# Does Infinity Really Exist? **URL:** \ **Category:** Great Debates\ **Created:** [November 24, 2008, 4:56pm UTC](https://boards.straightdope.com/t/does-infinity-really-exist/474269 "2008-11-24T16:56:49Z")\ **Posts on this page:** 16\ **Page:** 6
**Author:** ![Liberal](https://avatars.discourse-cdn.com/v4/letter/l/848f3c/32.png) [@Liberal](https://boards.straightdope.com/u/Liberal)\ **Post date:** [November 26, 2008, 10:06pm UTC](https://boards.straightdope.com/t/does-infinity-really-exist/474269/101 "2008-11-26T22:06:20Z")
> [@cmyk](#): > > Small words and crayons might help me. 😉 I’m certainly not Indistinguishable, but I’m good at writing small words with crayons. I think he is basically saying that you can set up a number system and system of arithmetic to suit almost any purpose, even if that purpose includes nailing down things like zero or infinity. Think of math as a map. It’s tied to the real world in the same way that a squiggly line is tied to a real road. ---
**Author:** ![erislover](https://avatars.discourse-cdn.com/v4/letter/e/71e660/32.png) [@erislover](https://boards.straightdope.com/u/erislover)\ **Post date:** [November 27, 2008, 1:00am UTC](https://boards.straightdope.com/t/does-infinity-really-exist/474269/102 "2008-11-27T01:00:56Z")
> [@Indistinguishable](#): > > So your objection is only that the layman is not already familiar with these particular uses of mathematics to correspond to physical reality? That doesn’t quite seem like it addresses the same skepticism the OP is getting at. For a word to have such a meaning, I’m aware of no better criteria for having entered the language. Infinity is used all the time, but not in relation to phenomena, which _is_ the skepticism the OP has expressed. > [@](#): > > I would argue that saying infinity has a real analog means only that there is some natural analogy between mathematical systems containing infinity and the physical world; regardless of whether the layman is explicitly aware of them, exceeding simple such analogies are already in ubiquitous use, at least among those with such baseline scientific sophistication as to be worth looking at for this. The infinite is no more problematic to correspond with the physical world than -5 or 3/4; the trick, always, is to choose an appropriate system of correspondence. It would be some trick. How does one recognize an experience of infinity? How does one name an aggregate of experience “infinity”? Suppose I want to teach a child this word (\< 18 for the sake of argument), how do I do so? If your answer is “with math” then I must reassert my incredulity and share in the OP’s skepticism. I don’t suppose there is some unique circumstance that, once and for all, stands for the name “infinity.” Such a word might have many uses. I only mean that describing the world with mathematics is a far cry from asserting the reality of infinity. Mathematics is more of an abstraction of reality than a reification of concepts. ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [November 27, 2008, 2:06am UTC](https://boards.straightdope.com/t/does-infinity-really-exist/474269/103 "2008-11-27T02:06:03Z")
Hm… I don’t know. I suspect the point of difference between us is that I feel it is appropriate to tackle “infinity” as mathematical jargon in addressing the OP’s concern and you want to focus on some broader notion of ordinary language instead. It’s not so much that the two are separate, though. I’m not sure… It seems odd to me to focus on whether the layman is aware of analogs or not. > [@erislover](#): > > For a word to have such a meaning, I’m aware of no better criteria for having entered the language. Infinity is used all the time, but not in relation to phenomena, which _is_ the skepticism the OP has expressed. What exactly do you mean by “in relation to phenomena”? > [@](#): > > It would be some trick. How does one recognize an experience of infinity? How does one name an aggregate of experience “infinity”? Suppose I want to teach a child this word (\< 18 for the sake of argument), how do I do so? If your answer is “with math” then I must reassert my incredulity and share in the OP’s skepticism. I would not want to talk about “an experience of infinity” anymore than I would want to talk about “an experience of π”. But, yes, my answer is something like “with math”, for I am treating “infinity” as a technical term of mathematics (though only to the slight degree as I am restricting attention to its use by those with some mathematical sophistication (i.e., who are familiar with the language of mainstream mathematics)), the same as one would do if asked whether exponential growth or imaginary numbers or derivatives had physical analogs. > [@](#): > > I don’t suppose there is some unique circumstance that, once and for all, stands for the name “infinity.” Such a word might have many uses. I agree with this. Like most terms, even within mathematics, “infinity” denotes not a single, concrete concept, but rather a nebulous web of them, related through family resemblances, even if no single thread runs through them all. > [@](#): > > I only mean that describing the world with mathematics is a far cry from asserting the reality of infinity. As you said before, I am, as a default, inclined to respond to “is infinity real” as something like “Is there a grammatical place for infinity?”