consistency

## Idea

A system of [[formal logic]] is _consistent_ if [[false]] is not [[deduction|deducible]] in the system.

Otherwise one speaks of _[[inconsistency]]_. See there for more.

[[!redirects consistency]]
[[!redirects consistencies]]
[[!redirects consistent system]]
[[!redirects consistent systems]]
[[!redirects consistent axiom]]
[[!redirects consistent axioms]]
[[!redirects consistent theory]]
[[!redirects consistent theories]]


consistent histories approach to quantum mechanics

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## Idea

This approach to [[quantum mechanics]] assigns probabilities to various alternative histories of measurements on a system such that the probabilities for each history obey the rules of classical probability while being consistent with the [[Schrödinger equation]]. It is claimed to resolve what is generally taken to be paradoxical about quantum mechanics, such as the supposed collapse of the wave-function.

Measurements in this approach are typically understood in terms of [[quantum decoherence]], hence an alternative name _decoherent histories_.

## Related concepts

* [[quantum decoherence]]

* [[interpretation of quantum mechanics]]

## References

* [[Robert Griffiths]], [SEP: The Consistent Histories Approach to Quantum Mechanics] (http://plato.stanford.edu/entries/qm-consistent-histories/)

The consistent histories approach was introduced by

* [[Robert Griffiths]], *Consistent histories and the interpretation of quantum mechanics*, Journal of Statistical Physics **36** (1984) 219-272 [[doi:10.1007/BF01015734](https://doi.org/10.1007/BF01015734)]

with a comprehensive account in:

* [[Robert Griffiths]], *Consistent Quantum Theory*, Cambridge University Press (2001) [[doi:10.1017/CBO9780511606052](https://doi.org/10.1017/CBO9780511606052), [webpage](http://quantum.phys.cmu.edu/CQT/)]

It was also discussed by [[Roland Omnès]] in a series of papers beginning with

* [[Roland Omnès]] (1988) "Logical reformulation of quantum mechanics I. Foundations", Journal of Statistical Physics, 53: 893–932.

See also his books

* [[Roland Omnès]] (1999). _Understanding Quantum Mechanics_, Princeton University Press.

* [[Roland Omnès]] (1999). _Quantum Philosophy_, Princeton University Press, 1999.

The closely related *decoherent histories* approach first appeared in

* [[Murray Gell-Mann]] and [[James Hartle]] (1990) "Quantum mechanics in the light of quantum cosmology", in W. H. Zurek (ed.), Complexity, entropy and the physics of information. Redwood City, Calif.: Addison-Wesley, pp. 425–458.

[[!redirects consistent history]]
[[!redirects consistent histories]]


constant function

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## Definition

Given two sets $S$ and $T$, there is a function $c:T \to (S \to T)$ such that for every element $x:T$ there is a function $c(x):S \to T$ such that for every element $a:S$, $c(x)(a) = x$. For every element $x:T$, the function $c(x)$ is called the __constant function__ from $S$ to $T$ with value $x$.

Given two [[sets]] $S$ and $T$ and an element $x$ of $T$, the __constant function__ from $S$ to $T$ with value $x$ is the [[function]] $f$ defined by
$$ f(a) = x $$
for every element $a$ of $S$.

More generally, any function $f: S \to T$ is a __constant function__ if
$$ f(a) = f(b) $$
for every element $a$ and element $b$ of $S$.  Note that every constant function with particular value (as defined earlier) is constant (as defined here).

The converse is a little more complicated.  If $S$ is [[inhabited set|inhabited]], then every constant function from $S$ to $T$ is the constant function from $S$ to $T$ with some particular value, which is unique.  If $S$ is [[empty set|empty]] but $T$ is inhabited, then the unique function from $S$ to $T$ is constant with any particular value in $T$.  If $S$ and $T$ are both empty, then the unique function from $S$ to $T$ is constant, but not constant at any particular value.

Using [[excluded middle]], we can say that, if $T$ is inhabited, then any constant function from $S$ to $T$ is constant at some (possibly not unique) value, but this theorem fails in [[constructive mathematics]].

See also [[constant morphism]].

### In dependent type theory

In [[dependent type theory]], a **propositionally constant function** or **typally constant function** from type $A$ to $B$ at an element $b:B$ is a function $f:A \to B$ such that $f(x) =_B b$ for all $x:A$. Equivalently, it is an element of the dependent sum type

$$\sum_{b:B} \sum_{f:A \to B} \prod_{x:A} f(x) =_B b$$

For all $b:B$, there is always a propositionally constant function given by $\lambda x:A.b:A \to B$ with $\beta_{A, B}^{x:A.b}:\prod_{x:A} (\lambda x:A.b)(x) =_B b$ from the typal [[computation rules]] for [[function types]], or

$$(b, \lambda x:A.b, \beta_{A, B}^{x:A.b}):\sum_{b:B} \sum_{f:A \to B} \prod_{x:A} f(x) =_B b$$

When function types have judgmental computation rules than one has $\beta_{A, B}^{x:A.b} \equiv \lambda x:A.\mathrm{refl}_B(b)$.

Alternatively, a propositionally constant function $f:A \to B$ is a function which factors through the base generator $\mathrm{base}:A \to \mathrm{Cone}(A)$ of the [[cone type]] of $A$; there exists a function $b:\mathrm{Cone}(A) \to B$ such that for all $x:A$, $b(\mathrm{base}(x)) \equiv f(x)$. Since the [[cone type]] of $A$ is always a [[contractible type]], it is a [[terminal object]] in types.

Given a propositionally constant function $f:A \to B$ with $p:\prod_{x:A} f(x) =_B b$, by the recursion principle of the cone type, one has

$$\mathrm{rec}_{\mathrm{Cone}(A)}^B(b, f, p):\mathrm{Cone}(A) \to B$$

such that

$$\mathrm{rec}_{\mathrm{Cone}(A)}^B(b, f, p)(\mathrm{vertex}) \equiv b:B$$

$$\mathrm{rec}_{\mathrm{Cone}(A)}^B(b, f, p)(\mathrm{base}(x)) \equiv f(x):B$$

$$\mathrm{ap}_{\mathrm{rec}_{\mathrm{Cone}(A)}^B(b, f, p)}(\mathrm{edge}(x)) \equiv p:f(x) =_B b$$

## Related concepts

* [[constant of a theory]]

* **constant function** /  [[locally constant function]]

* [[constant sheaf]] / [[locally constant sheaf]]

* [[constant stack]] / [[locally constant stack]]

* [[constant ∞-stack]] / [[locally constant ∞-stack]]

* [[local system]]

* [[weakly constant function]]

* [[deterministic random variable]], [[almost surely]]

* [[cone type]]

[[!redirects constant function]]
[[!redirects constant functions]]
[[!redirects constant map]]
[[!redirects constant maps]]

[[!redirects propositionally constant function]]
[[!redirects propositionally constant functions]]
[[!redirects propositionally constant map]]
[[!redirects propositionally constant maps]]

[[!redirects typally constant function]]
[[!redirects typally constant functions]]
[[!redirects typally constant map]]
[[!redirects typally constant maps]]


constant functor

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## Definition

A **constant functor** $\Delta d: C\to D$ is a [[functor]] that maps each [[object]] of the [[category]] $C$ to a fixed object $d\in D$ and each morphism of $C$ to the identity morphism of that fixed object.

(The notation $\Delta d$ is suggested by the fact that if $d: 1 \to D$ names an object $d$ of $D$, then the composite

$$1 \stackrel{d}{\to} D \stackrel{\Delta}{\to} D^C$$

names a functor $\Delta d: C \to D$ which is the constant functor at $d$. Here $\Delta: D \to D^C$ denotes a [[diagonal functor]].)

Viewing $D \cong [1,D]$, it is possible to see that $\Delta$ is a base change of the functor categories $D^1$ and $D^C$, given the functor $!_C: C \to 1$.

More generally, we may say that a functor $F:C\to D$ is **essentially constant** if it is [[naturally isomorphic]] to a constant functor.

## Properties

Note that a constant functor can be expressed as the composite
$$C \stackrel{!}{\to} 1 \stackrel{[d]}{\to} D.$$
Here $1$ is a [[terminal category]] (exactly one object and exactly one morphism, namely the identity), and $[d]$ denotes the unique functor from $1$ with $F(\bullet) = d$ and $F(Id_\bullet) = Id_d$.  It follows that $F:C\to D$ is essentially constant if and only if it factors through $1$ up to isomorphism.

If $C$ has at least one object, then there is a [[codescent object]]
$$ C\times C\times C \underoverset{\to}{\to}{\to} C\times C \rightrightarrows C \to 1. $$
Therefore, in this case a functor $F:C\to D$ is essentially constant if and only if we have a natural isomorphism between the two composites $C\times C \rightrightarrows C \xrightarrow{F} D$ (i.e. isomorphisms $F(c_1)\cong F(c_2)$ natural in $c_1,c_2\in C$) that satisfies a [[cocycle condition]].  This is a categorified version of the statement that a [[function]] with inhabited domain is a [[constant function]] if and only if the images of any two elements are equal.  Just as in that case, there is a distinction in the trivial case: the [[identity functor]] of the [[empty category]] satisfies this weaker condition, but is not essentially constant because there is no object for it to be constant at.

## Examples

* For $F$ any functor, a [[natural transformation]]

  $\Delta_d \Rightarrow F$

  from a constant functor into $F$ is precisely a [[cone]] over $F$. Similarly a natural transformation

  $F \Rightarrow \Delta_d$

  is a [[cocone]].

## Related concepts

* [[constant presheaf]], [[locally constant sheaf]]
* [[diagonal functor]]
* [[constant morphism]], [[constant function]]

[[!redirects constant functors]]
[[!redirects essentially constant functor]]
[[!redirects essentially constant functors]]

[[!redirects constant diagram]]
[[!redirects constant diagrams]]


constant infinity-stack

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## Definition

A constant [[∞-stack]]/[[(∞,1)-sheaf]] is the [[∞-stackification]] of a [[(∞,1)-presheaf]] which is constant as an [[(∞,1)-functor]].

With the [[global section]] [[(∞,1)-functor]] the constant $\infty$-stack functor $LConst$ forms the [[terminal object|terminal]] [[(∞,1)-geometric morphism]]

$$
  (LConst \dashv \Gamma)
  :
  Sh_{(\infty,1)}(C)
  \stackrel{\overset{LConst}{\leftarrow}}{\underset{\Gamma}{\to}}
  \infty Grpd
  \,.
$$

## On $Top$

Notice that in the special case of [[∞-stack]]s on [[Top]], hence of [[topological ∞-groupoid]], which may be thought of as [[Top]]-valued presheaves on [[Top]](!), there are _two different_ obvious ways to regard a topological space $X$ as an [[∞-stack]] on [[Top]]:

* there is the [[∞-stack]] $\bar const_X$ _constant_ on $X$, meaning constant on the [[Kan complex]] that is  the [[fundamental ∞-groupoid]] $Sing X = \Pi(X)$ of $X$;

* there is the [[Yoneda embedding]] $Y(X)$ of $X$ into [[∞-stack]].

The first regards $X$ really as an [[∞-groupoid]], forgetting its topology, the second regards $X$ as a [[locale]], not caring about the homotopies that are inside.

For any [[(∞,1)-category]] $S$, there is the obvious embedding of [[∞-groupoid]]s into [[(∞,1)-presheaf|(∞,1)-presheaves]] on $S$

$$
  const : \infty Grpd \to [S^{op}, \infty Grpd]
$$

where of course

$$
  const_K : U \mapsto K
$$

for all $U$.

This is all very obvious, but deserves maybe a special remark in the case that [[∞-groupoid]]s are modeled as (compactly generated and weakly Hausdorff) [[topological space]]s: in particular in the case that $S = Top$ itself, there are then two different ways to regard a topological space as an $\infty$-stack, and they have very different meaning.

In particular, with $X$ a [[topological space]], the $\infty$-stack _constant_ on $X$ has the property that its [[loop space object]] $\Lambda X$ is indeed the $\infty$-stack constant on the free loop space of $X$, while the [[loop space object]] of $X$ regarded as a representable $\infty$-stack is just $X$ itself again.

