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Analytic-synthetic vs priori/ posteriori

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Analytic-synthetic vs priori/ posteriori

180 Proof
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Posted 07/09/09 - 05:51 AM:	#31
keda wrote: This is quite a common objection leveled against Kant however Kant never denied the existence of non euclidean geometries, but on the contrary his theory anticipated non euclidean geometries and contrary to the charge does their discovery only confirm the syntheticity of Kant's claim that euclidean geometry is synthetic, for this entails that non euclidean geometries can be concieved. Explain how Kant's "Transcendental Aesthetic" specifically "anticipated non-Euclidean geometries". The denial of the parallel postulate does not lead to a contradiction. With respect to Euclid's system, your statement is false, keda. The meaning of any postulate is determined by the system to which it belongs; absent that system it is meaningless, that is to say neither analytic nor synthetic. Denial within the system is contradictory. Regarding Einstein's theory if relativity, if this is what you refer to by post-Newtonian physics, it is merely a semantic disagreement of what is to be called spatial and temporal measurement. Again, incorrect. "Space and time" have been demonstrated to be relative to inertial reference frames (i.e. masses in motion) and therefore cannot merely be "a priori forms of intuition". Einstein's counterintuitive notions of space and time and consequently the use of non euclidean geometry is a consequence from ... ... being far more precise & accurate in modelling phenomena than Newtonian-Euclidean models. Philosophically, one would expect "counterintuitive notions" to be more applicable to reality than "intuitive notions" because reality is not limited to human scale "experience". Kant's "transcendental project" is epistemologically inadequate because it's too anthropocentric (e.g. vide Schopenhauer). Space is still euclidean in the intuitive sense it is normally used despite what those who may be under the semantic trance that Einstein's non standard use of language have induced, confusedly suggests. I suppose Aristotlean-Ptolemians also whined that Newtonians were just semantically entranced by Copernican-Cartesian jargon. Special pleading gibberish. Edited by 180 Proof on 07/11/09 - 10:33 AM. Reason: spelling ...
The question isn't "Which explanations do I believe?" but rather "Which explanations do I least disbelieve?" Absence of evidence THAT MUST BE THERE (i.e. implied by any claim, concept, or (its) predicates, that affects changes in/to the world) entails evidence of absence. [What cannot be done?[What cannot be hoped?[What cannot be known?]]]

movinartist
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Posted 07/09/09 - 06:33 AM:	#32
I believe information can have truth in anything. Anything can also be contradictory and still be true in information. Information can be attached to anything. It can be taken out of context. Words are information. What makes information more or less right depends on its relationship to objects. Once information is not connected to the comparison of objects, information becomes less definable. All this talk of words gives words too much credit. Their context dictates the meaning, not the other way around. Even the method should reflect the order of objects. In order to be accurate, the method should be specific to the circumstance. Postulating about postulating is a mirror in a mirror.

