phase semantics

# Phase semantics

* table of contents
{: toc}

## Idea

Girard's *phase semantics* is a way of building [[star-autonomous category|star-autonomous]] [[posets]] with [[exponential modalities]], hence models of classical [[linear logic]].

## Definition

Let $M$ be a commutative [[monoid]], and $\bot\subseteq M$ a specified subset.  ($M$ equipped with $\bot$ is sometimes called a *phase space*.)  Then $P M$, the [[powerset]] of $M$, is a commutative [[quantale]], with tensor product

$$ X\otimes Y = \{ m n \mid m\in X \wedge n \in Y \}$$

and internal-hom

$$ X\multimap Y = \{ m \mid \forall n \in X, m n \in Y \}. $$

Indeed, if we regard $M$ as a discrete [[poset]], hence as a category enriched over $\mathbf{2} = \{0\le 1\}$, then $P M$ is its $\mathbf{2}$-enriched [[presheaf category]], and this is its [[Day convolution]] monoidal structure.

Now $\bot$ is an object of $P M$, hence induces a contravariant self-[[adjunction]] $(-\multimap\bot) \dashv (-\multimap \bot)$ of $P M$, which is [[idempotent adjunction|idempotent]] since $P M$ is a poset.  We define a **fact** to be a [[fixed point of an adjunction|fixed point]] of this adjunction, i.e. a subset $X\in P M$ of the form $Y\multimap \bot$, or equivalently such that $X = (X\multimap\bot)\multimap\bot$.

+-- {: .un_theorem}
###### Theorem
The poset of facts is [[star-autonomous category|star-autonomous]].
=--
+-- {: .proof}
###### Proof
It is closed under $\multimap$, since $(X\multimap (Y\multimap\bot)) = (X\otimes Y\multimap \bot)$.  And it is reflective, with reflector $(-\multimap\bot)\multimap\bot$.  Thus, it is closed symmetric monoidal with tensor product $((X\otimes Y)\multimap\bot)\multimap\bot$.  Since it also contains $\bot$, as $\bot = (I\multimap \bot)$, it is therefore star-autonomous by construction.
=--

The poset of facts is also, of course, a [[complete lattice]], since it a reflective sub-poset of the complete lattice $P M$.  In addition, it admits [[exponential modalities]] $!$ and $?$.  There are different ways to obtain these, but perhaps the simplest (see [here](#GirardSS)) is to take

$$ !X = ((X \cap Idem \cap 1)\multimap\bot)\multimap\bot $$

where $Idem= \{ m \mid m m = m \}$ is the set of idempotents in $M$, and $1 = (\{1\}\multimap\bot)\multimap\bot$ is the unit object of the monoidal category of facts.

## Properties

Phase semantics is complete for provability in linear logic, i.e. if a sequent of linear logic is valid in the phase semantics for all choices of $M$ and $\bot$, then it is provable.  This follows from a construction of a particular $M$ and $\bot$ out of the syntax: let $M$ be the free commutative monoid on the formulas of linear logic subject to the relation that formulas of the form $?A$ are idempotent, and let $\bot$ be the set of $\Gamma\in M$ that are provable when regarded as one-sided sequents $\vdash\Gamma$, and interpret a formula $A$ by the set of $\Gamma\in M$ such that $\vdash \Gamma,A$ is provable.  See [Girard](#Girard) for details.

## Generalizations

When phrased categorically as above, there is an obvious generalization to the case when $M$ is a commutative monoidal *poset*, with $P M$ replaced by its $\mathbf{2}$-enriched presheaf category, i.e. the poset $D M$ of downsets in $M$.  We can also generalize to other enrichments, although in that case the idempotence of the adjunction $(-\multimap\bot) \dashv (-\multimap \bot)$ is no longer automatic but has to be assumed.

If $M$ is assumed only to be a [[promonoidal category|promonoidal]] $\mathbf{2}$-enriched category, i.e. equipped with a suitable [[relation]] $M\times M   ⇸ M$, then it is a [[ternary frame]].  This can be used to construct models of more general [[substructural logics]].

## References

*  [[Jean-Yves Girard]], _Linear logic_,   Theoretical Computer Science 50:1, 1987.  ([pdf](http://iml.univ-mrs.fr/~girard/linear.pdf))
 {#Girard}

* [[Jean-Yves Girard]], _Linear logic, its syntax and semantics_ ([pdf](http://www.cs.brandeis.edu/~cs112/2006-cs112/docs/girard-intro.pdf))
 {#GirardSS}

[[!redirects phase space semantics for linear logic]]
[[!redirects linear logic phase space]]
[[!redirects linear logic phase spaces]]
[[!redirects phase space for linear logic]]
[[!redirects phase spaces for linear logic]]
[[!redirects Girard phase space]]
[[!redirects Girard phase spaces]]


phase space

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Variational calculus
+--{: .hide}
[[!include variational calculus - contents]]
=--
#### Physics
+--{: .hide}
[[!include physicscontents]]
=--
=--
=--

\tableofcontents

## Idea
 {#Idea}

The _covariant phase space_ of a system in [[physics]] (a [[field theory]], in general) is

* the [[space]] of all of those [[field (physics)|fields]] on [[spacetime]] which solve the [[equations of motion]], hence

* the space of [[on-shell]] [[field histories]], hence

* the _[[space of trajectories]]_ of the system.

Often one considers a parameterization of this by [[initial value data]] of fields on a [[Cauchy surface]] inside [[spacetime]]. This parameterization is what traditionally is just called a "phase space", or *canonical phase space*, for emphasis. Its elements are typically described by

1. "[[canonical coordinates]]": the value of the [[field (physics)|fields]] themselves, on the Cauchy surface

1. "[[canonical momenta]]": essentially the temporal [[derivatives]] of the [[field (physics)|fields]], on the Cauchy surface.

The "covariant" in "covariant phase space" is to indicate that it is obtained without any (necessarily un-[[natural transformation|natural]], hence in physics jargon: non-covariant) choices of [[foliation]] of spacetime by [[Cauchy surfaces]]. But if a canonical phase space exists, then time evolution of initial value data is an [[isomorphism]] from the "canonical" to the "covariant" phase space.

\begin{imagefromfile}
    "file_name": "CanonicalAndCovariantPhaseSpace-240124.jpg",
    "width": 700,
    "unit": "px",
    "margin": {
        "top": -40,
        "bottom": 20,
        "right": 0,
        "left": 10
    }
\end{imagefromfile}

For a system described by [[Lagrangian]] mechanics, the covariant phase space comes canonically equipped with a [[presymplectic structure]]. A proper _phase space_ or _[[reduced phase space]]_ is a [[quotient space]] of the covariant phase space on which the presymplectic structure refines to a [[symplectic manifold|symplectic structure]] or [[Poisson manifold|Poisson strucure]].

Typically these phase spaces are (locally) naturally parameterized by the suitable boundary conditions which uniquely determine the corresponding history of the physical system. Much of the literature on phase spaces deals with parameterizing these boundary conditions.

For instance for a non-relativistic [[particle]] propagating on a [[Riemannian manifold]] $X$ with the usual [[action functional]], a trajectory is uniquely fixed by the position $x \in X$ and the momentum $p \in T^*_x X$ of the particle at a given time. Correspondingly the space of all solutions and hence the (covariant) phase space of the system may be identified with the [[cotangent bundle]] $T^* X$ of $X$.

(The term "phase" in "phase space" can be related to the phase of [[complex numbers]] in this example, see at _[[phase and phase space in physics]]_.)

However, even [[reduced phase spaces]] are not all cotangent bundles, typically not, for instance, if they are obtained by [[symplectic reduction]]. This way a finite-dimensional phase space can sometimes describe continuous systems (e.g. in hydrodynamics) whch have infinitely many degrees of freedom; that phase space is however not a cotangent bundle of something in general.

## Covariant phase space

There are two main routes to the construction of the covariant phase space,

* (S) via [[presymplectic structure]];

* (P) via [[Poisson algebra]] structure.

\linebreak

### (S) Via pre-symplectic structures
 {#ViaSymplecticStructure}

We describe the canonical [[presymplectic structure]] on the covariant phase space of a [[local action functional]]. The _covariant phase space_ is defined as the space of critical points of an action functional or, equivalently, the space of solutions of its [[Euler-Lagrange equation]]s, also known as the _shell_. The shell is naturally embedded as a subset of the space of all field configurations. Below, terminology and notation are as in the discussion at [[variational bicomplex]].

Let $E \to X$ be a smooth [[fiber bundle]] over an $n$-dimensional [[spacetime]] $X$ and $j_\infty E$ the corresponding [[jet bundle]]. The total [[de Rham differential]] on $j_\infty E$ decomposes, $\mathbf{d} = d + \delta$, into the anti-commuting horizontal and vertical differentials, $d$ and $\delta$. The [[local action functional]]

$$
  S : \Gamma(E) \to \mathbb{R}
$$

is by definition given by a [[Lagrangian]]

$$
  L : \Gamma(j_\infty E) \to \Omega^{n}(X)
$$

as

$$
  S : \phi \mapsto \int_X L(j_\infty(\phi))
  \,.
$$

The [[variational calculus|variation]] of the action $S$ can be written in terms of the vertical differential:

$$
  \delta S (\phi) = \int_X (\delta L)(j_\infty(\phi)) \,,
$$
where the first $\delta$ the variational differential (as interpreted in the [bicomplex definition](variational+bicomplex#BicomplexDefinition)) and the second $\delta$ is the vertical differential. Thus, for the purposes of variational calculus, we can concentrate purely on the vertical differential of the Lagrangian, which canonically decomposes as follows:

$$
  \delta L = EL(\phi)\delta\phi - d \theta(\phi) \,,
$$

where $EL(\phi) = 0$ is the [[Euler-Lagrange equation]] on $\phi$, and where $EL(\phi)\delta\phi$ is a degree-$(n,1)$ and $\theta$ is a degree-$(n-1,1)$ element in the [[variational bicomplex]] (one variational form degree and, respectively, $n$ or $(n-1)$-spacetime form degrees).

+-- {: .num_remark #CanonicalThetaDensityInLocalCoordinates}
###### Remark

On a local [[coordinate]] patch $\{x^i\}$ for $X$ the form $\theta$ here is given by

$$
  \theta(\phi)
    =
  (\iota_{\partial_i} \frac{\delta L}{\delta  \phi^a_{,i}} )
  \wedge
   \delta \phi^a
  \,.
$$

If $L = L_{kin} + L_{pot}$ is the sum of a standard [[kinetic Lagrangian]] for a [[free field theory]] and a potential term that only depends on the fields themselves and not on their derivatives,  then this is at the same time the canonical [[multisymplectic form]] for the given [[field bundle]]. See at _[[multisymplectic geometry]]_ the section _[Examples -- Free field theory](#CanonicalThetaDensityInLocalCoordinates)_ for more on this relation.

=--

The above definition of $EL$ and $\theta$ in terms of $\delta L$ yields the following identity upon taking another exterior variational derivative of both sides

$$
  \delta(\delta L) = 0
  = \delta(EL(\phi)) \wedge \delta\phi
    + d \delta \theta(\phi)
  \,,
$$

where the first term on the right clearly vanishes when pulled back to the shell, $EL(\phi)=0$ on $X$. The above wedge product between variational forms is inherited from the wedge product on $\Omega^\bullet(j_\infty E)$. This implies that the [[presymplectic current]], $\omega(\phi) = \delta(\theta(\phi))$, is horizontally closed on shell:

$$
\begin{aligned}
  d \iota^* \omega(\phi)
  &= d \iota^* (\delta \theta(\phi)) \\
  &= \iota^* d (\delta \theta(\phi)) \\
  &= -\iota^* \delta(EL(\phi)) \wedge \delta\phi \\
  &= 0
  \, ,
\end{aligned}
$$

where $\iota: \mathcal{S}_L \to \Gamma(E)$ is the embedding of the solutions of the [[Euler-Lagrange equation]]s  (the shell) into the space of all field configurations. The reason is that $\iota^* \delta(EL(\phi)) = 0$ for each $x\in X$, since the functions $EL(\phi)$ are all constant (in fact $0$) on solutions.

The variational 1-form on the space of field configurations

$$
  \Theta = \int_{X|_{in}} \theta(\phi)
$$

given by an [[integration]] over a [[Cauchy surface]] $X|_{in}$ is the potential for the presymplectic form

$$
  \Omega = \delta \Theta = \int_{X|_{in}} \delta(\theta(\phi))
  = \int_{X|_{in}} \omega(\phi)
$$

on the space of field configurations. Since $\omega(\phi)$ is horizontally closed on shell, the the pullback of the presymplectic form $\iota^* \Omega$ is independent of the choice of the surface $X|_{in}$ (provided the choice is restricted to a single homology class of surfaces).

\begin{proposition}

The presymplectic form $\iota^* \Omega$ on the covariant phase space is symplectic iff the linearized [[Euler-Lagrange equation]]s, $EL(\phi)=0$, have a locally well-posed initial value problem on $X|_{in}$. In particular, in the presence of [[gauge symmetries]], due to the failure of uniqueness of solutions for given initial data on $X|_{in}$, the form $\iota^*\Omega$ is only presymplectic.
\end{proposition}

However, the infinitesimal actions of gauge symmetries exhaust the kernel of the $\iota^* \Omega$ and upon performing [[symplectic reduction]], we obtain the space of orbits of solutions under the action of gauge symmetries, which is the _physical_ or _[[reduced phase space]]_.

