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Synthetic A Priori math. Kant!
Need someone who knows Kant well

Synthetic A Priori math. Kant!

Dranu
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Posted 04/23/08 - 02:42 PM: Subject: Synthetic A Priori math. Kant!	#1
Some help from someone who believes and/or understands Kant would be of the most help to me. Why 'he is wrong'(not that I am claiming he is or isn't) would be useful to know, but I am really trying to understand what he is getting at first. Ok, I am being very slow in picking up on Kant's 12=7+5 being synthetic a priori. This may sound odd, but I think it is obvious 7+5=12 is synthetic. No matter how many times you put seven ones next to five ones, you still only have seven ones and five ones. Somehow intuition enables you to count beyond seven or five. However (to flip it), to say seven and five is not contained in twelve, I do not understand at all. If the concept of gold is 'yellow and metal', and I analytically derive 'metal' from it, why can't I do the same with 5 from 12 (or 7 from 12)? Anyhow maybe this question being answered would help: 1.)Is deriving 'blade' from the concept of fan analytical or synthetic? And why?(you see I can say the gold is metal, but I can't say the fan is blade.)

Tritium
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Posted 04/23/08 - 09:13 PM:	#2
I'm no Kant expert, but I think I can add something to the 7+5=12 confusion. I've read an idea to understand this statement in a matter of identical objects. 7 lines and 5 lines on one side and 12 lines on the other side. Observed as a whole you see that there are 12 lines on one side and 12 lines on the other side. Therefor you get a system in which you observe the statement, A is A, an analytical statement. This helps me understand that mathematics is really analytical knowledge and I think leads to the idea that Kant was wrong about synthetic a priori knowledge. I've heard it said that the two terms synthetic and a priori are impossible by mere concept, though I couldn't reproduce the argument for you.

Dranu
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Posted 04/23/08 - 10:24 PM:	#3
Tritium wrote: read an idea to understand this statement in a matter of identical objects. 7 lines and 5 lines on one side and 12 lines on the other side. Observed as a whole you see that there are 12 lines on one side and 12 lines on the other side. True, but that is not looking at the concepts of 5 and 7, but rather the synthesis of them as one. So going that direction it seems synthetic to me. However, going in the opposite direction (12=5+7), I do not see how 5 and 7 are synthesized rather than analyzed.

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Posted 04/24/08 - 02:04 AM:	#4
I guess you are misunderstanding what containment means in this context. As an example, the predicate man is contained in the predicate bachelor, as it follows from the meaning of bachelor that anyone who is a bachelor is also a man. The judgment that all bachelors are men is thus analytic. If we on the other hand say that all tables are yellow, then we see that yellow is not contained in tables i.e. the meaning of table does not necessitate it being yellow (it is possible to have for instance red tables). Note that analytic judgments are not informative i.e. they give us no new information, in contrast to synthetic ones that actually say something about something. What Kant is saying when he says 7+5 does not contain 12, is that it doesn't follow from the meaning of 7 + 5 but that the former represents the result of adding 7 and 5, the information of which we represent by that judgment. The syntheticity of arithemtics has been confirmed by the developement of non-standard mathematics: http://www.earlham.edu/~peters/writing/synth.htm However as have been shown by Hector Neri Castañeda (can't remember the name of the essay at the moment) reversing any of the peano postulates result in inability to count things we would normally be able to, but in fact that any working non-standard arithmetics will have to rely on the peano postulates on a meta level.
_____________________ Free Europe Now The man who trades freedom for security does not deserve nor will he ever receive either. -Benjamin Franklin If my sons did not want wars, there would be none - Gutle Rothschild It's not the people who vote that count, it's the people who count the votes - Josef Stalin Sometimes Satan comes as a man of peace - Bob Dylan A prolonged peace favours the predominance of a mere commercial spirit, and with it a debasing self-interest, cowardice, and effeminacy, and tends to degrade the character of the nation. - Immanuel Kant

