To be honest I am enthusiastic about whatever I have learnt till now. Sometimes I am attracted to Topology, sometimes to abstract algebra and now with this discussion I have started liking this subject too. Why this happens?
@user400188 That sounds like practical application to me. It is widely believed by most proof theorists that ACA suffices for nearly all practical mathematics. =)
Note that the constructive incompleteness theorem (which we didn't discuss) shows that ACA can't prove its own consistency, which we should believe if we use it!
I think you will find it consistent with your belief that there is a unique reality and "it doesn't lie" in the sense of every sentence about reality is either true or false and not both.
Your question has two main facets. The first is that you did not grasp the way logic does not fall to the liar paradoxes. The second is that there are deeper reasons as to why we have such apparently innocuous sentences in natural language that seem to defy assimilation into formal logic systems....
@Secret <− And the statement you made here is also a fact that is not caused by anything.
@Secret This is possible, however it is not the simplest explanation. That is the reason for favouring the belief in existence of a conscious lawmaker.
For example, it is possible that your computer is a product of random putting together of parts.
Btw, I am ok to accept that it is a fact that A exists, but I am more happy if we can show it is the case, and not just believe it
More generally, if something has some property A (be it undecidable, true, false etc.) it is always nice we can show it, or show that we cannot did so either because it is the negation, or a limiting rule such as incompleteness theorem prevents it.
I am also ok with, known known, known unknown, and unknown unknowns, as long I am somehow aware it.
However... there exists one single concept that I detest a lot about this reality...
The only entity I hate (let's hope I actually wrote this properly because my logic kinda sucks) can be expressed as follows:
sorry, I accidentally gone off a tangent because the discussion reminds me the thing I hate the most (which has nothing to do with the discussion). Let's get back on topic then
Well, reality is probably more than any formal system can capture, otherwise we would have known all the unknown unknowns already
and unknown unknowns are precisely the exciting things that reality has
Unknown unknown is actually pretty weird if you look closely: It can have any truth value we can and cannot imagine, or no true value at all, or something more bizarre than words can capture, and we will not know until we make them known
@Secret I don't know about that, but formal systems cannot access reality even if it can capture it. For example even if you believe in an infinite world with classical mechanics, you still can't show that PA captures reality even if it does, because you need an interpretation mapping from sentences over PA to sentences about reality.
It is possible that some formal system captures reality but whatever the case we will never be able to non-circularly show that.
As for unknowns, you may be referring to quantum effects. It's a common idea that quantum mechanics has convincingly shown that the world is not deterministic, but that is not at all a valid conclusion. Note that Bell's inequality only rules out certain classes of local hidden variables, and only under the assumption that particles actually have classical quantities that we can measure.
In my opinion (not sure how many physicists share it) every particle is literally its wavefunction, not a point entity.
So Bell's inequality says nothing and it is entirely possible that everything is deterministic.
In that case there would be no sufficiently precise sentence that does not have a truth value. Here the phrase "sufficiently precise" is in fact very vague, but I can't help it.
And yet I do not believe that everything is deterministic, because we can't exclude the possibility of certain kinds of free will.
nah quantum is nothing compared to unknown unknowns. Unknown unknown is by definition, something we don't have ideas on until we discover them and make them known (e.g. can you imagine some stone age people talk about the internet)
The physics community agrees on that particles are not pointlike entities. For $\psi$-ontic interpretations, the wavefunction is a physical entity uniquely associate to any quantum state including particles. But there are also $\psi$-epistemic interpretations which said $\psi$ is just a summary of our knowledge about the quantum system
They mostly do, but they dared not to say anything about hard consciousness, though they believe they will eventually pin it down as some brain function
I'm also not sure how convincing it would be to you, but I personally believe that the same Occam's razor (choosing the simplest accurate explanation) points strongly towards the existence of free will.
Just considering the nature of consciousness may not be sufficient for this.
But suppose the very slim possibility that consciousness is nonphysical, then we need to start worry about when beliefs directly interact with each other as if they are individuals, it could be exciting but I am not sure (I actually have discussed about this with a philosopher in a NewScientist panal event back in August)
My viewpoint in general is that if the supernatural exists, our primary concern is not to (insert suitable sentence), but to figure out how we can coexist and work together
> Basically, we talk about conscious beings here. Don't ask me to define "conscious"; use your 'common sense'. We define a being to be moral iff it does not intentionally cause harm to moral beings, and minimizes harm to immoral beings. This is a recursive and not obviously well-founded definition, so it gives us conclusions in some cases and is inconclusive in others.
> Now "harm" is defined slightly differently in the two contexts. "harm to moral beings" is defined to be what those moral beings perceive to be harm to themselves. "harm to immoral beings" is defined to be what the being in question perceives to be harm to them. Namely, moral beings decide what is harm to them but immoral beings don't have that right.