This is because

* the $\infty$-stack represented by $X$ regards $X$ as a [[discrete category|categorically discrete]] topological groupoid;

* while the $\infty$-stack constant on $X$ regards $X$ as a topologically discrete groupoid which however may have nontrivial morphisms.

## Pattern

* A [[locally constant function]] is a section of a [[constant sheaf]];

* a [[locally constant sheaf]] is a section of a [[constant stack]];

* a [[locally constant stack]] is a section of (... and so on...)

* a [[locally constant ∞-stack]] is a section of a **constant $\infty$-stack**.

A locally constant sheaf / $\infty$-stack is also called a [[local system]].

[[!redirects constant infinity-stacks]]
[[!redirects constant ∞-stack]]
[[!redirects constant ∞-stacks]]

[[!redirects constant (∞,1)-sheaf]]
[[!redirects constant (∞,1)-sheaves]]


constant morphism

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## Idea

The notion of a _constant morphism_ in a [[category]] generalises the notion of [[constant function]].

## Definition

+-- {: .num_defn #ConstMorph}
###### Definition
A __constant morphism__ in a [[category]] $\mathcal{C}$ is a [[morphism]] $c\colon B \to C$ with the property that if $f,g\colon A \to B$ are morphisms in $\mathcal{C}$ then $c \circ f = c \circ g$.  In other words, for every object $A$, *at most* one morphism from $A$ to $C$ factors through $c$.
=--

Thus, $c$ is a constant morphism if the function $c_* \colon \mathcal{C}(A,B) \to \mathcal{C}(A,C)$ given by composition with $c$ is a [[constant function]] for every object $A$.

Another definition that is sometimes used is the following.

+-- {: .num_defn #ConstTerm}
###### Definition
If $\mathcal{C}$ has a [[terminal object]], then $c:B\to C$ is __constant__ iff it factors through this terminal object.

If $\mathcal{C}$ does not have a terminal object, then we can reformulate this by saying that the image of $c$ under the [[Yoneda embedding]], $(c\circ -) \colon \mathcal{C}(-,B)\to \mathcal{C}(-,C)$, factors through the [[terminal object|terminal]] [[presheaf]].  In elementary terms, this means that we can choose for each object $X$ a morphism $f_X:X\to C$ such that (1) $f_B=c$ and (2) the $f_X$ are natural in $X$, i.e. for any $g:Y\to X$ we have $f_X g = f_Y$.
=--

This second definition implies the first, but they are not equivalent in general.  In the [[category of sets]], the first implies the second if the set $B$ is [[inhabited set|inhabited]], or if the set $B$ is [[empty set]] and the set $C$ is inhabited.  (By [[excluded middle]], $B$ is either inhabited or empty, so it suffices to assume that $C$ is inhabited with no assumption about $B$.)

More generally, if $c\colon B \to C$ is a morphism in a category $\mathcal{C}$, then the two definitions are equivalent if $\mathcal{C}(A,B)$ is inhabited for every $A$, since we can then define $f_X = c g$ for some (hence any) $g:A\to B$.  If $\mathcal{C}$ has a terminal object $1$, then this is equivalent to the existence of a [[global section]] $b\colon 1 \to B$.

See the [forum](https://nforum.ncatlab.org/discussion/2689/constant-morphism/) for further discussion of this.

## Relation to (sub)terminal objects

The [[identity morphism]] on an object $B$ satisfies definition 1 if and only if $B$ is [[subterminal object|subterminal]]; it satisfies definition 2 iff $B$ is [[terminal object|terminal]].  In particular, the identity function on the [[empty set]] satisfies definition 1 but not definition 2.

## Examples

Using the two-point set, it is simple to show that the constant morphisms in [[Set]] are precisely the constant functions.

## Related concepts

* [[constant function]]

* [[steady function]]

[[!redirects constant morphism]]
[[!redirects constant morphisms]]


constant of a theory

\tableofcontents

## Idea

In [[logic]], particularly in the syntactic notion of a (first-order) [[theory]], a *constant* is a term without free variables, modulo equivalence imposed by equational axioms of the theory.

## Definition

Recall that a [[signature]] $\Sigma$ of a theory (let us say a one-sorted theory, for the sake of simplicity) consists of constant symbols together with function symbols and relation symbols with assigned arities. Then

* A constant symbol $c$ of $\Sigma$ is considered a constant term of the theory, and

* If $f$ is a function symbol of arity $n$ and $c_1, \ldots, c_n$ are constant terms, then $f(c_1, \ldots, c_n)$ is a constant term of the theory.

[Note that in some accounts, constant symbols are considered as function symbols of arity zero.]

For instance, the theory of rings is typically presented by a signature $(R, +, -, \cdot, 0, 1)$ where $R$ denotes the sort, with binary operations $+, -, \cdot$ and constant (symbols) $0, 1$, whence other constant terms may be derived such as $(1 + 1) \cdot ((0-1) -1)$.

Constant terms are syntactic expressions derived from the signature, but the axioms of the theory may impose equations between constant terms; for example, the equational axioms of ring theory force equations like

$$(1 + 1) \cdot ((0-1) -1) = 0 - (1 + ((1 + 1) + 1))$$

so that what one is really after is equivalence classes of syntactic expressions. A *constant* of the theory is such an equivalence class of constant terms.

\begin{example}
Let $R$ be a fixed commutative ring. The theory of $R$-algebras may be presented just as the theory of rings, but including a constant symbol $c_r$ for each element $r \in R$, and adding equations $c_0 = 0$, $c_1 = 1$, $c_r + c_s = c_{r+s}$, $c_{r-s} = c_r - c_s$, and $c_r \cdot c_s = c_{r s}$.
\end{example}

## Constants of algebraic theories

For a more semantic description, suppose $T$ is a [[Lawvere theory]], and let

$$Set^T = \mathbf{Prod}(T, Set)$$

denote the category of $T$-algebras = [[product-preserving functors]] $T \to Set$. Then there is a forgetful functor

$$U: Set^T \to Set,$$

and constants of $T$ may be identified, up to natural isomorphism, with natural transformations of the form

$$1 \to U$$

(where $1 = U^0$ here denotes the terminal functor $Set^T \to Set$). Alternatively, constants of $T$ may be naturally identified with morphisms in the hom-set $T(i(0), i(1))$ where $i: FinSet^{op} \to T$ is the product-preserving functor given as one of the data for $T$.

The same idea applies to any infinitary [[algebraic theory]].

From this point of view, identifying (equivalence classes of) $n$-ary operations of $T$ with natural transformations $f: U^n \to U$, one also often says that $f$ is constant if it factors through the unique map $U^n \to 1$, as in [[constant morphism]].

For example, for the Lawvere theory of commutative $R$-algebras, $n$-ary operations $U^n \to U$ are in natural bijection with polynomials $p \in R[x_1, \ldots, x_n]$, and such a $p$ is constant iff it belongs to the image of the canonical inclusion map $R \to R[x_1, \ldots, x_n]$. See also [[constant polynomial]].

## Related concepts

* [[constant function]]

[[!redirects constants of a theory]]


constant polynomial

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## Definition

A **constant polynomial** is a [[polynomial]] where not having [[degree of a polynomial|degree]] $0$ implies that it is the zero polynomial. (In [[classical mathematics]] this is equivalent to saying that a constant polynomial is a polynomial which either has [[degree of a polynomial|degree]] $0$ or is the zero polynomial.)

Equivalently, given a [[commutative ring]] $R$, [[polynomial rings]] $R[S]$ are [[free object|free]] [[commutative algebra|commutative $R$-algebras]] on an [[inhabited set|inhabited]] [[finite set]] of [[generators]] $S$; and are characterized by the universal property that it is the [[initial object]] with a [[ring homomorphism]] $h:R \to R[S]$ and a [[function]] $i:S \to R[S]$. A **constant polynomial** of $R[S]$ is a polynomial which is in the [[image]] of the canonical [[ring homomorphism]] $h:R \to R[S]$. As a result, the subring of constant polynomials is in [[isomorphism]] with the [[ground ring]].

A related point of view, falling under the rubric "structure-semantics duality" in the old terminology of Lawvere, emanates from the recognition that if $F: Set \to CommRing$ is the left adjoint of the forgetful functor $U: CommRing \to Set$, so that $F$ takes a set $S$ to the polynomial ring $R[S]$, then a polynomial in (say) $n$ variables can be naturally identified with any of the following:

* An element $1 \to U F(n)$;

* A ring map $F(1) \to F(n)$;

* A natural transformation between representable functors $CommRing(F(n), -) \to CommRing(F(1), -)$;

* A natural transformation $U^n \to U$.

From that point of view, a polynomial in $n$ variables is constant if the corresponding transformation $U^n \to U$ is constant in the sense of [[constant morphism]], i.e., factors through the terminal functor $U^0$.

## Related concepts

* [[polynomial]]

* [[degree of a polynomial]]

* [[constant morphism]]

* [[constant of a theory]]

## References

* Wikipedia, [Constant polynomial](https://en.wikipedia.org/wiki/Constant_polynomial)

[[!redirects constant polynomial]]
[[!redirects constant polynomials]]


constant presheaf

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## Definition

A [[presheaf]] is just a [[functor]] $C^{op} \to Set$ and it is a _constant presheaf_ if that functor is a [[constant functor]].

The [[sheafification]] of a constant presheaf is not in general constant anymore, but is called a [[locally constant sheaf]] (in fact it is also sometimes called just a [[constant sheaf]], beware).

## Related concepts

* [[locally connected topos]]

[[!redirects constant presheaves]]


constant sheaf

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## Definition

A **constant sheaf** is a [[sheaf]] (on some [[site]] C) that is [[isomorphism|isomorphic]] to the [[sheafification]] of a constant [[presheaf]] (a [[constant functor]]).

Together with the [[global section]] functor, the constant sheaf functor is a [[geometric morphism]]

$$
  \Gamma : Sh(C) \stackrel{\leftarrow}{\to} Set : const
$$

from the [[Grothendieck topos|sheaf topos]] to the [[topos]] [[Set]].

## Examples

* The [[natural numbers object]] in any [[Grothendieck topos]] is the constant sheaf on the set $\mathbb{N}$ of natural numbers.

## Pattern

* A [[locally constant function]] is a section of a **constant sheaf**;

* a [[locally constant sheaf]] is a section of a [[constant stack]];

* a [[locally constant stack]] is a section of (... and so on...)

* a [[locally constant ∞-stack]] is a section of a [[constant ∞-stack]].

A locally constant sheaf / $\infty$-stack is also called a [[local system]].

See also at _[[constant presheaf]]_.

category: sheaf theory
[[!redirects constant sheaves]]


constant stack

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## Definition

A **constant stack** is a [[stack]] that is equivalent to the [[stackification]] of a presheaf that is constant (as a [[functor]]).

## Pattern

* A [[locally constant function]] is a section of a [[constant sheaf]];

* a [[locally constant sheaf]] is a section of a **constant stack**;

* a [[locally constant stack]] is a section of (... and so on...)

* a [[locally constant ∞-stack]] is a section of a [[constant ∞-stack]].

A locally constant sheaf / $\infty$-stack is also called a [[local system]].


