From birth, children are taken to this daycare+school+apartment+prison complex to learn life skills such as trigonometry. — Scarecow

If only. It's the gender theory taught to five year olds that worries people about the government schools these days.

I'm taking the Google Maps directions/map and making them more 'mathematical'. Let me try iteration 0 and tell me if this is clear:

Iteration 0
1) Start at 6445-6451 Peel Regional Rd 1
2) Travel the road Erin Mills Pkwy/Peel Regional Rd 1 N towards McDonalds
3) Arrive at intermediate destination: McDonalds
4) Travel the road Millcreek Dr towards 6335-6361 Millcreek Dr
5) Arrive at destination: 6335-6361 Millcreek Dr
— keystone

I find this incredibly annoying. Can't you get to the point?

Do you honestly not see how this relates to the Google Maps screenshot I sent a few posts back? — keystone

I don't see the point.

I'm developing a framework that applies topological metric spaces to describe continua with arbitrarily fine precision. [ /quote[

Oh. Ok. I understand that. I appreciate this clear, simple statement of what you are doing.

Question: Don't the standard real numbers already do a fine job of exactly that?
— keystone

This might seem esoteric, — keystone

It's not esoteric, it's basic high school math. The real number line.

but achieving this involves turning everything upside down—without dismissing any past mathematical progress. This approach offers a powerful new perspective on mathematics. — keystone

You aren't making a case for that.

It begins with this map example because I want to (1) describe the continuous journey using intervals and (2) show how those intervals can be described by a topological metric space. However, you're not even letting me do step (1). — keystone

I"ll stipulate to the real number line.

I'm still confused by your describing points on the real line with two coordinates.

Please tell me which iteration you are tripping up on: 0, 1, 2, 3, or 4? — keystone

— keystone

I get that you start at 0, land at .5, and end up at 1. Is that sufficient for your purposes?

I still don't see why you use two coordinates to describe a point on the real line.

I'm using interval notation. It's an interval. — keystone

It's an interval?? What? You are labeling locations on the real line as intervals? That makes little sense. Google maps doesn't do that.

No, that's not right. I referred back to your picture. You described the origin on the line as (0,0). What is the meaning of that?

You described the point commonly notated as .5 as (.5, .5). What am I supposed to take from that?

In fact the notation (.5, .5) is a degenerate open interval. It denotes the empty set. If I take (.5, .5) as interval notation, there are no points at all in it. Do you see that?

So we have two specific questions on the table.

1) What does (.5, .5) represent? In standard mathematical notation, it's the empty set. At best, [.5, .5] would simply be the point .5. But why do that?

2) Don't the standard real numbers already "describe continua with arbitrarily fine precision?" [

Although I didn't plan to start with directions and maps, I'm glad we ended up here. It's an excellent starting point. — keystone

I don't even get a mention now? That's the only way I know when someone's talking to me.

I don't know what you are talking about. I swear to God, I do not understand what you are doing, what you're talking about, why you're doing this. I am totally lost. I went back over the thread, i simply don't understand what you are talking about. And you flat out refuse to tell me. At one point you were talking about Achilles, so is this something to do with one of Zeno's paradoxes?

Why can't you just give me the top-line summary of what you are doing? A while back we were talking about metric spaces and topological spaces, that at least made some sense. The rest of this, the map, the grid, I just don't know what you are doing. I don't know what is the overall point being made, what I'm supposed to be getting from this. It's very frustrating.

I'll stipulate that you can traverse a grid. Or a line. Your coordinates have two components yet appear on a straight line. That's a little odd. What is your point?

I do have one specific question. Why do your points on a straight line have two coordinates? What does that denote?

The word is "logic", and I think it's pretty important to a discussion like this, to have good agreement as to what this word means. — Metaphysician Undercover

Well, it's not logical to reject a possibility on the grounds that you don't like it.

If I simply assert, as if a true proposition, "chocolate is better than vanilla", there is not logic here. But if I state my premises, I am allergic to vanilla, and to have an allergic reaction is bad, then my stated preference "i prefer chocolate to vanilla" is supported by logic and is logical. Do you agree? . — Metaphysician Undercover

You're allergic to randomness? Is that the argument you're making?

If you said that believing in randomness makes you break out in a rash, and therefore you prefer to not believe in randomness, that would be logical. But it would be a logical argument for why you hold that belief. It would not be a logical argument against randomness.

I've been trying to build towards a more important point but I feel like I have to keep going simpler and simpler to find a common ground with you. I'm hoping interpreting a map is the common ground where we can start from. If you acknowledge that you understand how directions and maps work then I will advance with my point. — keystone

Please start any time. I simply have no idea what your overall point is, nor have I understood any of your examples. Start from the top. "I wish to reform the entire corpus of modern mathematics." Then tell me what are numbers, sets, functions, relations, etc.

I just can't figure out what you are doing.

Explain to me as you would if I were standing in front of you, what point you are making with the map.

It seems that you're either unable or unwilling to acknowledge even the most basic points I've raised. — keystone

I'm unable to understand their point.

I apologize if this appears to diverge from your interests, but focusing on the image below, can you see how the instructions on the left relate to the image on the right? (This is not a trick question) — keystone

McDonalds, Sushi, Wok and Roll. Now I'm hungry.

Once again you leave me utterly baffled as to why you posted this.

Slow down, you are not taking the time to understand what I said. In the application of logic, there is two aspects to soundness, the truth or falsity of the premises, and the validity of the logical process. — Metaphysician Undercover

We're just arguing about a word. If you want to claim that "I prefer chocolate to vanilla" is an example of logical reasoning, what is the point of my arguing with you about a thing like that?

Therefore, we must respect the fact that moral arguments can proceed with valid logic, — Metaphysician Undercover

In the sense that we can argue a conclusion from moral premises, I agree.

But "I find such and so repugnant, therefore such and so is logically false," is simply bad logic. Or not even logic. As you prefer. But it's not good logic. I'm certain of that.

I appreciate that you wrote a lot. I haven't the heart to continue this line of discussion, my apologies. "Randomness is repugnant to me therefore it's false" is not good logic. As to whether it's bad logic, or not even logic, I'll leave open.

I really don't see how there could be a staircase which is not physical. — Metaphysician Undercover

If I understood the OP, the walker spends arbitrarily small amounts of time on each step, 1/2 second, 1/4 second, etc. That violates the known laws of physics. So it's not a physical situation. It's a cognitive error to think we're contrasting math to physics. There is no physics in this problem.

That really makes not sense. However, just like in the case of the word "determine", we need to allow for two senses of "physical". You seem to be saying that to be physical requires that the thing referred to must obey the laws of physics. — Metaphysician Undercover

Well yeah. To be a fish a thing has to obey the known laws of fishes. Note that I include the word "known." Biologists could discover a new fish that extends our concept of what's a fish, just as physicists refine their laws from time to time. But a physical thing must obey the known laws of physics. This seems a very trivial point, i can't imagine what you mean by questioning it.

But the classic definition of "physical" is "of the body". — Metaphysician Undercover

Wasn't that a classic Star Trek episode? "Are you of the body?" And if you weren't, they zapped you with an electric stick.

And when a body moves itself, as in the case of a freely willed action, that body violates Newton's first law. — Metaphysician Undercover

Sorry, what? Given me an example of something that violates Newton's laws, unless it's an object large enough, small enough, or going fast enough to be subject to quantum or relativistic effects.

A freely willed action? Can you give me an example? You mean like throwing a ball? You kind of lost me here.

Therefore we have to allow for a sense of "physical" which refers to things which are known to violate the laws of physics, like human beings with freely willed actions. — Metaphysician Undercover

I'll be happy to consider any specific examples you have of human beings whose actions violate the laws of physics. If you mean actions caused by mentation, that's a bit of a puzzler, but I'm not sure how to violate the laws of physics. If I set out today to violate Newton's laws, I don't know how I could do that.

What is implied here is that the laws of physics are in some way deficient in their capacity for understanding what is "physical" in the sense of "of the body". — Metaphysician Undercover

Do you mean something like, "I think about raising my right hand and my right hand goes up, how does that happen?" If so, I agree that nobody understands the mechanism.

That's why people commonly accept that there is a distinction between the laws of physics and the laws of nature. — Metaphysician Undercover

That's why I included the word "known." I allow that the laws of physics are historically contingent approximations to the laws of nature.

The laws of physics are a human creation, intended to represent the laws of nature, that is the goal, as what is attempted. — Metaphysician Undercover

Agreed.

And, so far as the representation is true and accurate, physical things will be observed to obey the laws of physics, but wherever the laws are false or inaccurate, things will be observed as violating the laws of physics. — Metaphysician Undercover

Still waiting for specific examples. I believe the muons were misbehaving a while back and it made the news. Of course there are things we don't understand, like dark matter, dark energy, a quantum theory of gravity.

Evidently there are a lot of violations occurring, with anomalies such as dark energy, dark matter, etc., so that we must conclude that the attempt, or goal at representation has not been successful. — Metaphysician Undercover

Ok. Scary that you and I are thinking along the same lines. What is your point here with respect to the subject of the thread?

