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searchaux" style="display:none">American mathematician (1928–1983)</div> <style data-mw-deduplicate="TemplateStyles:r1257001546">.mw-parser-output .infobox-subbox{padding:0;border:none;margin:-3px;width:auto;min-width:100%;font-size:100%;clear:none;float:none;background-color:transparent}.mw-parser-output .infobox-3cols-child{margin:auto}.mw-parser-output .infobox .navbar{font-size:100%}@media screen{html.skin-theme-clientpref-night .mw-parser-output .infobox-full-data:not(.notheme)>div:not(.notheme)[style]{background:#1f1f23!important;color:#f8f9fa}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .infobox-full-data:not(.notheme) div:not(.notheme){background:#1f1f23!important;color:#f8f9fa}}@media(min-width:640px){body.skin--responsive .mw-parser-output .infobox-table{display:table!important}body.skin--responsive .mw-parser-output .infobox-table>caption{display:table-caption!important}body.skin--responsive .mw-parser-output .infobox-table>tbody{display:table-row-group}body.skin--responsive .mw-parser-output .infobox-table tr{display:table-row!important}body.skin--responsive .mw-parser-output .infobox-table th,body.skin--responsive .mw-parser-output .infobox-table td{padding-left:inherit;padding-right:inherit}}</style><table class="infobox biography vcard"><tbody><tr><th colspan="2" class="infobox-above" style="font-size:125%;"><div class="fn">Errett A. Bishop</div></th></tr><tr><th scope="row" class="infobox-label">Born</th><td class="infobox-data"><span style="display:none">(<span class="bday">1928-07-14</span>)</span>July 14, 1928<br /><div style="display:inline" class="birthplace"><a href="/wiki/Newton,_Kansas" title="Newton, Kansas">Newton, Kansas</a></div></td></tr><tr><th scope="row" class="infobox-label">Died</th><td class="infobox-data">April 14, 1983<span style="display:none">(1983-04-14)</span> (aged 54)<br /><div style="display:inline" class="deathplace"><a href="/wiki/San_Diego,_California" class="mw-redirect" title="San Diego, California">San Diego, California</a></div></td></tr><tr><th scope="row" class="infobox-label">Nationality</th><td class="infobox-data category">American</td></tr><tr><th scope="row" class="infobox-label">Alma mater</th><td class="infobox-data"><a href="/wiki/University_of_Chicago" title="University of Chicago">University of Chicago</a></td></tr><tr><th scope="row" class="infobox-label">Known for</th><td class="infobox-data"><a href="/wiki/Bishop_set" class="mw-redirect" title="Bishop set">Bishop set</a>, <a href="/wiki/Constructive_analysis" title="Constructive analysis">Constructive analysis</a></td></tr><tr><td colspan="2" class="infobox-full-data"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1257001546"><b>Scientific career</b></td></tr><tr><th scope="row" class="infobox-label">Fields</th><td class="infobox-data category">Mathematics</td></tr><tr><th scope="row" class="infobox-label">Institutions</th><td class="infobox-data"><a href="/wiki/University_of_California_at_San_Diego" class="mw-redirect" title="University of California at San Diego">University of California at San Diego</a></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Doctoral_advisor" title="Doctoral advisor">Doctoral advisor</a></th><td class="infobox-data"><a href="/wiki/Paul_Halmos" title="Paul Halmos">Paul Halmos</a></td></tr><tr style="display:none"><td colspan="2"> </td></tr></tbody></table> <p><b>Errett Albert Bishop</b> (July 14, 1928 – April 14, 1983)<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> was an <a href="/wiki/Americans" title="Americans">American</a> <a href="/wiki/Mathematician" title="Mathematician">mathematician</a> known for his work on analysis. He is best known for developing <a href="/wiki/Constructive_analysis" title="Constructive analysis">constructive analysis</a> in his 1967 <i>Foundations of Constructive Analysis</i>, where he <a href="/wiki/Mathematical_proof" title="Mathematical proof">proved</a> most of the important <a href="/wiki/Theorem" title="Theorem">theorems</a> in <a href="/wiki/Real_analysis" title="Real analysis">real analysis</a> using "<a href="/wiki/Constructivism_(mathematics)" class="mw-redirect" title="Constructivism (mathematics)">constructivist</a>" methods. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Life">Life</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Errett_Bishop&action=edit&section=1" title="Edit section: Life"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Errett Bishop's father, Albert T. Bishop, graduated from the <a href="/wiki/United_States_Military_Academy" title="United States Military Academy">United States Military Academy</a> at <a href="/wiki/West_Point" class="mw-redirect" title="West Point">West Point</a>, ending his career as professor of mathematics at <a href="/wiki/Wichita_State_University" title="Wichita State University">Wichita State University</a> in Kansas. Although he died when Errett was less than 4 years old, he influenced Errett's eventual career by the math texts he left behind, which is how Errett discovered mathematics. Errett grew up in <a href="/wiki/Newton,_Kansas" title="Newton, Kansas">Newton, Kansas</a>. Errett and his sister were apparent math prodigies. </p><p>Bishop entered the <a href="/wiki/University_of_Chicago" title="University of Chicago">University of Chicago</a> in 1944, obtaining both the BS and MS in 1947. The doctoral studies he began in that year were interrupted by two years in the <a href="/wiki/US_Army" class="mw-redirect" title="US Army">US Army</a>, 1950–52, doing mathematical research at the <a href="/wiki/National_Bureau_of_Standards" class="mw-redirect" title="National Bureau of Standards">National Bureau of Standards</a>. He completed his Ph.D. in 1954 under <a href="/wiki/Paul_Halmos" title="Paul Halmos">Paul Halmos</a>; his thesis was titled <i>Spectral Theory for Operations on Banach Spaces</i>. </p><p>Bishop taught at the <a href="/wiki/University_of_California" title="University of California">University of California</a>, 1954–65. He spent the 1964–65 academic year at the <a href="/wiki/Miller_Institute" title="Miller Institute">Miller Institute for Basic Research</a> in <a href="/wiki/Berkeley,_California" title="Berkeley, California">Berkeley</a>. He was a visiting scholar at the <a href="/wiki/Institute_for_Advanced_Study" title="Institute for Advanced Study">Institute for Advanced Study</a> in 1961–62.<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> From 1965 until his death, he was professor at the <a href="/wiki/University_of_California_at_San_Diego" class="mw-redirect" title="University of California at San Diego">University of California at San Diego</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Work">Work</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Errett_Bishop&action=edit&section=2" title="Edit section: Work"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1251242444">.mw-parser-output .ambox{border:1px solid #a2a9b1;border-left:10px solid #36c;background-color:#fbfbfb;box-sizing:border-box}.mw-parser-output .ambox+link+.ambox,.mw-parser-output .ambox+link+style+.ambox,.mw-parser-output .ambox+link+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+style+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+link+.ambox{margin-top:-1px}html body.mediawiki .mw-parser-output .ambox.mbox-small-left{margin:4px 1em 4px 0;overflow:hidden;width:238px;border-collapse:collapse;font-size:88%;line-height:1.25em}.mw-parser-output .ambox-speedy{border-left:10px solid #b32424;background-color:#fee7e6}.mw-parser-output .ambox-delete{border-left:10px solid #b32424}.mw-parser-output .ambox-content{border-left:10px solid #f28500}.mw-parser-output .ambox-style{border-left:10px solid #fc3}.mw-parser-output .ambox-move{border-left:10px solid #9932cc}.mw-parser-output .ambox-protection{border-left:10px solid #a2a9b1}.mw-parser-output .ambox .mbox-text{border:none;padding:0.25em 0.5em;width:100%}.mw-parser-output .ambox .mbox-image{border:none;padding:2px 0 2px 0.5em;text-align:center}.mw-parser-output .ambox .mbox-imageright{border:none;padding:2px 0.5em 2px 0;text-align:center}.mw-parser-output .ambox .mbox-empty-cell{border:none;padding:0;width:1px}.mw-parser-output .ambox .mbox-image-div{width:52px}@media(min-width:720px){.mw-parser-output .ambox{margin:0 10%}}@media print{body.ns-0 .mw-parser-output .ambox{display:none!important}}</style><table class="box-More_citations_needed plainlinks metadata ambox ambox-content ambox-Refimprove" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><a href="/wiki/File:Question_book-new.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/50px-Question_book-new.svg.png" decoding="async" width="50" height="39" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/75px-Question_book-new.