Curse

Divinitys Creature · Feb 05, 2006, 03:54 PM // 15:54

Quote:

Originally Posted by PieXags

*sigh* I don't expect anyone to accept what I've said, people have a bad habit of sticking only to what they've been taught. You can "prove" such theories only because you've been taught how to do so within the given existing realm of numbers that make up our mathematics. I can guarantee that what I've said about 0.0(...)1 existing is in fact true in a certain light, just not by the means you people are looking at it. My explaination earlier about pi. We assume that it never ends, don't we? But can we really prove that it is without an end so far as we know? Can we prove that 0.9(...) even exists? If you can prove to me that infinite exists at all, then by all means try to do so and I'll say alright. Infinite is an IDEA, and only an idea. It is not a fact, you cannot show me in any sense that 0.9(...) even exists. We only know of the idea, of 0.9(...) and can only REPRESENT 0.9 recurring by labeling it as such---but we can't prove it at all, can we?

If 0.9(...) equals one, then how do we know infinite exists? Since 0.9(...) extends forever, and according to you, is also equal to one---doesn't that mean there is no infinite? Afterall, one has an end, doesn't it? One is just one, it has an end and there's no recurring at all. To say that an infinite number is also equal to a number that has a one is to say that 0.9(...) might not as well mean anything at all. Also, keep in mind that just because you can't prove something doesn't by any means mean it isn't (or can't be) true. If we go back to my example of imaginary numbers---what we really have is a simple representation of something that can't exist in any other form. Would it be so wrong to consider 0.0(...)1 an "imaginary number"? We know it can't really exist, but what it represents is something that would by all means make enough sense to represent a difference between what would then be two entirely different numbers.

It's easy to prove when you declare everything else to be impossible. Of course, the "proof" we have is based on the ACCEPTANCE that "infinite" can actually exist. Of course if it couldn't people wouldn't give a damn anyways, they'd just use it to represent something that would exist otherwise. (Which might be what we're doing right now.)

...this is a really confusing thing to read I'm sure, but do you understand what I'm saying? Lets pretend for a moment that math actually does make a bit of sense: We've got imaginary numbers, representing the square root of negative one in the simplest of senses. We know you can't have a square root of negative one, but apparently mathematicians didn't care to stop there so they decided to just tack on an i to represent what would OTHERWISE be impossible. You accept that, even though it's absolute "bs". Probably because it's in textbooks.

Now lets say for example (assuming we've accepted the possiblity of infinite) you've got 0.0(...)1---the difference between one and 0.9(...). We know that in any reasonable sense, 0.0(...)1 can't exist ---but lets be like mathematicians of old and not let that stop us. I would label 0.9(...), as something like...^1. Which would represent 0.0(...)1 away from one. Do you see what I'm getting at? So you'd have 0.9(...) = ^1, not 1. It doesn't make any sense at all and represents something we know you can't have, but that's EXACTLY what something like imaginary numbers does. Represents something that doesn't exist outside the little symbol, that doesn't exist in the real world. (Hence we call them imaginary.) In the given example, a ^ might indicate a number tacked on with infinite. You could then right the difference as 0.0^1.

...now granted, that was just an example I tossed out of my mind, it might or might not make much sense but that's alright. Think of it as another axiom, you can't prove or disprove it, it's a matter of acceptance. (Of course you could tell me that by other means it would have conflicting ideas, but then you'd have to do a little bit of rethinking, wouldn't you?) You guys are going by the book, PieXags never liked the book because it had way too much shit that didn't even exist in real life that was represented by symbols. You people accept those only because they're in those books and because people before you accepted them as truths. What made me so angry about things like that were simply that you can throw out ANY bit of information, declare it an axiom, and base a system around it. (What do you think fiction books are?) I see absolutely no problem with an alternative solution to expressing the difference between 1 and 0.9(...), because that's all ALL of it is, is just ways to express things, that may or may not exist. Math is full of both.

Ah, now I'm going to go do something a bit more useful, like get a drink. Drinks do exist, and have nothing to do with trying to prove infinite.

Edit: If it WASN'T a matter of acceptance and how you percieve things, do you really think this topic would've been created?

I go along with PieXags here.

Eclair · Feb 05, 2006, 08:30 PM // 20:30

Quote:

Originally Posted by PieXags

*sigh* I don't expect anyone to accept what I've said, people have a bad habit of sticking only to what they've been taught. You can "prove" such theories only because you've been taught how to do so within the given existing realm of numbers that make up our mathematics. I can guarantee that what I've said about 0.0(...)1 existing is in fact true in a certain light, just not by the means you people are looking at it. My explaination earlier about pi. We assume that it never ends, don't we? But can we really prove that it is without an end so far as we know? Can we prove that 0.9(...) even exists? If you can prove to me that infinite exists at all, then by all means try to do so and I'll say alright. Infinite is an IDEA, and only an idea. It is not a fact, you cannot show me in any sense that 0.9(...) even exists. We only know of the idea, of 0.9(...) and can only REPRESENT 0.9 recurring by labeling it as such---but we can't prove it at all, can we?

