ETCS

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### Context
#### Topos Theory
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#### Foundations
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[[!include foundations - contents]]
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# Contents
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## Idea

The **Elementary Theory of the Category of Sets**, or _ETCS_ for short, is an axiomatic formulation of [[set theory]] in a [[category theory|category-theoretic]] spirit.  As such, it is the prototypical [[structural set theory]]. Proposed shortly after [[ETCC]] in ([Lawvere 64](#Lawvere64)) it is also the paradigm for a [[foundations|categorical foundation]] of mathematics.[^Law]

The theory intends to capture in an invariant way the notion of a (constant) _'abstract set'_ whose elements lack internal structure and whose only external property is cardinality with further external relations arising from _mappings_. The membership relation is _local_ and _relative_ i.e. membership is meaningful only between an element of a set and a subset of the very same set. (See [Lawvere (1976, p.119)](#Lawvere76) for a detailed description of the notion 'abstract set'.[^cantor] [^dedekind] [^schoenflies] [^vonneumann])

[^cantor]: It has been pointed out by John Myhill that Cantor's concept of 'cardinal' as a set of abstract units should be viewed as a structural set theory and a precursor to Lawvere's concept of an 'abstract set'. This view is endorsed and expanded in [Lawvere 1994](#Lawvere94).

[^dedekind]: [[Richard Dedekind| R. Dedekind's]] views are also anticipating 'abstract sets' e.g. Bernstein reports in Dedekind's works vol.3 (1932, p.449) that Dedekind gave as his intuition of a set: "a closed bag, containing determinate things that one can not see and of which one knows nothing beyond their existence and determinateness".

[^schoenflies]: The first axiomatic set theory without primitive membership relation $\in$ was presumably proposed by A. Schoenflies in 1920: he modeled elements of sets as indecomposable subsets. See A. Schoenflies, _Zur Axiomatik der Mengenlehre_ , Math. Ann. **83** (1921) pp.173-200; and _Bemerkung zur Axiomatik der Grössen und Mengen_ , Math. Ann. **85** (1922) pp.60-64.

[^vonneumann]: The first axiomatic set theory based on the notion of function was [[John von Neumann|von Neumann]]'s 1925 version of what later became the set based [[NBG]] theory of classes.

[^Law]: For a comparative discussion of its virtues as foundation see [[foundations of mathematics]] , the [texts by Todd Trimble](#ExpositionByTrimble) or the informative paper by [McLarty (2004)](#McLarty04).

More in detail, ETCS is a [[first-order theory]] axiomatizing [[elementary toposes]] and specifically those which are [[well-pointed topos|well-pointed]], have a [[natural numbers object]] and satisfy the [[axiom of choice]]. The theory omits the [[axiom of replacement]], however.

The idea is, first of all, that much of traditional mathematics naturally takes place "[[internal logic|inside]]" such a topos of _constant_ sets, and second that this perspective generalizes beyond ETCS proper to toposes of _variable_ and _cohesive_ sets by varying the axioms: for instance omitting the [[well-pointed topos|well-pointedness]] and the [[axiom of choice]] but adding the [[Kock-Lawvere axiom]] gives a [[smooth topos]] inside which [[synthetic differential geometry]] takes place.

That is, ETCS locates the category of sets by the well-pointedness axiom as the discrete zero point on a 'continuous' range of toposes eligible for foundations. In particular, whereas ZF mainly provides 'substance' for mathematics, ETCS lives as a special type of form within the continuum of mathematical form itself.

## Definition

The axioms of ETCS can be summed up in one sentence as:

+-- {: .num_defn}
###### Definition

The [[Set|category of sets]] is [[generalized the|the]] [[topos]] which

1. is a [[well-pointed topos]]

1. has a [[natural numbers object]]

1. and satisfies the [[axiom of choice]].

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For more details see

* [[fully formal ETCS]].

## A constructive view

[[Erik Palmgren]] ([Palmgren 2012](#Palmgren)) has a [[constructive mathematics|constructive]] [[predicative mathematics|predicative]] variant of ETCS, which can be summarized as:

$Set$ is a [[well-pointed topos|well-pointed]] $\Pi$-[[Π-pretopos|pretopos]] with a [[NNO]] and [[enough projectives]] (i.e. [[COSHEP]] is satisfied). Here "well-pointed" must be taken in its constructive sense, as including that the [[terminal object]] is indecomposable and projective.

## A contemporary perspective

Modern mathematics with its emphasis on concepts from [[homotopy theory]] would more directly be founded in a similar spirit by an axiomatization not just of [[elementary toposes]] but of [[elementary (∞,1)-toposes]]. This is roughly what [[univalence|univalent]] [[homotopy type theory]] accomplishes -- for more on this see at _[relation between type theory and category theory -- Univalent HoTT and Elementary infinity-toposes](relation+between+type+theory+and+category+theory#HomotopyWithUnivalence)_.

Instead of increasing the [[higher category theory|higher categorical dimension]] [[(n,r)-category|(n,r)]] in the first argument, one may also, in this context of elementary foundations, consider raising the second argument.
The case $(2,2)$ is the elementary theory of the 2-category of categories ([[ETCC]]).

## Todd Trimble's exposition of ETCS
 {#ExpositionByTrimble}

[[Todd Trimble]] has a series of expository writings on ETCS which provide a very careful introduction and at the same time a wealth of useful details.

* Todd Trimble, _ZFC and ETCS: Elementary Theory of the Category of Sets_ ([[Trimble on ETCS I|nLab entry]],  [original blog entry](http://topologicalmusings.wordpress.com/2008/09/01/zfc-and-etcs-elementary-theory-of-the-category-of-sets/))

* Todd Trimble, _ETCS: Internalizing the logic_ ([[Trimble on ETCS II|nLab entry]], [original blog entry](http://topologicalmusings.wordpress.com/2008/09/10/etcs-internalizing-the-logic/))

* Todd Trimble, _ETCS: Building joins and coproducts_
([[Trimble on ETCS III|nLab entry]], [original blog entry](http://topologicalmusings.wordpress.com/2008/12/15/etcs-building-joins-and-coproducts/))

## Related entries

* [[fully formal ETCS]]
* [[ETCS with elements]]
* [[structural set theory]]
* [[ZFC|Zermelo Fraenkel set theory]]
* [[Cohesive Toposes and Cantor's "lauter Einsen"]]
* [[axiom of replacement]]
* [[ETCC]]
* [[practical foundations of mathematics]]
* [[foundations of mathematics]]
* [[ETCR]]
* [[Mostowski set theory]]

## References

ETCS grew out of Lawvere's experiences of teaching undergraduate foundations of analysis at Reed college in 1963 and was originally published in

* {#Lawvere64} [[William Lawvere]], _An elementary theory of the category of sets_ , Proceedings of the National Academy of Science of the U.S.A **52** pp.1506-1511 (1964). ([pdf](http://www.ncbi.nlm.nih.gov/pmc/articles/PMC300477/pdf/pnas00186-0196.pdf))

A more or less contemporary review is

* C.C. Elgot, _Review_,  JSL **37** no.1 (1972) pp. 191-192.

A longer version of Lawvere's 1964 paper appears in

* [[William Lawvere]], [[Colin McLarty]], _An elementary theory of the category of sets (long version) with commentary_ , Reprints in Theory and Applications of Categories **11** (2005) pp. 1-35. ([TAC](http://tac.mta.ca/tac/reprints/articles/11/tr11abs.html))

An undergraduate set-theory textbook using it is

* {#LawvereRosebrugh} [[William Lawvere]], [[Robert Rosebrugh]], _[[Sets for Mathematics]]_ , CUP  2003. ([web](http://books.google.de/books?id=h3_7aZz9ZMoC&pg=PP1&dq=sets+for+mathematics))

Lawvere explains in detail his views on constant and variable 'abstract sets' on pp.118-128 of

* {#Lawvere76} [[William Lawvere]], _Variable Quantities and Variable Structures in Topoi_ , pp.101-131 in Heller, Tierney (eds.), _Algebra, Topology and Category Theory: a Collection of Papers in Honor of Samuel Eilenberg_ , Academic Press New York 1976.

See also ch. 2,3 of

* [[William Lawvere]], _Variable Sets Entendu and Variable Structure in Topoi_ , lecture notes University of Chicago 1975.

On the anticipation of 'abstract sets' in [[Georg Cantor|Cantor]]:

* {#Lawvere94}[[William Lawvere]],  _[[Cohesive Toposes and Cantor's "lauter Einsen"]]_, Philosophia Mathematica **2** no.3 (1994) pp.5-15. ([[LawvereCohesiveToposes.pdf:file]]) {#law94}

A short overview article on ETCS:

* [[Tom Leinster]], _Rethinking set theory_ [arXiv](http://arxiv.org/abs/1212.6543).

An insightful and non-partisan view of ETCS can be found in a section of:

* [[Andreas Blass]], Yuri Gurevich, _Why Sets ?_ , Bull. Europ. Assoc. Theoret. Comp. Sci. **84** (2004) 139-156.  ([draft](http://www.math.lsa.umich.edu/~ablass/set.pdf))

An extended discussion from a philosophical perspective is in

* {#McLarty04}[[Colin McLarty]], _Exploring Categorical Structuralism_ , Phil. Math. **12** (2004) 37-53.

* Øystein Linnebo, Richard Pettigrew, _Category theory as an autonomous foundation_, Phil. Math. **19** (2011) 227–254 ([doi:10.1093/philmat/nkr024](https://doi.org/10.1093/philmat/nkr024),  [preprint version (pdf)](http://philsci-archive.pitt.edu/5392/1/onlyuptoiso.pdf))

For a more recent review from a critical perspective containing additional recent references see

* [[Solomon Feferman]], _Foundations of Unlimited Category Theory: What Remains to be Done_ , Review of Symbolic Logic **6** no.1 (2013) 6-15. ([pdf](http://math.stanford.edu/~feferman/papers/FCT-RSL-2013.pdf))

An informative discussion of the pros and cons of Lawvere's approach can be found in

* [[Jean-Pierre Marquis]], _Kreisel and Lawvere on Category Theory and the Foundations of Mathematics_ . ([pdf-slides](http://www.math.mcgill.ca/rags/seminar/Marquis_KreiselLawvere.pdf))

Palmgren's ideas can be found here:

*  [[Erik Palmgren]], _Constructivist and Structuralist Foundations: Bishop's and Lawvere's Theories of Sets_ , Ann. Pure. App. Logic **163** no.10 (2012) 1384-1399. ([pdf](http://www.math.uu.se/~palmgren/cetcs.pdf)) {#Palmgren}

For the relation between the theory of well-pointed toposes and **weak Zermelo set theory** as elucidated by work of [[Julian Cole|J. Cole]], [[Barry Mitchell]], and [[Gerhard Osius|G. Osius]] in the early 1970s see

* [[Peter Johnstone]], _Topos Theory_ , Academic Press New York 1977 (Dover reprint 2014). (sections 9.2-3)

* [[Barry Mitchell]], _Boolean Topoi and the Theory of Sets_ ,  JPAA **2** (1972) pp.261-274.

* [[Gerhard Osius]], _Categorical Set Theory: A Characterization of the Category of Sets_,  JPAA **4** (1974) 79-119.

[[!redirects ETCS]]
[[!redirects etcs]]
[[!redirects elementary theory of the category of sets]]


ETCS with elements

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# Contents
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In usual presentations of categorical set theories such as [[ETCS]], the set theory is usually a theory of [[sets]] and [[functions]] in an abstract [[category]], with [[elements]] being defined as the [[global elements]], the [[morphisms]] out of the [[terminal object]]. However, this approach poses a few conceptual and practical issues, namely that in ordinary mathematical practice, elements are not the same as functions out of the terminal set $\mathbb{1}$, although $A$ is in bijection with the function set $A^\mathbb{1}$, and function evaluation of elements defined as function composition of functions out of the terminal set is an abuse of notation.

There are other [[structural set theories]], such as [[SEAR]], which explicitly put in the elements of a set as a primitive of the theory. In such a theory involving sets and functions, function evaluation would be a primitive of the theory, rather than derived from function composition of [[global elements]], and the axiom of function extensionality is likewise defined directly on the elements. Such a theory reads more like a traditional presentation of set theory in terms of sets and elements, rather than category theory. However, in contrast to [[SEAR]], this theory is essentially the same structurally as [[ETCS]]: as a theory of sets and functions.

In this presentation, we will be adapting [[Tom Leinster]]'s presentation of ETCS.

