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<div>Sorry to continue this thread.<br>
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<div>On 21/08/2018, at 13:49, "Markus Klyver" <<a href="mailto:markusklyver@hotmail.com">markusklyver@hotmail.com</a>> wrote:<br>
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<p style="margin-top:0;margin-bottom:0">Definitions *are* <span>equivalences.</span></p>
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<div>But most equivalences are certainly not definitions.</div>
<div>So the same symbols should not be used.</div>
<div>(For example, 5 is equivalent to 2 modulo 3. But it would be silly to use 5 as a canonical example of its equivalence class mod 3.)</div>
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<div>A definition should be introducing the *words* that are going to be used to refer to a mathematical concept. Frequently a notation to represent the concept will also be presented.</div>
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<div>In a sense the definition is necessarily a tautology, hence an equivalence.</div>
<div>But to a student who has not met the terminology already, such a characterization is completely useless. </div>
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<div>After further study it becomes apparent that there can be many alternative ways to see that a particular mathematical property holds, as in the det A = 0 example. But these are not alternative definitions, despite the fact that some authors try to say
that they are. That they are equivalent *properties* is usually presented as a theorem, and each requires a calculation to demonstrate the equivalence.</div>
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<div>Texts that introduce new concepts by means of a definition, without previous motivating examples, are notoriously difficult to learn from. That's why Bourbaki never really gained much traction — if you are not already an expert, just forget it.</div>
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<div id="divRplyFwdMsg" dir="ltr"><font face="Calibri, sans-serif" style="font-size:11pt" color="#000000"><b>Från:</b> MacOSX-TeX <<a href="mailto:macosx-tex-bounces@email.esm.psu.edu">macosx-tex-bounces@email.esm.psu.edu</a>> för Ross Moore <<a href="mailto:ross.moore@mq.edu.au">ross.moore@mq.edu.au</a>><br>
<b>Skickat:</b> den 21 augusti 2018 13:21<br>
<b>Till:</b> TeX on Mac OS X Mailing List<br>
<b>Kopia:</b> TeX on Mac OS X Mailing List<br>
<b>Ämne:</b> Re: [OS X TeX] Latex symbol for "define equal"</font>
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<div>Hi all.</div>
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<div>I agree with using "when", or better "precisely when", in definitions.</div>
<div>Do not use these words in theorems. There are many alternative ways to express what you want.</div>
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<div>But didn't this thread start out asking for a symbolic notation?</div>
<div>In that context, equivalence is *not* the word you want for a definition, as equivalence depends on a particular relation, so is weaker than a definition.</div>
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<div>There are nuances in mathematical concepts and notations that are not always correctly appreciated by those who work primarily in other (mostly applied) fields.</div>
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On 21/08/2018, at 13:03, "Markus Klyver" <<a href="mailto:markusklyver@hotmail.com" id="LPlnk774387" class="OWAAutoLink" previewremoved="true">markusklyver@hotmail.com</a>> wrote:<br>
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<p style="margin-top:0; margin-bottom:0">But how would the reader know if it's a theorem or a definition? Consider the statement "f(x)=1 when x=2". Do we mean f(x)=1 <---> x=2 or just f(x)=1 <--- x=2?</p>
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<div id="x_divRplyFwdMsg" dir="ltr"><font face="Calibri, sans-serif" color="#000000" style="font-size:11pt"><b>Från:</b> MacOSX-TeX <<a href="mailto:macosx-tex-bounces@email.esm.psu.edu" id="LPlnk556178" class="OWAAutoLink" previewremoved="true">macosx-tex-bounces@email.esm.psu.edu</a>>
för Josep Maria Font <<a href="mailto:jmfont@ub.edu" id="LPlnk835669" class="OWAAutoLink" previewremoved="true">jmfont@ub.edu</a>><br>
<b>Skickat:</b> den 21 augusti 2018 12:15<br>
<b>Till:</b> TeX on Mac OS X Mailing List<br>
<b>Ämne:</b> Re: [OS X TeX] Latex symbol for "define equal"</font>
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<div class="x_PlainText">That's why I prefer to use "when" for definitions: [...this NEW thing happens...] WHEN [...such-and-such known, understandable condition holds...].<br>
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JMaF</div>
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<div>Please excuse this indulgence.</div>
<div>Being on holiday I was not in a position to reply earlier.</div>
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<div>Cheers,</div>
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<div> Ross</div>
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