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Cont: Deeper than primes - Continuation 2

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doronshadmi said:
I wish to clarify something.

Given a non-composed endless straight line and a point not on that line, there are endless larger non-composed circles that are smaller than that line, and there are endless smaller non-composed circles that are larger than that point.

Yet pi is an invariant proportion among the endless larger and the endless smaller non-composed circles.

So a fixed value can be related to infinitely many things as long as it is not used to define their amount, their sum are any other fixed value that contradicts their property of being endless larger or endless smaller things.

For more details, search for Nicholas of Cusa.
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You can rest assured that no one here will question the constant value of pi, unless you decide to do it yourself.
 
jsfisher said:
Endlessly smaller values would be an endless sequence of smaller and smaller values.

Or did you mean infinitesimals? For that, I'll refer you to the wikipedia entry for Archimedean Property, since it provides a good functional definition for infinitesimal as it relates to the Archimedean Property, and it will also expose you to the definition of, you know, the Archimedean Property.

Two birds with one stone as it were.
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The Archimedean Property is "the property of having no infinitely large or infinitely small elements." (https://en.wikipedia.org/wiki/Archimedean_property)

"In mathematics, infinitesimals are things so small that there is no way to measure them." (https://en.wikipedia.org/wiki/Infinitesimal)

In other words, infinitesimals are exactly endless smaller things that can't be measured by fixed values that contradict the property of being endless smaller things > 0.
 
doronshadmi said:
The Archimedean Property is "the property of having no infinitely large or infinitely small elements." (https://en.wikipedia.org/wiki/Archimedean_property)
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You left out something important from that.

"In mathematics, infinitesimals are things so small that there is no way to measure them." (https://en.wikipedia.org/wiki/Infinitesimal)
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Yep. Individual things, each an infinitesimal.

In other words, infinitesimals are exactly endless smaller things
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No. The sources you cite say nothing about a sequence of smaller and smaller values. An infinitesimal is a single thing.
 
jsfisher said:
You left out something important from that.
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What something important, exactly?


jsfisher said:
Yep. Individual things, each an infinitesimal.
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Yet, they are infinitely many individual things that are > 0 as their common property.


jsfisher said:
An infinitesimal is a single thing.
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Yet, it belongs to collection of infinitely many individual things that can't be but > 0 as their common property ( https://en.wikipedia.org/wiki/0.999...#Infinitesimals ), which contradicts the The Archimedean Property (which disallows value > 0 between infinitely many smaller individual things and some fixed value, know as their limit).
 
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jsfisher said:
Yes, and so? The reals have infinitely many values that are greater than zero. None are infinitesimals.



The reals, remember? Not the hyper-reals, et alia.
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Some proofs that 0.999… = 1 rely on the Archimedean property of the real numbers: that there are no nonzero infinitesimals. Specifically, the difference 1 − 0.999… must be smaller than any positive rational number, so it must be an infinitesimal; but since the reals do not contain nonzero infinitesimals, the difference is therefore zero, and therefore the two values are the same.
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( https://en.wikipedia.org/wiki/0.999...#Infinitesimals )

In other words, jsfisher, you have no argument, 0.999… = 1 rely on the Archimedean property, which disallows value > 0 between infinitely many smaller individual things (0.9+0.09+0.009+..., in this case) and some fixed value (1, in this case) known as their limit.

So the nonsense that disallows value > 0 between infinitely many smaller individual things and some fixed value, rely on the Archimedean property.
 
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doronshadmi said:
In other words, jsfisher, you have no argument, 0.999… = 1 rely on the Archimedean property
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The proof can rely on the property, can. But that's ok, since the group (R,+) has the property. Actually, we don't need the reals. The group of rationals, (Q,+), is sufficient. Luckily, that group has the Archimedean Property, too.
 
jsfisher said:
The proof can rely on the property, can. But that's ok, since the group (R,+) has the property. Actually, we don't need the reals. The group of rationals, (Q,+), is sufficient. Luckily, that group has the Archimedean Property, too.
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There is no such proof by using symbolic-only brain skills, exactly as defined in http://www.internationalskeptics.com/forums/showpost.php?p=11708547&postcount=2451 by using visual AND symbolic brain skills.

Symbolize it as (R,+) , (Q,+) or whatever, still a fixed value can be related to infinitely many things as long as it is not used to define their amount, their sum or any other fixed value that contradicts their property of being endless larger or endless smaller individual things.


Also you continue to skip on http://www.internationalskeptics.com/forums/showpost.php?p=11735133&postcount=2610 , http://www.internationalskeptics.com/forums/showpost.php?p=11735408&postcount=2638 and http://www.internationalskeptics.com/forums/showpost.php?p=11735016&postcount=2605.
 
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jsfisher said:
It is simply a property--somethings have the property and other things don't.
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You can't ignore your brain skills in order to know it, in the first place.

And by using both your visual AND symbolic brain skills you are able to know that infinitely many smaller individual things > 0 (including (R,+) or (Q,+)) do not have this property, as already given in http://www.internationalskeptics.com/forums/showpost.php?p=11736509&postcount=2649 (including all of its links).
 
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doronshadmi said:
There is no such proof by using symbolic-only brain skills, exactly as defined in http://www.internationalskeptics.com/forums/showpost.php?p=11708547&postcount=2451 by using visual AND symbolic brain skills.
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Many of us remember your grand ruse of "hidden assumption." You'd make grandiose, semi-incoherent claims. We, of course, were blinded by this hidden assumption and thus prevented from comprehending your obviously correct claims. Any counters we offered to your claims were flawed by the hidden assumption and therefore could (and were) rejected out of hand.

Many of us remember your grand ruse of "hidden assumption" evolving into your equally grand ruse of "direct perception." You'd make grandiose, semi-incoherent claims. We, of course, were incapable of direct perception and thus prevented from comprehending your obviously correct claims. Any counters we offered to your claims were flawed by omission of direct perception and therefore could (and were)) rejected out of hand.

Some things never change. They just get recycled under new names.


The group (R,+) has no infinitesimals. It matters not to Mathematics that you choose to dismiss this fact out of hand. Mathematics doesn't care.

If you want to challenge that, do so with Mathematics, not these offensive and hollow claims of superior cognitive skills.
 
doronshadmi said:
Nonsense, given a straight line with finite length X, it is trivial that finitely many smaller straight lines can be added into some length > X (which is the Archimedean property).
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Back in my day, something with finite length could only be a line segment and not a line. You kids get off my analytic geometry!
 
doronshadmi said:
Nonsense, given a straight line with finite length X, it is trivial that finitely many smaller straight lines can be added into some length > X (which is the Archimedean property).
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That is not the Archimedean Property.

The Archimedean Property for the group, (R,+), is that for every pair of positive elements A and B of R, there is a multiple of A that is greater than B. (Or, if you prefer, for some integer n, nA > B.)

If you want to disprove that the reals are Archimedean, then you simply need to show us a pair of real numbers A and B for which no multiple of A exceeds B.
 
CriticalThanking said:
Back in my day, something with finite length could only be a line segment and not a line. You kids get off my analytic geometry!
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A line-segment is first of all a line. If you define some part of a line, it is called line-segment.

Now you can take finitely many smaller (w.r.t each other) multiple line-segments and add them such that they are > than a line-segment with length X, and walla, we have the Archimedean Property even if no nA is involved.
 
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doronshadmi said:
That is exactly the Archimedean Property.

Your symbolic-only definition of it does not change this fact.
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If you want to challenge a point in Mathematics, do so with Mathematics, not these offensive and hollow claims of superior cognitive skills.
 
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