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this.value = '' : true" onblur="this.value == '' ? this.value = 'Search' : true" /></fieldset> </form> </div> <div id="revision"> <html xmlns="http://www.w3.org/1999/xhtml" xmlns:svg="http://www.w3.org/2000/svg" xml:lang="en" lang="en"> <head><meta http-equiv="Content-type" content="application/xhtml+xml;charset=utf-8" /><title>Contents</title></head> <body> <blockquote> This page is about <a class="existingWikiWord" href="/nlab/show/computability">computability</a> of fundamental <a class="existingWikiWord" href="/nlab/show/physics">physics</a> in the sense of <a class="existingWikiWord" href="/nlab/show/computable+mathematics">computable mathematics</a> and <a class="existingWikiWord" href="/nlab/show/constructive+analysis">constructive analysis</a>. It is not about <a class="existingWikiWord" href="/nlab/show/computational+physics">computational physics</a>, though of course there is a relation. For computational <a class="existingWikiWord" href="/nlab/show/complexity+theory">complexity theory</a> in physics see at <a class="existingWikiWord" href="/nlab/show/computational+complexity+and+physics">computational complexity and physics</a>. </blockquote> <div class="rightHandSide"> <div class="toc clickDown" tabindex="0"> <h3 id="context">Context</h3> <h4 id="physics">Physics</h4> <div class="hide"><div> <a class="existingWikiWord" href="/nlab/show/physics">physics</a>, <a class="existingWikiWord" href="/nlab/show/mathematical+physics">mathematical physics</a>, <a class="existingWikiWord" href="/nlab/show/philosophy+of+physics">philosophy of physics</a> <h2 id="surveys_textbooks_and_lecture_notes">Surveys, textbooks and lecture notes</h2> <ul> <li> <a class="existingWikiWord" href="/nlab/show/higher+category+theory+and+physics">(higher) category theory and physics</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/geometry+of+physics">geometry of physics</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/books+and+reviews+in+mathematical+physics">books and reviews</a>, <a class="existingWikiWord" href="/nlab/show/physics+resources">physics resources</a> </li> </ul> <hr /> <a class="existingWikiWord" href="/nlab/show/theory+%28physics%29">theory (physics)</a>, <a class="existingWikiWord" href="/nlab/show/model+%28physics%29">model (physics)</a> <a class="existingWikiWord" href="/nlab/show/experiment">experiment</a>, <a class="existingWikiWord" href="/nlab/show/measurement">measurement</a>, <a class="existingWikiWord" href="/nlab/show/computable+physics">computable physics</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/mechanics">mechanics</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/mass">mass</a>, <a class="existingWikiWord" href="/nlab/show/charge">charge</a>, <a class="existingWikiWord" href="/nlab/show/momentum">momentum</a>, <a class="existingWikiWord" href="/nlab/show/angular+momentum">angular momentum</a>, <a class="existingWikiWord" href="/nlab/show/moment+of+inertia">moment of inertia</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/dynamics+on+Lie+groups">dynamics on Lie groups</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/rigid+body+dynamics">rigid body dynamics</a></li> </ul> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/field+%28physics%29">field (physics)</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/Lagrangian+mechanics">Lagrangian mechanics</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/configuration+space">configuration space</a>, <a class="existingWikiWord" href="/nlab/show/state">state</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/action+functional">action functional</a>, <a class="existingWikiWord" href="/nlab/show/Lagrangian">Lagrangian</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/covariant+phase+space">covariant phase space</a>, <a class="existingWikiWord" href="/nlab/show/Euler-Lagrange+equations">Euler-Lagrange equations</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/Hamiltonian+mechanics">Hamiltonian mechanics</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/phase+space">phase space</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/symplectic+geometry">symplectic geometry</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/Poisson+manifold">Poisson manifold</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/symplectic+manifold">symplectic manifold</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/symplectic+groupoid">symplectic groupoid</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/multisymplectic+geometry">multisymplectic geometry</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/n-symplectic+manifold">n-symplectic manifold</a></li> </ul> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/smooth+Lorentzian+manifold">smooth Lorentzian manifold</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/special+relativity">special relativity</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/general+relativity">general relativity</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/gravity">gravity</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/supergravity">supergravity</a>, <a class="existingWikiWord" href="/nlab/show/dilaton+gravity">dilaton gravity</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/black+hole">black hole</a> </li> </ul> </li> </ul> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/classical+field+theory">Classical