Cheers. Useful stuff. When someone makes such obvious mistakes, it's probably not worth giving detailed responses, because chances are they will not be able to recognise or understand the argument. The result will be interminable.

I think part of what’s driving the disagreement here is that two different notions of “same size as” are in play, and they come apart precisely in the infinite case. — Esse Quam Videri

Yes, but this far too charitable. There are compelling reasons for rejecting Magnus's account. The notion of "same size" he work with is inadequate to deal with infinities coherently - using it results in inconsistencies.

Here's a formalisation of Magnus's account.

• Proper Subset Principle
  If $A \subset B$ and $A \neq B$, then $A$ is smaller than $B$.
• Subtraction Principle
  If $B \setminus A \neq \varnothing$, then $B$ is larger than $A$.
• Transitivity of Size
  If $A$ is smaller than $B$ and $B$ is smaller than $C$, then $A$ is smaller than $C$.

These principles are all valid for finite sets.

Let's look at a few contradictions that result.

Contradiction 1: ℕ vs Even Numbers
Let

$\mathbb{N} = {1,2,3,4,\dots}$
$E = {2,4,6,\dots}$

• $E \subset \mathbb{N}$ and $E \neq \mathbb{N}$
  ⇒ by (N1), $E$ is smaller than $\mathbb{N}$.
  
• $\mathbb{N} \setminus E = {1,3,5,\dots}$ is infinite
  ⇒ by (N2), $\mathbb{N}$ is larger than $E$.

But define the pairing:

$f(n) = 2n$

This is a one-to-one correspondence between $\mathbb{N}$ and $E$.

So:

• $E$ and $\mathbb{N}$ are the same size.
• $E$ is strictly smaller than $\mathbb{N}$.

Thus:

$E \lt \mathbb{N} \quad \text{and} \quad E = \mathbb{N}$

This violates antisymmetry.

Contradiction 2: ℕ vs ℤ
Let

$\mathbb{N} = {1,2,3,\dots}$
$\mathbb{Z} = {\dots,-2,-1,0,1,2,\dots}$

• $\mathbb{N} \subset \mathbb{Z}$, proper subset
  ⇒ by (N1), $\mathbb{N} \lt \mathbb{Z}$.
  
• $\mathbb{Z} \setminus \mathbb{N}$ is infinite
  ⇒ by (N2), $\mathbb{Z} \gt \mathbb{N}$.

But define a pairing:

$0 \leftrightarrow 1,\quad<br /> -1 \leftrightarrow 2,\quad<br /> 1 \leftrightarrow 3,\quad<br /> -2 \leftrightarrow 4,\dots$

So:

• $\mathbb{N}$ and $\mathbb{Z}$ are the same size.
• $\mathbb{N}$ is strictly smaller than $\mathbb{Z}$.

Again:

$\mathbb{N} \lt \mathbb{Z} \quad \text{and} \quad \mathbb{N} = \mathbb{Z}$

Contradiction.

Contradiction 3: Self-Subtraction
Let $A = \mathbb{N}$.

Partition $A$ into two disjoint infinite subsets:

$A = E \cup O$

where

$E = {2,4,6,\dots}$
$O = {1,3,5,\dots}$

By (N1):

• $E \lt A$
• $O \lt A$

But:

$A = E \cup O$

So $A$ is the union of two sets each strictly smaller than $A$.

This is impossible under the naïve size rules, which are now mutually inconsistent.

Contradiction 4: Hilbert’s Hotel
Let hotel $H$ have rooms ${1,2,3,\dots}$, all occupied.

Define:

$f(n) = n+1$

This moves each guest up one room, freeing room 1.

• No guests are removed.
• A new room becomes available.
• The hotel is both “the same size” and “larger”.

Under subtraction-based size:

• Adding capacity without increasing size is impossible.
• Removing nothing yet gaining space is impossible.

The governing rules of “size” break down.

Conclusion
Once infinite sets are admitted, the principles:

• proper subset ⇒ smaller,
• remainder ⇒ larger,
• antisymmetry and transitivity,

cannot all be maintained. The naïve notion of “same size” does not merely yield counter-intuitive results — it generates outright contradictions.

This is the sense in which the mathematical objection applies: the concept fails to define a coherent ordering on infinite collections.

