>If you really are interested in the structure of proofs in
>pure logic, I reccommend G.H. Hardy's little book "A
>Mathematician's Apology." He wrote it for the general
>audience, but he also walks through a few of the classic
>proofs of mathematics.
>

This sounds really interesting.

>>>>First, Descartes must prove that he himself exists, and he
>>>>does so through his famous statement, "I think therefore I
>>>>am", as it is a clear and distinct statement that is self
>>>>evident, no proof beyond the act of saying it is required.
>>>
>>>Well, that's a question of epistemology, not a tenet of
>>>logic.
>>>
>>>>The reasoning behind this statement is as follows; if I am
>>>>able to percieve my own existence from within, I must
>>>>therefore exist.
>>>
>>>And that's not reasoning, it's just repetition.
>>>
>>
>>Hmm, I disagree with you. Please explain how it is
>>repetition and not reasoning.
>
>I disagree right back. Read the two statements again. You
>didn't really state anything new, you just took the original
>statement and stretched it out. You inserted more words,
>the words don't serve a purpose, except to satiate those
>people who judge the value of a statement by the number of
>words. Some words are more valuable than others. (and I
>admit this in full knowledge of the fact that I've been
>known to write some very long posts)

I wasn't attempting to state anything new. Hmm, I find your analysis of what I wrote very strange, as I don't see how I could have given that impression.

Stating this;
""I think therefore I
>>>>am", as it is a clear and distinct statement that is self
>>>>evident, no proof beyond the act of saying it is required."

and;

"The reasoning behind this statement is as follows; if I am
>>>>able to percieve my own existence from within, I must
>>>>therefore exist."

This is not repetition. The second statement is the explanation of the first statement. To me, the first statement alone does not say; "if i am able to percieve my own existence from WITHIN, i must therefore exist", the first statement is a reference to "clear and distinct" which is a concept created by descartes to descibe statements which are statemenuts that you can trust, that is why i said they are self evident. I didn't even make any reference to existence in the first statement. I merely provided the process, and in the second statement, I provided the point of thought or study, which is existence.

I think your analysis is incorrect.

>
>>>>Next, Descartes will discuss the relationship between
>>>>physical objects, and himself. He is not certain of physical
>>>>objects in his environment, but of his relationship to them.
>>>
>>>And that much is very commendable. Descartes's
>>>relationalism is probably his greatest achievement, though
>>>it is still disputed today, in some quarters (I sit solidly
>>>on his side, though).
>>>
>>>>In an experiment, Descartes takes a piece of wax, and allows
>>>>it to melt. He then, observes what has changed, and what has
>>>>not changed. Through his experiment, Descartes arrives at
>>>>the conclusion that the wax, may melt in many different
>>>>ways, or in inifinite ways. The state of change between a
>>>>solid piece of wax and a melted piece of wax is infinite. A
>>>>simple example of the infinite are numbers.
>>>
>>>And here's where the age of the argument really begins to
>>>drag it down. In his time, it was natural to assume that
>>>the number, N, of ways in which a block of wax could melt
>>>is infinite. In fact, it is now known physically that N is
>>>not infinite! It is an enormous number, but finite
>>>nonetheless. This fact is a simple result of modern quantum
>>>mechanics, and it underlies all of our understanding of
>>>statistical thermodynamics. This is not to say that we
>>>"know" N is finite. There are likely some errors in the
>>>present formulation of quantum theory. Nonetheless, it is
>>>far more reasonable for us to assume N is finite.
>>
>>You state that is it reasonable to assume that N is finite.
>
>In fact more reasonable than assuming it is infinite! We
>must assume something somewhere.
>
>>Also, if you do not like the wax block experiment, you could
>>just think of numbers, and counting from 1, to infinity.
>>Correct? Are you stating that even that is finite?
>
>Haha! No, I'm not stating that the set of positive integers
>is finite. But note that Descartes did not cite the
>positive integers as his example. He used wax for a reason.
> It's easy to argue that wax "exists," that is, that it is
>an artifact of the larger universe, and not of our
>imaginations; that it retains its reality regardless of
>whether people are considering its existence. The same is
>not true of the positive integers. Despite their name, the
>real numbers are an invention of the mathematician's mind.
>This might be hard to accept, since the real numbers are so
>familiar. But consider other number systems: the complex
>numbers, the quaternions, the octonions, the supernumbers.
>We have invented all sorts of number systems over the
>centuries. You want a number system, I'll give you one
>right now. The "integers mod 2." This consists of the set
>{0,1}, and the arithmetic DEFINED by:
>
>0+0=0
>0+1=1
>1+0=1
>1+1=0 (notice that one)
>
>0*0=0
>0*1=0
>1*0=0
>1*1=1
>
>This number system is just as valid, philosophically, as the
>real number field. In fact, in recent decades, it has
>become exceedingly important, for an obvious reason. You
>will note, also, that it does not involve an infinite number
>of elements. There are only two.

