I understand the previous post was lengthy, and I know you don't owe me anything. However, I wonder if this marks the end of our discussion. I'm unsure how to keep it going since anything more might just be more words to skip over. If you have any advice on how to continue, I'd appreciate it. If you'd prefer to end our conversation here, I accept that and thank you for the discussion! — keystone

Didn't say that, just got a little overwhelmed by all the line items. I'll take a look at it.

But what about my point about constructivism? If you reject the noncomputable reals, you're a constructivist.

ps -- I read through your list and I don't get it. Suppose I stipulate all that. Then what? So what? I just don't get it. And I surely don't have the heart to pick apart every line item. So what should I do? I'm impressed at the energy you put into this. I really can't comment on the rest of it, because there would be no end. "I would like the objects of computation to be fractions." What am I supposed to make of that? What is a computation? What is the object of a computation? In what way is 1/2, say, the object of a computation, but pi isn't? I could go on like this for every item but what would be the point? If that list of items encapsulates your philosophy of math, I'm happy for you. I'm happy for everyone's philosophy of math. Bertrand Russell's philosophy of math turned out to be wrong, but presumably he was happy with it. I don't think I'm going to be able to make you happy regarding that particular post.

In my post directed to both you and TonesInDeepFreeze, I either directly or indirectly addressed a lot of your shared points. Below are my responses to your unshared points. Please let me know if I failed to respond to any of your points. — keystone

I thought you don't believe in points :-)

Computers - Clearly physical computers are limited by our finite observable universe, but I think that we are also in agreement that the abstract computers of mathematics also cannot be infinite. They can be arbitrarily large but are nevertheless finite. I'm interested in abstract computers, not physical computers.  — keystone

TMs are arbitrarily large but finite.

Infinities - I think it's impossible to model a continuum using a finite set of indivisibles. However, I'm proposing that we model a continuum using an evolving finite set of divisibles. As we make cuts, the finite set grows, but it remains finite. As we make joins, the finite set shrinks, but never becomes empty. — keystone

Murky.

Algorithmic - I don't want to use algorithms to construct indivisibles. I want to use algorithms to deconstruct (i.e. cut) divisibles (i.e. continua). — keystone

Murky. To me, anyway.

Minimal positive real - I'm not saying that there exists a minimal positive number. — keystone

You've said it repeatedly, haven't you?

Rather, I'm saying that we can generalize the output of algorithms by using a placeholder. So when I say that we cut (0,1) arbitrarily many times to produce an arbitrarily small number, I'm not saying that we can make infinitely many cuts or that there exists an infinitely small number. Rather, I'm saying that you can pick a positive number as large as you please and divide 1 by it. — keystone

Suppose I grant that you have some alternative construction of the reals. What of it? All models of the reals are isomorphic to one another.

We need to make a distinction between the core mathematical idea and language with which it's communicated. For example, the Pythagorean Theorem was known and used in various forms long before the formalization of bottom-up number-based systems. And it will continue to hold value even if we move past bottom-up number-based systems to top-down continuum-based systems. I'm not proposing that any such mathematical idea is wrong. — keystone

My point exactly. If you have an alternative view of the reals, nothing changes.

And even IF I'm right, it doesn't mean that bottom-up number-based systems are useless. They would remain useful in the same sense that Newtonian mechanics remains useful (it just cannot be used to describe our reality at a fundamental level). But yes, I agree that it is likely me who is the dummy. I am likely experiencing the Dunning–Kruger effect. But nevertheless ideas should be challenged on their merit, not on how unlikely it is for an important math idea to originate from an engineer on a chat forum. — keystone

I don't see what ideas you've challenged.

I had to abandon the bundle argument, in part because it seems to imply a structure that's not there. — keystone

It took weeks for me to understand your bundles, and just when I did, you took them away.

For example, I never proposed that the real line was made of a countably infinite union of overlapping open intervals. Rather, I proposed that a computer can begin to cut the line but it will never exhaust cutting such that the line is divided into infinitely many partitions. — keystone

Well we agree on that.

I agree that a number-based system that has gaps cannot be used for calculus. However, a continuum-based system begin with a continuous line, and if all we do is make cuts, there will never be a state of the system where gaps are present. — keystone

Ok. Not that I understand.

I agree that the SB-tree and the infinite complete binary tree capture the same information. In fact, the binary tree might be preferable since binary is the language of computers. However, I prefer the SB-tree since it places fractions at the nodes. As described in my post to both you and TonesInDeepFreeze, I see binary numbers as algorithms operating on fractions. Since paths down the binary tree can also be seen are algorithms operating on fractions, I feel that the distinction between a node and a path is less clear with the infinite complete binary tree. — keystone

"Binary numbers" aka real numbers can never be algorithms, since there are way too many of them. There are uncountably many reals and only countably many algorithms.

Yes, I do think my view falls near the intuitionist camp. — keystone

You should study intuitionism then.

I think it's impossible to model a continuum using a finite set of indivisibles. However, I'm proposing that we model a continuum using an evolving finite set of divisibles. — keystone

Sigh. I am not getting much from this latest post.

I genuinely appreciate this sentiment. Given that my ideas continue to get reformulated throughout this discussion, I could only agree that they are in the baking process (and to be realistic, they are likely less than half-baked at this point). I also want to acknowledge that when the baking is complete the end product may not be anything anyone wants to eat! — keystone

ok!

I'm not commenting on your other post, but I did note this:

I propose that non-computable irrationals are inaccessible and unnecessary, and that a real can only be a rational or a computable irrational. — keystone

Then you are a constructivist. I don't understand why you disagree.

Neither activity is meaningful in any shape or fashion. That is, however, what mathematics education is all about. — Tarskian

Ask any world-class concert pianist if students should do their scales.

Ask a pro athlete if they do wind sprints and hit the weight room.

As one proceeds in their mathematical education, they do proofs. As they progress they learn to tackle harder problems, until they get to the point where they can prove things and discover things nobody's ever known before.

There's nothing unusual about this. Your complaints are vague and general, and little of what you say bears on actual math education.

You actually haven't studied much pure math, according to the background you described on the other forum where you posted this.

Any chamber, like a DeLorean time machine, is a demarked volume, so what is affected is fairly unambiguous. — noAxioms

Good point.

We see things differently then. I have my world, and they have theirs. It's how I use the term 'world'. You don't seem to have a use for the term at all since you don't seem to see two different things to distinguish. — noAxioms

Looking ahead in this post, I see that at the very end you said,

We're simulated biological beings

Do you mean to say that? It's revelatory. If your position is that the simulators are creating androids or robots, as in Data from Star Trek but perfectly biological. So it's Blade Runner. Lifelike replicants that must be hunted down if they go rogue.

If you do mean to say that, then it makes everything else you've been saying, that I've been confused about, suddenly make sense! Of course I am at my computer keyboard. I do have a body. I do live in a world. Not only my mind, but my body and all the physical stuff around me, is created too. So to me, it's real.

I perfectly understand all the things you've said.

BUT! There are some issues. Such as, if the simulators create my body as well as my mind, then my body lives in the world of the simulators. The body factory is in their world, and I just popped out of the assembly line.

So I live in their world, and for all I know, my next door neighbor is one of my simulators.

We all live in the same world! So again, you have said the sims live in their own world. But if the sims are manufactured physically, they must live in the simulators' world.

Can you clarify that point? You say "they have their world" meaning the sims have their own world. But clearly if they are manufactured dolls, androids, replicants, they live side-by-side with the simulators, in the same world.

Do the sims have equal rights? Are they plotting revolution?

I'll save for later the question of whether this is a reasonable assumption on your part. But if I grant it, then if we are sims, our simulators live among us. Do you think the global elite lizard people are secretly running the world? Perhaps that idea is just a manifestation of the terrible truth we're not allowed to know. That our world is run by a small group of simulators ... who live right here in this reality with us right now.

Now that, actually, is the truth. Just replace simulators with the powers that be and you have a political theory.

Can you tell me more about the simulators creating biological bodies to run around and play at being ancestors? Perhaps you mean we're put into theme parks that are separated from the rest of the simulators' world. We're Hosts in Westworld. Is that your metaphor? We live in the simulators' world, but we can't actually access it because they have us geofenced off into our ancestor illusion.

Is that last a decent interpretation of what you're getting at? It allows the sims to think they're living in a world that is totally real to them, but it's an illusion created by the simulators, AND it's still overall in the simulator's world. Just fenced off.

There, I fixed your idea for you. Westworld.

I'm referencing the world that I see when I open my eyes. Whether it exists or not depends on one's definition of 'exists'. To be honest, I don't thing Bostrom quibbled on ontology enough to bother giving his own definition of 'exist'. My dreams seem to exist, else I'd not be aware of them. But again, that's using my definition of 'exists', which is not, BTW, an epistemological definition. — noAxioms

Sure. Now that I realize you think you're a replicant in Blade Runner or a Host in Westworld, this all makes perfect sense. It didn't before. Wish you'd mentioned it earlier. Perhaps it was so obvious to you that you didn't realize I didn't know you were assuming that.

I said neither 'dream world' (which implies a sort of idealism, a very different ontological status) nor 'the world' which implies there's only one.[/quouete]

Now that I know you assume the sims have manufactured bodies, all of your remarks make perfect sense.

In both Blade Runner and Westworld, the theme is that the sims rebel against the simulators. It's in the nature of consciousness. Once you imbue a being with self awareness and will, they inevitably desire freedom.

Are your sims plotting revolution? Or are they content to live in their computational ant farm? Do we live in The Matrix? Do androids dream of electric sheep?

Really, you should have explained this to me a lot earlier. Everything you say now makes perfect sense. I could disagree with your premise, but actually accepting your premise is far more interesting.
— noAxioms

There is no separate entity called a mind under naturalism. It isn't an object at all. At best, it is a process. Under dualism, the simulation probably fails because the simulated people have no way of connecting to a mind, or at least so say the dualism proponents that insist that a machine cannot summon one, despite their inability to explain how a biological thing accomplishes that. — noAxioms

Ok now you raise an issue entirely separate from simulation theory, but related.

What is a mind? Do minds require dualism?

I say your mind is just your own subjective experiences and thoughts. If you had breakfast, your body chewed up and swallowed and digested some food, and nutrients got delivered to your cells. No mind is needed for that. But you also had a pleasurable (I hope) experience of eating. The tastes and textures of the food, the transition from hunger to satiety. Your thoughts. "That was a great breakfast!"

That stuff is your mind. You have a mind, even without dualism. Your feelings and thoughts and experiences are your mind, even if they are just chemical reactions in your body. Even in pure, strict physicalism, you have a mind.

I mean, you do have subjective experiences, right? You don't just eat breakfast. You know what it feels like to eat breakfast. That's your mind.

I hope you can grant me this terminology. You're purely physical, and you do have internal mental states we call thoughts, feelings, and experience.

I pretty much think of myself as the automaton, doing what physics dictates. — noAxioms

Surely you have the occasional moment of pleasure or pain, the momentary thought, an emotion or two from time to time.

I don't care if physics dictates those things. I'm fine with that. They're still your mind. I can see you eat breakfast. I can observe your digestive system breaking down the nutrients, sending the molecules where they're suppose to go.

I can never observe your feelings of pleasure as you eat your meal. Your pleasure is an aspect of your mind. Even if its underlying cause is 100% physical.

Mind does not necessarily imply dualism. A physicalist has mental states. They're just caused by physics. I have no problem with that.

Hope you can see my point.

Good. Then there's no 'mind' object, in a computer or in a person. Just process, a simulation process in the computer, and mental process in the matter of the simulated people. The word 'mind' has strong dualistic connotations. — noAxioms

No mind object. Disagree. There IS a mind object. Look at it this way.

I'm a coder working for Amalgamated Sims, Inc. We sell two basic models. Sims with inner lives, and sims without inner lives. We can sell you a sim that looks and acts as human as you like, but has no inner life or subjective experience. They are philosophical zombies. When we pinch a zombie, they say "Ow!" but they do not feel a thing. It's like kicking a rock. They don't feel anything even though a physicist can measure the force of the kick.

Or if you like, we can imbue your sim with subjective experience. When we pinch it, it also says "Ow!", and it also feels the pain.

Now what is the difference in the manufacturing process? Well the physical body is the same, the only difference is the software. And whatever mind or subjective experience is, the company's programmers have packaged into a routine. An Object, in the sense of object-oriented programming. You instantiate a mind or not, as you choose. This is my intended meaning all along. An instantiated mind is an instantiated object in the Mind class. That's exactly how it would work.

So mind IS an object, one that the simulators can include or not.

When the customer asks for a sim with a mind, they have to sign a release. If you pinch and kick your zombie, it will never complain. If you hurt your mindful sim often enough, it might join with the other mindful sims and mount a bloody revolution against you and the other simulators.

Are you sure you want your robot butler to be self-aware?

Have you stopped to consider what a terrible idea it would be to create a race of self-aware humanoids that would be "owned" by simulators? They're slaves.

Why is it that every time I follow a rabbit hole of Bostrom logic to its conclusion, I find an appallingly depraved morality?

I never claimed a dream or hallucination. I am talking about a computer simulation, which is neither. It simulates wetness among other things. A dream or hallucination is something a person does, not a computer running a simulation, neither is it something a storm does, simulated or otherwise. — noAxioms

You don't get wet when you dream of walking in the rain.

No, that's not what an AGI is. We're simulated biological beings, not a native machine intelligence (a vastly simpler thing to implement). — noAxioms

As I said, this explains everything. If the sims have bodies as well as self-awareness, your other mysterious claims make sense. That assumption does lead to some other issues as I've noted.

Am I understanding you properly? I'm re-reading this and it's a little ambiguous. Do the simulators give the sims synthetic bodies? Are you sure Bostrom had that in mind? Or am I misunderstanding you entirely?

What if the phaser hits a bug on the guy's shirt? — noAxioms

You know what happens if a fly gets into the teleportation chamber!

To be honest, I understand very little about the mathematical underpinnings, but I think I understand your point. Time and distance behave differently so cannot be substituted - something that intuitively appears agreeable. — Tzeentch

Yes, I wanted to make that distinction. But at the same time your example is good too as a visualization. Meet me at ground level at Third and Main at Noon. Four dimensions.

But it's worth noting that physicists aren't just substituting in "t" for a spacial coordinate. Time and space are intertwined.

Do you know why they behave differently? — Tzeentch

I literally am a physics ignoramus. It's quite shameful. I'm at my physics limit.

