Irrational Numbers And Reality As A Simulation
I must warn you that what I have to say in this post is probably not as well thought out as I'd have liked it to be as I'm neither a mathematician nor a coder but my intention isn't really to prove a point as much as it is to explore the topic which I will make clear in the following paragraphs.
As far as I'm aware, the idea that reality could be an illusion has been around since Descartes' deus deceptor. In the modern computer age, this theory has morphed into what we call simulated reality, the gist of this being that what we experience as reality could be generated by a sufficiently complex program, similar to, although more advanced than, the ones we use on a daily basis, run on a powerful enough computer.
For this theory of reality being a simulation to fly, it's necessary that the program that codes the simulation be finite for if not the program can't be completed/finished let alone executed on a computer.
In light of the above necessary truth, consider the matter of irrational numbers. We know that,
1. Irrational numbers exist
2. Irrational numbers have an infinite decimal expansion
3. There's no repetition of number sequences in irrational numbers
I'm somewhat aware that some irrational numbers can be calculated using a formula e.g. square roots of prime numbers can be calculated with a formula and a formula can easily be encoded in a program. However, there are some irrational numbers that can't be calculated with a formula. Such numbers, satisfy the three conditions I mentioned above but that means a program that codes for such numbers would require a separate line of code for each digit in such numbers as the digits aren't reducible to a pattern that can be encoded with a loop statement in the code. Plus, the digits in such numbers are infinite which means that such numbers would require an infinite number of statements in the code but if that's the case, the program, as I said earlier, can't be completed and so would never be actually run on any computer.
To sum up, the existence of irrational numbers that aren't formula-based proves that the reality we're living in isn't a simulation because the program required to encode for them would have to be infinite.
If I've made any errors please cut me some slack and redirect your attention to the notion of patternless infinities, like irrational numbers that aren't formula-friendly, as pertains to reality being a computer simulation for the simple reason that a program that encodes patternless infinities can never be finished as each element in such infinities will require a separate line of code and that translates into a program that has to be infinite. An infinite program can't be completed so can't be run. Reality, given the existence of patternless infinities, can't be a simulation.
As far as I'm aware, the idea that reality could be an illusion has been around since Descartes' deus deceptor. In the modern computer age, this theory has morphed into what we call simulated reality, the gist of this being that what we experience as reality could be generated by a sufficiently complex program, similar to, although more advanced than, the ones we use on a daily basis, run on a powerful enough computer.
For this theory of reality being a simulation to fly, it's necessary that the program that codes the simulation be finite for if not the program can't be completed/finished let alone executed on a computer.
In light of the above necessary truth, consider the matter of irrational numbers. We know that,
1. Irrational numbers exist
2. Irrational numbers have an infinite decimal expansion
3. There's no repetition of number sequences in irrational numbers
I'm somewhat aware that some irrational numbers can be calculated using a formula e.g. square roots of prime numbers can be calculated with a formula and a formula can easily be encoded in a program. However, there are some irrational numbers that can't be calculated with a formula. Such numbers, satisfy the three conditions I mentioned above but that means a program that codes for such numbers would require a separate line of code for each digit in such numbers as the digits aren't reducible to a pattern that can be encoded with a loop statement in the code. Plus, the digits in such numbers are infinite which means that such numbers would require an infinite number of statements in the code but if that's the case, the program, as I said earlier, can't be completed and so would never be actually run on any computer.
To sum up, the existence of irrational numbers that aren't formula-based proves that the reality we're living in isn't a simulation because the program required to encode for them would have to be infinite.
If I've made any errors please cut me some slack and redirect your attention to the notion of patternless infinities, like irrational numbers that aren't formula-friendly, as pertains to reality being a computer simulation for the simple reason that a program that encodes patternless infinities can never be finished as each element in such infinities will require a separate line of code and that translates into a program that has to be infinite. An infinite program can't be completed so can't be run. Reality, given the existence of patternless infinities, can't be a simulation.
Comments (51)
Good post. I especially appreciate your pointing out that some irrationals like sqrt(2) or pi are computable and actually encode only a finite amount of information.
Now your point would stand IF (big if) you can demonstrate that any noncomputable real number is instantiated in nature. Till you can do that (and you can't, nobody can), you have no argument. I don't happen to believe the universe is a computer, but I still can't endorse your argument that would seemingly support my belief. Because for all we know, noncomputable real numbers are nothing more than an artifact of our system of mathematics. Constructive mathematicians don't even believe in them.
As an analogy, a story about Pegasus, the flying horse, does not show that our theories of biology are wrong. Rather, Pegasus exists only in fiction; as do, according to some, noncomputable numbers.
That doesn't refute my argument for everything, including fiction, has to be coded if reality is a simulation and if noncomputable irrational numbers exist in fiction, that too requires to be coded and we run into the same problem of a program that's got to be infinite in size and that means it'll never be finished/completed and so can't be compiled/translated into an executable file. Reality can't be a simulation.
The fact that the mathematical existence of noncomputable numbers follows from the rules of standard math, doesn't imply that any noncomputable process is instantiated in the real world.
This is priceless what you wrote. Its both good and bad news. "The foolishness of God is greater than the wizdom of Men". Keep up the good fight. I believe the Jade King is related to the Man associated with 444. Berylium is the 4th element on the periodic table. He is somewhat the opposite of Superman but only in that sense. I doubt he is white but he could be. He was physically ugly according to the Holy book i subscribe too.
adding to my Journal.