If 0.9(...) equals one, then how do we know infinite exists? Since 0.9(...) extends forever, and according to you, is also equal to one---doesn't that mean there is no infinite? Afterall, one has an end, doesn't it? One is just one, it has an end and there's no recurring at all. To say that an infinite number is also equal to a number that has a one is to say that 0.9(...) might not as well mean anything at all. Also, keep in mind that just because you can't prove something doesn't by any means mean it isn't (or can't be) true. If we go back to my example of imaginary numbers---what we really have is a simple representation of something that can't exist in any other form. Would it be so wrong to consider 0.0(...)1 an "imaginary number"? We know it can't really exist, but what it represents is something that would by all means make enough sense to represent a difference between what would then be two entirely different numbers.

It's easy to prove when you declare everything else to be impossible. Of course, the "proof" we have is based on the ACCEPTANCE that "infinite" can actually exist. Of course if it couldn't people wouldn't give a damn anyways, they'd just use it to represent something that would exist otherwise. (Which might be what we're doing right now.)

...this is a really confusing thing to read I'm sure, but do you understand what I'm saying? Lets pretend for a moment that math actually does make a bit of sense: We've got imaginary numbers, representing the square root of negative one in the simplest of senses. We know you can't have a square root of negative one, but apparently mathematicians didn't care to stop there so they decided to just tack on an i to represent what would OTHERWISE be impossible. You accept that, even though it's absolute "bs". Probably because it's in textbooks.

Now lets say for example (assuming we've accepted the possiblity of infinite) you've got 0.0(...)1---the difference between one and 0.9(...). We know that in any reasonable sense, 0.0(...)1 can't exist ---but lets be like mathematicians of old and not let that stop us. I would label 0.9(...), as something like...^1. Which would represent 0.0(...)1 away from one. Do you see what I'm getting at? So you'd have 0.9(...) = ^1, not 1. It doesn't make any sense at all and represents something we know you can't have, but that's EXACTLY what something like imaginary numbers does. Represents something that doesn't exist outside the little symbol, that doesn't exist in the real world. (Hence we call them imaginary.) In the given example, a ^ might indicate a number tacked on with infinite. You could then right the difference as 0.0^1.

...now granted, that was just an example I tossed out of my mind, it might or might not make much sense but that's alright. Think of it as another axiom, you can't prove or disprove it, it's a matter of acceptance. (Of course you could tell me that by other means it would have conflicting ideas, but then you'd have to do a little bit of rethinking, wouldn't you?) You guys are going by the book, PieXags never liked the book because it had way too much shit that didn't even exist in real life that was represented by symbols. You people accept those only because they're in those books and because people before you accepted them as truths. What made me so angry about things like that were simply that you can throw out ANY bit of information, declare it an axiom, and base a system around it. (What do you think fiction books are?) I see absolutely no problem with an alternative solution to expressing the difference between 1 and 0.9(...), because that's all ALL of it is, is just ways to express things, that may or may not exist. Math is full of both.

Ah, now I'm going to go do something a bit more useful, like get a drink. Drinks do exist, and have nothing to do with trying to prove infinite.

Edit: If it WASN'T a matter of acceptance and how you percieve things, do you really think this topic would've been created?

Well, as you already said, 0.999 repeating = 1 is based on the assumption that infinity exists, which is entirely true. If the notion of infinity is faulty, then 0.999 repeating may or may not = 1. However, 0.999 repeating in and of itself is a representation of a number derived from the accepted definition of infinity. It only exists because of what mathematicians deem it to be based on the concept of infinity. If infinity does not exist, then 0.999 must end somewhere, and yes, it would not be equal to 1, however, that number would not be 0.999 repeating.

Which then gets to your second point about the pointlessness of representing things that do not exist in real life through math. On the contrary, many things in real life can only be explained and modelled through the completely unintuitive and non-sensical portions of math. Imaginary numbers do have their uses, for example, in signals processing for electric systems, the study and application of electricity and magnetism, quantum mechanics, etc.

Pretty much everything in the books is useful for something, in fact, many technological advances would not be possible if some mathematician didn't crank out some axiom or another. I agree with you that there is still quite a bit of math that just seems completely pointless. However, there could very well be a real use for them sometime in the near or distant future.

Rayndeon · Feb 07, 2006, 07:10 AM // 07:10

NOTE-EDIT: This forum does not accept symbols. Thus, in Section I, where it is said "to every w-consistent recursive class K of formulae there correspond recursive class signs r, such that neither v Gen r nor Neg(v Gen r) belongs to Flg(K) (where v is the free variable of r)," w represents omega and k represents kappa. Also, when mentioning "Kurt Godel" the e represents a latin e with two dots on top. I apologize for forgetting the name of the last symbol.

I:

Quote:

Originally Posted by PieXags

*sigh* I don't expect anyone to accept what I've said, people have a bad habit of sticking only to what they've been taught. You can "prove" such theories only because you've been taught how to do so within the given existing realm of numbers that make up our mathematics. I can guarantee that what I've said about 0.0(...)1 existing is in fact true in a certain light, just not by the means you people are looking at it.

