Freethought & Rationalism Archive

CardinalMan · 02-01-2002, 04:49 AM

Quote:

Just for the record: Cauchy sequences, Dedekind cuts *) etc. may be more elegant, but defining reals as infinite decimals (i.e. formal sequences of decimal digits) is perfectly OK. You just have to exclude those which have only a finite number of digits differing from 9.

Yes, this is true, but it is rather sloppy and artificial. Why use base 10 instead of base 47 when writing out your sequence of symbols? One could also define the natural numbers as finite sequences of symbols from 0,1,2,...,9 that do not start with a zero, but that seems kind of silly to me. If you define either the real numbers or natural numbers in this manner, then the definition of addition and multiplication becomes a mess, along with the checks that they satisfy associative, commutative, and distributive laws. Also, since the infinite decimal expansion of real numbers suggest that you are taking the sum of an infinite series, it seems much more pleasing to me to develop the properties of the real numbers and convergence before introducing them.

Furthermore, I think that the definition using Cauchy sequences or Dedekind cuts are much more insightful since they suggest the manner in which the real numbers are a completion of the rational numbers.

Quote:

What a real number is, should be left to everyone's private conjectures .....

I agree to a point. Surely, claims about the fundamental Platonic nature of the real numbers are just philosophical speculations, not mathematical arguments. However, since the base of decimal expansion is arbitrary, it seems to me to be much more natural to think of decimal expansions as representations of real numbers.

CardinalMan

[ February 01, 2002: Message edited by: CardinalMan ]

FarSeeker · 02-05-2002, 01:25 PM

Quote:

Originally posted by Photocrat:

Of course, it's true that, as CardinalMan has just pointed out for us, the real truth is just a tad more complicated... :] I could spell out a more full formulation, but it would be long & probably boring for most people... I mean, you don't usually think of a real number as the limit of a Cauchy sequence... Even so, with all our definitions, we can show that distinct sequences (.999..., 1.000...) can represent the same rational number, as I was pointing out :] Not very intiutive, is it?

Can I pop in on this and suggest what might be a simpler explanation, tho I don't know how logical it might be:

1/3 = .3333...(infinite series)

1 = 1/3 +1/3 +1/3 (QED)
ERGO
1 = .3333... +.3333... +.3333...
1 = .9999...(infinite series)
At least this is how I reasoned it out when I took calculus. Is that too simplistic? <img src="confused.gif" border="0">

Wizardry · 02-05-2002, 09:33 PM

Quote:

Originally posted by FarSeeker:


Can I pop in on this and suggest what might be a simpler explanation, tho I don't know how logical it might be:

1/3 = .3333...(infinite series)

1 = 1/3 +1/3 +1/3 (QED)
ERGO
1 = .3333... +.3333... +.3333...
1 = .9999...(infinite series)
At least this is how I reasoned it out when I took calculus. Is that too simplistic?

Actually, that's wrong. The series you suggest diverges rather than converging to 1. What you might be thinking of is

1 = 9/10 + 9/100 + 9/1000 + 9/10000 ...

1 = .9 + .09 + .009 + .0009 ... = .999...

.9(1/10)^n The sum of this series is (9/10)/(1-1/10) = (9/10)/(9/10) = 1.

Peace out.

FarSeeker · 02-06-2002, 03:03 PM

Quote:

Originally posted by Wizardry:
Actually, that's wrong. The series you suggest diverges rather than converging to 1. What you might be thinking of is

1 = 9/10 + 9/100 + 9/1000 + 9/10000 ...

1 = .9 + .09 + .009 + .0009 ... = .999...

.9(1/10)^n The sum of this series is (9/10)/(1-1/10) = (9/10)/(9/10) = 1.

Peace out.

Exqueege me? How does that diverge?

