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Jul 17 2005, 02:13 AM
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Member Group: Members Posts: 15 Joined: 14-June 05 Member No.: 491 |
Topic Description: A general discussion of Objectivist ideas of mathematics, especially regarding mathematical logic and set theory.
/ I'd like to know more about Objectivist views of mathematics, especially regarding mathematical logic and set theory. I have a number of questions. I encourage anyone with insight to address any of these questions or to offer their thoughts irrespective of my questions or to ask additional questions. I hope for a productive discussion that will probe for strengths and weaknesses of classical mathematical logic and set theory, intuitionist mathematics, Objectivist proposals for mathematics, and other approaches to mathematics. 1. What is the state of the art in Objectivist philosophy of mathematics? What Objectivist writers have made significant progress in this area, and what publications may one view? 2. What synopsis can be given of Objectivist views on mathematics? Has a systematic and rigorous statement of an Objectivist philosophy of mathematics been offered? 3. Are there Objectivist mathematicians who have published mathematics that embody particularly Objectivist mathematical concerns, approaches, or philosophy? Are there proposals for new systems of mathematical logic or for new axioms for set theory or mathematics? Or do Objectivist mathematicians find already existing systems and axioms satisfactory so that Objectivist alternatives are not needed? 4. How would mathematicians implement Objectivist ideas about mathematics? If the ideas are not suggested for implementation in mathematical theories, then in what way is it hoped that the ideas will affect mathematical practice? 5 How do Objectivists view mathematical logic and set theory? Is there a preference for classical mathematical logic, for intuitionist mathematical logic, or for other systems, approaches, or philosophies? Most particularly, if there is an Objectivist critique of classical mathematical logic and set theory, what is the critique and what alternatives do Objectivists offer? 6. Probably the most common definition of 'finite' and 'infinite' among mathematicians these days is: x is finite if and only if x is equinumerous with a natural number. x is infinite if and only if x is not finite. (Note: The subject of the empty set had been raised in another thread. The empty set is finite, according the above definitions, since the empty set is a natural number.) Also, set theories such as Zermelo-Fraenkel set theory, have an axiom that there exists a set w such that 0 is a member of w, and if n is a member of w, then the union of n and the singleton of n is a member of w. What objections, if any, do Objectivists have to these formulations? 7. While constructivist and/or intuitionist mathematics might not be a perfect fit for Objectivism, these might be closer to Objectivism than mathematics set in classical mathematical logic. Though I know of Heyting arithmetic, I have not been able to find intuitionist axioms for other concerns, especially the real numbers or for set theory. What are some intuitionist systems for real numbers or for set theory and how do Objectivists regard these proposals? 8. Would non-standard analysis be preferred by Objectivists or does non-standard analysis carry assumptions that are more disagreeable to Objectivism than notions of infinity in standard analysis? 9. I've heard a little about mathematics that allows only finite sets (if I'm putting this correctly). How does such mathematics express the real numbers? How does such mathematics present its own meta-theory? Do such efforts offer alternatives for Objectivists? / It is not required that in this thread we address all of these questions, but these questions do give us plenty to talk about. And I hope that all participants of this thread are allowed to comment or even digress upon anything that falls in the general area of mathematics or other subjects, such as epistemology or philosophy, that contribute to our understanding of mathematics. As well, I hope participants will be allowed to discuss the particulars of certain matters of mathematics as these may arise in the course of the larger subject. Thanks in advance for whatever light anyone may shed upon this subject. |
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PhilE.Stein Mathematics, Logic, Set Theory, and Objectivism Jul 17 2005, 02:13 AM
Ed from OC One quick note: there is no such thing as an Objec... Jul 17 2005, 02:30 AM
PhilE.Stein Thank you, Ed from OC. You point is well taken and... Jul 17 2005, 02:53 AM
Stephen Speicher QUOTE(PhilE.Stein @ Jul 16 2005, 07:13 PM)Whi... Jul 17 2005, 03:11 AM
PhilE.Stein Thanks, Stephen Speicher for your reply.
Just to ... Jul 17 2005, 04:37 AM
Stephen Speicher QUOTE(PhilE.Stein @ Jul 16 2005, 09:37 PM)By ... Jul 17 2005, 04:18 PM
Capitalism Forever QUOTE(PhilE.Stein @ Jul 17 2005, 04:13 AM)x i... Jul 17 2005, 02:02 PM
PhilE.Stein Thanks, Free Capitalism, for your remarks.