Through the process of inductive reasoning, I have concluded that you are speaking of the limitations and flaws of the axiomatic system. The eminent mathematician Kurt Godel formulated his famous "incompleteness theorem" in 1931. There were two portions to the theorem; a first and a second part although both had essentially the same informal message. The first theorem was that "to every w-consistent recursive class K of formulae there correspond recursive class signs r, such that neither v Gen r nor Neg(v Gen r) belongs to Flg(K) (where v is the free variable of r)," whilst the second theorem was that "for any formal theory T including basic arithmetical truths and also certain truths about formal provability, T includes a statement of its own consistency if and only if T is inconsistent." The incompleteness theorem, both portions, approximately stated that within any formal axiomatic system, such as mathematics, there will always exist situations in which things cannot be proved nor disproved; there are items which cannot be solved by an procedure or method within that system. This of course set limits on mathematics, which sent the community into an uproar. One must rememeber: this time was the death of the classical absolutist Newtonian scientific determinism. However, Godel's theorem was appreciated. Many mathematicians have judged the field on the basis of the constituent axioms, although some are proponents of the quasi-Neoinstitutionist reverse mathematics.

Your argument that "you can 'prove' such theories only because you've been taught how to do so within the given existing realm of numbers that make up our mathematics" is a paradigm of mathematical solipsistic nihilism, infusing mathematics and philosophy under an unholy logical pretense; proudly proclaiming the implications of "Cogito ergo sum" and other ontologically Cartesian philosophies without understanding the fundamental core of mathematics and holistically, of the entire world. Anything can be viewed in a said manner according to a said light; that is known as relativism. Many a mathematical Platonist would disagree with your sentiment on mathematical relativism. You are essentially implicating that due to relativism, mathematicians should adopt a mathematically nihilistic solipsist stance; one essentially becomes mathematically fatalistic. The flaws of mathematical fatalistic relativism are apparent; if no one system is no greater than the other, then how precisely is one justified in the aforementioned argument against the validity of the contented mathematical statement? Adopting a fatalistic relativist view is not advantageous nor does it support your argument anymore than it would support mine. However, you are correct in your assertion that axiomatic systems are dependent on their constituent axioms; however, apparently, mathematics and philosophy were designed to keep away from each other. The constituent axioms must satisfy in themselves the definition of axioms: in that they must be as painfully obvious as possible. In certain instances, it is not so painfully obvious, such as is the case with Euclid's 5th Postulate, concerning Parallelism. However, always in axiomatic systems are the constituent axioms few in number.

II:

Quote:

Originally Posted by PieXags

My explaination earlier about pi. We assume that it never ends, don't we? But can we really prove that it is without an end so far as we know? Can we prove that 0.9(...) even exists? If you can prove to me that infinite exists at all, then by all means try to do so and I'll say alright. Infinite is an IDEA, and only an idea. It is not a fact, you cannot show me in any sense that 0.9(...) even exists. We only know of the idea, of 0.9(...) and can only REPRESENT 0.9 recurring by labeling it as such---but we can't prove it at all, can we?

Pi has been proven to be irrational. Irrational numbers are interminably non-repeating. Ergo, pi has already been proven to be without end. It is interesting to note that the implications of your statement almost seems like the shock the Greeks endured upon discovering the incommensurability of the square root of 2. 0.9(...) has been proven to exist; it is simply an infinite convergent geometric series. The concept of infinity of itself is an axiom. You are again moving back to a generally fatalistically solipsistic nihilistic view, along with ontological Cartesian sentiments. Apparently, even tautologies are no longer valid. What you are confusing here is existence in terms of mathematics and existence in terms of reality. Mathematics is for the most part Platonic whilst reality seems to fashion after it in a rightfully Aristotelian manner, while still reflecting the Platonic roots. In fact, "Platonic" mathematics may even be reality; although that is to be contended. 0.9(...) is mathematically existent and thus, in an Aristotelian fashion, it exists in reality.

III:

Quote:

Originally Posted by PieXags

If 0.9(...) equals one, then how do we know infinite exists? Since 0.9(...) extends forever, and according to you, is also equal to one---doesn't that mean there is no infinite? Afterall, one has an end, doesn't it? One is just one, it has an end and there's no recurring at all. To say that an infinite number is also equal to a number that has a one is to say that 0.9(...) might not as well mean anything at all. Also, keep in mind that just because you can't prove something doesn't by any means mean it isn't (or can't be) true.

The argument is fallacious. All rational numbers are infinitely geometrically convergent. One is succeeded by an infinity of zeroes. In fact, all numbers are infinite decimal expansions. However, you are correct in your assertion that "just because you can't prove something doesn't by any means mean it isn't (or can't be) true," as the incompleteness theorem implies. It is effectively the boundary of axiomatic systems, in this case mathematics, although that boundary is not frequently tested. However, according the current model of mathematics, this is not the case.

IV:

Quote:

Originally Posted by PieXags

It's easy to prove when you declare everything else to be impossible. Of course, the "proof" we have is based on the ACCEPTANCE that "infinite" can actually exist. Of course if it couldn't people wouldn't give a damn anyways, they'd just use it to represent something that would exist otherwise. (Which might be what we're doing right now.)

