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Author | Topic: Question.... (Processes of Logic) | |||||||||||||||||||||||||||
Rrhain Member (Idle past 330 days) Posts: 6351 From: San Diego, CA, USA Joined: |
Since when did it become impossible to prove a negative?
I've been doing it for quite some time. It's called "indirect proof." You start with an assumption, lead yourself to a contradiction, and thus you prove the negation of the assumption. For example, let's prove there is no largest prime number. Assume there is a largest prime number, p. Thus, we can produce a list of all primes: p1, p2, ... , pn-1, pn where pn is the largest prime. Now, construct q as follows: q = p1 * p2 * ... * pn-1 * pn + 1 So the question then becomes, is q prime? Well, we have a list of all the primes but we find that none of them divide q. Therefore, we are left with one of two possibilities: 1) A number between p and q is prime2) q is prime 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. Well, that's a contradiction. Therefore, the assumption is not true: There is no largest prime. Notice, the negative that was just proven isn't the "no" in "no largest prime." It was the negation of the assumption: There is a largest prime. Proving a negative is quite simple to do. However, it requires well-defined objects behaving in well-defined ways. If you deviate from that, such as the methods of science which rely upon observation rather than definition, then things become more difficult. But, this doesn't mean it cannot happen. Take, for example, the following biology experiment: Take a single K-type E. coli bacterium. Grow it to a lawn and infect the lawn with T4 phage. The vast majority of the lawn will die though there will be a colony or two surviving. They have evolved to be immune to T4 phage and are now called K/4. But if we take one of these K/4 bacteria and let it reproduce to a lawn and then infect that lawn with T4 phage, we see plaques forming. So the question is, which one mutated? A little thought shows that it can not be the bacteria that mutated. If it were the bacteria that mutated, then they would be surrounded by K/4 bacteria that are immune to T4 phage. Thus, as soon as any of these K-type bacteria died, they'd be replaced with K/4 bacteria and we'd never see a plaque. But since we do see a plaque, we necessarily conclude that it is not the bacteria that mutated but rather the phage. Bingo! We just proved a negative. Once again, we had well-defined objects behaving in well-defined ways, but it does happen in the real world every now and again. ------------------Rrhain WWJD? JWRTFM!
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Rrhain Member (Idle past 330 days) Posts: 6351 From: San Diego, CA, USA Joined: |
John responds to me:
quote:quote: No, it is, indeed, proving a negative. There is no largest prime. The result is not that the individual number before us isn't the largest prime (though that, too, would be proving a negative) but that there is no prime anywhere no matter how you try to construct one that is the largest. The problem with "proving a negative" is not that it cannot be done but that the techniques available to do so require the ability to look at every single possibility. That isn't always possible. I can prove the negative that something doesn't exist in a particular room. Depending upon the specifics, it can be because the object in question is too large to fit in the room or that the object is definitively somewhere else or by doing an exhaustive search of all the objects in the room and determining that none of them are the object in question. The problem with something like proving god doesn't exist is that the definition of "god" is not very specific and the arena which needs to be examined is too large for a brute force method such as an exhaustive search as well as being too vaguely defined to be amenable to abstract methods. It isn't that you can't prove a negative. You can. It's that many things aren't amenable to the techniques involved. On the flip side, not all positives are amenable to proof, either.
quote: Didn't I say that? I'm sure I did. Yep...right here: "However, it requires well-defined objects behaving in well-defined ways. If you deviate from that, such as the methods of science which rely upon observation rather than definition, then things become more difficult." The fact that you have data to work with doesn't mean you aren't proving a negative. The fact that there is a positive means there is a negative, too. By the fact that an object is X, that means it is not Y.
quote: No, it's still the same thing. It's just that the scenarios are dissimilar. Someone tells you that there is a largest prime but this person offers no proof but instead challenges you to disprove the existence of the largest prime. Quite simple: Assume there is a largest prime, lead yourself to a contradiction, and thus logically conclude the negative: There is no largest prime. The only difference between your situation and mine is that my scenario has well-defined objects behaving in well-defined ways. If you refuse to define what a "giant" is and how it behaves, not only does it become impossible to prove that there is no giant, but also it becomes impossible to prove that there is one. Until we can get a concrete definition, one cannot prove anything. Take, for example, the proof of the four-color theorem. For those who don't know, the four-color theorem states that any map drawn on a contiguous plane surface (that is...a plain sheet of paper) can be colored using only four colors such that no two areas that share an edge have the same color. You might be able to get away with fewer, but you don't need more than four. Well, how do we prove that? For the longest time, we could handily show that for any map anybody had ever drawn, it could be colored using at most four colors. It's because we could come up with a concrete definition (this particular map), we could come up with an actual result (yes, it can be colored using no more than four colors). But, we couldn't come up with any progress on the larger question because we couldn't define what a "map" was in any concrete method. Eventually, through a lot of work, a method was developed that could generate every possible map you could draw. It was noticed that some maps are equivalent. For example, suppose you have a map that consists of a circle within a circle such that the two don't touch. Well, that is the same as a map that consists of just a circle: The color of the inner circle can be anything because it is completely surrounded by a single color. Therefore, we can remove that inner circle completely and not affect the number of colors needed to color the rest of the map. The technique allows us to get a much more concrete definition of "map." By using the greater comprehension, we can generate an exhaustive list of all possible maps. We can then go through every single one and determine if it can be colored using only four (this is where the computer came in which caused the controversy...did anybody bother to check if the computer did it right? Yes, they did.) If it turns out that the entire set of all possible maps can be colored using only four colors, then we have proven that there is no map that requires five. By proving the positive, each map can be colored using four colors, we also prove a negative, each map requires no more than four colors to color.
quote: Indeed. But if the definition of "giant" requires a necessary result that does not appear, then the giant can be proven not to exist. This is proving a negative. In fact, that's exactly what science does: Prove negatives. Nothing in science can ever be proven true. What is done all the time, however, is prove things not to be true. Newtonian mechanics, for example, necessarily requires a linear universe. The universe, however, is relative and not linear, therefore we prove Newtonian mechanics to be not true. Oh, it's very close to being true for many instances, but there is always an error between what the theory claims and what we actually observe. It may require extremely sensitive equipment to actually measure that discrepancy, but that doesn't mean the discrepancy isn't there.
quote: Actually, it's also proving a negative: The evidence is not a giant. If a giant and only a giant results in X and what we have before us is Y, then we prove that it is not from a giant. But again, it requires well-defined objects behaving in well-defined ways. When things become fuzzy, the ability to prove anything, positive or negative, becomes much more difficult.
quote: Only if the definition of "giant" is so vague as to be incapable of providing a definitive result. What, precisely, is a "giant"? How does it behave? What actions can be ascribed to a giant? What actions must be excluded from the definition? Are there any that are unique to a giant? Until these questions are answered, not only can we not prove the non-existence of the giant, we also can't prove the existence of the giant. We are left with an ill-formed premise behaving in an ill-formed way. No wonder we are at a loss to make any sort of acceptable statement.
quote: That's where the logical part of burden of proof comes along. The claimant is the one that needs to show evidence. And yes, that means those that say there is no giant. But, since this started with the person making the claim that there is a giant, it is his burden of proof to show that it is there. If my response is that there is no giant there, then it is my burden to justify my claim. I can easily do that by defining what a "giant" is and then showing that the mountains do not contain any evidence that is necessarily required for there to be a giant. If the other person then claims that my definition of "giant" is insufficient, then he needs to provide a better one. No proof in either direction can be provided unless a specific definition of "giant" is forth-coming.
