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.

He's saying that 2 isn't a thing. — frank

Yes, indeed he is.

And the counterpoint is that "being a thing", especially in mathematics, consists in being the value of a bound variable. As in "for any whole number, if it is between one and three then it is two"

This is not saying numbers are spooky abstract objects. It's saying that whatever our theory quantifies over, exists according to that theory. The nominalist’s complaint about “thinghood” simply misses the target.

It's a modifier like pink. You can't count pinks because it's not a thing you count. It's nominalism. — frank

“Pink” is not something that can be the value of a bound variable in arithmetic. “2” is. The analogy fails.

The other error here is to think that If something is countable, it must be a concrete or abstract thing. it ain't so. And note the equivocation of "thing".

This supposition that you have, that there are numbers between numbers is very problematic. — Metaphysician Undercover

So your argument is that 2 is not between 1 and three.

Righto.

I did it all ready in this thread, numerous times. — Metaphysician Undercover

Well, no. You claimed there is a contradiction, repeatedly, but never showed what it was. So go ahead and quote yourself.

"There exists a bijection with N" is explicitly saying "N is countable". Are you kidding me in pretending that you don't see this? — Metaphysician Undercover

"There exists a bijection of N" is the conclusion, not an assumption.

Yep.

Again, "integer" is a faulty concept, because it assumes that "a number" is a countable object. That's exactly the problem I explained to you. We ought not treat an idea as an individual object. Providing more examples of the same problem will not prove that the problem does not exist. The problem of Platonism is everywhere in western society, even outside of mathematics, so the examples of it are endless. — Metaphysician Undercover

We can make it simpler for you: How many whole numbers are there between one and three?

I say one. You say, they can't be counted.

A rule can contradict another rule within the same system. Saying that there is a rule which allows a specific bijection doesn't necessarily mean that there is not another rule which disallows such. — Metaphysician Undercover

Then show us that other rule. Pretty simple. Set the supposed contradiction out.

That's false... — Metaphysician Undercover

“Countable” is defined as “there exists a bijection with ℕ (or a subset of ℕ).” I bolded it for you. Again, if you think there is a contradiction in that, it is up to you to show it.

Begging the question would look like this:

• Assume N is countable.
• Therefore
• N is countable.

But what actually happens is:

• Definition: “countable” = “there exists a bijection with N”.
• Construction: Exhibit such a bijection (the identity).
• Conclusion: Therefore N is countable

.
That is definition + witness, not circularity.

You of course do not have to respond to my posts. Keep in mind that everything I have set out here is standard ZFC, and has been examined by countless mathematicians, yet remains solid. It is your account that is eccentric.

@Hanover, how goes your reading of Austin? I saw your interaction with @Michael, and suspect we have some agreement as to where his account falls over.

And yes, Austin is showing that we don't need talk of mental images in some form in order to explain perception, hallucination, illusion and so on. That doesn't seem to me to be dogmatic.

The first half of the book is an assault on Ayer's essay on perception, a pretty extreme example.
See Austin: Sense and Sensibilia

...surely no one would willingly go against God if they had certain knowledge or faith? — Tom Storm

Ask Lucifer...

That made me laugh. I'll need to think about it. — Tom Storm

Worth considering in terms of "flourishing", to see how it doesn't help. We could feed the pup or eat it. Both incur flourishing. Which is obligatory?

Nice.

Consider this: "acting rightly is acting in accordance with the structure of reality itself". This is much stronger than the simple "X is good IFF god commands X". Suppose we accept the whole Thomistic framing here, do we have an ought from an is?

"Acting rightly is acting in accordance with the structure of reality itself is descriptive. To make it proscriptive, we need to add "One ought to act in accordance with the structure of reality." Even if an act helps to realises the ends built into reality, it remains open to ask if we ought realises the ends built into reality.

You've moved to a teleological account. Teleology explains what counts as flourishing. It does not explain why flourishing is obligatory.

In addition, one cannot act otherwise than in accord with the structure of reality. Both kicking the pup and feeding it are possible; Either is "in accordance with the structure of reality itself". "Acting in accordance with the structure of reality itself" tells us nothing about which to choose.

Of course, why would say that? it's defined as infinite. That's the whole point. It is infinite and infinite is defined as boundless, endless, therefore not possible to count. So any axiom which states that it is countable contradicts this. — Metaphysician Undercover

"Infinite" means "not finite", not "not countable".

And to be countable is to have an injection into ℕ; or equivalently, to have a bijection with a subset of ℕ. For infinite items, that subset is ℕ itself.

ℕ is both infinite and countable. ℝ is infinite and uncountable.