. If I am pressed to respond differently, and asked if infinity has physical analogs, I can only say that it has physical analogs to precisely the same degree as any other mathematical concept, including such basic things as counting numbers. Granted, counting is a much more mundane and ubiquitous activity for most humans than describing lines by their slopes or measuring sound in decibels or what have you, but as far as what’s in and not in the physical universe goes, the analogs are there, which is what I suspect is of importance to the OP. Infinity is not any more “fake” than other mathematical concepts. > [@](#): > > Mathematics is more of an abstraction of reality than a reification of concepts. I often make grand pronouncements about the nature of mathematics myself, but maybe I shouldn’t, since it is also a wide variety of practices related only through loose family resemblances. At any rate, I would not say that I am treating mathematical concepts as reified objects outside their use in the language game of mathematics, as such, but just pointing out that, if one is inclined to say that some of them have physical analogs in the ways that the OP is, the situation is not fundamentally different for various uses of infinity. ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [November 27, 2008, 2:19am UTC](https://boards.straightdope.com/t/does-infinity-really-exist/474269/104 "2008-11-27T02:19:10Z")
> [@Indistinguishable](#): > > Hm… I don’t know. I suspect the point of difference between us is that I feel it is appropriate to tackle “infinity” as mathematical jargon in addressing the OP’s concern and you want to focus on some broader notion of ordinary language instead. It’s not so much that the two are separate, though. I’m not sure… It seems odd to me to focus on whether the layman is aware of analogs or not. To clarify, it’s not the layman’s employment of the term “infinity” which bothers me, I suppose, just his ignorance of how that employment relates to physical phenomena, or at least does so as well as any other mathematical concepts he may care to employ. I mean, I do understand that you’re pointing out that the OP is aware that infinities can occur in physical equations, and merely harbors skepticism that these equations are direct descriptions of “reality”, or something like that, but I still feel odd about it. I think complex numbers really are the best example. If the layman is unaware of how complex numbers naturally correspond to scaling and rotation, and are thus in ubiquity in physics, should we say by virtue of this that complex numbers have no physical analog? Do the failings of mathematicians and scientists to make their observations and practices sufficiently well-known to the average man impinge upon the question of whether a mathematical concept can be taken to correlate with the physical world in some way or not? Well, I have to run off to pre-Thanksgiving plans, so I’ll have to make my thoughts more coherent later. ---
**Author:** ![cmyk](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/cmyk/32/3353_2.png) [@cmyk](https://boards.straightdope.com/u/cmyk)\ **Post date:** [November 27, 2008, 4:06am UTC](https://boards.straightdope.com/t/does-infinity-really-exist/474269/105 "2008-11-27T04:06:39Z")
To no one in particular: My original angle in all of this is that the intuitive concept of infinity is there, aside from mathematics. There is also a philosophical infinity. It happens that infinity arises in math and philosophy in many different ways and flavors, but this does not denote its origin as a concept or settle its existence in the physical world. So many ways we can dream of the infinite; through numbers, expressions, words and ideas. Math is here to guide us and predict. Philosophy is here to open us and reason. Language is here to communicate and share. Science is here to observe and test. It’s a 4-way system that helps us uncover the truth about our universe – the _physical reality_ that trumps the experts and the laymen, our tools and analogical systems, in all these areas. So, does