https://en.formulasearchengine.com/index.php?title=Coercive_function&feed=atom&action=historyCoercive function - Revision history2024-03-28T16:56:53ZRevision history for this page on the wikiMediaWiki 1.42.0-wmf.5https://en.formulasearchengine.com/index.php?title=Coercive_function&diff=252506&oldid=preven>Yobot: WPCleaner v1.34b - Fixed using WP:WCW (Internal link written as an external link)2014-12-06T07:58:31Z<p><a href="/index.php?title=En:WP:CLEANER&action=edit&redlink=1" class="new" title="En:WP:CLEANER (page does not exist)">WPCleaner</a> v1.34b - Fixed using <a href="/index.php?title=WP:WCW&action=edit&redlink=1" class="new" title="WP:WCW (page does not exist)">WP:WCW</a> (Internal link written as an external link)</p>
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<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">{{Italic title}}</del></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">Hello</ins>. <ins style="font-weight: bold; text-decoration: none;">Allow me introduce </ins>the <ins style="font-weight: bold; text-decoration: none;">writer</ins>. <ins style="font-weight: bold; text-decoration: none;">Her title </ins>is <ins style="font-weight: bold; text-decoration: none;">Refugia Shryock</ins>. <ins style="font-weight: bold; text-decoration: none;">Hiring has been my occupation for </ins>some <ins style="font-weight: bold; text-decoration: none;">time </ins>but <ins style="font-weight: bold; text-decoration: none;">I</ins>'<ins style="font-weight: bold; text-decoration: none;">ve already applied </ins>for <ins style="font-weight: bold; text-decoration: none;">another 1</ins>. <ins style="font-weight: bold; text-decoration: none;">California </ins>is our <ins style="font-weight: bold; text-decoration: none;">birth location</ins>. <ins style="font-weight: bold; text-decoration: none;">Body developing is what my family and I appreciate</ins>.<<ins style="font-weight: bold; text-decoration: none;">br</ins>><<ins style="font-weight: bold; text-decoration: none;">br</ins>><ins style="font-weight: bold; text-decoration: none;">Feel free </ins>to <ins style="font-weight: bold; text-decoration: none;">visit my blog post </ins>- [http://www.<ins style="font-weight: bold; text-decoration: none;">xn</ins>--<ins style="font-weight: bold; text-decoration: none;">80aeerfhjjg1bv8c</ins>.<ins style="font-weight: bold; text-decoration: none;">xn--p1ai</ins>/<ins style="font-weight: bold; text-decoration: none;">%D0%A1%D0%BE%D0%B4%D0%B5%D1%80%D0%B6%D0%B0%D0%BD%D0%B8%D0%B5</ins>/<ins style="font-weight: bold; text-decoration: none;">cures-yeast</ins>-<ins style="font-weight: bold; text-decoration: none;">infection</ins>-<ins style="font-weight: bold; text-decoration: none;">tips</ins>-<ins style="font-weight: bold; text-decoration: none;">use</ins>-<ins style="font-weight: bold; text-decoration: none;">now std testing at home</ins>]</div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">'''''Die Grundlagen der Arithmetik''' (The Foundations of Arithmetic)'' is a book by [[Gottlob Frege]], published in 1884, which investigates the philosophical foundations of arithmetic</del>. <del style="font-weight: bold; text-decoration: none;">In a tour de force of literary and philosophical merit, Frege demolished other theories of number and developed his own theory of numbers. The ''Grundlagen'' also helped to motivate Frege's later works in [[logicism]].</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">The book was not well received and was not read widely when it was published. It did, however, draw </del>the <del style="font-weight: bold; text-decoration: none;">attentions of [[Bertrand Russell]] and [[Ludwig Wittgenstein]], who were both heavily influenced by Frege's philosophy</del>.</div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">==Criticisms of predecessors==</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">===Psychologistic accounts of mathematics===</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">Frege objects to any account of mathematics based on psychologism, that is the view that math and numbers are relative to the subjective thoughts of the people who think of them. According to Frege, psychological accounts appeal to what is subjective, while mathematics is purely objective: mathematics are completely independent from human thought. Mathematical entities, according to Frege, have objective properties regardless of humans thinking of them: it </del>is <del style="font-weight: bold; text-decoration: none;">not possible to think of mathematical statements as something which evolved naturally through human history and evolution</del>. <del style="font-weight: bold; text-decoration: none;">He sees a fundamental distinction between logic (and its extension, according to Frege, math) and psychology. Logic explains necessary facts the order of ideas, whereas psychology studies certain thought processes in individual minds.