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Jim considers a circuit with one input and one feed back.
Jim submits a single value to the circuit. Jim waits a moment...
The circuit now contains two values, n and n+1. Jim waits a moment...
The circuit now contains four values... and so on.
The circuit is limited (only) by bandwidth.
Wikipedia: Perceptron
http://en.wikipedia.org/wiki/Perceptron
The same idea, only explained more clearly. [zen_tom, Jun 10 2011]
[link]
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What you are describing is called a perceptron, and
was invented in 1957. |
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Incorrect --- the individual values of a perceptron cannot be distinguished. [The operator + must be reversible and this would require storage of an infinite number of partial sums, contradition :)] |
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As an asside the (Infinite) Zeno Machine is a contradiction. Which I think is as the namesake would have intended. Since Zeno is known to have proposed contraditions against dividing space and time infinitely. |
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Now I especially like the arrow contradition. Or more generally the contradition of motion itself. It (the existence of physical motion) states that time cannot be divided infintely. |
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If I was to extend the notion without basis --- in mathematics the existance of a limit contradicts the existence of infinite series. So again as an aside, not the axiom of infinity. |
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More generally not infinity itself although it is not clear what the contradiction is ... |
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Right, so you're saying that the Infinite Zeno
Machine is a contradiction, because a circuit
cannot hold more than one 'value' at any one
time. |
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You're also saying that the Perceptron can't have
its individual values distinguished either, but that
that is an altogether different contradiction to
the one outlined in the idea. |
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So, contradictions all round then! |
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A kind of contradiction confection, one over
which you can both assert your property rights,
and consume at the same time. |
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I'm still fairly sure that you can't say that the existence of a limit in mathematics contradicts
the existence of an infinite series. How exactly do
you get from one statement to the next? And in
doing so, let's for the sake of argument limit
ourselves to the realms of mathematics since, as
Zeno pointed out, it's tricky to explain in real-
world terms without getting hung up on paradoxes
caused by (cleverly) misinterpreting what some of
these terms mean. |
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The Infinite Zeno Machine is a contradiction because the finite machine exists and to define the infinite machine the bandwidth restriction is removed from the finite machine --- but this is absurd. |
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The perceptron value is a sum. In simple mathematics the product (*) and coproduct (+) are irriversible. For example 5 = 1 + 1 + 3 or 1 + 2 + 2... In general a product and coproduct are reversible if the injection and projections to the product are single valued. For example a pair (which is both a product and coproduct) has single valued left and right injections and left and right projections. |
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(edit) In the Finite Zeno machine the value can be viewed as pairs of pairs, perhaps a set... Take your pick what is important is that the individual (physical) frequencies in the circuit can be distinguished. Now there is an oportunity for the wouldbe quantum mechanics out there to drop in something about entanglement...(/edit) |
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Cantor 'proved' the existence of infinity in mathematics by showing the correspondence of elements in the set of naturals and the set of naturals plus 1. He did this as follows: 0->1, 1->2, 2->3, ... *forever*. Cantors arguement for the existence of an infinite set is based on a correspondence that will take an infinite amount of time. This is why the axiom of infinity is an axiom... |
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In the case of a limit in mathematics there is nothing absurd about the sum of an infinite series of fractons. This is because the axioms of mathematics allow it. |
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(edit)Now it is quite simple to define the limit of the following finite series:
2 - 1/2 = 1 + 1/2
2 - 1/4 = 1 + 1/2 + 1/4
2 - 1/8 = 1 + 1/2 + 1/4 + 1/8
....
2 - 0 = the sum of the infinite series (which can never exist)
In reality the sum is finite and the limit is a defined bit less than 2.(/edit) |
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If Zeno claimed only one thing (correctly) it is that physical attributes cannot be divided in half adinfinitum. So what purpose does the infinite model serve...? |
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I expected a misunderstanding --- and that is a shame... |
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He who expecteth nothing shall not be disappointed. |
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However I don't understand any of this. (well I understand many of the phrases but they are not joined together in a way that has meaning for my brain.) |
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What is amusing is that all of Zenos arguments are presented "as the paradox, contradiction, absurdity" and this pissed everyone off. |
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For example Aristotle was said to have gotten up and walked across the room when presented with the 'paradox of motion'... |
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//In reality the sum is finite and the limit is a defined bit less than 2// OK mad one, when do you stop adding, and how close to 2 can you get? |
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I can't claim to have misunderstood this idea, as I
don't understand it at all. (And I complain that it's
not written clearly.) How is a value
represented within the circuit? From the mention
of bandwidth, I suppose by a current oscillating at
a
particular frequency and phase? |
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The circuit would have to be ideally linear over
some frequency range, no? And would not the
energy stored in this circuit increase with
the amount of information stored? For some
amount of energy, presumably the linearity
assumption
would break down. If so, then, contra Jim
bandwidth is not the only limitation. |
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*Seems like the mathematics might be that of a
finite but unbounded oscillating string. (And
there's certainly plenty of *that* mathematics
lying about these days.) If so, then destructive
interference restricts possible frequencies to
quantum values, does it not? |
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PS: Did Cantor prove (or "'prove'") the
existence of infinity, or was it the existence of
multiple cardinalities? |
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// finite but unbounded mathematics |
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Cool what is that? (actually dont bother) |
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