Cont: Deeper than primes - Continuation 2

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Dessi said:
<snip>

The Doron Machine 2.0 Super Deluxe Ultra.

It's an extension of a normal Doron Machine with a better implementation of numbers which, by some miracle, doesn't store approximate quantities but rather stores quantities exactly. It also makes lattes and has a decent text editor. But the important thing is that any two quantities that are algebraically equal on paper really are equal in the Super Deluxe Ultra, and vice versa.
<snip>
Click to expand...

(I presume from your foto's etc. that you are quite a bit younger than I am).

And *that* is why modern computing is broken! What is wrong with normal black coffee and vi?

You see, whenever a 'new and hip paradigm' is discussed, I get passed by all them youngsters, but whenever I enter a competition I hand them their behinds on an epic scale...

All you need is the patience to think (and caffeine to stay awake during that process) and the ability to find all the letters on the keyboard in an acceptable timespan.

I never needed a debugger or profiler either and still my code runs stable and blazing fast.

</old fart mode off>
EDIT: ^^^ that is for fun, I am not a *grumpy* old man...(I am not even *that* old ahem)
 
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doronshadmi said:
It is a self-evident truth to those how understand that all whole rationals will never be < 1 AND > 0.
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No, it is not self-evident. I am interested to see your proof that 0.999... converges to a rational < 1 and > 0.
 
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Dessi said:
Even in Doron's very limited model of mathematical computation, the Doron Machine, the equivalence of 0.999... = 1 is inescapable. Intuition be damned.
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In my model (which is finer and therefore stronger than your model) 0.999... = 1 if the real-line is observed from |N| cardinality, or 0.999... < 1 if the real-line is observed from |R| cardinality, and no intuition is involved in http://www.internationalskeptics.com/forums/showpost.php?p=10332081&postcount=110.

Moreover, by observing the real-line from |N| cardinality 0.999...10 is an exact value along the real line that is < value 1 by the exact value 0.000...110.
 
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doronshadmi said:
It is self-evident if you observe the real-line by cardinality |R| exactly as done in http://www.internationalskeptics.com/forums/showpost.php?p=10332081&postcount=110.
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No, it isn't self-evident. I would also say the following statement:

"Exactly as some finite series that is observed from a convergent sequence with |N| values < some given limit value"

requires justification. Your statement implies that the sum S of a series Σ(n = 0, n -> ∞) f(n) has a limit L, then S < L. How do you know S < L? Can you show a proof?

Let me narrow the scope so we aren't talking about very broad generalities: you agree that S = Σ(n = 0, n -> ∞) (9/10)(1/10n) has a limit L = 1, how do you show S < 1?
 
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doronshadmi said:
In my model (which is finer and therefore stronger than your model) 0.999... = 1 if the real-line is observed from |N| cardinality, or 0.999... < 1 if the real-line is observed from |R| cardinality, and no intuition is involved in http://www.internationalskeptics.com/forums/showpost.php?p=10332081&postcount=110.
Click to expand...

You keep saying this as if it were sensible. It isn't. The real number line does not enter the discussion at any point, yet you keep referring to it and you keep accusing others of "observ[ing it] from |N| cardinality", which no one has done, even if we knew what exactly you meant by that particular confluence of words.

The only thing under consideration is the valuation of Sum(n=1 to infinity, 9/10^n).

The valuation of that summation, by the way, is a matter of definition, not point of view. You cannot disprove Mathematics by redefining its terms, although you persist in trying.
 
Dessi said:
No, it isn't self-evident. I would also say the following statement:

"Exactly as some finite series that is observed from a convergent sequence with |N| values < some given limit value"

requires justification. Your statement implies that the sum S of a series Σ(n = 0, n -> ∞) f(n) has a limit L, then S < L. How do you know S < L? Can you show a proof?
Click to expand...

Dear Dessi, it is so simple, all you have to do is to see the difference between, for example, 0.99910 < 1 and 0.999...10 = 1 by observing the real-line from cardinality |N|.
 
doronshadmi said:
Dear Dessi, it is so simple, all you have to do is to see the difference between, for example, 0.99910 < 1 and 0.999...10 = 1 by observing the real-line from cardinality |N|.
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I would like to do that, but however one "observes the real line from cardinality |N|" is undefined. I just don't know what you are talking about. You comment could be profoundly insightful, or simply gibberish, and I wouldn't know the difference.

