Gödel’s Infinite Candy Store

[Eugene I]

[Shajan624]

Consciousness having a hidden mathematical structure is an intriguing idea but I don’t see how Gödel’s theorem and Idealism can be connected. Incompleteness theorems applies to formal systems where the primitive symbols have no connection with the world of experience - the symbols are not ‘about anything’. Consciousness has to be about something so it is difficult to see how theorems of formal systems apply.

[Eugene I]

Consciousness having a hidden mathematical structure is an intriguing idea but I don’t see how Gödel’s theorem and Idealism can be connected. Incompleteness theorems applies to formal systems where the primitive symbols have no connection with the world of experience - the symbols are not ‘about anything’. Consciousness has to be about something so it is difficult to see how theorems of formal systems apply.

As Donald said, the formal is the "bones" that together with the "meat" of non-formal constitute the "body" of the totality of the experiential reality, so the totality is definitely not reducible to formal, yet there is a certain connection, or we can say, "inclusion".

[Cleric K]

Searching for answers through mathematics has fundamental limitation. To grasp this we must take a step backwards and appreciate the fact that first of all, doing mathematics is really doing thinking. At the moment we imagine that mathematical thoughts tell us in themselves something about reality, we are making a great presupposition. Implicitly we assume that reality is created by (conforms to) mathematical thoughts and we try to mirror the external mathematical reality process in our local math thoughts.

In the last few centuries this was justified approach but now we must go further. Instead of trying to make an intellectual model of the supposed reality through mathematical thoughts, we should turn our attention to the very intimate process of producing the thoughts in the first place. Why try to abstractly model (and thus avoid) that which we can directly investigate in its true nature, instead of confronting it directly?

Once our spiritual activity becomes the object of experience we begin to view formal mathematical systems as set of rules that tells us what is possible and what is forbidden to think. Consider this:



To put that in a simpler analogy, we can imagine an axiomatic mathematical system as rules for walking. A simple such system could be that we're allowed only to take one step at a time and turn at right angles. In this way we can quickly assess that the only reachable places we can step onto, form a kind of grid with one step size. This would correspond to the white tree above - the provable theorems. To prove a theorem means to find a combination of steps and turns which reaches at a specific spot (the theorem). Spots in between the grid are false theorems because it may (or it may not) be possible to show that we can never step there if we comply to the axiomatic rules. If we do step there we must have violated the rules.

This is a crude analogy, not taking everything into account. Now we must translate to thinking. To think within a formal system is similar to accepting some rules (like stepping and turning) but for our thinking. Yet our thinking is very flexible and we can continuously step out of the rules and encompass 'what we've been stepping through' from a higher level. This allows us to make a map of mathematical statements as the one above.

Today we're at a critical threshold of human development where we're bumping into this limit - where the intellectual thinking turns upon itself. Contemporary science and philosophy (even widespread spirituality) don't at all want to approach this point where thinking encounters itself. It is somewhat understandable - it's much more difficult to investigate something incessantly twisting and morphing. This is the great dilemma of the intellect. If it has to investigate itself in the way it feels comfortable with, it must deaden itself - it must freeze itself into immobile mineral forms which are convenient to look at. But this means that all thinking must cease! The other alternative - where thinking livingly experiences itself in mobility and constant metamorphosis is quite impossible to grasp in static concepts and thus it's considered unworthy for scientific exploration. Yet it is precisely there that we must look. We are indeed capable of beholding the mobile and living nature of thinking but we need concepts of another kind, which are fluid, living. Just as we can't learn to ride a bicycle by just holding on to abstract rules but must turn them into living, flowing experience, so the Imaginative experience of thinking is a skill that must be developed. Through it we begin to uncover higher order spiritual processes that ordinarily lie hidden behind the intellect. It's like the intellect is a result of a standing wave. Normally we are conscious only of the static points (corresponding to intellectual thoughts). The other parts of the 'vibrating medium' don't rise to consciousness. When we begin to glimpse into Imaginative consciousness a whole world of processes and beings becomes apparent - a world of unceasing metamorphosis. There, in the stationary nodes of this world we experience our ordinary ego with its rigid thoughts but now from a higher perspective of our true "I"-being, which we can't really say that we posses but it possesses us.

My whole point is that we should be quite careful when using mathematical conclusions for speculating about the nature of reality. We should never forget that after all we're exercising thinking in this way, and we are voluntarily locking ourselves within certain patterns and shapes of thinking spiritual activity.