field theory</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/classical+physics">classical physics</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/classical+mechanics">classical mechanics</a></li> <li><a class="existingWikiWord" href="/nlab/show/waves">waves</a> and <a class="existingWikiWord" href="/nlab/show/optics">optics</a></li> <li><a class="existingWikiWord" href="/nlab/show/thermodynamics">thermodynamics</a></li> </ul> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+mechanics">Quantum Mechanics</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/quantum+mechanics+in+terms+of+dagger-compact+categories">in terms of ∞-compact categories</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+information">quantum information</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Hamiltonian+operator">Hamiltonian operator</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/density+matrix">density matrix</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Kochen-Specker+theorem">Kochen-Specker theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Bell%27s+theorem">Bell's theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Gleason%27s+theorem">Gleason's theorem</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantization">Quantization</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/geometric+quantization">geometric quantization</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/deformation+quantization">deformation quantization</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/path+integral">path integral quantization</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/semiclassical+approximation">semiclassical approximation</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+field+theory">Quantum Field Theory</a> <ul> <li> Axiomatizations <ul> <li> <a class="existingWikiWord" href="/nlab/show/AQFT">algebraic QFT</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/Wightman+axioms">Wightman axioms</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Haag-Kastler+axioms">Haag-Kastler axioms</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/operator+algebra">operator algebra</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/local+net">local net</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/conformal+net">conformal net</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Reeh-Schlieder+theorem">Reeh-Schlieder theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Osterwalder-Schrader+theorem">Osterwalder-Schrader theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/PCT+theorem">PCT theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Bisognano-Wichmann+theorem">Bisognano-Wichmann theorem</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/modular+theory">modular theory</a></li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/spin-statistics+theorem">spin-statistics theorem</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/boson">boson</a>, <a class="existingWikiWord" href="/nlab/show/fermion">fermion</a></li> </ul> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/FQFT">functorial QFT</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/cobordism">cobordism</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2Cn%29-category+of+cobordisms">(∞,n)-category of cobordisms</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/cobordism+hypothesis">cobordism hypothesis</a>-theorem </li> <li> <a class="existingWikiWord" href="/nlab/show/extended+topological+quantum+field+theory">extended topological quantum field theory</a> </li> </ul> </li> </ul> </li> <li> Tools <ul> <li> <a class="existingWikiWord" href="/nlab/show/perturbative+quantum+field+theory">perturbative quantum field theory</a>, <a class="existingWikiWord" href="/nlab/show/vacuum">vacuum</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/effective+quantum+field+theory">effective quantum field theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/renormalization">renormalization</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/BV-BRST+formalism">BV-BRST formalism</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/geometric+%E2%88%9E-function+theory">geometric ∞-function theory</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/particle+physics">particle physics</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/phenomenology">phenomenology</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/model+%28in+particle+phyiscs%29">models</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/standard+model+of+particle+physics">standard model of particle physics</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/fields+and+quanta+-+table">fields and quanta</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/GUT">Grand Unified Theories</a>, <a class="existingWikiWord" href="/nlab/show/MSSM">MSSM</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/scattering+amplitude">scattering amplitude</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/on-shell+recursion">on-shell recursion</a>, <a class="existingWikiWord" href="/nlab/show/KLT+relations">KLT relations</a></li> </ul> </li> </ul> </li> <li> Structural phenomena <ul> <li> <a class="existingWikiWord" href="/nlab/show/universality+class">universality class</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+anomaly">quantum