I'm just going to focus on this little part of your post as I think the complex thinking you've performed can be explained more simply in the following way: How can you even say "reality could be simulated by a sufficiently complex program". How can that sentence make any sense? The only reality we have to rely upon in conceiving something like a program is the material universe that we ourselves are dynamically emergent from. Everything "informational" about what we are are basically "intra-actions" within a world composed of material objects. The information in our "heads" (i.e. perceptions, cognitions) is itself emergent, or dynamically contiguous with (in a neurological sense - the later and more evolved structures distant from the brainstem regulate the structures beneath it; these structures are not simply physical interactions, but also semiotic events between a physical system and the objects in the environment that either upregulate (toxin) or downregulate (nutrients) the homeostasis of the organism) the regulating rhythms of the physical organism itself. Feelings are these informational 'traces' that the higher level reflective mind is interfacing with. First something is felt; then its reflected upon; then its augmented within reflection by a focused deliberation on the objects significance. Now, how or where within this general stream is there an epistemological basis provided for the statement "reality could be simulated a sufficiently complex program". Where would this program be? What would it be made out of? How could anyone claim to know of anything existing that is not material in origin that can create a simulated reality? It boggles my mind how someone can make a claim like this, and not realize how poorly explicated it all is.
What is being assumed here? That a physical computer can generate a "universe" within the computer system itself? And that from within that universe (the computer) a universe could be created that looks like our universe? That is an extravagantly loaded set of assumptions to make about the world. Think about that means in terms of the potential infinity of it. If the universe is a computer simulation, then that means within the universe that the computer exists is itself caught up in its own simulation, ad infinitum. At which point does someone finally admit, "this is too stupid a claim to take seriously because it fails to evoke the sort of feelings that normally make someone take existence as a serious subject."
Feeling wise, nothing in me is released or produced by the slipshod idea that I'm in a computer program, because the philosophical dead end it creates (infinity) necessitates a being in another world controlling the program. It fails to accept that there is an unknown metaphysical principle which exceeds all human contemplation, and that whatever humans can known about this Other, all we have to rely upon are the the forms of the world itself. This is the typical and anthropologically common way human beings experience Nature: as a metaphor of some vaster Being. The computer idea fails to evoke the sense of magical "participation mystique" that the normal human relationship between self and world evokes.
What you seem to be underemphasizing is the possibility that the idea of a computer simulated reality is really your bodies need to make sense of this existential awareness of self and being but within the terms of what you presently value as a self i.e. according to the logic of computers and programming. In this situation, you have naively failed to realize the deeper reality of metaphor, itself reflecting the ontological situation of a superior Being communicating Itself to a being within itself. Why is this idea typically spurned? Because the traumas and pains - particularly the myriad times you've felt shame in your social existence as a self - that have occurred within your development as a person prevents you from experiencing the connection i.e. the feeling, that ordinarily exists between the self aware organism and the universe itself.
Green eyes are sometimes associated with witch craft but green eyes actually appears among all races. Merit goes far deeper than the color of someone's eyes.
What do you mean by instantiation? Remember that everything, E-V-E-R-Y-T-H-I-N-G, has to be simulated if the universe is and irrational numbers that are non-computable do exist in the same sense as the numbers 1, 2, 0, 1/2, 0.3333... exist and so must be coded in but that's not possible for to do that would require a program of infinite size that for that reason can't be finished. If so how can the program be executed? The universe is not a simulation.
Quoting Semiotic
Haven't you played video games? Haven't you heard of sim games? Video games are simulated realities and if you pay close attention, the do a mighty fine job of capturing real-world physics. Simulating the universe isn't, as far as I can tell, a question of IF but only of WHEN. Have look at what Nick Bostrom (1973 - ) has to say about reality being a simulation.
Quoting Semiotic
I don't get why you're putting up such a resistance to what is an idea that's both old and pops up regularly in philosophical discussions: Plato's allegory of the cave, Zhuangzhi's dream argument, Descartes' deus deceptor, the brain in a vat thought experiment; simulated reality is nothing but the modern incarnation of this nearly 2000 year old idea. It's still alive and well for the simple reason that people haven't been able to refute it and that says a lot in my world.
Quoting Semiotic
Google Nick Bostrom's trilemma and take a second look at what I've said. I've been looking around for good movies that are coming out in 2021 and The Matrix 4 is scheduled for release - that got me thinking about this whole simulated reality idea. If you give it some thought, the possibility that reality is a simulation actually provides more mystique than knowing reality is a WYSIWYG deal. The sense of mystery, the possibility that there's more to this universe than what we perceive through our senses and analyze with our minds, are central themes of all human endeavors that revolve around the pressing matter of the meaning of life which encompasses the relationship between us and the universe at large.
Quoting Semiotic
I don't know what you're on about. All I can say is if reality were nothing more than what we currently know or think we know it is then where's the fun in that? Unfortunately, it seems, for someone who's extremely fond of mysteries and hidden secrets, I've shot myself in the foot by proving, in my own small way, that there's nothing beyond what we're immediately aware of. So :sad: Reality isn't a simulation and there's nothing behind the curtains if there are any curtains at all.
It's funny that you talk of metaphors and then dismiss the idea of a simulated reality because if reality were a simulation you'd expect more metaphors and more interesting ones at that - coders are known to leave clues to their identity in hidden rooms, secret levels, easter eggs, and whatnot.
:ok:
Why? You seem to assume that whatever meta-reality "programs" our reality is subject to the same laws and processes that occur in our world. Perhaps our notion of time does not exist there, nor the physical laws of our universe. In that case your argument concerning the irrationals is meaningless. Just a thought. :chin:
The ‘illusory nature of reality’ has a very long history in philosophy Eastern and Western. But what your post is completely missing, is that this has a meta-cognitive and meta-ethical dimension. What I mean by that, is that, for instance, in Indic religions, the illusory nature of existence - m?y?, in Hindu systems, or Sa?s?ra, in the Buddhist world - is caused primarily, first and foremost, by avidya, which is normally translated as ‘ignorance’. However that will invariably prompt the question ‘ignorance of what? The second law of thermodynamics? Big Bang cosmology?’
What this misses is the meta-ethical meaning of ‘illusion’. In Eastern religions, beings are trapped in the cycle of sa?s?ra because of craving. And they crave the delusory objects of sensory experience because of avidya - which is ignorance, not of the periodic table, or the Big Bang theory of cosmology, but of (in one illustrative example) their ‘original nature’, which is ‘primordially blissful, pure, free, and not subject to death’ in one of the formulaic expressions.
