Date: Jun 28, 2013 12:13 AM
Author: GS Chandy
Subject: Is logic part of mathematics - or is mathematics part of logic?
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 ['in a sense'] 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. Peirce?s 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).
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).
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Message was edited by: GS Chandy
Message was edited by: GS Chandy