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/100px-Question_book-new.svg.png 2x" data-file-width="512" data-file-height="399" /></a></span></div></td><td class="mbox-text"><div class="mbox-text-span">This section <b>needs additional citations for <a href="/wiki/Wikipedia:Verifiability" title="Wikipedia:Verifiability">verification</a></b>.<span class="hide-when-compact"> Please help <a href="/wiki/Special:EditPage/Errett_Bishop" title="Special:EditPage/Errett Bishop">improve this article</a> by <a href="/wiki/Help:Referencing_for_beginners" title="Help:Referencing for beginners">adding citations to reliable sources</a> in this section. Unsourced material may be challenged and removed.<br /><small><span class="plainlinks"><i>Find sources:</i> <a rel="nofollow" class="external text" href="https://www.google.com/search?as_eq=wikipedia&q=%22Errett+Bishop%22">"Errett Bishop"</a> – <a rel="nofollow" class="external text" href="https://www.google.com/search?tbm=nws&q=%22Errett+Bishop%22+-wikipedia&tbs=ar:1">news</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.google.com/search?&q=%22Errett+Bishop%22&tbs=bkt:s&tbm=bks">newspapers</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.google.com/search?tbs=bks:1&q=%22Errett+Bishop%22+-wikipedia">books</a> <b>·</b> <a rel="nofollow" class="external text" href="https://scholar.google.com/scholar?q=%22Errett+Bishop%22">scholar</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.jstor.org/action/doBasicSearch?Query=%22Errett+Bishop%22&acc=on&wc=on">JSTOR</a></span></small></span> <span class="date-container"><i>(<span class="date">December 2023</span>)</i></span><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table> <p>Bishop's work falls into five categories: </p> <ol><li>Polynomial and rational approximation. Examples are extensions of <a href="/wiki/Mergelyan%27s_approximation_theorem" class="mw-redirect" title="Mergelyan's approximation theorem">Mergelyan's approximation theorem</a> and the theorem of <a href="/wiki/Frigyes_Riesz" title="Frigyes Riesz">Frigyes Riesz</a> and <a href="/wiki/Marcel_Riesz" title="Marcel Riesz">Marcel Riesz</a> concerning measures on the unit circle orthogonal to polynomials.</li> <li>The general theory of <a href="/wiki/Function_algebra" class="mw-redirect" title="Function algebra">function algebras</a>. Here Bishop worked on <a href="/wiki/Uniform_algebra" title="Uniform algebra">uniform algebras</a> (commutative <a href="/wiki/Banach_algebra" title="Banach algebra">Banach algebras</a> with unit whose norms are the <a href="/wiki/Spectral_norm" class="mw-redirect" title="Spectral norm">spectral norms</a>) proving results such as antisymmetric decomposition of a uniform algebra, the <a href="/w/index.php?title=Bishop%E2%80%93DeLeeuw_theorem&action=edit&redlink=1" class="new" title="Bishop–DeLeeuw theorem (page does not exist)">Bishop–DeLeeuw theorem</a>, and the proof of existence of <a href="/w/index.php?title=Jensen_measure&action=edit&redlink=1" class="new" title="Jensen measure (page does not exist)">Jensen measures</a>. Bishop wrote a 1965 survey "Uniform algebras," examining the interaction between the theory of uniform algebras and that of several complex variables.</li> <li><a href="/wiki/Banach_space" title="Banach space">Banach spaces</a> and <a href="/wiki/Operator_theory" title="Operator theory">operator theory</a>, the subject of his thesis. He introduced what is now called the <a href="/w/index.php?title=Bishop_condition&action=edit&redlink=1" class="new" title="Bishop condition (page does not exist)">Bishop condition</a>, useful in the theory of <a href="/wiki/Decomposable_operator" class="mw-redirect" title="Decomposable operator">decomposable operators</a>.</li> <li>The theory of functions of <a href="/wiki/Several_complex_variables" class="mw-redirect" title="Several complex variables">several complex variables</a>. An example is his 1962 "Analyticity in certain Banach spaces." He proved important results in this area such as the <a href="/w/index.php?title=Biholomorphic_embedding_theorem&action=edit&redlink=1" class="new" title="Biholomorphic embedding theorem (page does not exist)">biholomorphic embedding theorem</a> for a <a href="/wiki/Stein_manifold" title="Stein manifold">Stein manifold</a> as a closed <a href="/wiki/Submanifold" title="Submanifold">submanifold</a> in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {\mathbb {C} } ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {\mathbb {C} } ^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8570f53647f626a6ce111fab2520925c9929ccf5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {\mathbb {C} } ^{n}}"></span>, and a new proof of <a href="/wiki/Reinhold_Remmert" title="Reinhold Remmert">Remmert</a>'s <a href="/w/index.php?title=Proper_mapping_theorem&action=edit&redlink=1" class="new" title="Proper mapping theorem (page does not exist)">proper mapping theorem</a>.