Thanks to ChatGPT for help with the formatting, but even so the time taken to respond to the sort of nonsense promulgated by maths sceptics is far more than the net benefit.

Once that distinction is on the table, the question isn’t really “who is right,” but what we want the concept of “same size” to do in this context. Mathematics answers that one way; ordinary language answers it another. — Esse Quam Videri

The question is, "who is right?", and the answer is, the contradictions above show that Magnus' ideas cannot be made consistent. Formal language is nothing more than tight use of natural language - it is not unnatural. What is shown by the contradictions is not a conflict between natural and formal languages, but a lack of adequate tightness in Magnus's argument. Magnus’s argument lacks sufficient precision to handle the case he wants it to handle.

Notice also that the arguments stand alone, they are not appeals to authority.

The correct diagnosis is not conceptual pluralism, but logical failure.

ChatGPT will summarise a discussion:


Summarise the argument and responses at https://thephilosophyforum.com/discussion/16296/disability/p1 and the next few pages, by topic.


Result: https://chatgpt.com/share/6962116e-63d8-800f-ad10-77d861ed4b8a


This seems to have given a reasonable summary of the conversation and offered to do the next few pages.

Sorry, I hadn't noticed this:

That a bijective function exists, cretin, does not mean that the two sets can be put into a one-to-one correspondence. — Magnus Anderson

:lol:

Oh, well. :roll:

Reading isn't thinking. — Magnus Anderson

Nor is your making shit up.

Reading a maths book isn’t just passive; it’s fuel for precise thinking, especially when you’re debating infinite sets. It shows how folk have thought about these issues in the past, and the solutions they came up with that work.

Your responses are now a bit too sad to bother with. Thanks for the chat.

What you provided is the definition of the countable infinity. That's not the same as infinity. — Magnus Anderson

Well, it's one infinity amongst a few others...

If you want to prove that my definition is false — Magnus Anderson

Your "definition" of infinity is not a definition of infinity. It's not false, it's just an intuitive approximation.

Simply asserting that my definition is a heuristic that is useful for intuition is not an argument. — Magnus Anderson

Yep. So I went the step further, presenting one of the standard definitions.

That goes against what Cantor said. — Magnus Anderson

It seems then that you haven't understood Cantor, either.

And I am pretty sure you won't be able to prove it — Magnus Anderson

A bijection exists between N and A — e.g., $f(n) = \frac{1}{n+1}, \quad n \in \mathbb{N}$


You really should take 's advice and read a maths book.

Let A be a finite set that is { 1, 2, 3, ..., 100 }.
Let B be a finite set that is { 1, 2, 3, ..., 99 }. — Magnus Anderson

Matching one to one from the left, the one left out is the 100. :meh:

With your
A = { 1/2, 1/3, 1/4, ... }
and
N = { 1, 2, 3, .. . }

There isn't last element. Nothing is left out.

They aren't the same size. The set of even numbers has two times smaller. Doesn't matter what Cantor and mathematical establishment say. They aren't reality. — Magnus Anderson

Yep, the evens only has every second number, so it must be half the size... Thanks for the giggle!

...a number that is larger than every integer... — Magnus Anderson

...is not the definition of infinity. “Larger than every integer” is a heuristic, useful for intuition, but the mathematical definitions depend on limits or cardinality. Something like:

S is countably infinite ⟺∃f:N→S that is bijective (one-to-one and onto).

A heuristic for sets is the Infinite means the set never ends; there’s no last element. That allows for sets with transfinite elements.

And adding four to an integer is still an integer. — Magnus Anderson

Sure. Infinities are not integers.

If "add" means "increase in size" — Magnus Anderson

But it doesn't.

Adding four to infinity is still infinity.

By definition, to add an element X to an existing set of elements S means to increase the size of that set. — Magnus Anderson

Not for infinite sets. For obvious reasons.

ℕ and ℕ ∪ {0} really are the same size
Take:

ℕ = {1,2,3,…}
ℕ₀ = {0,1,2,3,…}

here:

f(n) = n - 1

This is:

• injective (no collisions)
• surjective (every element of ℕ₀ is hit)
• Total

That is a proof of equal cardinality. Nothing is “pretended”.
The fact that this offends finite intuition is exactly what “infinite” means in modern mathematics.