Notice that the form $\Omega$, on the field configuration space, does depend on the choice of Cauchy surface. Performing [[symplectic reduction]] gives the symplectic space of equivalence classes of solutions of equations of motion modulo [[gauge transformation]]s, and hence also the _[[reduced phase space]]_. Thus, the end point of the reduction no longer depends on the choice of the Cauchy surface.

\begin{remark}
\label{InverseProblem}
**Application to the inverse problem of the calculus of variations**
\linebreak

We discuss the _inverse problem_ of [[variational calculus]]:

given a [[presymplectic form]] on the locus of solutions of a system of [[partial differential equations]], when is it the covariant phase space of a [[local action functional]]?

(This section follows [Bridges, Hydon & Lawson](#BridgesHydonLawson).)

We use same notation as above. Namely, dependence on $\phi$ in local forms really means dependence on finitely many components of the infinite jet $j^\infty(\phi)$. Also, $\iota$ denotes the embedding of the space of solutions in the space of field configurations. Moreover, we presume to work on a sufficiently small neighborhoods in the space of solutions and field configurations that the Poincar\'e lemma applies.

Consider a system of partial differential equations $P(\phi)=0$, together with a local presymplectic form $\Omega = \int_{X|_{in}} \omega(\phi)$, where $\omega(\phi)$ is a degree-$(2,dim X-1)$ element of the [[variational bicomplex]], that is $\delta \omega = 0$. Suppose further that [[presymplectic current]] density $\omega(\phi)$ is horizontally conserved on solutions:

$$
  d  \iota^* \omega(\phi) = \iota^* d \omega(\phi) = 0
  \quad \implies \quad
  d \omega(\phi) = P(\phi) \lambda(\phi) + \delta(P(\phi)) \wedge \mu(\phi)
    = P(\phi) (\lambda(\phi) - \delta\mu(\phi)) + \delta (P(\phi)\mu(\phi))
  \, ,
$$
where $\lambda$ and $\mu$ are systems (suitably contracted with the $P(\phi)$ system) of degree-$(2,dim X)$ and degree-$(1,dim X)$ elements of the [[variational bicomplex]]. Using the variational closure of $\omega(\phi)$, we can conclude that
$\delta(P(\phi)(\lambda(\phi)-\delta\mu(\phi))) = 0$ and thus, locally, $P(\phi)(\lambda(\phi)-\delta(\phi)) = \delta(P(\phi)\lambda'(\phi))$. (...Justify why the representative on the last RHS can be chosen proportional to $P(\phi)$...)

On the other hand, the variational closure of $\omega(\phi)$ also implies the existence of a degree-$(1,dim X-1)$ form $\theta(\phi)$, such that $\delta(\theta(\phi)) = \omega(\phi)$. The following identity then allows us to (locally) reconstruct a Lagrangian whose [[Euler-Lagrange equation]]s are satisfied by solutions to $P(\phi)=0$.

$$
\begin{aligned}
  \delta(d \theta(\phi))
    &= - d(\delta\theta(\phi)) \\
    &= - d \omega(\phi) \\
    &= - \delta[P(\phi)(\lambda'(\phi)+\mu(\phi))] \\
  \implies d\theta(\phi) - \delta L(\phi)
    &= -P(\phi)\lambda''(\phi)
  \, ,
\end{aligned}
$$

where $\lambda''(\phi)$ is a degree-$(1,dim X)$ form and $L(\phi)$ is a Lagrangian degree-$(0,dim X)$ form. Rearranging the last equality as

$$
  \delta L(\phi) = P(\phi)\lambda''(\phi) + d\theta(\phi) \, ,
$$
we conclude that the [[Euler-Lagrange equation]]s of $L(\phi)$ are satisfied on solutions of $P(\phi)=0$, since $EL(\phi)\delta\phi = P(\phi)\lambda''(\phi)$.

\end{remark}

### (P) Via Poisson structures
 {#ViaPoissonStructure}

The covariant phase space can be embedded into the
[[space of field configurations]] as a subspace of the set of solutions that transversely intersects [[gauge orbit]]s. This embedding is characterized as the [[zero locus]] of the [[equations of motion]] and some [[gauge fixing]] conditions. The non-degenerate [[Poisson structure]] on the algebra of functions on the covariant phase space is given by the [[Peierls bracket]].

The [[Peierls bracket]] of two functions $A$ and $B$ is the antisymmetrized influence on $B$ of an infinitesimal perturbation of a [[gauge fixing|gauge-fixed]] [[action]] by function that restricts to $A$ on the embedding. The algebra of functions on the space of field configurations becomes a [[Poisson algebra]] in the following way. Pick a set of functions on the space of field configurations that restrict to a non-degenerate coordinate system on the embedded covariant phase space. These functions, together with the equations of motion and gauge fixing conditions define a Poisson bivector by being declared canonical, such that the [[kernel]] of the bivector coincides with the ideal generated by the equations of motion and the [[gauge fixing]] conditions. Obviously the Poisson structure thus constructed on the algebra of functions on field configurations is not unique and depends on the above choice of coordinates; the same non-uniqueness may be parametrized instead by a choice of a [[connection on a bundle|connection]] on the space of field configurations. The embedded covariant phase space becomes a leaf of the symplectic [[foliation]] of the space of field configurations.

### Via the BV-complex
 {#ReducedCovariantPhaseSpaceViaBV}

The [[BV-BRST complex]] of a [[local action functional]] is (the [[Isbell duality|formal dual]] to) a [[resolution]] of the _reduced_ covariant phase space (the quotient of the covariant phase space by symmetries). As discussed in more detail at _[[BV-BRST complex]]_ the ghost sector of that complex is a model for the quotient by the symmetries, whereas the antifield/antighost sector is a model for the [[critical locus]] of the action functional.

Moreover, by the nature of its construction, the BV-complex is canonically equipped with a _graded_ symplectic form $\Omega$, whose [[Gerstenhaber algebra|(-1)-graded Poisson bracket]] is called the _antibracket_ (essentially the canonical [[Schouten bracket]] on graded derivations, see at _[[schreiber:derived critical locus]]_). This is not the canonical symplectic form  $\int_\Sigma \omega$ on the [[reduced phase space]], as discussed above, but it is  something like a potential for it.

We want to claim the following

\begin{proposition}

Given a [[local action functional]] on a space of fields over a spacetime $X$. Let $d_{BV}$ denote the [[differential]] of the [[BV-BRST complex]] and let $d$ denote the horizontal de Rham differential on $X$. Then

$$
  d_{BV} \int_\Sigma \Omega = \int_X d \omega
  \,.
$$

If $X$ is a [[globally hyperbolic spacetime]] of the form $\Sigma \times [0,1]$ then this is

$$
  d_{BV} \int_\Sigma \Omega
    =
  \int_{\Sigma_{out}} \omega
   -
  \int_{\Sigma_{in}} \omega
  \,.
$$

\end{proposition}

We discuss this now in more detail. (The stament then also appears in [Cattaneo-Mnev-Reshetikhin 12, equation (9)](#CattaneoMnevReshetikhin12)).

Write

* $X$ for the [[spacetime]] on which the fields are defined;

* $Conf = Maps(X,V) = \Gamma_X(X \times V)$ for the [[configuration space]] of fields on $X$, where $V$ is
some [[vector space]] (for simplicity);

* $Conf_{BV} $ for the BV-extension; $V_{BV}$ for the corresponding [[graded vector space]];

* $\{v_a\}$ for a chosen [[basis]] of $V_{BV}$;

* $\Phi^a(x) \in \Omega_{loc,BV}^{0,0}(X \times Conf_{BV})$ for the functional whose value on $\phi^a v_a \in Conf$ is $\phi^a(x)$ (for $x \in X$);

* $\Omega_{loc,BV}^{\bullet,\bullet}(X \times Conf_{BV})$ for the corresponding [[variational bicomplex]].

* $\bar \Phi_a(x) \in \Omega^{dim X,0}_{loc}(X \times Conf_{BV})$ for the antifield corresponding to $\Phi^a(x)$ (which is an _antighost_ if $\Phi^a(x)$ is a ghost).

Then the symplectic form density $\Omega$ for the antibracket

$$
  \Omega \in \Omega^{(n,2)}_{loc, BV}(X \times Conf_{BV})
$$

is

$$
  \Omega = \int_X d_{var} \bar \Phi_a(x) \wedge d_{var} \Phi^a(x)
  \,.
$$

Then,

$$
  \begin{aligned}
    d_{BV} \int_\Sigma \Omega
    & =
    d_{BV} \left(
      \int_\Sigma
      d_{var} \bar \Phi_a(x) \wedge d_{var} \Phi^a(x)
    \right)
    \\
    & =
    - d_{var} (d_{BV} \int_\Sigma \bar \Phi_a(x)) \wedge d_{var} \Phi^a(x)
  \end{aligned}
$$

We observe that the term in parenthesis is -- in the notation at _[[schreiber:derived critical locus]]_ (so we are assuming now the assumptions made there) --

$$
  [\hat S + d_{BRST}, d_{W(BRST)}]
  =
  [\hat S, d_{W(BRST)}]
  \,.
$$

Hence

$$
\begin{aligned}
    \cdots & =
    - d_{var} EL_a(x) \wedge d_{var} \Phi^a(x)
    \\
    & = - \int_X d_{var}  EL
  \end{aligned}
  \,,
$$

where $EL \in \Omega^{dim X, 1}_{loc, BV}(X \times \mathrm{Conf}_{BV})$ is the [[Euler-Lagrange equation|Euler-Lagrange density]] as in [Zuckerman, p. 267](#Zuckerman). By equation g) there, this is

$$
  \cdots = - \int_X d \omega
  \,.
$$

## Examples

### Relativistic particle

The covariant phase space of the [[relativistic particle]] on a [[pseudo-Riemannian manifold]] $X$ is the space of [[geodesic]]s of $X$ (in the absence of a [[background gauge field]]).

### Chern-Simons theory

For [[Chern-Simons theory]] corresponding to a non-degenerate bilienear [[invariant polynomial]]
$\langle -,-\rangle$ on a [[Lie algebra]] $\mathfrak{g}$ the

* configuration space is the space $\Omega^1(\Sigma, \mathfrak{g})$ of [[Lie algebra valued forms]];

* covariant phase space is the space of field configurations $A$ whose [[curvature]] 2-form vanishes

  $$
    F_A = 0
  $$

* the [[presymplectic structure]] is

  $$
    \Omega(\delta A_1, \delta A_2) =
   \int_{\Sigma_0} \langle \delta A_1 \wedge \delta A_2\rangle
    \,.
  $$

See [[Chern-Simons theory]] and [[schreiber:∞-Chern-Simons theory]] for more details.

### In dg-geometry

(...)

Let the ambient context be that of [[dg-geometry]]. Let $C$ be an ordinary [[smooth manifold]], assumed finite dimensional for the moment, and $\exp(i S) : C \to \mathbb{R}/\mathbb{Z}$ an ordinary [[smooth function]] such that its 0-locus is an sub-manifold.

Then a presentation for the homotopy fiber of $\exp(i S)$ is given by the formal dual of the [[dg-algebra]]

$$
 (\wedge^{-\bullet}_{C^\infty(X)}\Gamma(T X), \iota_{(-)} d \exp(i S))
  =
 \left[
  \cdots \to \wedge_{C^\infty(X)}^2\Gamma(T X) \stackrel{{\iota_{(-)}d \exp(i S)}}{\to} \Gamma(T X) \stackrel{\iota_{(-)}d \exp(i S)}{\to} C^\infty(X)
  \right]
$$

concentrated in non-positive degree, which in degree $k$ has the $k$th exterior powers of the [[tangent vector]]s of $X$ and whose differential is given by contracting a tangent vector with the 1-form $d \exp(i S)$.

This is a [[Koszul resolution]]-type resolution of the 0-locus of $\exp(i S)$. More generally, the homotopy fiber is given by a [[Koszul-Tate resolution]]-type complex. This is known as the _antifield complex_ in the [[BV-BRST formalism|BV-BRST formulation]] of derived phase spaces.

(...)