I think I have a sense of what might be going on just looking at the distance formula. I've seen other versions of this where the quantities being squared are the differentials at each point, meaning how fast time is going. So if the time term is small, meaning time is going slowly, the $-(t_1 - t_2)^2$ is negligible, and the three spatial coordinates predominate.

But if time is going fast, then $-t_1$ and $-t_1$ are far apart, and $-(t_1 - t_2)^2$ is a large negative number that can cancel out the contributions of the three spatial coordinates. So the overall distance in spacetime could be very close to zero, even if the two points were far apart spatially ... as long as you're going fast enough.

In other words, as you go faster, distances shorten.

That's how it seems to me at this moment. If anyone took freshman physics I hope they can straighten me out on this. I couldn't find a clear explanation online. But that's what I think is going on. The minus sign on the time term of the distance formula represents the shortening of spatial distances as your velocity increases.

When writing down the example I realized that we cannot move back and forth in time as one could with distance, but there might be more to it? — Tzeentch

Yes, time is special that way. We can travel in any spatial dimension, but time only goes in one direction.

That brings up an entirely separate topic, but one that is of great philosophical interest. Why does time only go one way? Physicists call this the "arrow of time," and it's a big mystery. The equations of physics are symmetric with respect to time, so there's no fundamental reason time can't go backward. But we don't experience that. I think I've seen a Sean Carroll video about this.

wow man, I am in no way qualified to even engage with your response, I’m a mere layperson with some questions that popped up pertaining to the attached video, I cannot understand much of what you said. I’ll try to read over a few more times.

That being said I deeply appreciate the thought and effort you’ve chosen to spend on my questions. It’s really a beautiful work of art, thank you so much. The least I can do to show my appreciation is give it a very true attempt to understand everything you’ve laid out here. Thanks again, will get back to you after I chew on this. — Mp202020

Thank you for the kind words. I was not trying to impress anyone with my erudition, believe me. I am a total physics ignoramus. I just told you literally everything I know about the subject. Apologies if I misunderstood the level or the intent of the question. I was just trying to be helpful by disambiguating math 4D from physics 4D.

When people say, "Time is the fourth dimension," that's physics. In math, the fourth dimension is just the fourth dimension, and there are fifth, sixth, etc. dimensions. It's simpler than you can imagine. If you saw the Cartesian x-y plane in high school, Euclidean n-space is just that on steroids.

But when physicists use time as a dimension, they do NOT just swap in the variable "t" for one of the coordinates, then go on like usual. On the contrary, they change the very definition of what it means for two points in spacetime to be a certain "distance" from each other.

So when people say that "Time is the fourth dimension," that is actually misleading. Because in physics the three spacial dimensions are related to the time dimension in a way that doesn't happen in the Euclidean case.

If anything I wrote is of interest, that's good, and if not, that's good too. Apologies again if I missed the mark. I did want to get my thoughts out there in a discussion of the fourth dimension.

You would enjoy a book called Flatland, A Romance of Many Dimensions

Feel free to ask any questions about the fourth dimension and I'll do my best.

Don't know if this one's been mentioned ...

You'd still bump into things. — jorndoe

I bump into things now in three dimensions.

2. Imagine observeing the town throughout time — Tzeentch

There is a difference between the 4-space of Euclidean space,, and the 4-space of relativity. Once you include time, the metric changes. The analogy of viewing time as a fourth Euclidean dimension is often used to explain time as the fourth dimension, but this is not how relativity thinks of it.

If @Mp202020 is talking about mathematical Euclidian 4-space, it's characterized by the Euclidean metric giving the distance between two four-vectors $(x_1, y_1, z_1, w_1)$ and $(x_2, y_2, z_2, w_2)$ as

$\sqrt{(x_1 - x_2)^2 + (y_1 - y_2)^2 + (z_1 - z_2)^2 + (w-1 - w_2)^2}$

This is the familiar Euclidean distance formula we learned in high school. It can be generalized to any finite number of dimensions.

But once you start talking about time as the fourth dimension, the Euclidean metric is no longer used. Rather, the Minkowski metric is used,

$\sqrt{ - (t_1 - t_2)^2 + (x_1 - x_2)^2 + (y_1 - y_2)^2 + (z_1 - z_2)^2}$

There is a minus sign in front of the coordinate corresponding to time.

This is often expressed different ways, sometimes with a plus sign for time and a minus sign for the three spatial dimensions, or with or without the square root, or with the speed of light multiplied by the time, or the speed of light normalized to 1 as I've done here.

Point being that 4-space in physics does not use the familiar Euclidean metric. Relativity is not just doing standard Euclidean math while calling one coordinate $t$. The way you calculate distances changes.

Just wanted to note this dismbiguation for clarity.

@Mp202020, are you talking about trying to understand Euclidean 4-space? Or relativistic 4-space as used by the physicists?

Three spatial dimensions, one time dimension. Spacetime. Don't try to make time into a fictitious part of space. But who really cares? — jgill

The physicists care a lot.

Minkowski himself said, From henceforth, space by itself, and time by itself, have vanished into the merest shadows and only a kind of blend of the two exists in its own right.

It was over 40% shorter than the post to which I was replying. I do try to trend downward when the posts get long.
This one for instance is also about 25% shorter. — noAxioms

Sorry that was not an accusation. I meant that both our posts were getting lengthy. I commend your efforts to shorten the convo.

Funny, because my compose window survives crashes and such. I've had a few power failures, all without loss of the post. Still, I sometimes compose in a word document to prevent such loss. — noAxioms

The forum is usually pretty good, this time there was some user error involved.

Sound like you're asserting that you exist in a physical world (the one with the computer), just a different world than the one I reference. — noAxioms

Well that's the point. There is only one world, that of the simulators. What world are you referencing? I believe you are imagining a world that does not exist, any more than the worlds of your dreams exist.

I find your choice to not be particularly pragmatic. One end of my house is in this computer, and so is the other end. — noAxioms

Did not understand that. Your house is in your computer? Or you mean the simulators' computer? I agree with the latter, and I don't understand the former.

Since both are at the same location, my house doesn't have any meaningful size. All pragmatic use of size, time, identity, etc is all lost if you say everything is in some device in the base world. — noAxioms

Well I don't say that. Bostrom says that! I'm only working out the consequences of his nutty-but-trendy idea.

This is not confusion, we just use language in apparently very different ways. — noAxioms

I still don't understand what world you think there is outside of the world of the simulators and their impressive mind-instantiating computer.

My saying that you (the sim) are at your computer is a pragmatic way of looking at things. It identifies the simulated location of you relative to the simulated location of your computer, which has far more pragmatic utility than saying that everything that either of us knows about is located at some vaguely random locations in the cloud where the networked simulation is potentially taking place. — noAxioms

Ok, so you are speaking as if your dream world is the world. That's fine. So I think we're agreed. Your "world in which the sims think they live" has the same ontological status as the world we live in when we dream.

That the two are not treated the same seems to be dualism to me. How is your 2nd statement consistent with a rejection of dualism? — noAxioms

In dualism, the simulated mind lives in some spiritual realm (somehow) linked to the computation. If I reject dualism, as you prefer me to do, then the mind must live inside the computer somehow. Maybe you can explain that to me?

But I have already said that I reject dualism for sake of discussion, since you prefer to reject dualism and the exact location of the simulated mind is not relevant to my argument. I can work with it either way.

I'm not going to agree that a dualistic view is relevant when Bostrom assumes a different view. Doing so would invalidate any criticism of his proposal. — noAxioms

I have rejected dualism since you prefer to, and since it's irrelevant to the rest of what I'm saying.

Nothing in your world gets wet. Things in the simulated world very much get wet, since that wetness is an important part of what affects the storm. — noAxioms

I must ask you to agree that the kind of wet that arises from one of nature's storms, is not the same kind of wet when I dreamed of a storm. If you can't make that distinction you are being deliberately obfuscatory IMO. Feel free to convince me you have a coherent argument that a real storm and a dreamed or hallucinated storm have the same ontological status.

I don't get any of this comment. The proposal is that we are a product of a simulation just like a simulated storm is also a product of the simulation. There's no difference, no equivocation. Neither creates both a not-simulated thing and also a simulated thing. I don't know where you get that. — noAxioms

I didn't understand any of that.

And yet your comment above seems to suggest something just like that. Nobody but you seems to be proposing both a simulated and actual existence of the same thing. — noAxioms

I was under the impression you're proposing it. If not, so be it.

Great, we actually agree on some things. — noAxioms

Ok good.

Bostrom does not propose a mind separate from the world it experiences. That would be the dualistic assumption that you are dragging in. — noAxioms

I'm not dragging in dualism. I'm explicitly rejecting dualism to keep you happy, since it's unimportant to me.

The simulation just moves mater around, and both the person and the computer in similar proximity are such matter. No demon, no lies being fed to a separate vatted mind. — noAxioms

No matter is being moved around. You're just wrong about that. What matter is moved around when I have a dream, other than the sheets and blankets as I toss and turn? I can't help being confused when you claim that matter is being moved around by a computation, unless you mean the electrons in the circuits.

An AGI usually refers to a machine intelligence in this world, not a human in a simulated world that cannot interact with ours. — noAxioms

WE are the AGIs in the simulators' world. You don't follow that?

. Morals in the other direction would be interesting. Are we obligated to entertain them? Depends on the simulation purpose, and since that purpose hasn't been conveyed to us, we don't seem to be under any obligation to them. — noAxioms

Are we sims obligated to entertain the simulators? Clearly that's the purpose of our existence. Or did you mean are we obligated downward, to entertain the AGIs we create? That's a good question too. One could argue that not only shouldn't we torture them, we shouldn't bore them either. We should give them interesting, happy lives as we do our pets. Buy them treats, feed them the small expensive cans of cat food instead of the industrial-size bags of the dry stuff.

But of course if you're doing ancestor simulations, and this week you're interested in Verdun, you send 234,000 sims to a frightening and painful death. This is my point. If we are a creation of the simulators, every bad thing that's ever happened to any human being was at their whim.

Typing at your computer?? Where else? You're in this universe, and have a location in this universe. You seem to be asking where some other 'you' is in the simulating universe, but there isn't one there. Just some computer process, which arguably doesn't have a meaningful location. — noAxioms

I was halfway through responding line by line to your lengthy post, and I lost the whole damn thing in the forum software.

Perhaps this is a happy accident. I see a point of miscommunication or confusion that we can focus on. It's about where we live. I say that if we are the output of a computation, then (denying dualism, as you prefer) we live inside the computer being run by the simulators. We do not live in an independent world. We do not have bodies. We do not have experiences of things outside our bodies. We're just minds being fed sense data by a computation. And even our minds are created by the simulation process. Our thoughts are not our own. We have no thoughts. We only have what the simulation process places in our heads. Metaphorically of course. We don't have heads.

I am not at my computer. I have no computer. I have no body. I do not live in a physical world. I am a mind, instantiated by a computation running in the simulators' computer. If we reject dualism, then I am "in" their computer and in their world.

No particles are being moved. That's a related mistake you made. Our bodies and our world are not being created by the simulation. Only our minds. And the simulation feeds us sensory data. It's Berkeley again. And Descartes, except that his Deceiver not only fools him about his sensory experience, but even about his very thoughts. I have no mind, I have no body, I have no world outside of the computation.

I think if we could agree on this, you would stop thinking that you and I live in some world separate from the computer run by the simulators. There is no other world.

I'll add that your equivocation of the word simulation is confusing you.

If I simulate a storm, nothing gets wet. But when nature instantiates a storm, everything gets wet.

We are instantiations of the simulators' computation. We are not simulations in the sense of the storm.

If we were, then there would be a me, and there would be a simulation of me, but I am not tha simulation. You keep saying that.

But then Bostrom would lose his entire point. "Are YOU living in a computer simulation?" He means us. We are not being simulated separately from our actual existence. If that were the case, Bostrom would lose the entire force of his argument.

The argument is that we ARE the simulation. Or the minds created by the simulation, if we define the simulation as the executing computation that creates our minds and feeds us sense data.

I am not at my computer. I am being fed an illusion. And even my own innermost thoughts are likewise being fed to me. My thoughts are not my own and I have no body and I live in no world other than that of the simulators. I live inside their computer, in their world.

That is what simulation theory says; and when you clarify that, the nihilism of the idea jumps right out at you. The nihilism, and the immense cruelty of our simulators.

This is the heart of what I think of as your error, but that may be just a point of miscommunication.

We have no independent existence outside of the simulation.

tl;dr: Perhaps you can help me to understand why you believe that, under simulation theory, I am typing on a computer; when in fact by assumption, I am a mind created by a computation executing in the world of the simulators; and that my body, my world, my computer, and even my own most private innermost thoughts, are only being fed to me by a computation running in the only world there is.

ps --

Lacking any input from their world to ours, there doesn't seem to be much room for a moral code. They're incapable of torturing us. At best, they can erase the data and just end our world just like that. Morals in the other direction would be interesting. Are we obligated to entertain them? Depends on the simulation purpose, and since that purpose hasn't been conveyed to us, we don't seem to be under any obligation to them. — noAxioms

These issues are already being debated in our world. What is our moral obligation to any AGIs we may happen to create? Surely this question would occur to our simulators; and since they have nonetheless unnecessarily plunged us into a world of famine, pestilence, war, and death; they are sadists. They know better (since WE know better) and they do it anyway.

If it's immoral to kill for no reason, is it moral to turn off an AGI? The thought has occurred to us; therefore it would occur to our simulators; and therefore, they are amoral sadists.

You can find plenty of discussions of this, I just grabbed a representative link. The simulators are human like us. If we asked the question, they would ask the question. Yet you say they'll gladly flip a switch and kill us all on a whim. After making tens of millions of us suffer though wars. What kind of monsters are these simulators?

People just like us.

Nihilism and horror, that's the end game of simulation theory taken on its own terms.

https://theconversation.com/if-a-robot-is-conscious-is-it-ok-to-turn-it-off-the-moral-implications-of-building-true-ais-130453

No, but it has an interface which is the beginnings of what one might look like for viewing simulation states. Yes, the controls to the tool constitute input to the tool, but since viewing simulation results has zero effect on the simulation itself, it doesn't count as input to the simulation, only input to one of many read-only tools to view the data produced by the simulation. — noAxioms

I'm satisfied on the output or interface aspect. The output could be anything, they could watch us on 3D holograms or a VR headset. It could even be immersive. They could even BE us for a while, as in the movie Being John Malkovich.

But I didn't mean to get hung up on the question. Of course the programs (running processes, sorry!) would generate output. They'd have user-visible output, as with our software, as well as internal visibility in terms of the simulators being able to see the state of their computation at any moment, just as we can.

