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## Re: [ontolog-forum] Axiomatic ontology

 To: "[ontolog-forum] " Pat Hayes Thu, 7 Feb 2008 11:06:02 -0600
 At 10:18 AM -0600 2/7/08, Chris Menzel wrote: On Wed, 6 Feb 2008, John F. Sowa wrote: > Avril, > ... > On the other hand, I would not recommend the following approach > for number theory: To say the least. > > Axiom 9. can be maintained, but the meaning of 'every' has to be > > interpreted to denote a totality of something around 10**120, > > or Ackermann(5 5), or something finite that can be written down > > or understood in some way. The view isn't even coherent.  If Ackerman(5 5) exists, why not Ackerman(Ackerman(5 5) 5) -- a massively larger number?  And of course if *that* number exists, well, you get the idea. > Many 19th century mathematicians strongly objected to that way of > talking, and I sympathize with them.  But those mathematicians would > *never* agree to a fixed upper bound on the integers, such as 10**120, > Ackermann(5 5), or any other finite integer. Indeed -- which of course means that there are infinitely many finite integers, and hence that there is a set that contains them, hence a power set of that set, and off we go down the Cantorian bunny trail! :-) It is kind of fun to see if there is a coherent alternative, though. How could one make sense of the thesis that there are only finitely many integers? The counterproof seems very simple and is probably one of the oldest mathematical proofs ever devised. 1 Suppose there were a largest integer. 2 Call it N. 3 Consider N+1. 4 N+1 is larger than N: contradiction. 5 Ergo, there is no largest integer. Where can one object to this? There are several possibilities. a. Maybe the largest integer can't be named, so step 2 is illegitimate (but why not?) b. Maybe N+1 isn't larger than N for that N (how? What happens to arithmetic up at the large end?) c.  Maybe N+1 doesn't exist at all, its impossible. That seems more intuitive. But what would it mean to say that N exists but N+1 doesn't? After all, if N exists than the set {1, ... ,N} seems to exist, and the power set of this set has more than N elements. Its very hard to cleave to the strict finitist intuition and still do any kind of set theory. d. This is an argument by contradiction, and so is inherently suspicious. I find this quite unconvincing, myself, even though its popular. The actual logic of this argument seems quite sound to me. But my main point is that in order to maintain a coherent strict-finitist position one does need to consider arguments/debates like this and to think hard about the consequences of various positions. Its not enough to just announce as an obvious doctrine that infinity is wrong; still less to seem to link conventional mathematics to some kind of dark political conspiracy. Mathematicians tend to be Platonists because they are driven to it by following chains of thought which seem to be inevitable and conclusive. If you want to announce an alternative, you have to tell us where the less travelled paths branch off the mathematical highway. Pat You may not like where that leads, but it is very hard to argue that there is a nonarbitrary point at which you can stop that line of reasoning. -chris   _________________________________________________________________ Message Archives: http://ontolog.cim3.net/forum/ontolog-forum/  Subscribe/Config: http://ontolog.cim3.net/mailman/listinfo/ontolog-forum/  Unsubscribe: mailto:ontolog-forum-leave@xxxxxxxxxxxxxxxx Shared Files: http://ontolog.cim3.net/file/ Community Wiki: http://ontolog.cim3.net/wiki/ To Post: mailto:ontolog-forum@xxxxxxxxxxxxxxxx   ```-- ``` --------------------------------------------------------------------- IHMC               (850)434 8903 or (650)494 3973   home 40 South Alcaniz St.       (850)202 4416   office Pensacola                 (850)202 4440   fax FL 32502                     (850)291 0667    cell http://www.ihmc.us/users/phayes      phayesAT-SIGNihmc.us http://www.flickr.com/pathayes/collections ``` _________________________________________________________________ Message Archives: http://ontolog.cim3.net/forum/ontolog-forum/ Subscribe/Config: http://ontolog.cim3.net/mailman/listinfo/ontolog-forum/ Unsubscribe: mailto:ontolog-forum-leave@xxxxxxxxxxxxxxxx Shared Files: http://ontolog.cim3.net/file/ Community Wiki: http://ontolog.cim3.net/wiki/ To Post: mailto:ontolog-forum@xxxxxxxxxxxxxxxx    (01) ```
 Current Thread Re: [ontolog-forum] Axiomatic ontology, (continued) Re: [ontolog-forum] Axiomatic ontology, Avril Styrman Re: [ontolog-forum] Axiomatic ontology, Christopher Menzel Re: [ontolog-forum] Axiomatic ontology, Barker, Sean (UK) Re: [ontolog-forum] Axiomatic ontology, paola . dimaio Re: [ontolog-forum] Axiomatic ontology, Avril Styrman Re: [ontolog-forum] Axiomatic ontology, Avril Styrman Re: [ontolog-forum] Axiomatic ontology, Christopher Menzel Re: [ontolog-forum] Axiomatic ontology, Pat Hayes Re: [ontolog-forum] Axiomatic ontology, John F. Sowa Re: [ontolog-forum] Axiomatic ontology, Chris Menzel Re: [ontolog-forum] Axiomatic ontology, Pat Hayes <= Re: [ontolog-forum] Axiomatic ontology, Ed Barkmeyer Re: [ontolog-forum] Axiomatic ontology, John F. Sowa Re: [ontolog-forum] Axiomatic ontology, Pat Hayes Re: [ontolog-forum] Axiomatic ontology, John F. Sowa [ontolog-forum] Visualization of ontologies for business, Mills Davis Re: [ontolog-forum] Visualization of ontologies for business, matthew.west Re: [ontolog-forum] Visualization of ontologies for business, Mills Davis Re: [ontolog-forum] Visualization of ontologies for business, Adrian Walker Re: [ontolog-forum] Visualization of ontologies for business, Elisa F. Kendall Re: [ontolog-forum] Visualization of ontologies for business, Ed Barkmeyer