Suppose I told you that 0.999... = 1 because its a monad which is a monoid in the category of endofunctors, but I never told you the meaning of any of those words.

Suppose you asked me what a monoid is, and I said is a point-set anamorphism on the category of monads, you would not find that helpful.

Suppose you asked me to explain what these words meant, explain the meaning of "point-set anamorphism" so that mere mortals could understand it. And I replied that an anamorphism is obviously an injective catamorphism in the disjoint-union of topological categories... so what the problem? You might find that unhelpful too.

How would you know if my original statement was right? or wrong? or mu? How would you know speaking truthfully or speaking gibberish?

That's how I feel right now. Your proof is wrapped up in so much obscure jargon that no one, no matter their mathematical background, understands a word of what you say. I simply don't know what you are describing by the words "observing the real-line from cardinality |N|", I can't even guess at the meaning.

I would be most appreciative if you could communicate your thoughts in clear, precise, meaningful language. This isn't an unreasonable request. I would love if you could just clarify and define your jargon, without piling more and more jargon.

Pretend that I'm not a mathematician, pretend I'm a random person off the street or maybe a student in a high school class. How do you explain "observes the real line from cardinality |N|" so that a layman without any specialized mathematical training could understand it?
 
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Dessi, a naive question: is it reasonable (or even helpful) to ask for Doron to reproduce a simple living example in code (e.g c# or f#)? I'm thinking this would give us some insight into the process by eliminating the jargon.

Edit: no disrespect to Doron is intended, I'm just trying to get my head around what he's trying to communicate.
 
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alex04 said:
Dessi, a naive question: is it reasonable (or even helpful) to ask for Doron to reproduce a simple living example in code (e.g c# or f#)? I'm thinking this would give us some insight into the process by eliminating the jargon.
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It certainly couldn't hurt. I'd also recommend MathNet.Numerics.FSharp for arbitrary precision arithmetic.
 
For the naive and the eternally hopeful, I offer this link wherein Doron provides new insights and understandings into Zeno's Paradox, complete with three programming examples.
 
doronshadmi said:
In my model (which is finer and therefore stronger than your model) 0.999... = 1 if the real-line is observed from |N| cardinality, or 0.999... < 1 if the real-line is observed from |R| cardinality, and no intuition is involved in http://www.internationalskeptics.com/forums/showpost.php?p=10332081&postcount=110.

Moreover, by observing the real-line from |N| cardinality 0.999...10 is an exact value along the real line that is < value 1 by the exact value 0.000...110.
Click to expand...

When I observe from the real-line from |N| cardinality, I find 0.999... = 1.

Clearly I've made a mistake, or maybe you have. Can you show me the steps involved in your calculation? I read the post you linked and could not understand a word of it, so I would appreciate if you could provide a layman-friendly description for my benefit.

I also could not define any function f(x) = 0.000...1. How would you define f(x), using a layman-friendly description?
 
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jsfisher said:
For the naive and the eternally hopeful, I offer this link wherein Doron provides new insights and understandings into Zeno's Paradox, complete with three programming examples.
Click to expand...

Cheers.. wow. I'm out of my depth here, I may as well be listening to Bud Haggart.
 
jsfisher said:
For the naive and the eternally hopeful, I offer this link wherein Doron provides new insights and understandings into Zeno's Paradox, complete with three programming examples.
Click to expand...
This post nails it: Doron can only think of mathematical expressions in terms of a computer program. I am not at all surprised his analysis of Zeno's paradox involves an algorithm in pseudocode.

That said, I notice an error in Doron's analysis of Case B, in which he tries to determine Achille's position as the sum of shorter and shorter time intervals:

Code: 
Position X1 = 0
Position X2 = 10
Achilles Speed = Aspeed = 10
Tortoise Speed = Tspeed = 1
Time = 1

Do Loop K from 1 to ∞
    Achilles position = position X1 + distance ( = Aspeed * Time)
	Tortoise position = position X2 + distance (= Tspeed * Time)
	Position X1 = Achilles position
	Position X2 = Tortoise position
	If X1 ≥ X2 then STOP
	
	Time = Time / Aspeed (Achilles Speed = Aspeed = 10)
Next Loop K

Note that the first interval of time t1 = 1, the loop partitions it into infinite subintervals, where each subinterval tk is 1/ASpeed the length of its predecessor tk-1:

