Very large telescopes
When you need a large reflective surface with the shape that happens to be defined by the inverse-square law, why not let gravity do the work for you? If it's low gravity and near-vacuum, the reflective surface can be very thin and light.
Failure
The weird thing is, the sticking point for the mother of the Baltimore high school senior seems not to be that he's 18 now and hasn't learned anything, or that she's not allowed to send him to a school that actually functions, but that he's being "punished" by being sent back to 9th grade after being socially promoted for years.
“Why would he do three more years in school? He didn’t fail, the school failed him. The school failed at their job. They failed. They failed, that’s the problem here. He didn’t deserve that. He’s a good kid. Where’s the mentors? Where is the help for him? I hate that this is happening to my child,” said an emotional France.She never minded before that he wasn't learning anything and failed nearly all of his classes. It's just that now he doesn't get a diploma.
The 'Praetorian' Guard
The formerly-"National" Guard to receive a new award and a new service ribbon.
Tens of thousands of National Guard troops who deployed to Washington, D.C., ahead of a 2021 inauguration under threat of violence are eligible for a brand-new award in recognition of their service..."In recognition of their service as part of the security mission at the U.S. Capitol and other facilities in Washington, D.C., before, during and after the 59th Presidential Inauguration, the District of Columbia National Guard plans to present all Soldiers and Airmen who took part in the mission one or both of the following decorations: the District of Columbia National Guard Presidential Inauguration Support Ribbon and/or the District of Columbia Emergency Service Ribbon," Air Force Lt. Col. Robert Carver, spokesman for the Virginia Air National Guard and director of Joint Task Force-DC Joint Information Center, said in a statement.
Sounds familiar.
The Praetorian Guard (Latin: cohortes praetoriae) was an elite unit of the Imperial Roman army whose members served as personal bodyguards and intelligence for Roman emperors. During the era of the Roman Republic, the Praetorians served as a small escort force for high-ranking officials such as senators or provincial governors like procurators, and also serving as bodyguards for high-ranking officers within the Roman legions. With the republic's transition into the Roman Empire, however, the first emperor, Augustus, founded the Guard as his personal security detail.
The American Republic may well have ended with the 'fortified' election of 2020. In retrospect, we may mark this passage as having been as firm a transition to something else as we now mark Augustus' rise as the end of the Roman Republic.
More Dolly
Gringo misplaced a comment, but it deserves to be raised to the main page anyway because he has a much better version of the 'Dolly Parton at 14' video.
Gringo said...So watch this old video Instapundit found. That's her at 14, playing for one of Cas Walker's shows.The link, to a TV news short, had the video but had most of her singing erased in favor of announcer comments. Here is the the video with all of Dolly's singing. Much better than listening to a talking head's blather. WIVK-Radio Remote with Cas Walker and Dolly Parton 1961.
Like Aggie, I don't listen much to Dolly Parton-I prefer Western Swing- but have a lot of respect for her. (I worked with an accountant who had Dolly as a client.She had nothing but good to say about her interactions with Dolly.) That being said, Dolly's soulful singing at age 14 floored me. That is talent!
Plato's Parminedes, Preface: Zeno III, Aristotle II
I'm not going to deal with Zeno's 'moving columns' approach because we don't have enough to know what exactly it was he said; I think our Stanford author is correct in saying that Aristotle's take on it depends on reading a falsehood into it, which may not be fair. Also, these issues of simultaneity of events in motion prove to have much more interesting characteristics once relativity is discovered, which may be worth your time as a reader.
The rejection of the reconstruction of Zeno's argument in Physics 8 depends on so much of Aristotle's own furniture -- we are in the last book of the Physics, here, so he has laid out a complete vision of how movement works in nature -- that it would require more than a blog post to explain it. It might do as a beginning to say that Aristotle is wrong about this part. He begins by positing that there might be a kind of infinite motion if it were circular, by which he means the movement of the stars in heaven. We have no reason to believe that now.
Some of his proofs that rectilinear motion cannot be infinite end up applying to the 'circular' motion he intends to consider infinite. He admits this in one case: if two objects are moving in a line, one from A to B and the other from B to A, they will arrest each other. Two trains on the same track, headed in opposited directions, will crash into each other and stop. Yet this would be true of two trains on circular tracks, too, should they meet while headed in opposite directions.
He does not admit that the logic applies to another of his proofs, which is the proof about a thing turning back on its course. If you're moving from A to B, and halfway you decided you prefer A after all, you must come to a stop in the process of reversing your course. Circular motion does not do that because, after all, it's a circle: you can get back to A just by continuing on your course. Yet if you were to reverse course, you would in fact have to stop; Aristotle says you could simply 'turn back' at B without stopping, which implies something other than continuous motion in a circle.
