The series of 1/2 + 1/4 + 1/8 + ... equals 1. The sequence an=1−0.5n
...
converges to 1, yet 1 is not part of the sequence. As you agreed, there is no ∞-th item. Cool. — Lionino
The issue that I see is:
1 – if we admit that time is infinitely divisible;
2 – and we admit that an=1−0.5n
[bad markup omitted]
gives us the lenght covered by Achilles in the Zeno Walk at each step;
the walk only finishes if it accomplishes an infinite amount of steps. Right? — Lionino
If it is indeed accomplishing an infinite amount of steps, is there not a step where the sequence gives us 1? — Lionino
If not, how is the walk ever completed? — Lionino
; if so, is there not a corresponding state for the mechanism when the full time elapses? — Lionino
Nobody knows the answer to any of these questions.
In other words, by admitting that the result of an infinite series is necessarily true¹, how do you justify at the same time that the state is really undefined at 1 while also defending that Achilles can finish the run? — Lionino
I want to emphasise that I am not arguing about the mathematics, but about the (meta)physical meaning of some mathematical concepts. — Lionino
Does that make sense? — Lionino
1 – Is that also the case for non-standard analysis and arithmetic? — Lionino
OK, that other meaning of 'count'.
I think we're talking past each other. When asked for the difference between a mathematical and physical supertask, you seem to focus on two different definitions of countable: The assignment of a bijection, and calling or writing down each of the numbers. — noAxioms
I'm talking about a physical supertask as described by Zeno, which arguably has countably (first definition) steps performed in finite time. Nobody is posited to vocalize the number of each step as it is performed. — noAxioms
Bit off on the lore. It turns into a pumpkin, and at the 12th stroke, where presumably midnight is the first stroke, but I googled that and could not find an official ruling on the topic. — noAxioms
Point is you can define the state at the limit of a sequence to be anything you want. The lamp could turn into a pumpkin too. The premises of the problem don't forbid it.
I like Bernadete's Paradox of the Gods because it doesn't make those mistakes, and thus seems very much paradoxical since motion seems prevented by a nonexistent barrier. — noAxioms
For educational purposes concerning how infinity works, I like Littlewood-Ross Paradox because it is even more unintuitive, but actually not paradoxical at all since it doesn't break any of the above rules. It shows a linear series (effectively 9+9+9+...) being zero after the completion of every step. — noAxioms
And because, even if he doesn't have children, he's helping to train up the well taxed caregivers and inventors of helpful products for his old age. — Vera Mont
I think you are both mistaken to rely on physics to define what one wants to get at in this context. Physics is not only limited by the current state of knowledge, but also by its exclusion of much that one would normally take to be both physical and real. Somewhere near the heart of this is that there is no clear concept that will catch what we might mean by "whatever exists that is not mathematics" or by "whatever applied mathematics is applied to". — Ludwig V
I'm sorry. I didn't mean to gross you out. Perhaps if you think of death as a least upper bound, you'll be able to think of it differently. It is, after all, an everyday and commonplace event - even if, in polite society, we don't like to mention it. — Ludwig V
Yes ok
Yes. I was just drawing out the implications. You might disagree. — Ludwig V
Not too strenuously. As I mentioned I don't place as much metaphysical import on these puzzles.
Yes. In the context of the Achilles problem that's fine and I understand that you are treating that and the natural numbers as parallel. — Ludwig V
It's not clear to me that it really works. It makes sense to say that "1" limits "1/2, 1/4, ..." But I'm not at all sure that it makes sense to say that <omega> limits the sequence of natural numbers. "+1" adds to the previous value. "<divide by 2>" reduces from the previous value. The parallel is not complete. There are differences as well as similarities. — Ludwig V
How can it be out of reach? I went to the supermarket today. I walked from one end of the aisle to the other. I reached the end. I did indeed evidently sum a convergent infinite series.
— fishfry
Did you "get to the limit by successors" or "get there by a limiting process"? I don't think so. You are just not applying that frame to your trip. — Ludwig V
But if I did apply that frame, then Zeno would have a good point. I did somehow either 1) accomplish infinitely many tasks in finite time; or b) The world's not continuous like the real numbers.
I think Zeno had a very good point, and I don't accept the common wisdom that summing an infinite series solves the problem.
