I was not jesting at all! Sorry, I had the mistaken memory that you had explicitly welcomed me, because this has happened too often (people here are very welcoming). Then I thought your welcome to be warm because you paid attention to my remarks and also said they were elucidative. :-)

Thanks for the walm welcome. Illusionism is indeed a hard sell. It is, however, at least conceivable that there could be cognitive machines (functional minds) outputting false beliefs about there being ineffable experiences. This makes some sense when we consider that conscious experiences involve numerous cognitive judgments, rather than being (purely) some form of raw feeling. It is this feature which gives me hope that perhaps consciousness is just a cognitive illusion. However, like you, it does not fully convince me either.



It is alright to be dismissive with an inept interlocutor. Flat-earthers are very bad at physics, for instance. Illusionists, on the other hand, bring arguments and insights from cognitive science on the table, equipped with contemporary analytic philosophy of mind. They are not denying scientific evidence. What they are denying is that we have introspective evidence of qualia, and they do so by providing a somewhat detailed cognitive theory of how that comes about. I think their case is sufficiently well-argued for us to take them seriously.

At any rate, thank you for the cordial exchange. I enjoyed reading your first-hand account of what it is like to be an idealist. I have already consumed some of Kastrup's work, it is interesting indeed.

It's fine, I can see you're a nice guy.

Hey, to you I'm just words on a screen, but I'm an actual person. Sorry if I misunderstood you, there are a lot of comments in this thread and I'm not up to speed with the whole context.

Cardinality and Size aren't the same. (..) The interval between 1 and 3 is 2 and thus larger than the interval between 1 and 2 which is only 1.

Can you provide a formal criterion for what constitutes the size of an infinite set, beyond its cardinality?

When taking about intervals in ℝ, the cardinalities of the [1,2] and the [1,3] intervals are exactly the same, namely, the cardinality of the continuum. The reason is that there are bijective functions linking the two. Take, for instance, a function f : [1,3] → [1,2] with the rule f(x) = (x+1) / 2.

"Let us not forget: mathematician's discussions of the infinite are clearly finite discussions. By which I mean, they come to an end." - Philosophical grammar, p483. Wittgenstein.

Thanks for the welcome!

As regards Wittgenstein's remark, we use finite statements to fixate reference on infinite objects and work out their properties. There is no contradiction in that.

Here is a finite definition of an infinite set: "A given set S is infinite iff there exists a bijective function between S and a proper subset of S." Furthermore, such a bijective function can be stated finitely.

Here is an example. Take the set of natural numbers ℕ = { 0, 1, ··· }. Now take a proper subset of ℕ containing only even the numbers, ℙ = { 0 , 2 , ··· }. These two are equinumerous because there is a bijective function f : ℕ → ℙ, given by f(n) = 2n.

The proof that "f" is bijective is finite. So is the proof that ℙ is a proper subset of ℕ.

One can talk about infinity conceptually, as one does in mathematics, without reference to its empirical verifiability.

When it comes to the empirical application of the concept of infinity, it is indeed reasonable to think that it is fundamentally unverifiable whether something is infinite. So we couldn't know whether spacetime is continous or discrete, because our measurements have finite resolution. The same would go for whether the Universe is infinite in extension or not.

However, humans can be quite ingenious, and we shouldn't rule out any possibility apriori, just from armchair thinking. Perhaps the supposition that the Universe is continuous rather than discrete has different consequences for our finite observations; I don't know. The same goes to the cosmological hypothesis where the Universe is infinite, which is thought to hold in case the Universe's matter density equals its dark matter density, a possibiity referred to as Ω = 1. (I'm pretty much quoting Wikipedia on the expansion of the Universe.)

Thanks for the warm welcome and the thoughtful reply

What is the proper interpretation of the cosmological constant Λ? I understand that it corresponds to a vacuum energy density, pervading all reality. Such energy is called dark energy, I gather. Since I'm sketchy on field theory, I don't know how this goes, but somehow this energy density produces a repulsive force beween any two objects in spacetime (within each other's lightcones?). Matter remains cohesive because Λ is very small compared to other forces, so that its effects really only show at an intergalactical scale (megaparsec).

Now, somehow this leads to the expansion of the Universe even in the case where the Universe is finite and bounded, which is a possibility considered by cosmologists. In this case, the Universe is increasing in total size, but not increasing *into* anywhere, so it becomes bigger because it has more internal spatial structure. This is what I meant. Why do you think this is incorrect?