> I welcome any probing or critique of my ethical framework, as I apply it in real life and wouldn't want to make ethical errors. Though rather tangential, I believe to a certain extent logical reasoning helps to dissect ethical frameworks for possible inconsistencies.
Notice that nowhere does my moral code involve the lawmaker that I personally believe exists!
However, from observation of the laws over the physical world I come to the conclusion that the lawmaker himself is moral according to the above definition.
Nevertheless there are tons of immoral beings and hence the lawmaker does not always get what he wants.
My bottomost line for morality (I have other standards of course) is as long reality don't bore me, I will consider it as moral
Btw, I remember in a philosophy society I am in in uni, I once talked about what I think universal evil might be like:
It is something that suppress other worldviews (beliefs, individuals, objects etc.) to oblivion, and it will continue to do so until it is the only existence left, then it destroys nothingness and finally itself
The moral standard I used when talking about this entity is deutonological ethics, which said every individual has the right to fullfill their purpose or task, and something is considered immoral if they are prevented from completing their purpose
@Secret I don't think that makes much sense. No entity can destroy itself without relying on some more fundamental entity or laws by which it triggers and achieves that destruction.
That is the whole point of the lawmaker argument; it shows clearly that ultimately there is no valid explanation for the laws than a lawmaker, and also it shows a clear asymmetry between stability and chaos.
@Secret No. Even in the computing world you know that is false. The program has to run on the hardware. That hardware is the thing that carries out the command.
That said, I think it makes sense to believe that the lawmaker can voluntarily restrict himself.
Not in an absolute way, but as a matter of choice.
That is one of the numerous reasons to believe in free will, because it seems the lawmaker did indeed restrict himself in order to permit other conscious beings to have free will.
Interesting, so metaphysically speaking (assuming there are no alien logic that is incompatible with the logic we humans used) then it is consistent for our universe to have a law that cannot be violated: Thalt shall not destroy itself
But anyway that conclusion doesn't follow. Just because one being cannot destroy itself does not imply that everyone else cannot.
As you stated, a program can very well erase itself, by relying on lower-level entities.
@Secret I'm not sure what "universal" here is intended to denote, but I believe that my moral code is close to the the unique framework that maximizes stability preservation in the human world.
The universal moral code that extends beyond the human world is simply that a moral being is one that maximizes stability preservation.
I believe the universal moral code more or less logically leads to the human-specific one when we give high priority to mental and emotional stability.
@Secret Aha. Yes I believe that the evidence quite clearly points to at least two opposing conscious beings, one being the lawmaker and one being a conscious being that tries to mess things up.
That's why everything at the lowest levels seem to be stability preserving, while a lot of things at slightly higher levels seem chaos inducing.
For instance the deepest physical laws seem to make it such that one needs to continuously expend energy to maintain instability.
Whereas if one expends zero energy then the entire system will settle into a (possibly new) stable state.
For the physical world one could use a rigorous definition of physical stability to make sense of what I'm saying. I'm not too sure how to pin down this vague notion for the non-physical world.
@LeakyNun You would encode the consistency of TC as a sentence about strings. If you carefully work through the section about encoding program execution as a string, you would be able to do this!
Well, for some systems, such as far from equlibrium system, there is simply more ways to make orderly things than randomise homogenous things, hence entropy production lead to emergent patterns (cf Belousov-Zhabotinsky chemical reaction)
and life itself, at least partly, is an entropy production machine
Which comes full circle to one of my first few comments that I don't believe it's necessary for entropy to continually increase to have conscious life with free will.
My original idea of universal evil (which is now debunked by the lawmaker argument) is partly modelled based on the heat death, the concept of nothingness and then make it eternal. Imagine a heat death where no quantum fluctuation exists, thus no possibility to restart back into a lively state and cause another big bang
This is one example I commonly quote to my friends and tell them no matter what, don't let your future kids to get involve in bullying, because it will permanently scar your mind
My mind is scarred in that I actually really want to simultaneously destroy the world, to observe the world, and to maintain the world
@Secret You can turn your thoughts about destroying the world to good use; namely you can devise ways to avert the methods of destruction that you can foresee.
and there are neuroscience researches showed how bully victim and bullies have permanent changed in their brain structure, so... bullying is one of those things which everyone loses
They actually pop up together, and in fact, a lot of my personality traits are synergistic relations between my dark and light personalities
For example, my dark side desires to destroy and torture those who wronged me allows me to came up with very unusual monster barring techniques that allow me to explore mathematical structures in interesting ways
I wouldn't say you have a dark personality; that may be a bad way to view yourself that causes you to be unable to become completely free from the effects of the past.
