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From: Torkel Franzen on 9 Apr 2005 07:06 Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> writes: > It gave rise to the > illusion that real numbers are really numbers, and I looked in vain for > any benefit except for income of those who edited his schizophrene and > mystical rather than logically retraceable theory, wrote pertaining > books on set theory and got lucratice posts. This is fine ranting, even if somewhat baffling in its language.
From: Matt Gutting on 9 Apr 2005 10:25 Eckard Blumschein wrote: > On 4/8/2005 6:07 PM, Matt Gutting wrote: > > >>>>>>"There are neither more nor less nor equally many real numbers >>>>>>as compared to the rational ones." >>>> >>>>Did you mean to say the _number of_ real numbers is the same as >>>>the _number of_ rational numbers? There are some fellows over in >>>>comp.ai.philosophy and sci.philosophy.meta you should talk to. >>> >>> >>>Thank you for the hint. I was not aware of there groups. >>>No. I argue that the number of rational numbers exceeds any limit as >>>does the number of real numbers. The operation oo-oo is not reasonable >>>because oo is a quality , not a quantity. >> >>I don't see what subtraction has to do with it. > > > It depends on the result of subtraction whether a number is larger, > equal or smaller than the other one. I think I see your point here. But since, as you point out, oo is not a number, I don't see the relevance of your statement. > > I agree that oo is not > >>a *number* in the same sense that, e.g., pi is a number. > > > oo is just a quality, not a number at all. > I agree (as I said above) that it is not a number. But I don't understand what you mean by "quantity". > >>I don't think one can talk meaningfully about the number of numbers > > > Well, In German language I would rather say the Anzahl (amount) of > numbers. See above: Originally I wrote "more numbers". > That makes sense; in English "number" and "amount" can be used in the same sense (unlike, I believe, "Nummer" and "Anzahl" - is the same true for "Zahl" and "Anzahl"?). > > >>in either of these sets - that is, I don't think one can answer the >>question "How big is this set?" for either set. > > > That is indeed the point. > > >>I do think, however, that >>one can find a measure of comparison between the two sets - that it is >>possible to answer the question "Is this set bigger (in some sense) than >>that one?" > > > I see this the basic fallacy. Without a quantitative measure there is no > possibility of comparison. The quality is the same: Just infinite. > I disagree. Even with finite sets, it is still possible to determine which of two sets is bigger without assigning a quantity to either of them. In particular, for a finite set one can determine whether it is as big as another either by counting to answer the question "how many?", or by another agreed-on method. (I do have a specific method in mind.) For infinite sets, we obviously cannot use the first approach, since we cannot answer the question "how many?", but we can design the second approach so that it works for infinite sets as well as finite. > >>By the measures generally adopted by mathematicians we decide >>that the reals are "bigger" in a very specific sense. > > > Do you really decide? I rather found out that you are following Cantor's > split way of thinking/believing. > In the first place, I should not have used the word "we" - it would be presumptuous of me to claim to be a mathematician. I do indeed decide to follow Cantor's understanding of "infinity" and "bigger than" (I'm not sure why you describe it as "split"), because the notion of "bigger than" applies both to infinite and to finite sets. > >>But "bigger" has >>nothing to do with a numerical answer to the question "How many?" for >>either set - since this question cannot, in fact, be answered. > > > "Bigger" is the result of a quantitative comparison. But it has nothing > to do with countable or not. "Bigger", as I said above, is not necessarily the result of a quantitative comparison, nor need it be. If it is not particularly helpful to mathematicians to consider it so, there is no reason to ordain that it must be so. > > Eckard > > > >
From: Eckard Blumschein on 11 Apr 2005 02:12 On 4/9/2005 1:06 PM, Torkel Franzen wrote: > Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> writes: > >> It gave rise to the >> illusion that real numbers are really numbers, and I looked in vain for >> any benefit except for income of those who edited his schizophrene and >> mystical rather than logically retraceable theory, wrote pertaining >> books on set theory and got lucratice posts. > > This is fine ranting, even if somewhat baffling in its language. > >>>> Whateverever particular approach we favor in the teaching of logic >>>> and mathematics, we should take into account that it is bound to >>>> have limitations and weak spots of which we are most likely >>>> unaware. Those who are teaching Cantorian split thinking are per definition not aware of teaching nonsense. Perhaps one has to encrypt the necessary for belief baffling into deep sophistication like the hard to grasp assertion that some larger infinity is somehow required as to make some statement about some solvability of some diophantine equation. >>>> It is a good idea to enlist unsympathetic observers to help us >>>> evaluate our approach, and to help us separate the effects of >>>> enthusiastic and dedicated teaching from the effects of the >>>> particular choices of methods and material. I agree. I will enlist. Eckard