Total Time = tn + tn-1/ASpeed, t1 = 1.
Total Time = 1 + 1/10 + 1/100 + 1/1000 + . . .
Total Time = Σ(k = 0, k -> ∞) 1*(1/10k)
Total Time = (1 / (1 - 1/10) ) [ see this identity ]
Total Time = 1 / (9/10)
Total Time = 10/9

Doron's article says "The Race continues forever", but that's not true. After infinite iterations of his loop, about 1 second of the race has actually elapsed, falling short of forever by a considerable margin.

The program runs indefinitely only by accident, because Achilles is just too slow to overtake the tortoise in such a small interval of time. If ASpeed = 11 and the program maintains the loop invariant tk = tk-1 / ASpeed, the program halts:

- Total elapsed time = Σ(k = 0, k -> ∞) 1*(1/11k) = 1 / (1 - (1/11) ) = 11/10 seconds
- Achille's position = start position + speed * elapsed duration = 0 + 11 * 11/10 = 121/10 units
- Tortoise position = start position + speed * elapsed duration = 10 + 1 * 11/10 = 100/10 + 11/10 = 111/10 units

In fact, the program is guaranteed to halt for any ASpeed > 10.
 
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Dessi said:
This post nails it: Doron can only think of mathematical expressions in terms of a computer program. I am not at all surprised his analysis of Zeno's paradox involves an algorithm in pseudocode.
Click to expand...

Yup. Always a process. Even his imaginary parallel summation is a process, albeit a process of just one non-deterministic step.

In addition to process, notation is important. Doron had maintained that 1/4 and 0.25 were different numbers (though he may have finally abandoned that absurdity). There is also something of a concrete instance requirement the peeks out now and then. According to Doron, 2 is not a member of {2}, apparently because those must be different instantiations of 2.

Dessi said:
Doron's article says "The Race continues forever", but that's not true. After infinite iterations of his loop, about 1 second of the race has actually elapsed, falling short of forever by a considerable margin.
Click to expand...

Curiously, too, at the very beginning of his Zeno paper, Doron explicitly states that 1 + 1/2 + 1/4 + 1/8 + ... = 2. That is in stark contrast to his stated position, here, regarding 0.111... in base-2.

Contradiction is no stranger to Doron.
 
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jsfisher said:
Dessi said:
This post nails it: Doron can only think of mathematical expressions in terms of a computer program. I am not at all surprised his analysis of Zeno's paradox involves an algorithm in pseudocode.
Click to expand...
According to Doron, 2 is not a member of {2}, apparently because those must be different instantiations of 2.
Click to expand...
I am not familiar with Doron's other posts, but if this statement is true, it tells me he thinks of mathematical series only in terms of computer programs, and apparently thinks of programs only in terms of Java.

This deserves an explanation: a handful of built-in types in Java are not objects, but are primitives. For example, Java has an int primitive, but it also has an Integer class which is a wrapper around ints. This wrapper class exists to makes Java's implementation of parametric polymorphism / generics work.

Normally, Java's autoboxing usually makes the distinction between primitives and their object wrappers fairly transparent. Usually. There are exceptions:

Code: 
Integer smallX = 10;
Integer smallY = 10;

if (smallX == smallY)
    System.out.println("smallX equals smallY");
else
    System.out.println("smallX does not equal smallY");

Integer bigX = 1000;
Integer bigY = 1000;

if (bigX == bigY)
    System.out.println("bigX equals bigY");
else
    System.out.println("bigX does not equal bigY");

// output:
// smallX equals smallY
// bigX does not equal bigY

Wat? Remember, in Java, '==' checks two objects for referential equality (whether they point to the same address in memory), not structural equality (whether they represent the same value). The Integer class interns or caches Integer values < 128, meaning smallX and smallY are referentially equal. For all other values, like bigX and bigY, checking for equality with '==' fails because the values being compared do not point to the same address in memory. Although they are structurally equal, they are in fact different instances of the same type. It's just a quirk of the Java language.

In a mental model where mathematics and Java programs are "one and the same", it completely makes sense that the numeric value in '2' and '{2}' are different instances of the same type: they point to different addresses in memory. Obviously ;)
 
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Dessi sure provided the influx of a new breath of comic relief in this 7 year old discussion. Mostly due to the fact that she thinks herself a smart enough cookie not to read back and so being prone to repeating the errors of all that have gone before.

