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Analytic-synthetic vs priori/ posteriori

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Analytic-synthetic vs priori/ posteriori

jtoma
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Posted 07/18/09 - 12:23 AM:	#71
Right, thanks for clarifying. Regarding (2), '5+7=12' is true whether or not the real world ever instantiates that claim with an example. For another example, the claim 'there is no largest prime number' is true, but not empirically verifiable. Maybe Kant's intellectual intuition could verify that. Anyway, I don't see how mathematical statements need to refer to some aspect of reality--at least in the empirical sense. I am not sure I understand what you think an element of a mathematical statement would refer to in reality. Like, is the referent of '0' a physical object existing in the real world? Does '0's referent only exist in our heads--if so, then would '0' refer some object of thought? Is an object of thought part of what you consider reality? Based on your passage on Kant above, it looks like you think 'sense' is reserved for things passively received, not like zero which I conceive of (if unclearly) at will. You must think that you can sense (in your sense) the referents of terms in some mathematical statements. For example, picking 1 petal off a flower that starts with 5 leaves 4 petals. You can sense those petals and the picking. But what does that sense perception have to do with the mathematical statement that '5-1=4'? Well, it is an instance of the general statement implied (to some) by '5-1=4'. But that the flower can serve as an instance of the mathematical statement DOES NOT MEAN that in any utterance of '5-1=4', '5' refers to the original number of petals, '1' the number of petals picked, and so on. If you are with me there, then we can agree that the knowledge you have of the petals and the flower is known a posteriori. Likewise, the knowledge that you have of the mathematical statement instantiated by the flower is known a priori and the proposition that the statement represents is analytic. gravy? //(in math it does not seem as necessary as it is with common language to appeal to propositions as entities different than the statements--so long as propositions don't get an entirely new language, the referent's of mathematical terms are well defined and contextually constant, at least compared to common language). I can't tell if you think it is bad that there may be empirical evidence for your a priori synthetic claim. Yea, its too bad I left the critique of pure reason at school because I could really use some transcendental aesthetics right now... keda wrote: The only other possible way of knowing objects is when the reverse causal relationship holds, namely our knowledge of the object causes the existence of the object i.e. the intellect creates the object of its knowledge. is this what you think about 0?

keda
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1 of 2 people found this post helpful Posted 07/18/09 - 07:20 AM:	#72
The referent of 0 is not a physical object, or any number for that matter, but is nonetheless a thinkable object, which is thought when we count physical objects. The way in which it relates to reality is according to Kant, in the anticipations of perception in the critique of pure reason: The principle of these is: In all phenomena the Real, that which is an object of sensation, has Intensive Quantity, that is, has a Degree ... 1572. Now, a gradual transition from empirical consciousness to pure consciousness is possible, inasmuch as the real in this consciousness entirely vanishes, and there remains a merely formal consciousness (a priori) of the manifold in time and space; consequently there is possible a synthesis also of the production of the quantity of a sensation from its commencement, that is, from the pure intuition = 0 onwards up to a certain quantity of the sensation. 1573. Now as sensation in itself is not an objective representation, and in it is to be found neither the intuition of space nor of time, it cannot possess any extensive quantity, and yet there does belong to it a quantity (and that by means of its apprehension, in which empirical consciousness can within a certain time rise from nothing = 0 up to its given amount), consequently an intensive quantity. What this does mean is that reality is quantifiable, and not just that but it has to be, and therefore subject to arithmetics. This suggests also that there is an extensive relation to reality, in particular the objective reality. It is this sense in which it has to apply to flower petals, as these are objects. The homogenous in these objects (petalhood) is successively added to one another until we have determined the full magnitude (5 petals) subtract one and we have 4. The number itself is abstracted from what this homogenous unit is. In this there is nothing empirical left, neither in the concept number or concept of sum which is the reason why we can know this a priori. jtoma wrote: is this what you think about 0? 0 is an artifact of the senses, or rather the mind's access to reality through them. An intellectual intution has no need for numbers, but the properties of objects are immediately cognized as they are in themselves in the object.
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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1 of 1 people found this post helpful Posted 07/18/09 - 03:38 PM:	#73
Keda, The definition of a set can, and indeed must, specify which elements are instantiated in that set. To say that this is not a definition is nonsensical, because it is nothing more than specifying what the set is. Question: Why is it nonsensical to say it isn't a defintion? Answer: Because it specifies what the set is. The answer is insufficient to answer the question, if you can specify a set without it being a definition. I provided a counterexample showing that you can specify a set without it being a definition. Therefore the answer is insufficient to answer the question. No, as I already said, you provided a definition that implicitely includes a synthetic claim. My "correct" I was referring to the fact that the definition is not just a definition. Something I thought I made clear it the follow-up text, and something that I have clarified now at couple of times. The fact that some concepts cannot be defined without making implicit assumptions about synthetic judgements just reflects the fact that some concepts are about the real world. They are not concepts in Kant's sense. According to Kant concepts are universals and cannot therefore make direct references to objects such as my cat, or 0. I really could not care less whether Kant chooses to call it a "concept" or not. The sense that I am using the term "concept" above should be quite clear from the context. If it is not, then please ask for clarification. Your logic teacher is absolutely correct, as long as he/she means "existence" in the sense of being part of reality. But as I have repeatedly pointed out, the statement "0 is a natural number" does not require that 0 exist in this sense, as the statement "my cat is an animal" does. He did not make such a distinction. In fact I don't remember anyone ever making such a distinction except you. I am certainly not the first person to recognize that saying something like "there exists a natural number X such that X + 1 = 5" is a completely different sense of "exists" than saying something like "My cat exists". As I mentioned before, I cannot even imaging how one could meaningfully claim that the number 2 exists in the same sense that a rock or a cat does. Even the Platonic notion of numbers being somehow "real" doesn't make such an absurd and nonsensical claim. I don't understand the question. There are many relationships between various number systems and physical objects. Every time we use mathematics to model reality, we are theorizing about such relationships. Can you give an example? Quantum Mechanics theorizes that complex wave-functions that evolve according to Schrodinger's equation accurately describe the probabilities of making various observations under various conditions. That is a pretty clear example, I would say. I define it mathematically, depending on which formal system I am refering to. How do you define it in Peano artithemtics? Why are you asking me? You can just look it up yourself. Of course not. It makes absolutely no sense to say that an object is a 0. If so, then 0 is not a concept, at least not in the Kantian sense. If something is a concept e.g. "cat" then you should be able to say "a cat". Then it's not a concept in the Kantian sense. So what? They are not real, so by your reasoning, all mathematical statements would have to be false, which is ridiculous. Not by my reasoning, since I do not assume they are not real. You don't have to assume that they are not real, because numbers are not defined in such a way that it would make any sense to say that they are. The closest you could come to saying that they are real is to say that they accurately describe real things in some way. The underlined sentence is ambiguous. In order for it to mean anything, additional clarification must be provided. This clarification can be explicit, as it was when I expanded in into "according to the axioms of arithmetic, 5 + 7 = 12", or it can be implied by the context. You said this: Likewise, when you say "5 + 7 = 12", unless you specify some units or specific context, I would interpret this to be saying "according to the axioms of arithmetic, 5 + 7 = 12 I suggest this is no more clarifying of the underlined than of what G is in interpreting GNU to mean GNU is not Unix. You end up running in circles or G will remain undefined and your interpretation is incomplete. In your case: according to the axioms of arithmetic, according to the axioms of arithmetic, according to the axioms of arithmetic, .... or according to the axioms of arithmetic, "5 + 7 = 12" but "5 + 7 = 12" is left uninterpreted. No, that is not what I am saying at all. Look, anytime I say anything, there is a context to what I am saying. Sometimes that context is explicitely established in the sentence, and sometimes it is inferred by the person reading the sentence. For example, If I say to my mother "Jim stopped by yesterday", she knows that I am referring to my brother Jim. The context is implicit. If I was saying it to somebody else, I would have to say something like "My brother Jim stopped by yesterday". Now, it should be pretty clear that when I say to my mother "Jim stopped by yesterday", she interprets "Jim stopped by yesterday" to mean "My brother Jim stopped by yesterday". It should be equally clear that this does not mean that this expands into "My brother My brother My brother..." In set theory it is true that X, but what is X? Also "In set theory it is true.."