## Basic primitives ##

Our theory has the following primitives:

* Some things called **sets**;

* For every set $A$, these things called **elements in $A$**, with elements $a$ in $A$ written as $a \in A$;

* For every set $A$ and $B$, these things called **functions from $A$ to $B$**, with functions $f$ from $A$ to $B$ written as $f:A \to B$;

* For every set $A$ and $B$, an operation called **function evaluation** assigning every element $a \in A$ and function $f:A \to B$ an element $f(a) \in B$;

* For every set $A$, a function $id_A:A \to A$ called the **identity function of $A$**;

* For every set $A$, $B$, and $C$, an operation called **function composition** assigning every function $f:A \to B$ and $g:B \to C$ a function $g \circ f:A \to C$;

## Equality of elements and functions ##

For every set $A$ and elements $a \in A$ and $b \in A$, there is a relation $a = b$ called **equality of elements**, such that

* for every element $a \in A$, $a = a$
* for every element $a \in A$ and $b \in A$, $a = b$ implies that $b = a$
* for every element $a \in A$, $b \in A$, and $c \in A$, $a = b$ and $b = c$ implies that $a = c$.

For every set $A$ and $B$ and functions $f:A \to B$ and $g:A \to B$, there is a relation $f = g$ called **equality of functions**, such that

* for every function $f:A \to B$, $f = f$
* for every function $f:A \to B$ and $g:A \to B$, $f = g$ implies that $g = f$
* for every function $f:A \to B$, $g:A \to B$, and $h:A \to B$, $f = g$ and $g = h$ implies that $f = h$.

## Basic axioms for functions ##

**Axiom of identity functions.** For every set $A$ and for every element $a \in A$, $id_A(a) = a$.

**Axiom of composition/evaluation equivalence.** For every set $A$, $B$, and $C$, and for every element $a \in A$, $(g \circ f)(a) = g(f(a))$.

**Axiom of extensionality.** For every set $A$ and $B$ and for every function $f:A \to B$ and $g:A \to B$, if $f(x) = g(x)$ for all elements $x \colon A$, then $f = g$.

The associativity and unit laws of function composition follow from the axioms:

* For every set $A$ and $B$, function $f:A \to B$, and element $a \in A$,
$$(g \circ id_A)(a) = g(id_A(a)) = g(a)$$
and extensionality implies that $g \circ id_A = g$.

* For every set $A$ and $B$, function $f:A \to B$, and element $a \in A$,
$$(id_B \circ g)(a) = id_B(g(a)) = g(a)$$
and extensionality implies that $id_B \circ g = g$.

* For every set $A$, $B$, $C$, and $D$, function $f:A \to B$, $g:B \to C$, and $h:C \to D$, and element $a \in A$,
$$(h \circ (g \circ f))(a) = h((g \circ f)(a)) = h(g(f(a)) = (h \circ g)(f(a)) = ((h \circ g) \circ f)(a)$$
and extensionality implies that $h \circ (g \circ f) = (h \circ g) \circ f$.

Thus, these axioms imply that the collection of sets, functions, and elements form a [[category]].

## Other axioms ##

**Axiom of singletons.** There is a set $\mathbb{1}$, called a **singleton**, with a unique element $* \in \mathbb{1}$, called a **point**.

**Axiom of Cartesian products.** For every set $A$ and $B$, there is a set $A \times B$, called a **Cartesian product** of $A$ and $B$, with a function $p_A:A \times B \to A$ called the **projection onto $A$** and a function $p_B:A \times B \to B$ called the **projection onto $B$**, such that given two elements $a \in A$ and $b \in B$ there is a unique element $a, b \in A \times B$ such that $p_A(a, b) = a$ and $p_B(a, b) = b$.

**Axiom of fibers.** For every set $A$ and $B$, element $b \in B$, and function $f:A \to B$, there is a set $f^{-1}(b)$ called the **fiber** of $f$ over $b$ with a function $i:f^{-1}(b) \to A$, such that for every element $a \in f^{-1}(b)$, $f(i(a)) = b$, and for every other set $C$ and function $g:C \to B$ such that for every element $c \in C$, $f(g(c)) = b$, there is a unique function $j:C \to f^{-1}(b)$ such that for every element $c \in C$, $g(c) = i(j(c))$. A fiber of $f$ over $b$ is also called the **solution set** of the equation $f(x) = b$.

An **injection** is a function $f:A \to B$ such that for every element $a \in A$ and $b \in A$, if $f(a) = f(b)$, then $a = b$.

**Axiom of truth values.** There is a set $\Omega$ called a **set of truth values** with an element $\bot \in \Omega$ called **true** such that for every set $A$ and $B$ and injection $f:A \to B$, there is a unique function $\chi_A:B \to \Omega$ called the **classifying function** of $A$ such that $A$ is the fiber of $\chi_A$ over $\bot$.

**Axiom of function sets.** For every set $A$ and $B$ there is a set $B^A$ called the **function set** with a function $(-)((-)):B^A \times A \to B$ such that for every set $C$ and function $f:C \times A \to B$ there is a unique function $g:C \to B^A$ such that for all elements $c \in C$ and $a \in A$, $g(c)(a) = f(c, a)$.

**Axiom of natural numbers.** There exists a set $\mathbb{N}$ called a **set of natural numbers** with an element $0 \in \mathbb{N}$ and a function $s:\mathbb{N} \to \mathbb{N}$ such that for every other set $A$ with an element $a \in A$ and a function $g:A \to A$, there is a unique function $f:\mathbb{N} \to A$ such that $f(0) = a$ and $f(s(a)) = g(f(a))$.

A **surjection** is a function $f:A \to B$ such that for every $b \in B$ the fiber of $f$ at $b$ is inhabited. A **right inverse** of $f$ is a function $g:B \to A$ such that for all elements $a \in A$, $f(g(a)) = a$. A **choice set** is a set $B$ such that all surjections into $B$ have right inverses.

**Axiom of choice.** Every set is a choice set.

These axioms imply that the collection of sets, functions, and elements form a model of [[ETCS]].

## See also ##

* [[ETCS]]

* [[Trimble on ETCS I]]

* [[SEAR]]

## References ##

* [[Tom Leinster]], Rethinking set theory, ([arxiv:1212.6543](https://arxiv.org/abs/1212.6543))


Etera Livine

__Etera R. Livine__ works in quantum gravity, especially Ponzano-Regge type models, spin-foam models in general and connections to noncommutative geometry.

* [[Laurent Freidel]], E. R. Livine, Phys. Rev. Lett. __96__ (2006), no. 22, 221301, 4 pp.; [MR2007j:83051](http://www.ams.org/mathscinet-getitem?mr=2231432)
* Livine, Etera R. Matrix models as non-commutative field theories on $\mathbf{R}^3$. Classical Quantum Gravity 26 (2009), no. 19, 195014, [doi](http://dx.doi.org/10.1088/0264-9381/26/19/195014)
* [[Laurent Freidel]], Etera R. Livine, _The fine structure of $SU(2)$ intertwiners from $U(N)$ representations_, J. Math. Phys. __51__ (2010), no. 8, 082502, [MR2011j:81140](http://www.ams.org/mathscinet-getitem?mr=2683530), [doi](url=http://dx.doi.org/10.1063/1.3473786)
* F. Girelli, E. R. Livine, _Field theories with homogenous momentum space_, in Proceedings of 25th Max Born Symposium: The Planck Scale (Wroclaw, Poland, 2009), AIP Conf. Proc. 1196, (2009), 115–123, [arXiv:0910.3107](http://arxiv.org/abs/0910.3107)
* F. Girelli, E. R. Livine, _Scalar field theory in Snyder space-time: alternatives_, [arXiv:1004.0621](http://arxiv.org/abs/1004.0621)
* Florian Girelli, Etera R. Livine, _A deformed Poincaré invariance for group field theories_, Classical Quantum Gravity __27__ (2010), no. 24, 245018 [MR2011j:83018](http://www.ams.org/mathscinet-getitem?mr=2739974), [doi](http://dx.doi.org/10.1088/0264-9381/27/24/245018)
* Florian Girelli, Etera R. Livine, Daniele Oriti, _Four-dimensional deformed special relativity from group field theories_, Phys. Rev. D 81 (2010), no. 2, 024015, 14 pp. [MR2610868](http://www.ams.org/mathscinet-getitem?mr=2610868)

[[!redirects E. R. Livine]]
[[!redirects Etera R. Livine]]


eternal cosmic inflation > history

> see at _[[chaotic cosmic inflation]]_


eternal inflation

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## Idea

In the context of [[cosmic inflation]] one speaks of _eternal inflation_ if the inflationary expansion process perpetuates, in space and/or time.

Many classes of models of [[cosmic inflation]] in fact exhibit eternal inflation, notably the observationally preferred plateau-models, such as the [[Starobinsky model of cosmic inflation]] ([Barenboim-Kinney-Park 16](#BarenboimKinneyPark16)). (Also models of [[chaotic inflation]] do, but these are now strongly disfavoured by [[experiment|observation]]).

## Implications

The idea of eternal cosmic inflation has been argued to provide a possible way to conceptualize the measured values of dimensionless  "physical constants", such as the [[fine structure constant]], the [[Yukawa couplings]] and notably the [[cosmological constant]]: for if in the fundamental [[theory (physics)|theory]] these parameters are not really constants but are dynamical [[field (physics)|fields]] that just happen to have constant value ("moduli fields") over large scales, then they might, so the argument, still vary from one "inflationary bubble" to the next. Thereby the idea of eternal inflation combined with that of a fundamental theory that has [[moduli fields]] (such as [[string theory]] with its [[landscape of string theory vacua]]) has been argued to put "physical constants" on the same footing as other more or less random phenomenological parameters, such as for instance the distance of our planets from the sun, etc.

For more on this see at _[[multiverse]]_.

## Related concepts

* [[false vacuum eternal inflation]]

## References

* [[Alexander Vilenkin]], _Birth of inflationary universes_, Phys. Rev. D 27, 2848 ([doi:10.1103/PhysRevD.27.2848](https://doi.org/10.1103/PhysRevD.27.2848))

* [[Aleksei Starobinsky]], _Stochastic de sitter (inflationary) stage in the early universe_, In: de Vega H.J., Sánchez N. (eds) Field Theory, Quantum Gravity and Strings. Lecture Notes in Physics, vol 246. Springer, Berlin, Heidelberg ([doi:10.1007/3-540-16452-9_6](https://doi.org/10.1007/3-540-16452-9_6))

* [[Alan Guth]], _Eternal inflation and its implications_, J.Phys.A40:6811-6826, 2007 ([arXivL:hep-th/0702178](https://arxiv.org/abs/hep-th/0702178))

* {#BarenboimKinneyPark16} Gabriela Barenboim, [[William Kinney]], Wan-Il Park, _Eternal Hilltop Inflation_,  Journal of Cosmology and Astroparticle Physics, Volume 2016, May 2016 ([arXiv:1601.08140](https://arxiv.org/abs/1601.08140))

See also

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

[[!redirects eternal cosmic inflation]]


Ethan Bernstein

* [MathematicsGenealogy page](https://www.mathgenealogy.org/id.php?id=273367)

## Selected writinga

Introducing the notion of [[quantum complexity theory]]:

* [[Ethan Bernstein]], [[Umesh Vazirani]], *Quantum Complexity Theory*, Siam Journal of Computation **26** 5 (1997) 1411-1473 [[doi:10.1137/S0097539796300921](https://doi.org/10.1137/S0097539796300921), [pdf](https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.655.1186&rep=rep1&type=pdf)]

category: people


Ethan Devinatz

* [webpage](http://www.math.washington.edu/~devinatz/)

## Selected writings

Introducing the [[nilpotence theorem]] in [[stable homotopy theory]]:

* {#DevinatzHopkinsSmith88} [[Ethan Devinatz]], [[Michael Hopkins]], [[Jeffrey Smith]], _Nilpotence and Stable Homotopy Theory I_, Annals of Mathematics Second Series, Vol. 128, No. 2 (Sep., 1988), pp. 207-241 ([jstor:1971440](https://www.jstor.org/stable/1971440))

* {#DevinatzHopkinsSmith98} [[Ethan Devinatz]], [[Michael Hopkins]], [[Jeffrey Smith]], _Nilpotence and Stable Homotopy Theory II_, Annals of Mathematics Second Series, Vol. 148, No. 1 (Jul., 1998), pp. 1-49 ([jstor:120991](https://www.jstor.org/stable/120991))

## Related entries

* [[nilpotence theorem]]

category: people


Ethan Siegel

* [webpage](http://www.phys.ufl.edu/siegel/)

## related $n$Lab entries

* [[cosmology]]

* [[dark matter]]

category: people


Ethereum

__Ethereum__ is a public [[blockchain]] which is established as a sequel to [[bitcoin]] with intention to include [[smart contract]]s and distributed applications as envisioned by [[Vitalik Buterin]] in 2013. This platform is considered as the main platform of blockchain 2.0 stage in blockchain technology development; as of 2019 it still uses proof of work.
As of 2019, Ethereum mainnet, which supports the Ether cryptocurrency is still officially beta and many improvements in 2019 and 2020 are planned, called Ethereum 1.x, including replacing the present Ethereum virtual machine (EVM) with ewasm virtual machine for [[WebAssembly]] bytecode. Somewhere in 2020 or 2021, a new release of Ethereum based on proof of stake with sharding and many other improvements (including a rent for smart contracts) is planned to be created; the new platform is called Ethereum 2.0 and will qualify for 3rd generation blockchain platform in terms of scalability and speed, and possibly some level of interoperability. Migration of users from the Ethereum 1.x beta chain will be performed via smart contracting, without hard forks. The old net is likely to stay as a read only record of old history and be attached to the new net as a shardchain. The implementations of Ethereum 2.0 are being coded simultaneously by 9 different teams in several different programming languages.

