Meta's errors include only thinking of something being either in the world or in the mind. So money, property and number, amongst other things, cause him great difficulty because they rely on communal intent. We might be tempted to express this as "they exist between minds", but that's not quite it, either. Some - many - things owe their existence to public rules, practices and recognition, and these need both minds (plural) and the world. Meta is trapped, as notes, because if numbers are only in the world, he owes us a story about where they are; and if they are only in the mind, he owes us a story about how we manage to do things with them in the world.

Numbers are not like rocks, nor are they like sensations.

That's part of the reason that he can't make sense of logical precedence, restricting himself to temporal or spatial precedence. His metaphysical picture cannot represent logical priority at all, since it's neither purely mental or purely of the world. And along with that go other things that rely on public standards for correctness, such as normative dependence, and rule-dependence.

The following makes his error particularly clear:

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

Notice that this odd position is blandly asserted, not supported by any argument.

He relies on presuming that all reference must be object-reference, that object-reference must be either mental or Platonic, and that public sameness requires numerical identity of a referent. Meta relies on an unargued slide: “same object” → “same referent” → “same use” He treats these as equivalent, but they are not. What is required for reference to function is not that we talk about the same object but that we have a public criteria for correctness. It's learning that public criteria that so clearly portrays; learning to count is learning to participate in public activities involving fingers and toy cars and slices of pizza. Numerals get their identity from roles in activities, not from reference to entities.

the first abstract object — frank

I might have said property - this counts as being mine. Basic idea is right.

Washington Post shows video of the gun being removed by an ICE agent prior to the murder.

It was tucked in to the back of his belt. It was not being brandished.

It's showing that this pattern applies to fingers and to toy cars and lollies and so on - divorcing the pattern from the things being counted. Only after this pattern is understood does the child begin to ask about bigger numbers, and eventually to realise there is no biggest number.

So we get "One counts as a number" and "every number has a subsequent number" and discover that the pattern does not end, and then learn to talk of the whole as being unbounded and that infinite counts as being unbounded... iterating the "...counts as..." to invoke more language games.

But the Wittgensteinian idea is that this isn't a metaphysical ascent to a realm of completed entities. It's a reworking of our practice (what we do), still embedded in human activity and a form of life. The novelty comes from what we now allow as a correct move, not from discovering a new kind of object behind the calculus. — Sam26

Yep.

It's not platonic.

Is this the sort of thing you're getting at? — Srap Tasmaner

Pretty much. So we have "Any number has a subsequent number", a procedure - if something is a number, then there is a subsequent number. But we need another step - "1 counts as a number" - to get the procedure moving.

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

And I just don't suppose that Wittgenstein, a clever chap, had missed this point as was saying that all we need in maths is procedures.

I very explicitly said that John and Jane agree that the bath water is 37°C but disagree as to whether this 37°C water is hot or cold.

You seem to be intentionally engaging with a strawman. — Michael

No straw man - I was questioning why the topic came up... it is the fact of their disagreement that is salient.

You appear to have stoped addressing the actual material before you.

he is diagnosing a philosophical temptation — Sam26

Yep, and that diagnosis applies to the foundations of maths - the area in which he thought he had made the greatest contribution.

A rule does not interpret itself. Yet we have rules that set up novel interpretations. Following a rule can involve treating something as if it were something more. The move is essentially to build a new language game on the back of another. And something like this seems implicit in a form of life. The whole remains embedded in human activity, in a form of life.

One follow on question is the extent to which this is a reflection of what Wittgenstein is getting at in PI  §201. @Sam26 may well insist that Wittgenstein had no such thing in mind. I'm not so sure. While he didn't use the "...counts as..." terminology, it seems to me implicit in his continuation of the account.

So we have :

A. Field structure (algebraic axioms)
B. Order axioms
C. Completeness (least upper bound property) — Banno

Now the field structure and the order axioms are the rules that @Sam26 and @Ludwig V have been discussing, that set up the sequence of numbers in order.

We've already left Meta behind, since he has claimed numbers are not ordered...

The completeness axiom is a second-order statement (because of the quantification over subsets S), and it expresses completeness of ℝ.

