Mersenne's Miscelleny

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Re: Mersenne's Miscelleny

Post by Mersenne »

With you bro. Sometimes I wish computers had lapels so I could grab them and shout in their faces.

Er, and faces- I wish they had faces.
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Re: Mersenne's Miscelleny

Post by James Strom »

Can you believe it happened again? "General error" came up when I attempted to submit my post. Fortunately I got smart and copied my post before submitting. So here it is.

Computers have faces. See? :twisted: So do I. :cry:

I will wager that any definition of infinity that you can come up with will be recursive. That is it will have a hidden infinity somewhere within the definition. That or it will involve some subjective element.
If you are adding up an infinite number of infinitesimal components to create a finite quantity so as to solve a practical problem you may not be quite doing what you're thinking. Another way of looking at it is that you're approximating. One-third plus one-third of one-third plus... (a million times) is so close to one-half that we can treat it as such for any practical purpose that I can think of. No infinities are necessary to do calculus.

"Apparent paradox, such as the AP, is resolved with a proper understanding of infinities."

Why bother when the problem can be resolved simply by not allowing their use. Occam's razor. But if infinity hasn't been exactly defined then the paradox hasn't been resolved by the proper understanding of them.

"Well, your definition is! The Peano axioms are a little firmer."

How many natural numbers does Peano claim there are?

" Penrose (Emperor's New Mind) argues that the mind is non-algorithmic because it addresses infinities (strictly, because it can draw information from utside any finite system)"

A computer draws information from outside its system all the time. I think Penrose is being a bit humanocentric. Maybe he fears obsolesence.

"I think you're wrong to identify the nonmaterial with the subjective, but this statement does puzzle me. Perhaps you cold enlarge on it?"

Their is both subjective and objective reality. Both are equally important. The objective is the world outside of our immediate selves and therefore measurable. I would define material as that which can be detected or known. Following the axiom that we can't see or know ourselves then the subjective will be immaterial. That doesn't mean that we don't exist! Others can see us. It just becomes part of our subjective world. We often visualize it by ascribing our own traits and such to things outside of ourselves. This is why I would guess that you believe numbers can be infinite. Although infinity may be disallowed in the objective world it is common, if not mandatory, in the subjective one.
What constitutes ourselves, though, can be very difficult to determine. Are our children ourselves? Our lovers? That would explain why love is blind. They are us so we cannot see them clearly.

"Exact models use ininitesimal calculus and so admit the use of infinities."

No exact model can be infinitely detailed as that would take an infinity of time to create. Therefore, at some point, the infinitesimal calculus will only be approximating. Which is all it can do anyway. 9.99999.... is not 10, it's just close enough to round up to it without causing problems.

"The programmer could, but the machine he/she programs couldn't. The programmer has god-like abilities, i.e. spirit."

Correct.

"There are no hidden variables, is what aspect/Bel proved experimentally. Radiocative decay is indeed random. A Geiger counter is a very good analogue random number generator."

Allowing for infinity in the subjective world makes radioactive decay actually random because the particles are so small that we cannot observe them without making them at least in some way a part of ourselves.

"This makes sense, but doesn't fit in with your "checkerboard". That's black and white, and a piece is either in one or the other."

Imagine that some super computer somewhere is playing a game of 'life' and that at some point intelligent life evolved on it. The universe in which these beings exist is very large but quite finite. One of them is a philosopher who proves through some logic that there are infinities in the universe. Suppose it were the same logic that you've been following. Would logic itself then be wrong?
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Re: Mersenne's Miscelleny

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I will wager that any definition of infinity that you can come up with will be recursive.
All numbers have recursive definition in the sense that they are deined in terms of other numbers. Why should infinity be any different?
No infinities are necessary to do calculus.
They're there in the engine of calculus. All our algeraic short-cuts all require them for justification.
"Apparent paradox, such as the AP, is resolved with a proper understanding of infinities."

