Here's my summary/reconstruction of lecture III on Achilles and the tortoise:-

The first part of the third lecture is about a real chestnut. But it is rather hard to follow, in the sense that it is hard to see where he is going. I think it helps to start with his conclusion, his diagnosis of the problem.

His final remark is not a surprise – “Similarly (i.e. to the fatalist’s dilemma) here we have been talking, so to speak, in one breath with the sporting reporter of a newspaper, and in another breath with our mathematics master, and so find ourselves describing 1) a sprint in terms of numerators and denominators and 2) of relations between fractions in terms of efforts and despair.” p. 53 (numbers and strikethrough mine).

On the previous page (52), we find the specifics – “We decide factual questions about the length and duration of a race by one procedure, namely measurement; we decide arithmetical questions by another procedure, namely calculation. But then, given some facts about the race (such as whether Achilles will win) established by measurement, we can decide other questions about that race (such as where and when Achilles will overtake the tortoise) by calculations applied to these measurements. The two procedures of settling the different sorts of questions intertwine, somehow, into a procedure for establishing by calculation concrete, measurable facts about this particular race. We have the pony in the harness that was meant for any such pony, yet we can mismanage the previously quite manageable pony in its previously quite manageable harness.” His summary his helpfully simpler – “Two separate skills do not, in the beginning, intertwine into one conjoint skill.”

(I think this is his gesture towards the mathematical solution of the problem by application of the calculus which demonstrates that we can calculate when Achilles will overtake the tortoise to any level of accuracy that we desire).

Going back a bit further he acknowledges the common ground between the two skills (p.48) “… in an important way we are, in all applications, thinking in terms of or operating with the same overarching notions of part, whole, fraction, total, plus, minus and multiplied by.” He articulates the question (p.50), as “How is (what we know quite well about the stages of an athlete's victorious pursuit) to be married with (what we also know quite well about the results of adding together a fraction of a whole, that fraction of the remainder, that fraction of the next remainder, and so on)?”

He partially answers this question by pointing out:-

1) that “never” in this scenario is ambiguous between the harmless truism “To say this (sc. that the sum of 1/2, 1/4, 1/8, 1/16 etc., never amounts to unity). is simply to utter the general proposition that any particular remainder-bisection leaves a remainder to bisect.” and the alarming prophecy that “if a silly computer were to attempt to continue bisecting remainders until he had found one which was halved but had no second half, his attempt would then go on to all eternity.” p. 50/51.
He also, less transparently, finds an ambiguity between “all” as in the total when all the parts are added up and “any”. I don’t quite understand it and cannot find a suitably brief quotation.

So that’s my backwards summary of the part of his lecture that begins on p. 48 with:- “Now let us draw some general lessons from this dilemma.” Returning to the beginning, Ryle’s aim is getting us to see that the paradox hypnotizes us into seeing it only as an endless series. We need to appreciate two distinct points of view. One is the overview of the whole event (by a non-participant) and the other is the narrow view of a competitor in the race. He approaches this by considering dividing up a cake alongside dividing up the race. I think the point is that in dividing up the race, we tend to forget the overview of the whole; it is easier to keep the whole cake in mind because it is not a temporal process. Some of the points that I found helpful:-

1) On p. 42, he imagines that we might mark out the course by planting a flag at each point of the calculation. At the half-way mark, the quarter-way mark and so on. The method itself guarantees that there will always be a place for another flag, so we think that Achilles will never reach the tortoise. But if we reversed the process, would we be convinced that the race did not have a beginning?

2) “Similarly Zeno, in his mentions of the successive leads to be made up by Achilles, is, though surreptitiously and only by implication, referring to the total two-mile course run by Achilles in overtaking the tortoise; or in other words, his argument itself rests on the unadvertised premiss that Achilles does catch the tortoise in, say, precisely two miles and in precisely one hour.” p.44.

3) We need to see that there is a crucial difference between two questions – “To put a central point very crudely, we have to distinguish the question' How many portions have you cut off the object?' from the question 'How many portions have you cut it into?’”. (p. 46) In the first question, we have partly cut the cake and there is always part of it left. In the second question, we have cut the whole cake up.

Oh by the way, what I am discussing versus a specific identity versus a general future event, is not so indirectly related to this passage in Ryle: — schopenhauer1

Yes. That puts a different perspective on things. Very helpful.