∀ S ⊆ ℝ, (S ≠ ∅ ∧ ∃ M ∈ ℝ: ∀ s ∈ S, s ≤ M) ⇒ ∃ L ∈ ℝ: (∀ s ∈ S, s ≤ L) ∧ (∀ L' < L, ∃ s ∈ S, L' < s)

∀ S ⊆ ℝ says the axiom quantifies over subsets of ℝ, and does so without specifying which subsets.

(S ≠ ∅ ∧ ∃ M ∈ ℝ: ∀ s ∈ S, s ≤ M) is the antecedent in the axiom. "S ≠ ∅" discounts an empty domain. ∃ M ∈ ℝ: ∀ s ∈ S, s ≤ M specifies that there be numbers bigger than or equal to those in S; mathematically, S is bounded above.

So the antecedent is "if there is a non-empty set of real numbers with some upper bound..."

∃ L ∈ ℝ: (∀ s ∈ S, s ≤ L) says that there is some number that is larger than or equal to every number in S.

and

(∀ L' < L, ∃ s ∈ S, L' < s) says that there is also always some number that is less than that larger number, but still a part of S. That is, we have an L such that we can't lower L even slightly and still have the upper bound, and yet anything smaller than L fails to give us that upper bound.

Putting it together, For every non-empty set of real numbers that is bounded above, there exists a real number which is the smallest number greater than or equal to every element of the set.

By way of an example, If S were {.9, 0.99, 0.999...}. And S is not empty; S is bounded above by 1; and so by the completeness axiom, there is a real number which is the smallest number greater than or equal to every element of the set. Again, int his case, 1.

With S: {.9, 0.99, 0.999...}, we would have a process, "keep adding another 9", that would be acceptable to a Wittgensteinian - a rule that allows arbitrarily many interruptions. But the true-blue Wittgensteinian would deny that we thereby have a whole set; we have a rule for constructing an arbitrarily long string, and nothing more. To get to the compete set we need S={x∈R∣x=1−10−n for some n∈N}, which presumes ℝ and so presumes already that we can talk about infinite sets.

From a modal perspective, S is a subset of ℝ with 1 as its supremum. From a Wittgensteinian perspective the rule “add another 9” never produces 1, “approaching 1” is not “being bounded by an element”, and the talk of a completed S is a projection of grammar. So the statement "The supremum of S is 1” is treated as a useful way of talking, not a statement reporting a fact about a completed domain.

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.

And I would add that this amounts to no more than following more rules. We have the definiendum ℝ on the left, and on the right we have a rule setting out what counts as ℝ.

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

______________
My apologies for that post, it's sloppy, and under argued, and I moved from limits to ℝ as I went through the argument. The whole needs reworking, but I'll let it stand because it sets out the direction of my thinking in response to the last page or so from @Sam26, @Ludwig V, @Srap Tasmaner and @frank; that we can legitimately reify a procedure with a "...counts as..." constitutive rule. In this case the axioms count as setting up ℝ.

Yep. there is a difference between being cold and feeling cold, as is shown by the fact that we have that very grammatical structure.

Another interesting point is how this comes to light only when there is a disagreement. If John and Jenny had agreed that the water was just right, that's an end to the discussion. And what this shows is the opposite of what Michael is suggesting: that all there is to hot and cold is the sensation.

...could be... — Christoffer

:confused:

You haven't refuted the public-use account of meaning so much as rejected it in favour of a phenomenalist error theory — and then acted surprised that others won’t follow him there.

If “hot”, “cold”, “painful”, “harmful”, etc. were mere fictions, then safety thresholds, medical advice, engineering tolerances and so on would all lose their point. Science would be answering questions no one had.

That John and Jane disagree as to the temperature of the bath is not a fiction; it's something to be explained. This is lost in your account.

We have on the one hand: Science refines and explains ordinary concepts; meaning is fixed by public use; sensations play a role without grounding semantics.

And on the other: Ordinary predicates are systematically mistaken; disagreement is illusory; language misrepresents reality; only physical descriptions are real.

Now I think this latter view is plainly impoverished.