keda
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Posted 07/10/09 - 07:11 AM:	#33
Death Monkey wrote: Well, I suppose this is as good a place to start as any: http://en.wikipedia.org/wiki/Peano_axioms In a nutshell, the symbols "5", "7", and "12" are defined as natural numbers, which are in turn defined in terms of succession. The statement "5+7=12" is a statement about a particular relationship between those natural numbers. That this particular relationship between these numbers obtains, logically follows from the definition of the natural numbers. It cannot follow from the definition of natural numbers alone, but must also follow from the definition of +, which according to the site is defined as: a+0 = a, a+S(b) = S(a+b) Now this can't be analytic judgment because a+0 = a makes a singular judgment about 0. A definition is merely suppose to have general judgments, namely the defining characteristics of a type of thing e.g. defining X as Peter's friend, is not a definition, because it assumes the existence of Peter. If Peter doesn't exist, the judgment is false, which no analytic judgment can be. The point is that in order to make such an abstraction, you have to formally define what you mean by, for example, the size of a set of imaginary objects. How about the size being an equivalence class of sets where the equivalence relation is specified by there being a bijection between the sets. x is a successor number of y iff there exist an x' and y', such that y' is a member of y, x' is not a member of y', and the union of y' and x' is a member of x. x is a sum of y and z iff there exist an x', y' and z' such that x' is a member of x, y' is a member of y, z' is a member of z, there exist no member w such that w is a member of x' and w is a member of y', and x' is a union of y' and z'. The way in which 5+7=12 can be shown a priori is demonstrating that there are non empirical elements, namely successive units of time, which can be the x' in the above definitions. We do this automatically when counting since we do it by cognizing a succession of units of time. Sets containing empirical elements are deemed to be members of the classes synthetic a posteriori, while sets containing only pure elements are deemed as members of the classes synthetic a priori. This idea was insipired by this document I read although I don't fully agree what he is saying, but perhaps it's mostly because him not using entirely Kantian terminology. The truth-value of statements like "5+7=12" directly follow from those definitions. As defined above, it does not follow. No, the meaning of the sentence "this flower is blue" is specified by the English language. A series of symbols is not a statement until such time as its meaning is specified. Change the meaning of any of the words in that sentence, and it is still the same sentence, but is no longer the same statement. Indeed, it may no longer be a statement at all. Personally I would have called it sentence too, but you seemed to indicate that even statements have to be interpreted here: Change those rules, and you change the meaning of the statement In any case I hope you got the point there that the truth value of "this flower is blue" is not fixed by the definitions of the words in the sentence But let's say what you meant to say was this: Either way, the point is that mathematics defines the rules specify the meaning of the mathematical sentence. Change those rules, and you change the meaning of the sentence. Therefore the truth-value of the sentence is fixed by its meaning. It still doesn't specify any feature that distinguishes math from the english language in fixing the truth value by its meaning. 180 Proof wrote: Explain how Kant's "Transcendental Aesthetic" specifically "anticipated non-Euclidean geometries". rolling eyes I thought I just did. Non euclidean geometries can be concieved according to Kant because Euclidean geometry is synthetic per definition. What cannot be done however is to imagine non euclidean geometries. With respect to Euclid's system, your statement is false, keda. The meaning of any postulate is determined by the system to which it belongs; absent that system it is meaningless, that is to say neither analytic nor synthetic. Denial within the system is contradictory. The parallell postulate is not just a string of symbols, but an interpretation of such. What I'm saying is that negating this interpretation, does not result in a contradiction. In any case I think you've just admitted the semantic problem yourself. If the parallell postulade does not have the same meaning in Einstein's theory of relativity as it has for Euclid and Kant, simply because it is not an analytic truth in Einstein's theory of relativity while it is in Euclid's, then one cannot use the negation of the parallell postulate used by Einstein as a refutation of Euclid or Kant. ... of being far more precise & accurate in modelling phenomena than Newtonian-Euclidean models. While this is true, I made no mention of Newton, but Lorentz, who did make the same predictions as Einstein regarding special relativity, except with the use of Euclidean space.
All about making money Free Europe Now How to fix your country In thought, men distance themselves from nature in order thus imaginatively to present it to themselves--but only in order to determine how it is to be dominated - Adorno and Horkheimer

