Truth (Was:Re: PROOF INFINITY DOES NOT EXIST!...)
in [Math]
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From: Wolf K on 19 Jul 2010 10:29 On 19/07/2010 07:20, R. Srinivasan wrote: > On Jul 17, 8:09 am, c...(a)kcwc.com (Curt Welch) wrote: >> "K_h"<KHol...(a)SX729.com> wrote: >>> "Curt Welch"<c...(a)kcwc.com> wrote in message >>> news:20100715013832.531$Gi(a)newsreader.com... >> > [...] >> >>> Well, this is simply wrong. The truth underlying "1+1=2" is an absolute >>> truth. [...] 1+1=2 is a formal truth, which is absolute only in the sense that its domain of reference can be precisely defined. If 1 and 2 refer to integers, the statement is true. If they refer to vectors, it is true only for vectors with identical direction. If they refer to material quantities, its truth depends entirely on the materials in question, and will always include some error. Etc. > [...] >> >> Gee, how many times do I have to tell you that absolute truth doesn't exist [...] >> >> > I wouldn't go so far as to say that absolute truth does not exist. [...] I don't think "exist" is a good word to use about truth. I prefer "subsist" as the technical term. But that's a side issue. This sub-thread on truth is marred by an absence of definition. Exactly what do you mean by truth? What do Curt and the others mean? All the examples used are statements, which should be a clue. That is, an implicit stance in all the arguments so far is that truth is a property of statements. I don't think that is a good enough concept, as part two of this screed will I hope demonstrate. A) Formal (logical) and contingent truth I taught formal logic in high school, (I sneaked it in under the aim of "teach critical thinking".) As you might expect, some students twigged to the fact that "truth" is a vague, ambiguous, polysemous, slippery term. "Logical truth" is clearly defined: A statement is "logically true" when it has the form X = Y, where X and Y are well-formed statements in some language, and the rules of inference allow the transformation of X into Y, and vice versa. Note that this is a characterisation of a statement. However, it is not clear that X or Y are themselves true. A logical argument can demonstrate that some conclusion follows from some premises. If the premises are true, then so is the conclusion. But logic cannot demonstrate that the premises are true. You can show that the premises follow from some other premises, and so on, until you get to the axioms. But the truth of the axioms must be assumed. IOW, we need some means for agreeing on the truth of the premises. At this point in the discussion, students started invoking experience, common sense, obviousness, etc. And realised that "what is true for one person is not true for another." It was difficult to get them past that, but in the end most accepted that some replicable procedure could guarantee a limited truth: if we have the same experience, and say the same things about it, then the odds are that what we say is true, more or less. If we differ, then what we both have said is more or less wrong. Since someone can always disagree about what we have said, all statements about common experience are more or less wrong (and conversely more or less true). This too is a characterisation of statements: here we have contingent truth. B) Truth as a relationship So, what do we mean when conceive "truth" as a property of statements? A statement is an image of a concept. It has the same relationship to a concept as a photograph has to its subject. Of both we say that they are "true" if we apperceive some similarity between the statement and the concept, the photograph and its subject. Ditto for a theory (model) and the slice of universe it refers to. IOW, "truth" is a relationship between image and object, where "image" can be a sentence, a picture, a piece of music, an equation, etc, and "object" is whatever those images "are about". That relationship between image and object is an unanalysed given: we either get it or we don't. It rests on some formal equivalences, on patterns. We are a pattern-perceiving species, so much so that we perceive patterns "that aren't really there", in the sense that a slightly different point of view may destroy the pattern, while a "real" pattern can be perceived from several (sometimes drastically different) points of view. (Science has been characterised as the search for patterns that remain the same no matter how you look at them.) In a sense, we are democratic about truth, as Curt seems to be claiming: if a lot of people can see the same pattern from many different points of view, and/or if many people can replicate the pattern by some agreed-upon process, it is "really there." But we are also elitist: some patterns can be perceived only after more or less arduous training. But amongst those who have undergone this training, there is a pretty strong consensus on what the "real" patterns are, hence on what can be truthfully said about them. It should be obvious that "consensus" truths are contingent. They are also empirical: some unanticipated future experience may change our notion of what they refer to, of their limits as true statements. This is so even in the realm of formal truths, where we often do not know a priori whether any two statements are logically equivalent, or whether some set of premises implies some set of conclusions. Only the experiment of devising proofs can decide the question. And those proofs may show that the equivalence or conclusion is limited to a range of values (ie, objects that it refers to). In this respect, mathematics resembles empirical science. For more on how we arrive at some consensus about what's true, see Bas van Fraassen's "The Empirical Stance", Yale University Press, 2002. Disclosure: Bas and I were classmates many years ago, and discussed much of what I've distilled above. He discusses these themes much more expertly than I can. Hence my recommendation of his book. We do not entirely agree: ask two philosophers a question, and you'll get four answers. At least. ;-) cheers, wolf k.