Turning to Greek philosophy, you find in that tradition a much greater emphasis on the rational intellect (‘nous’) which is ‘that which grasps the eternal forms of things’. But you still find in the Parmenides, for instance, a cryptic allusion to the notion that Parmenides, ‘the sage’, voyages to where ‘the goddess resides in a well-known mythological space: where Night and Day have their meeting place.’ I take this to be a reference to the non-dualism that was also characteristic of Indian philosophy in the ‘axial age’ although as noted the Greeks placed much greater emphasis on reason and natural science. (This is how come we have these neat computer thingies. There’s a wonderful and under-appreciated book on the relationship of Ancient Greek and Indian philosophy called the Shape of Ancient Thought, Thomas McEvilly.)
But in all of these cases the key point is, being lost in the illusion of the world is a situation of moral blightedness or blindness, of not seeing ‘what truly is’. That is what I think many of these ruminations on ‘simulated worlds’ are not seeing. Why? Because suffering is real. It’s not on the big screen, it has real blood, people actually die. And I think the reason that movies like Matrix, Inception, and so on, are so compelling, is that they evoke the awareness of the possible illusory nature of what we normally understand as reality, albeit unconsciously. So awareness of the human plight, of being trapped in the round of Sa?s?ra, is depicted cinematically in these sci-fi stories. Just like the way that science fiction stories of interstellar conquest represent our sublimated longing for Heaven. But don’t think just because we have computers, we really understand the depths of the situation that is being posited as ‘a simulation’.
When I saw Matrix with my then-teenage sons, I found the red pill/blue pill scene almost blasphemous. Why? Because I thought, and still think, this touched on something of profound importance, and the cavalier way those brothers who made the Matrix treated it, for $Hollywood$, really gave me the shits, to be honest.
These ideas as subversive, loaded, they can be liberating, but they can also be the exact opposite. Understanding what is a metaphor for what, what is real and what is projection, in this cyber-age of a trillion screens - never has the promise of liberation been at once so immediate, and so far away.
Yes, but a code that simulates reality has to be finite. It's not just a spatio-temporal matter, it has to do with the nature of infinite randomness as something that can't be contained within a finite number of steps, and programs will consist of step-by-step rendering of the simulation.
Hi. I like the topic you've picked. I have two responses.
The first is Cipher's response. To me it doesn't much matter if my everyday reality is called a simulation or not. Pleasure and pain as I know them, the things I value, 'real' or not, just are what they are. I'm not offended by the idea that it's a simulation, but the question (as always?) is what does that really mean?
The second response is more technical. The so-called 'existence' of non-computable numbers seems to be a kind of fictional/conventional existence within a particular domain. What do we mean by 'existence' and 'infinite'? Within the game the players know well enough to keep the game going, but what are we to make of these tokens removed from that semi-controlled original context?
Great point. If this realm is fiction, then perhaps our math, physics, and biology (and so on) is just more worldbuilding.
Quoting Wayfarer
Interesting take on the reality as a simulation theory. Though thematically The Matrix movies and the, as you mentioned, philosophical notion of reality as an illusion are more or less identical, there's a subtle difference between Plato's allegory of the cave, the Buddhist Maya and The Matrix movies. In the case of the former, the illusion is the bad guy and we're advised to move away from it towards the light so to speak as if to say that knowing true reality will be a panacea for all our misery. In the case of the latter - The Matrix movies - this, what is a Platonic ideal, is turned on its head and the illusion of living normal lives in a simulated world's good, nay, far better than reality as living batteries for AI overlords.
Read f64's response below and you'll get an idea of how people might, after catching a glimpse of the real world chockablock with what most people know as "the hard facts of life" or what my father calls "bitter truths", come scurrying back to their AI masters begging to be plugged back into The Matrix.
Quoting f64
Here's what I suppose will be what most people will opt for in descending order of preference:
1. Real + happiness
2. Simulated + happiness
3. Simulated + suffering
4. Real + suffering
If given a full-option offer, people will chose the real over a simulation provided that in both cases the same level of happiness is guaranteed. If the first choice is taken away, people will happily choose a simulated reality [this is what I suspect Cypher/Cipher is going through]. Neo, Morpheus, and the rest of the human underground resistance chose 4 only because their victory is a gateway to 1. Had, option 1 been precluded for whatever reason, almost everyone would go for option 2 and ask to be reconnected to The Matrix.
Quoting f64
If say x, an non-computable irrational number, exists, I mean, limiting myself to the current domain of discourse, that it has the same ontolological status as, say, the number 2 or the square root of 2 or pi or e. If reality is a simulation, there should be some lines in the code that describe these numbers, these lines being executed to render the number to us in full detail.
The problem is numbers like x are patternless random infinities insofar as their digits matter. There's no pattern so the lines in the simulation code can't be a short, compact formula. The digits are infinite and so, in light of the preceding observation, there has to be infinite lines in the code, each line for each random digit. Thus, randomness and infinity, properties of numbers like x, will need a program of infinite length, length being a function of the number of instruction lines in the program. Being infinite, such a program can't be completed and if it can't be completed, it can't be run. Since numbers like x exist in our world, at least in the mathematical universe, reality can't be a simulation.
The latter are all computable and encode only a finite amount of information. In fact that's exactly why you can name and identify specific ones.
Can you name or identify any specific noncomputable number? If not, then you're wrong that they have the same ontological status as computable numbers. In this regard I find agreement with the constructivists. A number that requires an infinite amount of information to specify has a weaker ontological status than one that only requires a finite amount of information. Even you agree with this point. If you claim noncomputatlble numbers exist, name one.
Of course noncomputable numbers have mathematical existence in that we can prove (given the standard rules of math) that they exist; but that's only an existence proof that gives no clue of how to find one. That is exactly the constructivists' complaint.
I like your spiritual math here. Clearly there's something in us humans (or most of us) that thirsts for the 'real.' The first matrix was a 'utopia,' but the humans kept waking up. Why? Because humans thirst for conflict, drama, the 'real.' The Matrix is a film that would have been shown within the matrix. The idea that it's all a simulation has a kind of sexy violence. 'This is all a dream, all an illusion.' As you mentioned, this is an ancient thought. Maybe it's the philosophical thought.