Constantin Bachas

* [webpage](http://www.lpt.ens.fr/?page=membre&id=27&lang=en)

## Selected writings

On the [[Schwinger effect]] seen in [[open string|open]] [[string theory]]:

* [[Constantin Bachas]], [[Massimo Porrati]], _Pair Creation of Open Strings in an Electric Field_, Phys. Lett. B296 (1992) 77-84 ([arXiv:hep-th/9209032](https://arxiv.org/abs/hep-th/9209032))

On [[type I' string theory]] and [[geometric engineering of QFT]] on [[D3-D5 brane intersections]] subject to the [[s-rule]]:

* [[Constantin Bachas]], [[Michael Green]], Adam Schwimmer, Section 2.3 of: _$(8,0)$ Quantum mechanics and symmetry enhancement in type I' superstrings_, JHEP 9801 :006, 1998 ([arXiv:hep-th/9712086](https://arxiv.org/abs/hep-th/9712086))

and related to the [[Pauli exclusion principle]]:

* [[Constantin Bachas]], [[Michael Green]], _A Classical Manifestation of the Pauli Exclusion Principle_, JHEP 9801 (1998) 015 ([arXiv:hep-th/9712187](https://arxiv.org/abs/hep-th/9712187))

On [[higher curvature corrections]] to the (abelian) [[DBI-action]] for [[D-branes]]:

* [[Constantin Bachas]], P. Bain, [[Michael Green]], _Curvature terms in D-brane actions and their M-theory origin_, JHEP 9905:011, 1999 ([arXiv:hep-th/9903210](https://arxiv.org/abs/hep-th/9903210))

On [[D-branes]]:

* [[Constantin Bachas]], *D-branes* [[arXiv:2311.18456](https://arxiv.org/abs/2311.18456)]

category: people


Constantin Carathéodory

* [Wikipedia entry](http://en.wikipedia.org/wiki/Constantin_Carath%C3%A9odory)

## Related entries

* [[de Donder-Weyl formalism]]

category: people

[[!redirects Constantin Caratheodory]]
[[!redirects Carathéodory]]
[[!redirects Caratheodory]]


Constantin Năstăsescu

* [Wikipedia entry (ro)](https://ro.m.wikipedia.org/wiki/Constantin_N%C4%83st%C4%83sescu)

## Selected writings

Introducing the notion of [[separable functors]]:

* {#NăstăsescuVanDen BerghVanOystaeyen89} [[Constantin Năstăsescu]], [[Michel van den Bergh]], [[F. van Oystaeyen]], _Separable functors applied to graded rings_, J. Algebra __123__ (1989) 397-413 [doi:10.1016/0021-8693(89)90053-7]

category: people

[[!redirects Constantin Nastasescu]]


Constantin Teleman

* [website](http://math.berkeley.edu/~teleman/)

## Selected writings

On a [[de Rham theorem]] for [[algebraic stack|algbraic]] [[infinity-stacks|$\infty$-stacks]] via an [[infinitesimal shape modality]] realizing the [[de Rham stack]]:

* {#SimpsonTeleman} [[Carlos Simpson]], [[Constantin Teleman]], _deRham theorem for $\infty$-stacks_ (1997) [[pdf](http://math.berkeley.edu/~teleman/math/simpson.pdf), [[SimpsonTeleman-deRham.pdf:file]]]

On [[twisted equivariant K-theory]] with an eye towards [[twisted ad-equivariant K-theory]]:

* [[Daniel Freed]], [[Michael Hopkins]], [[Constantin Teleman]], _Twisted equivariant K-theory with complex coefficients_, Journal of Topology, Volume 1, Issue 1
([arXiv:math/0206257](https://arxiv.org/abs/math/0206257), [doi:10.1112/jtopol/jtm001](https://doi.org/10.1112/jtopol/jtm001))

On [[twisted ad-equivariant K-theory]] of [[compact Lie groups]] and the identification with the [[Verlinde ring]] of [[positive energy representations]] of their [[loop group]]:

* [[Daniel S. Freed]], [[Michael Hopkins]], [[Constantin Teleman]],

  1. _[[Loop Groups and Twisted K-Theory]] I_,

     J. Topology, 4 (2011), 737-789

     ([arXiv:0711.1906](http://arxiv.org/abs/0711.1906), [doi:10.1112/jtopol/jtr019](https://doi.org/10.1112/jtopol/jtr019))

  1. _[[Loop Groups and Twisted K-Theory]] II_,

     J. Amer. Math. Soc. 26 (2013), 595-644

     ([arXiv:0511232](http://arxiv.org/abs/math/0511232), [doi:10.1090/S0894-0347-2013-00761-4](https://doi.org/10.1090/S0894-0347-2013-00761-4))

  1. _[[Loop Groups and Twisted K-Theory]] III_,

     Annals of Mathematics, Volume 174 (2011) 947-1007

     ([arXiv:math/0312155](http://arxiv.org/abs/math/0312155), [doi:10.4007/annals.2011.174.2.5](http://dx.doi.org/10.4007/annals.2011.174.2.5))

* [[Daniel S. Freed]], [[Constantin Teleman]],

  _Dirac families for loop groups as matrix factorizations_,

  Comptes Rendus Mathematique, Volume 353, Issue 5, May 2015, Pages 415-419

  ([arxiv/1409.6051](http://arxiv.org/abs/1409.6051), [doi:10.1016/j.crma.2015.02.011](https://doi.org/10.1016/j.crma.2015.02.011))

On [[Gromov-Witten theory]]:

* [[Edward Frenkel]], Constantin Teleman, [[A. J. Tolland]], _Gromov-Witten Gauge Theory I_ ([arXiv](http://arxiv.org/abs/0904.4834))

On [[2d TQFT]]:

* Constantin Teleman, _The structure of 2D semi-simple field theories_ ([arXiv:0712.0160](http://arxiv.org/abs/0712.0160))

On [[quantization]] of Dijkgraaf-Witten theory and [[Chern-Simons theory]];

* [[Dan Freed]], [[Mike Hopkins]], [[Jacob Lurie]], Constantin Teleman, _[[Topological Quantum Field Theories from Compact Lie Groups]]_

On [[string topology]] and [[TCFT]]:

* [[Andrew Blumberg]], [[Ralph Cohen]], Constantin Teleman, _Open-closed field theories, string topology, and Hochschild homology_ ([arXiv:0906.5198](http://arxiv.org/abs/0906.5198))

On [[TQFT]]:

* {#Teleman14} [[Constantin Teleman]], *Five lectures on topological field theory*, in *Geometry and Quantization of Moduli Spaces*, CRM Advanced Courses in Mathematics, Birkhäuser (2016) [[doi:10.1007/978-3-319-33578-0_3](https://doi.org/10.1007/978-3-319-33578-0_3), [pdf](http://math.berkeley.edu/~teleman/math/barclect.pdf), [[Teleman-LecturesOnTFT.pdf:file]]]

category: people


Constantinos Papageorgakis

* [webpage](https://www.qmul.ac.uk/spcs/staff/academics/profiles/cpapageorgakis.html)

## Selected writings

On [[M2-M5 brane bound states]] in relation to [[Dp-D(p+2) brane intersection]] [[fuzzy funnels]], via the [[ABJM model]]:

* [[Horatiu Nastase]], [[Constantinos Papageorgakis]], [[Sanjaye Ramgoolam]], _The fuzzy $S^2$ structure of M2-M5 systems in ABJM membrane theories_, JHEP 0905:123, 2009 ([arXiv:0903.3966](https://arxiv.org/abs/0903.3966))

* [[Horatiu Nastase]], [[Constantinos Papageorgakis]], _Bifundamental Fuzzy 2-Sphere and Fuzzy Killing Spinors_, SIGMA 6:058, 2010 ([arXiv:1003.5590](https://arxiv.org/abs/1003.5590))

On multiple [[M2-branes]] and their worldvolume [[BLG model]]/[[ABJM model]]:

* {#BaggerLambertMukhiPapageorgakis13} [[Jonathan Bagger]], [[Neil Lambert]], [[Sunil Mukhi]], [[Constantinos Papageorgakis]], _Multiple Membranes in M-theory_, Physics Reports, Volume 527, Issue 1, 1 June 2013, Pages 1-100 ([arXiv:1203.3546](http://arxiv.org/abs/1203.3546), [doi:10.1016/j.physrep.2013.01.006](https://doi.org/10.1016/j.physrep.2013.01.006))

On relation of [[Dp-D(p+2)-brane bound states]] ([hence](Dp-Dp+2-brane+bound+states#ReferencesRelationToMonopoles)
[[Yang-Mills monopoles]]) to [[knot invariants]] via [[chord diagrams]] computing [[radii]] of [[fuzzy spheres]]:

* [[Simon McNamara]], [[Constantinos Papageorgakis]], [[Sanyaje Ramgoolam]], [[Bill Spence]], Appendix A of: _Finite $N$ effects on the collapse of fuzzy spheres_, JHEP 0605:060, 2006 ([arxiv:hep-th/0512145](https://arxiv.org/abs/hep-th/0512145))

* [[Constantinos Papageorgakis]], p. 161-162 of: _On matrix D-brane dynamics and fuzzy spheres_, 2006 ([[Papageorgakis06.pdf:file]])

On using [[machine learning]] to solve the [[conformal bootstrap]]:

* Gergely Kántor, Vasilis Niarchos, [[Constantinos Papageorgakis]], *Solving Conformal Field Theories with Artificial Intelligence* ([arXiv:2108.08859](https://arxiv.org/abs/2108.08859))

## Related $n$Lab entries

* [[M2-brane]]

  [[3d superconformal gauge field theory]]

  [[BLG model]], [[ABJM model]]

category: people


Constantinos Skordis

* [website](https://www.ceico.cz/team/leads/constantinos-skordis)

## Selected writings

On a [[relativistic field theory]] version of [[MOND]]:

* [[Constantinos Skordis]], [[Tom Zlosnik]], *A new relativistic theory for Modified Newtonian Dynamics*. Physical Review Letters. Volume 127, Issue 16. Published 15 October 2021 ([doi:10.1103/PhysRevLett.127.161302](https://doi.org/10.1103/PhysRevLett.127.161302), [arXiv:2007.00082](https://arxiv.org/abs/2007.00082))

* [[Constantinos Skordis]], [[Tom Zlosnik]], *Aether scalar tensor theory: Linear stability on Minkowski space*. Physical Review D. Volume 106, Issue 10. Published 15 November 2022. ([doi:10.1103/PhysRevD.106.104041](https://doi.org/10.1103/PhysRevD.106.104041), [arXiv:2109.13287](https://arxiv.org/abs/2109.13287))

category: people


Constanze Roitzheim

* [webpage](https://www.kent.ac.uk/smsas/personal/csrr/)

## Selected writings

Generalizing the [[model structure on chain complexes]] to [[bicomplexes]]:

* {#MuroRoitzheim19} [[Fernando Muro]], [[Constanze Roitzheim]], *Homotopy Theory of Bicomplexes*, Journal of Pure and Applied Algebra **223** 5 (2019) 1913-1939 [[arXiv:1802.07610](https://arxiv.org/abs/1802.07610), [doi:10.1016/j.jpaa.2018.08.007](https://doi.org/10.1016/j.jpaa.2018.08.007)]

On [[stable homotopy theory]]:

* [[David Barnes]], [[Constanze Roitzheim]], _Foundations of Stable Homotopy Theory_, Cambridge University Press 2020 ([doi:10.1017/9781108636575](https://doi.org/10.1017/9781108636575))

category: people


constituent quark model

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## Idea

By a _constituent quark_ one refers to a [[quark]] thought of as part of a [[hadron]] [[bound state]].

## Related concepts

* [[diquark]]

* [[exotic meson]]

## References

* [[Howard Georgi]], §17  in: *Lie Algebras In Particle Physics*, Westview Press (1999), CRC Press (2019) [[doi:10.1201/9780429499210](https://doi.org/10.1201/9780429499210)]

See also

* Wikipedia, _[Constituent quark](https://en.m.wikipedia.org/wiki/Constituent_quark)

[[!redirects constituent quark]]
[[!redirects constituent quarks]]


constrained mechanics

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## Idea

(...)

## Related cocepts

* [[first class constraint]]

* [[Gauss law]]

## References

The original discussion of [[Hamiltonian mechanics]] with constraints:

* [[Paul A. M. Dirac]], *Generalized Hamiltonian Dynamics*, Canadian Journal of Mathematics **2** (1950) 129-148 [[doi:10.4153/CJM-1950-012-1](https://doi.org/10.4153/CJM-1950-012-1)]

  reprinted in:

  Alexander S. Blum, Dean Rickles (eds.), Ch.34 in: *Quantum Gravity in the First Half of the Twentieth Century: A Sourcebook*, Edition Open Sources **10** (2018) 484-503 [[doi:10.34663/9783945561317-00](https://edition-open-sources.org/sources/10/), [pdf](https://edition-open-sources.org/media/sources/10/36/sources10chap34.pdf)]

  > (beware that Dirac never uses the term "constraint" but speaks of "equations")

Textbook account:

* [[Marc Henneaux]], [[Claudio Teitelboim]], Chapter One in: _[[Quantization of Gauge Systems]]_, Princeton University Press (1992) [[ISBN:9780691037691](https://press.princeton.edu/books/paperback/9780691037691/quantization-of-gauge-systems), [doi:10.2307/j.ctv10crg0r](https://doi.org/10.2307/j.ctv10crg0r), [jstor:j.ctv10crg0r](https://www.jstor.org/stable/j.ctv10crg0r)]

[[!redirects constraint]]
[[!redirects constraints]]


constructible set

#Contents#
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## Idea

 _Constructible sets_ are one of the central notions in
[[descriptive set theory]].

