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From: zuhair on 27 Dec 2009 17:45 On Dec 27, 12:51 pm, Marshall <marshall.spi...(a)gmail.com> wrote: > On Dec 27, 6:40 am, zuhair <zaljo...(a)gmail.com> wrote: > > > > > > > On Dec 26, 1:38 pm, Marshall <marshall.spi...(a)gmail.com> wrote: > > > On Dec 24, 7:41 pm, zuhair <zaljo...(a)gmail.com> wrote: > > > > > One of the facts of set theory is that the primitive relation > > > > "e" is totally vague relation, there is no way of understanding this > > > > relation at all, even at informal level! Which is actually a great > > > > drawback, since in set theory we are building hierarchy of objects > > > > using this relation, a relation that nobody understand what it stands > > > > for? > > > > This is completely untrue. A simple, "intuitive" understanding > > > of of the is-an-element-of relation is trivial, even for middle > > > school children; they are taught about sets and it is quite > > > easy stuff. That is completely incorrect, actually you can't, these things you are speaking about are not intuitive understandings as you might think, they are mostly circular approximations to the matter. At the formal level, we have the axioms. > > > > I can see no basis whatsoever for these claims. > > > Well why you don't just tell me what is this what you call "intuitive" > > meaning of the membership relation epsilon is? > > It means "is an element of." At the middle school level, we > can easily have a conception of sets as collections of > arbitrary elements, and we can define operations on > sets like union and intersection. Union and intersection > are no more difficult than logical "or" and "and." Yea, at primary school also you are taught that 1+1=2, in primary school you are taught about natural numbers, integers, rational numbers etc..,but did anybody understood what these stand for, until set theory came into place. That what you are not understanding Marshall, there is a big difference between circular approximations that are used to teach these matters and between real comprehensive intuitive treatment of these matters. > > But in any event, I reject the idea that studying what > one currently does or does not find intuitive is worth > spending any time on. Rather, it is productive to > train one's intuition to better match theory. It is theory > that represents the best understanding of mathematics, > not intuition; if they conflict, then intuition must improve. > > Of course, you may object that there are times when > new theories are born. However these times are not > as a rule the result of considering trivia such as whether > the existence empty set violates some unspecified intuitive > sense. Yes, they might indeed be. > > Note that it is not your statements merely that > I object to, but your very methodology. You are > thinking about these issues in the wrong way. Well that is your objection, which is built on personal grounds. > > The considerations that are likely to be useful in your > pursuit are things like simplicity, expressive power, > utility. Sometimes I hear the term "power-to-weight > ratio." Think about what you want to accomplish with > the theory, about what you want to be able to do. > Is it easier with your new approach or with the > established ways? Yea, these things are well known, but they are not the only things that matter. > > Unless you can give some practical benefits > to your approach, you haven't found anything > worth attention. "I like this way better" is > unlikely to be practical enough. Agreed. > > > Let me give you some issues: > > > If you say that a set is an aggregate of it's members (Review > > Bertrand Russell in Introduction to mathematical philosophy) > > which is by the way the trivial meaning of a "set" as used > > in customary everyday language, then you will end up with > > the difficulty of having a singleton that is not its member, and > > also you will have the catastrophic difficulty of the empty set. > > These issues were discussed almost one century ago, > > and nobody came to answer this issue. > > I reject the idea that these are issues. I reject the relevance > of everyday language. Well, no problem, that's your personal stance, I guess I should respect it, thanks for telling me. > > > Now if you say that a set is a "container" and the relationship > > epsilon stands for "being contained in" or " being inside" > > or something like that, you'll have the difficulty of why > > should containers be identical if there contents are identical? > > I see no difficulty. OK, that's up to you. . but I do see a difficulty at intuitive level, you cannot just come and put rules on matters as you like, there should be intuitive justification behind these rules. If you say sets are containers, then why these containers must be identical if they have the same contents, you