statement, ad hoc. Nor is introducing additional argument or producing a modified version of an argument, ad hoc. And pointing out the blatantly obvious fact that we do not know everything is damn sure not ad hoc. Now, my statement was:
Could be. Of two things which appear to us to be the same, or make no difference, one or the other may well be true and the other false. Whether we are ever able to sort this out is another thing. Do you actually deny that this is an accurate statement? Do you deny that appearances may deceive us? That our current understanding could be wrong? If you do deny these things, then... wow... what can I say? How do you wake someone from that dogmatic slumber? If you don't deny these things, it is absurd to insist on absolutes as you do.
quote: ummm... when I asked for examples did it not occur to you that I WAS ASKING FOR EXAMPLES? If you gave some and I missed them, sorry. I certainly didn't do it on purpose. More likely, you made some statement or statements which you feel qualify as the examples I am looking for, but which don't look that way to me to the point that I don't even know which ones it is to which you refer.
quote: Are you serious? Definitions constitute infallible knowledge? A definition is a tag. It is shorthand. Definitions are names. I can pick any combination of letters and make up a definition. How is that even knowledge, much less infallible knowledge? Need I direct you attention to where you claim that you do not confuse definitions with reality?
quote: I know that some textbooks have this proof in them. I don't know how many such textbooks there are, but I don't think it matters. And not having those books, I can't say why the question is asked. Am I sure, they think there is a point. But I don't think that matters either. Here is why. A postulate is "A statement, also known as an axiom, which is taken to be true without proof." ( mathworld.wolfram.com ) Yes? Now, ever heard of the identity postulate? Also know as the reflexive postulate? Now put the two together. Identity is a fundamental assumption of mathematics and logic. Proving it isn't possible within the system. The argument would be circular.
quote: I realize that. Which is why I asked the question. This is the point.
quote: Funny... I posted the full response you gave me. All you said was "How is that insufficient?" What kind of context do you want? Maybe the thread itself? Surely a smart guy like you can keep track? Personally, I think you are avoiding the fact that we invented math.
quote: Oh? Tell me. When you've been talking about geometry, is that a geometry to which Incompleteness Theorems apply? If so, it isn't me who is misunderstanding, it is you who are in denial.
quote: How do you know they have any logic at all?
quote: Doesn't follow. There may be no pattern at all, and hence no logic.
quote: ??????? Then it isn't randomness and chaos. A truly random series has no pattern. Logic is all about pattern.
quote: No. This isn't the argument. It isn't that you are incapable of disproving a finite number of things. It is that to make a statement that encompasses the whole universe you would have to disprove an infinite number of things. You can limit your set, but then you are not making a statement about the whole universe-- it is no longer a universal statement. This is what you did with the car key example. It is no longer a statement about the absolute existence, or non-existence, of the car keys. It is a conditional.
quote: I'm afraid you are. Notice every example you give of counting. Two apple + 2 apples = four apples. hmmm... try again. Same result. And again. Again. Same result. Eventually this pattern was generalized into a law, a maxim, an axiom or whatever you want to call it. This is a jump from the specific-- lots of specifics actually-- to the universal. This was the creation of mathematics.
quote: Deduction has the same failing as induction for the reasons given above. The axioms of the system had their origins in the observation of events. Those observations were generalized to universals to create the axiomatic system.
quote: A joke based on the Latin word for finger, perhaps? No. I don't see 'five.' I see fingers and I count five of them. I do not percieve any element of 'fiveness.'
quote: I get. A · ~A And. ~(A v ~A) which gives ~A · A Ok. Got it. ------------------
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John Inactive Member |
quote: A positive times a positive is a positive. A negative times a negative is a negative. i requires that a negative times a negative is a negative or that a positive times a positive is a negative.
quote: I didn't say that mathematical operators within the complex number system would be the same as within the real number system. What I said is that [/i]i[/i] itself is a problematic number, as exlained above. You've got one system with a contradictory number, or two systems incompatible with one another. The 'bridge' between the two is a number that can't exist in one system. Either way, the point is the same, there are holes in mathematics. It isn't complete. It isn't absolute.
quote: Yeah, I know that. And, as you say...
quote: Fine. Why are you avoiding the question? When you refer to geometry and mathematics and logic, are you not talking about systems sophisticated enough that the incompleteness theorems apply? In other words, the logic and mathematics which you have made out to be absolute and infallible are the very things which get screwed by Godel's proof.
quote: More whining... Sorry. I assumed you knew what you were talking about.
quote: The argument itself is true, else it would be a silly argument. You are talking about the right side of the if/then statement-- the conclusion. I am talking about the truth value of the whole statement, or the whole argument.
quote: Why are you changing the rules to suit yourself?
quote: You need to dig yourself out of math and into the real world.
quote: Because we made up the rules. The universe is not bound by our rules.
quote: Yes, in an absolute sense. You can always assume a starting point and go from there, which is exactly what we do. There is no other choice. But it is dishonest to make absolute statements when there are assumptions at the foundation.
quote: You just don't listen do you? I give you definition I used and you still don't get it? This is getting to be ridiculous. Half your argument has come to special pleading for your preferred definition.
quote: Jesus Christ!!! I rephrased the damn question for you!!!