</li> <li><a href="/wiki/Constructive_mathematics" class="mw-redirect" title="Constructive mathematics">Constructive mathematics</a>. Bishop became interested in foundational issues while at the Miller Institute. His now-famous <i>Foundations of Constructive Analysis</i> (1967)<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> aimed to show that a constructive treatment of analysis is feasible, something about which <a href="/wiki/Hermann_Weyl" title="Hermann Weyl">Weyl</a> had been pessimistic. A 1985 revision, called <i>Constructive Analysis</i>, was completed with the assistance of Douglas Bridges.</li></ol> <p>In 1972, Bishop (with Henry Cheng) published <i>Constructive Measure Theory</i>. </p><p>In the later part of his life, Bishop was seen as the leading mathematician in the area of constructivist mathematics. In 1966, he was invited to speak at the <a href="/wiki/International_Congress_of_Mathematicians" title="International Congress of Mathematicians">International Congress of Mathematicians</a> on that theme. His talk was titled "The Constructivisation of Abstract Mathematical Analysis."<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> The <a href="/wiki/American_Mathematical_Society" title="American Mathematical Society">American Mathematical Society</a> invited him to give four hour-long lectures as part of the Colloquium Lectures series. The title of his lectures was "Schizophrenia of Contemporary Mathematics." <a href="/wiki/Abraham_Robinson" title="Abraham Robinson">Abraham Robinson</a> wrote of Bishop's work in constructivist mathematics: "Even those who are not willing to accept Bishop's basic philosophy must be impressed with the great analytical power displayed in his work." <sup id="cite_ref-FOOTNOTEWarschawski1985_5-0" class="reference"><a href="#cite_note-FOOTNOTEWarschawski1985-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> Robinson, however, wrote in his review of Bishop's book that Bishop's historical commentary is "more vigorous than accurate". </p> <div class="mw-heading mw-heading2"><h2 id="Quotes">Quotes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Errett_Bishop&action=edit&section=3" title="Edit section: Quotes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>(A) "Mathematics is common sense";</li> <li>(B) "Do not ask whether a statement is true until you know what it means";</li> <li>(C) "A proof is any completely convincing argument";</li> <li>(D) "Meaningful distinctions deserve to be preserved".</li></ul> <dl><dd>(Items A through D are principles of constructivism from his <style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite class="citation book cs1"><a rel="nofollow" class="external text" href="https://books.google.com/books?id=fyUJGwAACAAJ"><i>Schizophrenia in Contemporary Mathematics</i></a>. <a href="/wiki/American_Mathematical_Society" title="American Mathematical Society">American Mathematical Society</a>. 1973.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Schizophrenia+in+Contemporary+Mathematics&rft.pub=American+Mathematical+Society&rft.date=1973&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DfyUJGwAACAAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3AErrett+Bishop" class="Z3988"></span> (Reprinted in Rosenblatt 1985.)</dd></dl> <ul><li>"The primary concern of mathematics is number, and this means the positive integers. . . . In the words of Kronecker, the positive integers were created by God. Kronecker would have expressed it even better if he had said that the positive integers were created by God for the benefit of man (and other finite beings). Mathematics belongs to man, not to God. We are not interested in properties of the positive integers that have no descriptive meaning for finite man. When a man proves a positive integer to exist, he should show how to find it. If God has mathematics of his own that needs to be done, let him do it himself." (Bishop 1967, Chapter 1, A Constructivist Manifesto, page 2)</li> <li>"We are not contending that idealistic mathematics is worthless from the constructive point of view. This would be as silly as contending that unrigorous mathematics is worthless from the classical point of view. Every