Dranu
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Posted 04/24/08 - 11:47 AM:	#5
Thanks for the article Keda. Perhaps I need to do some studying into modern predicate logic. Any chance someone could give an example of a 'true for an interpretation' and a 'logicaly valid/analytical' formula of predicate logic? I am still unsure how five and/or seven are not contained in the cocnept of 12. I can see how 12 is not contained in them, just not the reverse. It seems clear that 7 and 5 do not contain 12, but only their synthesis does. I am just having trouble on seeing how the reverse is not analytic, especially if one derived 12 from 7+5. Perhaps it is my understanding of Kant's analytical a priori. I understand 'man' is analyzed from 'bachelor', and 'metal' from 'gold.' What about blade from fan though (going back to my question in post #1)?

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Posted 04/25/08 - 09:50 AM:	#6
Dranu wrote: Thanks for the article Keda. Perhaps I need to do some studying into modern predicate logic. Modern predicate logic isn't perhaps the best way to understand Kant as it is formal, but Kant's logic is "contential" i.e. respects content as opposed to merely dealing with the form of judgments. A formal logic doesn't differentiate between the informativity of "No bachelor is married" and "No table is yellow". Any chance someone could give an example of a 'true for an interpretation' and a 'logicaly valid/analytical' formula of predicate logic? True for an interpretation could be "It is raining" while "It is either raining or not raining" is logically valid, analytic truth. I am still unsure how five and/or seven are not contained in the cocnept of 12. I can see how 12 is not contained in them, just not the reverse. It seems clear that 7 and 5 do not contain 12, but only their synthesis does. I am just having trouble on seeing how the reverse is not analytic, especially if one derived 12 from 7+5. Perhaps it is my understanding of Kant's analytical a priori. I understand 'man' is analyzed from 'bachelor', and 'metal' from 'gold.' What about blade from fan though (going back to my question in post #1)? Fans are not blades are they? Fans can be said to contain blades, but the predicate of "fan" cannot be said to contain the predicate "blade", perhaps the predicate "bladed thing" but not "blade". A car could be said to be a wheeled thing, but not a wheel. I hope you see the difference between the two senses of containment. 12 sticks may contain 7 sticks, but the predicate of being 12 doesn't contain the predicate of being 7, nor does it contain the predicate of the sum of 5 and 7, which indeed is 12, but not by virtue of containment. One can discover the synthetisity if one tries to bend the principles of mathematics and see where it leads you. Edited by keda on 04/25/08 - 09:58 AM
_____________________ Free Europe Now The man who trades freedom for security does not deserve nor will he ever receive either. -Benjamin Franklin If my sons did not want wars, there would be none - Gutle Rothschild It's not the people who vote that count, it's the people who count the votes - Josef Stalin Sometimes Satan comes as a man of peace - Bob Dylan A prolonged peace favours the predominance of a mere commercial spirit, and with it a debasing self-interest, cowardice, and effeminacy, and tends to degrade the character of the nation. - Immanuel Kant

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Posted 04/25/08 - 12:33 PM:	#7
??? I was under the understanding that 12=7+5 would be considered A Priori Synthetic knowledge if only it is something that can be synthesized by an individual without having first been exposed to him by an exterior source-- ie it is possible, if you understand the fundamental rules of whole numbers and addition, to discover 12=7+5 for yourself. Thus, would be synthetic a priori knowledge. Where does all the talk about predication come from?? =( =(

