An uncountable countable set
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From: Han de Bruijn on 2 Oct 2006 03:44 Virgil wrote: > In article <c87e0$451cc5b4$82a1e228$4275(a)news1.tudelft.nl>, > Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: > >>MoeBlee wrote: >> >>>Han de Bruijn wrote: >>> >>>>It's a priorities issue. Do axioms have to dictate what constructivism >>>>should be like? Should constructivism be tailored to the objectives of >>>>axiomatics? I think not. >>> >>>Fine, but if you don't give a formal system, then your mathematical >>>arguments are not subject to the objectivity of evaluation that >>>arguments backed up by formal systems are subject to. >> >>Exactly! Constructivism is not Formalism. > > Do constructivists have any statements which they accept as true without > proof? > If not how do they prove anything from nothing? > If so, then aren't those things they accept equivalent to axioms. Constructively valid proof. Intuition precedes axioms. A mathematician is like an architect who builds his mathematics. Take a look at some material concerning intuitionism and constructivism in the first place. That will answer your questions more effectively that HdB can ever do. Moreover, according to Torkel Franzen, I'm only a "pink" constructivist. Han de Bruijn
From: Han de Bruijn on 2 Oct 2006 03:48 Virgil wrote: > In article <51de2$451cc7e9$82a1e228$6256(a)news1.tudelft.nl>, > Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: > >>Virgil wrote: >> >>>In article <2e658$451b78ef$82a1e228$7519(a)news1.tudelft.nl>, >>> Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: >>> >>>>Tony Orlow wrote: >>>> >>>>>MoeBlee wrote: >>>>> >>>>>>Tony Orlow wrote: >>>>>> >>>>>>>Constructivism and Axiomatism are two sides of a coin. They can be >>>>>>>reconciled in larger framework, I think. >>>>>> >>>>>>I don't know what your definition of 'axiomatism' is, but there are >>>>>>axiomatic systems for constructive mathematics. >>>>> >>>>>I dunno. I was responding to Han's comment. I think he means >>>>>constructive concepts vs. axiomatic declarations. >>>> >>>>It's a priorities issue. Do axioms have to dictate what constructivism >>>>should be like? Should constructivism be tailored to the objectives of >>>>axiomatics? I think not. >>> >>>But if you cannot clearly state what you are assuming/accepting as true, >>>all you have is a morass of ambiguity. >> >>Ambiguity does not necessarily comprise a morass. > > How do constructivists deduce new things if there is nothing that they > can say is true to start with? Constructivism starts with counting. All the rest is architecture. Han de Bruijn
From: Han de Bruijn on 2 Oct 2006 03:50 Virgil wrote: > In article <8cabe$451ccd62$82a1e228$12622(a)news1.tudelft.nl>, > Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: > >>Virgil wrote: >> >>>In article <9fd0$451b7e7b$82a1e228$8977(a)news1.tudelft.nl>, >>> Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: >>> >>>>Virgil wrote: >>>> >>>>>In article <451a8f41(a)news2.lightlink.com>, >>>>>Tony Orlow <tony(a)lightlink.com> wrote: >>>> >>>>>>For purposes of measure on the finite scale, infinitesimals can be >>>>>>considered nilpotent. That's all. Do you disagree? >>>>> >>>>>I disagree that scale changes can convert between zero and non-zero. >>>>> >>>>>There are approximation methods is which products of small quantities >>>>>are regarded as negligible in comparison to the quantities themselves, >>>>>but they are always just approximations. >>>> >>>>Crucial question: are those "approximation methods" part of mathematics? >>>>I'll take Yes or No as a sufficient answer. >>> >>>They are a part of the applications of mathematics to things other than >>>mathematics, so they are marginal. >> >>That sounds like a smart answer, but I don't buy it. >>Again: are those "approximation methods" part of mathematics? Yes or No. > > They are attempts to bend the mathematics to accommodate the needs of > the sciences, so one would have to say "Yes and No". Contradiction? Han de Bruijn
From: Han de Bruijn on 2 Oct 2006 03:56 Virgil wrote: > In article <e4ca4$451cd0dd$82a1e228$14108(a)news1.tudelft.nl>, > Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: > >>Virgil wrote: >> >>>In article <1159437062.473100.294820(a)k70g2000cwa.googlegroups.com>, >>> mueckenh(a)rz.fh-augsburg.de wrote: >>> >>>>Virgil schrieb: >>>> >>>>>Several sets may all have the common property of being pairwise >>>>>bijectable, but if any of their members are distinguishable from those >>>>>of another set then the sets are equally distinguishable. >>>> >>>>Each one of the sets expresses, represents, and *is* the same >>>>(cardinal) number. >>> >>>Then one apple and one orange are the same because they have the same >>>cardinality. >> >>In _that_ respect, with respect to counting: definitely, yes! > > But not necessarily in any other respect whatsoever, so that to say an > apple is an orange or an orange is an apple, as some have been saying, > is foolishly wrong. Why? Give me one piece of fruit. I don't care whether it is an orange or an apple .. Han de Bruijn
From: Han de Bruijn on 2 Oct 2006 04:06
Virgil wrote: > In article <2b79d$451cd38b$82a1e228$17494(a)news1.tudelft.nl>, > Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: > >>Virgil wrote: >> >>>In article <76b59$451ba0bd$82a1e228$18077(a)news2.tudelft.nl>, >>> Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: >>> >>>>mueckenh(a)rz.fh-augsburg.de wrote: >>>> >>>>>Virgil schrieb: >>>> >>>>>>>>You stated that you needed counting to determine the successor. That is >>>>>>>>false. The successor is defined without any reference to counting. >>>>>>> >>>>>>>The successor function *is* counting (+1). >>>>>> >>>>>>Not to those who can't count. Successorship does not require numbers, it >>>>>>only requires "next". >>>>> >>>>>How far would those who cannot count be able to find "the next"? >>>> >>>>And how do you distinguish "the next" from something previous? >>> >>>By pointing at them separately. >>> >>>>This is >>>>not a joke. Many young children don't find it trivial that you shouldn't >>>>count a thing twice. >>> >>>But they are much less prone to mistaking who has more marbles, or >>>whatever, which argues that injection, surjection and bijection are more >>>basic than counting. >> >>Have two bags with say a hundred marbles in it and _make_ the bijection. >>I wish you good luck. And, BTW, I would like to have a computer program >>which does the job, properly. Video circuit attached. >> > The age at which children would be able to compare bags of about n > marbles successfully ins an increasing function of the age of the > children. There is an age in which they could not even compare empty > bags. A long time ago, I was reading a pictures book with one of my children. There was a flower on page one, a bear on page two, a fly on page three and so on and so forth. And then we turned the last page: what's there? After a while I suggested: "nothing". I remained silent for some time .. But then, very enthousiastic: nothing, nothing, nothing !! Han de Bruijn |