</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">===Kant===</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">Frege greatly appreciates the work of [[Immanuel Kant]]. He criticizes him mainly on the grounds that numerical statements are not synthetic-a priori, but rather analytic-a priori.</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">Kant claims that 7+5=12 is a synthetic statement. No matter how much we analyze the idea of 7+5 we will not find there the idea of 12. We must arrive at the idea of 12 by application to objects in the intuition. Kant points out that this becomes all the more clear with bigger numbers. Frege, on this point precisely, argues towards the opposite direction. Kant wrongly assumes that in a proposition containing "big" numbers we must count points or </del>some <del style="font-weight: bold; text-decoration: none;">such thing to assert their [[truth value]]. Frege argues that without ever having any intuition toward any of the numbers in the following equation: 654,768+436,382=1,091,150 we nevertheless can assert it is true. This is provided as evidence that such a proposition is analytic. While Frege agrees that geometry is indeed synthetic a priori, arithmetic must be analytic.</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">==Development of Frege's own view of a number==</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">Frege makes a distinction between particular numerical statements such as 1+1=2, and general statements such as a+b=b+a. The latter are statements true of numbers just as well as the former. Therefore it is necessary to ask for a definition of the concept of number itself. </del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">Frege investigates the possibility that number is determined in external things. He demonstrates how numbers function in natural language just as adjectives. "This desk has 5 drawers" is similar in form to "This desk has green drawers". The drawers being green is an objective fact, grounded in the external world. But this is not the case with 5. Frege argues that each drawer is on its own green, </del>but <del style="font-weight: bold; text-decoration: none;">not every drawer is 5. </del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">Frege urges us to remember that from this it does not follow that numbers may be subjective. Indeed, numbers are similar to colors at least in that both are wholly objective. </del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">Frege tells us that we can convert number statements where number words appear adjectivally (e.g., 'there are four horses') into statements where number terms appear as singular terms ('the number of horses is four</del>'<del style="font-weight: bold; text-decoration: none;">). Frege recommends such translations because he takes numbers to be objects. It makes no sense to ask whether any objects fall under 4. After Frege gives some reasons </del>for <del style="font-weight: bold; text-decoration: none;">thinking that numbers are objects, he concludes that statements of numbers are assertions about concepts</del>. </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">Frege takes this observation to be the fundamental thought of ''Grundlagen''. For example, the sentence "the number of horses in the barn </del>is <del style="font-weight: bold; text-decoration: none;">four" means that four objects fall under the concept ''horse in the barn''. Frege attempts to explain </del>our <del style="font-weight: bold; text-decoration: none;">grasp of numbers through a contextual definition of the cardinality operation ('the number of</del>..<del style="font-weight: bold; text-decoration: none;">.', or </del><<del style="font-weight: bold; text-decoration: none;">math</del>> <del style="font-weight: bold; text-decoration: none;">Nx: Fx </del><<del style="font-weight: bold; text-decoration: none;">/math</del>><del style="font-weight: bold; text-decoration: none;">). He attempts to construct the content of a judgment involving numerical identity by relying on Hume's principle (which states that the number of Fs equals the number of Gs if and only if F and G are equinumerous, i.e. in one-one correspondence). He rejects this definition because it doesn't fix the truth value of identity statements when a singular term not of the form 'the number of Fs' flanks the identity sign. Frege goes on </del>to <del style="font-weight: bold; text-decoration: none;">give an explicit definition of number in terms of extensions of concepts, but expresses some hesitation.</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">===Frege's definition of a number===</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">Frege argues that numbers are objects and assert something about a concept. Frege defines numbers as extensions of concepts. 