The Principle of Computational Equivalence: A New Kind of Science

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<header id="header"> <div class="table width-full height-full"> <div class="heirs-vertical-align-m"> <div class="padding-lr-1"> <h1><a href="/nks/"><img src="/nks/img/a-new-kind-of-science-logo.png" alt="Stephen Wolfram's A New Kind of Science | Online" width="372" height="43" class="fill"></a></h1> </div> <div class="padding-r-1"> <nav id="nav" class="table"> <div class="nav-main heirs-vertical-align-t"> <div class="size-picker"> <i class="icon"></i> <ul> <li class="size-auto checked"><span>auto</span></li> <li class="size-small"><span>small</span></li> <li class="size-medium"><span>medium</span></li> <li class="size-large"><span>large</span></li> <li class="size-x-large"><span>x-large</span></li> </ul> </div> <div class="search"> <div><i class="icon"></i></div> </div> <div class="toc"> <div><a href="/nks/"><i class="icon"></i></a></div> </div> <div class="breadcrumbs"> <div> <span class="chapter"><span>Chapter 12</span></span> </div> </div> </div> <div class="nav-burger"> <div> <div 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Outline of Basic Ideas</span> </a> </li> <li class="section section-2"> <a href="/nks/p7--relations-to-other-areas/"> <span>2</span> <span>Relations to Other Areas</span> </a> </li> <li class="section section-3"> <a href="/nks/p12--some-past-initiatives/"> <span>3</span> <span>Some Past Initiatives</span> </a> </li> <li class="section section-4"> <a href="/nks/p17--the-personal-story-of-the-science-in-this-book/"> <span>4</span> <span>The Personal Story of the Science in This Book</span> </a> </li> </ul> </li> <li class="chapter chapter-2"> <a href="/nks/chap-2--the-crucial-experiment/"> <span>2</span> <span>The Crucial Experiment</span> </a> <ul class="sections"> <li class="section section-1"> <a href="/nks/p23--how-do-simple-programs-behave/"> <span>1</span> <span>How Do Simple Programs Behave?</span> </a> </li> <li class="section section-2"> <a href="/nks/p39--the-need-for-a-new-intuition/"> <span>2</span> <span>The Need for a New Intuition</span> </a> </li> <li class="section section-3"> <a href="/nks/p42--why-these-discoveries-were-not-made-before/"> <span>3</span> <span>Why These Discoveries Were Not Made Before</span> </a> </li> </ul> </li> <li class="chapter chapter-3"> <a href="/nks/chap-3--the-world-of-simple-programs/"> <span>3</span> <span>The World of Simple Programs</span> </a> <ul class="sections"> <li class="section section-1"> <a href="/nks/p51--the-search-for-general-features/"> <span>1</span> <span>The Search for General Features</span> </a> </li> <li class="section section-2"> <a href="/nks/p53--more-cellular-automata/"> <span>2</span> <span>More Cellular Automata</span> </a> </li> <li class="section section-3"> <a href="/nks/p71--mobile-automata/"> <span>3</span> <span>Mobile Automata</span> </a> </li> <li class="section section-4"> <a href="/nks/p78--turing-machines/"> <span>4</span> <span>Turing Machines</span> </a> </li> <li class="section section-5"> <a href="/nks/p82--substitution-systems/"> <span>5</span> <span>Substitution Systems</span> </a> </li> <li class="section section-6"> <a href="/nks/p88--sequential-substitution-systems/"> <span>6</span> <span>Sequential Substitution Systems</span> </a> </li> <li class="section section-7"> <a href="/nks/p93--tag-systems/"> <span>7</span> <span>Tag Systems</span> </a> </li> <li class="section section-8"> <a href="/nks/p95--cyclic-tag-systems/"> <span>8</span> <span>Cyclic Tag Systems</span> </a> </li> <li class="section section-9"> <a href="/nks/p97--register-machines/"> <span>9</span> <span>Register Machines</span> </a> </li> <li class="section section-10"> <a href="/nks/p102--symbolic-systems/"> <span>10</span> <span>Symbolic Systems</span> </a> </li> <li class="section section-11"> <a href="/nks/p105--some-conclusions/"> <span>11</span> <span>Some Conclusions</span> </a> </li> <li class="section section-12"> <a href="/nks/p108--how-the-discoveries-in-this-chapter-were-made/"> <span>12</span> <span>How the Discoveries in This Chapter Were Made</span> </a> </li> </ul> </li> <li class="chapter chapter-4"> <a href="/nks/chap-4--systems-based-on-numbers/"> <span>4</span> <span>Systems Based on Numbers</span> </a> <ul class="sections"> <li class="section section-1"> <a href="/nks/p115--the-notion-of-numbers/"> <span>1</span> <span>The Notion of Numbers</span> </a> </li> <li class="section section-2"> <a href="/nks/p117--elementary-arithmetic/"> <span>2</span> <span>Elementary Arithmetic</span> </a> </li> <li class="section section-3"> <a href="/nks/p128--recursive-sequences/"> <span>3</span> <span>Recursive Sequences</span> </a> </li> <li class="section section-4"> <a href="/nks/p132--the-sequence-of-primes/"> <span>4</span> <span>The Sequence of Primes</span> </a> </li> <li class="section section-5"> <a href="/nks/p136--mathematical-constants/"> <span>5</span> <span>Mathematical Constants</span> </a> </li> <li class="section section-6"> <a href="/nks/p145--mathematical-functions/"> <span>6</span> <span>Mathematical Functions</span> </a> </li> <li class="section section-7"> <a href="/nks/p149--iterated-maps-and-the-chaos-phenomenon/"> <span>7</span> <span>Iterated Maps and the Chaos Phenomenon</span> </a> </li> <li class="section section-8"> <a href="/nks/p155--continuous-cellular-automata/"> <span>8</span> <span>Continuous Cellular Automata</span> </a> </li> <li class="section section-9"> <a href="/nks/p161--partial-differential-equations/"> <span>9</span> <span>Partial Differential Equations</span> </a> </li> <li class="section section-10"> <a href="/nks/p167--continuous-versus-discrete-systems/"> <span>10</span> <span>Continuous Versus Discrete Systems</span> </a> </li> </ul> </li> <li class="chapter chapter-5"> <a href="/nks/chap-5--two-dimensions-and-beyond/"> <span>5</span> <span>Two Dimensions and Beyond</span> </a> <ul class="sections"> <li class="section section-1"> <a href="/nks/p169--introduction/"> <span>1</span> <span>Introduction</span> </a> </li> <li class="section section-2"> <a href="/nks/p170--cellular-automata/"> <span>2</span> <span>Cellular Automata</span> </a> </li> <li class="section section-3"> <a href="/nks/p184--turing-machines/"> <span>3</span> <span>Turing Machines</span> </a> </li> <li class="section section-4"> <a