## Related concepts

[[!include symplectic reduction - table]]

## References

A a popular introduction to the notion of phase space with some comments on the history of the concept are in

* {#Nolte10} David Nolte, _The tangled tale of phase space_, PhysicsToday (April 2010) [[web](https://works.bepress.com/ddnolte/2)]

### Covariant phase space

Textbook account with an eye towards [[BRST-BV formalism]]:

* [[Marc Henneaux]], [[Claudio Teitelboim]], 17.1 in: _[[Quantization of Gauge Systems]]_, Princeton University Press (1992) [[ISBN:9780691037691](https://press.princeton.edu/books/paperback/9780691037691/quantization-of-gauge-systems), [jstor:j.ctv10crg0r](https://www.jstor.org/stable/j.ctv10crg0r)]

An article reviewing much of the content of the following references:

* {#Khavkine14} [[Igor Khavkine]], _Covariant phase space, constraints, gauge and the Peierls formula_, International Journal of Modern Physics A **29** (2014) [[arXiv:1402.1282](http://arxiv.org/abs/1402.1282), [doi:10.1142/s0217751x14300099](https://doi.org/10.1142/S0217751X14300099)]

Further review:

* {#Vitagliano} L. Vitagliano, *Secondary calculus and the covariant phase space* [[arXiv:0809.4164](https://arxiv.org/abs/0809.4164)]

* M. J. Gotay, J. Isenberg, R. Montgomery, J. E. Marsden, _Momentum Maps and Classical Fields, Part I: Covariant Field Theory_ [pdf](http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.6.6137&rep=rep1&type=pdf)

A discussion in the context of the [[variational bicomplex]] with further pointers to the use in [[physics]] is in

* E. Reyes, _On Covariant Phase Space and the Variational Bicomplex_ , Int. J. Theor. Phys. 43, no 5 (2004) 1267-1286

A discussion in the language of [[D-modules]], following the book [[Chiral Algebras]] and leading up to the derived covariant phase space by [[BRST-BV formalism]] is in section 8.3 of

* {#Paugam} [[Frédéric Paugam]], _Towards the mathematics of quantum field theory_ ([pdf](http://www.math.jussieu.fr/~fpaugam/documents/enseignement/master-mathematical-physics-impa-v01-2011.pdf))

Discussion in the context of [[smooth sets]]:

* {#GiotopoulosSati23} [[Grigorios Giotopoulos]], [[Hisham Sati]], §7.3 in: *Field Theory via Higher Geometry I: [[schreiber:Smooth Sets of Fields]]* [[arXiv:2312.16301](https://arxiv.org/abs/2312.16301)]

On the relation between covariant phase space methods and [[multisymplectic geometry]]:

* {#Helein} [[Frédéric Hélein]], _Multisymplectic formalism and the covariant phase_, in  Roger B ielawski, Kevin Houston, Martin Speight, J. Martin Speight (eds.) _Variational problems in differential geometry_, London Mathematical Society Lecture Note Series (2012) ([arXiv:1106.2086](http://arxiv.org/abs/1106.2086))

* {#BridgesHydonLawson} Thomas J. Bridges, Peter E. Hydon and Jeffrey K. Lawson, _Multisymplectic structures and the variational bicomplex_ [[doi](http://dx.doi.org/10.1017/S0305004109990259)]

* {#ForgerRomero04} [[Michael Forger]], [[Sandro Romero]], *Covariant Poisson brackets in geometric field theory*, Commun. Math. Phys. **256** (2005) 375-410 [[arXiv:math-ph/0408008](http://arxiv.org/abs/math-ph/0408008), [doi:10.1007/s00220-005-1287-8](https://doi.org/10.1007/s00220-005-1287-8)]

### Original articles on covariant phase space technology

In the following we give a commented list of references following the historical development.

It seems that there have been two independent lines of development,
somewhat independent of each other, but also with some
crosspollination along the way. One stream [(P)](#ViaPoissonStructure) was concerned with the
construction of a [[Poisson structure]] on functions of solutions, while the other [(S)](#ViaSymplecticStructure) was concerned with the construction of a  [[symplectic form]]
on the space of solutions. Obviously, the differences between the two
kinds of constructions show up precisely in [[gauge theory]] systems.

The earliest papers are those of Peierls and
Bergmann-Schiller in the early 50's. They seem to have been
self-motivated. Below, we try to list papers that have established key results or have served as important popularizers together with their
main influences from previous works. The constructions of the (S)
stream seem to have appeared nearly independently several times over
several decades, until Lee-Wald and followups of Ashtekar's papers
became standard references. The (P) stream flowed slowly, but
consistently from original idea of Peierls. The papers of
Fredenhagen-Dütsch-et-al are a most readable modern formulation.

* R. Peierls, _The commutation laws of relativistic field theory_ (1952) ([jstor](http://www.jstor.org/pss/99080))

  (P)

  Influence: Appears to be self-motivated. The result may have been guessed from the known unequal-time, covariant Poisson brackets between free fields, which involve the causal Green function in an
explicit way.

  Result: The [[Peierls bracket]] on the covariant phase pace of a non-gauge system is defined and equivalence with the Hamiltonian phase space symplectic structure is proved. Gauge systems are treated by example of electrodynamics. The solution to degenerate equations of motion is either adding gauge fixing terms to the action or consider non-degenerate equations of motion satisfied by gauge invariant fields. Fermions are handled by multiplying fermi fields by formal odd parameters.

* Bergmann-Schiller, _Classical and quantum field theories in the Lagrangian Formalism_ Phys. Rev. 89 4-16 (1953)  ([prola](http://prola.aps.org/abstract/PR/v89/i1/p4_1))

  (S)

  Influence: Appears to be motivated by Bergmann's long term and "brute force" analysis of the gauge invariance structure of relativity and structurally similar theories. Another motivation is the possibility of writing divergence-less stress energy tensors as "curls" of "superpotentials".

  Result: The pre-symplectic potential form-current makes an appearance. The definitions of the covariant phase space and the resulting symplectic form on it are obscure (at least to me), but later papers make reference to them.

* Bergmann-Goldberg-Janis-Newman, _Canonical Transformations and Commutators in the Lagrangian Formalism_ Phys. Rev. 103, 807–813 (1956) ([prola](http://prola.aps.org/abstract/PR/v103/i3/p807_1))

  (P&S)

  Influence: Peierls, Bergmann-Schiller

  Result: Proof of the equivalence between the Peierls bracket and the bracket obtained by Bergmann-Schiller for non-gauge systems. The
treatment of gauge systems and many of the calculations are maybe a  bit obscure.

* Komar, _Commutators on Characteristic Surfaces_
  Phys. Rev. 134, B1430–B1440 (1964)
  (proceedings w/ Bergmann 1962)

  (S)

  Influence: Bergmann-Schiller

  Result: Uses the methods of Bergmann-Schiller to define Poisson
brackets on initial data for linearized gravitational waves on null
hypersurfaces, including at null infinity.

* DeWitt B.S., DeWitt C. (eds.) _Relativity, Groups, and Topology_
  (Les Houches 1963, Gordon and Breach, 1964), pp.585-820
  (1964; lecture notes 1963)

  (P)

  Influence: Peierls

  Result: A more detailed exposition of Peierls' idea, with a bit more generality in the treatment of gauge systems. Becomes a standard reference along with Peierls' original paper.

* Friedman, _Generic instability of rotating relativistic stars_
  Commun. math. Phys. 62, 247--278 (1978) ([springer] (http://dx.doi.org/10.1007/BF01202527))

  (S)

  Influence: Bergmann-Schiller, Komar, ... (?)

* Ashtekar-Magnon, _On the symplectic structure of general relativity_ Communications in Mathematical Physics
Volume 86, Number 1, 55-68, DOI: 10.1007/BF01205661 (1982) ([springer](http://dx.doi.org/10.1007/BF01205661))

  (S)

  Influence: Komar, also some more obscure papers by Fridmann and Palmer

  Result: Write down the explicit pre-symplectic current and the
corresponding pre-symplectic form on the space of solutions of
linearized gravitational wave equations. This symplectic form on the reduced space of solutions is shown to coincide with the ADM symplectic structure and with the Komar symplectic structure at null
infinity. The motivation explicit pre-symplectic current formula appears from the desire to interpolate between the above two formalisms.

* [[Edward Witten]], §5 in: *Interacting field theory of open superstrings*, Nuclear Physics B **276** 2 (1986) 291-324 [<a href="https://doi.org/10.1016/0550-3213(86)90298-1">doi:10.1016/0550-3213(86)90298-1</a>, [inspire:228073](https://inspirehep.net/literature/228073)]

  (S)

  Influence: Appears to be self-motivated. Witten needed a covariant way to define Poisson brackets from the action of open string field theory. Acknowledges parallel, independent work of [Zuckerman](#Zuckerman).

  Result: Witten writes down the pre-symplectic form-current "by inspection" for a scalar field and for a Chern-Simons theory, then he explicitly shows that the current is conserved on the space of solutions, and that its integral over a Cauchy surface is variationally closed.

* Ashtekar, Bombelli, Koul, YS Kim, WW Zachary (eds.), The Physics of Phase Space: Nonlinear Dynamics and Chaos, Geometric Quantization, and Wigner Function
(p.356) ISBN 0-387-17894-5
(1987; preprint 1986)

  (S)

  Influence: Ashtekar-Magnon

  Result: Explicit formula for the symplectic form of full, nonlinear
[[general relativity]] as an integral of a conserved form-current over an arbitrary Cauchy surface. Discussion of conversion of symmetries into conserved quantities using this form. Some of the aspects related to the construction of such conserved have apparently been treated in earlier works by Friedman, Sachs and possibly some others.

* {#Zuckerman} [[Gregg Zuckerman]], _Action Principles and Global Geometry_, in Mathematical Aspects of String Theory, S. T. Yau (Ed.), World Scientific, Singapore, 1987, pp. 259-284. ([[ZuckermanVariation.pdf:file]])

  (S)

  Influence: Appears to be self-motivated. Possible motivation is a more general construction of conserved currents in variational problems, a la Noether. Acknowledges that he reproduces parallel, but applied to specific theories, work of [Crnković & Witten (1987)](#CrnkovićWitten87) and Ashtekar-Bombelli-Koul.

  Result: The construction of the pre-symplectic potential form-current and pre-symplectic form-current are given in essentially modern terms. Not all proofs are given. Little discussion of symplectic reduction.

* {#CrnkovićWitten87} [[Čedomir Crnković]], [[Edward Witten]], *Covariant Description of Canonical Formalism in Geometrical Theories*, chapter 16 in: S. W. Hawking and W. Israel (eds.) *Three Hundred Years of Gravitation*, Cambridge University Press (1987) 676-684 [[pdf](https://www.ias.edu/sites/default/files/sns/files/CovariantPaper-1987.pdf), [ISBN:9780521379762](https://www.cambridge.org/us/universitypress/subjects/physics/cosmology-relativity-and-gravitation/three-hundred-years-gravitation?format=PB&isbn=9780521379762)]

  (S)

  Influence: Witten

  Result: Applies the "by inspection" methods of Witten's earlier paper to write down conserved, variationally-closed pre-symplectic currents for Yang-Mills and general relativity.

* [[Čedomir Crnković]], *Symplectic Geometry of the Covariant Phase Space*, Class. Quant. Grav. **5** (1988) 1557-1575 [[doi:10.1088/0264-9381/5/12/008](https://doi.org/10.1088/0264-9381/5/12/008), [inspire:247290](https://inspirehep.net/literature/247290)]

* J. Lee, [[Robert Wald]], _Local symmetries and constraints_ (1989) J. Math. Phys. 31, 725 (1990); doi:10.1063/1.528801  ([jMP](http://jmp.aip.org/resource/1/jmapaq/v31/i3/p725_s1?isAuthorized=no))

  (S)

  Influence: Crnković-Witten, Friedman

  Result: Essentially gives the modern formulation as described above. The exposition is quite similar to Zuckerman (though no connection with that work is made), but with complete proofs.

* Ashtekar-Bombelli-Reula, _The Covariant Phase Space of Asymptotically Flat Gravitational Fields_ in Mechanics, Analysis and Geometry: 200 Years after Lagrange edited by M. Francaviglia and D.~Holm, North-Holland, Amsterdam, 1990. ([web](http://www.famaf.unc.edu.ar/~reula/Papers/200_Lagrange.pdf))

  (S)

  Influence: Ashtekar-...

  Result: Becomes standard reference. (?)

* [[Donald Marolf]],

  _Poisson Brackets on the Space of Histories_ Annals of Physics Volume 236, Issue 2, December 1994, Pages 374-391
  ([arXiv:hep-th/9308141](http://arxiv.org/abs/hep-th/9308141))

  _The generalized Peierls bracket_ ([arXiv:hep-th/9308150](http://arxiv.org/abs/hep-th/9308150))

  (1994; preprints 1993, thesis 1992)

  (P)

  Influence: Peierls, DeWitt; mentions but does not use the material
from Lee-Wald, Crnkovic-Witten, Ashtekar

  Result: Gives the construction outlined above.

* [[Stefan Hollands]], [[Donald Marolf]], _Asymptotic generators of fermionic charges and boundary conditions preserving supersymmetry_ ([arXiv](http://arxiv.org/abs/gr-qc/0611044)) (2007, preprint 2006)

  (S)

  Influence: Lee-Wald

  Result: Extends analysis to classical fermi fields.

* Dütsch [[Klaus Fredenhagen]], _The Master Ward Identity and Generalized Schwinger-Dyson Equation in Classical Field Theory_ ([arXiv:hep-th/0211242](http://arxiv.org/abs/hep-th/0211242)) (2003),

  Brennecke-Duetsch, _Removal of violations of the Master Ward Identity in perturbative QFT_ (2008) ([arXiv:0705.3160](http://arxiv.org/abs/0705.3160)),

  Rejzner, _Fermionic fields in the functional approach to classical field theory_  (2011) ([arXiv:1101.5126](http://arxiv.org/abs/1101.5126))

  (P)

  Influence: Peierls, DeWitt, Marolf; also some ideas related to retarded products that go back to Glaser-Lehmann-Zimmermann in the
1950s (more precisely: Glaser, V., Lehmann, H. and Zimmermann, W., "Field Operators and Retarded Functions" _Nuovo Cimen._ **6** (1957) 1122)

  Result: Modern, elementary treatment of the Peierls bracket, including for classical fermi fields. The discussion is simplified by sticking to non-degenerate (or gauge fixed) actions. The choice of coordinates on the space of field configurations is simplified by considering only vector bundle valued fields.