anomaly</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/Green-Schwarz+mechanism">Green-Schwarz mechanism</a></li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/instanton">instanton</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/spontaneously+broken+symmetry">spontaneously broken symmetry</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Kaluza-Klein+mechanism">Kaluza-Klein mechanism</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/integrable+systems">integrable systems</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/holonomic+quantum+fields">holonomic quantum fields</a> </li> </ul> </li> <li> Types of quantum field thories <ul> <li> <a class="existingWikiWord" href="/nlab/show/TQFT">TQFT</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/2d+TQFT">2d TQFT</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Dijkgraaf-Witten+theory">Dijkgraaf-Witten theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Chern-Simons+theory">Chern-Simons theory</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/TCFT">TCFT</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/A-model">A-model</a>, <a class="existingWikiWord" href="/nlab/show/B-model">B-model</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/homological+mirror+symmetry">homological mirror symmetry</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/QFT+with+defects">QFT with defects</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/conformal+field+theory">conformal field theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/%281%2C1%29-dimensional+Euclidean+field+theories+and+K-theory">(1,1)-dimensional Euclidean field theories and K-theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/%282%2C1%29-dimensional+Euclidean+field+theory">(2,1)-dimensional Euclidean field theory and elliptic cohomology</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/CFT">CFT</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/WZW+model">WZW model</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/6d+%282%2C0%29-supersymmetric+QFT">6d (2,0)-supersymmetric QFT</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/gauge+theory">gauge theory</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/field+strength">field strength</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/gauge+group">gauge group</a>, <a class="existingWikiWord" href="/nlab/show/gauge+transformation">gauge transformation</a>, <a class="existingWikiWord" href="/nlab/show/gauge+fixing">gauge fixing</a> </li> <li> examples <ul> <li><a class="existingWikiWord" href="/nlab/show/electromagnetic+field">electromagnetic field</a>, <a class="existingWikiWord" href="/nlab/show/QED">QED</a></li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/electric+charge">electric charge</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/magnetic+charge">magnetic charge</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/Yang-Mills+field">Yang-Mills field</a>, <a class="existingWikiWord" href="/nlab/show/QCD">QCD</a></li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/Yang-Mills+theory">Yang-Mills theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/spinors+in+Yang-Mills+theory">spinors in Yang-Mills theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/topological+Yang-Mills+theory">topological Yang-Mills theory</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/Kalb-Ramond+field">Kalb-Ramond field</a></li> <li><a class="existingWikiWord" href="/nlab/show/supergravity+C-field">supergravity C-field</a></li> <li><a class="existingWikiWord" href="/nlab/show/RR+field">RR field</a></li> <li><a class="existingWikiWord" href="/nlab/show/first-order+formulation+of+gravity">first-order formulation of gravity</a></li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/general+covariance">general covariance</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/supergravity">supergravity</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/D%27Auria-Fre+formulation+of+supergravity">D'Auria-Fre formulation of supergravity</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/gravity+as+a+BF-theory">gravity as a BF-theory</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/sigma-model">sigma-model</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/particle">particle</a>, <a class="existingWikiWord" href="/nlab/show/relativistic+particle">relativistic particle</a>, <a class="existingWikiWord" href="/nlab/show/fundamental+particle">fundamental particle</a>, <a class="existingWikiWord" href="/nlab/show/spinning+particle">spinning particle</a>, <a class="existingWikiWord" href="/nlab/show/superparticle">superparticle</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/string">string</a>, <a class="existingWikiWord" href="/nlab/show/spinning+string">spinning string</a>, <a class="existingWikiWord" href="/nlab/show/superstring">superstring</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/membrane">membrane</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/AKSZ+theory">AKSZ theory</a> </li> </ul> </li> </ul> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/string+theory">String Theory</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/string+theory+results+applied+elsewhere">string theory results applied elsewhere</a></li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/number+theory+and+physics">number theory and