Reductio ad absurdum is a valid form of argumentation. Proofs are based on assumptions. Even "proofs" are based on assumptions. In this case, axioms are the assumptions. However, as discussed earlier, axioms are designed to be "painfully obvious" and kept to a minimum in number. One cannot prove anything without assumptions, and to stay in such a state would essentially leave you in a pure solipsistic nihilisitc shock. However, the acceptance of the existence of infinity is essential in maintaining the integrity of mathematics and there is evidence pointing towards its existence, such as the work of Georg Cantor.

V:

Quote:

Originally Posted by PieXags

...this is a really confusing thing to read I'm sure, but do you understand what I'm saying? Lets pretend for a moment that math actually does make a bit of sense: We've got imaginary numbers, representing the square root of negative one in the simplest of senses. We know you can't have a square root of negative one, but apparently mathematicians didn't care to stop there so they decided to just tack on an i to represent what would OTHERWISE be impossible. You accept that, even though it's absolute "bs". Probably because it's in textbooks.

Now lets say for example (assuming we've accepted the possiblity of infinite) you've got 0.0(...)1---the difference between one and 0.9(...). We know that in any reasonable sense, 0.0(...)1 can't exist ---but lets be like mathematicians of old and not let that stop us. I would label 0.9(...), as something like...^1. Which would represent 0.0(...)1 away from one. Do you see what I'm getting at? So you'd have 0.9(...) = ^1, not 1. It doesn't make any sense at all and represents something we know you can't have, but that's EXACTLY what something like imaginary numbers does. Represents something that doesn't exist outside the little symbol, that doesn't exist in the real world. (Hence we call them imaginary.) In the given example, a ^ might indicate a number tacked on with infinite. You could then right the difference as 0.0^1.

I believe that I understand you. However, imaginary numbers have been used, as Eclair helpfully pointed out, in electronic fields and other scientific areas such as quantum mechanics and in doing Fourier transforms. Also, imaginary numbers were proved if I recall correctly. They are tautologies in themselves: idealistically ideal, and Platonically Platonic.

However, the main problem with your proposition is that the idea of the existence of 0.0(...)1 is contradictory to evidence in mathematics today; such as the axiom of infinity and the intermediate value theorem. 0.0(...)1 cannot exist. Imaginary numbers however violate nothing.

VI:

Quote:

Originally Posted by PieXags

...now granted, that was just an example I tossed out of my mind, it might or might not make much sense but that's alright. Think of it as another axiom, you can't prove or disprove it, it's a matter of acceptance. (Of course you could tell me that by other means it would have conflicting ideas, but then you'd have to do a little bit of rethinking, wouldn't you?) You guys are going by the book, PieXags never liked the book because it had way too much shit that didn't even exist in real life that was represented by symbols. You people accept those only because they're in those books and because people before you accepted them as truths. What made me so angry about things like that were simply that you can throw out ANY bit of information, declare it an axiom, and base a system around it. (What do you think fiction books are?) I see absolutely no problem with an alternative solution to expressing the difference between 1 and 0.9(...), because that's all ALL of it is, is just ways to express things, that may or may not exist. Math is full of both.

However, it can be disproved and no, it is not an axiom. You cannot assert thus unless you declare infinity to be false, which seems to be against the strong evidence for infinity. Your comment on "rethinking" is fallacious; the definition of infinity would not be altered; instead the assertion of the existence of 0.0(...)1 would be declared false, as of which it already has been due to reductio ad absurdum.

I find nothing wrong with the rigor and method of mathematics. If anything, it is the most objective thing that has ever existed. Even relativists will admit this. To dismiss mathematics is to essentially dismiss a massive portion of human accomplishments in theoretical and practical fields. If it were not for mathematics, the world would be a much more different place.

You make a fallacious argument by stating that "you people accept those only because they're in those books and because people before you accepted them as truths." The only things that are accepted are axioms, and the current axioms are universally accepted by the mathematical community. Everything else is proved. Intuition, acceptance, and like are frowned upon in mathematics and it is the last sentiment you want to have to even contemplate succeeding in mathematics.

Also, on the comment that one can come up with "ANY bit of information, declare it an axiom, and base a system around it." This is wrong. You cannot come up with "ANY" bit of information. Axioms are to be free of contradictions, and "painfully obvious." They must be fundamental truths. After that, all deductions are based on these axioms. The difference between fiction books and mathematics is great. Mathematics is a rigorous field that heavily demands of syntax, semantics, etc. One cannot produce determinably nonsensical axioms.

Your argument that there is "absolutely no problem with an alternative solution to expressing the difference between 1 and 0.9(...), because that's all ALL of it is, is just ways to express things, that may or may not exist. Math is full of both," is contendable. Mathematically, 0.0(...)1 cannot exist. You are committing a violation by doing this. It will not work. Unless the axiom of infinity will change, along with the entire field of mathematics, 0.0(...)1 will remain non-existent.

VII:

Quote:

Originally Posted by PieXags

Ah, now I'm going to go do something a bit more useful, like get a drink. Drinks do exist, and have nothing to do with trying to prove infinite.

Edit: If it WASN'T a matter of acceptance and how you percieve things, do you really think this topic would've been created?

There is a difference between existence in mathematics and existence in reality. Interestingly though, with an eerie regularity, mathematics has proven itself in reality in a fittingly Platonic-Aristotelian fashion.

Also, on the thread, the creator seemed to be inquiring about the validity of 0.999(...) = 1 This seemed to be a case of proof. Of course, this is a point of contention that isn't really worth arguing about between us.