must seek an intuitive justification, which is unfortunately missing with the mere container intuitive understanding of sets. > > > that doesn't make any sense at intuitive level, intuitively > > speaking different containers of the same size can > > contain the same elements, there is no reason whatsoever > > to believe that two containers having the same contents cannot be > > different, which is what axiom of Extensionality says. > > These "difficulties" are actually virtues, and whether you > want to work with a system that has these properties > or not, until you see why they are virtues, you are not > going to get anywhere. No this is teleological thinking, that is not enough. it might help as a temporizing measure, but it must not be the aim. > > Consider a system where, in addition to its elements, > every instance of a set also has identity. Then we > go from having one kind of equality to two kinds: > equality of members and equality of identity. The > system is now vastly more complicated. Is there > some corresponding vast improvement in some > other dimension to make up for this increase in > complexity? No there is not. > > What math does, what logic does, is to abstract. > To abstract is fundamentally to leave things out; > this is inescapable. If you want to keep everything > in, use the actual universe itself as your system. > This is the smallest system that will capture *all* > the behavior of the world around you. If you want > to represent three oranges, then you must use > three oranges to do that, in fact you must specify > which specific three oranges you mean, because > you apparently care about identity. If you want to > know how many oranges you will have when you > add another orange, you must first specify which > particular orange you mean. The result, then, is > those exact oranges you specified, and we cannot > say anything less than that. Generalization is > impossible, because generalization necessitates > leaving things out. > > On the other hand, if you just want to know what > 3+1 is, then you are necessarily leaving out a lot > of details about the oranges. You are abstracting > away those details. This is a virtue; it means your > results will be widely applicable. > > > On the other hand, having an *empty* set, cannot be explained > > intuitively other than by saying that what is called as "set" is in > > reality some sort of a "container" since only "containers" can be > > empty! > > Unless you have some trouble with the idea that 0 is a number, > I see no reason for you to have any difficulty with the idea that > the empty set is a set. It seems that you are not reading what I said above, I was saying that the container theory provides intuitive grounds for the empty set, so what are you objecting to here, I am in agreement with you on that point. > > > Same thing is applicable with singleton sets that are not identical to > > their members, also these can be though of as "containers" containing > > one member. > > > There is another difficulty with the "container" intuitive background > > of sets, and that is:_ we have strong theories allowing 'sets' to be > > in themselves like Aczel's set theory, and this is somewhat difficult > > to explain by the container theory, since this entail the existence of > > a container that contain itself, although this is not an absolute > > difficulty against the container concept, but yet it demands intuitive > > justification of containers containing themselves and the example of a > > snake that swallows itself is an example of how a container might be > > in itself, but yet it is not so clear, and it needs further treatment. > > I reject the suggestion that anything "demands intuitive > justification" and say again that your thinking in these > terms means that your thinking is fundamentally broken. > (But entirely fixable.) Ok, that is your personal view, thanks for telling me so. > > Again, you are thinking in excessively concrete terms. Oh, yes I do. If the > only way you are able to think about containers is by analogy > with a grocery bag, then you are in serious conceptual trouble. > Writing a computer program in which a data structure contains > a reference to itself is a trivial exercise. So why should this > present any difficulty to set theory? > > > [...] > > On the other side, it might not be useful as you said. > > I propose that you *start* with utility or simplicity or some > such virtue and proceed from there. See below. > > Marshall Well I know what you are trying to say very well Marshall, I actually don't have full objection to it. However intuitive formal theories would be easier to understand, and easier to interact with, and by then you can work with them easily, and they will definitely be fruitful. The system that I wrote above is actually an axiomatic theory (although I didn't compete it yet) with three primitives binary relations: identity "=", trivial membership " e' " and epsilon membership "e", and one primitive constant symbol V. , and the language is first order logic with identity theory. There is no doubt that this system is complex, and more complex than the traditional ones, though it might be nearer to our common intuitive reasoning. However it is not this theory in particular that matters, the general idea is the most important, and I will say it again, Oh yes trying to build formal theories having well non circularly understandable primitives at intuitive level is a fruitful thing Oh yes it is. Zuhair
