<div dir="ltr"><div class="gmail_default" style="font-family:comic sans ms,sans-serif;font-size:large;color:#000000">The Xrays on my bad knee showed that it was bone on bone - no whatever kind of tissue was there is not there anymore. Maybe osteoarthritis kills it slowly. Overuse does too: consider football linemen: tremendous pressure every play on the knees and hips, and ergo many hip and knee replacements in that group. Spike - it's just age, old buddy. You can try glucosamine/chondroitin and it probably won't help. Avoid injections if at all possible: there is a very limited number of them you can have, and to boot, they damage tissue too, as does any steroid. You want to save the injection until the pain just begs you to do something. And when the injections have to stop it's time to consider replacement. Getting older is dealing with pain, no way around it. You could also try some anti-inflammatories, such as Naproxen, curcurmin, aspirin, and so on. bill w</div></div><br><div class="gmail_quote"><div dir="ltr" class="gmail_attr">On Mon, Jan 24, 2022 at 11:01 AM spike jones via extropy-chat <<a href="mailto:extropy-chat@lists.extropy.org">extropy-chat@lists.extropy.org</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"><br>
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-----Original Message-----<br>
From: <a href="mailto:spike@rainier66.com" target="_blank">spike@rainier66.com</a> <<a href="mailto:spike@rainier66.com" target="_blank">spike@rainier66.com</a>> <br>
Sent: Monday, January 24, 2022 7:56 AM<br>
To: 'ExI chat list' <<a href="mailto:extropy-chat@lists.extropy.org" target="_blank">extropy-chat@lists.extropy.org</a>><br>
Cc: <a href="mailto:spike@rainier66.com" target="_blank">spike@rainier66.com</a><br>
Subject: RE: [ExI] How Infinite Series Reveal the Unity of Mathematics<br>
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-----Original Message-----<br>
From: extropy-chat <<a href="mailto:extropy-chat-bounces@lists.extropy.org" target="_blank">extropy-chat-bounces@lists.extropy.org</a>> On Behalf Of BillK via extropy-chat<br>
Subject: [ExI] How Infinite Series Reveal the Unity of Mathematics<br>
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Infinite sums are among the most underrated yet powerful concepts in mathematics, capable of linking concepts across math’s vast web.<br>
By Steven StrogatzJanuary 24, 2022<br>
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<<a href="https://www.quantamagazine.org/how-infinite-series-reveal-the-unity-of-mathematics-20220124/" rel="noreferrer" target="_blank">https://www.quantamagazine.org/how-infinite-series-reveal-the-unity-of-mathematics-20220124/</a>><br>
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The most compelling reason for learning about infinite series (or so I tell my students) is that they’re stunning connectors. They reveal ties between different areas of mathematics, unexpected links between everything that came before. It’s only when you get to this part of calculus that the true structure of math — all of math — finally starts to emerge.<br>
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Fascinating story! Recommended!<br>
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BillK<br>
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>...Great article BillK thanks!<br>
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>From BillK's Strogatz article:<br>
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>...which tells us a = 35 × 20 = 12 miles. Similar reasoning reveals that the legs shrink by a ratio of r= 15 each time the fly turns around. Von Neumann saw all of this instantly and, using the a/(1–r) formula above, he found the total distance traveled by the fly:<br>
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>...S = 121−15 = 1245 = 604 = 15 miles...<br>
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It is entirely plausible that Von Neumann really did use the infinite series approach to solve the fly problem. Engineers will nearly always see the shortcut but mathematicians know the hell outta their infinite series formulas and tend to think in those terms. My son and I had a similar experience with the fly problem: he solved in about a minute using the a/(1–r) formula. I got it in about 10 seconds using engineering insight.<br>
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Speaking of which... a new puzzle (of sorts) occurred to me this morning as my teenage mathematician bounded down the stairs baDOOMP ba DOOMP baDOOMP and the thought came to mind: I don't bound down the stairs like that. So... why don't I bound down the stairs like that? Because it hurts my knees to bound down like that. So... why does it hurt my knees but not his, and why didn't it hurt mine thirty years ago? If I damaged my knees with running, well... that was over with thirty years ago. I stopped running daily about in my early 30s, but now I am paying the price for that youthful indiscretion. But thirty years ago, after I stopped running, I would still bound down the stairs without pain.<br>
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For the last 30 years, I have taken good care of my knees, but still they remind me early and often of the decades of youthful indiscretion, of the common paradigm of the day: no pain no gain. OK, we can arrange pain. In retrospect it should have been no brain no gain, but we didn't know. We thought it had to hurt to do any good. So... we made it hurt. Now we don't want it to hurt anymore, and even if it does, it appears to be a no gain anyway.<br>
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So... why do old knees hurt but young knees do not? What physiological change took place? <br>
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spike<br>
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