If “X is cold₂” just means “X causes cold₁ sensations in me”, then:

• instruments don’t measure cold, only predict feelings
• disagreements are merely parallel reports
• learning temperature terms requires introspection
• correction becomes impossible except as etiquette

That is not how the language works, and it is not how science or ordinary life proceeds.

That is orthogonal to the earlier dispute.

I'm getting sick and tired of repeating myself. — Michael

I'm not surprised.

If you do not think reference fixes meaning, then you agree with Hanover and I and the objection collapses. If you do think reference fixes meaning, then you are committed to a private-language-style semantics and have to face all the familiar problems — which you have been carefully avoiding.

The reason you keep repeating yourself is that your claim is orthogonal to the debate.

No one is claiming that "headache" does not refer to a sensation. What is being pointed out is that the sensation is not what fixes the meaning of "headache". Again, the meaning is found in the behaviour, avowals, characteristic causes, typical remedies, patterns of use in diagnosis, excuses, and so on, that surround the use of the word.

Your argument is that the meaning of "headache" is fixed by reference to a sensation. It isn't. If it is 'fixed" at all, its fixed by use. And that use is communal.

No one is claiming:

• No word may refer to a sensation.

What is being claimed is:

• Reference to a private sensation cannot be what fixes meaning.

I am only saying that the word "headaches" refers to headaches, and that headaches are a sensation. — Michael

First, can you see that the grammar of "headache" and the grammar of "cold" are very different?

If just one word refers to private sensations then this argument that you and Hanover keep pushing that meaning is just public use, that private sensations must drop out of consideration because we can't know each other's experiences, etc. is shown to fail. — Michael

The meaning if "headache" is not fixed only by some collective for private sensations. It is found in behaviour, avowals, characteristic causes, typical remedies, patterns of use in diagnosis, excuses, and so on. It's not private in the sense that is required.

But further, there's something your account seems not to acknowledge. Your headaches differed markedly from being cold or being red, in that while we can check the truth of what you claim about the latter, yet the claim you have a headache is much the only evidence we have that you do indeed have a headache. The grammar for headaches is quote different to that for looking, touching, smelling.

The argument about “meaning = public use” never claims that words cannot be about inner states. It claims that reference to an inner state cannot be what fixes meaning.

Understanding is displayed in correct use, not in possession of a matching quale. This is demonstrated by the fact that someone who has never had a headache may nevertheless use the term correctly.

Some resources:

Axioms of Real Number System: an explanation of field structure, order and completeness.

An Introduction to Real Analysis: a University of California text, as a PDF. We're looking at getting to 6.1 and perhaps 8.1.

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.

If you are following along, I'd encourage you to drop the text in to ChatGPT and have it explain any complexities.

So we have first order logic, =, +, <, and the Reals. We have the necessities for doing calculations, and a relation "<" and appropriate rules, and unlike the rational numbers, no gaps.

Stealing from ChatGPT,

• A. Field structure (algebraic axioms)
  There are two operations, + and ·, and two distinguished elements, 0 and 1, satisfying:
  
  • Addition is associative: (x + y) + z = x + (y + z)
  • Addition is commutative: x + y = y + x
  • Additive identity: x + 0 = x
  • Additive inverse: for every x, there exists −x such that x + (−x) = 0
  • Multiplication is associative: (x · y) · z = x · (y · z)
  • Multiplication is commutative: x · y = y · x
  • Multiplicative identity: x · 1 = x
  • Multiplicative inverse: for every x ≠ 0, there exists x⁻¹ such that x · x⁻¹ = 1
  • Distributive law: x · (y + z) = x · y + x · z
• B. Order axioms
  There is a relation < on ℝ satisfying:
  
  • Transitivity: if x < y and y < z, then x < z
  • Trichotomy: exactly one of x < y, x = y, or x > y holds
  • Compatibility with addition: if x < y, then x + z < y + z
  • Compatibility with multiplication: if x < y and 0 < z, then x · z < y · z
• C. Completeness (least upper bound property)
  Every non-empty subset of ℝ that is bounded above has a least upper bound (supremum) in ℝ. Formally:
  
  • ∀ S ⊆ ℝ, (S ≠ ∅ ∧ ∃ M ∈ ℝ: ∀ s ∈ S, s ≤ M) ⇒ ∃ L ∈ ℝ: (∀ s ∈ S, s ≤ L) ∧ (∀ L' < L, ∃ s ∈ S, L' < s)
  This is what allows limits, convergent series, and calculus to exist without introducing actual ∞.