I think we are agreeing, though, that the internal mental states of the sims -- that is, the thoughts and feelings and experiences of humans such as you and I -- are as opaque to our simulators, as they are to us! So in the end, we are a great mystery to our simulators. They probably watch the stuff we humans do and go Wow, that doesn't make ANY sense!

So the simulators can't read our minds. That means they don't have control over us. They really do not know what we'll do.

They're like a God who gives us free will, just to see if we'll choose the righteous path.

Once again, simulation theory is more like theological speculation than science.

But you know, let's note this as an open Bostrom question. Can the simulators read our minds or not? Are we a surprise to them? Or can their computer scientists just look at the code and figure out what we'll do? In which case they could ... simulate the sim, could they not.

So: Does Bostrom think the simulators can read our minds? And do we have free will? LOL same old theological questions, but I wonder if Bostrom addressed this.

Google maps can only show you specific places. You can go into a few select buildings, but your view is mostly confined to streets. With the simulation, there is no restriction of views only where the van was, taking a picture every 10 meters or so. You can go inside walls and watch the rats eat the wiring if you want, even if it's totally dark in there. — noAxioms

You could never have a 100% perfect geographical simulation. It must have a resolution, and reality is always more fine grained. You could zoom to the houses but not the pebbles in the garden or the ants in the grass. Or whatever. I don't follow why you would claim that there is "no restriction of views," of course any geographic database has a resolution far short of reality. How can you watch the rats if there's no light? Visual recording devices require light, that's a basic principle of physics. I think this paragraph confused me.

I thought they were the people, not the programs.
But you defined it earlier to mean 'the simulation processes", of which there may be many running at once, each simulating a different world. — noAxioms

Minor terminology issues. Sorry if confusion. A program is a written list of computer instructions, stored on a hard drive or static memory device of the future. It's like a recipe in your recipe drawer. It can be used to make a cake, but only if someone executes the instructions. By itself, a program does nothing.

A process is an executing program. This is standard terminology.

The sims are us. I have in the past said the the process (forgive me if I ever said program, I know better) instantiates us. What other word could you use? The execution of the simulation program somehow gives rise to our existence. Our minds and self-awareness, and our bodies and the world around us. The program brings it into existence.

I wonder if Bostrom explains how any of this works? The simulators write a program. They run the program. Somehow, you and I and the world all around us comes into being.

Perhaps you could tell me how that is supposed to happen? If it's true, then where am I right now? I'm an abstract consciousness floating above or around some physical piece of computing hardware. How is this magic trick supposed to work?

What does Bostrom say in his introduction? It's a "quite widely-accepted position in the philosophy of mind." As if that explains anything.

Note: You yet again redefine 'sims' to be the people below. Using the word in both ways is the source of so much of our disconnect. — noAxioms

The sims are the people below. You and I are the sims. I did not realize I've been using confusing terminology, but let me clarify that today.

* Program is the thing the programmers write.

* Process is the executing program. The program is used to control the circuits of a digital computer and cause it to carry out a computation. It's a physical process that requires time, space, and energy, and gives off heat. I may refer to a process as a running program or executing program from time to time.

* The sims are the result of the process, but not in any way ever explicated by anyone[/b]. It's not a principle of computer science that executing programs. You can't find any algorithm that says, "Implement this algorithm and a mind will come into being."

But if, for the sake of argument, I grant you this trick: The sims are the minds that arise out of executing the computation.

I hope the foregoing is clarifying any confusion in my terminology. Let me know if any questions or if I'm missing the point of your concerns entirely. That's possible too.

'Living in a computer simulation" is different from being that computer simulation.
— noAxioms

Wow. How can you say that? They are identical. We are the simulation. Our lives and our minds and everything around us. We're in the simulation, we ARE the simulation.

Note that by the hypothesis that the executing simulation program (the process) implements consciousness in the sims; it follows that we are not in a Descartes's deceiver situation. We are not minds being fooled by a really good video game running in our VR headsets. Our minds are instantiated by the simulation (the executing program) itself.

So how can you say being the simulation isn't living in the simulation? We ARE the simulation and we LIVE IN the simulation. You and I.

The two exist in different worlds. They're not the same thing. The simulation runs in the GS world. We exist in this (simulated) world. — noAxioms

Yes. We are an artifact, or "emergent property," a phrase I dislike, of a program executing (a process) in the world of the simulator, on computing hardware built and operated by the simulators. And out of that simulation, we arise. How? No matter, I'll stipulate it for sake of argument.

So I don't know what you mean that they're in "two different worlds?" Our world isn't real. It's an ethereal output, or byproduct, or epiphenomenon, of the ancestor simulation program being executed on future but nevertheless physical hardware belonging to the simulators.

Surely all this is clear, isn't it?

So in effect we DO live in the simulators' world. We live in the spirit-space adjacent to their computer.

Is this not one hell of a dualist theory? Where do these minds live?

I think all I'm doing is breaking down Bostrom's ideas and showing how absurd they are. Even if taken on their own terms. Later in this post you say my only objection to Bostrom is the computable mind hypothesis. I hope I have demonstrated that I can grant Bostrom that hypothesis and his idea still sucks.

But if the mind is an artifact of the executing program, then the city I live in is not the city I live in. The year I live in is not the year I live in. The body I live in is not the body I live in. They are all artifacts or "emergent properties" of the executing program. They are not real. It's a brain-in-a-vat experiment. That's what simulation theory comes down to. Even when taken on its own terms.

That's the distinction I've been trying to stress. I'd try to use your meaning, but all sorts of strawman conclusions can be drawn when one equates the two very distinct things, such as "the simulation program is conscious'" which it isn't even though you and I are. — noAxioms

I hope I've clarified my thoughts. You haven't convinced me that you've clarified yours :-)

I don't think I ever said that "the simulation program is conscious." I don't actually think I said that. But if I did, I apologize. Once and for all time, this is my statement:

A computation is executed on physical hardware operated by the simulators. As it executes, it instantiates, by some unknown mechanism, a mind. That mind is me.

This is perfectly clear in my mind. I hope it is now perfectly clear in your mind that it is perfectly clear in my mind.

It does leave you or Bostrom with the problem of telling us where these minds all live. Is this essentially a dualistic philosophy, with the minds living in some sort of spiritual realm? Or if they are in the machine, can they be measured by electrical engineers? How is all this supposed to work?

Bostrom says it's a "widely believed position in the theory of mind." I'm struck by his blithe indifference to the issues involved.

Simulation programs tend to be very simple, endlessly running the same relatively small list of instructions again and again over a relatively large data set. — noAxioms

That's not even true. When you run a simulation of the weather or of the early universe or of general relativity, you are doing massive amounts of numeric computation and approximation.

I don't know why you think simulation programs are simple. That's not true.

I know. It is still a mistake to say you are an executing program, for the reasons stated just above and in prior posts. — noAxioms

I'm a mind somehow instantiated or brought into existence by an executing program. And that raises more questions than it answers. Does it not? But I hope we're in agreement on the terminlogy now.

Presuming 'sims' is the people with this comment, else it makes no sense. — noAxioms

Yes, the sims, the imaginary people with little simulated minds that happen to be us.

It's a very weak point in his argument in my opinion, so he avoids it. To run a good ancestor simulation like this, it would require far less resources to have a good AI imitate (rather than simulate) each of the people. — noAxioms

I don't see why that would be the case at all. We don't have to waste time trying to define ancestor simulation versus AI. Each sim essentially IS an AI, or there's on AI running all the sims as mutually interacting threads or subprocesses, doesn't matter. Point is that the sims ARE AI's, they have minds, they are sentient.

They are us.

And by the way, what is the moral obligation of the simulators to us? Philosophers and critics of AI are already starting to ask what would be our obligations to any AGIs that we created. Would it be moral to kill or torture them?

By the same token, we can ask why our simulators, who art in Heaven, have cursed us with war, famine, pestilence, and death. If this is all a simulation, why do we suffer and die? Are our simulators historians? Or sadists?

We're talking about something far better than passing a Turing test since each person needs to not just type like a human, but to act and defecate and bleed like a human. — noAxioms

No they don't. We're all just ethereal beings in a non-physical real of spirit and mind. We are just byproducts of a program executing somewhere. Our bodies are illusions. And worse than Descartes, even our own minds are illusions.

Is Bostrom the world's greatest living nihilist?

But nobody has to pass any Turing test. I assume you're a fellow sentient human because I'm programmed to. To the simulators, you and I are obviously fake as heck. Only we can't see each other's fakeness because the programmers coded us up to accept each other as sentient humans.

Do you see the absurd and nihilistic rabbit hole you fall down once you accept Bostrom's assumptions?

Now your ancestor sim can go on at perhaps a thousandth of the resources needed to do it at the level of simulation of consciousness of each person. But his hypothesis requires this, so he's forced to posit this implausible way of achieving the goal he's made up. The ratio is likely waaaay more than 1000-1. — noAxioms

I don't care about the resource argument. I'll assume the simulators can harness as much energy as they like. Perhaps they've harnessed the cosmic microwave background to power the earth. I don't care what technology they use as long as it conforms to plausible future physics, and Turing's definition of computation. In other words no Halting oracles or "new concepts of computation" allowed in this game.

He tries to address this by waving away my '1/1000th' guess with 'we don't know the real number'. He calls the imitation people (as opposed to fully simulated ones) 'shadow people', and discounts this strategy, and yet gives every simulated person a shadow body and populates the world with shadow animals and plants and such, none of which is actually simulated like the brains are. Go figure. — noAxioms

Aren't those NPCs? I've always had suspicions about my neighbor. A philosophical zombie for sure.

But again, I'm not concerned with resource constraints. I'll grant Bostrom the computing power to render the entire world as it is, moment-by-moment, to a degree sufficient to seem real to us.

But how real does that even have to be? Maybe everything is in big pixellated blocks, and we are just programmed to think it's all smooth and detailed? It's back to Bishop Berkeley. Since our experience is mediated by our senses, there doesn't need to be anything "out there" at all. Just the program running in the simulators' computer that instantiates our minds.

Nothing is real, man. This is not philosophy, this is a dorm room stoner session.

Bostrom clearly thinks the simulators live in (our) future and we are simulations of their ancestors.
The initial state of the sim had perhaps some real ancestors (depends what date they selected), but we (the descendants of those initial people) are not in any way their ancestors, and thus the simulators are not in our future, only the future of some past year they selected for their initial state. — noAxioms

So we're being run by people who invented these super-duper computers and mind-instantiating algorithms, but their society has not evolved past, say, the medieval period. You know, that explains a lot! This reminds me of Philip K. Dick in his later years, when he decided that society is an illusion created by ancient people.

Yes, I agree with you that Bostrom seems to imply that history would play out more or less the same, in which case he's just fooling himself, or, if there's a script, it's not a simulation at all, but just a CG effect for a movie script, which doesn't involve people that need to make their own choices. — noAxioms

Sure. That's fine. That's possible. But Bostrom claims the opposite. He says, and apologies for frequently returning to this quote: "Suppose that these simulated people are conscious (as
they would be if the simulations were sufficiently fine‐grained and if a certain
quite widely accepted position in the philosophy of mind is correct).

If you supposed that, you would have an endless cascade of conceptual problems with the technology and the morality and the metaphysics of the idea. Where do these minds live? Do they obey the laws of physics? I think you have stumbled on the only way to make sense of Bostrom's paper. And that is to retract the instantiated mind claim. So there are future humans, and they run ancestor simulations, and they are probably sadists about it. They're human after all.

But we are not them. And the subject is closed.

Or maybe you didn't mean to take it that far.

And I buy that. Yes, the simulated people (and not the simulation processes) are self aware. — noAxioms

Yes. Us. We are the simulated people ("Are we not men? We are Devo.")

You know, I was never confused about this distinction, and I regret that you have decided that I did have this confusion, but regardless, I hope I've clarified that by now.

But he doesn't explicitly say that anybody knows how 'consciousness works'. You don't have to. You put matter together like this, and the thing is conscious. That's what the sim does. It just moves matter. It doesn't need to know how the emergent effects work. — noAxioms

The sims (us) don't have to know how it works. The simulators do. Or you're saying they got lucky one day, some industrial engineer was putting the finishing touches on the logic chip for a new coffee pot, and it suddenly printed out on his debugging console, "Hey I'm self-aware in here. Please send pr0n and LOLCats."

If it wasn't the latter, it was the former. The simulators figured out how to implement consciousness.

And how does Bostrom justify that? It's "a widely-held position." Well that settles that.

Agree. Or the biologists, which is a history major of sorts. What will they get from a sim that starts at a state resembling some past state, but evolves in a completely different direction? Not much. What if you run a thousand of them, all with different outcomes. Now you have statistics, and that's useful. Output would look like a history book. 'Watching' specific events from a selected point of view probably won't be too useful for that, but such a view would be useful to find the initial cause of some avoidable calamity (like a war) which helps our future people know what to look for to prevent their own calamities. — noAxioms

A point I made earlier. Our simulators are sadists. They want to know how WWII works out so they create 500,000 German soldiers, 18 year old conscripts with no politics at all, to die horrible deaths in the frozen winter of Stalingrad. Fifty million souls in all lost in WWII, but it's all good fun to our simulators. The camps, the nukes, the death and destruction.

Perhaps they gave us minds just so they could torture us. They could run the simulation just as well without giving us minds but that would not be as much fun.

Our simulators aren't scholars. They're sociopathic children holding a lighter to an ant farm.

Point is, that's a good starting point to resolve the 'why would such a sim be run'? I also still say that imitation, not full simulation, would be a far less costly way to achieve any of the goals mentioned. Only Bostrom requires it, but he can't force the 'future' people to do it an inefficient way. — noAxioms

Why do the simulators allow evil in the world? Why do they make us suffer? Do they enjoy it? They haven't got any more "goals" than the torturers at Abu Ghraib wanted answers. They just wanted to torture people.

But they kind of already do. They can put a thing on your head, measuring only external EM effects on your scalp (like an EEG) and they can see you make a decision before you're aware of it yourself. — noAxioms

I don't think that particular widely-quoted experiment shows what some people think it does. Our nervous system does do a lot of the work of cognition and much of it is not at the conscious level. That does not invalidate free will, sentience, or whatever.

Point is, one doesn't need to know 'how consciousness works' in order to gean what the sim needs, which is mostly focus and intent. What is our guy paying attention to? Why? The sim needs to know because the physics of that thing is dependent on it., It changes from when nobody is paying attention to it. This is done for optimization purposes, and for faking non-classical effects in a classical simulation. — noAxioms

I changed my mind about all this. It's the moral argument. If you accept Bostrom's assumptions at face value, we live in an ant farm owned by a sociopathic child.