Nice summary.

This is my reaction to Lecture III.

Ryle writes at the very beginning of Dilemmas, “One familiar kind of conflict is that in which two or more theorists offer rival solutions of the same problem. In the simplest cases, their solutions are rivals in the sense that if one of them is true the others are false.” It is strange that Ryle would not include the conflict found in Achilles and the Tortoise as an instance of rival theories having a clear winner. He says on p 36, “It is quite certain that a fast runner following a slow runner will overtake him in the end” and “Nothing could be more decisively settled” if you consider the speeds of each contestant and the ground in which they cover. How closer to the truth must we get? Our experience of such an event as well as our mathematical description of such an event is decisive. Clearly, we have an answer to the problem of who will win the race between Achilles and the Tortoise.

However, Ryle has something else in mind. He says, “Yet there is a very different answer which also seems to follow with equal cogency from the same data.” But what “data” is that? Surely not the data experimentally collected by watching a race between two such opponents. He must mean the data generated based on the hypothetical of an infinite number of steps bisecting and never reaching unity. Now if Ryle stopped here, I could understand because in a way one theory is beholden to an outcome of an actual race, while the other is beholden to the thought experiment where coherence and consistency rule the day. So, this goes along with Ryle’s idea that “There often arise quarrels between theories, or, more generally, between lines of thought, which are not rival solutions of the same problem, but rather solutions or would-be solutions of different problems, and which, none the less, seem to be irreconcilable with one another.”

So, why did Ryle not just declare a winner and be done with it? I believe Ryle is committed to showing that Zeno’s paradox does give us some kind of knowledge, although a more rationalistic kind. However, during this analysis, Ryle seems to oscillate between rationalism and empiricism. For example, take his interesting example of dividing up a cake, p 39:

“But now suppose that the mother of a family chooses instead to circulate an uncut cake round the table, instructing the children that each is to cut off a bit and only a bit of what is on the plate; i.e. that no child is to take the whole of what he finds on the plate. Then, obviously so long as her instructions are observed, however far and often the cake circulates, there is always a bit of cake left. If they obey her orders always to leave a bit, then they always leave a bit. Or to put it the other way round, if they obey her orders never to take the whole of the last fragment, a fragment always remains untaken.”

What is Ryle referring to here? To actual cake, or some abstract object call “a cake”? This is where I think Ryle presents a confusing picture. From p 40, “The plate never stops circulating. After each cut there remains a morsel to be bisected by the next child. Obviously, the children’s patience or their eyesight will give out before the cake gives out. For the cake cannot give out on this principle of division.” In one breath he seems to present an example that should reflect actual reality of bisecting a cake by children and is practically limited by the child’s eyesight and patience. Yet, in the very next sentence talks about “a cake” that cannot give out on this principle of division. However, if it was an actual cake, and depending on how heterogeneous the cake was and how easily it can be divided, each morsel may not be the cake anymore, but a piece of fruit, sugar, salt, etc. At some point, it may become the actual ingredients we used to make the cake to start with. So, from this perspective, “the cake” gives out on the principle of division. But an idea of “a cake” that represents the general concept of “object” may be another story. Similarly, to how I described the Zeno paradox “dilemma”, one is beholden to the concept of division, and the other to the experiment of cutting up a heterogenous mixture of stuff.

Ryle concludes his analysis of the Zeno paradox with “…nor can he grasp the other abstract platitude that the portions cut off something at no stage amount to the whole of that thing. (p48)” So, I guess we ended up in the same place in that Zeno is not answering the question “who will win the race.” But Ryle wants to say something additional, Zeno is putting forth an abstract platitude. But I say Zeno parades a metaphysical fiction disguised as a scientist hypothesis.

My overall response is that Lecture 2 has some confusion in regard to the modal considerations

I've previously set out the story that the ordinary language philosophy in use here gave way to a return to more formal considerations, mostly as a result of developments in Logic. Part of that was Kripke's development of a formal semantics for modal logic. Ryle's position here seems to be reliant on a descriptive notion of naming, and would need some considerable re-writing in the light of the development of rigid designators. Some idea of the complexity involved can be gleaned from The Possibilism-Actualism Debate. I doubt it's a road we would want to go down here. @schopenhauer1's new thread shows how convolute that area becomes.

