I think its possible your description of hte Sun there lands us in the same position: If that, to you, is 'direct awareness' I don't understand the claim. It is not "the object" in any sense - it is light ferried across one AU, bringing with it information about the Sun. We call this 'seeing the sun' because its easier and better for "getting on with it". — AmadeusD

I'm not clear what "it" refers to in "it is not 'the object' in any sense". These examples scramble our intuitions - our common sense. The problem is that there is a slippery slope here. Under normal circumstances, we have no hesitation about saying that we see the computer screen on which we are typing. And yet, there is a time lag between light leaving the screen and it arriving at our eyes. But when we find cases where the time lag is longer, we don't quite know what to say. Nothing wrong with that.
Let's add in the phenomenon of nova stars, which are stars that explode in a brilliant flash, easily seen on earth, even though the phenomenon is light years away. So far that the star has usually disappeared by the time we see it.
So, what is it that we see - the computer screen, the sun, the exploding star? If I allow that we don't see the star, I'll need to admit we don't see the sun and we don't see the computer screen. I prefer to work the other way. I insist that we see the computer screen, the sun and the exploding star. I accept it's a choice, but that's the point. There's no right way to go here.
I don't think that we normally see light, except as reflected or emitted from things - and even then, what we see is the objects from which the light is reflected or emitted. So I was most uncomfortable when I needed to describe what travels from the sun to the earth. Information?, an image? I don't know the right term. But I need to be clear that I see the sun, just as I see the computer screen on which I am writing. Light is the medium that enables me to see, not something that I see in its own right.

Right. I've been considering exactly this is recent days - ..... I am sorry if this isn't directly on point, but it seems clear to me "error" comes in different kinds, and the one I mean (related to the latter example) cannot be adjudicated by further looking at the object: It can change from red to grey as I see fit, in some sense. I am not bound by the object to see it as a certain colour in that case. — AmadeusD

There's a lot in here. I agree that there are many different sources of error. I would hate to have to create a taxonomy. However, there is one key point here and that is the concept of interpretation. Many errors are errors of interpretation and so do not require positing any kind of intermediary object. That's what is left out of this debate.

We don't argue about whether "watching the game" on recording is direct awareness of the game, or the recording (well, it seems to me we dont?). I don't quite see a difference here. — AmadeusD

I'm not sure about that. I agree most people will happily say that they are watching the game under all those conditions. But I think most people will differentiate between watching the game live and watching a recording. They will likely not talk of "direct" or "indirect", but still...

The apple doesn't exist during the second ten seconds and so cannot be a constituent of the experience, and so the conclusion is false. Therefore, one of the premises is false. Given that I agree with P1a, my conclusion is that P2a is false. — Michael

So the experience of an apple in the first ten seconds was not an experience of an apple. H'm.

In the first interval, the act is fulfilled by the apple; in the second, it is not. — Esse Quam Videri

Why not? There is no relevant difference between the information carried by the light in the first ten seconds and the second ten seconds. The presence or absence of the apple when the light arrives is irrelevant. IMO.

That asymmetry is not captured by describing the light alone, and it’s precisely what distinguishes veridical perception from residual or empty intentionality. — Esse Quam Videri

The destruction of the apple is too late to influence what has gone out; it cannot have any effect until tn seconds have elapsed, i.e. until the third ten seconds. You seem to think that the disappearance of the apple after the light has been sent on its way makes a difference to what is seen. But the apple was there when the light started its journey and so it carries the information that was accurate at the time of dispatch.

Most direct realists say that we have direct visual perception of apples and trees and everything else that emits or reflects light into our eyes, whereas your account is that we only have direct visual perception of the light. Yours is a strange kind of direct realism. — Michael

I must have drafted something very badly. My position is that I only see objects that reflect or emit light. I don't know what it would be to see light as such - in transit, so to speak.

P1. If the apple is not a constituent of the experience during the second 10 seconds then it is not a constituent of the experience during the first 10 seconds
P2. The apple is not a constituent of the experience during the second 10 seconds
C1. Therefore, the apple is not a constituent of the experience during the first 10 seconds — Michael

If that's a good argument, then what's wrong with this?
P1a. If the apple is a constituent of the experience during the first ten seconds, then it is a constituent of the experience during the second ten seconds.
P2a. The apple is a constituent of the experience during the first ten seconds.
C1a. Therefore, the apple is a constituent of the experience during the second ten seconds.

Exactly. So you're admitting you're seeing light which presents the sun as it was eight minutes ago. I shall leave this there and just see if it lands. — AmadeusD

I think its patently clear that there is no way to assess error beyond error as a mathematical/statistical exercise or a purely practical one (trial and error, i guess) no matter which theory you prefer. The DRist, I think, wants to say that a mediated perception is direct enough to capture error. I just disagree. — AmadeusD

I had in mind the ordinary ways in which we realize we didn't see what we thought we saw. Which essentially means an inconsistency in the stream of perceptions that we experience. (This is a very rough sketch, because I expect you know what I'm talking about.) Philosophers tend to look for decisions on the spot. In real life, sometimes additional information comes in later or from a wider perspective.

So you're admitting you're seeing light which presents the sun as it was eight minutes ago — AmadeusD

Not quite. I'm seeing light from the sun that carries information about it as it was eight minutes ago.

Which is batshit insane on the facts, to my mind. Not concluding error might be reasonable, but denying any reason for doubt is just... good god. Not sure i'm cut out for such a wild claim. The following doesn't help, because its entirely recursive. — AmadeusD

Well, perhaps I over-stated the point. I can see the reason for doubt but don't think that it carries much weight.

I have to argue this point. — Athena

I wasn't arguing that secular morality is not part of how we bring up our children. So I don't disagree with you at all. My perception of the political landscape in which we leave is that religious leaders attract more publicity and exercise more influence than you would expect, given the size of their congregations. That's all.
I don't know how many supporters White Christian Nationalists have, but I'm willing to bet that their leaders have more political clout than you would expect from those numbers.

Democracy depends on the philosophy of Hellenism and Rome. Not the Bible and German philosophers. — Athena

We certainly think we are the heirs of Greece in the matter of democracy. Rome's democracy is, I would think, less influential, given that it was an autocracy for so long. The Bible is certainly not a democratic document. But, in the history of European institutions, there was an ancient German tradition that was very influential.

A thing also known as a folkmoot, assembly, tribal council, and by other names, was a governing assembly in early Germanic society, made up of the free people of the community presided over by a lawspeaker. — Wikipedia - Thing, assembly

The reports of this institution go back to the 1st century CE.
The oldest democratic institution still surviving is almost certainly the Icelandic parliament, which owes nothing to the Greco-Roman tradition and everything to the German-Scandinavian tradition. The Althing was founded in 930 CE.
(Did you ever wonder why the person who chairs the debates in the UK parliament is known as the Speaker? Now you know.)