I've met other mathematicians who agree that Achilles is not interesting. But I'm fascinated that you think the arrow is interesting. I don't. Starting is a boundary condition and so not part of the temporal sequence, any more than the boundary of my garden is a patch of land. End of problem.[or/quote]
If time is made of instants, then from instant to instant, how do things know what to do next? Where is the momentum and velocity information stores? It's like a computer program where an object has associated with it a data structure containing information about the object. If I shoot an arrow, where is the arrow's data structure stored? I think it's a good question. But I've never really given a lot of thought to the matter. It all seems to work out.
— Ludwig V
But this may be interesting in the context of what we are talking about. A geometrical point does not occupy any space. It is dimensionless. One could say it is infinitely small. But it is obvious that there is no problem about passing an infinite number of them. It is a question of how you think about them. This is not quite the same as Zeno's problem, but it is close. — Ludwig V
That is a perfectly sensible answer to the question, "What is the state at the limit?" It's perfectly sensible because the conditions of the problem don't specify the value at the limit. And since the lamp is not physical, it can turn into anything we like at the limit. It's no different than Cinderella's coach, which is a coach at 1/2 second before midnight, 1/4 second before midnight, and so on, and turns into a coach at midnight.
— fishfry
I agree with that. — Ludwig V
Perhaps then, these problems are not mathematical and not physical, but imaginary - a thought experiment. (The Cinderella example shows that we can easily imagine physically impossible events) That suggests what you seem to be saying - that there are no rules. (Which is why I posited another infinite staircase going up). But if there are no rules, what is the experiment meant to show? — Ludwig V
The only restriction I can think of is that it needs to be logically self-consistent - and the infinite staircase is certainly that. I guess the weak spot in the supertask is the application of a time limit. — Ludwig V
However, I also want to say that I cannot imagine an endless staircase, only one that has not ended yet - once I've imagined that, I can wave my hand and say, that is actually an infinite staircase. — Ludwig V
I'm sorry about this rant, but I don't know how else to respond.
It depends on your philosophy of education. The thinking behind all education is a mess; the thinking behind higher education is even more of a mess; and the thinking about adult education is practically non-existent. You can think about in terms of vocational (career) benefits and non-vocational ("for fun") programmes and a combination of private benefits (for the student) and public benefits (for society in general). There's also an issue about benefits to employers, but these are rarely thought about in their own right. — Ludwig V
I appreciate your heartfelt thoughts about education, but the subject is the morality and economics of student loan "forgiveness," which is a gaslighting euphemism for passing the costs on to the taxpayers.
If I, as a competent adult, borrow $100k and sign a legally binding contract to pay it back; and then the government declares that YOU should pay it back instead; I take it you would object. One, it's not your debt; and two, it creates a moral hazard. Why shouldn't I go out and borrow another $100k in the expectation that the government will favor me again?
Clearly the morality and economics are no different if the cost of covering my $100k debt "forgiveness" is spread out among a few tens of millions of taxpayers, instead of you personally.
In the case of Biden's recent loan forgiveness, it also happens to be illegal. Only Congress can profligately waste taxpayer money. The House of Representatives has the "power of the purse." The Supreme Court ruled that Biden's earlier loan "forgiveness" program was illegal. He's just flouting the law because nobody will call him on it.
Underwater basket weaving looks like a bad career choice, but possibly a good choice for fun. Either way, the student should pay. — Ludwig V
Some programmes, like IT skills (and mathematical ones) lead to extremely profitable careers in the finance industry; again, the student should pay. But if there's a serious shortage of welders, such that various industries cannot find the workers they need, there's good reason why employers, and/or the state, might want to pay. — Ludwig V
Then there are programmes like social work and nursing, which require specialized professional training, but don't pay well. Isn't there a good case for state support? What abaout high-level professional careers which could be financed by students, but where that is impractical because of their high costs whether in infrastructure or time required; again, public subsidy makes sense. Another category is risky careers, like acting or archaeology or philosophy; again, there's a case for public subsidy, not only to ensure a supply for the labour market, but because the existence of those careers is a public good. — Ludwig V
If you thought that was a mess, consider the non-vocational subjects, or those subjects which can be studied for vocational reasons and can also be studied for fun. The catch here is that all the specific vocational careers presuppose some level of basic, general skills and knowledge, which enables people to function in society in general, both within and without their vocations; these skills are also the basis of good citizenship. These include reading, writing, and arithmetic, but also extend (In the UK) to Science, Technology, Engineering and Mathematics (practice needs theory, after all) and to various skills under the heading of good citizenship - philosophy, literature and history and the arts. Those last four are often regarded as purely for fun, so I don't claim that the idea that they are not just for fun is uncontentious. Perhaps the most effective argument for them is that democracy cannot function properly without them. J.S. Mill recognized this, but it seems now to be ignored, which is a pity. Mind you, the idea that an understanding of the humanities was essential for a decent society took a very serious knock in WW2. But it is far from dead. — Ludwig V
Underwater basket weaving? Probably not. Philosophy? Fine Art? There's at least a case to think about, isn't there? — Ludwig V
PS. I forgot to explain how students should pay when they need to. Through the tax system. If their career choice pays off, they will pay increased taxes, so the public purse will benefit and their debt "repaid" - or, if you prefer, the public investment in their career pays off. Where their career does not pay off in that way, the public (and employers) will benefit from an increased supply of highly qualified labour. Where their career is not directly developed by their qualification, it will have been helped by the "transferable skills" developed in their programme and by the improved contribution they can make by their contribution to social and political life.
In other words, payment through the tax system is perfectly well justified by the multiple benefits provided by higher education. Nobody has a problem with that way of paying for schools. Why would higher education be any different? — Ludwig V
That process - what was liberal and new, becomes old hat, and conservative. That what's happened to feminism, etc. The agenda has moved on. It's very disappointing to those of us who thought the problems were solved. But there are unsolved and unconsidered issues and big gaps in even the basic rights that one thought had been established. — Ludwig V
There's no rehabilitation going on. There's a revolving door of people committing violent crimes, being put back on the street, and re-offending.
— fishfry
If that's so, there is a problem. — Ludwig V
I dealt with this already. If you restrict the meaning of "physical" to that which abides by the law of physics, then every aspect of what we would call "the physical world" which violates the laws of physics, dark energy, dark matter, for example, and freely willed acts of human beings, would not be a part of the "physical" world. — Metaphysician Undercover
That's not true at all. It does not correctly represent how we use the word "physical". "Physical" has the wider application than "physics". We use "physical" to refer to all bodily things, and "physics" is the term used to refer to the field of study which takes these bodily things as its subject. Therefore the extent to which physical things "obey the known laws of physics" is dependent on the extent of human knowledge. If the knowledge of physics is incomplete, imperfect, or fallible in anyway, then there will be things which do not obey the laws of physics. Your claim "a physical thing must obey the known laws of physics" implies that the known laws of physics represents all possible movements of things. Even if you are determinist and do not agree with free will causation, quantum mechanics clearly demonstrates that your statement is false. — Metaphysician Undercover
I gave you an example. A human body moving by freely willed acts violates Newton's first law. — Metaphysician Undercover
"Newton’s first law states that every object will remain at rest or in uniform motion in a straight line unless compelled to change its state by the action of an external force. This tendency to resist changes in a state of motion is inertia." — Metaphysician Undercover
There is no such "external force" which causes the freely willed movements of the human body. We might create the illusion that the violation can be avoided by saying that the immaterial soul acts as the "force" which moves that body, but then we have an even bigger problem to account for the reality of that assumed force, which is an "internal force". Therefore Newton's first law has no provision for internal forces, and anytime such forces act on bodies, there is a violation of Newton's laws. — 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.
— fishfry
If you understand this, then you ought to understand that being physical in no way means that the thing which is physical must obey the laws of physics. — Metaphysician Undercover
It is not the case that we only call a thing "physical" if it obeys the laws of physics, the inverse is the case. We label things as "physical" then we apply physics, and attempt to produce the laws which describe the motions of those things. Physical things only obey the laws of physics to the extent that the laws of physics have been perfected. — Metaphysician Undercover
Ok, now we're getting somewhere. The point, in relation to the "paradox" of the thread is as follows. There are two incompatible scenarios referenced in the op. Icarus descending the stairs must pass an infinite number of steps at an ever increasing velocity because each step represents an increment of time which we allow the continuum to be divided into. In the described scenario, 60 seconds of time will not pass, because Icarus will always have more steps to cover first, due to the fact that our basic axioms of time allow for this infinite divisibility. The contrary, and incompatible scenario is that 60 seconds passes. This claim is supported by our empirical evidence, experience, observation, and our general knowledge of the way that time passes in the world. — Metaphysician Undercover
What I believe, is that the first step to understanding this sort of paradox is to see that these two are truly incompatible, instead of attempting to establish some sort of bridge between them. The bridging of the incompatibility only obscures the problem and doesn't allow us to analyze it properly. Michael takes this first step with a similar example of the counter ↪Michael, but I think he also jumps too far ahead with his conclusion that there must be restrictions to the divisibility of time. I say he "jumps to a conclusion", because he automatically assumes that the empirical representation, the conventional way of measuring time with clocks and imposed units is correct, and so he dismisses, based on what I call a prejudice, the infinite divisibility of time in Icarus' steps, and the counter example. — Metaphysician Undercover
I insist that we cannot make that "jump to a conclusion". — Metaphysician Undercover
We need to analyze both of the two incompatible representations separately and determine the faults which would allow us to prove one, or both, to be incorrect. So, as I've argued above, we cannot simply assume that the way of empirical science is the correct way because empirical science is known to be fallible. And, if we look at the conventional way of measuring time, we see that all the units are fundamentally arbitrary. They are based in repetitive motions without distinct points of separation, and the points of division are arbitrarily assigned. That we can proceed to any level, long or short, with these arbitrary divisions actually supports the idea of infinite divisibility. Nevertheless, we also observe that time keeps rolling along, despite our arbitrary divisions of it into arbitrary units. This aspect, "that time keeps rolling along", is what forces us to reject the infinite divisibility signified by Icarus' stairway to hell, and conclude as Michael did, that there must be limitations to the divisibility of time. — Metaphysician Undercover
Now the issue is difficult because we do not find naturally existing points of divisibility within the passage of time, and all empirical evidence points to a continuum, and the continuum is understood to be infinitely divisible. So the other option, that of empirical science is also incorrect. Both of the incompatible ways of representing time are incorrect. What is evident therefore, is that time is not a true continuum, in the sense of infinitely divisible, and it must have true, or real limitations to its divisibility. This implies real points within the passage of time, which restrict the way that it ought to be divided. The conventional way of representing time does not provide any real points of divisibility. — Metaphysician Undercover
"Real divisibility" is not well treated by mathematicians. — Metaphysician Undercover
The general overarching principle in math, is that any number may be divided in any way, infinite divisibility. — Metaphysician Undercover
However, in the reality of the physical universe we see that any time we attempt to divide something there is real limitations which restrict the way that the thing may be divided. Furthermore, different types of things are limited in different ways. This implies that different rules of division must be applied to different types of things, which further implies that mathematics requires a multitude of different rules of division to properly correspond with the divisibility of the physical world. — Metaphysician Undercover
Without the appropriate rules of divisibility, perfection in the laws of physics is impossible, and things such as "internal forces" will always be violating the laws of physics. — Metaphysician Undercover
The Planck limitations are just as arbitrary as the rest, being based in other arbitrary divisions and limitations such as the speed of light. The Planck units are not derived from any real points of divisibility in time. — Metaphysician Undercover
No, the point of the puzzle is to demonstrate that the sum is always less than one, and that the mathematician's practise of making the sum equivalent to one is just an attempt to bridge the gap between two incompatible ways of looking at the theoretical continuum. — Metaphysician Undercover
The assumption that the sum is equivalent to one is what creates the paradox. — Metaphysician Undercover
the completeness axiom of the real numbers is one of the crowning intellectual achievements of humanity.
— fishfry
I hope you're joking, — Metaphysician Undercover
but based on our previous discussions, I think you truly believe this. What a strangely sheltered world you must live in, under your idealistic umbrella. — Metaphysician Undercover
That was attempted back in 2016 with the whole "DESTROYS sjw with facts and logic", it only got worse. Some of the people on that "side" are victims, I imagine they don't even have an inner monologue so they can't even filter what information is fed to them.
Regardless of whether they even know what they are saying, the time to be subtle with people who want you gone and your culture burned was long ago, nobody cares about being called racist/sexist/theosophist any longer. There is no god anymore, everything goes. — Lionino
However, how do you arrive at that conclusion? — Lionino
The two options that I can think of is by admitting that the sum of an infinite series is an approximation instead of the exact value, — Lionino
or by casting some doubt on the idea of an ∞-th item of a series. — Lionino
The latter seems to cause more problems than solve them for me. Did you use a different reasoning? — Lionino
OK. I remembered WIttgenstein's oracular remark that death is not a part of life. My concern that the limit is not generated by the defining formula isn't the problem I thought it might be. — Ludwig V
I don't really believe in "possible" without qualification. There's logically possible, (is mathematically possible the same or something different? Does is apply here?), physically possible, and a range of others, such as legally possible. So what kind of possibility is a supertask? — Ludwig V
So your reply is that it is neither. It suggests a combination of physical and mathematical rules which is incoherent but generates an illusion. — Ludwig V
But then you say
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
Obviously, as each stage gets smaller, I will complete it more quickly. But still, it will take some period of time, and the final step looks out of reach. That looks like a combination of physical and mathematical rules. — Ludwig V
It isn't a real problem because I can analyze the task in a different way. I can complete the first yard, the second yard.... When I have completed 1760 yards, I have completed the task. But the supertasks seem not to permit that kind of analysis. Is that the issue? — Ludwig V
To count a set means to place it into bijection with:
— fishfry
OK, that meaning of 'count'. In that case, I don't see how mathematical counting differs from physical counting. That bijection can be done in either case. In the case with the tortoise, for any physical moment in time, the step number of that moment can be known. — noAxioms
I am saying that Zeno describes a physical supertask, that Achilles must first go to where the tortoise was before beginning to travel to where the tortoise is at the end of that prior step.
Zeno goes on to beg the impossibility of the task he's just described, so yes, he ends up with a contradiction, but not a paradox. — noAxioms
I also would hate to have to talk about the poor kilometerage that Bob's truck gets. — noAxioms
It [the even-oddness of ω]is neither, and who's asking such a thing?
— fishfry
The lamp scenario asks it, which is why the comment was relevant. — noAxioms
See here:
As Salmon (1998) has pointed out, much of the mystery of Zeno’s walk is dissolved given the modern definition of a limit. This provides a precise sense in which the following sum converges:
Although it has infinitely many terms, this sum is a geometric series that converges to 1 in the standard topology of the real numbers. — Michael
A discussion of the philosophy underpinning this fact can be found in Salmon (1998), and the mathematics of convergence in any real analysis textbook that deals with infinite series. — Michael
From this perspective, Achilles actually does complete all of the supertask steps in the limit as the number of steps goes to infinity. — Michael
]Suppose we switch off a lamp. After 1 minute we switch it on. After ½ a minute more we switch it off again, ¼ on, ⅛ off, and so on. Summing each of these times gives rise to an infinite geometric series that converges to 2 minutes, after which time the entire supertask has been completed. — Michael
I have been arguing that it is a non sequitur to argue that because the sum of an infinite series can be finite then supertasks are metaphysically possible. — Michael
The lack of a final or a first task entails that supertasks are metaphysically impossible. — Michael
I think this is obvious — Michael
if we consider the supertask of having counted down from infinity, and this is true of having counted up to infinity as well. — Michael
We can also consider a regressive version of Thomson's lamp; the lamp was off after 2 minutes, on after 1 minute, off after 30 seconds, on after 15 seconds, etc. We can sum such an infinite series, but such a supertask is metaphysically impossible to even start. — Michael
Not so. I don't know what data is available to you, but perhaps you should look around. All I'm saying is that you cannot assume that every vocational programme provides marketable qualifications nor that every non-vocational programme does not. It's up to the market to decide what it wants.
Equally, it is up to students to decide what they want, even if they make choices that you think are unwise. It's not as if we can predict and provide what the economy wants. — Ludwig V
Point being that pipefitters shouldn't be shouldering the cost of the loans forgiven for social justice majors.
— fishfry
Well, if the cost is funded by general taxation, the contribution will depend on their income. That doesn't seem unreasonable - unless you think that people should not study social justice. But I think it is a good thing that as many people as possible should understand what social justice is. — Ludwig V
It's complicated. In the UK, liberals in the 19th century were, by and large, members of the elite. They were never particularly enthusiastic about supporting the working classes. They were much more interested in free trade, political issues like voting rights and moral/social issues like divorce, gay rights &c. (Conservatives supported protection and social conservatism). The working classes, by and large, had to fight their own battles, which they did through the Trade Unions.
But I'm sure the alignments were different in the USA. — 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.
— fishfry
People can't re-offend if they're locked up.
— fishfry
You can't imprison violent criminals forever, unless you can prove them criminally insane. Sooner or later, they have to hit the streets again. That's why rehabilitation is so important. — 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.
— fishfry
Nothing wrong with that, so long as you are open to new ideas occasionally. — Ludwig V
I don't know the details, but my instinct is to suggest that if the rehabilitation programmes in NYC aren't working, find a better programme, don't give up on the attempt. Money spent on effective programmes to keep people out of prison is a good investment. Back that up by improving detection and arrest, which is by far the most effective deterrent. Tossing people out of prison into the general population will not work and putting them back in prison later on is very expensive, not only in running the prisons, but also in the damage inflicted on families and children. — Ludwig V
I appreciate you asking a specific question about my explanation instead of dismissing it outright. I believe this has helped us move forward. — keystone
What does (.5, .5) represent?
