Clearhat - Tag - infinity
2024-10-11T11:39:09-05:00
urn:md5:681a01897c8b427643d25b22ede9bc11
Dotclear
Euclid's clever solution to the problems of the void and the infinitesimal
urn:md5:347c0fbde07a18980a048c9a69c97483
2024-09-02T12:00:00-05:00
2024-10-05T01:01:46-05:00
Clearhat
Mathy Stuff and Ternary Logic
Euclid
geometry
infinitesimal
infinity
point
<h4>Euclid's Definition 1</h4>
<p>We normally think of a geometrical point as being smaller than anything, because of where Euclid placed it in the hierarchy of geometry. Building up from simple to more complex, his first definition is the simplest: "A point is that which has no part." Clearly, Euclid intended for anyone reading to be thinking of something simple, like a pebble in the hands of a sculptor, with "parts" being removed, getting smaller and smaller until it has no parts.</p>
<p>Mathematics is considered to happen outside the bounds of physics, but it is clear that his definition presupposes the existence of <em>something like matter</em> which he then eliminates, leaving a point so small that it no longer has no material existence. Since the matter (the "part") is eliminated from existence even as it is mentioned into existence, there is no need to discuss it further. In other words, although matter is being used to <em>define</em> the point, <em>the point</em> itself has no matter within it.</p>
<p>At the time, this approach was a clever way of managing two problems with Greek mathematical understanding.</p>
<h4>Problem 1: <em>Horror vacui</em></h4>
<p>First was a strong opposition to the idea of "the void." Wikipedia <a href="https://en.wikipedia.org/wiki/Horror_vacui_(physics)" hreflang="en" title="Horror vacui (Wikipedia)">says</a>:</p>
<blockquote>
<p>In philosophy and early physics, <strong>horror vacui</strong> (Latin: horror of the vacuum) — commonly stated as "nature abhors a vacuum", for example by Spinoza — is a hypothesis attributed to Aristotle that nature contains no vacuums because the denser surrounding material continuum would immediately fill the rarity of an incipient void. Aristotle also argued against the void in a more abstract sense: since a void is merely nothingness, following his teacher Plato, nothingness cannot rightly be said to exist. Furthermore, insofar as a void would be featureless, it could neither be encountered by the senses nor could its supposition lend additional explanatory power.</p>
</blockquote>
<p>Following Aristotle, Euclid would have understood that geometry could <em>never</em> be founded upon the void, even though it would be the one thing conceptually more simple than "that which has no part." Remember this was in the age where <a href="https://en.wikipedia.org/wiki/Hippasus" hreflang="en" title="Hippasus of Metapontum (Wikipedia)">a man was drowned at sea for telling the hidden truth</a> about irrational numbers. Because the <em>horror vacui</em> was strong enough (including the fact that there were negative religious connotations involved) it was likely <em>conceptually impossible</em> for Euclid to understand the point that "that which has no part" and "the void" were identical. In our age, we can see they are essentially talking about the same thing, but Euclid likely imagined that he was creating a technical description of a partless mathematical entity which was unrelated to the void.</p>
<h4>Problem 2: Measuring infinitesimals</h4>
<p>The second problem with Greek mathematical understanding would have helped hide the first problem even from a penetrating mind like Euclid. This problem required twenty more centuries to pass before it began to be resolved with the invention of calculus by Newton and Leibniz. It's the problem of the infinitesimal. This is the problem that Euclid was solving by saying "A point is that which has no part" instead of "A point is that which has the least-measurable-sized part," which would be an awkward but precise way of capturing the basic intuition that happens when we normally think of a point. Until we learn geometry, we do not think of point-like things as "having no part," but Euclid does, and he does it to avoid any reference to a measurable part.</p>
<p>Why? Infinitesimals are problematic. Euclid would have known about Zeno's paradoxes which involve an infinite number of infinitesimals to measure any distance -- the existence of which can be used to prove that motion is impossible. Infinitesimals in his time were understood as the <em>mathematical inversion of infinity</em> -- the opposite of infinity -- and they were already known to be logically nontrivial. Euclid knew he could not rely on an infinitesimal for the Definition 1 of geometry.</p>
<p>So this is why "that which has no part" is used instead of "that which has the tiniest possible size" or anything like it. And in using this phrasing, Euclid sidestepped the problem with infinitesimals. Curiously, the <em>similarity</em> of this description to infinitesimals helped cover up Problem #1 by drawing attention away from any obvious void analogies.</p>
<h4>The Solution: <em>Neither here nor there</em> is better than <em>either</em> here <em>or</em> there</h4>
<p>So those are the two problems that Euclid faced. He invented a clever solution which evaded both horns of the dilemma simultaneously.</p>
<p>At heart is a semantic trick which, when unraveled by logic, is not itself a better solution than either <strong>the void</strong> or <strong>the infinitesimal</strong>, being a hybrid of both which is also neither. For the purpose of comparing with the two problematic solutions, let's call this semantic trick <strong>the neither void nor infinitesimal</strong> because it's not really a "third" alternative so much as it's a clever combination of the first two in a way that pits them against each other and makes it possible to think a genuine third alternative exists.</p>
<p>All three solutions are roughly equal to the task of defining <em>the simplest thing upon which to build the rest of geometry</em>. That was Euclid's aim while creating Definition 1. But Euclid's choice, "that which has no part," has the paradoxical advantage that it brings an infinitesimal "that" into being and then immediately eliminates "that" by saying <em>it has no being</em>. We know it has no being because it has "no part." We're supposed to politely ignore the fact that something which has no part necessarily has no whole. This all happens within a simple phrase.</p>
<p>Any arguments talking about the simple phrase's similarity to the void can be easily dismissed, as well as any arguments talking about its similarity to the infinitesimal.</p>
<p>This hybrid solution which is half-infinitesimal and half-void while also being neither infinitesimal nor void <em>while also being essentially simple</em> must have seemed so clever to Euclid. It worked well enough to be the first definition in geometry for many centuries. However, by the time we get to our era, we've developed delicate logical scalpels which are sharper than the more concrete-minded ancient Greek mathematical tools, and it frankly doesn't work any more, although few actually take the time to question it.</p>
<p>Even though armed with greater precision, we now have a problem that Euclid's geometry didn't have when it was created: It has been around for a couple dozen centuries and its logical assumptions have pervaded the way we think in fundamental ways. It's hard to think outside of its constraints. The advent of non-Euclidean geometry in the 1800s was a good start in shaking these solid foundations. But we can do better.</p>
<p>Now it's time to unravel something even <a href="https://en.wikipedia.org/wiki/Parallel_postulate" hreflang="en" title="Parallel postulate (Wikipedia)">more fundamental than the 5th postulate</a>: the 1st definition.</p>
<p> </p>
https://www.clearhat.org/post/A-point-is-that-which-has-no-part-is-another-way-of-saying-a-point-is-infinite-and-this-realization-blows-a-hole-in-quantum-physics-which-talks-about-pointlike-particles#comment-form
https://www.clearhat.org/feed/atom/comments/293
A whole new way into heart meditation reveals... light... on Thanksgiving morning
urn:md5:e2545e736b377246ba5d4960932c09e6
2023-11-23T08:14:00-06:00
2024-04-12T16:15:15-05:00
Clearhat
Pursuing Happiness via Meditation
heart
infinity
meditation
mindful
projective geometry
<h4><img alt="" class="media media-right" src="https://www.clearhat.org/public/light_streaming_upward_from_darkness.png" style="width: 50%; margin: 10px; float: right;" />A year of mindful<sup data-footnote-id="8rnuz"><a href="https://www.clearhat.org/post/a-whole-new-way-into-heart-meditation-reveals-light-on-Thanksgiving-morning#footnote-1" id="footnote-marker-1-1" rel="footnote">[1]</a></sup> meditation</h4>
<p>Here it is early the morning of Thanksgiving 2023 and I just discovered a rather deep heartful meditation which took about a year to learn. The essential idea is quite simple and took only a few minutes to comprehend, but it took a year of preparation so that I could understand just the right approach.</p>
<p>For anyone new to meditation, a good way to understand it is that it is <em>a way to listen to God</em> instead of the usual <em>talking to God</em> commonly known as prayer.</p>
<p>For about a year now, I've been learning a new approach for <em>mindful</em> meditation, not heartful. They are two very different flavors of listening.</p>
<p>As a matter of background context, I've been learning to meditate for a couple decades, with no teachers, no books, nor school.</p>
<h4>Prayer that slowly became meditation</h4>
<p>Unlike most people who meditate, instead of teachers or any other externally-developed techniques, I started without realizing I was even headed in the direction of meditation. The long journey began as a new form of prayer. I started seeking to <em>listen</em> to God instead of only <em>talk</em> to him. That simple desire changed everything.</p>
<p>Over the years I've developed relationships with sensitive inner awarenesses, inner dimensions of the soul, so to speak, which are like teachers I encounter within my own thoughts and feelings during meditation<sup data-footnote-id="mqigk"><a href="https://www.clearhat.org/post/a-whole-new-way-into-heart-meditation-reveals-light-on-Thanksgiving-morning#footnote-2" id="footnote-marker-2-1" rel="footnote">[2]</a></sup>.</p>