QUOTE(... Jul 18 2005, 01:22 AM
OliverComputing Some time ago I had an insight which I think is in... Jul 18 2005, 01:53 AM
Burgess Laughlin QUOTE(PhilE.Stein @ Jul 17 2005, 06:22 PM)Per... Jul 18 2005, 03:24 AM
Capitalism Forever QUOTE(PhilE.Stein @ Jul 18 2005, 03:22 AM)As ... Jul 18 2005, 04:02 PM
PhilE.Stein Stephen Speicher, that's some interesting hist... Jul 18 2005, 02:50 AM
Stephen Speicher QUOTE(PhilE.Stein @ Jul 17 2005, 07:50 PM)By ... Jul 18 2005, 03:57 AM
PhilE.Stein OliverComputing, your interesting ruminations are ... Jul 18 2005, 03:01 AM
PhilE.Stein NOTE: In an earlier post I mistakenly responded as... Jul 18 2005, 06:01 AM
Burgess Laughlin QUOTE(PhilE.Stein @ Jul 17 2005, 11:01 PM)In ... Jul 18 2005, 11:59 AM
Tom Rexton QUOTE(PhilE.Stein @ Jul 17 2005, 11:01 PM)... Jul 18 2005, 07:38 PM
Capitalism Forever QUOTE(PhilE.Stein @ Jul 18 2005, 03:22 AM)Mor... Jul 18 2005, 04:03 PM
Stephen Speicher QUOTE(Capitalism Forever @ Jul 18 2005, 09:03... Jul 18 2005, 09:18 PM
Capitalism Forever QUOTE(Stephen Speicher @ Jul 18 2005, 11:18 P... Jul 19 2005, 11:14 AM
PhilE.Stein I don't take responsibility for a definition o... Jul 18 2005, 04:43 PM
PhilE.Stein Note: Above post is addressed to Burgess Lauglin, ... Jul 18 2005, 04:44 PM
PhilE.Stein Capitalism Forever,
Your formalization is a defin... Jul 18 2005, 06:57 PM
Stephen Speicher QUOTE(PhilE.Stein @ Jul 18 2005, 11:57 AM)The... Jul 18 2005, 09:33 PM

ewv QUOTE(Stephen Speicher @ Jul 18 2005, 04:33 P... Jul 23 2005, 04:32 AM

Stephen Speicher QUOTE(ewv @ Jul 22 2005, 09:32 PM)Can I ask w... Jul 23 2005, 04:39 AM
Capitalism Forever QUOTE(PhilE.Stein @ Jul 18 2005, 08:57 PM)The... Jul 19 2005, 11:10 AM
Fesapo Correct me if I'm wrong, but mathematics went ... Sep 15 2005, 09:32 PM
ewv QUOTECorrect me if I'm wrong, but mathematics ... Sep 15 2005, 11:15 PM
Fesapo QUOTE(ewv @ Sep 15 2005, 03:15 PM)Despite the... Sep 16 2005, 04:51 PM
ewv QUOTEFesapo: So, what I would like to know from Ob... Sep 16 2005, 03:41 AM
Fesapo QUOTE(ewv @ Sep 15 2005, 07:41 PM)The essence... Sep 16 2005, 05:01 PM
dondigitalia QUOTE(Fesapo @ Sep 16 2005, 12:01 PM)So an Ob... Sep 16 2005, 06:31 PM
ewv QUOTEFesapo: Now, Ayn Rand in ItOE cites Russell a... Sep 16 2005, 04:33 AM
Stephen Speicher One interesting quote by Ayn Rand in ITOE which re... Sep 16 2005, 05:07 AM
ewv I think she was aware of that notorious quote. As ... Sep 16 2005, 06:34 AM
ewv The reason Ayn Rand did not formulate a philosophy... Sep 17 2005, 06:40 AM
Fesapo QUOTEAs for what to do with set theory, in thinkin... Sep 19 2005, 05:14 PM
ewv QUOTEewv: As for what to do with set theory, in th... Sep 19 2005, 10:15 PM
Fesapo QUOTEYou do more than question it; you start with ... Sep 20 2005, 09:33 PM
ewv QUOTEFesapo: I went through college Calculus. I... Sep 21 2005, 01:50 AM
Fesapo QUOTE(ewv @ Sep 20 2005, 05:50 PM)If you have... Sep 21 2005, 03:39 PM
Ed from OC QUOTE(Fesapo @ Sep 21 2005, 07:39 AM)I have a... Sep 27 2005, 04:13 PM
Fesapo QUOTE(Ed from OC @ Sep 27 2005, 08:13 AM)I th... Sep 28 2005, 04:51 PM
ewv QUOTEQUOTE"Foundations" of science isn... Sep 21 2005, 06:41 PM
Fesapo QUOTE(ewv @ Sep 21 2005, 10:41 AM)You are in ... Sep 22 2005, 09:46 PM

ewv QUOTEsoftware engineeringWhat kind of algorithms d... Sep 22 2005, 10:15 PM

Fesapo QUOTE(ewv @ Sep 22 2005, 02:15 PM)What kind o... Sep 26 2005, 11:23 PM
Fesapo QUOTE(ewv @ Sep 21 2005, 10:41 AM)........ Oct 7 2005, 11:01 PM
ewv QUOTEFesapo: Well, I'm CTO of a not-at-present... Sep 27 2005, 02:30 AM