Using this account, the naive realist must accept that the apple is not a "constituent" of the experiential episode during the second 10 seconds — because no such apple exists — and so is not the direct object of perception. — Michael

Are you saying that the apple is a constituent of the episode during the first 10 seconds? I would then point out that the relationship of the apple to the light signal during the first 10 seconds and the second 10 seconds is identical. You have no ground for distinguishing between the two.

I'd say I directly perceive pain, colours, smells, tastes, etc. — Michael

Interesting. There is the introspective perception, in which whatever seems to be so, is so. But truth and falsity don't apply in the usual ways. Perception of actual objects is different, of course, in that our experiences can be corrected. But our perceptions of colours etc. can also be corrected. "That grass isn't really brown - it just looks that way."

Strictly speaking, insofar as the apple has disintegrated, there is no direct object of perception during the second interval. So while the intentional content persists, the perceptual act goes unfulfilled. — Esse Quam Videri

Speaking even more strictlly, the undisintegrated apple stands in exactly the same relationship to the light during the first interval and in the second interval.

Given that "I see X" is true if "I indirectly see X" is true, it is a non sequitur to argue that if "I see X" is true then "I directly see X" is true. — Michael

But surely "I see X" is also true if "I directly see X" is true.

In any case, this is one of hte uncomfortable realities of, at least leaning, IR. How can we explain actual error in perception? — AmadeusD

I thought the selling point of IR is that it can explain error in perception where DR cannot.

We do not need direct access to objects for that system to work. — AmadeusD

Well, we need to assess whether given indirect perceptions are veridical by some means that is independent of them. What do you suggest?

But that would be admitted that you're essentially looking at a pale imitation — AmadeusD

I certainly am not. Ex hypothesi, the light waves are derived from the sun and demonstrate to us exactly what the state of the sun was eight minutes ago. There's no better way of knowing what's going there.

If humans are, as this seems to make clear, restricted to an experience of light reflected from the sun eight minutes ago, we can never be sure and that's fine. — AmadeusD

I don't think there's any reasonable ground for doubt - and we can be sure that if we are wrong, we will know all about it in the next eight minutes.

We are able to flexibly attend to phenomenology, or to object. But our attentional stance does not speak to the epistemological relationship between phenomenology and object. — hypericin

Yes, we can attend to either. But I don't understand the second sentence.

Neural nets of course do not function by representing one thing as another. they function by modifying weightings. It’s just a pattern of activations and weights, with no intrinsic “aboutness” or semantic content. — Banno

That's why scrutinizing brain waves is not likely to tell us much about how perception works. The computer analogy does not help with this.

As in, what is the apple in the noumena? — Hanover

On my understanding, it is unknowable and therefore not perceivable. That's why I think that Kant may have had a point here, but went wrong in suggesting that the noumena is a class of objects. Almost everything that we know about is only partially known. Very few things are either completely known or not known at all.

My grandmother never talked about religion, — Athena

I think that both religion and politics were tabu, except on specific occasions. I don't know about wider society, but it seems to be still observed in most of the circles I move in. Perhaps there are generational differences here. There's a good reason for a ban. Those are both topics that are likely to disrupt social and business occasions.

Franklin Roosevelt was crippled by polio, and the media kept that secret. I am quite sure Kennedy was a womanizer, and the press kept that secret. — Athena

Yes. It was, let us say, tactful of them. Roosevelt's polio didn't interfere much with his work as President. The issues about the Kennedys are more serious. I think that cases like that are part of the reason for today's, perhaps over-done, openness.

I don't think the state here would care about a gang of church-goers. — jkop

I think that religious lobbying tends to punch above its weight. In spite of the various scandals, religion still tends to command the high moral ground.

It's unrealistic to expect everyone else to comply to one particular religious belief. Freedom of religion means that people are free to practice their beliefs on condition that they don't violate each other. — jkop

Yes, that's the rule. But the difficulty is getting agreement on what violates people. It can't work unless there is consensus and mutual respect. A large religious community is always going to be at least visible. There's no need to push religious communities into a ghetto, where they can be ignored.

But that personal sense of liberty does not represent many communities that were established to preserve a separate space that excludes outsiders on various levels. — Paine

Yes. It's the other side of the coin from the worldly churches. Those sequestered communities can become a problem for wider society. It's a difficult balance.

Is the separation of church and state even possible? — Athena

A lot depends on what you mean by "separation". I was tempted to say that it is relatively simple to sort out at the institutional level. Quite a lot of states have done that in one way or another. However, it is a comparatively modern invention. For most of human history, church and state have been very closely aligned. Separating them was hugely controversial and complex. Nonetheless, many states have achieved it and it seems to me that it is not longer a hugely divisive issue in most states that I know of. (Iran is a prominent exception.)

Unlike private belief, a church is a means to practice shared belief in large groups, which then becomes an opportunity for its leaders to control people. A state can therefore use a church to control entire populations. — jkop

That's right. A national church is a powerful instrument of control for the state. So it is remarkable that the Roman Catholic church sustained its position in so many countries for so long. The key point is the question of loyalty, and the independent church could easily become a force to undermine the state. Very few states would put up with that, and during the Reformation in Europe, the dam burst in country after country.

However, even if church and state are institutionally separated, that doesn't mean the the church(es) will not exercise power in different ways. A church is a large body of organized people, so, at an institutional level, it seems inevitable that churches will have a considerable influence on the state, alongside all the other lobby groups that vie for the opportunity to exercise their influence.

Desirable as it is, it seems to me asking a lot to ask people to keep their religious beliefs as a private matter. Religions are about how to live life and pursue happiness. Religious beliefs are hugely important to people. It seems to me most unrealistic to expect people to keep their most important beliefs, not only about their own lives, but about the lives of everyone else as well, entirely to themselves.

I say that whatever is the direct object of perception between 10:00:20 and 10:00:30 is also the direct object of perception between 10:00:10 and 10:00:20, — Michael

H'm. Perhaps we agree, then. What is perceived is the same object in both time periods. I see the apple during the first time period, so I also see the apple in the second time period.

if I'm watching something on CCTV then the thing I'm watching is the object (or "event" if you prefer) of perception but not the direct object of perception. — Michael

OK. So you are really watching the TV, not the event shown on the TV? It sounds a bit daft. A TV just sits there and does nothing. In other words, to describe the object of perception as the TV in this case excludes the point of the exercise, which is not to watch the TV, but to watch the match. So I'll agree that I'm watching the match by means of the TV, if you'll agree that to say that one is watching it indirectly misrepresents the point of the exercise. To repeat, watching the match is the point - the TV is just the means to an end.

Michael has used a bit of rhetoric to put those opposed to indirect perception on the back foot. They feel obliged to defend "direct" realism. — Banno

Yes. I like the quote marks. I've decided that calling it direct realism is not helpful.

I would say that given the speed of the light and the distance of the apple that you see an intact apple for 20 seconds between 10:00:10 and 10:00:30 — even though an intact apple doesn’t exist after 10:00:20. — Michael

So would I, except that I would specify that you see the apple placed in front of you. The delay in transmission does not affect this. I don't see what all the fuss is about.