There are other issues. So I suspect the intersection example suffers from confusing modality with probability. We can't name the individual accidents that were avoided, but can still maintain that the overall probability of an accident was reduced.

We can't name the individual accidents that were avoided, but can still maintain that the overall probability of an accident was reduced. — Banno

Yes. That's because, of course, there are, ex hypothesi no individual (actual) accidents to be averted. I don't see that Ryle is at all confused here.

Clearly, we have an answer to the problem of who will win the race between Achilles and the Tortoise. — Richard B

Surely, you are missing the point here. No-one doubts who will win the race. The question is how Zeno makes it appear that there is some question about that. The answer is that he considers the race from a certain, misleading, point of view. Ryle's project here is to understand how that illusion is created. Wittgenstein speaks of conjuring tricks. Austin, in Sense and Sensibilia has similar, but less brutal, descriptions of the process.

He says, “Yet there is a very different answer which also seems to follow with equal cogency from the same data.” But what “data” is that? — Richard B

Ryle is not always precise in his language. "Data" just means the set-up of Achilles racing the tortoise

which are not rival solutions of the same problem, but rather solutions or would-be solutions of different problems, and which, none the less, seem to be irreconcilable with one another.” — Richard B

Yes, I think that's exactly what Ryle is saying about this problem.

So, why did Ryle not just declare a winner and be done with it? — Richard B

Well, he wants to diagnose why anyone would have taken Zeno's problem seriously - and, by the way, Zeno also took this problem seriously in that he believes that all change, including motion, is an illusion.
I think there is a real problem here, and it needs to be acknowledged. You can calculate the time it takes for Achilles to complete the race and for the tortoise to complete the race, you; you can then compare the times and see that Achilles will win. But if you ask when (or where) Achilles will catch up and pass the tortoise, you can't - not accurately, as you can with the first calculation. The consolation prize is that you can calculate it to any degree of accuracy you like; but that didn't become possible until the calculus of infinitesimals was invented in the 17th century CE.

To actual cake, or some abstract object call “a cake”? This is where I think Ryle presents a confusing picture. — Richard B

Yes, Zeno's problem is purely theoretical not, in some sense of the word, real. Which is why it is so tempting to simply declare the winner.

But Ryle wants to say something additional, Zeno is putting forth an abstract platitude. But I say Zeno parades a metaphysical fiction disguised as a scientist hypothesis. — Richard B

Well, yes. Zeno does have a metaphysical solution to the problem, which is to declare motion impossible. Philosophy has progressed to the point where we don't need to argue about that any more. Who says philosophy never makes any progress?

Yes. That's because, of course, there are, ex hypothesi no individual (actual) accidents to be averted. I don't see that Ryle is at all confused here. — Ludwig V

I should have been clearer, yes, the confusion is not Ryle's, but those who mistake the modal for the probable.

Some idea of the complexity involved can be gleaned from The Possibilism-Actualism Debate. I doubt it's a road we would want to go down here. schopenhauer1's new thread shows how convolute that area becomes. — Banno

Yes indeed. My argument seems to parallel Ryle's that the future is more along the lines of "possiblism", and that even looking in the past, there could have been actual counterfactuals that could have happened. However, there are some things which are logically impossible because they require necessity. For example, prior to your conception, if there was any slight change to the gametes meeting, there was no person that was you. Most circumstances would lead to the outcome that if any slight circumstance changed prior to conception, then the current you would not exist to look back upon these counterfactual possibilities.

I'm not too keen on going in to that, due to the effort required and the small payout. If you ask whether you might have had different gametes, then that's a question about you, using a rigid designation. it's obviously not impossible that you could have had somewhat different genetics. As to whether your genetics might have been completely different, that will depend on how you understand the designation. It's a minefield, and intuition is a poor guide.

it's obviously not impossible that you could have had somewhat different genetics. — Banno

I know you don't want to get in the weeds, but that is exactly what I am contesting. Even if it was a slight second earlier or later, whatever that person becomes, it was/is not you.