In any case, this proof-by-standstill is important to his last rejection of Zeno. (Note that the points on the line Aristotle is using go A, B, G because "Gamma" is the third letter in Greek.)
We may start as follows: we have three points, starting-point, middle-point, and finishing-point, of which the middle-point in virtue of the relations in which it stands severally to the other two is both a starting-point and a finishing-point, and though numerically one is theoretically two. We have further the distinction between the potential and the actual. So in the straight line in question any one of the points lying between the two extremes is potentially a middle-point: but it is not actually so unless that which is in motion divides the line by coming to a stand at that point and beginning its motion again: thus the middle-point becomes both a starting-point and a goal, the starting-point of the latter part and the finishing-point of the first part of the motion. This is the case e.g. when A in the course of its locomotion comes to a stand at B and starts again towards G: but when its motion is continuous A cannot either have come to be or have ceased to be at the point B: it can only have been there at the moment of passing, its passage not being contained within any period of time except the whole of which the particular moment is a dividing-point.
Here Aristotle is using his potential/actual distinction in a curious way. The 'middle point' on the line is potentially but not actually a destination; we may discuss or think of it as the finishing of the first half of a continuous motion from A to G, but it isn't actually so. The proof is that the motion doesn't actually stop at B, the midpoint, and then resume. Rather, the motion from A to G, being continuous, just happens to pass over B.
If you recall that Zeno's Stadium argument was built on the need to pass an infinity of midpoints to complete a finite motion, you can probably see where Aristotle is going with this. The time it takes to cross B isn't an actual period of time spent passing B, just a potential division of the actual time it took to go from A to G. Thus, just as we can speak of B as 'numerically one but theoretically two,' i.e., both the ending of the first half of the motion to G and the beginning of the second half, we don't have to speak of B at all. It's not actually there; it's just conceptually so.
The way I said this the last time was that the points on the line don't exist in our three dimensional world, being one dimensional, in quite the same way that the three dimensional ground does. You may cross endless infinities of one dimensional objects in any simple motion through the third (fourth?) dimension.
So, in any case, Aristotle returns to Zeno.
The same method should also be adopted in replying to those who ask, in the terms of Zeno’s argument, whether we admit that before any distance can be traversed half the distance must be traversed, that these half-distances are infinite in number, and that it is impossible to traverse distances infinite in number-or some on the lines of this same argument put the questions in another form, and would have us grant that in the time during which a motion is in progress it should be possible to reckon a half-motion before the whole for every half-distance that we get, so that we have the result that when the whole distance is traversed we have reckoned an infinite number, which is admittedly impossible. Now when we first discussed the question of motion we put forward a solution of this difficulty turning on the fact that the period of time occupied in traversing the distance contains within itself an infinite number of units: there is no absurdity, we said, in supposing the traversing of infinite distances in infinite time, and the element of infinity is present in the time no less than in the distance.
You will recall that I described this as the infinities of time and space being 'geared together,' so that there was always enough time to cross the space because whatever divisions exist fit each other like the teeth of two gears (one space, and one time). There can be more divisions or fewer, but however many there are, there are exactly as many on both sides. Since motion in space is always motion in time, turning the one gear turns the other in the same way and to the same degree.
Aristotle is satisfied, but wants a theoretical answer and not just a practical one.
But, although this solution is adequate as a reply to the questioner (the question asked being whether it is possible in a finite time to traverse or reckon an infinite number of units), nevertheless as an account of the fact and explanation of its true nature it is inadequate. For suppose the distance to be left out of account and the question asked to be no longer whether it is possible in a finite time to traverse an infinite number of distances, and suppose that the inquiry is made to refer to the time taken by itself (for the time contains an infinite number of divisions): then this solution will no longer be adequate, and we must apply the truth that we enunciated in our recent discussion, stating it in the following way.
By abandoning distance and focusing on time alone, the 'gearing' solution is no longer viable. So what then?
In the act of dividing the continuous distance into two halves one point is treated as two, since we make it a starting-point and a finishing-point: and this same result is also produced by the act of reckoning halves as well as by the act of dividing into halves. But if divisions are made in this way, neither the distance nor the motion will be continuous: for motion if it is to be continuous must relate to what is continuous: and though what is continuous contains an infinite number of halves, they are not actual but potential halves. If the halves are made actual, we shall get not a continuous but an intermittent motion. In the case of reckoning the halves, it is clear that this result follows: for then one point must be reckoned as two: it will be the finishing-point of the one half and the starting-point of the other, if we reckon not the one continuous whole but the two halves. Therefore to the question whether it is possible to pass through an infinite number of units either of time or of distance we must reply that in a sense it is and in a sense it is not. If the units are actual, it is not possible: if they are potential, it is possible.