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Posted 04/25/08 - 12:54 PM:	#8
Findley wrote: ??? I was under the understanding that 12=7+5 would be considered A Priori Synthetic knowledge if only it is something that can be synthesized by an individual without having first been exposed to him by an exterior source-- ie it is possible, if you understand the fundamental rules of whole numbers and addition, to discover 12=7+5 for yourself. Thus, would be synthetic a priori knowledge. Where does all the talk about predication come from?? =( =( Rather than without having first been exposed by an exterior source, a judgment priori is one that is not justified on the basis of experience. In other words mathematics is not verified or falsified on the basis of experience, but is true independent of it. That it is synthetic, means that there is a synthesis yes, of a concept and an intuition (which is really a bad translation from the german Anschauung, which means something like "look at") as opposed to analytic which means splitting up concepts into their parts. If the predicate is contained, the analysis exposes a particular predicate that is contained, while if it is not, then we need a singular immediate reference (the "intuition") to a particular object of which by means of the concept we extends our knowledge. As an a posteriori example, "This is a cat" the "this" corresponds to the intuition, while "is a cat", corresponds to the concept. This intuition in the case of mathematics is according to Kant the forms of sensibility, namely space (in the case of geometry) and time (in the case of arithmetics).
_____________________ Free Europe Now The man who trades freedom for security does not deserve nor will he ever receive either. -Benjamin Franklin If my sons did not want wars, there would be none - Gutle Rothschild It's not the people who vote that count, it's the people who count the votes - Josef Stalin Sometimes Satan comes as a man of peace - Bob Dylan A prolonged peace favours the predominance of a mere commercial spirit, and with it a debasing self-interest, cowardice, and effeminacy, and tends to degrade the character of the nation. - Immanuel Kant

Dranu
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Posted 04/25/08 - 02:29 PM:	#9
Keda wrote: A car could be said to be a wheeled thing, but not a wheel. I hope you see the difference between the two senses of containment. 12 sticks may contain 7 sticks, but the predicate of being 12 doesn't contain the predicate of being 7, Ok, cool thanks Keda, I think I got it. Just to make sure, 'blade' cannot be analyzed from 'fan'(assuming all fans are bladed), but synthesized only? The only things that can be analyzed from something, is where saying that 'thing' is not X results in a contradiction? So for instance, I cannot analyze 'blade' from 'fan' since saying a fan is not a blade is not a contradiction? So somehow (even though we must say 5+7 is twelve on pain of contradiction) contradiction or law of identity is not what allows us to see 5+7=12 and 12=5+7. If that is right, then I guess my problem was that I was thinking that all breaking down of anything into parts was analysis and synthesis was always building up a concept. (Maybe that will help me figure out his proof for innateness of space in the hand argument too.) I guess it just seems weird since my conventional thought of containment refers to the parts of an object than to what it means to be the object. Clearly 12 contains 7 and 5, but (like you point out) being 12 doesn't contain being 7. 12 is not 5 and 12 is not 7, contrasted with a bachelor is a man and a bachelor is unmarried. As such, it does not need 5 and 7 to be 12, just like a line does not need infinite points to be a line as it is prior to its division? I think what is really odd to me is that one can synthesize bachelor from man and unmarried, then analyze man and unmarried from bachelor, yet one can synthesize 12 from 5 and 7, yet not analyze 5 and 7 from 12. Man this is mind boggling stuff (for me at least)!