href="/nks/p187--substitution-systems-and-fractals/"> <span>4</span> <span>Substitution Systems and Fractals</span> </a> </li> <li class="section section-5"> <a href="/nks/p193--network-systems/"> <span>5</span> <span>Network Systems</span> </a> </li> <li class="section section-6"> <a href="/nks/p204--multiway-systems/"> <span>6</span> <span>Multiway Systems</span> </a> </li> <li class="section section-7"> <a href="/nks/p210--systems-based-on-constraints/"> <span>7</span> <span>Systems Based on Constraints</span> </a> </li> </ul> </li> <li class="chapter chapter-6"> <a href="/nks/chap-6--starting-from-randomness/"> <span>6</span> <span>Starting from Randomness</span> </a> <ul class="sections"> <li class="section section-1"> <a href="/nks/p223--the-emergence-of-order/"> <span>1</span> <span>The Emergence of Order</span> </a> </li> <li class="section section-2"> <a href="/nks/p231--four-classes-of-behavior/"> <span>2</span> <span>Four Classes of Behavior</span> </a> </li> <li class="section section-3"> <a href="/nks/p250--sensitivity-to-initial-conditions/"> <span>3</span> <span>Sensitivity to Initial Conditions</span> </a> </li> <li class="section section-4"> <a href="/nks/p255--systems-of-limited-size-and-class-2-behavior/"> <span>4</span> <span>Systems of Limited Size and Class 2 Behavior</span> </a> </li> <li class="section section-5"> <a href="/nks/p261--randomness-in-class-3-systems/"> <span>5</span> <span>Randomness in Class 3 Systems</span> </a> </li> <li class="section section-6"> <a href="/nks/p266--special-initial-conditions/"> <span>6</span> <span>Special Initial Conditions</span> </a> </li> <li class="section section-7"> <a href="/nks/p275--the-notion-of-attractors/"> <span>7</span> <span>The Notion of Attractors</span> </a> </li> <li class="section section-8"> <a href="/nks/p281--structures-in-class-4-systems/"> <span>8</span> <span>Structures in Class 4 Systems</span> </a> </li> </ul> </li> <li class="chapter chapter-7"> <a href="/nks/chap-7--mechanisms-in-programs-and-nature/"> <span>7</span> <span>Mechanisms in Programs and Nature</span> </a> <ul class="sections"> <li class="section section-1"> <a href="/nks/p297--universality-of-behavior/"> <span>1</span> <span>Universality of Behavior</span> </a> </li> <li class="section section-2"> <a href="/nks/p299--three-mechanisms-for-randomness/"> <span>2</span> <span>Three Mechanisms for Randomness</span> </a> </li> <li class="section section-3"> <a href="/nks/p301--randomness-from-the-environment/"> <span>3</span> <span>Randomness from the Environment</span> </a> </li> <li class="section section-4"> <a href="/nks/p304--chaos-theory-and-randomness-from-initial-conditions/"> <span>4</span> <span>Chaos Theory and Randomness from Initial Conditions</span> </a> </li> <li class="section section-5"> <a href="/nks/p315--the-intrinsic-generation-of-randomness/"> <span>5</span> <span>The Intrinsic Generation of Randomness</span> </a> </li> <li class="section section-6"> <a href="/nks/p327--the-phenomenon-of-continuity/"> <span>6</span> <span>The Phenomenon of Continuity</span> </a> </li> <li class="section section-7"> <a href="/nks/p337--origins-of-discreteness/"> <span>7</span> <span>Origins of Discreteness</span> </a> </li> <li class="section section-8"> <a href="/nks/p342--the-problem-of-satisfying-constraints/"> <span>8</span> <span>The Problem of Satisfying Constraints</span> </a> </li> <li class="section section-9"> <a href="/nks/p351--origins-of-simple-behavior/"> <span>9</span> <span>Origins of Simple Behavior</span> </a> </li> </ul> </li> <li class="chapter chapter-8"> <a href="/nks/chap-8--implications-for-everyday-systems/"> <span>8</span> <span>Implications for Everyday Systems</span> </a> <ul class="sections"> <li class="section section-1"> <a href="/nks/p363--issues-of-modelling/"> <span>1</span> <span>Issues of Modelling</span> </a> </li> <li class="section section-2"> <a href="/nks/p369--the-growth-of-crystals/"> <span>2</span> <span>The Growth of Crystals</span> </a> </li> <li class="section section-3"> <a href="/nks/p374--the-breaking-of-materials/"> <span>3</span> <span>The Breaking of Materials</span> </a> </li> <li class="section section-4"> <a href="/nks/p376--fluid-flow/"> <span>4</span> <span>Fluid Flow</span> </a> </li> <li class="section section-5"> <a href="/nks/p383--fundamental-issues-in-biology/"> <span>5</span> <span>Fundamental Issues in Biology</span> </a> </li> <li class="section section-6"> <a href="/nks/p400--growth-of-plants-and-animals/"> <span>6</span> <span>Growth of Plants and Animals</span> </a> </li> <li class="section section-7"> <a href="/nks/p422--biological-pigmentation-patterns/"> <span>7</span> <span>Biological Pigmentation Patterns</span> </a> </li> <li class="section section-8"> <a href="/nks/p429--financial-systems/"> <span>8</span> <span>Financial Systems</span> </a> </li> </ul> </li> <li class="chapter chapter-9"> <a href="/nks/chap-9--fundamental-physics/"> <span>9</span> <span>Fundamental Physics</span> </a> <ul class="sections"> <li class="section section-1"> <a href="/nks/p433--the-problems-of-physics/"> <span>1</span> <span>The Problems of Physics</span> </a> </li> <li class="section section-2"> <a href="/nks/p435--the-notion-of-reversibility/"> <span>2</span> <span>The Notion of Reversibility</span> </a> </li> <li class="section section-3"> <a href="/nks/p441--irreversibility-and-the-second-law-of-thermodynamics/"> <span>3</span> <span>Irreversibility and the Second Law of Thermodynamics</span> </a> </li> <li class="section section-4"> <a href="/nks/p458--conserved-quantities-and-continuum-phenomena/"> <span>4</span> <span>Conserved Quantities and Continuum Phenomena</span> </a> </li> <li class="section section-5"> <a href="/nks/p465--ultimate-models-for-the-universe/"> <span>5</span> <span>Ultimate Models for the Universe</span> </a> </li> <li class="section section-6"> <a href="/nks/p472--the-nature-of-space/"> <span>6</span> <span>The Nature of Space</span> </a> </li> <li class="section section-7"> <a href="/nks/p475--space-as-a-network/"> <span>7</span> <span>Space as a Network</span> </a> </li> <li class="section section-8"> <a href="/nks/p481--the-relationship-of-space-and-time/"> <span>8</span> <span>The Relationship of Space and Time</span> </a> </li> <li class="section section-9"> <a href="/nks/p486--time-and-causal-networks/"> <span>9</span> <span>Time and Causal Networks</span> </a> </li> <li class="section section-10"> <a