Possible lingual solution to paradoxes.

[studiot]

The website prometheus linked to notes that there are several ways to resolve Zeno.

 

In fact you don't need 'infinity' at all when considering the sequence.

 

Further the website makes the usual sleight of hand switch from time to distance in its presentation.

[Acme]

...It does not make sense to me that one infinity can be larger than another, because infinity already is at the max of quantity. You cannot beat that, you cannot exceed that, because if you could, your first claim of being at the maximum possible count, would have been incorrect.

 

Regards, TAR

Arguing from personal incredulity is a logical fallacy. Just because you can't imagine it does not mean it can't be imagined or proved by others.

 

See Cardinality @ Wiki

 

Infinite sets

 

Our intuition gained from finite sets breaks down when dealing with infinite sets. In the late nineteenth century Georg Cantor, Gottlob Frege, Richard Dedekind and others rejected the view of Galileo (which derived from Euclid) that the whole cannot be the same size as the part. One example of this is Hilbert's paradox of the Grand Hotel. Indeed, Dedekind defined an infinite set as one that can be placed into a one-to-one correspondence with a strict subset (that is, having the same size in Cantor's sense); this notion of infinity is called Dedekind infinite. Cantor introduced the cardinal numbers, and showed that (according to his bijection-based definition of size) some infinite sets are greater than others. The smallest infinite cardinality is that of the natural numbers (aleph 0). ...

[tar]

 

Throughout this immersion in physics, Hilbert worked on putting rigor into the mathematics of physics. While highly dependent on higher math, physicists tended to be "sloppy" with it. To a "pure" mathematician like Hilbert, this was both "ugly" and difficult to understand. As he began to understand physics and how physicists were using mathematics, he developed a coherent mathematical theory for what he found, most importantly in the area of integral equations. When his colleague Richard Courant wrote the now classic Methods of Mathematical Physics including some of Hilbert's ideas, he added Hilbert's name as author even though Hilbert had not directly contributed to the writing. Hilbert said "Physics is too hard for physicists", implying that the necessary mathematics was generally beyond them; the Courant-Hilbert book made it easier for them.

 

 

Acme,

 

I acknowledge that others, many others have and have had superior intellect to mine. I do not say in the least that the ideas and concepts related to the size of a particular infinite set are not possible to have. It was taught to me in school that there were more real numbers, than integers yet they were both infinite in quantity. i, back then, imagined the number line, as containing all the numbers. Every number was on the line, even irrationals and trancendentals had their place on the line.

 

Infinity to me, was the fact that not only did the line extend forever in both directions, but there was no limit to the precision with which you could divide and label any choosen interval on the line. Given the decimal system as a guide, one could easily divide any interval into 10 equal slices by adding a digit. Since the line was infinite, in both scope and precision, there was an infinite number of divisions available between any two numbers. But with this scheme of understanding infinity, the "new" numbers that Cantor said he was making with the second diagonal scheme, were already accounted for. They were not new. Just numbers already on the line in a different place. They were not "new" or additional numbers that one could then consider part of a larger set than the original, infinite one.

 

My question was on the arbitrary nature of the second diagonal of Cantor. I was not suggesting it was impossible for him to think the thing. I was questioning if it made sense and worked out.

 

Regards, TAR

Perhaps a topic for another thread but within the set of humans that have been alive in the last 100 years there are subsets such as these.

 

1. Persons who cannot understand the concept.

 

2. Persons who can understand the concept but have not yet learned the thing or had the insights required.

 

3. Persons who can understand the concept, but find it lacking in some regard (purpose, intent, assumptions, rigor, application, scope, meaning, direction, or whatever.)

 

4. Persons who not only can, but do understand the concepts and agree with them as peers of concept describer.

 

5. Persons who don't understand the concepts, but agree with them anyway, because they were stated by persons of greater intellect and reviewed by persons of greater intellect and are acceptable on the basis of "trust the experts".

 

I think I am personally in the 2 category, with desires to be in the 4 category, but too lazy and ilinformed to get there.

But that does not negate my question. Nor require that my question is not deserving of an answer, simply because I do not see it Cantor's way.

 

There are some basic "hard" questions, related to Zeno, and Hilbert, and infinity and existence and such, that if "we" truely knew the answers to, "we" would not be having this discussion. So, in the context of this thread, I think a direct comment on my "problem" with the second diagonal example, would be better than a link, suggesting that I will find the answer to my question if I read and understood what I read. My question came "while" I was reading a previous link.

 

Regards, TAR

[Acme]

Acme,

 

I acknowledge that others, many others have and have had superior intellect to mine. I do not say in the least that the ideas and concepts related to the size of a particular infinite set are not possible to have.