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Posted 04/26/08 - 04:41 AM:	#10
Dranu wrote: Ok, cool thanks Keda, I think I got it. Just to make sure, 'blade' cannot be analyzed from 'fan'(assuming all fans are bladed), but synthesized only? Given that a fan needs a hub, and that blades do not contain hubs, it cannot be said that fans are blades even synthethically i.e. it is analytically true that hubs are not blades. That a proposition is not analytically true does not mean it is synthetic; it can be analytically false as well i.e. a contradiction in terms. For a more complete view of what constitues various types of judgments and judgment in general, you could read this article: http://plato.stanford.edu/entries/kant-judgment/ The only things that can be analyzed from something, is where saying that 'thing' is not X results in a contradiction? So for instance, I cannot analyze 'blade' from 'fan' since saying a fan is not a blade is not a contradiction? That is correct. So somehow (even though we must say 5+7 is twelve on pain of contradiction) contradiction or law of identity is not what allows us to see 5+7=12 and 12=5+7. Not alone; the propositions have to be free of contradictions to be cognized. If that is right, then I guess my problem was that I was thinking that all breaking down of anything into parts was analysis and synthesis was always building up a concept. (Maybe that will help me figure out his proof for innateness of space in the hand argument too.) I guess it just seems weird since my conventional thought of containment refers to the parts of an object than to what it means to be the object. I guessed that was the case. I suppose after all a good understanding of predicate logic can be helpful, at least to an extent in understanding Kant, since he is talking about these things, but note that analytic or synthetic are terms that have been used in different senses after Kant, and should be kept distinct from his usage or you'll easily get confused again. As such, it does not need 5 and 7 to be 12, just like a line does not need infinite points to be a line as it is prior to its division? I am not sure what you mean here. A line consists of infinite number of points, but this is again the part-whole type of containment, and not predicate containment. I think what is really odd to me is that one can synthesize bachelor from man and unmarried, then analyze man and unmarried from bachelor, yet one can synthesize 12 from 5 and 7, yet not analyze 5 and 7 from 12. Man this is mind boggling stuff (for me at least)! The concept of "bachelor" is not synthesized at all. The result of synthesis is always a complex cognition consisting of an intuition (anshauung) and a concept (begriff). All knowledge consists of subsuming particulars (intuitions) under universals (concepts) the process which is called synthesis.
_____________________ Free Europe Now The man who trades freedom for security does not deserve nor will he ever receive either. -Benjamin Franklin If my sons did not want wars, there would be none - Gutle Rothschild It's not the people who vote that count, it's the people who count the votes - Josef Stalin Sometimes Satan comes as a man of peace - Bob Dylan A prolonged peace favours the predominance of a mere commercial spirit, and with it a debasing self-interest, cowardice, and effeminacy, and tends to degrade the character of the nation. - Immanuel Kant

Dranu
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Posted 04/26/08 - 02:28 PM:	#11
Keda wrote: That a proposition is not analytically true does not mean it is synthetic; it can be analytically false as well i.e. a contradiction in terms. For a more complete view of what constitues various types of judgments and judgment in general, you could read this article: http://plato.stanford.edu/entries/kant-judgment/ For a synthetic a priori (form the article)"its denial is logically consistent." I am still unsure how this is possible, for I cannot even cognize denying it (I am almost sure this has to do with modern predicate logic ). Maybe this will clear up my confusion on the next thing as well: The concept of "bachelor" is not synthesized at all. The result of synthesis is always a complex cognition consisting of an intuition (anshauung) and a concept (begriff). All knowledge consists of subsuming particulars (intuitions) under universals (concepts) the process which is called synthesis. How do we come to the concept of bachelor then? It seems that it is an example of a synthetic judgment of putting man and unmarried together. I do understand (I think) how for Kant one needs empirical sensation in order to recognize the intuitive concepts at work, but I am just talking about the arrival at the concept of bachelor seems to be made synthetically. However, perhaps I am just mistaking synthetic again (actually I'm pretty sure I am). If one takes two concepts and calls those two concepts one, can that be analytical for Kant? If so I think that clears up my confusion. If not, what is it if it is neither synthetic nor analytical? I am not sure what you mean here. A line consists of infinite number of points, but this is again the part-whole type of containment, and not predicate containment. My example was that one could not define a line by its parts in that instance, because the whole must be prior to the division. Likewise, a plane cannot be defined by infinite lines, even though it contains infinite lines, and 12 can contain infinite number of division, and cannot be defined by that division unless one part speaks of the whole. That is, metal speaks to the whole of gold, but 5 does not speak to the whole of 12, nor blade to the whole of fan. This is very helpful btw, so thanks. Kant's work is frustratingly difficult for me to understand . I can pick up his whole argument for the system that he establishes, but it is the arguments for the crucial fundamentals like the what, why, and how of synthetic a priori judgements, that seem to elude me.