But doesn't an expanding universe mean that this process is infinite and thus the universe itself is limitless? It is not much different than if we consider the universe as being static, in which case it can also be infinite. — Alkis Piskas

Infinitiness and expansion are independent. The Universe could be infinite and expanding; finite and expanding; infinite and static; and finite and static. All combinations are possible.

It is important to understand what it means to say that the Universe is expanding.

If the Universe is infinite, such expansion does not mean that the Universe is increasing in cardinality (set-theoretic size). Infinity is infinity (of a given size: aleph-0, aleph-1, aleph-2, and so on).

What the Universe's expansion means, whether it is infinite or not, is that its local energy density is decreasing. In other words, there is more spatial structure between each of its internal field excitations (particles, energy).

(Note that, if the Universe were infinite, its global density would remain constant. This is because global density would be calculated by dividing two infinite sets of the same cardinality.)

This touches a related point: the Universe is undergoing an internal expansion. It is not expanding *into* something. There is no space external to the Universe. The Universe is just acquiring more internal structure. The cosmological details are sure to be complicated and relevant (and I'm no cosmologist), but that is the general gist.

There is also a related point: the Universe could be finite but still unbounded, without an edge. It could just be twisted unto itself, as in a loop, like the surface of the Earth. On our planet, if you keep going North, you'll eventually just change hemispheres and start going South.

Suppose that the universe has infinite space, and let's also say that there is an infinite number of particles in this space. For there to be space between the particles, would that not make space a bigger infinity than the infinite number of particles in the infinite space?

It would not! Are you familiar with injective, surjective, and bijective functions?

Suppose there are two sets of objects, A and B, whose size (cardinality) we wish to compare. That is, we want to know which is bigger (or equal): size(A) or size(B)? This is also written as card(A) and card(B).

If there is an injective function f : A → B mapping the objects of A into the objects of B, then for every distinct object in A one can find a distinct object in B. This entails that size(A) ≤ size(B).

If there is a surjective function f : A → B, then for every distinct object of B one can find a distinct object in A. This entails that size(A) ≥ size(B).

If there is a bijective function f : A → B, this just means that "f" is both injective and surjective, so that there is a one-to-one correspondence between elements of A and elements of B. This entails that size(A) = size(B).

This is the toolkit that defines the notion of size (cardinality) in set theory, and it must be used to compare sizes among sets, including infinite sets.

So take the positive natural numbers ℕ = { 1, 2, ··· }, which is infinite. Also take the non-zero integers ℤ = { ··· –2, –1, +1, +2, ··· }, which is also infinite. I have excluded zero for convenience.

Now, ℤ might seem bigger than ℕ. However, one can construct a bijective map between the two, proving that they have in fact the same size. There are many such possible maps. Any one of them suffices.
→ One such map begins by mapping all odd numbers in ℕ to the positive numbers in ℤ, like so: 1 maps to +1, 3 maps to +2, 5 maps to +3, and so on. The mapping rule is: 2k+1 → k+1 (where k starts from 0).
→ It continues by mapping all even numbers in ℕ to the negative numbers in ℤ, like so: 2 maps to –1, 4 maps to –2, 6 maps to –3. The mapping rule is: 2k → –k (where k starts from 1).
This covers all numbers both in ℕ and in ℤ, so they have the same size.

Infinite sets have this weird property where one can rearrange their items in many different ways, leading to surprising one-to-one correspondences. This is well illustrated by Hilbert's hotel.

In the case you provided, we could have 100 units of empty space for every 1 particle, but both would still be equinumerous, for there would still be a map between them. Just map the first 100 particles to the first 100 units of space, then the next 100 particles to the next 100 spaces, and so on. (This is an abstract mapping: you are not in fact shuffling particles around.) You will never run out of particles to map to some unit of space. Every particle will be mapped somewhere; every spatial unit will be designated a particle. So they might well be infinities of the same size.

This would be different if the space in question were continuous (like ℝ). Since particles are discrete (countable), they have the same cardinality as ℕ, which is strictly less than that of ℝ.

It is not virtuous to be dismissive. I believe onlookers to our debate will agree.

As with most age-old philosophical questions, any answer to the problem of consciousness will be deeply counter-intuitive; otherwise, it wouldn't have resisted solution for so long.

For reasons we could debate, idealism, dualism, panpsychism, emergentism, and non-reductive physicalism all face serious issues in connecting qualia to the functional properties of physical objects.

Illusionism is deeply counter-intuitive in that it explains away what seems to be the most given; but that is not to be rejected apriori, but only upon theoretical and empirical reflection. There might be conceptual and empirical reason to think that qualia are incoherent posits. Here is an argument outline.