So you agree that if there is a largest prime then our system of mathematics is contradictory? That was the question. How could you miss that? BTW, there is a little button right at the botom of the message that will take you back to all the context you could want.
quote: Do I have proof that this statement is true? Yes, and you know what that proof would be, or you wouldn't have agreed that the statement is true. Do I have proof that there is a largest prime and thus mathematics is contradictory? Do I have proof that any number exists? Nope. Kind-of a silly question really. We made them up to record quantities. They are mnemonics. But what does it matter? The point is that your 'absolutes' are conditional upon our understanding of mathematics.
quote: And? That still leaves the parts that cannot be decided. And you've already equated 'incomplete' and 'inconsistent.' Looks like you've cut your own throat.
quote: Can we stop playing games here? When I say 'math' do you think I mean some minimal system like Presburger arithmatic which only models addition? Or do you think I mean the mathematics kids are taught in grade school, high school and college? Why are you trying to wiggle around this by continually referencings some interesting but virtually non-functional mathmatical system?
quote: Until you leave the abstract you haven't really proven anything. Why do you think physicists look for observational evidence to support what has been proven in theory?
quote: Read you statement carefully and think 'generalization from the specific to the universal.' I could't have said it more plainly. Thanks.
quote: I am aware that many mathematicians share your belief. For the vast majority of people on Earth, God-- of some variety-- is real. Is that adequate reason for the belief?
quote: Don't know. If I have no sensory input of, or from, my fingers, I can't really say what they are doing or how many they are. I assume they stay the same, but I know that is assumption.
quote: Precisely where did I say I believe that by thinking really really hard I can change the number of fingers on my hand? This is pitiful, Rhhain.
quote: This little mantra is becoming very tiresome. You seem to pull it out when you don't have an response. Sad... Now....
quote: ( Copy your part ) Just because observation is consistent with prediction doesn't mean ( add my part ) contrary or variant theories are false. Why is that so hard to put together? And why is it so hard to answer?
quote: BS. You are back-pedaling. Look at the context. My post # 53:
[qs]Same with circles, same with triangles. And we end up with something we call geometry. It is the generalization from specific cases to the universal and as such will always be questionable.[/b][/quote] To which you respond in post # 64:
I hate to break it to you, but there was this guy named Euclid and he collected the works of other mathematicians, as well as coming up with some work on his own, and came up with an axiomatic system of geometry by which circles, triangles, and squares were not a "generalization from the specific to the universal" but rather were definitions. You respond to my analysis of the creation of geometry by saying that Euclid made up definitions. What does this tell me? Hint: You consider Euclid's work to be analogous with but contrary to my analysis. You consider Euclid's systematizing geometry to somehow negate the source of those axioms-- the experience of countless forgotten engineers stretching deep into the past. Despite the rant, you don't actually touch the issue. The point has been quoted above, but perhaps I should restate? The axioms did not come from nowhere. Euclid, and whomever he drew from, simplified known geometric figures into axioms-- into abstractions. This is why he constructed Euclidian geometry, instead of some other geometry. And this is why no one questioned these axioms for centuries. Plane geometry is blatantly obvious to someone living on a 'flat' surface, building square houses, etc.
quote: Don't be a child.
quote: Mathematically, but we were talking about a singularity in the real world. The universe appears to be jump-y. This is the essense of quantum theory. The world doesn't follow this mathematical law. Care to try again?
quote: Now you just being stubborn. Did you miss the part where I explained that...
[qs]If you gave some and I missed them, sorry. I certainly didn't do it on purpose. More likely, you made some statement or statements which you feel qualify as the examples I am looking for, but which don't look that way to me to the point that I don't even know which ones it is to which you refer.[/b][/quote] Grow up.
quote: Equivocation. This is absurd. Two grids of squares with numbers in them. Compare them. If the numbers match up, the two squares are the same. Don't pretend this proves the principle of identity. It is an application of that principle.
quote: So much for a straight answer. Lets see if we can't construct what you won't state plainly. "The various geometries taken individually are internally consistent." Fine. Taken individually, they are also inadequate. Each is very limited in scope, applying as each does to a particular type of surface. They are also mutually incompatible and there is no bridge between them, which there must be if we are to get a complete description of the real world, or even a complete ( colloquial ) abstract geometry for that matter.
quote: How do you know that everything must behave logically? This is nothing but 'I can't imagine it to be different.' Incredulity at its finest.
quote: Certainly. And it is horribly ill-named. I found the following definition:
Chaos is the study of deterministic systems that are so sensitive to measurement that their output appears random.