Ethereum 1.0 platform has largely influenced development of many subsequent platforms; in particular, [[zoranskoda:Hyperledger]] Fabric is a permissioned blockchain  developed by IBM and Linux Foundation on the basis of Ethereum.

Most important clients for Ethereum are (see also [here](https://github.com/ethereumbook/ethereumbook/blob/develop/03clients.asciidoc))

* Metamask  (Chrome extension) [metamask.io](https://metamask.io); source at [github](https://github.com/MetaMask/metamask-plugin)
* Mist browser (web3.js based browser for DApps) is now [deprecated](https://medium.com/@avsa/sunsetting-mist-da21c8e943d2); [github](https://github.com/ethereum/mist), [investopedia](https://www.investopedia.com/terms/m/mist-browser.asp)
* parity (node software written in [[Rust]]) [setup](https://wiki.parity.io/Setup)
* Geth (Go Ethereum, node software written in GoLang)

See also about setting your own private Ethereum net in {this tutorial](https://www.edureka.co/blog/ethereum-private-network-tutorial). A comprehensive book on Ethereum is

* Andreas M. Antonopoulos, Gavin Wood, _Mastering Ethereum: building smart contracts and DApps_, O’Reilly Media, 2018. The book is also on [github](https://github.com/ethereumbook/ethereumbook) together with the sourcecode

## Literature

See also [[blockchain]], [[smart contract]] and wikipedia article [Ethereum](https://en.wikipedia.org/wiki/Ethereum).

Official page [ethereum.org](https://www.ethereum.org) and [developer resources](https://www.ethereum.org/developers) there.

### General

*  V. Buterin, _Ethereum in 25 minutes_, [youtube](https://www.youtube.com/watch?v=66SaEDzlmP4); _Blockchain and Ethereum security on the higher level_, [youtube](https://www.youtube.com/watch?v=UFDAtStVXbc) 1:21h
* [[Vitalik Buterin]], Ethereum: A next-generation smart contract and decentralized application platform, [pdf](https://github.com/ethereum/wiki/wiki/White-Paper.pdf)
* Gavin Wood, _Ethereum: a secure decentralised generalised transaction
ledger_, Ethereum, Tech. Rep. 2017. [pdf](https://ethereum.github.io/yellowpaper/paper.pdf)
* [parity](https://www.parity.io) blockchain infrastructure for the decentralised web
* [Enterprise Ethereum Alliance](https://entethalliance.org)

Plans for Ethereum future from Ethereum Foundation as of May 2019:

* [blog.ethereum.org/2019/05/21/ethereum-foundation-spring-2019-update](https://blog.ethereum.org/2019/05/21/ethereum-foundation-spring-2019-update)
* A journey through phase 2 of Ethereum 2.0 [medium](https://medium.com/@william.j.villanueva/a-journey-through-phase-2-of-ethereum-2-0-c7a2397a36cb)
* Phase 2 proposal 1 [hackmd](https://notes.ethereum.org/s/HylpjAWsE#)
* V. Buterin, [A layer-1-minimizing phase 2 state execution proposal](https://ethresear.ch/t/a-layer-1-minimizing-phase-2-state-execution-proposal/5397/4), [Proposed further simplifications/abstraction for phase 2](https://ethresear.ch/t/proposed-further-simplifications-abstraction-for-phase-2/5445)

### Protocol improvements

* Vlad Zamfir, Nate Rush, Aditya Asgaonkar, Georgios Piliouras, _Introducing the “Minimal CBC Casper” family of consensus protocols_, [pdf](https://github.com/cbc-casper/cbc-casper-paper/blob/master/cbc-casper-paper-draft.pdf)
* [how ethereum works](https://www.coindesk.com/information/how-ethereum-works)
* Andrew Ashikhmin, Alexey Akhunov, _Red queen's sync protocol for Ethereum_, github/[pdf](https://github.com/yperbasis/silkworm/raw/master/doc/sync_protocol_v1.pdf)
* [cointelegraph.com/news/an-ethereum-20-proof-of-stake-testnet-blockchain-is-now-live](https://cointelegraph.com/news/an-ethereum-20-proof-of-stake-testnet-blockchain-is-now-live)

### Whisper

According to Mukhapadhyay, _Ethereum smart contract development_

> When [[Gavin Wood]] laid down the specifications for whisper as a peer-to-peer communication protocol, in his whitepaper on [GitHub](https://github.com/ethereum/wiki/wiki/Whisper-PoC-2-Protocol-Spec), he designed one of the most interesting technologies in the field of cryptography. At a considerable cost of bandwidth and latency, whisper is able to deliver a 100% dark operation. By completely dark operations, we mean that there is zero leakage of metadata during peer-to-peer By technical definition, whisper is a messaging system with multi distributed hash table (DHT), with routing privacy acting as a
companion protocol to the Ethereum blockchain.

> Whisper operates in a user-configurable manner with regard to how much information the communicating nodes are willing to leak concerning the decentralized application content that ultimately tracks the user activities.

> Whisper is based on two key concepts: messages and envelopes. If whisper is considered a datagram messaging service, then an envelope represents an un-encrypted data format, comprehensible by a node, which carries the encrypted message datagram inside it. An envelope consists of original time to live (TTL, in seconds), the absolute time to expiry (UNIX system time), the encrypted message data field, which is the actual payload, topics (cryptographically secure, probabilistic partial classifications of message), and nonce, an arbitrary value. This nonce is used for the PoW to judge the efforts of a peer. The message has a binary flag for signature with an unfixed payload.

### Ethereum virtual machine, storage and smart contracting

* EVM [storage layout](https://medium.com/@hayeah/diving-into-the-ethereum-vm-part-2-storage-layout-bc5349cb11b7)
* [2018/03/09/understanding-ethereum-smart-contract-storage](https://programtheblockchain.com/posts/2018/03/09/understanding-ethereum-smart-contract-storage)
* solidity [docs](https://solidity.readthedocs.io)
* solidity "guide", [bitdegree](https://www.bitdegree.org/learn/solidity)
* [stm](https://abhiroop.github.io/BlockChain-STM), [smart contract misconceptions](https://www.coindesk.com/three-smart-contract-misconceptions), [more storage](https://applicature.com/blog/blockchain-technology/ethereum-smart-contract-storage), [contract call at later time](https://ethereum.stackexchange.com/questions/42/how-can-a-contract-run-itself-at-a-later-time), Where do smart contracts reside in blockchain [stackoverflow](https://stackoverflow.com/questions/42081194/where-do-smart-contracts-reside-in-blockchain-ethereum-or-hyperledger), [chronos](https://www.stateofthedapps.com/dapps/chronos)
* [viper](https://ethereumclassic.github.io/blog/2017-03-13-viper)
* A. Unterweger et al. _Lessons learned from implementing a privacy-preserving
smart contract in Ethereum”_, 2018 9th IFIP Intern. Conf. on
New Technologies, Mobility and Security (NTMS) Feb. 2018, pp. 1–5
[doi](https://doi.org/10.1109/NTMS.2018.8328739)
* Ethereum Alarm Clock [documentation](http://docs.ethereum-alarm-clock.com/en/latest)
* _How can a contract run itself at a later time?_, [
ethereum.stackexchange](https://ethereum.stackexchange.com/questions/42/how-can-a-contract-run-itself-at-a-later-time)
* how-data-is-stored-in-ethereum [hackernoon](https://hackernoon.com/getting-deep-into-ethereum-how-data-is-stored-in-ethereum-e3f669d96033)

### Others

Vitalik proposed a privacy mixer for Ethereum to enhance anonymity via groupings

* _Minimal mixer design_, proposal at [hackmd](https://hackmd.io/s/rJj9hEJTN); [news at coindesk](https://www.coindesk.com/vitalik-roposes-mixer-to-anonymize-one-off-transactions-on-ethereum)


Etsuro Date

* [InSpire page](https://inspirehep.net/authors/2283631)

## Selected writings

On the [hypergeometric construction](Knizhnik-Zamolodchikov+equation#BraidRepresentationsViaTwisteddRCohomologyOfConfigurationSpaces) of solutions to the [[Knizhnik-Zamolodchikov equation]]:

* [[Etsuro Date]], [[Michio Jimbo]], [[Atsushi Matsuo]], [[Tetsuji Miwa]], *Hypergeometric-type integrals and the $\mathfrak{sl}(2,\mathbb{C})$-Knizhnik-Zamolodchikov equation*,  International Journal of Modern Physics B **04** 05 (1990) 1049-1057   $[$[doi:10.1142/S0217979290000528](https://doi.org/10.1142/S0217979290000528)$]$

On [[solitons]]:

* [[Michio Jimbo]], [[Tetsuji Miwa]], [[Etsuro Date]], (transl. from Japanese by [[Miles Reid|M. Reid]]), *Solitons: differential equations, symmetries and infinite dimensional algebras*,  Cambridge Tracts in Mathematics **135** (2012) [[ISBN:9781107404199](https://www.cambridge.org/us/universitypress/subjects/mathematics/geometry-and-topology/solitons-differential-equations-symmetries-and-infinite-dimensional-algebras)]

category: people


Ettore Aldrovandi

* [website](http://www.math.fsu.edu/~ealdrov/)

category: people


Euclid

# Euclid
* table of contents
{: toc}

## Life and work

__Euclid__ (Εὐκλείδης) of Alexandria was the ancient author of the _[[Elements]]_, a treatise consisting of thirteen books covering [[Euclidean geometry|plane and solid geometry]], arithmetic (elementary [[number theory]]), and an early theory of [[real numbers]]. He is regarded as the preeminent organizer of the [[mathematics]] of the Hellenistic world, and his work exerted a decisive influence, through the centuries to come, on the understanding and cultural image of mathematics as a rigorous deductive discipline.

Euclid\'s rigour became a byword for mathematics (and specifically for geometry) until the 19th century, particularly in contrast to the slapdash approach of [[infinitesimal calculus]].  By the end of that century, however, mathematical rigour had progressed beyond Euclid, and [[David Hilbert]] had to reorganise the foundations of geometry to fill in some of Euclid\'s gaps.

There is little hard data on Euclid as a person; it seems to be generally agreed that he worked and taught at the library at Alexandria during the reign of Ptolemy I (*circa* 300 BC), having earlier received his training in mathematics from students of Plato at Plato's Academy in Athens.

One could say that Euclid was the ancient version of [[Bourbaki]], who quite consciously fancied and presented themselves as a modern-day Euclid, collecting and reorganizing the mathematics of the day in rigorous logical form. (The naming of Bourbaki's _magnum opus_, the _[[Elements of Mathematics]]_, is of course no coincidence.) One notable stylistic difference: Euclid freely employed pictures as visual aids to the logical demonstrations in the _Elements_, whereas Bourbaki wrote in rather a puritanical spirit, sternly forbidding the use of pictorial representations, perhaps to avoid the kinds of errors that Hilbert patched in Euclid\'s work.

## Related entries

*  _[[Elements]]_ (Euclid's book on what is now called [[axiom|axiomatic]]/[[synthetic geometry|synthetic]] [[Euclidean geometry]])

* [[Euclidean domain]]

* [[Euclidean geometry]]

* [[Euclidean relation]]

* [[Euclidean space]]

* [[Euclidean topology]]

* [[monad terminology]]

## External links

*  [English Wikipedia](http://en.wikipedia.org/wiki/Euclid)

*  [An English translation of _Elements_ ](http://www.mathcs.clarku.edu/~djoyce/java/elements/elements.html)

* Online article on Euclid ([web](http://www.mathopenref.com/euclid.html))

* [St. Andrews history article](http://www-groups.dcs.st-and.ac.uk/history/Biographies/Euclid.html)

category: people

[[!redirects Euclid]]
[[!redirects Euclidean]]
[[!redirects euclidean]]
[[!redirects Εὐκλείδης]]
[[!redirects Ευκλείδης]]
[[!redirects Ευκλειδης]]


Euclidean domain

+-- {: .rightHandSide}
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###Context###
#### Arithmetic
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#### Algebra
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# Contents #
* table of contents
{:toc}

## Idea

A Euclidean domain is an integral domain which admits a form of the Euclidean quotient-and-remainder algorithm familiar from school mathematics, as well as a form of the extended Euclidean algorithm for calculating the [[greatest common divisor]] and [[Bézout coefficients]] in a [[Bézout domain]].