the physical universe harbor anything at all that can be said to be infinite? Is the universe itself infinite? If you think so, then demonstrate it. I still can’t think of a way to convince myself, show me how you have. As an aside to **Indistinguishable,** you’re apparently coming from the side of mathematics, which is fine, despite my OP. You sound as if you’re making the argument that you can’t even talk about the infinite without raising mathematics. They are necessarily bonded to each other. It’s meaningless to pull the two apart. For some reason I’m not convinced they’re mutually dependent. Certainly there are lots of ways to express infinity in math, but only in the same way there are lots of ways to describe a color in words that you see with your eyes (although, decidedly not as subjective). I hope you keep explaining your thoughts further, and have a great holiday! ---
**Author:** ![cmyk](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/cmyk/32/3353_2.png) [@cmyk](https://boards.straightdope.com/u/cmyk)\ **Post date:** [November 27, 2008, 4:24am UTC](https://boards.straightdope.com/t/does-infinity-really-exist/474269/106 "2008-11-27T04:24:17Z")
> [@Liberal](#): > > I’m certainly not Indistinguishable, but I’m good at writing small words with crayons. I think he is basically saying that you can set up a number system and system of arithmetic to suit almost any purpose, even if that purpose includes nailing down things like zero or infinity. Think of math as a map. It’s tied to the real world in the same way that a squiggly line is tied to a real road. Right, thank you. I love maps! But I never confuse them with the actual road I’m driving on. And math people of the world. Don’t get me wrong. I love math too (even if I’m not very good at it, or don’t understand much). This post isn’t meant as some sort of personal affront I have with math. I adore it. Yet, I’ve always made a distinction between math and reality. There are many things in nature we can look at and say, this equation is what’s _driving_ that. Such as the Fibonacci sequence driving the branching of a plant. But I’ve always seen it as, this equation is what _describes_ that as far as we can resolve it. But maybe we resolve _too_ far, sometimes? Hence, infinity? ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [December 2, 2008, 3:09am UTC](https://boards.straightdope.com/t/does-infinity-really-exist/474269/107 "2008-12-02T03:09:09Z")
So… How was everybody’s Thanksgiving? 🙂 ---
**Author:** ![Half\_Man\_Half\_Wit](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/half_man_half_wit/32/21766_2.png) [@Half\_Man\_Half\_Wit](https://boards.straightdope.com/u/Half_Man_Half_Wit)\ **Post date:** [December 2, 2008, 9:37am UTC](https://boards.straightdope.com/t/does-infinity-really-exist/474269/108 "2008-12-02T09:37:20Z")
> [@Indistinguishable](#): > > So… How was everybody’s Thanksgiving? 🙂 I found an infinite amount of turkey, but I eated it. ☹ ---
**Author:** ![Liberal](https://avatars.discourse-cdn.com/v4/letter/l/848f3c/32.png) [@Liberal](https://boards.straightdope.com/u/Liberal)\ **Post date:** [December 2, 2008, 11:52am UTC](https://boards.straightdope.com/t/does-infinity-really-exist/474269/109 "2008-12-02T11:52:24Z")
> [@Indistinguishable](#): > > So… How was everybody’s Thanksgiving? 🙂 Easy. I simply existed as it came and went. 🆒 ---
**Author:** ![erislover](https://avatars.discourse-cdn.com/v4/letter/e/71e660/32.png) [@erislover](https://boards.straightdope.com/u/erislover)\ **Post date:** [December 2, 2008, 2:20pm UTC](https://boards.straightdope.com/t/does-infinity-really-exist/474269/110 "2008-12-02T14:20:15Z")
> [@Indistinguishable](#): > > So your objection is only that the layman is not already familiar with these particular uses of mathematics to correspond to physical reality? That doesn’t quite seem like it addresses the same skepticism the OP is getting at. Hmm. My objection simply has a criteria which is that this should be the case, or at a minimum, that we could plausibly teach a layman to use the word in such a way. > [@](#): > > I would argue that saying infinity has a real analog means only that there is some natural analogy between mathematical systems containing infinity and the physical world; regardless of whether the layman is explicitly aware of them, exceeding simple such analogies are already in ubiquitous use, at least among those with such baseline scientific sophistication as to be worth looking at for this. So does 2 contain the concept of infinity because there is some infinite series which sums to 2? I don’t understand what this brings to the table. > [@](#): > > The infinite is no more problematic to correspond with the physical world than -5 or 3/4; the trick, always, is to choose an appropriate system of correspondence. So you keep saying, but so far all I have is your word for it. How does infinity correspond to real phenomena? This is the question. ---