— fishfry
Yes, it represents the point we would conventionally label 0.5. — keystone
Step one involves defining the journey through the use of intervals. — keystone
Step two entails describing these intervals within the framework of a topological metric space. — keystone
To successfully carry out step two, it's crucial that all elements involved are of the same type. For instance, I assume that defining a metric on a set that includes both points and intervals is not straightforward. — keystone
As mentioned earlier, rather than defining continua in terms of points, I am defining points in terms of continua, utilizing intervals (at least in the 1D case). — keystone
Don't the standard real numbers already "describe continua with arbitrarily fine precision?
— fishfry
Before I answer your question, I want to ensure we are on the same page. Do you understand how each of the five steps along the journey from 0 to 1 is represented by intervals, and that the union of these five intervals describes a continuous journey from 0 to 1? — keystone
From birth, children are taken to this daycare+school+apartment+prison complex to learn life skills such as trigonometry. — Scarecow
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
Do you honestly not see how this relates to the Google Maps screenshot I sent a few posts back? — keystone
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
but achieving this involves turning everything upside down—without dismissing any past mathematical progress. This approach offers a powerful new perspective on mathematics. — keystone
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
Please tell me which iteration you are tripping up on: 0, 1, 2, 3, or 4? — keystone
— keystone
I'm using interval notation. It's an interval. — keystone
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
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
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
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
It seems that you're either unable or unwilling to acknowledge even the most basic points I've raised. — keystone
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
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
Therefore, we must respect the fact that moral arguments can proceed with valid logic, — Metaphysician Undercover
I really don't see how there could be a staircase which is not physical. — Metaphysician Undercover
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
But the classic definition of "physical" is "of the body". — Metaphysician Undercover
And when a body moves itself, as in the case of a freely willed action, that body violates Newton's first law. — Metaphysician Undercover
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
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
That's why people commonly accept that there is a distinction between the laws of physics and the laws of nature. — Metaphysician Undercover
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
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
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
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
just like in Michaels example of the counter, such a counter is a physical object, — Metaphysician Undercover
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
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
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
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
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
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
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
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
By the way, ω is the "point at infinity" after the natural numbers
— fishfry
What does "after" mean here? — Ludwig V
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
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
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
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
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
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
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
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 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
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
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
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
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
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
I have more or less dropped out due to the repetitive assertions not making progress, but thank you for this post. — noAxioms
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
This is not radical. The rational numbers are countable, but not if counted in order, so it's not a new thing. — noAxioms
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
and we inquire about the final state at ω
Which works until you ask if ω is even or odd. — noAxioms
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
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
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
Sorry, but what? I still see no difference. What meaning of 'count them' are you using that it is easy only in mathematics? — noAxioms
That doesn't follow at all since by this reasoning, 'as far as we know' we can do physically infinite things. — noAxioms
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
Octonians shows signs of this sort of revolution. — noAxioms
Physicists are vague on this point, but if time is eternally creating new universes, why shouldn't there be infinitely many of them. — noAxioms
It is a mistake to talk about 'time creating these other universe'. — noAxioms
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
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
And two, the many-world interpretation of quantum physics.
That's the type III. — noAxioms
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
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
Everett's work is technically philosophy since, like any interpretation of anything, it is net empirically testable. — noAxioms
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
Ah, local boy. I am more used to interacting with those who walk a km. There's more of em. — noAxioms
And suppose that in the first bubble universe, somebody says "1".
The universes in eternal inflation theory are not countable. — noAxioms
Yes, each step in a supertask can and does have a serial number. That's what countably infinite means. — noAxioms
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
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
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
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
But it is also perfectly true that a recitation of the natural numbers cannot end. — Ludwig V
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
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
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
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
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
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
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
But Michael did not show that supertasks are philosophically repugnant. — Metaphysician Undercover
He showed that they are inconsistent with empirical science, — Metaphysician Undercover
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
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
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
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
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
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
Your blurred analysis — Relativist
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
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
So I treat supertasks as a reductio ad absurdum against the premise that time is infinitely divisible. — Michael
Quite so. That's why these puzzles are not simply mathematical and why I can't just walk away from them. — Ludwig V
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 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
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
I have no reason to give a flying fig about New York politics. — Vera Mont
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
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
They do, they are just playing dumb. — Lionino
"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
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
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
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
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
I believe that paradoxes such as Zeno's demonstrate an incompatibility between empirical knowledge, and what is logically possible. — Metaphysician Undercover
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