A different but related approach is

* {#KijowskiTulczyjew05} [[Jerzy Kijowski]], [[Wlodzimierz M. Tulczyjew]], *A Symplectic Framework for Field Theories*, Lecture Notes in Physics **107** (1979, 2005) [[doi:10.1007/3-540-09538-1](https://doi.org/10.1007/3-540-09538-1)]

also discussed (with focus on application to [[gravity]]) in:

* [[Alberto S. Cattaneo]], *Phase space for gravity with boundaries*,  Encyclopedia of Mathematical Physics (2023) [[arXiv:2307.04666](https://arxiv.org/abs/2307.04666)]

for the case of [[manifolds with boundary]]:

* [[Daniel Harlow]], [[Jie-qiang Wu]], *Covariant phase space with boundaries*,  J. High Energ. Phys. **2020** 146 (2020). [[arXiv:1906.08616](https://arxiv.org/abs/1906.08616), <a href="https://doi.org/10.1007/JHEP10(2020)146">doi:10.1007/JHEP10(2020)146</a>]

* Juan Margalef-Bentabol, Eduardo J. S. Villaseñor, *Geometric formulation of the covariant phase space methods with boundaries*, Phys. Rev. D **103** (2021) 025011 [[doi:10.1103/PhysRevD.103.025011](https://doi.org/10.1103/PhysRevD.103.025011), [arXiv](https://arxiv.org/abs/2008.01842)]

* Valle Varo, *The Covariant Phase Space of Gravity with Boundaries* [[arXiv:2301.12418](https://arxiv.org/abs/2301.12418)]

* [[Alejandro Corichi]], Juan D. Reyes, Tatjana Vukasinac, *On covariant and canonical Hamiltonian formalisms for gauge theories* [[arXiv:2312.10229]](https://arxiv.org/abs/2312.10229)]

### Reduced phase space

Standard textbooks on [[classical mechanics]] include

* [[Vladimir Arnold]], _Mathematical methods of classical mechanics_

* H. Goldstein, _Classical mechanics_

* L. D. Landau, E. M. Lifshitz, _Mechanics_, (vol. I of [[Landau-Lifschitz]] Course on theoretical physics)

* R. Abraham, J. E. Marsden, _Foundations of mechanics_

* J. E. Marsden, T. Ratiu, _Introduction to mechanics and symmetry_

* wikipedia: [phase space](http://en.wikipedia.org/wiki/Phase_space)

### BV formalism

Discussion via [[BV-formalism]] includes

* {#CattaneoMnevReshetikhin12} [[Alberto Cattaneo]], [[Pavel Mnev]], [[Nicolai Reshetikhin]], _Classical BV theories on manifolds with boundary_ ([arXiv:1201.0290](http://arxiv.org/abs/1201.0290))

[[!redirects phase spaces]]

[[!redirects covariant phase space]]
[[!redirects covariant phase spaces]]

[[!redirects canonical phase space]]
[[!redirects canonical phase spaces]]


phase space of pendulum


phase transition

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## Idea

(...)

## Related concepts

* [[Landau theory of phase transitions]]

* [[thermodynamic limit]]

* [[phase of matter]]

* [[quantum phase transition]]

* [[topological phase of matter]]

## References

Reviews:

* Alexander Umantsev, *Landau Theory of Phase Transitions*, in: *Field Theoretic Method in Phase Transformations*, Lecture Notes in Physics **840** (2012) 7–35 $[$[doi:10.1007/978-1-4614-1487-2_2](https://doi.org/10.1007/978-1-4614-1487-2_2)$]$

See also:

* Wikipedia, _[Phase transition](https://en.m.wikipedia.org/wiki/Phase_transition)_

Application of [[topological data analysis]] ([[persistent homology]]) to analysis of [[phase transitions]]:

* [[Alex Cole]], Gregory J. Loges, [[Gary Shiu]], _Quantitative and Interpretable Order Parameters for Phase Transitions from Persistent Homology_ ([arXiv:2009.14231](https://arxiv.org/abs/2009.14231))

[[!redirects phase transitions]]


phenomenology (philosophy)

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## Idea

**Phenomenology** is a philosophical movement initiated by [[Edmund Husserl]].

## References

* Stanford Encyclopedia of Philosophy, [Phenomenology](https://plato.stanford.edu/entries/phenomenology/)

* Wikipedia, Phenomenology (philosophy)


phenomenology (physics)

[[!redirects phenomenology]]

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## Idea

__Phenomenology__ is a part of [[theoretical physics]] concerned with a *selection* of a [[model (in particle physics)]] or parameters of a physical model for some physical process on the basis of finding consistency with diverse experimental data, but not necessarily modelling the regimes far from the situation in a considered range of experiments.

Therefore the phenomenology is not concerned with being fundamental, though it may help selecting fundamental models as well; it is interdisciplinary as it selects the models on the basis of various kinds of experimental data simulatenously; it is not concerned with models which are theoretically interesting but can not be measured or there is no concrete physical system known which behaves consistently with the model; phenomenological models are sometimes just the models of averaged or approximated level of description, even when more fundamental description is in principle known but too difficult to study or compare with the experiment.

For example, while so-called [[standard model of particle physics]] has inconsistencies if extended to very high energies (failure of the unitarity of [[S-matrix]], problems with the [[cosmological constant]]), the model is considered to agree to large extent with a low-energy limit of a true theory. Phenomenological models have to be in accordance with basic principles of physics, like thermodynamical laws, but do not need to pretend to go to the fundamental or microscopic description. For example, the Landau-Ginzburg equations for the description of superconductors do not describe the microscopic origin of supersymmetry, but are rather phenomenological, good enough desciption for certain range of mesoscopic phenomena.

## Related entries

* [[model (in particle physics)]]

  * [[standard model of particle physics]]

  * [[standard model of cosmology]]

* [[hierarchy problem]]

* [[LHC]]

* [[string phenomenology]]

* [string theory FAQ - Did string theory provide any insight relevant in experimental particle physics?](string+theory+FAQ#InsightInParticlePhysics)

* [[phenomenology (philosophy)]]


Phenomenology of Spirit

,
This page collects material related to

* [[Georg Hegel]]

  _Phänomenologie des Geistes_

  ( _The Phenomenology of Spirit_ )

  1807

  ([Wikipedia entry](http://en.wikipedia.org/wiki/The_Phenomenology_of_Spirit))

A brief version of roughly this material is in _[[Encyclopedia of the Philosophical Sciences]]_, third part, first section, B.

#Contents#
* table of contents
{:toc}

## Das Prinzip ist nicht die Vollendung

## Vorrede (Preface)

([German full text](https://www.marxists.org/deutsch/philosophie/hegel/phaenom/vorrede.htm#p3), [English translation](https://www.marxists.org/reference/archive/hegel/works/ph/phprefac.htm))

* {#Vorrede5} § 5 Daran mitzuarbeiten, daß die Philosophie der Form der Wissenschaft näher komme – dem Ziele, ihren Namen der Liebe zum Wissen ablegen zu können und wirkliches Wissen zu sein –, ist es, was ich mir vorgesetzt.

* To help to bring philosophy nearer to the form of science – that goal where it can lay aside the name of love of knowledge and be actual knowledge – that is what I have set before me.

(See also [SoL, Vorr. §8](Science+of+Logic#8).)

> {#ArbeitDesBegriffs} Wahre Gedanken und wissenschaftliche Einsicht ist nur in der Arbeit des Begriffs zu gewinnen.

> Er allein kann die Allgemeinheit des Wissens hervorbringen,  welche weder die gemeine Unbestimmtheit und Dürftigkeit des gemeinen Menschenverstandes, sondern gebildete und vollständige Erkenntnis,  noch die ungemeine Allgemeinheit der durch Trägheit und Eigendünkel von Genie sich verderbenden Anlage der Vernunft,  sondern die zu ihrer einheimischen Form gediehene Wahrheit, - welche fähig ist, das Eigentum aller selbstbewußten Vernunft zu sein.

## Vom wissenshcaftlichen Erkennen

## Jetziger Standpunkt des Geistes

### 12 Historical and mathematical proof

section _[12 Historical and mathematical proof](https://www.marxists.org/reference/archive/hegel/works/ph/phprefac.htm#12)_

the following complaint about [[proof]] in [[mathematics]] might be read as being a complaint about the traditional non-[[constructive mathematics|constructive]] concept of proof and about disregard of [[proof relevance]]:

> 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.

* {#Vorrede55} § 55 Das Dasein ist Qualität, sich selbst gleiche Bestimmtheit oder bestimmte Einfachheit, bestimmter Gedanke; dies ist der Verstand des Daseins. Dadurch ist es Nus, als für welchen Anaxagoras zuerst das Wesen erkannte.

* § 55 Existence is Quality, self-identical determinateness, or determinate simplicity, determinate thought: this is existence from the point of view of Understanding. On this account it is Nous, as Anaxagoras first thought reality to be.

[[Dasein]] is [[Nous]]

### 17 -- in its positive attitude; its subject

On "speculative sentences" as opposed to the usual "rational sentences"

* {#Vorrede63} § 63 die Meinung erfährt, daß es anders gemeint ist, als sie meinte, und diese Korrektion seiner Meinung nötigt das Wissen, auf den Satz zurückzukommen und ihn nun anders zu fassen.

* § 63 The common view discovers that the statement is intended in another sense than it is thinking of, and this correction of its opinion compels knowledge to recur to the proposition and take it now in some other sense.

* {#Vorrede64} § 64 erst diejenige philosophische Exposition würde es erreichen, plastisch zu sein, welche strenge die Art des gewöhnlichen Verhältnisses der Teile eines Satzes ausschlösse.

* § 64 only that philosophical exposition can manage to become plastic in character which resolutely sets aside and has nothing to do with the ordinary way of relating the parts of a proposition.

## related Entries

* [[Georg Hegel]], _[[Science of Logic]]_

## References

[[!redirects Phänomenologie des Geistes]]

category: reference


phi-meson

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## Idea

In [[quantum hadrodynamics]], the _$\Phi$-[[meson]]_ is a
the [[Lorentz group|Lorentz]] [[vector representation|vector]] and [[isospin]] [[scalar]] [[bound state|bound]] from

1. an [[up-quark|up]] or [[down-quark]] (from the [[first generation of fundamental particles]])

1. a [[strange quark]] (from the [[second generation of fundamental particles]].

Hence a [[heavy meson]] with respect to 2-flavor [[chiral perturbation theory]].

\begin{imagefromfile}
    "file_name": "LightAndChiralPartnerMesonFieldsPinIII.jpg",
    "width": 600,
    "unit": "px",
    "margin": {
        "top": 0,
        "right": 10,
        "bottom": 0,
        "left": 20
    }
\end{imagefromfile}

[[!include flavors of fundamental particles -- table]]

## Related concepts

* [[vector meson]]

* [[vector meson dominance]]

## References

* Rajesh Kumar, Arvind Kumar, _$\phi$-meson decay width in strange hadronic matter_ ([arXiv:2005.05133](https://arxiv.org/abs/2005.05133))

See also

* Wikipedia, _[Phi meson](https://en.wikipedia.org/wiki/Phi_meson)_

[[!redirects phi-mesons]]

[[!redirects phi meson]]
[[!redirects phi mesons]]

[[!redirects ϕ-meson]]
[[!redirects ϕ-mesons]]
[[!redirects ϕ meson]]
[[!redirects ϕ mesons]]


phi^n interaction

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## Idea

In [[quantum field theory]] of the [[scalar field]] $\Phi$, the canonical [[local observable|local]] [[interaction]] term is a [[Lagrangian density]] of the form

$$
  \mathbf{L}_{int}
  \;=\;
  \phi^n \, dvol_\Sigma
$$

(with notation as at _[[A first idea of quantum field theory]]_).

For $g_{sw} \in C^\infty_{cp}(\Sigma)$ any [[bump function]] on [[spacetime]], the corresponding [[adiabatic switching|adiabatically switched]] [[local observable]] is

$$
  \begin{aligned}
  S_{int}
  & =
  \underset{\Sigma}{\int}
  \underset{
     n \, \text{factors}
  }{
  \underbrace{
    \mathbf{\Phi}(x)
     \cdot
    \mathbf{\Phi}(x)
     \cdots
    \mathbf{\Phi}(x)
     \cdot
    \mathbf{\Phi}(x)
  }
  }
  \,
  dvol_\Sigma(x)
  \\
  & =
  \underset{\Sigma}{\int}
  :
  \underset{
     n \, \text{factors}
  }{
  \underbrace{
    \mathbf{\Phi}(x)
    \mathbf{\Phi}(x)
      \cdots
    \mathbf{\Phi}(x)
    \mathbf{\Phi}(x)
  }
  }
  :
  \,
  dvol_\Sigma(X)
  \end{aligned}
  \,,
$$

where in the first line we have the [[integral]] over a pointwise product ([this def.](A+first+idea+of+quantum+field+theory#Observable)) of $n$ [[field observables]] ([this def.](A+first+idea+of+quantum+field+theory#PointEvaluationObservables)), which in the second line we write equivalently as a [[normal ordered product]], by the discusssion at _[[Wick algebra]]_ ([this def.](Wick+algebra#NormalOrderedProductNotation)).