<p>In the language of a non-meditating person, learning from such an inner teacher feels similar to how intuition operates. The practice of years of quiet listening makes the experience feel more like a conversation, a dialogue instead of a monologue. Given room to breathe, intuition develops into a more distinct personality within the soul, and speaks with deep wisdom.</p>
<p>If that doesn't make sense, perhaps the best way to frame the idea of "distinct personality within the soul" is to think of the common concept of angels. In other words, if you begin with the sole intention to <em>Listen</em> instead of simply <em>Talk</em> to heaven, and you keep at it for years, eventually you begin to Hear. If you are inclined to believe in angels, this hearing naturally arrives in the form of angels -- which is simply the Greek word for 'messengers' -- who will guide you into increasingly more pure ways of listening. Remember the experience is more like a conversation with well-developed intuition than any kind of glowing person with a message from heaven. Perhaps there are such experiences in the future for me, but til now, such conversations tend to operate within my own intuitive awareness.</p>
<h4>Mindful and heartful meditations are different from each other</h4>
<p>In the "outer" world, we learn to listen <em>in order to speak</em>. Our whole education system is oriented around teaching this kind of listening. However, this is a mixed motive that we must unlearn by simplifying our listening. This is a process which takes years. I've been at it for two decades and still have much to unlearn.</p>
<p>There are many ways to meditate. From time to time I experiment with different variations, but generally tend to keep things simple as a matter of principle. About a year ago intuition taught me a new way to meditate by focusing ahead, with the eyes, instead of the usual focus down, into the heart. How that happened is hard to put into words. Rather than tell the story of how I learned that step, I'll simply say the process was similar to what I'm going to write about below: a combination of small occurrences in normal daily life along with some well-placed intuitive promptings in my inner life which led me to a new way of meditating.</p>
<p>This change from heart to mind happened over the course of a few weeks and was a notable difference for me, for I had long learned and preferred to focus more on the heart. What I did not know then was that, a year ago, I was focusing on <em>the mind's version of the heart</em>. Today I discovered my actual heart, which is much deeper within than I realized.</p>
<h4>A particular kind of mathematical infinity</h4>
<p>So for the past year, that's where I've been, slowly learning the art of mindful meditation.</p>
<p>A few days ago something new came up while I was studying math, which I do often because I love the beautiful concepts found there. In a <a href="https://www.youtube.com/watch?v=NYK0GBQVngs" hreflang="en" title="Projective geometry | Math History | NJ Wildberger">video</a> on Projective Geometry by professor Wildberger I learned about the "infinite horizon" and "line at infinity" or "point at infinity" concepts. I'm studying this because physicist Paul Dirac once said he was able to use this kind of geometry to inform his intuition about quantum physics. He had the best intuition about quantum physics -- no one has matched his insights yet -- so it's useful to consider his method, if you want to learn about the quantum realm.</p>
<p>Projective Geometry is a form of geometry developed by artists during the Renaissance to help guide drawing perspective in artwork and architecture. Leonardo da Vinci was an influential proponent of this method. It changed the world of visual art, which became much more realistic. Since then, projective geometry has slowly become more mathematically sophisticated because it turns out to have serious mathematical uses as well.</p>
<p>To get right to the point -- literally -- here is an example of the main concept, drawn by artist and mathematician <a href="https://en.wikipedia.org/wiki/Leon_Battista_Alberti" hreflang="en" title="Renaissance artist and mathematician Leon Battista Alberti">Leon Battista Alberti</a> in the 1400s:</p>
<p><img alt="Leon Battista Alberti, Of Painting in three books, “Book I”" src="https://www.clearhat.org/public/Leon_Battista_Alberti_-_projective_geometry.png" style="width: 100%;" /></p>
<p>See that point on the horizon, straight ahead? Remember that. But also note the similar point off to the left, which artists know about, but which most people don't. These two points, and many others, form a "line at infinity" which is directly in front of us every day, if only we turn our attention to it. Here is another image of railroad tracks, to make it even more clear what I mean about "right in front of us every day":</p>
<p><img alt="Parallel railroad tracks form a vanishing point" class="media media-left" src="https://www.clearhat.org/public/.800px-Railroad-Tracks-Perspective_m.jpg" style="width: 35%; float: left;" />Although I'm only just learning about it, I've actually written before about Projective Geometry without realizing it, because it overlaps with other concepts I've studied. For example here is <a href="https://www.clearhat.org/post/upon-realizing-i-independently-discovered-new-theory-mathematics" hreflang="en" title="Upon realizing I independently discovered a new theory in mathematics">something I wrote</a> about Wheel Theory a year and a half ago.</p>
<p><img alt="" class="media media-right" src="https://www.clearhat.org/public/.480px-Real_Projective_Line__RP1__s.png" style="width: 20%; float: right;" />In simplest form, Wheel Theory looks at what happens when the <em>horizon line</em> in Alberti's image above is placed upon a circle, where it becomes a single point (see the image to the right). So I've heard about Projective Geometry for years but never looked into it. I finally took some time to watch a video, and was pleasantly surprised to find it easy to understand.</p>
<h4>Applying math insights to meditation</h4>
<p>During meditation, I playfully turned my focus toward the projective geometry idea of infinity. It is kind of like finding a spot on a distant horizon and focusing on it as a way of quieting the inner landscape.</p>
<p>I've actually done this before while walking down an old country road. Because it was a remote gravel road with no traffic, it was safe to "tune out" of the immediate awareness and focus instead on a tree about a mile up the road. What happened after about a minute of walking like this became similar to meditation, but it happened while my eyes were open. I found myself in a very pleasant but somewhat strange, altered state of awareness. I later learned it was because my left brain let my right brain take over for awhile<sup data-footnote-id="tmxg2"><a href="https://www.clearhat.org/post/a-whole-new-way-into-heart-meditation-reveals-light-on-Thanksgiving-morning#footnote-3" id="footnote-marker-3-1" rel="footnote">[3]</a></sup>. I really ought to do that again some time, because the experience was quite enjoyable, and it's the only way I know for how to politely disengage the left brain for a little while.</p>
<h4>Simple does not always mean easy</h4>
<p>But continuing with my experiment in applying projective geometry inward, this ended up being a gentle yet powerful meditation technique, only slightly different from what I'd been learning every day for the preceding year. Note that I really mean "every day," since I've been encouraged by inner guidance <em>every day</em> to keep my focus forward whenever it wanders. It's hard work to keep the focus in a single direction. You might think because it's simple, it's easy. It's not.</p>
<p>Only today I finally understand how such a simple thing required so much learning because <em>it is not easy</em> but must be learned repeatedly <em>until it is easy</em>. And, because of the delicate nature of meditation, it is not something that can be forced, or it would become something else. Learning through repetition instead of through force of will (as some teachers teach), thus it took a year for me to get to the point where this next step could happen.</p>
<p>The main difference between the year-long training technique and the new, more playful technique is to move the focus of attention from about 1-2 foot ahead of the eyes to infinity, while remaining oriented straight ahead. Although this happens in darkness with eyes closed, it's comparable to turning one's attention from a page number at the top of a book held at arm's length to a tree on a distant horizon <em>behind</em> the book. The direction of the gaze doesn't change but the point of focus is projected much further. But note that seeking too deeply would be forceful. This is a mistake easy to make in darkness, but whenever I push too hard, the inner teacher guides me to relax a little. So this is not a distant infinity, but more like a relaxed, gentle focus, as though beholding the details of a beautiful sunset, far away but <em>easy</em> to view.</p>
<p>There is a delicate balance point between seeking and relaxing which is the essence of a good meditative state.</p>
<p>Cool experiment. What I did not know was, in doing this yesterday, I received the final lesson of the year-long "mindful" meditation.</p>
<h4>Light within the heart</h4>
<p>This morning as I began meditating a thought occurred to me: "What happens if you go inward now, like you've been learning to go outward?" If you know me, you know this is a meaningful, maybe even provocative question, as it is something I've approached from many angles over many years, so much so that to hear the question surprised me. It's like asking a bowling pro if he ever tried using a heavy bowling ball. I knew the question deserved more than a brief dismissal, though.</p>
<figure style="background-color:#ddddff;padding:10px;float:left;margin:5px 10px 0px 0px"><img alt="" class="media media-left" src="https://www.clearhat.org/public/.Hopf_Fibration-clearbg_s.png" />
<figcaption><span style="font-size:10px">Hopf fibration</span></figcaption>
</figure>