## Definition

A **Euclidean domain** is an [[integral domain]] $A$ for which there exists a function $d: A \setminus \{0\} \to \mathbb{N}$ to the [[natural numbers]], often called a *degree function*, such that given $f, g \in A$ with $g \neq 0$, there exist $q, r \in A$ such that $f = q \cdot g + r$ and either $r = 0$ or $d(r) \lt d(g)$. (One may harmlessly stipulate that $d(0) = 0$; what to do with the zero element varies from author to author.) There is no uniqueness requirement for $q, r$.

One could posit, instead of the [[property]] that for all $f \in A$ and $g \in A \setminus \{0\}$, there exist $q, r \in A$ such that $f = q \cdot g + r$ and either $r = 0$ or $d(r) \lt d(g)$; that the integral domain has the [[structure]] of a function $(-)\div(-):A \times A \setminus \{0\} \to A$ called the *[[division]] function*, and a function $(-)\ \%\ (-):A \times A \setminus \{0\} \to A$ called the *remainder function*, such that for all $f \in A$ and $g \in A \setminus \{0\}$, $f = (f \div g) \cdot g + (f\ \%\ g)$ and either $f\ \%\ g = 0$ or $d(f\ \%\ g) \lt d(g)$. There might be multiple such division and remainder functions for the integral domain $A$.

Some authors also add the requirement that $d(a) \leq d(a b)$ for all nonzero $a, b$. There is no loss of generality in assuming it; every Euclidean domain admits such a degree function $d'$, defining $d'(a) = \min \{d(a b): b \in A, b \neq 0\}$. We'll use it freely below, if and when we need to.

### In constructive mathematics ###

In constructive mathematics, there are different types of integral domains, yielding different types of Euclidean domains: the degree function, division, function, and remainder function are no longer valued in $A \setminus \{0\}$ in one variable, but in $\{x \in A \vert x \neq 0\}$, $\{x \in A \vert x \# 0\}$, or some other definition, depending on what the base [[integral domain]] ends up being (classical, Heyting, discrete, residue, et cetera).

The definition of the Euclidean domain further bifurcates in [[constructive mathematics]] due to the disjunctive condition above. These statements, which are equivalent in the presence of [[excluded middle]], include:

* $(r = 0) \vee (d(r) \lt d(g))$
* $\not((r \neq 0) \wedge (d(r) \geq d(g)))$
* $(d(r) \geq d(g)) \to (r = 0)$

or equivalently,

* $(f\ \%\ g = 0) \vee (d(f\ \%\ g) \lt d(g))$
* $\not((f\ \%\ g \neq 0) \wedge (d(f\ \%\ g) \geq d(g))$
* $(d(f\ \%\ g) \geq d(g)) \to (f\ \%\ g = 0)$

## Examples

* The (rational) [[integers]] $\mathbb{Z}$.

* Any field (trivially).

* The ring of [[polynomials]] $k[x]$ over a [[discrete field]] $k$, using the ordinary [[polynomial degree]].

* On the other hand, if $k$ is a [[Heyting field]] which is not a discrete field, not every polynomial in $k[x]$ apart from zero can be proven to have a well-defined degree, and so is not an [[Euclidean domain]].

* The real polynomial ring $\mathbb{R}[x]$ is only an Euclidean domain in the presence of the [[analytic limited principle of omniscience]], which states that it has decidable apartness.

* Any [[valuation ring|discrete valuation ring]]: letting $\pi$ be a generator of the maximal ideal, put $d(x) = n$ where $x = u \pi^n$, with $u$ a unit.

* The [[Gaussian integers]] $\mathbb{Z}[i]$, with $d(a + b i) = a^2 + b^2$ (the norm).

* The [[Eisenstein integers]] $\mathbb{Z}[\omega]$, where $\omega$ is a primitive cube [[root of unity]], with $d(a + b \omega) = a^2 - a b + b^2$ (the norm).

## Properties

+-- {: .num_prop}
###### Proposition
A Euclidean domain is a [[Bézout domain]].
=--

+-- {: .proof}
###### Proof
Let $I \subseteq A$ be a nonzero finitely generated ideal, and suppose $d(g)$ is the minimum degree taken over all nonzero $g \in I$. For such a $g$ and any $f \in I$, we may write $f = q g + r$ where either $r = 0$ or $d(r) \lt d(g)$ (which is impossible since $r \in I$ and $d(g)$ is minimal). So $r = 0$ it is, and thus $I = (g)$.
=--

This proof uses the [[well-order|well-orderedness]] of $\mathbb{N}$. It also suggests the practical procedure known as the **extended Euclidean algorithm**: given two elements $a, b$ in a Euclidean domain $A$, this algorithm provides a generator $d$ of the ideal $I = (a) + (b)$, called a [[greatest common divisor]] of $a, b$, as follows. One constructs a sequence of pairs $(a_n, b_n)$ of elements of $I$, with $(a_0, b_0) = (a, b)$, and $(a_{n+1}, b_{n+1}) = (b_n, r_n)$ where $a_n = b_n q_n + r_n$ for some choice of $q_n, r_n$ such that $d(r_n) \lt d(b_n)$ and $r_n \neq 0$. If no such choices for $q_n, r_n$ exist, then the sequence terminates at $(a_n, b_n)$, and $b_n$ generates $I$, i.e, $(b_n) = I$. Notice that the sequence terminates after finitely many steps since we cannot have an [[infinite descent]] of natural numbers

$$d(r_0) \gt d(r_1) \gt \ldots,$$
again by well-orderedness of $\mathbb{N}$.

The proof that this algorithm produces a gcd consists in the observation that at each step we have

$$(a_{n+1}) + (b_{n+1}) = (a_n) + (b_n),$$

and that from the definition of Euclidean domain, the only reason why the sequence would terminate at $(a_n, b_n)$ is that $a_n = b_n q_n + 0$ for some $q_n$, and in that case $(a_n) + (b_n) = (b_n)$.

## Generalisations

Euclidean domains could be generalised to the case where the structure is only a [[rig]] instead of a ring; these objects could be called __Euclidean cancellative rigs__.

## See also

* [[integral domain]]
* [[GCD domain]]
* [[Bézout domain]]
* [[field]]
* [[Euclidean ring]]

## References

See also:

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

* Wikipedia, _[extended Euclidean algorithm](https://en.wikipedia.org/wiki/Extended_Euclidean_algorithm)

[[!redirects Euclidean domains]]
[[!redirects Euclidean algorithm]]
[[!redirects extended Euclidean algorithm]]


Euclidean field

> This entry is about a notion in [[algebra]] and [[analysis]]. For the notion in [[quantum field theory]] see at *[[Euclidean field theory]]*.

***

+-- {: .rightHandSide}
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### Context
#### Algebra
+-- {: .hide}
[[!include higher algebra - contents]]
=--
#### Analysis
+-- {: .hide}
[[!include analysis - contents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

## Definition

A **Euclidean field** is an [[ordered field]] $F$ with a [[principal square root]] [[function]] $\mathrm{sqrt}:[0, \infty) \to F$ which satisfies the [[functional equation]] $\mathrm{sqrt}(x^2) = \vert x \vert$ for all $x \in F$, where $x^2$ is the [[square function]] and $\vert x \vert$ is the [[absolute value]]. Here $[0,\infty)$ denotes the non-negative elements of $F$, hence, either $\{0\}\cup P$, or $(-P)^c$, for $P=\{x\in F \mid x \gt 0\}$ the subset of positive elements, or $\{x\in F\mid x\geq 0\}$ (in [[constructive mathematics]] some care will need to be taken in how this is defined.)

## Examples

* Every [[real closed field]] is a Euclidean field.

* The real numbers constructible as lengths (or their negatives) via straightedge and compass from rational numbers.

## See also

* [[square root]]

* [[real square root]]

* [[Euclidean space]]

## References

See also:

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

[[!redirects Euclidean fields]]


Euclidean field theory

+-- {: .rightHandSide}
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### Context
#### Algebraic Quantum Field Theory
+--{: .hide}
[[!include AQFT and operator algebra contents]]
=--
#### Measure and probability theory
+-- {: .hide}
[[!include measure theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

### General

In [[field theory]] one speaks of _Euclidean field theory_ if the underlying [[spaces]] on which the [[field (physics)|fields]] are defined are [[Riemannian manifolds]], as opposed to [[Lorentzian  manifold|Lorentzian]] [[spacetimes]] used in [[relativistic field theory]], hence locally are _[[Euclidean spaces]]_ instead of [[Minkowski spacetimes]], whence the name "Euclidean field theory."

Concretely this means that in Euclidean field theory the [[local field theory|locality]] condition on the [[net of observables|net of]] [[quantum observables]] requires observables $A, B$ to [[commutator|commute]] as soon as their [[spacetime supports]] are [[disjoint subset|disjoint]] at all

$$
  supp(A) \cap supp(B) = \emptyset
  \;\;\;\;\Rightarrow\;\;\;\;
  [A,B] = 0
  \,.
$$

This is in contrast to the analogous condition in [[relativistic field theory]] whose _[[causal locality]]_ requires this implication only if the two [[supports]] are in addition [[spacelike]]-separated.

Equivalently this means that the [[n-point functions]] of Euclidean field theories are [[distributions of several variables]] with [[singularities]] on the [[fat diagonal]] (instead of on all of the relative [[light cones]], as for [[relativistic field theory]]). This means that Euclidean $n$-point functions [[restriction of distributions|restrict]] to [[non-singular distributions]] on the [[configuration spaces of points|configuration space of n points]], allowing to express [[correlators as differential forms on configuration spaces of points]]. Systematic discussion of [[perturbative quantum field theory|perturbative]] [[renormalization|renormalized]] Euclidean field theory from this perspective is due to [Bergbauer-Brunetti-Kreimer 09](#BergbauerBrunettiKreimer09), [Berghoff 14a](#Berghoff14a), [Berghoff 14b](#Berghoff14b).

This Euclidean locality property applies in particular in [[statistical mechanics]], where the "[[field (physics)|fields]]" of the field theory are not thought of as encoding the [[spacetime]]-behaviour of [[fundamental particles]] as governed by [[quantum physics]], but instead the spatial [[expectation values]] (at any given time) of [[equilibrium]] [[thermodynamics|thermodynamic]]-processes governed by [[classical physics]].

An archetypical example of a Euclidean field theory in this thermodynamic sense is the [[Ising model]]. Generally, most [[2d conformal field theories]] considered are Euclidean field theories and (should) have the interpretation of describing [[thermodynamics|thermodynamical]] systems "at criticality".

However, other Euclidean [[2d CFTs]] are not necessarily regarded as thermodynamical systems, notably the [[worldsheet]]-field theories defining a [[string perturbation series]] in [[perturbative string theory]].

Another broad class of examples of Euclidean field theories are [[topological quantum field theories]] after their [[diffeomorphism]]-[[symmetry]] is partially [[gauge fixing|gauge fixed]] by a choice of [[Riemannian metric]]. The archetypical example here is [[perturbative quantum field theory|perturbative]] [[Chern-Simons theory]], see [there](Chern-Simons+theory#FeynmanPerturbationSeries) for more.

### Wick rotation to Relativistic Field theory

Despite this superficially stark contrast between Euclidean and relativistic field theory, the two turn out to be tightly related to each other, at least under some conditions, in a subtle way that involves and generalizes the concept of [[analytic continuation]] from [[complex analysis]], here this is called _[[Wick rotation]]_.

Roughly this says that [[propagators]] and hence [[n-point functions]] of [[relativistic field theory]] on [[Minkowski spacetime]] $\mathbb{R}^{d,1}$ (may) have [[analytic continuation]] to [[complex number|complex values]] of the [[time]]-[[coordinates]], such that replacing [[real number|real]] time with [[imaginary number|imaginary]] time turns these [[n-point functions]] into those of a [[Euclidean field theory]] on $\mathbb{R}^{d+1}$, and vice versa.

Precise formulation of the conditions that go into this [[Wick rotation]] between [[relativistic field theory]] and Euclidean field theory is the content of the [[Osterwalder-Schrader theorem]].