From: zuhair on 27 Dec 2009 17:46 On Dec 27, 5:02 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > zuhair wrote: > > Hi all, > > > One of the facts of set theory is that the primitive relation > > "e" is totally vague relation, there is no way of understanding this > > relation at all, even at informal level! Which is actually a great > > drawback, since in set theory we are building hierarchy of objects > > using this relation, a relation that nobody understand what it stands > > for? > > > Here I will present a simple trial to understand this relation "e" > > and to give it some informal intuitive background. > > > First I shall speak about the concept of set as used in ordinary > > language. > > I think it's a bad idea to assert that there's a unique "set" binary relation > you've referred to as 'this relation "e"' that doesn't any intrinsic > vagueness in it. > > So long as we talk about a kind of set formal systems that could be used > to encode the naturals (which most common set theories - as well as your > alluded set theories - are) then there's always vagueness in these > set concepts. For instance, is there any clarity of the truth of > "There are infinite counter examples of GC" in these set systems? > > One certainly could make the set concept more implicated (e.g. by introducing > multiple binary epsilon relations) but I think one should abandon the idea > that one would have "better" explanation (as you've tried in this thread) > of what set concept is than the existing set formal systems, given the > constraint/limitation mentioned above. Well one can try! > > > > > > > When we say we have a set of objects x,y,z , then we usually mean that > > there is an entity that represent the *Whole* of > > x,y,z, so although x,y,z are distinct objects, but yet > > the *whole* that is constituted from these objects is considered the > > set of these objects, or the *collection* of these objects. > > > However this *Whole* of x,y,z is in a sense these objects themselves. > > > So a set is a form of describing multiplicity by unity, that unity is > > taken to be the totality of the Whole of these multiple objects. > > > However if we collect ONE object, then it is pretty intuitive to say, > > that the set of all objects collected IS that object. > > > Also it is pretty intuitive to say that if there are no objects > > collected, then there is NO set of objects collected, in other words > > there do not exist an empty collection. > > > Of course the traditional membership relation "e" doesn't capture all > > these pretty intuitive concepts of the customary use of the word > > membership and of the word "set" or the word "collection". > > > To try to put all of the above informal account into rigorous > > mathematical context, we can define a theory in FOL with identity > > and the primitive binary relation symbol e' , which stands for > > what I call "intuitive membership", or in reality "membership", since > > epsilon doesn't capture the intuitive concept of membership. > > > So we can have the following axioms that characterize this binary > > relation e'. > > > (1) Extensionality: > > For all z ( z e' x <-> z e' y ) -> x=y > > > Define(singleton): > > x is singleton <-> Exist z for all y ( y e' x <-> y=z ) > > > (2) Membership: x e' y -> x is singleton > > > (3) Singletons: x is singleton -> x e' x > > > theorem: x is singleton <-> x e' x > > Proof: we have x e' x -> x is singleton (2) > > and we have x is singleton -> x e' x (3) > > thus x is singleton <-> x e' x > > QED > > > (4) Non Emptiness: For all x Exist y ( y e' x) > > > (5) Comprehension: if Phi(y) is a formula in which at least y is free, > > and in which x is not free, then all closures of > > > Exist y ( y is singleton Phi(y) ) -> > > Exist x for all y ( y e' x <-> ( y is singleton Phi(y) ) ) > > > are axioms. > > > Define[y|Phi]: > > Exist y ( y is singleton Phi(y) ) -> > > x=[y|Phi] <-> for all y ( y e' x <-> ( y is singleton Phi(y) ) ) > > > Now we come to explain what "sets" in set theory mean: > > > The approach here is that objects in a set theory like Zermelo's set > > theory, are actually nothing but "containers", that is their reality. > > > However