my way to the "number reality" was convoluted, but in looking back maybe
two books could give you the central idea:
Lakoff and Nunez: Where does mathematics come from,
which argues that numbers arise from evolutionary considerations
(materialist in tenor, Platonia etc ruled out).
The next step then is to realize that modern physics gives us only
relational knowledge of the world
Ladyman Et Al. Every thing must go.
(for an excellent overview and discussion), and that matter is indeed
not "needed" (this was the crossing point into number-reality for me,
not the Maudlin thought experiment, because I am somewhat skeptical of
thought experiments (you never know if you've forgotten hidden
Computatations (that's the transition to pure number) then give a more
well defined picture than "all of mathematics", which gives no handle
whatever on white rabbits etc.
But then book one (Lakoff Et Al) fits again nicely into the bigger
picture, explaining how certain structures can evolve to see numbers
(one simply drops the materialist tenor).
John Mikes wrote:
> Günther and Bruno,
> am I sorry for not being ~30-40 years younger! I could start to study
> all those excellent books in diverse kinds of logic (what I missed) and
> could even have a chance to learn all those advancing ideas over the
> next 30 or so years...
> Makes me think of it: 30-40 years ago I WAS that young and did not start.
> I was busy making 20+ more practical polymer related patents without
> even thinking of the futility of "physical World" illusions. I just
> lived (in it)/(them).
> I am happy in my scientific agnosticim and would love to read something
> to bring me closer to the idea that 'numbers' consitute the world and
> not "are the mental products of us, eventuel travellers in this (one)
> Bruno used the word 'axiomatic', in my vocabulary an axiom is an
> unjustifiable belief (illusion?) necessary to maintain the validity of a
> theory - in this case the 'physical world'. Like: 2 + 2 = 4 -
> Br:"> AUDA is based on the self-reference logic of axiomatizable or
> > recursively enumerable theories, of machine...."
> Who is self-referencing, or even acknowledging self-reference? Or 'Self'
> for that matter? 'Recursively' I agree with, it is 'within'. Machine
> (limited capability) is 'us', so the 'enumerable theories' are OK.
> With such handicap in my thinking it is hard to fully follow the flow of
> the (A)UDA dicussions. I try.
> Best regards
> John M
> On Wed, Jan 28, 2009 at 12:01 PM, Günther Greindl
> <guenther.grei...@gmail.com <mailto:guenther.grei...@gmail.com>> wrote:
> Dear Bruno,
> thanks for the good references, I will integrate them on the resource
> page (or on a separate page).
> Some of these books I have already read (Boolos), others are on my list
> Smullyan's Forever Undecided is unfortunately out of print, but I am on
> the lookout for used copies ;-)
> Best Wishes and thanks for your time in thinking about the best
> P.S.: I agree with you that the best way to convey knowledge is
> discussion - I will keep bugging you with questions concerning COMP and
> UDA *grin*
> Bruno Marchal wrote:
> > Günther,
> > AUDA is based on the self-reference logic of axiomatizable or
> > recursively enumerable theories, of machine. Those machines or
> > must be rich enough. In practice this means their theorems or beliefs
> > are close for induction.This is the work of Gödel and followers,
> > Löb, who found a nice generalzation of Gödel's theorem and
> Solovay who
> > proves the arithmetical completeness of the logic he will call G
> and G'.
> > Here is the key paper:
> > Solovay, R. M. (1976). Provability Interpretation of Modal
> > Logic. /Israel Journal of Mathematics/, 25:287-304.
> > I follow Boolos 1979 and Smullyan "Forever Undecided" in calling such
> > system G and G*. G has got many names K4W, PrL, GL.
> > There are four excellent books on this subject:
> > Boolos, G. (1979). /The unprovability of consistency/. Cambridge
> > University Press, London.
> > This is the oldest book. Probably the best for AUDA. And (very
> > event) it has been reedited in paperback recently; I ordered it,
> and I
> > got it today (o frabjous day! Callooh! Callay! :). It contains a
> > chapter on the S4 intensional variant of G, and the theorem (in my
> > notation) that S4Grz = S4Grz*. The first person is the same from the
> > divine (true) view and the terrestrial (provable) view.
> > I have it now in three exemplars but two are wandering.