## Definition

In [[set theory]], a (Gödel) __constructible set__ is any set in a [[constructible universe]], see there.

One can generalize to constructible elements in more general [[Boolean lattices]].

## Examples

* In [[general topology]], given a [[topological space]] $(X,\tau)$, a __constructible subset__ of $X$, sometimes (when $(X,\tau)$ is fixed) often simply referred as a constructible set, is an element of the smallest Boolean algebra of sets containing the topology $\tau$; the constructible sets are precisely the finite unions of locally closed sets.

* In the case of [[theory]] ACF of [[algebraically closed fields]], for every [[model]] $K$, the constructible sets in the [[affine space]] $K^n$ are precisely the [[definable sets]] for the [[language in model theory|language]] of [[fields]].

## Related concepts

* [[constructible universe]]

* [[recursive subset]], [[definable set]]

* [[constructible sheaf]]

## References

* Wikipedia, _[Constructible set](http://en.wikipedia.org/wiki/Constructible_set_%28topology%29)

[[!redirects constructible sets]]

[[!redirects constructible subset]]
[[!redirects constructible subsets]]


constructible sheaf

#Contents#
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## Idea

A [[sheaf]] on an [[étale site]] is _constructible_ if its restriction to a suitable decomposition into [[constructible subsets]] is a [[locally constant sheaf]].

## References

Original articles include

* [[Pierre Deligne]], _La conjecture de Weil. II_. Inst. Hautes ´Etudes Sci. Publ. Math., (52):137–252, 1980.

* [[Torsten Ekedahl]], _On the adic formalism_. In The Grothendieck Festschrift, Vol. II, volume 87 of Progr. Math., pages 197–218. Birkhäuser Boston, Boston, MA, 1990.

* Alexander Beilinson, _Constructible sheaves are holonomic_, Selecta Mathematica 22 (2016) 1797-1819; a slightly updated version (with respect to the published one) is at [arxiv/1505.06768](http://arxiv.org/abs/1505.06768)

An introductory survey is in

* Florian Klein, Gerrit Begher, _Constructible Sheaves and their derived category_ ([pdf](http://www.mathematik.uni-stuttgart.de/~geckmf/skyefiles/talk10.pdf))

A list of relevant definitions and facts is at

* [[Günter Tamme]], section II 9.3.2 _[[Introduction to Étale Cohomology]]_

* [[James Milne]], section 17 _[[Lectures on Étale Cohomology]]_

* [[The Stacks Project]], _[44.62. Constructible sheaves](http://stacks.math.columbia.edu/tag/05BE)_

and with more on [[chain complexes of sheaves]] and [[abelian sheaf cohomology]] in

* Bhargav Bhatt, [[Peter Scholze]], section 6.3 of _The pro-étale topology for schemes_ ([arXiv:1309.1198](http://arxiv.org/abs/1309.1198))
 {#BhattScholze13}

with an eye towards the application in [[l-adic cohomology]]/the [[pro-étale topos]].

See also

* [[Jacob Lurie]], _Constructible sheaves_ ([pdf](http://www.math.harvard.edu/~lurie/287xnotes/Lecture18.pdf))

[[!redirects constructible sheaves]]


constructible universe

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# The constructible universe
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## Definition

__Gödel's constructible universe__ $L$ is a [[proper class|subclass]] of the [[von Neumann universe]] $V$ of well-founded [[pure sets]]. It is defined by [[transfinite induction]] as $L = \bigcup_\alpha L_\alpha$ where

$$L_0 = \emptyset$$

$$ L_{\alpha +1} = P(L_\alpha) \cap I(L_\alpha \cup \{L_\alpha\}) $$

and if $\alpha$ is a limit ordinal:

$$ L_\alpha = \bigcup_{\beta\lt \alpha} L_\beta$$

Here, for $X$ a pure set, by $I(X)$ we denote the smallest set containing $X$ and closed with respect to the operations of [[Cartesian product]], [[relative complement|set difference]], [[unordered pair]], [[pairing|ordered pair]], taking the [[domain]] of a [[binary relation]], and performing a [[permutation]] of an ordered triple. This definition, while less technical, offers different levels than the standard set-theoretical definition, for example the ordinal $\alpha$ need not be a member of $L_{\alpha+1}$.

Alternatively, we may say that
$$ L_\alpha = \bigcup_{\beta \lt \alpha} P(L_\beta) \cap I(L_\beta \cup \{L_\beta\}) $$
for any ordinal $\alpha$ ($0$, successor, or limit).

The elements of the constructible universe are called __[[constructible sets]]__; the idea is similar to the [[constructible sets]] in topology and algebraic geometry.

[[AnonymousCoward]]: Does there exist a [[structural set theory]] definition of a constructible set and the constructible universe?

## Properties

$L$ is a [[transitive set|transitive]] [[big class]] containing all the ordinals.

There is a standard definition of a well-ordering of $L$, <${}_L$, therefore the sets in this class can be effectively enumerated by von Neumann ordinals, via the ranking function of <${}_L$.

## As inner model
In fact, $L$ is a model of the set theory (consider ZF), namely the smallest transitive class containing all the ordinals. Indeed, we can well-order all sets by a well-ordering <${}_L$ and therefore the axiom of choice holds in $L$. The generalized continuum hypothesis also holds in $L$.
Note that properties that hold in $V$ need not hold in $L$.

## Relation to the von Neumann universe

$L=V$ is an axiom which states that every set is in $L$. This collapses the [[von Neumann universe]] $V$ to the "thinner" inner model $L$. The question of whether $L\neq V$ can not be decided in ZF. If $L\neq V$ then we still do not know how, without an axiom of choice, to produce specific sets that are not constructible.

Given the independence, one may add $L=V$ as an axiom. However, note that formally stating that axiom is more technical (involving ordinals, etc.) than the standard set theory axioms. Moreover, by its restrictiveness, that axiom also rules out many other popular studied axioms (e.g. it is inconsistent with the existence of "large" [[large cardinals]] such as [[measurable cardinals]] and above).

## References

* [[Richard Matthews]], [[Michael Rathjen]], _Constructing the Constructible Universe Constructively_ [[arxiv:2206.08283](https://arxiv.org/abs/2206.08283)]

See also:

* Wikipedia, *[Constructible universe](http://en.wikipedia.org/wiki/Constructible_universe)*

[[!redirects constructible universe]]

[[!redirects constructible universes]]

[[!redirects Gödel's constructible universe]]
[[!redirects Gödel's constructible universe]]
[[!redirects Godel's constructible universe]]
[[!redirects Godel's constructible universe]]


construction in philosophy

[[!redirects empty 154]]
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## Idea
**Construction in philosophy** is the name of an article by F. W. J. Schelling in   '_Kritisches Journal der Philosophie_' (1802) that was the preliminary culmination point of a development in philosophy going back to Descartes and Spinoza and forcefully updated by [[Immanuel Kant|I. Kant]] in '_Metaphysische Anfangsgründe der Naturwissenschaft_' (1786) , namely the attempt to philosophize '_more geometrico_'.

In the 20th century this tradition has been revitalized partly by the school of phenomenology and in the context of analytical philosophy by M. Dummett.

## The rationalist roots

Of course, the expression 'more geometrico demonstrata' was the programmatic slogan of [[Spinoza|Spinoza's]] system, but the philosopher had (somewhat surprisingly) purloined the decisive idea from Thomas Hobbes.

Descartes' renewal of philosophy was fuelled by the desire to achieve in metaphysics the same level of certainty as in geometry. Consequently, he was asked in the second objections to his 'Meditationes de prima philosophia' (1641) to give geometric versions of his proof of the existence of god, a challenge he met in his replies.

Whereas Descartes saw in this merely a convenient way of exposition, Hobbes proposed to view this proof 'by bringing about' as an adequate cognition of the essence of the thing constructed by the mind as a 'causa efficiens' and suggested to use it for political theory which also deals in human artifacts.

The next step was Spinoza's proposal to take construction as the model for the cognition of all objects not necessarily only man made. This had two results:

* 1) It forced the finite human subject to 'see things in god' as the absolute cause i.e. the cartesian ego is plunged into the absolute which starts to play the role of space as infinite background into which finite objects are constructed, a picture that very much underlies Schelling's intuitions during his constructivist period around 1802. One could view the type-theoretical mathematics done in a cohesive topos using a universe as a late descendent of the architecture envisioned by Spinoza and Schelling.

* 2) The coincidence of proof and object identifies being and thinking so that this absolute approach to cognition resurrects the Greek logos concept. This ontological corollary of Spinoza's system proved to be important for Hegel onwards from his cooperation with Schelling in Jena. His logic though contains in the later parts a severe criticism of the constructive method in philosophy since it runs against the primary importance of mediation in his philosophy by relying on immediate intuition.

## Peirce on construction in mathematics and philosophy

>Kant is entirely right in saying that, in drawing those consequences, the mathematician uses what, in geometry, is called a 'construction', or in general a diagram, or visual array of characters or lines. Such a construction is formed according to a precept furnished by the hypothesis. Being formed, the construction is submitted to the scrutiny of observation, and new relations are discovered among its parts, not stated in the precept by which it was formed, and are found, by a little mental experimentation, to be such that they will always be present in such a construction. Thus the necessary reasoning of mathematics is performed by means of observation and experiment, and its necessary character is due simply to the circumstance that the subject of this observation and experiment is a diagram of our own creation, the condition of whose being we know all about.

>But Kant, owing to the slight development which formal logic had received in his time, and especially owing to his total ignorance of the logic of relatives, which throws a brilliant light upon the whole of logic, fell into error in supposing that mathematical and philosophical necessary reasoning are distinguished by the circumstance that the former uses constructions. This is not true. All necessary reasoning whatsoever proceeds by constructions; and the difference between mathematical and philosophical necessary deductions is that the latter are so excessively simple that the construction attracts no attention and is overlooked.

[[Charles Peirce|C. S. Peirce]] _'The Logic of Mathematics in Relation to Education'_ (1898, CP iii. 350)

## Related entries

* [[constructivism]]

* [[intuitionism]]

* [[Attempt to Introduce the Concept of Negative Quantities into Philosophy|Concept of Negative Quantities]]

* [[Aufhebung]]

* [[absolute conclusion]]

* [[Science of Logic]]

* [[geometric theory]]

## References

* W. Bartuschat, _Baruch de Spinoza_ , Beck München 1996. (Section III.1.a)

* O. Becker, _Mathematische Existenz: Untersuchungen zur Logik und Ontologie mathematischer Phänomene_ , Jahrbuch für Philosophie und phänomenologische Forschung **VIII** (1927) pp.439-768. ([pdf](http://www.digizeitschriften.de/download/PPN827944462_0008/PPN827944462_0008___LOG_0009.pdf))

* F. W. J. Schelling, _Vorlesungen über die Methode des akademischen Studiums_ , Meiner Hamburg 1974[1803]. (Fourth lecture: _'Über das Studium der reinen Vernunftwissenschaften: der Mathematik, und der Philosophie im Allgemeinen'_)

* F. W. J. Schelling, _Konstruktion in der Philosophie_ , pp.201-224 in Schelling, Hegel (eds.), _Kritisches Journal der Philosophie_ , Reclam Leipzig 1981[1802].