Yes, I think I understand the argument, and it's a fascinating point. An noncomputable real number contains an infinite amount of information. Fair enough. But this result depends on various human conventions. So what is the ontological status of such a number? As another poster has mentioned, other mathematical conventions are possible for which such numbers do not exist. All you need is a group of people to set up some rules, control who gets funding, who gets published, etc., and you have yourself a version of mathematics. (So the weak part of your argument in my eyes is that it takes a particular human conception as absolute.)
Suppose the following is the complete list of computable irrational numbers between e and pi
2.71828...[e]
2.71829...
2.71832...
.
.
.
3.14158...
3.14159...[pi]
Using Cantor's diagonal argument I can show that there's a number not on this list x such that e < x < pi. In other words there exists a non-computable irrational number between e and pi, existing in the same sense as e or pi.
Now that I think about it, I believe an infinite random sequence of numbers can be generated using a simple algorithm:
1. Display v [a number, any number]
2. Calculate character length of display = c
3. Change one/all digits in the display of character length c and assign it to v
4. Go to 1
Note: After the first display operation for v, subsequent v's are attached to the previous v. So if the first v = 2, the second v = 23, the third v = 2345 or 2325, the fourth v = 23451267 or v = 23251246, ad infinitum.
And that's as far as I managed to get...comments?!
It's much simpler to show that there are uncomputable numbers in [e,pi] (neglecting some techincal issues with your proof.) The measure of the computable subset C of [e,pi] is 0, so the measure of the rest of [e,pi] is pi - e > 0. So there are uncountably many uncomputables in [e,pi].
But behind this argument is mainstream measure theory and everything it is built on. You say 'existing in the same sense as e and pi.' Well, yes. But how do they exist? Like pieces in a game. There are certain rules that allow to put new pieces on the board. Your argument might work for mathematical platonists...or on anyone who thinks of math as a kind of ultimate physics.
You gave an existence proof without naming any specific noncomputable number. And in order to do so you needed a cardinality or a measure theoretic argument, neither of which are physically meaningful.
The point is that a number whose existence is shown only through an existence proof has a lesser claim on mathematical existence than one one built by construction.
Of course his post is finite so it's not likely that he's specified any particular noncomputable real. But the larger point is that a number that encodes an infinite amount of information has a lesser claim to mathematical existence than one that encodes only a finite amount of information.
And either way, mathematical existence is not physical existence, A computer could put in our minds the idea of a flying horse, Captain Ahab, Captain Kirk, and noncomputable numbers. But since those things don't exist in the physical world, they are not evidence that the world is not a computer.
Of course numbers in general are abstract and even fractions like 2/3 are not instantiated in the world. You can't measure 2/3 of anything, unless you're going to refer to a quark of 2/3 spin or charge. In which case I'll just use 3/4 as an example of a number whose mathematical existence is on solid ground but whose physical existence is doubtful.
Please remember that all physical measurement is approximate. Even the positive integers are murky. I can show you three oranges or three planets but I can't show you the number three. Numbers have only abstract existence; so the mathematical existence of any type of esoteric number can never tell us anything about the physical world.
ps -- Just to anticipate @TheMadFool's objection: Just as a computer could put in our minds the idea that the world is or isn't a computation; why couldn't it put in our minds the idea that noncomputable numbers might or might not exist? I truly don't follow your argument that just because the great computer in the sky puts some contradictory idea in our head, that this is evidence of the nonexistence of the great computer. After all, God created atheists!
Quoting TheMadFool
Fine, name one. All you have is an existence proof; and an existence proof is a weaker class of metaphysical existence than a constructive proof like showing that 2/3 or pi exists.
Quoting TheMadFool
You mean an infinite decimal representation of a number? I'm afraid I didn't follow your algorithm at all. Perhaps you can give an example or explain it more clearly.
Quoting TheMadFool
As your final number is the output of an algorithm, it's surely not noncomputable. Though I'm not sure I really understand the details of your idea. Would like to see a more clear exposition.
But if you are generating a number from an algorithm, you haven't generated a noncomputable.
Let me leave you with an interesting example.
Suppose someone claims they have the following algorithmic procedure to generate a noncomputable number.
* Enumerate the computable numbers. We can do this because they are countably infinite.
* Form the antidiagonal according to a deterministic rule. Replace each digit n with n+1 (mod 10). That is, 0 is replaced by 1, 1 is replace by 2, ..., and 9 is replaced by 0.
Since we have enumerated all the computable numbers, and the antidiagonal is not on the list,
we have seemingly devised a perfectly deterministic procedure that has generated a noncomputable number!
What is the flaw? It's subtle. There is an enumeration of the computable numbers; but there is no computable enumeration of the computable numbers! That is, by a cardinality argument there is a bijection from the positive integers to the computable numbers. But that bijection can not itself be computable! Why not? Because to form such a list we have to look at every Turing machine and generate each digit of the corresponsing computable number by successively inputting 0, 1, 2, 3, ... and seeing what digits it outputs. But how do we know which TMs will halt and which will loop or go on forever without outputting a digit? We can't, because the Halting problem is unsolvable. Turing worked this out in 1936
There is no computable function that enumerates all and only those TMs that halt. So there is no computable enumeration of the computable numbers. And our "deterministic" generation of a noncomputable number doesn't work.
Finally: If you want to prove that we are not computations, all you have to do is figure out how to solve the Halting problem. We already know that no computation can solve it. If a human can, then we are not computations. The problem is that nobody's ever figured out how to solve the Halting problem.
This idea is intimately related to Gödel's first incompleteness theorem. No mechanical procedure can determine all mathematical truth. Penrose thinks this shows that we are not computations. Nobody buys his argument; but everyone agrees that Penrose's bad ideas are better than most people's good ones.
What do you mean? Any point on the well-known number line exist in the same metaphysical sense as another point. There's e and there's pi and there's a non-computable irrational number between them which is a point on the number line. Are you saying some points on the number line are different from other points on the number line? If yes, never heard that before but that's probably just me. Care to clarify your metaphysical objection?
