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Featured Math, Assumptions, and the Supernatural

Discussion in 'General Religious Debates' started by Polymath257, Feb 12, 2020.

  1. Polymath257

    Polymath257 Think & Care
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    I'm not sure how this analysis will go over in this forum, but here goes.

    First, some background. Mathematics is proof based. The original example
    of how math is done is Euclid's Elements, the first geometry 'textbook'.
    What Euclid did was give a number of axioms and postulates: basic assumptions
    of how things work that he considered to be 'intuitively obvious'. he then
    proceeded to use those assumptions to prove various things about geometry,
    numbers, etc.

    But there was one postulate, the parallel postulate, that just didn't seem to
    be as 'intuitively obvious' as the rest of Euclid's assumptions. Many people
    over many centuries tried to prove this postulate from the other assumptions,
    but failed to do so. So, while nobody actually believed the parallel postulate
    to be false, nobody was able to prove it either.

    Then, about 200 years ago, people started working with a system of geometry
    that assumed the parallel postulate was false. And, what they eventually found
    is that geometry with the parallel postulate and geometry with its negation
    are *equally* consistent internally. In fact, using either, you can construct
    a 'model' where the assumptions of the other hold.

    This shook the foundations of mathematics. Math had always been held as the
    most certain area of study. It was used by Plato as a crucial example showing the
    existence of his 'Forms'. It had inspired Descartes to re-examine his assumptions,
    leading to his break with Aristotelian philosophy.

    But now, there were *two* equally consistent theories of geometry. Neither could
    be discarded solely on the basis of logic. They were equally consistent internally.

    One aspect of this revolution was that mathematicians started looking much closer
    at the basic assumptions of *all* of math, even those underlying basic arithmetic.

    One goal was to find a collection of assumptions that is consistent--where it could
    be *proved* that no inconsistencies would ever be derived. It was also a goal that
    it could be proved that these assumptions could answer any (mathematical) question,
    this being called 'completeness'. It was felt that, even if we didn't have all the
    'correct' axioms, there should be a collection of axioms that is both consistent
    and complete and finding that collection of assumptions became a central goal for
    many mathematicians in the early 1900's.

    Then, a mathematician named Kurt Godel came along. He *proved* that any sysem of
    axioms that is unambiguous (in a well defined sense) could NOT be both consistent
    and complete if it was strong enough to deal with ordinary arithmetic.

    So, the problem that had been raised by the parallel postulate in geometry was shown
    to exist even in ordinary, elementary school arithmetic: it is simply not possible
    to even prove arithmetic is consistent without making assumptions stronger than
    arithmetic itself.

    Moreover, any system of axioms strong enough to deal with arithmetic *must* have
    questions it cannot answer. If consistent, it simply cannot be complete (and unambiguous
    in the sense above). There MUST be questions that are unanswerble without making
    additional assumptions. We say that such questions are independent of our axioms

    And we know of many independent questions in mathematics: questions that simply
    cannot be answered one way or the other from the axioms we use for modern math.
    Some have been subsequently assumed (the Axiom of Choice). Others are still being hotly
    debated (Continuum Hypothesis). Others are obscure even to most mathematicians (Martin's
    axiom).

    For ALL of these, we get different theories of *mathematics* by either assuming them
    OR assuming them to be false. And *both* ways are equally consistent internally.

    In a sense, these questions have no actual truth value, even though they are meaningful
    and some are even important. You can use either the statement or its negation equally
    well and have a mathematics that is equally consistent either way.

    Bummer. Well, sort of.

    Now, I have been pondering. We all make assumptions about the 'real world'. Among those
    is that we are not the only conscious being, that there is an external world that our
    senses can perceive, however imperfectly, that our memories are reliable to at least
    some extent, etc.

    We use these assumptions to move around in the world, learn how that world works, etc.
    We develop science and technology, etc. We make art, music, and love others.

    Then comes the issue of religion.

    People have many different views of what things are true in their religions. There are
    multitudes of different religions (especially through time) with huge variations on what
    was believed, thought important, etc.

    What if the existence of a supernatural is simply inconsistent?

    What if the existence of a supernatural is neither provable nor disprovable from our other
    assumptions? What if the question really doesn't have a truth value at all?

    Now, if this is the case, what are the advantages and disadvantages of the assumptions either way?
     
    #1 Polymath257, Feb 12, 2020
    Last edited: Feb 12, 2020
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  2. Maximus

    Maximus the Confessor

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    Define supernatural.
     
  3. Revoltingest

    Revoltingest Christine's Uncle Fergus
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    I like the term, "pustulate", but what is it?

    As for making assumptions, it all boils down to usefulness.
    We can construct all sorts of systems. I like the ones that
    can model reality. Others can be fun too, eg, modular
    arithmetic used in cryptography.
     
    #3 Revoltingest, Feb 12, 2020
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  4. Ayjaydee

    Ayjaydee Active Member

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    The assumptions do seem to allow us to operate in the mundane world. Why is it necessary to explore the possibility of a supernatural world unless we enter one. I believe the monk Buddhdassa called the supramundane world I think he felt that through insight meditation we would discover new assumptions to enable us to see it clearly
     
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  5. Daemon Sophic

    Daemon Sophic Avatar in flux

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    Nice post.

    I would agree with the first two sentences of @Ayjaydee ’s response...
    The various “intuitively obvious assumptions” that you discuss in the OP work for the universe that we dwell in, with the observed phenomena that exist here.
    If/when some new phenomena occur and are observed/recorded (like a god shows up and says “Hi!”, or a ghost does anything, or a magician starts teleporting around the planet).....THEN we can come up with new “assumptions” based upon our new observations.