* J. Weber, _Begriff und Konstruktion - Rezeptionsanalytische Untersuchungen zu Kant und Schelling_ , PhD Göttingen 1998. ([link](http://ediss.uni-goettingen.de/handle/11858/00-1735-0000-0022-5D40-7))


constructive analysis

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## Idea

Constructive analysis is the incarnation of [[analysis]] in [[constructive mathematics]].  It is related to, but distinct from, _[[computable analysis]]_; the latter is developed in [[classical logic]] explicitly using [[computability theory]], whereas constructive analysis is developed in [[constructive logic]] where the computation is implicitly built-in. One can compile results in constructive analysis to computable analysis using [[realizability]].

In applications in [[computer science]] one uses for instance the _[[completion monad]]_ for _exact_ computations with [[real numbers]] (as opposed to floating point arithmetic). Therefore one also sometimes speaks of _exact analysis_. See also at _[[computable real number]]_.

## Related concepts

[[!include computable mathematics -- table]]

* [[computable real number]]

* [[exact real computer arithmetic]]

* [[computable physics]]

* [[constructive set theory]]

* [[constructive algebraic topology]]

## References
 {#References}

Introducing the formulation of [[analysis]] in [[constructive mathematics]], together with the notions of [[Bishop set]]/[[setoid]]:

* {#Bishop} [[Errett Bishop]]: *[[Foundations of Constructive Analysis]]*, McGraw-Hill (1967)

* [[Errett Bishop]], [[Douglas Bridges]]: *[[Constructive Analysis]]*, Grundlehren der mathematischen Wissenschaften **279**, Springer (1985) [[doi:10.1007/978-3-642-61667-9](https://doi.org/10.1007/978-3-642-61667-9)]

Introduction and review:

* [[Douglas Bridges]], *Constructive mathematics: a foundation for computable analysis*, Theoretical Computer Science **219** 1–2 (1999) 95-109 [doi:10.1016/S0304-3975(98)00285-0]

* [[Herman Geuvers]], [[Milad Niqui]], [[Bas Spitters]], [[Freek Wiedijk]], _Constructive analysis, types and exact real numbers_, Mathematical Structures in Computer Science **17** 01 (2007) 3-36 [[doi:10.1017/S0960129506005834](https://doi.org/10.1017/S0960129506005834)]

* [[Helmut Schwichtenberg]], *Constructive analysis with witnesses* (2013) [[pdf](https://www.mathematik.uni-muenchen.de/~schwicht/seminars/semss13/constr13.pdf), [[Schwichtenberg-ConstructiveAnalysis.pdf:file]]]

* {#Johansson19} Fredrik Johansson, *Reliable real computing*, talk at Mathematical Institute of Oxford (2019) [[pdf](http://fredrikj.net/math/oxford2019.pdf)]

See also:

* [[Helmut Schwichtenberg]], *Program Extraction in Constructive Analysis*, in *Logicism, Intuitionism, and Formalism*, Synthese Library, **341** Springer (2009) [[doi:10.1007/978-1-4020-8926-8_13](https://doi.org/10.1007/978-1-4020-8926-8_13)]

An undergraduate real analysis textbook taking a constructive approach using interval analysis is

* {#Bridger2019} Mark Bridger, _Real Analysis: A Constructive Approach Through Interval Arithmetic_, Pure and Applied Undergraduate Texts **38**, American Mathematical Society, 2019.

Implementations of constructive [[real number]] analysis in [[type theory]] implemented in [[Coq]] via the [[completion monad]]:

* [[Russell O'Connor]], *A Monadic, Functional Implementation of Real Numbers*, Math. Struc. Comp. Sci. **17** 1 (2007) 129-159 [[arXiv:cs/0605058](https://arxiv.org/abs/cs/0605058), [doi:10.1017/S0960129506005871](https://doi.org/10.1017/S0960129506005871)]

* [[Russell O'Connor]], _Certified exact transcendental real number computation in Coq_, in *Theorem Proving in Higher Order Logics. TPHOLs 2008*,  Lecture Notes in Computer Science **5170** (2008) 246-261 [[arXiv:0805.2438](https://arxiv.org/abs/0805.2438), [doi:10.1007/978-3-540-71067-7_21](https://doi.org/10.1007/978-3-540-71067-7_21)]

* [[Russell O'Connor]], _Incompleteness and Completeness: Formalizing Logic and Analysis in Type Theory_, PhD thesis, Radboud University Nijmegen (2009) [[ubn:2066/76032](https://repository.ubn.ru.nl/handle/2066/76032), [author webpage](http://r6.ca/Thesis/)]

* {#KrebbersSpitters11} [[Robbert Krebbers]], [[Bas Spitters]], *Type classes for efficient exact real arithmetic in Coq*, Logical Methods in Computer Science, **9** 1 (2013) lmcs:958 [[arXiv:1106.3448](http://arxiv.org/abs/1106.3448/), doi:10.2168/LMCS-9(1:1)2013]

> (via [[ufias2012:Type classes]] in [[Coq]])

Exposition in:

* [[Bas Spitters]], *Verified Implementation of Exact Real Arithmetic in Type Theory*, talk at *Computable Analysis and Rigorous Numeric* (2013) [[pdf](https://users-cs.au.dk/spitters/CARN.pdf), [[Spitters-ExactRealArithmetic.pdf:file]]]

With emphasis on the use of the [[univalence axiom]]:

* {#Booij17} [[Auke Booij]], _Constructive analysis in univalent type theory_, 2017 ([pdf](https://www.cs.bham.ac.uk/~abb538/slides/2018-02-darmstadt.pdf))

* {#Booij18} [[Auke Booij]], _Extensional constructive real analysis via locators_, Mathematical Structures in Computer Science **31** 1 (2021) 64-88  [[arXiv:1805.06781](https://arxiv.org/abs/1805.06781), [doi:10.1017/S0960129520000171](https://doi.org/10.1017/S0960129520000171)]

* {#Booij20} [[Auke Booij]], *Analysis in Univalent Type Theory* (2020) [[etheses:10411](http://etheses.bham.ac.uk/id/eprint/10411), [pdf](https://etheses.bham.ac.uk/id/eprint/10411/7/Booij2020PhD.pdf), [[Booij-AnalysisInUF.pdf:file]]]

Implementation in [[Agda]] of [[Cauchy real numbers]] according to [Bishop (1967)](#Bishop67):

* [[Martin Lundfall]], *Formalizing real numbers in Agda* (2015) [pdf, [[Lundfall-RealNumbersInAgda.pdf:file]], [github](https://github.com/MrChico/Reals-in-agda)]

* [[Zachary Murray]], *Constructive Analysis in the Agda Proof Assistant* [[arXiv:2205.08354](https://arxiv.org/abs/2205.08354), [github](https://github.com/z-murray/honours-project-constructive-analysis-in-agda)]

review:

* [[Zachary Murray]], *Constructive Real Numbers in the Agda Proof Assistant*, [talk at *CQTS*](Center+for+Quantum+and+Topological+Systems#MurrayFeb2023), (Feb 2023) [video:[YT](https://www.youtube.com/watch?v=7Q_sjfyPJqU)]

For the alternative [[Dedekind real numbers]] or [[HoTT book real numbers]], see the references there.

[[!redirects exact analysis]]


constructive Gelfand duality theorem

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#Contents#
* table of contents
{:toc}

## Idea

There is a statement and proof of [[Gelfand duality]] in [[constructive mathematics]]. This therefore makes sense in any [[topos]].

## Examples

* An application of constructive Gelfand duality in the [[presheaf topos]] over the [[semilattice of commutative subalgebras]] of a [[C-star algebra]] produces an internal [[locale]] from any noncommmutative $C^*$-algebra. See [[semilattice of commutative subalgebras]].

## References

A proof of Gelfand duality claimed to be constructive was given in.

* Bernhard Banaschewski; [[Christopher J. Mulvey]], _A globalisation of the Gelfand duality theorem_ Annals of Pure and Applied Logic, 137(1–3):62–103, 2006

This uses however [[Barr's theorem]] which requires itself non-constructive logic and requires the ambient topos to be a [[Grothendieck topos]]

A fully constructive proof is claimed in

* [[Thierry Coquand]], [[Bas Spitters]], _Constructive Gelfand duality for $C^*$-algebras_ , Mathematical Proceedings of the Cambridge Philosophical society , Volume 147, Issue 02, September 2009, pp 323-337 ([arXiv:0808.1518](http://arxiv.org/abs/0808.1518))

A review of some aspects of constructive Gelfand duality (with an eye towards [[Bohr toposes]]) is in

* {#HeunenLandsmanSpitters} [[Chris Heunen]], [[Klaas Landsman]], [[Bas Spitters]], section 2 of _A topos for algebraic quantum theory_, Comm. Math. Phys. 291:63–110, 2009, free access [doi](http://dx.doi.org/10.1007/s00220-009-0865-6), [arXiv:0709.4364](http://arxiv.org/abs/0709.4364)

Further disussion includes

* [[Simon Henry]], _Constructive Gelfand duality for non-unital commutative $C^\ast$-algebras_ ([arXiv:1412.2009](https://arxiv.org/abs/1412.2009))

* [[Simon Henry]], _Localic Metric spaces and the localic Gelfand duality_ ([arXiv:1411.0898](https://arxiv.org/abs/1411.0898))

[[!redirects Stone-Gelfand-Naimark theorem]]
[[!redirects constructive Gelfand duality]]


constructive mathematics

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# Constructive mathematics
* table of contents
{: toc}

## Idea

Broadly speaking, __constructive mathematics__ is [[mathematics]] done without the [[principle of excluded middle]], or other principles, such as the full [[axiom of choice]], that imply it, hence without "non-constructive" methods of [[formal proof]], such as [[proof by contradiction]]. This is in contrast to _[[classical mathematics]]_, where such principles are taken to hold.

There are variations of what exactly is regarded as constructive mathematics, for instance [[intuitionistic mathematics|intuitionism]] or [[predicative mathematics|predicativism]], see the list of schools [below](#OriginsAndSchools). But beware the ambiguity in terminology: In [[Brouwer]]-style [[intuitionistic mathematics]] one includes [[axioms]] that _[[contradiction|contradict]]_ [[classical logic]], while other authors use "intuitionistic" to mean what elsewhere is called "constructive", and referring only to rejection of [[excluded middle]] and [[axiom of choice|choice]]. Some authors (particularly [[material set theory|material set theorists]]) use "constructive" to mean *[[predicative]]* constructive and "intuitionistic" to mean impredicative constructive.
Other authors emphasize the necessity that constructive theories be [[proof relevance|proof relevant]], with denial of the excluded middle's universality subordinate to this requirement; see ([Harper 2013](#Harper13)).

__Constructivism__ is the philosophy that such mathematics is useful, or (more strongly) that non-constructive mathematics is wrong.  Historically, constructive mathematics was first pursued explicitly by mathematicians who believed the latter.  However, many modern mathematicians who do constructive mathematics do it not because of any philosophical belief about the wrongness of non-constructive mathematics, but because constructive mathematics is interesting in its own right.  In the '[pluralist](http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.93.9892)' approach to the [[foundations]] of mathematics, a constructive proof (when it exists) is better because it is valid in more versions of mathematics, but a classical proof remains valid for [[classical mathematics]].

Another motivation for modern mathematicians ---especially category theorists like those on the nLab--- is that the study of constructive mathematics has potential applications to non-constructive mathematics.  For example, even if one believes the principle of excluded middle to be true, the "[[internal logic|internal]]" version of excluded middle in many interesting [[category|categories]] is still false; thus constructive mathematics can be useful in the study of such categories, even if mathematics is "globally" non-constructive.  This is the neutral motivation for constructive mathematics from the [nPOV](/nlab/show/nPOV#Logic).

Here we write mostly about the mathematics, trying to be mostly neutral philosophically.

## Origins and schools
 {#OriginsAndSchools}

During the "[[foundations|foundational]] crisis" in mathematics around the beginning of the 20th century, a number of mathematicians espoused philosophies that are generally grouped together and labeled **constructivist**. The common feature of these philosophies was a rejection of axioms and principles of logic that lead to nonconstructive proofs.  There was much talk at the time about potential vs absolute infinity, but from an axiomatic perspective, it turns out that (if one stops short of finitism) the two main culprits are

* the [[axiom of choice]] and

* the principle of [[excluded middle]].