Why bother when the problem can be resolved simply by not allowing their use. Occam's razor. But if infinity hasn't been exactly defined then the paradox hasn't been resolved by the proper understanding of them.
Infinity has been properly defined. The many different types of infinity have been properly defined. We know what infinites are and how to use them. The Achilles paradox isn't resolved by ignoring infinities because it deals with an infinity, and is a paradox only because Zeno failed to understand how infinities behave. Occam's razor requires the simplest possible answer, but to ignore infinity is to fail to answer, or even to address the paradox.
How many natural numbers does Peano claim there are?
Prove, not claim there are. An infinite number.
A computer draws information from outside its system all the time. I think Penrose is being a bit humanocentric. Maybe he fears obsolesence.


In my experience, a computer may be fed information from outside its system, but doesn't look for it. But let's suppose a computer has an interface that it can use to explore its environment. Suppose it is asked to compose an arithmetical proof, one of those which in Godel's terms is true, can be stated using data already accessible to the computer, and which cannot be proved in terms of this data. Could the computer generate the hypothesis that its current data is insufficient? If it could generate the hypothesis, could it recognise the missing elements if it encountered them in its environment? Bear in mind, it will have nothing to compare the missing elements to. There won't even be an apparent "gap" in the data it has, because its data is sufficiently complete to state the problem.

Penrose probably does feel threatened, however. Personally I think technology is a cocoon we will one day outgrow.
Their is both subjective and objective reality. Both are equally important. The objective is the world outside of our immediate selves and therefore measurable. I would define material as that which can be detected or known. Following the axiom that we can't see or know ourselves then the subjective will be immaterial. That doesn't mean that we don't exist! Others can see us. It just becomes part of our subjective world. We often visualize it by ascribing our own traits and such to things outside of ourselves. This is why I would guess that you believe numbers can be infinite. Although infinity may be disallowed in the objective world it is common, if not mandatory, in the subjective one.
What constitutes ourselves, though, can be very difficult to determine. Are our children ourselves? Our lovers? That would explain why love is blind. They are us so we cannot see them clearly.
I see the confusion. We agree that there is both subjective and objective (though I also subscribe to the veiw that both are expressions of a single reality, a Tao if you will). Infinities however exist in the subjctive and objective worlds. Not only in terms of an infinite extension of space or time; what about the infinite curvature of a singularity?
"Exact models use infinitesimal calculus and so admit the use of infinities."

No exact model can be infinitely detailed as that would take an infinity of time to create. Therefore, at some point, the infinitesimal calculus will only be approximating. Which is all it can do anyway. 9.99999.... is not 10, it's just close enough to round up to it without causing problems.
Analytical infinitesimal calculus does not approximate. It uses integration to find an exact solution- just as in Achilles, a finite time step can be considered an infinite series of the infinitely small. It is discrete, finite-difference calculus which approximates, because it employs finite time steps instead of infinitesimal.
"There are no hidden variables, is what aspect/Bel proved experimentally. Radiocative decay is indeed random. A Geiger counter is a very good analogue random number generator."

Allowing for infinity in the subjective world makes radioactive decay actually random because the particles are so small that we cannot observe them without making them at least in some way a part of ourselves.
Interesting, but doesn't invalidate my point.
"This makes sense, but doesn't fit in with your "checkerboard". That's black and white, and a piece is either in one or the other."

Imagine that some super computer somewhere is playing a game of 'life' and that at some point intelligent life evolved on it. The universe in which these beings exist is very large but quite finite. One of them is a philosopher who proves through some logic that there are infinities in the universe. Suppose it were the same logic that you've been following. Would logic itself then be wrong?
I think you're anwering a different question- I'm asking how there can be "grey" between the black and white squares of your discrete universe.