The [[interacting field theory]] with [[Lagrangian density]] that of the [[free field|free]] [[scalar field]] plus interactions of the form $\phi^k$ as above, up to order $n$, is often called simply "$\Phi^n$-theory".

## Examples

The [[mass]] term of the [[free field theory|free]] [[scalar field]] is a $\Phi^2$-interaction.

The [[Higgs field]] involves a quadratic and quartic interaction of this form.

The potential for the [[inflaton field]] in [[chaotic cosmic inflation]] is a $\Phi^2$-interaction.

## Related concepts

* [[perturbative quantum field theory]]

* [[non-perturbative quantum field theory]]

* [[Higgs field]]

* [[inflaton field]]

## References

An introduction to $\Phi^4$ theory could be found in lecture 13 of

* Sourav Chatterjee, _Introduction to Quantum Field Theory for Mathematicians_, [pdf](https://statweb.stanford.edu/~souravc/qft-lectures-combined.pdf)

The [[weak adiabatic limit]] for [[mass]]-less $\Phi^4$ theory was established in

* {#BlanchardSeneor75} P. Blanchard and R. Seneor, _Green’s functions for theories with massless particles (in perturbation theory)_, Ann. Inst. H. Poincaré Sec. A
23 (2), 147–209 (1975) ([Numdam](http://www.numdam.org/item?id=AIHPA_1975__23_2_147_0))

See also

* Wikipedia, _[Quartic interaction](http://en.wikipedia.org/wiki/Quartic_interaction)_

[[!redirects phi^n interactions]]

[[!redirects Phi^n interaction]]
[[!redirects Phi^n interactions]]

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[[!redirects phi^2 interactions]]

[[!redirects Phi^2 interaction]]
[[!redirects Phi^2 interactions]]

[[!redirects phi^3 interaction]]
[[!redirects phi^3 interactions]]

[[!redirects Phi^3 interaction]]
[[!redirects Phi^3 interactions]]

[[!redirects phi^n theory]]
[[!redirects phi^n theories]]
[[!redirects phi^n-theory]]
[[!redirects phi^n-theories]]

[[!redirects Phi^n theory]]
[[!redirects Phi^n theory]]
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[[!redirects Phi^n-theory]]

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[[!redirects phi^4 theories]]

[[!redirects phi^4-theory]]
[[!redirects phi^4-theories]]

[[!redirects Phi^4 theory]]
[[!redirects Phi^4 theories]]

[[!redirects phin theory]]


Phiala Shanahan

* [website](https://physics.mit.edu/faculty/phiala-shanahan/)

* [Wikipedia](https://en.wikipedia.org/wiki/Phiala_E._Shanahan)

## Selected publications

On the [[pressure]] inside the [[proton]] computed via [[lattice gauge field theory]]:

* [[Phiala Shanahan]], [[William Detmold]], *Pressure Distribution and Shear Forces inside the Proton*, Physical Review Letters **122** 7 (2019) [[doi:10.1103/PhysRevLett.122.072003](https://doi.org/10.1103/PhysRevLett.122.072003), [arXiv:1810.07589](https://arxiv.org/abs/1810.07589)]

category:people


Phil Ehlers

Phil Ehlers was a postgraduate student (1989-1993) at Bangor, working in the team of [[Ronnie Brown]] and [[Tim Porter]], who was Phil's doctoral supervisor.

## Selected writings

On [[simplicial groupoids]] (thanks are due to [[Stephen Gaito]] for preparing a scanned version with Phil's permission):

* [[Philip J. Ehlers]], _Simplicial groupoids as models for homotopy type_, Master's thesis (1991) [[pdf](https://ncatlab.org/nlab/files/Ehlers-MSc-thesis.pdf)]

also known as [[sSet]]-[[enriched groupoids]]:

* [[Philip J. Ehlers]], *Algebraic Homotopy in Simplicially Enriched Groupoids*, PhD thesis, University of Wales Bangor (1993) [[[Ehlers-thesis.pdf|pdf:file]]]

He has published several papers based on his work (jointly with Tim Porter).  These include:

* [[Phil Ehlers|P. J. Ehlers]] and [[Tim Porter]], _Joins for (Augmented) Simplicial Sets_,  Jour. Pure Applied Algebra, 145 (2000) 37-44
([arXiv](http://arxiv.org/abs/math.CT/9904039)) .

* [[Phil Ehlers|P. J. Ehlers]] and [[Tim Porter]], _Varieties of simplicial groupoids, I, Crossed complexes_, Jour. Pure Applied Algebra, 120 (1997) 221 - 233; Erratum: Jour. Pure Applied Algebra, 134 (1999) 221-233.

* [[Phil Ehlers|P. J. Ehlers]] and [[Tim Porter]], _Ordinal subdivision and special pasting in quasicategories_, Advances in Math. 217 (2007), No 2. pp 489-518.

category: people

[[!redirects Philip Ehlers]]

[[!redirects Philip J. Ehlers]]


Phil Hanlon

* [Wikipedia entry](https://en.wikipedia.org/wiki/Philip_J._Hanlon)

## Selected writings

On [[representation theory]] and [[geometric group theory]] in [[probability theory]] and [[statistics]] ([[Cayley distance]], [[Kendall distance]], ...):

* [[Persi Diaconis]], [[Phil Hanlon]], *Eigen Analysis for Some Examples of the Metropolis Algorithm*, in Donald Richards (ed.) *Hypergeometric Functions on Domains of Positivity, Jack Polynomials, and Applications*, Contemporary Mathematics Vol. 138, AMS 1992 ([doi:10.1090/conm/138](http://dx.doi.org/10.1090/conm/138), [EFS NSF 392](https://statistics.stanford.edu/research/eigen-analysis-some-examples-metropolis-algorithm), [pdf](https://statweb.stanford.edu/~cgates/PERSI/papers/eigen-metalg.pdf))

On something similar to (but different from) [[n-Lie algebras]]:

* [[Phil Hanlon]], [[Michelle Wachs]], _On Lie $k$-Algebras_, Advances in Mathematics Volume 113, Issue 2, July 1995, Pages 206–236 ([doi:10.1006/aima.1995.1038](https://doi.org/10.1006/aima.1995.1038))

category: people

[[!redirects Philip Hanlon]]


Phil Saad

* [webpage](https://physics.stanford.edu/people/phil-saad)

## Selected writings

On relation of [[random matrix theory]]

to [[Jackiw-Teitelboim gravity]] (via [[AdS/CFT]] and [[topological recursion]]):

* {#SaadShenkerStanford19} [[Phil Saad]], [[Stephen Shenker]], [[Douglas Stanford]], _JT gravity as a matrix integral_ ([arXiv:1903.11115](https://arxiv.org/abs/1903.11115))

and to the [[SYK model]]:

* Jordan S. Cotler, Guy Gur-Ari, Masanori Hanada, [[Joseph Polchinski]], [[Phil Saad]], [[Stephen Shenker]], [[Douglas Stanford]], Alexandre Streicher, Masaki Tezuka, _Black Holes and Random Matrices_,  JHEP 1705:118, 2017 ([arXiv:1611.04650](https://arxiv.org/abs/1611.04650))

category: people


Philip Argyres

* [webpage](https://homepages.uc.edu/~argyrepc/)

## Selected writings

Proposal for a non-abelian [[DBI-action]] for [[intersecting brane|coincident]] [[D-branes]]:

* [[Philip Argyres]], [[Chiara Nappi]], _Spin-1 effective actions from open strings_, Nuclear Physics B Volume 330, Issue 1, 22 January 1990, Pages 151-173 Nuclear Physics B (doi:10.1016/0550-3213(90)90305-W)

On [[orientifolds]] lifted to [[M-theory]]:

* {#ArgyresMaimonPelland02} [[Philip Argyres]], [[Ron Maimon]], Sophie Pelland, _The M theory lift of two O6 planes and four D6 branes_, JHEP 0205 (2002) 008 ([arXiv:hep-th/0204127](https://arxiv.org/abs/hep-th/0204127))

category: people


Philip Ball

* [personal page](https://www.philipball.co.uk/)

## Selected writings

An interview with [[Xiao-Gang Wen]] on [[topological order]]:

* [[Philip Ball]], *Making the world from topological order*, National Science Review **6** 2 (2019) 227–230 $[$[doi:10.1093/nsr/nwy116](https://doi.org/10.1093/nsr/nwy116)$]$

On claims of experimental realization of [[quantum supremacy]]:

* [[Philip Ball]], *Physicists in China challenge Google’s "quantum advantage"*, Nature **588** 380 (2020) [[doi:10.1038/d41586-020-03434-7](https://doi.org/10.1038/d41586-020-03434-7)]

category: people


Philip Boalch

* [webpage](http://www.math.u-psud.fr/~boalch/)

## related $n$Lab entries

* [[icosahedral group]]

category: people


Philip Candelas

Philip Candelas is professor of mathematical [[physics]] at Oxford.

* [webpage](http://www.maths.ox.ac.uk/people/profiles/philip.candelas)

## related $n$Lab entries

* [[heterotic string theory]]

* [[supersymmetry and Calabi-Yau manifolds]]

* [[string phenomenology]]

* [[Calabi-Yau varieties]] over [[finite fields]]

category: people


Philip Gibbs

Philip Gibbs is an independent thinker, working on physics and mathematics.

* [G+ page](https://plus.google.com/u/0/109950732766352929935)

He is the founder of the [viXra](http://vixra.org/) preprint archive. For more on this see:

* Philip Gibbs, _[Quora reply to: Are there any serious papers on viXra?](https://www.quora.com/Are-there-any-serious-papers-on-viXra/answer/Philip-Gibbs-1)_

* Wikipedia, _[ViXra](https://en.wikipedia.org/wiki/ViXra)_

## Writings

Gibbs seems to have been the first to publically discuss the detection of the [[Higgs particle]] at a [[rest mass]] of 125 [[GeV]], a little before the official announcement by the [[LHC]] collaborations:

* {#Gibbs11a} [[Philip Gibbs]], _Seminar Watch (Higgs Special), Rumoured Higgs at 125 GeV and What Would a Higgs at 125 GeV Tell Us?_, Prespacetime Journal, December 2011, Vol. 2 Issue 12 pp. 1899-1905 ([web](http://prespacetime.com/index.php/pst/article/view/308/298))

In this article Gibbs already highlights the near-criticality of the resulting Higgs [[vacuum]], first published on his blog as

* _[What would a Higgs at 125 GeV tell us?](https://vixra.wordpress.com/2011/12/04/what-would-a-higgs-at-125-gev-tell-us/)_

For more on this see at _[[Higgs field]]_ the section _[Mass and vacuum stability](Higgs+field#MassAndVacuumInstability)_.

## related $n$Lab entries

* [[Higgs field]]

category: people


Philip Hackney

[[!redirects Phillip Hackney]]

* [webpage](http://phck.net/)

## related $n$Lab entries

* [[PROP]]

category: people


Philip Higgins

Philip J. Higgins (born 1926, died March 25, 2015) was a British  mathematician, who  worked in areas of algebra and algebraic topology.  He is well known for his book

## Selected writings

On [[categories]] and [[groupoids]]:

* [[Philip J. Higgins]], *Categories and Groupoids*, Mathematical  Studies **32**, van Nostrand New  York (1971), Reprints in Theory and Applications of Categories **7** (2005) 1-195 [[tac:tr7] (http://www.tac.mta.ca/tac/reprints/articles/7/tr7abs.html), [pdf](http://www.tac.mta.ca/tac/reprints/articles/7/tr7-2a4.pdf)]

For his long standing collaboration with [[Ronnie Brown]] on [[nonabelian algebraic topology]], see [here](http://groupoids.org.uk/nonab-a-t.html).

His earlier paper

* P. J. Higgins, _Algebras with a scheme of operators_, Math. Nachr. 27 (1963) 115--132.

 has been influential: it deals with partial operations, following on from his paper

* P. J. Higgins, _Groups with multiple operators_, Proc London Math. Soc. VI (1956) 366-416.

His paper

* P. J. Higgins, _The fundamental groupoid of a graph of groups_, J. London Math. Soc. (2), 13 (1976)  145--149,

developed methods that were later on used by [[Haefliger]] in assigning a groupoid to a complex of groups.