Thus, constructivists (including many still active today) reject proofs that make use of either of these.  (In fact, it was realized in 1975 by Diaconescu that the axiom of choice implies the principle of excluded middle.  Excluded middle is precisely the "finitely indexed" axiom of choice; see [[excluded middle]] for a proof.)

There are, however, differences among constructivists as well, even discounting the pluralists.

* Some, like [[Fred Richman]] (see (Richman 2000)), simply remove choice and excluded middle from [[classical mathematics]] with nothing to replace them. This is called [[neutral constructive mathematics]]. However, this makes it difficult to define a satisfactory notion of [[continuous function]], even from the [[real line]] to itself, without using [[locales]]; see (Waaldijk 2003).

* Others accept weaker versions of choice, such as [[countable choice]] or even [[dependent choice]]. [[Toby Bartels]] argues that the intuition behind accepting these justifies the (yet stronger) [[presentation axiom]] (that [[the category of sets]] has [[enough projectives]]).

* Others add "non-classical" axioms which contradict choice or excluded middle, but which are consistent in their absence, such as "all total functions $[0,1] \to \mathbb{R}$ are continuous" (the [[continuity axiom]] of the "[[intuitionism|intuitionistic]]" school of [[L. E. J. Brouwer]]) or "all partial functions $\mathbb{N} \to \mathbb{N}$ are computable" (the [[computability axiom]] of the "[[Russian constructivism|Russian]]" school of [[Andrey Markov Jr|A. A. Markov]], which is also called "constructive recursive analysis").

* [[Errett Bishop]] had his own branch of constructive mathematics, known as [[Bishop's constructive mathematics]] or [[BISH]], where [[equality]] and [[sets]] are defined notion rather than a primitive on [[sets]].

* Still others, following [[Hermann Weyl]], go even further and refuse to allow "impredicative" constructions; see [[predicative mathematics]].  Most of the work done by the schools of Brouwer, Bishop, and Markov (but not Richman) is also predicative, even though those founders were not adamant about it; as a result, predicativism often appears in the [[foundations]] of constructive mathematics (in particular, in those of [[Peter Aczel|Aczel]] and [[Per Martin-Löf|Martin-Löf]], but not those of [[Harvey Friedman|Friedman]] or [[Thierry Coquand|Coquand]]).

* An extreme form of constructive [[predicative mathematics]] is to reject the use of [[implication]] and the [[universal quantifier]] completely and work in [[coherent logic]], resulting in [[coherent mathematics]]. Excluded middle and double negation cannot be formed, since negation can only be formed as a [[sequent]] rather than a [[formula]] in the [[coherent logic]]. Examples of [[coherent mathematics]] include [[dependent type theory]] which does not make use of [[function types]] and [[dependent function types]], as well as mathematics done inside an [[arithmetic pretopos]].

* One also has [[geometric mathematics]], which is coherent mathematics with all [[colimits]] but no [[function sets]].

* Very extreme constructivists, like [[Doron Zeilberger]], reject the existence of infinite sets; see [[finitism]].  (Kronecker, the 'father of constructivism', is often considered a finitist, but as he came before the foundational crisis, it is difficult to classify him accurately.)  Ironically, this means that excluded middle and choice, in their naïve forms, may become acceptable, or even true.  (Even constructivists believe that they are true [[internal logic|in]] the category [[FinSet]] of [[finite sets]].)

* The most extreme position of all is to deny even the existence of very large finite sets.  This is called [[ultrafinitism]].  This has been studied largely by [[Alexander Yesenin-Volpin]] and (more recently) [[Edward Nelson]]; foundational systems such as [[soft linear logic]] can also be argued to have an ultrafinitist flavor.  Zeilberger also claims sympathy with ultrafinitist philosophy, although his actual work fits well into a (merely) finitist framework.

Many constructivists (like many classical mathematicians) believe in an absolute mathematical sense of "truth," and that in this sense choice and excluded middle are simply _wrong_.  (Some constructivists, using classically false axioms, can even refute them; others merely claim that no possible correct reasoning could ever prove them.  See [Truth versus assertability](#truevassert) below.)  To most mathematicians, this makes them seem quite strange.  Other constructivists adopt a wait-and-see attitude, or even a relative notion of truth (which can seem strange in another way).

The following is a partial list of schools and subtheories of constructive mathematics:

* [[neutral constructive mathematics]]

* Brouwer's [[intuitionistic mathematics]]

* [[Russian constructivism]]

* [[Bishop's constructive mathematics]]

* [[constructive reverse mathematics]]

* (at least related) [[explicit mathematics]]

## Topos theory

With the invention of [[topos|topos theory]] in the second half of the 20th century, a new sort of constructivism arose.  It was observed (by [[Bill Lawvere|Lawvere]] and others) that any topos with a [[natural numbers object]] has an [[internal logic]] which is powerful enough to interpret most of mathematics, but that this logic in general fails to satisfy choice and excluded middle.  This meant that even for a mathematican who likes to use choice and excluded middle (and _a fortiori_ for one who believes them to be "true"), there is a reason to care about what can be proven without them, because only if a proof is constructive can it be interpreted in an arbitrary topos with NNO.

Even starting from a "completely classical" world of mathematics, many interesting toposes arise naturally (such as the category of [[sheaf|sheaves]] on any [[topological space]], or more generally any [[site]]) whose internal logic is not [[classical logic]].  Then by internal reasoning in such a topos (which is constructive reasoning), one can prove various useful facts, which can then be reinterpreted as external statements about the behavior of the topos itself.  For example, the constructive theorem that every [[one-sided real number]] that is both a lower real and an upper real must be located (which, classically, is an utter triviality) becomes the theorem that any [[semicontinuous function]] (from any [[topological space]] to the [[real line]]) that is both upper and lower semicontinuous must be [[continuous map|continuous]] (which is at least somewhat nontrivial).

By now it is known that many of the non-classical axioms used by the early constructivists have natural models in particular toposes.  For instance, in the topos of sheaves on the real numbers $R$, the [[continuity axiom]] that "all total functions $R \to R$ are continuous" holds.  And in the [[effective topos]], the [[computability axiom]] that "all partial functions $N\to N$ are computable" holds.

However, there are no non-classical (or classical) axioms beyond "pure constructivism" that are true in _all_ toposes with NNO.  In particular, there is a [[free topos]] with NNO such that a statement is true in the free topos precisely when it is provable in pure (Richman-school) constructive mathematics.  This means that for an argument to apply in all toposes, even mild assumptions such as countable or dependent choice are unacceptable (but impredicativity is fine).  However, topos theory has also provided ideas that solve many of the problems with pure constructivism.  For example, a well-behaved notion of "continuous function" can be recovered by using [[locale|locales]] instead of topological spaces, which was first discovered in the context of toposes and is closely related to them in any case.

## Some features of constructive mathematics

### Rephrasing of classical ideas

Sometimes, all that is necessary to make a piece of mathematics constructive is careful use of language.  It is common in [[classical mathematics]] to define things with an unnecessary amount of [[negation]].  This doesn't work so well constructively, since a statement can be not false without being true, but we can sometimes do perfectly well by just removing unnecessary [[double negations]].

For example, classically one often speaks about "nonempty" sets, meaning a set which "does _not_ have _no_ elements."  Constructively it is much better to say that a set "*does* have at least one element"; constructivists often call such a set _[[inhabited set|inhabited]]_ to remind themselves that it is a "positive" notion to replace the negative one of "nonempty". Others continue to use the word "non-empty" but understand it as a term of art that really means "inhabited".

### Bifurcation of notions

On the other hand, differences in axiomatization or definition that make no difference classically can result in actual differences in behavior constructively.  Therefore, classically equivalent notions often "bifurcate" (or "trifurcate" or worse) into multiple inequivalent constructive ones.  This tends to happen whenever a concept involves _[[negation]]_ and, to a lesser degree, _[[disjunction]]_ and _[[existential quantification]]_.  In some cases there is a "correct" constructive version of the definition, although it may take some thought to uncover it, but in many cases more than one of the resulting concepts is important and useful.  For example:

* There are multiple inequivalent constructive definitions of a [[field]], because of the axioms "every _nonzero_ element has an inverse" and $0 \neq 1$.

* In pure constructive logic, there are multiple inequivalent definitions of [[real numbers]].
  * the [[Dedekind real numbers]] and the [[Cauchy real numbers]] need no longer coincide.  The Cauchy reals sit inside the Dedekind reals, but in general not every Dedekind real need be approximable by a _[[sequence]]_ of rationals.  From a topos-theoretic viewpoint, the Dedekind reals are usually the "correct" notion to study (if not the [[locale of real numbers]] as a whole). However, the weak [[limited principle of omniscience]] suffices to ensure that every Dedekind real is a Cauchy real, and hence that the two notions coincide; see ([[Univalent Foundations Project]] 2013) and (Booij 2020). Weak countable choice implies the weak limited principle of omniscience, which in turn is implied by excluded middle and the axiom of choice; see (Bridges et al. 1993).
  * Similarly, the [[Cauchy real numbers]] are not sequentially [[Cauchy complete]]. There is an intermediate set of [[real numbers]] between the Cauchy reals and the Dedekind reals that are sequentially Cauchy complete called the [[HoTT book real numbers]], such that there are embeddings from the Cauchy reals to the HoTT reals, and from the HoTT reals to the Dedekind reals, but there are no embeddings in the reverse direction.
  * A real number with a [[locator]] is equivalent to a real number represented by a [[Cauchy sequence]] with [[modulus of convergence]], which implies that it is [[sequentially complete]]. That every [[modulated Cauchy real number]] has a [[locator]] or is a [[Cauchy sequence]] with [[modulus of convergence]] implies that the Cauchy real numbers coincide with the HoTT real numbers.
  * That every real number with a locator in the [[unit interval]] can be represented by a [[sequence]] of (positive) digits is equivalent to the lesser [[limited principle of omniscience]], so it is no longer true that every such real number has an infinite decimal representation. The set of all real numbers with infinite decimal representations are called [[prealgebra real numbers]].

* There are at least three different constructive notions of [[ordinal number]]; see (Taylor 1996) and (Joyal--Moerdijk 1995).

* Without the axiom of choice, [[functor|functors]] and [[anafunctor|anafunctors]] become distinct, and often the latter is more appropriate; see (Makkai, 1996).

* Perhaps most disturbingly of all to the classical mathematician, one must distinguish between _finite sets_, _subfinite sets_, _finitely-indexed sets_, and even _subfinitely indexed sets_; see [[finite set]] for definitions.  However, in practice it is usually either finite or finitely-indexed sets that are important, and with practice a little bit of thought suffices to show which is the relevant concept.

### Negative translation

This allows one to translate classically valid theorems into intuitionistically valid theorems. See [[double negation translation]].

### Truth versus assertability
{#truevassert}

Already in [[classical mathematics]], there is a difference between saying that something is true and saying that something is [[proof|provable]]. If you adopt [[ZFC]] because you believe it correct, then presumably you believe that ZFC is consistent even though (assuming that you\'re aware of certain theorems) you also know that you cannot prove it so. In that case, you also believe that the [[continuum hypothesis]] (for example) is either true or false; you may or may not have an opinion on which it is, but in any case again you know that you cannot prove it either way.

A constructive mathematician can be even subtler. If you belong to the Bishop school, then you accept no classically false axioms, and anything that you can prove is valid also in classical mathematics. Even so, you can believe that the principle of excluded middle is false (even though, of course, you know that you cannot prove it false). So you can really confuse the classical mathematicians by saying, on the one hand, that they can safely accept all of your theorems as valid, and then saying, on the other hand, that you know excluded middle to be false. The resolution, of course, is that you never claimed that this was a *theorem*.

This way of talking has even been formalised in (Bishop, 1967) with a convention adopted by many (but not all) of his followers. In this convention, the word 'not', used normally in a vernacular sentence whose mathematical content would otherwise be the proposition $p$, changes to the content to $\neg p$, or $p \to \bot$, as usual. (This follows all of the rules of [[intuitionistic logic]]; in particular, any statement that is 'not' true is false (just as in [[classical logic]]).) However, the word 'not', in italics as shown here, changes the content to $p \to x$, where $x$ is some statement that is known (although not proved!), according to Bishop, to be false. Now a statement that is 'not' true may be false, but this may be unknown, and it is even possible that it is also 'not' false (so $\neg p \to x$, possibly for a different $x$).