Another problem with the Gödel's candy shop is that it implicitly assumes a certain fundamental character of time. It is indeed true that at our stage of evolution it is like we're only seeing a tiny aperture of the Spiritual potential at a given time. Yet it is a preconception that this aperture will always stay of the same size, so to speak, and will be able to probe only that much of the potential at a time. Even the most preliminary glimpses in the Imaginative realm already present us with the 'vertical' aspect of the potential and the fact that the aperture actually grows. This really changes the way we view Time. This aperture grows all the time through the integrative process of memory. In our ordinary consciousness we can only think about the memories but in Imaginative consciousness we can really see that Time-memory is a growth process and the past exists within the metamorphic organism that we have turned into. The more the evolution proceeds, the more the living aperture of our being encompasses the Eternal, which so to say inflows in our being. This is what DH dismisses - that there's a 'vertical' integration of potential. It's assumed that the potential is explored only 'horizontally' in bits as large as the aperture allows. But the forms of higher cognition available to us through the proper training, clearly present us with the vertical aspect. This actually makes the whole evolution much more profound. We not only explore the candies in a given horizontal plane but at certain stages we rise above and encompass as a whole the domains which previously we were forced to explore sequentially. Needless to say, this completely transforms our self-image. We see our ordinary self as being spread out in a labyrinth and how it was gradually cohering towards the point where it can see itself as proceeding from a higher order self. This latter part is the main obstacle in our age. People just don't want to even consider that there could be anything of higher order that lives behind their ordinary thoughts and feelings.

[Eugene I]

Cleric, we should extrapolate the Godel's theorem to the whole reality only metaphorically and as a simplified model/example. But the core idea is that the variety of ideal forms is inexhaustible and there is no end to their exploration and evolution, both into horizontal and vertical/hierarchical dimensions.

I'm still skeptical about the actual existence of eternity/infinity though, for me this is an undecidable question. I can have an idea of infinity/eternity, but I have no actual conscious experience of it, so for me it remains an abstraction. If you do have an actual experience of infinity then good for you

[AshvinP]

Cleric, we should extrapolate the Godel's theorem to the whole reality only metaphorically and as a simplified model/example. But the core idea is that the variety of ideal forms is inexhaustible and there is no end to their exploration and evolution, both into horizontal and vertical/hierarchical dimensions.

I'm still skeptical about the actual existence of eternity/infinity though, for me this is an undecidable question. I can have an idea of infinity/eternity, but I have no actual conscious experience of it, so for me it remains an abstraction. If you do have an actual experience of infinity then good for you

Eugene,

When you say "there is no end to their exploration and evolution", do you mean that every individual perspective will experience every ideal form in the course of its evolution?

Also. how can there be "no end", "inexhaustible", etc. yet also no eternal-infinite aspect to ideation? How can you have an idea of something which is not within the set of all potential inexhaustible ideation?

[Eugene I]

Eugene,

When you say "there is no end to their exploration and evolution", do you mean that every individual perspective will experience every ideal form in the course of its evolution?

Well, that's the point of the Godel "candy shop": it is impossible to ever experience every possible ideal form from either individual or overall/collective perspectives. The candy shop is in principle inexhaustible.

Also. how can there be "no end", "inexhaustible", etc. yet also no eternal-infinite aspect to ideation? How can you have an idea of something which is not within the set of all potential inexhaustible ideation?

May be Santeri will explain better The idea of the actual infinity is an abstraction, what we know from experience, as well as from computational mathematics, is only endless sequences of phenomena or events. To my knowledge nobody has ever experienced the whole infinity of events. For example, the sequence of the events of iterating addition x(n+1) = x(n) + 1 has no end, it never stops, however, the computational process will never reach the whole infinity of all x(n) numbers. So, every specific number x(n) actually exists because there is an algorithm (path) to reach to it (experience it). However, there is no algorithm/path to reach to the infinite wholeness of all numbers. In mathematical terms, the infinity of all numbers is incomputable, and similarly, the set of all ideal forms is un-experienceable. So, the set of all potential ideations is simply an abstraction, an idea, the actual set of all ideations it is not possible for us to experience, and therefore there is no experiential evidence that it actually exists.

Now, if you assume a hypothesis that the set of all ideation still actually exists, you will run into the Russel self-referencing paradox. The set of all ideations is obviously itself an ideation, so it must contain itself. But that is not possible and leads to contradiction, here is a simple math proof that goes like this (but you need to know the set theory to decode it):
Suppose there were a set U of all sets (for example, an ideation of all ideations). Let A be defined by A={S∈U|S∉S} Since A is a set, it's an element of U , and you can ask if A∈A. Following Russel's argument, you can show that A∈A if and only if A∉A. Since the assumption of the existence of a set U of all sets leads to a contradiction, therefore such a set doesn't exist.

[AshvinP]

Eugene,

When you say "there is no end to their exploration and evolution", do you mean that every individual perspective will experience every ideal form in the course of its evolution?

Well, that's the point of the Godel "candy shop": it is impossible to ever experience every possible ideal form from either individual or overall/collective perspectives. The candy shop is in principle inexhaustible.

Right, let me rephrase the question this way - will you or I, under the "candy shop" view, eventually get to experience all ideal forms that other perspectives have experienced?

Also. how can there be "no end", "inexhaustible", etc. yet also no eternal-infinite aspect to ideation? How can you have an idea of something which is not within the set of all potential inexhaustible ideation?