quote: Not a problem if the definition of "giant" allows there to be a scenaro that necessarily results in a certain situation. If that situation does not exist, then the absence of evidence is, indeed, evidence of absence. The definition of ice requires the existence of a solid. If we can determine that there are no solids anywhere in the system, then we can necessarily conclude that there is no ice. The absence of solids is evidence of the absence of ice.
quote: Then the definition of "giant" is so vague that not only can we not show it doesn't exist, we also can't show that it does. If the giant leaves no trace, then there is also no evidence for the existence. Of course, a difference that makes no difference is no difference. If the evidence that we have is perfectly consistent with both the presence and the absence of the giant, then we logically conclude that there is no giant since there is no difference between the two. Note, this assumes that we have complete information. Again, the problem is not the act of proving the negative. The problem is the lack of definitive information. If the objects are not well-defined and we do not have a concrete definition of how they behave, then we become susceptible to overlooking something.
quote: You've just stated my point: With no information to go off of, you can't say anything either way. We aren't stuck with proving positives. We don't even have the ability to prove a positive. Nothing in science is ever proven true. Instead, science concerns itself with proving things false. The function of every experiment is to try and disprove the theory because a successful experiment doesn't prove the theory true, it merely shows it to be accurate. It may be, like Newtonian physics, that our instruments are sensitive enough to detect the error. ------------------Rrhain WWJD? JWRTFM!
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Rrhain Member (Idle past 330 days) Posts: 6351 From: San Diego, CA, USA Joined: |
Mister Pamboli writes:
quote: Correct...sorta. In that case, I was showing a specific negative case. Given the scenario, we know that there was some sort of change. The bacteria could have mutated or the phage could have. We end up proving the positive by proving the negtive and leaving the positive behind. In this case, we can list all the possible outcomes: The bacteria mutated or the phage mutated. And, in fact, the process of proving that the phage mutated is accomplished by proving that the bacteria couldn't be the one that mutated. We eliminated the impossible leaving behind the truth, to paraphrase Holmes. My point is that we shouldn't rail against the idea of "proving a negative." It can be done and is commonly done in science. What we should be railing against are the poorly defined objects that we are being asked to manipulate; the shifting definitions when the consequences prove to be disliked by the claimants; the equivocation of terms and ad hoc adjustments that revive statements that have no justification; the shifting of burden of proof. The problem isn't proving a negative. It's the demand to prove a negative when there isn't even enough information to make a definitive statement in the first place. [This message has been edited by Rrhain, 05-07-2003]
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Rrhain Member (Idle past 330 days) Posts: 6351 From: San Diego, CA, USA Joined: |
John responds to me:
quote: Which is the negative of the assumption: There is no prime number larger than the assumed largest prime.
quote: Incorrect. In the real world, objects are not always ill-defined. Again, I gave a specific example of showing that an object is not in a room. We can show that the object is too big to fit in the room or that the object is located elsewhere or do an exhaustive search of all the objects in the room. Or are you saying that, say, your car keys are ill-defined? Indeed..."god" is an ill-defined object but that doesn't mean everything is ill-defined.
quote:quote: Um... yes, it does. If an object is X, that means it is not Y (assuming that there is no overlap of X and Y...that is, squares are rectangles so if an object is a square, then it is also a rectangle, but squares are not circles so if an object is a square, then it is not a circle. By showing the positive, X is a square, we necessarily show a negative, too, X is not a circle.)
quote: No, I think that's what you're doing.
quote: No, that is not the negative I had in mind. You're confusing the logic and the mathematics. If John exists, then John does not not-exist. If I can show that X is a square, I automatically get X not being a circle as well. ------------------Rrhain WWJD? JWRTFM!
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Rrhain Member (Idle past 330 days) Posts: 6351 From: San Diego, CA, USA Joined: |
John responds to me:
quote: Is a negative. If you have a statement and prove its negative, then you have proven a negative. If you claim that X is a circle and I show that it is a square, then I have proven a negative: X is not a circle.
quote:quote: Are you saying you don't have absolute knowledge about some things like what your car keys look like? I agree that for a lot of things, we don't have absolute knowledge, but that doesn't mean we don't have it for some things.
quote:quote: I most certainly am. They are just highly specific.
quote: I know. But it is still an existential room. There exists no element X in the set. That is an existential statement.
quote: So? What does that have to do with anything?
quote: If it is not detectable no matter what, then it is the same as if it doesn't exist.
quote: If it doesn't exist, wasn't that what we were trying to demonstrate? If we have examined everything that does exist, then that only leaves the things that don't exist. But doesn't that mean we've just succeeded in showing that it doesn't exist? Isn't it a truism that A v ~A?
quote: Why the restriction?
quote: Of course. You're trying to challenge me to do something mathematically without letting me use mathematics to do it.
quote: Sure we can. We do it all the time. There is no way to square the circle using straightedge-and-compass. There is no planar object that is both a square and a circle. We can prove lots of things don't exist.
quote:quote: (*chuckle*) I think you're confusing position and momentum. ![]() quote:quote: I most certainly am! X exists and is a member of a set: $ x such that x S.
quote: More formally, an existential statement is one that uses the existential operator: "There exists." This is in contrast to the universal operator: "For all."
quote: I know. But that doesn't mean "X is not Y" can not be cast as an existential statement. There does not exist x X such that x Y.
quote: Not at all.
quote: Definitely not. There is no largest prime number.
quote: Agreed. But you shouldn't hold that against the other sections of the universe.
quote: Oy...they equivocated. They shifted from the object to the name of the object. That is, they shifted from "Santa Claus does not exist" to "The name 'Santa Claus' does not exist" and then acted as if the two statements were equivalent.
quote:quote: I think I get to be the arbiter of what negative I was trying to prove. If you are concerned about a different negative statement, then don't hold that against me. I wasn't dealing with that one.
quote: You're trying to explain that because I was disproving X and not Y, even though I said I was only disproving X and not Y, that it is somehow my fault that I didn't disprove Y?
quote: If the original statement is that god is a Buick, then the negative is that god is not a Buick. If Y is the only thing that can be, then if X is not Y, then X is not.
quote: No! The relevant formulation is that "god is not a Buick." You're not talking about the existential operator...you're talking about the universal. Not all negative universal statements can be demonstrated (even though the example I gave of no largest prime is a negative universal).
quote: Sure it can. There does not exist a largest prime number, remember?
quote: No, it couldn't. There is no prime number. You don't even need to look at everything to know that there isn't. Simply assume that it could be and lead yourself to a contradiction.
quote: No, you need a solid definition to prove they don't. There is no such thing as a planar object that has the properties of both a square and a circle. For you see, one of the properties of a circle is that all line segments that cross the center are the same length. However, assuming that one can define the "center" of a square as that point where the diagonals cross, not all line segments that cross the center of a square are the same length. The diagonals are longer than the laterals. Thus, by definition, there can be no planar object that has the properties of both a square and a circle.
quote: Not at all. You simply need to have a solid definition of the purple people eaters (now, I've always wondered...are the purple people eaters beings that are purple and who eat people or are they beings that eat purple people?) such that you can put forward a definitive test: If there are purple people eaters, then X must occur. If X does not occur, then we necessarily conclude that there are no purple people eaters.