Your finite philosopher might exist in a finite universe, but even a finite universe can contain infinities. That's the idea of the intgral calculus again. Infinite points to an inch, infinite line segments to a square inch, infiniies 2D squares to an inch cubed, and so on.
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Re: Mersenne's Miscelleny

Post by James Strom »

We seem to be having a dozen conversations at once so I'll try to stick to one topic at a time. Since almost all your arguments depend upon infinity being identifiable in an objective sense I'll stay with that.

"All numbers have recursive definition in the sense that they are deined in terms of other numbers. Why should infinity be any different? "

All words are defined in terms of words but that doesn't make all definitions recursive. If that were the case then no definitions would have any worth.
By recursive I mean that the term being defined is put into the definition. Defining two as a pair would be recursive because the definition of a pair itself would require the use of the number two. Defining 5 as 2+3 is not recursive because no 5 was used. Sure, it is the number five being equal to the number two plus the number three, but we're not defining the term 'number' here.

I can define infinity but it is a subjective one. It is a number that is so large that to use an even larger number will not change the result of our calculation. This is subjective because it depends upon what I am calculating along with the degree of precision.
One divided by infinity is equal to zero. In this instance I am using a eight-digit pocket calculator. Infinity can then be equal to one billion.
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Re: Mersenne's Miscelleny

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We seem to be having a dozen conversations at once...
Yes, isn't it fun?
Since almost all your arguments depend upon infinity being identifiable in an objective sense I'll stay with that.
Which it is, so at least those of my arguments depending on this are sound!
All words are defined in terms of words but that doesn't make all definitions recursive. If that were the case then no definitions would have any worth.
Verbal definition proceeds by synonym, so it is recursive. If the definition doesn't identify with the defined, it isn't a definition. What we take to be definition is in fact exemplification or description;

CAT: Felis domestica or Felis catus (definition, right back at the cat)
a small domesticated carnivore (description)
any of several carnivores of the family Felidae, as the lion, tiger, leopard, jaguar, etc. (exemplification)
By recursive I mean that the term being defined is put into the definition. Defining two as a pair would be recursive because the definition of a pair itself would require the use of the number two. Defining 5 as 2+3 is not recursive because no 5 was used. Sure, it is the number five being equal to the number two plus the number three, but we're not defining the term 'number' here.


The set must be a subset of itself. 5 must be equal to itself. Equality must possess the reflexive quality! To define 2 as a pair is not recursion but a tautology.

But even 5 = 2 + 3 evinces this, since 2 = 5 - 3 and 3 = 5 - 2. Hence 5 = (5-3) + (5-2). To admit of numbers is to define numbers recursively.
I can define infinity but it is a subjective one. It is a number that is so large that to use an even larger number will not change the result of our calculation. This is subjective because it depends upon what I am calculating along with the degree of precision.
One divided by infinity is equal to zero. In this instance I am using a eight-digit pocket calculator. Infinity can then be equal to one billion.
That's a good working procedure but not a definition.

But let's not forget that infinity isn't actually a number, it's a quantity greater than any number in the set of reals R. If it were a number we could add one on to get a greater number. There is, however, a formal definition of infnity using two numbers, 1 and 0;

"infinity is the limit as x goes to zero (from the positive side) of the quantity one divided by x".

This isn't the same as 1/0 whch is simply undefined.

Even a subjective infinity can represent an objective infinity. Consider the projection of the North Pole onto a plane parallel through the equator; this gives the single point at infinity, which can be reached by moving in any direction.
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Re: Mersenne's Miscelleny

Post by James Strom »

Mersenne wrote:
We seem to be having a dozen conversations at once...
Yes, isn't it fun?

It is. But my time on the computer is limited. They only give me an hour at the library. By the time I think of a good reply I almost don't have enough time left to get my post deleted, er, submitted.

Verbal definition proceeds by synonym, so it is recursive.

Not necessarily.

Cat def. An animal with four legs, pointy ears, eats mice,...

No synonym of cat involved.