Quoting fishfry
The algorithm I posted is just something that popped into my mind and isn't one that's ready for prime time as they say. What's the problem with it though? It's got only 4 instructions.
Let's go over it together.
Assume that a substitition-cipher-like process is involved and 0 is substitited with 9, 1 with 8, 2 with 7, 3 with 6 , and 4 with 5
1. The first step is to print a number n [e.g. 29]
2. The second step is to find how many digits n has, say it has d digits [d = 2]
3. The third step is to create a d digit number with all digits substituted/changed from n and assign it to n [n = 70 as 2 is replaced with 7, 0 is swapped with 9]
4. Go to 1
The first iteration of this algorithm using the examples I gave will print 2970
The second iteration would look like this: 29707029
The third iteration would look like this: 2970702970292970
The fourth iteration would like this :29707029702929707029297029707029
Is there are repetition in the sequence of digits? No. So, no pattern
Is the sequence random?
This is a difficult question for me to answer but here's what I think:
(i). If only four digits (0, 2, 7, 9) are being considered, the sequence is random as each digit appears the same number of times as the other digits, making their appearance in the sequence equiprobable (that's randomness right)
However,
(ii). If we consider all 10 digits available to us (0, 1, 2, 3, 4, 5, 6, 7, 8, 9), the digits are not random; only 0, 2, 7, 9 make an appearance
As I admitted at the outset, I'm neither a coder nor a mathematician so kindly cut me some slack.
Quoting fishfry
If you notice the algorithm isn't mathematical. It's more like a cipher but I don't know whether that, in itself, suffices to make the output of the algorithm non-computable. It is irrational thought as the digits are infinite and don't repeat.
Quoting fishfry
I found this on wikipedia:
[quote=wikipedia]While the set of real numbers is uncountable, the set of computable numbers is classically countable and thus almost all real numbers are not computable[/quote]
:chin:
Quoting fishfry
I don't know to whom I said this to but I'll say it again for your benefit: E-V-E-R-Y-T-H-I-N-G is a simulation if reality is a simulation and that non-computable irrational numbers exist in some space (mental/platonic/mathematical, you decide), it must be accounted for in the code that creates the simulation.
Quoting fishfry
:rofl: I'm voting for you if you ever contest elections! You should be president.
All that out of the way, I'd like to run something by you. I have this notion of infinite randomness in my mind. To me it means the existence of an infinity that is completely devoid of all patterns. If such infinite randomness were discovered to exist (I don't care as to where) can we infer the impossibility of reality being an illusion based on the premise that to code infinite randomness would require an infinite set of instructions, a task that can't be completed, and if so, such a code can't ever be actually executed?
Have you seen an irrational number? You’ve seen a symbol for it, you have thought about the idea of it, but you haven’t seen one, with its whole infinite decimal expansion. So it could be argued that they don’t exist as more than an idea, and then your reasoning doesn’t apply.
And if they do exist ... why couldn’t the program that runs the simulation be infinite? It can’t be infinite within the simulation, but beyond the simulation you don’t know that.
So in both cases your reasoning doesn’t prove we aren’t in a simulation.
But if we’re in a simulation, it’s a simulation that has the power to give us consciousness, feelings, thoughts, ... so it’s more than a mere computer simulation.
Yup. As I think I stated or strongly implied. It's just a game with rules that a group of humans agree on well enough to keep playing. Those without training in it take it too seriously or 'metaphysically.' They've never watched the sausage being made or seen long, boring proofs.
Quoting fishfry
Intuitively I agree. Though I think you'd agree that existence is just existence in terms of proof. Certain conventions guarantee a single notion of existence, even if constructive proofs encourage us to take the extra-mathematical existence (in some sense) of this or that number more seriously.
Quoting fishfry
I tend to agree with you here, but I allow for the possibility of some philosopher arguing that mathematical existence is also some kind of extra-mathematical existence. What the game means beyond the game is not decided within or by the game. People could claim that integers are more real than chairs or clouds. What are supposed to make of that is another issue.
My initial issue with simulation theories (and philosophy in general) is semantic. What does it even mean to say that this is simulation ? I guess we are supposed to picture ourselves as characters in a video game created by aliens of some kind. But maybe some human has visions and claims to see these aliens and this video game. How is that distinguishable from delusion or just more simulation?
I know this is for @fishfry, but I'm caffeinated and here, so I'll play too. Here's what you seem to be talking about: https://en.wikipedia.org/wiki/Kolmogorov_complexity
My understanding that such 'objects' have been discovered to mathematically exist. Most real numbers are not computable, by the simple measure-theoretic argument given above. So within one tradition (which happens to be dominant) there exist non-computable numbers. But by definition they can never be looked at directly. They are a byproduct of measure theory, you might say. But some thinkers might look at this byproduct and doubt the system that produced them. Real numbers might be useful fictions. Continuity might be an 'illusion.' (The semantic issues what that are the usual semantic issues with all interesting philosophy. Do people ever know exactly what they mean? Or do the generations come and go, muddling through with their conventional noises somehow?)
A system of real numbers is any system that satisfies certain axioms. So if you were interested in the metaphysical ramifications of math, you'd probably want to look at constructions of the real numbers, set theory, etc., to see how much weight you'd give these human creations outside the system of conventions in which they conventionally exist. You can argue that pi exists in the same way a chair exists, or that pi is more real, etc. But once you leave the chessboard and its rules....you're another improviser trying to synthesize a big picture in a language you cannot control.
That there are noncomputable irrational numbers is certain: Chaitin's constant
If there's no algorithm that can compute a number, each digit will, if the universe is a simulation, require a separate line in the code and that means such a program will be infinite, can't be finished, ergo, can't be run, hence, the universe can't be a simulation because of the existence of such numbers (noncomputable irrational numbers).