href="/nks/p497--the-sequencing-of-events-in-the-universe/"> <span>10</span> <span>The Sequencing of Events in the Universe</span> </a> </li> <li class="section section-11"> <a href="/nks/p504--uniqueness-and-branching-in-time/"> <span>11</span> <span>Uniqueness and Branching in Time</span> </a> </li> <li class="section section-12"> <a href="/nks/p508--evolution-of-networks/"> <span>12</span> <span>Evolution of Networks</span> </a> </li> <li class="section section-13"> <a href="/nks/p516--space-time-and-relativity/"> <span>13</span> <span>Space, Time and Relativity</span> </a> </li> <li class="section section-14"> <a href="/nks/p525--elementary-particles/"> <span>14</span> <span>Elementary Particles</span> </a> </li> <li class="section section-15"> <a href="/nks/p530--the-phenomenon-of-gravity/"> <span>15</span> <span>The Phenomenon of Gravity</span> </a> </li> <li class="section section-16"> <a href="/nks/p537--quantum-phenomena/"> <span>16</span> <span>Quantum Phenomena</span> </a> </li> </ul> </li> <li class="chapter chapter-10"> <a href="/nks/chap-10--processes-of-perception-and-analysis/"> <span>10</span> <span>Processes of Perception and Analysis</span> </a> <ul class="sections"> <li class="section section-1"> <a href="/nks/p547--introduction/"> <span>1</span> <span>Introduction</span> </a> </li> <li class="section section-2"> <a href="/nks/p548--what-perception-and-analysis-do/"> <span>2</span> <span>What Perception and Analysis Do</span> </a> </li> <li class="section section-3"> <a href="/nks/p552--defining-the-notion-of-randomness/"> <span>3</span> <span>Defining the Notion of Randomness</span> </a> </li> <li class="section section-4"> <a href="/nks/p557--defining-complexity/"> <span>4</span> <span>Defining Complexity</span> </a> </li> <li class="section section-5"> <a href="/nks/p560--data-compression/"> <span>5</span> <span>Data Compression</span> </a> </li> <li class="section section-6"> <a href="/nks/p572--irreversible-data-compression/"> <span>6</span> <span>Irreversible Data Compression</span> </a> </li> <li class="section section-7"> <a href="/nks/p577--visual-perception/"> <span>7</span> <span>Visual Perception</span> </a> </li> <li class="section section-8"> <a href="/nks/p585--auditory-perception/"> <span>8</span> <span>Auditory Perception</span> </a> </li> <li class="section section-9"> <a href="/nks/p588--statistical-analysis/"> <span>9</span> <span>Statistical Analysis</span> </a> </li> <li class="section section-10"> <a href="/nks/p598--cryptography-and-cryptanalysis/"> <span>10</span> <span>Cryptography and Cryptanalysis</span> </a> </li> <li class="section section-11"> <a href="/nks/p606--traditional-mathematics-and-mathematical-formulas/"> <span>11</span> <span>Traditional Mathematics and Mathematical Formulas</span> </a> </li> <li class="section section-12"> <a href="/nks/p620--human-thinking/"> <span>12</span> <span>Human Thinking</span> </a> </li> <li class="section section-13"> <a href="/nks/p632--higher-forms-of-perception-and-analysis/"> <span>13</span> <span>Higher Forms of Perception and Analysis</span> </a> </li> </ul> </li> <li class="chapter chapter-11"> <a href="/nks/chap-11--the-notion-of-computation/"> <span>11</span> <span>The Notion of Computation</span> </a> <ul class="sections"> <li class="section section-1"> <a href="/nks/p637--computation-as-a-framework/"> <span>1</span> <span>Computation as a Framework</span> </a> </li> <li class="section section-2"> <a href="/nks/p638--computations-in-cellular-automata/"> <span>2</span> <span>Computations in Cellular Automata</span> </a> </li> <li class="section section-3"> <a href="/nks/p642--the-phenomenon-of-universality/"> <span>3</span> <span>The Phenomenon of Universality</span> </a> </li> <li class="section section-4"> <a href="/nks/p644--a-universal-cellular-automaton/"> <span>4</span> <span>A Universal Cellular Automaton</span> </a> </li> <li class="section section-5"> <a href="/nks/p656--emulating-other-systems-with-cellular-automata/"> <span>5</span> <span>Emulating Other Systems with Cellular Automata</span> </a> </li> <li class="section section-6"> <a href="/nks/p664--emulating-cellular-automata-with-other-systems/"> <span>6</span> <span>Emulating Cellular Automata with Other Systems</span> </a> </li> <li class="section section-7"> <a href="/nks/p674--implications-of-universality/"> <span>7</span> <span>Implications of Universality</span> </a> </li> <li class="section section-8"> <a href="/nks/p675--the-rule-110-cellular-automaton/"> <span>8</span> <span>The Rule 110 Cellular Automaton</span> </a> </li> <li class="section section-9"> <a href="/nks/p690--the-significance-of-universality-in-rule-110/"> <span>9</span> <span>The Significance of Universality in Rule 110</span> </a> </li> <li class="section section-10"> <a href="/nks/p691--class-4-behavior-and-universality/"> <span>10</span> <span>Class 4 Behavior and Universality</span> </a> </li> <li class="section section-11"> <a href="/nks/p694--the-threshold-of-universality-in-cellular-automata/"> <span>11</span> <span>The Threshold of Universality in Cellular Automata</span> </a> </li> <li class="section section-12"> <a href="/nks/p706--universality-in-turing-machines-and-other-systems/"> <span>12</span> <span>Universality in Turing Machines and Other Systems</span> </a> </li> </ul> </li> <li class="chapter chapter-12 selected"> <a href="/nks/chap-12--the-principle-of-computational-equivalence/"> <span>12</span> <span>The Principle of Computational Equivalence</span> </a> <ul class="sections"> <li class="section section-1"> <a href="/nks/p715--basic-framework/"> <span>1</span> <span>Basic Framework</span> </a> </li> <li class="section section-2"> <a href="/nks/p716--outline-of-the-principle/"> <span>2</span> <span>Outline of the Principle</span> </a> </li> <li class="section section-3"> <a href="/nks/p719--the-content-of-the-principle/"> <span>3</span> <span>The Content of the Principle</span> </a> </li> <li class="section section-4"> <a href="/nks/p726--the-validity-of-the-principle/"> <span>4</span> <span>The Validity of the Principle</span> </a> </li> <li class="section section-5"> <a href="/nks/p735--explaining-the-phenomenon-of-complexity/"> <span>5</span> <span>Explaining the Phenomenon of Complexity</span> </a> </li> <li class="section section-6"> <a href="/nks/p737--computational-irreducibility/"> <span>6</span> <span>Computational Irreducibility</span> </a> </li> <li