There is a description for that behaviour, but let's not dwell on that. This thread is about Doron's Discoveries.

My prediction (and ok, it is rather a repeat of earlier predictions, all of which have come true for the most part) is that *any* discussion with Doron about this subject will go like this:

A = n00b poster in the thread, B = jsfisher or other veteran, C = Doron

A: Look, C, I am not rude like these others, I offer you logic in formal language X, Y or Z
B: That's no use, we went through all of them over the course of years
A: *pedantic* I am smarter than you guys, look C, here it is, infallible ironclad logic
B: No, not smarter, just more intelligent, there's a difference...
C: *ignore B* Ah, fresh blood... eh... hello welcome newcomer, let me see what you have written...
C: Ah, yes, but if we twist meaning f into slot g then your logic fails you because you do not know how to twist. In fact, that is nowhere to be seen in your so-called logic.
A: Well, then show me how to twist f into g, then I will gladly point out where you went wrong.
C: Ok, if you use a rapuctor to boodlezwam into infinity then all the real numbers take off their plastic coating and become all natural.
A: Wut? You can't use a rapuctor for that! What the heck *is* a rapuctor anyway? Look, real numbers are not the same as natural numbers, here is the ironclad logic, backed up by bulletproof math.
B: Been there, done that, got the hat.
C: *ignore B* You see, that is where you are wrong, your logic simply does not allow you to boodlezwam! And therefore, anything you want to prove in your logic is wrong.
A: *ignore B* Ok, so then show me how you define boodlezwam.
C: Please respond to rapuctor! Show me why it can not beedlebork!
A: Beedlebork? Boodlezwam? That's not even real math!
C: You don't get it! Please respond to Boodlebork!
A: *silently ignores the thread, never to be heard from again*
B: Cheers Doron, here's to another year!
 
Dessi said:
Can you show me the steps involved in your calculation?
Click to expand...
Dear Dessi, as long as you think only in terms of serial steps, you have no chance to conclude anything of what I say.

Dessi said:
Pretend that I'm not a mathematician, pretend I'm a random person off the street or maybe a student in a high school class. How do you explain "observes the real line from cardinality |N|" so that a layman without any specialized mathematical training could understand it?
Click to expand...

Let's say that you wish to build an infinitely long line of curbstones along a trail, such that each curbstone is marked by a unique symbol, for example {1,2,3,...}.

If there is a one worker that puts the curbstones step-by-step, the mission is never accomplished.

If there is a bijection between workers and curbstones, each worker puts exactly one curbstone along the trail together with the rest of the workers in the same time, and the mission is accomplished in one step (this working style is done in parallel).
 
realpaladin said:
Dessi sure provided the influx of a new breath of comic relief in this 7 year old discussion. Mostly due to the fact that she thinks herself a smart enough cookie not to read back and so being prone to repeating the errors of all that have gone before.

There is a description for that behaviour, but let's not dwell on that. This thread is about Doron's Discoveries.

My prediction (and ok, it is rather a repeat of earlier predictions, all of which have come true for the most part) is that *any* discussion with Doron about this subject will go like this:

A = n00b poster in the thread, B = jsfisher or other veteran, C = Doron

A: Look, C, I am not rude like these others, I offer you logic in formal language X, Y or Z
B: That's no use, we went through all of them over the course of years
A: *pedantic* I am smarter than you guys, look C, here it is, infallible ironclad logic
B: No, not smarter, just more intelligent, there's a difference...
C: *ignore B* Ah, fresh blood... eh... hello welcome newcomer, let me see what you have written...
C: Ah, yes, but if we twist meaning f into slot g then your logic fails you because you do not know how to twist. In fact, that is nowhere to be seen in your so-called logic.
A: Well, then show me how to twist f into g, then I will gladly point out where you went wrong.
C: Ok, if you use a rapuctor to boodlezwam into infinity then all the real numbers take off their plastic coating and become all natural.
A: Wut? You can't use a rapuctor for that! What the heck *is* a rapuctor anyway? Look, real numbers are not the same as natural numbers, here is the ironclad logic, backed up by bulletproof math.
B: Been there, done that, got the hat.
C: *ignore B* You see, that is where you are wrong, your logic simply does not allow you to boodlezwam! And therefore, anything you want to prove in your logic is wrong.
A: *ignore B* Ok, so then show me how you define boodlezwam.
C: Please respond to rapuctor! Show me why it can not beedlebork!
A: Beedlebork? Boodlezwam? That's not even real math!
C: You don't get it! Please respond to Boodlebork!
A: *silently ignores the thread, never to be heard from again*
B: Cheers Doron, here's to another year!
Click to expand...