You should get on well with Meta.

I'll leave you to it.

The topic attracts cranks.

See The Enumeration of the Positive Rationals

It should be pretty clear.

It lacks exactly one element. — Magnus Anderson

Which element is missing?

I didn't take Cantor's word for it, I read his diagonal argument.

Consider A = { 0, 1, 2, 3, 4, 5 } and B = { 1, 2, 3, 4, 5 }.

0↔︎1
1↔︎2
2↔︎3
3↔︎4
4↔︎5
5↔︎?

There are not enough items in your second set to map one-to-one to the first set. Hence the cardinality of the first is larger than that of the second. Looks pretty convincing to me.

We should take your word for this?

I gave an argument - albeit briefly. Fractions can be placed in a sequence, and so are no more than countably infinite.

Were did I go wrong?

The paperback finally arrived today.

Might be some revision for this thread ensuing.

And I'd still like to get back to How to Prove Hume’s Law.

See how we go.

Here's an article from a few months back - starting with quote from Charlie Kirk - about disability as the canary in the coal mine of social policy.

The US right is coming for disabled people. Here’s why that threatens everyone

It makes pretty sad reading.

This quote struck me as salient to this thread:

“Disabled people were not always marginalized; we were incorporated into society in the ancient past,” said Dr Alexandra F Morris, a lecturer in classical studies at the University of Lincoln who studies disability in ancient Egypt. “We have the means to create and return to a more equitable society if we wish to, but it is our modern-day thinking that sees disability as marginalized … and a burden.”

I know.

You are right that there are infinite infinities, but even with all those fractions, there are still only the same number as there are integers - ℵ₀, the smallest infinity - countably many. You can list them in a sequence, 1/1,1/2, 1/3, 2/3, 1/4, and so on, and so you can count them - line them up one-to-one with the integers.

Cantor showed that some infinities are larger, uncountably infinite. And then there are more than that. It's an interesting, curious area of maths. Check out Cantor's diagonal argument.

Yep. Consider these:

Trans

word-forming element meaning "across, beyond, through, on the other side of; go beyond," from Latin trans (prep.) "across, over, beyond," perhaps originally present participle of a verb *trare-, meaning "to cross," from PIE *tra-, variant of root *tere- (2) "cross over, pass through, overcome" [Watkins].

Besides its use in numerous English words taken from Latin words with this prefix, it is used to some extent as an English formative .... It is commonly used in its literal sense, but also as implying complete change, as in transfigure, transform, etc. [Century Dictionary]

In chemical use indicating "a compound in which two characteristic groups are situated on opposite sides of an axis of a molecule" [Flood].

Many trans- words in Middle English via Old French arrived originally as tres-, due to sound changes in French, but most English spellings were restored later; trespass and trestle being exceptions. — https://www.etymonline.com/search?q=trans

transgender(adj.)
also trans-gender, by 1974 in reference to persons whose sense of personal identity does not correspond with their anatomical sex, from trans- + gender (n.). Related: Transgendered.

cisgender(adj.)
also cis-gender, "not transgender," in general use by 2011, in the jargon of psychological journals from 1990s, from cis- "on this side of" + gender. — Etymonline

So it's indicative of a "crossing over, passing through, overcoming" of binary gender identities.

I knew you were going to say that...

...and yet we get on, regardless. Yep. It's what we do.

Cheers.

That it has a use doesn't mean it can be had. — Hanover

Not at all sure what that means.

The choice here is between on the one hand an account that divides the world into mind and object, then finds itself unable to explain objects; and an account that makes no such presumption.

One account begins by dividing mind from world and then cannot recover the world. The other refuses that division and never loses it.

Perception is a activity within the world, not an impassible bridge between worlds.

And here's the thing: we do manage to talk about ships, cups, walls and each other.

"Troll" for you is an effective, articulate debater with an opposing viewpoint.

Please! Distract me! — Ecurb

Here you go:



The world is still a good place.