Everybody has a dark and light side. what is important is how not to be consumed by the darkness and learn how to face both and (ideally) make them work together
For example, a lot of weird ideas that Leaky and the main knew about me stems from the dark side, but with the actual darkness harming stripped bare, leaving behind the neutral part
It is true that everyone does some bad things, but if you look at the real desires alone, some people's deepest desire is for a perfect world in which no immorality resides.
That's true. As for my deepest desires, it is a neutral one: To mix ideas and different thinking, generate new ideas, and not get bored
That's what I called Idea Mixing, and it can arguably be called the most powerful part of my light side, as it allows me to naturally acquire a lot of traits that the society said a good person should have
such as teamwork, communication, open mindness etc.
Or more precisely, most of my good personality traits are a byproduct of pursuing this ideal
You can thought through some of them, but there are many which you cannot anticipate until when it happens. And when it does, it is a split second decision. I'll let my intuition (and whatever left of my rationality at those points) to deal with that one
Anyway I don't wish to push you in any direction unless you are personally convinced that it is correct. And even then, it doesn't exclude the possibility of changing one's mind in the future.
< 2011: Scientism 2011: First time truely open up to interpersonal emotions 2014: Agnostic 2015: More artistic and intuitive thinking opens up 2016: Esoterics, and how not to get in trouble with the sciences
(and tbh I have not read much about it, so its true nature I don't understood)
I will be able to talk more about charkras, crystals auras, buddhist spin offs, newage, quanutm mysticism and others (and I know how not to mix them with science) than scientology or satanism
@Secret There is one more curious thing I thought about when you mentioned this. The whole underlying framework of science is based on physical laws being there and remaining the same (stable world).
Rather, if the supernatural exists, then since it is so hard to reproduce it, science can say nothing about it and thus excluded from science, so science naturally filters only the phenomenon that are stable in some sense
@Secret This is why I defer to logic instead of empirical science when discussing non-physical issues. Because the argument for one reality is sufficient to justify classical logic and hence we can use it for reality even regarding the non-physical aspects.
Since I am agnostic, I do not deny nor confirm the existence of the supernatural, but when working with scientists, unless we are having a chat on worldbuilding and philosophy, supernaturals will be out of the discussion.
The logical viewpoint does not readily came to me, partly because I am so bad at logic until in recent years when you and leaky and others trained me from the discussions
@LeakyNun: Just so that I don't forget, all the examples I gave you for sentences about strings that TC cannot prove are actually also unprovable in TC+induction.
The reason is that TC+induction is as strong as PA. TC itself is as strong as PA−. "As strong as" meaning that there is a computable translation from one to the other that preserves provability.
So if you want an even simpler one that TC cannot prove but TC+induction can, just choose some suitably hard instance of the induction axiom.
If you recall we previously discussed how to define unary.
Then ∀x ( unary(x) ⇒ ∃y ( x=y+y ∨ x=y+y+"1" ) ) cannot be proven by TC! (I think!)
I will sketch how to prove that it cannot be proven. Remember we had this model of TC that I called CBLLO (countable binary labelled linear orders)?
A countable binary labelled linear order (CBLLO) is a relation < on a subset S of N that satisfies reflexivity and transitivity and anti-symmetry plus a binary labelling of S (namely a function from S to {0,1}). Let C be the set of all CBLLOs. Given any X,Y in C, we define X~Y iff there is an isomorphism from X to Y (namely a bijection f from X to Y such that for any a,b in X we have f(a).label = a.label and ( f(a) < f(b) iff a < b )). Then ~ is an equivalence relation on C.
Given any G,H in C/~ define G+H = { X+Y : X/~ = G and Y/~ = H }. Then you can prove that G+H is a member of C/~. So now L is C/~ with the + we have just defined on it.
If you're lazy to think in terms of equivalence classes (like me), then just think in terms of the original CBLLOs and show that the axioms hold if you interpret "=" as "~", and then apply the general theorem that you can mod out ~ to get a first-order model as long as + respects ~.
CBLLO does not satisfy cancellation, namely the sentence "∀x,y,c ( x+c = y+c ⇒ x=y )". And I just realized that cancellation can be proven by TC+induction. So there you have another concrete example!
But CBLLO also does not satisfy ∀x ( unary(x) ⇒ ∃y ( x=y+y ∨ x=y+y+"1" ) ), because the linear order W (for "weird") on the rationals in (0,1] ⋃ {2} ⋃ [3,4) all labelled with a "1" is a counter-example.
Namely, in CBLLO we have unary(W) is true but ¬∃y ( W=y+y ∨ W=y+y+"1" ).
The reason is that any such y must have a gap (two elements with nothing in-between), and W has exactly 2 gaps, so y must have exactly one gap. But then you can check the cases and both fail.
Though it's true that ∃y ( W=y+"1"+y ).
@LeakyNun: Interesting? But I shall really be off now. We can continue discussing TC next time. =)