I think we need to distinguish between "object of perception" and "direct object of perception". — Michael

This begs the question. One can only distinguish two objects of perception of the same thing if one has already accepted indirect realism.

We naively think of this phenomenal quality as being one of the properties that the bird has even when nobody is looking at it, but our science has confirmed that it isn't. — Michael

Back when modern science was being invented, a decision was taken to ignore anything that could not be included in mathematical representations. That is not the same as proving that colours don't exist. All it proves is that modern science cannot recognize them.
Just as we hear sounds as being located at the origin of the sound waves, so we see colours as being located on the surface that is reflecting them. That's part of the phenomenal quality. A system that did not give that information would be pretty useless, don't you think?
The bird is reflecting the light waves that we see as red. We see not only the colour, but where the relevant light waves are coming from. The phenomena are not accurately described unless we acknowledge that the bird is red and red is not in our head.

I would argue that the fault-line in the debate runs all the way through how the subject-as-conscious-subset is to be best understood—specifically, whether it must be characterized as an observer standing behind a curtain of phenomenal intermediaries, or as an embodied mode of world-directed access. — Esse Quam Videri

It is possible to think of the subject as a dis-embodier observer. That happens when we think about the observer in a picture as we are deciphering the perspective in the picture. It's also implicit in the concept of the "point of view" in cinematography. That's the concept that allows this problem to get hold of us. The embodied subject allows us to see perception as part of a system, linked to other activities as part of an internal control system, which cannot sensibly be thought of in the same breath as anything going on outside or beyond or independently of the system. This avoids the temptation to think of perception as a process with a terminus - the "experience". I admit this is all a bit rough-and-ready, but I have little doubt that it is more constructive that trying to establish a direct-indirect distinction in a conceptual vacuum.

Couldn't it just be that we tend to favor one perspective over the other in our daily lives? That one of them is viscerally lived, while the other is more intellectual abstraction? — hypericin

There may be some level at which our personality favours one kind of theory over others. Scepticism seems to be a good candidate - a yearning for certainty.
I'm not sure how one would describe the personalities that make for direct or indirect realism. Perhaps indirect realists like a safe distance between themselves and the world?

Perhaps those with a more integrated default feeling of selfhood tend towards direct realism. How about you? — hypericin

A few days ago, I would have said I was a direct realist - possible even a naive one. Now, I'm not so sure. It turns out that I don't really know what direct realism is - and consequently I don't know what indirect realism is. I think I may be a survival from the good old days when almost all philosophy was thought to be meaningless nonsense.
I wouldn't have described myself as an integrated person. I spend most of my life muddling through. Perhaps it is not an accident that I like clarity a lot more than I like solutions.

It is wrong to allow for multiple answers to Type B questions. — hypericin

Wittgenstein would certainly agree with that.
In the Blue Book, he is clear that he is looking for a diagnosis of our philosophical temptations, but he seems to see that as a matter of temptations to misunderstand language. He doesn't, so far as I know, ever get into personality types.

Surely if I see an intact apple 10m in front of me but there is not an intact apple 10m in front of me then the direct object of perception is not an intact apple 10m in front of me? — Michael

There is a problem from the start here. You think that "the direct object of perception" refers to something. If I've understood you, you don't know what the something is. But I'm not convinced that the phrase does refer to anything. But it seems you are looking for something that it exactly what it seems to be, about which I cannot be wrong. That sounds like introspection of a phenomenal object, so that's how I am interpreting you.
It is clear that if there is not an intact apple in front of me, I am not perceiving or seeing an apple. I think that is true whether we are talking about direct or indirect perception. But it is very tempting to think that even if I am not seeing an apple, if I believe that I am seeing an apple, then I must be seeing something apple-like.
I am a bit confused about this example, so I will propose another case. When I watch the start of a horse-race, I can see the tape go up and hear the starting gun. But, if I am some distance away, I may hear the gun some time after the tape goes up. But I see no problem in saying that I hear it. Similarly, in the case of the sun, I see no problem in saying that I see the sun as it was eight minutes before I saw it. I can't see any problem that is resolved by proposing that an image is involved in the process.

I think indirect realism is best understand in contrast to the naive realism it disputes. Whereas "semantic" direct realists might mean something else by "direct" I think both naive and indirect realists mean the same thing, and our perception of distal objects is not direct in the way that naive realism says it is. — Michael

This is a bit confusing. Direct and indirect realism are opposites, but linked in that direct and indirect are defined in opposition to each other. So you would have thought that they could agree on what the issue is. But I don't really understand what naive realism is. (Nor do I know what "semantic" direct realists are.) So I doubt that I can say anything much about this. But what is the thing that both naive and indirect realists agree about?

I think there is a perception; it's what exists/occurs when the visual cortex is active in the right kind of way. Although whether this thing is physical or a non-physical emergent phenomenon is the biggest question in the philosophy (and science) of mind. — Michael

You are moving between thinking of a perception as an entity and as a process, which makes this rather hard to understand. I guess everyone agrees that there is a physical process involved, and it is worth noting that when this debate started, with Bishop Berkeley, those processes were more or less completely unknown. I think that the issue here is how we regard the internal processing that goes on.

I’d be interested in understanding what ulterior motive lies behind their promotion. What do we stand to lose if we lose these concepts? I suspect it’s something like losing Zeus when we came to better understand the skies. — NOS4A2

I expect you know about the scene in Shakespeare's play about Macbeth in which he thinks he sees a dagger in front of him and makes a long speech about how guilty he feels about the murders he has committed. It's a hallucination, so he doesn't see a dagger. But yet, we want to say, he must be seeing something dagger-like.

Neither of these perspectives on the subject is intrinsically wrong. Am I the organism, or the conscious agent? They are both valid ways of looking at what counts as the subject. And so neither direct nor indirect realism is intrinsically wrong. If so, the debate will never end until both sides understand this fact. — hypericin

That's very plausible. But I think there is a bit more to be said about how and why the debate arises and why one position or the other is more attractive to adherents.

This part matters, Banno. When you cast your eyes to the Sun, you literally are not seeing the Sun. You're seeing light from the sun which is eight minutes old. Nothing interesting about this, except trying to get around it to say you're directly aware of hte Sun in any given moment. Just stupid. — AmadeusD

It's certainly stupid if "direct awareness" is defined as "by introspection" - perceptions that are guaranteed correct, even if they are wrong. But, if all perception is by introspection, how do we ever know that it is wrong?
I have another problem with this. I don't think I see light as such. Surely, light is what enables me to see whatever it is that I see. Compare the role of air when we hear sounds. We don't hear the air; it is what enables us to hear.
Why can't I just say that I see the sun as it was eight minutes ago?
I'm sure you are aware that a similar argument applies to everything that we see (or hear).