    ————————
    Sorry, but the OP reminds me of an old joke my mom (a physicist/mathematician) told me.
    Three guys are applying for a job with the local baseball league as umpire. The first is an experienced umpire, the second is a physicist, and the third is a mathematician.
    When each was asked “How do you make your calls as an umpire?”,
    the umpire says “I call ‘em like I see ‘em.”
    the physicist says “I call them like they are.”
    and the mathematician says “I call them as the ought to be.”

    Hilarity ensues.
     
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  6. Sunstone

    Sunstone De Diablo Del Fora
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    I am unfamiliar with some of your terms, Poly. Could you please elaborate on how you are defining "strong" here. As in "a strong assumption". Do you simply mean a strong assumption assumes more (and is less evidence based) than a weak assumption? Or is there more to it than that? In other words, how does one measure the strength of an assumption?
     
  7. Sunstone

    Sunstone De Diablo Del Fora
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    Of possible interest here, I believe George Berkeley might have shown that the notion things have no existence apart from our minds is just as logically consistent internally as is the notion that things have an existence apart from our minds. I'm not absolutely sure he did, though, since I have never studied his theory of immaterialism in depth.

    On a side note, one of my professors was of the opinion that Berkeley actually believed in materialism -- that things have an existence apart from our minds -- but advanced his theory of immaterialism merely as an exercise to show that the opposing view was just as internally consistent as the other view.
     
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  8. Sunstone

    Sunstone De Diablo Del Fora
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    Loved the OP, by the way. Made it worth getting out of bed this morning all by itself. Thanks, Poly!
     
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  9. Sunstone

    Sunstone De Diablo Del Fora
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    It's all very good and well to indulge oneself in the American taste for immediately practical thinking, but have you ever taken a moment to wonder where humanity would be without people willing to to engage in "What if" speculative thinking? Not all human progress can be attributed to hewing close to the obvious. Not trying to be argumentative. Just sayin'.
     
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  10. Jayhawker Soule

    Jayhawker Soule <yawn> ignore </yawn>
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    Loved Goedel, Escher, Bach. :)
     
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  11. Ayjaydee

    Ayjaydee Active Member

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    Assuming we've progressed
     
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  12. Sunstone

    Sunstone De Diablo Del Fora
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    Why have I not been informed of this book before? I demand an investigation!
     
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  13. Sunstone

    Sunstone De Diablo Del Fora
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    Excellent point! :D
     
  14. SoyLeche

    SoyLeche meh...

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    It’s really excellent. A lot of the last part isn’t really relevant anymore (a lot of AI stuff that turned out to be wrong), but the first 3/4 of the book is very interesting. I have a copy of it somewhere, but can’t find it. I like to re-read it every 5 years or so, and I’m overdo.
     
    #14 SoyLeche, Feb 12, 2020
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  15. Sunstone

    Sunstone De Diablo Del Fora
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    Thank you and @Jayhawker Soule for the heads up! I just added it to my shopping list.
     
  16. exchemist

    exchemist Well-Known Member

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    I must admit I rather like the suggestion that seems to be implied, that in any closed universe it is not possible to know everything. That feels aesthetically right, somehow. But maybe it is my early exposure to the Uncertainty Principle that prejudices me.

    If the question about the existence or not of a supernatural has no truth value, I suppose it will come down to the practical utility, for each individual, of assuming one or other of the two possibilities. Which seems to work out differently for different people, according to culture and upbringing.
     
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  17. WhyIsThatSo

    WhyIsThatSo Well-Known Member

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    "what if the existence of a supernatural is simply inconsistent? "

    "supernatural" = "supra" ( above, beyond ) "natural" (nature). In other words, intelligent design and life above and beyond the "natural" realm of existence we share with all the other biological (and other) life forms here on earth in what we call... "mother nature".

    I'm no rocket scientist but it doesn't take one to see that this "world" is a whole lot more complex and complicated than little men in white coats studying numbers all day can even begin to realize.
     
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  18. Brickjectivity

    Brickjectivity Veteran Member
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    "What if the existence of a supernatural is simply inconsistent?

    What if the existence of a supernatural is neither provable nor disprovable from our other
    assumptions? What if the question really doesn't have a truth value at all?

    Now, if this is the case, what are the advantages and disadvantages of the assumptions either way?
    "

    We're talking about whether we can contemplate something that is unnatural. I think we cannot contemplate everything that could be unnatural, but we can inquire about some things and whether they are natural or not. That is a start.

    Assuming that there were supernatural things the only way I could come up with to interact with them would be guessing about them, but this might not be as useless as it sounds. We can wonder if human love is a shadow of something supernatural and change the world by doing so. Does this mean the greater love exists? We can't prove it. Nevertheless it impacts us.
     
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  19. Daemon Sophic

    Daemon Sophic Avatar in flux

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    Of course. :)
    I was considering how to reply to you, when I read @Brickjectivity ’s post. :thumbsup:....so I’ll use his final two paragraphs (with one small, but very important edit).

    [QUOT="Brickjectivity, post: 6531512, member: 41354"]"....
    We're talking about whether we can contemplate something that is unnatural. I think we cannot contemplate everything that could be unnatural, but we can inquire about some things and whether they are natural or not. That is a start.

    Assuming that there were supernatural things the only way I could come up with to interact with them would be guessing about them, but this might not be as useless as it sounds. We can wonder if human love is a shadow of something supernatural and change the world by doing so. Does this mean the greater love exists? We can't prove it. Nevertheless Perhaps it impacts us.[/QUOT]
     
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  20. Polymath257

    Polymath257 Think & Care
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    I actually meant to ask what if it is *independent*? In other words, if it can be consistently assumed either way?
     
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