Sure, it's a conceptual thought experiment, but the interpretation must follow the description. A staircase is a staircase, which is a described physical thing, — Metaphysician Undercover

The walker spends ever smaller amounts of time on each step, and that eventually violates the Planck scale.

just like in Michaels example of the counter, such a counter is a physical object, — Metaphysician Undercover

Which counter? The lamp? The lamp is not physical. No physical circuit can switch in arbitrarily small intervals of time.

and in the case of quantum experiments, a photon detector is a physical object. And of course we apply math to such things, but there are limits to what we can do with math when we apply it, depending on the axioms used. The staircase, as a conceptual thought experiment is designed to expose these limits. — Metaphysician Undercover

It's designed to confuse people who mis-learned a little calculus and don't know what's allowed or disallowed by the laws of physics.

OK sure, but that's a limit created by the axioms of the mathematics. So it serves as a limit to the applicability of the mathematics. The least upper bound is just what I described as "the lowest total amount of time which the process can never surpass". Notice that the supposed sequence which would constitute the set with the bound, has already summed the total. This is not part of the described staircase, which only divides time into smaller increments. It is this further process, turning around, and summing it, which is used to produce the limit. The limit is in the summation, not the division. — Metaphysician Undercover

Ok I guess. No walker can traverse a staircase as described by the premises of the problem. So if I said the staircase was not physical, I should have said the walker is not physical. Better?

It is very clear therefore, that the bound is part of the measurement system, a feature of the mathematical axioms employed, the completeness axiom, not a feature of the process described by the staircase descent. The described staircase has no such bound, because the total time passed during the process of descending the stairs is not a feature of that description. This allows that the process continues infinitely, consuming a larger and larger quantity of tiny bits of time, without any limit, regardless of how one may sum up the total amount of time. Therefore completeness axioms are not truly consistent with the described staircase. — Metaphysician Undercover

I don't see why not. The whole point of the puzzle is to sum 1/2 + 1/4 + ... = 1, and then to ask what is the final state. Which, as I have pointed out repeatedly, is not defined, but could be defined to be anything you like.

However, since our empirical observations never produce a scenario like the staircase, that inconsistency appears to be irrelevant to the application of the mathematics, with those limitations inherent within the axioms. The limitations are there though, and they are inconsistent with what the staircase example demonstrates as logically possible, continuation without limitation. Therefore we can conclude that this type of axiom, completeness axioms, are illogical, incoherent. — Metaphysician Undercover

I'm sorry that you fine the completeness axiom of the real numbers incoherent. On the contrary, the completeness axiom of the real numbers is one of the crowning intellectual achievements of humanity.

The real problem is that as much as we can say that the staircase scenario will never occur in our empirical observations, we cannot conclude from this that the incoherency is completely irrelevant. — Metaphysician Undercover

The premises violate the known laws of physics, specifically the claim that we can know the walker's duration on each step even though that duration is below the Plancktime.

We have not at this point addressed other scenarios where the completeness axioms might mislead us. Therefore the incoherency may be causing problems already, in other places of application. — Metaphysician Undercover

Modern math is incoherent. Is it possible that you simply haven't learned to appreciate its coherence?

Yes. I got enough from it to realize a) that ω is one of a class of numbers and b) that it comes after the natural numbers (so doesn't pretend to be generated by "+1") — Ludwig V

Yes exactly. $\omega$ comes into existence via a limiting process. The idea is that the natural numbers are generated by successors, and the higher ordinals are generated by successors and limits. So we're adding a new rule of number formation, if you like. We go 1, 2, 3, ... by successors, and then to $\omega$ by taking a limit, then $\omega `=+ 1$, $\omega + 2$, etc., then eventually we get to $\omega + \omega$ by taking a limit, then we keep on going. I don't want to go too far afield, but the idea is that we can take successors and limits to get to all the higher ordinals.

This business about actions is what confuses people.
— fishfry
Certainly. That's what needs to be clarified, at least in my book. There's a temptation to think that actions must, so to speak, occur in the real world, or at least in time. But that's not true of mathematical and logical operations. Even more complicated, I realized that we continually use spatial and temporal terms as metaphors or at least in extended senses:- — Ludwig V

Right. A lamp that cycles in arbitrarily small amounts of time is not physical. A staircase that we occupy for arbitrarily small intervals of time is not physical. So trying to use physical reasoning is counterproductive and confusing. That's my objection to all these kinds of puzzles. People say there's a conflict between the math and the physics ... but as i see it, there's no physics either.

By the way, ω is the "point at infinity" after the natural numbers
— fishfry
What does "after" mean here? — Ludwig V

Follows in order. Given 1, 2, 3, 4, ..., we can adjoin $\omega$ "at the end." What do I mean by that? I mean that we extend the "<" symbol so that

1 < $\omega$, 2 < $\omega$, 3 < $\omega$, and so forth. So that conceptually, every natural number is strictly smaller than $\omega$. Does that make sense?

If you want to think about the sequence 1/2, 3/4, 7/8, ... "never ending," that's fine. Yet we can still toss the entire sequence into a set, and then we can toss in the number 1. That's how sets work
— fishfry
Yes, but it seems to me that this is not literally true, because numbers aren't objects and a set isn't a basket. (I'm not looking for some sort of reductionist verificationism or empiricism here.) — Ludwig V

I can always form a set out of a collection of objects. Not following your objection.

{1/2, 3/4, ...} is a set, and {1} is a set, and I can surely take the union of the two sets, right?

{1/2, 3/4, ..., 1} is just a particular subset of the closed unit interval [0,1].

If you are not sure about what I'm saying we should stay on this point. I can definitely form a set out of any arbitrary collection of other sets. And each of 1, 2, 3, ... and $\omega$ can be defined as particular sets.

Just think about {1/2, 3/4, 7/8, ..., 1}. It's the exact same set, with respect to what we care about, namely the property of being an infinite sequence followed by one extra term that occurs after the sequence.
— fishfry
In that respect, yes. But I can't help thinking about the ways in which they are different. — Ludwig V

Of course {1, 2, 3, ..., $\omega$} is a different set that {1/2, 3/4, ..., 1}. But strictly in terms of their order, they are exactly the same. And with ordinals, all we care about is order.

That's a confusing way to think about it. It "ends" in the sense that we can conceptualize all of the natural numbers, along with one extra thing after the natural numbers.
— fishfry
Yes. But it doesn't end in the sense that we can't count from any given natural number up to the end of the sequence. — Ludwig V

The sequence is endless, and there's an extra point that's defined to be strictly greater than all the others. We can't get to the limit by successors, but we can get there by a limiting process.

I try not to mention this in public, but the fact is that I never took a calculus class, nor was I ever taught to think about limits or infinity in the ways that mathematicians sometimes do. I did a little formal loic in my first year undergraduate programme. Perhaps that's an advantage. — Ludwig V

You're far better off. People who take calculus and then engineering math end up confused about limits and the nature of the real numbers. Taking logic and not calculus is actually helpful, in that you haven't mis-learned bad ideas about limits.

Calculus is focussed on the computational and not the philosophical aspects of limits, and calculus students often end up a little confused about some of the technical details. I was actually referring to the other poster who you noted was talking past me and vice versa.

I have the impression that you don't think that they are mathematically possible either. (I admit I may be confused.) So does that mean you don't think that supertasks are possible? — Ludwig V

I've convinced myself both ways. On the one hand we can't physically count all the natural numbers, because there aren't enough atoms in the observable universe. We're finite creatures.

On the other hand, supertasks are possible, because I can walk a mile, meaning I walked 1/2 a mile, 1/4 mile, dot dot dot.

I have no strong belief or opinion about supertasks. I have strong opinions about some of the bad logic and argumentation around supertasks.

H'm. In principle, that is a valid complaint. But, back when I was involved, something like 60% of vacancies for graduates (i.e. those requiring a BA degree or higher) did not specify the subject. That may have changed. But you might be surprised at where Eng. Lit. and Fine Arts graduates end up. — Ludwig V

In the DEI departments of university administrations I imagine.

I'm not sure how education for professions and trades differs now; there's a lot of emphasis on training all the way up to BA level and higher. Many Universities are re-casting their non-vocational qualifications as vocational and there's effort going in to tracking what level of job graduates actually get. I've heard anecdotes that some vocational programmes don't do very well. It's complicated. I suspect that the identity of the awarding institution is more important than the subject. Whether it is question of reputation, prestige or snobbery depends on how polite I'm feeling. — Ludwig V

Point being that pipefitters shouldn't be shouldering the cost of the loans forgiven for social justice majors.

Oh, I wondered why that business about the student loans was happening now. Not pretty, but then, one has to please one's voters. — Ludwig V

It's a scandal. The executive branch (Biden) actually has no authority to forgive those loans and foist them on the taxpayers. The Supreme Court already ruled on that. Biden's actions are illegal. Just election year pandering. And of course we're seeing this week who those students are.

It has happened gradually over two or three decades. I hesitate to get too detailed. It's mainly about social liberalism/conservativism - abortion, gay rights &c. Curiously, the Conservative party now seems to be at least as socially liberal as the Labour party, if not more so. There is certainly an issue in the Labour party that the liberal metropolitan elite now vote for Labour and this often clashes with the conservative social values of many "working class" people (not a politically correct classification any more.) — Ludwig V

Right. The "liberals" used to be for the working classes. Now the liberals support the elite against the working classes. Bit of a puzzler.