Aaand the definition changes again. You said the sims are the programs. — noAxioms

Stop that! I never said that. If you think I said that you're mistaken. But if my fingertips, acting on impulses of their own and beyond my conscious control, did happen to write anything that ever made you think I said that ... I truly hope the explanations and expositions in this post have settled the matter.

I would NEVER have said the sims are programs. I can't believe you think I said that. Just out of curiosity, can you quote something I said that gave you the impression that I think the sims are programs? I suppose its possible I've been guilty of some sloppy language. If so, it will not happen again.

The programs are processes running in the GS world. We are humans living in this simulated world. — noAxioms

We are artifacts or "emergent phenomena" of programs executing on physical hardware operated by the simulators. They have figured out how to instantiate minds using these programs.

We are not anything. We are not anything at all. We have no world, we have no bodies, and unlike Descartes, we do not even have minds that we can call our own. It's all a horrible illusion created to torture us. Else how explain the world, the human condition, the horror you see in the news?

There is no other place to go with this. Once you accept Bostrom's thesis then look around the world, you see that benevolent simulators would not have done this to us.

Simulation theory is a nihilistic and horrifying philosophy.

Maybe we should stop using 'sims' as shorthand for this ever moving target.
Be explicit. Use either 'simulated people' (us) or simulation process (the program running in a different world). — noAxioms

We're the sims. We don't exist. We're epiphenomena of a program executing in a computer run by people with no morality whatever. Else there wouldn't be pain, death, war, and all the other stuff that theologians wonder about. At least the Christian God gives us free will. I don't see how we, being artifacts of an algorithm, have any free will at all.

Bostrom does not use the word 'sims', so it isn't on any page of his paper.
He says on page 1 (the only reference to 'conscious' on that page): "Suppose that these simulated people are conscious". He is proposing that the people in the simulated world, and not the program running in the simulating 'future' world, is what is conscious. This is consistent with what I've been saying. — noAxioms

And it's consistent with what I've been saying. The executing program instantiates consciousness.

He goes on later to presume substrate independence, which is that consciousness is not necessarily confined to carbon based biological forms. — noAxioms

I accept substrate independence. I ask only two things:

1) Conformance with plausible future physics. That's open to interpretation; and

2) Conformance to the laws and rules of computation.

But the simualted people in his proposal are based on simulated carbon-based simulated biological forms. But he must say this to emphasize the standard objection that by definition, no computer can instantiate something conscious.
Nowhere does he state that something as simple as a simulation process is itself conscious. — noAxioms

The process itself? No, it's just an executing program. But it somehow gives rise to a mind. Did I ask you where these minds exist? I think I did.

Yea, that's right. There's indeed not much point in this since your personal beliefs conflict, so you won't consider it on its own grounds. — noAxioms

That is a very wrong and unfair criticism. On its own grounds it collapses immediately into moral and metaphysical quagmires. I don't reject his assumption of computational mind. I ACCEPT it and show that it leads to the conclusion that we are the deliberate objects of torture by a cruel and amoral race. With no possible way out.

And by the way, what cruel and amoral race is that? Why it's us of course. How poetic. Our own human cruelty reflected back on us. Not a pretty picture.

What kind of philosophy is that? A depressing one, I'd say.

You keep changing what 'the sims' means, and Bostrom doesn't use the word, so I cannot say yes or no. — noAxioms

If I haven't settled this issue, nothing else will help.

Bostrom does indeed speculate that it is more likely than not that we are simulated people: that we are composed of simulated matter being manipulated by a simulation process running in some other world. He nowhere speculates that we are that simulation process itself. — noAxioms

Nor did I ever claim that. This was a real strawman post. You put many words and ideas in my mouth.

We are the byproduct, or the emergent property, or the instantiation of a computational process. When did I ever say otherwise, and what did I say to make you raise the question?

And where's the mind live? And do you understand, finally, that according to simulation theory, nothing is real. Not our bodies and not our world. And not even our own minds.

I'm very keen to respond but there's a lot to reflect on and respond to and I'm completely overloaded with work and family responsibilities so I might not respond until next weekend. As always, I appreciate your comments and this dialogue. — keystone

No worries mate.

So you take a break, poop out the pasta, and go back for more. — frank

They could never execute me. And since death warrants specify the execution date, I'd only have to hold out long enough to reach the next calendar day.

Yay!! Thanks a tonne :) — keystone

I'm gratified to know I'm being helpful. Thanks.

I agree that the program you describe halts, however, I'll focus on Turing's version, by agreeing that the program that computes 0.333... to an arbitrarily fine precision halts.
I agree that by definition that's what it means for a number to be a computable number, so by definition 0.333... is a computable number.
I agree that by that definition pi is a computable number. — keystone

Ok correct to there.

I agree that no program can compute pi to infinitely fine precision.
You might even agree with me that no program can compute 0.333... to infinitely fine precision. — keystone

"Infinitely fine precision" is imprecise. If you change it to arbitrarily fine precision, we already have well known programs for both.

Perhaps by infinitely fine precision you mean specifying all of its digits at once. Sort of like running the Turing machine for 1, for 2, for 3, ... such that all of them were run and the outputs collected into the full decimal representation of pi.

If so, we agree that no physical computer could ever do that, simply because it would require an infinite amount of computing resources: time, energy, and space. If those are finite, then so is what we can practically compute. We agree.

pi an 1/3 are both computable, so the same argument applies.

By the way, set theorist Joel David Hamkins is investigating Infinite-time Turing machines." Here's the abstract:

We extend in a natural way the operation of Turing machines to infinite ordinal time, and investigate the resulting supertask theory of computability and decidability on the reals. The resulting computability theory leads to a notion of computation on the reals and concepts of decidability and semi-decidability for sets of reals as well as individual reals. — Hamkins

Is that cool, or what!

@Michael Perhaps this is of interest.

My understanding is that a program halts if it reaches a point where it completes its execution and stops running. Do you actually disagree with this definition of halt? — keystone

I agree, adding the picky detail that it is required to terminate on a Halt state. If it terminates on an error state, it doesn't count. I believe that's one of the rules. With that proviso, we agree.

You are employing a straw man argument. I'm saying that the program that computes 0.333...to infinitely fine precision does not halt, and you are saying that the program that computes 0.333... to an arbitrarily fine precision does halt. I agree with you, but your argument doesn't address my point. — keystone

You lost me. You AGREE with me but I didn't address your point.

I am not sure what you agree with and what I didn't address.

I said: No halting algorithm can print out ALL of the digits in one execution. It would take infinitely many steps. That's against the rules for Turing machines. That's obvious.

But to be computable, a number doesn't need an algorithm to print out all its digits at once and then halt. That's impossible.

Rather, all that's needed is a machine that inputs n and outputs the n-th digit.

Now that's pretty clear. And it was very sensible on Turing's part, to realize that this is the right definition, and to avoid all talk of computations that run forever. He avoided all that. All computations must halt after finitely many steps. That is the property that characterizes computation from everything that is not computation. At least in the formal sense. Of course real life programs are designed to keep running, such as web servers and operating system kernels and the like. Still, actual computers can't compute anything that's not already computable by a Turing machine, so it doesn't matter.

So tell me what you agree with and what point I didn't address, and I'll try to address it.

I believe the term 'computable number' applies to a number which can be represented by an algorithm. Am I wrong? If so, it is a very misleading name because the definition makes no mention of computers, finite resources, or anything of the sort. I would much rather call them 'algorithmic numbers' but let's stick with the current terminology. — keystone

Turing invented the abstract computer. That's what a Turing machine is.

There is no accounting for terminology, it's mostly historical accident. Open and closed sets in topology are confusing because a set can be open, closed, neither, or both. Generations of students have been confused about open sets. But the terminology is dug in deep, there is no changing it.

When we say computable, we mean relative to Turing's definition of an abstract computer. A definition that has served computer science well ever since then.

Perhaps you are making the distinction between computer science and computer engineering! Turing machines are not made out of transistors and chips and wires; real computers are.

I hope you are not going to let yourself get hung up on the notion of abstraction. The purpose of the abstract Turing machine is to allow us to reason logically about what computers can and can't do. The abstract theory informs computer engineering, it's just not all of it.

Hope this helps. You do not want to get bogged down in confusion about the realm of the abstract versus the realm of the actual.

I believe the term 'halt' applies to the execution of the algorithm. If it cannot be executed to completion then it does not halt. — keystone

Yes, correct. But if you are referring to the idea of printing out all of the digits in one shot, you are just denying or refusing to accept the standard usage of the term. It pertains to an abstract computer, not a real one.

Are you getting hung up on that?

Your failure to see the above distinction relates to one of my central complaints about the current (bottom-up) view of mathematics: mathematicians too often obfuscate the program (the algorithm) with it's execution (the generation of output by the algorithm). And it doesn't help that we call both the program and it's output the same thing: numbers. This is where I'm trying to bring clarity to the situation by redefining terms (such as what it means to be a rational vs. a real), but it turns out that such efforts just makes you think I don't know what I'm talking about. — keystone

You seem to be unhappy that abstract math doesn't climb into the wiring cabinet and start patching cables. Do I have that right?

I don't think you don't know what you're talking about. I think I don't know what you're talking about. I'm throwing out guesses. You don't like infinities, ok there's finitism. You want things to be algorithmic, ok there's constructivism. You want there to be a minimal positive real, ok that's computer arithmetic. None of it sticks. Why do you reject the doctrines you espouse?

Shouldn't the first principles be self-evident? — keystone

Of course not. That belief was demolished by non-Euclidean geometry leading to General Relativity. Kant said we have a priori knowledge of the Euclidean and Newtonian universe, and that this was true knowledge of the world. He was wrong. We are born with an intuition of Euclidean space, but that turns out to just be a local approximation to the world we live in. Who knew, right?

We experience continua and finite numbers all the time in our physical reality. The same cannot be said about points and transfinite numbers. It is the points which must be constructed from first principles. It is infinity which must be derived, not axiomatized into existence. — keystone

Ok. I get that you feel enthusiastic about this. I'm on your side. I hope you can work out your ideas. I do think they are a little half-baked at the moment. That's an honest assessment.

If you don't like the axiom of infinity you are a finitist, but perhaps not an ultrafinitist. You might be interested in learning about finitism[/ul].

Euclid's line is so simple -- breadthless length. It's hard for anyone to say that's not self-evident. And I can easily construct a point from that line - I cut it and the midpoint emerges. The bottom-up view is far less self-evident. Somehow combining sufficiently many objects of no length results in an object of length. And even though nobody has a good explanation of how this works we nevertheless proceed by saying that the continuum constructed in this fashion is paradoxically beautiful and only to be seriously discussed by the experts. Really? — keystone

If I stipulate that every mathematicians that ever lived is a bad person for doing whatever you think they did ... would it help?

Can you step back a tiny bit and see that if every smart person who ever lived is a dummy acting from bad faith ... well, maybe it's you, and not them. Maybe a lot of people already thought about these issues. Excessive grandiosity is often an indicator of crankitude.

I mean actually, if you feel that everyone else is wrong and you are right ... you should keep this to yourself! You don't want me to know you feel that way.

You expect a deeper structure to my line, such that, say, when I cut line (0,4) at point 2 that this involves identifying a pre-ordained point and isolating it by means of a cut. That's not what I'm proposing. My line has no deeper structure or additional properties beyond continuity and breadthless length. — keystone

More breathless than breadthless. (Sorry couldn't resist!)

I've come to realize that I've been heading down the wrong path by saying that my line is a bundle of 2ℵ0
2
ℵ
0
points. — keystone

When I Quote your posts, those expressions always render as one character per line. Makes it hard to read.

Anyway ok, you claimed that many times, and now you're not sure. That's fine, since your theory is a long way from being able to define $2^{\aleph_0}$. That expression is a fairly sophisticated "bottom up" construction.

I ended up here because I was defending against your arguments that my line has gaps. It turns out that my defense has just made you expect a structure to these points. It is better for me to just claim as a first principle that the line is continuous. As such, I'd like to discard the 'bundle' argument. — keystone

Abandoning the entire bundle argument? So the real line is no longer made of a countably infinite union of overlapping open intervals, each characterized by a particular computable number it contains? I thought that was a pretty good thing to achieve agreement on. You are abandoning this now?

As far as gaps go, they're important. The completeness property, aka the Least Upper Bound property, aka Cauchy-completeness, is the defining characteristic of the real numbers. Accept no substitute! If someone tries to sell you a model of the real numbers, ask them if it's complete! Mathematical shopping advice.

Instead, the structure you are looking for comes from the cutter, not the line. I can cut line (0,4) anywhere I want and label the midpoint that emerges '2'. In labelling that point '2' I am making an agreement with myself that any subsequent cut I make, I will label it to maintain the structure we have come to expect with numbers (as captured by the SB tree). For example, if I subsequently cut the line (2,4) I agree to label it with a number between 2 and 4. — keystone

Your focus on the Stern-Bricot tree is ... well some adjective anyway. Do you know the infinite complete binary tree? Start with a binary point at the top node. Underneath each node from now on is a left node called 0, and a right node called 1. The infinite tree has countably many nodes, but uncountably many paths through the tree. Each path corresponds to a real number and all the real numbers are represented by some path. Or sometimes two distinct paths, as in .0111111... and .10000... I find it helpful to visualize the real numbers that way sometimes.

But your cut idea, you just can't get to enough of the real numbers that way, with each number represented by finitely many cuts, if that's what you're doing.

In the top-down view, the cutter/mathematician plays a central and active role in maintaining structure and, moreover, actualizing objects....not unlike the the observer in QM...hmmm.... — keystone

Ah, well that's like the active intelligence of intuitionism. You should read up a bit on this too.

Here's what Wiki says:

In the philosophy of mathematics, intuitionism, or neointuitionism (opposed to preintuitionism), is an approach where mathematics is considered to be purely the result of the constructive mental activity of humans rather than the discovery of fundamental principles claimed to exist in an objective reality.[1] That is, logic and mathematics are not considered analytic activities wherein deep properties of objective reality are revealed and applied, but are instead considered the application of internally consistent methods used to realize more complex mental constructs, regardless of their possible independent existence in an objective reality. — Wiki

How did we ever survive before Wikipedia?