I see your point. I am curious, why is:
">1+1=0 (notice that one) " philosophically valid?


>
>>>But Descartes's argument is not about wax. It's about the
>>>nature of infinity in the physical world. In fact many of
>>>the quantities of the physical world, once assumed to be
>>>infinite, have been rendered finite by the modern quantum
>>>viewpoint. The extent of this is not yet completely clear.
>>>Many quantum systems admit finite bases (that is, a finite
>>>number of "states"). Many more admit "countable" bases
>>>(that's a designation that might be considered "in between"
>>>the standard definitions of finite and infinite), and some,
>>>truly, uncountably infinite bases. In fact, most systems
>>>admit more than one of the above, simultaneously. That is,
>>>the question of how many states a system might take, depends
>>>on how the observer chooses to differentiate them. And in
>>>fact, in the most mathematically rigorous treatments of
>>>quantum theory yet made, uncountably infinite bases are
>>>avoided like the plague. The problem is that they usually
>>>lead to logical inconsistencies, a sign that the "methods of
>>>distinction" which lead to infinite bases are generally
>>>unnatural, and amount to the philosopher forcing too much of
>>>his own opinions upon the system.
>>>
>>
>>You are using quantum theory to disprove infinity, ofcourse,
>>within the bounds of quantum theory, which are clearly
>>"finite", just like any study, you are going to find finite
>>solutions. That is exactly the point Descartes is attempting
>>to make. Human beings are incapable of the infinite. We are
>>not discussing quantum theory here, we are discussing the
>>infinite, the infinite is not subjected to quantum theory as
>>it is just that, the infinite, and therefore you cannot
>>possibly disprove it's existence. Also, if I understand you
>>correctly, you mentioned that what was considered infinite
>>is now proven to be finite. At one point in our world's
>>history, we thought the world was flat. Do you see where I
>>am going with this? Also, within the context of quantum
>>theory, I see that there has been a great deal of assumption
>>that is now beginning to be clarified. Your counter argument
>>is irrelevant to this discussion, as it is subjected to a
>>scientifical field of study. How can science possibly be
>>infinite? It is the study of the finite is it not?
>>Therefore, this only helps prove Descartes argument, that we
>>are incapable of the infinite.
>
>So what is it you are referring to when you describe the
>infinite? Descartes referred to wax, clearly a system bound
>by the laws of physics. You want to dismiss the scientific
>approach, apparently because it no longer supports your
>"worldview." It's quite disingenuous to cite the scientific
>viewpoint when it is convenient, but to summarily dismiss it
>when it becomes inconvenient.
>
>So you use numbers instead. My contention is that number
>systems are human inventions. If you must cite numbers to
>argue that the infinite "exists", you are only arguing that
>the infinite exists in the mind of a mathematician, which
>seems to contradict your original thesis (that we are
>incapable of the infinite).
>
>See, I'm saying you've got it all backwards! The physical
>world is finite. The imagination is infinite.

I understand much better now. Thanks.

>
>>>I doubt I'm making any sense. My point is that it now
>>>appears unlikely that any physical phenomena permit truly
>>>infinite variation. It seems more likely that the concept
>>>of infinity is an invention of man, and is forced upon the
>>>physical world now far too quickly.
>>>
>>
>>You are making sense. You mentioned "physical phenomena", I
>>am not speaking about physical phenomena, I am speaking
>>about God.
>
>Oh, okay. What the hell is that?
>
>>Is God physical?
>
>You tell me. I don't even know what you're talking about.

If God exists? is it a physical phenomena? how do you define physical?


