https://machinelearning.subwiki.org/w/index.php?action=history&feed=atom&title=User%3AIssaRice%2FReflections_on_working_through_Tao%27s_AnalysisUser:IssaRice/Reflections on working through Tao's Analysis - Revision history2026-07-09T09:50:08ZRevision history for this page on the wikiMediaWiki 1.41.2https://machinelearning.subwiki.org/w/index.php?title=User:IssaRice/Reflections_on_working_through_Tao%27s_Analysis&diff=3095&oldid=prevIssaRice at 04:43, 23 May 20202020-05-23T04:43:40Z<p></p>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:Similarly, he never writes <math>(a_n)</math>, which many people do (and which leaves the bound variable <math>n</math> looking like a free variable).</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:Similarly, he never writes <math>(a_n)</math>, which many people do (and which leaves the bound variable <math>n</math> looking like a free variable).</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* some of the notation, like n++, a--b, a//b, LIM, <math>p.c.\int</math>, and some of the terms, like eps-close, eps-adherent, etc. are non-standard. But i like this since it tells you that you can be creative in your terms/notation. you can build things up in an idiosyncratic way, and still reach results that everybody else knows. you build everything up by hand so you know ''everything'' about how these things work.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* some of the notation, like n++, a--b, a//b, LIM, <math>p.c.\int</math>, and some of the terms, like eps-close, eps-adherent, etc. are non-standard. But i like this since it tells you that you can be creative in your terms/notation. you can build things up in an idiosyncratic way, and still reach results that everybody else knows. you build everything up by hand so you know ''everything'' about how these things work.</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>* also, here's a note that may restore the cosmic balance: in the book, we show that <math>a--b = a-b</math>, <math>a//b = a/b</math>, and <math>\mathrm{LIM}_{n\to\infty} a_n = \lim_{n\to\infty} a_n</math>. But what is the analogous result for ++? It's <math>n++ = n+1</math>, but the notation doesn't seem to match like the others! Here's one way to think about it: if you want the increment operation to visually resemble <math>+1</math>, it could be <math>{+}\!\!{-}\!\!{-}\!\!\!\!{1}</math>, i.e., a + and a 1 stuck together. But that actually looks just like ++ (especially in handwriting).</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>* also, here's a note that may restore the cosmic balance: in the book, we show that <math>a<ins style="font-weight: bold; text-decoration: none;">{</ins>-<ins style="font-weight: bold; text-decoration: none;">}\!\!{</ins>-<ins style="font-weight: bold; text-decoration: none;">}</ins>b = a-b</math>, <math>a//b = a/b</math>, and <math>\mathrm{LIM}_{n\to\infty} a_n = \lim_{n\to\infty} a_n</math>. But what is the analogous result for ++? It's <math>n<ins style="font-weight: bold; text-decoration: none;">{</ins>+<ins style="font-weight: bold; text-decoration: none;">}\!\!{</ins>+<ins style="font-weight: bold; text-decoration: none;">} </ins>= n+1</math>, but the notation doesn't seem to match like the others! Here's one way to think about it: if you want the increment operation to visually resemble <math>+1</math>, it could be <math>{+}\!\!{-}\!\!{-}\!\!\!\!{1}</math>, i.e., a + and a 1 stuck together. But that actually looks just like ++ (especially in handwriting<ins style="font-weight: bold; text-decoration: none;">). Now all of the formal operations have a pattern to them: the formal version visually resembles the actual version (but it's "busier"/visually messier</ins>).</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* Definition, example, theorem, proof: I had previously thought that I didn't like this style of writing mathematical texts, but somehow I like it in this text, so that tells me I need to introspect on this practice more, to see what specifically I like/don't like about it. It might be the case that I don't like terse/formal writing, and terse/formal writing in math textbooks tends to accompany definition, example, theorem, proof style. Informal/conversational/intuitive writing tends to be in blog posts or short articles and these tend not to have def/ex/thm/proof style. Tao does this weird thing where he adopts the def/thm/proof style but does it in a more conversational/motivated way. at least, that's one explanation i could give for why i like it.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* Definition, example, theorem, proof: I had previously thought that I didn't like this style of writing mathematical texts, but somehow I like it in this text, so that tells me I need to introspect on this practice more, to see what specifically I like/don't like about it. It might be the case that I don't like terse/formal writing, and terse/formal writing in math textbooks tends to accompany definition, example, theorem, proof style. Informal/conversational/intuitive writing tends to be in blog posts or short articles and these tend not to have def/ex/thm/proof style. Tao does this weird thing where he adopts the def/thm/proof style but does it in a more conversational/motivated way. at least, that's one explanation i could give for why i like it.