quote: But that is the case only if we have no definitive method of verification. I agree that not everything can create a definitive method of verification. But we shouldn't hold that against those things that can.
quote: Logical error: Ad hoc.
quote: Logical error: Incredulity.
quote: And you just proved a negative. Newton's mechanics are not true.
quote: Incorrect. As crashfrog said, an inductive conclusion might be wrong. A deductive conclusion cannot be.
quote: I didn't say I could. You seem to be confusing my claim that you can prove a negative with some sort of idea that one can prove everything. Well, no. Inductive statements (at least in the physical sense, not the mathematical) are not things that can be proven, merely shown to be accurate.
quote: No, not "without absolute knowledge." Without a sufficient definition.
quote:quote: I know, it seems trivial, but it isn't. One of the first questions from my Linear Algebra class was "Show that A = A." One would think that would be inherent, but you have to actually show it.
quote: You're assuming that there is something else that can be. F'rinstance, if we show that there is no workshop at the North Pole, that all those presents you got in the morning were planted there by your parents, etc., it does no good to say that, "Well, then 'Santa Claus' is really your dad." To do so is to shift your definition. You started with the definition that "Santa Claus" was an object that lived at the North Pole and delivered presents. We've just shown that there isn't anything up there and that all those presents were delivered by somebody else. You don't get to make an ad hoc shift of definition to allow "Santa Claus" to be something else. You don't get to make an ad hoc shift that the workshop is "invisible" or that you apparently weren't "nice" and thus Santa skipped your house. ------------------Rrhain WWJD? JWRTFM!
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Rrhain Member (Idle past 330 days) Posts: 6351 From: San Diego, CA, USA Joined: |
crashfrog responds to me:
quote:quote: Sure they do. I think we've got a philosophical argument here. Question: Have you heard of the term "Platonist"? What you've just said is one of the common questions that strikes mathematicians: Do the objects that we study really exist? Most mathematicians wind up saying yes, they do. "2" really does exist just as much as "red" exists. If you agree that apples are real and if you have two of them, how can you say that "2" doesn't exist when you can see the two of them right there? Now, can you construct a physical circle? No. But that doesn't mean it isn't real. Is love real? It isn't something that you can bottle. It isn't a smell or a sight. It has no tangible qualities, and yet it is very real, isn't it? So yes, squares and circles do exist. Those mathematicians that agree with this line of thinking are called "Platonists." It hearkens back to Plato's Parable of the Cave. But, some mathematicians disagree. They are only abstracts, conventions that we make to describe things we observe, not "real."
quote: Not at all. Some things in real life are axiomatic.
quote: You mean when I look at a red object, it might not really be red?
quote: I agree. Are you saying you can't deduce anything in the real world?
quote: I guess you are. Forgive me if I disagree. If Johnny has 5 apples and gives 3 of them to Suzie, I think we can deduce that Johnny has 2 apples left. ------------------Rrhain WWJD? JWRTFM!
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Rrhain Member (Idle past 330 days) Posts: 6351 From: San Diego, CA, USA Joined: |
crashfrog responds to me:
quote:quote: Incorrect. If I'm standing here looking at my keys and I watch them for 10 minutes and they don't go anywhere, then I necessarily conclude that you did not take them. For if you had taken them, I would have seen. And since I did not see you take them, then you did not.
quote: But I just did. You didn't take my keys.
quote: Only if there is no way to construct a definitive test. If there is, like me watching my keys, then absence of evidence is evidence of absence.
quote: Indeed. It's the question between induction and deduction. You've already said you don't think deduction can happen in the real world. I think it can. Ergo, you can prove a negative. It still requires well-defined objects behaving in well-defined methods. ------------------Rrhain WWJD? JWRTFM!
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Rrhain Member (Idle past 330 days) Posts: 6351 From: San Diego, CA, USA Joined: |
crashfrog responds to me:
quote: My friends and I really talk like that. My apologies. That said, I didn't give a one-word dismissal. I continued on.
quote: Ad hoc scenarios are illogical. Ergo, they are dismissed as unjustified. If what I'm seeing doesn't really exist and I'm just "plagued by demons" as Descartes put it, then I'd never know the difference and therefore, it can be treated as if it really did exist. A difference that makes no difference is no difference.
quote: I've never said otherwise. Euclidean geometry, for example, rests upon the axioms and postulates. Change those, and the rest of the geometry changes. Take the Fifth Postulate. It states that if two lines are crossed by a traversal such that the interior angles on one side are less than two right angles, then the lines, when extended indefinitely, will meet on that side. Lots of people thought it could be derived from the other postulates and axioms. But, it turns out, it can't. So all you have to do to change the geometry is rewrite the Fifth Postulate. You do that, and you get different kinds of geometry. Euclidean geometry is what you get on a flat, planar surface. Thus, triangles always have exactly 180 degrees. But move to the surface of a sphere, and things are different...triangles have more than 180 degrees. Deduction is always premised upon the axioms. But if we assume the axioms are true (thus, the term "axiom"), then it necessarily follows that all logical deductions from them are true. Including negatives. ------------------Rrhain WWJD? JWRTFM!
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Rrhain Member (Idle past 330 days) Posts: 6351 From: San Diego, CA, USA Joined: |
DBlevins:
quote: How did it manage to get to the entire lawn? As soon as those bacteria died, the K/4 would move in. We'd never see plaques. I'm not saying that it is impossible for the bacteria to experience a mutation that would make them susceptible. I'm saying that the reason we see plaques is because the phage has mutated. Bacterial mutations in this scenario don't cause plaques because the K/4 fill in any susceptible bacteria. ------------------Rrhain WWJD? JWRTFM!
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Rrhain Member (Idle past 330 days) Posts: 6351 From: San Diego, CA, USA Joined: |
John responds to me:
quote:quote: And? How does this negate the fact that by being a square, it is necessarily not a circle? Are you saying that it is possible to be both a square and a circle?
quote: Yes and no. Indeed, showing that something is not a circle does not mean it is a square. However, showing that something is a square does mean it is not a circle. You have to keep your implications going in the correct direction.
quote: And thus, it is not a circle. Or are you saying that it is possible for a planar object to be both a square and a circle?
quote: You've reversed the implications. A -> B does not imply B -> A. X being a square implies X is not a circle. X is not a circle does not imply X is a square.
quote: And how is that not proving a negative? When you draw a boundary, you are also defining the outside as well as the inside. It's a common technique in drawing when you're having trouble with an object: Don't draw the object but instead draw the space around it. By defining everything that is not the object, you'll be left with the object. In mathematics, a similar thing happens. It is often difficult to describe the elements of a set but much easier to describe the elements that are not in the set. By accounting for all of the non-elements, you are left with the elements and having proven what you were trying to prove in the first place.
quote: So? How is that not proving a negative? Once again, you're saying that I must prove something mathematically without being allowed to use mathematics to do so.
quote:quote: Who said anything about the subatomic level? You don't know what your keys look like?
quote:quote: Incorrect. If the statement uses the existential operator, then it is an existential statement. That is by definition.
