On Ultrafinitism
in [Logic]
|
Prev: Modal Logic
Next: The Definition of Points
From: MoeBlee on 30 Oct 2006 13:51 Rupert wrote: > Edward Nelson, on the other hand, has done > some interesting research about certain weak axiomatic theories in > arithmetic, which may embody his stance. See his book "Predicative > Arithmetic". Thanks. I just downloaded it for free as a PDF file. And the first chapter of his unfinished book on IST too. Do you have any other recommendations? What do you think of the system Shaughan Lavine gives in his 'Understanding The Infinite'? (I've read some high praise for this book.) I only saw the book briefly, so I didn't have a chance to ponder his system. But ab initio in my thinking about this, I can't imagine how an ultrafinitist can give a truly rigorous axioimatization in which there is largest natural and block us from adding 1 to that largest natural (the paradox of the heap, as it were). But, of course, I need to study the systems to see for myself, since otherwise my questions are uninformed. And, aside from that one chapter by Nelson, do you know of a good explanation (rigorous but not so difficult that it starts with advanced concepts).of axioms for IST? I saw a book by Alain Robert but it's not what I'd like; it's more of sketchbook of ideas than it is a tight theorem by theorem treatment. And some of the other books I found in the QA299 section of the library jump right into more advanced stuff way too fast for me. MoeBlee
From: MoeBlee on 30 Oct 2006 14:03 Bill Taylor wrote: > Is the following a reasonable point of view, do people think? > > I'm still kind of wondering where Yessenin-Volpin, Edward Nelson, > and other ultrafinitists are coming from. > > They purport to find, or rather take the public stance of finding, > that the concept of "all the naturals" is confusing and vague, > whereas it is indeed *crystal-clear* to the rest of us. To be fair, for example, I just read an essay by Abraham Robinson in which he denies that he can conceive of an infinite set. He does endose working with infinite sets as ideal objects in a formal theory, but he makes crystal clear that to him the concept of an infinite set is meaningless otherwise. > Some time ago, back in the late seventies to early eighties, > there was a brief flurry of interest from fringe mathematicians > in "fuzzy math". It was never quite clear what this was, but it > still has a small amount of library shelf space, though perhaps > little or no presence in math departments in academia. > It seemed to be (AFAICT), basically, that joke that > used to go around about "Generalized Mathematics" - Yet I see more and more new books on the shelves every time I visit the library that have titles with the word 'fuzzy' in them, and these are usually Springer Verlag books, including textbooks and collections of articles and reportings of conference proceedings. Some of the material seems to be computer science, but plenty of it is categorized as mathematics. Well, my just saying that there are a lot of new books, written by professional mathematicians, with the words 'fuzzy' in the title is not an argument that fuzzy is an important area of study, but it at least gives me some reason to doubt claims that it is not currently an important area of study. MoeBlee
From: Dave L. Renfro on 30 Oct 2006 16:11 Bill Taylor wrote (in part): >> Some time ago, back in the late seventies to early eighties, >> there was a brief flurry of interest from fringe mathematicians >> in "fuzzy math". It was never quite clear what this was, but it >> still has a small amount of library shelf space, though perhaps >> little or no presence in math departments in academia. >> It seemed to be (AFAICT), basically, that joke that >> used to go around about "Generalized Mathematics" - MoeBlee wrote (in part): > Yet I see more and more new books on the shelves every time > I visit the library that have titles with the word 'fuzzy' > in them, and these are usually Springer Verlag books, > including textbooks and collections of articles and > reportings of conference proceedings. Some of the material > seems to be computer science, but plenty of it is categorized > as mathematics. Well, my just saying that there are a lot > of new books, written by professional mathematicians, > with the words 'fuzzy' in the title is not an argument > that fuzzy is an important area of study, but it at > least gives me some reason to doubt claims that it is > not currently an important area of study. If you glance through some of the bound volumes of Mathematical Reviews, looking at subject areas 26, 28, and 54 from the mid 1970's to the late 1990's (maybe glance at one month's issue every two or three years), you'll find a huge increase in titles with "fuzzy" in them as the years go by. This might be a bit harsh, but it almost seems as if certain mathematicians, who otherwise had done nothing very deep or interesting, decided they could get a lot of mileage by going through and fuzzifying certain areas of pure mathematics (those areas they had graduate courses in). In many cases, it appears to me that the people writing the papers are coming from just outside of mathematics (economics, computer science, etc.) and the papers tend to reflect where they're at in learning standard graduate mathematics (i.e. learn about the open mapping theorem, then write a paper where the theorem and proof receive straightforward fuzzified translations). Try googling "fuzzy" with various math words and phrases from analysis and topology: http://www.google.com/search?q=fuzzy-topology http://www.google.com/search?q=fuzzy+open-mapping-theorem http://www.google.com/search?q=fuzzy+Baire-category http://www.google.com/search?q=fuzzy+tychonoff-theorem http://www.google.com/search?q=fuzzy-measure-theory http://www.google.com/search?q=fuzzy-uniform-space http://www.google.com/search?q=fuzzy+Banach-limits http://www.google.com/search?q=fuzzy+convergence-in-measure http://www.google.com/search?q=fuzzy+stone-weierstrass-theorem http://www.google.com/search?q=fuzzy-Banach-space You get the idea, I think. Dave L. Renfro