### Temporal compactification to Thermal relativistic field theory
  {#TemporalCompactificationToThermalRelativisticFieldTheory}

In fact this relation goes deeper still: Under suitable conditions the Euclidean field theory not on $\mathbb{R}^{d+1}$ but on $\mathbb{R}^d \times S^1_\beta$, with the [[circle]]-[[product space|factor]] $S^1_{\beta}$ of [[length]] $\beta$, corresponds to relativistic field theory on [[Minkowski spacetime]] $\mathbb{R}^{d,1}$ in a [[vacuum state]] that represents [[thermal equilibrium]] at [[inverse temperature]] $\beta = 1/T$. (The previous case of Euclidean field theory on $\mathbb{R}^{d+1}$ may be thought of as the special case $\beta \to \infty$, hence $T \to 0$.)

This curious relation of [[Wick rotation]] with "compact peridodic Euclidean time" makes, when it applies, Euclidean field theory be a unification of [[relativistic field theory]] with [[statistical mechanics]]/[[thermodynamics]], then called _thermal quantum field theory_ or _quantum statistical field theory_ or similar.

$$
  \array{
  \left.
  \array{
     \text{relativistic field theory}
     \\
     \text{on Minkowski spacetime}
     \\
     \mathbb{R}^{d,1}
     \\
     \text{in a thermal equilibrium state}
     \\
     \text{at temperature}\; T
  }
  \right\}
  &
  \;\;\;\;
  \overset{ \text{Wick rotation} }{\leftrightarrow}
  \;\;\;\;
  &
  \left\{
  \array{
    \text{Euclidean field theory}
    \\
    \text{on Euclidean space}
    \\
    \mathbb{R}^d \times S^1_{\beta}
    \\
    \text{with compact/periodic Euclidean time}
    \\
    \text{of length} \; \beta = 1/T
  }
  \right.
  \\
  \phantom{A}
  \\
  \underset{
    {
    \text{equal-time n-point function}
    \atop
    \text{of relativistic fields}
    }
    \atop
    \text{ in thermal equilibrium state } \; \vert T\rangle
  }{
  \underbrace{
    \left\langle  T\vert  :\mathbf{\Phi}(x_1,t) \mathbf{\Phi}(x_2,t) \cdots \mathbf{\Phi}(x_n,t) : \vert T \right\rangle_{\mathbb{R}^{d,1}}
  }}
  &\;=\;&
  \underset{
    \text{correlator of Euclidean fields}
    \atop
    \text{ for "Euclidean time" of periodicity}\; \beta = 1/T
  }{
  \underbrace{
  \left\langle 0 \vert \mathbf{\Phi}(x_1,t) \mathbf{\Phi}(x_2,t) \cdots \mathbf{\Phi}(x_n,t)  \vert 0 \right\rangle_{\mathbb{R}^{d} \times S^1_{\beta}}
  }}
  }
$$

> graphics grabbed form [Frolov-Zelnikov 11](#FrolovZelnikov11)

Notice that the evident [[symmetry breaking|breaking]] of [[Lorentz group|Lorentz symmetry]] on the right side of this correspondence is perfectly consistent with what happens on the left hand: A thermal vacuum state in Minkowski spacetime also singles out a preferred Lorentz frame.

The basic idea of this relation seems to go back to [Bloch 58](#Bloch58). The physics literature often states this suggestively but informally in terms of [[path integral]]-imagery, see e.g. [Moore 03, section 1.1](#Moore03).

The first precise formulation seems to be due to [Høegh-Krohn 74](#HoeghKrohn74) (in 1+1 dimensions) and a more comprehensive discussion in view of the [[Osterwalder-Schrader theorem]] for compact Euclidean time is due to [Klein-Landau 81](#KleinLandau81).

The use of a "periodic Euclidean time coordinate" is also known as _[[Matsubara formalism]]_ (e.g. [Landsman-vanWert 87, section 2.3.1](#LandsmanVanWert87)) and specifically the condition that the periodicity has to be $\beta \coloneqq 1/T$ is known as the _[[KMS conditions]]_ for a _[[KMS state]]_ (For _Kubo-Martin-Schwinger_, due to [Kubo 57](#Kubo57), [Martin-Schwinger 59](#MartinSchwinger59) with its final form due to [Haag-Hugenholtz-Winnink 67](#HaagHugenholtzWinnink67), see [Fulling-Ruijsenaars 87, section 3.1](#FullingRuijsenaars87)).

Beware that literature discussing the KMS-condition often does not make the periodicity of Euclidean time explicit, and vice versa. This is clarified in [Fulling-Ruijsenaars 87, sections 2 and 3](#FullingRuijsenaars87).

General introduction to Euclidean and thermal field theory includes [Thoma 00, section 2.2](#Thoma00), [Peeters-Zamaklar 09, section 1.3](#PeetersZamaklar09).

## Examples

* [[Ising model]]

## Related concepts

* [[inverse temperature]]

* [[instanton]], [[caloron]]

* [[2d CFT]]

* [[relativistic field theory]]

* [[lattice gauge theory]]

* [[infinite-temperature thermal field theory]]

## References

### General

General introduction to Euclidean field theory includes

* {#PeetersZamaklar09} [[Kasper Peeters]], [[Marija Zamaklar]], _Euclidean Field Theory_, Lecture notes 2009-2011 ([web](http://maths.dur.ac.uk/users/kasper.peeters/eft.html), [pdf](http://maths.dur.ac.uk/users/kasper.peeters/pdf/eft.pdf))

A systematic development of Euclidean [[renormalization|renormalized]] [[perturbative quantum field theory]] with [[correlators as differential forms on configuration spaces of points]] is described in

* {#BergbauerBrunettiKreimer09} [[Christoph Bergbauer]], [[Romeo Brunetti]], [[Dirk Kreimer]], _Renormalization and resolution of singularities_ ([arXiv:0908.0633](https://arxiv.org/abs/0908.0633))

* {#Bergbauer09} [[Christoph Bergbauer]], _Renormalization and resolution of singularities_, talks as IHES and Boston, 2009 ([pdf](http://www.emg.uni-mainz.de/Dateien/bergbauer.pdf))

* {#Berghoff14a} [[Marko Berghoff]], _Wonderful renormalization_, 2014 ([pdf](http://www2.mathematik.hu-berlin.de/%7Ekreimer/wp-content/uploads/Berghoff-Marko.pdf), [doi:10.18452/17160](https://doi.org/10.18452/17160))

* {#Berghoff14b} [[Marko Berghoff]], _Wonderful compactifications in quantum field theory_,  Communications in Number Theory and Physics Volume 9 (2015) Number 3 ([arXiv:1411.5583](https://arxiv.org/abs/1411.5583))

### Thermal quantum field theory

A good introduction is in

* {#FullingRuijsenaars87} S.A. Fulling, S.N.M. Ruijsenaars, _Temperature, periodicity and horizons_, Physics Reports Volume 152, Issue 3, August 1987, Pages 135-176 ([pdf](https://www1.maths.leeds.ac.uk/~siru/papers/p26.pdf), doi:10.1016/0370-1573(87)90136-0)

The idea of Wick rotating thermal relativistic field theory to compact periodic Euclidean time apparently goes back to

* {#Bloch58} Claude Bloch, _Sur la détermination de l'état fondamental d'un système de particules_,  Nucl. Phys. 7 (1958) 451

This has maybe first been made precise, for the case of 1+1 dimensions, in

* {#HoeghKrohn74} [[Raphael Høegh-Krohn]], _Relativistic Quantum Statistical Mechanics in two-dimensional Space-Time_, Communications in Mathematical Physics 38.3 (1974): 195-224 ([pdf](https://www.duo.uio.no/bitstream/handle/10852/44072/1973-22.pdf))

A systematic discussion of the [[Osterwalder-Schrader theorem]] on [[Wick rotation]] for the case of thermal field theory/periodic Euclidean time is in

* {#KleinLandau81} Abel Klein, Lawrence Landau, _Periodic Gaussian Osterwalder-Schrader positive processes and the two-sided Markov property on the circle_, Pacific Journal of Mathematics, Vol. 94, No. 2, 1981 ([
DOI: 10.2140/pjm.1981.94.341](https://msp.org/pjm/1981/94-2/p12.xhtml), [pdf](https://msp.org/pjm/1981/94-2/pjm-v94-n2-p12-s.pdf))

See also

* [Fulling-Ruijsenaars 87, section 2](#FullingRuijsenaars87)

The formulation of the KMS condition is due to

* {#Kubo57} R. Kubo _Statistical-Mechanical Theory of Irreversible Processes  I. General Theory and Simple Applications to Magnetic and Conduction Problems_, Journal of the Physical Society of Japan 12, 570-586 1957

* {#MartinSchwinger59} Paul C. Martin, [[Julian Schwinger]], _Theory of Many-Particle Systems.  I_, Physical Review 115, 1342-1373 (1959)

and found its final, now generally accepted, form in

* {#HaagHugenholtzWinnink67} [[Rudolf Haag]], N. M. Hugenholtz, M. Winnink, _On the equilibrium states in quantum statistical mechanics_, Comm. Math. Phys. Volume 5, Number 3 (1967), 215-236 ([euclid:1103840050](https://projecteuclid.org/euclid.cmp/1103840050))

Review of thermal field theory via Euclidean field theory includes

* [Fulling-Ruijsenaars 87, sections 2 and 3](#FullingRuijsenaars87)

* {#LandsmanVanWert87} [[Klaas Landsman]], Ch.G.van Weert, _Real- and imaginary-time field theory at finite temperature and density_, Physics Reports Volume 145, Issues 3–4, January 1987, Pages 141-249 (doi:10.1016/0370-1573(87)90121-9)

* [[Jean Zinn-Justin]], _Quantum Field Theory at Finite Temperature: An Introduction_ ([arXiv:hep-ph/0005272](https://arxiv.org/abs/hep-ph/0005272))

* {#Thoma00} Markus H. Thoma, _New Developments and Applications of Thermal Field Theory_ ([arXiv:hep-ph/0010164](https://arxiv.org/abs/hep-ph/0010164))

* Yuhao Yang, _An Introduction to Thermal Field Theory_, 2011 ([pdf](https://www.imperial.ac.uk/media/imperial-college/research-centres-and-groups/theoretical-physics/msc/dissertations/2011/Yuhao-Yang-Dissertation.pdf))

* Yi-Cheng Huang, _Field Theory in the Imaginary-time Formulation_ ([arXiv:1311.1990v4](https://arxiv.org/abs/1311.1990v4))

* [[Roberto Longo]], _Kubo-Martin-Schwinger, Non-Equilibrium Thermal states, and Conformal Field Theory_, 2016 ([pdf](https://www.mat.uniroma2.it/~longo/Slides_files/Harvard16.pdf))

Further discussion:

* Christian D. Jäkel, _The Reeh–Schlieder property for thermal field theories_, Journal of Mathematical Physics 41, 1745 2000 ([doi:10.1063/1.533208](https://doi.org/10.1063/1.533208))

* {#JaehelRobl11} Christian D. Jäkel, Florian Robl, _The relativistic KMS condition for the thermal $n$-point functions of the $P(\phi)_2$ model_ ([arXiv:1103.3609](https://arxiv.org/abs/1103.3609))

With an eye towards [[lattice gauge theory]]:

* {#Moore03} Guy Moore, _Informal lectures on lattice gauge theory_, 2003 ([pdf](https://theorie.ikp.physik.tu-darmstadt.de/qcd/moore/latt_lectures.pdf))

In application to [[black hole thermodynamics]]:

* [Fulling-Ruijsenaars 87, section 4](#FullingRuijsenaars87)

* {#GibbonsPerry78} [[Gary Gibbons]], Malcolm J. Perry, _Black Holes and Thermal Green Functions_, Vol. 358, No. 1695 (1978) ([jstor:79482](https://www.jstor.org/stable/79482))

* {#FrolovZelnikov11} [[Valeri Frolov]], Andrei Zelnikov, section F4.4 of _Introduction to black hole physics_, Oxford 2011

Discussion of thermal [[Wick rotation]] on global [[anti-de Sitter spacetime]] (which is already periodic in _real_ time) is in

* {#AllenFolacciGibbons87} B. Allen, A. Folacci, [[Gary Gibbons]], _Anti-de Sitter space at finite temperature_, Physics Letters B Volume 189, Issue 3, 7 May 1987, Pages 304-310 (doi:10.1016/0370-2693(87)91437-7)

The expansion of thermal field theory around the [[infinite-temperature thermal field theory|infinite-temperature-limit]] (i.e. around [[inverse temperature]] $\beta = 1/T = 0$, i.e. [[KK-reduction]] in compact/periodic Euclidean time) is discussed in

* {#Ginsparg80} [[Paul Ginsparg]], _First and second order phase transitions in gauge theories at finite temperature_, Nuclear Physics B Volume 170, Issue 3, 15 December 1980, Pages 388-408 (doi:10.1016/0550-3213(80)90418-6)