Bishop gives in his introduction several statements, suitable for use as $x$ above, including:

* excluded middle itself, which Bishop call the 'principle of omniscience';
* the '[[limited principle of omniscience]]': any infinite sequence in $\{0,1\}^N$ is either all $0$ or has at least one $1$;
* the '[[lesser limited principle of omniscience]]': for any two infinite sequence in $\{0,1\}^N$ that do not both have at least one $1$, at least one of these sequences does not have at least one $1$;
* others.

At one point in his book, while discussing the possibility of a pointwise-continuous function $[0,1] \to R$ that is not uniformly continuous (a [[Specker function]], whose existence Markov\'s school claims as a theorem), Bishop seems to assert that this theorem is both not true and not false; he does not put it this way, so this may not be exactly what he meant, but there is no contradiction if it is. (But it *is* a contradiction, even in intuitionistic logic, if a statment is both not true and not false; indeed, a definition of 'false' may be taken to be 'not true'.)

This practice can be understood through a careful distinction between [[object language]] and [[metalanguage]]. A mathematical statement $p$ (such as the continuum hypothesis, or that a Specker function exists) belongs to the object language, as does $\neg p$ or $p \to x$ for any specific mathematical statement $x$. But the statement that $p$ is provable in some formal system belongs to the metalanguage (although it can also phrased internal to any object language suitable for mathematics), and as such may be written $\vdash p$. The metalanguage has its own logic (the metalogic, which for the sake of argument we may even assume to be classical), but notice that $\neg \vdash p$ and $\vdash \neg p$ are different; the first claims that $p$ cannot be proved, while the second claims that $p$ can be refuted, which (in any [[consistent logic|consistent]] formal system, even a classical one) is strictly stronger. Although Bishop did not commit to any formal system, if we assume for the sake of argument that he settled on one, then we may write $\vdash \neg p$ as our interpretation of his meaning when he asserts 'not $p$' but $\neg \vdash p$ as his meaning when he asserts 'not $p$'.

Even without the notational convention of the Bishop school, it is important when reading constructive mathematics to remember the difference between what can be refuted (proved false) and what merely cannot be proved true. Although Brouwer\'s and Markov\'s schools will sometimes claim to prove statements that are (classically) outright false (such as that every function $[0,1] \to R$ is pointwise-continuous), it is more common (and can happen in any school) that a constructive mathematician makes a claim that merely *sounds* false, when what they really mean is only that they don\'t accept what you say as true. Often modal phrases like 'not necessarily' will be used where Bishop would use 'not', as a clue that we\'re shifting to a metalanguage, or at least merely remaining agnostic rather than outright disagreeing.

The distinction between object language and metalanguage exists even in [[classical mathematics]], but it seems that most classical mathematicians are not used to remembering it, although it is not entirely clear why this should be so.  One possibly relevant observation is that even if $P$ is a statement which is neither provable nor disprovable (like the continuum hypothesis), in classical mathematics it is still provable that "$P$ is either true or false."  Moreover, classical [[model theory]] often restricts itself to [[two-valued model|two-valued models]] in which the only truth values are "true" and "false," although classical logic still holds in [[Boolean algebra|Boolean-valued]] models, and in such a case the truth value of $P$ may be neither "true" nor "false," although the truth value of "$P$ or not $P$" is still "true."  Certainly when talking about classical truths which fail constructively, such as excluded middle, it is important to remember that "fails to be provable" is different from "is provably false."

## Prehistory

In

* [[Georg Hegel]], _[[Phenomenology of Spirit]]_  (1807)

the following comment about mathematical proof (in section _[12 Historical and mathematical proof](https://www.marxists.org/reference/archive/hegel/works/ph/phprefac.htm#12)_ of the Preface) might be read as being a complaint about the traditional non-constructive concept of proof and about the traditional lack of [[proof relevance]]:

> {#PoSOnProofRelevance} All the same, while proof is essential in the case of mathematical knowledge, it still does not have the significance and nature of being
a moment in the result itself; the proof is over when we get the
result, and has disappeared. The process of mathematical proof does not belong to the object; it is a function that takes place outside the matter in hand.

> [footnote 42](https://www.marxists.org/reference/archive/hegel/help/finpref.htm#m042): Mathematical truths are not thought to be known unless proved true. Their demonstrations are not, however, kept as parts of what they prove, but are only our subjective means towards knowing the latter. In philosophy, however, consequences always form part of the essence made manifest in them, which returns to itself in such expressions.

See also earlier conceptions of proofs expressing the 'cause' of a theorem, where *reductio* proofs in particular were taken generally to fail. Such an idea goes back to Aristotle for whom a proper answer to the question "Why is the angle in a semicircle a right-angle?" gives its cause.

* Paolo Mancosu, _Philosophy of Mathematics and Mathematical Practice in the Seventeenth Century_, OUP, 1996.

## Related entries

See also

* [[intuitionistic mathematics]]

* [[realizability]]

* [[computability]]

* [[constructive set theory]]

* [[constructive analysis]]

* [[constructive algebraic topology]]

* [[taboo]]

Concepts that usually arise in constructive mathematics, often because they are classically trivial:

*  [[proof relevance]]
*  [[anafunctor]] (classically equivalent to a [[functor]])
*  [[apartness relation]] (classically complementary to an [[equivalence relation]])
*  [[comparison]] (classically complementary to a [[transitive relation]])
*  [[decidable equality]] (classically trivial)
*  [[decidable subset]] (classically trivial)
*  [[inhabited set]] (classically equivalent to a non-[[empty set]])
*  [[pseudo-order]] (classically complementary to a [[total order]])
*  [[locale]] (classically similar to but not equivalent to a [[topological space]])
*  [[connected irreflexive comparison]] (classically complementary to a [[partial order]])
*  [[subsingleton]] (classically equivalent to the empty set or a [[singleton]])
*  [[Heyting field]], [[weak Heyting field]], [[discrete field]] (classically all equivalent to a [[field]])
*  [[weak local ring]], [[local ring]], [[discrete local ring]] (classically equivalent to a [[local ring]])
*  [[Cantor real numbers]], [[HoTT book real numbers]], [[Dedekind real numbers]] (classically all equivalent as the [[real numbers]])

Some of these are also useful internally or even classically.

Topics relevant to the foundations of constructive mathematics:

*  [[axiom of choice]]
*  [[Bishop set]]
*  [[centipede mathematics]]
*  [[choice object]]
*  [[COSHEP]]
*  [[excluded middle]]
*  [[finite mathematics]]
*  [[internalization]]
*  [[internal logic]]
*  [[intuitionistic logic]]
*  [[Markov's principle]]
*  [[power set]]
*  [[predicative mathematics]]
*  [[pretopos]]
*  [[set theory]]
*  [[topos]]
*  [[truth value]]
*  [[type theory]]
   *  [[intuitionistic type theory]]
   *  [[homotopy type theory]]

Other articles with content relating to constructive mathematics (rather incomplete):

*  [[axiom of foundation]]
*  [[biproduct]]
*  [[Cantor's theorem]]
*  [[cardinal number]]
*  [[complement]]
*  [[complete lattice]]
*  [[countable set]]
*  [[cyclic order]]
*  [[direct sum]]
*  [[extended natural number system]]
*  [[field]]
*  [[filter]]
*  [[finite set]]
*  [[Grothendieck topos]]
*  [[Hausdorff space]]
*  [[hereditarily finite set]]
*  [[infinite set]]
*  [[injection]]
*  [[local ring]]
*  [[metric space]]
*  [[partial function]]
*  [[pure set]]
*  [[real number]]
*  [[sequence]]
*  [[Set]]
*  [[Sierpinski space]]
*  [[simple object]]
*  [[sober space]]
*  [[topological space]]
*  [[Tychonoff theorem]]
*  [[uniform space]]
*  [[well-founded relation]]
*  [[well-order]]
*  [[well-ordering theorem]]
*  [[well-pointed topos]]
*  [[Zorn's lemma]]

In principle, every article could explain how it applies to constructive mathematics, but that will probably never happen.

There is also

* [[constructivism and idealism]]

## References
 {#References}

See also the references at _[[intuitionistic mathematics]]_ for more.

Original texts:

* {#Bishop} [[Errett Bishop]], *[[Foundations of Constructive Analysis]]*, McGraw-Hill (1967)

rewritten as:

* [[Errett Bishop]], [[Douglas Bridges]] *[[Constructive Analysis]]*, Grundlehren der mathematischen Wissenschaften **279**, Springer (1985) [[doi:10.1007/978-3-642-61667-9](https://doi.org/10.1007/978-3-642-61667-9)]

Early monographs:

* [[Anne Sjerp Troelstra]], [[Dirk van Dalen]], *Constructivism in Mathematics -- An introduction*, Vol 1, Studies in Logic and the Foundations of Mathematics **121**, North Holland (1988) [[ISBN:9780444702661](https://www.elsevier.com/books/constructivism-in-mathematics-vol-1/troelstra/978-0-444-70266-1)]

Gentle introductions:

* [[Douglas Bridges]], _Introducing constructive mathematics_, Lecture Notes, Nis 2013 ([pdf](http://www2.masfak.ni.ac.rs/cmfp2013/Nis%20lecture%20170113.pdf))

* [[Andrej Bauer]], _Five Stages of Accepting Constructive Mathematics_,
Bull. Amer. Math. Soc. 54 (2017), 481-498 ([pdf](http://dx.doi.org/10.1090/bull/1556)]. See also a talk at IAS March 18, 2013  ([video](http://video.ias.edu/members/1213/0318-AndrejBauer)).

* [[Fred Richman]], _[Interview with a constructive mathematician](http://math.fau.edu/richman/docs/intrview.html)_,  Modern Logic 6 (1996), no. 3, 247–271 ([MathSciNet](http://www.ams.org/mathscinet-getitem?mr=1400617))

* {#Blechschmidt15} [[Ingo Blechschmidt]], _Double-negation translation and CPS transforms_, 2015 ([pdf](http://rawgit.com/iblech/talk-constructive-mathematics/master/negneg-translation-notes.pdf))

* Stanford Encyclopedia of Philosophy, _[Constructive mathematics](http://plato.stanford.edu/entries/mathematics-constructive/)_

An more technical introduction to constructive reasoning in mathematics is (with an eye towards [[homotopy type theory]]) in the introduction of:

* [[Univalent Foundations Project]], _[[Homotopy Type Theory -- Univalent Foundations of Mathematics]]_

Other accounts:

* [[Michael J. Beeson]], *Foundations of Constructive Mathematics*,  Ergebnisse der Mathematik und ihrer Grenzgebiete **3** 6, Springer 1985 ([doi:10.1007/978-3-642-68952-9](https://link.springer.com/book/10.1007/978-3-642-68952-9), [pdf](https://link.springer.com/content/pdf/10.1007%2F978-3-642-68952-9.pdf))

* [[Douglas Bridges]] and [[Fred Richman]], _Varieties of constructive mathematics_ (1987)

* [[Fred Richman]], [[Douglas Bridges]], Peter Schuster, *A weak countable choice principle*. Proceedings of the American Mathematical Society 128(9):2749-2752, March 2000. [[doi:10.1090/S0002-9939-00-05327-2](http://dx.doi.org/10.1090/S0002-9939-00-05327-2)]

*  [[Michael Makkai]] (1996). [Avoiding the axiom of choice in general category theory](http://www.math.mcgill.ca/makkai/anafun/).

* [[Fred Richman]], *Constructive Mathematics without Choice*,
in: *Reuniting the Antipodes -- Constructive and Nonstandard Views of the Continuum*, Synthese Library **306**, Springer (2001) 199-206 [[doi:10.1007/978-94-015-9757-9_17](https://doi.org/10.1007/978-94-015-9757-9_17)]

*  [[Paul Taylor]] (1996). Intuitionistic Sets and Ordinals. Available (with several other references) at [Induction, recursion, replacement and the ordinals](http://www.PaulTaylor.EU/ordinals/index.php).