It looks like Zeno wins a point here: Aristotle admits that it is impossible to travel through an actual set of infinities. Now Aristotle thinks that, since continuous motion can be observed to exist, this proves that the divisions aren't actual, but merely potential. Yet Aristotle himself has maintained, especially in the Physics, that potentiality is first actuality; for example, that lumber is potentially a house because it is actually the kind of thing that could become a house. To say that there are potential divisions is thus to say that there are, in a way, actual divisions. And if so, an apparently-observed continuous motion is impossible -- which is exactly what Zeno wanted to prove. Q.E.D., Aristotle.
We can rescue Aristotle here by suggesting that he means 'potential' in a different way here; perhaps one that is closer to the way he said 'theoretically' when he was talking about one point being spoken of as both a start and an end. Let's see how that move works with the rest of what he has to say.
For in the course of a continuous motion the traveller has traversed an infinite number of units in an accidental sense but not in an unqualified sense: for though it is an accidental characteristic of the distance to be an infinite number of half-distances, this is not its real and essential character.
Aristotle is now attempting to discuss this in terms of another distinction that was important to his physics and metaphysics, which is the distinction between essence and accidents. This doesn't look like it works to me for the same reason that the potential/actual distinction does: accidents are indeed accidental, in the sense that they happen-to-be but might-be-otherwise, but they nevertheless are are. Indeed, they are actual: a blue table might be blue accidentally, but it is actually blue. Stating that the divisions are accidental but not essential will not save Aristotle from Zeno.
It is also plain that unless we hold that the point of time that divides earlier from later always belongs only to the later so far as the thing is concerned, we shall be involved in the consequence that the same thing is at the same moment existent and not existent, and that a thing is not existent at the moment when it has become. It is true that the point is common to both times, the earlier as well as the later, and that, while numerically one and the same, it is theoretically not so, being the finishing-point of the one and the starting-point of the other: but so far as the thing is concerned it belongs to the later stage of what happens to it.
This argument is closer to the 'theoretical, not real' move. If we make all these theoretical divisions, Aristotle says, we fall into logical contradictions. For example, if every moment that is numerically one is treated as 'really' two, both a start and an end, then the moment at which a thing finishes coming to be is also a moment at which it isn't, quite yet.
But this is no answer to Zeno! His whole point was that our account of motion (including any sort of coming-to-be, which can be discussed as a kind of motion) leads to logical contradictions. Aristotle argues that the fact that the contradictions pop up is a reason to dismiss the idea that these divisions as fully real; Zeno's point is that the contradictions come up whenever we try to discuss the ways motion could occur. They're only differing over whether to dismiss the motion as a consequence of the logical contradictions, or to dismiss the reality of our theoretical framework.
This does not get better in the longer explication of it, in which Aristotle briefly introduces 'time atoms' of the sort he rejected in the Physics 6 argument I treated the last time.
Let us suppose a time ABG and a thing D [i.e. "Delta"; and note that for some reason Gamma has to be between Alpha and Beta for this argument to work as a line --Grim], D being white in the time A and not-white in the time B. Then D is at the moment G white and not-white: for if we were right in saying that it is white during the whole time A, it is true to call it white at any moment of A, and not-white in B, and G is in both A and B. We must not allow, therefore, that it is white in the whole of A, but must say that it is so in all of it except the last moment G. G belongs already to the later period, and if in the whole of A not-white was in process of becoming and white of perishing, at G the process is complete. And so G is the first moment at which it is true to call the thing white or not white respectively. Otherwise a thing may be non-existent at the moment when it has become and existent at the moment when it has perished: or else it must be possible for a thing at the same time to be white and not white and in fact to be existent and non-existent. Further, if anything that exists after having been previously non-existent must become existent and does not exist when it is becoming, time cannot be divisible into time-atoms. For suppose that D was becoming white in the time A and that at another time B, a time-atom consecutive with the last atom of A, D has already become white and so is white at that moment: then, inasmuch as in the time A it was becoming white and so was not white and at the moment B it is white, there must have been a becoming between A and B and therefore also a time in which the becoming took place. On the other hand, those who deny atoms of time (as we do) are not affected by this argument: according to them D has become and so is white at the last point of the actual time in which it was becoming white: and this point has no other point consecutive with or in succession to it, whereas time-atoms are conceived as successive. Moreover it is clear that if D was becoming white in the whole time A, the time occupied by it in having become white in addition to having been in process of becoming white is no more than all that it occupied in the mere process of becoming white.
It turns out that Aristotle's final answer to Zeno is much weaker than his earlier one. Yes, the contradictions he discusses arise, and they arise whether or not time can be divided into indivisibles, i.e., time atoms. But that was Zeno's point all along.
