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Conceptual structure of spacetimes, and category of concept algebras

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Abstract

We explore the first-order logic conceptual structure of special relativistic spacetime. Namely, we describe the algebra of explicitly definable relations of Minkowski-spacetime, and we draw conclusions such as “the concept of lightlike-separability can be defined from that of timelike-separability by using 4 variables but not by using three variables”, or “no non-trivial equivalence relation can be defined in Minkowski-spacetime”, or “there are no interpretations between the classical (Newtonian) and the relativistic spacetimes, in either direction”. After this, we generalize the notion of a concept algebra from first-order logic to any logic. A duality between algebra and category theory emerges here quite nicely. In particular, category theoretic properties of the category of all concept algebras shed light on definability properties of the logic. For example, “all implicitly definable concepts are explicitly definable (Beth definability property) if and only if epimorphisms are surjective in the category of concept algebras”, or “all existence-requiringly implicitly definable concepts are explicitly definable (weak Beth definability property) if and only if there is no proper epi-reflective subcategory of the category of concept algebras that contain the so-called full concept algebras”. Connections with category theoretic injectivity logic also show up.
Conceptual structure of spacetimes
and
Category of concept algebras
Hajnal Andr´eka, Judit Madar´asz, Istv´an N´emeti, Gergely Sz´ekely
MTA R´enyi Institute
Categorical Philosophy of Science, Munich, April 26, 2019.
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Plan of the talk
1The concept algebra of a concrete physical theory: Special
relativistic spacetime.
2The category of concept algebras of a(n arbitrary) language:
Universal algebraic logic.
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PART I
Concept algebra of a concrete physical theory:
special relativistic spacetime
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What is special relativistic spacetime SR?
Definition (Relativistic Spacetime SR)
SR is the system of timelike straight lines:
SR =R4,colt
colt(p,q,r)p,q,rare on a timelike straight line.
We will show that from timelike collinearity one can define the
full-fledged scale-invariant Minkowski spacetime: lightlike
connectedness, Minkowski-equidistance, Minkowski-orthogonality,
etc.
What is the concept algebra of SR?
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The concept algebra of SR
Definition (Concept)
Aconcept in SR is the extension of any open formula.
If ϕ(x1,...,xn) is a formula in the language of SR with free
variables x1,. . . , xn, its extension in SR is
ϕ(x1,...,xn)SR ={ha1,...,ani:SR |=ϕ(a1,...,an)}.
Definition (Concept Algebra)
The concept algebra of SR is the natural algebra of these
concepts, where the operations are defined by the connectives of
our language:
CA(SR) = DnϕSR :ϕis in the language of SRo,,¬,xnEnN,
where
ϕSR ψSR = (ϕψ)SR,¬ϕSR = (¬ϕ)SR ,xnϕSR = (xnϕ)SR.
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You get two in one
Definition (Classical non-Relativistic Spacetime NT)
NT is the system of non-horisontal straight lines:
NT =R4,col
col(p,q,r)p,q,rare on a slanted straight line.
Definition (Relativistic Spacetime SR)
SR is the system of timelike straight lines:
SR =R4,colt
colt(p,q,r)p,q,rare on a timelike straight line.
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Summing up:
Newton spacetime
NT =R4,col
Einstein spacetime
SR =R4,colt
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Newton spacetime
NT =R4,col
The grey parts form an
equivalence relation
Einstein spacetime
SR =R4,colt
Transitive closure of the gray
parts is everything
Theorem
No nontrivial equivalence relation can be defined in SR.
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Theorem
No nontrivial equivalence relation can be defined in SR.
Proof. First show that any two timelike connected pairs of events
can be taken to each other by an automorphism of SR.
Lorenz transformations are
automorphisms of SR.
Key players in relativity theory.
Einstein spacetime
SR =R4,colt
Do the same for spacelike and lightlike connected pairs of events.
Then show that the transitive closure of each of these relations
have one block. Q.E.D.
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Newton spacetime
NT =R4,col
Einstein spacetime
SR =R4,colt
Theorem (Corollary)
NT cannot be interpreted in SR.
Interpretations are homomorphisms between the concept algebras.
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Newton spacetime
NT =R4,col
Einstein spacetime
SR =R4,colt
Theorem
SR cannot be interpreted in NT, either.
Reason: SR is conceptually richer, more relations can be defined
in SR than in NT (even when NT is enriched with more
structure).
Let’s see.
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Structure of binary relations
Newton spacetime
NT =R4,col
Einstein spacetime
SR =R4,colt
We get the same picture if we use a stronger language, e.g., SOL.
Except that we have concepts concerning subsets of R4, too!
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Theorem
Lightlike connectedness can be defined from timelike
connectedness in SR by using 4 variables.
Proof.
Einstein spacetime
SR =R4,colt
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Theorem
Lightlike connectedness cannot be defined from timelike
connectedness in SR by using only 3 variables.
Proof.
In the relation algebra of the binary definable relations, timelike
connectedness does not generate lightlike connectedness. By a
theorem from algebraic logic, this implies that lightlike
connectedness cannot be defined with a FOL-formula using only 3
variables.
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Newton spacetime
NT =R4,col
Einstein spacetime
SR =R4,colt
these are all definitionally equivalent
Theorem
NT is not definitionally equivalent to any structure that has only
binary relations.
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Structure of ternary relations
Newton spacetime
NT =R4,col
Eucl circles cannot be defined.
Einstein spacetime
SR =R4,colt
Mink circles can be defined
Both:
Col can be defined in the grey part.
Have infinitely many atoms.
For reals:
Atomic, we know the atoms.
There are non-trivial subalgebras.
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Converse can be defined with 3 variables
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Converse cannot be defined with 2 variables
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Some properties of converse cannot be proved with 3 variables
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Concept algebras have geometric aspects.
Concept algebras have algebraic aspects.
Concept algebras have logical aspects.
They have categorical aspects, too!
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PART II
Duality between algebra and category theory
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Universal Algebraic Logic
We depart from first-order logic: we deal with any logic,
second-order logic, many-sorted logic, modal logics,. . .
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Category of concept algebras
Concept algebras for an arbitrary language in the framework of
general language theory
Definition (Algebraizable language)
Algebraizable language:L=hF,M,mngi
(1.) F=W(P,Cn) is a context-free language.
(2.) Compositionality: the meaning of a compound term depends
only on the meanings of the compounds.
Examples: FOL, SOL, modal logic, propositional logic, . . .
Non-examples: equational logic, injectivity logic, . ..
These two conditions are exactly what are needed for forming
concept algebras!
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Algebraic version of L
F=hW(P,Cn),cicCn is the word-algebra c(w1,w2) = cw1w2.
Definition (concept algebra)
CA(M) = hCAM,cicCn
is the concept algebra of M, where
CAM={mng(ϕ, M) : ϕF}
the set of concepts (meanings) of Mand
c(mng(ϕ, M),mng(ψ, M)) = mng(cϕψ, M).
Forgetting P! Category theoretic logic
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Definition (class of concept algebras)
Algm(L) = {CA(M) : MM}, the class of concept algebras.
Definition (class of Lindenbaum-Tarski algebras)
Alg(L) = I{F/K:KM},where
ϕKψmng(ϕ, M) = mng(,M) for all MK,
the class of Lindenbaum-Tarski algebras, up to isomorphism.
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Introducing truth: L=hF,M,mng,|=i, language becomes logic
Definition
(3.) We can “code” mng(ϕ) = mng(ψ) with formulas, by using
derived connectives ,>as
M|=ϕM|=ϕ↔ >, and
M|=ϕψmng(ϕ, M) = mng(ψ, M).
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Algebraic logic is bridge between Logic and Algebra:
Logic LAlgebra Alg(L)
formulas equations
Taut(L) Eq(Alg(L))
Lis complete Alg(L) is axiomatizable by . . .
Lis compact Alg(L) is closed under ultraproducts
Definability properties of Lcategory theoretic properties of Alg(L)
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Definability theory of general languages and Category of
concept algebras
We need varying the vocabulary/signature P:
Definition (general logic)
Ageneral logic is L={LP:P∈ P} where
LP=hF,M,mng,|=isatisfies (1.)-(3.) for all P∈ P, and
(4.) Cn, ,>are the same for all LP,P∈ P.
(5.) Some conditions between LPand LQfor P,Q∈ P such as:
there are arbitrary large signatures in P;
if PQ, then LPis the natural restriction of LQto LP;
all meanings can be chosen to be the meanings of atomic
formulas in Pfor some P∈ P.
FOL, SOL, modal logic, propositional logic are general logics.
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Algebraic version of a general logic
Let L={LP:P∈ P} be a general logic.
Algm(L) = [
P∈P
AlgmLP
Alg(L) = [
P∈P
Alg LP
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Category of Alg(L)
Objects: elements of Alg(L),
Morphisms: homomorphisms fbetween A,BAlg(L), (A,f,B).
Af B
Objects correspond to theories of L, and Morphisms correspond to
interpretations between these theories.
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Categories of Algebras
Objects: all algebras of some similarity type,
Morphisms: homomorphisms between these algebras.
Internal properties External properties
Onto maps Epimorphisms: left cancellative morphisms
fg =fh =g=h
ffg
h
One-to-one maps Monomorphisms: right cancellative ones
gf =hf =g=h
ff
g
h
Direct product Universal (smallest) cone
AA
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Categories of Algebras
Objects: some algebras of some similarity type,
Morphisms: homomorphisms between these algebras.
External properties are sensitive to context!
??? Epimorphisms: left cancellative morphisms
fg =fh =g=h
ffg
h
In many well-behaved classes of algebras, there are epimorphisms
that are not surjective.
Well investigated question in algebra: In which classes of algebras
are epimorphisms surjective.
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Beth definability property of general logics
Let L={LP:P∈ P} be a general logic.
Let P,Q∈ P, and R=Q\P. Let Σ LQ.
Definition (implicit definition)
Σdefines Rimplicitly in Qiff
for all Q-models M,Nof Σ if all P-formulas have the same
meanings in M,Nthen all Q-formulas have the same meanings in
them.
Definition (explicit definition)
Σdefines Rexplicitly in Qiff
for all rR, there is ϕrFPsuch that Σ |=rϕr.
Definition
Lhas the Beth definability property iff
for all P Q,Rand Σ as above, if Σ defines Rimplicitly, then Σ
defines Ralso explicitly.
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Theorem
Lhas Beth definability property Epis are surjective in Alg(L)
if Lhas the patchwork property for models.
(FOL, SOL,. . . all have the patchwork property for models).
Logic Category
Beth Definability Property Epimorphisms are surjective
Idea of proof. Morphisms correspond to definitions.
Epimorphisms correspond to implicit definitions.
Surjections correspond to explicit definitions.
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weak Beth definability property of general logics
Let L={LP:P∈ P} be a general logic.
Let P,Q∈ P, and R=Q\P. Let Σ LQ.
Definition (strongly implicit definition)
Σdefines Rstrongly implicitly in Qiff
Σ defines Rimplicitly in Q, and in addition for all P-models Mof
the P-consequences of Σ there are Q-models Nof Σ such that all
P-formulas have the same meanings in Mand N.
Definition
Lhas the weak Beth definability property iff
for all P Q,Rand Σ as above, if Σ defines Rstrongly implicitly,
then Σ defines Ralso explicitly.
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K-injective morphism, Full model
Definition (K-injectivity)
Let Cbe a category, Kbe a subcategory and fa morphism in C.
fis K-injective iff all morphisms from the domain of finto an
object of Kfactor through f. (Validity in injectivity logic.)
Definition
Let Lbe a general logic. CA(M)Algm(L) is maximal iff it is not
a proper subalgebra of any member of Algm(L).
Full(L) denotes the class of all maximal members of Algm(L) in
this sense.
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Theorem
Let Lbe a general logic in which each model can be expanded to a
maximal one (FOL, SOL,. . . are like this). Then (i) - (iii) below are
equivalent, and perhaps with (?) also.
(i) Lhas weak Beth definability property
(ii) Full(L)-injective epis are surjective in Alg(L).
(iii) Alg(L)has no proper reflective subcategory containing Full(L).
(?) Alg(L)has no proper limit-closed subcategory containing
Full(L).
Equivalence of (i) and (?) may be independent of set theory, but it
cannot be false!
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Conclusion
What have we learnt?
What have we learnt about the world?
What have we learnt about the physical world?
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Logic, categories and philosophy of mathematics
Budapest 20 June - 21 June 2019
Michael Makkai is turning 80 in 2019. We are pleased to announce
that the Alfr´ed R´enyi Institute of Mathematics, the Department of
Logic,Institute of Philosophy, E¨otv¨os University, and the Faculty of
Science, E¨otv¨os University are organizing a conference celebrating
this occasion. The main topics of the conference are logic,
category theory, model theory, philosophy of mathematics.
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Thank you for your attention!
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