You have yet to show a contradiction in bijection; indeed, you have yet to show what that might even mean. Mathematics on the other hand takes a bijection between two sets A and
B to mean there is a rule f such that each element of A is paired with exactly one element of B, and each element of B is paired with exactly one element of A. Here is a proof that ℕ has a bijection with ℕ, and so is countable:

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

• Well-defined:
  For every $n \in \mathbb{N}$, we have $f(n) = n \in \mathbb{N}$.
  Hence $f$ is well-defined.
• Injective:
  Suppose $f(n_1) = f(n_2)$. Then
  $n_1 = n_2$, so $f$ is injective.
• Surjective:
  Let $m \in \mathbb{N}$. Define $n = m \in \mathbb{N}$. Then
  $f(n) = f(m) = m$, so every element of $\mathbb{N}$ is hit by $f$.
  Hence $f$ is surjective.

Conclusion:
The function $f(n) = n$ is a bijection from $\mathbb{N}$ to $\mathbb{N}$.
Therefore $\mathbb{N}$ is countable.

The bijection is not assumed, it is demonstrated.

A theist might say that god as goodness itself functions as a brute fact. — Tom Storm

Indeed, and in so doing hope to close themselves off from the Euthyphro by asserting a supposed brute fact that god's will and what is good are the very same thing. But the result is to remove any normative value from what is good, and to make it a mere fact - the will of god. The account fails to explain normativity.

The point is that a number is not a thing which can be counted, it is something in the mind, mental. — Metaphysician Undercover

Now many integers are there between zero and five?

Did you choose to be born? Do you choose to die? Not everything is of your own choosing. — Wayfarer

Yep.

But some things are of your own choosing. And convincing yourself that you had no choice when you plainly might have done otherwise is... unwise? A recipe for disfunction.

You made me say it.

The religious only follow their god because they so choose.
— Banno

My conscience is captive to the word of God. Here I stand. I can do no other. God help me. Amen.
— Martin Luther — Wayfarer

Yes, there is something unsettling in such certainty... the denial that one might have chosen otherwise. Luther excusing his own sins.

Here's the "argument" from the OP:

This is simply the plain truth. For rhetorical purposes, they will try to avoid the plain truth but it is what it is and when you break down what they say when they're being honest- you will see that for all their noble-sounding talk which is meant to propound the alleged morality of their position.... they lack of a basis for morality and are moral relativists. They don't believe in morality. Morality from such a stance is whatever you think it is- if one is consistent. — Ram

It invokes an impressive number of non sequiturs. But let's set those aside and instead note that choosing to follow god's will does not absolve us from choice. That is, as they themselves will profess, it remains up to each of us to choose what to do and what not to do. Those who profess that there is an objective good decided by their god also admit that it can be chosen or rejected.

The fact of choice, and the issues of direction of fit, are ineliminable.

That is, the problem alluded to in the OP applies as much to the religious as to the secular. The religious only follow their god because they so choose.

...if we adopt the Christian language game in which God is the embodiment of goodness... — Tom Storm

Yep. It's a common Christian response to the Euthyphro.

Why ought we adopt that game?

The argument you present relocates the normative element into a definition. It, and the is/ought gap, are still there, just shuffled sideways a bit.

You've read previously about Anscombe's shopping list. This is much the same thing; the difference between a shopping list and an itemised receipt is not found in the items on the list, but the intent we attach to it - to what we do with the list.

Metaphysician Undercover believes it is illegitimate in some way. — Ludwig V

If you think Meta has convincingly shown that numbers do not exist, then I suppose that's an end to this discussion. And to mathematics.

But I hope you see the incoherence of his position.

You presume the sequence "this is the case" is followed by a judgement "this ought be the case", then show that this is muddled.

Yep.

That sequence is added by you, not inherent in my post.

Even if we could demonstrate that God exists, it does not follow that we ought to act in any particular way. — Tom Storm

Yep.

Even if we had before us is the undoubted word of god, it does not follow that we ought do as he says.

It remains open for us to do as the book says, or not.

There's a cut missing in these considerations. One that has been known to philosophers for a centuries, but hasn't transferred to the general consciousness.

We can look around the world and see how things are. And broadly, we find ourselves in agreement that there are purses and puppies and clouds. We agree as to how things are.

We can also look around the world and think about how things ought be. Again, broadly, we find ourselves in agreement that it's best not to steal stuff or kick puppies.

Now what we want is dependent on what is the case. One can't steal a purse if there are no purses, nor kick the pup if there are no pups.

But that things are indeed arranged in a certain way says nothing about how they ought be arranged. That there are purses tells us nothing about how those purses ought be distributed. That there are puppies tells us nothing about how we ought treat them.

And generally, that the world is arranged in a certain way does not tell us about how it ought be arranged.

Two aspects of this are salient to this thread.