'The number of F's' is defined as the extension of the concept ''G is a concept that is equinumerous to F''. The concept in question leads to an equivalence class of all concepts that have the number of F (including F). Frege defines 0 as the extension of the concept ''being non self</del>-<del style="font-weight: bold; text-decoration: none;">identical''. So, the number of this concept is the extension of the concept of all concepts that have no objects falling under them.</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">== Legacy ==</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">The book was fundamental in the development of two main disciplines, the foundations of mathematics and philosophy. Although Bertrand Russell later found an apparent flaw in Frege's work (this apparent flaw is known as [[Russell's paradox]], which is resolved by [[axiomatic set theory]]), the book was influential in subsequent developments, such as ''[[Principia Mathematica]]''. The book can also be considered the starting point in analytic philosophy, since it revolves mainly around the analysis of language, with the goal of clarifying the concept of number. Frege's views on mathematics are also a starting point on the philosophy of mathematics, since it introduces an innovative account on the epistemology of numbers and math in general, known as logicism.</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">==Notes==</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">{{Reflist}}</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">==External links==</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">*</del>[http://www.<del style="font-weight: bold; text-decoration: none;">ac</del>-<del style="font-weight: bold; text-decoration: none;">nancy</del>-<del style="font-weight: bold; text-decoration: none;">metz</del>.<del style="font-weight: bold; text-decoration: none;">fr</del>/<del style="font-weight: bold; text-decoration: none;">enseign</del>/<del style="font-weight: bold; text-decoration: none;">philo/textesph/Frege.pdf Original German text.] Maintained by Alain Blachair, Académie de Nancy</del>-<del style="font-weight: bold; text-decoration: none;">Metz. </del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">*[[Stanford Encyclopedia of Philosophy]]: "[http://plato.stanford.edu/entries/frege</del>-<del style="font-weight: bold; text-decoration: none;">logic/ Frege's Logic, Theorem, and Foundations for Arithmetic,]" </del>-- <del style="font-weight: bold; text-decoration: none;">by [[Edward Zalta]].</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">*{{SpringerEOM| title=Number | id=Number | oldid=11869 | first=V.I. | last=Nechaev }}</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">{{DEFAULTSORT:Foundations Of Arithmetic, The}}</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">[[Category:1884 books]]</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">[[Category:Books by Gottlob Frege]]</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">[[Category:Logic books]]</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
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</table>en>Yobothttps://en.formulasearchengine.com/index.php?title=Coercive_function&diff=16073&oldid=preven>John of Reading: Typo/general fixing, replaced: necesarily → necessarily using AWB2013-12-21T17:46:15Z<p>Typo/<a href="/index.php?title=WP:AWB/GF&action=edit&redlink=1" class="new" title="WP:AWB/GF (page does not exist)">general</a> fixing, replaced: necesarily → necessarily using <a href="/index.php?title=Testwiki:AWB&action=edit&redlink=1" class="new" title="Testwiki:AWB (page does not exist)">AWB</a></p>
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<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">Alyson Meagher </del>is the <del style="font-weight: bold; text-decoration: none;">title her parents gave her but she doesn</del>'<del style="font-weight: bold; text-decoration: none;">t like </del>when people <del style="font-weight: bold; text-decoration: none;">use her complete title</del>. <del style="font-weight: bold; text-decoration: none;">Alaska </del>is <del style="font-weight: bold; text-decoration: none;">where he</del>'s <del style="font-weight: bold; text-decoration: none;">usually been residing</del>. <del style="font-weight: bold; text-decoration: none;">Distributing production has been his profession </del>for <del style="font-weight: bold; text-decoration: none;">some time</del>. <del style="font-weight: bold; text-decoration: none;">What me and my family members love </del>is to <del style="font-weight: bold; text-decoration: none;">climb </del>but <del style="font-weight: bold; text-decoration: none;">I</del>'<del style="font-weight: bold; text-decoration: none;">m </del>thinking <del style="font-weight: bold; text-decoration: none;">on starting something new</del>.<<del style="font-weight: bold; text-decoration: none;">br</del>><<del style="font-weight: bold; text-decoration: none;">br</del>><del style="font-weight: bold; text-decoration: none;">my homepage </del>... [http://www.<del style="font-weight: bold; text-decoration: none;">khuplaza</del>.