From: Eckard Blumschein on 11 Apr 2005 03:22 On 4/9/2005 4:25 PM, Matt Gutting wrote: >> oo is just a quality, not a number at all. >> > > I agree (as I said above) that it is not a number. But I don't > understand what you mean by "quantity". A physical quantity is something that can be measured in principle, for instance ten apples or all just living people. Mathematics abstracts from the subjects and just counts the entities. Primarely, any quantity is based on counting. This point of view is close to Kronecker's finitism. Is pi a quantity? I think so even if one cannot numerically approach it. However, sometimes it is not reasonable to count entities or define quantities by means of belonging mathematical connections. English language does not count continuous things like milk or sugar. While we do not ask how many but how much in this case, we are nonetheless aware of the possibilities to count all pieces of a small amount of sugar or to use the measurable mass instead. Abstract physical quantities like mass or length are based on the idea of unlimited divisibility and also unlimited enlargement. This is quite a different mathematical basis than counting, not ouside Mathematics but strictly separated from the realm of numbers. One has to perform border crossing from numbers (which are always finite without any exception) to the qualities infinite alias continuous or vice versa. > That makes sense; in English "number" and "amount" can be used in > the same sense (unlike, I believe, "Nummer" and "Anzahl" - is the > same true for "Zahl" and "Anzahl"?). Take Russell's paradox of the set of all sets for an example how stupid people can stumble about their own terminology. >>>I do think, however, that >>>one can find a measure of comparison between the two sets - that it is >>>possible to answer the question "Is this set bigger (in some sense) than >>>that one?" >> >> >> I see this the basic fallacy. Without a quantitative measure there is no >> possibility of comparison. The quality is the same: Just infinite. >> > > I disagree. Even with finite sets, it is still possible to determine > which of two sets is bigger without assigning a quantity to either of > them. Finite sets have the quality to allow comparison. If something exceeds any limit, then it does not offer this opportunity. > In particular, for a finite set one can determine whether it is as big > as another either by counting to answer the question "how many?", Yes. > or by another agreed-on method. (I do have a specific method in mind.) > For infinite sets, we obviously cannot use the first approach, since we > cannot answer the question "how many?", but we can design the second > approach so that it works for infinite sets as well as finite. It was G. Cantor who introduced the principle of bijection in combination with complete induction in order to be surprized himself because not just the potentially infinite series of natural numbers but also the rational numbers are potentially countable. Realizing that the whole universe does mathematically seen not contain more points than a tiny linear interval, Cantor wrote: "Je le vois. Mais je ne le crois pas." Intuition of stupid thinkers including G. Cantor and B. Russell goes wrong when it suggests that the infinite can be enlarged. The notion of infinity is best expressed by oo+a=oo. > >> >>>By the measures generally adopted by mathematicians we decide >>>that the reals are "bigger" in a very specific sense. >> >> >> Do you really decide? I rather found out that you are following Cantor's >> split way of thinking/believing. >> > > In the first place, I should not have used the word "we" - it would be > presumptuous of me to claim to be a mathematician. I am just an engineer myself. . > I do indeed decide > to follow Cantor's understanding of "infinity" and "bigger than" (I'm > not sure why you describe it as "split"), because the notion of "bigger > than" applies both to infinite and to finite sets. I see this Cantors misleading intuition. Cantor ignored oo+a=oo without any justification except for his own intuition and ambitionism. Intuitively oo+a is bigger than just oo. However, oo cannot be enlarged. >> "Bigger" is the result of a quantitative comparison. But it has nothing >> to do with countable or not. > > "Bigger", as I said above, is not necessarily the result of a > quantitative comparison, nor need it be. If it is not particularly > helpful to mathematicians to consider it so, there is no reason to > ordain that it must be so. The word bigger is much elder than Cantorian nonsense. Everybody has a feeling for "bigger". However, nobody has a likewise correct feeling for the notion of infinite like something that cannot be enlarged except maybe for mathematicians, logicians, or engineers like me who developed a sufficient ability of abstract imagination. Eckard
From: Matt Gutting on 11 Apr 2005 10:06
Eckard Blumschein wrote: > On 4/9/2005 4:25 PM, Matt Gutting wrote: > > > >>>oo is just a quality, not a number at all. >>> >> >>I agree (as I said above) that it is not a number. But I don't >>understand what you mean by "quantity". > > > A physical quantity is something that can be measured in principle, for > instance ten apples or all just living people. Mathematics abstracts > from the subjects and just counts the entities. Primarely, any quantity > is based on counting. This point of view is close to Kronecker's > finitism. Is pi a quantity? I think so even if one cannot numerically > approach it. If any quantity is based on counting, how can one call anything not a natural number a quantity? And I apologize, my second sentence contains an error. To correct that error, let me ask: What do you mean by saying that "oo is just a quality?" > > However, sometimes it is not reasonable to count entities or define > quantities by means of belonging mathematical