Now that I think of it, humans have struggled greatly with the concept of infinity. Basically, infinity DOESN'T COMPUTE! for humans. Last I checked, it all "started making sense" in the 1870's with Georg Cantor's work. This, at some level, suggests that the universe doesn't contain actual infinities and that our brains can't handle what is essentially infinite information. The universe could be a simulation for that reason - no algorithm can manage infinity: infinity + 1 = infinity; infinity + infinity = infinity; and so on. We hit a wall and things stop making sense: IT DOESN'T COMPUTE!
The other side of this story is that non-computable irrationals (Chaitin's constant for example) exist. In other words, the universe does contain instances of infinite randomness and these can't be reduced to finite algorithms. Ergo, the universe isn't a simulation.
Even Cantor's work was hugely controversial. What's strange is that the infinite does compute, within certain systems that give it a formal meaning. The sideways 8 is used correctly or incorrectly in the game of mainstream math. And for set theory experts or grad students there are more complicated rules and more than one flavor of infinity and even more than one mathematical tradition.
Quoting TheMadFool
Perhaps. For me the issue is semantic. What does infinity even mean? It has various meanings in various contexts, and we kinda-sorta prove that we understand these meanings by our mumblings being tolerated in these contexts. The student gets an A. The journal publishes the professor's latest paper. Nobody has to know exactly what is going on. Ultimately they need to be housed and fed, treated as worthy people. (I think 'infinity' is just one version of this. We can also talk about 'good' and so on. )
You mention e and pi. It might be easier to talk about the square root of 2. What could be more classic? Some positive rationals when squared are too small. Others are too big. We can endlessly zero in on the hole where sqrt(2) should be. If a person didn't know that root(2) was irrational, they might spend their life trying to finally get that magical rational number that finally squares to 2.
Basically we can 'see' the diagonal of a unit square, so we decided there was a hole in the number system. In fact there was more hole than non-hole, at least once we got a system up and running. Worse, there was more incomputable 'super-hole' than 'hole.' The computable reals are like the rationals in that they have a finite expression, except it's a finite program instead of a pair of integers. But what's a few bits here and there, as long as the description is finite? On the other hand, the formal existence of a boatload of super-hole incomputables is also the result of something finite, namely a computer checkable proof. So the blob of all of them is tied to something finite.
Quoting TheMadFool
Chaitin's Metamath is pretty great. I think @fishfry would say that we only know that our human imagination contains infinite randomness. And I'd add that we have some notion of infinite randomness. We can make certain arguments. But I remember Chaitin gently suggesting that maybe real numbers aren't real. A person might decide that the mainstream continuum is fiction indeed because it is mostly an unnameable hole.
To play Devil's advocate: maybe aliens who exist in the hidden 'real' world actually do understand infinity and write infinite programs. But they programmed us with finite minds. Perhaps it amused them to make us capable of a glimpse of our limitations. If this is a simulation, why should the computer that runs it have the limitations of our simulated 'Flatland' computers?
So, what's infinity + 1? How does your answer, which must be infinity, square with the answer to 2 + 1?
Infinity DOES NOT COMPUTE!
What's 1 ÷ infinity? If it's 0 then infinity × 0 = 1??
Infinity DOES NOT COMPUTE!
Quoting f64
Let's not get our knickers in a twist. Take a simple instance of infinity, Whole numbers = {0, 1, 2,...}
Then take a part of it, a subset, Even numbers = {0, 2, 4,...}
We know, from the great Cantor's work, the cardinality of the set of Even numbers = cardinality of the set of Whole numbers. A part = The whole.
Infinity DOES NOT COMPUTE! [ :joke: ]
Quoting f64
I tried. The precision, as per my calculations, can be infinite.
x = sqrt(2) = 1.4142135624...
Assume, x = 1.414...
1000x = 1414.414414...
999x = 1413
x = 1413/999 = sqrt(2) correct to 3 decimal places
y = 1.4142135...
10000000y = 14142135.4142135...
9999999y = 14142134
y = 14142134/9999999 = sqrt(2) correct to 7 decimal places.
In this way we can achieve arbitrary precision (infinite) on the value of the sqrt(2). Just saying. My relationship eith math is love-hate. I love math but I think she hates me!
Quoting f64
A proof of the existence of noncomputables is not the same as an algorithm that can generate noncomputables.
Quoting f64
Insofar as the universe being a simulation is the issue, the distinction real-unreal is irrelevant. The real numbers can be accessed through our minds and that means they have to be encoded in the simulation unless the universe is a partial simulation like a cyborg or thereabouts.
Quoting f64
DOES NOT COMPUTE!
Thanks for the stimulating discussion. I'm out of my depth here so thanks for indulging me and my bizarre ideas.
I'm rusty at this stuff, but basically let's consider the set N* = N-union-{x}. That's the set of natural numbers (let's exclude 0 and say 1,2,3,...) with the addition of some non-natural element x. Then to get a bijection from N to N*, we set f(1) = x and f(n) = n - 1 for n >= 2. Basically that's x,1,2,3,4,5...
Clearly we can add any finite numbers to N and get the same cardinality, by tinkering with our bijection. I have this book right beside me: Cantor Book
The whole cleverness or charm of Cantor is that he actually made this stuff work. He extended the concept of cardinality so that folks could play with an infinite tower of infinities --in a way that makes sense to mathematicians. Of course people can always say that's not the infinity that I mean. Fair enough. But I'd say: well, what infinity do you mean? If it's just vague metaphysical speculation, that's fine. But then it's the usual opinion-mongering. A person can joyfully wallow in suggestive ambiguity and the impossibility of a consensus or they can at least come to a consensus about the rules of a particular discourse. Cantor's work is full of surprises and ingenuity. Does it matter much when it comes to engineering? As far as I know, not really.
Quoting TheMadFool
True, but there can't be an algorithm that generates a particular noncomputable, by definition. My point is something like: everything we humans do is finite. Even our idea of these dark matter numbers pops out of a finite construction.
Quoting TheMadFool
OK, that's a fascinating point. So if our imaginations are part of the simulation, then who cares if the black and seamless sea of incomputables is pure fiction? All that we can dream is part of the program. OK. But what exactly do we dream when we dream of noncomputables? A finite proof, and the vague and questionable interpretation of that finite proof. Even on the level of fiction and I am saying that the concept is slippery and ambiguous. I think this applies to whatever is plucked out of the game of math.