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* plain variables work just fine: Tao rarely uses bolding for variables, does not use calligraphic letters, does not use arrows over variables, etc. e.g. vectors are <math>v,w,u\dots</math> rather than <math>\mathbf v</math> or <math>\vec v</math>. This does not make his writing any less clear, because he gives the type of each variable and takes care to use groups of letters (like a,b,c vs x,y,z) for separate things.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* plain variables work just fine: Tao rarely uses bolding for variables, does not use calligraphic letters, does not use arrows over variables, etc. e.g. vectors are <math>v,w,u\dots</math> rather than <math>\mathbf v</math> or <math>\vec v</math>. This does not make his writing any less clear, because he gives the type of each variable and takes care to use groups of letters (like a,b,c vs x,y,z) for separate things.</div></td></tr>
</table>IssaRicehttps://machinelearning.subwiki.org/w/index.php?title=User:IssaRice/Reflections_on_working_through_Tao%27s_Analysis&diff=3094&oldid=prevIssaRice at 04:40, 23 May 20202020-05-23T04:40:44Z<p></p>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:Similarly, he never writes <math>(a_n)</math>, which many people do (and which leaves the bound variable <math>n</math> looking like a free variable).</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:Similarly, he never writes <math>(a_n)</math>, which many people do (and which leaves the bound variable <math>n</math> looking like a free variable).</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* some of the notation, like n++, a--b, a//b, LIM, <math>p.c.\int</math>, and some of the terms, like eps-close, eps-adherent, etc. are non-standard. But i like this since it tells you that you can be creative in your terms/notation. you can build things up in an idiosyncratic way, and still reach results that everybody else knows. you build everything up by hand so you know ''everything'' about how these things work.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* some of the notation, like n++, a--b, a//b, LIM, <math>p.c.\int</math>, and some of the terms, like eps-close, eps-adherent, etc. are non-standard. But i like this since it tells you that you can be creative in your terms/notation. you can build things up in an idiosyncratic way, and still reach results that everybody else knows. you build everything up by hand so you know ''everything'' about how these things work.</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;">* also, here's a note that may restore the cosmic balance: in the book, we show that <math>a--b = a-b</math>, <math>a//b = a/b</math>, and <math>\mathrm{LIM}_{n\to\infty} a_n = \lim_{n\to\infty} a_n</math>. But what is the analogous result for ++? It's <math>n++ = n+1</math>, but the notation doesn't seem to match like the others! Here's one way to think about it: if you want the increment operation to visually resemble <math>+1</math>, it could be <math>{+}\!\!{-}\!\!{-}\!\!\!\!{1}</math>, i.e., a + and a 1 stuck together. But that actually looks just like ++ (especially in handwriting).</ins></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* Definition, example, theorem, proof: I had previously thought that I didn't like this style of writing mathematical texts, but somehow I like it in this text, so that tells me I need to introspect on this practice more, to see what specifically I like/don't like about it. It might be the case that I don't like terse/formal writing, and terse/formal writing in math textbooks tends to accompany definition, example, theorem, proof style. Informal/conversational/intuitive writing tends to be in blog posts or short articles and these tend not to have def/ex/thm/proof style. Tao does this weird thing where he adopts the def/thm/proof style but does it in a more conversational/motivated way. at least, that's one explanation i could give for why i like it.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* Definition, example, theorem, proof: I had previously thought that I didn't like this style of writing mathematical texts, but somehow I like it in this text, so that tells me I need to introspect on this practice more, to see what specifically I like/don't like about it. It might be the case that I don't like terse/formal writing, and terse/formal writing in math textbooks tends to accompany definition, example, theorem, proof style. Informal/conversational/intuitive writing tends to be in blog posts or short articles and these tend not to have def/ex/thm/proof style. Tao does this weird thing where he adopts the def/thm/proof style but does it in a more conversational/motivated way. at least, that's one explanation i could give for why i like it.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* plain variables work just fine: Tao rarely uses bolding for variables, does not use calligraphic letters, does not use arrows over variables, etc. e.g. vectors are <math>v,w,u\dots</math> rather than <math>\mathbf v</math> or <math>\vec v</math>. This does not make his writing any less clear, because he gives the type of each variable and takes care to use groups of letters (like a,b,c vs x,y,z) for separate things.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* plain variables work just fine: Tao rarely uses bolding for variables, does not use calligraphic letters, does not use arrows over variables, etc. e.g. vectors are <math>v,w,u\dots</math> rather than <math>\mathbf v</math> or <math>\vec v</math>. This does not make his writing any less clear, because he gives the type of each variable and takes care to use groups of letters (like a,b,c vs x,y,z) for separate things.</div></td></tr>