This last says, informally, that every non-empty subset of R that is less than some number has a smallest number that is bigger than it... and is what distinguishes the reals from the rationals. SO it's perhaps where our attention might dwel.

Ok, you first.

:wink:

Would it help for us to consider the axiomatisation of calculus?

It just assumes first order logic and extensional equivalence over a domain of the reals. We could go into the difference between a limit and a least upper bound, between ∞ and ω, and how infinity never appears in the axiomatisation - except as shorthand for a process.

Might be novel to consider this stuff in detail. I have only a hazy memory of it from first year pure maths.

Good to have you drop past, even in AI form.

Yep; and bringing Wittgenstein in explicitly is interesting. There's a fine line between quantifying over infinities and hypostatising them.

The case of the bumble bee, if true, shows that the theory of flight was incomplete, and now, if the account of how they fly works, has wider application.

One is Zeno's way, the other is simple arithmetic... — Ludwig V

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.

What we don't have is a proof. Or how it fits everything else. — ssu

A proof of what, and to what ends? We know it's consistent and we do have rigorous axiomatisations...

I'm not following this, since

And we do use it. It is, well, essential. — ssu

seems to be saying that it does fit in with everything else...

What would count as success here? hat doubt would be removed, what practice would change if the "proof" were given?

Actually I just assumed my views would bore you. — frank

Never!

(If the ever did, I'd just not respond.)

Norman Malcolm — Richard B

Another escape from Oxbridge natural language philosophy. Yes, good stuff. The formalisation of this came with Davidson, and then the partial dissolution. Davidson disarmed the metaphysics, but the itch remains.

"People universally believe that objects look colored because they are colored, just as we experience them. The sky looks blue because it is blue, grass looks green because it is green, and blood looks red because it is red. As surprising as it may seem, these beliefs are fundamentally mistaken. Neither objects nor lights are actually “colored” in anything like the way we experience them. Rather, color is a psychological property of our visual experiences when we look at objects and lights, not a physical property of those objects or lights. The colors we see are based on physical properties of objects and lights that cause us to see them as colored, to be sure, but these physical properties are different in important ways from the colors we perceive." (Palmer 1999: 95) — Michael

You keep bringing this quote up. It shows pretty clearly the confusion of the epistemic and the causal accounts that you rely on, ignoring the difference between "It looks red" to "It is red", treating these as if all we ever have is "It looks red" and never "It is red". This is the account given by the "indirect realist", who then supposes that anyone who disagrees with them must think that if it looks red then it is red, and calls these folk "direct realists".

The rest of us see the muddle, and recognise the difference between "It looks red" and "It is red".

Yep.

Yep. @Michael doesn't seperate the epistemically and causal chains.

The word "headache" refers to the sensation we tend to feel after a heavy night of drinking, the word "cold" refers to the sensation we tend to feel in low temperatures, the word "hot" refers to the sensation we tend to feel in high temperatures, and the word "pain" refers to the sensation we tend to feel if stabbed. — Michael

Again, "headache" is not like "hot". John and Jane can disagree as to the water being hot, but not as to John having a headache: We get John saying "the water is hot" and Jane saying "no it isn't", but not John saying "I have a headache" and Jane saying "no I don't".

Your analogy misfires.

There is more to the meaning of these words than just their "public use". There is also the sensations they refer to. — Michael

This is an interesting move, in that you here allow for a public use as well as reference to sensations. Your previous accounts have insisted that words such as "hot" refer to the sensation alone. That's progress. No one here, so far as I can see, is saying that we do not have sensations. They are pointing out that we can refer to how things are, as well as how things feel: that there is a difference between "The water is hot" and "The water feels hot"; between "The ship looks red" and "The ship is red"; between "That is my wife's voice" and "That sounds like my wife's voice". And that we can talk about how things are as well as about how things appear to us. Indirect realism has to do work in order to explain this, usually by saying we make an inference from the sensation to the fact; but this is nonsense. We certainly do not actively, consciously infer from "That feels hot" to "That is hot", or from "That ship looks red" to "That ship is red". And if it is supposed that the inference is made somewhere beneath consciousness, then we must have a discussion about why we should call it an inference at all. What we feel is the water, what we see is the ship.

