Gaudere:
Okay, maybe it can’t be a religion, but, it can be a philosophy. There is no reason an atheist cannot step across the line into antitheism, particularly if he perceives some calling (like O’Hair) or some ontological necessity (like Rand).
“It is lucky for rulers that men do not think.” — Adolf Hitler
Howso? All atheism means is “I lack belief in God”. Even Strong Atheism says only “there is no God”. Now try to formulate a system of philosophical concepts based on that alone. It’s simply a negative statement; like “there is no soul”. All very well and good, but for a complete philosophy you need some additional concepts not included in the atheism package. If you decide to be a materialist, fine, but that’s not necessarily a part of atheism. Look at Glitch: just as much of an atheist as me, yet he believes in Osu (the power of the human spirit).
Lib:
I try, Lib, I reallt try. You, yourself, noted that I often ask you to explain in detail what you mean by a statement; that is my way of discovering “why [you] think teh way you do.” Unfortunately, as this thread provides further evidence, you often ignore my questions. Apparently, you usually believe that you have already supplied the answers. I wonder, then, why do you think I keep asking? And do you, for instance, feel like you answered my questions in this thread concerning your use of the … notation?
The best lack all conviction
The worst are full of passionate intensity.
*
All right, I hereby formally remove my foot from my mouth and take a seat.
Now I see what Lib was trying to do and I give him kudos.
Gaudere and Lib, and whatever others may be involved, please try to stick to the math, the Atheist (non)Religion thread is still there for that, unless Phaedy’s invaded it.
Firefly:
Okay, I see what you mean, and you do make a lot of sense. If you went over the proofs line by line, then you deserved more than a brush-off. Maybe I just had a chip on my shoulder, but if you will look back, you can see a number of remarks from you that, taken without knowing your intentions, might be interpreted as sanctimonious.
I think that’s one of the reasons the Atheist Religion thread has been so successful at being a relatively peaceful discussion. People approach it fairly delicately, usually saying things up front like, “I don’t mean this in any way to demean your point of view, but…”. Anyway, thanks for explaining yourself in your last post.
Gaudere:
An Ad Hoc Atheist Philosophy
Epistemology: Science
Metaphysics: None
Ethic: The Golden Rule
Aesthetic: Subjective interpretation
“It is lucky for rulers that men do not think.” — Adolf Hitler
Libertarian
Considering your handle, you’ve probably figured this out already: I was practicing. Pretty selfish, huh?
Anyway, the proper way wasn’t all that clear. I guess it was worth a shot, though.
Now, let me ask you, what are your motives? What are you going to do after you clean up Dodge?
.
RM:
My motive was, as I said, to provide a forum for a nonreligious, nonpolitical discussion in response to a request made by PeeQueue in another thread, where he had logged on for the first time and lamented the spate of Jesus threads.
Whatever else happened was a happy accident.
As to what I’ll do now, I’ll just get back to the business of swatting at tyranny wherever I find its defenders, and trying to understand the atheist mindset. But I might start a new thread for the purpose of debating the chronosynclasticinfindibulum in the context of cosmology.
“It is lucky for rulers that men do not think.” — Adolf Hitler
Spiritus:
It is often hard to get a man to answer your question when his own is pending. Recheck the OP.
“It is lucky for rulers that men do not think.” — Adolf Hitler
Atheist Tangent: Guadere, there are many who take atheism as the crucial foundation of their personal philosophy. It is much more than simple lack of belief in GOD. It is actively having faith that there is no GOD or LOKI, ENLIL, ANUBIS, et al.
Faith implies something beyond the rational. What that something is, differs from atheist to atheist, but it certainly is one of the principles for the conduct of my life, making it a philosophy of sorts. Atheism has a direct impact on behavior and mental balance.
Hell is Other People.
Lib - I do understand how the epsilon post especially might have come across that way, and I apologize. I also apologize for, and retract, my unwarranted accusation of trolling. And, needless to say, your apology is accepted.
I think we can find ways of being able to post to the same threads, and get along as respectful adversaries, at least.
Lib:
You have lost me there. I ansered the OP in my first post. You now claim that you were purposely ignoring repeated inquiries because you felt that we all owed you something first? Is this how you demonstrate to us the open and respectful discourse that you so often request?
