
Re: Is logic part of mathematics  or is mathematics part of logic?
Posted:
Jun 28, 2013 1:03 PM


You already said it
The answer to both questions is the not very profound "YES". Nearly always emphasis on one rather than the other  almost exclusion of one in deference to the other  is implicitly obvious from the context so the real question is: Why waste anymore time on it?
Wayne
At 09:13 PM 6/27/2013, GS Chandy wrote: >The questions in the subjectline have long >interested me most keenly during my decadeslong >efforts to develop practical means that could >help people at large design and improve existing >'systems' of various kinds'  'individual'; 'organisational; and 'societal'. > >Over the years, I've invested a fair amount of >time and effort contemplating and considering >the above questions. I must confess I do not >yet have a definitive answer by any means except >the not very profound insight that the answer >has to be 'YES' to both questions, i.e.: > >i) Logic (in a sense) "IS INCLUDED IN" math; and > >ii) Math (in a sense) "IS INCLUDED IN" logic. > >(Of course, both statements demand that the >transitive relationship "IS INCLUDED IN" would >need to be clearly specified. Also, the 'sense' >of the qualifier would need to be adequately clarified). > >Of course, the above statements imply that: > >a) Logic (in a sense) "INCLUDES" math; and >b) Math (in a sense) "INCLUDES" logic. > >[None of the above statements is 'universally' true). > >ALL of the above would become somewhat clearer >if we could in parallel use what I call >'structural graphics' [i.e. use 'prose + >structural graphics'' (p+sg)] to extend the 'pure prose' I am using here. > >(I note that 'p+sg' would include Venn diagram >illustrations of various prose statements  >these are most useful indeed. The greatest >particular benefit of p+sg is that it enables >people at all levels of knowledge and >understanding to communicate with considerable >clarity on the most complex and troubling >issues. It's generally a lack of clarity in >communications between different parties to any >discussion that leads to fruitless >argumentation, anger, litigation  even war!!! > >Unfortunately such needed facilities to use p+sg >effective do not currently exist  I hope to >have a website developed in a few months, at >which I shall be making such facilities available. > >All of the above would become considerably more >clear if we were to construct (and discuss) >structural representations of a few of our >'mutual mental models' when we discuss these >questions  representations of my mental models >integrated with the readers' mental >models. This requires a little learning (of >Warfield's approach to systems science) along >with a little 'unlearning' that our conventional >education systems have stuffed into our minds since we were children. > >In particular, the (transitive) relationship I >have found that most helps clarify (to the >inquiring mind) the structure of a system is: > >"CONTRIBUTES TO" (in various strengths: 'may'; >'would'; 'does'). We do need to use 'prose _ >structural graphics' (p+sg) for effective discussion of these issues. >[The question will, most likely, never be >clarified to the mind that does not construct an >adequate number of Interpretive Structural >Models (ISMs) using the "CONTRIBUTION" relationship]. > >The transitive relationship "PRECEDES"  much >beloved by managers, 'management experts' and >'conventional thinkers' is not very useful to >clarify 'system structure': in fact, using >'PRECEDENCE' as the primary system relationship >may well lead to major misunderstandings about >the nature of the systems under consideration. > >Probably the most profound insights on issues >related to the subject title have developed from: > >i) Bertrand Russell (and Alfred North Whitehead) in "Principia Mathematica"; >ii) Ludwig Wittgenstein in Tractatus >LogicoPhilosophicus, and the works of his students and followers; >iii) Benjamin Peirce (BP) particularly in "The >Science of Necessary Conclusions"; >iv) Charles Sanders Peirce (BP's son). > >[I must confess I have not by any means studied >the works of the abovenoted >mathematicians/philosophers as closely as I >probably should have to gain needed clarity on >the interrelationships between math and logic. > >John N. Warfield (a fair bit of whose work I >HAVE studied and applied in some depth and >fairly extensively) had developed a great many >of his logical, mathematical AND 'systems' >insights from the works of Charles Sanders >Peirce  and his works have led to what I claim >is probably THE most effective way to real >progress