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Thread: On the philosophy of Cantor

  1. #1 On the philosophy of Cantor 
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    Here is a quote from wikipedia:http://en.wikipedia.org/wiki/Georg_C...7s_mathematics

    "Philosophy, religion and Cantor's mathematics

    The concept of the existence of an actual infinity was an important shared concern within the realms of mathematics, philosophy and religion. Preserving the orthodoxy of the relationship between God and mathematics, although not in the same form as held by his critics, was long a concern of Cantor's. He directly addressed this intersection between these disciplines in the introduction to his Grundlagen einer allgemeinen Mannigfaltigkeitslehre, where he stressed the connection between his view of the infinite and the philosophical one. To Cantor, his mathematical views were intrinsically linked to their philosophical and theological implications—he identified the Absolute Infinite with God, and he considered his work on transfinite numbers to have been directly communicated to him by God, who had chosen Cantor to reveal them to the world.

    Debate among mathematicians grew out of opposing views in the philosophy of mathematics regarding the nature of actual infinity. Some held to the view that infinity was an abstraction which was not mathematically legitimate, and denied its existence. Mathematicians from three major schools of thought (constructivism and its two offshoots, intuitionism and finitism) opposed Cantor's theories in this matter. For constructivists such as Kronecker, this rejection of actual infinity stems from fundamental disagreement with the idea that nonconstructive proofs such as Cantor's diagonal argument are sufficient proof that something exists, holding instead that constructive proofs are required. Intuitionism also rejects the idea that actual infinity is an expression of any sort of reality, but arrive at the decision via a different route than constructivism.
    Firstly, Cantor's argument rests on logic to prove the existence of transfinite numbers as an actual mathematical entity, whereas intuitionists hold that mathematical entities cannot be reduced to logical propositions, originating instead in the intuitions of the mind.
    Secondly, the notion of infinity as an expression of reality is itself disallowed in intuitionism, since the human mind cannot intuitively construct an infinite set. Mathematicians such as Brouwer and especially Poincaré adopted an intuitionist stance against Cantor's work. Citing the paradoxes of set theory as an example of its fundamentally flawed nature, Poincaré held that "most of the ideas of Cantorian set theory should be banished from mathematics once and for all."
    Finally, Wittgenstein's attacks were finitist: he believed that Cantor's diagonal argument conflated the intension of a set of cardinal or real numbers with its extension, thus conflating the concept of rules for generating a set with an actual set."

    Cantors definition of God is rather sofisticated...Any comments?


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    Wasn't Cantors work instumental in the foundation of set theory? Anyway with regard to his concept of God, I think the idea of communicating directly with God through mathmatics is probarbly a little far out there. That said the use of maths in a search for greater understanding of everything is one most us can relate to. I would also have to say a share the doubts of some of his critics when it comes to infinity.

    I particularly like this quote from the Wiki page bio, "Intuitionism also rejects the idea that actual infinity is an expression of any sort of reality".


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    Quote Originally Posted by Chrisgorlitz View Post
    Wasn't Cantors work instumental in the foundation of set theory? Anyway with regard to his concept of God, I think the idea of communicating directly with God through mathmatics is probarbly a little far out there. That said the use of maths in a search for greater understanding of everything is one most us can relate to. I would also have to say a share the doubts of some of his critics when it comes to infinity.
    I particularly like this quote from the Wiki page bio, "Intuitionism also rejects the idea that actual infinity is an expression of any sort of reality".
    A cool thing with Cantors concept of God = Absolute infinity is that the question whether god exists or not gets tricky! If we say that god exists then we make a contradiction...BUT his principle of Reflection then guarantees the existence of a "lesser" god= Ordinary Infinity that is not self inconsistent ...So by denying the highest god Cantor lets in all the rest of the gods!? Since his religious research is expressed in the language of mathematics it is overlooked and forgotten But that is as expected...few innovators gets full credit for their work just see what happened to Parmenides: He discovered the first law of thermodynamics and gave the first model of block time...eh...the modern view of space&time. And is that recognized? Nope.