More recently he had worked with [[Kirill Mackenzie]] on [[Lie groupoids]] and [[Lie algebroids]]

category: people
[[!redirects Higgins]]

[[!redirects P. J. Higgins]]
[[!redirects P.J. Higgins]]
[[!redirects Philip J. Higgins]]


Philip Hirschhorn

* [website](http://math.mit.edu/~psh/)

## Selected writings

On [[model categories]] and [[Bousfield localization of model categories|Bousfield]] [[localization of model categories]]:

* {#Hirschhorn02} [[Philip Hirschhorn]], _[[Model Categories and Their Localizations]]_, AMS Math. Survey and Monographs Vol 99 (2002) ([ISBN:978-0-8218-4917-0](https://bookstore.ams.org/surv-99-s/), [pdf toc](http://www.gbv.de/dms/goettingen/360115845.pdf), [pdf](https://people.math.rochester.edu/faculty/doug/otherpapers/pshmain.pdf), [pdf](http://www.maths.ed.ac.uk/~aar/papers/hirschhornloc.pdf))

On [[derived functors]] (such as [[homotopy limit]]-functors) on [[model categories]] and more general [[homotopical categories]]:

* [[William Dwyer]], [[Philip Hirschhorn]], [[Daniel Kan]], [[Jeff Smith]], **[[Homotopy Limit Functors on Model Categories and Homotopical Categories]]**, Mathematical Surveys and Monographs **113**, AMS 2004 ([ISBN: 978-1-4704-1340-8](https://bookstore.ams.org/surv-113-s), [pdf](http://dodo.pdmi.ras.ru/~topology/books/dhks.pdf))

On the [[classical model structure on topological spaces]]:

* {#Hirschhorn15} [[Philip Hirschhorn]], _The Quillen model category of topological spaces_, Expositiones Mathematicae Volume 37, Issue 1, March 2019, Pages 2-24 ([arXiv:1508.01942](http://arxiv.org/abs/1508.01942), [doi:10.1016/j.exmath.2017.10.004](https://doi.org/10.1016/j.exmath.2017.10.004))

On [[slice model structures]]:

* {#Hirschhorn21} [[Philip Hirschhorn]], *Overcategories and undercategories of model categories*, J. Homotopy Relat. Struct. **16**  (2021) 753–768 [[arXiv:1507.01624](https://arxiv.org/abs/1507.01624), [doi:10.1007/s40062-021-00294-4](https://doi.org/10.1007/s40062-021-00294-4)]

## related entries

* [[model category]]

* [[model structure on topological spaces]]

category: people


Philip Russel Wallace

* [Wikipedia entry](https://en.wikipedia.org/wiki/P._R._Wallace)

## Selected writings

Predicting the [[electronic band structure]] of [[graphene]] (long before that term was coined):

* [[Philip Russel Wallace]], *The Band Theory of Graphite*, Phys. Rev. **71** (1947) 622 ([doi:10.1103/PhysRev.71.622](https://doi.org/10.1103/PhysRev.71.622))

category: people

[[!redirects R. P. Wallace]]


Philip S. Hirschhorn

* [personal page](http://www-math.mit.edu/~psh/)

## Selected writings

On [[model categories]] on their [[localization of model categories|localization]]:

* [[Philip Hirschhorn]], _[[Model Categories and Their Localizations]]_, AMS Math. Survey and Monographs **99** (2002) [[ISBN:978-0-8218-4917-0](https://bookstore.ams.org/surv-99-s/), [pdf toc](http://www.gbv.de/dms/goettingen/360115845.pdf), [pdf](https://people.math.rochester.edu/faculty/doug/otherpapers/pshmain.pdf), [pdf](http://www.maths.ed.ac.uk/~aar/papers/hirschhornloc.pdf)]

[[!redirects Philip Hirschhorn]]
[[!redirects Hirschhorn]]


Philip Scott

> Not to be confused with [[Dana Scott]].

Philip Scott (1947 - 2023) was Professor of Mathematics at University of Ottawa, with research interests in [[formal logic]], [[category theory]], [[foundations]] of mathematics and computing, and in theoretical [[computer science]]. His monograph [Lambek & Scott 1986](#LambekScott86) laid the foundation of modern [[categorical logic]].

* [homepage](http://www.site.uottawa.ca/~phil/)

## Selected writings

On [[categorical logic]] and [[topos theory]] (and establishing the [[relation between category theory and type theory|relation]] between [[simply-typed lambda calculus]] and [[cartesian closed categories]]):

* {#LambekScott86} [[Joachim Lambek]], [[Philip J. Scott]], *Introduction to higher order categorical logic*, Cambridge Studies in Advanced Mathematics **7** (1986) [[ISBN: 0-521-24665-2](https://www.cambridge.org/ae/academic/subjects/mathematics/logic-categories-and-sets/introduction-higher-order-categorical-logic?format=PB&isbn=9780521356534), [pdf](https://raw.githubusercontent.com/Mzk-Levi/texts/master/Lambek%20J.%2C%20Scott%20P.J.%20Introduction%20to%20Higher%20Order%20Categorical%20Logic.pdf)]

On [[categorical semantics]] of [[linear type theory]]:

* [[Richard Blute]], [[Philip Scott]], *Category theory for linear logicians*, in: *Linear Logic in Computer Science*, Cambridge University Press (2004) 3-64 [[doi:10.1017/CBO9780511550850.002](https://doi.org/10.1017/CBO9780511550850.002), [pdf](https://www.site.uottawa.ca/~phil/papers/catsurv.web.pdf)]

category: people

[[!redirects Philip Scott]]

[[!redirects Philip J. Scott]]
[[!redirects Phil Scott]]


Philip Tanedo

* [webpage](https://theory.ucr.edu/flip/)

## Selected writings

On [[flavour anomalies]] via exotic [[Higgs field]] couplings to a [[hidden sector]]:

* Jared A. Evans, [[Philip Tanedo]], Mohammadreza Zakeri, _Exotic Lepton-Flavor Violating Higgs Decays_ ([arxiv:1910.07533](https://arxiv.org/abs/1910.07533))


Philip W. Anderson

* [Wikipedia entry](https://en.wikipedia.org/wiki/Philip_W._Anderson)

## Selected writings

Predicting [[quantum spin liquid]] (under the name "[[resonating valence bond states]]"):

* [[Philip W. Anderson]], *Resonating valence bonds: A new kind of insulator?*, Materials Research Bulletin
**8** 2 (1973) 153-160 [doi:10.1016/0025-5408(73)90167-0]

## Related entries

* [[condensed matter physics]]

category: people

[[!redirects P. W. Anderson]]
[[!redirects Philip Anderson]]


Philip Wadler

* [webpage](http://homepages.inf.ed.ac.uk/wadler/)

## Selected writings

On [[monads in computer science]]:

* [[Philip Wadler]], *Comprehending Monads*, in _Conference on Lisp and functional programming_, ACM Press (1990) [[[WadlerMonads.pdf:file]], [doi:10.1145/91556.91592](https://doi.org/10.1145/91556.91592)]

* [[Philip Wadler]], *Monads for functional programming*, in M. Broy (eds.) *Program Design Calculi* NATO ASI Series, **118** Springer (1992) [[doi;10.1007/978-3-662-02880-3_8](https://doi.org/10.1007/978-3-662-02880-3_8), [pdf](https://homepages.inf.ed.ac.uk/wadler/papers/marktoberdorf/baastad.pdf)]

* [[Philip Wadler]], *The essence of functional programming*, *POPL '92: Principles of programming languages* (1992) 1-14 [[doi:10.1145/143165.143169](https://doi.org/10.1145/143165.143169), [pdf](https://jgbm.github.io/eecs762f19/papers/wadler-monads.pdf)]

Introducing the [[IO-monad]] (then adopted in [[Haskell]]):

* [[Simon L. Peyton Jones]], [[Philip Wadler]], *Imperative functional programming*, Principles of programming languages (1993) 71–84 [[doi:10.1145/158511.158524](https://doi.org/10.1145/158511.158524), [pdf](https://www.microsoft.com/en-us/research/wp-content/uploads/1993/01/imperative.pdf)]

On [[syntax]] and [[categorical semantics]] of [[linear logic]] with an [[exponential modality]]:

* {#Wadler93} [[Philip  Wadler]], *A syntax for linear logic*, in: *Ninth  International  Coference  on  the Mathematical Foundations of Programming Semantics*, Lecture Notes in Computer Science **802** Springer (1993) [[doi:10.1007/3-540-58027-1_24](https://doi.org/10.1007/3-540-58027-1_24)]

On the history of the development of the [[Haskell]] [[programming language]]:

* [[Paul Hudak]], [[John Hughes]], [[Simon Peyton Jones]], [[Philip Wadler]], *A history of Haskell: being lazy with class*, History of Programming Languages (2007) 12-1 - 12-55 [[doi:10.1145/1238844.1238856](https://doi.org/10.1145/1238844.1238856)]

On [[propositions as types]]:

* [[Philip Wadler]], *Propositions as Types*, Communications of the ACM **58** 12 (2015) 75–84 [[doi:10.1145/2699407](https://doi.org/10.1145/2699407), [pdf](https://homepages.inf.ed.ac.uk/wadler/papers/propositions-as-types/propositions-as-types.pdf), [video](https://www.youtube.com/watch?v=IOiZatlZtGU)]

## Related entries

* [[monad (in computer science)]]

* [[adjoint logic]]

category: people


Philip Welch

* [webpage](https://people.maths.bris.ac.uk/~mapdw/)

## Selected publications

* [[Philip Welch]], *Proving Theorems from Reflection*, in: *[[Reflections on the Foundations of Mathematics]]*, Synthese Library **407** Springer (2019) [[doi:10.1007/978-3-030-15655-8_4](https://doi.org/10.1007/978-3-030-15655-8_4), [pdf](https://people.maths.bris.ac.uk/~mapdw/Welch%20Proving%20Theorems%20from%20Reflection-revised%2014-08-18.pdf)]

category: people


Philipp G. Haselwarter

* [personal page](https://haselwarter.org/)

## Selected writings

On [[dependent type theory]]:

* {#BauerHaselwarterLumsdaine20} [[Andrej Bauer]], [[Philipp G. Haselwarter]], [[Peter LeFanu Lumsdaine]], *A general definition of dependent type theories* [[arXiv:2009.05539](https://arxiv.org/abs/2009.05539)]

category: people

[[!redirects Philipp Haselwarter]]


Philipp Joram

* [website](https://taltech.ee/en/contacts/philipp-joram)

## Selected writings and talks

* [[Philipp Joram]], *Final Coalgebras of Analytic Functors in Homotopy Type Theory*, [[Homotopy Type Theory Electronic Seminar Talks]], 15 December 2022 ([slides](https://www.uwo.ca/math/faculty/kapulkin/seminars/hottestfiles/Joram-2022-12-15-HoTTEST.pdf), [video](https://www.youtube.com/watch?v=nsgztQ5UbLw))


Philippe Christe

* [WorldCat page](http://worldcat.org/identities/lccn-n93012523/)

## Selected writings

A precursor result to the [[hypergeometric construction of KZ solutions]], specifically of [[conformal blocks]] for [[affine Lie algebra]]/[[WZW-model]]-[[2d CFTs]]:

* [[Philippe Christe]], [[Rainald Flume]], *The four-point correlations of all primary operators of the $d = 2$ conformally invariant $SU(2)$ $\sigma$-model with Wess-Zumino term*, Nuclear Physics B **282** (1987) 466-494 $[$doi:10.1016/0550-3213(87)90693-6$]$

category: people


Philippe Di Francesco

* [institute page](https://math.illinois.edu/directory/profile/philippe)

## Selected writings

On [[integrable system|integrable]] [[lattice models]] for [[2d CFTs]]:

* [[Philippe Di Francesco]], *Integrable lattice models, graphs and modular invariant conformal field theories*, International Journal of Modern Physics A **7** 03 (1992) 407-500 $[$[doi:10.1142/S0217751X92000223](https://doi.org/10.1142/S0217751X92000223)$]$

On ([[2d CFT|2d]]) [[conformal field theory]]:

* [[Philippe Di Francesco]], Pierre Mathieu, David Sénéchal, *Conformal field theory*, Springer (1997) $[$[doi:10.1007/978-1-4612-2256-9](https://doi.org/10.1007/978-1-4612-2256-9)$]$

category: people


Philippe Flajolet

[[!redirects Phillipe Flajolet]]
Philippe Flajolet was a mathematician who, in particular, together with R. Sedgewick wrote an influential textbook entitled "Analytic Combinatorics" which has natural connections to [[polynomial functors]], and to the theory of [[species]] of [[Andre Joyal|André Joyal]].

## related $n$Lab entries

* [[species]]
* [[generating function]]

## References

* [personal website](http://algo.inria.fr/flajolet/)

* [ Algorithms Project webpage](http://algo.inria.fr/topics.html)

* [Wikipedia](https://en.wikipedia.org/wiki/Philippe_Flajolet)

* Philippe Flajolet, Robert Sedgewick, _Analytic Combinatorics_, 2009 ([pdf](http://algo.inria.fr/flajolet/Publications/book.pdf))

category: people

[[!redirects Phillipe Flajolet]]


Philippe Gaucher

* [Webpage](http://www.irif.fr/~gaucher)

## Selected writings

On [[Delta-generated topological spaces|$\Delta$-generated]] and [[compactly generated topological spaces]]:

* {#Gaucher07} [[Philippe Gaucher]], Section 2 of: *Homotopical interpretation of globular complex by multipointed d-space*, Theory and Applications of Categories, vol. 22, number 22, 588-621, 2009 ([arXiv:0710.3553](https://arxiv.org/abs/0710.3553))

On [[model structures on topological spaces]] and the notion of $\Delta$-Hausdorff $\Delta$-generated space (a natural [[separation condition]] for [[Delta-generated topological spaces|$\Delta$-generated spaces]]):

* {#Gaucher21} [[Philippe Gaucher]], Section 2 and Appendix B of: *Left properness of flows*, Theory and Applications of Categories, **37** 19 (2021) 562-612 ([arXiv:1907.01454](https://arxiv.org/abs/1907.01454), [TAC:37-19](http://www.tac.mta.ca/tac/volumes/37/19/37-19abs.html))

On [[directed homotopy theory]] and [[higher dimensional transition systems]]:

* [[Philippe Gaucher]], _Directed algebraic topology and higher dimensional transition system_, New-York Journal of Mathematics 16 (2010), 409 - 461 ([arxiv:0903.4276](http://arxiv.org/abs/0903.4276))

*  [[Philippe Gaucher]], _Towards a homotopy theory of higher dimensional transition systems_, Theory and Applications of Categories, Vol. 25, No. 12, 2011, pp. 295–341 ([arxiv:1011.0918](http://arxiv.org/abs/1011.0918))

On [[left-determined model categories]]:

* [[Philippe Gaucher]], _Left determined model categories_, New York Journal of Mathematics **21** (2015), 1093-1115 ([nyjm:j/2015/21-50](http://nyjm.albany.edu/j/2015/21-50.html), [arXiv:1507.02128](https://arxiv.org/abs/1507.02128))

## Related $n$Lab pages

* [[PPS]]

category: people


Philippe Gille

* [personal page](http://math.univ-lyon1.fr/homes-www/gille/)

* [Mathematics Genealogy page](https://www.genealogy.math.ndsu.nodak.edu/id.php?id=86443)

## Selected writings

On [[Galois cohomology]]:

* {#GilleSzamuely06} [[Philippe Gille]], [[Tamás Szamuely]], *Central Simple Algebras and Galois Cohomology*, Cambridge University Press  2006 ([doi:10.1017/CBO9780511607219](https://doi.org/10.1017/CBO9780511607219), [pdf](http://www.math.ens.fr/~benoist/refs/Gille-Szamuely.pdf))

category: people


Philippe Malbos

Philippe Malbos is a French mathematician at the Institut Camille Jordan,
Université Claude Bernard Lyon1, but actually attached to [[PPS]] in Paris at the moment. He is interested in homological algebra, algebraic semantics and rewriting systems.  In particular he has worked with [[Yves Guiraud]] on finite convergence conditions on rewriting systems.