Originally the Labour party was explicitly a party for the working class - it was founded by the Trade Union movement. The Conservative Party tended also to have foundations in the "higher" parts of the class system; but now it's more about economics - free market vs state intervention (not Socialism as such). It does seem that many people in what used to be the working class who might well have voted Labour in the past now vote Conservative. This is all not very reliable. I'm not an expert.[/qgge.uote]

Me either, I was making a much more limited point earlier, and the poster I was making it to has chosen not to engage.
— Ludwig V

Compassion for criminals is anti-compassion for their victims.
— fishfry
I don't see why it has to be. Except, of course, that a victim may be more vengeful than the system is. But I don't see that as a question of compassion or not. Support for victims (in the UK at least) has been pathetic, but is now improving (but not nearly perfect). — Ludwig V

Violent criminals are being put back on the street to re-offend. That's not fair to the victims. Violent criminals belong behind bars.

I think the first duty of civic authorities is to provide for civic order.
— fishfry
Of course that's true. Part of the argument is that sympathetic ("humane") treatment of criminals and addicts gets better results in preventing recidivism - and a huge proportion of crime is recidivism. There's empirical evidence for that. — Ludwig V

People can't re-offend if they're locked up.

Another part is that more severe sentences are not effective in preventing crime. Effective detection and police work is much more effective. It makes sense. 20 years in jail is not much of a deterrent if you aren't going to get caught. But if you know you won't get away with, you know also that you won't benefit much, whatever the penalty. (Some crimes are not deterred even by the high likelihood of getting caught, but those are unlikely to be deterred by severe penalties.) I know, I know, justice demands.... That, in my book, is not about justice; it is about revenge. Prevention is more important than revenge. — Ludwig V

Perhaps I just spend to much time following NYC politics. They're having a problem with soft-on-crime politicians leading to a great decrease in public safety.

And as I keep explaining, the issue with supertasks has nothing to do with mathematics. Using mathematics to try to prove that supertasks are possible is a fallacy. — Michael

But I'm not doing that. I haven't been doing that. Are you deliberately misunderstanding me or am I being unclear?

"Using mathematics to try to prove that supertasks are possible is a fallacy"

Who did that? Are they in the room with us right now?

I do not think it is some secret plan. They are anti-regulation, anti-LGBT rights, pro-discrimination on the basis of religious freedom, and pro-gun. — Fooloso4

But then why did they saddle the GOPs with a five to ten point deficit in every election at every level everywhere in the country for years, by overturning Roe? Now abortion's always on the ballot. That hurts every cause you listed. That's my point. Dobbs was profoundly counterproductive for the right.

I have more or less dropped out due to the repetitive assertions not making progress, but thank you for this post. — noAxioms

Thanks.

the set {1/2, 3/4, 7/8, ..., 1}
— fishfry
Interesting. Is it a countable set? I suppose it is, but only if you count the 1 first. The set without the 1 can be counted in order. The set with the 1 is still ordered, but cannot be counted in order unless you assign ω as its count, but that isn't a number, one to which one can apply operations that one might do to a number, such as factor it. That 'final step' does have a defined start and finish after all, both of which can be computed from knowing where it appears on the list. — noAxioms

Of course it's a countable set. It's a subset of the rationals, after all. You are right that it's not order-isomorphic to 1, 2, 3, ...

This is not radical. The rational numbers are countable, but not if counted in order, so it's not a new thing. — noAxioms

Right. Exactly right. Point being that contemplating a set that includes an infinite sequence along with an extra point is nothing strange at all. And it serves as a nice conceptual model for supertask puzzles.

If Zeno includes 'ω' as a zero-duration final step, then there is a final step, but it doesn't resolve the lamp thing because ω being odd or even is not a defined thing. — noAxioms

There is no final step. There is a point at infinity. Not quite the same. Unless you allow the limiting process itself as a step. It's just semantics.

and we inquire about the final state at ω
Which works until you ask if ω is even or odd. — noAxioms

It's neither, and who's asking such a thing? Even and odd apply to the integers.

Anyway if this is repetitive feel free to not reply. I just go through my mentions everyday trying to reply best I can. And I do have a thesis, which is that the ordinal $\omega + 1$ is the proper setting for the mathematical analysis of supertask puzzles. So I'll repeat that every chance I get.

Constructive or healthy modes of competition. We cannot eliminate our desire to win or outcompete one another. We like reward, acknowledgement and status. All we can do is steer the compulsion away from competition that worsens the the wellbeing or basic rights of the losing group. — Benj96

What ruling body decides on that? Steer the compulsion away if it "worsens the wellbeing?" Are their commissars for that? Your idea sounds like top-down authoritarianism in the guise of being caring. "We're crushing your competitiveness for your own good, Comrade. Enjoy your stay at the Gulag."

The relativity thing was more of a refinement and had little practical value for some time. Newtonian physics put men on the moon well over a half century later.
QM on the other hand was quite a hit, especially to logic. Still, logic survived without changes and only a whole mess of intuitive premises had to be questioned. Can you think of any physical example that actually is exempt from mathematics or logic?[/quot]

Relativity more of a refinement? Not a conceptual revolution? I don't think I even need to debate that. In any even it's a side issue. It's clear that the universe doesn't care what mathematics people use. In that sense, the laws of nature are exempt from mathematics. Historically contingent human ideas about the world are always playing catch up to the world itself. But if you disagree that's ok, it's a minor sidepoint of the discussion.
— noAxioms

QM is also the road to travel if you want to find a way to demonstrate that supertasks are incoherent.
Zeno's primary premise is probably not valid under QM, but the points I'm trying to make presume it is. — noAxioms

I don't really care much about supertasks and haven't argued that they're coherent or incoherent. I'm mostly trying to clarify some of the bad reasoning around them.

If you mean mentally ponder each number in turn, that takes a finite time per number, and no person will get very far. That's one meaning of 'count'. Another is to assign this bijection, the creation of a method to assign a counting number to any given integer, and that is a task that can be done physically. It is this latter definition that is being referenced when a set is declared to be countably infinite. It means you can work out the count of any given term, not that there is a meaningful total count of them. — noAxioms

Ok. I think some of the quoting got mangled since things I said ended up as part of your post.

But if anyone thinks I can't count all the natural numbers 1, 2, 3, ... by mathematical means, please identify the first one I can't count.

Sorry, but what? I still see no difference. What meaning of 'count them' are you using that it is easy only in mathematics? — noAxioms

To count a set means to place it into bijection with:

a) A natural number; or

b) the set of natural numbers, to establish countability; or

c) some ordinal number, if one is a set theorist or logician or proof theorist.

That doesn't follow at all since by this reasoning, 'as far as we know' we can do physically infinite things. — noAxioms

Lost me. As far as we know takes into account the great conceptual revolutions of the past, as evidence that there will be more such in the future.

They've been a possibility already, since very long ago. It's just not been proven. Zeno's premise is a demonstration of one. — noAxioms

Ok. I think I'm a little lost in the quoting and not actually sure what we are talking about here. I'm not strenuously defending whatever ideas you're concerned with.

Octonians shows signs of this sort of revolution. — noAxioms

Well ... ok.

Physicists are vague on this point, but if time is eternally creating new universes, why shouldn't there be infinitely many of them. — noAxioms

But that's exactly my point. If speculative physics is starting to take physically instantiated infinity seriously, then it's perfectly reasonable that in the future, physically instantiated infinity may become a core aspect of physics; in which case supertasks may be on the table.

It is a mistake to talk about 'time creating these other universe'. — noAxioms

Was this for me? I never said any such thing nor quoted anyone else saying it.

Time, as we know it, is a feature/dimension of our one 'universe' and there isn't that sort of time 'on the outside'. There is no simultaneity convention, so it isn't meaningful to talk about if new bubbles are still being started or that this one came before that one. — noAxioms

I'll have to plead ignorance on the question of whether there's a meta-universal time that transcends the bubble universes. Good question though.

All that said, the model has no reason to be bounded, and infinite bubbles is likely. This is the type-II multiverse, as categorized by Tegmark. Types I and III are also infinite, as is IV if you accept his take on it. All different categories of multiverses. — noAxioms

You are completely agreeing with my point. That if speculative physic already includes infinity, then mainstream physics may include infinity in the future.

And two, the many-world interpretation of quantum physics.
That's the type III. — noAxioms

You are agreeing with my point.

Observation for one is a horrible word, implying that human experience of something is necessary for something fundamental to occur. This is only true in Wigner interpretation, and Wigner himself abandoned it due to it leading so solipsism. — noAxioms

Nothing to do with my point, which is that speculative physics already includes infinity, therefore mainstream physics may include infinity after the next scientific revolution.

I don't buy into MWI, but bullshit is is not. It is easily the most clean and elegant of the interpretations with only one simple premise: "All isolated systems evolve according to the Schrodinger equation". That's it. — noAxioms

You're agreeing with me again. Why are you typing this stuff in? You've kind of lost me.

Everett's work is technically philosophy since, like any interpretation of anything, it is net empirically testable. — noAxioms

Ok.

I would have loved to see Einstein's take on MWI since it so embraces the deterministic no-dice-rolling principle to which he held so dear. — noAxioms

Ok.

Ah, local boy. I am more used to interacting with those who walk a km. There's more of em. — noAxioms

Depends on the exchange rate.

And suppose that in the first bubble universe, somebody says "1".
The universes in eternal inflation theory are not countable. — noAxioms

Wow. You have evidence for that? My understanding is that it is an open question in eternal inflation as to the cardinality of the bubbles: finite, countably infinite, or uncountable. But either way my point about reciting the integers stands. I don't actually get the sense that you're engaging with anything I wrote.

Yes, each step in a supertask can and does have a serial number. That's what countably infinite means. — noAxioms

That's not the definition of supertask others are using. But I used the example of bubble universes to illustrate the possibility of counting the natural numbers physically.

Anyway sorry if I got lost in the quoting and didn't really understand some of your responses.

The issue behind the student loan question is the question how far state-funded free education should go. If you want a level playing field in careers, everyone who can benefit should get higher education - and that means that almost everybody should be entitled to have a go. At the same time, if people benefit financially, there is a good case for saying that some of that benefit should go back to whoever funded it. Ironically, in the UK, the financial benefit from higher education is rapidly shrinking and, some say, has disappeared, mainly because it has been extended so widely. The proportion of student loans that is actually repaid is astonishingly low. (I can't remember the actual figures.) — Ludwig V

That's right. So the students majoring in unmarketable majors are subsidized by people who skipped school and went into the trades. That doesn't seem fair. It's just that the college grads vote for Democrats and the tradesmen vote for Republicans, so the Democratic administration forgives billions in student loans -- illegally, as the Supreme Court has already ruled -- in an election year.

And not just that. The Democratic party use to be the party of the tradesmen and no longer is. When did the left abandon the workers, and why? I gather the Labour party in the UK has undergone a similar transition, is that right?

So is it possible that a different version of the social justice approach might be more effective? Is it possible that other places may be implementing it in a better way? — Ludwig V

Compassion for criminals is anti-compassion for their victims. New York City is a great lesson in restorative justice gone too far. I think the first duty of civic authorities is to provide for civic order. What's weird is that the voters themselves vote for the faux-compassion that ends up hurting them.

So the real solution to our problems is better voters!

I've watched this debate for a long time - though I don't claim to have understood all of it. But I think those two quotes show that you are talking past each other. — Ludwig V

He'll come around :-)

I didn't like ω at all, when it was first mentioned. I'm still nowhere near understanding it. But the question whether a mathematical symbol like ω is real and a number is simply whether it can be used in calculations. That's why we now accept that 1 and 0 are numbers and calculus and non-Euclidean geometries. ω can be used in calculations. So that's that. See the Wikipedia article on this for more details. — Ludwig V

This paragraph gratified me. If you are struggling to understand my posts then I'm getting through to at least one person. My talk about $\omega$ is something most people haven't seen, but the ideas aren't that hard. For what it's worth there's a Wiki page on ordinal numbers. The page itself isn't all that enlightening, but it does at least show that the ordinal numbers really are a thing in math, I'm not just making it all up.

You have a great insight that what makes a mathematical concept real is, in the end, its utility. Sometimes not even to anything practical, but just to math itself. We want to solve the equation x + 5 = 0 so we invent negative numbers. That kind of thing. In that sense, the ordinals exist.

But another way to think about it is that it's just an interesting new move in a game. As if you were learning chess and they told you how the knight moves. You don't say, "Wait, knights slay dragons and rescue damsels, they don't move like that." Rather, you just accept the rules of the game. You can think of ordinals like that. Just accept them, work with them, and at some point they become real to you. Just like the moves of the chess pieces any other formal game.

But I have tried to give a very concrete, down to earth example of how this works.

Suppose that we have the sequence 1/2, 3/4, 7/8, ... It converges to 1.

Now we can certainly form the set {1/2, 3/4, 7/8, ..., 1}. It's just some points in the closed unit interval.

But it gives us a model, or an example, of a set that contains an entire infinite sequence that "never ends" blah blah blah, and also contains its limit.

If you believe in the set {1/2, 3/4, 7/8, ..., 1}, then you should have no trouble at all believing in the set

{1, 2, 3, ..., $\omega$}. That's also just a set that contains an entire infinite sequence, along with its limit. We typically don't encounter this concept in the math curriculum that most people see, but it's perfectly standard once you go a little further. Also a lot of people have seen the extended real numbers with $\pm \infty$ and nobody complains about that, or do they?

It's true that the distances are different inside the two sets. But in terms of order, the two sets are exactly the same: an infinite series, along with its limit.

Anyway, this framework is very handy for understanding supertask type problems. That's why I'm mentioning it.

So if you don't like $\omega$, that' s no problem. Just think about {1/2, 3/4, 7/8, ..., 1}. It's the exact same set, with respect to what we care about, namely the property of being an infinite sequence followed by one extra term that occurs after the sequence.

Does that help?

But it is also perfectly true that a recitation of the natural numbers cannot end. — Ludwig V

That's a confusing way to think about it. It "ends" in the sense that we can conceptualize all of the natural numbers, along with one extra thing after the natural numbers.

And if we can't imagine that, we can certainly imagine {1/2, 3/4, 7/8, ..., 1}. There's nothing mysterious about that. An entire infinite sequence is in there, along with an extra point. It's a legitimate set.

If you want to think about the sequence 1/2, 3/4, 7/8, ... "never ending," that's fine. Yet we can still toss the entire sequence into a set, and then we can toss in the number 1. That's how sets work. They are containers for infinite collections of things.

By the way, $\omega$ is the "point at infinity" after the natural numbers. And $\omega + 1$ is the name for the set {1, 2, 3, 4, ..., $\omega$}.

$\omega + 1$ is the natural setting for all supertask puzzles. We have the state at each natural number, and we inquire about the final state at $\omega$.

That's why I like $\omega + 1$ as a mental model for these kinds of problems.

As I said earlier, it is remarkable that we can prove it. Yet we cannot distinguish between a sequence of actions that has not yet ended from one that is endless by following the steps of the sequence. So we are already in strange territory. — Ludwig V

This business about actions is what confuses people. They set up scenarios that violate the laws of physics, like the lamp that switches in arbitrarily small intervals of time, and then they try to use physical reasoning about them. Then they get confused.

In the way I'm describing this, you may think that the difference is between the abstract world (domain) of mathematics and another world, which might be called physical, though I don't think that is right. — Ludwig V

Well yes, you are correct to feel that it's not quite right. Because there is nothing physical about the lamp or the staircase. So it's a category error to try to use everyday reasoning about the physical world. That's why people get confused.

I'm very puzzled about what is going on here, but I'm pretty sure that it is more about how one thinks about the world than any multiverse. — Ludwig V

I think it all comes down the fact that calculus classes care about computation and not theory. That, and the fact that we don't know the ultimate nature of the world, and there's are good reasons to think it's not anything like the mathematical real numbers.

So on the one hand, the continuity of the world is an open question. And two, calculus classes are not designed to teach people how to think about limits in the more general ways that mathematicians sometimes do. Put those together with quasi-physical entities like physics-defying lamps, and you have a recipe for confusion.

There is no such thing as "going by pure logic", toward understanding the nature of reality. [/quore]

Agreed. But that does not justify using some means OTHER than logic to understand reality, and calling it logic! That's @Michael's fallacy. Saying something's a logical contradiction when it merely makes no sense to him. You agreed with me earlier that this is a fallacy. But you defend it when YOU do it.

To be clear: I have no objection to using extra-logical means of understanding reality. But then don't turn around and all it logic.

— Metaphysician Undercover

"Pure logic" would be form with no content, symbols which do not represent anything. All logic must proceed from premises, and the premises provide the content. And premises are often judged for truth or falsity. But as explained in the passage which ↪wonderer1 referenced, in the case of an "appeal to consequences", there is no fallacy if the premises are judged as good or bad, instead of true or false. That's why I said that this type of logic is very commonly employed in moral philosophy, religion, and metaphysical judgements of means, methods, and pragmatics in general. So for example, one can make a logically valid argument, with an appeal to consequences, which concludes that the scientific method is good. No fallacy there, just valid logic and good premises. — Metaphysician Undercover

Call it anything you like, but not logic! Logic means something else. That term is already taken. You are using extra-logic. Morality, right or wrong, productive/nonproductive. All well and good, but not logic. If logic is to mean anything, it has to mean something.

Therefore it is not the case that the reasoning is "extra-logical", it employs logic just like any other reasoning. What is the case is that the premises are a different sort of premises, instead of looking for truth and falsity in the premises we look for good and bad. So this type of judgement, the judgement of good or bad, produces the content which the logic gets applied to. — Metaphysician Undercover

Let's agree to disagree on that point.

No, that is not the case, because there are two very distinct senses of "determined". One is the sense employed by determinism, to say that all the future is determined by the past. The other is the the sense in which a person determines something, through a free will choice. In this second sense, a choice may determine the future in a way which is not determined by the past. And, since it is a choice it cannot be said to be random. Therefore it is not true that if the world is not random then it's determined (in the sense of determinism), because we still have to account for freely willed acts which are neither determined in the sense of determinism, nor random.[/qouote]

You can't have determinism and free will. Frankly if the world is random and we have some kind of influence on it through our will, or spirit, I find that much more hopeful than a universe in which I'm just a pinball clanging around a well-oiled machine.

Determinism is the nihilistic outlook, not randomness. In randomness there is hope for freedom. Say that's a pretty catchy saying. The church of Kolmogorov. In randomness lies the hope of freedom.
— Metaphysician Undercover

As I said above, it is not a matter of transcending logic, the conclusions are logical, but the premises are judged as to good or bad rather than true or false. So from premises of what is judged as good (rejecting repugnant principles), God may follow as a logical conclusion. — Metaphysician Undercover

God's going to hurl thunderbolts at you for so blithely enlisting him on your side to make such a specious argument. If I'm choosing good versus bad I'm not using logic, I'm using feelings. Logic says kill the one rather than the million. But if the one's you or yours, you kill the million. It's been done. Feelings trump logic. But your feelings are not logic!!

No I was not arguing that. In that case I was arguing that the idea ought not be accepted (ought to be rejected) unless it is justified. In the case of being repugnant, that in itself is, as I explained, justification for rejection. You appear unwilling to recognize what wonderer1's article said about the fallacy called "appeal to consequences". It is only a fallacy if we are looking for truth and falsity. If we are talking principles of "ought", it is valid logic. Therefore the argument that the assumption of randomness ought to be rejected because it is philosophically repugnant, cannot be said to be invalid by this fallacy, and so it may be considered as valid justification. — Metaphysician Undercover

Yes but the contrary proposition of determinism is even more repugnant, as I've noted. Shouldn't we (logically!) choose the lesser of two repugnancies?

But Michael did not show that supertasks are philosophically repugnant. — Metaphysician Undercover

And you have not shown randomness philosophically repugnant. By the time I thought about it a little, I realized that randomness is our only hope for salvation. It's the only way we're not automatons. Clockwork oranges. So you haven't made your point here. I am a proud randomite.

He showed that they are inconsistent with empirical science, — Metaphysician Undercover

There's no empirical science in these silly omega sequence paradoxes like the effing lamp and the effing staircase. That's the massive category error everyone makes. They posit these physics-defying scenarios then claim they're talking about the physical world.

and his prejudice for what is known as "physical reality" (reality as understood by the empirical study of physics) influenced him to assert that supertasks are impossible. — Metaphysician Undercover

I believe I made the same claim, but qualified it to "presently known physics."

As I explained in the other thread, in philosophy we learn that the senses are apt to mislead us, so all empirical science must be subjected to the skeptic's doubt. So it is actually repugnant to accept the representation of physical reality given to us by the empirical sciences, over the reasoned reality which demonstrates the supertask. And this is why that type of paradox is philosophically significant. It inspires us to seek the true reasons for the incompatibility between what reason shows us, and what empirical evidence shows us. We ought not simply take for granted that empirical science delivers truth. — Metaphysician Undercover

This is way past the lamp. The lamp is not a physical thing. These puzzles have no bearing on physical reality. That's a cognitive error everyone makes about them.

Also, there's more bad reasoning than "reason" in the discussions about these problems.

As explained above, I am not taking a standpoint of determinism. There are two very distinct senses of "determine", one consistent with determinism, one opposed to determinism (as the person who has a very strong will is said to be determined). I allow for the reality of both. — Metaphysician Undercover

You say randomness and determinism are compatible, and your justification is to use an alternate and unrelated meaning of the word determined?

But as of now, in this very post, I've convinced myself that I'm a randomist. But then again I've always suspected I'm a Boltzmann brain, and that's how randomists come into existence.

So...you're thinking of a limit in a vauge way ("symbolic"), and vaugely asserting the series "reaches" infinity, and then rationalize this with a mathematical system that defines infinity as a number. — Relativist

No. My thinking about limits is extremely precise and perhaps a bit more general than what you're accustomed to. I have never said that a series (or sequence if that's what you mean here) reaches infinity. I would not say that, and I did not say that.

What I said was that there is a mathematical view that sheds light on the subject, and makes it clear in where the limit of a sequence lives. The sequence 1/2, 3/4, 7/8, ... has the limit 1. Of course it never "reaches" 1. But you would have no objection to my putting {1/2, 3/4, ..., 1} into a set together. After all, I am allowed to take unions of sets: and {1/2, 3/4, ...} U {1} = {1/2, 3/4, ..., 1}. So it's a legit set.

Now 1 is in no way a "point at infinity," after all it's just the plain old number 1. And no member of the sequence ever "reaches" it. But it does live there as the limit; as the result of a well-defined limiting process.

I have suggested this mathematical model as a thought aid to these kinds of paradoxes. If you find it helpful all to the good, but if not, that's ok too. I find it helpful.

For what it's worth, in math, the natural numbers have an upward limit, called $\omega$, that plays the same role for the sequence 1, 2, 3, ... that the number 1 is for the sequence 1/2, 3/4, ...

It's the limit. It's more general notion of limit, one that allows us to reason about a "point at infinity." Which is exactly what these puzzles are about. That's why it's a handy framework for thinking about these kinds of puzzles.

You have a sequence that's defined (on/off, on a step, whatever) at each member of a convergent sequence; and you want to speculate on the definition at the limit. $\omega$ is exactly what you need; or rather, a set called $\omega + 1$, which is like the set {1/2, 3/4, ,,,, 1}. It's a set that contains an entire infinite sequence and its limit. It's exactly what we need to analyze these problems.

If it helps, here's the Wiki page on ordinals, at least so that you know they're a real thing. You can "keep counting past the natural numbers," and you get some very cool mathematical structures. Ordinals find application in proof theory and mathematical logic.

Although it's true that there are such mathematical systems, it doesn't apply to the supertask. Time is being divided into increasingly smaller segments approaching, but never reaching, the 1 minute mark. — Relativist

I'm going to defer talking about supertasks today, had enough for a while.

There is a mathematical (and logical) difference between the line segments defined by these two formulae:
A. All x, such that 0<=x < 1
B. All x, such that 0<=x <= 1 — Relativist

Please reread what I wrote. This is not on topic if you understand what I'm saying.

Your blurred analysis — Relativist

I'm doing my best to fit you with a sharper pair of mathematical eyeglasses to unblur your vision ... but you keep making a spectacle of yourself!!

conflates these, but it is their difference that matters in the analysis. The task maps exactly to formula A, but not to formula B (except in a vague, approximate way). Mathematics is about precise answers. — Relativist

You might consider using words like "reach" and "approach" with precision. They are not part of the mathematical definition of a limit. They're casual everyday synonyms that you are allowing to confuse you.

Then rather than recite the natural numbers I recite the digits 0 - 9, or the colours of the rainbow, on repeat ad infinitum.

It makes no sense to claim that my endless recitation can end, or that when it does end it doesn't end on one of the items being recited – let alone that it can end in finite time. — Michael

The natural numbers do not end, yet they have a successor in the ordinal numbers, namely $\omega$. This is an established mathematical fact.

I regard this as a helpful point of view when analyzing these kind of puzzles. I've explained it as best I can.

"It makes no sense" is not a logical argument. It's only a description of your subjective mental state. Once, violating the parallel postulate or the earth going around the sun or splitting the atom made no sense. You are not making an argument.

So I treat supertasks as a reductio ad absurdum against the premise that time is infinitely divisible. — Michael

If you only demonstrated the reductio. All you have is "it makes no sense," and that is not an argument.

Quite so. That's why these puzzles are not simply mathematical and why I can't just walk away from them. — Ludwig V

I think a lot of people feel that way.

The problem with Margaret Thatcher is that she thought that a dumb quip is a substitute for serious thinking. But then, she was a politician. She also believed that there is no such thing as society. — Ludwig V

I thought it was on point. People in the US like "forgiving student debt." But every nickel is just passed on to the taxpayers. Government doesn't have any money that it doesn't take from someone else. Or borrow and print, that's a nice game that has to end at some point too.

I agree that equality of outcome is not a reliable index of equality of opportunity and that people often talk, lazily, as if they were. But if equality of opportunity does not result in changes to outcomes, then it is meaningless. The only question is, how much change is it reasonable to expect? If 50% of the population is female and only eight of UK's top 100 companies are headed by women (Guardian Oct. 2021), don't you think it is reasonable to ask why? I agree that it doesn't follow that unfair discrimination is at work, but it must be at least a possibility. No? — Ludwig V

I agree. We need a balance between trying to homogenize society, and old-fashioned notions of merit.

Perhaps it's a matter of pendulum swinging and patience.

There are always issues with the NHS in the UK. But that's not about universal health care or not. It's about what can be afforded, what priority it has. Difficult decisions, indeed, but anyone with sense knows they must be made. That's why we have the national institute of clinical excellence. It is not perfect, but it is an attempt to make rational decisions; other systems do not even attempt to do that.
Of course, when my life, or my child's life, is at stake, I will put the system under as much pressure as I can to try everything. And to repeat, it's not about charity or robbing the rich. It's about insurance. — Ludwig V

Health care policy's hard, I agree. I've only heard anecdotal evidence about NHS.

I have no reason to give a flying fig about New York politics. — Vera Mont

It's a beautiful living experiment in what's known as restorative justice.

Crime is rampant and the DA is busy prosecuting the victims. People don't feel safe. It's going to sink Mayor Adams's once-promising political career.

I would think that many people interested in politics do follow New York City politics. But if you don't, that's cool. Not sure you are qualified to comment on the social justice approach to crime, though. It's failing in New York City in a very obvious way.

I can explain it very easily. There is two different senses of "limit" being used here. One is a logical "limit" as employed in mathematics, to describe the point where the sequence "converges". And "unlimited" is being used to refer to a real physical boundary which would be place on the process, preventing it from proceeding any further. There is no such "limit" to a process such as that described by the op. The appearance of paradox is the result of equivocation. — Metaphysician Undercover

Mathematicians would just refer to it as an "upper bound."

But you talk about a "real physical boundary." Here you imagine that the staircase is physical. It's not. The conditions of the problem violate known laws of physics.

It's only a conceptual thought experiment. And why shouldn't math apply to that?

But anyway, it's an upper bound. If it's a least upper bound, it's a limit.

I do think that there are members of the court who have an agenda. It is not that they are on Trump's side but that they see Trump as useful to their side. An expedient for attaining their conservative goals. — Fooloso4

Maybe. Dobbs certainly. But that's been a disaster for the GOP. It cost them the 2022 red wave and many local and special elections. It put abortion back on the table as a political issue. Centrist voters that could trend GOP on the economy, crime, and immigration, now have to vote Dem for abortion.

The percentage of the public for whom abortion is their top issue, represents a certain percent of the vote lost to the GOP and won by the Dems in every election at every level of politics forever, till Congress hashes out a law everyone can live with. And good luck with that.

In the long run, the conservative justices have done more harm to the conservative cause than if they'd left Roe in place. The Dems are going to win a lot of races they'd otherwise have lost, as long as abortion is on the ballot. And overturning Roe put abortion back on every ballot.

If this is part of some secret plan by the conservative Supes, I wonder what it is.

They do, they are just playing dumb. — Lionino

LOL. I'm just trying to take the subtle approach.

"Repugnant", is a commonly used word in philosophy. The argument I gave is logical, but what is concluded is that the assumption, "there is ontological randomness" is philosophically repugnant, because it would be counter-productive to the desire to know. Therefore it's more like a moral argument. The desire to know is good. The assumption of ontological randomness hinders the desire to know. Therefore that assumption is bad and one ought not accept it. — Metaphysician Undercover

I can agree with your reasoning that one "ought" not to accept it, but the reason is extra-logical. That is, if we are going by pure logic, you have not argued against it. It's like solipsism. Can't refute but pointless to believe it.

But consider: If the world is not random, then it's determined. And is that not equally repugnant? Nothing matters because we have no choice.

What do you say to that? It's repugnant either way. Either there's no meaning or ... there's no meaning. Is there a way out?

Since the argument concerns an attitude, the philosophical attitude, or desire to know, you're right to say that it is an argument concerning "feelings". But that's what morality consists of, and having the right attitude toward knowledge of the universe is a very important aspect of morality. This is where "God" enters the context, "God" is assumed to account for the intelligibility of things which appear to us to be unintelligible, thereby encouraging us to maintain faith in the universe's ability to be understood. Notice how faith is not certainty, and the assumption that the universe is intelligible is believed as probable, through faith — Metaphysician Undercover

God transcends logic, fair enough. But again, that's not a logical argument.

Not only is it pointless to believe it, but I would say it is actually negative. Choosing the direction that leads nowhere is actually bad when there are good places to be going to. — Metaphysician Undercover

Determinism is worse.

I agree that it is very important to leave as undecided, anything which is logically possible, until it is demonstrated as impossible. Notice what I argue against is the assumption of real randomness, that is completely different from the possibility of real randomness. — Metaphysician Undercover

In that case we are entirely in agreement. I never pretend to know the ultimate nature of the world. It may be random, it may be determined, it may be a combination of both, or it may be something entirely else such that the random/determined dichotomy is rendered meaningless.

That we ought to leave logical possibilities undecided was the point I argued Michael on the infinite staircase thread. Michael argued that sort of supertask is impossible, but I told him the impossibility needed to be demonstrated, and his assumption of impossibility was based in prejudice. — Metaphysician Undercover

But yes!! Here you are arguing that just because an idea is repugnant is no logical reason to reject it! So you should apply the same reasoning to randomness.

I believe that paradoxes such as Zeno's demonstrate an incompatibility between empirical knowledge, and what is logically possible. — Metaphysician Undercover

I think it's highly unlikely that the world will turn out to be a mathematical continuum like the real numbers. The real numbers are far too strange.

Most people will accept the conventions of empirical knowledge, and argue that the logically possible which is inconsistent with empirical knowledge is really impossible, based on that prejudice. But I've learned through philosophy to be skeptical of what the senses show us, therefore empirical knowledge in general, and to put more faith and trust in reason. So, to deal with the logical possibility presented in that thread, we must develop a greater intellectual understanding of the fundamental principles, space and time, rather than appeal to empirical knowledge. Likewise, here, to show that the logical possibility of ontological randomness is really impossible, requires a greater understanding of the universe in general. — Metaphysician Undercover

I agree with you there. I agree with most of what you wrote. Still I do want to understand why you see that @Michael is wrong to say that supertasks are logically impossible, when they are merely repugnant; yet you seem to reject that same reasoning when applied to randomness.

Also, don't you think determinism is at least as repugnant as randomness?

Imagine the nerve of somebody demanding fair treatment for all kinds of people, even the designated victims! Appalling, innit? — Vera Mont

Indeed it is. I quite share your sensibilities, or at the very least I have great sympathy for them.

But the larger point is that you have heard about people these days who prefer equity to equality, equality of outcome over equality of opportunity. You in fact seem to happen to be one of those folks.

But earlier, you claimed there were no such people.

So I take it that you have conceded my point. I'm not arguing the point of view pro or con; only that the point of view exists. That in fact you exemplify and represent it. So what you initially said, that you did not believe there were many of these people, was not quite true. Have I got that right? I don't want to presume, I may have misunderstood you.

Secondly, and again purely for conversation, on the issue of criminal justice. Do you follow New York City politics and current events? Do you support Alvin Bragg? Can you see how some people might think that compassion to criminals, no matter how well intentioned, can end up becoming a pronounced lack of compassion for their victims? Some of the folks pushed onto subway tracks by individuals previously treated gently by the criminal justice system might see it that way. Can you at least see that?

Indulge me in an analogy.

I see the Matrix (pictures): — keystone

This entire idea was completely lost on me.

Both perspectives accurately correspond to the simulation. So I agree that sets are fundamental, and I could even be convinced that digital rain is more fundamental than the Matrix. — keystone

Digital rain is more fundamental than the Matrix. That's very poetic.

But Let's not go there. I'm specifically talking about the (continuous version of the) Matrix where I believe continua are more fundamental than points. But I don't even want to debate this further, I'd rather show you what could be done with a Top-down approach and let you decide. — keystone

You know, it might be better if you would write a draft then edit it. This seems like stream of consciousness. It has a groovy vibe to it but it doesn't say anything.

I bring up the Matrix because, I want you to know that I recognize the unique purity and precision of the digital rain, but there are times, especially in discussions on geometry, when it's more effective to visually interpret the geometry from within the Matrix. Please allow yourself to enter the Matrix, try to understand my visuals, just for a little while. End of Matrix analogy. — keystone

As it happens I hate that stupid movie. It's a kung-fu flick with silly pretensions to pseudo-intellectuality. Also someone did the calculation and it turns out that humans make lousy batteries. Very inefficient.

Where is the line between your indulging yourself, and your trying to communicate a clear idea to me?

Okay, I lost you because I made a mistake. Let me try again:

Set: { (0,0) , (0,0.5) , (0.5,0.5) , (0.5,1) , (1,1) } where x1 and y1 in element (x1,y1) is a rational number

Metric: d((x1,y1),(x2,y2)) = | (x1+y1)/2 - (x2+y2)/2 | — keystone

No idea what you are trying to do, what you are doing, why you are doing it, and why you are telling me about it.

Upon further consideration, I've decided to significantly restrict my focus to a smaller enclosing set. I am now interested only in what I want to call 'continuous sets' which are those sets where, when sorted primarily by the x-coordinate and secondarily by the y-coordinate, the y-coordinate of one element matches the x-coordinate of the subsequent element. For example, we'd have something like: — keystone

Like a triangular section of the plane? Why?

You're right, |x-y| doesn't qualify as a metric. Let me try again. Forget about Universal Set. Instead, I aim to define a Continuous Exact Set. A set is defined as an exact set if all elements satisfy |x-y|=0. I propose that within my enclosing set, the only Exact Set is the trivial set, containing just one element. Once again, this isn't a groundbreaking revelation; I am simply emphasizing that rational numbers by themselves are insufficient for modeling a continuum. — keystone

I just don't know what you're doing. You seem to be having fun, and I don't mind because this like taking a rest after the mortal combat of the staircase thread.

Zeno greatly inspires me, yet from my viewpoint, his paradoxes serve merely as an aside. I assure you, the core thesis I'm proposing is much more significant than his paradoxes. But to save me from creating a new picture, please allow me to reuse the Achilles image below as I try again to explain the visuals.


The story: Achilles travels on a continuous and direct path from 0 to 1.
The bottom-up view: During Achilles' journey he travels through infinite points, each point corresponding to a real number within the interval [0,1].
The top-down view: In this case, where there's only markings on the ground at 0, 0.5, and 1, I have to make some compromises. I'll pick the set defined above and describe his journey as follows:

(0,0) -> (0,0.5) -> (0.5,0.5) -> (0.5,1) -> (1,1) — keystone

Wasted on me, hope you got something from it.

In words what I'm saying is that he starts at 0, then he occupies the space between 0 and 0.5 for some time, then he is at 0.5, then he occupies the space between 0.5 and 1 for some time, and finally he arrives at 1. — keystone

No idea, eyes glazed long ago.

Inconsistent systems allow for proving any statement, granting them infinite power. While debating the consistency of ZFC is beyond my current scope and ability, my goal is to develop a form of mathematics that not only achieves maximal power but also maintains consistency. Furthermore, I aim to show that this mathematical framework is entirely adequate for satisfying all our practical and theoretical needs. — keystone

Quite a tall order.

I haven't studied his original work, so I can't say with certainty, but I don't believe I'm referring to Euclid's formulation. — keystone

Well Euclid considered points fundamental, along with lines and planes. But modern set-theory based math takes sets as fundamental. In fact there is nothing other than sets. You start with the empty set and the rules of set theory and build up everything else.

The word point is only used casually, to mean an element of some set, or a tuple in Euclidean space, or a function in a function space.

I'm familiar with these methods. 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. — keystone

I'm just throwing things out that seem related to what you're saying.

I'm getting there, and your feedback has been instrumental in enhancing my understanding of this 'digital rain'. Up until now, my approach has primarily been visual. — keystone

I'm very glad I can help. What is the digital rain? Do you remember the Church of the Cathode Ray from the movie Videodrome?

Aside: Please note that I will have a house guest for several days, which may cause my responses to be slower than usual. — keystone

No problem, take your time. I hope you and your guest have a lovely visit.

That's not quite what I'm saying. The process described by the op has no limit. — Metaphysician Undercover

Oh I had no idea we were still talking about the OP. This thread's gone way beyond that.

I thought you were making a more general point, that the limit lives in a different kind of conceptual space than the sequence itself, or that the limit was imposed on the sequence by observers.

If I misunderstood then nevermind. I've long forgotten the staircase problem. I don't think I ever actually understood it.

That should be clear to you. It starts with a first step which takes a duration of time to complete. Then the process carries on with further steps, each step taking half as much time as the prior. The continuity of time is assumed to be infinitely divisible, so the stepping process can continue indefinitely without a limit. Clearly there is no limit to that described process — Metaphysician Undercover

Well 1/2 + 1/4 + 1/8 + ... is a well known convergent sequence. It converges to 1. And surely we've all experience one second going by. So that's the paradox, right?

I think what's confusing you into thinking that there is a limit, is that if the first increment of time is known, then mathematicians can apply a formula to determine the lowest total amount of time which the process can never surpass. Notice that this so-called "limit" does not actually limit the process in any way. The process carries on, unlimited, despite the fact that the mathematician can determine that lowest total amount of time which it is impossible for the process to surpass. — Metaphysician Undercover

It has not been productive in the past for us to discuss mathematics, and your misunderstanding of limits is not my job to fix. I gave at the office. Nothing personal but you know we have been down this road. I sort of get what you're saying but mostly not. "The process carries on, unlimited, even though there's a limit." I haven't the keystrokes to untangle the myriad conceptual difficulties with that statement, and the beliefs and mindset behind it; even if I had the inclination. I hope you'll forgive me, and understand.

Clearly, the supposed "limit" is something determined by, and imposed by, the mathematician. — Metaphysician Undercover

LOL. And the meaning of Moby Dick is only because of what we all determined the symbols to mean. Man and His Symbols, Jung. Yes we are symbolic beasts.

But within the sphere of math, the definition of a limit is as objective as can be. We lay down a definition, you know the business with epsilon and L, and we confirm that the sum converges; just as in the sphere of the English language, Moby Dick is a story about a bunch of guys who go whaling and it mostly doesn't end well.

To understand this, imagine the very same process, with an unspecified duration of time for the first step. The first step takes an amount of time, and each following step takes half as much time as before. In this case, can you see that the mathematician cannot determine "the limit"? All we can say is that the total cannot be more than double the first duration. But that's not a limit to the process at all. It's just a descriptive statement about the process. It is a fact which is implied by an interpretation of the described process. As an implied fact, it is a logical conclusion made by the interpreter, it is "not inherent to the sequence", but implied by it. — Metaphysician Undercover

I'm sorry, I can't really talk about the staircase problem specifically, I never paid much attention to it at the beginning. I mostly got interested in this thread when other issues were introduced. But mathematicians are very good at determining limits, and the one in question is perfectly well known to everyone who ever took a year of calculus. You might take a look at the Wiki page on limits.

That it is not inherent, but implied, can be understood from the fact that principles other than those stated in the description of the process, must be applied to determine the so-called "limit". — Metaphysician Undercover

You don't need any esoteric "principles other than those stated in the description of the process" to determine the sum of a geometric series as a particular limit.

We must subvert our tendency to compete — Benj96

I just noticed this. What means would you use to bring this about?

This appears to be the case. — Vera Mont

Just for light conversation ... when I say that a lot of people these days are advocating for equality of outcome rather than equality of opportunity ... you do not know what I am referring to? The DEI movement, social justice, wokitude, and the like? Disciplinary standards relaxed in schools, admission criteria relaxed in universities, the criminal justice system biased in favor of criminals, massive social change for the purpose of balancing out racial categories?

This news has not yet reached your province?

No. What Trump says and does and what the Supreme Court says and does are not the same. — Fooloso4

I only mentioned it because this is a bit of Trumpy thread. A lot of people think the court's on Trump's side and not being judicially impartial. And opinions about that correlate with people's opinions on Trump. So this is really a Trump thread. Or at least a Trumpy thread. That was my thought process anyway. But I'm not actually participating in the thread, so I haven't got any strong feelings, I had just noted that there's a zillion-page long Trump thread, and I assumed that was there to soak up the gusher of opinion on the guy.

Gotta say, the man was on reality tv for ten years, he knows what the American people love, or love to hate. The historians will have a field day, if we all live that long.

You're pointing to the limit of a mathematical series. A step-by-step process does not reach anything. There is no step that ends at, or after, the one-minute mark. Calculating the limit does not alter that mathematical fact. — Relativist

You can think of it that way. Or you can think of it "reaching" its limit at a symbolic point at infinity. Just as we augment the real numbers with plus and minus infinity in calculus, to get the extended real numbers, we can do something analogous with the natural numbers, and augment them with a symbolic point at infinity $\omega$, so that the augmented natural numbers look like this:

1, 2, 3, 4, ..., $\omega$

Now a sequence is just a function that for each of 1, 2, 3, ..., we assign the value of the sequence, the n-th term. And we can simply assign the limit as the value of the function at $\omega$. It's perfectly legitimate. We can define a function with ANY set as its domains. So a sequence is a function on $\omega$, and a sequence augmented with its limit (or any other value!) is just a function on $\omega + 1$.

This is a key point. I've stated it a number of times recently and I'm not sure I'm getting through. The natural numbers augmented with a point at infinity is a perfectly good domain for a function; and we can use such a function to identify each of the points of a sequence, along with the limit.

I also think you are misinterpreting the meaning of limit. — Relativist

On a forum our words must speak for themselves. But in this instance I can assure you that nothing could possibly be farther from the truth.

This article describes it this way:
In mathematics, a limit is the value that a function (or sequence) approaches as the input (or index) approaches some value...

The formal definition intuitively means that eventually, all elements of the sequence get arbitrarily close to the limit, since the absolute value |an − L| is the distance between an and L. — Relativist

Wiki is not necessarily a good source for mathematical accuracy or subtlety of expression, and in this case they have led you astray.

I hope very much that you will give some thought to what I wrote about defining the limit of a sequence as the value of some function on the naturals augmented with a symbolic point at infinity; or more concisely, as a function on $\omega + 1$. A sequence is just a function on $\omega$, which is a synonym for $\mathbb N$. You can "attach" the limit to the sequence by extending the same function to one on $\omega + 1$ .

I hope this is clear. I find it an extremely clarifying mental model of what's going on with a sequence and its mysterious limit. "Where does the limit live?" I get that it's kind of confusing. The limit lives at the point at infinity stuck to the right end of the natural numbers.

1, 2, 3, 4, ... $\omega$

That's how people need to learn how to count in order to better understand supertasks and limits.

You just said to me that one second of time can't pass; and this, I reject. Am I understanding you correctly?
— fishfry
No, I didn't. I said the stair-stepping PROCESS doesn't reach the 1 second mark. Are you suggesting it does? — Relativist

Sure, after 1 second. It's perfect obvious from daily existence. When I got up to make a snack I did first walk halfway to the kitchen then halfway again. So now I'm arguing for supertasks. But I could just as well argue against supertasks. So whatever you said, I could probably convince myself to agree with it.

I think the word "reach" is being abused in this conversation. It comes out of badly taught calculus classes, and Wikipedia.

There is no limiting process in the premises of the op, nor in what is described by ↪Relativist . The "limiting process" is a separate process which a person will utilize to determine the limit which the described activity approaches. Therefore it is the person calculating the limit who reaches the limit (determines it through the calculation), not the described activity which reaches the limited. — Metaphysician Undercover

Wow that's deep. Deep and wrong at the same time. That's interesting.

If I am understanding you: You say that if we have a sequence; that if that sequence happens to have a limit, then the limit is not inherent to the sequence, but is rather imposed by the observer.

I suppose the analogy is color, which is in the eye-brain system of the observer, not in the object or even in the light.

But actually, the limit can be considered part of the sequence. Just as a sequence is a function defined on the natural numbers; a sequence along with its limit can be defined as a function on the natural numbers augmented with a point at infinity, which I've been calling $\omega$.

It's really no different than taking the set {1/2, 3/4, 7/8, ...} and augmenting it with the number 1, to yield the new set {1/2, 3/4, 7/8, ..., 1}. Surely you can see that 1 is a perfectly sensible number on the number line. In many ways it's the ONLY sensible number. All other numbers are derived from it. That and 0. Give me 0 and 1 and I'll build all the numbers anyone needs.

So if that's what you're saying, I find that a very interesting thought. But there is no reason to imbue limits with mysticism. They're very straightforward. They're just the value of a sequence at the augmented point at infinity; which, if you don't like calling it that, is just adding the number 1 to the 1/2, 3/4, ... sequence.