class="section section-7"> <a href="/nks/p750--the-phenomenon-of-free-will/"> <span>7</span> <span>The Phenomenon of Free Will</span> </a> </li> <li class="section section-8"> <a href="/nks/p753--undecidability-and-intractability/"> <span>8</span> <span>Undecidability and Intractability</span> </a> </li> <li class="section section-9"> <a href="/nks/p772--implications-for-mathematics-and-its-foundations/"> <span>9</span> <span>Implications for Mathematics and Its Foundations</span> </a> </li> <li class="section section-10"> <a href="/nks/p822--intelligence-in-the-universe/"> <span>10</span> <span>Intelligence in the Universe</span> </a> </li> <li class="section section-11"> <a href="/nks/p840--implications-for-technology/"> <span>11</span> <span>Implications for Technology</span> </a> </li> <li class="section section-12"> <a href="/nks/p844--historical-perspectives/"> <span>12</span> <span>Historical Perspectives</span> </a> </li> </ul> </li> <li class="chapter notes"> <a href="/nks/notes-for-chapters/"> <span></span> <span>Notes</span> </a> </li> </ul> <div class="search"> </div> </div> </div> </nav> </div> </div> </div> <div id="search"> <div class="text-align-c__600"> <div class="inline-grid cols-3 cols-1__600"> <form id="search-index-form" action="/nks/index/"><div><label for="search-index-input">Look up in index</label><input type="search" id="search-index-input" name="search-index-input"></div></form> <form id="search-all-form" action="/nks/search/"><div><label for="search-all-input">Text search</label><input type="search" id="search-all-input" name="search-all-input"></div></form> <form id="search-jump-form"><div><label for="search-jump-input">Jump to page</label><input type="search" id="search-jump-input" name="search-jump-input"></div></form> </div> </div> </div> </header>  <main id="main"> <div class="thin-wrapper"> <div id="previous"><a href="/nks/chap-11--the-notion-of-computation/"><span>‹</span></a></div> <div id="next" style="display:none;"><a href="/"><span>›</span></a></div> <div id="content" class="thin-padding"> <div class="ribbon-wrapper top"> </div> <section class="chapter-title"> <h1 class="chapter-heading"><a href="/nks/p715--basic-framework/">Chapter 12: <span>The Principle of Computational Equivalence</span></a></h1> <hr> </section> <section class="chapter-content"> <div> <h2 id="sect-12-1--basic-framework" class="section-heading"><a href="/nks/p715--basic-framework/">Section 1: <span>Basic Framework</span></a><span class="section-anchor"><img src="/nks/img/icon-hyperlink.svg" alt="" width="20" height="20"><span class="clipboard-inline clipboard-sectionlink">https://www.wolframscience.com/nks/chap-12--the-principle-of-computational-equivalence#sect-12-1--basic-framework</span></span></h2> <div class="pages"> <a href="/nks/p715--basic-framework/"><img src="/nks/pages/p715--basic-framework/image-tiny.png" alt="Page 715" width="70" height="86"></a><a href="/nks/sect-12-1--basic-framework--notes/" class="notes-thumb"><img src="/nks/img/notes-thumb.png" alt="Notes" width="70" height="86"></a> </div> </div> <hr> <div> <h2 id="sect-12-2--outline-of-the-principle" class="section-heading"><a href="/nks/p716--outline-of-the-principle/">Section 2: <span>Outline of the Principle</span></a><span class="section-anchor"><img src="/nks/img/icon-hyperlink.svg" alt="" width="20" height="20"><span class="clipboard-inline clipboard-sectionlink">https://www.wolframscience.com/nks/chap-12--the-principle-of-computational-equivalence#sect-12-2--outline-of-the-principle</span></span></h2> <div class="pages"> <a href="/nks/p716--outline-of-the-principle/"><img src="/nks/pages/p716--outline-of-the-principle/image-tiny.png" alt="Page 716" width="70" height="86"></a> <a href="/nks/p717--outline-of-the-principle/"><img src="/nks/pages/p717--outline-of-the-principle/image-tiny.png" alt="Page 717" width="70" height="86"></a> <a href="/nks/p718--outline-of-the-principle/"><img src="/nks/pages/p718--outline-of-the-principle/image-tiny.png" alt="Page 718" width="70" height="86"></a><a href="/nks/sect-12-2--outline-of-the-principle--notes/" class="notes-thumb"><img src="/nks/img/notes-thumb.png" alt="Notes" width="70" height="86"></a> </div> </div> <hr> <div> <h2 id="sect-12-3--the-content-of-the-principle" class="section-heading"><a href="/nks/p719--the-content-of-the-principle/">Section 3: <span>The Content of the Principle</span></a><span class="section-anchor"><img src="/nks/img/icon-hyperlink.svg" alt="" width="20" height="20"><span class="clipboard-inline clipboard-sectionlink">https://www.wolframscience.com/nks/chap-12--the-principle-of-computational-equivalence#sect-12-3--the-content-of-the-principle</span></span></h2> <div class="pages"> <a href="/nks/p719--the-content-of-the-principle/"><img src="/nks/pages/p719--the-content-of-the-principle/image-tiny.png" alt="Page 719" width="70" height="86"></a> <a href="/nks/p720--the-content-of-the-principle/"><img src="/nks/pages/p720--the-content-of-the-principle/image-tiny.png" alt="Page 720" width="70" height="86"></a> <a href="/nks/p721--the-content-of-the-principle/"><img src="/nks/pages/p721--the-content-of-the-principle/image-tiny.png" alt="Page 721" width="70" height="86"></a> <a href="/nks/p722--the-content-of-the-principle/"><img src="/nks/pages/p722--the-content-of-the-principle/image-tiny.png" alt="Page 722" width="70" height="86"></a> <a href="/nks/p723--the-content-of-the-principle/"><img src="/nks/pages/p723--the-content-of-the-principle/image-tiny.png" alt="Page 723" width="70" height="86"></a> <a href="/nks/p724--the-content-of-the-principle/"><img src="/nks/pages/p724--the-content-of-the-principle/image-tiny.png" alt="Page 724" width="70" height="86"></a> <a href="/nks/p725--the-content-of-the-principle/"><img src="/nks/pages/p725--the-content-of-the-principle/image-tiny.png" alt="Page 725" width="70" height="86"></a><a href="/nks/sect-12-3--the-content-of-the-principle--notes/" class="notes-thumb"><img src="/nks/img/notes-thumb.png" alt="Notes" width="70" height="86"></a> </div> </div> <hr> <div> <h2 id="sect-12-4--the-validity-of-the-principle" class="section-heading"><a href="/nks/p726--the-validity-of-the-principle/">Section 4: <span>The Validity of the Principle</span></a><span class="section-anchor"><img src="/nks/img/icon-hyperlink.svg" alt="" width="20" height="20"><span class="clipboard-inline