His webpage is [here](http://math.univ-lyon1.fr/~malbos/).


Philippe Roche

* [spire page](https://inspirehep.net/authors/1046024?ui-citation-summary=true)

## Selected writings

On [[Drinfeld doubles]] of [[quasi-Hopf algebras]], motivated from [[rational conformal field theory|rational]] [[orbifold]] [[conformal field theories]]:

* [[Robbert Dijkgraaf]], [[Vincent Pasquier]], [[Philippe Roche]], *QuasiHopf algebras, group cohomology and orbifold models*, Nucl. Phys. B Proc. Suppl **18** (1990) 60-72 [doi:10.1016/0920-5632(91)90123-V]

* [[Robbert Dijkgraaf]], [[Vincent Pasquier]], [[Philippe Roche]], _Quasi-quantum groups related to orbifold models_, *International Colloquium on Modern Quantum Field Theory* (Bombay, 1990), 375-383, World Sci. (1991) [[cds:206306](https://cds.cern.ch/record/206306?ln=ka), [[DPR-QuasiQuantumGroups.pdf:file]]]

category: people


Philippe Tondeur

* [webpage](https://faculty.math.illinois.edu/~tondeur/)

* [Mathematics Genealogy page](http://www.genealogy.ams.org/id.php?id=4722)

## Selected writings

* {#LamberTondeur67} F. Kamber, [[Philippe Tondeur]], _Flat Bundles and Characteristic Classes of Group-Representations_, American Journal of Mathematics, Vol. 89, No. 4 (Oct., 1967), pp. 857-886 ([doi:10.2307/2373408](https://www.jstor.org/stable/2373408))

On [[Kuo Tsai Chen]]:

* [[Richard Hain]], [[Philippe Tondeur]], _The life and work of Kuo-Tsai Chen_,  Illinois J. Math. Volume 34, Issue 2 (1990), 175-190. ([euclid:ijm/1255988263](http://projecteuclid.org/euclid.ijm/1255988263))

## Related $n$Lab entries

* [[characteristic class of a linear representation]]

category: people


Phillip Griffith

* [Wikipedia entry](https://en.wikipedia.org/wiki/Phillip_Griffith)

* [institute page](https://math.illinois.edu/directory/profile/pgriffit)

## Selected writings

On [[periods]] in [[algebraic geometry]]:

* J. Carlson, [[Phillip Griffith]], *What is a period domain?*,  AMS Notices **55** 11 (2008) 1418-1419 [[pdf](http://www.ams.org/notices/200811/tx081101418p.pdf), full issue:[pdf](http://www.ams.org/notices/200811/200811FullIssue.pdf)]

category: people


Phillip Griffiths

* [Wikipedia entry](http://en.wikipedia.org/wiki/Phillip_Griffiths)

## Selected writings

On [[rational homotopy theory]] of [[Kähler manifolds]] and the statement that _[[the co-binary Sullivan differential is the dual Whitehead product]]_:

* {#DeligneGriffithsMorganSullivan75} [[Pierre Deligne]], [[Phillip Griffiths]], [[John Morgan]], [[Dennis Sullivan]], _Real homotopy theory of Kähler manifolds_, Invent Math (1975) 29: 245 ([doi:10.1007/BF01389853](https://doi.org/10.1007/BF01389853))

On [[rational homotopy theory]]:

* {#GriffithMorgan13} [[Phillip Griffiths]], [[John Morgan]], _Rational Homotopy Theory and Differential Forms_,  Progress in Mathematics Volume 16, Birkhäuser (2013)

## Related entries

* [[intermediate Jacobian]]

* [[Whitehead product]],

  [[the co-binary Sullivan differential is the dual Whitehead product]]

category: people

[[!redirects Phillip A. Griffiths]]


Phillip James E. Peebles

* [Institute page](https://phy.princeton.edu/people/p-james-peebles)

* [Wikipedia entry](https://en.wikipedia.org/wiki/Jim_Peebles)

## Selected writings

On [[cosmology]]:

* [[P. J. E. Peebles]], *Principles of Physical Cosmology*, Princeton University Press (1993, 2020) [[ISBN:9780691209814](https://press.princeton.edu/books/paperback/9780691209814/principles-of-physical-cosmology), [pdf](http://fma.if.usp.br/~mlima/teaching/PGF5292_2021/Peebles_PPC.pdf)]

On observed deviations from the [[standard model of cosmology]] (such as large-scale [[inhomogeneous cosmology|inhomogeneties]] like the [[KBC void]]):

* {#Peebles22} [[Phillip James E. Peebles]], *Anomalies in Physical Cosmology* [[arXiv:2208.05018](https://arxiv.org/abs/2208.05018)]

category: people

[[!redirects Jim Peebles]]
[[!redirects Phillip J. E. Peebles]]
[[!redirects P. J. E. Peebles]]
[[!redirects Phillip Peebles]]


Phillip Kaye

* [institute page](https://creativedestructionlab.com/mentors/phil-kaye/)

## Selected writings

On [[addition]] in the context of [[reversible computation]] and [[quantum computation]]:

* {#Kaye04} [[Phillip Kaye]], *Reversible addition circuit using one ancillary bit with application to quantum computing* [[quant-ph/0408173](https://arxiv.org/abs/quant-ph/0408173)]

On [[quantum computation]]:

* [[Phillip Kaye]], [[Raymond Laflamme]], [[Michele Mosca]], *An Introduction to Quantum Computing*, Oxford University Press (2007) [[pdf](http://mmrc.amss.cas.cn/tlb/201702/W020170224608149125645.pdf), [ISBN:9780198570493](https://global.oup.com/academic/product/an-introduction-to-quantum-computing-9780198570493?cc=at&lang=en&)]

category: people


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#### Physics
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[[!include physicscontents]]
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=--
=--

> For [[Bertrand Russell|Russell]] & [[Alfred Whitehead|Whitehead]]'s *[[Principia Mathematica]]* see there.

***

This page compiles links related to:

* [[Isaac Newton]]:

  \linebreak

  **Philosophiæ naturalis principia mathematica**

  \linebreak

  (1687)

  [Camb. Univ. Press page](https://cudl.lib.cam.ac.uk/view/PR-ADV-B-00039-00001/17)

  [Wikipedia entry](https://en.wikipedia.org/wiki/Philosophi%C3%A6_Naturalis_Principia_Mathematica)

  [Stanf. Enc. Phil. entry](https://plato.stanford.edu/entries/newton-principia/)

laying foundations of modern [[physics]] (in contrast to the previous *[[natural philosophy]]*) in the guise of what came to be known as *[[Newton's laws of motion]]* and *[[Newtonian mechanics]]*.

category: reference

[[!redirects Philosophiæ Naturalis Principia Mathematica]]

[[!redirects Philosophiae naturalis principia mathematica]]
[[!redirects Philosophiae Naturalis Principia Mathematica]]

[[!redirects Newton's Principia]]


philosophy

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#### Philosophy
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# Philosophy
* table of contents
{: toc}

## Scope

In the philosophical part of $n$Lab we already discuss [[higher algebra]], [[homotopy theory]], [[type theory]], [[category theory]], and [[higher category theory]] and its repercussions in philosophy. More widely, the entries on philosophy in $n$Lab would be nice to contain [[philosophy of mathematics]] in general, and of [[logic and foundations]] in particular. Similarly about [[philosophy of physics]] including interpretation and foundational issues of [[quantum mechanics]]. As it is usual for philosophy and the study of thought, it is usefully carried on via study of historical thinkers and their ideas, hence some idea-related aspects of the [[history of mathematics]] are welcome.

## Nonscope

There are many articles which are not directly philosophical, but rather essays on general mathematics, often opinion pieces on what is important and so on. Although mathematicians will often speak of their 'philosophy', this is not philosophy _per se_, but it may be relevant to an understanding of the nature of mathematics through its practice, see, for instance, [[development and current state of mathematics]].

## Idea of relevance of higher mathematical structures

Philosophical interest in higher mathematical structures may be characterised as belonging to one of two kinds.

* **[[metaphysics|Metaphysical]]**: The formation of a new language which may prove to be as important for philosophy as predicate logic was for [[Bertrand Russell]] and the [[analytic philosophy|analytic philosophers]] he inspired (see, e.g., [Corfield 20](#Corfield20)).

* **Illustrative of mathematics as intellectual enquiry**: Such a reconstitution of the fundamental language of [[mathematics]] reveals much about the discipline as a tradition of enquiry stretching back several millennia, for instance, the continued willingness to reconsider basic concepts (see, e.g., [Corfield 12](#Corfield19), [Corfield 19](#Corfield19)).

## Metaphysics

[[metaphysics]]

* [[higher category theory|Higher category theory]] and its associated type theories, such as [[homotopy type theory]] provides a new foundation for mathematics - [[foundations|logical]] and [[foundations and philosophy|philosophical]].

* [[higher category theory|Higher category theory]] refines the notion of sameness to allow more subtle variants, adhering to the [[principle of equivalence]].

* There ought to be a [[vertical categorification|categorified]] [[internal logic|logic]], or 2-logic. There are some suggestions that existing work on modal logic is relevant. Blog discussion: [I](http://golem.ph.utexas.edu/category/2007/11/concrete_groups_and_axiomatic_1.html), [II](https://golem.ph.utexas.edu/category/2008/01/2toposes.html), [III](http://golem.ph.utexas.edu/category/2008/02/2galois_and_2logic.html), [IV](http://golem.ph.utexas.edu/category/2008/03/worrying_about_2logic.html), [V](http://golem.ph.utexas.edu/category/2008/11/the_internal_language_of_a_2to.html). Mike Shulman's project: [2-categorical logic](http://ncatlab.org/michaelshulman/show/2-categorical+logic).

* [[Homotopy type theory]] may be thought of as a vertical categorification of logic to $(\infinity,1)$.

* [[higher category theory|Higher category theory]] may provide the right tools to take [[physics]] forward. [A Prehistory of n-Categorical Physics](http://math.ucr.edu/home/baez/history.pdf) See also [[physics]].

* [[higher category theory and physics]], [[geometry of physics]]

* More speculatively, [[category theory]] may prove useful in [[biology]].

## Enquiry

>"Mathematical wisdom, if not forgotten, lives as an invariant of all its (re)presentations in a permanently self–renewing discourse." (Yuri Manin)

To [[vertical categorification|categorify]] mathematical constructions properly, one must have understood their essential features. This leads us to consider what it is to get concepts 'right'. Which kind of 'realism' is suitable for mathematics? Which virtues should a mathematical community possess to further its ends: a knowledge of its history, close attention to instruction and the sharing of knowledge, a willingness to admit to what is currently lacking in its programmes?

### Research programs in mathematics

* [[Thomas Kuhn]]

* [[Imre Lakatos]]

* [[paradigm]]

* [[paradigm shift]]

* [[research program]]

This entire subject about past research programs, paradigms in mathematics and paradigm shifts could be expanded on in the nLab. Examples include the shift from Euclidean geometry to non-Euclidean geometries in the 19th century, and the dominance of the material set theory paradigm in the 20th century and its failure with higher structures, the evolution of analytic concepts such as the differential, the integral, the real numbers, over the course of the 20th century, but there are surely others out there.