physics</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/Riemann+hypothesis+and+physics">Riemann hypothesis and physics</a></li> </ul> </li> </ul> <div> <a href="/nlab/edit/physicscontents">Edit this sidebar</a> </div></div></div> <h4 id="constructivism_realizability_computability">Constructivism, Realizability, Computability</h4> <div class="hide"><div> <a class="existingWikiWord" href="/nlab/show/constructive+mathematics">constructive mathematics</a>, <a class="existingWikiWord" href="/nlab/show/realizability">realizability</a>, <a class="existingWikiWord" href="/nlab/show/computability">computability</a> <a class="existingWikiWord" href="/nlab/show/intuitionistic+mathematics">intuitionistic mathematics</a> <a class="existingWikiWord" href="/nlab/show/propositions+as+types">propositions as types</a>, <a class="existingWikiWord" href="/nlab/show/proofs+as+programs">proofs as programs</a>, <a class="existingWikiWord" href="/nlab/show/computational+trinitarianism">computational trinitarianism</a> <h3 id="constructive_mathematics">Constructive mathematics</h3> <ul> <li> <a class="existingWikiWord" href="/nlab/show/topos">topos</a>, <a class="existingWikiWord" href="/nlab/show/%28infinity%2C1%29-topos">homotopy topos</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/type+theory">type theory</a>, <a class="existingWikiWord" href="/nlab/show/homotopy+type+theory">homotopy type theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/canonical+form">canonical form</a>, <a class="existingWikiWord" href="/nlab/show/univalence">univalence</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Bishop+set">Bishop set</a>, <a class="existingWikiWord" href="/nlab/show/h-set">h-set</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/decidable+equality">decidable equality</a>, <a class="existingWikiWord" href="/nlab/show/decidable+subset">decidable subset</a>, <a class="existingWikiWord" href="/nlab/show/inhabited+set">inhabited set</a>, <a class="existingWikiWord" href="/nlab/show/subsingleton">subsingleton</a> </li> </ul> <h3 id="realizability">Realizability</h3> <ul> <li> <a class="existingWikiWord" href="/nlab/show/realizability+topos">realizability topos</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/realizability+model">realizability model</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/realizability+interpretation">realizability interpretation</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/effective+topos">effective topos</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Kleene%27s+first+algebra">Kleene's first algebra</a>, <a class="existingWikiWord" href="/nlab/show/Kleene%27s+second+algebra">Kleene's second algebra</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/function+realizability">function realizability</a> </li> </ul> <h3 id="computability">Computability</h3> <ul> <li> <a class="existingWikiWord" href="/nlab/show/computability">computability</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/computation">computation</a>, <a class="existingWikiWord" href="/nlab/show/computational+type+theory">computational type theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/computable+function">computable function</a>, <a class="existingWikiWord" href="/nlab/show/partial+recursive+function">partial recursive function</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/computable+analysis">computable analysis</a>, <a class="existingWikiWord" href="/nlab/show/constructive+analysis">constructive analysis</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/Type+Two+Theory+of+Effectivity">Type Two Theory of Effectivity</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/computable+function+%28analysis%29">computable function (analysis)</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/exact+real+computer+arithmetic">exact real computer arithmetic</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/computable+set">computable set</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/persistent+homology">persistent homology</a>, <a class="existingWikiWord" href="/nlab/show/effective+homology">effective homology</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/computable+physics">computable physics</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Church-Turing+thesis">Church-Turing thesis</a> </li> </ul> </div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> The following idea or observation or sentiment has been expressed independently by many authors. We quote from <a href="#Szudzik10">Szudzik 10, section 2</a>: <blockquote> The central problem is that <a class="existingWikiWord" href="/nlab/show/model+%28physics%29">physical models</a> use <a class="existingWikiWord" href="/nlab/show/real+numbers">real numbers</a> to represent the values of <a class="existingWikiWord" href="/nlab/show/observable">observable quantities</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[...]