Death Monkey
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Posted 07/10/09 - 10:51 AM:	#34
keda, It cannot follow from the definition of natural numbers alone, but must also follow from the definition of +, which according to the site is defined as: a+0 = a, a+S(b) = S(a+b) Now this can't be analytic judgment because a+0 = a makes a singular judgment about 0. A definition is merely suppose to have general judgments, namely the defining characteristics of a type of thing e.g. defining X as Peter's friend, is not a definition, because it assumes the existence of Peter. If Peter doesn't exist, the judgment is false, which no analytic judgment can be. You are mistaking the '+' symbol as being a subject of the statement along with the numbers, but it is not. The subjects are the numbers '5', '7', and '12'. The predicate that is being attributed to the subjects is that these numbers exhibit a certain relationship with each other. That said, the fact that the particular relationship between those three numbers obtains, does directly follow from the definition of the natural numbers. The fact that you have to define the terms you use to describe that relationship in terms of the natural numbers, is not a problem. After all, how could it be otherwise? The natural numbers are a wholly made-up set to begin with. The only properties that can be attributed to them are those that they are defined to have, so it necessarily follows that any terms we use to describe relationships between natural numbers will, themselves, have to be defined in terms of the definition of the natural numbers. Those three numbers exhibit that relationship that those operators are used to describe regardless of whether you ever bother to define those specific operators or not. In fact, the relationship can be expressed in many different ways, using any number of different operators. The relationship itself, though, directly follows from the definition of the natural numbers. It is not logically possible for that relationship to not hold for those three numbers. Thus the predicate (that this particular relationship obtains) is contained in the subject (those three numbers). The point is that in order to make such an abstraction, you have to formally define what you mean by, for example, the size of a set of imaginary objects. How about the size being an equivalence class of sets where the equivalence relation is specified by there being a bijection between the sets. x is a successor number of y iff there exist an x' and y', such that y' is a member of y, x' is not a member of y', and the union of y' and x' is a member of x. x is a sum of y and z iff there exist an x', y' and z' such that x' is a member of x, y' is a member of y, z' is a member of z, there exist no member w such that w is a member of x' and w is a member of y', and x' is a union of y' and z'. The way in which 5+7=12 can be shown a priori is demonstrating that there are non empirical elements, namely successive units of time, which can be the x' in the above definitions. We do this automatically when counting since we do it by cognizing a succession of units of time. Sets containing empirical elements are deemed to be members of the classes synthetic a posteriori, while sets containing only pure elements are deemed as members of the classes synthetic a priori. This idea was insipired by this document I read although I don't fully agree what he is saying, but perhaps it's mostly because him not using entirely Kantian terminology. OK, here you are talking about set theory, but the same argument holds. There is nothing synthetic about sets containing only pure elements, because the only properties that can be attributed to such sets and their elements, are those that the sets and elements are defined to have. In any case I hope you got the point there that the truth value of "this flower is blue" is not fixed by the definitions of the words in the sentence Of course not. It is a synthetic statement. Either way, the point is that mathematics defines the rules specify the meaning of the mathematical sentence. Change those rules, and you change the meaning of the sentence. Therefore the truth-value of the sentence is fixed by its meaning. It still doesn't specify any feature that distinguishes math from the english language in fixing the truth value by its meaning. Like I said above, in mathematical statements the objects being talked about have only those properties that they are defined to have. This is not true in ordinary language. The definition of the natural numbers completely specifies all properties of the natural numbers. The definition of the word "cat" does not specify all of the properties that a cat can have, or even all of the properties that cats in general have. Statements about numbers are therefore analytic, while a statement about a cat could be analytic (if the property being attributed follows from the definition), or could be synthetic (if it does not). Non euclidean geometries can be concieved according to Kant because Euclidean geometry is synthetic per definition. There is no truth about any of the objects defined in euclidean geometry that does not logically follow from the definition of euclidean geometry. Therefore any predicate a statement could attribute to objects defined in euclidean geometry, are necessarily contained in euclidean geometry. Thus it is analytic. DM
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keda
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Posted 07/10/09 - 01:30 PM:	#35
Death Monkey wrote: You are mistaking the '+' symbol as being a subject of the statement along with the numbers, but it is not. The subjects are the numbers '5', '7', and '12'. The predicate that is being attributed to the subjects is that these numbers exhibit a certain relationship with each other. I did not rely on + being a subject or part of it. Suppose + is part of the predicate. It still makes a singular judgment about 0. That said, the fact that the particular relationship between those three numbers obtains, does directly follow from the definition of the natural numbers. The so called definition of natural numbers apparently suffer from the same problem i.e. it makes a singular judgment about 0 (axiom 5). Such judgments cannot be analytic. OK, here you are talking about set theory, but the same argument holds. There is nothing synthetic about sets containing only pure elements, because the only properties that can be attributed to such sets and their elements, are those that the sets and elements are defined to have. I didn't say there is something synthetic about sets containing only pure elements. I was talking about sets containing pure elements, being part of a type of equivalence class I defined above. Like I said above, in mathematical statements the objects being talked about have only those properties that they are defined to have. I don't see why. a statement about a cat could be analytic (if the property being attributed follows from the definition) As explained earlier, an analytic statements cannot fix particulars, such as a particular cat. This would make it synthetic. E.g. My cat is an animal, is syntehtic, because it posits the existence of a cat. While it doesn't say anything about the cat (other than that it is mine), it can be divided up into a complex of judgments that are presupposed in the claim. There is no truth about any of the objects defined in euclidean geometry that does not logically follow from the definition of euclidean geometry. Therefore any predicate a statement could attribute to objects defined in euclidean geometry, are necessarily contained in euclidean geometry. Thus it is analytic. It seems like we are back to the "is analtyic because I define it so"-style argument. What if there is a pure science of geometry that you have simply ignore much like Leibnitz ignore your car by defining "your car" as having the property grey? It doesn't make the science disappear no more than the car does.
All about making money Free Europe Now How to fix your country In thought, men distance themselves from nature in order thus imaginatively to present it to themselves--but only in order to determine how it is to be dominated - Adorno and Horkheimer