quote: Incorrect. Didn't you read my response? Oy...they equivocated. They shifted from the object to the name of the object. That is, they shifted from "Santa Claus does not exist" to "The name 'Santa Claus' does not exist" and then acted as if the two statements were equivalent. In short, the article made a fundamental error. It confused an object for its name. An object and the name of the object are separate objects. To say that the statement "Santa Claus does not exist" cannot be true because the physical words "Santa Claus" exist is to completely misunderstand the point behind the statement, "Santa Claus does not exist."
quote: I have. There is no largest prime number. It does not exist.
quote: So? One can show that an infinite number of things are of a certain set by showing that they are not members of the complement. Take, for example, the definition of an infinite set: "A set that is not finite is infinite." From one of my Analysis texts (Principles of Mathematical Analysis, Walter Rudin, McGraw-Hill, 1976): 2.4 Definition For any positive integer n, let Jn be the set whose elements are the integers 1, 2, ..., n; let J be the set consisting of all positive integers. For any set A, we say: (a) A is finite if A ~ Jn for some n (the empty set is also considered to be finite).(b) A is infinite if A is not finite. (c) A is countable if A ~ J. (d) A is uncountable if A is neither finite nor countable. (e) A is at most countable if A is finite or countable. You will notice that both "infinite" and "uncountable" are defined in terms of not being something else. It is much easier to describe a finite set than it is to define an infinite set. Ergo, infinite sets are defined as that which is not finite. One aspect of sets is that the complement of an open set is closed. From the same source: 2.23 Theorem A set E is open if and only if its complement is closed. Proof First, suppose Ec is closed. Choose x element of E. Then x not element of Ec, and x is not a limit point of Ec. Hence there exists a neighborhood N of x such that Ec intersection N is empty, that is, N subset E. Thus x is an interior point of E, and E is open. Next, suppose E is open. Let x be a limit point of Ec. Then every neighborhood of x contains a point of Ec, so that x is not an interior point of E. Since E is open, this means that x element Ec. It follows that Ec is closed. Corollary A set F is closed if and only if its complement is open. From this, we can then talk about the "closure" of a set in a metric space. One aspect of a closure of a set is that it is closed. The way we prove that is by taking a point p that is in the metric space but not an element of the closure, E-bar. This means that p is neither a point in E nor a limit point of E. This means that p has a neighborhood which does not intersect E. This means that the complement of E-bar is open (since neighborhoods are open) and therefore, E-bar is closed. In other words, you often prove things by showing that only certain possibilities necessarily exist and that the specific scenario which we have necessitates that all possibilities but one are impossible. Therefore, you are left necessarily concluding that the specific scenario is the one that is left behind. As Holmes put it, when you have eliminated the impossible, whatever remains, however improbable, must be the truth.
quote: Why not? I can tell you everything about what a "square-circle" is: It has all the properties of a square and a circle. But such an object still doesn't exist.
quote: Incorrect. Information only exists for what actually is defined. If you leave something undefined, then you lose information.
quote: No, what is meant by "you cannot prove a negative" is acceding to the logical error of ad hoc argumentation. It surrenders to the vague definition rather than meeting it head on to directly state that the definition is vague and therefore not only can we have a problem proving negatives about it, but we will also have problems proving positives about it.
quote:quote: You mean Descartes was wrong? A difference that makes no difference really is a difference?
quote: Not quite. It isn't proving a negative. It is, however, a misuse of the "not" operator. Argumentum ad ignorantiam goes both ways and is just as invalid when applied to a positive: You cannot prove that X > Y, therefore X <= Y. The problem is the inability to prove something does not give us information about it. Argumentum ad ignorantiam is a special instance of the error of False Dilemma. However, if the dilemma is not false, then there is no error of False Dilemma. And with no error of false dilemma, there is no argumentum ad ignorantiam.
quote: The final statement simply isn't true and is directly contradicted by the statement immediately preceding: There are cases where lack of evidence for S is relevant to the truth or falsity of S. There is no largest prime. Or are you saying that my proof of it is unjustified? Or that I wasn't proving the non-existence of something?
quote:quote: You mean I didn't prove the non-existence of a largest prime? I have reviewed the fallacies you've quoted. There is an error. You can prove a negative. Again, if X is a square, then it is not a circle is a logical statement. It's reversal is not, however.
quote:quote: You mean there is a largest prime? Or that I cannot prove that there isn't a largest prime? Strange...I thought I could prove there isn't a largest prime.
quote: But if the thing's existence requires that we have evidence and we don't, then we necessarily conclude that it doesn't exist. Again, it is nothing more than a special case of the False Dilemma. If the dilemma is not false, then absence of evidence is evidence of absence.
quote: Agreed. But the problem is not proving a negative. It's that the definitions are too vague. They allow possibilities that have yet to be elucidated. But if you can elucidate all possibilites, then it is sufficient to prove that something is of one of them by showing that it is none of the others. F'rinstance, all integers are either odd or even. If I can show that it is not odd, then I have necessarily shown that it is even. There is nothing else for it to be. The dilemma here is not false...there really are only two possibilities.
quote:quote: That's what I've been saying. If you have all possibilities staked out (which requires well-defined objects behaving in well-defined ways), then by showing that something is not of all but one, then you necessarily have shown that it is that last one. It may be difficult to show that something is of a particular set. It may be easier to show that it is not of its complement. Therefore, if we do show that it is not of the complement, we have shown that it is of the set.
quote: No...you have A, therefore A. Similarly, A, therefore ~(~A). And notice in your first statement, you said "~p." How can you possibly say "~p" without proving a negative? You just negated p!
quote: It is tautological, but it is not pointless. It is not trivial.
quote: Unless evidence is necessarily required. If evidence must necessarily exist and it doesn't, then we necessarily conclude absence. The problem is not the ability to prove a negative in general. It is the ability to do it in specific. That is, the error of argumentum ad ignorantiam is not "If evidence must necessarily exist and it doesn't," it is "If evidence must necessarily exist and it doesn't that we know of." The question immediately then becomes, "How do you know you've taken everything into account"? For many processes, we can't because the objects which we are examining aren't sufficiently defined. But if they are, we can show precisely why the evidence that must necessarily exist doesn't.
quote:quote: That is because that's precisely what happens. By proving what something is, you necessarily prove what it isn't. Something that is a square is necessarily not a circle. What you are complaining about, and it is a legitimate complaint, is that proving what something is not does not necessarily prove what it is. That only happens when you can account for all possibilities for what it might not be. That can often be quite difficult if not impossible.
quote: You mean if I show that something is a square, there is a possibility that it might be a circle, too?
quote: Actually, it does. Are you seriously saying that there can be a planar object that has the properties of both a circle and a square? Some properties of a circle: The longest chord of a circle is its diameter.All diameters pass through the center. If a chord passes through the center, it is a diameter. Two non-identical diameters intersect at the center. All diameters are the same length. Let's see what happens when we look at a square, now: The longest chord of a square are its diagonals.The diagonals of a square cross at a certain point, which we'll call the center as defined above. It is possible to draw a chord through the center that is not the diagonal. Therefore, since we know that all chords through the center of a circle are of the same length, this must mean that the line through the center of the square must be the same length as the diagonal. However, it isn't. Therefore, there can be no planar object that has the properties of both a circle and a square. By having one, it necessarily cannot have the other.
quote: How is that insufficient?