connections. English > language does not count continuous things like milk or sugar. While we > do not ask how many but how much in this case, we are nonetheless aware > of the possibilities to count all pieces of a small amount of sugar or > to use the measurable mass instead. That is true. Similarly, mathematics has other tools, such as measure, to speak of the "size" of a set. These other tools do not necessarily yield the same results as counting or bijections. They are used for different purposes. > > Abstract physical quantities like mass or length are based on the idea > of unlimited divisibility and also unlimited enlargement. This is quite > a different mathematical basis than counting, not ouside Mathematics but > strictly separated from the realm of numbers. One has to perform border > crossing from numbers (which are always finite without any exception) to > the qualities infinite alias continuous or vice versa. > > > >>That makes sense; in English "number" and "amount" can be used in >>the same sense (unlike, I believe, "Nummer" and "Anzahl" - is the >>same true for "Zahl" and "Anzahl"?). > > > Take Russell's paradox of the set of all sets for an example how stupid > people can stumble about their own terminology. > > > >>>>I do think, however, that >>>>one can find a measure of comparison between the two sets - that it is >>>>possible to answer the question "Is this set bigger (in some sense) than >>>>that one?" >>> >>> >>>I see this the basic fallacy. Without a quantitative measure there is no >>>possibility of comparison. The quality is the same: Just infinite. >>> >> >>I disagree. Even with finite sets, it is still possible to determine >>which of two sets is bigger without assigning a quantity to either of >>them. > > > Finite sets have the quality to allow comparison. If something exceeds > any limit, then it does not offer this opportunity. Why not? > > > >>In particular, for a finite set one can determine whether it is as big >>as another either by counting to answer the question "how many?", > > > Yes. > > >>or by another agreed-on method. (I do have a specific method in mind.) >>For infinite sets, we obviously cannot use the first approach, since we >>cannot answer the question "how many?", but we can design the second >>approach so that it works for infinite sets as well as finite. > > > It was G. Cantor who introduced the principle of bijection in > combination with complete induction in order to be surprized himself > because not just the potentially infinite series of natural numbers but > also the rational numbers are potentially countable. Realizing that the > whole universe does mathematically seen not contain more points than a > tiny linear interval, Cantor wrote: "Je le vois. Mais je ne le crois pas." > Do you disagree that the principle of bijection does not answer the question "Is this set as big as that one?" for finite sets in exactly the same way as counting would? If you do (that is, if you can think of a pair of sets which have the same count of objects but do not have a bijection between them, or vice versa) I'd appreciate an example. If you do not disagree, then what hinders one from extending this notion of bijection to infinite sets, as long as it gives useful answers? > Intuition of stupid thinkers including G. Cantor and B. Russell goes > wrong when it suggests that the infinite can be enlarged. The notion of > infinity is best expressed by oo+a=oo. That is your opinion - but see my final comment below for a full response. > > >>>>By the measures generally adopted by mathematicians we decide >>>>that the reals are "bigger" in a very specific sense. >>> >>> >>>Do you really decide? I rather found out that you are following Cantor's >>>split way of thinking/believing. >>> >> >>In the first place, I should not have used the word "we" - it would be >>presumptuous of me to claim to be a mathematician. > > > I am just an engineer myself. . > > >>I do indeed decide >>to follow Cantor's understanding of "infinity" and "bigger than" (I'm >>not sure why you describe it as "split"), because the notion of "bigger >>than" applies both to infinite and to finite sets. > > > I see this Cantors misleading intuition. Cantor ignored oo+a=oo without > any justification except for his own intuition and ambitionism. > Intuitively oo+a is bigger than just oo. However, oo cannot be enlarged. > Intuitively, since oo is not a number, oo + a doesn't make sense - it's like saying "apple + seventeen". And what, precisely (*very* precisely) do you mean by "cannot be enlarged"? > >>>"Bigger" is the result of a quantitative comparison. But it has nothing >>>to do with countable or not. >> >>"Bigger", as I said above, is not necessarily the result of a >>quantitative comparison, nor need it be. If it is not particularly >>helpful to mathematicians to consider it so, there is no reason to >>ordain that it must be so. > > > The word bigger is much elder than Cantorian nonsense. Everybody has a > feeling for "bigger". However, nobody has a likewise correct feeling for > the notion of infinite like something that cannot be enlarged except > maybe for mathematicians, logicians, or engineers like me who developed > a sufficient ability of abstract imagination. In the first place, infinity isn't defined as "something that cannot be enlarged"; in non-mathematical terms it might be described as "something that cannot be exhausted" or "something that cannot end". Even with that issue cleared up, what people have a "feeling" for has no bearing on how it is (or should be) used in mathematics. Mathematical usage is governed by formal definition, agreed on by mathematicians, and by strict logical consequences. > > Eckard > > > |