* {#AppelquistPisarski81} Thomas Appelquist, [[Robert Pisarski]], _High-temperature Yang-Mills theories and three-dimensional quantum chromodynamics_, Phys. Rev. D 23, 2305 (1981) ([doi:10.1103/PhysRevD.23.2305](https://doi.org/10.1103/PhysRevD.23.2305))

* {#Nadkarni83} Sudhir Nadkarni, _Dimensional reduction in finite-temperature quantum chromodynamics_, Phys. Rev. D 27, 917 (1983) ([doi:10.1103/PhysRevD.27.917](https://doi.org/10.1103/PhysRevD.27.917))

* {#Nadkarni88} Sudhir Nadkarni, _Dimensional reduction in finite-temperature quantum chromodynamics. II_, Phys. Rev. D 38, 3287 (1988) ([doi:10.1103/PhysRevD.38.3287](https://doi.org/10.1103/PhysRevD.38.3287))

* {#Jourjine84} Alexander N Jourjine, _Quantum field theory in the infinite temperature limit_, Annals of Physics Volume 155, Issue 2, July 1984, Pages 305-332 (doi:10.1016/0003-4916(84)90003-4)

* [[Klaas Landsman]], _Limitations to dimensional reduction at high temperature_, Nuclear Physics B Volume 322, Issue 2, 14 August 1989, Pages 498-530 (doi:10.1016/0550-3213(89)90424-0)

* T. Reisz, _Realization of dimensional reduction at high temperature_, Z. Phys. C - Particles and Fields (1992) 53: 169 ([doi:10.1007/BF01483886](https://doi.org/10.1007/BF01483886))

* {#Braaten95} Eric Braaten, _Solution to the Perturbative Infrared Catastrophe of Hot Gauge Theories_, Phys. Rev. Lett. 74, 2164 (1995) ([doi:10.1103/PhysRevLett.74.2164](https://doi.org/10.1103/PhysRevLett.74.2164))

* {#KajantieaLaineRummukainenShaposhnikov96} K. Kajantiea, M. Laine, K. Rummukainen, M. Shaposhnikov, _Generic rules for high temperature dimensional reduction and their application to the standard model_, Nuclear Physics B Volume 458, Issues 1–2, 1 January 1996, Pages 90-136 (doi:10.1016/0550-3213(95)00549-8)

and specifically with an eye to discussion of the [[quark-gluon plasma]] in

* {#BlaizotIancuRebhan03} Jean-Paul Blaizot, Edmond Iancu, [[Anton Rebhan]], _Thermodynamics of the high temperature quark gluon plasma_, Quark–Gluon Plasma 3, pp. 60-122 (2004) ([arXiv:hep-ph/0303185](https://arxiv.org/abs/hep-ph/0303185), [spire:615570](http://inspirehep.net/record/615570))

General discussion for [[QCD]]:

* Jacopo Ghiglieri, Aleksi Kurkela, Michael Strickland, Aleksi Vuorinen, _Perturbative Thermal QCD: Formalism and Applications_ ([arXiv:2002.10188](https://arxiv.org/abs/2002.10188))

See also

* Wikipedia, _[Thermal quantum field theory](https://en.wikipedia.org/wiki/Thermal_quantum_field_theory)_

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

* Wolfram Math World, _[KMS condition](http://mathworld.wolfram.com/KMSCondition.html)_

[[!redirects Euclidean field theories]]

[[!redirects thermal quantum field theory]]
[[!redirects thermal quantum field theories]]
[[!redirects thermal QFT]]

[[!redirects thermal field theory]]
[[!redirects thermal field theories]]

[[!redirects Matsubara formalism]]
[[!redirects Matsubara formalisms]]

[[!redirects Matsubara frequency]]
[[!redirects Matsubara frequencies]]

[[!redirects KMS condition]]
[[!redirects KMS conditions]]


Euclidean G-space

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Topology
+--{: .hide}
[[!include topology - contents]]
=--
#### Representation theory
+-- {: .hide}
[[!include representation theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

Let $G$ be a [[compact Lie group]] and let $V \in RO(G)$ be an [[orthogonal group|orthogonal]] $G$-[[linear representation]] on a [[real vector space]] $V$. Then the underlying [[Euclidean space]] $\mathbb{R}^V$ inherits the [[mathematical structure|structure]] of a [[topological G-space]]

\begin{xymatrix}
   \mathbb{R}^V\ar@(ul,ur)^G
   &
   \in G TopSpaces
\end{xymatrix}

We may call this the _[[Euclidean G-space]]_ associated with the linear representation $V$.

## Properties

### Relation to representation spheres

The [[one-point compactification]] of a Euclidean $G$-space is the corresponding [[representation sphere]]:

$$
  \big( \mathbb{R}^V \big)^{cpt}
  \;\simeq\;
  S^V
  \,.
$$

### Relation to representation tori

Let $V \in RO(G)$ be an [[orthogonal group|orthogonal]] [[linear representation]] of a [[finite group]] $G$ on a [[real vector space]] $V$.

If $G$ is the [[point group]] of a [[crystallographic group]] inside the [[Euclidean group]]

$$
  N \rtimes G
  \hookrightarrow
  Iso(\mathbb{R}^V)
$$

then the $G$-[[action]] on the Euclidean $G$-space $\mathbb{R}^V$ descends to the [[quotient]] by the [[action]] of the translational [[normal subgroup]] [[lattice in a vector space|lattice]] $N$ ([this Prop.](crystallographic+group#InducedPointGroupActionOnTorus)). The resulting $G$-space is an [[n-torus]] with $G$-action, which might be called the _[[representation torus]]_ of $V$

\begin{xymatrix}
   \mathbb{R}^V\ar@(ul,ur)^G
   \ar@{}[r]|-{:=}
   &
   \big( \mathbb{R}^V\ar@(ul,ur)^G\big)/N
\end{xymatrix}

> graphics grabbed from [SS 19](equivariant+Hopf+degree+theorem#SatiSchreiber19)

### Equivariant configurations in Euclidean $G$-spaces

(...)

## Related concepts

* [[Euclidean space]]

* [[super-Euclidean space]]

## References

Discussion of equivariant [[configuration spaces of points]] in Euclidean $G$-spaces:

* {#RourkeSanderson00} [[Colin Rourke]], [[Brian Sanderson]], _Equivariant Configuration Spaces_, J. London Math. Soc. 62 (2000) 544-552 ([arXiv:math/9712216](https://arxiv.org/abs/math/9712216))

[[!redirects Euclidean G-spaces]]


Euclidean geometry

+-- {: .rightHandSide}
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### Context
#### Geometry
+-- {: .hide}
[[!include higher geometry - contents]]
=--
#### Differential geometry
+-- {: .hide}
[[!include synthetic differential geometry - contents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

## Idea

_Euclidean geometry_ ([Euclid 300BC](#Euclid300BC)) studies the [[geometry]] of [[Euclidean spaces]].  These spaces come equipped with a canonical [[metric space|metric]] in the sense of [[analysis]], in fact a [[Riemannian metric]], and Euclidean geometry may be regarded as the _local model_ for [[Riemannian geometry]], in the sense of [[Cartan geometry]].

In distinction from [[non-Euclidean geometry]] ([[hyperbolic geometry]] or [[elliptic geometry]]), Euclidean geometry is [[Riemannian geometry]] of [[Riemannian manifolds]] with [[zero]] [[sectional curvature]].

## Properties

* [[Pythagorean theorem]]

* [[pentagon decagon hexagon identity]]

* [[conic section]]

* [[Hadwiger's theorem]]

* [[Non-Euclidean geometry]]

## Axiomatizations
 {#Axiomatizations}

Euclidean geometry in the sense of the Idea section makes# reference to the extrinsic notion of [[real number]] (to delineate the notion of Euclidean space) and whatever surrounding structure is used to found that notion, but it has also received more intrinsic [[axiom|axiomatic]] treatments in the spirit of [[synthetic geometry]]. Notable among these are [[Hilbert|Hilbert's]] axioms, as set out in his *Grundlagen der Geometrie*, and [[Alfred Tarski|Tarski]]'s axioms, which form a classical first-order axiomatic theory.

### Hilbert's axioms

Here we will just give an extremely sketchy description (for now, see [Wikipedia](http://en.wikipedia.org/wiki/Hilbert%27s_axioms) for a fuller account). It should be said right away that two of the Hilbert axioms are *not* first-order.

Hilbert's [[theory]] has

* Three sorts: points, line, planes (or if you like: tables, chairs, and beer mugs[^1]);

[^1]: After a famous saying attributed to Hilbert which in some sense heralds the axiomatic spirit: '_Man muß jederzeit an Stelle von "Punkte, Geraden, Ebenen" "Tische, Stühle, Bierseidel" sagen können_'. The 1891 anecdote is reported by O. Blumenthal in Hilbert, _Gesammelte Abhandlungen III. Band_ , Springer Berlin 1935, p.403. The three classes became '_Liebe, Gesetz, Schornsteinfeger_' (love, law, chimney-sweep) in Hilbert's letter to [[Gottlob Frege|Frege]] (Dec. 29th 1899). The question raised in Frege's rejoinder (Jan. 6th 1900) whether his pocket watch is a point is still an open problem.

* Six relations: betweenness (a ternary relation on points), three incidence relations (one for points and lines, one for points and planes, one for lines and planes), and two congruence relations (a relation $L(a, b, c, d)$ on points whose intuitive meaning is that the the line segment $a b$ is congruent to the line segment $c d$, and a relation $A(a, b, c, d, e, f)$ on points whose intuitive meaning is that the angle $a b c$ is congruent to the angle $d e f$);

* A bunch of [[axioms]] (20 or 21 depending on which specific edition or translation or exposition one is working with; Wikipedia says 20 but actually has, confusingly, 21 -- see the Talk page for more information). They are organized as
  * Axioms of incidence (8),
  * Axioms of betweenness (4),
  * Axioms of congruence (6),
  * Axiom of parallels (the parallel postulate) (1),
  * Axioms of completeness (2), mirroring the archimedean property and the completeness properties of the real numbers. These are not first-order.

I ([[Todd Trimble]]) assume Hilbert's theory is, or is at least intended to be, [[categoricity|categorical]] in the model-theoretic sense, at least considered against a fixed set-theoretic background. In other words I assume the real numbers as complete ordered field would be a definable structure and the axioms are uniquely modeled by $\mathbb{R}^3$ with its Euclidean structure.

### Tarski's axioms
 {#TarskiAxioms}

Tarski's [[theory]] EPG (Elementary Plane [Euclidean] Geometry) is a one-sorted theory in classical first-order logic with equality, with variables representing points. There are no constants or operations in the signature, and just two relations:

* A ternary relation $B$ (betweenness), with $B(x, y, z)$ meaning $y$ is between $x$ and $z$ ($y$ is on the line segment between $x$ and $z$) -- we will write instead $B x y z$ to conserve space;

* A 4-ary relation $C$ (congruence), with $C(x, y, z, w)$ meaning that a line segment $x y$ is congruent (of the same length) as $z w$. For readability, we write this as $x y \equiv z w$.

The theory can be presented (in some sense very parsimoniously) by 10 axioms and one first-order axiom scheme. Axioms 8 and 9 constrain the dimension to $n = 2$ (Euclidean plane geometry), but there are analogues of 8 and 9 that constrain the dimension to any chosen number. Axiom 10 is an extremely disguised version of the [[parallel postulate|Parallel Postulate]], with a complicated history behind it. The axiom scheme 11 plays the role of a Dedekind cut axiom expressed in first-order terms.

1. $\vdash \; a b \equiv b a$,

1. $a b \equiv p q, a b \equiv r s \;\; \vdash \;\; p q \equiv r s$,

1. $a b \equiv c c \vdash a = b$,

1. $\vdash \; (\exists x)\; B q a x \wedge a x \equiv b c$ (segment construction: given a line segment $b c$ and a line segment from $q$ to $a$, one can extend that line segment in that direction by a segment $a x$ congruent to $b c$),

1. $a \neq b, B a b c, B a' b' c', a b \equiv a' b', b c \equiv b' c', a d \equiv a' d', b d \equiv b' d' \;\; \vdash \;\; c d \equiv c' d'$ (5 segment axiom),

1. $B a b a \; \vdash \; a = b$,

1. $B a p c, B b q c \;\; \vdash \;\; (\exists x)\; B p x b \wedge B q x a$ (Pasch axiom: roughly, if a line goes through one side of a triangle and not through a vertex, then it must go through another side),

1. $\vdash \; (\exists a, b, c)\; \neg (B a b c \vee B b c a \vee B c a b)$ (there exist three non-collinear points: the dimension is at least 2),

1. $p \neq q, a p \equiv a q, b p \equiv b q, c p \equiv c q \;\; \vdash \;\; B a b c \vee B b c a \vee B c a b$ (if three points are each equidistant from two given distinct points $p$ and $q$, then those three points are collinear: the dimension is at most 2),

1. $B a d t, B b d c, a \neq d \;\; \vdash \;\; (\exists x, y)\; B a b x \wedge B a c y \wedge B y t x$ (Parallel Postulate (!)),

1. (Cut schema) Let $\phi(x)$ be a formula in which variables $a, b, y$ do not occur freely, and let $\psi(y)$ be a formula in which $a, b, x$ do not occur freely. Then
$$(\forall x, y)\; (\phi(x) \wedge \psi(y) \Rightarrow B a x y) \;\; \vdash \;\; (\exists b)(\forall x, y)\; (\phi(x) \wedge \psi(y) \Rightarrow B x b y)$$
(if every element $y$ in the set defined by $\psi$ lies beyond every element $x$ in the set defined by $\phi$ as seen from the perspective of $a$, then there is some $b$ between every such $x$ and every such $y$).