The Sun is not what we see when we look at the sun. — AmadeusD

I expect you mean that what we see is an image of the sun. But an image of the sun is not an entity that exists independently of the sun. It is defined by its relationship to the sun. So I can only know that I'm seeing an image of the sun if I know what the sun looks like. Scrutinizing images will never tell me that.

I would much rather know what mind-dependent thing or substance the light or thick air or any other environmental mediator is supposed to represent in these analogies, because that is what the indirect realist proposes he is directly perceiving. What are their properties, their mass, their speed. Give us a thought-experiment about those things, if you wouldn’t mind. — NOS4A2

I expect you know that there is no answer to that. These objects go by many names, which have in common that they are not reality, but are defined by their relationship to reality. To get anywhere with this debate, we have to look more closely at these various objects (concepts) and understand how they work, what jobs they do.

Therefore, the direct object of perception during the first 10 seconds when the light travels at 299,792,458m/s is not an intact apple 10m in front of me — Michael

I agree that the exact time it takes for the light to travel to my eye is not really relevant. But this looks to me like a fancy way of saying that I do not see the apple in front me instantly. It does not follow that I don't see the apple, but something else. Compare how we deal with the time it takes for sound to travel to my ears.

1. The direct objects of perception are distal objects
2. The direct objects of perception are proximal stimuli
3. The direct objects of perception are mental phenomena — Michael

Any of these might be acceptable, depending on how "direct" (and "indirect") are defined. Perception is a complicated process, which can be be analyzed in many different ways. A major difficulty is that there is no physical entity - a perception - that is the product of the process.

I had to double-check but I never posted this! A couple times I wrote a post which contained exactly this point. — Srap Tasmaner

I think it's just a coincidence. I used this example because it occurred to me at the time, not because I had read it before.

When did shepherds start using notched sticks or knotted strings to count cattle? How on earth did they come up with such an idea? — Srap Tasmaner

I imagine that there was a problem on the second day that someone took someone else's sheep out and came back with fewer. There has to be an agreed record of how many sheep went out.

Zeno insists that we count the sheep — that is, the rational numbers — as we find them, in their natural order. — Srap Tasmaner

You are making me very curious about the rationals, reals, etc. But I think I'll leave them for another occasion. Thank you for your help. .. and you for yours.

Where a function will have exactly one result for each input, a procedure need not. — Banno

Thanks for that distinction. I wasn't aware.

I hadn't considered that someone would suppose that logical procedures are somehow temporal. I find that idea quite odd. — Banno

I'm glad you agree with me. I had noticed that people often speak as if the procedure (or function) somehow executed itself. Obviously a procedure or function only achieves the result if someone follows the instructions. In that case, talk of a function yielding a result is short-hand, omitting the proviso "when someone follows the instructions. Would that be right? The problem is the idea that the rule executes itself in advance of our following it.

do we want natural numbers or counting numbers? — Banno

OK. It depends on what you are doing. I was thinking of the point of origin on a graph, but that's not quite the same as counting numbers.

The difference between numerals and numbers is not ontological, it is grammatical. — Banno

So the numeral is the number in the way that lump of wood is the king in chess? Yes, that's much neater.
Ockham would be pleased.

The confusion here is between differing language games; to think that "object" only means tables and chairs and not 7 or fully incorporated companies. — Banno

Oh dear. I obviously made my point very badly. I was trying to get at the point that there are different kinds of object, that's all.

Eh. A procedure, as I'm using the term here, accepts some input and yields some output. You show me a natural number, and I can show you another. — Srap Tasmaner

OK. In that case, you carry out the procedure. What bothers me is the idea that a formula like S(n)=n+1 is not a set of instructions about how to do something, but actually does it. So someone might say that formula generates the infinity of numbers. That's not at all the same thing.

What I was suggesting was that we can replace our pre-theoretical understanding of counting with this system, consisting of exactly two rules (that 1 is a natural number, and every natural number has a successor), and we will (a) lose nothing, and (b) gain considerably in convenience for doing things that build on counting. — Srap Tasmaner

I don't have a problem with that. Something like regularizing, tidying up, making explicit - even get a whole new perspective on something entirely familiar. I can see a point to that.

But it doesn't necessarily tell you what counting actually is. — Srap Tasmaner

Yes. One would need a demonstration of the written instructions as well. It's the gesture of adding one to the total, letting one sheep through the gate, and one more, let through the next one and so on.

I've been thinking a little, as we've gone along, about the most famous "primitive" counting systems, — Srap Tasmaner

Yes. I do like half-way houses. They can be very instructive.

we might ask whether people using one counting system are doing something psychologically different from people using another, — Srap Tasmaner

It would depend on the details.

if being is reduced to value, that's idealism, not necessarily platonist though, but most cases yes. — Metaphysician Undercover

Who said anything about reducing being to value?

A place in an order, or hierarchy is a value. — Metaphysician Undercover

Hierarchy, yes. Order not necessarily. Alphabetical order doesn't imply value.

What we were discussing was the act of assigning value, counting. — Metaphysician Undercover

Oh, dear. How can one assign a value without assigning it to something? In any case, counting chickens, for example, answers the question "How many" and assigns a value to the brood, if you like. But it doesn't assign any particular value to any of the chickens.

Why do you allow that sometimes when words refer to ideas (two, three, for example), they refer to things, but sometimes when words refer to ideas (dragons, present king of France), they do not refer to things? — Metaphysician Undercover

When I say that the President is bold, I am talking about the President, not the idea of the President. When I say that the President has executive power, I'm talking about the idea of the President. The idea of something is a different entity (if it is an entity at all) from the something that it is an idea of.

Whether they say their minds or not, a statement has clear meanings. Behaviors can have many different interpretations. And even if you interpreted with mos likely reasonable way, they could say, I didn't mean that at all, or how could you possibly imagined that? — Corvus

Statements do not always have clear meanings and sometime people deliberately mislead us and sometimes we just get it wrong. But not always. The fact that it is possible to get it wrong does not mean that we never get it right, nor does it mean that we cannot correct our mistakes. You are a victim of philosophical scepticism.

Too broad claim to be meaningful I am afraid. I am not denying philosophy of action. But just saying it doesn't seem to go well with this thread. :) — Corvus

I'm afraid we'll have to agree to disagree, then. I agree that it's a bit on the fringe of this topic.

Let me provide another example, this one from the biological world. — Richard B

I like that example. I'm also fond of the case of our balance perception. Sometimes we are aware of sensations from it, but most of the time it works without our perceiving any sensations at all.

Plus, folks don't always show their minds via behavior or actions. — Corvus

... and they don't always show their minds via what they say. Feeling the water and reporting feedback is one thing. Putting on (or taking off) clothes is another. Shivering, sweating. All sorts of clues.

Behavior and actions would be more of psychological topic. — Corvus

What we say is also behaviour. I don't understand why you regard non-verbal behaviour as outside the scope of philosophy.