</table>IssaRicehttps://machinelearning.subwiki.org/w/index.php?title=User:IssaRice/Reflections_on_working_through_Tao%27s_Analysis&diff=3046&oldid=prevIssaRice at 07:45, 13 April 20202020-04-13T07:45:27Z<p></p>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* excessive use of numbers to refer to past results. this was actually good for exercise hints but not good for other things.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* excessive use of numbers to refer to past results. this was actually good for exercise hints but not good for other things.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* I sort of find amusing that the errata for the text is so long. Is this a fact about mathematics in general? about analysis in particular? about Tao (e.g. perhaps the way he writes in LaTeX)? But none of the errors seem very "serious" (in the sense that they can be patched up quickly/locally); this seems like an instance of the phenomenon discussed in https://www.gwern.net/The-Existential-Risk-of-Mathematical-Error</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* I sort of find amusing that the errata for the text is so long. Is this a fact about mathematics in general? about analysis in particular? about Tao (e.g. perhaps the way he writes in LaTeX)? But none of the errors seem very "serious" (in the sense that they can be patched up quickly/locally); this seems like an instance of the phenomenon discussed in https://www.gwern.net/The-Existential-Risk-of-Mathematical-Error</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>* i wish there was a solutions manual for the text.</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>* i wish there was a solutions manual for the text. <ins style="font-weight: bold; text-decoration: none;">EDIT: i felt this so strongly that I decided to start writing it myself: https://taoanalysis.wordpress.com/</ins></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* for definitions that he states using his idiosyncratic terminology, i wish he gave the equivalent (more standard) way of phrasing it as well. (he does this for a while, but gives up for e.g. definition 9.9.2)</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* for definitions that he states using his idiosyncratic terminology, i wish he gave the equivalent (more standard) way of phrasing it as well. (he does this for a while, but gives up for e.g. definition 9.9.2)</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* there are no pictures in the text. in fact, at several points during the text, Tao remarks how wonderful it is that we can do everything analytically without relying on geometry. The best we get is the piston analogy for sup/limsup.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* there are no pictures in the text. in fact, at several points during the text, Tao remarks how wonderful it is that we can do everything analytically without relying on geometry. The best we get is the piston analogy for sup/limsup.</div></td></tr>
</table>IssaRicehttps://machinelearning.subwiki.org/w/index.php?title=User:IssaRice/Reflections_on_working_through_Tao%27s_Analysis&diff=3045&oldid=prevIssaRice at 07:43, 13 April 20202020-04-13T07:43:27Z<p></p>
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</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l16">Line 16:</td>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* For each definition that "clobbers up" a notation, check consistency with old definition (See inverse image for functions).</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* For each definition that "clobbers up" a notation, check consistency with old definition (See inverse image for functions).</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* Definitions: Disambiguate similar-seeming concepts: example of distinct vs disjoint set. strictly monotone vs monotone (one implies the other, but the converse doesn't hold).</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* Definitions: Disambiguate similar-seeming concepts: example of distinct vs disjoint set. strictly monotone vs monotone (one implies the other, but the converse doesn't hold).</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>* "(Why?)": this feels similar to a [https://docs.python.org/3/library/exceptions.html#NotImplementedError "not implemented"] error in programming. it's also like the usual practice of breaking things off to lemmas, to make a proof more modular.</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>* "(Why?)": this feels similar to a [https://docs.python.org/3/library/exceptions.html#NotImplementedError "not implemented"] error in programming. it's also like the usual practice of breaking things off to lemmas, to make a proof more modular<ins style="font-weight: bold; text-decoration: none;">. You get to see the higher-level thoughts, rather than seeing one higher-level thought, then being forced to think through low-level details, then coming back for the next higher-level though, then going down again, ... . The latter can get tiring, so these "(why?)"s help to force the low-level details to go away. I think if this book was available as a web page, it would be cool if you could click on the "(why?)" to see the answer in pop-up window (sort of like how some websites do footnotes now)</ins>.