on the issues relating mathematics and >logic, specifically in matters relating to societal problems and issues. > >I am not certain I have all that is relevant to >the abovenoted questions. I provide in the >following a partial list of useful references >follows: it is highly eclectic, and eclectically arranged. > >If you need something better organised, you will >have to approach me when my OPMS book is >published  or you can check the book itself, >which will contained an organised list of references. > >(This is in NO PARTICULAR ORDER, items have just >been put down as they came to mind). The >simplest way now available of gaining insights >into the above questions would be by using the >'OPMS', described at No. 2 below  and to create usable models for yourself. > >(Note: "SEP"  'Stanford Encyclopedia of >Philosophy'  a most useful, largely free resource). > >1. John N. Warfield  website: >http://www.jnwarfield.com and the "John N. >Warfield Collection"  >http://ead.lib.virginia.edu/vivaxtf/view?docId=gmu/vifgm00008.xml;query=; > >2. 'One Page Management System' (OPMS)  a >simple, practical aid to problem solving and >decision making developed on the seminal >contributes to 'systems science' from John N. >Warfield. OPMS is briefly described at the >attachments to my post at the thread "Democracy: >how to achieve it"  http://mathforum.org/kb/thread.jspa?threadID=2419536 > >3. Charles Sanders Peirce  >http://www.peirce.org/ . Contains quite >extensive archive of his writings on a variety >of issues, including math and logic. Arisbe  >The Peirce Gateway  http://www.cspeirce.com/ >3a. Peirceâs Philosophy of Logic, Jay Zeman  >http://web.clas.ufl.edu/users/jzeman/csphiloflogic.htm > >4. Benjamin Peirce and "The Science of Necessary >Conclusions"  >http://www.lib.noaa.gov/noaainfo/heritage/coastandgeodeticsurvey/Peircechapter.pdf >"Linear Associative Algebra"  >http://ia600306.us.archive.org/4/items/linearassociati00peirgoog/linearassociati00peirgoog_djvu.txt >(Readable online in .txt format and this is the inadequate way I've read it. >  Benjamin Peirce in SEP  >http://plato.stanford.edu/entries/peircebenjamin/ > >5. Ludwig Wittgenstein  'Tractatus...'  >'Philosophy of Mathematics'  http://plato.stanford.edu/entries/wittgenstein/ >5a. The Cambridge Wittgenstein Archive  http://www.wittgencam.ac.uk/ > >6. A Crash Course in Arrow Logic, Yde Venema  >http://staff.science.uva.nl/~yde/papers/arrow.pdf > >7. Logic and Mathematics, Stephen G. Simpson  >http://www.math.psu.edu/simpson/papers/philmath/ > >8. "Creativity and Logic" >http://www.wetcanvas.com/forums/archive/index.php/t231129.html > >9. Jan Brouwer   >http://plato.stanford.edu/entries/brouwer/  see also "Intuitionism" in SEP >  "The Development of Intuitionistic Logic" >(in SEP)  http://plato.stanford.edu/entries/intuitionisticlogicdevelopment/ > >9a. Russell and Whitehead: "Principia >Mathematica"  promotes 'logicism': the view >that (some or all of) mathematics can be reduced >to (formal) logic  a view somewhat contrary to >the position from which OPMS is developed  >http://plato.stanford.edu/entries/principiamathematica/. > >10. Gaurav Tiwari  "A trip to mathematics"  >http://gauravtiwari.org/2011/10/25/atriptomathematicspartilogic/ >(A simple overview for beginners) > >11. Zermello's The Axiom of Choice; The Axiom >of Choice and Logic; Zorn's Lemma; Maximal >principles  SEP >http://plato.stanford.edu/entries/axiomchoice/ >(et seq)  the Axiom of Choice has several 'deep >connections' with the approach of the OPMS, not >all of which I've explored adequately. > >12. "Phenomenology"  Avi Sion  in The >Logician  >http://www.thelogician.net/2b_phenome_nology/2b_chapter_08.htm >(Based on an approach 'somewhat' different from >that of the OPMS  but not logically contrary to it). > >13. Lotfi A. Zadeh: "Fuzzy Sets and Fuzzy >Mathematics"; 'Soft Computing'  >https://en.wikipedia.org/wiki/Lotfi_A._Zadeh  >an approach to math philosophically quite >congruent to that of OPMS  very much against >the 'traditionalist view of math. > >14. "Logical Foundations of Fuzzy Mathematics"  >based on the 'fuzzy mathematics' approach >initiated by Lotfi A. Zadeh  >http://www.mathfuzzlog.org/images/4/4d/BehounekPhD.pdf >(the doctoral thesis of one of the students of >one of the followers of Lotfi A. Zadeh). > >GSC >P.S.: On rereading the above, I find that, on >each issue and question discussed, I have left >out much more than I have discussed. This >deficiency may be rectified (at least to an >extent) when I bring out the OPMS book). > > End of Forwarded Message
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