>
>>I do not think infinity is an
>>invention of man, as it is easy to disprove that by simply
>>counting from 1 to infinity.
>
>Hmmm . . . when I count in the integers mod 2, I get
>0,1,0,1,0,1,0,1. We can see the pattern here. If you want
>me to, I can give a quick inductive proof that I'll never
>get anything else. I was done the first time I reached 1.
>
>When I count in the complex number field, I get . . . I get
>. . . wait a second! There's no such thing as "counting" in
>the field of complex numbers! I can't count a damn thing
>until I restrict attention to a "denumerable subset." Let's
>just grab two elements, "A" and "B", I don't have time to
>play around here. Now we need to give them an arithmetic.
>I already wrote one down above, let's just use that one
>again, to save some space, for a change. Okay, now I can
>count: A,B,A,B,A,B . . . DAMMITALL! Now I'm all up on some
>isomorphic shit!
>
>

lol. That was funny, I don't blame you for writing it, it was humerous and I see where my error was in using integers as proof, I agree it is created by the imagination.

Okay, I want to ask you a question. How do you feel about Mathematics as a divine language? In a philosophical statement, Saukrates asks a slave boy, to draw a triangle, and the boy was able to, without knowing the formula for it as he did not have an education. Or, pythagoream theory in showflakes. What do you see mathematics as? And, is it logical to say that math itself is infinite, as it is possible to do anything with it? Not in the sense that it has a finite number in it's calcuation, but the calculation itself?


>>>>I am not detailing each point of study as I want to arrive
>>>>at the proof of God as quickly as possible for the sake of
>>>>this conversation, so please excuse me for leaving
>>>>information out if you have already studied Descartes.
>>>>
>>>>Next, Descartes invents the idea of the "Evil Genius" which
>>>>may be percieved as something that is decieving to his
>>>>senses and himself. In order to disprove the existence of
>>>>the "Evil Genius", Descartes is faced with the challenge of
>>>>proving God's existence. He does this as follows;
>>>>
>>>>Human beings are finite beings, meaning, our physical bodies
>>>>will eventually die.
>>>
>>>Sure.
>>>
>>>>Also, as finite beings, we are
>>>>incapable of the infinite, yet we can be certain the
>>>>infinite exists by observing a melting a piece of wax.
>>>
>>>Again, that's not really true. We ASSUME the infinite
>>>exists, or rather, Descartes did.
>>>
>>
>>I'd like to hear your proof against the infinity when faced
>>with simple counting.
>
>Again, I don't doubt that infinity exists. I know it
>exists. We invented it!
>
>By the way, I'm not just being snarky here. You cite the
>positive integers as an example of the infinite, and you are
>under the impression that this number system was somehow
>"God given." However, it is clear from any study of early
>math history that the invention of the positive integers was
>a long and tedious process of HUMAN logic. As far as we can
>tell, it was first completed in ancient mesopotamia. They
>seem to have invented the first number system which
>incorporates concepts of cardinality and "base." While our
>number system is base 10, theirs was base 60. That, by the
>way, is the reason there are 360 degrees in the circle.
>Back then, there were 12 signs of the zodiac, and each was
>subdivided into 60 parts.
>

Very interesting, I understand.

>>>>We
>>>>can grasp the concept of the infinite, but we are incapable
>>>>of it's excecution.
>>>
>>>That seems to me to be a sign that it was a figment of our
>>>imagination all along.
>>
>>I disagree.
>>
>>>>If you feel otherwise, please state how
>>>>as human beings, we can excecute the infinite. Descartes
>>>>concludes that God is a supreme being capable of the
>>>>infinite.
>>>
>>>And this is a HUGE leap! Even if we were to assume the
>>>physical existence of the infinite, there is absolutely no
>>>reason this requires a "being capable of the infinite."
>>>
>>
>>It does not have to be a being, or "God". It can simply be a
>>force of the infinite, and truly, that takes higher ground
>>over human beings incapable of executing the infinite. The
>>fact that Descartes called it God, is probably because he
>>was brought up in a religious environment and took it upon
>>himself to prove the existence of "GOD", which does not
>>actually have to be labeled that.
>
>Oh, so now you say Descartes was not arguing about the
>supernatural, but rather only about the existence of
>infinity. You can change the rules midgame if you like, but
>I'll point out again that the details of his arguments have
>been completely superseded by ensuing developments in
>mathematics and physics. Some form of the argument still
>stands, but now the only reasonable conclusion (if you
>insist on defining "God" by a command of the infinite) is
>that "Descartes is God." Now there's a conclusion I'm
>tempted to "believe!"

lol, Peace, thanks for taking the time to respond. Looking forward to see your thoughts on what I asked. Have a good one, peace.