<p>So I thought of the heartful meditation which I learned long ago, back when I first began journeying "into the heart." I also thought of the fascinating mathematical object known as the <a href="https://en.wikipedia.org/wiki/Hopf_fibration" hreflang="en" title="Hopf fibration">Hopf fibration</a>, which contains another mathematical form of infinity. (See the image to the left. It's like a Mobius strip grew up into a sphere. Animations of this mathematical concept are beautiful, and reveal an intriguing sense of inward which is also outward.) I've also been studying this recently, again because it is related to Paul Dirac's intuition for quantum physics.</p>
<p>Going into the heart was familiar, but with the new mathematically-inspired intuitions about how to see infinity I was thinking of going <em>deeper</em> within. In other words, in the same way that "seeing the tree on the distant horizon" is deeper than "reading a book at arm's length." So I tried it. And sure enough, after a little while of reorienting <em>downward</em> toward the heart instead of <em>ahead</em>, before my eyes, meditative focus went much deeper into my heart than it ever did before.</p>
<p>I eventually arrived at the wordless region, which is of course hard to describe with words, although I try in <a href="https://www.clearhat.org/post/Awaken-the-deep-insights-from-a-meditative-adventure-on-the-nature-of-Speaking-and-Being" hreflang="en" title="Awaken the deep: insights from a meditative adventure on the nature of Speaking and Being">another recent post</a>. The best way to summarize the experience is with these three words:</p>
<p>"I found light."</p>
<h4>The beginning of a whole new world</h4>
<p><img alt="" class="media media-right" src="https://www.clearhat.org/public/heart-woodcut-colored-by-jdjs.png" style="width: 35%; border-width: 1px; border-style: solid; margin: 10px; float: right;" />I found light within my heart. And it was so beautiful, this light within the darkness. It was not a transitory flash like the afterimage of some memory in mind's eye -- a common experience during meditation. It was not that tiny pinpoint of light which sometimes appears and then fades when you try to follow it. It was not even the light which interpenetrates the darkness of a meditation, which I've experienced a few times<sup data-footnote-id="e2hlg"><a href="https://www.clearhat.org/post/a-whole-new-way-into-heart-meditation-reveals-light-on-Thanksgiving-morning#footnote-4" id="footnote-marker-4-1" rel="footnote">[4]</a></sup>.</p>
<p>This was a whole new realm of light, which I could see on the other side of the darkness. It looked kind of like... um... whales of light moving deep in the dark ocean, if that makes any sense. I tried to capture a sense of it by inverting an underwater picture (see the first image for this article, above), but like trying to describe the wordless region, it's a little hard to portray accurately, though the image above comes close.</p>
<p>While it happened, it seemed rather ordinary because I was in a meditative state which is non-judgmental, but after the meditation concluded I was deeply moved. I realized what had just happened was the beginning of a whole new world. This time, I could see the light in the distance, but this is <em>the kind of light I can go into</em>, as I continue developing the focal point which I've been learning for the past year.</p>
<h4>The method of the inner teacher thus proved</h4>
<p>Once I discovered this new world, knowing that I had just found a way to get to that light at will in the future, I immediately realized something wonderful, confirming the value and method of the way I learn about meditation; from within.</p>
<p>The inner teacher of intuition was teaching me daily for <em>a solid year</em> in order to find a very specific kind of... outward infinity. It's a kind of infinity which is hidden in plain sight until you realize how it works. Then as soon as I found it, it was time to go inward to the heart.</p>
<p>Only <em>then</em> I found the inward infinity, which I never would have found without this outward guidance. This is unlike any previous heartful meditation. Indeed, I've experienced hundreds of heart meditations, but I never saw light before today, light coming from the center of my heart, in a way that I can now go toward at will.</p>
<p>[Update, a few days later: I've been seeking this experience during meditation for a few days now. Although I now know what to do, it looks like it could be weeks or months before I experience the inner light again. But I know it will happen, and then eventually it will happen again, and again until I've fully opened a door into this inner world of light.]</p>
<p>I am in awe at how intuition revealed this to me so carefully, for so long, in order to reveal a point that took only minutes to comprehend. But could not have been comprehended if I hadn't taken that year.</p>
<p>Thankful, you might say.</p>
<section class="footnotes">
<header>
<h4>Note(s)</h4>
</header>
<ol>
<li data-footnote-id="8rnuz" id="footnote-1"><sup><a href="https://www.clearhat.org/post/a-whole-new-way-into-heart-meditation-reveals-light-on-Thanksgiving-morning#footnote-marker-1-1">^</a> </sup><cite>I use the word "mindful" in a non-standard way throughout this article, which I only realized a couple months later while researching another article that goes into the standard way of practicing mindfulness. What I mean in this article is "of the mind" as compared to "of the heart." What is a more standard usage of "mindful" is talking about a set of practices which increase self-awareness, and "living in the moment."</cite></li>
<li data-footnote-id="mqigk" id="footnote-2"><sup><a href="https://www.clearhat.org/post/a-whole-new-way-into-heart-meditation-reveals-light-on-Thanksgiving-morning#footnote-marker-2-1">^</a> </sup><cite>Most of the time meditation is pretty boring, but occasionally I have some cool adventures in the inner realms. Here is a recent example written a couple months ago: "<a href="https://www.clearhat.org/post/Awaken-the-deep-insights-from-a-meditative-adventure-on-the-nature-of-Speaking-and-Being">Awaken the deep</a>: insights from a meditative adventure on the nature of Speaking and Being."</cite></li>
<li data-footnote-id="tmxg2" id="footnote-3"><sup><a href="https://www.clearhat.org/post/a-whole-new-way-into-heart-meditation-reveals-light-on-Thanksgiving-morning#footnote-marker-3-1">^</a> </sup><cite>In the first TED talk which went megaviral with many millions of viewers: "<a href="https://www.youtube.com/watch?v=UyyjU8fzEYU">My stroke of insight</a>" by Jill Bolte Taylor, the speaker is a neuroscientist who had a stroke one day. Because half her brain stopped working, she discovered how the left brain and right brain operate from a first-person view. This was unknown to science before. This video is stunning if you haven't seen it already, highly recommended if you want to understand how your left brain rarely, if ever, lets go of control. She now teaches people how to integrate both halves to work in better harmony than our culture normall teaches.</cite></li>
<li data-footnote-id="e2hlg" id="footnote-4"><sup><a href="https://www.clearhat.org/post/a-whole-new-way-into-heart-meditation-reveals-light-on-Thanksgiving-morning#footnote-marker-4-1">^</a> </sup><cite>The "light which interpenetrates the darkness of a meditation" is an elusive description if you haven't experienced it, so I'll give you an image to help understand what I mean. It's kind of related to Platonic forms, if you understand that concept, but even if not, try this: Imagine a scene you enjoy, say a path through a forest or a field of flowers, or even just a single flower. Now imagine that same scene made out of light. Now imagine both simultaneously, and you can understand a little what this particular meditative state is like.</cite></li>
</ol>
</section>
https://www.clearhat.org/post/a-whole-new-way-into-heart-meditation-reveals-light-on-Thanksgiving-morning#comment-form
https://www.clearhat.org/feed/atom/comments/277
The origin of synarché and why it is an infinity greater than all infinities
urn:md5:0e7d711bee21bbf3f56369d8f867f646
2021-04-22T21:20:00-05:00
2024-05-19T13:23:23-05:00
Clearhat
Mathy Stuff and Ternary Logic
Ein Sof
infinity
ternary logic
zero
<p>The word <em>infinite</em> is from the Latin roots for "un-limited" or another way to say it: "without-end." Structurally, this word is similar to Greek <em>a-perion</em>, Hebrew <em>ein-sof</em>, Chinese <em>tai-ji</em>, and many others, all referring to the same underlying concept: that which is so great it is beyond ability to measure, endless. (Like un-measureable, "endless" is of course another example of the same "negate something-limited" structure, which combines into one word <em>a negative</em> with <em>an ultimate limit</em> of some kind.)</p>
<p>It's one of those little things most people don't notice or care about, but I believe there is actually a really deep question going right to the heart of the way we perceive what we call reality on a fundamental level:</p>
<p><strong>Why is a concept which is the utmost, the greatest, the ultimate, the infinite, the "beyond-measure," consistently defined by reference to <em>what it is not</em>, rather than by what it is, <em>no matter what language you choose?</em></strong></p>
<p>I recently faced this paradoxical question directly when I was working on a thought experiment which took the two endpoints of positive and negative infinity, brought them together, and merged them with zero where it lives at the midpoint of the number line (I turned the ordinary number line into a circle). That may sound like a peculiar idea at first, but it's a thought experiment I have entertained numerous times over the past decade or more because doing so answers a number of other questions.</p>
<p>This particular thought experiment reveals hidden insights about fundamental math and logic structures, often hidden because we take our foundation for granted and don't dig deeper in some of its riddles. Hence I was contemplating this idea the other day when some missing pieces involving complex numbers and the Riemann Sphere fell into place. Suddenly, before my eyes for the first time ever, was a coherent visual representation drawing on multiple separate thought experiments coming together into a single intuitively simple form.</p>
<h4>A coherent way to visualize the union of several paradoxes</h4>
<p>Here is a preliminary image I created <a href="https://www.clearhat.org/index.php?post/imagining-riemann-sphere-rotated-new-insights">while writing about this insight the other day</a>. It's the well-known Riemann Sphere rotated 90 degrees backward so that <strong>Infinity</strong> (which is usually at the top) is at the backside. This rotation also brings the bottom <strong>Zero</strong> to the front (and also perceptually nearer to the ∞ than I'd ever seen it before, which is important. That small shift in perspective may have been the trigger for what happened next.) To do the thought experiment, I started with <a href="https://en.wikipedia.org/wiki/Riemann_sphere">this image</a> from Wikipedia and brought the ∞ and the 0 together <em>into the center</em> where the question mark is pointing (see illustration):</p>
<p align="center"><img src="https://www.clearhat.org/public/RiemannSphereCorrectedWithQuestion.png" width="200" /></p>
<p>For the first time, probably because I was using a sphere instead of a linear number line, it all worked! Not only could ∞ and 0 coherently unite at the center, but doing this move left two smaller question marks: What goes in the places where ∞ and 0 were (i.e. where they can be seen in the image above)? I thought about this for a while, but it didn't take long to see there was now room for complex "j" and "k" to fall neatly into place where the ∞ and the 0 had been, without disrupting anything else. I confess I'm not smart enough to know why this shouldn't work, so I continued with the thought experiment since it seemed to fit and was getting profound. I continued making similar sketches using Paint.Net to quickly draft the ideas, and eventually developed things to the point where I wanted a new name for the point at the center where ∞ and 0 are combined.</p>
<p><img src="https://www.clearhat.org/public/strongs_arche.png" style="width: 40%; margin: 10px; float: left; border-width: 1px; border-style: solid;" />Searching for an apt name, I experimented with numerous options over several days, whenever I had an idle moment. I began researching things like the word coined by ancient Greek Anaximander: <em>aperion,</em> and its predecessor, the word <em>arché</em>, said to be coined by his predecessor Thales (see illustration). I began looking into the concept of infinity in other languages, as well as looking at different possible symbols. At one point, I drew a symbol which combined zero and the lemniscate ∞ into a single shape.</p>
<p>As symbols go, it's a pretty one which conveys its meaning quickly, in spite of looking like something from a Prince album cover. I especially liked the fact that this new symbol for infinity had three parts instead of the usual two, which corresponded well with my understanding of infinity as it is perceived using ternary logic. Many is the time I've pointed out that the main exclusion of binary logic's "excluded middle" is infinity itself. Likewise, I have lamented the resulting hard link between binary logic and finity.</p>
<p>However, much though I liked the new zerofinity symbol, I did not like the resulting complexity of the symbol -- its shape ought be more simple, more elegant, to convey the essential simplicity, or singularitiness, in this new kind of infinity. I also knew that one of the fundamental stumbling-blocks to understanding ternary logic in its own terms is the tendency to think that ternary "three" is simply "one" plus "two." The true ternary "three" arises in a non-linear manner, not simply serially after "two." Therefore merging a lemniscate symbol with a zero might be a nice way to begin, but the stumbling-block of linearity would be right there in front of everyone, so I sought something deeper and more intrinsically elegant.</p>
<h4>Infinity in different languages leads to <em>what is infinity?</em></h4>
<p>Working on etymologies and intrinsic rules like this, about then I noticed that I kept encountering references to infinity in different ancient languages, but they were all structurally similar: they <em>all</em> could be translated to something very close to "without-end." Yet they were also all talking about something enormous, vast, encompassing, boundless, so immense as to contain everything everywhere. As I thought about this, it seemed obvious that some unknown perceptual dynamic had led humans to repeatedly arrive at a word for <em>immeasureable</em> which was rooted in the <em>measureable</em>. I began to wonder, how many cultures defined infinity by saying what it is not? And why not call it <em>what it is</em>, instead of <em>what it is not</em>?</p>
<p>This led to the question: if we're going to go that direction, what is it? What is infinity, really?</p>
<p>Thought experiments in this direction were fruitful right away. For years, I've been studying the history of a curious phrase: "<a href="https://www.clearhat.org/blog/posts/god-is-an-infinite-sphere-the-center-of-which-is-everywhere-the-circumference-nowhere">God is an infinite sphere, the center of which is everywhere, the circumference nowhere</a>." I well know the infinite is not simply "out-there," "far-away," "at-the-end-of-the-numbers," "beyond-all-that-is," but it is <em>right here</em> -- and everywhere else also. Everywhere in space, everywhere in time, everywhere in all dimensions. It is true that what we call infinite has several branches of meaning (like "potential" vs. "actual" and "qualitative" vs. "quantitative", "countable" vs. "uncountable," etc.), but the trunk of its meaning is consistently a reference to something endlessly unbounded.</p>
<p>If infinity is so unbounded at its root essence, then why does it have boundaries? Why does it stop at the end of mathematics? Why can it not penetrate everything everywhere always? Suffice it to say that I have long thought confining infinity to a limited place in mathematics -- at the end of countable numbers -- reveals a weak understanding of how truly vast "endless" infinity actually is.</p>
<p><img src="https://www.clearhat.org/public/true_endless_infinity_is_everywhere_always.png" style="width: 50%; margin: 10px; float: right;" />In short, I well understand that <strong>infinity exists not only "after" all countable numbers but it also penetrates all of them from beginning to end</strong> -- although I will admit that I only realized the clarity of that thought just now as I wrote it. I promptly bolded it because it is clear and succinct. Here, I made an image to convey the idea (see illustration):</p>
<p>If Georg Cantor's ideas on infinity "created a paradise" in the words of the great mathematician David Hilbert, what describes the beyond-paradisaical beauty of the <em>significantly larger</em> form of "infinity" -- the largest possible -- which encompasses all of Cantor's paradise, plus all the rest of mathematics, <em>plus all of physics</em> as well? (For a visualization of what I mean, consider the illustration above. Instead of one of the black-bordered boxes <em>containing</em> infinity, how about the whole illustration, or better still, how about this whole article? The whole Internet? Where do we stop measuring infinity?)</p>
<p>If we're going to call something "without-end" then why are we giving it an end? Is the "set" of mathematical Set Theory not <em>itself</em> contained within the greater infinity which is the set of real, physical, life itself? What exactly forbids mathematics from touching physics... forever... even at the outermost periphery of infinity which supposedly goes on forever? Clearly it doesn't go on forever if it is limited to the boundaries of mathematics.</p>
<p>This true "deep infinity" so utterly dwarfs the already-stupendous mathematical infinities of Cantor's paradise, there is most certainly a need for a new word describing it which is <em>not built out of a reference to what it is not</em>, pardon the double-negative. An "endless" infinity which "ends" and is defined by what it is not is just as paradoxical as Euclid's "a point is that which has no part," which I have discovered in other thought experiments is ironically the founding-stone that inevitably leads to this paradox with infinity.</p>
<h4>We need something new, built from the ground-up with an etymological reference to its everywhereness</h4>
<p><img src="https://www.clearhat.org/public/beyond_infinity.jpg" style="width: 30%; margin: 10px; float: left;" />I began researching this concept. Soon I was reading about <em>Ein Sof</em> (again), I was reading about <em>Ayin and Yesh</em> (that was a new one). Then I was back to <em>apeiron</em> and Anaximander, then over to Aristotle, and forward again to Hilbert and Einstein, then back to (yet another) <a href="https://web.archive.org/web/20210922142326/https://www.math.tamu.edu/~don.allen/history/infinity.pdf">PDF on the history of infinity</a> in case I might find something new.</p>
<p>I've read a dozen of these kinds of PDFs over the years because the history of infinity is equally as fascinating as the history of zero, both of which have met fierce resistance within mathematics but are accepted as fundamental elements these days which even the youngest schoolchildren learn. In fact, while randomly at a bookstore two days ago, I of course bought one thing: a book on infinity, ironically titled "Beyond Infinity" as if there could be something <em>beyond that which is endless</em> (see illustration). Maybe we gravitate to such "beyond beyond-end" metaphors because we sense that infinity is not yet "the ultimate" which it pretends?</p>
<p>I was up in the middle of the night, learning that <em>Tai Chi</em> was not just an elegant form of boxing but a philosophical concept "tai-ji" meaning something akin to "Supreme Beyond-Pole," which is not quite the same as the "Supreme Ultimate" I learned back when I took a semester of Tai Chi years ago. My mind wandered often to resolving this quandary for several days.</p>
<p>I found myself in a curious research paper reading about the structure of what happens beneath Planck scale, where physics breaks down and intuitively <em>everything becomes one with everything</em> but nobody talks about it because they think the Planck measure, at 10<sup>-35</sup> meters, is as small as anything can ever be.</p>
<p>Coincidentally, <a href="https://www.clearhat.org/index.php?post/rather-curious-analysis-and-assessment-gateway-process-and-its-long-lost-page-25">that particular research paper</a> was due to another line of study which fell into my lap at the same time I was researching infinity. Such coincidences happen often when I am studying math, so much that they should not be called coincidences, but synchronicities. Confirmation that it was a synchronicity came from a rather literally random direction that I should briefly explain.</p>