Don't forget D! The guy who successfully predicts the debacle. Yes, yes. We have seen them come and go.

And then there's E. The Lurker, who went silent but still watches in morbid fascination whenever new blood tries to engage Doron.

This E. thanks the As, especially the current one, for helping illuminate Doron's frame of thinking.
 
Apathia said:
Don't forget D! The guy who successfully predicts the debacle. Yes, yes. We have seen them come and go.

And then there's E. The Lurker, who went silent but still watches in morbid fascination whenever new blood tries to engage Doron.

This E. thanks the As, especially the current one, for helping illuminate Doron's frame of thinking.
Click to expand...

Thank you for putting me in my place, I needed that/had it coming :)
 
doronshadmi said:
Dear Dessi, as long as you think only in terms of serial steps, you have no chance to conclude anything of what I say.



Let's say that you wish to build an infinitely long line of curbstones along a trail, such that each curbstone is marked by a unique symbol, for example {1,2,3,...}.

If there is a one worker that puts the curbstones step-by-step, the mission is never accomplished.

If there is a bijection between workers and curbstones, each worker puts exactly one curbstone along the trail together with the rest of the workers in the same time, and the mission is accomplished in one step (this working style is done in parallel).
Click to expand...

The logic error in this example is of course that this will only accomplish the laying of the curbstones.

It will not tell you how many there are, since no single worker knows how many others there are to the left or to the right of him.
EDIT: actually, the first worker knows that on one side there is nobody there...

They'd need to sound off, like 1, 2, 3... like... oh... for want of a better word... sequentially?
 
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Dessi said:
Doron can only think of mathematical expressions in terms of a computer program.
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Dear Dessi, you are wrong in this case, please do not try to understand me only according to your remarkable Programming Background (
1997: VB5
1998: VB6
1999: HTML
1999: JavaScript
1999: VBScript/Classic ASP
2000: PHP
2001: C++*
2001: Java
2002: Perl*
2003: VB.Net
2004: C#
2006: Python
2007: Delphi
2007: OCaml
2007: F#
)
 
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doronshadmi said:
Dessi said:
Pretend that I'm not a mathematician, pretend I'm a random person off the street or maybe a student in a high school class. How do you explain "observes the real line from cardinality |N|" so that a layman without any specialized mathematical training could understand it?
Click to expand...
Let's say that you wish to build an infinitely long line of curbstones along a trail, such that each curbstone is marked by a unique symbol, for example {1,2,3,...}.

If there is a one worker that puts the curbstones step-by-step, the mission is never accomplished.

If there is a bijection between workers and curbstones, each worker puts exactly one curbstone along the trail together with the rest of the workers in the same time, and the mission is accomplished in one step (this working style is done in parallel).
Click to expand...

If an infinite number of workers placing an infinite number of stones completes the mission, can I say that one worker who places an infinite number of stones infinitely fast also completes the mission? I think so. Are an infinite number of workers equivalent to a single infinitely productive worker? Seems like it.

Does it make a difference whether we set all stones in the set simultaneously, or set stones one after the other at infinite speed? No, not really. It's hard for me to see the difference.

Is my thinking here correct?

--

Also, out of curiosity can you explain how one " observes the real-line is observed from |R| cardinality", in layman-friendly terms?
 
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realpaladin said:
A: *silently ignores the thread, never to be heard from again*
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I not generally invested in threads for very long anyway.

Actually, I am rarely invested in online communities longer than a year either. ISF is a very exceptional case, it's active and interesting. I never thought I'd have more than a few dozen posts after I joined.
 
realpaladin said:
Thank you for putting me in my place, I needed that/had it coming :)
Click to expand...

Well, you're certainly not the only one who has played the D role in this fine Human interest story. There's Zooterkin and others.