quote: I wasn't trying to. Instead, I was trying to prove the non-existence of "circle-squares." Have you checked the definition of "strawman" in those logic sites of yours? That's where you take an argument that isn't the one your interlocutor is trying to make, and show that it isn't valid.
quote:quote: Then you can understand why I refuse to concede to such a request. By your logic, if you were to say, "It is physically impossible for a heavier-than-air object to fly," and I were to show you an airplane and explain the concept of Bernoulli's Principle, for you to come back and say, "Yeah, but try and do that without using any physics," you would understand why I'd blink at you. How can I show something physically without using physics? So how can I show you something mathematically without using any mathematics?
quote: Why? I am arguing the validity of deduction and how one can logically deduce negative propositions. You're arguing whether or not the axioms from which we make our deductions can be accepted. I wholeheartedly agree that if there is something wrong with the axioms, then the deductions we make from them cannot be trusted. But I am not arguing that the axioms are faulty. I am arguing that deductive statements are logically valid, even when those deductions are negative.
quote: Sure I can. If apples are inconsistent, then apples don't exist. I agree that for this specific example, it's going to be very hard to do so. But, that depends upon the definition of "apple." If the definition allows it to be anywhere in the universe, then I'm going to have an extremely tough time, conceivably impossible. But if the definition requires it to be sitting right here on my desk, then the task will be much easier.
quote: Incorrect. A banana, for example, is something that is not an apple. It is not the only thing that is not an apple, but that's something. Again, you've got your directions reversed. If it's a square, then that necessarily means it is not a circle. That is what I'm arguing.
quote: Of course not. That's why they're called "axioms." Those are the statements that are true without being proven and form the basis by which all are statements are derived.
quote:quote: But it's my argument. Therefore we use my definitions. You're arguing a strawman.
quote: Strawman. That is not the argument I am making.
quote: Since when were we switching geometries? You've just argued yet another strawman. That things are true for non-Euclidean geometry does not mean they are true for Euclidean. For you to switch to non-Euclidean geometry in the middle of a sentence is a logical error.
quote: Incorrect. Even at singularities, logical things happen. Do not confuse our personal knowledge of something with its ability to exist or not exist. You just committed argumentum ad ignorantiam. Question: Do you know about the mathematical concept of "Platonism"? Here's an example of a question which would define a Platonist from a non-Platonist: Does the Continuum Hypothesis have an answer? The Continuum Hypothesis has to do with the size of the Real Numbers. We know they're bigger than aleph-null, but we don't know if they're aleph-one. We do know that assuming they're aleph-one does not lead to a contradiction. But, we also know that assuming they're not aleph-one does not lead to a contradiction. A Platonist would then say that there is an answer to the question, we just don't know what it is. The size of the Reals does exist, we just have no tools to let us know what it is. Thus, inside the event horizon of a black hole, at atomic scales, etc., things happen according to their internal logic...we just don't know what it is.
quote: No, we discovered math. Mathematics would still exist, even if there were no people around to think about it.
quote: I didn't say we did. However, you were saying that there is something wrong with a deductive system. But that simply isn't true. Deductive systems are necessarily logical. If you want to argue the validity of the axioms by which the deductive statements are made, that's another question entirely.
quote:quote: But you prove that by demonstrating a negative: Any prime that you think is the largest one really isn't.
quote:quote: Strange...just above you were saying I hadn't read the article at all.
quote:quote: Yes, I do. Crashfrog was stating a universal. The way to disprove a universal (there's that "proving a negative" thing again) is to prove an existential. That is, if you say, "for all," then I can disprove that by showing, "there exists for which it isn't true." Therefore, if crashfrog has a specific negative in mind, then it is quite possible that I won't be able to prove it. Instead, however, he claimed that all negatives were impossible to prove. Therefore, I simply need to show one negative that can be proven in order to show that his universal statement is false. And in the process, I end up proving two negatives: Not only do I end up proving my negative, but I also prove the negative that "You can't prove a negative" is false. 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: Strawman. Was the phrase that started this exchange, "You can't prove a negative"? Therefore, wouldn't it be sufficient to demonstrate a negative you can prove in order to show that statement false? Is it not true that there is no largest prime? Ergo, I just proved a negative. Ergo, "You can't prove a negative" is false.
quote: Strange...I've been saying the same thing to you.
quote: Isn't that sufficient to show that you can prove a negative? What I have been saying all along is that proving a negative requires well-defined objects behaving in well-defined manners.
quote: That just proves my point. The reason why you don't hear that is because there is no well-defined description of "god." You seem to be arguing that because I cannot disprove the existence of any possible description of god you might be able to come up with, that means I am incapable of disproving the existence of any specific description of god. I, however, am arguing the other direction: Given a specific description of god, it may be possible to show that god as defined by that description does not exist. If you then change your definition of "god," then my proof of non-existence may no longer work, but we wouldn't expect to work since, after all, you changed your definition. You seem to want me to prove a universal without a definition. I am insisting that you give me a definition before I try to do anything. I may or may not be able to do so, but until you give me that definition, I can't do anything. 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: No, it means making a definition. I don't hve to "observe and record everything that is, was, and will be across all of space and any other spaces there might be" in order to define a square. Or are you saying that there is no definition of a square?
quote: Sure I can. I have my definition. If my definition leads to a contradiction, then I necessarily conclude that it does not exist.
quote: Whose talking about other things? I thought we were talking about the one thing that we were specifically trying to disprove. You're absolutely right that my ability to disprove the existence of square-circles says nothing about my ability to prove or disprove the existence of triangles. But then again, we weren't talking about triangles. Why are you setting up a strawman?
quote: Only if you are engaging in False Dilemma. If you aren't, then lack of evidence isn't a fallacy.
quote:quote: Now I admit I am confused. Before you said you were talking about existentials. Now you say you're not. So which is it? Are you talking about existentials or not? Are you talking about the existential operator or not? The negation of the universal is the existential and vice versa.
quote:quote: Yes.
quote: Perhaps...perhaps not. It is, however, like saying love exists. Does love exist?
quote: So? How is that not proving a negative? If something is either X or Y and it is X, then it is necessarily not Y. You're arguing a strawman again.
quote: Strawman. Stop arguing your point and start arguing mine.
quote:quote: And that means there is no largest prime number because of what, precisely?
quote:quote: And that means we haven't proven a negative because of what, precisely? Definitions exist, too. You seem to be arguing that because axioms may be questionable, the ability to make valid deductive arguments based upon those axioms is impossible. That simply is true. Indeed, if the axioms are questionable, then the results of the deductive process may not be accurate, but the process, itself, is still valid. The problem is not the deductive process, it is the axioms that aren't.
quote: Who cares about the plane as we don't define it? We're not talking about that. If you want to talk about that, then we'll shift gears and deal with that other definition. But until you come up with that other definition, we can't say anything at all, positive or negative.
quote: Who cares? We're not dealing with those other possibilities. We're only concerned with the definition before us. It does not matter that triangles have more than 180 degrees on a sphere when we're dealing with a plane. Because on a plane, triangles have exactly 180 degrees, they always will, and there is no way they can have anything other than 180 degrees. The question of whether or not we are in a plane is a completely different question from how triangles behave in a plane. Too, the existence of triangles that are consistent with a planar triangle does not mean we are in a plane (unless we also know that "planar triangles" can only happen in a plane...which isn't true, they can happen in non-planes.)