Our ability to perform conscious judgments are strongly connected to our brain processes. What happens to our brain affects our attention, object detection, object identification, object tracking, pattern detection, similarity judgment, distance judgment, duration perception, proprioception, and so on.

This is evidenced by perceptual impairments caused by brain damage, such as hemispatial neglect (seeing but ignoring objects without noticing), cortical blindness (unconscious seeing), visual anosognosia (denial of blindness), prosopagnosia (no detection of faces), akinetopsia (no detection of motion), mixed transcortical aphasia (where a person can sing but not talk), and the effects of psychedelics in perception, proprioception, ego fragmentation, and ego dissolution. The work of Oliver Sacks and the work of V. S. Ramachandran are very interesting in this regard.

From the above, some conclude that qualia are just brain processes (reductive physicalists), where others conclude that they are caused by brain processes (non-reductive physicalists, dualists), and still others believe that they partially constitute brain processes (panpsychists, dual-aspect monists, idealists). Either way, we must accept that the mind and the brain are deeply connected.

Having said so, here are some direct motivations for illusionism.

1. Consciousness seems unified, but it is not. Our brain processes are temporally and spatially distributed. There is no tiny interval in spacetime where our brain perceptual judgments coalesce so as to possibly form a unified conscious state. I like Dennett's multiple drafts hypothesis on this regard, which receives empirical support in his paper "Time and the observer" (cf. color phi phenomenon, cutaneous rabbit pheomenon). There is also something to say about the unity of consciousness when reflecting on split-brain patients; more on this in the succeeding item.

2. Our access to conscious states seems infallible, but it is not. Access to conscious states requires a physical process connecting qualia to memory, action, and speech, but such a physical connection coud aways fail. We could form false memories or simply forget what we just felt. We could feel something but not be able to think about it, act based upon it, or talk about it. This happens with split-brain patients: the right hemisphere is able to detect objects alright (and even draw them), but it cannot *talk* about it. What's worse, the right hemisphere does not notice that it cannot talk about anything. How does that conscious state (or "soul") function? Was the person's soul divided?

3. There is even an argument from the philosophy of time. The standard Minkowski interpretation of Einsteinian relativity in terms of a 4D spacetime seemingly entails eternalism – that there is no objective present and that time does not objectively pass. Reality is static; time is a static relation between static events; the flow of time is an illusion. Yet, conscious states seem intrinsically dynamic, although they are in fact static.

These statements show that conscious states might not be what they appear, contradicting Berkeley's principle "esse est percipi". And if there can be a partial cognitive illusion about qualia, why not a complete cognitive illusion?

Consciousness remains a mystery, for physicalists and non-physicalists alike.

To support my claim, I will reframe the problem of consciousness in the way I see it. The problem is to explain how qualia interact with non-qualia in a way that reflects its qualitative content. For example, why do aversive qualia (e.g. suffering) cause aversive physical reactions?

There are three possible solutions. One is to explain away qualia, as illusionists do. Another is to explaim away non-qualia, as idealists do. The third is to explain the bridge between qualia and non-qualia, as most people try to do (e.g. dual-aspect monists, panpsychists, orchestrated objective reduction theorists, information integration theorists).

For lack of a better alternative, I am drawn towards illusionism, which sees qualia as cognitive illusions. We are all in fact philosophical zombies, but our cognitive apparatuses couldn't possibly believe that on an intuitive level. Qualia are just our cognitive judgments about ourselves and the world around us. I find the arguments in Dennett's work elucidating in this aspect, although they are not decisive (cf. "Quining qualia" and "Time and the observer").

My view is that there might be no single concept of infinity. People talk about infinities using informal language, using mathematical language, and in the context of physics. If all coherent concepts of infinity turn out to be equivalent, this will be a surprise, and it must be demonstrated.

The statement that some infinities are bigger than others comes from set theory. The OP talks about infinities in the context of counting procedures. These are two different concepts of infinity.

In set theory, infinities are just infinite sets. In turn, infinite sets are those equinumerous to one of its proper subsets. That is, they have the same cardinality, which is defined by the existence of a bijective map between the two.

In set theory, an infinite set is bigger than another when there exists a surjective function from one to the other, but not vice-versa. What Cantor has proven is as simple as that: one cannot construct a surjective map from ℕ to ℝ.

Perhaps you would like to work with a different definition of cardinality, infinite set, or infinity (scraping sets altogether). That is fine, but keep in mind that you would be changing subjects, rather than disagreeing with set theory in general or Cantor in specific.