One divided by three equals a decimal point followed by and infinite number of threes.
1/3 = 0.333�
Multiply one-third (1/3) by three (3) you get one (1).
(1/3)*3 = 1
Multiply a decimal point followed by an infinite number of threes by three and you get a decimal point followed by an infinite number of nines.
(0.333�)*3 = 0.999�
Ergo, 1 = 0.999�
QED

Given A = B
Given A*3 = C
Then B*3 = C

Unless you dispute the laws of common mathematics.

tommyc · 02-11-2002, 03:38 AM

The writers of "Blackadder" have a similar
confusion about significance of the holy ghost:

The king (on declaring the new archbishop of Canterbury)

King: Members of the Court and, uh, Clergy: I have, at last -- after careful consultation with the Lord God; His Son, Jesus Christ; and His Insub
stantial Friend, the Holy Ghost -- decided upon the next Archbishop.

FarSeeker · 02-13-2002, 05:24 PM

Anunnaki January 29, 2002 07:44 AM

Quote:

Complex mathematical equations and countless hours pondering how a circle could be a triangle is a complete waste of time since what you`re trying to figure out will NEVER make any logical sense.
It didn`t make sense when they made it up,it doesn`t now and it never will.
Perhaps this will help.

Why? Are conic sections illogical? That is the basis of my argument; the other is Sagan's "visit" to Flatland on the "Cosmos" video. Look at the end of a equal sided cylinder, it's a circle. Now look at side, it's the shape of a square.

And a HTML reference to Murphy and Ingersol? Perhaps you get your info on Jews from Hitler and David duke? Any non-bias sources?

If it's a complete waste of time in your opinion, then why waste my time with your opinion.

Originally posted by FarSeeker:
quote:
>Sorry, 2 out of 5 is not passing.
>2. How can a circle be a square?
>Spin a square on an axis running from the middle of one side to the middle
>of the opposing parallel side. Now >spin the square. It sweeps out the
>volume of a cylinder; the ends of which are circles. (ok, so I missed, there
>are 2 circles) The cylinder will cast both a square and a circular shadow
>into/on to a 2D "Flatland"/surface

daemon23

Quote:

posted January 29, 2002 10:40 AM
Okay, so how does your example prove that the circle is a square? You can use a variety of transformations to create them, but this seems to be an attempt to confuse the issue. This is like saying 1 = 2 because 1+1=2. It's neither mathematically or semantically correct.
Your question 3 is simply a repetition of the same exercise. Please speak more clearly--word games are no substitute for thought.

(emphasis added)

You must be purposely misrepresenting my statements; I can't see any other reason for you not understanding. Try to think about this without your closed-mindedness getting in the way.

Quote:

For a long time the sphinx had guarded the city allowing no one to enter unless they could solve its riddle, and as yet none had. This situation was fine by the sphinx as it provided the sphinx a constantly full belly. But then two brave bodies, by the name of daemon23 and Bill Snedden, came and challenged the sphinx. "Tell us your riddle," they said, "And we shall show you what a fool you are." So the sphinx smiled and spoke the riddle to them.

After hearing the riddle, the two consulted among themselves, and then laughing, turned and spoke to the sphinx. "The answer that any typical fool would give to you is 'a man,' but in truth that cannot be so, for men never walks on anything but 2 legs." They continued, "Babies crawl on HANDS and knees, and old men walk on 2 legs with A CANE, but these things cannot legitimately be called legs. Therefore 'man' is an equivocating answer."

At first a look of shock flashed onto the sphinx's face, but it vanished as quickly as it appeared. The sphinx grabbed the two fellow travelers up in to huge mouth and devoured them. Then, cleaning itself with its giant paws, the sphinx muttered to itself, "nobody likes smart asses."

The point of a riddle is to get you to "think outside of the box." It is strange that you nitpick my riddle, but did not nitpick the sphinx's riddle: men never walks on anything but 2 legs because babies crawl on HANDS and knees, and old men walk on 2 legs and A CANE. You manipulate my riddle in order to accuse me of wrongdoing, but you're dissembling. Get a grip. <img src="graemlins/banghead.gif" border="0" alt="[Bang Head]" />

I didn't ask "how is a square a circle." I asked, "How can a circle be a square?" In other words, under what conditions can a square also have the attributes of a circle? The Triangle and circle riddle followed the same pattern.

