I'm afraid if I answer this, our disagreement will disappear. :grin: — frank

And that would be bad? End of the thread, I suppose.

So now, in order to maintain your account, you must say their difference is a fiction? you deny that John is hot and Jane is cold? But disagreement exists only because words do not refer solely to private sensations. Physics and physiology explain how perception works, not what words mean.

Again, we might agree entirely as to the physics and psychology. None of which serves to fix the reference of "cold", in the way your account insists. Jane and John can disagree only because the words they use function in their shared world.

So the paradox involves confusing a way of talking, the maths, with a description of how things are, the ontology. We can be pretty confident that space is not infinitely divisible and yet still use calculus to plot satellite orbits.

(And all of this makes sense only if we agree that there is a whole number between one and three.)

Indirect realism isn't disputing this. Remember that it is realism. — frank

Yes, and the question is, is your use of "cold" only about some mental image, or about the water? The disagreement only makes sense if we are talking about the bath water and not just our sensations.

Yep.

Yes, so the words "hot" and "cold" refer to the sensations they feel (and even though they predicate them of the bath). — Michael

Saying “the bath is hot” is world-directed. The word “hot” functions as a normative, public concept. What each person feels merely mediates access to that standard — it is not the referent. Again, if two people report opposite sensations, hot and cold, and if “hot” and “cold” referred to private sensations, disagreement would be impossible. The very notion of conflict about the bath would evaporate. But disagreement does occur. You again slide from “experience influences word use” to “words refer to experience.”

If it were not public, it would be as if you said "I have a headache" and I replied "No I don't!" — Banno

So what they sought was an argument not only that Zeno posed no threat to the mathematics of infinity but also that that mathematics correctly describes objects, time and space. — SEP

See how explicit the admixture of two differing language games is here?

One can argue that calculus doesn't solve Zeno's paradoxes as we don't have yet a clear understanding of infinity. — ssu

What we have are ways of talking, language games, a grammar, or a paradigm - whatever you want to call it. Infinity is a mathematical notion that we can use to calculate physical results. It is not an ontology.

Two people can disagree about whether the bath is hot or cold. It does not then follow that the bath either "really is" hot or "really is" cold, and that one of them is wrong. The reality is that the bath causes one to feel hot and the other to feel cold, and the words "hot" and "cold" are referring to their private sensations. — Michael

Read that again, carefully.

Yes, one person feels hot, the other cold. Yes, that experience is caused by the bath. But the example does not support the phenomenalist claim. It only illustrates that words are influenced by perception and physiology, not that influence is reference.

Disagreement, surprise, and correction remain intelligible because meaning is not fixed by private sensations, but by shared, world-involving practices. That they disagree makes no sense unless the bath has a temperature that both feel. If it were not public, it would be as if you said "I have a headache" and I replied "No I don't!"


Get out the thermometer.

If we want calculus to solve Zeno's paradox, we have to assume that the math is telling us something about space and time. — frank

It doesn't solve, it dissolves.

The paradoxes only appear to work because they slide between the mathematical and ontological games.

Pick one, and set it out, and we can see how this happens in the detail.

The trivial truth you trading on is that different viewers of the dress have different visual experiences, that Visual experience plays a causal role in what people say. That is not in question.

Then, with "they are using the words that they associate with the character of their visual experiences", you slide from experience causally influences word use to experience fixes meaning and reference.

If causal influence were sufficient for reference, then “loud” would refer to cochlear activity, “heavy” to muscle strain, and “square” to retinal stimulation. But they do not. We hear the sound, feel the weight, and see the shape.

You suppose that any “disagreement”, be it bent stick, visor, dress or whatever, is a tacit endorsement of the naive view that the phenomenal character of their experience is a mind-independent property. But disagreement presupposes shared norms of correctness, not naive metaphysics. People disagree about whether something is funny, whether a painting is balanced, whether a sound is too loud, without supposing that their experiences are literally properties of objects. The disagreement is about how public concepts apply under given conditions, not about whose inner screen mirrors reality.

The very intelligibility of the disagreement shows that “white”, “gold”, “black”, and “blue” are not names for private qualia. If they were, each speaker would be infallible. Your example works against your account. We can ask “Which colour is it really?”, “What lighting was it under?” “Why do cameras show it differently?” “What colour is the fabric?” only because colour terms are world-directed and corrigible. If they were nothing but private sensations, these questions would be useless.