So you've moved slightly. Now you must either retreat further and accept world-directed predicates, or
double down and deny ordinary disagreement, authority, and usage.

So here

Both John and Jane agree on the temperature. Is 37°C hot or cold? What do the words "hot" and "cold" mean in either case? I think it quite obvious that they refer to the different sensations that 37°C water causes John and Jane to feel. — Michael

Michael asks if 37°C is hot or cold. Now if being hot or cold is exactly a sensation, this would be the same as asking "Does 37°C feel hot or does it feel cold?" But it isn't. Therefore the presumption that "hot" refers to a sensation is mistaken.

ou appear to accept that the sensations occur but then for some reason think that they have nothing to do with the meaning of the words we use. — Michael

(1) as sensation reports or (2) as world-directed predicates. — Esse Quam Videri

We differentiate quite simply between the bath being hot and it's feeling hot.

We might add that it sometimes makes sense to say that the water is cold but feels hot - when you have been out in the snow, perhaps, or due to erythromelalgia.

If @Michael's view were accepted, such that "hot" refers only to the sensation and not the water temperate, then this would not be possible. For him, if the water feels hot, "hot" refers to a sensation, and not to a fact about the water. So for him if the water feels cold, it is cold. And this is so regardless of wha the thermometer shows.

So he could have water at 5℃ and it still be true that the water is hot, because "hot" refers to a sensation, not a temperature.

Now this might even be consistent, at least with itself, and might explain why Michael is so enamoured with this view. Except that it is not how we do talk.

I don’t deny that sensations occur — Esse Quam Videri

Nor do I. The point here is that "the water is hot" is about the water, not about how the water feels. The re is a difference between "The water is hot" and "The water feels hot" this cannot be made in your account, Michael.

The example neatly shows a natural language differentiation between two locations for the Markov Blanket. The language track two loci, world-side features and subject-side sensations, effortlessly.

Cheers.

insisting 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.

But you are right that 's response is just his way of excusing his own views from critique... :wink:

It's a belief if one thinks that it's so. It's faith if one believes it is so despite the evidence.

But this is about method rather than belief. What is suggested is that if there is an inconsistency we reconsider what it is we are saying about how things are, rather than deciding that the world must be inconsistent.

There presumably is a point at which the world is so confusing that our reconsidering of what we say is insufficient to explain what is going on. But I hope we're not there yet.

The risks are that we are hiding behind grammar, using it as a shield against metaphysical inquiry rather than engaging it, or that we miss phenomena that actually resist conceptual capture. I acknowledge that.

How rude! :razz:

Not faith so much as care and attention.

Have you some alternative? :wink:

It never ends. :smile: — frank

It certainly won't, if all can do is repeat his assertion that "hot refers to the sensation".

John and Jane's disagreement is not a fiction; perhaps they really do differ in their beliefs. That Michael denies this shows the poverty of the insistence that "hot" is nothing more than a sensation. Treating it as a fiction does not help decide what to do.

Zeno wasn't arguing that we can't plot satellite orbits with acceptable precision. — frank

Well, he was, from what we know, arguing that motion was not real.

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

The words "hot" and "cold" refer to the sensations that John and Jane feel when sitting in the bath and yet they disagree on whether the bath is hot or cold. — Michael

Here you are not presenting an argument, but blandly restating your opinion. That's not a proof af anything.

The bottom line is that John and Jane are not disagreeing as to their sensations. Jane can agree that John feels the water is hot, and maintain that she feels the water is cold, and vice versa, without inconsistency.

What happens next? Do they add hot, or add cold, or wait? Or dot hey talk about why one feels cold and the other hot? The incident is not isolated, it's part of an ongoing and historical discussion. Sensation varies across people, while meaning, reference, and disagreement persist because words are embedded in a public, normative, world-directed practice. It’s not an isolated utterance; it’s a system of practices over time.