Cont: Deeper than primes - Continuation 2

[realpaladin]

<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>

(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)

 

[doronshadmi]

Show a proof please.

It is a self-evident truth to those how understand that all whole rationals will never be < 1 AND > 0.

 

[Dessi]

It is a self-evident truth to those how understand that all whole rationals will never be < 1 AND > 0.

No, it is not self-evident. I am interested to see your proof that 0.999... converges to a rational < 1 and > 0.

 

[doronshadmi]

Even in Doron's very limited model of mathematical computation, the Doron Machine, the equivalence of 0.999... = 1 is inescapable. Intuition be damned.

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.

 

[doronshadmi]

No, it is not self-evident. I am interested to see your proof that 0.999... converges to a rational < 1 and > 0.

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.

 

[Dessi]

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.

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 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?

 

[jsfisher]

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.

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.

 

[doronshadmi]

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 has a limit L, then S < L. How do you know S < L? Can you show a proof?

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|.

 

[zooterkin]

When you deal with infinity, using step-by-step (serial) thinking style gives the illusion that some kind of "process" is involved.

Which, bizarrely, is what you seem to be prone to.

 

[Dessi]

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|.

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?

 

[alex04]

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.

 

[Dessi]

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.

It certainly couldn't hurt. I'd also recommend MathNet.Numerics.FSharp for arbitrary precision arithmetic.

 

[jsfisher]

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.

 

[Dessi]

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.

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?

 

[alex04]

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.

Cheers.. wow. I'm out of my depth here, I may as well be listening to Bud Haggart.

 

[jsfisher]

Cheers.. wow. I'm out of my depth here, I may as well be listening to Bud Haggart.

If Doron's writings confuse you, the fault is likely not with you.

 

[Dessi]

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.

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:

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 t_k is 1/ASpeed the length of its predecessor t_k-1:

Total Time = t_n + t_n-1/ASpeed, t₁ = 1.
Total Time = 1 + 1/10 + 1/100 + 1/1000 + . . .
Total Time = Σ(k = 0, k -> ∞) 1*(1/10^k)
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 t_k = t_k-1 / ASpeed, the program halts:

- Total elapsed time = Σ(k = 0, k -> ∞) 1*(1/11^k) = 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.

 

[jsfisher]

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.

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.

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.

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.

 

[Dessi]

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.

According to Doron, 2 is not a member of {2}, apparently because those must be different instantiations of 2.

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:

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

 

[doronshadmi]

Which, bizarrely, is what you seem to be prone to.

Please support you claim (which according to it, when I deal with infinity, I am using only step-by-step (serial) thinking style) according to what is written in http://www.internationalskeptics.com/forums/showpost.php?p=10332081&postcount=110.