quote:quote: Carefully. A "square-circle" is a planar object that has the properties of both a square and a circle. The "largest prime" is the prime number for which there is no larger. Neither one exists, but both have solid definitions.
quote:quote: Incorrect. The largest prime number doesn't exist, but we know exactly what must happen if it did. That's how we go about proving that it doesn't: Assume it does. The existence of a "largest prime number" means that there are a finite number of prime numbers. That means we can list them (the Sieve of Eratosthenes is sufficient to get them all.) That means we can construct a number that is the product of all the primes...plus 1. That means that either this new number is prime or that there is a prime between the "largest prime" and this new number. Things that don't exist can easily have definitions. That's the only way we can know they don't exist: Something about their definitions makes them disallowable.
quote: I don't know. Can we? You'll need to define that. If they can, then yes. But if they can't always be filmed, then no. The term "ghost," in and of itself, is not sufficient. Therefore, since we have an ill-defined definition, we are going to have a hard time saying anything, positive or negative, about ghosts.
quote: That we have no evidence of ghosts doesn't mean we don't have a definition of ghosts. And even if the definition of ghosts is that they can be captured on film, that we don't have any pictures of them is not sufficient to say that they don't exist. If we are going to be using pictures as a test, then we need to be able to set up a scenario whereby a picture of a ghost must necessarily result.
quote:quote: It is when you are making an ad hoc addition to the defintion. When making absolute claims, one must include every case imaginable within the context of the definition. If we're going to talk about the angular sum of planar triangles, it doesn't matter what spherical triangles are like because we're not talking about spherical geometry. We can say with absolute certainty that all triangles have 180 degrees in the plane.
quote:quote: For some things, yes. For other things, no. We should not hold the instances where we don't have infallible knowledge against those instances where we do.
quote: Hah! I'm a mathematican by training. Do we really need to go through credentialing before we hit the argument from authority?
quote:quote: Descriptions are things, too.
quote: Indeed, you don't. Now that we've gotten the catty remarks out of the way, can we return to the issue at hand?
quote:quote: Incorrect. Wrote about him in 10th grade.
quote: Still have the paper I wrote on it somewhere.
quote: Incorrect. What Godel showed is that for an axiomatic system complex enough to model simple arithmetic, there will be statements within the system that can not be proven within the system. Not all axiomatic systems are complex enough to model simple arithmetic and indeed, for some of them, there are no undecideable propositions. By the way, "inconsistent" is equivalent to "incomplete." It all depends upon how you look at it.
quote:quote: But only so far as required to create the definition. I don't need to know anything about spherical geometry in order to make definitions in planar.
quote: So?
quote:quote: But the thing is its definition. You are perfectly free to say that this other definition is what you really meant by "Santa Claus," that's fine. It's ad hoc and thus does not alter the fact that we showed the non-existence of the original object as described by the original definition. Your new definition may require a new proof and we will have to examine it to be certain.
quote: Why not? Why is it impossible to define Santa? I thought we had one: A "jolly old elf," male, white, beard, overweight, lives at the North Pole at the surface, not invisible, delivers presents to all of the good boys and girls (seemingly only the Christian ones) on Christmas Eve. Well, we've been to the North Pole. There's nothing there. In fact, the ice moves at the northern polar ice cap so quickly that it is impractical to put a "pole" there like we have at the South Pole. If Santa built a workshop at the North Pole, it would no longer be there within hours. I recall a program when Hugh Downes was at the North Pole talking about what is happening in each section of the world as he walked around the North Pole and he mentioned that by the time the program is over, he'd be nowhere near the pole. Another program showed a baseball game that took place at the North Pole with the pitcher's mound at the pole...and again, the comment that by the time the game was over, they'd be nowhere near the pole. Ergo, we've shown that Santa Claus doesn't exist. His definition does not correspond to reality. Now, if you want to come along and change the definition...workshop is at the bottom of the ocean, not the surface, he can make himself invisible, whatever...then you're dealing with something else entirely. The fact that you're calling this new thing "Santa Claus" does not make it the same thing as the old thing.
quote: Incorrect. I only need the definition. I do not need to have a Santa to investigate. We don't have any electrons to directly investigate...they've always been detected by inference...and yet we have a definition of them. ------------------Rrhain WWJD? JWRTFM!
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Rrhain Member (Idle past 330 days) Posts: 6351 From: San Diego, CA, USA Joined: |
I think, crashfrog, that this entire discussion is that I am a Platonist and you aren't. I say this because of this exchange:
quote:quote: From my understanding, math is not "just a symbols game." Are you telling me that if you have two apples, you don't really have two apples? That you would have some other number of apples if the symbol we happened to use was something other than "2"? That the concept of "number" is just a figment of our imagination and no more real than the color of the apples?
quote: In simple existence, the same way that the objects, themselves, would exist were there nobody to look at them. Are you saying that if a tree falls in the forest and there's nobody around to hear it, there isn't even a forest?
quote: Right next to the tree that fell without you hearing it. Was there some other number of planets orbiting the sun until humans came along and counted them?
quote: Are you saying that if you have two apples, you don't have two apples? That number isn't real but that color is? That unless there is a humann being to formalize the concept of number, there is no such thing?
quote: And you've made the mistake of confusing a formalism with a fantasy. As Shakespeare said, "That which call a rose, by any other name, would smell as sweet."
quote:quote: Only if one denies the existence of mathematical reality. I don't, ergo, it happens all the time. There is no way to use a straightedge and compass to square the circle. Those two things are physical objects and it doesn't matter how clever you think you are, it simply cannot be done. But if you deny the physical existence of squares and circles, then it doesn't matter to you. If you add one apple to one apple, you don't get three apples, you get two. But if you deny the existence of "two," then it doesn't matter to you.
quote: Really? Try me.
quote: Actually, some people are. ------------------Rrhain WWJD? JWRTFM!
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Rrhain Member (Idle past 330 days) Posts: 6351 From: San Diego, CA, USA Joined: |
crashfrog responds to me:
quote:quote: Briefly, yes. The keyboard that I am writing this message upon exists despite the fact that I call it a "keyboard" as a speaker of English. The word "keyboard" is a completely arbitrary name and, indeed, the specific symbols that we use to describe mathematical actions are arbitrary. The the fact that we write "2 + 2 = 4" doesn't mean that 2, 4, addition, and equation don't exist any more than the fact that we write "keyboard" means the object my fingers are striking at this moment doesn't exist.
quote:quote: So you could conceivably have three apples? How is this any different from the color of the apple? Would there be no such thing as color if we were all blind?
quote: You're talking language. I'm talking existence. Indeed, there are different methods of describing number in language. But from a purely behavioural concept: Do two apples behave the same way as three apples?
quote:quote: You're confusing the name of the object and the object, again. I can't see into the infrared, but I can distinguish between the infrared and microwaves. Linguistically, there are languages that only have two pure color terms (and it turns out that every single one of them has the two being, if translated into English, "black" and "white.") This doesn't mean that they don't have any way to describe other colors...it's just that the terms used are derived from objects. To use an English example, "turquoise" is not a pure color term...it is based upon the rock. So the fact that I, as a speaker of English, tends to divide the visible spectrum around 400 nm to be "red" doesn't mean that there isn't really something different between electromagnetic wavelengths of 400 nm and those of 700 nm. In fact, we can detect a physical difference between the two: The photoelectric effect has a threshhold. If you aren't of a certain wavelength, then you simply don't knock any electrons off.
quote: This is completely personal experience and me engaging in armchair psychology, so take in that light, but my experience has been that those who claim to have no use for philosophy actually have a very utilitarian philosophy, though they may not have the vocabulary to describe it...at least, not in the terms commonly used by those who study philosophy. F'rinstance, you may not know of the supposed "grue/bleen" paradox or even consider it a useful thing to know, but if it were explained to you, you'd probably be able to find something in the way you experience the world that is related to it. In fact, the "grue/bleen" paradox is seemingly highly appropriate to this very discussion: What do words mean and how do they relate to the objects they describe?
quote:quote:quote: Really? Try me.
quote: If I have one apple and I add one apple, I end up with two apples. What other result is there?
quote: How do ninjas, aliens, and fairies adjust the result of adding one apple to one apple giving us two apples? Is there some other result possible?
quote: In an induction, yes. Not all things in the real world are inductive, though. Some things are deductive. ------------------Rrhain WWJD? JWRTFM!