? It would not be possible to interpret such no more than GNU can be intepreted in the way I mentioned. It is certainly not analytical in Kant's sense. Oddly enough mathematicians have no trouble interpreting it. They just don't interpret it to mean anything more than that a particular formal system is defined in a specific way. As for whether Kant would call it "analytic", I neither know, nor care, what he would have chosen to call it. I am interested in arguments and reasoning. It is thus a completely self-contained formal system, which does not make any claims about, or even references to, anything real. If so we can throw in the garbage can because it is worthless. Arithmetics isn't worthless because it isn't selfcontained but refers to reality. No, it is useful because it can be used to describe reality. But again, the fact that this is the case is known to us a posteriori. Nothing about the definition of any mathematical system in any way implies that the system will be useful in any way. That is something we discover by trying to use it, and succeeding. We have no a priori justification for thinking that mass should be conserved under any circumstances, much less all of them. That is not the point. You have defined the terms in such a way that 5kg+7kg=12kg can be true only a posteriori. Yes, I know. That was the whole point. When you add the units it ceases to be an analytic statement about made-up objects called "numbers", and becomes a synthetic claim about massive objects in the real world. This is fine, but this goes beyond counting to describe a physical process, which isn't what one normally would call a mathematical statement. Also exactly my point. As soon as the context implies that the statement is about actual objects in reality, it is not a mathematical statement anymore. Normally when it is said that 5kg+7kg=12kg, it is meant that the sum of masses of anything with the mass of 5kg and anything with the mass of 7kg, is 12kg. This does not describe a physical process, but merely of counting together 5kg and 7kg. How is counting physical objects not a physical process? You can't make any a priori synthetic judgements about it, because your only source of information about how mass works in the real world is your observations. This is irrelevant, because according to my interpretation, the mass is freezed in its representation during the calculation, so however mass works, it doesn't do it in my representation of it during calculation. Great. So now you are including in your interpretation of the sentence the implicit qualification of "as long as mass works according to the rules of arithmetic". But then it just becomes analytic again. Sort of like "If Bob exists, and unmarried, then Bob is a bachelor". This is also analytic. You can think of this calculation as operating on sets. You have two sets, each containing elements of each 1kg. One contains 5 of them while the other 7. Now even if the masses would alter while counting, you are not refering to the altering masses but to the masses in the sets which refer to the masses at whatever particular time they exist in. When you do this, no sense input can prove 5kg+7kg=12kg is wrong. Sorry, that doesn't help. If you are talking about sets in the mathematical sense, then your are making no sense, because a set cannot contain a real physical object. It can contain a representation of a real object. But then your statement is analytic again. It would only become synthetic when you then try to conclude from the statement that if you were to physically combine the objects that are referred to by the elements of those sets, that you would have 12kg. And of course that conclusion could only be justified a posteriori. We can generalize this for any units, not just kg, and thus we get the proposition 5+7=12 which is to say 5 of any unit and 7 of that unit counted together is 12 of that unit. Which is analytic, because your "units" are no longer real objects. And and any inference we could make from the mathematical statement about real objects would have to be a posteriori. Nope. That is not an example of me telling you anything about the world outside of my mind. It is an example of me telling you about what I can know about the world outside of my mind. Knowing is a mental process. So in fact, I am just telling you something about my mind. Knowing is a relation between the mind and the world, so it is also say something about the