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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Abstract structure</span></span>
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</style><div role="note" class="hatnote navigation-not-searchable">This article is about a type of abstraction in science, mathematics, and philosophy. For the structure of an abstract (summary), see <a href="Abstract_(summary)#Structure" title="Abstract (summary)">Abstract (summary) ยง Structure</a>.</div><p>In <a href="Mathematics" title="Mathematics">mathematics</a> and related fields, an <b>abstract structure</b> is a way of describing a set of mathematical objects and the relationships between them, focusing on the essential rules and properties rather than any specific meaning or example.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup>
</p><p>For example, in a game such as <a href="Chess" title="Chess">chess</a>, the rules of how the pieces move and interact define the structure of the game, regardless of whether the pieces are made of wood or plastic. Similarly, an abstract structure defines a framework of objects, operations, and relationships. These structures are studied in their own right, revealing fundamental mathematical principles. While a real-world object or computer program might <i>represent, instantiate,</i> or <i>implement</i> an abstract structure, the structure itself exists as an abstract concept, independent of any particular representation.
</p><p>This abstraction allows to see common patterns across seemingly different areas of mathematics and to apply the same reasoning and tools to analyze them. Abstract structures are studied not only in <a href="Logic" title="Logic">logic</a> and <a href="Mathematics" title="Mathematics">mathematics</a> but in the fields that apply them, as <a href="Computer_science" title="Computer science">computer science</a> and <a href="Computer_graphics" title="Computer graphics">computer graphics</a>, and in the studies that reflect on them, such as <a href="Philosophy" title="Philosophy">philosophy</a> (especially the <a href="Philosophy_of_mathematics" title="Philosophy of mathematics">philosophy of mathematics</a>).
</p><p>An abstract structure has a richer structure than a <a href="Concept" title="Concept">concept</a> or an <a href="Idea" title="Idea">idea</a>. An abstract structure must include precise rules of behaviour which can be used to determine whether a candidate implementation actually matches the abstract structure in question, and it must be free from <a href="Contradiction" title="Contradiction">contradictions</a>. Thus we may debate how well a particular government fits the concept of <a href="Democracy" title="Democracy">democracy</a>, but there is no room for debate over whether a given sequence of moves is or is not a valid game of chess (for example <a href="Kasparov" class="mw-redirect" title="Kasparov">Kasparovian</a> approaches).
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<div class="mw-heading mw-heading2"><h2 id="Examples">Examples</h2></div>
<ul><li>A <a href="Sorting_algorithm" title="Sorting algorithm">sorting algorithm</a> is an abstract structure, but a <a href="Recipe" title="Recipe">recipe</a> is not, because it depends on the properties and quantities of its ingredients.</li>
<li>A simple <a href="Melody" title="Melody">melody</a> is an abstract structure, but an <a href="Orchestration" title="Orchestration">orchestration</a> is not, because it depends on the properties of particular instruments.</li>
<li><a href="Euclidean_geometry" title="Euclidean geometry">Euclidean geometry</a> is an abstract structure, but the theory of <a href="Continental_drift" title="Continental drift">continental drift</a> is not, because it depends on the geology of the <a href="Earth" title="Earth">Earth</a>.</li>
<li>A <a href="Formal_language" title="Formal language">formal language</a> is an abstract structure, but a <a href="Natural_language" title="Natural language">natural language</a> is not, because its rules of grammar and syntax are open to debate and interpretation.</li></ul>
<div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2></div>
<div class="mw-references-wrap"><ol class="references">
<li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">
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</style><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://news.cnrs.fr/articles/bourbaki-and-the-foundations-of-modern-mathematics">"Bourbaki and the Foundations of Modern Mathematics"</a>. <i>CNRS News</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2025-01-26</span></span>.</cite></span>
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<li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><cite id="CITEREFMac_Lane2010" class="citation book cs1">Mac Lane, Saunders (2010). <i>Categories for the working mathematician</i>. Graduate texts in mathematics (2nd. ed., Softcover version of original hardcover edition 1998 ed.). New York, NY: Springer. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1-4419-3123-8</bdi>.</cite></span>
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<div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2></div>
<ul><li><a href="Abstraction_(computer_science)" title="Abstraction (computer science)">Abstraction in computer science</a></li>
<li><a href="Abstraction" title="Abstraction">Abstraction in general</a></li>
<li><a href="Abstraction_(mathematics)" title="Abstraction (mathematics)">Abstraction in mathematics</a></li>
<li><a href="Abstract_object" class="mw-redirect" title="Abstract object">Abstract object</a></li>
<li><a href="Deductive_apparatus" class="mw-redirect" title="Deductive apparatus">Deductive apparatus</a></li>
<li><a href="Formal_science" title="Formal science">Formal sciences</a></li>
<li><a href="Mathematical_structure" title="Mathematical structure">Mathematical structure</a></li></ul></div><!--htdig_noindex--><div><div class="zim-footer">
This article is issued from <a class="external text" title="Last edited on 2025-01-26" href="https://en.wikipedia.org/wiki/?title=Abstract_structure&oldid=1272034426">Wikipedia</a>. The text is available under <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en">Creative Commons Attribution-Share Alike 4.0</a> unless otherwise noted. Additional terms may apply for the media files.
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