<div dir="ltr">Reuben had an article in Issue 65 of Eureka Magazine titled 'Solving Problems by "Cheating": Operational Calculi, Function Theory, and Differential Equations'. The article is a compilation of tricks that he ran across during his career that seemed to apply in a general way to solving problems. The theme is that you doodle with methods that you have no right to assume would work in this particular case, and if you get something worthwhile, then go back and prove it.<div><br></div><div>Towards the end of his life he became more interested in the metaphors that are at the basis of mathematical thinking, the bodily actions that have been abstracted into mathematical concepts. Yuri I. Manin also spoke of Mathematics as Metaphor is a slightly different way in his essays. </div><div><br></div><div>-Roger</div><div><br></div></div><br><div class="gmail_quote"><div dir="ltr" class="gmail_attr">On Mon, Sep 6, 2021 at 8:34 PM Frank Wimberly <<a href="mailto:wimberly3@gmail.com">wimberly3@gmail.com</a>> wrote:<br></div><blockquote class="gmail_quote" style="margin:0px 0px 0px 0.8ex;border-left:1px solid rgb(204,204,204);padding-left:1ex"><div dir="auto">Our late friend Reuben Hersh was interested in these questions.<br><br><div>---<br>Frank C. Wimberly<br>140 Calle Ojo Feliz, <br>Santa Fe, NM 87505<br><br>505 670-9918<br>Santa Fe, NM</div></div><br><div class="gmail_quote"><div dir="ltr" class="gmail_attr">On Mon, Sep 6, 2021, 7:58 PM Eric Charles <<a href="mailto:eric.phillip.charles@gmail.com" target="_blank">eric.phillip.charles@gmail.com</a>> wrote:<br></div><blockquote class="gmail_quote" style="margin:0px 0px 0px 0.8ex;border-left:1px solid rgb(204,204,204);padding-left:1ex"><div dir="ltr">As I said a few days ago: I think traditionally, "mathematical" would have been synonymous with "rigorous deduction from a minimal number of axioms", but I doubt that approach is clear cut anymore.<div><br></div><div>I am pretty confident that modern mathematics is WAY more open-field than that. The Stanford Encyclopedia of Philosophy seems to agree with that intuition, though I think it is an even broader topic than implied by just this entry:
<a href="https://plato.stanford.edu/entries/mathematics-nondeductive/" rel="noreferrer" target="_blank">Non-Deductive Methods in Mathematics (Stanford Encyclopedia of Philosophy)</a> <br><div><div><br clear="all"><div><div dir="ltr"><div dir="ltr"><div dir="ltr"><div dir="ltr"><div dir="ltr"><br><div></div></div><div dir="ltr"><a href="mailto:echarles@american.edu" rel="noreferrer" target="_blank"></a></div></div></div></div></div></div><br></div></div></div></div><br><div class="gmail_quote"><div dir="ltr" class="gmail_attr">On Mon, Sep 6, 2021 at 11:19 AM Barry MacKichan <<a href="mailto:barry.mackichan@mackichan.com" rel="noreferrer" target="_blank">barry.mackichan@mackichan.com</a>> wrote:<br></div><blockquote class="gmail_quote" style="margin:0px 0px 0px 0.8ex;border-left:1px solid rgb(204,204,204);padding-left:1ex"><u></u>
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<div style="font-family:sans-serif"><div style="white-space:normal"><p dir="auto">Briefly, and in my opinion, mathematics can only make claims like ‘if A is true then B is true’. To say B is true, you must also say A is true. Eventually you have to go back to the beginning of the deductive chain, and the truth of the initial statement is inductive, not deductive or mathematics. You can predict the time and place of an eclipse, and this prediction is based on mathematics and a mathematical model of reality — Newton’s laws in this case. But the truth of this prediction is inductive since the initial positions and velocities for the calculation are inductive, as is the applicability of Newton’s laws to reality, and even the ‘fact’ that mathematics can describe the universe is inductive.</p>
<p dir="auto">And Einstein showed that the applicability of Newton’s laws was in fact wrong and offered a new model — which we inductively accept as true, if only provisionally.</p>
<p dir="auto">Mathematics cannot prove any statement about the real world. Any such statement will depend at some point on an inductive truth or a definition.</p>
<p dir="auto">—Barry</p>
<br><p dir="auto">On 3 Sep 2021, at 18:10, <a href="mailto:thompnickson2@gmail.com" rel="noreferrer" target="_blank">thompnickson2@gmail.com</a> wrote:</p>
<blockquote style="border-left:2px solid rgb(119,119,119);color:rgb(119,119,119);margin:0px 0px 5px;padding-left:5px"><p dir="auto">Ok, is mathematics (logic, etc.) a way of arriving at true propositions distinct from observation or are mathematical truths different from empirical truths? </p>
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