...

I think I am personally in the 2 category, with desires to be in the 4 category, but too lazy and ilinformed to get there.

But that does not negate my question. Nor require that my question is not deserving of an answer, simply because I do not see it Cantor's way.

 

...

Regards, TAR

To be clear, Hilbert did not construct Hilbert's paradox. I put the incorrect link with his name. The correct link is here: >> Hilbert's paradox @ Wiki

 

It was first described by George Gamow in his 1947 book One Two Three ... Infinity and jokingly attributed to David Hilbert.[1][2]

While being lazy may be the reason your are arguing as you do, it is no excuse.

[tar]

Acme,

 

But not having a degree, does not precude one from thinking about stuff.

 

I was following along with Cantor in an attempt to understand how the term cardinality was being used, what was an aleph and what was a null and so on, but there were aspects of pairing up numbers in one set an another, where an understanding of the "complete" set was required. Any infinite quanity is by definition incomplete, as you can not arrive at the last member of the set...ever. If we are to be playing games with infinite sets, concieving of what set of numbers has a companion in the other set, whether there are any extras or duplicates and whether the match is covered in one direction and not in the other, and so on, it would be first required for one to understand the second diagonal argument. As that I thought I understood it, but did not think it was correctly identifying "extra" unaccounted for numbers, the whole argument was not valid in my estimation. I have to get that part squared away, before I can go to more complicated combinations of metamath he is dealing in pretending I understand what aspect of infinity Cantor is suggesting is sizable, when I don;t see it

 

Can't we just talk about the idea of infinity...as in what is its definition, what does it mean to you, without concentrating on the fact that I have not read the book someone wrote.

 

What is interesting to me, or problematic for me, is trying to understand when it is that someone is sufficiently able to write a book. Any tom dick or harry can write a book. And even the best, most well read most agreed upon book is subject to error and the ideas within, open for discussion.

 

The main thought I wished to address with this thread, was that a paradox is usually self induced, as that the universe is not prone to contradicting itself. If one is going merrily along, matching assumptions up with evidence and basing ones next assumptions on what has already been found to work, then everything moves along nicely. If carrying forward two lines of thought, results in a conflict, or impossibility, then something is wrong. Maybe with one idea, maybe with the other, or maybe they are both good, but some aspect or another has not been completely thought out, or tested out, and needs some work. Or maybe the analogy just does not hold all the way through, from concept to reality. Some adjustment needs to be made, some back checking, as to where could we have gone wrong, what is it we need to redefine or look at another way, inorder to have the thing work, with no contradictions or impossible components being required inorder to make it work.

 

You don't like the way I argue. Or perhaps you don't like that I argue, without bringing any quotes to the discussion. But that does not matter. What matters more is if you think I am wrong or right about which parts of my arguments, or if you know the answer to a question I have. Prometheus offered a link, suggesting it gave a good account of our understanding of infinity, which would help in our understanding of the ancient greek paradoxes. I am trying to talk about "our understanding" to get a handle on what that understanding is, and how it is different, better or worse than an understanding we used to have that was somehow defective.

 

Regards, TAR

[Acme]

Acme,

 

But not having a degree, does not precude one from thinking about stuff.

...

Regards, TAR

No, but it does preclude one doing stuff. By your argument I should let you tune my automobile engine even though you tell me you don't really know how but have thought about it.

[studiot]

 

as you can not arrive at the last member of the set...ever.

 

Whilst I have offered you alternatives to thinking about infinities and you have eschewed them, the above is not true.

 

So it is up to you if you wish to know more about what I mean or just continue to condemn the thought processes of others.

[tar]

Acme,

 

OK, granted, I am a jack of all trades, master of none, but I am a pretty good troubleshooter. I had a joke saying when we were faced with a tough problem, "it's got to be something." It was a joke, but it always turned out to be true. If something does not work as it was designed to work, then something is wrong. Sometimes with the design, sometimes with the material, sometimes something just fails, or breaks or gets loose or comes up missing.

 

In the case of a paradox, something is wrong, and "its got to be something" that is causing the problem.

 

And not everyone agreed with everyone that ever came up with a good idea.

 

Studiot,

 

Well sure, I would like to know the last member of an infinite set. There is however not a way you could deliver such a thing. The smallest number you could show me could be halved and the largest could be doubled.

