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practical foundation of mathematics (changes) in nLab

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Context

Type theory

natural deduction metalanguage, practical foundations

judgement
hypothetical judgement, sequent
- antecedents ⊢\vdash consequent, succedents

type theory (dependent, intensional, observational type theory, homotopy type theory)

calculus of constructions

syntax object language

theory, axiom
proposition/type (propositions as types)
definition/proof/program (proofs as programs)
theorem

computational trinitarianism = \linebreak propositions as types +programs as proofs +relation type theory/category theory

logic	set theory (internal logic of)	category theory	type theory
proposition	set	object	type
predicate	family of sets	display morphism	dependent type
proof	element	generalized element	term/program
cut rule		composition of classifying morphisms / pullback of display maps	substitution
introduction rule for implication		counit for hom-tensor adjunction	lambda
elimination rule for implication		unit for hom-tensor adjunction	application
cut elimination for implication		one of the zigzag identities for hom-tensor adjunction	beta reduction
identity elimination for implication		the other zigzag identity for hom-tensor adjunction	eta conversion
true	singleton	terminal object/(-2)-truncated object	h-level 0-type/unit type
false	empty set	initial object	empty type
proposition, truth value	subsingleton	subterminal object/(-1)-truncated object	h-proposition, mere proposition
logical conjunction	cartesian product	product	product type
disjunction	disjoint union (support of)	coproduct ((-1)-truncation of)	sum type (bracket type of)
implication	function set (into subsingleton)	internal hom (into subterminal object)	function type (into h-proposition)
negation	function set into empty set	internal hom into initial object	function type into empty type
universal quantification	indexed cartesian product (of family of subsingletons)	dependent product (of family of subterminal objects)	dependent product type (of family of h-propositions)
existential quantification	indexed disjoint union (support of)	dependent sum ((-1)-truncation of)	dependent sum type (bracket type of)
logical equivalence	bijection set	object of isomorphisms	equivalence type
	support set	support object/(-1)-truncation	propositional truncation/bracket type
		n-image of morphism into terminal object/n-truncation	n-truncation modality
equality	diagonal function/diagonal subset/diagonal relation	path space object	identity type/path type
completely presented set	set	discrete object/0-truncated object	h-level 2-type/set/h-set
set	set with equivalence relation	internal 0-groupoid	Bishop set/setoid with its pseudo-equivalence relation an actual equivalence relation
	equivalence class/quotient set	quotient	quotient type
induction		colimit	inductive type, W-type, M-type
higher induction		higher colimit	higher inductive type
-		0-truncated higher colimit	quotient inductive type
coinduction		limit	coinductive type
	preset		type without identity types
	set of truth values	subobject classifier	type of propositions
domain of discourse	universe	object classifier	type universe
modality		closure operator, (idempotent) monad	modal type theory, monad (in computer science)
linear logic		(symmetric, closed) monoidal category	linear type theory/quantum computation
proof net		string diagram	quantum circuit
(absence of) contraction rule		(absence of) diagonal	no-cloning theorem
		synthetic mathematics	domain specific embedded programming language

homotopy levels

semantics

Foundations

foundations

The basis of it all

Set theory

set theory

fundamentals of set theory
material set theory
presentations of set theory
structuralism in set theory
- material set theory
- structural set theory
  - categorical set theory
    - ETCS
    - structural ZFC
  - allegorical set theory
    - SEAR
class-set theory
constructive set theory
algebraic set theory

Foundational axioms

foundational axioms

Removing axioms

Mathematics

Idea

In the context of foundations of mathematics the term practical foundations (following a term introduced in (Taylor)) refers to emphasis on conceptually natural formalizations.

In my own education I was fortunate to have two teachers who used the term “foundations” in a common-sense way (rather than in the speculative way of the Bolzano-Frege-Peano-Russell tradition). This way is exemplified by their work in Foundations of Algebraic Topology, published in 1952 by Eilenberg (with Steenrod), and The Mechanical Foundations of Elasticity and Fluid Mechanics, published in the same year by Truesdell. The orientation of these works seemed to be “concentrate the essence of practice and in turn use the result to guide practice”. (Lawvere 2003: 213)

Formal systems of interest here are natural deduction in type theories, which allow natural expressions for central concepts in mathematics, notably via their categorical semantics and the conceptual strength of category theory (see Harper).

References

Paul Taylor, Practical Foundations of Mathematics, Cambridge University Press (web)

William Lawvere, Foundations and applications: axiomatization and education, Bulletin of Symbolic Logic 9 (2003), 213-224 (ps)

Under computational trinitarianism this corresponds to a practical foundation in programming, laid out in

Robert Harper, Practical Foundations for Programming Languages (web)

A foundation for algebraic topology in this practical spirit is laid out in

Samuel Eilenberg, Norman Steenrod, Foundations of Algebraic Topology

Last revised on October 18, 2022 at 05:06:33. See the history of this page for a list of all contributions to it.