As to whether your genetics might have been completely different, that will depend on how you understand the designation. It's a minefield, and intuition is a poor guide. — Banno

Indeed it is a minefield, but now there's a whole thread devoted to it, if you want to take a stab at it. Obviously, with that article you referenced, this opens a tremendous can of worms and encompasses a whole lot of ideas in metaphysics, causality, necessity, identity, and the like.

One of the things that makes this hard to be a definite "rigid designation" is that it is conceivable that there is the ever so distant possibility that the same set of gametes could have been selected in some artificial way that was exactly the same as the ones that comprised the non-artificial version. So, is it conceivable that someone could still come about in a way that was different than the instant of the two gametes coming together? Perhaps. But then this brings up ideas of different causes for the same outcome... In other words, it may refute the claim that everything would have to happen as is prior to conception for you to have existed. However, in 99.99999999 cases, the circumstances would have had to be the same for you to have been conceived. How much does the limit have to reach 100% for it to considered a necessity that everything had to be exactly the same?

Well, he wants to diagnose why anyone would have taken Zeno's problem seriously - and, by the way, Zeno also took this problem seriously in that he believes that all change, including motion, is an illusion. — Ludwig V

Yes, Zeno's problem is purely theoretical not, in some sense of the word, real. Which is why it is so tempting to simply declare the winner. — Ludwig V

I do not believe Ryle should have taken this paradox serious, nor anyone else for that matter. And if we are going to talk about "temptation", it should be to ask why would anyone be tempted to take this serious to begin with. Just because one presents a picture that is cogent does not mean it has any application in the real world. And it was Ryle who describe one kind of dilemma as "In the simplest cases, their solutions are rivals in the sense that if one of them is true the others are false." Here I argue that that this is a simple case of one line of thought as "true" and the other "false". How can anyone argue it has any application to the world we experience? Why is the appeal of our experiences not the deciding factor that drive us to say that this is a simple case that easily decides on "which is true" and "which is false"? Do weneed Ryle to take the extra step to "clarify the language" that this is actually an "abstract platitiude"? We can't trust our experience to dismiss Zeno, we need the extraordinary insight of Ryle to show us out of the fly-bottle so to speak.?

If I show Ryle a film of a race between Achilles and the Tortoise in which the Tortoise gets a lead, and by strategic camera angles and editing shows the Tortoise winning even though Achilles looks faster. The film certainly is not a logical impossibility. Do I need Ryle to help us understand that there should not be any confusion with what actually happens in such event?

Lastly, to say Zeno's problem is purely theoretical, I think is a bit incomplete. Zeno's argument is to show that all change/motion is an illusion, but this is not some fantasy world for Zeno, it is the actual world around us that seems to have change but is illusionary.

All I am saying is experience settles some questions not just lingustic analysis. And in this case, experience should be arbiter.

this is not some fantasy world for Zeno, — Richard B

That's certainly true. I didn't distinguish carefully enough between Zeno's thinking and ours. We have the benefit of an established distinction between theory and practice, which didn't exist in Zeno's time.

All I am saying is experience settles some questions not just lingustic analysis. And in this case, experience should be arbiter. — Richard B

That's true. It would be interesting to know why you think that experience should be the arbiter in this case. By the way, I don't think that anyone thinks that Achilles won't overtake the tortoise.
Experience isn't a given. It needs interpreting. You experience the sun coming up over the horizon on Monday morning. You have the same experience on Tuesday morning. What tells you that it is the same sun and not a new one every day? How do you know that the sun doesn't rise, but the earth turns?

it should be to ask why would anyone be tempted to take this serious to begin with. — Richard B

Well, Zeno did. So have many other people. If you want to know why, read Ryle.

I think that the best answer to what you are saying is that the paradox isn't a problem. It's a puzzle. Whether it's a serious puzzle or not is another question. Whether it's an interesting puzzle is yet another question.

But then this brings up ideas of different causes for the same outcome. — schopenhauer1

Yes. Isn't that implicit in "necessary but not sufficient"?

How much does the limit have to reach 100% for it to considered a necessity that everything had to be exactly the same? — schopenhauer1

I would say it has to reach at least 100%. But maybe you don't?

using a rigid designation. — Banno

I wish I had thought of that days ago. But I'm not sure it applies. Doesn't Ryle's argument about the future mean that rigid designators cannot be rigid in the future tense?