No webpage found at provided URL: http://www.gweep.net/~rocko/sufficiency/node10.html In other words, chaos theory deals with systems so complex they appear random. It is the search for pattern in apparent chaos. It doesn't follow that what appears chaotic actually has a pattern. We won't know that until a pattern is found.
quote: In the dream world of mathematics where you get to make up all the rules, you might be able to do this. The real world doesn't have to cooperate. Why can you not see this?
quote: I didn't say it was an axiom. Pay attention. It is a conclusion-- more of an observation really-- based on experience. Axioms are/were generalized from this experience. Think science. You make a bunch of observations and generalize to a rule or a mechanism-- an 'axiom.' The same process underlies the origin of mathematics.
quote: Because I don't see 'five.' I see five fingers. I see four books. I don't see 'four.'
quote: Your color analogy is ridiculous. 'Red' can be analyzed in several different ways. It has measurable properties-- like wavelength, energy, etc. 'Five' has none of those measurable properties. Build a pure 'raw number' detector then get back to me. Pass 'five' through a detector and give me a mass, frequency, particle size-- something. ------------------
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John Inactive Member |
quote: I have not once said that i is either positive or negative. If you read my posts you'd know that. You've built a nice big straw man, tossed in some irrelevant terms, and danced right around the issue. Pathetic. Why is it so hard to address the thing head on? Want another chance? i is defined as the square root of negative one. How does one get a square root? Hint: You find a number which multiplied by itself gives the number in question. Now, riddle me this... What number multiplied by itself gives a NEGATIVE number? i is a number which multiplied by itself gives -1. There isn't one. How is that hard to grasp? So what to do? We made up a new number-- one that cannot exist in the Natural, Integer, Rational, or Real number systems. And we made up a new number system within which we could use this new number. But that is just the point. We have incompatible number systems, or unconnected number systems. There are gaps. Consider Relativity and Quantum Mechanics. The two are mutually incompatible, yet both are very well supported and function just fine within limits. Still, there is a gap and while there is a gap you can't claim that our understanding of physics is not contradictory. The same is true of mathematics.
quote: You simply repeat this over and over. Don't you realize you are missing the point? For one, this is diversionary. You have refused to answer one fundamental question: When you perform some calculation do you use some minimal system like Presburger arithmetic or do you use something a wee bit more complicated-- like third grade mathematics? You keep bringing this up, but it is silly to defend one axiomatic system by pointing out that another one is complete and consistent. That tactic can't possibly make sense to you. Now, let's see... Which number systems are sophisticated enough to model arithmetic? How about... ARITHMETIC?
quote: Fine. Do you argue that the whole is not inconsistent because some of the component parts, taken individually, are consistent? It appears that you do. Do you think a car is reliable when some of the parts work, some are missing, some cancel each other out, and some you just don't know about?
quote: Because those statements are limited in scope.
On the other hand, the existence of a contradiction in a formal system need not completely vitiate the system. The only operative consistency in any system, mathematical or physical, is LOCAL. We can define a metric on the space generated by the axioms of a system, and find that there is consistency within a certain region of that space, even though there are global inconsistencies. No webpage found at provided URL: http://www.mathpages.com/home/kmath372.htm Yet you want to make global and universal claims? Such isn't reasonable.
quote: Lets see what that question was, shall we?
Why are you pretending that something significant can be constructed that does not violate that proof? This was a reaction to your continual repetition of the mantra that the Incompleteness theorems do not apply to all systems. You are, however, avoiding the fact that the Incompleteness Theorems do apply to a large number of, frankly, unavoidable components of mathematics.
quote: ummm.... no. It is a simple question. You have used mathematics numerous times in this thread. Are you not using about systems sophisticated enough that the incompleteness theorems apply? In other words, you bring up Presburger arithmetic over and over. Are you performing your calculations within such a system? Nope. So really, it is a red herring. You are making the error of which you accuse me, essentially, and are trying to buttress all of mathematics by reference to a tiny and barely functional axiomatic system which exists more outside of mathematics than inside it.
quote: Like arguing with a fundie... ------------------
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John Inactive Member |
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John Inactive Member |
Wow... thats better than it was where I grew up.
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John Inactive Member |
quote: The trick is that we understand the physical processes largely through our constructs. I think this is what Kant glimpsed when he divided the world into noumena and phenomena. Anyway... not argument-- just wanted to point out that most of us here seem to be defending that first sentence, while Rhhain does the reverse. ------------------
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John Inactive Member |
You have repeatedly claimed that mathematical entities are real things and that mathematics were discovered, not invented. Invented == constructed. Now you are going to say it ain't so?
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John Inactive Member |
quote: What of mathematics do we have that IS NOT our representation of it? ------------------
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