<del style="font-weight: bold; text-decoration: none;">com</del>/<del style="font-weight: bold; text-decoration: none;">dent</del>/<del style="font-weight: bold; text-decoration: none;">14869889 tarot card readings</del>]</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">{{Italic title}}</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">'''''Die Grundlagen der Arithmetik''' (The Foundations of Arithmetic)'' </ins>is <ins style="font-weight: bold; text-decoration: none;">a book by [[Gottlob Frege]], published in 1884, which investigates </ins>the <ins style="font-weight: bold; text-decoration: none;">philosophical foundations of arithmetic. In a tour de force of literary and philosophical merit, Frege demolished other theories of number and developed his own theory of numbers. The ''Grundlagen'' also helped to motivate Frege</ins>'<ins style="font-weight: bold; text-decoration: none;">s later works in [[logicism]].</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">The book was not well received and was not read widely </ins>when <ins style="font-weight: bold; text-decoration: none;">it was published. It did, however, draw the attentions of [[Bertrand Russell]] and [[Ludwig Wittgenstein]], who were both heavily influenced by Frege's philosophy.</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">==Criticisms of predecessors==</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">===Psychologistic accounts of mathematics===</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">Frege objects to any account of mathematics based on psychologism, that is the view that math and numbers are relative to the subjective thoughts of the </ins>people <ins style="font-weight: bold; text-decoration: none;">who think of them. According to Frege, psychological accounts appeal to what is subjective, while mathematics is purely objective: mathematics are completely independent from human thought. Mathematical entities, according to Frege, have objective properties regardless of humans thinking of them: it is not possible to think of mathematical statements as something which evolved naturally through human history and evolution. He sees a fundamental distinction between logic (and its extension, according to Frege, math) and psychology. Logic explains necessary facts the order of ideas, whereas psychology studies certain thought processes in individual minds.</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">===Kant===</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">Frege greatly appreciates the work of [[Immanuel Kant]]. He criticizes him mainly on the grounds that numerical statements are not synthetic-a priori, but rather analytic-a priori.</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">Kant claims that 7+5=12 is a synthetic statement. No matter how much we analyze the idea of 7+5 we will not find there the idea of 12. We must arrive at the idea of 12 by application to objects in the intuition. Kant points out that this becomes all the more clear with bigger numbers. Frege, on this point precisely, argues towards the opposite direction. Kant wrongly assumes that in a proposition containing "big" numbers we must count points or some such thing to assert their [[truth value]]. Frege argues that without ever having any intuition toward any of the numbers in the following equation: 654,768+436,382=1,091,150 we nevertheless can assert it is true. This is provided as evidence that such a proposition is analytic</ins>. <ins style="font-weight: bold; text-decoration: none;">While Frege agrees that geometry </ins>is <ins style="font-weight: bold; text-decoration: none;">indeed synthetic a priori, arithmetic must be analytic.</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">==Development of Frege</ins>'s <ins style="font-weight: bold; text-decoration: none;">own view of a number==</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">Frege makes a distinction between particular numerical statements such as 1+1=2, and general statements such as a+b=b+a</ins>. <ins style="font-weight: bold; text-decoration: none;">The latter are statements true of numbers just as well as the former. Therefore it is necessary to ask </ins>for <ins style="font-weight: bold; text-decoration: none;">a definition of the concept of number itself. </ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">Frege investigates the possibility that number is determined in external things. He demonstrates how numbers function in natural language just as adjectives</ins>. <ins style="font-weight: bold; text-decoration: none;">"This desk has 5 drawers" </ins>is <ins style="font-weight: bold; text-decoration: none;">similar in form </ins>to <ins style="font-weight: bold; text-decoration: none;">"This desk has green drawers". The drawers being green is an objective fact, grounded in the external world. But this is not the case with 5. Frege argues that each drawer is on its own green, </ins>but <ins style="font-weight: bold; text-decoration: none;">not every drawer is 5. </ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">Frege