Behavior is random and would be too subjective for interpretation. — Corvus

That's odd. That's exactly how I feel about what people say. I would much rather trust how they behave. Actions, as they say, speak louder than words.

This is why these traditional direct realists were naive colour realists. — Michael

I'm afraid that philosophers are not immune from the temptation to coin descriptions of doctrines they disagree with that have a rhetorical effect on those who believe in them.

We now know both that ordinary objects are not phenomenally present and that the world is radically different to how it appears, hence indirect realism being the scientific view of perception. — Michael

How do we know that the world is radically different from how it appears? From our senses, that is, from the way the world appears to us.

The perceiving is a mental event, but the cat is not. You see the cat, not a representation. — jkop

Exactly. The idea that the world is actually different from the way it appears does not come from comparing it with anything, which is impossible.

If one heard that statement, one can only conclude his/her body is feeling hot. That is all there is to it. — Corvus

Well, there is behaviour as well.

That's exactly the reality of translation. In most cases there is no true equivalence "across different systems". — Metaphysician Undercover

However, in the case of symbols used in calculation, an equivalence can be established.

My objection was to the hypocrisy of publicly rejecting platonism then employing platonist principles. — Metaphysician Undercover

So you think that "to be is to be the value of a variable" is a platonist principle? I know you sometimes use words in ways I find hard to understand. This seems to be another case.

When you count something publicly, you share your assignment of value. — Metaphysician Undercover

Very true. Except that ordinal numbers don't assign a value; that assigns a place in an order. Assigning a value in mathematics just means what you do when you substitute a specific number (or word or sentence) to a place in a formula that is designated for such "values".

This clearly is about ideas in our minds. — Metaphysician Undercover

No, it isn't. It is about whatever I am assigning a value to.

we distinguish noun and verb, object from subject, subject from predicate. — Metaphysician Undercover

In the context of traditional grammar, an object can be almost any noun, limited only by the specific subject and verb that you are talking about.

I think that would be an odd use of language, if every word referred to an object. — Metaphysician Undercover

Not all words refer to anything. That's why there's such a fuss about dragons and the present king of France.

Then do we have broad agreement? — Banno

I was referring to the previous two posts. Beyond that, there's much that I agree with, but I still have puzzles (questions), which is not quite the same as disagreement. Partly, they centre on the questions about what it is for a mathematical object, such as a number, to exist. Partly, they centre on what the timeless present means in this context.

And all this by way of showing that some rules are not procedural at all; they are constitutive norms. — Banno

I agree with that. I don't have a problem about the timeless present in the case of constitutive norms. But in relation to procedures, I do. For the obvious reason, that a procedure takes place in time.

But we need another step - "1 counts as a number" - to get the procedure moving. — Banno

Of course. You may care to know that, as I understand it, the reason the Pythagoreans did not count 1 as a number was, at least partly, because they saw it as the source of all the other numbers. But don't we also need 0, as the starting-point?

I have a procedure for producing one natural number from another, but more to the point is that the natural numbers just are what you get when you do that. — Srap Tasmaner

That's reassuring! But I'm not quite clear what it means to "produce" a number. It's not as if we say to ourselves "I need another number here" and so instigate the procedure. Does your procedure create the numbers it produces from scratch or does it just produce another copy of the number????

Numerals get their identity from roles in activities, not from reference to entities. — Banno

You are not wrong. But now we are getting into trouble with the difference between numerals and numbers. I have a feeling, however, that we may need numbers in order to identify correspondences between numeral systems and perhaps even number systems with different bases.
I'm also getting puzzled about "to be is to be the value of a variable", or, more expansively, the idea that existence is defined within language games and the rejection of single (absolute?) criterion of existence across language games. I think that approach has a great deal to be said for it.

What I said, is that if a numeral is taken to refer to an object, a thing called a number, that object must be a platonic object. This is supported by the argument above. However, I do not believe that a numeral refers to an object called a number. I believe that it refers to an idea called a value. I believe that values are not objects, yet they are referred to. Therefore, in no way do I believe that all reference is "object-reference". — Metaphysician Undercover

I think many people believe that if something is referred to, it counts as an object.
It is true that we equate numbers with values, in the mathematical sense. That's to do with the uses that we put numbers to. So you are right to foreground what we do with numbers - or numerals if you prefer. But I think you slip up when you say that the numeral refers to an idea. That just resuscitates that argument you gave about numbers as ideas. The assignation of value in this context is public and shared, so it cannot be about ideas in our individual minds.

The only way to assume that the numeral refers to the same object for distinct individuals, is to assume that the object is independent. That's Platonism. — Metaphysician Undercover

I'm getting the impression that your objection is simply to the concept of an abstract object, which you call platonism. Would that be fair?

An object in your mind is called a mental object. An object in your hand is a physical object. An abstract object is something that isn't physical, but it's not simply mental either. — frank

Yes. Though there are lots of different kinds of physical object, not all of which can be held in your hand. Shadows, reflections, clouds, lightning, colours, sounds, surfaces, centres of gravity and on and on. Similarly with mental objects. Abstract objects also come in lots of different kinds.

For example, we can do a bijection between the numerals and the things to be quantified. The presumption of "numbers" is superfluous in this case. — Metaphysician Undercover

In the Roman number system "V" counts as five. The Chinese system has 五 (wǔ) for the same number. The ancient greeks used the letters of their alphabet as numerals, so five was the letter epsilon. If you just talk about numerals, you lose the equivalences across different systems.

I don't think I can add anything to your replies. I would likely just confuse the issue.

Finally, you ask whether we're talking about a generalization or a rule, which sounds quite a bit like asking me if mathematics is discovered or invented. It's an unavoidable issue, and I've suggested before where my intuitions lie, which of course involves answering "neither". — Srap Tasmaner

You are right of course. Like you, I am disinclined to back either option. But I prefer to treat each claim as a comparison or analogy and to note similarities and differences between the language-games. This may appear to be a cop-out, but I think it is more judicious than drawing up battle-lines. The same goes for intuitions, and you give a good example. There is, I think, a similar phenomenon wherever people acquire in-depth expertise; it's not something we are born with, but something that is born of long and intimate acquaintance with the relevant skills.

I'll only add that I think too often we think we can fruitfully approach this issue by staring really hard at the natural numbers or at triangles and circles to figure out what they really are and where they came from, when we would do better to look at the practice of mathematics to see what's going on there. — Srap Tasmaner

Wittgenstein is very good on this, as I'm sure you know. It is important. I'm fond of the adage that a rich diet of examples is very helpful. That is also part of Wittgenstein's practice.

Now what I would maintain is that the two are for all intents and purposes the same. That is, the ellipsis as it stands does not tell us how to continue on, and so falls to the sort of view expressed by Kripke; but we dissolve this by insisting that there is a correct way to carry on, given by the model theoretical account. — Banno

It seems clear to me that Wittgenstein would agree with you:-

201"] That there is a misunderstanding here is shown by the mere fact that in this chain of reasoning we place one interpretation behind another, as if each one contented us at least for a moment, until we thought of yet another lying behind it. For what we thereby show is that there is a way of grasping a rule which is not an interpretation, but which, from case to case of application, is exhibited in what we call “following the rule” and “going against it”.