FarSeeker:

Quote:

5 seconds to get 2 answers? I think you are losing you touch. Especially with the Flatland hint. The electron (e- = C) (another missed hint) has "personalities". One is a particle; the other a wave. "A" Particle cannot "B" a Wave. They are two distinct objects. Yet the electron is both.

daemon23

Quote:

Again, you're playing word games. At least one of the claims here must logically be false; if an electron is both particle and wave, it is obvious that it is certainly possible for a particle to be a wave.
You're attempting to use your personal and provably incorrect assumptions as basis for logically proving logic invalid. Congratulations on your astounding lack of reason.

Well, some Atheists hold that infanticide isn't murder (isn't that right Bill?), so I guess you could believe a particle could be a wave. Unfortunately, lpetrich disagrees with you:

lpetrich

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As to wave-particle duality, that is abundantly verified experimentally, even though exactly why it happens is a murky question.

As one who has no respect for me, lpetrich must be absolutely livid that he has actually agreed with me.

daemon23

Quote:

posted January 30, 2002 06:11 PM
Triangle means three angles, not objects, and said angles are properties of the triangle.

Angles can exist independently, thus they are objects.

daemon23

Quote:

Actually, what you asked was

quote:
How can a circle be a square?

At no time did you indicate transformation was any part of the question; you asked how two different geometric objects could be identical, and the answer is that they cannot. Is my left hand my spleen simply because both are part of me?
I see no relevance to higher dimensional mathematics to the questions you posed. You asked questions about 2-dimensional objects, and you got answers about 2-dimensional objects. If you had intended on referring to higher dimensional geometries, perhaps you should have stated that, but you did not.

If you don't understand that Christians hold that God is transcendent (extra-dimensional), then perhaps you need to educate yourself about our beliefs.
At no time did the sphinx indicate that the "day" meant lifetime. Riddles work by leaving things out. GAC.

FarSeeker:
2.) A quantum particle is a discrete object. Consult linguistics and physics.
3.) A wave is a diffuse object. Drop a stone into a puddle. Look at a waveform on an oscilloscope.
If "it is obvious that it is certainly possible for a particle to be a wave," Please do so and solve one of the great QM mysteries of the =>20th centuries.

daemon23

Quote:

I find it amazing that you can ask such a bizarre question, and think this is such a great mystery, when you have already claimed to have solved it yourself! Is not an electron both particle and wave? I saw you say so yourself! Then, to continue and claim that it is an impossibility is laughably ludicrous! It quite obviously is the case that an electron exhibits both wave-like and particle-like behaviors, so just as obviously it is possible. If it were impossible, it would not be so, ne?
The more relevant question here is how it is, and that I cannot tell you.
[ January 30, 2002: Message edited by: daemon23 ]

What question did I ask there? I have offered no solution to the wave-particle duality nature of electrons. Daemon23 is clearly misrepresenting my words. I simply state the nature as revealed by scientific experimentation. A particle cannot be a wave by definition, but an electron can be both by observation. That is a mystery. I offer no solution, but I seem to remember reading that a possible solution does exist in higher dimensional String Theory.

But deamon23 does offer a cheap solution to the Trinity problem: "It quite obviously is the case that God exhibits both Father, Son and Spirit-like behaviors, so just as obviously it is possible. If it were impossible, it would not be so, ne?

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02-11-2002, 03:38 AM	#105
tommyc Veteran Member Join Date: Feb 2002 Location: London, England Posts: 1,206	The writers of "Blackadder" have a similar confusion about significance of the holy ghost: The king (on declaring the new archbishop of Canterbury) King: Members of the Court and, uh, Clergy: I have, at last -- after careful consultation with the Lord God; His Son, Jesus Christ; and His Insub stantial Friend, the Holy Ghost -- decided upon the next Archbishop.

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