Adventures in the Network of Theories: Salzburg Insights

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Delve into the world of many-sorted first-order logic with equality, exploring the intricate connections between theories, languages, axioms, and interpretations. Discover how compound sorts and relations play a crucial role in defining concepts like married couples and lines in this thought-provoking discourse from Salzburg, September 7, 2016.

  • Theories
  • Logic
  • Interpretation
  • Salzburg
  • Language
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  1. Adventures in the Network of Theories Hajnal Andr ka and Istv n N meti Page: 1 Network of theories Salzburg, September 7, 2016.

  2. We are in many-sorted first-order logic FOL with equality. Network = Theories + Connections Theory = Language + Axioms Language is determined by vocabulary = atomic sorts + atomic relations Axioms = some formulas of the language Page: 2 Salzburg, September 7, 2016 Network of theories

  3. Network = Theories + Connections Connection = Interpretation Interpretation = Atomic goes to compound T1=L1+Ax1 L1 T2=L2+Ax2 L2 Page: 3 Network of theories Salzburg, September 7, 2016

  4. Network = Theories + Connections Interpretation = atomic to compound What are compound sorts? Married couples are composed of people and to marry . Lines are coded as pairs of distinct points by equating those pairs that determine the same line. Pairs of points Equivalence class of pairs that determine the same line Pairs of distinct points Page: 4 Salzburg, September 7, 2016 Network of theories

  5. Network = Theories + Connections Interpretation = atomic to compound What are compound sorts Lines are coded as pairs of distinct points by equating those pairs that determine the same line. (p,r,p ,r ) : p r and p r and Coll(p,r,p ) and Coll(p,r,r ) S new sort for lines l q new function for quotient forming neck q (p,r)=l Equivalence class of pairs that name the same line Pairs of points p,r Pairs of distinct points = names for lines Page: 5 Salzburg, September 7, 2016 Network of theories

  6. Network = Theories + Connections Interpretation = atomic to compound Conceptual completion Tcof T=L+Ax We add the compound sorts S and their necks q to the language and we add their descriptions (i.e., definitions) to the axioms. Lc= L + S , q , for equivalences of T Axc= Ax + q is factoring by , There is a natural translation of the extended language to the original by substituting defined terms with their definitions. tr L Lc Page: 6 Network of theories Salzburg, September 7, 2016

  7. Network = Theories + Connections Interpretation = atomic to compound Compound sorts of T = sorts of Tc Compound relations of T = open formulas of Tc Page: 7 Salzburg, September 7, 2016 Network of theories

  8. Network = Theories + Connections Interpretation = atomic to compound T1=L1+Ax1 T2=L2+Ax2 Interpretation = atomic goes to compound F: atomic of L1 compound of L2 tr F L1 L2c L2 F tr Mod(Ax1) Mod(Ax2) Requirement: Ax2 proves translation of Ax1 Page: 8 Salzburg, September 7, 2016 Network of theories

  9. Network = Theories + Connections Interpretation = atomic to compound Interpretation = Structure preserving mapping of concepts of T1 into concepts of T2 CA(T2) F CA(T1) CA(T) = concept algebra of T = natural algebra of open formulas of Tc =Lindenbaum-Tarski algebra of Tc F is a homomorphism between the two structures Page: 9 Salzburg, September 7, 2016 Network of theories

  10. Network = Theories + Connections Interpretation = atomic to compound Formula-algebra of a many-sorted theory A two-sorted algebra. i / s Set of open formulas factorized by the theory = =ij Set of sorts i(s, / ) = = ( vis )/ = =ij(s) = = (vis= vjs)/ Page: 10 Salzburg, September 7, 2016 Network of theories

  11. Network = Theories + Connections Interpretation = atomic to compound T1 is generalized definitionally equivalent to T2 (T1 is conceptually equivalent to T2) F CA(T1) CA(T2) F is an isomorphism between the two structures J. Madar sz: PhD Dissertation 2002. Page: 11 Salzburg, September 7, 2016 Network of theories

  12. a tangible mathematical model for relationist view of Mach and Leibniz: Interpretation rich theory SpecRel Spacetime theory poor theory SigTh Experiment-oriented emergence Network of theories Salzburg, September 7, 2016 Page:12

  13. Signaling Theory SigTh Language: Sorts: experimenters, signals Relations: send, receive Send Experimenters Signals Receive James Ax s paper The elementary foundations of spacetime , Foundations of Physics 8 (1978), 507-546. Network of theories Salzburg, September 7, 2016 Page:13

  14. SigTh: Axioms all formulas valid in the intended model: Experimenters: straight lines of slope < 1 in R4 Signals: finite segments of lines of slope 1 in R4 Sends: starting point of segment lies on line Receives: end point of segment lies on line Receives(e ,s) s Sends(e,s) e e Salzburg, September 7, 2016 Network of theories Page:14