urges us to remember that from this it does not follow that numbers may be subjective. Indeed, numbers are similar to colors at least in that both are wholly objective. </ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">Frege tells us that we can convert number statements where number words appear adjectivally (e.g., 'there are four horses') into statements where number terms appear as singular terms ('the number of horses is four</ins>'<ins style="font-weight: bold; text-decoration: none;">). Frege recommends such translations because he takes numbers to be objects. It makes no sense to ask whether any objects fall under 4. After Frege gives some reasons for </ins>thinking <ins style="font-weight: bold; text-decoration: none;">that numbers are objects, he concludes that statements of numbers are assertions about concepts. </ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">Frege takes this observation to be the fundamental thought of ''Grundlagen''. For example, the sentence "the number of horses in the barn is four" means that four objects fall under the concept ''horse in the barn''. Frege attempts to explain our grasp of numbers through a contextual definition of the cardinality operation ('the number of..</ins>.<ins style="font-weight: bold; text-decoration: none;">', or </ins><<ins style="font-weight: bold; text-decoration: none;">math</ins>> <ins style="font-weight: bold; text-decoration: none;">Nx: Fx </ins><<ins style="font-weight: bold; text-decoration: none;">/math</ins>><ins style="font-weight: bold; text-decoration: none;">). He attempts to construct the content of a judgment involving numerical identity by relying on Hume's principle (which states that the number of Fs equals the number of Gs if and only if F and G are equinumerous, i.e. in one-one correspondence). He rejects this definition because it doesn't fix the truth value of identity statements when a singular term not of the form 'the number of Fs' flanks the identity sign. Frege goes on to give an explicit definition of number in terms of extensions of concepts, but expresses some hesitation.</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">===Frege's definition of a number===</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">Frege argues that numbers are objects and assert something about a concept. Frege defines numbers as extensions of concepts. 'The number of F's' is defined as the extension of the concept ''G is a concept that is equinumerous to F''. The concept in question leads to an equivalence class of all concepts that have the number of F (including F). Frege defines 0 as the extension of the concept ''being non self-identical''. So, the number of this concept is the extension of the concept of all concepts that have no objects falling under them.</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">== Legacy ==</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">The book was fundamental in the development of two main disciplines, the foundations of mathematics and philosophy. Although Bertrand Russell later found an apparent flaw in Frege's work (this apparent flaw is known as [[Russell's paradox]], which is resolved by [[axiomatic set theory]]), the book was influential in subsequent developments, such as ''[[Principia Mathematica]]''</ins>. <ins style="font-weight: bold; text-decoration: none;">The book can also be considered the starting point in analytic philosophy, since it revolves mainly around the analysis of language, with the goal of clarifying the concept of number</ins>. <ins style="font-weight: bold; text-decoration: none;">Frege's views on mathematics are also a starting point on the philosophy of mathematics, since it introduces an innovative account on the epistemology of numbers and math in general, known as logicism</ins>.</div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">==Notes==</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">{{Reflist}}</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">==External links==</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">*</ins>[http://www.<ins style="font-weight: bold; text-decoration: none;">ac-nancy-metz</ins>.<ins style="font-weight: bold; text-decoration: none;">fr</ins>/<ins style="font-weight: bold; text-decoration: none;">enseign</ins>/<ins style="font-weight: bold; text-decoration: none;">philo/textesph/Frege.pdf Original German text.] Maintained by Alain Blachair, Académie de Nancy-Metz. </ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">*[[Stanford Encyclopedia of Philosophy]]: "[http://plato.stanford.edu/entries/frege-logic/ Frege's Logic, Theorem, and Foundations for Arithmetic,]" -- by [[Edward Zalta]].</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">*{{SpringerEOM| title=Number | id=Number | oldid=11869 | first=V.I. | last=Nechaev }}</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">{{DEFAULTSORT:Foundations Of Arithmetic, The}}</ins></div></td></tr>
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