**Author:** ![erislover](https://avatars.discourse-cdn.com/v4/letter/e/71e660/32.png) [@erislover](https://boards.straightdope.com/u/erislover)\ **Post date:** [December 2, 2008, 2:49pm UTC](https://boards.straightdope.com/t/does-infinity-really-exist/474269/111 "2008-12-02T14:49:45Z")
Sorry about that, was meant to be part of a longer post addressing your last objection. Those were earlier quotes. > [@](#): > > To clarify, it’s not the layman’s employment of the term “infinity” which bothers me, I suppose, just his ignorance of how that employment relates to physical phenomena, or at least does so as well as any other mathematical concepts he may care to employ… I think complex numbers really are the best example. If the layman is unaware of how complex numbers naturally correspond to scaling and rotation, and are thus in ubiquity in physics, should we say by virtue of this that complex numbers have no physical analog? They’re not unfamiliar with scaling and rotation. They call it “scaling” and “rotation.” I could probably teach a 12 year old about it. There’s no need to reify complex numbers to cover this. I understand what the expressive power of complex numbers can bring to a study, but it’s one thing to use a complex number to describe reality and another to say that reality uses complex numbers… that is, we find complex numbers in reality. I’m trying to think of a way to explain my problem. I don’t think that complex numbers are “behind” scaling and rotation, fluid flow, or electrical phase. I think complex numbers can help us understand and explain what happens. I get the feeling your position is that if they serve this purpose, then one might as well say they’re there? If so, we probably agree, I’m just setting down a criteria for when we can say that, which is, when most people have such a sense of complex numbers. We’re not there, so “scaling” and “rotation” it is. So, all good. But then, what about infinity? What phenomena are you thinking of, specifically? ---
**Author:** ![erislover](https://avatars.discourse-cdn.com/v4/letter/e/71e660/32.png) [@erislover](https://boards.straightdope.com/u/erislover)\ **Post date:** [December 2, 2008, 6:20pm UTC](https://boards.straightdope.com/t/does-infinity-really-exist/474269/112 "2008-12-02T18:20:10Z")
After further consideration, perhaps I can explain my position a bit more clearly. The distinction between whether something is or isn’t real, in my view, begins to disappear when the word enters common vocubulary, at which point denial loses general sense (though can be maintained in specific counterfactuals: “Suppose the sun ceased to exist…”). So I’m imagining a sort of sloppy scale with [Humpty-Dumptyism](http://en.wiktionary.org/wiki/Humpty_Dumptyism) on one end, moving through pejoritive jargon to technical jargon and on to broader acceptance. I’m saying: until we move out of jargon, the reality of concepts is a non-starter. A role for “exists” has not been clearly laid out. ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [December 2, 2008, 7:24pm UTC](https://boards.straightdope.com/t/does-infinity-really-exist/474269/113 "2008-12-02T19:24:20Z")
> [@erislover](#): > > After further consideration, perhaps I can explain my position a bit more clearly. > > The distinction between whether something is or isn’t real, in my view, begins to disappear when the word enters common vocubulary, at which point denial loses general sense (though can be maintained in specific counterfactuals: “Suppose the sun ceased to exist…”). So I’m imagining a sort of sloppy scale with [Humpty-Dumptyism](http://en.wiktionary.org/wiki/Humpty_Dumptyism) on one end, moving through pejoritive jargon to technical jargon and on to broader acceptance. I’m saying: until we move out of jargon, the reality of concepts is a non-starter. A role for “exists” has not been clearly laid out. Ah. Very well. I can agree with “A role for ‘exists’ has not been clearly laid out.” But reading the OP’s posts, I don’t think this forcefully enough addresses the position he is in; he is not wondering about the grammar of the word “exists”. He is simply wondering about the makeup of the physical universe, and whether any measurement within it corresponds to the infinite. And, of course, he suspects/posits that the answer is no, in stark contrast to a multitude of measurements which correspond to small integers and rational numbers. Somewhat like you, my goal is to dissolve the very question. However, my particular