We have the benefit of an established distinction between theory and practice, which didn't exist in Zeno's time. — Ludwig V

You seriously underestimate Zeno and others, especially his teacher, Parmenides. Zeno knew full well that when he walked he was able to go from one place to another. The larger problem is the relationship between thinking and being. Rather than attempt to resolve the interpretive problems I will frame the problem in the form of a question and in light of this turn to Zeno's a priori puzzle.

Is thinking the way to being or an impediment? (See Parmenides poem and the way of truth - alethia)
The Eleatic philosophers were said to hold that all things are one. Parmenides does not simply accept this, he inquires dialectically, treating it as a hypothesis. If all things are one then what follows?

If all things are one then it follows that there can be no motion. Can we reasonably argue that there is no motion? Zeno provides the arguments. We might say that he was misled by treating this a priori, but if all is one and thereby thinking and being are one, then this should make no difference.

I wish I had thought of that days ago. But I'm not sure it applies. Doesn't Ryle's argument about the future mean that rigid designators cannot be rigid in the future tense? — Ludwig V

Rigid designation is a bit tangential perhaps, but it can be pulled into this debate. As I see it, rigid designators are about invariant necessities involved in something's name. Generally this is "proper names" but can be expanded to scientific kinds and other things as well. So when someone is rigidly designating "Ludwig V", that means this person is Ludwig V in all possible worlds. Ludwig V is causally "linked" through a dubbing process that cannot be invariant across worlds.

In this sense, we can start making connections to this notion of identity, gametes, and temporal-causes. That is to say, the gametes combining in such a way at such and such instant (they usually go together, though in a fantastical conceivable way I can think of a way they might not), that it must be this event (and combination of gametes), invariant across all possible worlds. It is in a way, "rigidly designated" as "you". Across these worlds, you might have different hair colors, different ways in which you interacted with the environment, but what has to be invariant was the gamete combination at that instance of coming together for it to be you, and not someone who is just similar. Just as there is a causal link in Ludwig V with the person Ludwig V. The person Ludwig V is linked "as an individual person" by way of causal instance of gametes combining.

The person Ludwig V is linked "as an individual person" by way of causal instance of gametes combining. — schopenhauer1

If the link is causal, it is empirical. Which means it is not necessary.

Zeno provides the arguments. — Fooloso4

Quite so. But if he was misled, doesn't that suggest that the conclusion of the argument is wrong, or at least may be wrong? Does that really make no difference?

If the link is causal, it is empirical. Which means it is not necessary. — Ludwig V

The causality is the necessity. That is similar to Kripke's causal-theory of proper names and use of rigid designators.

That is similar to Kripke's causal-theory of proper names and use of rigid designators. — schopenhauer1

You say that as if it settled the matter. Is there a universal consensus that Kripke is necessarily right? That would indeed be remarkable.

You say that as if it settled the matter. Is there a universal consensus that Kripke is necessarily right? That would indeed be remarkable. — Ludwig V

No it is not. Someone brought up rigid designators (and thus Kripke).

I pointed to the fact that generally his work on rigid designators (he invented the term I think), involves proper names.

It has also been extended to natural kinds (like H20 necessarily being the term "water").

I remarked that rigid designators can be tied into identity of individual personhood by way of causal necessity (these two gametes meeting at a certain instant of time whether natural or artificially).

I'd also like to add, that perhaps it falls less under his proper names "necessity by way of causality across all possible worlds", and could simply fall under "necessity by way of natural kinds". One has a causal aspect to it (someone dubs an object in a speech act that then becomes the origination of the name tied to that person in a chain of events). The other seems to be essentialist in terms of something akin to "substance" (H20 is water). Interestingly, both can be the case in terms of individual personhood.

And did Kripke invent causal necessity as well?

But if he was misled, — Ludwig V

If I am right in claiming that their methodological approach is dialectical then he was not misled. He was treating the claim that all is one as a hypothesis to be examined and if it was supported by reason. Pointing to experience in order to reject the claim begs the question of the unity of thinking and being. If they are the same then perhaps what should be rejected is what experience seems to show. Dialectical movement does not resolve things, it keeps them in play.