clipboard-sectionlink">https://www.wolframscience.com/nks/chap-12--the-principle-of-computational-equivalence#sect-12-4--the-validity-of-the-principle</span></span></h2> <div class="pages"> <a href="/nks/p726--the-validity-of-the-principle/"><img src="/nks/pages/p726--the-validity-of-the-principle/image-tiny.png" alt="Page 726" width="70" height="86"></a> <a href="/nks/p727--the-validity-of-the-principle/"><img src="/nks/pages/p727--the-validity-of-the-principle/image-tiny.png" alt="Page 727" width="70" height="86"></a> <a href="/nks/p728--the-validity-of-the-principle/"><img src="/nks/pages/p728--the-validity-of-the-principle/image-tiny.png" alt="Page 728" width="70" height="86"></a> <a href="/nks/p729--the-validity-of-the-principle/"><img src="/nks/pages/p729--the-validity-of-the-principle/image-tiny.png" alt="Page 729" width="70" height="86"></a> <a href="/nks/p730--the-validity-of-the-principle/"><img src="/nks/pages/p730--the-validity-of-the-principle/image-tiny.png" alt="Page 730" width="70" height="86"></a> <a href="/nks/p731--the-validity-of-the-principle/"><img src="/nks/pages/p731--the-validity-of-the-principle/image-tiny.png" alt="Page 731" width="70" height="86"></a> <a href="/nks/p732--the-validity-of-the-principle/"><img src="/nks/pages/p732--the-validity-of-the-principle/image-tiny.png" alt="Page 732" width="70" height="86"></a> <a href="/nks/p733--the-validity-of-the-principle/"><img src="/nks/pages/p733--the-validity-of-the-principle/image-tiny.png" alt="Page 733" width="70" height="86"></a> <a href="/nks/p734--the-validity-of-the-principle/"><img src="/nks/pages/p734--the-validity-of-the-principle/image-tiny.png" alt="Page 734" width="70" height="86"></a><a href="/nks/sect-12-4--the-validity-of-the-principle--notes/" class="notes-thumb"><img src="/nks/img/notes-thumb.png" alt="Notes" width="70" height="86"></a> </div> </div> <hr> <div> <h2 id="sect-12-5--explaining-the-phenomenon-of-complexity" class="section-heading"><a href="/nks/p735--explaining-the-phenomenon-of-complexity/">Section 5: <span>Explaining the Phenomenon of Complexity</span></a><span class="section-anchor"><img src="/nks/img/icon-hyperlink.svg" alt="" width="20" height="20"><span class="clipboard-inline clipboard-sectionlink">https://www.wolframscience.com/nks/chap-12--the-principle-of-computational-equivalence#sect-12-5--explaining-the-phenomenon-of-complexity</span></span></h2> <div class="pages"> <a href="/nks/p735--explaining-the-phenomenon-of-complexity/"><img src="/nks/pages/p735--explaining-the-phenomenon-of-complexity/image-tiny.png" alt="Page 735" width="70" height="86"></a> <a href="/nks/p736--explaining-the-phenomenon-of-complexity/"><img src="/nks/pages/p736--explaining-the-phenomenon-of-complexity/image-tiny.png" alt="Page 736" width="70" height="86"></a><a href="/nks/sect-12-5--explaining-the-phenomenon-of-complexity--notes/" class="notes-thumb"><img src="/nks/img/notes-thumb.png" alt="Notes" width="70" height="86"></a> </div> </div> <hr> <div> <h2 id="sect-12-6--computational-irreducibility" class="section-heading"><a href="/nks/p737--computational-irreducibility/">Section 6: <span>Computational Irreducibility</span></a><span class="section-anchor"><img src="/nks/img/icon-hyperlink.svg" alt="" width="20" height="20"><span class="clipboard-inline clipboard-sectionlink">https://www.wolframscience.com/nks/chap-12--the-principle-of-computational-equivalence#sect-12-6--computational-irreducibility</span></span></h2> <div class="pages"> <a href="/nks/p737--computational-irreducibility/"><img src="/nks/pages/p737--computational-irreducibility/image-tiny.png" alt="Page 737" width="70" height="86"></a> <a href="/nks/p738--computational-irreducibility/"><img src="/nks/pages/p738--computational-irreducibility/image-tiny.png" alt="Page 738" width="70" height="86"></a> <a href="/nks/p739--computational-irreducibility/"><img src="/nks/pages/p739--computational-irreducibility/image-tiny.png" alt="Page 739" width="70" height="86"></a> <a href="/nks/p740--computational-irreducibility/"><img src="/nks/pages/p740--computational-irreducibility/image-tiny.png" alt="Page 740" width="70" height="86"></a> <a href="/nks/p741--computational-irreducibility/"><img src="/nks/pages/p741--computational-irreducibility/image-tiny.png" alt="Page 741" width="70" height="86"></a> <a href="/nks/p742--computational-irreducibility/"><img src="/nks/pages/p742--computational-irreducibility/image-tiny.png" alt="Page 742" width="70" height="86"></a> <a href="/nks/p743--computational-irreducibility/"><img src="/nks/pages/p743--computational-irreducibility/image-tiny.png" alt="Page 743" width="70" height="86"></a> <a href="/nks/p744--computational-irreducibility/"><img src="/nks/pages/p744--computational-irreducibility/image-tiny.png" alt="Page 744" width="70" height="86"></a> <a href="/nks/p745--computational-irreducibility/"><img src="/nks/pages/p745--computational-irreducibility/image-tiny.png" alt="Page 745" width="70" height="86"></a> <a href="/nks/p746--computational-irreducibility/"><img src="/nks/pages/p746--computational-irreducibility/image-tiny.png" alt="Page 746" width="70" height="86"></a> <a href="/nks/p747--computational-irreducibility/"><img src="/nks/pages/p747--computational-irreducibility/image-tiny.png" alt="Page 747" width="70" height="86"></a> <a href="/nks/p748--computational-irreducibility/"><img src="/nks/pages/p748--computational-irreducibility/image-tiny.png" alt="Page 748" width="70" height="86"></a> <a href="/nks/p749--computational-irreducibility/"><img src="/nks/pages/p749--computational-irreducibility/image-tiny.png" alt="Page 749" width="70" height="86"></a><a href="/nks/sect-12-6--computational-irreducibility--notes/" class="notes-thumb"><img src="/nks/img/notes-thumb.png" alt="Notes" width="70" height="86"></a> </div> </div> <hr> <div> <h2 id="sect-12-7--the-phenomenon-of-free-will" class="section-heading"><a href="/nks/p750--the-phenomenon-of-free-will/">Section 7: <span>The Phenomenon of Free Will</span></a><span class="section-anchor"><img src="/nks/img/icon-hyperlink.svg" alt="" width="20" height="20"><span class="clipboard-inline clipboard-sectionlink">https://www.wolframscience.com/nks/chap-12--the-principle-of-computational-equivalence#sect-12-7--the-phenomenon-of-free-will</span></span></h2> <div class="pages"> <a href="/nks/p750--the-phenomenon-of-free-will/"><img src="/nks/pages/p750--the-phenomenon-of-free-will/image-tiny.png" alt="Page 