From: Curt Welch on 19 Jul 2010 16:15 greywolf(a)ruddy.moss wrote: > On 19/07/2010 07:20, R. Srinivasan wrote: > > On Jul 17, 8:09 am, c...(a)kcwc.com (Curt Welch) wrote: > >> "K_h"<KHol...(a)SX729.com> wrote: > >>> "Curt Welch"<c...(a)kcwc.com> wrote in message > >>> news:20100715013832.531$Gi(a)newsreader.com... > >> > > [...] > >> > >>> Well, this is simply wrong. The truth underlying "1+1=2" is an > >>> absolute truth. [...] > > 1+1=2 is a formal truth, which is absolute only in the sense that its > domain of reference can be precisely defined. If 1 and 2 refer to > integers, the statement is true. If they refer to vectors, it is true > only for vectors with identical direction. If they refer to material > quantities, its truth depends entirely on the materials in question, and > will always include some error. Etc. > > > [...] > >> > >> Gee, how many times do I have to tell you that absolute truth doesn't > >> exist [...] > >> > >> > > I wouldn't go so far as to say that absolute truth does not exist. > [...] > > I don't think "exist" is a good word to use about truth. I prefer > "subsist" as the technical term. But that's a side issue. > > This sub-thread on truth is marred by an absence of definition. Exactly > what do you mean by truth? What do Curt and the others mean? I took the time to clearly define it in at least one of my messages. It's a language statement that accurately describes some aspect of the universe. I claim there are no absolute truths because there is no way to build a machine to test if a given statement is an accurate description of some aspect of the universe with 100% certainty. Machines in this universe can't operate with 100% certainty of function. Humans are machines that are subject this this same uncertainty in their actions and beliefs. We can build machines to verify a suspected truth to very high levels of certainty, but never absolute certainty. > All the examples used are statements, which should be a clue. That is, > an implicit stance in all the arguments so far is that truth is a > property of statements. I don't think that is a good enough concept, as > part two of this screed will I hope demonstrate. Whether the statement is making a claim which is a truth of language alone (1+1=2), or a truth about a non-language feature of the universe (the ball is red), it's still a language description of some aspect of the universe that must be tested for accuracy. > A) Formal (logical) and contingent truth > > I taught formal logic in high school, (I sneaked it in under the aim of > "teach critical thinking".) As you might expect, some students twigged > to the fact that "truth" is a vague, ambiguous, polysemous, slippery > term. > > "Logical truth" is clearly defined: A statement is "logically true" when > it has the form X = Y, where X and Y are well-formed statements in some > language, and the rules of inference allow the transformation of X into > Y, and vice versa. Note that this is a characterisation of a statement. X is an accurate description of Y (which is an aspect of the universe), and vice versa. > However, it is not clear that X or Y are themselves true. If, per the rules of the language, X is a description of Y, it's true by the way I look at it. "1+1 = 2" is true because 1+1 is a valid description of 2 in that language. Whether either alone is a valid description of something else in the universe is not relevant to the truth of the description being test. At least that's how I look at it. > A logical > argument can demonstrate that some conclusion follows from some > premises. If the premises are true, then so is the conclusion. But logic > cannot demonstrate that the premises are true. You can show that the > premises follow from some other premises, and so on, until you get to > the axioms. But the truth of the axioms must be assumed. IOW, we need > some means for agreeing on the truth of the premises. > > At this point in the discussion, students started invoking experience, > common sense, obviousness, etc. And realised that "what is true for one > person is not true for another." It was difficult to get them past that, > but in the end most accepted that some replicable procedure could > guarantee a limited truth: if we have the same experience, and say the > same things about it, then the odds are that what we say is true, more > or less. If we differ, then what we both have said is more or less > wrong. Since someone can always disagree about what we have said, all > statements about common experience are more or less wrong (and > conversely more or less true). This too is a characterisation of > statements: here we have contingent truth. > > B) Truth as a relationship > > So, what do we mean when conceive "truth" as a property of statements? A > statement is an image of a concept. It has the same relationship to a > concept as a photograph has to its subject. Of both we say that they are > "true" if we apperceive some similarity between the statement and the > concept, the photograph and its subject. Ditto for a theory (model) and > the slice of universe it refers to. > > IOW, "truth" is a relationship between image and object, where "image" > can be a sentence, a picture, a piece of music, an equation, etc, and > "object" is whatever those images "are about". Yes, I would call your "image" language because it's always a representation of some other aspect of the universe. Truth is found