    So you got doubts about infinity?... well its an interesting subject ...But first note that intuitonism is a philosophy invented to escape from a problem! My view is that it came into existence as a "cowards way out"... But to be fair: its not uncommon to deny an axiom of a consistent theory to get a new theory, (see the denial of euclides paralell axiom), but claiming an axiom system is inconsistent and then attempting to create a consistent system by denying an axiom in the inconsistent system is another matter. (But only seen as discrete Mathematics it is an interesting theory. Lets by all means forget it was created in sin.)
    Last edited by sigurdW; July 10th, 2012 at 05:42 AM.
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    I wonder though why you set such store in the wisdom of someone who is somewhat largely over looked. What is about about Cantor's philosophy that's so captured your imagination I wonder.
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    Quote Originally Posted by Chrisgorlitz View Post
    I wonder though why you set such store in the wisdom of someone who is somewhat largely over looked. What is about about Cantor's philosophy that's so captured your imagination I wonder.
    Thanx for asking!

    First of all I think the invention of set theory is a grand intellectual achievement of Cantors. And it IS possible to consistently derive set theory from the idea of the Absolute Infinity together with the principle of Reflection. There are set theoricians actually doing that. This gives sufficient reason to examine also his idea to identify the Absolute Infinity with God. But I know of no such analysis. So im just trying to get someone to read ,think and post My way of interpreting Cantors Idea is as follows:

    It does not matter if there is a God or not since if there is no god then there is a Reality doing
    (or preferring not to do)
    whatever the assumed god was supposed to be doing, so therefore there actually is a God!

    Isnt this as close to a valid proof of the existence of god as you can get, asks sigurdW?
    Last edited by sigurdW; July 12th, 2012 at 02:18 AM.
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    I think this is interesting as it has the appeal of offering up the a solution to the idea of God not existing, though it seeks to redifine God as reality if God should prove not to exist.
    This requires some further examination, lets say we accept that God is proved not to exist so we replace the traditional theological idea of God with this new idea that reality is God, we are basically changing the nature of our understanding of God, we are really just usurping a concept thats already existed for thousands of years instead of just totally rejecting the idea and moving forward and accepting the actual specific quallities of reality it's self.

    Though it does serve to offer a way of understanding to the undecided, to true theists though I can't see this idea catching on as first they have to accept an idea of a disproved theological God and no matter how evidence is presented it unlikely they'll be convinced of such.

    The some aspects though I do find appealing in as much as this seeks to link the concept of God with the things we see around us, the things we can actually understand and work out though again I think others may have a problem with this.

    The one particular aspect I'm not so keen on is this idea of an 'Absolute Infinity', again though I'm not so keen on any idea or suggestion of infinity.
    Though while I contemplate it, it does raise rather an interesting question, Given the choice would you to exist, in some form or other, for infinity or would you end at some point?
    There is also a third option of not wishing to have the power to choose. I find this question interesting as it in some way puts the power of our destiny back in our own hands.
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    Quote Originally Posted by Chrisgorlitz View Post
    I think this is interesting as it has the appeal of offering up the a solution to the idea of God not existing, though it seeks to redifine God as reality if God should prove not to exist.
    This requires some further examination, lets say we accept that God is proved not to exist so we replace the traditional theological idea of God with this new idea that reality is God, we are basically changing the nature of our understanding of God, we are really just usurping a concept thats already existed for thousands of years instead of just totally rejecting the idea and moving forward and accepting the actual specific quallities of reality it's self.

    Though it does serve to offer a way of understanding to the undecided, to true theists though I can't see this idea catching on as first they have to accept an idea of a disproved theological God and no matter how evidence is presented it unlikely they'll be convinced of such.

    The some aspects though I do find appealing in as much as this seeks to link the concept of God with the things we see around us, the things we can actually understand and work out though again I think others may have a problem with this.

    The one particular aspect I'm not so keen on is this idea of an 'Absolute Infinity', again though I'm not so keen on any idea or suggestion of infinity.
    Though while I contemplate it, it does raise rather an interesting question, Given the choice would you to exist, in some form or other, for infinity or would you end at some point?
    There is also a third option of not wishing to have the power to choose. I find this question interesting as it in some way puts the power of our destiny back in our own hands.
    Ill not change anything in my quote of you. You begin to think in a way I like.
    The fact is that Cantor himself is not overlooked by mathematicians... but his religious idea is! And as for theologicians...BAH! (Here comes a set of words better censored.)
    The proof of gods existence is rather solid! As you know theres too many different concepts of god for making it possible for a god with ALL the claimed qualities to exist, so what Reality has to do is having all necessary qualities... in particular it has to create itself and thereby us.
    So if there is no other god then reality is god: if and only if there are necessary qualities in the god concept.
    (I doubt theologicians can agree what qualities something must have in order to be god.
    But again: if theres no god then Reality created itself and thereby us!)

    I myself have no big interest in the question
    I always denied anyone could prove theres a difference between God and Reality...
    Making me Pantheist from babyhood! But I found an ally in Cantor so I took it as my responsibility to explain his thinking to anyone interested. (Youre first, peace be unto you brother.)
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  9. #8  
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    Thanks for the mnemonic sigurd:

    Parmenedes "on nature"

    The steeds that bear me carried me as far as ever my heart
    Desired, since they brought me and set me on the renowned
    Way of the goddess, who with her own hands conducts the man
    who knows through all things. On what way was I borne

    along; for on it did the wise steeds carry me, drawing my car,
    and maidens showed the way. And the axle, glowing in the socket -
    for it was urged round by the whirling wheels at each
    end - gave forth a sound as of a pipe, when the daughters of the
    Sun, hasting to convey me into the light, threw back their veils

    from off their faces and left the abode of Night.
    There are the gates of the ways of Night and Day, fitted
    above with a lintel and below with a threshold of stone. They
    themselves, high in the air, are closed by mighty doors, and
    Avenging Justice keeps the keys that open them. Here did

    the maidens entreat with gentle words and skilfully persuade
    to unfasten without demur the bolted bars from the gates.
    Then, when the doors were thrown back,
    they disclosed a wide opening, when their brazen
    hinges swung backwards in the

    sockets fastened with rivets and nails. Straight through them,
    on the broad way, did the maidens guide the horses and the car,
    and the goddess greeted me kindly, and took my right hand
    in hers, and spake to me these words: -
    Welcome, noble youth, that comest to my abode on the car

    that bears thee tended by immortal charioteers ! It is no ill
    chance, but justice and right that has sent thee forth to travel
    on this way. Far, indeed, does it lie from the beaten track of
    men ! Meet it is that thou shouldst learn all things, as well
    the unshaken heart of persuasive truth, as the opinions of

    mortals in which is no true belief at all. Yet none the less
    shalt thou learn of these things also, since thou must judge
    approvedly of the things that seem to ...

    (hat in hand do i bow my head)
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    Hi Sculptor! I wasnt aware so much of Parmenides work survived... Admit it: Its you making poetry in the style of the Ancients?

    You know about my liking of the Ancient Philosophers working under harsh conditions, so Cheers to you!
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  11. #10  
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    Quote Originally Posted by sigurdW View Post
    ... Admit it: Its you making poetry in the style of the Ancients?
    would that i could sigurd, would that i could

    here's the link to the rest of what remains of this poem:
    Poem of Parmenides : on nature
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  12. #11  
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    Quote Originally Posted by sculptor View Post
    Quote Originally Posted by sigurdW View Post
    ... Admit it: Its you making poetry in the style of the Ancients?
    would that i could sigurd, would that i could

    here's the link to the rest of what remains of this poem:
    Poem of Parmenides : on nature
    Thank you VERY much!
    Im not sure you are aware of the importance I think Parmenides is. He is the first known to use logic to enquire into truth and nature.
    I never looked at the source...I sort of reasoned out through what others told me what MrP had done.
    First he proves (correctly in my opinion) the following:

    Suppose nothing is,
    Then it is so that nothing is.
    But since something is so,
    Then nothing is not!

    Then he defines change, here he goes wrong (He is a pioneer remember! Some mistakes can be excused.) and I dont care much of the details, but the result is a static view of The Everything and this is the favorite world view of physics still today! Theres a past, theres a future, no point of time is special, every point (so to say) "exists" simultaneously! Time does not pass...We cant change the future...Free will is an illusion.

    Against this view Heraclit protests by saying you cant step twice in the same river...

    And a correct theory of time must unite these two opposing theories
    since both are true in part of what they claim.
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    Whats left to mention about Ancient Philosophy (Remember its me speaking, I select only what is of personal interest!) is their definition of truth, their ideas of Logic and Classification and their repeated failure to understand Infinity. Lets jump to Galileo and Medieval Philosophy:

    1 There are the set of Natural numbers: 123...
    2 There are the set of squares of natural numbers: 149...
    3 Since not all natural numbers can be found in the set of squares
    then it seems there must be more natural numbers then there are squares of natural numbers.
    4 But to each and every natural number there corresponds one and only one square
    so there must be as many squares as there are natural numbers!

    This is known as Galileos Paradox... and I think he inspired Cantor! You do see how, dont you?
    Last edited by sigurdW; July 12th, 2012 at 01:19 PM.
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  14. #13  
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    love the paradox (through paradox do we truely learn)
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