And there are better Es who have more sense than to post in this thread. :wackylaugh:
 
Dessi said:
I not generally invested in threads for very long anyway.

Actually, I am rarely invested in online communities longer than a year either. ISF is a very exceptional case, it's active and interesting. I never thought I'd have more than a few dozen posts after I joined.
Click to expand...

And here you are, over 2.5k posts... :)

Btw, I see that you are mostly into the garbage collected languages?
(I count Bjarne's concoction not to the bare metal languages; I find it a hideous construct, but that is for another thread).

How about doing some real fun with ASM? Making your code trick the MMU so it fits completely in the L1 Cache and all of it's data in the L2?

As far as I can make out from your online presence, coding is a tool to express your mathematical abilities, not generic problem solving (as in hacking stuff).

But all of that is neither here nor there, we are talking about this 'Deeper than Primes' thingy.

The reason why most people stay away after some time is because Doron has the uncanny ability to trick people in defending their stance that his claim is wrong. He plays 'reductio ad absurdum' all the time.

But this is not the correct scientific way; Doron makes a claim, Doron needs to corroborate it.

The rest of us do not have to prove that he is wrong; he must prove he is right.
 
Apathia said:
Well, you're certainly not the only one who has played the D role in this fine Human interest story. There's Zooterkin and others.

And there are better Es who have more sense than to post in this thread. :wackylaugh:
Click to expand...

Yep. As I said before, Doron is like having a favourite toothache.

But it would be something if we ever would make *some* progress.

We had unity, Transcendental Meditation, Organic Mathematics, infinity, two islands, Cantor is wrong, Cantor is right, etc...

And I keep asking "ok, what *if* we *all* agreed that you are right, without need for any proof, then what?" and we never get an answer to that...

So my conclusion from the empiric evidence is that Doron's goal is kibitzing and nought else.
 
Direct perception is still my favorite. How can you go wrong with direct perception?

If Doron thinks it true, well, it must be true because that's what Doron perceives it to be...directly. My only regret is the direct perception is inaccessible to the rest of us because of that visual/spatial thing or something. Actually, we just don't get it.

Heck, I'm so backwards I believe the union of the members of {{A}, {B}} to be {A, B} instead of {{A}, {B}}....
 
Dessi said:
Does it make a difference whether we set all stones in the set simultaneously, or set stones one after the other at infinite speed? No, not really. It's hard for me to see the difference.
Click to expand...
Dear Dessi,

First, please look again at http://www.internationalskeptics.com/forums/showpost.php?p=10330277&postcount=102.



In terms of process, infinite sequential (step-by-step) speed is equivalent to moving all stones simultaneously (in parallel, by one step).

So in both cases the keyword is one step, or on other words: "mission is accomplished by one step", which means that there is no room for the notion of process of more than one step among the considered subject.

Again, infinite sequential (step-by-step) speed is actually one and only one step, or in other words, the notion of more than one step (for example: step-by-step) is not satisfied.

So the parallel model is better than the serial model as an explanation method for layman, because it is naturally lack of any potential illusion of process of more than one step among the considered subject.

Here is a concrete example of wrong conclusions by a person that is definitely not a layman, if she observes infinite collections only in terms of serial (step-by-step) observation: http://www.internationalskeptics.com/forums/showpost.php?p=10326945&postcount=66.

If you fail because you are using only step-by-step observation of infinite collections, it is clear that your only step-by-step observation is not a useful point of view to explain |N| to a given layman.

Once again, the cardinality of the natural numbers is |N| and this size is known in one step, no matter what complexity is involved among the natural numbers, as demonstrated, for example, in the following diagram:

5695547493_fbbe88a093_z.jpg


Dessi said:
Also, out of curiosity can you explain how one " observes the real-line is observed from |R| cardinality", in layman-friendly terms?
Click to expand...

Since the considered subject here is related to process in terms of one step (no processes of more than one step is used) one easily follows after one step |R| and one step |N| of the real-line, by using the fact that |N|<|R|, and this is exactly what I am doing in http://www.internationalskeptics.com/forums/showpost.php?p=10332081&postcount=110.

Dear Dessi, using ∞ in order to deduce conclusions in terms of infinity is not accurate enough, simply because it does not use the accurate observation of |N| < |P(N)| < |P(P(N))| < |P(P(P(N)))| < |P(P(P(P(N))))| < ... different levels of infinity, where each one of them is achieved in no more than one step.
 