by testing to see if the representation matches the universe it describes. This of course also adds the complexity of determining _what_ it describes - which requires language interpretation hardware which acts as the definition of what the language is attempting to describe - which is the same hardware which attempts to test if the universe matches the language. > That relationship between image and object is an unanalysed given: we > either get it or we don't. It rests on some formal equivalences, on > patterns. We are a pattern-perceiving species, so much so that we > perceive patterns "that aren't really there", in the sense that a > slightly different point of view may destroy the pattern, while a "real" > pattern can be perceived from several (sometimes drastically different) > points of view. (Science has been characterised as the search for > patterns that remain the same no matter how you look at them.) > > In a sense, we are democratic about truth, as Curt seems to be claiming: > if a lot of people can see the same pattern from many different points > of view, and/or if many people can replicate the pattern by some > agreed-upon process, it is "really there." But we are also elitist: some > patterns can be perceived only after more or less arduous training. But > amongst those who have undergone this training, there is a pretty strong > consensus on what the "real" patterns are, hence on what can be > truthfully said about them. > > It should be obvious that "consensus" truths are contingent. They are > also empirical: some unanticipated future experience may change our > notion of what they refer to, of their limits as true statements. This > is so even in the realm of formal truths, where we often do not know a > priori whether any two statements are logically equivalent, or whether > some set of premises implies some set of conclusions. Only the > experiment of devising proofs can decide the question. And those proofs > may show that the equivalence or conclusion is limited to a range of > values (ie, objects that it refers to). In this respect, mathematics > resembles empirical science. Our natural language is a socially negotiated standard. The "truth" about what the language standard adds yet another level of complexity to identifying truth that I didn't even think about. But that only results in whether truth is agreed between two or more "truth testing" machines. If we look at only one machine, then we can use it's definition of the language to determine the truth. If either the machine fails to evaluate truth the same way, or if it changes it's definition of the language over time, we have yet another reason to say that truth is not absolute. > For more on how we arrive at some consensus about what's true, see Bas > van Fraassen's "The Empirical Stance", Yale University Press, 2002. > > Disclosure: Bas and I were classmates many years ago, and discussed much > of what I've distilled above. He discusses these themes much more > expertly than I can. Hence my recommendation of his book. We do not > entirely agree: ask two philosophers a question, and you'll get four > answers. At least. ;-) > > cheers, > wolf k. All interesting stuff. -- Curt Welch http://CurtWelch.Com/ curt(a)kcwc.com http://NewsReader.Com/
From: casey on 19 Jul 2010 17:08 On Jul 20, 12:29 am, Wolf K <weki...(a)sympatico.ca> wrote: > On 19/07/2010 07:20, R. Srinivasan wrote: > > > On Jul 17, 8:09 am, c...(a)kcwc.com (Curt Welch) wrote: > >> "K_h"<KHol...(a)SX729.com> wrote: > >>> "Curt Welch"<c...(a)kcwc.com> wrote in message > >>>news:20100715013832.531$Gi(a)newsreader.com... > > > [...] > > >>> Well, this is simply wrong. The truth underlying "1+1=2" is an absolute > >>> truth. [...] > > 1+1=2 is a formal truth, which is absolute only in the sense that its > domain of reference can be precisely defined. If 1 and 2 refer to > integers, the statement is true. If they refer to vectors, it is true > only for vectors with identical direction. If they refer to material > quantities, its truth depends entirely on the materials in question, and > will always include some error. Etc. > > > [...] > > >> Gee, how many times do I have to tell you that absolute truth doesn't exist [...] > > > I wouldn't go so far as to say that absolute truth does not exist. > > [...] > > I don't think "exist" is a good word to use about truth. I prefer > "subsist" as the technical term. But that's a side issue. > > This sub-thread on truth is marred by an absence of definition. Exactly > what do you mean by truth? What do Curt and the others mean? > > All the examples used are statements, which should be a clue. That is, > an implicit stance in all the arguments so far is that truth is a > property of statements. I don't think that is a good enough concept, as > part two of this screed will I hope demonstrate. > > A) Formal (logical) and contingent truth > > I taught formal logic in high school, (I sneaked it in under the aim of > "teach critical thinking".) As you might expect, some students twigged > to the fact that "truth" is a vague, ambiguous, polysemous, slippery term.. > > "Logical truth" is clearly defined: A statement is "logically true" when > it has the form X = Y, where X and Y are well-formed statements in some > language, and the rules of inference allow the transformation of X into > Y, and vice versa. Note that this is a characterisation of a statement. > > However, it