these containers in Z, have the property of being identical if > > they have the same contents, which is not a property that is inherent > > in the concept of containers, since containers of equal size can have > > exactly the same contents in them but yet can be different from each > > other, however this is not allowed in Z, so how come I say that these > > objects are containers. > > > To understand that issue, lets assume that sets are containers, and > > lets assume the relation epsilon in the language of Z refers > > informally to "contained in" so x e y means informally > > x is contained in y. > > > Of course "e" here is a primitive concept also, but in this approach > > it has an informal background as I illustrated above, it refers to > > "containment". > > > Now lets have the following axiom: > > > (6) Containment: x e y -> ( x is singleton & y is singleton ) > > > were singleton is defined above ( note: the definition of singleton > > uses the relation e' so it is not the same as singletons present in > > customary set theories which uses the relation e). > > > e can be read as " is a content of" > > > so x e y is to be read as: x is a content of y. > > > Now we define any collection of containers in which no two distinct > > containers have the same contents as a "discriminable collection" > > > Now the union of two discriminable collections is not necessarily > > a discriminable collection. > > > So we need to define discriminable union as the union of any two > > discriminable collections that is itself discriminable. > > > Now lets define V as the largest discriminable union of > > discriminable collections (this notion needs further treatment as to > > what "largest" would mean, but for now it means that every > > discriminable union of discriminable collections would be strictly > > subnumerous to V) > > > Now a 'set' might be defined as a member of V. > > > So V here would stand as the model of that set theory. > > > Then other axioms of Z, or ZF can be added ( were the variables in > > these axioms are members of V ). > > > This way of understanding sets, although it is not a complete workup > > yet, but it does provide us with an insight to what are sets, what are > > those objects in ZFC set theory. > > > They are Containers discriminable by their contents, nothing more, > > nothing less. > > > The relation e' can be used here to build proper classes also. > > > So it is a theory of this theory that: > > > For all x,z,u,w ((u=[y|yex] & w=[y|yez] & u=w) ->x=z) > > > Zuhair
From: Marshall on 27 Dec 2009 19:12 Thank you for taking the time to read my lengthy critique, and for taking it seriously. I think we have managed to quickly and productively get to the heart of the disagreement, and that is something just by itself! A few points which may be repetitive: On Dec 27, 2:45 pm, zuhair <zaljo...(a)gmail.com> wrote: > On Dec 27, 12:51 pm, Marshall <marshall.spi...(a)gmail.com> wrote: > > > > Now if you say that a set is a "container" and the relationship > > > epsilon stands for "being contained in" or " being inside" > > > or something like that, you'll have the difficulty of why > > > should containers be identical if there contents are identical? > > > I see no difficulty. > > OK, that's up to you. . but I do see a difficulty at intuitive > level, you cannot just come and put rules on matters > as you like, there should > be intuitive justification behind these rules. Actually, I disagree with this. We may, in fact, simply come in and put rules on matters at our own whim. The only *objective* objection that might arise is inconsistency; if we avoid that we are entirely justified. Whether we have produced something useful is of course another matter. > If you say sets > are containers, then why these containers must be identical > if they have the same contents, you must seek an intuitive > justification, which is unfortunately missing with the mere > container intuitive understanding of sets. Again, I actually see the lack of a concept of identity here as a virtue. Consider that most any object oriented programming language provides us with a model in which this sort of identity is preserved, and consider the voluminous body of evidence which shows how much trouble this sort of identity can cause. > > > that doesn't make any sense at intuitive level, intuitively > > > speaking different containers of the same size can > > > contain the same elements, there is no reason whatsoever > > > to believe that two containers having the same contents cannot be > > > different, which is what axiom of Extensionality says. > > > These "difficulties" are actually virtues, and whether you > > want to work with a system that has these properties > > or not, until you see why they are virtues, you are not > > going to get anywhere. > > No this is teleological