Trying to say why categorical proof theory is fun

1. Edwardian Proofs
Futuristic Programs
Categorical Proof Theory
Categorical Proof Theory
for everyone
Valeria de Paiva
June, 2021
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2. Edwardian Proofs
Futuristic Programs
Categorical Proof Theory
a recommendation...
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3. Edwardian Proofs
Futuristic Programs
Categorical Proof Theory
Greeks
2.500 years hegemony
mathematics growing by leaps and bounds
The only logic...
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4. Edwardian Proofs
Futuristic Programs
Categorical Proof Theory
Greeks
2.500 years hegemony
mathematics growing by leaps and bounds
The only logic...
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5. Edwardian Proofs
Futuristic Programs
Categorical Proof Theory
Greeks
2.500 years hegemony
mathematics growing by leaps and bounds
The only logic...
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6. Edwardian Proofs
Futuristic Programs
Categorical Proof Theory
Mathematics is full of surprises
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7. Edwardian Proofs
Futuristic Programs
Categorical Proof Theory
Proofs?
The bulk of mathematics today got crystallized in the last years of
the 19th century, first years of the 20th century.
The shock is still being felt. A Revolution in Mathematics? What
Really Happened a Century Ago and Why It Matters Today Frank
Quinn (Notices of the AMS, Jan 2012)
The relationship between Algebra, Proofs and Programs
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8. Edwardian Proofs
Futuristic Programs
Categorical Proof Theory
Algebra
[...] a fundamental shift occurred in mathematics from
about 1880 to 1940–the consideration of a wide variety of
mathematical ”structures,”defined axiomatically and stu-
died both individually and as the classes of structures sa-
tisfying those axioms. This approach is so common now
that it is almost superfluous to mention it explicitly, but
it represented a major conceptual shift in answering the
question: What is mathematics?
The axiomatization of Linear Algebra, Moore, Historia Mathematica, 1995.
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9. Edwardian Proofs
Futuristic Programs
Categorical Proof Theory
Edwardian Algebra
Bourbaki on Abstract Algebra
The axiomatization of algebra was begun by Dedekind and
Hilbert, and then vigorously pursued by Steinitz (1910). It
was then completed in the years following 1920 by Artin,
Noether and their colleagues at Göttingen (Hasse, Krull,
Schreier, van der Waerden). It was presented to the world
in complete form by van der Waerden’s book (1930).
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10. Edwardian Proofs
Futuristic Programs
Categorical Proof Theory
Edwardian Proofs
Frege: one of the founders of modern symbolic logic put forward
the view that mathematics is reducible to logic.
Begriffsschrift, 1879
Was the first to write proofs using a collection of abstract symbols:
instead of B → A and B hence A
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11. Edwardian Proofs
Futuristic Programs
Categorical Proof Theory
Why Proofs?
Mathematics in turmoil in the turn of the century because of
paradoxes e.g. Russell’s Paradox
Hilbert’s Program: Base all of mathematics in finitistic methods
Proving the consistency of Arithmetic: the big quest
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12. Edwardian Proofs
Futuristic Programs
Categorical Proof Theory
Edwardian Turmoil...
Hilbert’s program:provide secure foundations for all mathematics.
How? Formalization
all mathematical statements should be written in a precise formal
language, and manipulated according to well defined rules.
There is no ignorabimus in mathematics.. .
Sounds good, doesn’t it?
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13. Edwardian Proofs
Futuristic Programs
Categorical Proof Theory
Hilbert’s Program
Consistent: no contradiction can be obtained in the formalism of
mathematics.
Complete: all true mathematical statements can be proven in the
formalism. Consistency proof use only “finitistic”reasoning about finite mathematical objects.
Conservative: any result about “real objects”obtained using
reasoning about “ideal objects”(such as uncountable sets) can be
proved without ideal objects.
Decidable: an algorithm for deciding the truth or falsity of any
mathematical statement.
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14. Edwardian Proofs
Futuristic Programs
Categorical Proof Theory
Gödel’s Incompleteness Theorems (1931)
Hilbert’s program impossible, if interpreted in the most obvious
way. BUT:
The development of proof theory itself is an outgrowth of
Hilbert’s program. Gentzen’s development of natural de-
duction and the sequent calculus [too]. Gödel obtained his
incompleteness theorems while trying to prove the consis-
tency of analysis. The tradition of reductive proof theory
of the Gentzen-Schütte school is itself a direct continua-
tion of Hilbert’s program.
R. Zach, 2005
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15. Edwardian Proofs
Futuristic Programs
Categorical Proof Theory
20th Century Proofs
To prove the consistency of Arithmetic Gentzen invented his
systems of
NATURAL DEDUCTION
(how mathematicians think)
SEQUENT CALCULUS
(how he could formalize the thinking to obtain the main result he
needed, his Hauptsatz. (1934))
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16. Edwardian Proofs
Futuristic Programs
Categorical Proof Theory
Church and lambda-calculus
Alonzo Church: the lambda calculus (1932)
Church realized that lambda terms could be used to express every
function that could ever be computed by a machine.
Instead of “the function f where f (x) = t”, he simply wrote λx.t.
The lambda calculus is an universal programming language.
The Curry-Howard correspondence: logicians and computer
scientists developed a cornucopia of new logics/program constructs
based on the correspondence between proofs and programs.
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17. Edwardian Proofs
Futuristic Programs
Categorical Proof Theory
Curry-Howard for Implication
Natural deduction rules for implication (without λ-terms)
A → B A
B
[A]
·
·
·
·
π
B
A → B
Natural deduction rules for implication (with λ-terms)
M : A → B N : A
M(N): B
[x : A]
·
·
·
·
π
M : B
λx.M : A → B
function application abstraction
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18. Edwardian Proofs
Futuristic Programs
Categorical Proof Theory
Proofs are Programs!
Types are formulae/objects in appropriate category,
Terms/programs are proofs/morphisms in the category,
Logical constructors are appropriate categorical constructions.
Most important: Reduction is proof normalization (Tait)
Outcome: Transfer results/tools from logic to CT to CSci
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19. Edwardian Proofs
Futuristic Programs
Categorical Proof Theory
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20. Edwardian Proofs
Futuristic Programs
Categorical Proof Theory
Categorical Proof Theory
Model derivations/proofs, not whether theorems are true or not.
proof-relevant version
Proofs definitely first-class citizens
How? Uses extended Curry-Howard correspondence
Why is it good? Modeling derivations useful in linguistics,
functional programming, compilers..
Why is it important? Widespread use of logic/algebra in CS means
new important problems to solve with our favorite tools.
Why so little impact on logic itself?
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21. Edwardian Proofs
Futuristic Programs
Categorical Proof Theory
How many Curry-Howard Correspondences?
Easier to count, if thinking about the logics:
Intuitionistic Propositional Logic, System F, Dependent Type
Theory (Martin-Löf), Linear Logic, Constructive Modal Logics,
various versions of Classical Logic since the early 90’s.
The programs corresponding to these logical systems are futuristic
programs.
The logics inform the design of new type systems, that can be used
in new applications.
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22. Edwardian Proofs
Futuristic Programs
Categorical Proof Theory
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Futuristic Programs
Categorical Proof Theory
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Futuristic Programs
Categorical Proof Theory
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Futuristic Programs
Categorical Proof Theory
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