Death Monkey
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Posted 07/10/09 - 03:10 PM:	#36
keda, I did not rely on + being a subject or part of it. Suppose + is part of the predicate. It still makes a singular judgment about 0. It specifies a mapping from pairs of natural numbers to single natural numbers. Again, the relationship between the three numbers in the statement holds independantly of how you define the symbol '+', and obtains even if you never define that symbol. The symbol is just used to describe that relationship. It is not even necessary to use that symbol to describe the relationship. That said, the fact that the particular relationship between those three numbers obtains, does directly follow from the definition of the natural numbers. The so called definition of natural numbers apparently suffer from the same problem i.e. it makes a singular judgment about 0 (axiom 5). Such judgments cannot be analytic. I don't understand this argument. The definition of the natural numbers defines each and every natural number. The fact that it defines zero one way, and then defines all the others in an iterative way, doesn't make any difference. Each of those numbers is exactly what it is stipulated to be by the definition. Nothing more and nothing less. I don't see why you think it should make a difference. After all, the issue is whether the predicate is "contained" in the subject, right? In this case, the subject is the natural numbers, and any property that follows from the definition of the natural numbers is therefore contained in the natural numbers. What difference does it make how the natural numbers are defined? All that matters is that the predicate of the statement in question is contained in the natural numbers. Like I said above, in mathematical statements the objects being talked about have only those properties that they are defined to have. I don't see why. Well, name a counter-example. If you think that a particular mathematical statement is true, but cannot logically derive that it is true from the axioms of the mathematical system the statement is formulated in, then point it out, and explain to me how you know the statement is, in fact, true. For example, if you do not agree that "5 + 7 = 12" is analytic, then explain to me how you know that it is true. You cannot claim that you can logically derive that it is true from some set of axioms, because then it would be analytic. And of course you cannot cite emprical evidence, because then it would not be a priori. You can try to cite "sensibility" as your source of information, but I don't see how you can justify such a claim. a statement about a cat could be analytic (if the property being attributed follows from the definition) As explained earlier, an analytic statements cannot fix particulars, such as a particular cat. This would make it synthetic. E.g. My cat is an animal, is syntehtic, because it posits the existence of a cat. While it doesn't say anything about the cat (other than that it is mine), it can be divided up into a complex of judgments that are presupposed in the claim. I didn't say anything about it being a specific cat. For example, the statement "cats are mammals" is analytic. Also, going back to the point about 0 being defined in a singular way, this argument does not apply because it is not logically possible for the natural number 0 to not exist. After all, in the context of natural numbers we are not talking about existence in the sense of being instantiated in this world. In that context, existence just means that it is a member of the natural numbers, which by definition of the natural numbers, it is. There is no truth about any of the objects defined in euclidean geometry that does not logically follow from the definition of euclidean geometry. Therefore any predicate a statement could attribute to objects defined in euclidean geometry, are necessarily contained in euclidean geometry. Thus it is analytic. It seems like we are back to the "is analtyic because I define it so"-style argument. No, the argument is that it is analytic because the predicate is clearly contained in the subject, which is how analytic is defined. I did not define it so. What if there is a pure science of geometry that you have simply ignore much like Leibnitz ignore your car by defining "your car" as having the property grey? It doesn't make the science disappear no more than the car does. I don't see your point. What does this "pure science of geometry" have to do with euclidean geometry? They are two completely different things. Euclidean geometry is what it is defined to be, regardless of whether some "pure geometry" exists, or whether it agrees with this "pure geometry" in any way. I am not ignoring anything by claiming that euclidean geometry is analytic. I am not making any claims about anything other than euclidean geometry by making that claim. In fact, this seems to me to be a pretty clear indication of it being analytic. It doesn't make any difference whether (or to what degree) reality conforms to euclidean geometry. It doesn't have to conform to it, because euclidean geometry is something we made up. It is what we have stipulated it to be. Nothing more and nothing less. DM
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180 Proof
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Posted 07/10/09 - 03:11 PM:	#37
keda wrote: I thought I just did. Not at all. You've completely lost me. Non euclidean geometries can be concieved according to Kant because Euclidean geometry is synthetic per definition. What cannot be done however is to imagine non euclidean geometries. So because Kant defines geometry as synthetic that makes it synthetic? Read Lewis Carroll much?
The question isn't "Which explanations do I believe?" but rather "Which explanations do I least disbelieve?" Absence of evidence THAT MUST BE THERE (i.e. implied by any claim, concept, or (its) predicates, that affects changes in/to the world) entails evidence of absence. [What cannot be done?[What cannot be hoped?[What cannot be known?]]]