That we have evolved in a certain way tells us nothing about how we ought behave. Even supposing we are disposed to act in a certain way by evolution, it does not follow that we ought act in that way. It remains open that we ought act in a way contrary to evolution.

The second is the more general point that while we can find out how things are by looking around at the world, we can't use that method to find out how things ought to be. More generally, while science tells us how things are, it cannot tell us how things ought be.

The area that examines how things ought be is ethics. And it's worth reading a bit bout it, especially in regard to the logic of ought sentences.

Well, I'm pleased you had a read. I suspect the trouble might be that you want a theory of inner states, while Austin wants to show that the philosophical pressure to construct such a theory is illusory. For you, perceptual explanation requires structured phenomenal states, but that is exactly the assumption under dispute, not a neutral starting point. For you, something like “phenomenal structure is explanatorily necessary” is axiomatic; Austin treats it as a philosophical posit whose motivation has evaporated once ordinary perceptual talk is taken seriously. He's showing that explaining a perceptual error does not require analysing an inner state into veridical and delusive components. he shows that nothing forces us into the ontology of phenomenology.

You were claiming that numbers "exist", and how to be, is to be a value. Now you've totally changed the subject to "assigning a value". — Metaphysician Undercover

Same thing. Again, not my problem that you don't understand this.

Sure, and those rules are axioms about "mathematical objects". When you were in grade school, were you taught that "1", "2", and "3" are numerals, which represent numbers? Notice, "2" is not a symbol with meaning like the word "notice" is. It's a symbol which represents an object known as a number. In case you haven't been formally educated in metaphysics, that's known as Platonism. — Metaphysician Undercover

Very sloppy work. Platonism is not the claim that symbols refer to something, but that mathematical objects exist independently of any theory, language, practice, or mind, and are discovered, not constituted, by mathematics. Nothing here commits to that. You are equivocating between reference and ontological independence.

And I'll opt to believe that you willfully deny the truth, rather than simply misunderstand. — Metaphysician Undercover

You are looking for a rhetorical dodge to get out of the mess you find yourself in.

Anytime a value has being, that's Platonism — Metaphysician Undercover

No, Meta. Quantification or assigning a value does not require Platonic commitment. A value can ‘have being’ within a formal system, a constructive framework, or a model, without existing independently as Plato would claim.

Do you recognize that set theory is based in Platonism? — Metaphysician Undercover

Sad. Formally, set theory is just a system of rules. Treating its sets as independently real is a Platonic interpretation, not a necessity.

Guess it's back to ignoring your posts.

As I said, Platonism, which is an unacceptable ontology. — Metaphysician Undercover

Platonism is indeed unacceptable, but quantification is not platonic. Sad you can't see that.

Quantification does not require Platonic commitment; it merely specifies the domain of discourse and what statements about it are true. This is consistent with nominalist or structuralist interpretations.

More often, having nailed their flag to the mast, they will double down.

What I have been trying to show is that science can only assist in helping us understand at a microlevel how humans have consistency in color judgment and how some may have divergent judgments (color blindness). Science relies on shared standards of color, consistency of color judgments, and shared language, not private introspection of sense data. So the metaphysics of indirect realism cannot find support from science. — Richard B

Nice work.

You have an awe deficit. — frank

...or you have an awe surfeit.

Awe is not an argument.

You completely missed my point. — frank

You want I should be awe struck into agreement? Nuh.

Take a moment to stop and take in the world around you: the sights, sounds, movements in time and space. Now take in that all of it is generated by your brain (possibly with some quantum magic). — frank

Well, no, it isn't. The bits and pieces around me have a place in there as well. Be they quantum fluctuations or cups and cats.

Your jump from "neural processes are necessary for perception" all the way to "the world is generated by the brain" is illegitimate.

Ok. but "More word smithing" says nothing. I suppose it's just more word smithing.

Or is it that you hung your flag on the "indirect realist" mast, then found that you basically agreed with what I had to say?

I've tried to follow what you are doing here, but scattered inaccuracies and errors make it very difficult. I gather you want to Cantor’s argument into a constructive or even computational lens. It’s valid in that framework, yet you seem to think it can be taken as refuting classical results about cardinality. I musty be misreading you.

, , yes, Maria Corina Machado is playing trump almost as well as does Putin.

Well put.

"I gave you a fact"

says the monkey — Banno

, not noticing how the "fact" is the result of his own attitudes and presumptions.

We experience (are aware of) something when we dream, when we hallucinate, (when we have synaesthesia?), etc., — Michael

That's the point at issue. The thing about an hallucination or dream is exactly that there is no something.

An hallucination is defined precisely by there being no object of which one is aware, only a belief-like state produced in a derivative way.

:meh: