That completely inverts the issue in the question of the OP — Fire Ologist

Good.

See

I think that just as the cosmological argument proves the existence of God from knowing the existence of tables and chairs, so too the moral argument proves the reality of God from knowing the reality of right and wrong. — BenMcLean

Oli, your craving for certainty is not a firm grounding for belief.

...Banno makes some seemingly random claims about the existence of numbers. — Metaphysician Undercover

To be is to be the value of a bound variable. ω and ∞ are cases in point. In maths, Quine's rule fits: existence is not discovered by metaphysical intuition but incurred by theory choice. Quantification, ∃(x)f(x), sets out what we can and can't discuss.

Yes., well phrased Although we might differ a bit on the extensibility of maths.

It's just extending the way we talk about numbers. What started with the Biggest Number game gets extended into infinity, both ∞ and ω, the difference being that while ∞+1=∞, ω+1>ω; The first reflecting the teacher's answer "infinity plus one is still infinity", the second, the player's answer "infinity plus one is bigger than infinity". What we have is a division in how we proceeded, in the rules of the game, not in what "exists" in any firm ontological sense. It's chess against checkers, not cats against dogs. Neither set of rules is "true" while the other is "false".

And the great thing about these games is that they are extensible, in that we add more rules as we go, keeping the game coherent, while being able to talk about more and different stuff.

Part of where Meta and Magnus have difficulty is in their insistence that one way of talking is right, the other, they call variously incoherent or inconsistent, both without providing an argument and in the face of demonstrations to the opposition effect. To establish incoherence, they would need to show a violated rule internal to the system, or an explicit contradiction derivable from its axioms.
Mere discomfort with plural rule-sets doesn’t suffice.

Within cardinal arithmetic, ∞+1=∞ is true; within ordinal arithmetic, ω+1>ω is true. Cross-applying the rules is what generates the illusion of contradiction.

I'd also relate this back to my essay Two ways to philosophise, and to the arguments in Logical Nihilism. It's better to have an incomplete theory that is coherent than a complete theory that is inconsistent or artificially restricted. And better to have many differing, incomplete logics than one, monolithic yet restricted logic. These allow for growth. Advocating for new rules, new distinctions, new domains of discourse gives us a normative standard that is neither realist nor relativist.

Critics may conflate pluralism with anything-goes relativism. But only because coherence is doing real work; incoherent extensions are still excluded. Others will insist that without a privileged logic, critique collapses. But critique is local, rules are criticised from within practices or at their interfaces, not from a mythical God’s-eye view.

I loved teaching this stuff to third and fourth grade kids. The Biggest Number game; they say "A hundred ", reply "A hundred and one"; they say "A million million", you reply "A million million and one"; someone says "infinity" and someone says "infinity and one"...

I was surprised, on enlisting in these fora, to find that there are folk who don't get to the stage of understanding that every natural number has a successor, that "...and one" works for any natural number. (not ordinals... another bit of the puzzle.)

And that infinity and one is still infinity. This hazy number play sets up the kid's intuitions. Especially where it doesn't work. Infinity is not part of the structure that lets us play the number game. It needs new rules.

There's an ontology which presumes that numbers exist — Metaphysician Undercover

We don't need much ontology. Quantification will suffice.

It seems very odd to need a proof that god exists in order to do the right thing.

You don't have access to your wife's voice. — frank

I'm not sure I know what that might mean; but I do hear my wife's voice, through the telephone. That's indirect, in comparison to when she is in the room, but perhaps more direct than listening to a recording...

maybe see https://thephilosophyforum.com/discussion/comment/1034678

The causal chain remains the same, but our attention (the blanket) can be placed in differing locations. So in one throw we can refer to your wife’s voice, in another to the electronically constructed reproduction, and so on.

Hence the similarity with the distribution board.

Returning to colourblindness: the basis for calling the judgment an error is not that the colourblind person’s experience fails to match mine, nor that it fails to match some phenomenal property instantiated by the object. The basis is that, within a shared practice of identifying and re-identifying objects across conditions, their judgments systematically fail to track features that figure in stable, publicly coordinated practices of correction and re-identification. That is an epistemic failure relative to those practices, not a phenomenal defect. — Esse Quam Videri

Interestingly, this is pretty much the reply I owe you from that other discussion.


Good reply.

"Is" and "of" are not the same word.

...he wasn't doing any philosophical work for us... — Esse Quam Videri

Well, he at the least served as a poor example, showing us that the theory that there are two populations does not have a truth value.

...word smithing... — frank

I prefer "conceptual clarification"... I clarify concepts, you smith words, he makes shit up... :wink:

My contribution to your word smithing would be that we do need to speak in terms of experience. Sight is not an isolated activity. It's integrated into a whole. And there is some functional entity we generally refer to as "you" which directs attention. — frank

Yes. I quite agree.

As Isaac may have mentioned to you... — frank

A moment for the departed; he and I had long conversations about this, and I think he introduced me to Markov Blankets; together we forged an agreement that pretty much bypassed the direct/indirect dichotomy. The main distribution board was part of that discussion, another place to throw the blanket. Would that he were here now to give his opinion.

When you hear your wife's voice on the phone, that's not really her voice. It's a computer generated representation. If the logic of that throws you for a loop, I guess we could work through it. I wouldn't advise rejecting it because sounds illogical, though. — frank

See the weasel word? Did you hear your wife's voice? what dis she say? Were have you thrown the Markov Blanket? Were else might you throw it?

Ben only asserts. Watch, you'll see.

I don't think experience has any particular location. — frank

Well, that's a start.

It's something creatures with nervous systems do. A flood of electrical data comes into the brain, and the brain creates an integrated experience. Are you denying that? — frank

No. I'm denying that what we experience is that flood of electrical data. Rather, having an experience is having that flood of electrical data. What you experience, if we must talk in that way, is the cat.

You see the cat, not your neural activity. Your neural activity is seeing the cat. At least in part.

There's a category error that involves thinking that because we can't start at one and write down every subsequent natural number, they don't exist.

1 is a number, and every number has a successor. That's enough to show that the natural numbers exist.

Sure. You experience the cat indirectly. You experience the ship indirectly. You experience the smell of the coffee indirectly. Welcome to indirect realism. — frank

SO your response not by presenting an argument but by reasserting your error.

Ok.

Having a "content of experience" presupposes a container–contained picture of mind: an inner arena where experiences “have” objects or qualities. That’s precisely the sort of framing being rejected. Once you reject the Given, the idea of content starts to feel artificial, a placeholder for a problem that doesn’t exist.

Instead of talking in terms of content, we can frame perception as engagement with the world, and neural processes as how that engagement happens. We drop any separate “object of experience” in the mind.

It will help if you reply to what I say, rather than what you want me to have said.

Ok. The content of your experience is neural representations. Happy? — frank

No. The content of my experience is the cat, the ship, the smell of coffee. Not my neural processes, and not my neural representations.

That, if we must make use of "content of experience".

You're an indirect realist. You allow that humans experience neural representations, whether we call that seeing, hearing, tasting/smelling, touching (pressure and texture sensing). — frank

No. Humans do not experience neural representations; experience is having neural representations.

You are not separate from your neural processes.

I don’t want to deny the coherence of these scenarios altogether, but I do want to deny that they carry the philosophical weight Michael wants them to carry. Once truth and error are located at the level of world-directed judgment, private inversion possibilities become explanatorily idle, even if they remain metaphysically conceivable. — Esse Quam Videri

Frank turns up at our laboratory, and we are unable to categorise him into one population or the other. Michael wants to maintain that there are nevertheless two populations, while I maintain that that the issue has no truth value. You, EQV, just refuse to commit. :wink:

“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.

What those who reject indirect realism are actually rejecting may not be indirectness as such, but the reification of something “given” — an object of awareness that is prior to, or independent of, our conceptual, practical, and normative engagement with the world. Once you posit sense-data, qualia as objects, appearances as inner items, you generate the “veil of perception” problem automatically. “Direct realism” then looks like the heroic attempt to tear down the veil. But if you never put the veil there in the first place, there is nothing to tear down.

You see the cat. Perhaps you see it in the mirror, or turn to see it directly. And here the word "directly" has a use. You see the ship indirectly through the screen of your camera, but directly when you look over the top; and here the word "directly" has a use. The philosophical use of ‘indirect’ is parasitic on ordinary contrasts that do not support the theory. “Directly” is contrastive and context-bound, it does not name a metaphysical relation of mind to object, it does not imply the absence of causal mediation.

What you do not see is a sense datum, a representation, an appearance, or a mental image. You might well see by constructing such a representation, and all the physics and physiology that involves. But to claim that what you see is that construct and not the cat is a mistake.

One can admit that neural representations exist and denying that such things are the objects of perception. These neural representations are our seeing, not what we see.

Starting by misunderstanding what is at issue, and then inventing for yourself the opposing case, makes the issue very easy to decide. Well done.

Thanks. I hope nothing I've said is at odds with this? Good feedback.

I suppose that while transfinite numbers are not much used in physics, continuum cardinality and so on are present as background commitments. So if the space-time manifold in General Relativity is continuous, then I suppose transfinite cardinals are included by default in that formalisation; or so I believe. Of course, quantum theories would involve granularity, but this is entering into speculative physics, a can of worms.

There are oddities. In particular, I've had discussions previously with "finitist" folk who denied limits and such, and so were unable to make sense of differential calculus, and so in turn were led to denying corresponding physical entities such as instantaneous velocity. @Metaphysician Undercover has been known to do something along these lines.

Oh well, no more analytic geometry. — Srap Tasmaner

Indeed. And not just that. Much of modern maths would be unavailable or need reworking, with no apparent gain.

Magnus's position appears incoherent, in that he makes use of ℕ and other infinities while disavowing the relations between them. Meta is perhaps more consistent in apparently simply rejecting any infinities - or something like that.

The onus of proof is always on the one making the claim. If you're making the claim that bijection between N and N0 exists, you have to show it, and that means, you have to show that such a bijection is not a contradiction in terms. That's what it means to show that something exists in mathematics. — Magnus Anderson

The very first line of the proof does exactly what you ask for here. A function maps a each individual in one domain with an individual in the other. Hence:

The function is Well-defined: For every $n \in \mathbb{N}$, we have $n \ge 1$, so $n-1 \ge 0$. Hence $f(n) \in \mathbb{N}_0$, and the function is well-defined.

If there is some other contradiction, then that is your claim, and up to you to demonstrate.

And that's not true.

The only thing that you have shown is that you can take any element from N and uniquely pair it with an element from N0. — Magnus Anderson

This is perverse. That is exactly what has been shown. That each element of ℕ can be paired with an element of ℕ₀, and that each element of ℕ₀ can be paired with an element of ℕ. The bijection is fully established.

It's as if someone were to say "A circle is a plain figure with every point equidistant from a given point", and you were to insist that such a thing cannot be spoke of until it is shown not to involve an inherent contradiction...

Why not work with the definition unless some contradiction is shown?

And in the cases of infinite sets, you have not shown a contradiction.

It's brilliant and convincing. — Srap Tasmaner

Yes!

The diagonal argument and its friends are amongst the most beautiful and impressive intellectual presentations. I pity those who do not see this. The exercise here is to show folk something extraordinary; but it seems that there are a small but vocal minority who for whatever reason cannot see.

A description close to Davidson's anomalous monism, the view that while thoughts and actions are physically grounded (monism), there are not governed by strict laws.

Yep.

Bijection does not mean that you can take ANY element from N and uniquely pair it with an element from N0. Of course you can do that with N and N0.

It means that you can take EVERY element from N and uniquely pair it with an element from N0. And that's what you can't do.

Do you see the subtle difference? — Magnus Anderson

The proof given shows that for each element in $\mathbb{N}$ there is exactly one element in $\mathbb{N}_0$.

Take any element of $\mathbb{N}$ and there is a corresponding element in $\mathbb{N}_0$. Take any element in $\mathbb{N}_0$ and there is a corresponding element in $\mathbb{N}$.

Since that works for any element, it works for every element. There is no gap. The rule at work here is Universal Generalisation.

Therefore the sets are the same cardinality.

What this means is that you have to show that f: N -> N0, f( n ) = n - 1 is not a contradiction in terms before you can conclude that it exists.

Has anyone done that? — Magnus Anderson

Well, yes.

In standard mathematics, we can define a function f: ℕ → ℕ₀, f(n) = n − 1, and check the definition. We saw that every n ∈ ℕ maps to exactly one element in ℕ₀.

Once the definition is satisfied, the function exists by construction. There is no need to “show it is not a contradiction.” A contradiction would arise only if the rule could not possibly assign outputs in the codomain, which is not the case here.

You suppose that before a function exists, we have to show it is not contradictory. But in mathematical thinking we define the function, check the definition, then if all requirements are satisfied, the function exists.

Again, it is up to you to show any contradiction, not up to us to show there isn't one. You have misplaced the burden of proof.

While it's good to see you using some formal notation, this isn't an example of a function. A function from A to B is a set of ordered pairs satisfying certain conditions. So writing f : A → B, f(n) = n − 1 is not merely symbolic stipulation; it is a claim that the rule maps every element of A into B. But in this case, as you point out, not every element of A maps to an element of B.

That's not a contradiction within a function, as you diagnose, but a failure to specify a function.

That is, <f : A → B, f(n) = n − 1, with A ={ 1, 2, 3, 4 } and B = { 0, 1, 2 }> does not construct a function. It's ill-formed.

In the argument we are considering, there is no such malformation. This is not just asserted, but demonstrated by the conclusion that the function is well-defined, injective and surjective.

Unlike the finite example, f: ℕ → ℕ₀, f(n) = n − 1 does satisfy the totality requirement: every natural number n has n−1 ∈ ℕ₀. Therefore the function exists and is indeed a bijection.

Thanks - your acknowledgement is appreciated.

Excellent use of the chess analogy.

And Banno, you were right. — ssu

Things would be so much easier if everyone just accepted this dictum. :wink:

Cheers.

It's a point that is missed in almost all the ethics hereabouts.

But the argument in this thread was particularly poor.

It's honestly quite surprising that you of all people are suggesting that something is true only if we can determine that it's true. That's very antirealist of you. — Michael

Interesting. I'm not saying it's not true, but that it's not even true, or false. It's not well formed enough to be true or false. Some strings of words fail to be truth-apt in the first place.

If <"the blugleberry is foo-coloured" is true if and only if the strawberry is red>, then we have some basis for assigning a truth value.

If there is no public content, the truth condition is not fixed; so unless "the blugleberry is foo-coloured" has some equivalent, such as "the strawberry is red", it has no truth value. After Davidson, we could not recognise "the blugleberry is foo-coloured" as a sentence. A string counts as a sentence only if it can be interpreted. Interpretation requires publicly identifiable conditions of truth. “The blugleberry is foo-coloured” lacks any such conditions. Therefore, it fails prior to truth and falsity: it is not even recognisable as a sentence.

I disagree with your assertion that we must be able to determine which group someone belongs to for there to be two different groups. — Michael

So in your scenario, it is not possible to assign Fred to one of the populations, but you maintain that the distinction is meaningful. That strikes me as absurd.

The same applies to your picture. How could you ever determine that what the chap on the left sees is different to what the chap on the right sees?

This, it seems, might be the core difference between our accounts. You insist that there are private phenomena while apparently agreeing that they make no difference, while I say that since there is no difference, there is no private phenomena.

The symbol we're talking about is this: — Magnus Anderson

That's a group of symbols... so you mean the $f$? And your claim is that the definition
$f: \mathbb{N} \to \mathbb{N}_0, \quad f(n) = n - 1$
"specifies what that symbol can be used to represent", but not that what it represents is not somehow contradictory? OK. Then over to you. If you think there is a contradiction here, it's up to you to show it. Exhibited it as derivations of ⊥.

Hence we come back to what it is for a mathematical object to exist, and the point you seem not to have accepted, that for "classical" maths ∃x P(x) is true iff P(x) is derivable in a consistent formal system - as here. You appear to be rejecting that rejection, while claiming not to reject it. All very difficult to follow. Hence, the impression of an intellectual drift on your part rather than any actual argument.

The first fallacious proof they use to show that N and N0 are of the same size is the observation that, if you add 1 to infinity, you still get infinity. — Magnus Anderson

Where do you think this claim appears in the proof?

The claim “infinity + 1 = infinity” does not appear anywhere in the proofs I have used.

The second fallacious proof they use is grounded in the premise that, if you can come up with a symbol that is defined as bijection between N and N0, it follows that a bijection between N and N0 exists ( i.e. it's not a contradiction in terms. ) — Magnus Anderson

The proof doesn't just "define a symbol for a bijection"; it provides an explicit function:

$f: \mathbb{N} \to \mathbb{N}_0, \quad f(n) = n - 1$

• f is the name of a function.
• ℕ is the set of natural numbers. Here, on convention, ℕ = {1, 2, 3, …}.
• ℕ₀ is the set of natural numbers including 0, i.e., ℕ₀ = {0,1,2,3,…}.
• → is read as “maps to” or “a function from … to …”.

So we can read the definition as:

f is a function from the natural numbers ℕ to the natural numbers including zero ℕ₀ such that for each natural number n, f(n) is equal to n minus 1.

What is defined here is a function, not a symbol. This is a concrete mapping, not a mere linguistic construct, and it suffices to show that a bijection exists.

That's a lie you've been shamelessly pushing forward. — Magnus Anderson

Well, it's not just me...

The definition you suggest cannot be used effectively with infinite sets. But enumeration, that is a surjection, will do everything that can be done with your definition and then do more - quite a bit more - with other sets. So your definition is effectively included in yet surpassed by enumeration.