<h4>Brief detour to breakdown a synchronicity</h4>
<p><img src="https://www.clearhat.org/public/far_away_from_rational_thought.png" style="width: 40%; margin: 10px; float: right; border-width: 1px; border-style: solid;" />Long ago, while studying similar ideas, I was meditating, and pretty far away from rational thought (see illustration) when I gained the purely intuitive (i.e. non-rational) understanding that true Infinity, true Oneness, and true Random -- all three of these -- were somehow <em>the same thing</em>. I made a note to myself during that meditative thought experiment, telling a future me in the strongest possible way: "Infinity, Oneness and Random are all the same. Remember that!"</p>
<p>Without going into more detail on a decidedly non-linear hard-to-explain process, I'll simply say that is why I went to the search engines looking for a symbol for random. To me, random is somehow related to infinity, so I cannot pursue one for long without bringing in the other. Hopefully one day, I'll find the more rational version of this insight so I can share it with others.</p>
<p>However, there is no reliable symbol for random.</p>
<p>Honest, try and search, you'll find fragments, some proposed ideas (card shuffling, dice-rolling, etc.) which are appropriate to specific contexts, and lots of debate, but nothing definitive like you find with Infinity or One or Zero, these kinds of big ideas. Random is a hard idea to symbolize, so... it remains to be symbolized.</p>
<p>I dug deeper. And that's where I first came across the Greek letter "ξ" pronounced "xsi," which Riemann used when writing about a variation of his famous zeta function (later the uppercase form of xsi "<b>Ξ</b>" was adopted for this purpose). According to Wikipedia, it's also a symbol used <a href="https://en.wikipedia.org/wiki/Xi_(letter)">when talking about random variables</a>, which is why it came up in my search.</p>
<p><img src="https://www.clearhat.org/public/lowercase_xsi_small.png" style="width: 100px; height: 189px; margin: 10px; float: left;" />I pondered the simplicity, elegance, and beauty of that symbol, thinking about its three curliques, which brought me back to ternary logic and how its "third pole" gently blends everything into infinity... and then remembered I was on the larger task of finding a symbol and name for Infinity and Oneness, not Random. No one would get this reference to random but me. So I dropped it, and continued the larger search.</p>
<p>Looking at Greek and Latin roots, I found myself back in Strong's concordance again, contemplating the meaning of the word <em>arché</em>, not because of its relationship with infinity, but this time because "beginning" is the opposite of "end" in the "without-end" of "infinity."</p>
<p>Arché means "beginning" and "origin" and even "sovereign authority," and it has the benefit of being the word that Thales used... long ago... when the idea of infinity was just beginning to be understood. Clearly, arché has a solid etymological root with which to begin.</p>
<p>Okay, the Greek root for "beginning" is going to work, so how about considering a structural opposite to "without-end" by looking at the etymology of "with"? A search engine quickly pulled up a list of <a href="https://www.englishhints.com/latin-prefixes.html">Greek and Latin prefixes</a>. I went to Latin first, but "com-arché" or "coarché" or "co-initio" didn't feel right. I went to the Greek roots for the prefix "with."</p>
<p><img src="https://www.clearhat.org/public/ksun_older_form_of_syn.png" style="width: 40%; border-width: 1px; border-style: solid; margin: 10px; float: right;" /><strong>Syn</strong>. That's the Greek root. As I searched around a little more, it was becoming clear that for what I was seeking, "synarché" would be the most accurate neologism. But happily, this time when I went to Strong's concordance for another look at arché, my search engine delivered a webpage that included a reference to Thayer's Greek Lexicon, and there it was (see illustration):</p>
<p>There is an older form of "syn" and it begins with the wonderful ξ! The word is "ξύν," pronounced "ksun" or "xsin." Excited to see the xsi again, I joined the two roots: <em>xsinarché</em>, then <em>ksunarché</em>, but alas, if I wanted anyone else to understand what I was saying, <em>xsinarché</em> was obscure, worse than the awkwardness of <em>cominitio</em> by a yard.</p>
<p>I thought for a little while about that. Wait, what about using the transliteration <em>synarché</em>, but still refering to the original root ξύν <em>xsun</em> when using the Greek?</p>
<p>And there you have it. The origin of "with-beginning" or synarché, spelled with a ξ. Etymologically sound, graphically elegant, a nice solid foundation upon which to build a truly paradisaical "infinity."</p>
<h4>Now the important step of making it easier for others to use</h4>
<p>With that solved, there remained the issue of whether to call this <em>synarché</em> or <em>synarche</em>, to make it easier for others to write it. Diacriticals like é are not always easy to enter into a computer, say, for example, in a URL. Or they can cause display problems in some older terminals (which display something like "synarchÃ©"). Should it be www.synarché.org or www.synarche.org? One should think about such consequences when one creates new words. And what about mathematicians who are already using <em>xsi</em> to refer to random variables? Are there other similar collisions? (Yes, for one example, a little bit of research on the similar word synarchy turns up a whole host of references that are not at all related to mathematical symbols for infinity.)</p>
<p>The answer will not please everyone, but there may not be a way to please everyone on such matters -- the experts who are to be pleased are often the most hair-splitting, since it is by splitting hairs that etymological experts learn what they know. So here's the answer:</p>
<h4>The point where mathematical infinity touches reality</h4>
<p>Synarché, a new word for what was formerly known as infinity -- well, actually, something much bigger -- can be spelled synarché or synarche; either one is acceptable, but synarchy is already taken, so we'll hold firm to the <em>e</em> at the end to keep things clear. The symbol xsi "ξ", which is to be used when referring to the center of the rotated Riemann Sphere with i, j, and k as described above, or to <em>the point where mathematical infinity touches reality</em>, or to all of creation, is just too beautiful to lay aside in pursuit of something more pedantically perfect.</p>
<p>If my intuition about it is correct, people who are already using xsi "ξ" for random variables may enjoy the fact that they're also touching infinity from time to time as they go about their work. (I have a future essay in mind regarding specifically what is meant by "touching infinity," or rather, we should now say "touching synarché," since mathematical infinity cannot be touched, but synarché can -- it is everywhere.) If I am incorrect, no one will care about what I write here so the point is moot.</p>
<p><img src="https://www.clearhat.org/public/whyarenumbersbeautiful.jpeg" style="width: 50%; margin: 10px; float: left;" />My appeal is to mathematical beauty, and I hope, as I continue researching this delightful discovery, it continues to astonish and inspire, and ultimately, it works.</p>
<h4>Some caveats</h4>
<p>I've been gloriously wrong before. Many have stood at the high precipice of defining a new concept within mathematics and discovered that they were wrong. Perhaps as many more never did figure out that they were wrong... but they were. Thus, I may be wrong, and hopefully I have the ability to walk down from a precipice of wrongness if need be. In which case, this is a fun adventure which will soon be forgotten, and nothing of consequence has happened.</p>
<p>For those who appreciate this idea but prefer to stay out of theological debates, the threefold vortex of lowercase xsi refers to "Infinity, Oneness, and Random," or, if that's still too far out there, it refers to the graceful blending-into-infinity nature of the third pole of ternary logic, or even simply the original approach of joining a lemniscate ∞ with a zero 0.</p>
<p>Last thing; a curious puzzle: as a letter, the original meaning of Xsi is "detached from" which is what the three dashes in the uppercase version "Ξ" represent -- the middle bar is <em>detached from</em> what is above and below. It's curious because the full prefix "ksun" or "syn" (ξύν) itself means "with."</p>
<p style="text-align: center;"><img src="https://www.clearhat.org/public/fancy_squiggle.png" /></p>
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Mathematical infinity at the beginning instead of at the ends
urn:md5:27ae3b68473bb496577771ded10131f5
2021-04-15T06:26:00-05:00
2024-05-19T13:27:54-05:00
Clearhat
Mathy Stuff and Ternary Logic
Aristotle
certainty
emptiness
empty set
horror vacui
infinity
negative infinity
zero
<p><em>The following is a "thinking-out-loud" kind of thought experiment which sort of went off the rails... and then got righted again.</em></p>
<h4>The simplicity of emptiness</h4>
<p><img alt="" class="media media-right" src="https://www.clearhat.org/public/glass-161034_640.png" style="width: 30%; float: right;" />We accept the simplicity of emptiness as a reliable foundation upon which to establish all of mathematics without question. The logic is plain: clearly, there can be nothing more simple than the empty set. We start there, and, knowing that we have begun with the most logically solid foundation possible, develop the rest of set theory. For a similar reason, Peano's axioms, widely understood to be fundamental, start with zero as the first number, and follow this same pattern; from simplicity to complexity.</p>
<p>Note that we entail a few little-considered intuitions about mathematics when we begin with emptiness in this way. For example, one of the little-realized implications of starting with emptiness is that it is then easier to accept the assumption that <em>none of mathematics has any physical weight whatsoever</em>. In other words, the full continuum of real numbers, in all its vast and infinitely divisible infinitude -- as well as its related dividing techniques like Dedekind cuts or Cantorian diagonals which we use to separate reals and infinities from each other -- all of this together weighs nothing.</p>