 

There was a commerical where the kid was challenged to come up with a number bigger than infinity and the answer was infinity plus one, which was trumped by infinity times infinity. We played that game in math class and the teacher just told us we had the concept wrong, and infinity was not a number, it was a concept. This, we were taught many years after Cantor and aleph and null and all. Set theory and matching stuff up, pairing things off suffers in my imagination from inappropriate grain size shifting. Considering an infinite set an object you can hold in your mind and manipulate and make judgements about. If you say you can do this, and I can't, I will simply call your method inappropriate to the case. You can not provide me with the final member of an infinite set. If you could, it would not be an infinite set. It would be a finite set, with a first, a bunch of others, and a final member.

 

Regards, TAR

[studiot]

 

Well sure, I would like to know the last member of an infinite set. There is however not a way you could deliver such a thing. The smallest number you could show me could be halved and the largest could be doubled.

 

No offence meant. but does T stand for ( the biblical) Thomas?

 

Nothing I like more than to think my reply will be received with an open mind.

 

OK

 

On your wrist you have a gadget that performs exactly this miracle every hour.

 

Let us take a line and connect the ends together into a circle.

Now I can divide the line into (as many) infinitely small bits as I like, but the end result is still a finite circle, with a finite circumference.

 

If I were to say I could not draw my circle or my watch hand could not pass the 12 because if I assemble an infinite number of these infinitely small bits of circumference will take an infinite amount of time, would I be right?

 

Now extend that idea to the infinite number line.

 

This time the circumference is infinite, but any point (as King Arthur said) on a round table can be taken as the start point, and is also the end point.

So I can see both ends of my infinite number line.

 

So there you have infinity without the maths.

 

This was a very quickly dashed off exposition since It is well past my midnight and I am toddling off now.

 

Final point the last step in an infinite sequence that reduces to zero must have zero length. I do not need to be able to count to infinity to know that.

 

I did ask, a while back, that since your original post did not contain a question, whether you wanted to discuss infinities or the deficiencies in Zeno's logic. The deficiencies are not about infinities.

Edited June 8, 2014 by studiot

[tar]

Studiot,

 

The circle you drew is only one aspect of an infinite number line. The real number line goes forever in both directions and is as well infinitely divisable.

 

The circle is not infinite in length, its circumference is Pi D. A finite length. If you can not decide on where to start the measurement (first element) and where to end it (same element) then you can be flakey about the infinite choices you have to choose from, but the length is finite, as soon as you choose a unit. The circle is a finite number of those units long. So it is not a good representative of both the first and last member of an infinite set. If you say it is, then anything is. A coastline or a comb or the space between your ears, can be divided imaginarily into any number of slices you desire. An infinite amount. But we are talking about the real number line here. That one that both goes forever in both directions, and has infinite divisions possible of any unit you wish to mark it off with. You can not give be the highest number of units possible, nor can you give me the reciprocal of that number. Because both the length of the line, in units, and the reciprocal of that length as in 1 divided by infinity, are not defined. They therefore cannot either be counted or not counted, depending on a whim or a device or a definition. If something is infinitely large, or infinitely small, you can't find anything larger or smaller than that, cause the last member is not defined. If it was defined, it would be a finite number, which one could then decide was comparable with another number. But since the infinite number line, has no end, and no arbitrary smallest unit, all infinities are exactly like each other, in that the quantity denoted is either infinitely large, or infinitely small.

 

Interesting that Cantor actually argued against infinitesimals, and according to the Wiki article on him was discredited with an erroneous proof against them.

 

So, although I did not ask a direct question in the OP, my question is can we solve paradoxes by talking them through, and seeing where the inappropriate grain size or perspective shift is being made?

 

Or the inappropiate assumption, or the slight of hand switch, or the failure to carry an analogy consistently through.

 

Regards, TAR

And yes T is for Thomas

A is for Ammon

R is for Roth

[md65536]

It does not make sense to me that one infinity can be larger than another, because infinity already is at the max of quantity. You cannot beat that, you cannot exceed that, because if you could, your first claim of being at the maximum possible count, would have been incorrect.

So, although I did not ask a direct question in the OP, my question is can we solve paradoxes by talking them through, and seeing where the inappropriate grain size or perspective shift is being made?

Note that you gave an incorrect definition of infinity, and then showed that it ("your first claim" which you made up) was incorrect.

 

Your description of infinity followed the rules of linguistics. Mathematically it was wrong.

 

If you use only linguistics, and are free to choose whatever meanings of words you want, you can write out Zeno's paradox or others in ways that can be solved depending on the meanings you choose, or that can't be solved by choosing other meanings that don't follow the math/physics/etc.

[Ophiolite]

Tar, you seem to be engaged in repeated physical abuse of an equine quadruped devoid of its vital functions, using a slim, flexible, extended leather implement. Might it be time to call it a day?