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Rrhain Member (Idle past 330 days) Posts: 6351 From: San Diego, CA, USA Joined: |
crashfrog responds to me:
quote:quote: Because when I take two apples and add two apples, I get four apples. Are you seriously suggesting that if I were to take two apples and add two apple I would get something other than four apples? That it is possible to get something other than four apples?
quote:quote: So you're saying you could conceivably have three apples. All you have to do is "disagree to play by the rules of numbers" and suddenly you could have any other amount.
quote: And that would change what, precisely? Is a red object no longer red when you close your eyes?
quote:quote: No, color is a property of all objects that emit or reflect light.
quote: So? If I take a piece that is painted red and strip it of the paint, nothing about it could tell me that it was ever red. If I take one of the two apples and examine it all by itself, everything about it tells me that it is one apple.
quote: But the apples constitute a set. Even nothingness is a set. The empty set. See...this is where the Platonist/non-Platonist division comes into play. You claim there is no such thing as a "set." I say there is. Existence is a set. If something exists, then the set of it necessarily exists, too.
quote: So as soon as you close your eyes, the apples don't exist anymore?
quote:quote: Yes. That's how you can tell that there is a difference among one, two, and three apples.
quote: But if color exists without anybody there to see it, why does number need a person to perceive it?
quote: But color and number go together. Red things behave differently than blue things. Two things behave differently than three things.
quote:quote: But my question is, isn't number part of that utility? Don't you behave differently to two than you do to three? Maybe you don't. But I know I do. That's why I think that two does exist.
quote:quote: Since when? I just cut it open and it seems to have a non-segmented meat, no pulp, tiny little seeds all concentrated in the center of the fruit, etc. Sure seems to me to be an apple, not an orange. Do you really think that by changing the scenario, that alters the validity of the original? Sure, two and two equals four, but two and three don't equal four! Well, of course they don't, but we weren't talking about two and three...we were talking about two and two. We weren't talking about one apple and one orange. We were talking about one apple and one apple. Now, if you want to abstract the objects a bit to talk about one fruit and one fruit, that's fine, but the result is the same.
quote: Nope. Still an apple. Here...have a bite.
quote: Nope. Still an apple. Have another bite. No, don't eat it all. We won't have any left if you eat it all. Oh...but isn't that irrelevant? If you eat the apple, we'll still have two apples left because you don't "play by the rules," right?
quote: Nope. Still an apple. Have another bite. It would seem to me that you're playing games.
quote:quote: Because mathematics exists. Logic exists. Unless you're going to embrace Cartesian Doubt, there are some things that can be known absolutely.
quote: So? You deduced, didn't you?
quote: Except for those things for which we have absolute knowledge. ------------------Rrhain WWJD? JWRTFM!
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Rrhain Member (Idle past 330 days) Posts: 6351 From: San Diego, CA, USA Joined: |
John responds to me:
quote:quote: Strange...I was going to say you had missed the point. But since you seem to be arguing my interjection rather than my explanation, I am at a loss for how to respond.
quote:quote: Says who? You're not about to invoke Godel, are you? Be careful, because the Incompleteness Theorems do not say that all axiomatics systems are necessarily incomplete or inconsistent. Instead, they say that all axiomatic systems sophisticated enough to model arithmetic are such. Not all axiomatic systems are that sophisticated and, indeed, they are both complete and consistent. When did we agree on the axiomatic system?
quote: I was going to say the same thing to you.
quote: I know. I have always said so. You can't prove anything without a premise. Are you about to embrace Cartesian Doubt?
quote: But by being true, it necessarily results in certain other things being false. Proof draws a boundary. Thus, there is a distinction.
quote: And that statement just might be "It is true that this is false."
quote: But we're not talking about all of those. We're only talking about this specific one. If we're having a discussion about the color of my hair, it doesn't matter how many statements, true or false, you can make about my sister's hair. We're not talking about my sister's hair. We're talking about mine.
quote: But sometimes the only way to show something to be true is by showing something else to be false.
quote:quote: Strange. I was going to say the same thing to you. Now that we have the ad hominem comments out of the way, can we get back to the matter at hand?
quote: No, it does convey some information. And if what we're looking for is circles, then we don't need to know any more. If I'm trying to show you that 2 + 2 != 5, it is not necessary for me to show that 2 + 2 = 4. While that would, indeed, be sufficient, it is not necessary. You do understand the difference between sufficient and necessary, yes?
quote: Sometimes that's all you need.
quote: I know. But you're arguing a completely different point.
quote: Who said anything about drawing anything? You're arguing something completely different. Please come back to the subject at hand. I have no idea where you're going.
quote:quote: Incorrect. You merely need to include all the necessary components. You do understand the difference between necessary and sufficient, yes?
quote:quote: Excuse me? We're talking about existential mathematical statements and somehow the existential operator isn't relevant? Did you really just say that? If you cannot understand how an existential mathematical statement is one that uses the existential operator, then I really don't know how we can continue.
quote: Strange...I was going to say the same thing to you.
quote:quote: Indeed. It is written as a negative existential statement: There does not exist x such that there is no prime number larger than x.
quote: Incorrect. As my Fundamental Concepts professor so often pointed out to me, a proof with an infinite number of steps isn't a proof. We do not even attempt to "fill in the blank" with a specific number because the specific number is not necessary. In fact, to give a specific number will defeat the purpose. Sure, you may have proven it for that specific number, but what about this other specific number? Until you manage to do so for all numbers, of which there is an infinite number, you haven't shown what you are trying to show. So you don't be specific. It doesn't matter what the number actually is. It is not necessary to know. The only thing that is necessary is that it have the trait of being the largest prime.
quote: Incorrect. It is a finite series of deductive statements leading to a contradiction which necessarily requires the conclusion of a negative existential statement.
quote:quote: We do this all the time. Take the set of me. There is me and the complement of me, "not-me." What are the elements of "not-me"? They are the infinite possible things that are not in the set of "me." Besides, I gave you some real world examples...or do you claim that the objects of mathematics aren't real?
quote:quote: But he's right. And in fact, that's precisely how science works: Science isn't about showing things to be true. It's about showing things to be false, leaving only the truth behind. Since science works as an observational, inductive method, it can never be sure that it has actually arrived at the truth. Just because observation is consistent with prediction doesn't mean the theory is "true." So instead, science sets out to chip away the false things so that what is left behind is the most consistent with observation. I'm afraid I'm going to have to be doing a lot of editing of your response because I literally do not understand where you are coming from. You have hacked my post down to such small statements, including single words, eliminating all context, that I am having a hard time following you.
quote:quote: Like I just did: It has all the properties of a square and a circle. What more do you need to know that isn't taken care of by the definition?
quote: By it's definition.
quote: No. Do I need to crack open Elements and give you the definition?
quote: 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. Perhaps you've heard of the book...it's the most popular book in the world after the Bible. It's called Elements. It provides definitions of circles (definition 15), triangles (20 and 21), and squares (22).
quote: And yet, I do. By definition, a "square-circle" is an object that has all the properties of a square and a circle.
quote: Sure I can. I simply need to provide a definition that is necessary and sufficient. And I have: A "square-circle" is a plane object that has both the properties of a square and a circle. Now, if you wish to backfill those definitions, here we go: Of quadrilateral figures, a square is that which is both equilateral and right-angled. A circle is a plane figure contained by one line such that all the straight lines falling upon it from one point among those lying within the figure equal one another. Ah, but that requires us to know what a "quadrilateral figure" is and what a "straight line" is. Rectilinear figures are those which are contained by straight lines, trilateral figures being those contained by three, quadrilateral those contained by four, and multilateral those contained by more than four straight lines. But, this requires us to know what "rectilinear" is. And when the lines containing the angle are straight, the angle is called rectilinear. But, this requires us to know what a straight line is. A straight line is a line which lies evenly with the points on itself. But, this requires us to know what a line is and what a point is. A line is breadthless length. A point is that which has no part. So there you go.
quote: Of course not. That's why they're called assumptions. If they were derived from some place else, they'd be conclusions.
quote: Are you saying planes don't exist in the real world? And no, let's not be disingenuous and talk about aeroplanes. I'm talking about the flat surfaces one might find in a discussion of plane geometry. After all, there's a whole bunch of mathematics surrounding the substitution of Euclid's Fifth Postulate with something else. But you end up with something that isn't plane geometry.
quote:quote: Nope. A circle is contained by a line. A line has more than one point. If space results in a single point, we have no circles, no squares, no lines. Simply a point.
quote: No, squares have four sides, not one. Even in a singularity, we can't have a "square-circle" because you can't have a square. And since "square-circles" need to have the properties of squares, then if squares can't exist, neither can "square-circles."
quote:quote: I'm not. You're reversing the direction again. All squares are rectangles, but not all rectangles are squares. Things that exist have definition, but not all things that have definition exist.
quote:quote: Strange...I was going to say the exact same thing to you. You keep refusing to deal with the question at hand but instead want to deal with all of these side issues, throwing ad hoc comments in as if they had any relevance.
quote:quote: See, this is what I mean about you hacking my post to shreds, completely devoid of context. Let's try it again: You mean Cartesian Doubt is logically valid?
quote:quote: How? There is no difference.
quote: Ad hoc. This argument is akin to the one I'm having with crashfrog where I am asking him if we have one apple and add one apple, do we not get two apples. His response? It isn't an apple! It's an orange! Excuse me? Since when did it become an orange? We weren't talking about oranges. We were talking about apples. Indeed, if we take one apple and and one orange, we don't get two apples (though we do get two fruits). But once again, we're not talking about oranges. We're talking about apples. If what I added was an orange, then the scenario of "take one apple and add one apple" is not satisfied and we should not be surprised to find that we've come up with a different result. If I take one apple and add one apple, is it possible for me to get something other than two apples?
quote:quote: I gave you examples. That you refused to read them is not my problem.
quote:quote: But I can state it negatively in mathematics, too: Given a factorization P of an integer n, there does not exist x such that x > max(p element P) for all n element I.
quote:quote: And why is it impossible to have absolute and infallible knowledge? Is that not what definitions are for?
quote:quote: Then why do I have so many textbooks that ask you to prove that A = A? Surely they think there is a point to it.
quote:quote: No, let's not. By turning around, you risk running into a different result. All squares are rectangles, but not all rectangles are squares. Sometimes it's easier to define something by what it is. Sometimes it's easier to define something by what it isn't. Take, for example, the example I gave you about finite and infinite sets. It's easier to define what a finite set is by describing what it is: A set whose elements can be mapped to Jn for some n. However, it's easier to define what an infinite set is by describing what it is not: A set that is not finite.
quote:quote: Once again, you have hacked my post to shreds so small that all context has been lost.
quote: Because you apparently do not understand the Incompleteness Theorems. They do not say that all axiomatic systems are necessarily inconsistent. Rather, they say that only those that are sophisticated enough to model arithmetic are. Not all axiomatic systems are that sophisticated and, sure enough, they are complete and consistent.
quote:quote: Yes, I am. The claim was a universal. It was not that you can't induct a negative. It was that you can't even deduct one.
quote:quote: Nope. Even randomness and chaos behaves logically.
quote:quote: Indeed, it is a philosophical concept, but it is a common discussion among mathematical circles. Please forgive my sloppy phrasing. That said, could you answer the question: Do you know about Platonism and how it relates to mathematics?
quote:quote: So? I'm not saying it's the most elegant example out there. I'm simply providing it as a way to describe the difference.
quote:quote: Excuse me? How does one get from the size of a set to the value of the elements within the set? You do understand that the bag you carry your apples home in is not the same as the apples it contains, yes?
quote:quote: Because there is no other way for them to behave. Are you saying they behave according to the rules of external logic? That we can think our way into forcing the behaviour of a black hole to conform to those thoughts?
quote: No, because logic says so. Black holes exist. Therefore, there is a behaviour on black holes. Therefore, there is an internal logic to that behaviour. Even randomness and chaos behave in a logical manner.
quote: Sure it is. What else could it be?
quote: I've got a better one: You not understanding.
quote: Is there a way it couldn't be?
quote:quote: So you are saying that because I can't disprove an infinite number of things, I am incapable of disproving a finite number of things. That seems to be the problem. You're looking for a disproof of all possible things while I'm simply referring to the specific one at hand. You don't have to do this "infinite series" of yours because we're not dealing with an infinite number of objects. It's only one.
quote:quote: No, I don't. I simply need to define the extent of my disproof. For example, the proof that my car keys are not in this room is only good for this room. It may or may not apply to other rooms.
quote: Who's jumping from the specific to the universal? I'm certainly not.
quote: I'm not talking about induction, though. I'mm talking about deduction.
quote:quote: Assuming you have a typical hand, take a look at it. You'll see five.
quote:quote: In the sense of the incompleteness theorems, yes. You can choose which way you want to go. You can write it out such that you get A and ~A or you can write it out such that you cannot say either A or ~A. They're the same thing. It's kinda like various statements of Euclid's Fifth Postulate. You can state it in a whole bunch of ways, but they all amount to the same thing. Non-Euclidean geometry is, primarily, about actually changing the substance of the Fifth Postulate. ------------------Rrhain WWJD? JWRTFM!
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Rrhain Member (Idle past 330 days) Posts: 6351 From: San Diego, CA, USA Joined: |
crashfrog:
quote: No, life intervening. Yeah, I know...shock and amazement that I have a life. ------------------Rrhain WWJD? JWRTFM!
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