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* rolling up concepts to make them more processable: epsilon-close, epsilon-adherent, continually epsilon-close, etc. see gowers blog post as well, where he does this for the definition of limit of a sequence.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* rolling up concepts to make them more processable: epsilon-close, epsilon-adherent, continually epsilon-close, etc. see gowers blog post as well, where he does this for the definition of limit of a sequence.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* psychologically speaking, I prefer "''Proof''. See Exercise n.m.p." to a textbook just ''mentioning'' a result in passing and using it, or just jumping in and using it without proof. I think I like seeing the formal statement of the result. Also, i think many books just don't even mention the proposition in the text -- they just put the whole thing in the exercises. but i like having at least the proposition in the main text, with a pointer to the exercise.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* psychologically speaking, I prefer "''Proof''. See Exercise n.m.p." to a textbook just ''mentioning'' a result in passing and using it, or just jumping in and using it without proof. I think I like seeing the formal statement of the result. Also, i think many books just don't even mention the proposition in the text -- they just put the whole thing in the exercises. but i like having at least the proposition in the main text, with a pointer to the exercise.</div></td></tr>
</table>IssaRicehttps://machinelearning.subwiki.org/w/index.php?title=User:IssaRice/Reflections_on_working_through_Tao%27s_Analysis&diff=3044&oldid=prevIssaRice at 07:39, 13 April 20202020-04-13T07:39:44Z<p></p>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* Definition, example, theorem, proof: I had previously thought that I didn't like this style of writing mathematical texts, but somehow I like it in this text, so that tells me I need to introspect on this practice more, to see what specifically I like/don't like about it. It might be the case that I don't like terse/formal writing, and terse/formal writing in math textbooks tends to accompany definition, example, theorem, proof style. Informal/conversational/intuitive writing tends to be in blog posts or short articles and these tend not to have def/ex/thm/proof style. Tao does this weird thing where he adopts the def/thm/proof style but does it in a more conversational/motivated way. at least, that's one explanation i could give for why i like it.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* Definition, example, theorem, proof: I had previously thought that I didn't like this style of writing mathematical texts, but somehow I like it in this text, so that tells me I need to introspect on this practice more, to see what specifically I like/don't like about it. It might be the case that I don't like terse/formal writing, and terse/formal writing in math textbooks tends to accompany definition, example, theorem, proof style. Informal/conversational/intuitive writing tends to be in blog posts or short articles and these tend not to have def/ex/thm/proof style. Tao does this weird thing where he adopts the def/thm/proof style but does it in a more conversational/motivated way. at least, that's one explanation i could give for why i like it.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* plain variables work just fine: Tao rarely uses bolding for variables, does not use calligraphic letters, does not use arrows over variables, etc. e.g. vectors are <math>v,w,u\dots</math> rather than <math>\mathbf v</math> or <math>\vec v</math>. This does not make his writing any less clear, because he gives the type of each variable and takes care to use groups of letters (like a,b,c vs x,y,z) for separate things.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* plain variables work just fine: Tao rarely uses bolding for variables, does not use calligraphic letters, does not use arrows over variables, etc. e.g. vectors are <math>v,w,u\dots</math> rather than <math>\mathbf v</math> or <math>\vec v</math>. This does not make his writing any less clear, because he gives the type of each variable and takes care to use groups of letters (like a,b,c vs x,y,z) for separate things.</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>* Tao manages prerequisites by explaining even the most basic things, which is why his explanations ''seem clearer'' to a larger audience than otherwise.</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>* Tao manages prerequisites by explaining even the most basic things, which is why his explanations ''seem clearer'' to a larger audience than otherwise<ins style="font-weight: bold; text-decoration: none;">. Eliezer used the same strategy in the Sequences</ins>.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* At each theorem, check necessity of hypotheses. Two ways of doing this: (1) remove each hypothesis individually to see that the theorem doesn't hold by coming up with counterexamples; (2) in the proof of the theorem, point out where each hypothesis is being used.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* At each theorem, check necessity of hypotheses. Two ways of doing this: (1) remove each hypothesis individually to see that the theorem doesn't hold by coming up with counterexamples; (2) in the proof of the theorem, point out where each hypothesis is being used.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* At each definition, give both positive and negative examples. (I wish there was more of an emphasis on giving "edge cases" though.)</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* At each definition, give both positive and negative examples. (I wish there was more of an emphasis on giving "edge cases" though.)</div></td></tr>