[md65536]

TAR, I had figured that you probably never would have bother along the lines of reasoning you were on, if you already understood existing ways of resolving the paradox. But you said you were reading up on the paradox, so there's always the opportunity to incorporate existing solutions. Anyway I think that it's more useful to research what's already been done (whether with the paradox or linguistics) than to start from scratch, and better to be meticulous and systematic than being vague and ambiguous... I mean like focussing on the rules of linguistics rather than divining meaning from heaps of words. For example, is it possible to implement a Turing machine using the rules of linguistics? (I don't know anything about linguistics but I assume it's not.) If you could, then you could use linguistics to solve a HUGE set of classes of problems that can be solved using Turing machines. Or similarly, if you define the words "not" and "and", and have enough rules to organize arbitrarily long combinations of the words, you might be able to describe the output of any binary logic circuit using only linguistic rules. Then anything computable by a computer can be described in words using only those rules and the few defined words.

 

This is the opposite of what you're aiming at I think. This might prove that a linguistic resolution is possible, but it might be super long and convoluted and not easily understood, only "technically" meaningful. But, if you build meticulously on what is known to be correct, you can end up with something correct. Like I said before I think, if you keep looking broader and adding more vaguely defined ideas, it just gets harder to find errors in reasoning, now not only in the "paradox" but also in everything you've described about it.

 

Anyway, if you could "compute" a linguistic solution, that might not be enough anyway. If the design of the sentences comes from the math etc, I think that's still math, not just linguistics. (But then again!, could you build a linguistic machine to generate and evaluate all possible linguistic machines, or something like that? If so then you might produce computable results of math without ever needing the math.)

 

Now I've going off on a ramble...

TL;DR I think you'll find the answers in the details and in what's already been figured out by others, not in the unlimitedly broad meaning of it all.

Edited June 11, 2014 by md65536

[tar]

MD65536,

 

But although I did not on purpose propose an incorrect definition so that I could prove it false, such an activity is probably not distant from providing a correct defintion, so you can prove it true.

 

What started this whole discussion, even before this thread, was the fact that I never thought the warrior and the tortoise "problem" was a problem. I never saw the issue. It was just a trick of words, of looking at what would happen to one infinity and inappropriately saying that should have any bearing on another. The warrior would certainly overtake the tortoise, so the question is a non question, the problem has been setup incorrectly, and the "problem" is to figure out where the mistake has been made.

 

It is a GIVEN to me that a mistake has been made or some attribute of reality has been overlooked, because the warrior absolutely overtakes the tortoise.

 

Whether the thing was solved once or a million times makes no difference. Whether I "understand" what makes it a non problem is immaterial. Whether it has been solved or not, 5000 years ago, today or tomorrow, makes no difference.

 

Its not a problem to begin with. Finding a way to solve it, convoluted or simple, machine wise or human wise, language or math, makes no difference if its a strawman argument to begin with.

 

Regards, TAR

[tar]

I think there is a huge area of misunderstanding possible, when it comes to "problems".

 

Turing machines and I have never gotton along.

I do not ever get the definiton, because they talk as if the machine is doing some sort of problem solving, when the person has just set it up, with all the right definitions and codes and algorithms so that the thing can be "solved".

 

You mention a huge class of problems that turing machines can solve. I suppose the best defintion for that huge class of problems would be that type of problem that turing machines can solve. Regardless, the program still has to be written, the code still has to be designed. The language the machine is going to operate in needs to be known, and the rules upon which the problem will be worded and the solution understood need to be understood.

 

Othewise, you have the monkeys typing out Shakespear situation. They probably can not, and nobody would be there to read it and verify its accuracy anyway, so its a mute point, even if it was possible, which it is not.

 

A machine designed by a human, is endowed with human logic to begin with. It already has a purpose, it already has a goal, it already has a problem that it is solving by its very existence.

Perhaps a thought for another thread, perhaps exactly the issue for this one, but I have a tendency to consider the world analogue in nature. Others have a tendency to view it as digital...or a Turing machine, or a simple Ideal Gas Equation.

I tend to think of a human being as a focus point, a self, a consciousness that is in and of the world, and has therefore senses and experiences and memories "of and about" the place.

Others tend to think that somehow we have gotten it wrong, and are not properly equipped to handle the experience. Really?

 

Like thinking it is impossible to move, and therefore movement is an illusion, is some sort of useful and remarkable idea.

[physica]

so we've gone off tangent here. So far people can describe the paradox lingually but this is far from a solution. Hopefully we have got over the notion that thinking a solution to the fact that an infinite amount of processes has a finite outcome is excessive is not a reasonable response.