750" width="70" height="86"></a> <a href="/nks/p751--the-phenomenon-of-free-will/"><img src="/nks/pages/p751--the-phenomenon-of-free-will/image-tiny.png" alt="Page 751" width="70" height="86"></a> <a href="/nks/p752--the-phenomenon-of-free-will/"><img src="/nks/pages/p752--the-phenomenon-of-free-will/image-tiny.png" alt="Page 752" width="70" height="86"></a><a href="/nks/sect-12-7--the-phenomenon-of-free-will--notes/" class="notes-thumb"><img src="/nks/img/notes-thumb.png" alt="Notes" width="70" height="86"></a> </div> </div> <hr> <div> <h2 id="sect-12-8--undecidability-and-intractability" class="section-heading"><a href="/nks/p753--undecidability-and-intractability/">Section 8: <span>Undecidability and Intractability</span></a><span class="section-anchor"><img src="/nks/img/icon-hyperlink.svg" alt="" width="20" height="20"><span class="clipboard-inline clipboard-sectionlink">https://www.wolframscience.com/nks/chap-12--the-principle-of-computational-equivalence#sect-12-8--undecidability-and-intractability</span></span></h2> <div class="pages"> <a href="/nks/p753--undecidability-and-intractability/"><img src="/nks/pages/p753--undecidability-and-intractability/image-tiny.png" alt="Page 753" width="70" height="86"></a> <a href="/nks/p754--undecidability-and-intractability/"><img src="/nks/pages/p754--undecidability-and-intractability/image-tiny.png" alt="Page 754" width="70" height="86"></a> <a href="/nks/p755--undecidability-and-intractability/"><img src="/nks/pages/p755--undecidability-and-intractability/image-tiny.png" alt="Page 755" width="70" height="86"></a> <a href="/nks/p756--undecidability-and-intractability/"><img src="/nks/pages/p756--undecidability-and-intractability/image-tiny.png" alt="Page 756" width="70" height="86"></a> <a href="/nks/p757--undecidability-and-intractability/"><img src="/nks/pages/p757--undecidability-and-intractability/image-tiny.png" alt="Page 757" width="70" height="86"></a> <a href="/nks/p758--undecidability-and-intractability/"><img src="/nks/pages/p758--undecidability-and-intractability/image-tiny.png" alt="Page 758" width="70" height="86"></a> <a href="/nks/p759--undecidability-and-intractability/"><img src="/nks/pages/p759--undecidability-and-intractability/image-tiny.png" alt="Page 759" width="70" height="86"></a> <a href="/nks/p760--undecidability-and-intractability/"><img src="/nks/pages/p760--undecidability-and-intractability/image-tiny.png" alt="Page 760" width="70" height="86"></a> <a href="/nks/p761--undecidability-and-intractability/"><img src="/nks/pages/p761--undecidability-and-intractability/image-tiny.png" alt="Page 761" width="70" height="86"></a> <a href="/nks/p762--undecidability-and-intractability/"><img src="/nks/pages/p762--undecidability-and-intractability/image-tiny.png" alt="Page 762" width="70" height="86"></a> <a href="/nks/p763--undecidability-and-intractability/"><img src="/nks/pages/p763--undecidability-and-intractability/image-tiny.png" alt="Page 763" width="70" height="86"></a> <a href="/nks/p764--undecidability-and-intractability/"><img src="/nks/pages/p764--undecidability-and-intractability/image-tiny.png" alt="Page 764" width="70" height="86"></a> <a href="/nks/p765--undecidability-and-intractability/"><img src="/nks/pages/p765--undecidability-and-intractability/image-tiny.png" alt="Page 765" width="70" height="86"></a> <a href="/nks/p766--undecidability-and-intractability/"><img src="/nks/pages/p766--undecidability-and-intractability/image-tiny.png" alt="Page 766" width="70" height="86"></a> <a href="/nks/p767--undecidability-and-intractability/"><img src="/nks/pages/p767--undecidability-and-intractability/image-tiny.png" alt="Page 767" width="70" height="86"></a> <a href="/nks/p768--undecidability-and-intractability/"><img src="/nks/pages/p768--undecidability-and-intractability/image-tiny.png" alt="Page 768" width="70" height="86"></a> <a href="/nks/p769--undecidability-and-intractability/"><img src="/nks/pages/p769--undecidability-and-intractability/image-tiny.png" alt="Page 769" width="70" height="86"></a> <a href="/nks/p770--undecidability-and-intractability/"><img src="/nks/pages/p770--undecidability-and-intractability/image-tiny.png" alt="Page 770" width="70" height="86"></a> <a href="/nks/p771--undecidability-and-intractability/"><img src="/nks/pages/p771--undecidability-and-intractability/image-tiny.png" alt="Page 771" width="70" height="86"></a><a href="/nks/sect-12-8--undecidability-and-intractability--notes/" class="notes-thumb"><img src="/nks/img/notes-thumb.png" alt="Notes" width="70" height="86"></a> </div> </div> <hr> <div> <h2 id="sect-12-9--implications-for-mathematics-and-its-foundations" class="section-heading"><a href="/nks/p772--implications-for-mathematics-and-its-foundations/">Section 9: <span>Implications for Mathematics and Its Foundations</span></a><span class="section-anchor"><img src="/nks/img/icon-hyperlink.svg" alt="" width="20" height="20"><span class="clipboard-inline clipboard-sectionlink">https://www.wolframscience.com/nks/chap-12--the-principle-of-computational-equivalence#sect-12-9--implications-for-mathematics-and-its-foundations</span></span></h2> <div class="pages"> <a href="/nks/p772--implications-for-mathematics-and-its-foundations/"><img src="/nks/pages/p772--implications-for-mathematics-and-its-foundations/image-tiny.png" alt="Page 772" width="70" height="86"></a> <a href="/nks/p773--implications-for-mathematics-and-its-foundations/"><img src="/nks/pages/p773--implications-for-mathematics-and-its-foundations/image-tiny.png" alt="Page 773" width="70" height="86"></a> <a href="/nks/p774--implications-for-mathematics-and-its-foundations/"><img src="/nks/pages/p774--implications-for-mathematics-and-its-foundations/image-tiny.png" alt="Page 774" width="70" height="86"></a> <a href="/nks/p775--implications-for-mathematics-and-its-foundations/"><img src="/nks/pages/p775--implications-for-mathematics-and-its-foundations/image-tiny.png" alt="Page 775" width="70" height="86"></a> <a href="/nks/p776--implications-for-mathematics-and-its-foundations/"><img src="/nks/pages/p776--implications-for-mathematics-and-its-foundations/image-tiny.png" alt="Page 776" width="70" height="86"></a> <a href="/nks/p777--implications-for-mathematics-and-its-foundations/"><img src="/nks/pages/p777--implications-for-mathematics-and-its-foundations/image-tiny.png" alt="Page 777" width="70" height="86"></a> <a