> > Boolos, G. (1993). /The Logic of Provability/. Cambridge University
> > Press, Cambridge.
> > This is the sequel, with the Russians' solutions to virtually all
> > problems in Boolos 1979. The main problem was the question of the
> > axiomatizability of the first-order extension of G and G* (which
> I note
> > sometimes qG and qG*). And the answers, completely detailed in
> > book, are as negative as they can possibly be. qG is PI_2
> complete, and
> > qG* is PI_1 complete *in* the Arithmetical Truth. The divine
> > intelligible of Peano Arithmetic is far more complex than Peano
> > Arithmetic's ONE, or God, in the arithmetical interpretation of
> > Smoryński, P. (1985). /Self-Reference and Modal Logic/. Springer
> > New York.
> > I have abandon this one sometimes ago, because of my eyes sight
> > but with spectacles I have been able to distinguish tobacco
> product from
> > indices in formula, and by many tokens, it could be very well
> suited for
> > AUDA. The reason is that it develops the theory in term of
> > function instead of assertions, showing directly the relation between
> > computability and SIGMA_1 provability. Nice intro from Hilbert's
> > to Gödel and Löb's theorem, and the Hilbert Bernays versus Löb
> > derivability conditions. It contains a chapter, a bit too much
> blazed in
> > the tone, on the algebraic approach to self-reference, which indeed
> > initiates originally the field in Italy (Roberto Magari).
> > It contains also chapter on the Rosser intensional variants.
> > Smullyan, R. (1987). /Forever Undecided/. Knopf, New York.
> > This is a recreative introduction to the modal logic G. I was
> used some
> > times ago in this list to refer to that book by FU, and I don't
> > to use some of Smullyan's trick to ease the way toward
> > It helps some, but can irritate others.
> > Note that Smullyan wrote *many* technical books around mathematical
> > self-reference, Gödel's theorems in many systems.
> > Modal logic is not so well known that such book can presuppose
> it, and
> > all those books introduce modal logic in a rather gentle way. But all
> > those books presuppose some familiarity with logic. Boolos Et Al.
> is OK.
> > It is difficult to choose among many good introduction to Logic.
> By some
> > aspect Epstein and Carnielly is very good too for our purpose.
> > Note that the original papers are readable (in this field). All
> this for
> > people who does not suffer from math anxiety which reminds me I
> have to
> > cure Kim soon or later. The seventh step requires some math. AUDA
> > requires to understand that those math are accessible to all
> > machine 'grasping" the induction principle, this is the work of Gödel
> > and Al.
> > I think the book by Rogers is also fundamental. Cutland's book is
> > but it omits the study of the Arithmetical Hierarchy (SIGMA_0,
> > PI_0, SIGMA_1, PI_1, SIGMA_2, PI_2, ...).
> > AUDA without math = Plotinus (or Ibn Arabi or any serious and
> > mystic). Roughly speaking.
> > I will think about a layman explanation of AUDA without math, and
> > different from UDA.
> > Best regards,
> > Bruno
> > On 25 Jan 2009, at 18:45, Günther Greindl wrote:
> >> Hi Bruno,
> >>>> Goldblatt, Mathematics of Modality
> >>> Note that it is advanced stuff for people familiarized with
> >>> mathematical logic (it presupposes Mendelson's book, or Boolos &
> >>> Jeffrey).
> >>> Two papers in that book are "part" of AUDA: the UDA explain to the
> >>> universal machine, and her opinion on the matter.
> >> I would like to add a "guide to AUDA" section on the resources page.
> >> Maybe you could specify the core references necessary for
> >> the AUDA (if you like and have the time)?
> >> Here a first suggestion of what I am thinking of:
> >> Boolos Et Al. Computability and Logic. 2002. 4th Edition
> >> Chellas. Modal Logic. 1980.
> >> Goldblatt, Semantic Analysis of Orthologic and
> >> Arithmetical Necessity, Provability and Intuitionistic Logic
> >> to be found in Goldblatt, Mathematics of Modality. 1993.
> >> What do you think?
> >> Best Wishes,
> >> Günther
> > http://iridia.ulb.ac.be/~marchal/
Department of Philosophy of Science
University of Vienna
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