## Philosophical positions

* [[pluralism]], [[reductionism]]

* [[Platonism]]

* [[idealism]]

  * [[objective idealism]]

  * [[subjective idealism]]

* [[positivism]]

* [[universal exceptionalism]]

## References and links

### Related entries

* [[foundations and logic]], [[logic]], [[reasoning]]
* [[category (philosophy)]], [[modality]]
* [[logos (in philosophy)]], [[Nous]]
* [[history of mathematics]], [[structuralism]], [[opinions on development of mathematics]]

* [[epistemology]]
* [[epistemology of mathematics]]
* [[metaphysics]]
* [[hermeticism]], [[mysticism]], [[gnosticism]], [[speculative philosophy]]
* _[[Science of Logic]]_
* [[analytic philosophy]]
* [[philosophy of mathematics]]
* [[constructive mathematics]], [[predicative mathematics]]
* [[philosophy of science]], [[philosophy of physics]]
* [[Judea Pearl]]
* [[Gravity and Grace]]
* [[Michael Friedman]]

### General

* wikipedia: [philosophy of mathematics](http://en.wikipedia.org/wiki/Philosophy_of_mathematics); wikipedia.ru [Философия математики](http://ru.wikipedia.org/wiki/%D0%A4%D0%B8%D0%BB%D0%BE%D1%81%D0%BE%D1%84%D0%B8%D1%8F_%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B8)
* [Internet Encyclopedia of Philosophy](http://www.iep.utm.edu), "A peer reviewed academic resource"
* Stanford Encyclopaedia of Philosophy [online](http://plato.stanford.edu), [contents](http://plato.stanford.edu/contents.html)
* Stewart Shapiro, _[[The Oxford handbook of philosophy of mathematics and logic]]_, Oxford University Press 2005
* [free-philosophy-ebooks](http://www.openculture.com/free-philosophy-ebooks) at openculture
* [Less Wrong Wiki](http://lesswrong.com) "a community blog devoted to refining the art of human rationality"; including Eliezer Yudkowsky's [Epistemology 101](https://wiki.lesswrong.com/wiki/Highly_Advanced_Epistemology_101_for_Beginners)

### Books

* [[Hegel]], _Wissenschaft der Logik_ ( _[[Science of Logic]]_ )
  * [[unity of opposites]]
    * [[becoming]] $\colon$ [[nothing]] $\dashv$ [[being]]
* [[Hegel]], _[[Lectures on the History of Philosophy]]_
* [[Ludwig Wittgenstein]], _[[Tractatus Logico-Philosophicus]]_
* [[Bertrand Russell]], _[[A History of Western Philosophy]]_

* [[Albert Lautman]], _Mathematics, ideas and the physical real_, 2011 translation by Simon B. Duffy; English edition of _Les Mathématiques, les idées et le réel physique_, Librairie Philosophique, J. VRIN, 2006

* Michael D. Potter, _Set theory and its philosophy: a critical introduction_, Oxford Univ. Press 2004

* [[Fernando Zalamea]], _Filosofía sintética de las matemáticas contemporáneas_, (Spanish) Obra Selecta. Editorial Universidad Nacional de Colombia, Bogotá, 2009. 231 pp. [MR2599170](http://www.ams.org/mathscinet-getitem?mr=2599170), ISBN: 978-958-719-206-3, [pdf](<http://files.acervopeirceano.webnode.es/200000065-18c1b19bb9/Zalamea-Fil-Sint-Mat-Cont.pdf>). Transl. into English by Zachary Luke Fraser: _Synthetic philosophy of contemporary mathematics_, Sep. 2011. [bookpage](http://www.urbanomic.com/pub_syntheticmath.php). Some excerpts [here](http://ifile.it/2c3qgz5).

* [[David Corfield]], _Towards a philosophy of real mathematics_, Cambridge University Press, 2003, [gBooks](http://books.google.hr/books?id=s37rhfWs73QC&lpg=PP1&ots=Zobx2bIWFh&dq=david%20corfield%20towards%20a%20philosophy%20of%20real%20mathematics&hl=en&pg=PP1#v=onepage&q&f=false)

* [[Saunders MacLane]], _Mathematics, form and function_, Springer-Verlag 1986, xi+476 pp. [MR87g:00041](http://www.ams.org/mathscinet-getitem?mr=816347), [wikipedia](http://en.wikipedia.org/wiki/Mathematics,_Form_and_Function)

* [[George Lakoff]], Rafael E. Núñez, _Where mathematics comes from_, Basic Books 2000, xviii+493 pp. [MR2001i:00013](http://www.ams.org/mathscinet-getitem?mr=1794854)

* [[Yuri Manin|Yuri I. Manin]], _Mathematics as Metaphor:
Selected Essays of Yuri Manin_, Amer. Math. Soc. 2007

* [[Ralf Krömer]], _Tool and object: A history and philosophy of category theory_, Birkhäuser 2007

* [[Jean-Pierre Marquis]], _From a geometrical point of view: a study of the history and philosophy of category theory_, Springer, 2008

* Ian Hacking, _Why is there philosophy of mathematics at all?_, Cambridge University Press 2014

* William Bragg Ewald, _From Kant to Hilbert, From Kant to Hilbert: Readings in the Foundations of Mathematics_, 2 vols. (original readings in English translation)

* [[Roland Omnès]], _[[Converging Realities -- Toward a common philosophy of physics and mathematics]]_, Princeton University Press, 2005

* {#Corfield20} [[David Corfield]], _Modal homotopy type theory_, Oxford University Press 2020 ([ISBN: 9780198853404](https://global.oup.com/academic/product/modal-homotopy-type-theory-9780198853404))

* Fernando Zalamea (editor), _Rondas en Sais. Ensayos sobre matemáticas y cultura contemporánea._ (Essays on mathematics and contemporary culture, by Moreno, Javier; de Lorenzo, Javier; Villaveces, Andrés; Pérez, Jesús Hernando; Restrepo, Gabriel; Cruz Morales, John Alexánder; Vargas, Francisco; Oostra, Arnold; Ferreirós, José; Zalamea, Fernando; Martín, Alejandro) Universidad Nacional de Colombia, Facultad de Ciencias Humanas 2012 [pdf](https://repositorio.unal.edu.co/bitstream/handle/unal/78631/9789587618693.pdf)

### Articles

* Glenn G. Parsons, James Robert Brown, _Platonism, metaphor, and mathematics_, Dialogue __43__ (2004), no. 1, 47–66, [MR2004k:00004](http://www.ams.org/mathscinet-getitem?mr=2053877)
* John Baldwin, _Model theoretic perspectives on the philosophy of mathematics_, [pdf](http://homepages.math.uic.edu/~jbaldwin/pub/amstersh.pdf)
* [[Yuri Manin|Yu. I. Manin]], _Mathematical knowledge: internal, social and cultural aspects_, [arXiv:math.HO/0703427](http://arxiv.org/abs/math.HO/0703427); _Georg Cantor and his heritage_, [arxiv/math.AG/0209244](http://arxiv.org/abs/math.AG/0209244); _Truth as value and duty: lessons of mathematics_, [arxiv/0805.4057](http://arxiv.org/abs/0805.4057)
* M. G. Katz, E. Leichtnam, _Commuting and noncommuting infinitesimals_, Amer. Math. Monthly 120 (2013), no. 7, 631-641 [arxiv/1304.0583](http://arxiv.org/abs/1304.0583)
* M. G. Katz, Thomas Mormann, _Infinitesimals as an issue in neo-Kantian philosophy of science_, [arxiv/1304.1027](http://arxiv.org/abs/1304.1027)
* [[Mikhail Gromov]], _Ergostructures, Ergologic and the Universal Learning Problem: Chapters 1, 2, 3._ (2013) [pdf](http://www.ihes.fr/~gromov/PDF/ergologic3.1.pdf); _Structures, Learning and Ergosystems: Chapters 1-4, 6_ (2011) [pdf](http://www.ihes.fr/~gromov/PDF/ergobrain.pdf)
* [[William Lawvere]], _[[Cohesive Toposes and Cantor's "lauter Einsen"]]_
* Jeremy Avigad, _Mathematics and language_, [arxiv/1505.07238](http://arxiv.org/abs/1505.07238)
* {#Corfield12} [[David Corfield]], _Narrative and the Rationality of Mathematical Practice_, in A. Doxiadis and B. Mazur (eds.), Circles Disturbed, Princeton, 2012, ([preprint](https://ncatlab.org/davidcorfield/show/Narrative+and+the+Rationality+of+Mathematical+Practice)).

Some philosophical aspects of the role of category theory are touched upon in some parts of the introductory paper

* [[Jean-Pierre Marquis]], _What is category theory?_, pdf can be found at [academia.edu](https://www.academia.edu/16339607/What_is_category_theory)

### Talks

* {#Corfield19} [[David Corfield]], _The Dynamics of Mathematical Reason_, [slides 1](https://ncatlab.org/davidcorfield/files/WIP2.pdf), [slides 2](https://ncatlab.org/davidcorfield/show/Vienna#DynamicsOfMathematicalReason)

category: philosophy

[[!redirects philosophy]]
[[!redirects Philosophy]]


philosophy - contents

**[[philosophy]]**

* [[ontology]]

  * [[logos (in philosophy)|logos]], _[[Science of Logic]]_

  * [[being]], [[becoming]]

### Schools

* [[objective idealism]]

### Of mathematics

* [[philosophy of mathematics]]

### Of physics

* [[natural philosophy]]

* [[philosophy of physics]]

[Edit this sidebar](/nlab/edit/philosophy+-+contents)


Philosophy > history

< [[Philosophy]]
[[!redirects Philosophy -- history]]


philosophy of mathematics

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#Contents#
* table of contents
{:toc}

## Idea

The [[philosophy]] of [[mathematics]].

## Related concepts

* [[pluralism]], [[reductionism]]

* [[Platonism]]

* [[finitism]], [[ultrafinitism]]

* [[constructivism]], [[predicative mathematics]]

* [[structuralism]]

* [[philosophy of science]]
  * [[philosophy of physics]]
* [[history of mathematics]]

* [[epistemology of mathematics]]

## References

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

* Wikipedia, _[Philosophy of mathematics](http://en.wikipedia.org/wiki/Philosophy_of_mathematics)_

* [[Albert Lautman]], _Mathematics, ideas and the physical real_, 2011 translation by Simon B. Duffy; English edition of _Les Mathématiques, les idées et le réel physique_, Librairie Philosophique, J. VRIN, 2006

* [[Fernando Zalamea]], _Filosofía sintética de las matemáticas contemporáneas_, (Spanish) Obra Selecta. Editorial Universidad Nacional de Colombia, Bogotá, 2009. 231 pp. [MR2599170](http://www.ams.org/mathscinet-getitem?mr=2599170), ISBN: 978-958-719-206-3, [pdf](<http://files.acervopeirceano.webnode.es/200000065-18c1b19bb9/Zalamea-Fil-Sint-Mat-Cont.pdf>). Transl. into English by Zachary Luke Fraser: _Synthetic philosophy of contemporary mathematics_, Sep. 2011. [bookpage](http://www.urbanomic.com/pub_syntheticmath.php). Some excerpts [here](http://ifile.it/2c3qgz5).

* [[David Corfield]], _Towards a philosophy of real mathematics_, Cambridge University Press, 2003 ([doi:10.1017/CBO9780511487576](https://doi.org/10.1017/CBO9780511487576), [gBooks](http://books.google.hr/books?id=s37rhfWs73QC&lpg=PP1&ots=Zobx2bIWFh&dq=david%20corfield%20towards%20a%20philosophy%20of%20real%20mathematics&hl=en&pg=PP1#v=onepage&q&f=false))

* [[Roland Omnès]], _[[Converging Realities -- Toward a common philosophy of physics and mathematics]]_, Princeton University Press, 2005

* [[Saunders MacLane]], _Mathematics, form and function_, Springer-Verlag 1986, xi+476 pp. [MR87g:00041](http://www.ams.org/mathscinet-getitem?mr=816347), [wikipedia](http://en.wikipedia.org/wiki/Mathematics,_Form_and_Function)

* [[George Lakoff]], Rafael E. Núñez, _Where mathematics comes from_, Basic Books 2000, xviii+493 pp. [MR2001i:00013](http://www.ams.org/mathscinet-getitem?mr=1794854)

* [[Yuri Manin|Yuri I. Manin]], _Mathematics as Metaphor:
Selected Essays of Yuri Manin_, Amer. Math. Soc. 2007

* Michael D. Potter, _Set theory and its philosophy: a critical introduction_, Oxford Univ. Press 2004

* [[Ralf Krömer]], _Tool and object: A history and philosophy of category theory_, Birkhäuser 2007

* [[Jean-Pierre Marquis]], _From a geometrical point of view: a study of the history and philosophy of category theory_, Springer, 2008

* Ian Hacking, _Why is there philosophy of mathematics at all?_, Cambridge University Press 2014

* William Bragg Ewald, _From Kant to Hilbert, From Kant to Hilbert: Readings in the Foundations of Mathematics_, 2 vols. (original readings in English translation)

* Paolo Mancosu, _Philosophy of mathematics and mathematical practice in the seventeenth century_, OUP, 1996.

* Imre Lakatos, _Proof and refutations_

* [[Hermann Weyl]], _Philosophy of Mathematics and Natural Science_, Princeton University Press, 2nd edition, Princeton University Press, 1950 ([ISBN:9780691141206](https://press.princeton.edu/books/paperback/9780691141206/philosophy-of-mathematics-and-natural-science))

* [[Jacques Hadamard]], _Psychology of invention in the mathematical field_, Princeton University Press 1945

The role of [[Heraclitus]]:

* {#Lambek} [[Joachim Lambek]], _The influence of Heraclitus on modern mathematics_, In _Scientific Philosophy Today: Essays in Honor of Mario Bunge_, edited by Joseph Agassi and Robert S Cohen, 111–21. Boston: D. Reidel Publishing Co.