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realpaladin said:
And here you are, over 2.5k posts... :)

Btw, I see that you are mostly into the garbage collected languages?
(I count Bjarne's concoction not to the bare metal languages; I find it a hideous construct, but that is for another thread).

How about doing some real fun with ASM? Making your code trick the MMU so it fits completely in the L1 Cache and all of it's data in the L2?
Click to expand...
But I can't even see the metal while floating 30,000 ft on my Haskell cloud.

As far as I can make out from your online presence, coding is a tool to express your mathematical abilities, not generic problem solving (as in hacking stuff).
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I'm a professional software engineer who tinkers with Arduino and functional programming ^_^

I prefer to express mathematical concepts using established, accepted mathematical notation. Unless I were specifically explaining how to encode certain concepts in code, like an implementation of a specific graphing algorithm, I would almost never use a source code to express a mathematical concept.

The snippet of pseudocode in this post is not meant to illustrate a mathematical concept. That piece of code was authored by Doron, not myself, in his paper on Zeno's paradox. I did a quick peer review and noticed his paper incorrectly states that the program models an infintely long race, when in fact in models a 1 second long race.

The snippet of code in this post is not meant to illustrate a mathematical either. I provided it to support my personal speculation on how Doron arrives at concepts like "different instantiations of 2". I hypothesize that, if this concept is meaningful to Doron, it must be because he is a Java programmer, and he generalizes the quirks in Java's type system (where "different instantiations of 2" is a meaningful statement) as an inherent feature in mathematics.

But all of that is neither here nor there, we are talking about this 'Deeper than Primes' thingy. [ . . . ]

But this is not the correct scientific way; Doron makes a claim, Doron needs to corroborate it.
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Doron can correct me if I'm wrong, but I gather from his posts that he's a programmer with an interest in mathematics.

I hypothesize that Doron interprets mathematical notation as a kind of source code. Mathematics is a kind of programming language with its own unique and flexible syntax. Computation is, conceptually, the result of executing mathematical source code on a computer. His comments actually make a lot of sense in this context:

  • The distinction between "serial-computation" and "parallel-computation". Obviously a computer program would never halt on inputs involving actual infinities, so we need to optimize its execution. Parallelization is the most straightforward optimization. Doron agrees that an infinitely fast serial process is an acceptable and equivalent optimization.

  • He presumably makes a distinction between "different instantiations of 2". This is the exact same verbiage that object-oriented programmers use to explain how different instances of a class/type may contain same value, but are not referentially equal because they are stored at different addresses in memory. Evidently numeric values and variables in a mathematical expressions imply allocating memory and new'ing up instances of the Number class.

  • On a computer, 0.999... not exactly equal to 1, because they have a different pattern of bits. 0.999... and 1 can't be equal because they are not bitwise equal.

  • 0.000...1 is defined to be the smallest interval between 0.999... and 1. This number serves the same purpose as machine episilon, which is the upperbound relative error in floating point arithematic on computers.

  • He wrote a paper analyzing Zeno's paradox. But, instead of using the standard mathematical notation of series and recurrence relations, he literally wrote source code in a BASIC-like pseudocode, complete with loops and mutable variables. Code is code, right? It should be readable. The dense and ugly syntax of mathematics, APL, or Perl, is almost never preferable to readable pseudocode.

  • He insists that parallel-thinking is not a "process", not an "operator", it's not really anything that can be expressed in mathematical notation. It's a different way of thinking about the objects modeled in code, it's a programming paradigm.
In this mindset, quirks and gotchas in computer arithmetic are inherent features in mathematics more generally.

That's just my hypothesis. I'm happy to be corrected if I've misinterpreted Doron entirely.

In the mean time, Doron, you will enjoy reading the evolution of a Haskell programmer :)
 
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Dessi said:
But I can't even see the metal while floating 30,000 ft on my Haskell cloud.
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Jenga... that's why I can teach my students to get into just about any system nowadays :)

Dessi said:
I'm a professional software engineer who tinkers with Arduino and functional programming ^_^
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Well coding is a meritocracy where designations mean little; I mean, I managed a company in India for 1.5 years and I got certified PhD guys in software engineering who I had to teach the basics of complexity theory.