</annotation></semantics></math> Careful consideration of this problem, however, reveals that the real numbers are not actually necessary in physical models. <a class="existingWikiWord" href="/nlab/show/natural+number">Non-negative integers</a> suffice for the representation of observable quantities because numbers <a class="existingWikiWord" href="/nlab/show/measurement">measured</a> in laboratory <a class="existingWikiWord" href="/nlab/show/experiments">experiments</a> necessarily have only finitely many digits of precision. </blockquote> Diverse conclusions have been drawn from this. One which seems useful and well-informed by the theory of <a class="existingWikiWord" href="/nlab/show/computability">computability</a> in <a class="existingWikiWord" href="/nlab/show/mathematics">mathematics</a> is the following (further quoting from <a href="#Szudzik10">Szudzik 10, section 2</a>) <blockquote> So, we suffer no loss of generality by restricting the values of all observable quantities to be expressed as <a class="existingWikiWord" href="/nlab/show/natural+numbers">non-negative integers</a> — the restriction only forces us to make the methods of error analysis, which were tacitly assumed when dealing with real numbers, an explicit part of each model. </blockquote> In type-I <a class="existingWikiWord" href="/nlab/show/computability">computability</a> the <a class="existingWikiWord" href="/nlab/show/computable+functions">computable functions</a> are <a class="existingWikiWord" href="/nlab/show/partial+recursive+functions">partial recursive functions</a> and in view of this some authors concluded (e.g. <a href="#Kreisel74">Kreisel 74</a>) (and we still quote <a href="#Szudzik10">Szudzik 10, section 2</a> for this): <blockquote> To show that a <a class="existingWikiWord" href="/nlab/show/model+%28physics%29">model</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math> of physics <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/computable+function">computable</a>, the model must somehow be expressed using <a class="existingWikiWord" href="/nlab/show/partial+recursive+function">recursive functions</a>. </blockquote> A similar sentiment is voiced by <a href="#GerochHartle">Geroch and Hartle</a>: <blockquote> We propose, in parallel with the notion of a computable number in mathematics, that of a measurable number in a physical theory. The question of whether there exists an algorithm for implementing a theory may then be formulated more precisely as the question of whether the measurable numbers of the theory are computable. We argue that the measurable numbers are in fact computable in the familiar theories of physics, but there is no reason why this need be the case in order that a theory have predictive power. Indeed, in some recent formulations of quantum gravity as a sum ver histories, there are candidates for numbers that are measurable but not computable. </blockquote> However, in <a class="existingWikiWord" href="/nlab/show/computability+theory">computability theory</a> there is also the concept of <a class="existingWikiWord" href="/nlab/show/computable+function+%28analysis%29">type-II computable functions</a> used in the field of “<a class="existingWikiWord" href="/nlab/show/constructive+analysis">constructive analysis</a>”, “<a class="existingWikiWord" href="/nlab/show/computable+analysis">computable analysis</a>”. This is based on the idea that for instance for specifying <a class="existingWikiWord" href="/nlab/show/computable+real+numbers">computable real numbers</a> as used in <a class="existingWikiWord" href="/nlab/show/physics">physics</a>, an <a class="existingWikiWord" href="/nlab/show/algorithm">algorithm</a> may work not just on single <a class="existingWikiWord" href="/nlab/show/natural+numbers">natural numbers</a>, but indefinitely on sequences of them, producing output that is in each step a finite, but in each next step a more accurate approximation. <div> <a class="existingWikiWord" href="/nlab/show/computability">computability</a> <table><thead><tr><th></th><th>type I computability</th><th>type II computability</th></tr></thead><tbody><tr><td style="text-align: left;">typical domain</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/natural+numbers">natural numbers</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">\mathbb{N}</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Baire+space+%28computability%29">Baire space</a> of infinite sequences <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔹</mi><mo>=</mo><msup><mi>ℕ</mi> <mi>ℕ</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{B} = \mathbb{N}^{\mathbb{N}}</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/computable+functions">computable functions</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/partial+recursive+function">partial recursive function</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/computable+function+%28analysis%29">computable function (analysis)</a></td></tr> <tr><td style="text-align: left;">type of <a class="existingWikiWord" href="/nlab/show/computable+mathematics">computable mathematics</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/recursive+mathematics">recursive mathematics</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/computable+analysis">computable analysis</a>, <a class="existingWikiWord" href="/nlab/show/Type+Two+Theory+of+Effectivity">Type Two Theory of Effectivity</a></td></tr> <tr><td style="text-align: left;">type of <a class="existingWikiWord" href="/nlab/show/realizability">realizability</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/number+realizability">number realizability</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/function+realizability">function realizability</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/partial+combinatory+algebra">partial combinatory