You would maintain that
1. Colour terms refer to private phenomenal character.
2. Speakers can disagree about colour.
3. Speakers can be wrong about colour.
4. Colour talk is public and communicable.
But (1) is incompatible with (2),(3) and (4).

My headache isn't public. — Michael

That's irrelevant. The question is not whether experiences are private. The question is whether the meaning and reference of colour words is fixed by those private experiences. “Red” is not like “my headache”, colour words are not avowals of inner states, they are world-directed predicates governed by norms of correctness.

If colour terms worked like headache reports, then disagreement, correction, and error about colour would be impossible. But as the dress demonstrates, plainly they aren’t.

I replied to your visor example here, here and here.
This is probably the most salient bit:

. But yours is a much imporved argument. Indeed, it supports direct realism by showing that we routinely and intelligibly “see through” intermediaries without reifying them as perceptual objects. — Banno

And

In your visor world, the visors drop out of the discussion when folk talk about ships. They are not seeing the image on the screen, they are seeing ship. — Banno

The words "gold" and "white" in the above sentence refer to the phenomenal quality of the experience that some people have when they look at the photo. — Michael

You simply keep repeating this same error. No, they do not refer to "phenomenal qualities", because such "qualities" are never just "phenomena", they are always public.

Your visor example has been adequately responded to, by myself and others. The visor story feels compelling because it trades on an illicit slide between disruption of discriminatory capacities and reference to private qualia. Once you separate those, the argument collapses.

What I need is for you to explain why you think calculus tells us something about space and time. It's in the article. — frank

I'm not claiming calculus tells us what space and time are; I'm denying that this is a coherent question.

Wittgenstein would say it worked, so, so what?

What does math have to do with the structure of space and time? — frank

:grin:

I just did. Did you? Which paradox would you like explained?

Maybe tomorrow.

And some can’t do the maths.

It's not well enough known, to summaries?

see .

No need to overcomplicate things.

Zeno's paradox is a convergent series, dude. It doesn't matter what order you sum it in. — frank

Yep.

$\frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \cdots$

is

$\sum_{n=1}^{\infty} \frac{1}{2^n}$

that is, 1.

there are an infinite number of steps in this description of the distance between 0 and 1, but that simply does not stop it being traversed in a finite time.

Zeno mistook an infinite description of motion for an infinite obstacle to motion.

:wink:

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

I think it is important to underline that the mapping between the sets is identified between the first few steps in the series... — Ludwig V

So we picture {2, 4, 6...} and carry on in this way.

But we know that there are innumerable ways to continue this sequence.

The mapping is not based on "carry on in this way" but on the function

$f: \mathbb{N} \to \mathbb{E}, \quad f(n) = 2n$

And this is what shows the bijection. At every n∈N, f assigns exactly one even number, and every even number is assigned to exactly one n.

We know exactly how to carry on.

Then replace "mind-independent" with "exists at a distance to my body and has such properties even when nobody is looking at it". — Michael

That' doesn't cut it. You continue to suppose that colour terms fundamentally refer to phenomenal qualities, while I and others maintain they are part of a public, world-involving practice.

So see

The bird certainly has properties even when nobody is looking at it, and one of these properties is to reflect 700nm light, but the word "red" as ordinarily understood doesn't refer to such a property. — Michael

The bird is still red after it flies away. You account is obliged to interpret this with the obtuse explanation that it would be red if it were being observed, even though it isn't being observed. But that's importing more philosophical hokum. The bird is red.

For you the colourblind would not be able to agree that the rosella is red, since they do not have the requisite experience. And yet they do, because they participate in the practice.

The disagreement as to that dress is a case in point for colour not being fundamentally phenomenal. Folk are arguing about what colour the dress is, not merely about how it appears to them. The dress was black and blue. It looked gold and white. Folk are willing to say “I was wrong about the colour once I saw it in different light”. That makes no sense if “white”, “blue”, “gold”, etc. name phenomenal qualities. The case shows that colour concepts are not anchored in private experience, because private experience alone cannot sustain disagreement, correction, or explanation.

Your talk of the "naive" view just shows you're continuing to respond on an argument that is not being made.

Cheers.

Half the problem here is that those who are advocating indirect realism think the only alternative is a naive direct realism.