href="/nks/p778--implications-for-mathematics-and-its-foundations/"><img src="/nks/pages/p778--implications-for-mathematics-and-its-foundations/image-tiny.png" alt="Page 778" width="70" height="86"></a> <a href="/nks/p779--implications-for-mathematics-and-its-foundations/"><img src="/nks/pages/p779--implications-for-mathematics-and-its-foundations/image-tiny.png" alt="Page 779" width="70" height="86"></a> <a href="/nks/p780--implications-for-mathematics-and-its-foundations/"><img src="/nks/pages/p780--implications-for-mathematics-and-its-foundations/image-tiny.png" alt="Page 780" width="70" height="86"></a> <a href="/nks/p781--implications-for-mathematics-and-its-foundations/"><img src="/nks/pages/p781--implications-for-mathematics-and-its-foundations/image-tiny.png" alt="Page 781" 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src="/nks/pages/p800--implications-for-mathematics-and-its-foundations/image-tiny.png" alt="Page 800" width="70" height="86"></a> <a href="/nks/p801--implications-for-mathematics-and-its-foundations/"><img src="/nks/pages/p801--implications-for-mathematics-and-its-foundations/image-tiny.png" alt="Page 801" width="70" height="86"></a> <a href="/nks/p802--implications-for-mathematics-and-its-foundations/"><img src="/nks/pages/p802--implications-for-mathematics-and-its-foundations/image-tiny.png" alt="Page 802" width="70" height="86"></a> <a href="/nks/p803--implications-for-mathematics-and-its-foundations/"><img src="/nks/pages/p803--implications-for-mathematics-and-its-foundations/image-tiny.png" alt="Page 803" width="70" height="86"></a> <a href="/nks/p804--implications-for-mathematics-and-its-foundations/"><img src="/nks/pages/p804--implications-for-mathematics-and-its-foundations/image-tiny.png" alt="Page 804" width="70" height="86"></a> <a href="/nks/p805--implications-for-mathematics-and-its-foundations/"><img src="/nks/pages/p805--implications-for-mathematics-and-its-foundations/image-tiny.png" alt="Page 805" width="70" height="86"></a> <a href="/nks/p806--implications-for-mathematics-and-its-foundations/"><img src="/nks/pages/p806--implications-for-mathematics-and-its-foundations/image-tiny.png" alt="Page 806" width="70" height="86"></a> <a href="/nks/p807--implications-for-mathematics-and-its-foundations/"><img src="/nks/pages/p807--implications-for-mathematics-and-its-foundations/image-tiny.png" alt="Page 807" width="70" height="86"></a> <a href="/nks/p808--implications-for-mathematics-and-its-foundations/"><img src="/nks/pages/p808--implications-for-mathematics-and-its-foundations/image-tiny.png" alt="Page 808" width="70" height="86"></a> <a href="/nks/p809--implications-for-mathematics-and-its-foundations/"><img src="/nks/pages/p809--implications-for-mathematics-and-its-foundations/image-tiny.png" alt="Page 809" width="70" height="86"></a> <a href="/nks/p810--implications-for-mathematics-and-its-foundations/"><img src="/nks/pages/p810--implications-for-mathematics-and-its-foundations/image-tiny.png" alt="Page 810" width="70" height="86"></a> <a href="/nks/p811--implications-for-mathematics-and-its-foundations/"><img src="/nks/pages/p811--implications-for-mathematics-and-its-foundations/image-tiny.png" alt="Page 811" width="70" height="86"></a> <a href="/nks/p812--implications-for-mathematics-and-its-foundations/"><img src="/nks/pages/p812--implications-for-mathematics-and-its-foundations/image-tiny.png" alt="Page 812" width="70" height="86"></a> <a href="/nks/p813--implications-for-mathematics-and-its-foundations/"><img src="/nks/pages/p813--implications-for-mathematics-and-its-foundations/image-tiny.png" alt="Page 813" width="70" height="86"></a> <a href="/nks/p814--implications-for-mathematics-and-its-foundations/"><img src="/nks/pages/p814--implications-for-mathematics-and-its-foundations/image-tiny.png" alt="Page 814" width="70" height="86"></a> <a href="/nks/p815--implications-for-mathematics-and-its-foundations/"><img src="/nks/pages/p815--implications-for-mathematics-and-its-foundations/image-tiny.png" alt="Page 815" width="70" height="86"></a> <a href="/nks/p816--implications-for-mathematics-and-its-foundations/"><img src="/nks/pages/p816--implications-for-mathematics-and-its-foundations/image-tiny.png" alt="Page 816" width="70" height="86"></a> <a href="/nks/p817--implications-for-mathematics-and-its-foundations/"><img src="/nks/pages/p817--implications-for-mathematics-and-its-foundations/image-tiny.png" alt="Page 817" width="70" height="86"></a> <a href="/nks/p818--implications-for-mathematics-and-its-foundations/"><img src="/nks/pages/p818--implications-for-mathematics-and-its-foundations/image-tiny.png" alt="Page 818" width="70" height="86"></a> <a href="/nks/p819--implications-for-mathematics-and-its-foundations/"><img src="/nks/pages/p819--implications-for-mathematics-and-its-foundations/image-tiny.png" alt="Page 819" width="70" height="86"></a> <a href="/nks/p820--implications-for-mathematics-and-its-foundations/"><img src="/nks/pages/p820--implications-for-mathematics-and-its-foundations/image-tiny.png" alt="Page 820" width="70" height="86"></a> <a href="/nks/p821--implications-for-mathematics-and-its-foundations/"><img src="/nks/pages/p821--implications-for-mathematics-and-its-foundations/image-tiny.png" alt="Page 821" width="70" height="86"></a><a href="/nks/sect-12-9--implications-for-mathematics-and-its-foundations--notes/" class="notes-thumb"><img src="/nks/img/notes-thumb.png" alt="Notes" width="70" height="86"></a> </div> </div> <hr> <div> <h2 id="sect-12-10--intelligence-in-the-universe" class="section-heading"><a href="/nks/p822--intelligence-in-the-universe/">Section 10: <span>Intelligence in the Universe</span></a><span class="section-anchor"><img src="/nks/img/icon-hyperlink.svg" alt="" width="20" height="20"><span class="clipboard-inline clipboard-sectionlink">https://www.wolframscience.com/nks/chap-12--the-principle-of-computational-equivalence#sect-12-10--intelligence-in-the-universe</span></span></h2> <div class="pages"> <a href="/nks/p822--intelligence-in-the-universe/"><img src="/nks/pages/p822--intelligence-in-the-universe/image-tiny.png" alt="Page 822" width="70" height="86"></a> <a href="/nks/p823--intelligence-in-the-universe/"><img src="/nks/pages/p823--intelligence-in-the-universe/image-tiny.png" alt="Page 823" width="70" height="86"></a> <a href="/nks/p824--intelligence-in-the-universe/"><img