Dranu
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Posted 05/07/08 - 10:21 AM:	#12
Anyone?

keda
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Posted 05/07/08 - 11:33 AM:	#13
Dranu wrote: For a synthetic a priori (form the article)"its denial is logically consistent." I am still unsure how this is possible, for I cannot even cognize denying it (I am almost sure this has to do with modern predicate logic ). Well here the predicate logic can come in handy because denial of something can be logically possible, yet not cognizable. How do we come to the concept of bachelor then? It seems that it is an example of a synthetic judgment of putting man and unmarried together. Note that synthesis is always putting together a concept and an intuition. The unification of which the predicate gives us information about the subject. As such we analysis is only possible after synthesis. We need to have some experience of bachelors, or at least imagine such. According to Kant concepts are constructed using three methods: Reflection, comparison and abstraction. In reflection we are trying to find the characteristics of a kind of thing, in comparison we look at a number of instances and try to find what they have in common, and in abstraction, we strip characteristics to produce more general concepts e.g. mammal from cat and animal from mammal. However, perhaps I am just mistaking synthetic again (actually I'm pretty sure I am). If one takes two concepts and calls those two concepts one, can that be analytical for Kant? If so I think that clears up my confusion. If not, what is it if it is neither synthetic nor analytical? It depends on the way you put together the concepts, and what exactly the concepts are. As you know, every sentence contains predicate and subject, such as "the dog barks" "the dog" being the subject and "barks" being the predicate. Since "the dog" doesn't contain that it barks, unlike the concept of mammal, which it does, that it barks gives you information about what it is doing. Synthesis is exemplified in the process is of putting together these two "the dog" which is already a complex cognition, that is it contains already an synthesis of the concept dog and a particular dog in question, and the concept of "barks" is further brining the particular dog under the concept barking things in a more complex union "the barking dog", that gives us more information about it. However if we form a sentence "The barking dog is an animal" then you shrug and say, well I already knew that, because the subject already contains being an animal. This is called analysis, as it does clarify some of the meaning of a barking dog. My example was that one could not define a line by its parts in that instance, because the whole must be prior to the division. Likewise, a plane cannot be defined by infinite lines, even though it contains infinite lines, and 12 can contain infinite number of division, and cannot be defined by that division unless one part speaks of the whole. That is, metal speaks to the whole of gold, but 5 does not speak to the whole of 12, nor blade to the whole of fan. That is correct. When you said the whole must be prior to the division, this is the right way of understanding Kant's intuition of space, because there are alternative ways in which parts of space can be put together into a whole, and likewise is this also Kant's intuition of time, and consequently arithmetics which is founded on the intuition of counting of units of time, which is a particular way of putting them together, as opposed to other ways of putting them together. If you put the blades of a fan with its hub together differently, it will not work anymore, because the blades will divert the flow of air in a different direction. Arithmetics is put together in a particular way, according to Peano postulates, that would not work anymore were we to do it differently. Unlike the fan, it is not empirical however, because our intuition of time is prior to all experience (we can imagine an empty time containing no objects). This is very helpful btw, so thanks. Kant's work is frustratingly difficult for me to understand . Kant is notoriously difficult, and you are not alone with the struggle, but if you can master his philosophical system, you would probably become philosophically invincible I can pick up his whole argument for the system that he establishes, but it is the arguments for the crucial fundamentals like the what, why, and how of synthetic a priori judgements, that seem to elude me. Well, a priori synthetic judgments are crucial cornerstones in his whole philosophical system, like the poles holding up a tent that no wind or rain can penetrate, so you better get those straight or the whole system collapses
_____________________ Free Europe Now The man who trades freedom for security does not deserve nor will he ever receive either. -Benjamin Franklin If my sons did not want wars, there would be none - Gutle Rothschild It's not the people who vote that count, it's the people who count the votes - Josef Stalin Sometimes Satan comes as a man of peace - Bob Dylan A prolonged peace favours the predominance of a mere commercial spirit, and with it a debasing self-interest, cowardice, and effeminacy, and tends to degrade the character of the nation. - Immanuel Kant

Dranu
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Posted 05/07/08 - 12:03 PM:	#14
Keda wrote: Well, a priori synthetic judgments are crucial cornerstones in his whole philosophical system, like the poles holding up a tent that no wind or rain can penetrate, so you better get those straight or the whole system collapses Heheh, indeed! Thanks for the responses, they are very helpful. I think I have a good base now to work with. I think I should move from the Prolegomena to the Critique of Pure Reason and gather a better understanding of modern predicate logic before I ask anything else . Hopefully I'll find the time to trudge through that soon.

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