thinking, that is not enough. it might help > as a temporizing measure, but it must not be the aim. I think your use of the word "teleological" here is justified, however I must still assert that until you can clearly see in what manner extensionality is a virtue, your analysis will be insufficient, *even if* (and perhaps *especially*) if you do not wish to subscribe to extensionality. One must understand one's enemies before one can defeat them. Also, I just want to repeat the below: > > What math does, what logic does, is to abstract. > > To abstract is fundamentally to leave things out; > > this is inescapable. because it is a really important point. > > I propose that you *start* with utility or simplicity or some > > such virtue and proceed from there. > > See below. > > Well I know what you are trying to say very well Marshall, I actually > don't have full objection to it. > > However intuitive formal theories would be easier to understand, and > easier to interact with, and by then you can work with them easily, > and they will definitely be fruitful. > > The system that I wrote above is actually an axiomatic theory > (although I didn't compete it yet) with three primitives binary > relations: identity "=", trivial membership " e' " and epsilon > membership "e", and one primitive constant symbol V. > > , and the language is first order logic with identity theory. > > There is no doubt that this system is complex, and more complex than > the traditional ones, though it might be nearer to our common > intuitive reasoning. > > However it is not this theory in particular that matters, the general > idea is the most important, and I will say it again, > > Oh yes trying to build formal theories having well non circularly > understandable primitives at intuitive level is a fruitful thing Oh > yes it is. Your position is well argued, and yet I cannot fully agree. I believe your affection for intuitiveness is misplaced, because it is so subjective, so personal, and so hard to evaluate effectively. I propose instead, for your consideration, simplicity; it is a quality that has most of what you value in intuitiveness, and it is much less subjective. My other objection to intuitiveness as a design criteria is that it works in the direction of causing a stasis in our understanding. If we don't understand or work well with a particular idea, we may simply brand it as unintuitive and thereby avoid learning anything new and perhaps better than what we currently know. On the other hand, if an idea or system is counterintuitive but still demonstrably simple, it is a good candidate for something that we ought to invest some time in learning. That has been my experience anyway. I hope you find these ideas useful or at least thought-provoking. And again, I thank you for taking them seriously. Happy New Year! Marshall
From: zuhair on 27 Dec 2009 20:18 On Dec 27, 7:12 pm, Marshall <marshall.spi...(a)gmail.com> wrote: > Thank you for taking the time to read my lengthy critique, > and for taking it seriously. I think we have managed to > quickly and productively get to the heart of the disagreement, > and that is something just by itself! > > A few points which may be repetitive: > > On Dec 27, 2:45 pm, zuhair <zaljo...(a)gmail.com> wrote: > > > On Dec 27, 12:51 pm, Marshall <marshall.spi...(a)gmail.com> wrote: > > > > > Now if you say that a set is a "container" and the relationship > > > > epsilon stands for "being contained in" or " being inside" > > > > or something like that, you'll have the difficulty of why > > > > should containers be identical if there contents are identical? > > > > I see no difficulty. > > > OK, that's up to you. . but I do see a difficulty at intuitive > > level, you cannot just come and put rules on matters > > as you like, there should > > be intuitive justification behind these rules. > > Actually, I disagree with this. We may, in fact, simply come > in and put rules on matters at our own whim. The only > *objective* objection that might arise is inconsistency; > if we avoid that we are entirely justified. Whether we > have produced something useful is of course another > matter. > > > If you say sets > > are containers, then why these containers must be identical > > if they have the same contents, you must seek an intuitive > > justification, which is unfortunately missing with the mere > > container intuitive understanding of sets. > > Again, I actually see the lack of a concept of identity > here as a virtue. Consider that most any object > oriented programming language provides us with > a model in which this sort of identity is preserved, > and consider the voluminous body of evidence > which shows how much trouble this sort of identity > can cause. > > > > > that doesn't make any sense at intuitive level, intuitively > > > > speaking different containers of the