world, specifically how it is known. No, it is not. Knowing is something you do. You are conflating the activity of holding a justified belief about the world with the relationships between the mind the world that are required in order for you to hold a justified belief about the world. Saying what you can know about the world is an a priori synthetic claim i.e. the following is a genuinely true a priori synthetic claim: No, it is not. Our sensory input is necessary in order for us to establish the conditions under which that sensory input is relaible. For example, you could imagine a sentient being that has not yet recieved any sensory input, but which understands how to reason logically, and is capable of reasoning about sensory input when it eventually recieves it. But before you turn on the sensory input, it is not going to be able to make any inferences about what it will be able to conclude from sensory input when it does recieve it. The being would have absolutely no way of knowing whether its sensory input will be in any way useful to it. Indeed, it would be entirely possible to just wire up the being's sensory input to random noise, in which case it would be totally useless to it. What it could do is draw some conclusions about what conditions would have to obtain in order for its sensory input to be useful. It could even potentially go so far as to completely figure out the scientific method, so that when its input is turned on it can immediately begin formulating falsifiable hypotheses and testing them. But until it starts recieving that input, it has no way of knowing that those necessary conditions will obtain, and no way of knowing that the scientific method will actually work. DM
Pseudoscience makes Baby Jesus cry.

keda
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Posted 07/19/09 - 03:27 AM:	#74
Death Monkey wrote: As for whether Kant would call it "analytic", I neither know, nor care, what he would have chosen to call it. I am interested in arguments and reasoning. Well, I suppose we could stop it here because this whole thread started with Kant's distinctions of analytic/synthetic and a priori/a posteriori. If your point is that mathematics is analytic in your own sense, not in Kant's sense, then it has no bearing on Kant's claim that mathematics is a priori synthetic. If you aren't at all interested in how a priori synthetic claims are possible in Kant's sense, then I guess I'm just wasting my time.
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

cosscos
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Posted 07/19/09 - 04:50 AM:	#75
Kant's cognition concerning object is determined by sense-data. This sense-data is produced by power of imagination in which transcendental deduction is completed under spontaneous judgement through sensibility and understanding, or intuition and concept. This power of imagination is nothing other than one's transcendental union of apperception about object which come into being located in the world of space and time. 5+7=12 is presumably a priori synthetic knowledge. As for this proposition, the object 12 is a priori which is located in noumenal realm in which one realize sensous commune by synthetic procedure called trascendental aesthetics by which 12 come to have the itness, being real. In my opinion, a priori synthetic proposition is pretty subjective, as I mentioned above. As for 5+7=12, I wonder how on earth it could be justified for all rather than specific individual as a priori synthetic proposition. cc

Death Monkey
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Posted 07/19/09 - 05:14 AM:	#76
keda, As for whether Kant would call it "analytic", I neither know, nor care, what he would have chosen to call it. I am interested in arguments and reasoning. Well, I suppose we could stop it here because this whole thread started with Kant's distinctions of analytic/synthetic and a priori/a posteriori. If your point is that mathematics is analytic in your own sense, not in Kant's sense, then it has no bearing on Kant's claim that mathematics is a priori synthetic. If you aren't at all interested in how a priori synthetic claims are possible in Kant's sense, then I guess I'm just wasting my time. I just told you, I am interested in arguments and reasoning. That includes Kant's arguments and reasoning, if you wish to present it. What I am not interested in is semantic arguments about what some particular word does or does not mean according to some particular definition. This is not about "my" sense of "analytic" vs Kant's or anybody else's. My understanding throughout this thread has been that we are all using the term "analytic" in the same sense, which is to be a statement that is true by virtue of its meaning. Anyway, the point is irrelevant because your assertion that Kant would not regard it as an analytic statement was based on your misinterpretation of what I meant by the statement. As I mentioned, your GNU analogy is not applicable to what I said. DM
Pseudoscience makes Baby Jesus cry.