theorem proved with idealistic methods presents a challenge: to find a constructive version, and to give it a constructive proof." (Bishop 1967, Preface, page x)</li> <li>"Theorem 1 is the famous theorem of Cantor, that the real numbers are uncountable. The proof is essentially Cantor's 'diagonal' proof. Both Cantor's theorem and his method of proof are of great importance." (Bishop 1967, Chapter 2, Calculus and the Real Numbers, page 25)</li> <li>"The real numbers, for certain purposes, are too thin. Many beautiful phenomena become fully visible only when the <a href="/wiki/Complex_number" title="Complex number">complex numbers</a> are brought to the fore." (Bishop 1967, Chapter 5, Complex Analysis, page 113)</li> <li>"It is clear that many of the results in this book could be programmed for a computer, by some such procedure as that indicated above. In particular, it is likely that most of the results of Chaps. 2, 4, 5, 9, 10, and 11 could be presented as computer programs. As an example, a complete separable metric space <i>X</i> can be described by a sequence of real numbers, and therefore by a sequence of integers, simply by listing the distances between each pair of elements of a given countable dense set. . . . As written, this book is person-oriented rather than computer-oriented. It would be of great interest to have a computer-oriented version." (Bishop 1967, Appendix B, Aspects of Constructive Truth, pages 356 and 357)</li> <li>"Very possibly classical mathematics will cease to exist as an independent discipline" (Bishop, 1970, p. 54)</li> <li>"Brouwer's criticisms of classical mathematics were concerned with what I shall refer to as 'the debasement of meaning<span style="padding-right:.15em;">'</span>" (Bishop in Rosenblatt, 1985, page 1)</li></ul> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Errett_Bishop&action=edit&section=4" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Bishop_set" class="mw-redirect" title="Bishop set">Bishop set</a></li> <li><a href="/wiki/Constructive_analysis" title="Constructive analysis">Constructive analysis</a></li> <li><a href="/wiki/Constructivism_(mathematics)" class="mw-redirect" title="Constructivism (mathematics)">Constructivism (mathematics)</a></li> <li><a href="/wiki/Criticism_of_non-standard_analysis" class="mw-redirect" title="Criticism of non-standard analysis">Criticism of non-standard analysis</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Errett_Bishop&action=edit&section=5" title="Edit section: Notes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <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"><a rel="nofollow" class="external text" href="http://texts.cdlib.org/view?docId=hb767nb3z6&doc.view=frames&chunk.id=div00011&toc.depth=1&toc.id=">UCSD Obituary</a></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://www.ias.edu/people/cos/">Institute for Advanced Study: A Community of Scholars</a></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFStolzenberg,_Gabriel1970" class="citation journal cs1">Stolzenberg, Gabriel (1970). <a rel="nofollow" class="external text" href="http://projecteuclid.org/euclid.bams/1183531480">"Review: Errett Bishop, <i>Foundations of Constructive Analysis</i>"</a>. <i><a href="/wiki/Bull._Amer._Math._Soc." class="mw-redirect" title="Bull. Amer. Math. Soc.">Bull. Amer. Math. Soc.</a></i> <b>76</b> (2): 301–323. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1090%2Fs0002-9904-1970-12455-7">10.1090/s0002-9904-1970-12455-7</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Bull.+Amer.+Math.+Soc.&rft.atitle=Review%3A+Errett+Bishop%2C+Foundations+of+Constructive+Analysis&rft.volume=76&rft.issue=2&rft.pages=301-323&rft.date=1970&rft_id=info%3Adoi%2F10.1090%2Fs0002-9904-1970-12455-7&rft.au=Stolzenberg%2C+Gabriel&rft_id=http%3A%2F%2Fprojecteuclid.org%2Feuclid.bams%2F1183531480&rfr_id=info%3Asid%2Fen.wikipedia.org%3AErrett+Bishop" class="Z3988"></span></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBishop" class="citation web cs1">Bishop, Errett. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20171107011802/http://www.mathunion.org/ICM/ICM1966.1/Main/icm1966.1.0308.0313.ocr.pdf">"The constructivization of abstract mathematical analysis"</a> <span class="cs1-format">(PDF)</span>. International Mathematical Union. Archived from <a rel="nofollow" class="external text" href="http://www.mathunion.org/ICM/ICM1966.1/Main/icm1966.1.0308.0313.ocr.pdf">the original</a> <span class="cs1-format">(PDF)</span> on 7 November 2017<span class="reference-accessdate">. Retrieved <span class="nowrap">1 November</span> 2017</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=The+constructivization+of+abstract+mathematical+analysis&rft.pub=International+Mathematical+Union&rft.aulast=Bishop&rft.aufirst=Errett&rft_id=http%3A%2F%2Fwww.mathunion.org%2FICM%2FICM1966.1%2FMain%2Ficm1966.1.0308.0313.ocr.