is not clear that X or Y are themselves true. A logical > argument can demonstrate that some conclusion follows from some > premises. If the premises are true, then so is the conclusion. But logic > cannot demonstrate that the premises are true. You can show that the > premises follow from some other premises, and so on, until you get to > the axioms. But the truth of the axioms must be assumed. IOW, we need > some means for agreeing on the truth of the premises. > > At this point in the discussion, students started invoking experience, > common sense, obviousness, etc. And realised that "what is true for one > person is not true for another." It was difficult to get them past that, > but in the end most accepted that some replicable procedure could > guarantee a limited truth: if we have the same experience, and say the > same things about it, then the odds are that what we say is true, more > or less. If we differ, then what we both have said is more or less > wrong. Since someone can always disagree about what we have said, all > statements about common experience are more or less wrong (and > conversely more or less true). This too is a characterisation of > statements: here we have contingent truth. > > B) Truth as a relationship > > So, what do we mean when conceive "truth" as a property of statements? A > statement is an image of a concept. It has the same relationship to a > concept as a photograph has to its subject. Of both we say that they are > "true" if we apperceive some similarity between the statement and the > concept, the photograph and its subject. Ditto for a theory (model) and > the slice of universe it refers to. > > IOW, "truth" is a relationship between image and object, where "image" > can be a sentence, a picture, a piece of music, an equation, etc, and > "object" is whatever those images "are about". > > That relationship between image and object is an unanalysed given: we > either get it or we don't. It rests on some formal equivalences, on > patterns. We are a pattern-perceiving species, so much so that we > perceive patterns "that aren't really there", in the sense that a > slightly different point of view may destroy the pattern, while a "real" > pattern can be perceived from several (sometimes drastically different) > points of view. (Science has been characterised as the search for > patterns that remain the same no matter how you look at them.) > > In a sense, we are democratic about truth, as Curt seems to be claiming: > if a lot of people can see the same pattern from many different points > of view, and/or if many people can replicate the pattern by some > agreed-upon process, it is "really there." But we are also elitist: some > patterns can be perceived only after more or less arduous training. But > amongst those who have undergone this training, there is a pretty strong > consensus on what the "real" patterns are, hence on what can be > truthfully said about them. > > It should be obvious that "consensus" truths are contingent. They are > also empirical: some unanticipated future experience may change our > notion of what they refer to, of their limits as true statements. This > is so even in the realm of formal truths, where we often do not know a > priori whether any two statements are logically equivalent, or whether > some set of premises implies some set of conclusions. Only the > experiment of devising proofs can decide the question. And those proofs > may show that the equivalence or conclusion is limited to a range of > values (ie, objects that it refers to). In this respect, mathematics > resembles empirical science. > > For more on how we arrive at some consensus about what's true, see Bas > van Fraassen's "The Empirical Stance", Yale University Press, 2002. > > Disclosure: Bas and I were classmates many years ago, and discussed much > of what I've distilled above. He discusses these themes much more > expertly than I can. Hence my recommendation of his book. We do not > entirely agree: ask two philosophers a question, and you'll get four > answers. At least. ;-) > > cheers, > wolf k. Interesting post. JC
From: Wolf K on 19 Jul 2010 22:14 On 19/07/2010 16:15, Curt Welch wrote: > greywolf(a)ruddy.moss wrote: [snip most fo Curt's remarks] >> For more on how we arrive at some consensus about what's true, see Bas >> van Fraassen's "The Empirical Stance", Yale University Press, 2002. >> >> Disclosure: Bas and I were classmates many years ago, and discussed much >> of what I've distilled above. He discusses these themes much more >> expertly than I can. Hence my recommendation of his book. We do not >> entirely agree: ask two philosophers a question, and you'll get four >> answers. At least. ;-) >> >> cheers, >> wolf k. > > All interesting stuff. > My first impulse was to say that your remarks miss the point, but in reflection I think they expand on it, albeit obliquely. cheers, wolf k.
From: Wolf K on 19 Jul 2010 22:15
On 19/07/2010 17:08, casey wrote: > On Jul 20, 12:29 am, Wolf K<weki...(a)sympatico.ca> wrote: [...]>> Disclosure: Bas and I were classmates many years ago, and discussed much >> of what I've distilled above. He discusses these themes much more >> expertly than I can. Hence my recommendation of his book. We do not >> entirely agree: ask two philosophers a question, and you'll get four >> answers. At least. ;-) >> >> cheers, >> wolf k. > > Interesting post. > > > JC > Thanks. wolf k. |