See [Tarski-Givant](#TG) for a thorough discussion of this and related axiom systems, and an extensive bibliography.

One important aspect of the meta-theory lies in the relation of models of this theory to [[real closed fields]]. If $R$ is a real closed field, then there is a corresponding model of EPG whose underlying set is $R \times R$ and where the relations $B$ and $C$ are interpreted by expected conditions:

$$C(x, y, z, w) \;\;\; iff \;\;\; (x_1 - y_1)^2 + (x_2 - y_2)^2 = (z_1 - w_1)^2 + (z_2 - w_2)^2$$

$$\,$$

$$B(x, y, z) \;\;\; iff \;\;\; (x_1 - y_1)(y_2 - z_2) = (x_2 - y_2)(y_1 - z_1) \; \wedge \; 0 \leq (x_1 - y_1)(y_1 - z_1) \; \wedge \; 0 \leq (x_2 - y_2)(y_2 - z_2).$$

It is mostly straightforward to verify that this structure on $R \times R$ satisfies the EPG axioms (all but the cut schema hold for any ordered field). More importantly,

+-- {: .num_theorem}
###### Theorem
For every model $M$ of EPG there is a real closed field $R$ such that $M \cong R \times R$ as models.
=--

Of course there is some arbitrariness involved in the choice of isomorphism; for example, any line segment $x y$ can be used to stipulate a unit length and a coordinate axis. The proof is lightly sketched in [Tarski](#Tar); again only the first ten axioms are needed to produce an ordered field structure on $R$, with the cut schema being used to verify the real closure.

The significance of this result is that EPG is a complete and decidable theory, just as the theory of real closed fields is complete and decidable. Hence there exists an algorithm which, on being fed a sentence in the language of Elementary Plane Geometry, will produce a proof of the sentence or a proof of its negation in the theory.

### Mac Lane's axioms
 {#MacLaneAxioms}

To be written, see [Mac Lane (1959)](#MacLane) for details.

## Related concepts

* [[trigonometry]]

{#BasicNotionsAndFact} **Basic notions and facts in ([[differential geometry|
differential]]) Euclidean geometry** (see also at: [[differential geometry of curves and surfaces]])

* [[parallel postulate]]

* [[length]], [[area]], [[volume]]

* [[curve]]

  * [[velocity]]

  * [[areal velocity]]

  * [[acceleration]]

  * [[jerk]]

* [[circle]]

  * [[circumference of a circle]]

  * [[area enclosed by a circle]]

* [[regular polygon]]

  * [[center of a regular polygon]]

* [[space form]]

\linebreak

**Other flavors of geometry**

| [[synthetic geometry]]  |
|-------------------------|
| [[Euclidean geometry]]  |
| [[hyperbolic geometry]] |
| [[elliptic geometry]]   |

[[Riemannian geometry]], [[Cartan geometry]]:

[[!include local and global geometry - table]]

## References

* {#Euclid300BC} [[Euclid]], _[[Elements]]_, 300BC

* {#Tarski} [[Alfred Tarski]], _What is elementary geometry?_, in _The axiomatic method. With special reference to geometry and physics_ Proceedings of an International Symposium held at the Univ. of Calif., Berkeley, Dec. 26, 1957-Jan. 4, 1958 (ed. L. Henkin, P. Suppes, and A. Tarski), Studies in Logic and the Foundations of Mathematics, Amsterdam: North-Holland (1959), pp. 16–29. ([pdf](http://www.geographicknowledge.de/SeminarLFG/Tarski_Whatiselementarygeometry.pdf))

* {#TG}  [[Alfred Tarski]] and Steven Givant, _Tarski's system of geometry_, Bull. Symb. Logic, Vol. 5 No. 2 (1999), 175–214. ([web](http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.27.9012))

* {#MacLane} [[Saunders Mac Lane]], _Metric postulates for plane geometry_, American Mathematical Monthly, Vol. 66 No. 7 (1959), 543–555. ([doi:10.2307/2309851](https://doi.org/10.2307/2309851), [web](https://www.jstor.org/stable/2309851))

Textbook account:

* {#Rees05} [[Elmer Rees]], _Notes on Geometry_, Springer 2005

Expository lecture notes:

* {#Armstrong17} Drew Armstrong, _Introduction to Geometry_, 2017 ([pdf](http://www.math.miami.edu/~armstrong/daVinci/HGnotes.pdf))

Textbook accounts of the axiomatization of Euclidean geometry include these:

* [[Wolfram Schwabhäuser]], [[Wanda Szmielew]], [[Alfred Tarski]], _Mathematische Methoden in der Geometrie_, Springer 1983

* {#Petrunin13} [[Anton Petrunin]], _Euclidean Plane and its Relatives; a minimalistic introduction_, ([arXiv:1302.1630](https://arxiv.org/abs/1302.1630))

Full formalization of this book in [[Coq]] (as [[synthetic geometry]] but following Tarski's work) is discussed at

* [[Gabriel Braun]], [[Pierre Boutry]], [[Julien Narboux]], _[GeoCoq](http://geocoq.github.io/GeoCoq/)_, _[La géométrie de Tarski en Coq](http://gabrielbraun.free.fr/Geometry/Tarski/)_

A formalisation of Euclid using diagrammatic reasoning is in

* J. Avigad, E. Dean, J. Mumma, _A formal system for Euclid's Elements_ , [arXiv:0810.4315](http://arxiv.org/abs/0810.4315v3) (2009).

Another formalization of Euclidean geometry in [[Coq]]:

* {#Ivashkevich19} Evgeny V. Ivashkevich, _On Constructive-Deductive Method For Plane Euclidean Geometry_ ([arXiv:1903.05175](https://arxiv.org/abs/1903.05175))

[[!redirects Euclidean geometry]]
[[!redirects Euclidean geometries]]


Euclidean gravity

## Idea

[[gravity]]+[[Wick rotation]]

## Related concepts

* [[spacetime signature]]


Euclidean quantum field theory

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## Idea

A _Euclidean quantum field theory_ is a [[quantum field theory]] given, as an [[FQFT]], by a [[functor]] on a [[category of cobordisms]] that are equipped with [[Riemannian structure]].

This is in contrast to the phenomenologically more realistic case where the cobordisms are equipped with [[pseudo-Riemannian manifold|pseudo-Riemannian structure]] and thus can be thought of as pieces of [[spacetime]].

It is also in contrast to the case where the cobordisms are equipped with no extra structure, which is the case of [[topological quantum field theory]].

## Related entries

* [[Wick rotation]]

* [[relativistic quantum field theory]]

* [[(1,1)-dimensional Euclidean field theories and K-theory]]

## References

Textbook account in the school of [[algebraic quantum field theory]]:

* [[Franco Strocchi]], §5 of: _An Introduction to Non-Perturbative Foundations of Quantum Field Theory_, Oxford University Press (2013) [[doi:10.1093/acprof:oso/9780199671571.001.0001](https://doi.org/10.1093/acprof:oso/9780199671571.001.0001)]


euclidean relation

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# Euclidean relations
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## Definitions

A (binary) [[relation]] $\sim$ on an [[abstract set]] $A$ is __left euclidean__ if every two elements both related to a third are also related to each other:
$$ x, y, z: A,\; x \sim z,\; y \sim z \;\vdash\; x \sim y .$$
A relation $\sim$ is __right euclidean__ if this works in the other order:
$$ x, y, z: A,\; x \sim y,\; x \sim z \;\vdash\; y \sim z .$$
One could also say __euclidean__ and __co-euclidean__ or __op-euclidean__, or similarly.

## Relationship to other kinds of relations

An [[equivalence relation]] is a relation that is both [[reflexive relation|reflexive]] and euclidean in either direction (in which case it\'s euclidean in both directions).  Defining equivalence relations this way (rather than using transitivity and symmetry) is analogous to defining a [[group]] using (one-sided) division instead of multiplication and inverse; both are special cases of an analogous definition of [[groupoid]].  One can also define an equivalence relation as a relation that is both [[entire relation|entire]] and left euclidean (or the reverse); this is analogous to defining a group using division and furthermore merely stating that some element exists rather than that there is an identity element.  (Perhaps more familiarly, one can define a [[subgroup]] of a given group as a subset that is closed under division and has at least one element.)

A _[[circular relation]]_ satisfies
$$ x, y, z: A,\; x \sim y,\; y \sim z \;\vdash\; z \sim x .$$
This is analogous to taking the reciprocal of the product in a group(oid).  Again, a reflexive circular relation is an equivalence relation.

## History

The above (reflexive and euclidean) is the earliest definition of equivalence, dating back (at least) to [[Euclid]]\'s Common Notion 1 (hence the name).  In Heath\'s translation,

>Things \[geometric figures\] which are equal to \[have the same magnitude as\] the same thing are also equal to one another.

It is not clear, however, if Euclid appreciated the distinction between this and transitivity; he may have taken it for granted that equality of magnitude is symmetric, which destroys the distinction.  (One can similarly argue whether Common Notion 4 expresses reflexivity or whether it merely means that [[congruence]] entails [[equality]] of magnitude, with the equivalence-relation nature of congruence again going unstated.  See [this discussion](http://aleph0.clarku.edu/~djoyce/java/elements/bookI/cn.html).)

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Euclidean ring

[[!redirects Euclidean rings]]

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## Idea ##

In the same vein that [[commutative rings]] are to [[integral domains]] and [[GCD rings]] are to [[GCD domains]], [[Euclidean rings]] are to [[Euclidean domains]].

## Definition ##

A **Euclidean ring** is a [[commutative ring]] $R$ for which there exists a [[function]] $d \colon \mathrm{Can}(R) \to \mathbb{N}$ from the [[multiplicative submonoid of cancellative elements]] in $R$ to the [[natural numbers]], often called a *degree function*, such that for all $f \in R$ and $g \in \mathrm{Can}(R)$, there exist $q, r \in R$ such that $f = q \cdot g + r$ and either $r = 0$ or $d(r) \lt d(g)$. There is no uniqueness requirement for $q, r$.

One could posit, instead of the [[property]] that for all $f \in R$ and $g \in \mathrm{Can}(R)$, there exist $q, r \in R$ such that $f = q \cdot g + r$ and either $r = 0$ or $d(r) \lt d(g)$; that the commutative ring has the [[structure]] of a function $(-)\div(-):R \times \mathrm{Can}(R) \to R$ called the *[[division]] function*, and a function $(-)\ \%\ (-):R \times \mathrm{Can}(R) \to R$ called the *remainder function*, such that for all $f \in R$ and $g \in \mathrm{Can}(R)$, $f = (f \div g) \cdot g + (f\ \%\ g)$ and either $f\ \%\ g = 0$ or $d(f\ \%\ g) \lt d(g)$. There might be multiple such division and remainder functions for the commutative ring $R$.

One could also add the requirement that $d(a) \leq d(a b)$ for all nonzero $a, b$. There is no loss of generality in assuming it; every Euclidean ring admits such a degree function $d'$, defining $d'(a) = \min \{d(a b): b \in \mathrm{Can}(R)\}$.

### In constructive mathematics ###

The definition of the Euclidean ring further bifurcates in [[constructive mathematics]] due to the disjunctive condition above. These statements, which are equivalent in the presence of [[excluded middle]], include:

* $(r = 0) \vee (d(r) \lt d(g))$
* $\not((r \neq 0) \wedge (d(r) \geq d(g)))$
* $(d(r) \geq d(g)) \to (r = 0)$

or equivalently,

* $(f\ \%\ g = 0) \vee (d(f\ \%\ g) \lt d(g))$
* $\not((f\ \%\ g \neq 0) \wedge (d(f\ \%\ g) \geq d(g))$
* $(d(f\ \%\ g) \geq d(g)) \to (f\ \%\ g = 0)$

## Properties ##

Every Euclidean ring is a [[Bézout ring]], and every [[prefield]] is an Euclidean ring.

