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<p class="MsoNormal">That depends on the type system of the language and the types that have been given.<o:p></o:p></p>
<p class="MsoNormal"><o:p></o:p></p>
<p class="MsoNormal">$ math<o:p></o:p></p>
<p class="MsoNormal">Mathematica 12.1.1 Kernel for Linux x86 (64-bit)<o:p></o:p></p>
<p class="MsoNormal">Copyright 1988-2020 Wolfram Research, Inc.<o:p></o:p></p>
<p class="MsoNormal"><o:p> </o:p></p>
<p class="MsoNormal">In[1]:= Expand[(x+1)^2] == x^2+2x+1<o:p></o:p></p>
<p class="MsoNormal"><o:p> </o:p></p>
<p class="MsoNormal">Out[1]= True<o:p></o:p></p>
<p class="MsoNormal"><o:p> </o:p></p>
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<p class="MsoNormal"><b>From:</b> Friam <friam-bounces@redfish.com> <b>On Behalf Of
</b>Frank Wimberly<br>
<b>Sent:</b> Monday, January 25, 2021 6:28 PM<br>
<b>To:</b> The Friday Morning Applied Complexity Coffee Group <friam@redfish.com><br>
<b>Subject:</b> Re: [FRIAM] Acronyms<o:p></o:p></p>
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<p class="MsoNormal">I would not expect the compilers I have used (Algol, Fortran, Java, Lisp (interpreter), Pascal, C, C++) to produce the same result to the last bit for (x + 1)^2 and x^2 + 2x + 1. Would you?<o:p></o:p></p>
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<p class="MsoNormal">On Mon, Jan 25, 2021 at 7:18 PM jon zingale <<a href="mailto:jonzingale@gmail.com">jonzingale@gmail.com</a>> wrote:<o:p></o:p></p>
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<p class="MsoNormal">Nick ,<br>
<br>
Please pardon a second attempt to address your question. Let me grant your<br>
definition of *rigor* as meaning *that which compiles*. *Clarity*, however,<br>
I would like to treat differently. What strikes me as a structural<br>
difference between *rigor* and *clarity* here is that the former (as<br>
narrowly defined above) depends only on the property of being compilable.<br>
Something, anything, was stated such that a program can run. The latter, to<br>
my mind, would require a concept of two programs being equivalent (or at<br>
least orderable). How else could we claim that one program was stated more<br>
clearly than another? This equivalence can be shoddy as in an optimizing<br>
function, ie. modulo some countable things I value, or actual equivalence.<br>
Many computer languages do not allow for actual functional equivalence in<br>
this sense, though there are some narrow examples. While you and I, and some<br>
algebraically focused languages, can immediately tell that *adding 1 to x<br>
and then squaring the result* is the "same thing" as *adding together a<br>
squared x to two times x and a 1*, many languages would require checking<br>
every value to determine such an equivalence. In this way, *clarity* appears<br>
to me to require more structure than a notion of *rigor* does. To some<br>
extent, I wish to reject the programmer/compiler dialectic, as it seems that<br>
it hides more useable observations.<br>
<br>
<br>
<br>
--<br>
Sent from: <a href="http://friam.471366.n2.nabble.com/" target="_blank">http://friam.471366.n2.nabble.com/</a><br>
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<o:p></o:p></p>
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<p class="MsoNormal"><o:p> </o:p></p>
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<p class="MsoNormal">-- <o:p></o:p></p>
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<p class="MsoNormal">Frank Wimberly<br>
140 Calle Ojo Feliz<br>
Santa Fe, NM 87505<br>
505 670-9918<o:p></o:p></p>
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<p class="MsoNormal"><o:p> </o:p></p>
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<p class="MsoNormal">Research: <a href="https://www.researchgate.net/profile/Frank_Wimberly2" target="_blank">https://www.researchgate.net/profile/Frank_Wimberly2</a><o:p></o:p></p>
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