</table>IssaRicehttps://machinelearning.subwiki.org/w/index.php?title=User:IssaRice/Reflections_on_working_through_Tao%27s_Analysis&diff=3043&oldid=prevIssaRice at 07:38, 13 April 20202020-04-13T07:38:59Z<p></p>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* some of the notation, like n++, a--b, a//b, LIM, <math>p.c.\int</math>, and some of the terms, like eps-close, eps-adherent, etc. are non-standard. But i like this since it tells you that you can be creative in your terms/notation. you can build things up in an idiosyncratic way, and still reach results that everybody else knows. you build everything up by hand so you know ''everything'' about how these things work.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* some of the notation, like n++, a--b, a//b, LIM, <math>p.c.\int</math>, and some of the terms, like eps-close, eps-adherent, etc. are non-standard. But i like this since it tells you that you can be creative in your terms/notation. you can build things up in an idiosyncratic way, and still reach results that everybody else knows. you build everything up by hand so you know ''everything'' about how these things work.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* Definition, example, theorem, proof: I had previously thought that I didn't like this style of writing mathematical texts, but somehow I like it in this text, so that tells me I need to introspect on this practice more, to see what specifically I like/don't like about it. It might be the case that I don't like terse/formal writing, and terse/formal writing in math textbooks tends to accompany definition, example, theorem, proof style. Informal/conversational/intuitive writing tends to be in blog posts or short articles and these tend not to have def/ex/thm/proof style. Tao does this weird thing where he adopts the def/thm/proof style but does it in a more conversational/motivated way. at least, that's one explanation i could give for why i like it.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* Definition, example, theorem, proof: I had previously thought that I didn't like this style of writing mathematical texts, but somehow I like it in this text, so that tells me I need to introspect on this practice more, to see what specifically I like/don't like about it. It might be the case that I don't like terse/formal writing, and terse/formal writing in math textbooks tends to accompany definition, example, theorem, proof style. Informal/conversational/intuitive writing tends to be in blog posts or short articles and these tend not to have def/ex/thm/proof style. Tao does this weird thing where he adopts the def/thm/proof style but does it in a more conversational/motivated way. at least, that's one explanation i could give for why i like it.</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>* plain variables work just fine: Tao rarely uses bolding for variables, does not use calligraphic letters, does not use arrows over variables, etc. e.g. vectors are <math>v,w,u\dots</math> rather than <math>\mathbf v</math> or <math>\vec v</math>. This does not make his writing any less clear, because he gives the type of each variable.</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>* plain variables work just fine: Tao rarely uses bolding for variables, does not use calligraphic letters, does not use arrows over variables, etc. e.g. vectors are <math>v,w,u\dots</math> rather than <math>\mathbf v</math> or <math>\vec v</math>. This does not make his writing any less clear, because he gives the type of each variable <ins style="font-weight: bold; text-decoration: none;">and takes care to use groups of letters (like a,b,c vs x,y,z) for separate things</ins>.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* Tao manages prerequisites by explaining even the most basic things, which is why his explanations ''seem clearer'' to a larger audience than otherwise.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* Tao manages prerequisites by explaining even the most basic things, which is why his explanations ''seem clearer'' to a larger audience than otherwise.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* At each theorem, check necessity of hypotheses. Two ways of doing this: (1) remove each hypothesis individually to see that the theorem doesn't hold by coming up with counterexamples; (2) in the proof of the theorem, point out where each hypothesis is being used.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* At each theorem, check necessity of hypotheses. Two ways of doing this: (1) remove each hypothesis individually to see that the theorem doesn't hold by coming up with counterexamples; (2) in the proof of the theorem, point out where each hypothesis is being used.</div></td></tr>