<p>Such weightlessness -- although a kind of emptiness -- may seem unrelated to the empty set -- another kind of emptiness, but consider: if we imagined for a moment that numbers or anything in mathematics weighed anything, a curious paradox about <em>something resting upon nothing</em> would immediately appear to imagination in stark relief to the elegance of the well-laid foundation.</p>
<p>Therefore we can be sure all of mathematics weighs nothing, even though it's not the way we normally think about math.</p>
<p>By means of this small thought experiment, the overlapping relationship between various forms of emptiness becomes a little more obvious; emptiness is emptiness, whether we call it zero, empty set, weightlessness, null, or any other name. It is the lack of properties which blends all these nomenclatures into one.</p>
<p>Much of this point is simply taken for granted. For example, no one talks about mathematics as having no weight; there is not even a need for an axiom; it is simply not questioned, being of such little consequence that it is easily ignored. "Everyone knows" mathematics has no weight.</p>
<p>Although emptiness is a relatively new idea within mathematics, such assumptions about the non-physical nature of mathematics are ancient; they can easily be traced back through Descartes' separation between mind and matter, to Plato's perfect ideals, or Euclid's extensionless points, and further.</p>
<h4>A paradox at the root of set theory</h4>
<p>Although the non-physical nature of mathematics is ancient, recent studies of the cognitive roots of mathematical concepts prove that mathematical intuition and mathematical structures originate in -- and cannot be separated from -- the same perceptual awareness we use to comprehend the physical world. There are other, similar studies, but Giuseppe Longo and Arnaud Viarouge pose a striking paradox regarding how we accept the formless empty set as the cornerstone of mathematics:</p>
<blockquote>"...there is no Mathematics without structure; its constitutive analysis must be the opposite of the unstructured assembly which is the primary foundation and the conceptual origin of Set Theory." -- <a href="https://www.researchgate.net/publication/224000251_Mathematical_Intuition_and_the_Cognitive_Roots_of_Mathematical_Concepts" hreflang="en" title="Mathematical intuition and the cognitive roots of mathematical concepts">Mathematical Intuition and the Cognitive Roots of Mathematical Concepts</a></blockquote>
<p>In other words, the structureless form of the empty set <strong>does not correspond with the cognitive roots of mathematical concepts</strong>. It is an artificial construction. The authors go into a fair amount of detail on this matter, saying: "In Set Theory, elements or points precede structures; the latter are conceptually secondary. In our views, gestalts, as structures, precede points, they are our primary, proto-mathematical relations to the world."</p>
<p>This "Foundation Paradox" is strengthened as the authors go on to discuss the remarkable invariance and high degree of certainty which are defining features of Mathematics. This is a significant point, not to be glossed over as we try to resolve the dilemma -- in fact, it points in the direction we should go with our answer: toward certainty, away from artificial structures.</p>
<p>This invariant nature embedded within how we understand and process mathematics can be said to shine a spotlight onto the importance of getting the foundation correct: If mathematics is the science of structures and mathematics exhibits the highest degree of invariance (indeed, the highest certainty in all of scientific inquiry) then it follows that the most stable portion of mathematics -- its foundation -- should reveal structures and parallels to origins in our perceptions and cognitions rooted in billions of years of evolutionary progress. However, we find the opposite; a complete lack of structure, an emptiness, at the root of mathematics.</p>
<p>How did we get here?</p>
<h4>Infinity and zero historically separated</h4>
<p>Without the existence of such a paradox, the novel structure proposed by this present essay would be easily disregarded, because on the surface "infinity at the beginning instead of the ends" is as absurd as claiming that anything mathematical has physical properties -- like weight, or mass, etc.</p>
<p>But the paradox clearly does exist. And as long as it does, we must look more closely at all possible resolutions; even those which may seem absurd at first glance.</p>
<p>As we shall see, the absurdity of our proposition is <em>not</em> an essential feature. In fact, it is more an artifact of how the history of mathematics unfolded than it is an <em>actual mathematical impossibility</em>.</p>
<p>The initial conceptual difficulty with fundamentally changing how we look at infinity is rooted in an idea which previously dominated mathematics for centuries -- but is no longer considered important, being now a footnote in the history books. This idea is known as <em>horror vacui</em>, or, an extremely strong prejudice against the idea of emptiness. Aristotle was the first to write of this, and his influence on this matter reached centuries into the future, until the <em>horror</em> began to dissipate with the arrival of the zero from India in about 600 A.D.</p>
<p>That dissipation was slow. For example, when the zero first came into being, nobody quite realized that it would grow from its role as a convenient, semantically meaningless, placeholder used within large numbers, into the embodiment of something substantial: the founding "nothing" or "emptiness" by which we know it today. The evolution of zero was necessarily slow, for if its emptiness as we know it had been known early, the <em>horror</em> would have prevented its arrival altogether.</p>
<p>Although avoiding emptiness, Aristotle did not avoid that which we today consider its opposite: infinity. Indeed, he had a fairly sophisticated concept of infinity, even separating infinity into two different kinds, "actual" and "potential." Thus the idea of infinity as being larger than <strong>the end</strong> of countable numbers was known and studied long before zero was placed where we have it today, at <strong>the beginning</strong> of countable numbers.<img src="https://www.clearhat.org/public/infinities.png" style="float: right; margin: 10px; width: 50%;" /></p>
<p>The separation in time of these two insights is important. After zero was finally accepted as a placeholder and then later as a numerical symbol for nothing, eventually negative numbers were discovered. Along with them came negative infinity, a mirror image of positive infinity. In this way, emptiness was incrementally but firmly established at the center of the wide spectrum we now know as the real number line, extending from negative infinity to positive infinity.</p>
<p>To summarize: The <em>horror vacui</em>, a fear of emptiness, was fading from its dominant place in our minds at the same time zero, or emptiness, was evolving into its now-central place within our mathematical understanding. This gradual transformation is the key to understanding why no one ever seriously considered the possibility that zero and infinity could be joined, in the manner that we will now explore.</p>
<p>Thus the seeming absurdity of our position is more an artifact of how the history of mathematics unfolded, than it is an actual mathematical impossibility as it appears on the surface. Now that zero and emptiness have been firmly embedded in their places within mathematics for the past century, we have the liberty to consider things which previously no one, or few, previously considered.</p>
<h4>When infinities merge and move to the center</h4>
<p>Let us consider, then, what happens when the two endpoints of positive and negative infinity are brought to the center of the number line, where zero firmly exists. This turns mathematics inside out. It's like starting with an ordinary 1992 Ford Taurus, dividing its engine into four pieces, and putting one fourth of the engine on each wheel.</p>
<p>In the process of imagining such a thing, how quickly does such an idea become incoherent? For example, can such a car even function? One answer is no, the most essential part of a car is broken. But another answer is, yes, a small motor at each wheel is how many electric cars operate. Sometimes what seems absurd can turn out to be sensible, if we give the idea a little patience.</p>
<p>Patience is required here. Immediately after embarking into the thought experiment about bringing two opposite infinities together, it will feel like the vastness of infinity cannot fit into the emptiness of zero. Nor can the emptiness of zero contain anything within it and still retain the essential simplicity of emptiness. On the surface, it appears one extreme will cancel the other out, either way we start. The inevitable collision feels like a zero-sum-game with only a single winner possible.</p>
<p><img src="https://www.clearhat.org/public/superposition_waves.png" style="float: left; margin: 10px; width: 30%;" />Upon further consideration however, it turns out that quantum mechanics provides a ready intuition for how to do this, by lending its concept of superposition. Superposition is where two or more separate particles inhabit the same space simultaneously. So what happens if we superpose both infinities into a single infinity, and then superpose that singularity of infinities with zero?</p>
<p>Before answering that, let's take a moment before going too deeply into the magical melting-pot of superposition.</p>
<p>Consider briefly what happens at the positive and negative ends of the number line, now that the two infinities are removed from consideration. Without infinities, both number lines simply stop when the countable numbers end. After that, nothing. Surprisingly after such a dramatic removal, the remaining structure is already well-known within mathematics, especially by computer scientists who figure out how computers should process mathematics:</p>
<p>By eliminating infinities from the ends of the number lines, we have not created a Frankenstein, but simply confined ourselves to computable mathematics.</p>