algebra</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Kleene%27s+first+partial+combinatory+algebra">Kleene's first partial combinatory algebra</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Kleene%27s+second+partial+combinatory+algebra">Kleene's second partial combinatory algebra</a></td></tr> </tbody></table> </div> This concept of <a class="existingWikiWord" href="/nlab/show/computable+analysis">type-II computability</a> is arguably closer to actual practice in <a class="existingWikiWord" href="/nlab/show/physics">physics</a>. Of course there is a wide-spread (but of course controversial) vague speculation (often justified by alluding to expected implications of <a class="existingWikiWord" href="/nlab/show/quantum+gravity">quantum gravity</a> on the true microscopic nature of <a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a> and sometimes formalized in terms of cellular automata, e.g. <a href="#Zuse67">Zuse 67</a>) that in some sense the <a class="existingWikiWord" href="/nlab/show/observable+universe">observable universe</a> is fundamentally “<a class="existingWikiWord" href="/nlab/show/finite+object">finite</a>”, so that in the end <a class="existingWikiWord" href="/nlab/show/computability">computability</a> is a non-issue in <a class="existingWikiWord" href="/nlab/show/physics">physics</a> as one is really operating on a large but <a class="existingWikiWord" href="/nlab/show/finite+set">finite set</a> of states. However, since fundamental physics is <a class="existingWikiWord" href="/nlab/show/quantum+physics">quantum physics</a> and since <a class="existingWikiWord" href="/nlab/show/quantum+mechanics">quantum mechanics</a> with its <a class="existingWikiWord" href="/nlab/show/wave+functions">wave functions</a>, <a class="existingWikiWord" href="/nlab/show/Hilbert+spaces">Hilbert spaces</a> and <a class="existingWikiWord" href="/nlab/show/probability+amplitudes">probability amplitudes</a> invokes (<a class="existingWikiWord" href="/nlab/show/functional+analysis">functional</a>) <a class="existingWikiWord" href="/nlab/show/analysis">analysis</a> and hence “non-finite mathematics” even when describing the minimum of a <a class="existingWikiWord" href="/nlab/show/physical+system">physical system</a> with only two possible configurations (a “<a class="existingWikiWord" href="/nlab/show/qbit">qbit</a>”) a strict <a class="existingWikiWord" href="/nlab/show/finitism">finitism</a> perspective on fundamental <a class="existingWikiWord" href="/nlab/show/physics">physics</a> runs into problems (highlighted for instance in (<a href="#Feynman81">Feynman 81, slide 15</a>)). Precisely this kind of issue is the topic of <a class="existingWikiWord" href="/nlab/show/computable+analysis">computable analysis</a> (“<a class="existingWikiWord" href="/nlab/show/constructive+analysis">constructive analysis</a>”, “<a class="existingWikiWord" href="/nlab/show/exact+analysis">exact analysis</a>”) with its type-II notion of <a class="existingWikiWord" href="/nlab/show/computable+functions">computable functions</a>, which would therefore seem to be the right mathematical context discussing <a class="existingWikiWord" href="/nlab/show/computability">computability</a> in fundamental (namely quantum) <a class="existingWikiWord" href="/nlab/show/physics">physics</a>. This point is made in (<a href="#WeihrauchZhong02">Weihrauch-Zhong 02</a>) for the <a class="existingWikiWord" href="/nlab/show/wave+equation">wave equation</a>. This matters: there are solutions to the <a class="existingWikiWord" href="/nlab/show/wave+equation">wave equation</a> with type-I computable initial values which are not themselves type-I computable (<a href="#PourEl83">Pour-El et al. 83</a>). If type-I computability were the right concept of computabuility in physics, this result would show a violation of the <a class="existingWikiWord" href="/nlab/show/Church-Turing+thesis">Church-Turing thesis</a>. But in (<a href="#WeihrauchZhong02">Weihrauch-Zhong 02</a>) it is argued that the correct concept to use is indeed type-II computability and it is shown (<a href="#WeihrauchZhong02">Weihrauch-Zhong 02, theorem 3.2</a>) that the solution operator to the wave equation is indeed type-II computable. The same is shown for the <a class="existingWikiWord" href="/nlab/show/Schr%C3%B6dinger+equation">Schrödinger equation</a> of <a class="existingWikiWord" href="/nlab/show/quantum+mechanics">quantum mechanics</a> in (<a href="#WeihrauchZhong01">Weihrauch-Zhong 01</a>, <a href="#WeihrauchZhong06">Weihrauch-Zhong 06</a>) where it is found that the Schrödinger operator is computable on the <a class="existingWikiWord" href="/nlab/show/Lebesgue+space">Lebesgue space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>L</mi> <mi>p</mi></msup></mrow><annotation encoding="application/x-tex">L^p</annotation></semantics></math> precisely for the physically relevant value <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>=</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">p = 2</annotation></semantics></math>. A similar conclusion is reached by <a href="#Baez">Baez</a>. In the vein of type-II computability the issue specifically of computable <a class="existingWikiWord" href="/nlab/show/quantum+physics">quantum physics</a>/<a class="existingWikiWord" href="/nlab/show/quantum+logic">quantum logic</a> has only been further considered in (<a href="#Streicher12">Streicher 12</a>), where it is shown that at