May be Santeri will explain better The idea of the actual infinity is an abstraction, what we know from experience, as well as from computational mathematics, is only endless sequences of phenomena or events. To my knowledge nobody has ever experienced the whole infinity of events. For example, the sequence of the events of iterating addition x(n+1) = x(n) + 1 has no end, it never stops, however, the computational process will never reach the whole infinity of all x(n) numbers. So, every specific number x(n) actually exists because there is an algorithm (path) to reach to it (experience it). However, there is no algorithm/path to reach to the infinite wholeness of all numbers. In mathematical terms, the infinity of all numbers is incomputable, and similarly, the set of all ideal forms is un-experienceable. So, the set of all potential ideations is simply an abstraction, an idea, the actual set of all ideations it is not possible for us to experience, and therefore there is no experiential evidence that it actually exists.

Now, if you assume a hypothesis that the set of all ideation still actually exists, you will run into the Russel self-referencing paradox. The set of all ideations is obviously itself an ideation, so it must contain itself. But that is not possible and leads to contradiction, here is a simple math proof that goes like this (but you need to know the set theory to decode it):
Suppose there were a set U of all sets (for example, an ideation of all ideations). Let A be defined by A={S∈U|S∉S} Since A is a set, it's an element of U , and you can ask if A∈A. Following Russel's argument, you can show that A∈A if and only if A∉A. Since the assumption of the existence of a set U of all sets leads to a contradiction, therefore such a set doesn't exist.

I am just referring to an apparent internal contradiction in your logic. You conclude that Gödel's candy shop theorem means Reality will unfold into inexhaustible, never-ending ideal forms. How can that conclusion be true if it is also true that "infinity" has no "actual existence"? I don't see how those can be logically reconciled.

[Eugene I]

I am just referring to an apparent internal contradiction in your logic. You conclude that Gödel's candy shop theorem means Reality will unfold into inexhaustible, never-ending ideal forms. How can that conclusion be true if it is also true that "infinity" has no "actual existence"? I don't see how those can be logically reconciled.

I gave you a simple illustration from mathematics: a never-ending iterative algorithm of adding +1 actually exists and you can compute and experience every possible iteration of it. According to this algorithm, the sequence of natural numbers is inexhaustible and the execution of the iterative algorithm is never ending (because you cannot reach a number x(n) to which you can not add 1 and compute a larger number x(n)+1). But the wholeness of infinity of all natural numbers in incomputable and un-experienceable and therefore there is no experiential evidence that it actually exists, because "actual existence" means something that can be experienced (or computed in mathematics). This is not to say that the infinity can not exist as an abstraction (idea) - you can think of an infinity, but you cannot actually experience or imagine the whole actual infinity of all natural numbers at once. And I remember you saying elsewhere "if something can not be experienced, then it does not exist".

The idea of actual infinity is akin to the idea of matter: you can have an idea of matter, but you can never actually experience it.

[AshvinP]

I am just referring to an apparent internal contradiction in your logic. You conclude that Gödel's candy shop theorem means Reality will unfold into inexhaustible, never-ending ideal forms. How can that conclusion be true if it is also true that "infinity" has no "actual existence"? I don't see how those can be logically reconciled.

I gave you a simple illustration from mathematics: a never-ending iterative algorithm of adding +1 actually exists and you can compute and experience every possible iteration of it. According to this algorithm, the sequence of natural numbers is inexhaustible and the execution of the iterative algorithm is never ending (because you cannot reach a number x(n) to which you can not add 1 and compute a larger number x(n)+1). But the wholeness of infinity of all natural numbers in incomputable and un-experienceable and therefore there is no experiential evidence that it actually exists, because "actual existence" means something that can be experienced (or computed in mathematics). This is not to say that the infinity can not exist as an abstraction (idea) - you can think of an infinity, but you cannot actually experience or imagine the whole actual infinity of all natural numbers at once. And I remember you saying elsewhere "if something can not be experienced, then it does not exist".

The idea of actual infinity is akin to the idea of matter: you can have an idea of matter, but you can never actually experience it.

How are you going from "incomputable" to "un-experiencable"? I don't think that really matters for my argument, but I am curious. Clearly we already have experiences which are not computable, as in a computer program could never replicate the entire gamut of experience. It seems to me Cleric's post was precisely to caution against equating what is "computable" with what is "experienceable". It would actually be rather terrifying if that were true.

I am trying to take a common sense approach here, without getting lost in the abstractions. Maybe someone else can comment on the mathematical arguments you are making. But it is ironic to me that you are using abstract mathematics to deny existence to the experience of what you call the "abstract" idea of infinity wholeness.

Sticking with experience - if your conclusion re: Godel theorem holds true, then is it not also true we will all actually experience an infinite number of ideal forms? If not, please explain why without resorting to mathematical abstractions. I also still would like to know your answer to this question, as I suspect it will be important:

Right, let me rephrase the question this way - will you or I, under the "candy shop" view, eventually get to experience all ideal forms that other perspectives have experienced?