</table>IssaRicehttps://machinelearning.subwiki.org/w/index.php?title=User:IssaRice/Reflections_on_working_through_Tao%27s_Analysis&diff=3042&oldid=prevIssaRice at 07:37, 13 April 20202020-04-13T07:37:53Z<p></p>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:Similarly, he never writes <math>(a_n)</math>, which many people do (and which leaves the bound variable <math>n</math> looking like a free variable).</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:Similarly, he never writes <math>(a_n)</math>, which many people do (and which leaves the bound variable <math>n</math> looking like a free variable).</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* some of the notation, like n++, a--b, a//b, LIM, <math>p.c.\int</math>, and some of the terms, like eps-close, eps-adherent, etc. are non-standard. But i like this since it tells you that you can be creative in your terms/notation. you can build things up in an idiosyncratic way, and still reach results that everybody else knows. you build everything up by hand so you know ''everything'' about how these things work.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* some of the notation, like n++, a--b, a//b, LIM, <math>p.c.\int</math>, and some of the terms, like eps-close, eps-adherent, etc. are non-standard. But i like this since it tells you that you can be creative in your terms/notation. you can build things up in an idiosyncratic way, and still reach results that everybody else knows. you build everything up by hand so you know ''everything'' about how these things work.</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>* Definition, example, theorem, proof: I had previously thought that I didn't like this style of writing mathematical texts, but somehow I like it in this text, so that tells me I need to introspect on this practice more, to see what specifically I like/don't like about it.</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>* Definition, example, theorem, proof: I had previously thought that I didn't like this style of writing mathematical texts, but somehow I like it in this text, so that tells me I need to introspect on this practice more, to see what specifically I like/don't like about <ins style="font-weight: bold; text-decoration: none;">it. It might be the case that I don't like terse/formal writing, and terse/formal writing in math textbooks tends to accompany definition, example, theorem, proof style. Informal/conversational/intuitive writing tends to be in blog posts or short articles and these tend not to have def/ex/thm/proof style. Tao does this weird thing where he adopts the def/thm/proof style but does it in a more conversational/motivated way. at least, that's one explanation i could give for why i like </ins>it.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* plain variables work just fine: Tao rarely uses bolding for variables, does not use calligraphic letters, does not use arrows over variables, etc. e.g. vectors are <math>v,w,u\dots</math> rather than <math>\mathbf v</math> or <math>\vec v</math>. This does not make his writing any less clear, because he gives the type of each variable.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* plain variables work just fine: Tao rarely uses bolding for variables, does not use calligraphic letters, does not use arrows over variables, etc. e.g. vectors are <math>v,w,u\dots</math> rather than <math>\mathbf v</math> or <math>\vec v</math>. This does not make his writing any less clear, because he gives the type of each variable.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* Tao manages prerequisites by explaining even the most basic things, which is why his explanations ''seem clearer'' to a larger audience than otherwise.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* Tao manages prerequisites by explaining even the most basic things, which is why his explanations ''seem clearer'' to a larger audience than otherwise.</div></td></tr>
</table>IssaRicehttps://machinelearning.subwiki.org/w/index.php?title=User:IssaRice/Reflections_on_working_through_Tao%27s_Analysis&diff=3041&oldid=prevIssaRice at 07:34, 13 April 20202020-04-13T07:34:40Z<p></p>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:Similarly, I like how Tao generalizes notation like that of limits to <math>\lim_{x\to x_0;\, x\in E} f(x)</math>, where the "<math>x\in E</math>" allows us to pass in a separate set <math>E</math>. This allows us to define e.g. left and right limits separately as instances of this general notation.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:Similarly, I like how Tao generalizes notation like that of limits to <math>\lim_{x\to x_0;\, x\in E} f(x)</math>, where the "<math>x\in E</math>" allows us to pass in a separate set <math>E</math>. This allows us to define e.g. left and right limits separately as instances of this general notation.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:Similarly, he never writes <math>(a_n)</math>, which many people do (and which leaves the bound variable <math>n</math> looking like a free variable).</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:Similarly, he never writes <math>(a_n)</math>, which many people do (and which leaves the bound variable <math>n</math> looking like a free variable).</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;">* some of the notation, like n++, a--b, a//b, LIM, <math>p.c.\int</math>, and some of the terms, like eps-close, eps-adherent, etc. are non-standard. But i like this since it tells you that you can be creative in your terms/notation. you can build things up in an idiosyncratic way, and still reach results that everybody else knows. you build everything up by hand so you know ''everything'' about how these things work.