VIII:

Quote:

Originally Posted by Mr.Style

I agree that for all practical purposes in completing a math problem etc. that 0.999...=1. But, you have to remember that we are dealing with mathematics, and in mathematics, rather than dealing with measured quantities, we are dealing with exact numbers. Therefore, it can be concluded that, in mathematics, 0.999... does not equal to one, simply because it is an exact number that we know is less than one. In order to demonstrate this we may take the two exact numbers, one and one, to get zero, because 1-1=0, and all numbers in the equation are exact, it is safe to say that in the equation 1-0.999... which contains all exact numbers that 1-0.999...=0.000...001 and not 0. This shows that although the number 0.999... is determined to be 1 for all practical purposes it in fact does not equal to one.

Mr. Style, 0.999(...) is 1, as shown by several proofs. It is a convergent infinite geometric series, converging to 1.

Also:

Let x = 0.9999999999* (assignment of a variable)
x = 0.9999999999* (given)
10x = 9.99999999999* (Multiplication Property of Equality)
10x-x = 9.9999999999* - x (Subtraction Property of Equality)
10x-x = 9.9999999999* - 0.9999999999* (Substitution Property of Equality)
9x = 9 (Subtraction Property of Equality)
x = 1 (Division Property of Equality)
0.99999999999* = 1 (Substitution Property of Equality)

Also:

1/9 = 0.111111111.....
2/9 = 0.222222222.....
3/9 = 0.333333333.....
4/9 = 0.444444444.....
5/9 = 0.555555555.....
6/9 = 0.666666666.....
7/9 = 0.777777777.....
8/9 = 0.888888888.....
9/9 = 0.999999999..... (by inductive reasoning)

I found an interesting website discussing this: http://qntm.org/pointnine He makes strong, valid arguments.

1-0.999(...) must equal zero; otherwise it is in violation of the intermediate value theorem and thus by reductio ad absurdum, it must equal 1. Also, to say that 1-0.999(...) = 0.0(...)1 is in violation of mathematics, such as the intermediate value theorem and the axiom of infinity. By deductive reasoning, it is 1.

IX:

Quote:

Originally Posted by Mathias Deathwater

The fact that depending on how you look at it .999... may or may not equal 1 just proves that we haven't really "figured out" math completely yet, and we maybe never will. If you look at the problem as a geometric sequence, then yes, .999... does equal 1. But if you take the idea that x-x=0, then no, .999 does not equal 1, because you would end up with .000...9 (gah, i really wish I had that line you put over partially repeating decimals, just imagine it's there)

You cannot use infinity in arithmetic operation, strictly speaking. It rarely happens when one is allowed to and generally there are special circumstances. Also, 0.0(...)9 is again in violation of the intermediate value theorem and the axiom of infinity.

Also, mathematics is as close to truth as possible. It attempts to be as objective as possible, assuming fundamental truths and deductively reasoning from there. It is certainly the most immortal field we have at least. Aeschylus may be forgotten in time, but the mathematical statements of Archimedes and others will remain immortal, or at least they should.

PieXags · Feb 07, 2006, 08:20 AM // 08:20

I get the feeling you're assuming I expect my theory and idea to go along with todays math system? No, I'm well aware that to accept MY idea or more importantly---the possibility of it, would be to change your entire outlook on mathematics as a whole. Which is why I stress that it is a matter of acceptance---to say that something is a matter of acceptance is to emphasize that it's only one way to view something. I understand entirely how in our current system for mathematics, 0.9(...) = 1, but I also accept the possibility that under certain circumstances and rationalities 0.9(...) can not equal one. Mathematics created itself. Humans might be the only creatures who ran with the idea, but mathematics as a whole is just part of the understanding of everything that is. Where would we be if we didn't believe in amounts? But obviously someone didn't just sit down one day and say "Hmm...I want to acknowledge that there's more than...one? One of something? Did I just say a number?" Math is necessary and in a certain light yes, as close to the truth as possible. But only up to a certain point. Once you move out of the math relative to life, you end up with things that are so heavily strung up with axioms that it becomes, as I've said, only one way of looking at it. As the idea I've proposed of the difference between one and 0.9(...) repeating, proves, by being an obvious example of another solution to 0.9(...) and one.

The difference between whether or not you believe 0.9(...) = 1, is entirely whether or not you accept possibilities outside our current (limited) system of mathematics. Most people, should be able to accept the possibility of both as I have.

Old Dood · Feb 07, 2006, 12:42 PM // 12:42

PieXags! You should change your rank title to Brainiac 5 from the old DC Action comics.

He was one smart dude!

unienaule · Feb 07, 2006, 05:23 PM // 17:23

To sum up what Rayndeon said: You are using fatalistic relativism. You can't do math that way, because you will keep saying there are other possibilities of solutions to problems and never get anything done.

Divinitys Creature · Feb 07, 2006, 07:44 PM // 19:44

I'm reading these last few posts with interest.

Rayndeon · Feb 08, 2006, 08:14 AM // 08:14

I:

Quote:

Originally Posted by PieXags

There we go, Deathwater! That's how people need to start thinking. So far we've seen mostly "proof" using only what we know about math, but what they fail to understand is that math is so many different things in so many different senses most of it IS how you accept things.