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Posted 07/10/09 - 10:38 PM:	#38
Concerning the truth-value 5+7=12, what I believe is of its formality, not of meaning. When seven was added to five it is twelve. When this flower was blue, it is ... When my car was green, it is ... When my cat was an animal, it is ... I assume these sentences are ready for uncoveredness by extra=sensibility or intelligence or transcendence, or whatever for the sake of truth-value. The thing is 'What it is' from the proposition. What a priori synthetic proposition is so special than others is because it reveals impersonal subject 'It'.

keda
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Posted 07/11/09 - 04:19 AM:	#39
Death Monkey wrote: I don't understand this argument. The definition of the natural numbers defines each and every natural number. The fact that it defines zero one way, and then defines all the others in an iterative way, doesn't make any difference. Each of those numbers is exactly what it is stipulated to be by the definition. Nothing more and nothing less. Then look at it this way. The definition of a bachelor is an unmarried man. Does this mean that any unmarried men exist? No, because definitions do not define things into existence. Otherwise it is no longer a definition then, because it assumes something exists, which is a synthetic claim. It goes beyond the concept of merely being a bachelor in saying that a bachelor exists. The same is true about natural numbers. If you say something about a specific number, such as 0, it is no longer a definition, but a synthetic claim. In this case, the subject is the natural numbers, and any property that follows from the definition of the natural numbers is therefore contained in the natural numbers. It is a complex of judgments, one of which is singular, namely that 0 is a natural number, as I already pointed out. Well, name a counter-example. If you think that a particular mathematical statement is true, but cannot logically derive that it is true from the axioms of the mathematical system the statement is formulated in, then point it out, and explain to me how you know the statement is, in fact, true. For example, if you do not agree that "5 + 7 = 12" is analytic, then explain to me how you know that it is true. You cannot claim that you can logically derive that it is true from some set of axioms, because then it would be analytic. And of course you cannot cite emprical evidence, because then it would not be a priori. You can try to cite "sensibility" as your source of information, but I don't see how you can justify such a claim. I already provided a simplified a priori synthetic arithmetic with synthetic a priori addition. Its truth relies on the ability to intuit a sequence of representations in time, in order to count them as defined by the successor operation as I defined it, or summation in gerneral. Without this ability, it cannot be said to be true. Specifically, it uses the existential quantifier to explicate that both of the operations need third party object(s) to exist to obtain its truth value. The truth of this claim is ovely evident in that we use arithmetics purely in representation of time, such as after 5 and further 7 hours 12 in total has passed. I didn't say anything about it being a specific cat. For example, the statement "cats are mammals" is analytic. Yes, you are right, but particular judgments i.e. Some x is y, is still synthetic. "a cat is on the table" is a particular judgment, which posits the existence of an object, yet it is not a singular judgment e.g. "my cat is on the table". No, the argument is that it is analytic because the predicate is clearly contained in the subject, which is how analytic is defined. I did not define it so. By "it" I meant defining "euclidean geometry". You defined it as a definition, which is more than just a set of axioms, so of course it has to be analytic because definitions are analytic by definition. After all, in the context of natural numbers we are not talking about existence in the sense of being instantiated in this world Well I was talking about it in that sense, even if you didn't. The article you referenced didn't suggest either way. I don't see your point. What does this "pure science of geometry" have to do with euclidean geometry? They are two completely different things. Euclidean geometry is what it is defined to be, regardless of whether some "pure geometry" exists, or whether it agrees with this "pure geometry" in any way. Euclidean geometry is just a bunch of axioms. not something to be defined. Euclid himself believed they were statements about reality, and not empirical but self evident, so he certainly believed it was a pure science. In fact, pardon my not noticing it before but the "definition of euclidean geometry" sounds like you treat euclidean geometry as a concept, and not axioms. I thought you mean that they are definitions, but Euclid did not say that. They certainly don't have the grammatical structure of definitions.
All about making money Free Europe Now How to fix your country In thought, men distance themselves from nature in order thus imaginatively to present it to themselves--but only in order to determine how it is to be dominated - Adorno and Horkheimer

keda
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Posted 07/11/09 - 06:44 AM:	#40
180 Proof wrote: Not at all. You've completely lost me. Synthetic means true in virtue of something further than the meaning of its constituent concepts. which means that the concepts are insufficient to determine the truth value, however a contradiction arising from them would mean it is sufficient to determine the truth value, namely that it is false. Thus its negation must be true. The negation of a contradiction is a tautology or analytic judgment, which means its negation is a contradiction. Now if Kant says Euclidian geometry is synthetic, which he does, then that means he thinks its negation, non eucliean geometry is at least logically possible. So because Kant defines geometry as synthetic that makes it synthetic? Read Lewis Carroll much? No, I meant the definition of "synthetic", not "geometry".
All about making money Free Europe Now How to fix your country In thought, men distance themselves from nature in order thus imaginatively to present it to themselves--but only in order to determine how it is to be dominated - Adorno and Horkheimer

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