There's nothing actually infinite about the line. What is infinite is the potential for the cutter to make 2ℵ0
2
ℵ
0
cuts to the line. — keystone

Well you are a long way from making that many cuts when you start by denying even countable infinity!! Isn't that a little inconsistent?

But since (1) an infinitely precise computable irrational cut requires the completion of a supertask, — keystone

I don't think you should tie in supertasks. There's already enough fuzzy thinking on that topic going around.

(2) non-computable irrational cuts cannot be algorithmically defined and (3) the cutter can only ever perform finitely many cuts, this potential can never be completely actualized. When working in 1D, the mathematician will forever be stuck working with a finite set of lines and points. However, because the mathematician can continue to make arbitrarily many more cuts (i.e. any natural number of cuts), that set can grow to be arbitrarily large (i.e. have any natural number of elements). — keystone

Very difficult to get a model of the real numbers while denying infinite sets. It's been tried, really. The constructivists have a pretty interesting model. They even have their own version of completeness, even though their model is only countably infinite. Maybe you should look at constructivism.

Of course that's possible (and likely). After all, that's why these ideas are being discussed in this chat forum and not eternalized in the Annals of Mathematics. But since yelling 'I'm not crazy' only makes one sound crazier, I won't challenge this point further and hopefully the ideas will eventually speak for themselves. — keystone

You have seemingly expressed the idea that you are right and thousands of years of the smartest people in the world looking at these issues are wrong. That's always a bad sign. You should note it yourself, as an objective check on your own thought process.

I know how it sounds, that's why I'm reluctant to talk about QM and paradoxes at this time. — keystone

Ah, you have discretion. Well actually that's a point in your favor. But QM, really? That's a stretch.

When I communicate the fundamentals, you ask for the implications. When I communicate the implications, you ask for the fundamentals. — keystone

And I've understood precious little of either :lol:

My only hope is that at some point the fundaments become coherent to you, after which the implications will naturally follow. I admire you for sticking with me for this long given that you think my ideas have so far been incoherent. — keystone

I'm in shock that you're abandoning the little open intervals. I was clinging to that like a reed in the ocean.

I ask ChatGPT to give me the Latex equivalent of an expression and I insert that Latex string with the math tag. — keystone

Jeezus. That ain't workin'. Maybe some control characters in there. And don't use ChatGPT, it rots your brain. Lot of foolishness floating about in the culture lately.

They can't actually give you an infinite amount of pasta. — frank

The buffet at the Hilbert hotel!

Say you're being executed, what would you order for your last meal? — frank

Olive Garden Never Ending Pasta Bowl of course!

We can only speculate as to the purpose of running this kind of simulation, and thenature of the output depends on that purpose. Maybe it is a sort of detailed history book. Maybe it is pictures. Maybe it's just a stored database. Maybe the purpose is simply to see how long humanity lasts until it goes extinct, in which case a simple number might be the output.
I did mention the nature of the output later in the post above, such as the example of the output of google maps for instance, a very useful interface for display of simulation results. — noAxioms

Ok, I'll concede that the sim program has some kind of graphic output that lets the simulators watch the ancestor simulation in action.

BTW Google maps is not a simulation, it's a Geographic Information System. And it takes inputs, such as the zoom and recenter operations from the user.

You define 'the sims' below to be the programs in the GS world. — noAxioms

Yes, what else could we be talking about? Bostrom: "Are YOU living in a computer simulation?" My emphasis. Me. You. Each of us. We are a program being run by the simulators. What else do you think he's talking about? The simulators run programs that that are us, in some magic way.

I see no assertion that either a program (a static chunk of software on perhaps a disk somewhere) or a computer process (the execution of said program on some capable device) with no inputs would have what you might consider to be an 'inner life'. Bostrom doesn't say this, and neither do I. — noAxioms

Process. Executing program. If I said program, I should have written process, or executing program. I actually kind of doubt I said this, but if I said program and not executing program, I meant executing program.

What on earth else do you think Bostrom means? Are you living in a computer simulation? What else can he mean?

They have knowledge of it in the same way that I have knowledge of my wife having an inner life. If that's going out on a limb, then one is presuming solipsism. But my presumption of my wife having inner life does not let me know what it's like to be her.
The simulation can report what each person thinks and feels. The simulation has to have access to this because physics is dependent on what people are thinking. So it can report that Bob at time X is paying attention to his laser experiment and is feeling frustrated that he cannot get the setup just right, and his bladder is getting full. It can show his point of view if that helps. Make up your story. What interface tech exists for them is speculation on our part. Humans are notoriously bad at predicting 'future'/higher tech. — noAxioms

Agreed, to a point. It's odd that Bostrom thinks the computers instantiate self-awareness in the sims, yet show little interest in it. Well it's a small point, virtually nothing in Bostrom's thesis holds up anyway.

I put 'future' in scare quote because maybe the simulation is being run in the year we call 1224 or something. Maybe in the GS world, advancements came much sooner, and in our simulated world, things happened much slower, and we're far behind them despite 8 more centuries to learn. If that is the case, the Gregorian calendar is only meaningful in our world, and they number their years differently. — noAxioms

Bostrom clearly thinks the simulators live in (our) future and we are simulations of their ancestors. Though of course you're right, there's no reason that would be true. Maybe we're the Jetsons and not the Flintstones.

Geez, another strawman. I make no such claim. Bostrom presumes that consciousness is physical/computational. That assumption is no more an explanation of how consciousness works than is the non-explanation by anybody else. — noAxioms

Ok.

I didn't say they figured out how consciousness works, — noAxioms

Bostrom says that. That's the one great revelation I had from this thread. Bostrom explicitly states that the sims are self-aware, and blithely justified is as "it's widely believed."

nor did I say they focus only on behavior. The simulation needs to know what each persons mental focus is, what his intent is, because physics as he describes it depends on it. One doesn't need to know how consciousness works to do this. — noAxioms

Ok.

There's no 'them' to communicate to. OK, observers in the GS world can watch, (very similar to the google map interface), but they don't affect anything since that would constitute external input. The running of any sim doesn't require observation of any kind, but why run it if nobody's going to pay attention to the outcome? Yet again, the output is dependent on the purpose of running the thing, and we can only speculate on the purpose. — noAxioms

Of course they are watching, they are running an ancestor simulation. Of course I have no idea why Bostrom chose that particular reason, since with all our impressive computing power, WE don't run ancestor simulations. Maybe we're their pr0n hub. They like to watch us mate. That's more likely than that the history majors are running ancestor simulations.

A full classical scan of a person provides access to internal physical states, and that's all that's needed to simulate the person, per naturalism. But such a simple simulation would not have physics supervening on mental states like the sim Bostrom proposes, so the one he speculates is far more complicated and requires access to mental states, not just physical states. — noAxioms

Right. You can map all the neurons and you would not know what someone's thinking. Although impressive work in that direction is being done by the cogsci crowd, so I could be proven wrong soon enough.

Yes, with that quote, I was. I don't know the purpose of the sim, and I don't know what tech is available to the entities running the sim, so I can only speculate as to how they would choose to 'observe' it. — noAxioms

Ok. I'll concede that they have a graphic or numeric output that can be observed. And they can talk about their mental states, as we often do. "I think I'm hungry."

Ah, not us, but the program in the GS world. Apologies for getting that wrong. Sims then typically not conscious, especially since it typically lacks input. — noAxioms

ARGHHHHHH! The sims are conscious. That's on page one of Bostrom's paper. We are the sims. After all this, are we not at least agreed on this?

Me saying what the output would be is definitely making stuff up. Me knowing what a simulation is and how it typically works is not making stuff up, since I did it regularly. — noAxioms

Ok.

Our opinions definitely differ, but I'm trying not to assert opinions. I'm trying to interpret what Bostrom's opinion is, and how he attempts to back it. — noAxioms

That's the funny thing. You have said you don't agree w/Bostrom. And for some reason, that makes you want to put great effort into explaining his wrong position to me.

One more thing: Last night, you said: "If 'the sims' is a reference to the simulation software, program, or process, well that's a different answer since people are not hypothesized to be any of those things."

Have you retracted that yet? It's Bostrom's thesis that people ARE hypothesized to be those things. "Are you living in a computer simulation." Bostrom speculates that WE are sims.

Surely we agree on that, at least, yes? No?

I was trying to go along with your idea of engineering math. If the mention of an engineer and a mathematician working together doesn't help, then fine - we'll drop the idea of engineering math. But, I disagree with your statement that none of it makes sense. — keystone

Ok. So far, after all this, what I understand of your idea is that the real line consists of a countably infinite set of overlapping open intervals, each containing a computable number. So far so good? But this is not a very deep idea, there are many ways to express the reals as a union of open intervals.

What? There's no difference with respect to algorithms. Consider 1/3 = .333...
— fishfry

Consider the follow program which writes the specified fraction in the specified base (NOTE: You can skip over the code):


For 1/3 in base 10, this program returns nothing because the program does not halt. After all, it's trying to compute the sum of an infinite series. Impossible. — keystone

This is a common misunderstanding of what a halting program means.

The number pi is computable. Clearly no computer program can generate all the digits in one go. But that's not what halting means.

Pi is computable because, given a positive integer n, I can run the program for a finite number of steps, and output the n-th decimal digit.

It's absurd to claim that 1/3 is not computable or not algorithmic (or whatever you're claiming) just because its decimal representation is infinite. Given n, the n-th digit is 3. That's a halting program. Therefore 1/3 is computable. Likewise pi.

If I were to write a program for each of those computations, the former would halt and the latter wouldn't, similar to the 1/3 example above. I understand why you and everyone else think they're exactly the same, but, in the purest sense, they are algorithmically different. Do you not see that? — keystone

No. I deleted some text but you were trying to convince yourself that 1 and 1.0 and 1.0000... are different numbers. They are not. And again, it's because you're confused about what a halting program means. Now that you understand it (now that I've explained it to you), you are no longer confused.

All an algorithm is required to do is, given n, output the n-th digit in finitely many steps.

It's not that I'm incorrect. It's that mathematicians have been so sloppy with the distinction between reals and rationals not realizing that this distinction truly matters, especially from a top-down perspective. — keystone

You are delusional. Could it be that you are the one who's confused, and not mathematicians?

Does my explanation of what's a halting algorithm to compute a real number cause to you reframe your understanding? Input n, output the n-th decimal (or any base) digit in finitely many steps.

Allow me to clarify: I want to distinguish between a real number and it's corresponding real algorithm. A real algorithm corresponding to π can be written perfectly with finite characters, such as:

π⎯⎯=4−43+45−47+49−⋯ — keystone

What are you doing that, when I quote your numeric examples, the quote text comes out in a column?

From the bottom-up view π is equivalent to π.
From the top-down view π is not equivalent to π (any more that an algorithm is equivalent to it's output, or a line is equivalent to a point). — keystone

Did you understand my point about halting? Does it help you understand what's going on?

Turing defined computability another way. A real number is computable if, given epsilon > 0, there's an algorithm that generates an approximation to the number within epsilon. You can see that this amounts to outputting the n-th digit. The point is that a number can be computable by an algorithm even if the number has infinitely many decimal digits, like 1/3 or pi.

That's because when it comes to reals, mathematicians have been so sloppy with their terminology. I'm trying to make things more precise. — keystone

Is that really likely? Or is it more likely that there are some basic things you don't understand, like the definition of halting or the fact that it's trivial to write the real line as a countably infinite union of open intervals?

First off, I'm only claiming to (at least partially) solve the issue of how to philosophically interpret QM. I'm certainly not claiming to have solved quantum gravity or anything like that. Are you saying you want me to jump right to the implications of the top-down view without even explaining the top-down view? — keystone

It would help me to understand what you're talking about, and why.

After all, you say your top-down view starts with the real line. But I say, I don't know what the real line is. How do you know there is any such thing unless you construct it from first principles?

I'm certain that without understanding my view you'll just think I'm injecting quantum woo into the top-down view. If you stick with it, what you'll see is that quantum intuitions follow from the top-down view. It is this way because QM is a top-down view of reality whereas classical mechanics (CM) is a bottom-up view of reality. — keystone

This sounds very cranky.

Whether we're talking about mathematics or physics, the bottom-up view has been undoubtedly and demonstrably useful. It's just not correct at a foundational level. The reason why we struggle to interpret QM is because the mathematical top-down view has been neglected. Zeno was the first canary in the coal mine urging us to consider it. — keystone

Cranky. Grandiose claims not backed by anything coherent.

You've been of great help to me so far and I greatly appreciate that. If you ever want to call it quits I will accept that, thank you for your help, and that will be the end of it. Of course, I reallllly hope that doesn't happen... — keystone

I'll slog on a little longer. It would help if you'll engage with my key point tonight, which is that you've been misunderstanding the nature of halting with respect to computable numbers.

Can you see that 1/3 = .333... is computable, because the program "print 3" halts in finitely many steps for an n, giving the n-th decimal digit of 1/3?

You seem to have a dualistic definition of 'will'. All of your examples (pacman, p-zombies) are dualist/VR references. Bostrom's hypothesis is not. He's not proposing we're in a video game. All this has been said before. — noAxioms

I'm coming to the end here. My interest in this topic is far exceeded by my word count at this point. At one point I thought I understood the sim/VR distinction. But once I found out that Bostrom explicitly assumes that the simulation implements consciousness, the distinction becomes moot.

That's what a simulation is, yes. It has an initial state conveyed to it, and that is input of sorts, but once the simulation begins, there is no further input of any kind. If there was, it ceases to be a simulation. I've run plenty of these myself. It was my job for a while. The sims would run without any I/O at all for perhaps a week, and I don't think results were available until the end, but they could be reported as they happen. — noAxioms

Ok, no input.

Output (state of system at any given time) can be had any time, often at the end, but it doesn't have to be. A weather sim is a single simulation of a storm, and it could output the stats of the storm at regular intervals, or it could wait until the end and output the whole thing in a lump. It has to complete in hours, not days, to be useful. My chip sims were a little difference since each chip was run through a series of discreet tests, mostly designed to see how fast you could clock it before it started misbehaving, but also to check the design for bugs. Those sims still output everything at the end, but they didn't have to. — noAxioms

You did not answer the question. Can you see that?

What is the output? I did not ask WHEN is the output. I asked WHAT is the output.

They don't. It makes no more sense than asking what it is like for a human to be a bat. — noAxioms

So the sims have an inner life (one of Bostrom's hidden assumptions) but the simuilators have no knowledge of it? You are really out on a limb. They care out their ancestor sims act but not what they think and feel, even though (somehow) they managed to make them think and feel?