This isn't the sense of "counting" I'm using. The sense I'm using is "the act of reciting numbers in ascending order". I say "1" then I say "2" then I say "3", etc. — Michael

Yes, I agree with you that math and physics use different definitions.

I apologize for getting crabby last night. As I went to bed I was thinking, Why am I snarling at someone about supertasks, I don't even care about supertasks.

You're right, I was not the one you were originally addressing. I jumped in because I was annoyed by your total lack of logic in claiming that supertasks are metaphysically impossible or logical contradictions. I agree with you that supertasks don't exist physically today, but I allow for the possibility of new physics in the future, just as there's always been new physics in the past. I don't think you've supported your metaphysical or logical arguments. That's why I jumped in.

Also it's perfectly clear that I can walk a mile, and I first walked the first half mile, etc., so if someone (not me, really!) wanted to argue that supertasks exist on that basis, I'd say maybe they have a point.

P1. It takes me 30 seconds to recite the first natural number, 15 seconds to recite the second natural number, 7.5 seconds to recite the third natural number, and so on ad infinitum.

P2. 30 + 15 + 7.5 + ... = 60

C1. The sequence of operations1 described in P1 ends at 60 seconds without ending on some final natural number.

But given that ad infinitum means "without end", claiming that the sequence of operations described in P1 ends is a contradiction, and claiming that it ends without ending on some final operation is a cop out, and even a contradiction. What else does "the sequence of operations ends" mean if not "the final operation in the sequence is performed"?

So C1 is a contradiction. Therefore, as a proof by contradiction:

C2. P1 or P2 is false.

C3. P2 is necessarily true.

C4. Therefore, P1 is necessarily false.

And note that C4 doesn't entail that it is metaphysically impossible to recite the natural numbers ad infinitum; it only entails that it is metaphysically impossible to reduce the time between each recitation ad infinitum. — Michael

I think "reciting natural numbers" is a red herring, because it's perfectly clear that there are only finitely many atoms in the observable universe, and that we can't physically count all the natural numbers.

But let me riddle you this. Suppose that eternal inflation is true; so that the world had a beginning but no end, and bubble universes are forever coming into existence.

And suppose that in the first bubble universe, somebody says "1". And in the second bubble universe, somebody says, "2". Dot dot dot. And bubble universe are eternally created, with no upper bound on their number.

Therefore: Under these assumptions, there is no number that doesn't get spoken. And therefore, all the numbers are eventually counted.

You see we don't have to "reach the end," since we can't do that. All we have to do is show that there is no number that never gets counted. Therefore they all do. It's a standard inductive argument. You show something's true for all natural numbers because there can't be a smallest number where it's not true.

I remind you that while eternal inflation is speculative but is taken seriously by a lot of smart people.

Therefore I claim that it is metaphysically possible to physically count the natural numbers; and that no logical contradiction is entailed. I'll grant you that I haven't yet shown how to do it in finite time, and so I have not refuted your point. I'm giving more of a plausibility argument that someday, there might actually be a finite-time supertask. We just don't know. You personally can not know. That's my real point, bottom line.

You cannot know what future physics will allow or conceptualize. That's my whole argument. That's why I say that supertasks violate contemporary physics, Planck scale and all that. But based on the shocking paradigm shifts of the past, there will be shocking paradigm shifts in the future; and physically actualized infinitary processes are as good a candidate as any for what comes next.

I wrote a response to @NoAxioms above in which I laid out my thoughts, it might be of interest ... https://thephilosophyforum.com/discussion/comment/900398

Thanks again for your good cheer in not firing back!

What is it about 'physical' that makes this difference? Everybody just says 'it does', but I obviously can physically move from here to there, so the claim above seems pretty unreasonable, like physics is somehow exempt from mathematics (or logic in Relativist's case) or something. — noAxioms

Well physics is of course exempt from math and logic. The world does whatever it's doing. We humans came out of caves and invented math and logic. The world is always primary. Remember that Einstein's world was revolutionary -- overthrowing 230 years of Newtonian physics. The world told us what new math to use. The world is not constrained by math, nor logic, nor by any historically contingent work of fallible man.

Math and even logic have always been drawn from looking at the world around us. So just as an aside to the main discussion, but responding to this one sentence that caught my eye ... physics IS exempt from math and logic. Meaning that historically, and metaphysically, physics is always ahead of math and logic and drives the development of math and logic.

But to the main question, the physical/mathematical distinction is important. I can never count all the integers in the physical world (as far as we know -- to be clarified momentarily); but in math I can invoke the axiom of infinity, declare the natural numbers to be the smallest inductive set guaranteed by the axiom, and count its contents by placing it into order-bijection with itself. That is: The identity map on the natural numbers is an order-preserving bijection that shows that the natural numbers are countable.

The former is a physical activity taking place in the world and subject to limitations of space, time, and energy. The latter is a purely abstract mental activity. How meat puppets such as ourselves come to have the ability to have such lofty abstract thoughts is a mystery. And if we are physical beings; and if thoughts are biochemical processes; are not our thoughts of infinity a kind of physical manifestation? That's another good question.

Perhaps our very thoughts of infinity are nature's way of manifesting infinity in the world.

So bottom line it's clear to me that we can't count the integers physically, but we can easily count them mathematically. And the reason I say that we can't physically do infinitely many things in finite time "as far as we know," is because the history of physics shows that every few centuries or so, we get very radically new notions of how the world works. Nobody can say whether physically instantiated infinities might be part of physics in two hundred years.

You italicize 'according to present physics', like your argument is that there's some basic flaw in current physics that precludes supertasks. How so? — noAxioms

Not a flaw, of course, any more than general relativity revealed a flaw in Newtonian gravity. Rather, I expect radical refinements, paradigm shifts in Kuhn's terminology, in the way we understand the world. Infinitary physics is not part of contemporary physics. But there is no reason that it won't be at some time in the future. Therefore, I say that supertasks are incompatible with physics ... as far as I know.

I utterly reject the notion that supertasks are a logical contradiction or metaphysical impossibility. They're only a historically contingent impossibility. We split the atom, you know. That was regarded as a metaphysical impossibility once too.

I mean, I can claim that there are no physical supertasks, but only by presuming say some QM interpretation for which there is zero evidence, one that denies physical continuity of space and time. — noAxioms

I'm not being specific like that. I'm only saying this:

There have been radical paradigm shifts in physics in the past;

There will certainly be radical paradigm shifts in the future; and

The next shift just may well incorporate some notion of physically instantiated infinities or infinitary processes; in which case actual supertasks may be on the table.


I analogize with the case of non-Euclidean geometry; at first considered too absurd to exist; then when shown to be logically consistent, considered only a mathematician's plaything, of no use to more practical-minded folk; and then shown to be the most suitable framework for Einstein's radical new geometry of spacetime.

Mathematical curiosities often become physicists' tools a century or more later. I think it's perfectly possible that physically instantiated infinities may become part of mainstream physics at some point in the future.

I will close with two contemporary examples of where speculative physics is starting to think about infinity.

One, eternal inflation. That's a theory of cosmology that posits a fixed beginning for the universe, but no ending. In this eternal multiverse are many bubble universes; either infinitely many, or at least a very large finite number. Physicists are vague on this point, but if time is eternally creating new universes, why shouldn't there be infinitely many of them.

And two, the many-world interpretation of quantum physics. Most people have heard of the Copenhagen interpretation, in which observing a thing causes the thing to be in one state or another; whereas before the measurement, it was neither in one state nor the other, but rather a superposition of the two states.

In Everett's many-world's interpretation, an observation causes the thing to be in both states in different universes. The universe splits in two, one in which the thing is in one state, and another universe it's in the other state. In some other universe I didn't write this. I know it sounds like bullshit, but Sean Carroll, a very smart guy and a prominent Youtube physicist (he's a real physicist too) is a big believer. He's recently moved away from mainstream physics, and more into developing a new philosophy of physics that incorporates many-worlds. How many worlds are there? Again this is a little vague, infinitely many or a large finite number.

These are just two areas I know about in which the idea of infinity is being taken seriously by speculative physicists. Would anyone really bet that they personally can predict the next 200 years of physics?

By definition a supertask, physical or otherwise, is completed. If it can't, it's not a supertask. — noAxioms

Well I can walk a mile, and I first walked the first half mile, and so forth, so it's a matter of everyday observation that supertasks exist. That would be an argument for supertasks. Zeno really is a puzzler. I don't think the riddle's really been solved.

Well that's for reading, there's been a lot of back and forth lately and I hope I was able to at least express what I think about all this.

Appeal to consequences — wonderer1

Thanks.

I agree — Benj96

I'll quit when I'm ahead here then :-)

A healthy society can have universal healthcare — Benj96

Many issues with long wait times at NIH in Great Britain. And in Canada, they offer assisted suicide for depression. I'd like to see some datapoints where universal health care has worked. Not an expert on health care policy, just repeating anecdotal evidence re Britain and Canada. Not necessarily defending the expensive US system, but it's a complicated issue. Just giving people free stuff is not a panacea. Who pays for the free stuff? As Margaret Thatcher once noted, "The problem with socialism is that you eventually run out of other people's money."

Which has no bearing on what I'm arguing. — Michael

You are not arguing, you're repeating your lack of argument. I'll let you have the last word, you are incapable of rational discussion.