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AErrett+Bishop" class="Z3988"></span></span> </li> <li id="cite_note-FOOTNOTEWarschawski1985-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEWarschawski1985_5-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFWarschawski1985">Warschawski 1985</a>.</span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Errett_Bishop&action=edit&section=6" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Bishop, Errett 1967. <i>Foundations of Constructive Analysis</i>, New York: Academic Press. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/4-87187-714-0" title="Special:BookSources/4-87187-714-0">4-87187-714-0</a></li> <li>Bishop, Errett and Douglas Bridges, 1985. <i>Constructive Analysis</i>. New York: Springer. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-387-15066-8" title="Special:BookSources/0-387-15066-8">0-387-15066-8</a>.</li> <li>Bishop, Errett (1970) Mathematics as a numerical language. 1970 Intuitionism and Proof Theory (Proc. Conf., Buffalo, New York 1968) pages 53–71. North-Holland, Amsterdam.</li> <li>Bishop, E. (1985) Schizophrenia in contemporary mathematics. In Errett Bishop: reflections on him and his research (San Diego, California, 1983), 1–32, Contemp. Math. 39, American Math. Society, Providence, Rhode Island.</li> <li>Bridges, Douglas, "Constructive Mathematics", The Stanford Encyclopedia of Philosophy (Winter 2004 Edition), Edward N. Zalta (ed.), <a rel="nofollow" class="external autonumber" href="http://plato.stanford.edu/archives/win2004/entries/mathematics-constructive/">[1]</a> - Online article by Douglas Bridges, a collaborator of Bishop.</li> <li>Rosenblatt, M., ed., 1985. <i>Errett Bishop: Reflections on him and his research</i>. Proceedings of the memorial meeting for Errett Bishop held at the University of California-San Diego, September 24, 1983. <i>Contemporary Mathematics 39</i>. AMS.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWarschawski1985" class="citation cs2">Warschawski, S. (1985), "Errett Bishop - In Memoriam", in Rosenblatt, M. (ed.), <i>Errett Bishop: Reflections on him and his research</i>, Contemporary Mathematics, vol. 39, American Mathematical Society</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Errett+Bishop+-+In+Memoriam&rft.btitle=Errett+Bishop%3A+Reflections+on+him+and+his+research&rft.series=Contemporary+Mathematics&rft.pub=American+Mathematical+Society&rft.date=1985&rft.aulast=Warschawski&rft.aufirst=S.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AErrett+Bishop" class="Z3988"></span></li> <li>Schechter, Eric 1997. <i>Handbook of Analysis and its Foundations</i>. New York: Academic Press. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-12-622760-8" title="Special:BookSources/0-12-622760-8">0-12-622760-8</a> — Constructive ideas in analysis, cites Bishop.</li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Errett_Bishop&action=edit&section=7" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFO'ConnorRobertson" class="citation cs2">O'Connor, John J.; <a href="/wiki/Edmund_F._Robertson" title="Edmund F. Robertson">Robertson, Edmund F.</a>, <a rel="nofollow" class="external text" href="https://mathshistory.st-andrews.ac.uk/Biographies/Bishop.html">"Errett Bishop"</a>, <i><a href="/wiki/MacTutor_History_of_Mathematics_Archive" title="MacTutor History of Mathematics Archive">MacTutor History of Mathematics Archive</a></i>, <a href="/wiki/University_of_St_Andrews" title="University of St Andrews">University of St Andrews</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Errett+Bishop&rft.btitle=MacTutor+History+of+Mathematics+Archive&rft.pub=University+of+St+Andrews&rft.aulast=O%27Connor&rft.aufirst=John+J.&rft.au=Robertson%2C+Edmund+F.&rft_id=https%3A%2F%2Fmathshistory.st-andrews.ac.uk%2FBiographies%2FBishop.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AErrett+Bishop" class="Z3988"></span></li> <li><a rel="nofollow" class="external text" href="https://mathgenealogy.org/id.php?id=6445">Errett Bishop</a> at the <a 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