The best lack all conviction
The worst are full of passionate intensity.
*
The math in this thread is so high it gave me a nose bleed when I tried to read it.
OK, I’ve been lurking on this thread for the past few days, and I’d like to echo DrMatrixs sentiments when he says:
Can anyone sum up? I must’ve read the whole thread 3 times now, and I’m still scratching my head.
If i get this right Libertarian introduced a new axiomatic system and a new notation for infinite decimal expansions. (sadly, it wasn’t clear to me until about page 2 or so that we weren’t talking about good ole regular arithmetic and just a different way of typing the bar over the number notation)
This new system was then found to be not very useful because the operation of division by a non-zero number wasn’t closed (as in the 3.1[n]/4 example).
So heres what I’m still scratching my head over. This new axiomatic system… from what I gather, it had a largest integer 10[n-1] which was larger than the greatest number of digits allowed in the decimal expansion of a number, right?
And is 3.1[n] ok, even though it has “n+1” digits? (i.e. wouldn’t 10 * .31[n-1] = 3.1[n-1]?)
Phew.
I’d also like to say that I can see what RTFirefly was trying to do. The proofs in the OP seemed a lot like what a former college roommate of mine tried when he heard of the whole .9… = 1 thing and didn’t like the idea of infinity (though he had a much weaker grasp of mathematics than Lib. does), so my conversation went roughly the same way.
Hunsecker
The two proofs in the OP were of interest because they seemed to prove contradictory hypotheses. Proof 1, with merely the slightest variation, specifically leaving out the definition that .9… = .9[n], has been for a long, long time a famous proof that .9… = 1. One implication of this equality is that the set of real numbers is an infinite set. If .9… did not equal 1, then the implication is that there exist two real numbers not equal, and with nothing in between them.
The same point can be made without the definition though, by asking, however naively, where the extra 9 comes from when we multiply .9… by 10 to get 9.9… In other words, 10 times 9.999 equals 99.99, and not 99.999.
Those here who held up Cantor (whose genius I certainly recognize) were not mindful of the paradox raised in his Mengenlehre (Epistemology of Sets), which Russell did recognize, and which he solved by his theory of type sets. Great minds, Zermelo, Fraenkel, Bernays, and von Neumann all saw the problem with Cantor’s transfinite ordinals in that he believed that the infinte aggregate of all ordinals, having stated that every set can be well ordered, is itself well ordered.
Russell illustrated the paradox this way: In a small village, the barber shaves all those men and only those men who do not shave themselves. Now, the question is, who shaves the barber? If he is one of those men who do not shave themselves, then he must be shaved by the barber. But he is the barber. So if he shaves himself, then he does not shave himself. Oops.
Russell introduced the theory of types, such that individual elements are of type 0, classes of individuals are type 1, classes of classes are type 2, and so on. Thus, individuals can be elements of a class, and a class can be an element of a class of classes, and so on. In general, anything of type n can be an element of a class of type n + 1, but a class cannot be a member of itself. That means that the barber paradox is a confusion of types of classes, and thus the paradox is resolved.
And that is the ultimate reason that .9[n] fails as a definition for .9…, namely that .9… is an element of a class of one type (a number whose digits represent a transfinite cardinality), wherease .9[n] is an element of a different kind of class (numbers whose digits are finite and countable).
I had originally developed Proof 2 as an inductive proof, but ran across a lot of understandable resistance on the basis of a deductive proof (like Proof 1) being “superior” to an inductive proof. So, I simply made what I had induced before into an axiom, which allowed me to develop the proof deductively. And it is a very useful tool for certain things. It is true that 4 times .7[n] equals 3.1[n-2]08, but only when n is finite.
Father Mentock was the one who zeroed right in on the problem of .9[n] times 10 not being .9[n-1]. He then proceeded, in a very gentlemanly way, to show how transfinite ordinalities are derived, not from counting sets, but from operations on counting sets that have transfinite solutions. In fact, it is common that operations on real numbers, where real numbers are the universal set, often produce solutions that are empty in the universal set. One example is x^2 = -1. If you solve for x, you will need to expand your universal set to be the set of complex numbers if you are not satisfied with {} as your answer.