That, and mistaking a causal chain for an epistemic chain, make up most of the conceptual difficulties.

But then at the same time Banno appears to agree with you even though my understanding of him is that he claims that the word "bird" refers to the mind-independent object and the word "red" refers to one of its mind-independent properties (e.g. a surface that reflects 700nm light?), and so that you and him are arguing for opposite positions, whereas I'm arguing for a middle ground. — Michael

Not quite. I don't use "mind independent", it's a term of philosophical art, not at all useful

"Bird" refers to the bird. Red is the colour of its head, chest and back, it being a male rosella.

These sentences are extensionally true.

We do not need a metaphysical contrast between “mind-dependent” and “mind-independent” to make sense of any of this. Doing so is philosophical hokum.

I agree that this is what indirect realism is saying — frank

but

“Direct realism” is not a position that emerged from philosophers asking how perception is best understood, so much as a reaction to dialectical pressure created by a certain picture of perception, roughly: the idea that what we are immediately aware of are internal intermediaries, be they sense-data, representations, appearances, mental images, from which the external world is inferred.

Once that picture is in place, a binary seems forced: either we perceive the world indirectly, via inner objects; or we perceive it directly, without intermediaries. “Direct realism” is then coined as the negation of the first horn. It is not so much a positive theory as a reactive label: not that. This already suggests the diagnosis: the term exists because something has gone wrong earlier in the framing. — Banno

An addition:

The Cantor–Schröder–Bernstein theorem states:
If there exists an injection f:A→B and an injection g:B→A, then there exists a bijection h:A↔B.

That's all.

CSB does not claim:

• that a surjection A→B implies a surjection B→A;
• that injections can be replaced by surjections;
• that such functions must be computable;
• that the bijection is constructively obtainable.

We know that we can construct an injection P(N) --> N via Turing machine encoding of decidable sets. (|P(N)| <= N) — sime

No, we don't. Encoding Turing machines only enumerates decidable subsets of N, not all of P(N).

N --> Pdec(N) — sime

P(N) and Dec(N) are different sets. Pdec(N) is an odd notation; I presume you mean it as the decidable subsets of P(N). I'll use Dec(N) there, to avoid any ambiguity.

We can inject Dec(N) into P(N) but not P(N) into Dec(N).

That is, there are undecidable subsets of N.

This is true computationally as well as classically.

Perhaps you read Kripke more closely than you read Wittgenstein.

I think we agreed here:

Hmm. What is a pattern, if not some sort of rule-following? OR perhaps, there are two ways of showing that you understand a pattern - by setting it out explicitly in words, and by continuing it.

So here's the problem. Consider "101010..."

Someone says "you are writing a one followed by a zero, and you intend us to understand this as continuing in perpetuity"

Someone else says "The complete pattern is "101010010101", a symmetrical placement of one's and zero's".

A third person says "The series continues as "101010202020303030..." and so on, up to "...909090" and then finishes".

Our evidence, "101010...", is compatible with all of these, and much more besides.

It's not the absence of rules that is puzzling, it's their abundance.

Yes, explicit rules are in a way post hoc. — Banno

But please, start another thread if you like.

Well, not quite. Skepticism of this sort hinges on the opening paragraph of §201:This was our paradox: no course of action could be determined by a rule, because every course of action can be brought into accord with the rule. "; but fails to read the conclusion: "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”."

Second that and add §201.

@Frank has apparently accepted Kripke's rule-skepticism, but also notes that it's the doing, the "carry on in this way..." that is the key.

Amen.

Nor is an ad hominem.

More to that, I'd say the various ways to describe the ship are all correct, with none getting priority as more accurate than the other, just using different descriptions for different purposes. — Hanover

Yep, this in keeping with the mention of Markov Blankets and also fitting in with Mary Midgley. Works for me.

I'd add that it is true that the ship is a galleon, and not that the pixels, painting or mental image is a galleon; and derive from that, using existential generalisation, that something is a galleon; and hence, that there are galleons.

That's a non sequitur.

The USA elected him, twice.

That does not meant they are all in agreement with him.

It does means they are fucked.

So the real reason for invading Greenland is that "Your country" didn't give him the Nobel Peace Prize.

The USA is fucked.

Anecdote is not argument.