Your statement that "math is so many different things in so many different senses" is fallacious. Mathematics is highly concerned with its own integrity; that is the reason why it is so rigorous and concentrates on syntax, semantics, etc. Your statement is almost ironic: this is what mathematics is famous for, its rigor, which leads to its own integrity.

Also, your argument that "most of it [math] IS how you accept things" contains flaws. It is again, opposite of your statement. Mathematics assumes as little as possible, in fact, the only things it ever assumes are axioms. Nothing else is assumed, intuition and skipping rigor is condemned by the mathematical community; everything is proved. Again, your statement is almost ironic: this is also what mathematics is famous for, its use of deductive logic in which it only assumes fundamental truths and effectively comes to conclusions.

II:

Quote:

Originally Posted by PieXags

If you could prove, without flaw, that one equals the other there would be absolutely no discussion over it.

Considering the rigor in mathematics, I cannot see how this is detrimental. You said yourself, if you could "prove without flaw." If there are no flaws in the argument, than it becomes irrefutable within the current axioms. Are we going to argue with "1 + 1 = 2?" The only discussion of which will come from such proofs are generally shortening the proof, polishing it, discussing its implications and how it can be used in other fields, and also, in philosophy, although philosophy and mathematics were not really supposed to mix. The statement "1 + 1 = 2" will most likely be argued in terms of epistemology, ontology, model mathematics, etc. This is similar to what you are doing now; challenging the framework of mathematics.

However, the problem is that you are advocating fatalistic relativism, with healthy bits of solipsistic nihilism. Such a combination truly leads to the breakdown of human progress. By your argument, we can't prove anything; we always have to assume. You are trying to argue that one must be able to prove something; assumption is not enough. However, the flaw is that everything requires at even its primitive levels an assumption, a fundamental truth. Otherwise, we can't really say anything or do anything at all. You are suggesting the stagnation of human progress. If you don't realize that, look back at the logical implications of your argument.

III:

Quote:

Originally Posted by PieXags

The fact that we can't prove infinite (we can only represent it) is enough in itself to throw the ENTIRE thing far beyond any sort of "proof" we have right now, because you have to accept many, many different things as true before you can accept the "proof", which is in fact only one sense of looking at it.

What do you mean by "we can't prove infinite [sic] (we can only represent it)?" Do you mean that we require a symbol to represent infinity? If that is the argument, that is completely fallacious. Infinity is a concept, not an actual value. Also, infinity is an axiom; we cannot prove it although we can find evidence hinting towards it, as Georg Cantor did. However, by that logic, again, we can't prove anything in the entire human experience. You are, as I said earlier, implying the stagnation of the human species. Everything requires an assumption at its basest level.

Your comment that "you have to accept many... different things as true before you can accept the 'proof', which is in fact only one sense of looking at it" is also fallacious. As I said earlier, if there was ever a field that assumed the least, mathematics is it. The only assumptions are the axioms; "many" is a gross hyperbole.

You also refer back to fatalistic relativism again, saying that it is only "one sense of looking at it." Well, as said earlier, it is true that axiomatic systems are dependent on their constituent axioms. However, as repeatedly mentioned everything uses assumptions. Also, such a mode of relativism again leads to the stagnation of the human species

IV:

Quote:

Originally Posted by PieXags

Math is just a great big bunch of things that exist only within itself. You can't even use half of it to figure out real life situations, only more math! This gives people like PieXags every right to run away from "inside the box" thinking, and create new possiblities and representations for people to look at.

Au contraire. If mathematics existed only within itself, it certainly would have proven itself with alarming regularity in a typical Platonic-Aristotelian fashion. Also, mathematics can be seen everywhere: economists use it in determing patterns, the same of statisticians, physicist live and die by mathematics, debaters rely on mathematical logic, engineers use geometry and calculus, and... It keeps going. Mathematics is useful in the world. Even the most tediously abstract and pure branches have applications; this is one of the most astonishing things in mathematics. Without it, the world would certainly be a different place.

Also, on your comment about "'inside the box' thinking" and "new possibilities and representations for people to look at," please refer back to my statements on fatalistic relativism, assumptions, and generally, my entire post from I-IV.

V:

Quote:

Originally Posted by PieXags

I get the feeling you're assuming I expect my theory and idea to go along with todays math system? No, I'm well aware that to accept MY idea or more importantly---the possibility of it, would be to change your entire outlook on mathematics as a whole. Which is why I stress that it is a matter of acceptance---to say that something is a matter of acceptance is to emphasize that it's only one way to view something. I understand entirely how in our current system for mathematics, 0.9(...) = 1, but I also accept the possibility that under certain circumstances and rationalities 0.9(...) can not equal one.

However, if there are equally valid rationalities, how precisely would anything maintain integrity and live on? You are prophesizing the stagnation of humanity. Acceptance only extends to axioms. If you were to change the notion of infinity, you would quite really change mathematics entirely. Analysis for example would be non-existent if the axiom of infinity was simply not there.