approach, in dissuading the OP from thinking in this way, is to show him that measurements/mathematical analogies can be drawn however one likes, and that, in fact, there are plenty of simple, uncontroversial ones in ubiquity in common scientific practice which happen to involve some mathematical notion of the infinite. If it is not stressed enough that the only difference between the physical analogies he already acknowledges and those which involve the infinite is the matter of how many people have been taught to see them, and that this is not actually any sort of substantive intrinsic property of the physical universe, then I fear he will come away with rather a misguided perspective: not “a role for ‘exists’ has not been clearly laid out”, but “the bar for existence is clear, and the infinite fails to clear it, distinguishing it from other, more physically real mathematical concepts”. Anyway, I typed up responses to some of your other posts before I saw this clarification, so I might as well submit them as well, antiquated as they may now be. ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [December 2, 2008, 7:30pm UTC](https://boards.straightdope.com/t/does-infinity-really-exist/474269/114 "2008-12-02T19:30:47Z")
> [@erislover](#): > > Hmm. My objection simply has a criteria which is that this should be the case, or at a minimum, that we could plausibly teach a layman to use the word in such a way. Is there any doubt that we could plausibly teach a layman to use words the way specialists/experts do? That is, after all, how specialists/experts are created. > [@](#): > > So does 2 contain the concept of infinity because there is some infinite series which sums to 2? I don’t understand what this brings to the table. The question isn’t whether one concept contains another. Rather, what I am saying is that, to ask whether some mathematical system has a physical analog is simply to ask whether humans have found a useful and natural way to correspond that system with physical phenomena, and that this is as much true for simple counting numbers as it is for complex numbers or infinity. So, no, I do not want to say “2 contains the concept of infinity because there is some infinite series which sums to 2”. I do not want to say this nor do I want to say its negation; it is a kind of speech I do not find useful right now. My argument is not based on any claim like this, at least not directly. Rather, my argument is like this: suppose a rock and another, and another still, lay on the ground. Is this a physical analog of the mathematical concept of 3? Well, we could say it is, for the obvious reasons (we have some correspondence in mind between collections of rocks and natural numbers, under which an empty collection corresponds to 0 and a collection augmented by a single rock sees its label rise by 1). But why should we not just as well say this is a physical analog of the concept of -3, having counted in the other direction? Or standardly count not the rocks within the collection but the subcollections of rocks, to call this an analog of 8. Or the vertices and edges in travelling between all the rocks, reaching 5. Or perhaps, labelling collections of rocks by their “interchange-symmetries”, we should say this is an analog of the permutation group of order 6. Going back to the obvious, we may be compelled to call this an example of a triangle… but perhaps, without lines drawn between the rocks, we shall not. We could just as well think of these points as describing a circle. [Hell, we could label collections of rocks by months of the calendar, starting with the empty collection in the spring of March, and thus finding that this one corresponds to June; or to December, counting the other way, or to July, considering the terminus of the journey which visits each rock for a month and takes a month to travel between rocks.] And all of this is true. And none of this is actually there. Which is to say, what we have is not the number 3, or -3, or 8, or June. What we have is some rocks on the ground. Analogies with mathematical objects only enter the picture when we choose to draw them. Some are drawn by any layman and some are drawn by few laymen but by many scientists and some aren’t drawn at all, being too contrived to be useful for any purpose or too obscure to have been yet discovered. But analogies that link 2 to the physical world are not fundamentally different from those which link irrationals, complex numbers, or infinity to the physical world, save perhaps for how many people have been taught to draw them. > [@](#): > > So you keep saying, but so far all I have is your word for it. How does infinity correspond to real phenomena? This is the question. As I have noted, infinity corresponds to real phenomena just as well as 100 does. A football field’s length is commonly described with reference to 100 yards in a manner every layman is familiar with, sure, but just as well, an absolutely quiet room is commonly described with reference to -infinity decibels (not every layman is familiar with this, but every scientist/engineer studying sound is, and surely it is their expertise that the OP considers relevant?). 