Fesapo QUOTE(ewv @ Sep 26 2005, 06:30 PM)QUOTE(Fesap... Sep 27 2005, 04:56 PM
ewv QUOTEFesapo: My suspicion is that my AP Calculus t... Sep 30 2005, 02:47 AM
Fesapo QUOTE(ewv @ Sep 29 2005, 06:47 PM)
QUOTE
What... Sep 30 2005, 05:42 PM
ewv QUOTEFesapo: Well the books arrived. But I don... Oct 8 2005, 03:27 AM
Fesapo QUOTE(ewv @ Oct 7 2005, 07:27 PM)QUOTE(Fesapo... Oct 11 2005, 08:57 PM
Fesapo QUOTE(ewv @ Oct 7 2005, 07:27 PM)Chapter X pp... Oct 11 2005, 10:29 PM
ewv QUOTEQUOTEewv: Chapter X pp.40-44. Be sure y... Oct 12 2005, 04:50 AM
Fesapo QUOTE(ewv @ Oct 11 2005, 08:50 PM)You are try... Oct 12 2005, 10:13 PM
ewv QUOTEFesapo: Is this a proper proof of Theorem 2? ... Oct 13 2005, 01:54 AM
Fesapo QUOTE(ewv @ Oct 12 2005, 05:54 PM)Be careful ... Oct 13 2005, 06:45 PM
ewv QUOTEFesapo: So it would be better to call it ... Oct 14 2005, 12:47 AM
Fesapo QUOTE(ewv @ Oct 13 2005, 04:47 PM)QUOTE(Fesap... Oct 18 2005, 07:06 PM
Fesapo QUOTE(Fesapo @ Oct 18 2005, 11:06 AM)
Oops. N... Oct 18 2005, 07:20 PM
Fesapo QUOTE(Fesapo @ Oct 18 2005, 11:20 AM)In Th. 4... Oct 18 2005, 11:12 PM
ewv QUOTEfesapo: "... this shows that (1) is not ... Oct 19 2005, 09:30 PM
ewv QUOTEfesapo: In Th. 4, I can't get the reason ... Oct 20 2005, 07:59 PM
Fesapo QUOTE(ewv @ Oct 20 2005, 11:59 AM)
To apply t... Nov 4 2005, 11:10 PM
ewv Your formulationQUOTEIf I have a function a(x) and... Nov 5 2005, 01:19 AM
Fesapo QUOTE(ewv @ Nov 4 2005, 05:19 PM)
It is so si... Nov 8 2005, 10:08 PM
ewv The recursive definition of addition does not requ... Nov 10 2005, 02:00 AM
Fesapo QUOTE(ewv @ Nov 9 2005, 06:00 PM)The recursiv... Nov 11 2005, 12:13 AM
Tom Rexton QUOTE(ewv @ Nov 9 2005, 07:00 PM)The recursiv... Nov 11 2005, 12:25 AM
Fesapo QUOTE(Tom Rexton @ Nov 10 2005, 04:25 PM)I... Nov 11 2005, 06:11 PM
Tom Rexton QUOTE(Fesapo @ Nov 11 2005, 11:11 AM)You are ... Nov 11 2005, 06:59 PM
Fesapo QUOTE(Tom Rexton @ Nov 11 2005, 10:59 AM)My c... Nov 11 2005, 07:26 PM
Tom Rexton QUOTE(Fesapo @ Nov 11 2005, 12:26 PM)Oh, I lo... Nov 11 2005, 08:28 PM
ewv QUOTEewv: The recursive definition of addition doe... Nov 18 2005, 03:41 AM
Fesapo QUOTE(ewv @ Nov 17 2005, 07:41 PM)x + y' ... Nov 21 2005, 09:47 PM
ewv QUOTEThomas Rexton: I've noted that the proofs... Nov 18 2005, 03:47 AM
ewv QUOTEThomas Rexton: My concern was that the ... Nov 18 2005, 05:21 AM
Tom Rexton QUOTE(ewv @ Nov 17 2005, 10:21 PM)
In... Nov 25 2005, 01:29 AM
ewv QUOTEQUOTEewv: x + y' = (x+y)', together w... Nov 28 2005, 02:23 AM
Fesapo QUOTE(ewv @ Nov 27 2005, 06:23 PM)It is an im... Jan 11 2006, 10:32 PM
ewv QUOTETom Rexton: I understand now. My professor cl... Nov 28 2005, 02:33 AM
ewv QUOTEFesapo: I'm still stuck on why it is vali... Feb 16 2006, 04:55 AM
Fesapo QUOTE(ewv @ Feb 15 2006, 08:55 PM)
You have t... Mar 11 2006, 01:45 AM
jcbaduk Hello everyone, I'm new to this forum. I hope... Jul 28 2007, 01:06 AM
jcbaduk Sorry, let me clarify. The reason why the points ... Jul 28 2007, 01:14 AM
ewv QUOTE(jcbaduk @ Jul 27 2007, 09:06 PM) He... Jul 28 2007, 03:00 AM
jcbaduk Thank you evw for the reply. It makes things a lo... Jul 29 2007, 02:40 AM
ewv QUOTE(jcbaduk @ Jul 28 2007, 10:40 PM) Th... Jul 30 2007, 03:59 AM
Paul's Here There seems to be an assumption being made about t... Jul 29 2007, 01:32 PM
ewv There is no such thing as an infinitely thin line,... Jul 29 2007, 06:24 PM
inventor QUOTE (Paul's Here @ Jul 29 2007, 09... Apr 15 2010, 09:08 AM![]() ![]() |
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