VI:

Quote:

Originally Posted by PieXags

Mathematics created itself. Humans might be the only creatures who ran with the idea, but mathematics as a whole is just part of the understanding of everything that is. Where would we be if we didn't believe in amounts? But obviously someone didn't just sit down one day and say "Hmm...I want to acknowledge that there's more than...one? One of something? Did I just say a number?"

Historically, the idea of numbers has been around for a long time, perhaps as long as humans themselves. Your point of "mathematics created itself" is contestable; in fact, it is one of key philosophical debates in mathematics.

VII:

Quote:

Originally Posted by PieXags

Math is necessary and in a certain light yes, as close to the truth as possible. But only up to a certain point. Once you move out of the math relative to life, you end up with things that are so heavily strung up with axioms that it becomes, as I've said, only one way of looking at it. As the idea I've proposed of the difference between one and 0.9(...) repeating, proves, by being an obvious example of another solution to 0.9(...) and one.

PieXags, do you realize that those incredibly abstract fields are the very backbone of mathematics? All of those simple things in mathematics are derived from those abstract concepts.

Also, if you simply consider it "one way to look at it" you are implying that any field is just as good as the other; this does not seem to be the case when axiomatic systems based on fundamentally wrong concepts come crashing to their knees. On another issue, it's interesting to note how much you sound like O'Brien during Part III of Orwell's 1984.

VIII:

Quote:

Originally Posted by PieXags

The difference between whether or not you believe 0.9(...) = 1, is entirely whether or not you accept possibilities outside our current (limited) system of mathematics. Most people, should be able to accept the possibility of both as I have.

Belief is something that is forbidden in mathematics. The only belief allowed is in axioms; and in fact, as I have stated, EVERYTHING requires assumptions, these fundamental truths. Without them, human thought would have perished long ago.

PieXags · Feb 08, 2006, 12:09 PM // 12:09

Heheh, and now you know why PieXags decided to stop math and move on to other things when he did. Too much thinking that is by my own measures, pretty illogical in my opinion. You're right, everything is based off assumptions---but mathematics is the only one I'm dealing with right now that tries to take an assumption and turn it into fact by running with the idea. Of course PieXags executes such "fatalistic relativism"---that's more useful to me than is ANYTHING that tries to ignore other possibilities in such a manner. To understand the possibilities is very important to me, and I'll not be blind to them for the sake of pleasing textbooks and mathematicians of old. And while normally yes, thinking in such a way as I have would result in tearing apart the entire foundation of mathematics, you're right you never would get anything done...

Except find other possibilities for things like this. This thread here, are we trying to get anything done? No, we're not. We're discussing something that is so far as we'll take it, opinionated. It all depends on whether you agree with certain ideas, or whether you don't. Either way's fine. We're discussing a number we can't prove in the first place, and I see absolutely no point in sticking to textbook theories because an engineer in Missouri won't get anything done if he thinks like I have. The ONLY people in this world limited by the set rules and concepts involved in mathematics is the people who need to use it for other purposes other than exploring it. And being that I, and several other people here on these forums, are not those people it is ENTIRELY acceptable to think in such a way that could very well be another possibility, or way of representing one. Personally I don't think it's illogical at all to have 0.0(...)1 repeating, despite what our math system thinks. I'd explain it further with examples, but it would only be referred to as "fatalistic relativism" and in a sense, it is. But for our purposes of discussing 0.9(...) and 1, that's perfectly alright, regardless as to whether or not every existing concept agrees with it.

And for the record, belief is not forbidden in anything. And being that it's entirely your belief as to whether or not you accept certain mathematical concepts yes, belief applies even in mathematics. Assumption is just a belief as a whole, and math is littered with them.

EchoSex · Feb 08, 2006, 12:18 PM // 12:18

My theory? VERY, very simple. The reason .999999999 equals 1 is because...

"One." Is a HELL of a lot easier than saying "Point nine, nine, nine, nine, nine, nine, nine, nine...."

Let me give you an example of it's usage in everyday situations:

Man Number 1: "Say, how big is your pig?"
Man Number 2: "Why, my pig is three point nine, nine, nine, nine, nine nine-"
Man number 1: "Just say four, please."

PieXags · Feb 08, 2006, 12:44 PM // 12:44

Haha, c'mon Echo, you could just say:

"My pig is one tacked onto an infinite amount of 0's behind a decimal, away from four."

Haha, not quite there.

EchoSex · Feb 08, 2006, 12:47 PM // 12:47

My pig is bigger than Pie's.

Anyways, that's a factor a lot of people don't really seem to realize: Simplicity. Sometimes things are actually done to make things easier. I'm sure you can find stuff like that in all these mathmatical theories.. Even if math was designed to be anything BUT simplified.

PieXags · Feb 08, 2006, 12:48 PM // 12:48

If it were about simplicity, they would've dropped it and said "We can't prove infinite, want to get a donut?" and be done with it.

EchoSex · Feb 08, 2006, 12:49 PM // 12:49

Eh, true. Those people broke off during the stone age when they couldn't make a block shaped wheel roll. Know who those people are? ...My people - the lazififions.

Maxiemonster · Feb 08, 2006, 12:56 PM // 12:56

It's not called endlessly for nothing, it doesn't have an end.

xxSilhouette · Feb 08, 2006, 01:40 PM // 13:40

How about....we all go to english now!!