  15. Network of theories Salzburg, September 7, 2016 Page:15

  16. Salzburg, September 7, 2016 Network of theories Page:16

  17. No clocks No meter rods Salzburg, September 7, 2016 Network of theories Page:17

  18. Special Relativity SpecRel Language: Sorts: observers, photons, quantities Relations: worldview, +, , o b Observers Photons yz x t Coordinatizing World-view relation W(o,b,t,x,y,z) Real numbers +, , Network of theories Salzburg, September 7, 2016 Page:17

  19. DRAWING THE WORLD-VIEW RELATION 4 W Ob Ph Q W(o, b,t x y z) photon b is present at coordinates t x y z for observer o t b (worldline) o x y Network of theories Page:19 Salzburg, September 7, 2016

  20. Network of theories Page:20 Salzburg, September 7, 2016

  21. SpecRel axioms Photon Axiom: the world-lines of photons are exactly the straight lines of slope 1, in each worldview Event Axiom: all observers coordinatize the same physical reality Number Axiom: the structure of our quantities forms an ordered field Network of theories Salzburg, September 7, 2016 Page:21

  22. Interpretation SpecRel Rich theory, of time and space, we have quantities SigTh Poor theory, no time no space no quantities Observers Photons Quantities ? ? ? Sorts composed of experimenters and signals Relations composed of send, receive +, , Worldview ? ? Salzburg, September 7, 2016 Network of theories Page:22

  23. Idea for interpreting quantities First we define a geometry: Points = events = signals factorized under a relation expressing that they have the same beginning s1 s2 Se Coll s3 events send event qe e s1 s2 s1 s3 s2 e experimenters Coll signals Collinear: three events are collinear iff there is an experimenter participating in all three of them. Network of theories Salzburg, September 7, 2016 Page:23

  24. What will quantities (elements of the field) be? Coordinatizing field: by Hilbert s coordinatization method x quantities a x triples a b triples such that Coll(a,b,x) and a b b iso(abx, a b x ) Equivalence classes of triples a,b,x such that Coll(a,b,x) and a b under the equivalence relation iso(abx, a b x ) which expresses that the natural isomorphism taking ab to a b takes x to x . Page:24 Salzburg, September 7, 2016 Network of theories

  25. CA(SpecRel+) CA(SigTh) CA(SigTh) CA(SigTh) F F F CA(SpecRel) CA(SpecRel) The interpretation F we have constructed is onto but not one-to-one. It is a conceptual equivalence between SpecRel+ and SigTh. SigTh is conceptually equivalent with Events, lightlike-separability , or with Minkowskian equidistance. Network of theories Salzburg, September 7, 2016 Page: 25

  26. Are not there too many conceptually equivalent theories? The theory of fields, special relativity, Newtonian mechanics are all conceptually nonequivalent theories. Salzburg, September 7, 2016 Network of theories Page:26

  27. Electrodynamics MaxWel Interpreting SpecRel in MaxWel Network of theories Salzburg, September 7, 2016 Page:25

  28. Electrodynamics MaxWel Language: Sorts: observers, phenomena, test-charges, quantities Relations: world-view, +, , o b Observers test-charges x yz t p Coordinatizing World-view relation W(o,p,b,t,x,y,z) Real numbers +, , phenomena Salzburg, September 7, 2016 . Network of theories Page:28

  29. Language of MaxWel DEFINING THE ELECTROMAGNETIC FIELD E,M Everything is done in a world-view of an observer o wrt a phenomenon F E(txyz) = acceleration of test-charge at rest, at txyz M(txyz) is obtained from accelerations of three test-charges at motion, at txyz Lorentz-force Law: acc(p,txyz) = E(txyz) + vel(p,txyz) M(txyz) t Charge at txyz = div E Current at txyz = rot M dE/dt Worldline of test-charge p o,F x y Salzburg, September 7, 2016 Network of theories Page:29

  30. MaxWel axioms Maxwell s equations for E,M defined previously Lorentz-force Law: acc(p,txyz) = E(txyz) + vel(p,txyz) M(txyz) --Accelerations of test-particles exist and are determined by their velocities --M as needed in the Lorentz-force Law exists --The quantities form a field, the usual axioms needed for dealing with analysis --Event axiom --World-view transformations between observers are independent of phenomena and linear --If a phenomenon F exists, then its translations in a coordinate system exist, too. --The domain of a phenomenon is an open subset Interpretation SpecRel MaxWel Network of theories Page:30 Salzburg, September 7, 2016

  31. Thank you for your attention! Network of theories Page:31 Salzburg, September 7, 2016

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