Dessi said:
I prefer to express mathematical concepts using established, accepted mathematical notation. Unless I were specifically explaining how to encode certain concepts in code, like an implementation of a specific graphing algorithm, I would almost never use a source code to express a mathematical concept.
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That's where we would differ; I would use anything that works. If that would mean using a cardboard difference engine or plastic bottles filled with sand, so be it.

Dessi said:
The snippet of pseudocode in this post is not meant to illustrate a mathematical concept. That piece of code was authored by Doron, not myself, in his paper on Zeno's paradox. I did a quick peer review and noticed his paper incorrectly states that the program models an infintely long race, when in fact in models a 1 second long race.
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Cool. I'll add that to the communities' services rendered to Mr. Shadmi; code reviewing.

Dessi said:
The snippet of code in this post is not meant to illustrate a mathematical either. I provided it to support my personal speculation on how Doron arrives at concepts like "different instantiations of 2". I hypothesize that, if this concept is meaningful to Doron, it must be because he is a Java programmer, and he generalizes the quirks in Java's type system (where "different instantiations of 2" is a meaningful statement) as an inherent feature in mathematics.
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I rather have him pegged as a (Turbo)Pascal user who now migrates to Java.

Dessi said:
Doron can correct me if I'm wrong, but I gather from his posts that he's a programmer with an interest in mathematics.

I hypothesize that Doron interprets mathematical notation as a kind of source code. Mathematics is a kind of programming language with its own unique and flexible syntax. Computation is, conceptually, the result of executing mathematical source code on a computer. His comments actually make a lot of sense in this context:

[*] The distinction between "serial-computation" and "parallel-computation". Obviously a computer program would never halt on inputs involving actual infinities, so we need to optimize its execution. Parallelization is the most straightforward optimization. Doron agrees that an infinitely fast serial process is an acceptable and equivalent optimization.
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Which is essentially correct. But he needs to get his analogies right, that is why I mentioned the weighing versus counting; weighing is basically a one-step parallel addition of all the weights, as opposed to weighing one stone after another and adding all their respective weights together.

Dessi said:
[*] He presumably makes a distinction between "different instantiations of 2". Obviously that means 2 instances which are not referentially equal, they happen to be the same value which are stored at different addresses in memory.


[*] On a computer, 0.999... not exactly equal to 1, because they have a different pattern of bits. 0.999... and 1 can't be equal because they are not bitwise equal.
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I think that one is rather his 'intuitive feeling that something is rotten in Kislev'... if the most significant digit is different then the numbers must be different. Something like that.

Dessi said:
[*] 0.000...1 is defined to be the smallest interval between 0.999... and 1. This number serves the same purpose as machine episilon, which is the upperbound relative error in floating point arithematic on computers.
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But you do know that in mathematics, there is no smallest interval in a limit.

Dessi said:
[*] He wrote a paper analyzing Zeno's paradox. But, instead of using the standard mathematical notation of series and recurrence relations, he literally wrote source code in a BASIC-like pseudocode, complete with loops and mutable variables. Code is code, right? It should be readable. The dense and ugly syntax of mathematics, APL, or Perl, is almost never preferable to easy-to-read pseudocode.
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Try doing that in Brainf*ck (which, as you know, *is* a programming language)...

Dessi said:
[*] He insists that parallel-thinking is not a "process", not an "operator", it's not really anything that can be expressed in mathematical notation. It's a different way of thinking about the objects modeled in code, it's a programming paradigm.[/list]
In this mindset, quirks and gotchas in computer arithmetic are inherent features in mathematics more generally.

That's just my hypothesis. I'm happy to be corrected if I've misinterpreted Doron entirely.
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You've done pretty well.

The part that is missing is his intentions; at one point everyone in the thread agreed with him for the sake of finally moving on.

Then we got to the two islands thought experiment and he got tangled up; since then he supposedly ignores me, but keeps reacting (not responding, mind you) to my posts.

As Apathia stated; the pattern repeats and repeats; whatever happens, his joy is in the discussing of something and not in the achieving of something.

Dessi said:
In the mean time, Doron, you will enjoy reading the evolution of a Haskell programmer :)
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Yes, now there is some bliss when you have known the horrors of COBOL...

EDIT: I kind of mangled the layout somewhat... too lazy to fix properly
 
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