From: Rupert on 30 Oct 2006 17:33 MoeBlee wrote: > Rupert wrote: > > Edward Nelson, on the other hand, has done > > some interesting research about certain weak axiomatic theories in > > arithmetic, which may embody his stance. See his book "Predicative > > Arithmetic". > > Thanks. I just downloaded it for free as a PDF file. And the first > chapter of his unfinished book on IST too. Do you have any other > recommendations? > I'm afraid I don't know all that much about it. There's a famous essay by Yessenin-Volpin called "The Ultraintuitionistic Criticism and the Antitraditional Program for Foundations of Mathematics", in "Intuitionism and Proof Theory, Proceedings of the Conference at Buffalo 1968, North-Holland, Amsterdam, 1970." Some people like that essay but it doesn't have enough of what I can recognize as mathematical content for my taste. He announced that he had a consistency proof for ZF with any finite number of inaccessible cardinals, but apparently that proof is hard to get hold of, which is a shame. These might be interesting: http://www.springerlink.com/content/v76473730365861x/ http://www.turpion.org/php/paper.phtml?journal_id=rm&paper_id=2870 http://math.ucsd.edu/~sbuss/ResearchWeb/nelson/paper.pdf I can't actually find the one I was looking for, which talked about "Nelson's Program". > What do you think of the system Shaughan Lavine gives in his > 'Understanding The Infinite'? I haven't seen it. > (I've read some high praise for this > book.) I only saw the book briefly, so I didn't have a chance to ponder > his system. But ab initio in my thinking about this, I can't imagine > how an ultrafinitist can give a truly rigorous axioimatization in which > there is largest natural and block us from adding 1 to that largest > natural (the paradox of the heap, as it were). I don't think they would go about it that way. They could use a very weak arithmetic in which only functions of very low computational complexity could be proved total. Then only fairly small numbers could be feasibly defined. They could even add an axiom saying that e.g. the exponential function is not total, which would give rise to a theory with nonstandard models. Nelson developed an alternative foundation for probability theory using such an arithmetic. > But, of course, I need > to study the systems to see for myself, since otherwise my questions > are uninformed. > > And, aside from that one chapter by Nelson, do you know of a good > explanation (rigorous but not so difficult that it starts with advanced > concepts).of axioms for IST? Sorry, no. > I saw a book by Alain Robert but it's not > what I'd like; it's more of sketchbook of ideas than it is a tight > theorem by theorem treatment. And some of the other books I found in > the QA299 section of the library jump right into more advanced stuff > way too fast for me. > > MoeBlee
From: galathaea on 31 Oct 2006 00:45
Eckard Blumschein wrote: > On 10/29/2006 3:04 AM, galathaea wrote: > > > ultrafinitsm takes a conceptual step beyond finitism > > by stressing that > > not only is mathematics a finite process > > but there exist hard limits > > > > there is no potential infinity > > Mueckenheim was blamed an ultrafinitist. However, he denies the actual > infinity while the potential infinity seems to be obvious to anybody. plutarch was caught writing "as the indifferent is the mean between good and evil so there is some mean between finite and infinite" > > or legitimate means to assume > > a process can be continued indefinitely > > > > in any derivation > > > > it is necessary to question > > for any process specification > > whether that process can complete > > "within the limits of resources" > > available to mathematics > > > > because > > they insist > > mathematics is a physical process > > and one day too may suffer the entropic decay > > Mueckenheim obviously shares this view. I do not understand why he > cannot accept mathematics like dealing with the two abstract ideas > number and continuum. > > One alternative after the other failed to unmask Cantors paradise as > what I consider the Dedekind-Cantor Utopia. Kronecker even called the > natural numbers given by the Lord. Brouwer even intended to improve set > theory. Weyl suggested an atomist continuum. > > Even Cantor and Hilbert started at some sound finitist views. Now > ultrafinitism is rumored to be the most silly counterpoint to formalism. > > Tell me please whether or not there is a drawer you may put me in? > I consider the world of (countable) numbers quite different from the > complementing world of (uncountable) continuum. In principle Cantor was > conjecturing almost the same when he believed that there is nothing > between aleph_0 and aleph_1. at some point we all must return to democrites and the dilemma of the cone -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=- galathaea: prankster, fablist, magician, liar |