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?
At 09:13 PM 6/27/2013, GS Chandy wrote: >The questions in the subject-line have long >interested me most keenly during my decades-long >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 >Logico-Philosophicus, 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 above-noted >mathematicians/philosophers as closely as I >probably should have to gain needed clarity on >the inter-relationships 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 above-noted 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/peirce-benjamin/ > >5. Ludwig Wittgenstein - 'Tractatus...' - >'Philosophy of Mathematics' - http://plato.stanford.edu/entries/wittgenstein/ >5a. The Cambridge Wittgenstein Archive - http://www.wittgen-cam.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/t-231129.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/intuitionistic-logic-development/ > >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/principia-mathematica/. > >10. Gaurav Tiwari - "A trip to mathematics" - >http://gauravtiwari.org/2011/10/25/a-trip-to-mathematics-part-i-logic/ >(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/axiom-choice/ >(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/Behounek-PhD.pdf >(the doctoral thesis of one of the students of >one of the followers of Lotfi A. Zadeh). > >GSC >P.S.: On re-reading 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