You are under no obligation to respond. or even to read, to my posts.

keep the discussion on topic and engaging with ideas instead of petty insults. — Philosophim

:lol:


Here's my contribution:
Page four
Page three
Page two
Page one

I've argued that the claim “trans women are women” can be true. We are not obligated to use only a single fixed biological definition of "woman". Words such as man and woman are polysemous—they have multiple legitimate meanings that vary with context (social, legal, everyday use) and are not rigidly fixed by biology. Hence in contexts of gender identity and social role, “trans women are women” is true; rejecting it by privileging one narrow biological sense is to misunderstand how language works. The idea that there is a single true or default meaning of these terms independent of context is faulty, and insisting on such a view is arbitrary and ignores existing usages. The aim is to show the opposition’s original claim (that the slogan is categorically false) collapses once we acknowledge legitimate linguistic contexts in which the slogan is true.

You just doubled down on your erroneous understanding of language use, and your fascination with genitalia.

Why not? — Hanover

Because at some stage the conversation has a use.

Explain it then. — Philosophim

26 pages of your obsession with the contents of other people's underwear and the supposition that those contents dictate which toilette they must use, shows that there is not much point.

You're seriously trying to redefine "direct perception" in such a way that even with these visors and their computer-generated images on a screen they still directly see their shared environment? — Michael

That's not a redefinition. What this shows is how you misdiagnose the the argument. 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.

Direct realism doesn't claim that direct perception is perception without mediation. That would rule out glasses, mirrors, telescopes, microscopes, hearing aids, telephones — and even eyes and nerves. Mediation is ubiquitous and trivial. What matters is not causal mediation but epistemic termination. This is why your visor world makes no difference. The visor drops out of consideration, much as the beetle in a box does in discussions of private language. The function of the visor is irrelevant, in that the moment you insist — as the indirect realist must — that what we really ever see is only the visor-image, the example collapses.

When you hear your mother on the phone, you do not hear sound waves and then infer your mother as a hypothesis about an inner item. You hear your mother speaking, by means of sound waves, transmission systems, speakers, etc. The fact that you do not experience the causal chain as such is irrelevant — because direct realism never required that.

Direct realism is not the thesis that perception includes awareness of causal links. It is the thesis that perceptual verbs take worldly objects as their logical objects. You hear your mother, not a sound-wave-token; you see the ship, not a retinal image. The mediation is causal, not epistemic.

We can talk about the image on the visor, but this is derivative, dependent on our being able to talk about an image of the ship, and hence being able to talk about the ship.

The argument here is not redefining “direct”; but refusing to accept a Cartesian picture in which perception must either be inner and certain or outer and inferential.

This is were a Markov Blanket helps. On the indirect realist construal, the Markov blanket is treated as epistemically opaque. On the direct realist construal, the blanket is only causally isolating. Information flows across it, but that does not lead to epistemic confinement. The organism’s perceptual capacities are attuned to environmental states across the blanket; perception is an interaction spanning the boundary, not an encounter with an inner surrogate. What is perceived is the ship, not a mental image that stands in for it.

The indirect realist uses the debunked picture of a theatre of consciousness; the homunculus, sitting inside only ever seeing the ship on the visor. The better picture is that we see the ship, using the visor. There is no phenomenal state that is what we see in the place of the ship; rather, the neural process that constructs vision constitutes your seeing the ship.

I think this thought process assumes a virtue that has not been earned. — Philosophim

You entirely misunderstood the argument. No surprise there.

↪Banno I was like... damn, I know over 100% of the trans population? — DifferentiatingEgg

Hey — good to hear we have so many mutual friends! :rofl:


Edit: But there is a serious point here. If the folk here objecting to trans folk do not know any, then that explains why they are treating real humans in abstract terms.

Perhaps nothing helped acceptance of the queer community as much as the "revelation" that gay, lesbian, and queer folk are all around you, and pretty much like you and I.

is why the physiology is irrelevant. Even when the physiology is added to somewhat radically, the direct realist point remains.

But we need to add, neither direct realism nor indirect realism is the whole story - we can talk about the beetle in the box, the mental image; in Michael's world, someone might complain that their visor is faulty - and thereby change from the game of talking about ships to the game of talking 'bout visors - but having a faulty visor is to admit that there are functioning visors, and so to give the visor a place int he game.

It can't be beetles all the way down.

I meant to say that I hold no stock in the argument that the PLA refutes indirect realism. You appear to be accepting that these people are talking about their shared environment even though none of them ever directly see it (even the direct realist must accept this given the visors). — Michael

It's hard to see how the visor example counts against the private language argument. That's how you set the account up. You now want to use it as an example of indirect perception.