But perhaps, as Plato seems to suggest, he lacked the subtlety of Parmenides. Perhaps he did not treat this dialectically but either as a truth to be defended or as where reason leads us necessarily. In which case it seems plausible that he was misled by a priori reasoning. By what Kant would call the pure reason of metaphysics.

Doesn't Ryle's argument about the future mean that rigid designators cannot be rigid in the future tense? — Ludwig V

Yes, but I see no reason to take such a view seriously. The sentence "Ludwig may reply to this post later" is about you.

Dialectical movement does not resolve things, it keeps them in play. — Fooloso4

I'm puzzled. I thought Socrates/Plato invented dialectic. What's the evidence that any pre-Socratics knew about dialectics?

And did Kripke invent causal necessity as well? — Ludwig V

I'm not sure what you are implying here. Is this supposed to be sarcastic or something? I simply stated he invented the term "rigid designator" for the idea that a word attaches to an object in all possible world. In his version, it is through causal necessity.

Roughly, statements in the future tense cannot convey singular, but only general propositions, where statements in the present and past tense can convey both. More strictly, a statement to the effect that something will exist or happen is, in so far, a general statement. When I predict the next eclipse of the moon, I have indeed got the moon to make statements about, but I have not got her next eclipse to make statements about. — p.27

This is quite problematic. I've been unable to follow what Ryle means here by "general" and "singular". The sentence "The next total lunar eclipse will occur on March 14, 2025" is about the Moon; and indeed it is about a single event. In what way is it general and not singular?

Yes, but I see no reason to take such a view seriously. — Banno

I've discovered that I'm a bit prone to being distracted by side-issues, so I won't ask what that means.

I mention this point because some people have got the idea from some of the professions though not, I think, the practices of philosophers, that doing philosophy consists or should consist of untying logical knots one at a time-as if, to burlesque the idea, it would have been quite proper and feasible for Hume on Monday to analyse the use of the term 'cause', and then on Tuesday, Wednesday and Thursday to move on to analyse seriatim the uses of the terms 'causeway', 'cautery' and ,caution', in alphabetical order. — p 31

An apparent dig at Austin...?

It was a genuine question. I don't know what causal necessity means. I know what "I inherited my fair hair from my parents means." But then, I've been reading Hume.

I thought Socrates/Plato invented dialectic. What's the evidence that any pre-Socratics knew about dialectics? — Ludwig V

I cannot give you a definitive answer on this. The following from the SEP entry on Zeno gives some indication why such an answer is not available:

The portrait of Zeno and his tactics that emerges from Plato’s references makes it seem natural that Aristotle, in one of his lost dialogues, entitled Sophist, spoke of Zeno as the inventor of dialectic (D.L. 8.57; cf. 9.25; S.E. M. 7.7). Precisely what Aristotle meant by this remains a matter of speculation, given that Aristotle also attributes the invention of dialectic to Socrates (Arist. Metaph. M.4, 1078b25–30) and to Plato (Metaph. A.6, 987b31–3); he says he himself invented the theory of it (SE 34, 183b34–184b8). There is also the question of whether Aristotle viewed Zeno’s arguments as more eristic than properly dialectical. The difference, according to Aristotle, is that dialectical arguments proceed from endoxa or “views held by everyone or by most people or by the wise, that is, by all, most, or the especially famous and respected of the wise,” whereas eristic arguments proceed from what only seem to be, or what seems to follow from, endoxa (Top. 1.1, 100a29–30, b22–5). Aristotle clearly believes that some of Zeno’s assumptions have only a specious plausibility (see Top. 8.8, 160b7–9, SE 24, 279b17–21, Ph. 1.2, 233a21–31, Metaph. B.4.1001b13–16), so that they would by Aristotle’s own criteria be examples of eristic rather than properly dialectical arguments. For Aristotle, then, Zeno was a controversialist and paradox-monger, whose arguments were nevertheless both sophisticated enough to qualify him as the inventor of dialectic and were important for forcing clarification of concepts fundamental to natural science. Aristotle’s view of Zeno thus seems largely in accordance with Plato’s portrayal of him as a master of the art of contradiction.

If the Eleatic philosophers held the opinion that all is one then Zeno's argument could justifiably be regarded as endoxa rather than eristic. I added bolding to the quote above to help explain this distinction.

If you do a search you will find several articles that credit Zeno. But all this may be tangential to Ryle.

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