You can't follow a rule or go against it until you start applying it. Kripke's mistake was to demand that everything is settled in advance. There's a lot of discussion of similar ideas in the Blue Book (see p.34, 36 etc.)

At the same time, youa'e right that we can introduce further rules that effectively stabilize new ways of speaking. We can take an earlier practice and add a counts as norm that extends it. In this sense, following a rule can include treating a construction as if it were something more, because we have adopted criteria that make that treatment correct within the extended game. — Sam26

Paper money is a good example.

Calling on procedure alone is insufficient. We need there to be stuff to perform the procedure on. — Banno

You are right that not all rules are of the same kind. In addition to procedural, there are constitutive rules.

Math as we know it piggy-backed the development of money. Money, first invented in Lydia, was the first abstract object, typifying value, but not specifying the value of what. — frank

Yes, I think that may well be fair. But I can't help observing the ancient Egyptians had ordinary arithmetic, which, it would seem, was primarily aimed at the logistics of huge work forces - rations, supplies, etc. Ancient Sumer, China and Lombardy all contributed. There's plenty of people to share in the credit and the blame.

What's difficult for us, in talking about mathematics, or about language, or about concepts, is that we want to pass over the generation upon generation of practice and refinement, to recreate the primordial scene in which someone, however far back, came up with a way of doing this sort of thing that worked, and we want to identify the features of the environment that enabled it to work, very much as if we expect there would only be one way. — Srap Tasmaner

Yes. I do like bits of history as a way of understanding something about our present practices. But I wouldn't want to treat history as sacrosanct in some way. There's nothing wrong with inventing language games to bring out one point or another. Wittgenstein does it all the time, so it can't be wrong, can it?

The only way that "1" can refer to an object called "a number", instead of referring to distinct ideas in the minds of individual subjects is platonism. Platonism is the only way that "1" can refer to the same thing (a number, an object) for multiple people. Otherwise "1" refers, for you, to the idea you have in your head, for me, to the idea I have in my head, and so on. This is the way that values such as mathematical values are presumed to be objective rather than being subjective like many other values. It's known as platonism. — Metaphysician Undercover

The problem with Plato's ideas is that he tries to apply the model of 3D physical objects to abstract objects. Both exist and can be referred to, but they are not the same kind of objects. Your idea that the only kind of object that is not a 3D physical object is an idea in the mind. Numbers are not just ideas in the mind, but are rooted as objects in our shared practices.

I don't have much more to say on the subject. Thanks. — Sam26

That's fair enough. Thank you for your comments.

That false premise is what creates Zeno's paradoxes. — Metaphysician Undercover

Do you mean the premiss that space can be infinitely divided, not merely conceptually, but also physically?
I think most people would accept some version of that. But a physical limit to the process of division doesn't undermine the conceptual description. The physical limit will allows the conceptual division to continue.
Zeno produces an paradoxical analysis of the race. We can brush it aside and stick with the conventional analysis. There is an alternative, which is not paradoxical. Simple arithmetic and the definition of speed and (distance/time) tells us when Achilles will overtake the tortoise. So it is only a question of how you look at it. But still, people get hung up on the paradox. However, I think the real problems emerge in the analysis, for example, of circles and ellipses, which are not so easily dealt with in that way.

Calling it “unfinished” need not mean a temporality is at work, it can mean the grammar contains no stopping point. — Sam26

OK.

On your Aristotelian comment, Wittgenstein might ask what “actual” and “potential” are doing in our language, and whether they clarify the use of symbols or just swap one picture for another. — Sam26

As I said, I don't think the Aristotelian account clarifies anything much. If anything, it deepens the mystery.

And on existence, I am not denying that numbers exist. I’m blocking a slide in what “exist” means here. In mathematics, “exists” is governed by proof and use, not by the idea of a completed infinite inventory sitting somewhere. So, the rule can be firm without that extra picture. — Sam26

Perish the thought of denying that numbers exist!

The philosophical problem isn’t infinity; it’s the pictures our words seem to imply when we remove them from the practice that gives them sense. When we keep the use fixed, the mystery largely disappears. — Sam26

Yes, but here, we need to deal with the adaptation of terms that already have a use in some contexts, but need adaptation for this specific context.

So now, there’s actually a risk of civil war or serious violent unrest. Trump is too stupid and narcissistic to draw back and the governor is calling for resistance, calling in the national guard. — Christoffer

You may be underestimating him (or his advisers). I've heard a suggestion that the plan is to provoke serious unrest so that the Insurrection Act can be invoked and the elections suspended. Much as I admire Minnesota, it might be wise to save the opposition until after the election.

Wittgenstein’s point is to be careful not to treat the infinite as a finished object sitting out there. What we really have is a rule and the proofs we proceed with. — Sam26

I wouldn't argue Wittgenstein's point, though doesn't that point us firmly in the direction of the Aristotelian distinction between actual and potential infinity? Which itself leans heavily on our actions in relation to infinity. The second sentence is true if we are talking about our activity in relation to mathematical formulae.

That leans constructive in spirit, but it isn’t a knockdown argument that constructivism must be true. — Sam26

Fair comment. I used to think that constructivism was the way to go. No longer. Now, I'm seriously bewildered and working things out. I have noticed how time and process show up so often in talk about infinity and am wondering how deeply rooted it is.
Perhaps it is a metaphor. Perhaps it is an application of terms in a new, stretched, language game. Notice, though that your talk of the infinite as unfinished implies a process.

A rule can fix the standards for correctness without implying that the entire infinite list exists as a finished thing. We often feel “it’s already there” because the rule is firm, but what’s “already there” is the method, not a completed infinite inventory. — Sam26

Aren't you leaning here on an idea of what exists and/or is real? Isn't it that idea that leads us into difficulties about the status of the sequence. In one way, you are right. In another, you seem to be saying that there are natural numbers that don't exist or aren't real (non-mathematical sense of real). Aristotelian talk of potential numbers tries to find a half-way house, though I think it is a most unhelpful concept.

"Next" here implies a relation, and mathematics is the study of the relations between its "objects," which it is happy to treat as effectively undefined. — Srap Tasmaner

Are you happy to defend an interpretation which regard S(n)=n+1 as a remark about the relations between numbers? It must be that, unless you are thinking of the number line, which is a spatial metaphor. But if is just a remark about the relations between numbers, it seems more like a generalization that a rule.

There is no such thing as empty space between objects. — Metaphysician Undercover

Empirically, that may be true - especially if you regard a field (gravity, magnetism) as a medium. But setting up a set of co-ordinates does not require a medium in addition, so far as I can see.