Math is utterly replete with the introduction of new fields that are formed all the time by simple rearrangement of other fields: dropping an axiom here, tweaking a definition there. Noneuclidean geometry arose in exactly this way, by simply dropping the parallel line postulate.
But despite what some have said in this thread, an awful lot remains to be done in number theory with respect to both cardinality and ordinality. Infinity, interestingly enough, is not the final word. Nor has infinity been “pinned down”. There is nothing wrong with questioning Russell now, for example, and seeking to formulate a more cogent resolution to the paradox of infinite ordinals.
If everybody just stops and says, “oh well, it’s all finished,” then we won’t ever learn anything new.
“It is lucky for rulers that men do not think.” — Adolf Hitler
Spiritus:
And I told you your answer was wrong.
There was absolutely nothing wrong with premise 7 in Proof 2. The value of .0[n-1]1 is not undefined, but merely has multiple solutions, like the value of 2x = y. As I pointed out then, an undefined value is of the type x / 0. The value of 2x / (y - 5) = 0 is defined for every value of y, except 5.
Well, as you know, the best lack all conviction. The worst are full of passionate intensity.
“It is lucky for rulers that men do not think.” — Adolf Hitler
[Daniel P. Bostaph Hat: ON]
<font size=+10>n = </font><font size=+10 face=symbol>¥</font>
Q.E.D.
[Daniel P. Bostaph Hat: OFF]
Libertarian:
Thanks for the (lengthy) explanation, that makes things quite a bit clearer. It looked to me like the error in reasoning was something along the lines of thinking you couldn’t have an endless decimal expansion because the number of digits isn’t a member of the set of reals (here I was about to insert some stuff about Z/nZ groups, but I figured that would backfire and make me dig out my algebra books). That or when looking for a cardinality A where A+1 = A, you were using the + operation that was defined on the field of reals with a cardinality that wasn’t a member of the field.
Now, heres a GD topic… You mentioned that you had originally phrased proof 2 as an inductive proof, but there was
Whats everyone got against inductive proofs? Is there any logical reason that a deductive proof should be preferred? I mean, sure, most people probably didn’t learn proof by induction till they got to calculus or something, so it carries with it bad memories, but is there a good reason to avoid them?
Libertarian:
You were, and still are, wrong. You introduced your notation with the definition:
.9… = .9[n].
Shall I remind you of your own words?
In proof 1, you used the 9[n] notation that was consistent with an infinite sequence, so the proof was correct so long as we took 9[n] to be simply a notation of convenience rather than carrying teh implication of a finite value for n.
In proof 2, you introduced the 0[n-1] notation and the implications of a finite n. This value is undefined because it relies upon an identification that fails.
So let us recap. In this thread you have compared yourself to Galileo, attacked others for hubris, refused to answer questions because you were certain (incorrectly) that your position was more mathematically correct than yur questioners’, admitted that you were in error and a breath later accused me of being wrong for pointing out your error. You continue to be a shining light for Christian humility and logic.
BTW:
The best lack all conviction
The worst are full of passionate intensity.
*
Hunsecker:
Yes, there is.
When you deduce, you are making observations about one or more elements of a set based on known facts about the whole set. That is, you are reasoning from the general to the particular. For example, if you know that you are dealing with the Redheaded Chess Players Club, then you may make certain reasonable assumptions about individual members of the club, namely, that each has red hair and each plays chess.
Induction, on the other hand, makes generalizations about a whole set based on information on some or all of the elements of the set. That is, it reasons from the particular to the general. Now, if you do indeed review every single element of the set, then you can draw perfectly valid conclusions about the whole set. But any partial review of the elements can lead to untrustworthy conclusions.
Faulty induction is responsible for a lot of ignorant viewpoints. For example, “I personally know of some black people who are bad or lazy, therefore black people as a whole must be bad or lazy.” And there are innumerable examples. I bet you can think of plenty.
“It is lucky for rulers that men do not think.” — Adolf Hitler
Spiritus:
Oh, be nice.
“It is lucky for rulers that men do not think.” — Adolf Hitler