2 may describe the cardinality of one hand and another, but an unsigned infinity serves just as aptly to describe the slope of a vertical line. Etc. ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [December 2, 2008, 7:47pm UTC](https://boards.straightdope.com/t/does-infinity-really-exist/474269/115 "2008-12-02T19:47:35Z")
> [@erislover](#): > > They’re not unfamiliar with scaling and rotation. They call it “scaling” and “rotation.” I could probably teach a 12 year old about it. There’s no need to reify complex numbers to cover this. I never meant to imply that the layman was not thoroughly aware of scaling and rotation, in their own terms; I meant only to point out his ignorance of the direct connection between these and the arithmetic of complex numbers. > [@](#): > > I understand what the expressive power of complex numbers can bring to a study, but it’s one thing to use a complex number to describe reality and another to say that reality uses complex numbers… that is, we find complex numbers in reality. Well, I would not find this to be a distinction worth caring about, which is to say, I would not ever feel compelled to say “We find [mathematical object X] in reality” except as a means of saying “We use [mathematical object X] to describe reality”. The distinction here, at any rate, is present with using complex number arithmetic to describe scaling and rotation to only the exact same degree that it is present with uncontroversially using natural number arithmetic to describe counting rocks or rational/real number arithmetic to describe measured lengths. > [@](#): > > I’m trying to think of a way to explain my problem. I don’t think that complex numbers are “behind” scaling and rotation, fluid flow, or electrical phase. I think complex numbers can help us understand and explain what happens. I get the feeling your position is that if they serve this purpose, then one might as well say they’re there? If so, we probably agree Yes, we probably do fundamentally agree. > [@](#): > > I’m just setting down a criteria for when we can say that, which is, when most people have such a sense of complex numbers. We’re not there, so “scaling” and “rotation” it is. So, all good. But then, what about infinity? What phenomena are you thinking of, specifically? Eh, I’m not particularly inclined to tie such criteria to “most people”, or rather, once “most people” differ from “most scientists”, I lose all inclination to simply say "Because most people do not have this sense of complex numbers, I will say ‘Complex numbers do not have physical analogs the way integers do’ ". Once there is such conflict in awareness/usage, I am motivated to try even harder to dissuade talk of which mathematical concepts are and aren’t “physically real”, via giving the full spiel I’ve been giving in this thread. And if such talk is replaced only by discussion of which concepts do and don’t commonly get used by scientists to describe the physical world, then little serious controversy exists. Anyway, as for what physical phenomena I am thinking of where scientists may invoke the infinite, we could start with, among other things, any physical equation involving a ratio or a logarithm in the limiting case where it approaches zero; thus, as I mentioned before, resistance of a copper wire of vanishing cross-sectional area, the volume of absolute quiet in decibels, the slope of a vertical line, etc. ---
**Author:** ![cmyk](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/cmyk/32/3353_2.png) [@cmyk](https://boards.straightdope.com/u/cmyk)\ **Post date:** [December 3, 2008, 7:57pm UTC](https://boards.straightdope.com/t/does-infinity-really-exist/474269/116 "2008-12-03T19:57:58Z")
I guess this all comes down to scale. Is there a limit in scalability to the universe? I understand how something in the physical can converge toward the infinite, but if the fabric of the universe isn’t infinitely scalable, then it’s not really infinite. Is it? How so? ETA: I understand we can create a mathematical model that would address this, if the answer were ‘yes, there is a limit’, but first we must determine whether it is so, or not. Right? [Previous page](https://boards.straightdope.com/t/does-infinity-really-exist/474269.md?page=5)