Well, I can't say I understand exactly what you are proposing, but it seems like you are saying the question of the medium is secondary, but then you explain why it must be primary. — Metaphysician Undercover

No, it is simpler than that. We are using "medium" is different ways. I think. For me, empty space is not a mediium. A medium is substance that fills a space. Space is a co-ordinate system, which defines the possibilities where certain kinds of object may be. Objects are distinct from mediums because the latter are found everywhere, but objects have a locating within space.

I agree with everything you say. But it is not easy to say it clearly.

A rule can fix the standards for correctness without implying that the entire infinite list exists as a finished thing. We often feel “it’s already there” because the rule is firm, but what’s “already there” is the method, not a completed infinite inventory. — Sam26

What bothers me is that we seem driven to talk about processes in connection with infinity, as you do in the first sentence. But does such a concept make sense in the context of mathematics? Or does it mean that constructivism must be true, at least in the context of infinity?

Personally, I don't think God's foreknowledge contradicts free will, but others disagree. — Ecurb

I agree with you.

If I'm right aht Milton's defence of God, that the Fall was, in the end, a Good Thing, seems to me the deepest problem here.

I recently read this book. It was a real treat. I don't often finish a book thinking that I should read it again immediately. But I do still resist reading the actual poem.
I realize this is too late for your book group, but I'm sure there was a lively discussion - I hope so, anyway.
Some random comments.

It does seem that Satan is the most interesting character, and I'm sure Milton intended that. Satan could not tempt us if it was always clear that he was a demon. I do wonder how some of the people that are in the book reconciled their fascination with him with the threat that the revolution could turn sour, let us say. It doesn't seem to have put any of them off their projects.

Good point, although the French Revolution suffered from some of the same problems as Satan's rebellion. — Ecurb

Yes. In some ways, the greatest danger is in the moment of success. It's interesting to reflect on how Wordsworth reflected on this problem, by comparison with Milton. Both ended up as conservatives, though in rather different ways.

Paradise ruled by a dictator seems almost an oxymoron to me. And, I find it hard to think of a place where certain knowledge is forbidden as a paradise. — Ciceronianus

In the case of Paradise, since God's commandments are always good and right, it will always be good and right to obey. The only problems are our failures to grasp that. (In the background, of course, we have the Euthyphro problem, whether God's commandments are good and right because they are his commandments, or he decides what to command on the basis of his knowledge of what is good and right.)
The prohibition against eating the apple is a puzzle. In the Bible story, God fears that Adam and Eve will acquire knowledge of what is good and right; he cites a concern that they would become like gods as a result. I can't see that this makes any sense. In any case, in the event, they ate and clearly did not become gods.

Are revolutions doomed to fail? — Ecurb

I think it is dangerous to generalize here. The American Revolution. The Scientific Revolution. The Glorious Revolution of 1688 in the UK. Though much depends on what you think success is.

"The world was all before them..." Was paradise lost? Or gained? — Ecurb

I think that those final lines strongly suggest that Milton wants us to think that it was gained. As I remember it God responds to Satan's defection by saying that He will turn this disaster to good use. We have to assume that He was more or less in control throughout.
But there are questions. Assuming that Adam and Eve do acquire knowledge of good and evil, are we happy to say that they were better off as a result -- bearing in mind the punishments that God inflicts on them. Is the innocence of animals and small children a better, happier state than human life? But we do not think of humans who lack moral sense in a radical way as "better off" or even "happier" than ordinary people. I really don't know how one might even think about how to answer that question. Though I think it very likely that Milton would answer that a moral sense is essential to being human.
If Satan is an essential part of human life, is God complicit with him. Is it possible that God was complicit in the initial rebellion?

Since the concept of "space", and its accompanying mathematics provide for infinite divisibility, and the proposed medium is simply conceptual, how could the medium be modeled in any way other than a way which is consistent with the concept "space", and the related mathematics, i.e. as infinitely divisible. — Metaphysician Undercover

It seems to me that the question of a medium in space is secondary. The first move is to set up a co-ordinates and rules for plotting the position of objects on those. (In other words, the concept is defined by the practice.) Once we have co-ordinate and objects, the question of a medium makes some sense. How non-mathematicians develop the concept is another question. But we can be pretty sure it is by interacting with the ordinary world. Mathematics, in my book, is a development of that.

Limits, as against calculating velocities? Let's be clear, these two descriptions are quite consistent with each other. If you are pointing out that Zeno's description is incomplete because he doesn't include the bit where Achilles passes the tortoise, I think we agree. — Banno

I never intended to suggest that they were in some way inconsistent. On the contrary, the point is that they are both in order. So the question is, why do we prefer to use one rather than the other. Your suggestion is plausible - narrow focus in an analysis can be very helpful, but also very misldeading. The paradox of Zeno's paradox, for me, is that Achilles is precluded from reaching a point that defines the system - the limit. The first step is to divided the distance from the start to the goal, limit, by 2, and so on. The limit is not an optional add-on, (as it seems to be in the case the natural numbers).

And if I put on a “Wittgenstein hat” for a second, he migft say: don’t let the word “infinite” hypnotize you. Most of the time it just means “this process can continue without end,” not “we’ve discovered a weird tower of endless infinities.” — Sam26

I'm sure he would. But it is not so easy to rest content with "this process can continue without end". On one hand, we think that the result of the function for each value is "always already" true. On the other hand, we feel that the result is not available until the function has been applied to each value. What makes this game even more puzzling, is that it seems we can know things about the whole sequence without working out the results of the whole sequence. The first example of this is that we can know that the process can continue without end.

I suppose the thought here is to show that the limit is not so much made up or defined, but sitting there waiting to be found within ℝ. We construct ℝ then find these interesting results. — Banno

We are not comfortable with the fact that rules have consequences when they are surprising or not what we want.

nsisting that Zeno's infinities are about how the world is and not how we talk about it is question begging. That's exactly what is in question. — Banno

Let me try to be a bit clearer. I cited the bumble bee just because it was a case where there isn't much, if any, doubt about how the world is as opposed to how we think about it. I wanted to contrast that with the issues about infinity. There are two ways of approaching Achilles & co. One is Zeno's way, the other is simple arithmetic, which one might think is how the world is. But that's not how we respond. I'm not sure I understand why, exactly, except that both are methods of calculation, so both come from the same stable. (Contrast the bumble bee). Possibly, we could choose to stick with simple arithmetic in the Zeno case. So perhaps the reason is that we need it for other calculations, such as the orbits of planets and other issues in geometry. In which case we need both. In other words, this choice cannot really be posed as between how the world is and how we talk about it.

We should remember that we unfortunately have lost Plato's original book, where likely the Eleatic school would have made their own viewpoint. Now we have just the texts of those who were against the Eleatic school, the "mainstream" Socratic-Platonic school. — ssu

That would indeed be of great interest. I wonder if we could construct a reply that they might have made?