keda
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Posted 07/19/09 - 06:32 AM:	#77
Death Monkey wrote: I just told you, I am interested in arguments and reasoning. That includes Kant's arguments and reasoning, if you wish to present it. What I am not interested in is semantic arguments about what some particular word does or does not mean according to some particular definition. This is not about "my" sense of "analytic" vs Kant's or anybody else's. My understanding throughout this thread has been that we are all using the term "analytic" in the same sense, which is to be a statement that is true by virtue of its meaning I don't have a problem with you using your particular terminology or any terminology for that matter, but the issue is that it becomes a red herring to the original topic, if you are arguing mathematics is analytic on the basis of your terminology, and not on Kant's. Now likewise with a posteriori, we agreed that it was based on sense, but it turned out you had a much broader definition of sense than I had, so just because you say something is true by virtue of meaning doesn't mean that it means what I mean when I say just that, and I began to notice you started using words like analytic, definition and concept in a much broader sense than I did, which, in the end would only mean that what you were saying right at the start hasn't got much to do with what Kant was saying. The issue isn't that about which semantics is correct or not, its about which is relevant to the topic. Anyway, the point is irrelevant because your assertion that Kant would not regard it as an analytic statement was based on your misinterpretation of what I meant by the statement. As I mentioned, your GNU analogy is not applicable to what I said. Even if I misinterpreted you, I didn't reply to it anymore, because your original claims have been shown to be red herrings due to your terminology not matching Kant's.
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 07/19/09 - 06:42 AM:	#78
cosscos wrote: As for this proposition, the object 12 is a priori which is located in noumenal realm Located in the noumenal realm? The rest of your post doesn't seem too make too much sense either...
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 07/19/09 - 06:59 AM:	#79
keda, I don't have a problem with you using your particular terminology or any terminology for that matter, but the issue is that it becomes a red herring to the original topic, if you are arguing mathematics is analytic on the basis of your terminology, and not on Kant's. I am arguing that mathematics is analytic on the basis of the fact that true mathematical statements are true by virtue of their meaning. Nothing more and nothing less. Now likewise with a posteriori, we agreed that it was based on sense, but it turned out you had a much broader definition of sense than I had, so just because you say something is true by virtue of meaning doesn't mean that it means what I mean when I say just that, and I began to notice you started using words like analytic, definition and concept in a much broader sense than I did, which, in the end would only mean that what you were saying right at the start hasn't got much to do with what Kant was saying. The issue isn't that about which semantics is correct or not, its about which is relevant to the topic. Well, I don't see any sense in which a purely mathematical statement could not be said to be true by virtue of its meaning. The examples you have given, whereby you argue that the various mathematical objects (such as numbers, sets, or elements) that the statement refers to, could fail to "exist", thereby rendering the statement false, simply makes no sense to me. Perhaps you could begin by explaining what you mean when you say, for example, that the number 0 could fail to exist? What would it mean to say that 0 does not exist? And in what way is this sense of "existence" of the number 0 required in order for mathematical statements about the number 0 to be true? DM
Pseudoscience makes Baby Jesus cry.

keda
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Posted 07/19/09 - 07:15 AM:	#80
Death Monkey wrote: keda, Perhaps you could begin by explaining what you mean when you say, for example, that the number 0 could fail to exist? What would it mean to say that 0 does not exist? And in what way is this sense of "existence" of the number 0 required in order for mathematical statements about the number 0 to be true? This is something I've already suggested in post #72, namely that numbers are artifacts of our mode of knowledge through the senses. If you had intellectual intuition, every property of the object would be immediately known, and no number would be needed to measure them. Mathematics is essentially a science of space and time which are the necessary conditions under which any objects of the senses can be represented to us. Without them no sense perception would be possible.
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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