Cheers. Your discussion with Michael mirrors my previous discussions with him. We understand the item you used to be a galleon, a painting, a collection of pixels, a series of 1's and 0's in a computer memory; and that these are in a sense the same. Michael appears not to see this, insisting instead that it's only pixels; that there is only one true description. I wasn't able to move him on this. Let's see how the conversation progresses.

, Given Meta's rejection of quantification, and now of numbers being ordered, it's about as clear as it could be that for Meta there is very little left of mathematics.

There is a point at which one's interlocutor's commitments collapse the subject matter under discussion.

That's where we are at with Meta.

As Frank points out,

It really comes down to which view best accommodates what we do with math. — frank

And Meta's view undermines most of mathematics, despite what we do with it.

Meta treats the ∃ of quantification, a logical move within the game of maths that understands there is a symbol n in the domain of discourse that satisfies P according to the rules of the theory, as if it implies n exists as an abstract object independent of language, symbols, or human conventions. That's just a muddle. At the core he perhaps does not understand the difference between syntax, semantics and ontology.

Given that Meta asserts that 2 is not between 1 and 3, I think I'm done here. I don't see any gain in showing further absurdities in his position.

Again, there seems to me to be a bunch of errors in what you have said here. The core one seems to be equating P(N) with the decidable sets.

The statement “We can construct an injection P(N)→N via Turing machine encoding of decidable sets”
would mean every subset of N can be uniquely encoded by a natural number. But that is equivalent to saying ∣P(N)∣≤∣N∣, which directly contradicts Cantor’s theorem. So if the statement were true, Cantor’s theorem would already be false.

There are undecidable subsets of N. We cannot construct an injection P(N)→N via Turing machine encoding of decidable sets

I'll stop there. I can't see that your account works.

Sure. But in addition to the usual thngs nominalism rejects, Meta rejects the notion that numbers as values of variables. while nominalists say numbers aren’t abstract objects, they undersntad that they can still be quantified over. Meta says that numbers aren’t things at all — they’re modifiers like “pink”. That blocks:

• ∀n …
• ∃n …
• n = 2
• n < 3

No mainstream nominalism does this, because it destroys the grammar of mathematics.

And so on. It's not nominalism as usually understood. Even predicativist or fictionalist views preserve quantificational structure.

It really comes down to which view best accommodates what we do with math. — frank

Ok. Here's some stuff that won't work if we accept Meta's ideas.

• Quantification
  If numbers are not admissible as values of bound variables, then statements like “for any natural number n” or “there exists a number such that” are illegitimate. This eliminates axiomatic arithmetic, algebraic generality, and proof by universal or existential instantiation.
• Identity and equality
  Arithmetic relies on identity conditions such as 2 = 2, 2 ≠ 3, and “if n is between 1 and 3, then n = 2”. If numbers are merely modifiers, they cannot enter identity statements, cannot be uniquely satisfiable, and cannot ground equality.
• Ordering relations
  Relations like less than, greater than, and between require relata. If numbers are not entities in any sense, then statements like “2 is between 1 and 3” are not well-formed, and order theory collapses.
• Counting finite collections
  Even finite arithmetic fails. Claims such as “there is exactly one whole number between 1 and 3” or “this set has three elements” require individuation, discreteness, and cardinality. These cannot be recovered without smuggling in what the view denies.
• Functions
  Functions are mappings (e.g. f : ℕ → ℕ). If numbers are not admissible values, then functions have no domain or codomain, expressions like f(2) are meaningless, and recursion is impossible.
• Proof by construction
  Mathematics routinely proves existence by exhibiting a value (“let n = 2”). If numbers cannot be introduced as values, constructive proofs and witness-based reasoning disappear.
• Set theory
  Set theory quantifies over elements (e.g. 2 ∈ {1,2,3}). If numbers are not legitimate elements, sets of numbers are incoherent, cardinality is undefined, and bijections cannot be stated.
• Algebraic structure
  Even structuralism requires positions in structures. If individuation is denied altogether, then groups have no elements, rings have no units, and fields have no values. Structure without positions is empty.
• Application of mathematics
  Physics, engineering, and statistics require numerical values, parameters, and measurements. Treating numbers as mere modifiers strips equations of semantic content and collapses measurement and prediction.
• Self-undermining practice
  The view relies on finite counting, numerical distinction, and identity (“one”, “two”, “numerous times”) in order to be stated at all. It presupposes the very arithmetic it rejects.

We could go on.