Tocc · Feb 08, 2006, 02:05 PM // 14:05

I remember my first year of mathematics at the poly, there was a funny demonstration based on the 0.999...=1 statement.
The final result was that 1=2=3=4=.... (no other mathematic rule was breaked).

EchoSex · Feb 08, 2006, 02:07 PM // 14:07

Quote:

Originally Posted by xxSilhouette

How about....we all go to english now!!

Yes, let's. 'cause it's "Broken" not "breaked" :P

BTW I've heard that demo before - I'm almost certian I've seen it..

Rayndeon · Feb 09, 2006, 10:23 PM // 22:23

I:

Quote:

Originally Posted by PieXags

Heheh, and now you know why PieXags decided to stop math and move on to other things when he did. Too much thinking that is by my own measures, pretty illogical in my opinion.

You're right, everything is based off assumptions---but mathematics is the only one I'm dealing with right now that tries to take an assumption and turn it into fact by running with the idea. Of course PieXags executes such "fatalistic relativism"---that's more useful to me than is ANYTHING that tries to ignore other possibilities in such a manner. To understand the possibilities is very important to me, and I'll not be blind to them for the sake of pleasing textbooks and mathematicians of old. And while normally yes, thinking in such a way as I have would result in tearing apart the entire foundation of mathematics, you're right you never would get anything done...

Except find other possibilities for things like this.

The "over-thinking" is required for rigor as to maintain integrity and objectivity.

Your argument that "mathematics is the only one I'm dealing with right now that tries to take an assumption and turn it into fact by running with the idea" is fallacious. First of all, this happens in any field. If anything, the only thing absolutely certain may be the fact that you exist. Everything else is an assumption.

"Fatalistic relativism" isn't useful. It won't do humanity benefit. We wouldn't get anything done.

We aren't ignoring other possibilities. Other fields with well-defined, sensible axioms are also used, albeit rarely in comparision to the standard system, in mathematics as well. When you being to accept any field as valid as an other, is where the problem begins. However, the problem with many of these new fields is their consistency and integrity.

Also, I am not trying to "please" textbooks. This ad hominem is not at all helpful.

II:

Quote:

Originally Posted by PieXags

This thread here, are we trying to get anything done? No, we're not. We're discussing something that is so far as we'll take it, opinionated. It all depends on whether you agree with certain ideas, or whether you don't. Either way's fine. We're discussing a number we can't prove in the first place, and I see absolutely no point in sticking to textbook theories because an engineer in Missouri won't get anything done if he thinks like I have. The ONLY people in this world limited by the set rules and concepts involved in mathematics is the people who need to use it for other purposes other than exploring it. And being that I, and several other people here on these forums, are not those people it is ENTIRELY acceptable to think in such a way that could very well be another possibility, or way of representing one. Personally I don't think it's illogical at all to have 0.0(...)1 repeating, despite what our math system thinks. I'd explain it further with examples, but it would only be referred to as "fatalistic relativism" and in a sense, it is. But for our purposes of discussing 0.9(...) and 1, that's perfectly alright, regardless as to whether or not every existing concept agrees with it.

The purpose of this thread was to find out if, and how, 0.9 recurring was equal to 1 in the standard system.

And as I have said, 0.9 recurring is a infinite geometric series of Sigma x=1 Lim-> infinity 9 x 10^(-x) We can also show its existence by multiplying 0.3 recurring by 3. It does exist.

Also, you are not observing rigor. You cannot simply invent a new system arbitarily. It must be well-formed and defined. Could you please state the axioms of this new system? I would like it to be noted that using intuition and philosophy in mathematics is discouraged; mathematics is pure logic alone.

I would like to see your examples. I would also like you to create a new axiomatic system that is well-formed and defined, observes proper rigor, and uses these new definitions.

Also, the purpose of this thread was to discuss the validity of 0.9 recurring being equal to 1 in the standard system.

PieXags · Feb 09, 2006, 11:21 PM // 23:21

Heh, first of all nowhere in the original post of this thread does it state that we're dealing entirely within the standard system of mathematics. So don't try to tell me that simply because PieXags enjoys new means of thinking that it doesn't work or is incorrect, when it's in this thread my options are entirely valid---whether they work within our standard system or not.

I could write on and on about how yes, my presented idea is valid and I could sit here and take my time to form a system around it. But for someone you cares so much about logic, you're not being very logical. Why would I sit here and take up ANY of my time thinking of examples and more ways to represent my idea just to please you, because you'll never dare to move from the standard system we've put in place. Thinking in such "dangerous" ways that contrast existing systems have in the past benefited mankind in so many ways I couldn't begin to list them all, so please refrain from trying to tell me such ways of thinking are bad. For the purpose of this thread, they are entirely valid. I'm free to move from the standard system all I want, to represent new ideas and theories so much as I can. Some people can accept them, and some people can't. Personally I think something much MUCH more pointless than a relative sense of thinking is spending time trying to convince someone it's not dangerous at all, so I won't bother. Trying to get you to accept the possibility is like trying to convert someone from their way of life to a new one after 60 years, it's not going to happen and it's a waste of time to try, even if if the new cultures just as acceptable as the old. So I'll consider this a finished matter, and both possibilities acceptable.