<p>Constructivists will appreciate this move immediately, as they've been saying this is the correct way for well over a century. We may quibble with finitists on the details of how this happens, but it is enough, for the moment, to realize that we need not worry too much about the ends of the number lines once we remove positive and negative infinity.</p>
<p>No big disaster has happened. We can set aside major structural worries for now, and safely come back to the ends later, once we've decided how to resolve what's happening in the center.</p>
<p>Back to superposing infinity and zero. How do we <em>do</em> this? Can we actually merge both infinities into a single super-infinity? Or, perhaps we place emptiness in the center of two infinities? Note, as soon as we put anything into emptiness it is no longer empty, so, should we include reference to time or do we do this instantly?</p>
<p>How do we get visuals on this? Beginning with what we have in the way of symbols, what happens if we place a single, merged, infinity <em>inside the little emptiness</em> carried visually within the number zero? That approach seems like as good a place to start as any, let's try it.</p>
<p>Emptiness we can imagine -- the shape of the zero comes from the shape a pebble would leave in sand after it was removed; a zero makes a good visual for emptiness -- but how do we visualize infinity? Is it like a sun, a sphere flowing outward with light? Or is it like a supernova, exploding infinitely outward, bursting all bonds, even the ability to imagine? Or somewhere between these extremes; what about a fountain... a mountain spring flowing with water into a pool...</p>
<p>Yes! That's it!</p>
<h4>A mountain spring flowing outward</h4>
<p><img src="https://www.clearhat.org/public/havasu_falls.png" style="width: 40%; margin: 10px; float: right;" />Imagine a mountain spring flowing outward from within a circle (say, of small stones laid around the spring in the shape of a zero, which defines the border of emptiness). The spring is overflowing the stone boundary, and a river of... numbers... are flowing outward in all directions... from emptiness?</p>
<p>So there's a raw visual for how to do this. It contains a paradox, of "something arriving from nothing," but we at least have a starting point. It's not too bad, as visuals go. Maybe we could use fire instead of water, an iron band instead of a few stones... but infinity is so immense, compressing it into a too-small space might cause it to explode, so let's remain with flowing water til the idea is more stable -- experiment with fires and supernovas later if the visual is moving too slow.</p>
<p>Is this a reasonable way to think about placing infinity within emptiness?</p>
<p>(Long pause).</p>
<p>No, not exactly. The outflowing water representing infinity has completely replaced any hope of the previously motionless, simple, emptiness that makes such a solid foundation for set theory. And all of this outward flowing is <em>too physical</em>. We've spent many centuries thinking of mathematics and numbers as having no physical properties and here we clearly have introduced motion, not as something studied by math, but as something mathematical; if we keep this up, we'll have to re-invent calculus, which slices physical motion into infinitesimally small, non-moving pieces. We'll need a calculus for calculus. Oh dear.</p>
<p>And what does <em>a flowing number</em> even look like?</p>
<p>These are problems. But then again, it's... somewhat workable. It's stable enough to hold its own while we consider some of the consequences of placing infinity at the center. It hasn't collapsed yet. It may not be perfect, but let's keep going for now.</p>
<p>With infinity safely flowing outward from the center, we see that the number line (going off to the right toward the greatest positive countable number, and off to the left toward the least negative countable number) can be visualized as coming out of the fountain. Little has changed for the countable integers.</p>
<p>The numbers start small: one, two, three, and "flow" to larger and larger... wait, larger? Larger is what happens as we get closer to infinity, and here we are going away from infinity... but getting larger...</p>
<p>Okay, that's a problem.</p>
<p>A quick fix for this is to temporarily invert the two number lines. In other words, instead of starting with the smallest countable numbers, 1,2,3,... let's imagine that the numbers flowing from the fountain at the center begin with the <strong>largest countable</strong> (whatever that is. Let's use "999" to represent it temporarily) and descend, one by one, down to a single <strong>one</strong> at the outer edge, where infinity used to be.</p>
<p>What happens if we do this, completing the original turn-everything-inside-out movement of infinit(ies) which began our journey? Is this even coherent?</p>
<p>Barely. It's getting awkward... but it's still possible to imagine, even to draw a simple line-drawing representation to help visualize what is being described here:</p>
<p align="center"><img src="https://www.clearhat.org/public/infinity_at_the_center_and_counting_toward_one_on_the_edges734x100px.png" /></p>
<p>Infinity is at the center, and "the largest countable number" is nearest it, with the counting descending as the number line goes outward. This means that 1 and -1 are at the outermost edges. Since we know we're working in the computable realm, let's give realistic boundaries and say that "the largest countable number" is at the limit of <em>whatever memory space we have available within a given computer</em>.</p>
<p>Is there anything useful with this visualization, or did that last number-line-inversion cause it to lose all coherence? Can we even do something as basic as counting with this new inside-out structure? What do we do with 1 and -1 dangling in their vulnerable smallness at the outer edges?</p>
<p>Maybe we should undo that last temporary inversion and try another path?</p>
<p>Lastly, is there is anything in Nature that looks like this structure, maybe something we can use as an analogy to help think about things? Maybe a solar system, with a sun at the center, and smaller planets off toward the outer edges?</p>
<p style="text-align: center;"><img src="https://www.clearhat.org/public/fancy_squiggle.png" /></p>
<p>(That was an even longer pause.)</p>
<p>Okay, so a few days have passed, and I've been thinking about this often, trying to solve these riddles that I've created for myself. I've almost given up on the ridiculousness of where I've gotten myself in this thought experiment several times, but intuition keeps pulling me back to the image that I created above and finally, moments ago (early a.m. May 3, 2021) I just realized <em>yes, there is something in Nature that works very much like what I just described in the illustration above!</em></p>
<p>The solution is in, quite literally, the last place I'd normally ever look: my own imagination!</p>
<h4>Is imagination like a fountain of information out of nowhere?</h4>
<p>Look at that illustration above and you should be able to see that it is an accurate structural representation of what your own imagination does when it is counting.</p>
<p>We tend to think that counting is a linear thing, but take a few minutes to think about what happens when you count. Is it linear? When you iterate a new number at the end of all that you've counted, say, as you have carefully, one-by-one, reached the number 999, and you're about to count the next number, 1,000. <em>Where does that new number come from?</em> Does it not come from within the center of your imagination, or, <strong>out of nowhere</strong> out of "an empty set" and suddenly it exists, matching all the patterned rules that define what number it should be (i.e. Peano's axioms). A new number is effectively a new creation, pushing all the numbers that have come before it out to the edges, like water flowing from a spring.</p>
<p>Now that is a dumbstrucking realization.</p>
<p>I've never seen the structure of my own thoughts in that way before. This is a big enough insight that I'm ready to draw this little essay to a conclusion and consider it complete enough for now, because it turned up something potentially useful.</p>
<p>As a conclusion, this is not where I was headed, but... I did find something coherent and insightful right when I thought I had created a mess, and this gem, of understanding what counting looks like -- what it really looks like -- is absolutely worth the journey.</p>
<p>Also, I have the rather pleasant confirmation that intuition was correct in drawing me back to this paradox over and over, not giving up, until it got solved. Good job, intuition.</p>
<p>Now I'm pretty certain I can take the ideas I was writing in the first half of this essay and drive them to a better conclusion.</p>
<p>Good Lord, this adventure (over the past few months contemplating these kinds of ideas and continuing to find gems like this) is truly off the rails and I can't wait to see what happens next.</p>
<p> </p>
https://www.clearhat.org/post/mathematical-infinity-beginning-instead-ends#comment-form
https://www.clearhat.org/feed/atom/comments/189
First qutrit teleportation! Complex high-dimensional quantum states go from zero to infinity in one nanosecond
urn:md5:f05d94170d78b3b64fe3b12e0aa20a74
2019-09-01T09:16:00-05:00
2024-03-31T10:31:54-05:00
Clearhat
CyberIntelligencia
infinity
ternary logic
trit
<p>I knew there was a link between pure logic and the real world, and I knew it was through a window we call infinity, but I had no idea that physical "teleportation" of quantum states would be where ternary logic touches the physical world.</p>
<p>The first qutrit had a single digit beyond normal binary quantum teleportation, but now that we've broken that barrier, it will be easy to explain to people how the qutrit is not just the addition of another "it" to the "bit" concept, but the addition of a whole "infinity."</p>
<p>As in, an entire mappable 3-D landscape, not unlike the physical world.</p>
<p>More later as the epiphany coheres.</p>
<p align="center"><img alt="" src="https://www.clearhat.org/public/Aleph.jpg" width="500" /></p>
<hr />
<p>Wow, everyone is writing about this. So glad to see:</p>
<ul>
<li><a href="https://science.slashdot.org/story/19/08/23/2212204/complex-quantum-teleportation-achieved-for-the-first-time">https://science.slashdot.org/story/19/08/23/2212204/complex-quantum-teleportation-achieved-for-the-first-time</a></li>