Overall I still like your argument. It inspires some fun thinking.
Quoting TheMadFool
My pleasure. I suspect that all humans are basically out of their depth. The generations come and go, talking of God and truth and infinity and good and evil. It passes the time.
It's more like f(part) = f(whole). The sets aren't equal. They are just have the same cardinality. Like Jefferson and Washington were both presidents.
Quoting TheMadFool
Consider that there are lots of systems in math. (Check out an abstract algebra textbook. ) It's possible to extend R with positive and negative infinity, but a few nice features must be sacrificed. I think it's uncontroversial that we can't just add the sideways 8 to ordinary arithmetic.
Quoting TheMadFool
That's a fascinating issue right there. Arbitrary precision as infinite...as opposed to the idea of a completed infinity. You wrote N = {0,1,2,...}. So we kinda know what that means, but do 'all' of the elements exist somewhere? What exactly do we mean by those ellipses? Obviously mathematicians can crank out papers and keep the tradition alive. Nuns can smack children's hands with rulers for putting apples in N. Professors can mock confused grad students at a higher level for more complicated 'misunderstandings' (errors in manners).
In other words: social 'games' within social games, wherein we discover what we can and can't get away with saying, without ever necessarily knowing just what we mean. Semantically we're in the fog, whatever that's supposed to mean. 'This is how it's done around here.' That's the spongy bedrock. (That's one vision of Socrates, a guy trying to show people that they don't exactly know what they are talking about, who found himself fatally unpopular, having written no books.)
On the other hand a simulation could be made in wich aparent infinities would exist because you only need to simulate the brains. More over it is likely we don’t have free will so it’s funny that the Universe may not necessarily be a simulation but a mere playback of brain activity. In my opinion it’s not possible to create or simulate free will other than an illusion
I have a point that might be of interest to you.
Wiki is correct that if you take the standard rules of math, you can logically deduce the existence of noncomputable numbers. You've quoted an outline of the proof. Cantor's diagonal argument shows that the reals are uncountable; but we can show that there are only countably many Turing machines. So there must be real numbers whose digits can not be cranked out by a Turing machine. QED.
But what is a logical deduction? You start from some axioms, which are strings of symbols. You accept some rules of inference. You mechanically apply the rules of inference to your axioms, and conclude the result. The entire process of deduction could be carried out by a computer. You input the axioms as strings of symbols; you program in the rules of inference; and the computer can determine whether or not that's a valid proof.
So that's the thing. The proof that noncomputable real numbers exist, is itself a computation. In general, proofs are computations. Symbol manipulation according to formal rules.
In other words: A computational intelligence would eventually prove to itself that noncomputable real numbers exist given the rules of standard math.
What do you think about that?
(By 'there exists a definable uncomputable real number' in this context, I mean the ordinary mathematical sense:
There exists a formula F such that
E!x(Fx & x is an uncomputable real number)
is a theorem of set theory.)
In the context of set theoretic cardinality, there is no object named 'infinity'. Rather, there is the adjective 'is infinite'. So expressions like '1/infinity', et. al are meaningless, and supposed arguments based on such usage are fallacious.
(This does not contradict that there are things such as the extended real number system in which there are points of positive infinity and negative infinity and arithmetic on them, since that is a different context from set theoretic cardinality.)
Moreover, set theory does not assert that there is a set that is equal to a proper subset of itself. Indeed, it is a trivial theorem that there is no set S such that S is equal to a proper subset of S. That does not contradict however that there are sets that have a 1-1 function from the set onto a proper subset of itself. Having a 1-1 function between sets (of which we say 'the sets have a bijection' or 'the sets are equinumerous') is plainly distinct from the fact that the sets are not equal, not identical, not having all the properties of each other.
Right from the start of your argument, the merely ostensive (and not specified by actual mathematical description) list you gave is either actually not defined or it's finite. It ends with a certain value, yet doesn't specify how to squeeze in a countably infinite number of values in between the first and the last.
There are proofs that there exist definable uncomputable reals, but you haven't given such a proof.
That's a lot of negations, but for clarity, there does exist a definable yet noncomputable real, namely Chaitin's Omega. I haven't mentioned it because I didn't want to further complicate the conversation.
https://en.wikipedia.org/wiki/Chaitin%27s_constant
Quoting GrandMinnow
No that's not right. There are a countably infinity of Turing machines hence a countable infinity of computable numbers, hence a bijection between the natural numbers and the noncomputable numbers.
As I indicated earlier, there is no computable bijection; but there is a bijection.
Definability is a much more subtle concept and is best left out of the discussion. For one thing, first-order definability is not itself formally first-order definable. There are models in which everything is definable. Joel David Hamkins has written on this topic. It's not relevant here and would take us far beyond the topic.
I only meant to add clarity. It's a small point. The OP has disappeared and I was hoping he'd return long enough to comment on my observation that the proof of the existence of a noncomputable real is itself computable. That is, proofs are computations, hence a computational intelligence would be able to prove the existence of noncomputable phenomena, given the standard axioms and inference rules of math.
Quoting GrandMinnow
I believe your assertion that there is no definable list of noncomputable reals may well be true but I'm not sure that the proof would be at all elementary. Do you happen to have such a proof? There's surely no computable list, since that would solve the Halting problem. But why not a definable list? I don't know if what you said is true, but I do know that it's not trivial either way. Definability is very murky as Hamkins points out. See his brilliant response in this thread.
https://mathoverflow.net/questions/44102/is-the-analysis-as-taught-in-universities-in-fact-the-analysis-of-definable-numb
I didn't make that assertion. I only commented on the poster's particular argument. (I later edited 'definable' to 'defined', but in either case, my comment pertained only to the poster's own argument.)
Anyway, there is no list (no enumeration) of the the set of uncomputable reals, since the set of uncomputable reals is uncountable.