Maybe I'm using the word 'recursive' incorrectly. Perhaps 'tautological' is in order.
I can define infinity but it is a subjective one. It is a number that is so large that to use an even larger number will not change the result of our calculation. This is subjective because it depends upon what I am calculating along with the degree of precision.
One divided by infinity is equal to zero. In this instance I am using a eight-digit pocket calculator. Infinity can then be equal to one billion.
That's a good working procedure but not a definition.

Why not? It isn't an objective definition, of course, but I don't believe one could be made anyhow. That's my argument. My challenge to you is to come up with a definition of 'infinity' that doesn't involve a tautology or subjectivity. I won't even try. I know it's impossible.

But let's not forget that infinity isn't actually a number, it's a quantity greater than any number in the set of reals R.

If it's a quantity it's a number. You can't have it both ways. Well, you could, but then we would have to abandon logic.
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Re: Mersenne's Miscelleny

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"it's a quantity greater than any number in the set of reals R."

This makes it subjective. It is not specified which quantity, just a quantity. I could define a number as "a number between one and ten" but this would require input from the definer, thereby making it subjective.
When I was younger I would have these contests to see who could think of the largest number. It was considered cheating to use a subjective definition such as "your number plus one."

"infinity is the limit as x goes to zero (from the positive side) of the quantity one divided by x".

No mention is made of how far x goes to zero, making it a subjective definition.
Either:
1) Infinity is a certain number and can therefore be calculated via some method.
or
2) Infinity is an uncertain number and therefore more information is required from the definer in order to calculate it; it is subjective.
or
3) Infinity is not a number and therefore cannot be a quantity.

quan·ti·ty
/ˈkwäntətē/
Noun

The amount or number of a thing not usually estimated by spatial measurement: "the quantity of the fruit can be controlled".
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Re: Mersenne's Miscelleny

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ME: "it's a quantity greater than any number in the set of reals R."
This makes it subjective. It is not specified which quantity, just a quantity. I could define a number as "a number between one and ten" but this would require input from the definer, thereby making it subjective.
When I was younger I would have these contests to see who could think of the largest number. It was considered cheating to use a subjective definition such as "your number plus one."
But it doesn't make it subjective, James. You're making the mistake of thinking there's just one infinity and that it's a number. Every infinity is greater than any number, so it can't be a number; and any quantity greater than any number is an infinity.

And your chums weren't cheating; if you think of a googleplex and your friend names a googleplex plus one, he's thought of a bigger number. But he's unlikely to have a conception of the associated quantity involved.

ME: "infinity is the limit as x goes to zero (from the positive side) of the quantity one divided by x".
No mention is made of how far x goes to zero, making it a subjective definition.
It goes to the limit. And this limit exists, because the function 1/x is differentiable.
Either:
1) Infinity is a certain number and can therefore be calculated via some method.
No,infinity is not a number.
or
2) Infinity is an uncertain number and therefore more information is required from the definer in order to calculate it; it is subjective.
Infinity, whichever one we're talking about, is not a number. But even if it were uncertain, this wouldn't make it subjective, particularly if the additional information, known or not, exists.
or
3) Infinity is not a number and therefore cannot be a quantity.

quan·ti·ty
/ˈkwäntətē/
Noun

The amount or number of a thing not usually estimated by spatial measurement: "the quantity of the fruit can be controlled".
This is not the mathematician's definition of a number. A number is a member of a set, the nature of the number being symbolic and relating in an ordered way to a quantity. Quantities are objective, they exist, they are "out there", and numbers, whch are symbols, can be associated with most of them. But the infinities are amongst these quantities; they too are objective, they exist, they are "out there" but no number can be allocated to any of them.
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Re: Mersenne's Miscelleny

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James's comment on Infinity:
Infinity.gif
Infinity.gif (913.44 KiB) Viewed 4692 times
"Don't let the illusions of your past or future rob you of the infiniteness of your present." [Unknown]
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Re: Mersenne's Miscelleny

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admin wrote:James's comment on Infinity
Ouroboros' origin revealed.
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