## Examples ##

* The [[integers]] $\mathbb{Z}$ are a Euclidean ring.

* A classical Euclidean domain $A$ is a Euclidean ring where $\mathrm{Can}(A)$ is the multiplicative monoid of elements not equal to zero (where inequality is [[denial inequality]])

$$\mathrm{Can}(A) \cong \{x \in A \vert x \neq 0\}$$

* A Heyting Euclidean domain $A$ is a Euclidean ring where $\mathrm{Can}(A)$ is the multiplicative monoid of elements apart from zero

$$\mathrm{Can}(A) \cong \{x \in A \vert x \# 0\}$$

* An ordered Euclidean domain $A$ is a Euclidean ring where $\mathrm{Can}(A)$ is the multiplicative monoid of elements with a positive absolute value

$$\mathrm{Can}(A) \cong \{x \in A \vert 0 \lt \vert x \vert\}$$

* The [[trivial ring]] $0$ is the unique Euclidean ring up to unique [[isomorphism]] such that $0 \in \mathrm{Can}(0)$. The [[trivial ring]] is also the [[terminal object|terminal]] Euclidean ring.

## See also ##

* [[commutative ring]]

* [[GCD ring]]

* [[Bézout ring]]

* [[prefield ring]]

* [[Euclidean domain]]


Euclidean space

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# Euclidean spaces
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## Idea

The concept of  _Euclidean space_ in [[analysis]], [[topology]], [[differential geometry]] and specifically _[[Euclidean geometry]]_, and [[physics]] is a fomalization in modern terms of the [[spaces]] studied in [Euclid 300BC](#Euclid300BC), equipped with the [[extra structure|structures]] that Euclid recognised his spaces as having.

In the strict sense of the word, Euclidean space $E^n$ of dimension $n$ is, up to [[isometry]], [[generalized the|the]] [[metric space]] whose underlying [[set]] is the [[Cartesian space]] $\mathbb{R}^n$ and whose [[distance]] function $d$ is given by the [[Euclidean norm]]:

$$
  d_{Eucl}(x,y) \coloneqq {\Vert x-y\Vert} = \sqrt{ \sum_{i = 1}^n (y_i - x_i)^2 }
  \,.
$$

In [Euclid 300BC](#Euclid300BC) this is considered for $n = 3$; and it is considered not in terms of [[coordinate functions]] as above, but via [[axioms]] of _[[synthetic geometry]]_.

This means that in a Euclidean space one may construct for instance the [[unit sphere]] around any point, or the shortest [[curve]] connecting any two points. These are the operations studied in ([Euclid 300BC](#Euclid300BC)), see at _[[Euclidean geometry]]_.

Of course these operations may be considered in _every_ (other) [[metric space]], too, see at _[[non-Euclidean geometry]]_. [[Euclidean geometry]] is distinguished notably from [[elliptic geometry]] or [[hyperbolic geometry]] by the fact that it satisfies the [[parallel postulate]].

In regarding $E^n = (\mathbb{R}^n, d_{Eucl})$ (only) as a [[metric space]], some [[extra structure]] still carried by $\mathbb{R}^n$ is disregarded, such as its [[vector space]] structure, hence its [[affine space]] structure and its canonical [[inner product space]] structure. Sometimes "Euclidean space" is
used to refer to $E^n$ with that further extra structure remembered, which might then be called _[[Cartesian space]]_.

Retaining the [[inner product]] on top of the [[metric space]] structure means that on top of [[distances]] one may also speak of [[angles]] in a Euclidean space.

Then of course $\mathbb{R}^n$ carries also non-canonical [[inner product space]] structures, not corresponding to the [[Euclidean norm]]. Regarding $E^n$ as equipped with these one says that it is a  __pseudo-Euclidean space__. These are now, again in the sense of [[Cartan geometry]], the local model spaces for [[pseudo-Riemannian geometry]].

Finally one could generalize and allow the [[dimension]] to be countably infinite, and regard separable [[Hilbert spaces]] as generalized Euclidean spaces.

## Remarks on terminology

Arguably, the spaces studied by Euclid were not really modelled on inner product spaces, as the distances were lengths, not [[real numbers]] (which, if non-negative, are *ratios* of lengths).  So we should say that $V$ has an inner product valued in some [[orientation|oriented]] [[line]] $L$ (or rather, in $L^2$).  Of course, Euclid did not use the inner product (which takes negative values) directly, but today we can recover it from what Euclid did discuss: lengths (valued in $L$) and angles (dimensionless).

Since the days of [[René Descartes]], it is common to identify a Euclidean space with a [[Cartesian space]], that is $\mathbb{R}^n$ for $n$ the dimension.  But Euclid\'s spaces had no coordinates; and in any case, what we do with them is still coordinate-independent.

## Euclidean spaces with infinitesimals

Instead of working in the real numbers $\mathbb{R}$ and $n$-[[dimension|dimensional]] real vector spaces $V$, one could instead work in a [[Archimedean ordered Artinian local ring|Archimedean ordered Artinian local $\mathbb{R}$-algebra]] $A$ and [[rank]] $n$ $A$-modules $V$. $A$ has [[infinitesimals]], and so the $A$-modules $V$ have [[infinitesimals]] as well. Nevertheless, it is still possible to define the Euclidean distance function on $V$; the only difference is that the distance function is a [[pseudometric]] rather than a [[metric]] here.

Since $A$ is an [[local ring]], the quotient of $A$ by its [[ideal]] of non-invertible elements $I$ is $\mathbb{R}$ itself, and the canonical function used in defining the [[quotient ring]] is the function $\Re:A \to \mathbb{R}$ which takes a number $a \in A$ to its purely real component $\Re(a) \in \mathbb{R}$. Since $A$ is an ordered $\mathbb{R}$-algebra, there is a [[strictly monotone]] [[ring homomorphism]] $h:\mathbb{R} \to A$.

The real numbers have [[lattice]] structure $\min:\mathbb{R} \times \mathbb{R} \to \mathbb{R}$ and $\max:\mathbb{R} \times \mathbb{R} \to \mathbb{R}$. This means that $A$ has a distance function given by the function $\rho:A \times A \to \mathbb{R}$, defined as

$$\rho(a, b) \coloneqq \max(\Re(a), \Re(b)) - \min(\Re(a), \Re(b))$$

as well as an [[absolute value]] given by the function $\vert-\vert:A \to \mathbb{R}$, defined as

$$\vert a \vert \coloneqq \rho(a, 0)$$

Since $\min(a, b) \leq \max(a, b)$, the [[pseudometric]] and multiplicative [[seminorm]] are always non-negative. In addition, by definition, the pseudometric takes any two elements $a \in A$ and $b \in A$ whose difference $a - b \in I$ is an infinitesimal to zero $\rho(a, b) = 0$.

Since $\mathbb{R}$ is an [[Euclidean field]], it has a [[metric square root function]] $\sqrt{-}:[0, \infty) \to [0, \infty)$. Every [[rank]] $n$ $A$-module $V$ with basis $v:\mathrm{Fin}(n) \to V$ thus has a **Euclidean pseudometric** $\rho_V:V \times V \to K$ defined by

$$\rho_V(a, b) \coloneqq \sqrt{\sum_{i \in \mathrm{Fin}(n)} \rho(a_i, b_i)^2}$$
for module elements $a \in V$ and $b \in V$ and scalars $a_i \in A$ and $b_i \in A$ for index $i \in \mathrm{Fin}(n)$, where
$$a = \sum_{i \in \mathrm{Fin}(n)} a_i v_i \quad b = \sum_{i \in \mathrm{Fin}(n)} b_i v_i$$

If $A$ is an [[ordered field]], then this reduces down to the Euclidean metric defined above.

## In constructive mathematics

In [[constructive mathematics]], the [[real numbers]] used to define Euclidean spaces are the [[Dedekind real numbers]] $\mathbb{R}_{D}$, as those are the only ones that are [[Dedekind complete]], in the sense of not having any gaps in the [[dense linear order]]. The Dedekind real numbers are also the real numbers that are geometrically contractible: whose [[shape]] is [[homotopy theory|homotopically]] [[contractible]] $\esh(\mathbb{R}_D) \cong \mathbb{1}$.

### In predicative constructive mathematics

In [[predicative mathematics|predicative]] constructive mathematics, the [[Dedekind real numbers]] are defined relative to a universe $\mathcal{U}$, and thus there are many different such Dedekind real numbers that could be used to define Euclidean spaces, one $\mathbb{R}_\mathcal{U}$ for each $\mathcal{U}$. However, each set of Dedekind real numbers $\mathbb{R}_\mathcal{U}$ would be large relative to the sets in the universe $\mathcal{U}$.

If the predicative constructive foundations does not have universes, then there doesn't exist any [[dense linear order]] that is actually [[Dedekind complete]] in the usual sense, and so the usual definition of Euclidean space does not work. Some mathematicians have proposed to use [[Sierpinski space]]  $\Sigma$, the [[initial object|initial]] [[sigma-frame|$\sigma$-frame]], for defining the real numbers, in place of the large set of all [[propositions]] in a universe $\mathrm{Prop}_\mathcal{U}$, but the real numbers in that case are only [[sigma Dedekind complete|$\Sigma$-Dedekind complete]], which is a weaker condition than being [[Dedekind complete]]. Furthermore, Lešnik showed that for any two $\sigma$-frames $\Sigma$ and $\Sigma^{'}$ that embed into $\mathrm{Prop}_\mathcal{U}$ such that $\Sigma \subseteq \Sigma^{'}$, if $A$ is the $\Sigma$-Dedekind completion of $\mathbb{Q}$ and $B$ is the $\Sigma^{'}$-Dedekind completion of $\mathbb{Q}$, then $A \subseteq B$, so the $\Sigma$-Dedekind real numbers are not complete.

## Lengths and angles

Given two points $x$ and $y$ of a Euclidean space $E$, their difference $x - y$ belongs to the vector space $V$, where it has a norm
$$ {\|x - y\|} = \sqrt{\langle{x - y, x - y}\rangle} .$$
This real number (or properly, element of the line $L$) is the __distance__ between $x$ and $y$, or the __length__ of the line segment $\overline{x y}$.  This distance function makes $E$ into an ($L$-valued) [[metric space]].

Given three points $x, y, z$, with $x, y \ne z$ (so that ${\|x - z\|}, {\|y - z\|} \ne 0$), we can form the ratio
$$ \frac{\langle{x - z, y - z}\rangle}{{\|x - z\|} {\|y - z\|}} ,$$
which is a (dimensionless) real number.  By the Cauchy--Schwartz inequality, this number lies between $-1$ and $1$, so it\'s the [[cosine]] of a unique angle measure between $0$ and $\pi$ radians.  This is the measure of the __angle__ $\angle x z y$.  In a $2$-dimensional Euclidean space, we can interpret $\angle x z y$ as a signed angle (so taking values anywhere on the [[unit circle]]) if we fix an [[orientation]] of $E$.

Conversely, knowing angles and lengths, we may recover the inner product on $V$;
$$ \langle{x - z, y - z}\rangle = {\|\overline{x z}\|} {\|\overline{y z}\|} \cos \angle x z y ,$$
and other inner products are recovered by linearity.  (We must then use the axioms of [[Euclidean geometry]] to prove that this is well defined and actually an inner product.)  It's actually possible to recover the inner product and angles from lengths alone; this is discussed at [[Hilbert space]].

## Related concepts

* [[Euclidean field]]

* [[Euclidean G-space]]

* [[super-Euclidean space]]

* [[Euclidean geometry]]

* [[synthetic geometry]]

* [[Euclidean field theory]]

* [[geometric algebra]]

## References

* {#Euclid300BC} [[Euclid]], _[[Elements]]_, 300BC

Textbook accounts:

* [[Igor R. Shafarevich]], [[Alexey O. Remizov]]: §7 in: *Linear Algebra and Geometry* (2012) [[doi:10.1007/978-3-642-30994-6](https://doi.org/10.1007/978-3-642-30994-6), [MAA-review](https://maa.org/press/maa-reviews/linear-algebra-and-geometry)]

On the use of the Dedekind real numbers in constructive and predicative constructive mathematics, such as for Euclidean spaces:

* [[Univalent Foundations Project]], [[Homotopy Type Theory – Univalent Foundations of Mathematics]] (2013)
* [[Mike Shulman]], Brouwer’s fixed-point theorem in real-cohesive homotopy type theory, Mathematical Structures in Computer Science Vol 28 (6) (2018): 856-941 ([arXiv:1509.07584](https://arxiv.org/abs/1509.07584), [doi:10.1017/S0960129517000147](https://doi.org/10.1017/S0960129517000147))
* Davorin Lešnik, Synthetic Topology and Constructive Metric Spaces, ([arxiv:2104.10399](https://arxiv.org/abs/2104.10399))

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