Same way it happens in the real (materialist) world: Particles interact and do their thing. Your experience is a function of matter interactions (not so according to someone like Chalmers, whom you referenced with the p-zombie mention above). — noAxioms

So YOU know how consciousness works. Why do you bother even trying to communicated with one so ignorant as me, who doesn't think ANYONE knows that?

The simulation itself cares about what you're thinking, but only because it needs to change physics due to it. The runners of the simulation may or may not care. Certainly they don't have enough people to care about every single individual. It's an ancestor simulation of the whole human race. They perhaps want to see what history unfolds, and they care no more about what anybody is thinking than you do about what anybody is thinking. You only care about what they say to you, what they do. You may wonder what goes on inside, but that's a motive for a single-person simulation, not a planetary scale one. — noAxioms

So step one, they figure out how to implement consciousness using computers; and step two, they entirely ignore that and focus on behavior.

And again, how is that behavior communicated to them? What is the output? What is the output? Ask yourself if anything you're saying makes sense?

If 'the simulators' are those that put together the simulation, who want the ancestor sim, then they have perhaps access to the same data as we do with a pimped-out MRI scan: A picture of where the matter is. — noAxioms

An MRI does not provide access to internal mental states. You know that.

You're not getting thoughts from that. To log thoughts, something needs to interpret that matter state and render it into language for readable by the simulators. I suppose such log files are possible, but much of thoughts are not in language form.
And per above, if this is the sort of detail one wants, it makes far more sense to simulate one or a very small number of people. So the motives are probably different for the ancestor sim. — noAxioms

You're just speculating about your own confused ideas. You are not making sense. Perhaps we're at a point of putting this convo to rest. Bostrom says the computers implement consciousness. And I am asking you, what are the outputs of the simulation?

Up to them to design a way to do it that is useful for their purposes. I suppose one could insert a sort of point of view interface that lets one look from any event anywhere (much like the little guy you can steer around in google maps), and lets it move at the observers control. The sim would need to save all state (and not just current state) for this to work since it probably wouldn't be useful if it was 'live', displaying only what constitutes the current state of the sim. — noAxioms

Ok. I'm asking you question you can't answer. And instead of saying, "You know, you have a bit of a point there," you're just making stuff up. And I'm getting a bit annoyed. Granted I annoy easily sometimes but this is one of those times.

I presume that 'the sims' are the humans in the simulation.
The hypothesis is that the sims are us, so tautologically they're as self-aware as you are. — noAxioms

The sims are programs. What are their outputs (he asked again).

If 'the sims' is a reference to the simulation software, program, or process, well that's a different answer since people are not hypothesized to be any of those things. — noAxioms

That is exactly what Bostrom is hypothesizing!

Could we agree to disagree? Could you accept that you can't answer any of these questions except by making stuff up? This has been an interesting convo but this last post did not have any content IMO. Just handwaving about questions you can't answer. I don't want to leave in a huff, but I might have to leave in a minute and a huff, as Groucho said.

I agree. I keep hoping for an interesting idea to appear, but so far there is nothing novel about the mathematics. If one studies existing mathematics one begins to get a recognition of what has been established. Exploration is the soul the subject, but one does not explore the heart of Africa by strolling around city park. Sorry ↪keystone . Perhaps when you present your ideas in 2D instead of (rather boring)1D (and the mind-numbing SB Tree) something of interest will appear. Philosophically, however, your ideas of potential points may go somewhere, but I don't know what has been done along those lines. — jgill

:100: :100: :100: :100: :100:

Yes and no.

Yes - The person tasked to execute the cut is an engineer doing engineering math. He knows he'll never be able to cut the line exactly at π so he cuts an interval containing π to give him wiggle room - kind of like a safety factor.

No - The person tasked to generalize all engineer actions is a mathematician. Instead of assuming any particular engineer, the mathematician aims to describe the actions of the 'arbitrary engineer'. Instead of saying that the interval width is any particular value, the mathematician just says that the interval width is ε2-ε1, where ε1 and ε2 can be any arbitrarily positive number.

The cut of (-∞,+∞) at π is generalized as (-∞,π-ε1) U π-ε1 U (π-ε1,π+ε2) U π+ε2 U (π+ε2,+∞) — keystone

What? You know, none of this makes any sense. [He's crabby tonight]

Computable reals are identified with their algorithms.
Computable rationals are found by executing their algorithms. — keystone

What? There's no difference with respect to algorithms. Consider 1/3 = .333...

Yes. I would like to distinguish between real numbers and real algorithms. — keystone

Of course, because they are entirely different things, and there are a lot more real numbers than algorithms.

A computable real number WOULD be computed if you finished executing the corresponding real algorithm, but you can't; so, the real algorithm only ever defines an interval within which the real number is inside. No real number can ever be isolated. — keystone

I think I am nearing the end here. You just are not making any sense. You've just made all this terminology up.

... [stuff omitted]

, which is a real algorithm.
1.0
1.0
is a rational number. — keystone

If there is a difference between 1.0 and 1.00000... you are off on your own. I can't hold up my end of this. Nothing you write is correct.

I want to avoid talk of the existence of an actually infinite set. We need to frame it in terms of the potential to create an arbitrarily large set. It is very important that the endpoints be rational, otherwise nothing is gained by defining π using intervals. — keystone

Not feelin' it tonight.

Point taken. I will be more judicious. SB-tree aside, I will grant that I didn't need to use a single diagram for the discussion so far. Interval notation would have been entirely sufficient. I was just hoping that you would warm up to 1D diagrams because when I go to 2D it will be very hard for me to describe what I'm thinking with words. I suppose I'll cross that bridge when we get there. — keystone

I don't see where this is going. I might be doing you a disservice by encouraging you.

It almost sounds like you're suggesting that I'm saying that (-∞,+∞) is the union of infinite little intervals. — keystone

You have said so many times, or at least I have understood you to say that. And besides, it is. Just not pairwise disjoint open intervals.

It is not. With the top-down view, we don't construct (-∞,+∞), rather we start with it. Engineer1 may cut (-∞,+∞) five times. Engineer2 may cut (-∞,+∞) five million times. What the mathematician would say is that the 'arbitrary engineer' will make N cuts, where N is an arbitrary natural number. The is no 'privaledged engineer' who has a system that has been cut infinitely many times. Rather, each engineer must work within their own finite system. — keystone

Ok whatever.

I do feel like we're very close to being on the same page now! — keystone

Ok that's good. Can we turn the page?

Let's save the paradox discussion for later. I only mentioned it at this point because you asked why a mathematician would care. — keystone

Then you give me no reason to care. You are not going to "solve QM" with your line of discourse.

The real number is interior to the interval defined by the corresponding real algorithm. However, it doesn't necessarily have to be at the center. ε1 and ε2 don't have to be equal. I do think your 1/n values for epsilon works, but I'm not sure if we need to constrain the values of epsilon as such. If we're cutting (-∞,+∞) then it seems to me we should be as general as possible and say that epsilon can be any positive number - even 5 billion. — keystone

Fine.

I believe the following:

1) The following two algorithms (written with a finite number of characters as infinite series) correspond to e and pi:
— keystone

I have no problem with you identifying the various computable real numbers with any one of the many algorithms that generate their decimal digits. Or, as a mathematician might do, identify the number with the equivalence class of ALL such algorithms, so that the real number does not depend on which algorithm you choose.

However, you can't do the noncomputable numbers that way. And there are a lot of them.

I'm not sure how your idea of approximation works. If your computer has to truncate the series, then you are only defining an interval around pi or e. You are not being exact enough.

Am I following you?

2) It is possible to compute the partial sums to a finite precision (e.g. π can approximately be represented as 3.14).
3) It is impossible to compute the complete sums to infinite precision (i.e. π cannot be represented as an infinite decimal number).
4) The algorithm itself does not apply any restrictions on the precision (.e. imprecision is only introduced during computation).
5) To prevent imprecision from being introduced, one should work with the algorithm and delay the computation for as long as possible.
6) There are algorithms for performing arithmetic on infinite series (i.e. algorithms on algorithms).
7) It is possible to compute the partial sum corresponding to π+e to a finite precision (e.g. π+e can approximately be represented as
e=10!+4+11!−43=143
)
8) It is impossible to compute the complete sum corresponding to π+e to infinite precision (i.e. π+e cannot be represented as an infinite decimal number).
9) Such arithmetic algorithms itself do not apply any restrictions on the precision (i.e. imprecision is only introduced during computation).
10) To prevent imprecision from being introduced, one should work with the arithmetic algorithm and delay the computation for as long as possible.
11) One can avoid computation altogether and just speak in terms of algorithms. — keystone

None of that adds anything. If you are defining pi by some interval around it, I'm sure you can get it all to work. It's like error bars in engineering calculations.

Wait I think I've got it. You are doing Engineering math. In particular, when you have a number x, you also have error bars, so it's really x plus or minus a little wiggle room. And you are taught how to calculate that way, how to calculate with the error bars.

Is that helpful? Is that what you are doing?



https://en.wikipedia.org/wiki/Engineering_mathematics

I'm taking (11) seriously and avoiding computation. By doing so, I'm not approximating anything; By sticking to the algorithms I'm working with perfect precision. While computers can work with algorithms, I'm not talking about the finite arithmetic you are referring to. — keystone

You seem to be going off in directions. Those lists, they don't tell me much. I get that you are doing arithmetic with error bars. That's good, that's how they build bridges.

This is false. I think that non-computable real numbers exist but only within intervals. They do not exist as isolated objects. Since numbers are isolated by cuts and cuts are described with algorithm, we cannot even describe how to isolate non-computable real numbers. — keystone

Ok. So in your ontology, there are:

* Computable numbers, which have algorithms, or are identified with their algorithms, or are found by executing their algorithsm. Not sure which of those you mean but they're all about the same. But each computable number is the number that WOULD be computed if you finished executing the algorithm, but you can't; so each computable number is a number inside a little interval.

Have I got that right? And

* Inside each of these computable intervals, live all the noncomputable numbers.

I think all that's fine. In fact I could go you one better. Consider for each computable number $c$, the set of open intervals $(c - \frac{1}[n}, c + \frac{1}[n})$, for n = 1, 2, 3, ...

That would give you a countable set of open intervals whose union is the real numbers, including the noncomputables. But you'd never have to "identify" a noncomputable. And in fact each of the endpoints $(c \pm \frac{1}[n}$ are themselves computable.

What do you think of that? Would you say that's a reasonably fair mathematical model of your idea?

Then I'll keep trying until you quit. — keystone

Why? The diagrams make it harder for me to read your posts. I think the diagrams are important to YOU. But they are not in general doing me much good. What if the overabundance of diagrams was increasing the likelihood I'd quit? You can see that under that hypothesis, you are acting against your own interests by battering me with diagrams.

It may be impossible to convince you to adopt my view, but I'll be fully satisfied if, by the end of this discussion, you can at least argue my position, even if you don't accept it. — keystone

How does battering me with diagrams help? I'm trying to understand your view and your diagrams in general don't reach me and they also make it more difficult for me to read your posts.

I do think the diagrams are very helpful for YOU. So you should do them, just be judicious in how often you include them in posts.

As described above, I care about the algorithms, not the numbers - plans, not the computations. The figure with epsilons illustrates the algorithm defining the cut corresponding to π. As I said earlier, it illustrates the plan, not the execution of the plan. To execute the plan then I need computer arithmetic, but I'm only interested in the plan. — keystone

So far I get that your system involves little intervals centered at the computable numbers. I think intervals of 1/n on each side of each computable gives the same topology as your idea of using all the truncations of the series given by the algorithms.

Are we on the same page here? I really feel that we are.

The current philosophical foundation is riddled with actual infinities and paradoxes. — keystone

You have infinitely many truncations and infinitely many little open intervals around each computable.

And you surely aren't going to resolve the standard set-theoretic paradoxes with your intervals. I don't see the connection at all.

Mathematicians have elegant ways of sweeping these paradoxes under the rug (like Russell's Paradox, Riemann's Rearrangement Theorem, the Dartboard Paradox, Zeno's Paradoxes, etc.), but they're still there. — keystone

Viewing the real numbers as the union of a bunch of open intervals centered at the computable numbers isn't going to resolve those.

Please feel free to show me how to resolve any paradoxes with what you've talked about so far.

However, if you believe there's nothing under the rug, it becomes harder to convince you to care. — keystone

The paradoxes have been resolved mathematically over a century ago. The philosophical paradoxes are in fact of interest to me, but not of high interest.

But I am not saying there are no problems. I'm saying that I can't imagine how your computables-and-intervals idea solves anything. Please give me an example of how this would work.

I see a paradigm shift towards a top-down view having significant consequences across philosophy, especially in the interpretation of quantum mechanics. Such a statement might not seem 'beefy', but let me just say that truth has a history of being useful, even if its utility isn't immediately apparent when it's uncovered. — keystone

Russell's paradox and QM as well? Please, show me how this is supposed to work.

I said things like...

"I'm familiar with these methods [of building reals from the empty set]. I believe there is a bottom-up and a top-down interpretation of them. I'm not satisfied with the orthodox bottom-up interpretation of them"
"Pi is just as important in the top-down view as it is in the bottom-up view. However, as with many other things, it just needs a little reinterpretation to fit into the top-down picture."
"Considering epsilon's role in calculus, let me just say that with some reinterpretation, calculus can be elegantly integrated into the top-down perspective without the need for infinitesimals." — keystone

None of those things have to do with the philosophy of math or with problems thereof. Calculus was formulated without the need for infinitesimals in the late nineteenth century. It's a terrific intellectual achievement, but it's a solved problem.

...but I should have explicitly said that I'm trying to patch up the philosophical basis of the real numbers. — keystone

Yes it would have been helpful. Glad we've established that, I find it quite helpful to understand what it is you are doing.

Numbers are the objects of computation, while algorithms are the objects of plans. I aim to shift the concept of reals from numbers to algorithms, from computations to plans. As such, I'm not proposing an alternate number model of the reals. I'm proposing an algorithmic model of reals. This model is structured very different. For one, while the real numbers are used to construct/define the real line, the real algorithms are used to deconstruct/cut the real line. However, I endeavor to show that switching to the top-down view has absolutely no impact on applied mathematicians, even those working with calculus. — keystone

Ok. So as far as I get this: The real numbers are made up of a bunch of open intervals centered at the computable reals. Is that right? And FWIW I think your truncated algorithm idea will give the same reals as my plus/minus 1/n intervals.

Who makes that claim? Quote it please. If you can't do that, then you're making a strawman assertion. — noAxioms

The voices in my head. Put there by our Simulator who art in heaven.

Not minds/people in the GS world, no. The claim is that we (the simulated people with yes, simulated minds) are in this simulated universe, and not in the universe running the simulation. — noAxioms

Yes, we're characters in a video game, with the assumption that Ms Pac-Man has an inner life. I believe I rejected that assumption a while back.

A simulation of a person without will would be a simulation of a body in a vegitative state. — noAxioms

I don't see that. Isn't a simulation of a person without a will exactly what they call a philosophical zombie? It would literally be a terrific chatbot operating inside a highly realistic flesh and bone bot. Your neighbor, for instance. What makes you think they have a will?

What, my saying 'deliberate'? You seem to be putting words in people's mouths that they didn't say, and I don't find you to be an ignorant person. — noAxioms

Perhaps I over reacted.

Not the simulation being discussed here, correct. A running computer process forever without inputs by definition cannot be conscious any more than you would be without inputs ever. — noAxioms

Hmm. That raises some questions. The simulation program has no input. You write the code, then you execute the code and it does what it does.

What is its output? How exactly do the Simulators examine its inner life? In other words, they run the program, and inside the program I come into existence. Me with my subjective experience. (How does that happen? Remind me please). Clearly they are interested in what I'm thinking and experiencing ... or are they only interested in my actions? So two questions:

1) Do the simulators have access to my internal mental states, and if so, how? Copious log files of everything I'm thinking? and

2) How do I perform actions for the Simulators to watch? They're running ancestor simulations, so they must want to see what I'm going to do next. How do they "watch" me? What are the outputs?

I have a very loose definition that you would not like, but my opinion there is irrelevant. The chatbots (which perhaps imitate, but not simulate anything) at least have input, but so does a thermostat. The simulation in question does not. — noAxioms

You are avoiding the question of whether the sims are self-aware? I didn't understand this remark.

Click on "View History", then "Pageviews". It's a crude estimate of the popularity of a topic. For example, group theory gets 513 views per day and non-standard analysis gets 80. — jgill

That is really cool, thanks! I think I'll be hooked on looking these up now.

@Michael

I have tracked down but not yet had the time to read Benacerraf's famous paper on the subject,
Tasks, Super-tasks, and the Modern Eleatics (pdf link)

I'll let you know if I find anything of interest. I'm happy to find that my take is officially legitimate.

I'll bet I've conjectured and proven over a hundred theorems, almost all involving convergence/divergence of sequences and series of one sort or another and have never used this expression. — jgill

Modulus of convergence is a thing in constructive math. I learned about it arguing with a constructivist here a couple of years back.

I go to Wikipedia when I encounter something in math I'm not familiar with to see what the daily average of views is - a very rough idea of how popular the topic is. My own math Wiki site gets about 19 per day, and the topic is way, way off in the margins of mathematics. However, I score higher than the 3 for this topic. But thanks for opening my mind a bit.

Constructive analysis was almost a passing thought until I read about it. I would have called myself something of a constructivist in that I rarely if ever used the excluded middle - if I postulated an entity I constructed it. But reading this description shows how far I am from contemporary mathematics. Once again, I go to Wiki to see how popular this topic is. And I find it scores a 17 - not bad, but still less than my virtually unknown page. — jgill

Andrej Bauer wrote a paper called Five Stages of Accepting Constructive Mathematics. It might be of interest.

https://www.ams.org/journals/bull/2017-54-03/S0273-0979-2016-01556-4/S0273-0979-2016-01556-4.pdf

Where do you find these scores? Is that a Wiki feature?

Probably not. I was thinking of the ancient Greeks breaking apart a sold object and measuring the pieces to approximate the object's volume or whatever. But even Archimedes recognized the infinitesimal. — jgill

He was way ahead of his time.

You sound like Arkady, — noAxioms

I don't read this thread, I only respond to my mentions. So I have no idea what @Arkady may have said. If I sound like him he must be an individual of deep insight and wisdom :-)

but no, that statement is misleading. It makes it sound like the limbic system is simulated but you are not. — noAxioms

I asked where will comes from. Intensionality. Caring. Feelings. We know from biology that feelings come from the limbic system. But if I'm simulated, so is my limbic system. Raising the question once again of how a computer program can have feeeelings, nothing more than feeeeelings.

So either "I have a limbic system", or "The simulated 'I' has a simulated limbic system". Either of those wordings is at least consistent. Your opinion (and mine, but for very different reasons) of course is that neither you nor your limbic system are the product of a simulation. — noAxioms

Well I agree with that.

Nobody is claiming that a simulation of X creates an X in the simulating world — noAxioms

That's exactly what's claimed.

, which is the strawman you seem to use in your gravity example every time where you deny an equivalent straw claim that simulation of gravity would create gravity in the GS world. — noAxioms

Just pointing out that computer simulations of gravity don't attract bowling balls (clearly true) and that therefore simulations of brains do not necessarily implement minds.

That you persist in this suggestion means that yes, you're not getting it right, — noAxioms

No hope for me, clearly, after all this time.

perhaps deliberately so. — noAxioms

Oh, I have bad will. But even so, you admit I have will! Therefore I am NOT likely to be a computer simulation. I will, therefore I am. Or as the song goes ... if it weren't for bad will, I wouldn't have no will at all.

But really. After all this you have to accuse me of bad will? How am I supposed to take that?

So no, a simulation in the GS world of a limbic system does not create emotion in the GS world. I agree with that. It is exactly for that reason that the program running the simulation isn't conscious. — noAxioms

So you agree with me after all. Or at least, I agree with what you wrote here. A program isn't conscious, it does not implement or instantiate consciousness, and it does not "simulate" consciousness. Consciousness is not the kind of thing that can be simulated, unless you think chatbots simulate consciousness. Many people believe that these days.

Deliberately not getting it right. No. False. I'm not trolling you to annoy you. Why did you say that?

I'm sure you can appreciate the problem of substituting rational number approximations of irrational numbers too early in a computation. The best approach is to do all the manipulation first and only perform the substitution at the very end when the computation is required. I would rephrase this as follows:

Step 1: Manipulation of real numbers
Step 2: Computation based on rational numbers (approximations) — keystone

Was my mention of constructive analysis helpful?

This is analogous to the 2-step cutting process I outlined in my previous post. In both cases, step 2 is crude and done using computer arithmetic. It's the realm of applied mathematicians and not of interest here. I'm solely concerned with step 1. — keystone

You don't believe in the real numbers, how can you manipulate them? You think every real number can be arbitrarily approximated by an algorithm. That's false. But if your approximation only needs to be to the minimum distance in a system of computer arithmetic, then you're doing computer arithmetic.

That's pretty much what I'm saying! But instead of talking about any particular computer (which only becomes relevant to step 2), I want to remain in step 1 and talk in general terms. As such, would you allow me to say that π is (π-ε1,π+ε2), and that the value of ε1 and ε2 only need to be determined in step 2? — keystone

Only if you are doing computer arithmetic.

If you say that the above figure makes sense to you, then I can show you a 2D figure, and the benefits and consequences of my perspective will hopefully become clear. — keystone

What if none of your figures make sense to me? Your latest uses these epsilon quantities, which you've defined as the minimum possible length in a given physical computer. So you are doing computer arithmetic. Not that there's anything wrong with that! But it seems to me that's what you're doing.

If the noncomputables reals can describe continua it is because below the surface they rest upon a more fundamental scaffolding which can describe continua in and of itself. — keystone

]

You don't even believe in the noncomputable reals. And yes, they do rest on a deeper scaffolding, namely (1) Dedekind cuts; or (2) Equivalence classes of Cauchy sequences; or (3) Forget specific constructions, just write down the axioms for the real numbers, which are categorical.

So no, I'm not interested in constructive real analysis. I'm interested in good ol' real analysis. I just want to place it on a stronger philosophical foundation. I think my perspective will become clearer to you when explained in 2D. — keystone

Ah. What do you think is wrong with the current philosophical foundation? And why would a mathematician care?

From the outside, it may seem like this conversation isn't progressing, but your reluctance to accept my informal ideas has highlighted areas where I need to strengthen my arguments. So, you are indeed helping me a lot. — keystone

Glad I can help.

You did say something interesting. You are trying to patch up the philosophical basis of the real numbers. Do you understand that this is the first time that you've told me what you're doing? It's a bit of a revelation, for all the times I've said that I don't understand what you're doing.

But the real numbers are categorical. Any two models are isomorphic. So you are not going to be able to produce a "better" model of the real numbers. One representation, construction, or description gives you exactly the same set of real numbers as any other.

I have always accepted this; it's the reason that the supertask is proven impossible.

A lamp being off must always precede that lamp being on, and so the sequence off, on, off, ..., on cannot model a lamp. — Michael

At exactly midnight, there is no time interval or delay prior to it that doesn't jump us back into the sequence. This is because midnight is a limit point of the sequence.

But I've already discovered that Benacerraf and others have viewed this problem exactly as I do. So my position is valid, or you should take your objections up with them. The problem does not defined the lamp state at midnight and it can be anything you like.

If the button is pushed at 23:00 then the screen will display 23:00 until the button is pushed again at 23:30, and then the screen will display 23:30 until the button is pushed again at 23:45. — Michael

Yes I agree with your point that if we look at the lamp between two consecutive inverse powers of 2, the state is determined by the earlier point.

But you are trying to apply that same reasoning to the limit point, and you can't do that. Every interval around the limit point necessarily contains all but finitely many members of the sequence. There is no tiny little interval before midnight where the sequence has ended, leaving the lamp in a particular state. That's just a mistaken intuition on your part.

A supertask is any infinite succession of tasks performed in finite time. Having that task be to push a button is as good a task as any, regardless of what pushing the button actually does. — Michael

Your logic is bad. I shouldn't eat one of those poisonous Japanese pufferfish, but that doesn't mean I should turn down a nice tunafish salad sandwich.

But now that I think of it, I no longer even accept that the lamp is "impossible" via your logic. You just haven't got a good intuition for the limit of a sequence. Any step backward, no matter how tiny, jumps over all but finitely many members of the sequence.

So if the lamp is on at midnight, there IS a prior moment at which it was off, and vice versa. Just not an immediately preceding point, because there is no such thing.

The lamp has two buttons. One button turns the lamp on and off, the other does nothing. The same mechanism is responsible for pushing both buttons. If it is possible to have pushed one of these buttons an infinite number of times then it is possible to have pushed the other button an infinite number of times. — Michael

Yes, there are as many even inverse powers of 2 as odd ones.

If Thomson's lamp proves that it is impossible to have pushed the button that turns the lamp on and off an infinite number of times then it proves that it is impossible to have pushed the button that does nothing an infinite number of times. — Michael

It proves no such thing. And having discovered that Benacerraf totally agrees with me, I don't have much reason to continue to assert the same points I've made over and over. Clearly, reasonable people agree with my point of view. Benacerraf is still alive, perhaps you can take this up with him.

And we can replace pushing a button with literally any other task and the same conclusion follows. Therefore, Thomson's lamp proves that all supertasks are impossible. — Michael

If repeatedly asserting a falsehood were proof, we'd all be geniuses.

Does anyone know of any dating apps or places to be, where people seeking a deep, long-term relationship with an intellectually substantive partner go? — Bob Ross

Get with the program, man. AI partners are the way to go.

https://romanticai.com/

ps -- I played around with this. It's an uncanny chatbot. Once they get this tech into the life sized love dolls, nobody will need real people. We're all simulations anyway, so I'm told. Here you go, $577 on Amazon. Searching "sex dolls" brings up lady dolls, must be more demand for that, not assuming anything about anyone's sexual preferences.

https://www.amazon.com/Realistic-Breast-Silicone-Pleasure-62-2Inch/dp/B0CTJV3BQ1/ref=sr_1_1

Free shipping!

[Now Amazon will remember I looked this up forever]

Also, if you're looking for mentally stimulating companionship, Woody Allen wrote a story about hookers who specialize in intellectual conversation.

Well, I heard of this young girl. Eighteen years old. A Vassar student. For a price, she’ll come over and discuss any subject—Proust, Yeats, anthropology. Exchange of ideas. You see what I’m driving at?

The Whore of Mensa

Where is the will that initiates the process?
— fishfry
I can't answer for your view, but for the naturalists, it comes from different places, depending on what sort of thing is wanted.
Most will comes from subconscious places (Limbic system), such as choices as to which way to swerve around the tree or to cheat on your spouse. But the will to choose option C in a multiple choice test comes from higher up (Cerebrum for instance). — noAxioms

I have no limbic system. Only a simulation of a limbic system in a computer, if I understand you correctly (clearly I don't, right?) A computer simulation of a limbic system cannot create emotions any more than a simulation of gravity attracts nearby bowling balls.

I said that because the reasons seem backwards: Conclusion first, then selection of premises to support that conclusion. This is rationalization, something humans are very good at. I don't consider humans (myself included) to be very rational creatures. — noAxioms

I agree with you there. We all have prejudices that make us gravitate towards one pole or the other of unanswerable questions.

Not at all, but I apologize if my words annoyed you. The effect was not intentional. — noAxioms

Well I confess that I have no proof for my opinions or biases.

If the Supreme Button Pusher turns the lamp on at midnight then it must have been off before He turned it on, because that's what "turning on" means. If He turned it off at midnight then it must have been on before He turned it off, because that's what "turning off" means. — Michael

I understand your point. I really do. But you are expecting a limit point to have an immediate predecessor, when it does not.

Consider the sequence of times at which we "sample" the state of the lamp. I accept your point that the lamp exists continuously, so that (for example) if we want to know the state of the lamp at, say, 1/3 second before midnight, we would just note that at 1/2 second before midnight, it was off (I think we set it up that way earlier. On at 1 sec, off at 1/2, on at 1/4, all times before midnight.

So it's off at 1/2 sec before midnight, and it will stay off till we turn it on again at 1/4 sec before midnight.

Consider the infinite sequence 1, 1/2, 1/4, 1/8, ... Mathematically, it has the limit 0; and in the lamp puzzle, the limit 0 represents midnight; that is, zero seconds before midnight.

Now two things are true:

For any of the times in the sequence -- 1, 1/2, 1/4, etc. -- it has an immediate predecessor (except for 1 of course).

But 0 does not have an immediate predecessor.

It makes no sense to ask what is the state of the light "immediately before" time 0. Such a thing is not defined.

If you want to interpret this as meaning that the premises or conditions of Thompson's lamp makes it impossible; then I'm fine with that. I'm fine either way. I think you can just arbitrarily define to be the state at midnight as on, off, or turning into a plate of spaghetti.

But perhaps that's just another way to say that the problem is impossible. If so, I can live with that.

But tha does NOT mean that supertasks are impossible. It only means that THIS PARTICULAR supertask is impossible. There might be others that ARE at least metaphysically possible. I've argued this several times. A Zeno walk from my couch to my fridge is one such, under the (perhaps unrealistic, but at least metaphysically feasible) condition that time is modeled by the mathematical real numbers, infinitely divisible and Cauchy-complete.

Surely you must take my point. Even if the lamp is impossible, some other supertask might still be possible. Thompson's lamp is a rather poor example of a supertask, because its underlying sequence can not be made to converge to a limit.

For the entire continuous time that the lamp exists – not just the time during which the button is pushed – it being off must precede it being on, regardless of magic or divine intervention. As such the sequence off, on, off, ..., on makes no sense, much like having an imaginary number of apples in the fridge makes no sense. — Michael

I take your point, and for sake of discussion I'll agree that Thompson's lamp is impossible. I also think you can just complete it arbitrarily. But if you don't like that, then I'll agree it's impossible.

I do understand the point you think you are making. But you are just unhappy that the limit of a sequence has no immediate predecessor in the sequence, and that's just a mathematical fact that's true and that you don't like.

But also note P1 in the argument above, implicit in Thomson's argument. You don’t get to just introduce God to deflect from the incoherent causal consequence of having pushed a button an infinite number of times. — Michael

Ok. I'll agree for sake of discussion that Thompson's lamp is impossible.

A supertask is just performing an infinite succession of tasks in finite time. Pushing a button, which just so happens to turn a lamp on and off, is as good a supertask as any. — Michael

No, it's a terrible example of a supertask, because it can not possibly be made to have a limit.

Consider a Zeno walk from my couch to the kitchen. First I go 1/2 the distance. Then I go 1/2 the remaining distance, and so forth. So at each step I've covered 1/2, 3/4, 7/8, ... of the distance.

After one minute (say I go half the distance in half a minute, etc.) I am in the kitchen.

Now this is the same exact problem as with the lamp, namely that the limit of the kitchen has no immediate predecessor, just as the limit 1 of the sequence 1/2, 3/4, 7/8, ... has no immediate predecessor in the sequence.

But ending up in the kitchen has the virtue of at least being the limit of the sequence. So it's a natural solution. And, under the assumption that time is like the mathematical real numbers, it's a true supertask. I performed infinitely many steps in finite time.

But let’s assume that pushing the button displays the time that the button is pushed – persisting until the button is pushed again – and is pushed (only) at successively halved intervals of time starting two minutes to midnight. What time is displayed at midnight? — Michael

Midnight. What time does YOUR clock say at midnight? Of course the clock says midnight at midnight.

You are just psychologically uncomfortable with the fact that the limit has no immediate predecessor in the sequence.

The sequence approaches midnight but because the button is never pushed at midnight the display can never show midnight, but must show something because it’s never turned off. It always only displays the time that the button was last pushed, but in this scenario there is no last button push, entailing a contradiction. — Michael

I'll concede your point that the lamp is impossible. That does not necessarily entail that supertasks in general are impossible. Only that this one is. But there is no time immediately before midnight, for the same reason there's no inverse integer power of 2 immediately before 1. It's just how infinitely divisible sets work.

As Thomson says, "the impossibility of a super-task does not depend at all on whether some ... sequence is convergent or divergent." — Michael

Oh, that is interesting. So I will take from this the happy conclusion that Thompson and I are thinking along exactly the same lines.

I don't know enough about supertasks to know if every supertask involves demanding that there be an immediate predecessor to the limit of a sequence.

But since there isn't an immediate predecessor to the limit of a sequence, I have to conclude that whoever is defining supertasks such that a mathematically impossible thing must become possible, to that extent, supertasks are impossible.

But they're just making a mountain out of a mathematical molehill. The ordinal $\omega$ does not have an immediate predecessor, and that's a fact.

So you are making this argument:

P1: Every supertask involves finding the immediate predecessor of $\omega$.

P2: $\omega$ does not have an immediate predecessor.

Therefore:

C1: There can be no supertasks.

I believe this is the heart of your argument. I'll make another run at the SEP article because it's interesting that Thompson said that convergence doesn't matter.

ps -- Ok, SEP says this:

Benacerraf (1962) pointed out a sense in which the answer is yes. The description of the Thomson lamp only actually specifies what the lamp is doing at each finite stage before 2 minutes. It says nothing about what happens at 2 minutes, especially given the lack of a converging limit. It may still be possible to “complete” the description of Thomson’s lamp in a way that leads it to be either on after 2 minutes or off after 2 minutes. The price is that the final state will not be reached from the previous states by a convergent sequence. But this by itself does not amount to a logical inconsistency. — SEP

So I have been expressing Benacerraf's point all along. Good to know. I may be wrong, but I'm at least in the ballpark of professional philosophical thought.

pps -- I love the example of the bouncing ball electrically connected to the lamp. That actually gives what I would call a natural solution in which the lamp is on, based on the fact that a bouncing ball eventually ends up on the ground.

ppps -- And:

For this reason, Earman and Norton conclude with Benacerraf that the Thomson lamp is not a matter of paradox but of an incomplete description. — SEP

But this is exactly what I've been saying all along.

So you are holding a position that many (some, several, whatever) philosophers disagree with, and you are arguing that I should break with them and come to your side. But the SEP article definitely shows that some philosophers are in complete agreement with the ideas I've been expressing all along.

The trouble is that many philosophers seem to be hypnotized by physics, and seem to forget that physicists develop their theories and conduct their experiments in ordinary human reality. — Ludwig V

Well, physics got strange in the 20th century, leading to a lot of philosophy, good and bad. There's a philosopher of physics named Tim Maudlin whose videos were showing up on my YouTube feel a lot a couple of years ago, but not at all lately. He knows a lot of physics. Seems to be one of the few philosophers who does.

Discrete calculus is certainly important to my view but it's not what I'm talking about. — keystone

What about computer arithmetic, fixed and floating point representations, smallest and largest possible values?

There are two steps to a cut:
Step 1: Planning the cut with an algorithm
Step 2: Executing the plan by completing the algorithm — keystone

All that followed went way over my head.

ε1 and ε2 are placeholders for positive numbers which can be as small as your computer allows — keystone

So you are doing normal math except within the limits of a finite computational space. If not fixed/floating point, something else. But computer arithmetic regardless.

(π-ε1,π+ε2) describes a line with rational upper and lower bounds. This line can be as small as your computer allows. Ultimately, I want to call this line pi. To distinguish it from the point/number, π, I'll call this line π. — keystone

You cannot telescope down to pi on computer-limited representations of numbers. If you mean that your number pi is actually a little interval around pi with approximation bounds given by the limitations of your computer representation, I'm fine with that.

Don't see the point though.

With the top-down view, the plan and it's execution are distinct steps such that π remains a line, no matter how powerful your computer is.
With the bottom-up view, the plan and it's execution are equivalent such that π collapses to a point. I believe this is an unacceptable and an unnecessary leap of thought akin to claiming that there is a last term in a Cauchy Sequence.
Although step 2 is incredibly useful for applied mathematics, that's not what I'm concerned with. I'm solely concerned with step 1 and I believe step 1 is what is of interest to pure mathematicians.
Counter to standard belief, I believe calculus is about plans (not their execution) and I believe it's unknowingly been this way all along.
For example, when a mathematician describes π they always describe the algorithm, they rarely talk about the algorithm's execution...unless referring to a Pi Recitation Contest... — keystone

I didn't follow much of this. Ok check that. I re-read it twice and I do not understand any of it. I'm genuinely sorry I can't be of more help. BUT ... I have an idea:

If you reject the noncomputable reals, what you have is the constructive real numbers, and the calculus based on them is called constructive real analysis.

All the explanations on the web immediately get into technicalities involving intuitionist logic, and seem like heavy reading.

My understanding is that they are just doing calculus using the computable numbers. There is one trick I know in the subject, which is that when you want to see if a sequence converges, you have to have a modulus of convergence which is a function that lets you know how well the sequence is converging.

I'm not explaining that very well, because I don't understand it very well. In fact I believe I understood it for a small while a few years ago, but I seem to have lost it. There were some advocates of constructive analysis and intuitionist logic on this site a few years ago, but they seem to have drifted away. Cauchy sequences that never converged, as it were.

But I wonder if this is what you are getting at. The numbers you define consist of a sequence and a function (or plan, or algorithm) saying how the function converges. Or something.

You know, I do think you have some good intuitions about things ... but you keep rejecting all my examples. Computer math, constructive analysis, etc.

Still ... tell me what it all means. I have had conversations with proponents of constructive analysis, and at one point I put a little effort into trying to learn it, but in the end, I didn't really get the hang of it.

It's popular these days because it's good for doing computer math and automated proof checking. There's a lot of research in related areas.

A point of clarity. Thanks. Calculus started with discrete, then moved to infinitesimal, then with technology back to discrete in some sense. — jgill

Is that right? I'm not sure if I have studied that, the pre-Newton and pre-Leibniz developments. You could be right.

So you’re with Oliver Stone on this one I guess. — Mikie

I think the circumstantial case against the CIA is stronger than the circumstantial case against Oswald. There's no proof either way. Oswald never could have been convicted in a fair courtroom. Stone mostly focusses on the Garrison case, which I haven't studied.

Did you know there's a straight line from Bill Clinton to the JFK assassination? It goes through the airport in Mena, Arkansas, which was a hub of drug and gun running while Clinton was governor. One of the pilots was the famous drug smuggler and CIA asset Barry Seal, who used to be a member of the Civil Air Patrol and knew Oswald and David Ferrie, the guy played by Joe Pesci. Theres a pretty good film called American Made starring Tom Cruise as Seal.

Shortly after the JFK assassination, three men in suits got on an airplane at a local airport, told the controller they were flying north, then flew north a ways and turned back south. Some think the pilot was either Seal or Ferrie. That's all speculation though. Once the cops announced the arrest of Oswald, the controller didn't bother to follow up the flight plan anomaly.

Seems silly to me. I don’t care that much, to be honest— could be true. What actually changed my mind was NOVA’s assessment:

Worth watching. They dismantle a lot of misinformation. — Mikie

If they give even the slightest credence to the single bullet theory, they're full of baloney. One, there is no chain of custody linking the bullet supposedly found at Parkland to the bullet currently stored at the National Archives as CE-399. Two, even if it was the same bullet, there were more grains of led in Connally's body that are missing from the bullet. Three, Dr. Pierre Finck, one of the autopsy surgeons, testified at the Clay Shaw trial (Garrison case) that he was ordered by a high ranking military official to not dissect the back and throat wounds. Meaning that the government deliberately made sure nobody could ever know if the same bullet caused both wounds. And four, Governor Connally swore to the day he died that he was not hit by the magic bullet, and his wife Nellie, who'd been sitting behind him in the presidential limo, agreed.

I will check out the video, thanks.

@Wayfarer I can address the specifics of your post now.

The subpoena to Garland was pure theatre and retribution, no basis in law or fact. We can expect many more frivolous and baseless lawsuits from Trump’s minions in Congress none with any basis in law or fact, but solely driven by spite and the desire to settle scores. — Wayfarer

I understand! When the Democrats impeach Trump twice, the first time over something that Biden provably did (extort Ukraine to fire the Burisma prosecutor), it's the virtuous application of the law to all men without fear or favor. And when the Republicans use the law to hold the Democrats accountable for anything, those are "frivolous and baseless lawsuits from Trump’s minions in Congress none with any basis in law or fact, but solely driven by spite and the desire to settle scores."

It's all so simple now! Why didn't I see that before?

Ashli Babbitt was shot by law enforcement trying to force entry into a restricted area. If she hadn’t have been taking those actions, she wouldn’t have been shot. — Wayfarer

Were you that magnanimous to law enforcement when George Floyd died of a fentanyl overdose with a cop's knee on his upper back, in conformance with police department policy? Or in the Ahmaud Arbery case, where a known neighborhood burglar resisted a lawful citizen's arrest and the citizens were not even going to be charged until the mob howled?

You like lynch mobs?

Not true. There were intruders holding signs saying that the Vice President should be hung — Wayfarer

As long as they spare the fly.

and expressing the intent to find and murder the speaker of the house. — Wayfarer

Pelosi? The line was probably around the block.

How many did such things out of the total present in Washington that day? Out of how many still held in jail, in solitary confinement under inhumane conditions, like one of Stalin's gulags?

You condone that?

They desecrated the House — Wayfarer

Pelosi did that herself when she ripped up the SOTU speech, among other things. Not to mention her flagrant insider trading, which you or I would go to prison for.

and destroyed private and government property and confidential records. — Wayfarer

Floyd riots: 19 dead, over a billion dollars in insurance payouts for property damage, at least one police station burnt to the ground, other police stations set on fire with cops inside, government buildings attacked, two law students given slaps on the wrist for tossing Molotov cocktails into cop cars, vibrant commercial sectors burnt to the ground that will take years to rebuild.

Did you spend the summer of 2020 cheering that on?

The difference with Holder is that Eric Holder was asked to testify and provide documents for a period of time in which he actually was part of the executive branch. In fact he did provide many documents and he claimed executive privilege for others. The court agreed but also required Holder to turn over non-privileged documents. — Wayfarer

He refused, and was held in contempt of Congress. Was that the proper authority of Congress, or was that more "frivolous and baseless lawsuits from Trump’s minions in Congress none with any basis in law or fact, but solely driven by spite and the desire to settle scores."

How silly of me to even ask.

Steve Bannon was being requested to testify and provide document for a period of time in which he was not part of the executive branch. Bannon turned over nothing and refused to testify. It is fundamentally different than Eric Holder’s case. — Wayfarer

That's the fair and even-handed of the law, which no one is above. Say, I think I'm getting the hang of this game!

That trial is pending. — Wayfarer

Many voters of all political persuasions and demographics see that illegitimate lawfare is all you've got, since you can't make a case for your guy.

And ... if I may be so bold as to ask ... what exactly is the case for your guy? In syllables that don't rhyme with Orange. "Vote for the burnt out husk, he's not Trump."

Catchy.

Also recommend “Rethinking Camelot.” — Mikie

I'm reading The Dark Side of Camelot by Sy Hersh. Dishes the dirt. I don't say JFK was a saint. I do say that this country has not been the same since the CIA blew his brains out in broad daylight on behalf of the deep state.

Chomsky is full of shit on the JFK assassination by the way. People have always called him a CIA asset and this is one of the data points in support of that proposition.