</ins></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* Definition, example, theorem, proof: I had previously thought that I didn't like this style of writing mathematical texts, but somehow I like it in this text, so that tells me I need to introspect on this practice more, to see what specifically I like/don't like about it.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* Definition, example, theorem, proof: I had previously thought that I didn't like this style of writing mathematical texts, but somehow I like it in this text, so that tells me I need to introspect on this practice more, to see what specifically I like/don't like about it.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* plain variables work just fine: Tao rarely uses bolding for variables, does not use calligraphic letters, does not use arrows over variables, etc. e.g. vectors are <math>v,w,u\dots</math> rather than <math>\mathbf v</math> or <math>\vec v</math>. This does not make his writing any less clear, because he gives the type of each variable.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* plain variables work just fine: Tao rarely uses bolding for variables, does not use calligraphic letters, does not use arrows over variables, etc. e.g. vectors are <math>v,w,u\dots</math> rather than <math>\mathbf v</math> or <math>\vec v</math>. This does not make his writing any less clear, because he gives the type of each variable.</div></td></tr>
</table>IssaRicehttps://machinelearning.subwiki.org/w/index.php?title=User:IssaRice/Reflections_on_working_through_Tao%27s_Analysis&diff=2392&oldid=prevIssaRice: /* External links */2019-08-08T23:06:11Z<p><span dir="auto"><span class="autocomment">External links</span></span></p>
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 23:06, 8 August 2019</td>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==External links==</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==External links==</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></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;">* https://math.stackexchange.com/questions/3303829/terence-tao-type-books-in-other-fields</ins></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>I've written about this sort of thing in some other places and I should consolidate/organize these all at some point.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>I've written about this sort of thing in some other places and I should consolidate/organize these all at some point.</div></td></tr>
</table>IssaRicehttps://machinelearning.subwiki.org/w/index.php?title=User:IssaRice/Reflections_on_working_through_Tao%27s_Analysis&diff=2260&oldid=prevIssaRice at 07:05, 25 July 20192019-07-25T07:05:02Z<p></p>
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 07:05, 25 July 2019</td>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* Definition, example, theorem, proof: I had previously thought that I didn't like this style of writing mathematical texts, but somehow I like it in this text, so that tells me I need to introspect on this practice more, to see what specifically I like/don't like about it.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* Definition, example, theorem, proof: I had previously thought that I didn't like this style of writing mathematical texts, but somehow I like it in this text, so that tells me I need to introspect on this practice more, to see what specifically I like/don't like about it.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* plain variables work just fine: Tao rarely uses bolding for variables, does not use calligraphic letters, does not use arrows over variables, etc. e.g. vectors are <math>v,w,u\dots</math> rather than <math>\mathbf v</math> or <math>\vec v</math>. This does not make his writing any less clear, because he gives the type of each variable.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* plain variables work just fine: Tao rarely uses bolding for variables, does not use calligraphic letters, does not use arrows over variables, etc. e.g. vectors are <math>v,w,u\dots</math> rather than <math>\mathbf v</math> or <math>\vec v</math>. This does not make his writing any less clear, because he gives the type of each variable.</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;">* Tao manages prerequisites by explaining even the most basic things, which is why his explanations ''seem clearer'' to a larger audience than otherwise.</ins></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* At each theorem, check necessity of hypotheses. Two ways of doing this: (1) remove each hypothesis individually to see that the theorem doesn't hold by coming up with counterexamples; (2) in the proof of the theorem, point out where each hypothesis is being used.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* At each theorem, check necessity of hypotheses. Two ways of doing this: (1) remove each hypothesis individually to see that the theorem doesn't hold by coming up with counterexamples; (2) in the proof of the theorem, point out where each hypothesis is being used.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* At each definition, give both positive and negative examples. (I wish there was more of an emphasis on giving "edge cases" though.)</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* At each definition, give both positive and negative examples. (I wish there was more of an emphasis on giving "edge cases" though.)</div></td></tr>
</table>IssaRice