A Cultural Misunderstanding
If you remember "Sose the Ghost," the outlaw biker who was talking about maybe supporting police against BLM/Antifa rioters, he's got a series going on now where he's trying to help newcomers to biker culture understand what patches not to sew on themselves if they want to avoid trouble. Mostly I think he's entirely well-meaning, and has good advice.
One patch that he's particularly concerned about is the diamond-shaped 1% patch. This patch is worn by several outlaw clubs, and they feel a certain degree of ownership about it. He regularly cautions in the series against wearing anything that might be mistaken for this diamond 1% patch: any sort of diamond patch, especially with a number on it, because it could be misunderstood. Outlaw clubs who see you wearing such things might make you take them off, and fight you if you won't. It's probably very helpful advice, telling people things to avoid so they don't get into trouble.
So about 5m and 20 seconds into that video, someone asks him about this patch:
He says, "This is a military patch that he got while out serving. My thing with this is, yes, military, nothing but respect. But I know that from brothers, in areas, that this won't fly. Just because you were in the military, they're not going to respect you walking around with a diamond patch... some states might have some military, and they know and they respect it, they're old timers or whatever, but there are some places where..."
I guess he's a New York guy, so maybe he's never seen that patch before. I've never heard of anyone having trouble for wearing it, probably because the guys out West know to respect the heraldry of the 1st Marine Division.
A Noteworthy Improvement
I don't think I've ever heard a word from any P.T.A., anywhere I've ever lived. Apparently they once thought they were worthy of making comments on the quality of local parents.
In general I think we should disband all public agencies that would dare to come between parents and their children, for any reason short of murder, and maybe for any reason whatsoever. Imagine the gall it would take to write a letter chiding one for wearing short skirts and -- reputedly! -- drinking. Not even an official agency, either: the P.T.A.
So things aren't all getting worse.
Dolly Parton is a Good Woman
I don't think much of the Vox piece either, but the Hot Air summary doesn't do it justice. They actually did have one good point, which is that the Pigeon Forge attraction Dixie Stampede was in amazingly bad taste. There are videos and photos at the link. It was awful, up to and including segregation jokes in the bathrooms.
Anyone who has ever been to Pigeon Forge, though -- and if you haven't, I strongly recommend that you never go -- knows that the whole town is in tremendously bad taste. The basic concept appears to have been to construct a Disneyland around hillbilly and Old South stereotypes. It's amazing to me that so much bad taste could exist, let alone be contained in a single place. Don't blame Dolly for Pigeon Forge; there's too much blame there for any one person to carry.
What Dolly has done for poor kids from that very poor part of Appalachia, though, deserves the highest respect. She grew up in really tough circumstances, and she hasn't forgotten those who are still doing so. Few escape such circumstances, but far fewer do well by those who come behind them.
So watch this old video Instapundit found. That's her at 14, playing for one of Cas Walker's shows. Now I can tell you a bit about Cas Walker, because Dad used to talk about him sometimes. He rain a chain of stores and was something of a politician in and around Knoxville in the old days. To further his political ambitions, he'd bring singers and musicians like Dolly Parton and others down from the nearby mountains to play at his radio and in-person shows, and later also a television show. That got his shows attention so he could put out his political message.
Dad's favorite one of these stories was about Cas Walker's railing against the enforcement of drunk driving laws. He called for the police to abandon one particular checkpoint, which they'd been working regularly. "Some of our best citizens," he said, "are getting caught up in these police stops."
My mother didn't have much to say about Cas Walker, because her mother was too virtuous by the standards of the day to allow her to listen to Dolly Parton and her ilk. "My mother didn't approve of 'string music,' as she called it,' Mom told me, meaning anything but a capella singing. Her mother was brought up Primitive Baptist, which didn't permit instruments in the church. Those old country songs that Dolly grew up with weren't quite pure enough for my grandmother, let alone my great-grandmother, who was apparently a terrifying figure who lived to 97 years' age.
On the other hand, from my mother's report, the Primitive Baptist singing was not that great and could have used some accompaniment to cover up its flaws. But I suppose that'd be like women using make-up, which is definitely forbidden according to certain quite similar readings of the Bible.
Even so, the old Primitive Baptists were more forgiving than the current woke lot. They may not have liked it, nor partook of it, but they did coexist with it. We're lucky, because it turns out there's a lot of value in things like Dolly Parton.
Philosophy Break
No Aristotle or Zeno or Plato today; I'm planning a trip into Asheville. It's a sad town these days, but hopefully the spring will sweep all that away. North Carolina ought to be one of the Free States, like Florida and Texas and South Dakota, but we have a divided government and a bad governor. Outside the cities, though, Western North Carolina is a very pleasant anarchy with almost no visible signs of government at all.
Still, I have business in town, and so I must go.
Another Big Think Piece on COVID
I lost interest in this a long time ago, but everyone I know is sending it to me today because they remember me saying this part of it way back when:
Sometimes, experts and the public discussion failed to emphasize that we were balancing risks, as in the recurring cycles of debate over lockdowns or school openings. We should have done more to acknowledge that there were no good options, only trade-offs between different downsides. As a result, instead of recognizing the difficulty of the situation, too many people accused those on the other side of being callous and uncaring.
Well, I guess it's good people are coming around to the idea now, I guess. There's some nice talk about how the open spaces of the world are probably pretty safe most of the time, which is good to hear said in the hope that the Karens of the world might come to believe it.
UPDATE: This piece, also sent me today, has an interesting claim about mass transit including about the Japanese trains we were interested in at one time.
As long as people wear masks and don’t lick one another, New York’s subway-germ panic seems irrational. In Japan, ridership has returned to normal, and outbreaks traced to its famously crowded public transit system have been so scarce that the Japanese virologist Hitoshi Oshitani concluded, in an email to The Atlantic, that “transmission on the train is not common.” Like airline travelers forced to wait forever in line so that septuagenarians can get a patdown for underwear bombs, New Yorkers are being inconvenienced in the interest of eliminating a vanishingly small risk.
Plato's Parminedes Preface: Zeno II & Aristotle I
Sticking with the same two sources as yesterday, the Stanford article (by one John Palmer) and Aristotle's Physics 6, I'll now walk through how Aristotle treats Zeno's arguments.
Note that the Stanford article doesn't seem to think Aristotle was fair to Zeno. He objects to Aristotle's "incomplete presentation," which doesn't offer any "indication of how these four arguments might have functioned within the kind of dialectical scheme indicated by Plato’s Parmenides." This is part of a general concern he raises about how these arguments are "reconstructed." A point I think is worth raising is that the "reconstruction" seems to have started immediately:
Furthermore, Aristotle implies that people were reworking Zeno’s arguments soon after they were first propounded. In Physics 8.8, after giving a basic reconstruction of the so-called Stadium paradox (see below, sect. 2.2.1) recalling its presentation in Physics 6.9, Aristotle then notes that some propound the same argument in a different way; the alternative reconstruction he then describes (Arist. Ph. 8.8, 263a7–11) is in effect a new version of the original argument.
Now, plausibly the reason for this rapid "reconstruction" was the lack of reliable accounts of exactly what Zeno said, given the mostly oral and somewhat limited writing culture of ancient Greece. I reject this as likely, however; the best exploration of the oral culture of ancient Greece I know is Albert Lord's The Singer of Tales, which demonstrates inter alia that these oral approaches worked very well at preserving important details. They could widely alter stories in length, judging the importance of audience attention and interest, but even the abbreviated versions would be accurate to the heart of the story.
Rather, I think it is likely that the original forms of Zeno's paradoxes were rapidly disposed of by the brilliant thinkers of Socrates' and Plato's generation. What most likely happened, and what I suspect Aristotle is noting, is that other thinkers were finding more plausible ways of arguing for the point that Zeno had made. "He who strives for the stars may stumble on a straw," and perhaps Zeno's striving at his highly original arguments missed a few things; but people who weren't satisfied with the easy out constructed sounder proofs of the same point.
In any case, take it as read that we only have the one thing (from yesterday) that we think is what Zeno really said; but also that these arguments are interesting enough that even if you find a way to 'resolve' them you shouldn't set them aside. Maybe someone could find a way to resolve your resolution, too; maybe there's another approach that makes the argument better. It seems to me as if that was probably a big part of the program in what was one of the most interesting times and places for debate in human history.
So, on to the first problem:
Aristotle begins this part of his Physics with a more basic approach to explaining how things function. He is going to need this furniture to reject some of Zeno's arguments, so it makes sense to lay it out. He begins the book with a discussion of the nature of a contiuum.
Now if the terms 'continuous', 'in contact', and 'in succession' are understood as defined above things being 'continuous' if their extremities are one, 'in contact' if their extremities are together, and 'in succession' if there is nothing of their own kind intermediate between them-nothing that is continuous can be composed 'of indivisibles': e.g. a line cannot be composed of points, the line being continuous and the point indivisible. For the extremities of two points can neither be one (since of an indivisible there can be no extremity as distinct from some other part) nor together (since that which has no parts can have no extremity, the extremity and the thing of which it is the extremity being distinct).
The number line is a standard contemporary example of a continuum, but again it can be conceptually distracting because it is different from the physical objects under discussion. For example, a number line has not got extremities; it is infinitely extensive in both directions. For Aristotle, the open air might constitute a continuum; a stretch of ground might be thought of that way (as indeed he shall use it in a moment). The stretch begins here and finishes there, but we can talk and think about it as one thing that stretches for however long it does, rather than a bunch of pieces of ground next to one another.
Nevertheless, Aristotle is definitely doing the thing I'm trying to be careful not to do, which is mixing mathematical and physical concepts A line cannot be composed of points, and a line drawn across the ground is a continuum that is composed of ground, not of the points on the line drawn across it.
So the first paradox is the paradox of motion. I won't block-quote the Stanford discussion of this paradox because it is easily linked, but it may be helpful to read it first because it's a good summary of the problem. Here is what Aristotle says about it.
Hence Zeno's argument makes a false assumption in asserting that it is impossible for a thing to pass over or severally to come in contact with infinite things in a finite time. For there are two senses in which length and time and generally anything continuous are called 'infinite': they are called so either in respect of divisibility or in respect of their extremities. So while a thing in a finite time cannot come in contact with things quantitatively infinite, it can come in contact with things infinite in respect of divisibility: for in this sense the time itself is also infinite: and so we find that the time occupied by the passage over the infinite is not a finite but an infinite time, and the contact with the infinites is made by means of moments not finite but infinite in number.
The passage over the infinite, then, cannot occupy a finite time, and the passage over the finite cannot occupy an infinite time: if the time is infinite the magnitude must be infinite also, and if the magnitude is infinite, so also is the time. This may be shown as follows. Let AB be a finite magnitude, and let us suppose that it is traversed in infinite time G, and let a finite period GD of the time be taken. Now in this period the thing in motion will pass over a certain segment of the magnitude: let BE be the segment that it has thus passed over. (This will be either an exact measure of AB or less or greater than an exact measure: it makes no difference which it is.) Then, since a magnitude equal to BE will always be passed over in an equal time, and BE measures the whole magnitude, the whole time occupied in passing over AB will be finite: for it will be divisible into periods equal in number to the segments into which the magnitude is divisible. Moreover, if it is the case that infinite time is not occupied in passing over every magnitude, but it is possible to ass over some magnitude, say BE, in a finite time, and if this BE measures the whole of which it is a part, and if an equal magnitude is passed over in an equal time, then it follows that the time like the magnitude is finite. That infinite time will not be occupied in passing over BE is evident if the time be taken as limited in one direction: for as the part will be passed over in less time than the whole, the time occupied in traversing this part must be finite, the limit in one direction being given. The same reasoning will also show the falsity of the assumption that infinite length can be traversed in a finite time. It is evident, then, from what has been said that neither a line nor a surface nor in fact anything continuous can be indivisible.
This conclusion follows not only from the present argument but from the consideration that the opposite assumption implies the divisibility of the indivisible. For since the distinction of quicker and slower may apply to motions occupying any period of time and in an equal time the quicker passes over a greater length, it may happen that it will pass over a length twice, or one and a half times, as great as that passed over by the slower: for their respective velocities may stand to one another in this proportion. Suppose, then, that the quicker has in the same time been carried over a length one and a half times as great as that traversed by the slower, and that the respective magnitudes are divided, that of the quicker, the magnitude ABGD, into three indivisibles, and that of the slower into the two indivisibles EZ, ZH. Then the time may also be divided into three indivisibles, for an equal magnitude will be passed over in an equal time. Suppose then that it is thus divided into KL, Lm, MN. Again, since in the same time the slower has been carried over Ez, ZH, the time may also be similarly divided into two. Thus the indivisible will be divisible, and that which has no parts will be passed over not in an indivisible but in a greater time. It is evident, therefore, that nothing continuous is without parts.
The basic point that Aristotle is making here is that time and space are both divisible magnitudes, and that they are what I would call "geared together." That is, because motion in space also entails motion in time, you don't get a paradox of the sort Zeno is trying to set up. However long it takes to travel across the infinite divisions occupies enough of the equally infinitely divisible magnitude of time to allow for it.
(Contemporary physics offers us "spacetime," which makes this point that time and space are geared together even more emphatically.)
The other point that Aristotle wants to clarify is that both of these 'infinitely divisible' magnitudes are not made up of indivisibles: "the line is not made up of points," and time is not made up of indivisible moments of 'now.' Properly a point doesn't belong to the same dimension as physical reality; it exists here only conceptually, as a one dimensional point on a two-dimensional line in what is actually three dimensional space (or four dimensional spacetime, perhaps). The error of assuming that the points are fundamental to the line drawn across the space is what gives rise to the error that Zeno is propounding.
This turns out to be Aristotle's resolution of another of Zeno's paradoxes, which he disposes of very rapidly with the same furniture.
Zeno's reasoning, however, is fallacious, when he says that if everything when it occupies an equal space is at rest, and if that which is in locomotion is always occupying such a space at any moment, the flying arrow is therefore motionless. This is false, for time is not composed of indivisible moments any more than any other magnitude is composed of indivisibles.
That paradox is 2.2.3 in the Stanford piece, which treats it more seriously than Aristotle does. His account of how the argument works is that, at any given moment of time, the arrow must occupy a space exactly equal to its length. Yet this means the arrow is resting, because it neither extends into space it does not occupy in this moment, nor does it leave space it does not occupy. If it is resting at any random point of time, given that all points of time are the same, at every point it is resting; and thus it cannot move, because there is no extension at any point in time that we could call motion.
A more natural way of saying this might be that a flying arrow, at a frozen moment in time, is motionless; and since every length of time is composed of an infinite number of frozen moments, the arrow cannot be flying at all. Motion is impossible because at each of the divisions (a 'point in time' rather than a physical point) has no ability to sustain motion because the points are not extended objects.
Aristotle's rejection is a rejection of the whole frame, as above. There are no unextended points, not actually in our three dimensional world (or four, etc). Zeno is wrong not merely mathematically, but metaphysically: he is wrong about the nature of reality, which cannot actually be divided into indivisible points. Neither space nor time can be, so says Aristotle.
Plato's Parmenides, Preface: Zeno I
As a preface to Tom's attempt on the Parminedes, let's look at an article about Zeno and then at Aristotle's Physics 6. The reason for this is that the Parminedes opens with Socrates talking with the famous Zeno, of 'Zeno's paradoxes of motion,' and Physics 6 is where Aristotle gives his account of how to solve some of them.
Now, as this Stanford article points out, we don't actually know what Zeno said in his own words for the most part. Only one of the paradoxes comes down to us in what seems like a verbatim form. Socrates was a young man when the discussion with Zeno happened; Plato was Socrates' student; Aristotle was Plato's. So it's possible that Zeno might have rejected Aristotle's settlement, or even Aristotle's reconstruction of what he was trying to say.
Zeno is tackling some of the hardest parts of philosophy, which are the parts that treat the most basic things. You might think the basic things would be the easiest parts, but it's the other way around. Everyone can carry off an opinion about politics, which is enwrapped in ethical concerns and practical concerns and emotions and reasons and all of that. Any two old men sitting on a bench outside the bait store can do it, and make themselves understood to each other and any listeners. We deal with these complex and composite objects all the time, and think we know how to do it -- certainly we know a way of talking about it that will make sense to everyone. Reading Plato's Laws was relatively easy, because it is a conversation like that -- three old men on the road, chewing the fat about how things out to be.
It's really difficult to talk about the basic things. For example, here's the one Zeno antimony that we seem to have intact:
If there are many things, it is necessary that they be just so many as they are and neither greater than themselves nor fewer. But if they are just as many as they are, they will be limited. If there are many things, the things that are are unlimited; for there are always others between these entities, and again others between those. And thus the things that are are unlimited.
If that didn't make sense on the first pass, it's because the ideas are so basic and fundamental that they seem to lack context that helps you understand what is going on. So let's walk through it.
"If there are many things..."
Possibly there aren't many things. That's actually what Zeno is going to be getting at, and Parminedes as well when we get there.
It seems like there are obviously many things, though. You can look around you and see what appear to be many different things. In my vision right now are this computer, a coffee cup with a skull and crossbones on it, and a Gerber Applegate-Fairbairn combat knife. It seems like these are several separate things, not just because they don't appear to be touching, but because my mind knows what each of these artifacts is for and it's not the same thing. Since each artifact has a distinct purpose, it must have a distinct reason for having come into being; and thus, since each thing was made at a different time for a different reason, it follows that they must be different things.
So Zeno is raising a problem for an idea that seems extremely plausible, and suggesting that it's impossible.
"If there are many things, it is necessary that they be just so many as they are and neither greater than themselves nor fewer."
Say that there are many different things. Then, necessarily, there is a number of such things. We may not actually be able to count to it, but in principle the number of things could be known. This is true even if you move from counting knives and cups to counting atoms, or electrons, or whatever you decide qualifies as a thing -- that is itself a very important question in metaphysics. Whatever that number is, we shall call it a.
Now, all Zeno is saying is that a=a. Whatever a is, it must be equal to itself. It cannot be greater nor less than itself.
But if they are just as many as they are, they will be limited.
Whatever a is, it isn't greater than a; for example it isn't equal to a+1. It is a limited number, too, not an infinite number, because each of the individual things is finite (specifically, each part is one thing). You can't get to the infinite by adding up a finite number of things, no matter how big that finite number might be.
If there are many things, the things that are are unlimited; for there are always others between these entities, and again others between those.
An aside that non-mathematicians might skip: some of you are mathematicians, and you probably recognize this move as the shift from countable to uncountable infinities. That's bringing in a lot of furniture from mathematical theory that doesn't quite belong here; it would be easy to get confused if you think of this as integers being 'things' and the infinitely divisible 'space' on the number line between the integers as being the uncountably infinite. One reason that may confuse you is that there really are an infinite number of integers, but there really aren't an infinite number of things in the world. If you make that analogy (all analogies always break) the breaking point will be there. Zeno is talking about the physical world, not about the conceptual number line.
Aside over: is it reasonable to say that, in the physical world, the 'things' between objects are infinitely divisible? Well, yes, it is: and not just because we can't really say much about the levels of it that are too small to see with an electron microscope. Even if you can't divide a pea practically into much more than eighths, and that with a sharp knife, we know you could continue to divide it down to the level of atoms in theory; and even those can be split; and even there, there's a lot of 'empty space' that could be divided conceptually into parts. This is different from the purely conceptual division of the number line, because it is a conceptual division of a physical space; but it is, conceptually, just as infinite. (Indeed uncountably so, for those who followed the aside.)
But if they are just as many as they are, they will be limited.... [yet] thus the things that are are unlimited.
So here we have a paradox. Just because of the nature of there being 'many things,' those things must have a limited number. Yet because there will also be 'things' between them, and those things can be divided infinitely, a world of many things must have an unlimited number of things in it.
If the world has many things in it, then, they must be both limited and unlimited in number. Since that appears to be a logical contradiction arising from consequences of the antecedent "if there are many things," we must reject the antecedent. Therefore: There are not many things.
Thus, there are either no things at all, or only one thing.
There are several ways to attack this paradox. I think they're fun, so mostly I'll leave it as an exercise for you to do in the comments if you want. My personal favorite is to attack the idea that a conceptual division (which gives rise to the infinity) actually gives rise to additional things. Probably a better method, but a harder one, is to be precise about the metaphysical stature required to constitute a thing at all.
On the other hand, you could accept the argument. It's quite logical. If a material conditional leads to a contradiction in formal logic, we do indeed reject the antecedent in just the way Zeno proposes.
Ethel
An estate is very dear to every man,
if he can enjoy there in his house
whatever is right and proper in constant prosperity.
So says the Anglo-Saxon Rune Poem about the rune called in that language 'ethel,' which was the subject of some controversy at CPAC this weekend. There are several surviving rune poems, but that particular rune doesn't come up in every version of the runic languages.
As the poem suggests the old rune was apparently associated with the homestead, wealth, peace, and prosperity. The controversy came from the fact that some SS units apparently used it as a unit insignia during the war. Germany is now wealthy, and peaceful, prosperous, and a stable home -- but not for them, who are gone from the world, unmissed and unmourned.
There's a question about whether a symbol means just what you intend it to mean, or whether things like words carry a meaning that transcends what we want them to be. Tolkien used ancient word roots like warg and ent and orc, in something like their original intent. Was there a lingering power in the old symbol, the old sound, though living men had forgotten what it really meant for a very long time? I always wonder about that.
Plato’s Republic: Confer
The American Mind has a helpful summary of Plato’s ideas in the Republic on the present difficulty. Since we’ve just finished reading the Laws, it’s a good opportunity to compare and contrast the treatments.
Non-instinctive thinking
Some kinds of probability puzzles are particularly difficult:
1% of women at age forty who participate in routine screening have breast cancer. 80% of women with breast cancer will get positive mammographies. 9.6% of women without breast cancer will also get positive mammographies. A woman in this age group had a positive mammography in a routine screening. What is the probability that she actually has breast cancer?
Over 80% of doctors get this wrong. They tend to estimate that the woman with a positive mammogram has an 70-80% of really having cancer. The real answer is about 7.8%:
Out of 10,000 women, 100 have breast cancer; 80 of those 100 have positive mammographies. From the same 10,000 women, 9,900 will not have breast cancer and of those 9,900 women, 950 will also get positive mammographies. This makes the total number of women with positive mammographies 950+80 or 1,030. Of those 1,030 women with positive mammographies, 80 will have cancer. Expressed as a proportion, this is 80/1,030 or 0.07767 or 7.8%.
Sure, the fraction of women with false positives is much lower than those with true positives, but the percentage of women without cancer is so high that the raw numbers of cancer-free women to which we apply the 9.6% false-positive rate swamp the low rate. Similarly the percentage of women with cancer is so low that the high 80% true-positive rate is undermined by the low raw numbers of cancer-suffering women.
H/t Slate Star Codex archives mentioned in an open thread this weekend at Astral Codex Ten.