*  [[André Joyal]] and [[Ieke Moerdijk]] (1995). _Algebraic set theory_.

*  [[Frank Waaldijk]] (2003). [On the foundations of constructive mathematics - especially in relation to the theory of continuous functions](http://www.fwaaldijk.nl/foundations%20of%20constructive%20mathematics.pdf) (PDF).

*   [[Auke B. Booij]], Analysis in univalent type theory ([pdf](https://etheses.bham.ac.uk/id/eprint/10411/7/Booij2020PhD.pdf))

In view of [[reverse mathematics]]:

* [[Hajime Ishihara]], _Reverse Mathematics in Bishop's Constructive Mathematics_, Philosophia Scientiæ, CS 6 (2006) ([doi:10.4000/philosophiascientiae.406](https://doi.org/10.4000/philosophiascientiae.406),  [pdf](https://philosophiascientiae.revues.org/pdf/406))

* [[Hannes Diener]], *Constructive Reverse Mathematics*, 2018 ([arXiv:1804.05495](https://arxiv.org/abs/1804.05495), [dspace:ubsi/1306](https://dspace.ub.uni-siegen.de/handle/ubsi/1306))

General comments on intuitionistic mathematics/logic as the natural language for [[physics]] are in

* [[Andrej Bauer]], _[Intuitionistic mathematics for physics](http://math.andrej.com/2008/08/13/intuitionistic-mathematics-for-physics/)_, August 2008

For more on [[physics]] formalized in intuitionistic mathematics (notably in [[topos theory]]) see at _[[geometry of physics]]_.

For an emphasis on [[proof relevance]], see:

* {#Harper13} [[Robert Harper]],  _[Constructive Mathematics is not Metamathematics](https://existentialtype.wordpress.com/2013/07/10/constructive-mathematics-is-not-meta-mathematics/)_ (2013)

Most books on [[topos theory]] include some discussion of toposes\' [[internal logic|internal]] constructive logic.  One good reference is:

*  [[Peter Johnstone]] (2003). _[[Elephant|Sketches of an elephant]]_. Part D (in volume 2).

A historical account is in

* {#Troelstra91} [[Anne Sjerp Troelstra]], *History of Constructivism in the Twentieth Century* (1991) [[pdf](https://www.illc.uva.nl/Research/Publications/Reports/ML-1991-05.text.pdf), [[Troelstra-HistoryOfConstructivism.pdf:file]]]

The relation to [[realizability]] and [[computability]] is discussed in

* {#Bauer05} [[Andrej Bauer]], _Realizability as connection between constructive and computable mathematics_, in T. Grubba, P. Hertling, H. Tsuiki, and [[Klaus Weihrauch]], (eds.) _CCA 2005 - Second International Conference on Computability and Complexity in Analysis_, August 25-29,2005, Kyoto, Japan, ser. Informatik Berichte, vol. 326-7/2005. FernUniversität Hagen, Germany, 2005, pp. 378–379. ([pdf](http://math.andrej.com/data/c2c.pdf))

On [[commutative algebra]] with constructive methods:

* [[Henri Lombardi]], [[Claude Quitté]] (2010): *Commutative algebra: Constructive methods (Finite projective modules)* Translated by Tania K. Roblo, Springer (2015) ([doi:10.1007/978-94-017-9944-7](https://link.springer.com/book/10.1007/978-94-017-9944-7), [pdf](http://hlombardi.free.fr/CACM.pdf))

[[!redirects constructive]]
[[!redirects constructive mathematics]]
[[!redirects constructivist mathematics]]

[[!redirects constructivist]]
[[!redirects constructivism]]

[[!redirects construction]]
[[!redirects constructions]]

[[!redirects constructive logic]]
[[!redirects constructivist logic]]


constructive model structure on simplicial sets

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#### Model category theory
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#### Homotopy theory
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#### Constructivism, Realizability, Computability
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#Contents#
* table of contents
{:toc}

## Idea

A [[model structure on simplicial sets]] valid in [[constructive mathematics]]
that reduced to the [[classical model structure on simplicial sets]]
in presence of the [[law of excluded middle]] and [[axiom of choice]].

The original [[proofs]] of the existence of the [[classical model structure on simplicial sets]] are based in [[classical mathematics]] as they use the [[principle of excluded middle]] and the [[axiom of choice]], and are hence not valid in [[constructive mathematics]]. This becomes more than a philosophical issue with the relevance of this [[model category]]-[[structure]] in [[homotopy type theory]], where [[internalization]] into the [[type theory]] requires [[constructive mathematics|constructive]] methods for interpreting [[proofs as programs]].

A constructively valid model structure on simplicial sets and coinciding with the [[classical model structure on simplicial sets|classical model structure]] if [[excluded middle]] and [[axiom of choice]] are assumed was found in [Henry 19](#Henry19). Alternative simpler proofs were found in [Gambino-Sattler-Szumiło 19](#GambinoSattlerSzumilo19)

## Properties

### Fibrations and lifting properties

[[Fibrations]] and [[acyclic fibrations]]
are defined using the [[right lifting property]]
with respect to the usual sets of [[horn inclusions]] and [[boundary inclusions]].
However, the meaning of this definition is more refined:
due to the constructive nature of the [[existential quantifier]],
being an (acyclic) fibration means we have a section
of the map from the set of solutions to lifting problems to the set of
lifting problems.

### Weak equivalences

[[Simplicial weak equivalences]] must also be defined more carefully.
Suppose $f\colon X\to Y$ is a [[simplicial map]].

We follow [GambinoSattlerSzumilo19](#GambinoSattlerSzumilo19).
They give two (equivalent) definitions.

In the first proof:

* If $X$ and $Y$ are [[cofibrant]] [[Kan complexes]], then $f$ is a weak equivalence if it is a [[simplicial homotopy equivalence]];
* If $X$ and $Y$ are [[Kan complexes]], then $f$ is a weak equivalence
if there is a [[simplicial map]] $f'\colon X'\to Y'$ and [[acyclic fibrations]] with [[cofibrant]] domains $X'\to X$ and $Y'\to Y$ such that $f'$ is a weak equivalence in the previous sense and the square with $f$ and $f'$ commutes;
* If $X$ and $Y$ are [[cofibrant]], then $f$ is a weak equivalence
if for any [[Kan complex]] $K$ the map $Hom(f,K)$ is a weak equivalence in the previous sense (notice that its domain and codomain are [[Kan complexes]]);
* If $X$ and $Y$ are arbitrary, then $f$ is a weak equivalence
if it has a replacement $f'\colon X'\to Y'$ like above, with $f'$ being a weak equivalence in the previous sense.

In the second proof:

* Within [[cofibrant]] simplicial sets, weak equivalences are those maps writing as an acyclic cofibration followed by an acyclic fibration.
* In general, weak equivalences are defined via (strong) cofibrant replacement.

### Cofibrant objects

While in the [[classical model structure on simplicial sets]] all [[objects]] ([[simplicial sets]]) are [[cofibrant object|cofibrant]], a [[simplicial set]] is [[cofibrant]] in the constructive model structure
if and only if the inclusion of degenerate $n$-[[simplices]]
into all $n$-simplices is a [[decidable inclusion]] of sets.

Likewise, [[cofibrations]] $A\to B$ are [[simplicial maps]]
such that $A_n\to B_n$ is a [[decidable inclusion]] of sets
and the inclusion of degenerate $n$-simplices as a subset into $B_n\setminus A_n$
is also a [[decidable inclusion]].

Since not all objects are [[cofibrant]], [[proper model category|left properness]] is no longer automatic, but can still be established through nontrivial arguments.

This model structure is also [[right proper]].

See [Henry 19](#Henry19) and [Gambino-Sattler-Szumiło 19](#GambinoSattlerSzumilo19)
for a proof of these claims.

## References

* [[Simon Henry]], _Weak model categories in classical and constructive mathematics_ ([arXiv:1807.02650](https://arxiv.org/abs/1807.02650))

* {#Henry19} [[Simon Henry]], _A constructive account of the Kan-Quillen model structure and of Kan's $Ex^\infty$ functor_ ([arXiv:1905.06160](https://arxiv.org/abs/1905.06160))

* [[Nicola Gambino]], [[Simon Henry]], _Towards a constructive simplicial model of Univalent Foundations_ ([arXiv:1905.06281](https://arxiv.org/abs/1905.06281))

* {#GambinoSattlerSzumilo19} [[Nicola Gambino]], [[Christian Sattler]], [[Karol Szumiło]], _The constructive Kan-Quillen model structure: two new proofs_ ([arXiv:1907.05394](https://arxiv.org/abs/1907.05394))

[[!redirects constructive model structures on simplicial sets]]


constructive quantum field theory

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#### Physics
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#Contents#
* table of contents
{:toc}

## Idea

In the broad sense of the word _constructive quantum field theory_ refers to the mathematically rigorous construction of full (i.e. [[non-perturbative quantum field theory|non-perturbative]]) [[quantum field theories]]. More specifically the term has come to be used mainly for attempts to rigorously construct [[path integral]] [[measures]] for [[Wick rotation|Wick rotated]] Euclidean field theories on [[Minkowski spacetime]] ([Jaffe](#Jaffe)). This approach has led to a construction of [[scalar field theory]] in spacetime dimension 3, and Yang-Mills theory on a four-dimensional torus.

While a mathematically rigorous construction of [[perturbative quantum field theory]] is given by [[causal perturbation theory]]/[[perturbative AQFT]], construction of [[non-perturbative quantum field theories]] has remained by and large elusive, except for toy example of [[free field theories]] or low [[spacetime]] [[dimension]] (e.g. [[2d CFTs]] or [[scalar field theory]] in 3d) or [[topological quantum field theories]]. In fact the non-perturbative [[quantization of Yang-Mills theory]]([[QCD]]) in 4d is listed as one of the open "Millennium Problems" by the Clay Mathematics Institute (see [here](mass+gap#ReferencesMassGapProblem)).

It might be noteworthy that for the established rigorous construction of [[perturbative QFT]] via [[causal perturbation theory]]/[[perturbative AQFT]] a) the [[path integral]] or any [[measures]] that could go with it plays no role at all (instead the [[causal additivity]] of the [[S-matrix]] is axiomatized directly) and b) the construction is a [[formal deformation quantization]] ([Collini 16](perturbative+quantum+field+theory#Collini16)). This might suggest that rigorous construction of [[non-perturbative quantum field theory]] ought to analogously proceed via [[strict deformation quantization]].

## Related concepts

* [[AQFT]], [[FQFT]]

## References

Review includes

* {#Jaffe} [[Arthur Jaffe]], _Constructive quantum field theory_ ([pdf](http://www.arthurjaffe.com/Assets/pdf/CQFT.pdf))

* Stephen J. Summers, _A Perspective on Constructive Quantum Field Theory_ ([arXiv:1203.3991](http://arxiv.org/abs/1203.3991))

Discussion specifically of [[path integrals]] includes

* [[James Glimm]], [[Arthur Jaffe]], _[[Quantum physics -- A functional integral point of view]]_,  535 pages, Springer

* Simon, _Functional Integration in Quantum Physics_ (AMS, 2005)

* [[Sergio Albeverio]], Raphael Høegh-Krohn, Sonia Mazzucchi. _Mathematical theory of Feynman path integrals - An Introduction_, 2 nd corrected and enlarged edition,
Lecture Notes in Mathematics, Vol. 523. Springer, Berlin, 2008 ([ZMATH](href="http://www.zentralblatt-math.org/zmath/en/advanced/?q=an:pre05233292&format=complete))

* Sonia Mazzucchi, _Mathematical Feynman Path Integrals and Their Applications_,
World Scientific, Singapore, 2009.

* Magnen, Rivasseau, Seneor, _Construction of YM4 with an infrared cutoff_ ([CMP] (https://projecteuclid.org/euclid.cmp/1104253284))

Discussion of the problem of [[quantization of Yang-Mills theory]] from the point of view of constructive field theory is in

* Sourav Chatterjee, _Yang-Mills for probabilists_ ([arXiv:1803.01950](https://arxiv.org/abs/1803.01950))

[[!redirects constructive QFT]]
[[!redirects constructive field theory]]