/
That thread you linked to includes an argument that uses 'least undefinable ordinal' to throw shade on the "naive" notion of definability. But one would not claim that 'definabie' itself is a predicate in the theory. I only mentioned a certain syntactical fact - regarding formulas of a certain form. I wouldn't say in set theory itself, about set theory, that there exists a definable something or other. To even speak of that "something' is to speak of an object that exists per a set theoretic model, but indeed, as we well know, set theory (if it is consistent, which I take as a "background" assumption) does not prove the existence of a model of set theory. (Though, I'm not expert enough to defend against possible other complications in the matter.)
So to make the argument work that there are only countably many definable real numbers, maybe something like this in a set theoretic meta-theory for set theory:
Let 'Rx' be the set theory formula 'x is a real number'. Let M be any model of set theory such that any subset S of the universe of M satisfies 's is countable' if and only if S is countable. Let D (the set of definable reals) be the subset of the universe of M by D = {d | there exists a formula F of set theory such that (E!x(Fx & Rx) is a theorem of set theory & d satisfies Fx and d satisfies Rx). Then D is countable.
Duty calls for other stuff this evening, will get back to you later tonight or tomorrow to continue this interesting convo. But who on earth ever suggested that the set of noncomputable reals was anything other than uncountable? Surely not me and not the OP as far as I could tell. The claims (mine anyway) are:
* The set of computable reals is countable.
* The set of computable reals is NOT computably countable; and
* The question of whether the set of computable reals is definable is Hamkins-murky and way beyond my pay grade. Though if I understand Hamkins correctly, there are models of the reals in which everything is definable. That's why it's murky.
Meanwhile, I changed my post above to pertain to definability rather than computability, It's not a problem to show in set theory that the set of computable reals is countable, but there are issues in showing the countability of the set of definable reals, as mentioned in the article you linked to. I tried to sketch a sense in which the notion of definability can be used without such contradictions as 'the least undefinable ordinal' that was mentioned in that article.
You mean there is a bijection between the naturals and the computable reals. And I didn't claim otherwise. I only pointed out the incoherence of a particular poster's argument.
Sorry. Am I missing something here?
Yes sorry typo.
Hamkins shows that (assuming set theory is consistent) there is a model of set theory -- uncountably many models, in fact -- in which everything is definable. Every element, every set, every collection of sets, etc. In particular, every real number in this model is definable. Of course if set theory is not consistent it has no models at all, so it's necessary to assume consistency to get the discussion off the ground.
Now such a model must necessarily be countable, because there are after all only countably many possible definitions or predicates. That's easy to see, there are only countably many finite strings over a countable alphabet.
But, in such a model, the real numbers are still uncountable! That's because Cantor's theorem is a theorem of ZF, and is therefore true in any model of ZF. Cantor's theorem says that the reals, being bijectively equivalent to the powerset of the natural numbers, are uncountable.
So what's going on? The trick is the famous Lowenheim-Skolem theorem, which says that if a collection of axioms has an infinite model, it has models of all cardinalities. In particular there are countable models of set theory. Yet in those models, the real numbers are still uncountable. How does that work?
What does it mean for a set to be uncountable? It means there is no bijection between the set and the natural numbers.
Conceptually here is what's going on. Suppose that we have some countable set that is a model of set theory. Remember that a bijection is just a special kind of function; and that a function is just a particular kind of set (namely a set of ordered pairs in which each first element only appears once). So what we can do is go into the model, and remove all the bijections between the natural numbers and our set of interest.
If we are careful to remove bijections in such a way that the remaining elements still form a model of set theory, we'll have a countable universe that contains a set that is uncountable. In other words there are no bijections between the set and the natural numbers, because we've carefully removed them. But the universe is still a countable set. Of course the "carefulness" involved in throwing out the bijections while still satisfying the axioms of set theory can be made rigorous.
As seen from the outside, our uncountable set is actually countable. But as seen from inside the model, it's uncountable, because we've removed the bijections. This shows that the notion of countability is a "relative" property. It depends on whether you're looking at it from inside or from outside a given model.
See https://en.wikipedia.org/wiki/Absoluteness
and in particular, https://en.wikipedia.org/wiki/Absoluteness#Failure_of_absoluteness_for_countability
So in a model where everything is definable, that model is must necessarily be countable as seen from the outside. But it does contain the real numbers (which exist in any model of set theory); and by Cantor's theorem, the real numbers are uncountable (as seen from within the model) yet all real numbers are definable.
I mention all this to provide context and hopefully some clarity to my following comments.
Quoting GrandMinnow
By "throwing shade" you mean "flat out falsifying," in the same way that Russell's paradox "throws shade" on Frege's notion of unrestricted set formation, right? In other words Hamkins is not just casting vague innuendo. He's providing a technical argument that falsifies a commonly held belief. Just to be clear about this.
Quoting GrandMinnow
Correct, that one of Hamkins's points.
Quoting GrandMinnow
I'm afraid I couldn't quite glean the meaning of this.
Quoting GrandMinnow
No, that's not right. Set theory is consistent if and only if there's a model. That's Gödel's completeness theorem. If it's consistent there's a model, and if there's a model it's consistent.
Quoting GrandMinnow
Remember, there are only countably many real numbers as seen from outside a Hamkins model; but within the model there are uncountably many reals (by Cantor's theorem) and they're all definable. Tricky stuff.
Quoting GrandMinnow
There is a model of set theory in which everything is definable. In particular, each real number is definable. In such a model the real numbers are countable as seen from outside the model; but they are uncountable as seen from inside the model. You are using the word countable without regard for the fact that countability is a relative property. I think your argument is ambiguous because of that. In any Hamkins model, the reals are uncountable (inside the model) yet they're all definable. And that's because they're "really" countable as seen from outside the model.
tl;dr: There is a model of set theory in which all real numbers are definable. Such a model is actually countable, as seen from the outside; nevertheless within the model, Cantor's theorem is true and the reals are uncountable.
No, it is right. Yes, set theory is consistent if and only if there is a model of set theory. But if set theory is consistent then set theory itself doesn't prove that it has a model;. That's Godel's 2nd Incompleteness Theorem.
Oh I see, you're right. My parser got confused.