least a fair bit of the <a class="existingWikiWord" href="/nlab/show/Hilbert+space">Hilbert space</a> technology of <a class="existingWikiWord" href="/nlab/show/quantum+mechanics">quantum mechanics</a>/<a class="existingWikiWord" href="/nlab/show/quantum+logic">quantum logic</a> sits inside the <a class="existingWikiWord" href="/nlab/show/function+realizability+topos">function realizability topos</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>RT</mi><mo stretchy="false">(</mo><msub><mi>𝒦</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">RT(\mathcal{K}_2)</annotation></semantics></math>. The question whether the <a class="existingWikiWord" href="/nlab/show/observable+universe">observable universe</a> or at least all <a class="existingWikiWord" href="/nlab/show/experiments">experiments</a> done within are computable is part of the (strong) <a class="existingWikiWord" href="/nlab/show/physical+Church-Turing+thesis">physical Church-Turing thesis</a>. See there for more, and see (<a href="#Waaldijk03">Waaldijk 03</a>). Experiments showing non-computability in quantum processes have been claimed in (<a href="#CDDS10">CDDS 10</a>). <h2 id="related_concepts">Related concepts</h2> <ul> <li> <a class="existingWikiWord" href="/nlab/show/interpretation+of+quantum+mechanics">interpretation of quantum mechanics</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/measurement+problem">measurement problem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+logic">quantum logic</a>, <a class="existingWikiWord" href="/nlab/show/quantum+computation">quantum computation</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/coordination">coordination</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/p-adic+physics">p-adic physics</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/hypercomputation">hypercomputation</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/constructive+mathematics">constructive mathematics</a>, <a class="existingWikiWord" href="/nlab/show/geometry+of+physics">geometry of physics</a> </li> </ul> <h2 id="references">References</h2> Decent discussion of computability in physics includes the following <ul> <li> Robert Rosen, Church’s thesis and its relation to the concept of realizability in biology and physics, Bulletin of Mathematical Biophysics 24 (1962), 375–393. </li> <li id="Zuse67"> <a class="existingWikiWord" href="/nlab/show/Konrad+Zuse">Konrad Zuse</a>: Rechnender Raum, Schriften zur Datenverarbeitung 1, Viehweg & Sohn (1969) [<a href="https://doi.org/10.1007/978-3-663-02723-2">doi:https://doi.org/10.1007/978-3-663-02723-2</a>, <a class="existingWikiWord" href="/nlab/files/Zuse-RechnenderRaum.pdf" title="pdf">pdf</a>] English translation: Calculating Space, MIT Technical Translation, AZT-70-164-GEMIT (1970) [<a class="existingWikiWord" href="/nlab/files/Zuse-CalculatingSpace.pdf" title="pdf">pdf</a>] reprinted as chapter 36 of: Hector Zenil (ed.) A Computable Universe – Understanding and Exploring Nature as Computation, World Scientific (2012) [<a href="https://doi.org/10.1142/8306">doi:10.1142/8306</a>] </li> <li id="Kreisel74"> <a class="existingWikiWord" href="/nlab/show/Georg+Kreisel">Georg Kreisel</a>, A notion of mechanistic theory, Synthese 29 (1974), 11–26. </li> <li id="Feynman81"> <a class="existingWikiWord" href="/nlab/show/Richard+Feynman">Richard Feynman</a>, Simulating physics with computers, talk at 1st conference on Physics and Computation MIT (1981), (talk notes by P. Birnbaum, E. Tromer: <a href="http://www.wisdom.weizmann.ac.il/~naor/COURSE/feynman-simulating.pdf">pdf</a>) </li> <li id="PourEl83"> Marian Boykan Pour-El, J. Ian Richards, Computability and noncomputability in classical analysis’, Trans. Amer. Math. Soc. 275 (1983) 539-560. Marian Pour-El, Ning Zhong, The wave equation with computable initial data whose unique solution is nowhere computable, Math. Logic Quart. 43 (1997) no. 4, 499-509. </li> <li id="Bridges95"> <a class="existingWikiWord" href="/nlab/show/Douglas+Bridges">Douglas Bridges</a>, Constructive mathematics and unbounded operators – a reply to Hellman, J. Philosophical Logic 24, 549–561 (1995) </li> <li> <a class="existingWikiWord" href="/nlab/show/Douglas+S.+Bridges">Douglas S. Bridges</a>: Can Constructive Mathematics Be Applied in Physics?, Journal of Philosophical Logic 28 5 (1999) 439-453 [<a href="https://www.jstor.org/stable/30226680">jstor:30226680</a>, <a href="https://doi.org/10.1023/A:1004420413391">doi:10.1023/A:1004420413391</a>] </li> <li id="WeihrauchZhong01"> <a class="existingWikiWord" href="/nlab/show/Klaus+Weihrauch">Klaus Weihrauch</a>, Ning Zhong, Is the linear Schrödinger Propagator Turing Computable? , in Jens Blanck et al (eds.) Computability and Complexity in Analysis: 4th International Workshop , CCA, Springer 2001 </li> <li id="WeihrauchZhong02"> <a class="existingWikiWord" href="/nlab/show/Klaus+Weihrauch">Klaus Weihrauch</a>, Ning Zhong, Is wave propagation computable or can wave computers beat the Turing machine?, Proc. of the London Math. Soc. (3) 85 (2002) (<a href="http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.95.5994">web</a>) </li> <li id="Waaldijk03"> <a class="existingWikiWord" href="/nlab/show/Frank+Waaldijk">Frank Waaldijk</a>, section 7 of On the foundations of constructive mathematics – especially in relation to the theory of continuous functions, 2003 (<a