<li><a href="https://phys.org/news/2019-08-complex-quantum-teleportation.html">https://phys.org/news/2019-08-complex-quantum-teleportation.html</a></li>
<li><a href="https://www.popularmechanics.com/technology/a28798458/quantum-teleportation/">https://www.popularmechanics.com/technology/a28798458/quantum-teleportation/</a></li>
<li><a href="https://arxiv.org/ftp/arxiv/papers/1906/1906.09697.pdf">https://arxiv.org/ftp/arxiv/papers/1906/1906.09697.pdf</a></li>
<li><a href="https://physics.aps.org/synopsis-for/10.1103/PhysRevLett.123.070505">https://physics.aps.org/synopsis-for/10.1103/PhysRevLett.123.070505</a></li>
<li><a href="https://medium.com/@teodorteofilov/first-three-dimensional-quantum-teleportation-achived-for-the-first-time-b99157252a78">https://medium.com/@teodorteofilov/first-three-dimensional-quantum-teleportation-achived-for-the-first-time-b99157252a78</a></li>
</ul>
https://www.clearhat.org/post/first-qutrit-teleportation-complex-high-dimensional-quantum-states-go-from-zero-to-infinity-in-one-nanosecond#comment-form
https://www.clearhat.org/feed/atom/comments/175
You cannot step outside infinity to perform an operation on or within infinity
urn:md5:684d224fc30691bc5745ca2f46b9c71e
2018-05-04T20:27:00-05:00
2024-03-31T12:21:08-05:00
Clearhat
Phlosiphy and Langwidge
infinity
ternary logic
<p>Okay, it's a pretty raw tangle of thought experiment, but here we go: You can't. There is no such thing as multiple infinities. Imagine if there were, all such multiple infinities would simply exist within a "larger" infinity that is ultimately singular. That ultimately singular one is the only true infinity, as any smaller ones are "not as infinite" in comparison to it, and cannot rightfully be considered actually infinite, although they could be considered as approaching infinity.</p>
<p>Consider having a cake, with a book beside the cake that contains 144 ways to cut the cake. Now cut the cake according to one of the descriptions. Once the cake is cut, one way is implemented, and 143 other ways are not implemented. The others may exist "in theory" within the book, but "in practice" only one can ever exist.</p>
<p>The only way around this is to assume that time can be manipulated, and after one way of cutting the cake is implemented, time is somehow reversed, or bent, and thus the 2nd way can be implemented, etc. But if time can be manipulated in order to define multiple infinities, then what is the point of making a definition?</p>
<p>Your adherence to the idea of multiple infinities is admirable, but unnecessary. Rather than chase your tail trying to encompass the idea of multiple infinities which are so truly "infinite" they have no reference to each other (except in the thought experiment which assumes their existence, the existence of which itself creates a set that contains them all), simply define infinity as the top-most infinity and start from there. It contains <strong>e v e r y t h i n g</strong>, not just Cantor's many numerical infinities, which drift off toward potentiality and eventually escape the bounds of actuality, but also all of Nature, the entire Universe, and all within. <em>Everything</em>. That is the only true Infinity, and <em>within</em> that infinity is the idea of multiple infinities (but not the actuality of multiple infinities.)</p>
<p>I believe the idea of multiple independent infinities is an expression of ego, an attempt to place oneself in a place which is properly the throne of God, a being who encompasses all, and can operate freely upon all. I believe this because all operations upon infinity necessarily assume there are multiple infinities: or at the very least, the one being operated upon, and the one doing the operation, since you cannot, for example, infinitely divide an infinity from within itself.</p>
<p>The infinity doing the operation must contain everything within the "smaller" one, and more. The only one who can contain all in this manner is God himself. He alone can define whether there are multiple infinities, and for his opinion on the subject, we have the ensample of Jesus Christ, who died "once" for all. If there are multiple infinities, then Jesus Christ would be dying multiple times (or else the infinities are inferior to this one).</p>
<p>Supposing the case of a recursive infinity, where the "top" and the "bottom" of infinity (the Singularity and the sub-Planck-length? particles) are one and the same, with the difference between the two being the way each view is perceived (a perceptual difference rather than an actual difference), you have 3 infinities: The top, the recursively linked bottom, and the perceptual one that allows for multiple simultaneous views of the same thing.</p>
<p>Bother, this hurts the brain to try and think through this stuff, as there is a switch between binary and ternary logic which I keep implying but haven't been explicit about, and it gets confusing if you don't do this kind of thinking very methodically. Well, the headline captured the point I wanted to make anyway. G'day.</p>
https://www.clearhat.org/post/you-cannot-step-outside-infinity-to-perform-an-operation-on-or-within-infinity#comment-form
https://www.clearhat.org/feed/atom/comments/126
On the delightful coherence and simultaneous incoherence of epiphanies
urn:md5:623a1adbaf236df5105ae189db44769c
2017-10-19T13:26:00-05:00
2024-03-31T12:25:48-05:00
Clearhat
Phlosiphy and Langwidge
infinity
meditation
ternary logic
zero
<p>Woke up early this morning and in the world between worlds where some of the best insights come into view, I discovered one of the greater epiphanies of my decades-long study of ternary logic, infinity, zero, and world peace. I was striding through the fields of pure light and clarity -- the miracle of clarity -- which surrounds such an insight, and even had a hard time settling my mind during morning meditation because I was so excited to share the revelation with the world. The insight had arisen out of a philosophical/mathy type conversation on social media the night before, but it was much larger than that small thread, since it bridged two previously-separated worlds in a profound manner. For years I've intuited these two worlds were related, but could not tell precisely how until this morning.</p>
<p>So there I was, suffused in the joy of epiphany, when I opened up social media to add the new insight to the existing conversation. I did so and went on researching around the topic for a while, taking notes, preparing to write something deeper on the subject than the brief comment in social media forum. After a couple hours I went back to the comment I had written, and found it to be almost completely incoherent.</p>
<p>I knew what it was trying to say, but was astonished to find how poorly it was written. Not only was there a completely missing word that made the first sentence unintelligible, but the whole sentence was inside-out: I was talking about a transition from one idea to another and used "to" the originating idea, instead of "from" it. The next few sentences were more grammatically coherent but equally insensible in meaning. The only part that made sense was the footnote after the insight, where I was reflecting on its importance:</p>
<blockquote>I have just as of 5:00 a.m. today bridged two very separate lines of thought experiments, both of which I have been pursuing for years, knowing they were related but not knowing how. This is exciting! When Archimedes did that one time, he leapt out of his bathtub and ran down the street naked hollering Eureka. I fortunately have my clothes on, but am nevertheless excited to realize I'm definitely on the right path here. Right paths.</blockquote>
<p>I've had this experience many times, too many to count, where the initial written notes around a given epiphany are almost perfect gibberish. I know why it happens, too: intuition organizes information in a completely different structure from rational thought. I would say intuitive information operates like the middle of the ocean, and rational information operates like the land; in one, there is a completely three-dimensional awareness with no reference to floors or ceilings, and gravity isn't very important, in the other, there is a vast 2-D reference plane dividing heaven and earth, and gravity is of more importance, holding everything firmly to the plane except the birds who live in a hybrid world which is kind of like the dolphin in the ocean and kind of like the lion on the Savannah.</p>
<p>I carefully edited the comment on social media so that it read more coherently, thankful that I was up early enough it may have escaped anyone else's notice, and it took me 30 minutes to do so, the final result bearing little resemblance to the original comment.</p>
<p>Intuitive insights do not make sense to the rational mind until they go through a process of being broken into pieces and stitched back together again almost linearly, like turning a balloon into a pencil. Imagine a dolphin sitting in the middle of a Savannah, with no ability to move, stuck firmly to the ground by gravity, desperately seeking the nearest pool of water for relief. Or a lion, 100 feet below sea level, all his regal splendor to no avail as he flails around, trying above all to get his head above water just to breathe. Two worlds. Poets have fun with blending these worlds with language, as metaphor gives us the ability to see one world from the other.</p>
<p>I suspect that I travel between these two worlds of intuition and rational thought more often and more deeply than the average bear, because I experience far more of these moments where I am incoherent to others than I see others experience, in general. I suppose it's part of the way artists engage with the world, to have these moments of madness which are not so much mad as they are... a very different way of seeing things... creativeness...</p>
<p>Anyway, I have SO MUCH to write about the actual epiphany itself, as well as another long-awaited one from yesterday, so I'll get back to that, but wanted to throw this note out there while I'm at it, perhaps to smooth over my earlier incoherence a little better.</p>
https://www.clearhat.org/post/on-the-delightful-coherence-and-simultaneous-incoherence-of-epiphanies#comment-form
https://www.clearhat.org/feed/atom/comments/108