So back to this:

A direct realist believes that when we, say, look at a veritable ship, what we see is the ship. They hold that light is reflected from the ship, focused by the eye and incites certain neural pathways associated with things of that sort, and that this process is what we call seeing a ship. — Banno

Your visor users talk about the ship, and not what they see on the visor.

An indirect realist, in contrast, holds that what we see is not the ship, but something else, sometimes called a "mental image" of the ship, that is presented to us by the process of light being reflected from the ship, focused by the eye and inciting certain neural pathways associated with things of that sort. — Banno

An indirect realist says that all they see is the stuff on the visor.

So if we wish to talk about images on screens, we can adopt the perspective of the indirect realist. If we wish to talk about ships, we must bypass the visor and admit that we see the ship.

Hence:

Indirect realism effectively treats the Markov blanket as opaque, the system having only access to internal states in the form of the mooted "mental image". External states are inferred, never directly encountered, and what is “perceived” is confined to what is inside the blanket (representations, images, models).

Direct realism treats the Markov blanket as causally, but not epistemically, isolated, the system having access to external states through the mediation of the blanket. Seeing the ship is an interaction, not an appearance, and perception is a skilled engagement with environmental states across the blanket;
there is no inner object that perception terminates on. — Banno

So we are back to were we were four pages and 3 days ago. 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.

Usage remains constant regardless of what's going on out there, which is the point of the Wittgenstinian enterprise. — Hanover

Not quite. Rather, what we use is what remains constant... with regard to "out there"; but note that we ought also reject the phenomenological/cartesian picture of out there and in here. Wittgenstein emphasises what we do with words, in the world. His is not a form of idealism.

That is, what about unicorns? How do I deal with the words without references? — Hanover

"Unicorns" has a use, if not a referent, and if only as an example in philosophy fora. See if you can turn that into an argument.

In asking me to assume the external object is a constant so that we can be sure our perceptions are similar across one another is also problematic because it's false. — Hanover

Not sure what this was - a reference to the quote from PI? You are not there being asked to assume the external object is constant, but to notice that you have no way of telling if your private object has changed.

What is important is that we all engage in a word game, play it according to rules we all comprehend, and we interact in the form of life we know. — Hanover

I'll agree with that, and note the corollary that private objects cannot have a use in the game.

You[/u] claimed that it's impossible to talk about things unless we can see them directly, — Michael

Well, no. Certainly not. I do agree with the private language argument in so far as talk about boxed beetles and images in brains is useless.

I hold no stock in the private language argument. A society of people born with unremovable visors on their head with sensors on the outside and a screen on the inside displaying a computer-generated image of the environment could develop a language, talk about the environment, and lives their lives just as well as we can. — Michael

How is that in any way contrary to the private language argument? These folk are talking about their shared environment, not their unshared screen time...

Yes? That's how indirect perception works. You directly perceive some X and because of that indirectly perceive some Y. Even the direct realist must accept that this is how television and telephones work. — Michael

You are losing me here.

Sure, when we use a telephone we hear someone indirectly. Are you suggesting that undermines direct realism?

The point is that we don't need to directly[/i] see him to talk about him, and we don't need to directly see ships to talk about them. — Michael

Yep. But he is not only a mental image, or a firing of brain cells. He is public in a way that whatever indirect realists say they see, isn't.

It appears to me that you have moved on to equivocating about what it is that indirect realists suppose it is that is perceived.

There are about 15 transpeople in the world — Fire Ologist

Perhaps you should broaden your social circle.

I didn't say only ever. I explicitly said here that "in the non-hallucinatory case there is both hearing voices-as-mental-phenomena and hearing voices-as-distal-stimulus", with the former satisfying the philosophical notion of directness — as explained here — and the latter not. — Michael

Pretty ad hoc. Now we have both direct and indirect perception happening in the same individual for the same event.

I object to this use of the word "really". — Michael

So do I. Take it out, if you like. If what one sees is always private — cortex states, sense-data, whatever — then nothing in experience can fix reference to a public object.

The objection stands.

You and I can both talk about Napoleon. — Michael

That's exactly right. We can talk about Napoleon because there is more to him than the firing of neutrons. He is not an hallucination.