href="http://www.fwaaldijk.nl/foundations%20of%20constructive%20mathematics.pdf">pdf</a>, <a href="http://fwaaldijk.wordpress.com/2014/01/14/experiment-to-disprove-strong-physical-church-turing-thesis/">web discussion</a>). </li> <li id="WeihrauchZhong06"> <a class="existingWikiWord" href="/nlab/show/Klaus+Weihrauch">Klaus Weihrauch</a>, Computing Schrödinger propagators on Type-2 Turing machines, Journal of Complexity Volume 22, Issue 6, December 2006, Pages 918–935 </li> <li id="Szudzik10"> <a class="existingWikiWord" href="/nlab/show/Matthew+Szudzik">Matthew Szudzik</a>, The Computable Universe Hypothesis, in <a class="existingWikiWord" href="/nlab/show/Hector+Zenil">Hector Zenil</a> (ed.) <a href="http://www.worldscientific.com/worldscibooks/10.1142/8306">A Computable Universe: Understanding and Exploring Nature as Computation</a> , World Scientific, 2012, pp. 479-523 (<a href="http://arxiv.org/abs/1003.5831">arXiv:1003.5831</a>) </li> <li id="CDDS10"> Cristian Calude, Michael Dinneen, Monica Dumitrescu, Karl Svozil, Experimental Evidence of Quantum Randomness Incomputability, Phys. Rev. A 82, 022102 (2010) (<a href="http://arxiv.org/abs/1004.1521">arxiv:1004.1521</a>, <a href="http://www.technologyreview.com/view/418445/first-evidence-that-quantum-processes-generate-truly-random-numbers/">web announcement</a>) </li> <li id="Streicher12"> <a class="existingWikiWord" href="/nlab/show/Thomas+Streicher">Thomas Streicher</a>, Computability Theory for Quantum Theory, talk at Logic Seminar Univ. Utrecht in July 2012 (<a href="http://www.mathematik.tu-darmstadt.de/~streicher/TALKS/qtdm.pdf">pdf</a>) </li> <li> <a class="existingWikiWord" href="/nlab/show/Stephen+Wolfram">Stephen Wolfram</a>: A New Kind of Science, Wolfram Media (2002) [<a href="https://www.wolfram-media.com/products/nks/">ISBN:978-1-57955-008-0</a>, <a href="https://www.wolframscience.com">www.wolframscience.com</a>, <a href="https://en.wikipedia.org/wiki/A_New_Kind_of_Science">Wikipedia entry</a>] </li> <li> Hector Zenil (ed.): A Computable Universe – Understanding and Exploring Nature as Computation, World Scientific (2012) [<a href="https://doi.org/10.1142/8306">doi:10.1142/8306</a>] foreword by <a class="existingWikiWord" href="/nlab/show/Roger+Penrose">Roger Penrose</a> [<a href="https://arxiv.org/abs/1205.5823">arXiv:1205.5823</a>] introduction by Hector Zenil [<a href="https://arxiv.org/abs/1206.0376">arXiv:1206.0376</a>] </li> <li> Stanford Encyclopedia of Philosophy, <a href="http://plato.stanford.edu/entries/computation-physicalsystems/">Computation in physics</a> </li> </ul> Some general remarks on the impact of <a class="existingWikiWord" href="/nlab/show/realizability">realizability</a> (<a class="existingWikiWord" href="/nlab/show/intuitionistic+mathematics">intuitionistic</a>/<a class="existingWikiWord" href="/nlab/show/constructive+mathematics">constructive mathematics</a>) on computability in physics and the <a class="existingWikiWord" href="/nlab/show/Church-Turing+thesis">Church-Turing thesis</a> are in <ul> <li><a class="existingWikiWord" href="/nlab/show/Andrej+Bauer">Andrej Bauer</a>: Intuitionistic Mathematics and Realizability in the Physical World, in A Computable Universe (2012) 143-157 [<a href="https://doi.org/10.1142/9789814374309_0008">doi:10.1142/9789814374309_0008</a>, <a href="https://math.andrej.com/wp-content/uploads/2014/03/real-world-realizability.pdf">pdf</a>, <a href="https://math.andrej.com/2014/03/04/intuitionistic-mathematics-and-realizability-in-the-physical-world/">webpage</a>]</li> </ul> A useful set of lecture notes on the mathematical background of <a class="existingWikiWord" href="/nlab/show/computability">computability</a> and <a class="existingWikiWord" href="/nlab/show/realizability">realizability</a> is in <ul> <li id="Bauer05"> <a class="existingWikiWord" href="/nlab/show/Andrej+Bauer">Andrej Bauer</a>, Realizability as connection between constructive and computable mathematics, in T. Grubba, P. Hertling, H. Tsuiki, and <a class="existingWikiWord" href="/nlab/show/Klaus+Weihrauch">Klaus Weihrauch</a>, (eds.) CCA 2005 - Second International Conference on Computability and Complexity in Analysis, August 25-29,2005, Kyoto, Japan, ser. Informatik Berichte, , vol. 326-7/2005. FernUniversität Hagen, Germany, 2005, pp. 378–379. (<a href="http://math.andrej.com/data/c2c.pdf">pdf</a>) </li> <li id="GerochHartle"> <a class="existingWikiWord" href="/nlab/show/Robert+Geroch">Robert Geroch</a>, <a class="existingWikiWord" href="/nlab/show/James+Hartle">James Hartle</a>: Computability and physical theories, Foundations of Physics, 16 6 (1986) 533-550 [<a href="http://dx.doi.org/10.1007/BF01886519">doi:10.1007/BF01886519</a>, <a href="http://link.springer.com/content/pdf/10.1007/BF01886519.pdf">pdf</a>] </li> <li id="Baez"> <a class="existingWikiWord" href="/nlab/show/John+Baez">John Baez</a>, Recursivity in Quantum Mechanics, <a href="http://math.ucr.edu/home/baez/recursivity.pdf">PDF</a> </li> </ul> Ye presents a constructive development of part of quantum theory and relativity theory. <ul> <li>Feng Ye, Strict Finitism and the Logic of Mathematical Applications <a href="http://www.phil.pku.edu.cn/cllc/people/fengye/finitismAndTheLogicOfMathematicalApplications.pdf">PDF preprint</a></li> </ul> </body></html> </div> <div class="revisedby"> Last revised on November 1, 2024 at 07:08:34. 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