I do admire your devotion to the practical. Detaching yourself from it and purely following the contours of the mind will set you out in front of contradictions. — frank

It may well do so. It may also set you in front of outright fantasies that have no connection with any kind of truth. The theoretical stance needs a grounding in ordinary life, if only because there is no escaping ordinary life. Not even philosophers can really escape from it.

Paradoxes occur when we say things incorrectly. The world cannot be wrong, but what we say about it can be. — Banno

There used to be a story that aerodynamics showed that bumble bees cannot fly. Did anyone doubt that bumble bees can fly? I don't think so. I understand that aerodynamics is now clear that bumble bees can fly. But in that case, it was clear how the world is, as opposed to how we thought about it, or described it. Why is it that we don't just point out that the arrow will leave the bow, and that Achilles will catch up with the tortoise? It seems that we cannot simply correct infinity, but have to learn to live with it. Calculus fits in to that project.

I'm not convinced that all paradoxes can be resolved. Some of them, like Zeno's, may be inherent in the project of saying things about the world. Self-reference is another part of our language that we struggle to escape from. Are we sure that we cannot just live with at least some of them?

Everybody grows the psychological structures they need to deal with the life they have. I can't tell you how you need to think in order to successfully be you. If deep suspicion about mental stuff, coupled with strong faith in the world is the outlook your psyche thrives with, then God bless it. — frank

Each to their own, I suppose. But is that how you think about your own views, as well? If that's what's going on, why do we bother arguing with each other?
It is not uncommon for people to believe that the ordinary world is not really real, but is some kind of dream or fantasy or shadow. I doubt they would welcome a pat on the head and permission to believe whatever they need to believe.

My apologies for my curtness. I'v'e in mind heading off a divergence into discussions of rules. — Banno

Of course. The ghost at the feast, perhaps.

Zeno saw himself as proving that all motion is an illusion. You're saying that he's wrong, but you aren't providing an argument. That's fine. — frank

Well, one could simply argue that the argument is not a proof, but a reductio of a certain approach to space, time and infinity.
We can compute when Achilles will achieve his goal as soon as we know how fast he is running and how large the distance is. That figure does not change as the race progresses. Unless Zeno can find a fault in that calculation, it proves that the issue is in the approach to the question, not in the situation as described.
In a fixed period of time, Achilles passes an increasing number of distances, culminating, no doubt, in his traversing infinitely many distances in an infinitely small amount of time. Zeno seems to think that he takes a non-infinitesimal amount of time to traverse an infinitesimally small distance.

We know exactly how to carry on. — Banno

Thanks, Banno. I knew I would not get it quite right.

The "number" of natural numbers (positive integers) equals the number of integers because a 1:1 mapping can be identified between the sets. — Relativist

I think it is important to underline that the mapping between the sets is identified between the first few steps in the series (I may have the exact terms wrong). But the written identification runs out at some point and is conventionally followed by .... The conclusion that it applies right through the sequence follows from the fact that there is nothing to stop it continuing on the principle that every step in the sequence is structurally identical. (You would need to give a reason why things should be different at some point.)
I've found that this is not obvious to everyone.

However, i see it as implicit, and unavoidable from the meaning given to those terms within the system. — Metaphysician Undercover

I realize that you see the contradiction as implicit and unavoidable. But you are not recognizing the meaning given to the terms within the system.
"countable" within the system means only that some of them can be counted and we cannot find any numbers in the sequence that cannot be counted. Actually, since we had that discussion, I've come across the term "countably infinite" which I think is much less misleading.
And I think that you are not aware of how the term "limit" is used within the system. A limit, in this context, is a value that the series gets closer to, but never reaches. It is not a value derived from the function. It is not the last term in the series.
It does not constrain the series at all. So, in Zeno's paradox, Achilles gets closer and closer to the tortoise but never reaches it. (Forgive my inexpert account.)
From my perspective, the adjusted meaning of terms within the system is one of the biggest differences between us.

The connection to theology which I get a glimpse of, is the ontological argument, which became unacceptable even in theology. — Metaphysician Undercover

Thanks for that. I hadn't thought about it. But that wasn't what got Cantor into trouble.

Some Christian theologians saw Cantor's work as a challenge to the uniqueness of the absolute infinity in the nature of God. In particular, neo-Thomist thinkers saw the existence of an actual infinity that consisted of something other than God as jeopardizing "God's exclusive claim to supreme infinity". — Wikipedia - Georg Cantor

I think that a dissection of the ontological argument here might be thought off-topic. So I won't go any further into territory that I don't understand anyway.

If we were talking at the level of metaphysical or empirical truth I would agree with you. But at the level of formal truth, either ZFC ⊢ ∃f (bijection) is true, or it isn't. I have a hard time making sense of the claim that it isn't. — Esse Quam Videri

Yes. I was thinking in terms of a truth that could be recognized unconditionally, as it were. But then, "either ZFC ⊢ ∃f (bijection) is true, or it isn't" is just that. So I missed that truth. The difference is, I suppose, is that I doubt if one could defend similar claims beginning "In metaphysics..." or "Empirically..."

For many of us on the thread, (1) is straight-forwardly true. So when someone denies "the bijection exists", we hear it as a denial of (1), since that is all we mean, whereas I think the people making the denial are (perhaps?) not intending it in this way. Hence all of the confusion.
What are your thoughts on this? — Esse Quam Videri

It is possible, and has been the case at times in this discussion, that both sides of a debate are thinking in terms of "straightforwardly true". But there is a case for saying that, in this instance, "straightforward truth" just isn't available. To put it one way, there is truth in the orthodox account of infinity, and the "Aristotelian" and nominalist accounts. Much of this debate has circled round this, without producing much in the way of mutual understanding. Classic philosophy.

Wittgenstein's approach seems to me much more likely to be fruitful - which is not to say that everything that he argues for is equally convincing - ref; "infinity". The aim of dismantling philosophical theories by showing that some term or other had not been effectively defined is an excellent test. But nonetheless, the comparison of a philosophical view with an interpretation of a picture suggest that a more laid-back approach is more likely to enable us to understand the issue and its difficulties, at least. Then the option of admitting that the problem is insoluble or has more than one solution may be open to us or that the issue

I suppose so, but the GPS in your phone was designed using math invented by Descartes. — frank

Indeed it was. And it is important to see and to notice that mathematical ideas apply to the physical universe, at least sometimes. Whether this is unreasonable or not, is not obvious to me. But reasonable or not, it is so. Pragmatic approaches often get short shrift around here and I don't equate "works" with "true". But where some idea or technique does work, that seems an important fact about it - just as whether it is verifiable or not is not the whole story, but is an important part of it.

Is this a form of pragmatism? Yes, I think it probably is. I am not adopting the axioms because they are "true" in any robust sense, but because they enable so much interesting, beautiful and indispensably useful mathematics. — Esse Quam Videri

Yes. That's why I prefer the classical approach here. But I don't rule out that perhaps there is something about the alternative approaches that has not yet come to light. There is a connection to theology, which might explain why those approaches survive, though I confess it would not recommend them to me.