same size can > > > > contain the same elements, there is no reason whatsoever > > > > to believe that two containers having the same contents cannot be > > > > different, which is what axiom of Extensionality says. > > > > These "difficulties" are actually virtues, and whether you > > > want to work with a system that has these properties > > > or not, until you see why they are virtues, you are not > > > going to get anywhere. > > > No this is teleological thinking, that is not enough. it might help > > as a temporizing measure, but it must not be the aim. > > I think your use of the word "teleological" here is justified, > however I must still assert that until you can clearly see > in what manner extensionality is a virtue, your analysis > will be insufficient, *even if* (and perhaps *especially*) > if you do not wish to subscribe to extensionality. > > One must understand one's enemies before one can > defeat them. > > Also, I just want to repeat the below: > > > > What math does, what logic does, is to abstract. > > > To abstract is fundamentally to leave things out; > > > this is inescapable. > > because it is a really important point. > > > > > > > > I propose that you *start* with utility or simplicity or some > > > such virtue and proceed from there. > > > See below. > > > Well I know what you are trying to say very well Marshall, I actually > > don't have full objection to it. > > > However intuitive formal theories would be easier to understand, and > > easier to interact with, and by then you can work with them easily, > > and they will definitely be fruitful. > > > The system that I wrote above is actually an axiomatic theory > > (although I didn't compete it yet) with three primitives binary > > relations: identity "=", trivial membership " e' " and epsilon > > membership "e", and one primitive constant symbol V. > > > , and the language is first order logic with identity theory. > > > There is no doubt that this system is complex, and more complex than > > the traditional ones, though it might be nearer to our common > > intuitive reasoning. > > > However it is not this theory in particular that matters, the general > > idea is the most important, and I will say it again, > > > Oh yes trying to build formal theories having well non circularly > > understandable primitives at intuitive level is a fruitful thing Oh > > yes it is. > > Your position is well argued, and yet I cannot fully agree. I believe > your affection for intuitiveness is misplaced, because it is so > subjective, so personal, and so hard to evaluate effectively. > I propose instead, for your consideration, simplicity; it is a quality > that has most of what you value in intuitiveness, and it is much > less subjective. > > My other objection to intuitiveness as a design criteria is that > it works in the direction of causing a stasis in our understanding. > If we don't understand or work well with a particular idea, > we may simply brand it as unintuitive and thereby avoid > learning anything new and perhaps better than what we > currently know. On the other hand, if an idea or system > is counterintuitive but still demonstrably simple, it is a > good candidate for something that we ought to invest > some time in learning. > > That has been my experience anyway. I hope you > find these ideas useful or at least thought-provoking. > And again, I thank you for taking them seriously. > > Happy New Year! > > Marshall Thanks a lot really, to me I see this line of discussion with you fruitful really. I do agree with your criteria somehow, of course simplicity cost\benefit, etc.. an even I agree that concrete intuitiveness is sometimes handicapping ,I generally agree with much of what you say, but still I do think that we can come with a system that combines most of the virtues you are saying and be simpler to understand at the intuitive level. Certainly being intuitive by itself is not enough to justify building a formal system, but as I said I do think we can come with a better system as a whole. Thanks a lot Marshall, and Happy new year! Zuhair
From: Aatu Koskensilta on 28 Dec 2009 04:32
Marshall <marshall.spight(a)gmail.com> writes: > Actually, I disagree with this. We may, in fact, simply come in and > put rules on matters at our own whim. The only *objective* objection > that might arise is inconsistency; if we avoid that we are entirely > justified. Whether we have produced something useful is of course > another matter. What exactly do you mean by "inconsistency" here?[1] > Happy New Year! Happy ditto to you and all and sundry! Footnotes: [1] I know you sometimes find my style a tad exasperating, but just bear with me -- there's a point to this question, honest. -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon man nicht sprechan kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus |