Do you understand that the way you insult people prevents them from explaining things to you that would broaden your understanding? I'd be happy to outline the proof that a real number is rational if and only if its decimal representation eventually has a repeating block. But I intensely dislike your personal remarks. Knock it off or don't bother to reply to me any more since I won't be replying back.
Can "Infinity" ever be more than a mathematical abstraction?
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Do you understand that the way you insult people prevents them from explaining things to you that would broaden your understanding? I'd be happy to outline the proof that a real number is rational if and only if its decimal representation eventually has a repeating block. But I intensely dislike your personal remarks. Knock it off or don't bother to reply to me any more since I won't be replying back.
Write4U
Valued Senior Member
Speakpigeon
Valued Senior Member
I'd be sceptical anyway that someone who admitted to not knowing what "counted" means in mathematics to be able to talk competently about non-computability. And there you go talking about computability when computability is not the issue. I understand computability well enough and didn't ask anything about it. My issue is with non-computability. It's not because we haven't computed a number or even that we don't know how to do it or that we're not finished computing it that it is not computable in any meaningful sense.
Too subtle to discuss, probably.
EB
Speakpigeon
Valued Senior Member
You're missing what the issue is, again and I guess there's no point repeating myself.
So it's just as well that we drop this conversation altogether.
EB
Speakpigeon
Valued Senior Member
If any number exists, then infinity does, and one number exists.
Proof by contradiction.
Assuming all measures are approximate, to prove experimentally that any kinds of number correspond to a physical reality, we would need to be infinitely precise in our measurement, i.e. to measure again and again to add precision to the measure to get to the limit and obtain an exact measure.
So, if you believe that all measures are approximate and that infinity doesn't exist in the physical world, then you should believe no number could possibly correspond to anything physical. However, this would imply that there isn't one reality and this would contradict the idea that no number could correspond to something real.
QED
EB
Proof by contradiction.
Assuming all measures are approximate, to prove experimentally that any kinds of number correspond to a physical reality, we would need to be infinitely precise in our measurement, i.e. to measure again and again to add precision to the measure to get to the limit and obtain an exact measure.
So, if you believe that all measures are approximate and that infinity doesn't exist in the physical world, then you should believe no number could possibly correspond to anything physical. However, this would imply that there isn't one reality and this would contradict the idea that no number could correspond to something real.
QED
EB
Consider that someone running along a stadium track doesn't run an approximate distance between any two points, including any nominal start or finish. A track or a part of a track will have a length that can be approximately determined with physical meaurement, but a runner moves an actual length.Speakpigeon said:
Then you would have to put calculus aside. Calculus says a particle (or moving object) has an instantaneous velocity at each point along the path it moves. The continuous path is the infinite sum of infinitesimal velocities, according to Newton.
p.s. oh yeah, an infinitesimal is a number whose absolute value is always less than 1/n, for all values of n. Thus no infinitesimal is computable (ever, even with any future technology we might ever have, say the size of a planet or the solar system). But we can still compute dy/dx, by using the concept of the limit of a function.
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Speakpigeon
Valued Senior Member
He'll be flying.
So true.
He'll be flying.
So true.
EB
Speakpigeon is acting like a child. His proof by contradiction isn't much of a proof, probably because he has no real idea how to lay out a proof.
This thread is officially fucked, probably has been for the last few pages.
This thread is officially fucked, probably has been for the last few pages.
This one had me going. It depends on how we define computability for infinitesimals. If we define it one way, every infinitesimal is computable. Define it another way, and none are.
Let's review the definition of computability. We say a real number $$x$$ is computable if, given a real number $$\epsilon > 0$$, there is a Turing machine (TM) that halts in a finite number of steps, outputing a decimal approximation of $$x$$ to within $$\epsilon$$.
Earlier I mentioned that a real is computable if a TM can crank out its decimal digits. If you think about it, these two definitions are equivalent.
When we ask if an infinitesimal can be computable, one answer is, "The question is meaningless." It's like asking if an apple is a Rastafarian. The concept of religious affiliation only applies to people, not to deciduous fruits. Computability only applies to reals, not to hyperreals. [The hyperreals are a nonstandard model of the reals that contains infinitesimals].
But we can ask ourselves if we can extend the definition of computability in some way. Suppose we use the same definition of above but let $$x$$ be an infinesimal element of the hyperreals.
Let $$x$$ be infinitesimal. There are two cases:
* We require $$\epsilon > 0$$ to be real. Then I claim that every infinitesimal is computable. Why is that? Well, if you give me $$\epsilon > 0$$, by definition $$x$$ is ALREADY within $$\epsilon$$ of $$0$$. So the Turing machine:
Code:
output 0
Haltapproximates $$x$$ within $$\epsilon$$. That's one way to look at it.
* Or, what if we allow $$\epsilon$$ itself to be infinitesimal? Then no infinitesimal can be computable. An infinitesimal looks like something of the form $$\frac{1}{10^N}$$ where $$N$$ is a hyperinteger; that is, an infinite integer in the hyperreals. Such a hyperinteger can not be reached or expressed in a finite number of steps I believe. [I would not stake my life on this but I'm pretty sure based on what I know about the hyperreals].
But frankly the best answer might be that the question is meaningless. Computability is about approximating real numbers with Turing machines. It wasn't meant to apply to infinitesimals and as you can see, it doesn't really give much insight either way. Either every infinitesimal is computable or none are, depending on the definition you choose. But no actual insight into the nature of infinitesimals is gained by this exercise.
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Speakpigeon
Valued Senior Member
Don't quote people when you have fuck all to say about what they say.
Still, I'm sure you won't learn this lesson.
EB
Yes, I think the idea is supposed to be a number that can never be printed out or stored because it's too big.someguy1 said:
https://www.iep.utm.edu/zeno-par/#SH5e
This is false no matter how I try to interpret it. In fact I find it kind of "not even false," since I can't find an interpretation that makes sense to me. In fact when you say "the idea" and "it", I am not sure exactly what's being referred to. Can you explain your remark?
By the way I'm interpreting your posts as expressing a desire to better understand infinitesimals in the hyperreals, and noncomputability. I'm happy to oblige but I need something to work with. I would not be responding at all if I thought you were a crank of some sort. I looked at your posting history. You're capable of understanding these things but they are technical. I'll go do something else if you think that's better. You said the thread was fucked but I always believe that if you want better content, post some. I did my part.
This I also find strange. I posted a technical and, if I say so myself, insightful analysis of the question, "Is an infinitesimal computable?" In passing I referenced the hyperreals, which are a model of the first-order real number axioms in which there are infinitesimals.
You responded by quoting the section on the hyperreals in the SEP page on Zeno's paradoxes ... without any other comment of your own!
arfa brane Can you place your comments in perspective for me? You made a claim, "All infinitesimals are noncomputable." I saw that your claim was based on some mistaken or misunderstood notions, yet it was in fact a very interesting question! I responded with a well-thought out analysis. I say well thought out because I went through several stages of convincing myself one way or another till I found the right point of view. It wasn't something I understood off the top of my head. I had to work it out. I had to drill it down to the technical definitions and I thought this would be of interest to you, since you brought it up.
And after all this, you responded by ... copy/pasting the definition of the hyperreals from SEP? And not even the article on the hyperreals or on infinitesimals, but from the article on Zeno's paradoxes, which is not specifically about the hyperreals and could only be about the philosophy of the hyperreals and not the actual math. It can't be helpful. The hyperreals are already taken for granted when you want to talk about mathematical infinitesimals. It makes no sense at all in the context of my post to copy/paste this paragraph. I can't make heads or tails as to why you copy/pasted this material without comment.
tldr: You said the thread was fucked, I tried to elevate the conversation. I've read your other stuff, you can do better. So you tell me.
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If all measures are approximate, then there is no basis in the measurement of reality for denying the real existence of an infinite sum of distances abstracted as 1+1/2+1/4 - - - = 2. (There are no holes in the reality corresponding to line segment being summed)
Speakpigeon
Valued Senior Member
Yeah, I would agree that my "proof" may be a little hard to follow.
Still, I sort of broadly agree with your point except in terms of the wording. I think you should have said: there can be no proof that infinity doesn't exist. And I guess most reasonable people understand that anyway.
Also, what goes for infinity, or your particular instance of infinity, goes for God, too, and indeed any lunatic thing that would be beyond measurement. It wouldn't make any difference in this respect that infinity should be regarded as plausible, unlike God and lunatic beliefs.
EB
I thought the question would have been answered much earlier.
'Infinity' is not a number, so it cannot be used in any process of measurement.