src="/nks/pages/p824--intelligence-in-the-universe/image-tiny.png" alt="Page 824" width="70" height="86"></a> <a href="/nks/p825--intelligence-in-the-universe/"><img src="/nks/pages/p825--intelligence-in-the-universe/image-tiny.png" alt="Page 825" width="70" height="86"></a> <a href="/nks/p826--intelligence-in-the-universe/"><img src="/nks/pages/p826--intelligence-in-the-universe/image-tiny.png" alt="Page 826" width="70" height="86"></a> <a href="/nks/p827--intelligence-in-the-universe/"><img src="/nks/pages/p827--intelligence-in-the-universe/image-tiny.png" alt="Page 827" width="70" height="86"></a> <a href="/nks/p828--intelligence-in-the-universe/"><img src="/nks/pages/p828--intelligence-in-the-universe/image-tiny.png" alt="Page 828" width="70" height="86"></a> <a href="/nks/p829--intelligence-in-the-universe/"><img src="/nks/pages/p829--intelligence-in-the-universe/image-tiny.png" alt="Page 829" width="70" height="86"></a> <a href="/nks/p830--intelligence-in-the-universe/"><img src="/nks/pages/p830--intelligence-in-the-universe/image-tiny.png" alt="Page 830" width="70" height="86"></a> <a href="/nks/p831--intelligence-in-the-universe/"><img src="/nks/pages/p831--intelligence-in-the-universe/image-tiny.png" alt="Page 831" width="70" height="86"></a> <a href="/nks/p832--intelligence-in-the-universe/"><img src="/nks/pages/p832--intelligence-in-the-universe/image-tiny.png" alt="Page 832" width="70" height="86"></a> <a href="/nks/p833--intelligence-in-the-universe/"><img src="/nks/pages/p833--intelligence-in-the-universe/image-tiny.png" alt="Page 833" width="70" height="86"></a> <a href="/nks/p834--intelligence-in-the-universe/"><img src="/nks/pages/p834--intelligence-in-the-universe/image-tiny.png" alt="Page 834" width="70" height="86"></a> <a href="/nks/p835--intelligence-in-the-universe/"><img src="/nks/pages/p835--intelligence-in-the-universe/image-tiny.png" alt="Page 835" width="70" height="86"></a> <a href="/nks/p836--intelligence-in-the-universe/"><img src="/nks/pages/p836--intelligence-in-the-universe/image-tiny.png" alt="Page 836" width="70" height="86"></a> <a href="/nks/p837--intelligence-in-the-universe/"><img src="/nks/pages/p837--intelligence-in-the-universe/image-tiny.png" alt="Page 837" width="70" height="86"></a> <a href="/nks/p838--intelligence-in-the-universe/"><img src="/nks/pages/p838--intelligence-in-the-universe/image-tiny.png" alt="Page 838" width="70" height="86"></a> <a href="/nks/p839--intelligence-in-the-universe/"><img src="/nks/pages/p839--intelligence-in-the-universe/image-tiny.png" alt="Page 839" width="70" height="86"></a><a href="/nks/sect-12-10--intelligence-in-the-universe--notes/" class="notes-thumb"><img src="/nks/img/notes-thumb.png" alt="Notes" width="70" height="86"></a> </div> </div> <hr> <div> <h2 id="sect-12-11--implications-for-technology" class="section-heading"><a href="/nks/p840--implications-for-technology/">Section 11: <span>Implications for Technology</span></a><span class="section-anchor"><img src="/nks/img/icon-hyperlink.svg" alt="" width="20" height="20"><span class="clipboard-inline clipboard-sectionlink">https://www.wolframscience.com/nks/chap-12--the-principle-of-computational-equivalence#sect-12-11--implications-for-technology</span></span></h2> <div class="pages"> <a href="/nks/p840--implications-for-technology/"><img src="/nks/pages/p840--implications-for-technology/image-tiny.png" alt="Page 840" width="70" height="86"></a> <a href="/nks/p841--implications-for-technology/"><img src="/nks/pages/p841--implications-for-technology/image-tiny.png" alt="Page 841" width="70" height="86"></a> <a href="/nks/p842--implications-for-technology/"><img src="/nks/pages/p842--implications-for-technology/image-tiny.png" alt="Page 842" width="70" height="86"></a> <a href="/nks/p843--implications-for-technology/"><img src="/nks/pages/p843--implications-for-technology/image-tiny.png" alt="Page 843" width="70" height="86"></a><a href="/nks/sect-12-11--implications-for-technology--notes/" class="notes-thumb"><img src="/nks/img/notes-thumb.png" alt="Notes" width="70" height="86"></a> </div> </div> <hr> <div> <h2 id="sect-12-12--historical-perspectives" class="section-heading"><a href="/nks/p844--historical-perspectives/">Section 12: <span>Historical Perspectives</span></a><span class="section-anchor"><img src="/nks/img/icon-hyperlink.svg" alt="" width="20" height="20"><span class="clipboard-inline clipboard-sectionlink">https://www.wolframscience.com/nks/chap-12--the-principle-of-computational-equivalence#sect-12-12--historical-perspectives</span></span></h2> <div class="pages"> <a href="/nks/p844--historical-perspectives/"><img src="/nks/pages/p844--historical-perspectives/image-tiny.png" alt="Page 844" width="70" height="86"></a> <a href="/nks/p845--historical-perspectives/"><img src="/nks/pages/p845--historical-perspectives/image-tiny.png" alt="Page 845" width="70" height="86"></a> <a href="/nks/p846--historical-perspectives/"><img src="/nks/pages/p846--historical-perspectives/image-tiny.png" alt="Page 846" width="70" height="86"></a><a href="/nks/sect-12-12--historical-perspectives--notes/" class="notes-thumb"><img src="/nks/img/notes-thumb.png" alt="Notes" width="70" height="86"></a> </div> </div> <hr> </section> <section 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Wolfram uses his approach to tackle a remarkable array of fundamental problems in science, from the origins of apparent randomness in physical systems, to the development of complexity in biology, the ultimate scope and limitations of mathematics, the possibility of a truly fundamental theory of physics, the interplay between free will and determinism, and the character of intelligence in the universe.", "author": { "@type": "Person", "name": "Stephen Wolfram" }, "publisher": { "@type": "Corporation", "name": "Wolfram Media, Inc.", "url": "http://www.wolfram-media.com/" }, "isbn": "1-57955-008-8", "datePublished": "2002", "numberOfPages": "1197", "bookFormat": "print, online, iPad", "genre": "Science, Cellular Automata, Computation, Biology, Physics", "audience": "Scientists, Researchers, Science Educators, Biologists, Physicists, Mathematicians, Programmers", "discussionUrl": "http://community.wolfram.com/content?curTag=wolfram%20science", "inLanguage": "en" } </script>   </div> <div 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The Principle of Computational Equivalence: A New Kind of Science | Online by Stephen Wolfram