1. Adventures in the Network of Theories Hajnal Andréka and István Németi
Network of theories is an efficient tool for organizing scientific knowledge.
We want to share our adventures in this network with you.
Hajnal Andréka and István Németi
Salzburg, September 7, 2016.
Network of theories

2. Language is determined by vocabulary = atomic sorts + atomic relations
We are in many-sorted first-order logic FOL with equality.
Theory = Language + Axioms
Network = Theories + Connections
When we design a theory, first we make it clear for ourselves what kind of objects we want to talk about, and what relations between them. In short, we design the abstraction level for our theory.
For simplicity, we assume that the language is countable, and we treat functions, partial functions and constants as special relations.
Sorts: universes of discourse, kinds of entities.
Formulas are generated from atomic sorts and relations via the logical connectives.
Axioms are sentences.
Language is determined by vocabulary =
atomic sorts + atomic relations
Axioms = some formulas of the language
Salzburg, September 7, 2016
Network of theories

3. Atomic goes to compound
Network = Theories + Connections
Connection = Interpretation
Interpretation =
Atomic goes to compound
T1=L1+Ax1
T2=L2+Ax2
But a time will come when we want to change the level of abstraction of our investigation. We may want to analyze our atomic objects and relations further.
In short, an interpretation between two theories is a function between their languages. It takes the atomic sorts and relations of the first theory to the compound sorts and relations of the other theory. L1 L2
Salzburg, September 7, 2016
Network of theories

4. What are compound sorts?
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.
Generic example is the compound sort of lines in a geometry that talks only about points and collinearity.
Pairs of points
Equivalence class of pairs that determine the same line
Pairs of distinct points
Salzburg, September 7, 2016
Network of theories

5.  (p,r,p’,r’) : pr and p’r’ and Coll(p,r,p’) and Coll(p,r,r’)
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
{ x : Epsilon(x,x) } = names for new entities
Epsilon(x,y) exactly when two names name/denote/code the same entity
T proves that Epsilon is symmetric and transitive. We may require that T proves that Epsilon is nonempty.
Sorts and necks
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
Salzburg, September 7, 2016
Network of theories

6. Conceptual completion Tc of T=L+Ax
Network = Theories + Connections
Interpretation = atomic to compound
Conceptual completion Tc of 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  “,…
Why completion?
New sorts are shorthands just as “union of sets” is a shorthand in set-theory where the elementhood relation is the only atomic relation.
Idea of translation: replace “married couple” with “pairs of people who are married”. In the case of geometry, replace “line” with “pair of distinct points”, “incident” with “collinear with the pair of points”, and equality of lines with the formula epsilon.
The idea is that we consider the new symbols as abbreviations and then the translation is just replacing an abbreviation with its “unpacked meaning”. Unpack the meaning of an abbreviation.
Ontology: It is meaningful to say that the ontological commitment of T is its compound, defined sorts, not only its atomic sorts. Thus, the ontological commitment of T is the ontological commitment of Tc in the old Quinean sense. Barrett-Halvorson, Molinini
There is a natural translation of the extended language to the original
by substituting defined terms with their definitions. Lc L tr
Salzburg, September 7, 2016
Network of theories

7. Compound sorts of T = sorts of Tc
Network = Theories + Connections
Interpretation = atomic to compound
Compound sorts of T = sorts of Tc
Compound relations of T = open formulas of Tc
Big and not Red is a compound relation composed of Big and Red.
Compound relations of a theory are open formulas of its conceptual completion.
For example, a compound relation in the theory of affine planes with atomic concept as points, a compound relation can talk about lines, too, as entities.
Salzburg, September 7, 2016
Network of theories

8. atomic goes to compound
Network = Theories + Connections
Interpretation = atomic to compound
T1=L1+Ax1
T2=L2+Ax2
Interpretation =
atomic goes to compound
F: atomic of L compound of L2
We use the perfect duality between syntax and semantics in FOL: we switch freely between talk about formulas and talk about models, without even warning. L1 L2
L2c F tr
Mod(Ax2)
F tr
Mod(Ax1)
Requirement: Ax2 proves translation of Ax1
Salzburg, September 7, 2016
Network of theories

9. CA(T2) CA(T1) Interpretation = Structure preserving mapping of
Network = Theories + Connections
Interpretation = atomic to compound
Interpretation =
Structure preserving mapping of
concepts of T1 into concepts of T2
CA(T1)
CA(T2) F
CA(T) is Lindenbaum-Tarski algebra of open formulas of completion of T. This algebra is explained in the next slide.
CA(Tc) is isomorphic to CA(Tcc) via a recursive isomorphism.
Interpretation is giving structure to the theory. E.g., assume that electric currents are flows of electrons. This creates predictions (certain concepts coincide under F), which we can check then.
Inverse: generalization, for analyzing structure of concepts. Everyday practice in modular thinking. Useful items get treated on their own. Structural thinking, finding structure.
Economizing the epistemic process, team-work. This is intensively done in math, we all stand on the shoulders of giants.
Scope of definability? Implicit definability, Beth-type theorem.
What do interpretations say? Why are they useful? What can interpretations be used for?
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
Salzburg, September 7, 2016
Network of theories

10. Formula-algebra of a many-sorted theory
Network = Theories + Connections
Interpretation = atomic to compound
Formula-algebra of a many-sorted theory
A two-sorted algebra.
Set of open formulas
factorized by the theory
Set of sorts s
/  i
ij 
The formula-algebra CA(T) is the natural algebra of open formulas of a many-sorted theory T = L+Ax. It is a two-sorted algebra. The sorts-relations duality appears in it explicitly. One sort S contains the sort-symbols of the language L of the theory. The other sort Fm contains the equivalence blocks of the set of open formulas of the language factorized by the equivalence-relation “provable by the theory Ax”. The operations of the formula-algebra are built from the logical connectives of many-sorted FOL. “And” maps Fm x Fm to Fm, “Neg” maps Fm to Fm, =ij maps S to Fm and Ei maps S x Fm to Fm. The unary function =ij assignes to a sort s the [equivalence block of] the formula vis=vjs vhere vis is the i-th variable of sort s . The binary function Ei assignes to a sort s and (the block of) a formula phi the (block of ) the formula phi quantified by the variable vis.
This formula-algebra is a structure of concepts of the theory. (The concepts are the elements of S and Fm, and the structure is formed by the operations.)
i(s, / )  (vis)/
ij (s)  (vis vjs )/
Salzburg, September 7, 2016
Network of theories

11. T1 is generalized definitionally equivalent to T2
Network = Theories + Connections
Interpretation = atomic to compound
T1 is generalized definitionally equivalent to T2
(T1 is conceptually equivalent to T2)
CA(T1)
CA(T2)
Treating dualities: e.g., calculation-oriented theories versus conceptual-oriented theories. Or: few atomic notions and many axioms versus more atomic notions and less number of axioms.
Excess structure? Conceptual equivalence. Measuring, comparing theories. Changing focus.
This notion is an extension of the old “definitional equivalence of theories” of Tarski. In fact, definability theory was begun by Reichenbach in connection with special relativity in the 1920’s already. M-equivalence, Morita-equivalence, definitional equivalence, Makkai-equivalence, Madarasz-equivalence
This much about the theory of the network of theories. We turn now to applications. Experience: workable (because of its modular character) in axiomatizing parts of physics. F
F is an isomorphism between the two structures
J. Madarász: PhD Dissertation 2002.
Salzburg, September 7, 2016
Network of theories

12. a tangible mathematical model
for relationist view of Mach and Leibniz:
rich theory
poor theory
Interpretation
We will connect two theories: a rich theory and a poor theory. The poor theory will be experiment-oriented (operational) theory.
One theory is rich, while the second one is poor, economical, has meager resources. The interpretation will show us how the rich theory emerges from the poor one.
An interpretation technically means a bunch of explicit definitions, one for each primitive notion of the rich language in terms of the poor one. Such an explicit definition can be considered to be an experiment to be made in the poor world to establish whether a notion of the rich theory holds in a particular situation or not. This way we give experimental, operational semantics for the rich theory.
The poor theory will be a theory of signals (or photons), a theory of communication, while the rich theory will be a spacetime theory, special relativity . The interpretation we are about to show you provides an experimental, operational semantics for special relativity . In the other direction, this interpretation shows us how a society living in the sparesome word with signaling as the only tool, can discover time and space as useful notions for survival. This society makes experiments, devises and discards concepts, sets up inductive hypotheses, revises its beliefs, and so on. In short: it conducts a cognitive process of knowledge acquisition.
In our story there will be a world without time and space, hence no motion no change, and then time and space will emerge as a result of a cognitive process. We will define time and space (over timeless objects) as derived notions.
Reduction and Emergence
The important thing is that the two theories have different languages, different primitive concepts.
Case study in the sense that we choose special relativity as an example only, the same has been done for general relativistic spacetimes, too.
The interpretation is a connection. It represents dynamics, of time and space arising via a cognitive process.
SpecRel
Spacetime theory
SigTh
Experiment-oriented
emergence
Salzburg, September 7, 2016
Network of theories

13. Signaling Theory SigTh
Language:
Sorts: experimenters, signals
Relations: send, receive
Experimenters
Signals
Send
Receive
The timeless theory has a language where we have two sorts of entities, experimenters (or observers) and signals (or photons), and we have two binary relations: an experimenter can send out a signal and an experimenter can receive a signal. So this theory is based on two binary relations only, between experimenters and signals.
James Ax’s paper “The elementary foundations of spacetime”,
Foundations of Physics 8 (1978),
Salzburg, September 7, 2016
Network of theories

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
That was the language. The axioms of Signaling Theory will be all the formulas valid in its intended model. The intended model is the following.
We are in four-dimensional Euclidean space R to the four, R is the set of real numbers. Experimenters are represented as straight lines of slope less than 1 (i.e., they are more vertical than horizontal), and signals are represented as finite nonzero segments of lines with slope 1. An experimenter sends out a signal if the starting point of the signal lies on the line representing the experimenter, and an experimenter receives a signal if the end point of the signal lies on the line representing the experimenter.
This is the intended model, and the axioms of SigTh are the formulas valid in this model (in the FO language consisting of Sends and Receives).
A few words about Signaling Theory. A line in four-dimensional space represents motion in three-dimensional space. Experimenters represented with vertical lines are motionless and experimenters represented with slanted lines are in motion. Our experimenters, however, have no senses for motion, or time, or space. All what they can perceive of their world is sending out and receiving signals. In this model, from the point of view of sending and receiving signals, all the experimenters are alike: any one of them can be taken to any other by an automorphism. Hence, no matter how clever our experimenters are, they cannot distinguish the motionless among them. This is why they will discover special relativity as a spacetime for their world, and not Newtonian absolute time. By the way, all the signals are like each other in this model, too. In this respect our Signaling Theory resembles Incidence Geometry, where all the lines and points are alike. However, in SigTh there is no duality between experimenters and signals, as in geometry, signals behave differently from experimenters.*** s e’ e
Receives(e’,s)
Sends(e,s)
Salzburg, September 7, 2016
Network of theories

15. A visual representation of this intended model
A visual representation of this intended model. Here are some motionless experimenters, they can send out and receive signals.
Salzburg, September 7, 2016
Network of theories

16. We have running experimenters, too, they also can send out and receive signals.***
There is a more dynamic, more complex view of this: Andrew, Laurenz, Peter, … are members of the human society who would like to learn the world around. They make and plan experiments, and they share the results of these experiments. These thus get stored in a collective consciousness (data-base). They devise new concepts, simplify, ramify, test and assign likelihood to regularities – they lead a cognitive process. If their world is such, they can create a “Theory of Everything”, a finite set of regularities/rules according to which they can predict the outcomes of all the possible experiments.
Salzburg, September 7, 2016
Network of theories

17. However, our experimenters have no clocks and no meter rods
However, our experimenters have no clocks and no meter rods. Yet, eventually they will discover time and motion.
No clocks
No meter rods
Salzburg, September 7, 2016
Network of theories

18. Special Relativity SpecRel
Language:
Sorts: observers, photons, quantities
Relations: worldview, +, , 
Real numbers
Coordinatizing
World-view relation
W(o,b,t,x,y,z)
+,, 
Observers
Photons o b t y z x
Let’s turn to our rich theory. This language has three universes or sorts. These are the sort of observers, the sort of photons and the sort of numbers. These three sorts are connected by a 6-place relation called worldview relation, this represents coordinatization. The worldview relation tells us which observer sees which bodies at what coordinates. One can view the language as consisting of a set of coordinate systems, a coordinate system for each observer. This language is quite convenient for talking about motion, time, space, velocity, acceleration, etc. ***
SigTh is a very disciplined, economical theory in which we cannot say whatever intuitive idea comes to our minds. It is strictly regulated what can be said in this theory. On the other hand, the theory SpecRel is designed to help expressing our intuitions. We may talk about coordinate systems (time-coordinate, space-coordinate etc). It is designed for helping imagination. So, in some sense the two theories aim for two different goals/ambitions. Despite of this difference, our theorem will claim that they are definitionally equivalent theories. This means that in some sense they are different wrappings for the same theory.
Salzburg, September 7, 2016
Network of theories

19. DRAWING THE WORLD-VIEW RELATION
W(o, b,t x y z)  photon “b” is present at coordinates “t x y z” for observer “o” t x y
b (worldline)
Axiom 0: Every inertial observer coordinates his world by 4 coordinates, 1 time coord. and 3 space coord.
This axiom is built into the language
worldview o
Salzburg, September 7, 2016
Network of theories

20. Basically, the language of SpecRel talks about a bunch of coordinate systems, one for each observer. It does not talk directly about the house, only about the several views of this house.
Salzburg, September 7, 2016
Network of theories

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
We presented the language, next come the axioms. We present SpecRel by listing its 3 axioms. 1: All observers see photons move with the same speed in all directions. (Hence there is such a notion as the speed of light.) 2:All observers see the same events (an event is meeting of two or more photons). 3:The numbers are the usual.
By this we presented SpecRel. We note that all the characteristic predictions of special relativity can be formulated and are proved from these three axioms. E.g., moving clocks slow down.
Having introduced our two theories, now we show an interpretation between them.***
Salzburg, September 7, 2016
Network of theories

22. SpecRel SigTh Interpretation
Poor theory, no time no space no quantities
Rich theory, of time and space, we have quantities
One can imagine that observers will correspond to experimenters and photons to signals, somehow. But how shall we get quantities? We concentrate on this step.
Observers  ?
Photons  ?
Quantities  ?
+, ,   ?
Worldview  ?
Sorts composed of experimenters and signals
Relations composed of
send, receive
Salzburg, September 7, 2016
Network of theories

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
event e
send Se qe
experimenters
signals
events
Coll s1 s2 s3 s2 s1 s3 e
Coll
We can express that the beginning of s1 is the same as that of s2 by saying that “each experimenter who sends s1 also sends s2”.
We have an affine geometry over a field. So the field can be recovered from the structure of lines via the Hilbert coordinatization method. Let’s turn to this step.
First, having selected two distinct points on a line, we can define – by using the structure of the lines – an addition and multiplication that will be isomorphic to our original field. But, which one to choose of this as our quantity sort? All lines are alike, no one of them is definable by itself.
Collinear: three events are collinear iff there is an experimenter
participating in all three of them.
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Network of theories

24. We do not know which one to choose, so we take all of them!
What will quantities (elements of the field) be?
Coordinatizing field: by Hilbert’s coordinatization method b b’ a’ a x x’
quantities
triples
triples such that Coll(a,b,x)
and ab
iso(abx, a’b’x’)
We do not know which one to choose, so we take all of them!
This is completely analogous to the generic example. The names of lines were pairs of distinct points, here, the names of quantities (real numbers) are triples of collinear points. Two such triplets denote the same entity if there is a suitable isomorphism between the two lines they specify.
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’ .
Salzburg, September 7, 2016
Network of theories

25. CA(SpecRel) CA(SigTh) CA(SpecRel) CA(SigTh) CA(SpecRel+) CA(SigTh)F
CA(SpecRel)
CA(SigTh) F
CA(SpecRel+)
CA(SigTh) F
SpecRel+ is finite schema axiomatized, so SigTh can also be finite schema axiomatized. (Axiom system for SigTh can be recovered from that of SpecRel+ and the interpretation.) [Schema because we took the field of real numbers. If we took, instead, Euclidean fields, we get finite axiomatizations.] Few atomic notions and many axioms versus more atomic notions and less number of axioms. This interpretation can be considered as refining the notion of an observer (or coordinate system).
SigTh is conceptually equivalent with light-like separability on four-tuples of reals. Minkowski-distance is not expressible in SigTh, but Minkowski-equidistance is. SigTh is conceptually equivalent to Minkowski-equidistance on four-tuples of reals.
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.
Salzburg, September 7, 2016
Network of theories

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

27. Electrodynamics MaxWel
Interpreting SpecRel in MaxWel
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Network of theories

28. Electrodynamics MaxWel
Language:
Sorts: observers, phenomena, test-charges, quantities
Relations: world-view, +, , 
Observers o b
test-charges
The language for MaxWel is almost the same as for SpecRel. The difference is that we have one new sort, the physical phenomena, and we call photons test-charges. t x y z p
Coordinatizing
World-view relation
W(o,p,b,t,x,y,z)
phenomena
Real numbers
+,, 
Salzburg, September 7, 2016 .
Network of theories

29. DEFINING THE ELECTROMAGNETIC FIELD E,M
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)
Acceleration is equivalent in this context with force (we assume that the test-charges have unit masses).
M can be defined either so that the Lorentz force law holds and then proving that this makes M unique, or else we can “compute” M from three independent velocities and then state the Lfl for this M.
From E,M we can define charge density and current . t x y
Worldline of test-charge p
Charge at txyz = div E
Current at txyz = rot M – dE/dt
o,F
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Network of theories

30. MaxWel axioms SpecRel MaxWel
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 …
The main axioms of MaxWel are Maxwell’s equations and the Lorentz-force Law. The rest are auxiliary axioms, they will be streamlined as we will have more experience with this ax system.
Idea of interpretation: state so many axioms that enables to prove that spherically symmetric waves move with speed of light. Then use the fact that SpecRel can be interpreted in light-like separability.
In more detail, a name of a photon is o,F,d where o is an observer, F is spherically symmetric with a wave-front, and d is a spatial direction. oFd and o’F’d’ name the same photon if the worldview-trafo between o and o’ takes d to d’ and F=F’.
This way, photons will have colors (wavelength), energy, but we still do not take into account quantum-mechanical aspects.
SpecRel
MaxWel
Interpretation
Salzburg, September 7, 2016
Network of theories

31. Thank you for your attention!
Famous physicist Richard Feynman said in his Nobel Prize lecture that it is extremely important to have many different equivalent theories for the same known portion of physical reality, because these different theories will suggest different ideas when moving towards the unknown. ***
Salzburg, September 7, 2016
Network of theories