"Infinity' cannot even be a mental concept, since the mind has no capacity to process it.
George Cantor cannot help you at this point.
Physical measurements require material objects as references. A 'real'number specified as a length may not coincide with an atom, and thus be an approximation.
The 'infinite' sum given in #553 never equals 2, which is why it was defined as a limit.
Zeno never needed an 'infinite' sum, since the motions were constant.
'Infinity' is not a number, so it cannot be used in any process of measurement.
"Infinity' cannot even be a mental concept, since the mind has no capacity to process it.
George Cantor cannot help you at this point.
Physical measurements require material objects as references. A 'real'number specified as a length may not coincide with an atom, and thus be an approximation.
The 'infinite' sum given in #553 never equals 2, which is why it was defined as a limit.
Zeno never needed an 'infinite' sum, since the motions were constant.
Speakpigeon
Valued Senior Member
The question has been answered but it's been answered many times and in very different ways. We just all disagree, which isn't surprising since mankind has disagreed about the concept of infinity since the word first came out. Your post is just one more opinion on the subject.
The rest of your post is inane so I really don't feel motivated to waste my time addressing each point. Read the thread, somebody will have competently replied in advance.
EB
What I said goes a step farther: there is evidence that it does - such as the existence of motion, and the accuracy of the equations of motion incorporating these infinities. It's possible that an infinite division of space into distances does not exist - that a distance abstractable as "2" does not in fact exist - but as yet there's no equivalently accurate mathematically abstract model of motion, force propagation, cause and effect sequences, etc, incorporating an alternative assumption.
We are, after all, asking a question of fact. We're looking for evidence, not proof.
I say this because infinitesimals are supposed to be non-zero numbers which can't be measured. I'm equating measurement with computation.someguy1 said:
But there's the two things: measuring the distance something moves, and the actual distance it moves. If I think about why these are different, perhaps just different kinds of 'computation' (that is to say, you measure or compute distances by moving; if you happen to have a standard ruler with you, you can 'gauge' your movement, but you have to accept there will be measurement errors), and I also invoke the concept of an infinitesimal distance, then motion is supposed to be a sum of these infinitesimal distances (a large sum, actually an infinite sum), none which can be individually measured=computed, seems to imply motion is impossible and we have one of Zeno's paradoxes.
So what really, is this sum of infinitesimals, like an integral from -∞ to +∞? Calculus doesn't seem to tell us much about motion after all; viz Wikipedia
And note that neither Leibniz nor Newton could really justify them, except that they did the job--in equations they determine the motions of planets etc. So what are they really?
Speakpigeon
Valued Senior Member
It certainly didn't go further explicitly. It certainly did implicitly, which is why is suggested an edit to clarify. You just did, in the wrong direction, though, I think.
As to motion, I think you're just a victim of our intuitive sense of motion whereby motion would be continuous. Some guy here is very keen on the model not being the reality and I would agree with him. Our brain somehow provides us with a model of motion that's continuous and there's nothing I guess we could do to shake it off, just as we cannot not see the redness of a flower. However, we can move beyond our intuitions by conceiving alternative models. One would be discontinuous motion, something which definitely sounds impossible and absurd, at least as long as you allow you intuition to sort of comment on the concept. So, learn to tell it to shut up. Discontinuous motion is perfectly conceivable and logical. We already have a model of it on board games, like chess, go & al. Of course, in actual fact, we seem to move the pieces in a continuous manner but it's the concept that matters. The logic of movement is discontinuous. Locations on a chess board are countable in the same sense that you can count integers. And real space could be like that, too. And time. Locations would only be able to take discrete values, not continuous one. That's just one basic model and there's nothing in our observations that disprove such a model precisely because our measures are all approximate.
As to having an equivalently accurate mathematical model of motion, it's obvious that we could produce not only one but the beginning of a countable infinity of them. Just strip all decimals in our current model beyond some digit rank that's big enough to not make any practical difference in terms of measured and measurable predictions. Or some something a little less rash but with same effect. I could do it myself!
So, no, there's no evidence, I don't think. In principle, if space is continuous, there can't be any evidence nor proof. Unless we could suddenly measure exactly.
But we probably can't get beyond our own coarseness. We're too clunky to actually touch infinity.
Still, we have the concept of it. Something we have as yet to explain how we did it.
EB