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Thread: For Spun Silver -- Was Thomas Aquinas wrong?

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    Custom Title Mathman's Avatar
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    For Spun Silver -- Was Thomas Aquinas wrong?

    Thomas Aquinas posed to himself the question (true story), is there anything God can’t do? The Angelic Doctor pondered for a while and came up with a list. God cannot commit a sin. God cannot forget anything. God cannot stop being God…And NOBODY, not even God, can draw a triangle whose angles do not add up to 180 degrees. (God was God, but Euclid was EUCLID!)

    As fortune would have it, I own a 1052-acre farm that is shaped like a triangle. One day I measured the three angles and found to my horror that they were

    A = 30.000001 degrees, B = 60.000002 degrees, and C = 90.000003 degrees.

    What is the radius of the Earth?

    >

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    Solution: For a triangle drawn on a sphere of radius R, the three angles of a triangle add up to a little bit more than 180 degrees. The exact formula, with angles measured in radians, is

    A + B + C = Pi + (Area of triangle)/R^2

    In this problem the three angles add up to 180.000006 degrees, or Pi + .0000001047 radians. The area of the field is 1052 acres, or 1.64375 square miles. Substitute these values into the equation and solve for R to obtain the radius of the Earth, R = 3962 miles.

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    Figure Skating Is A Dangerous Sport Dee4707's Avatar
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    Math, your a genius at math but I think you need to go to the circus!!!

    Dee

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    Rooting for the divas with Kwanford Spun Silver's Avatar
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    OK, I knew bringing in Thomas Aquinas was opening a can of worms, but I didn't expect this. Eew, numbers.

    --Spun Silver


    Larry,

    It would seem that either you have faulty instruments or mathematics has made remarkable advances in the past 750 years. The latter would rather confirm than shake my faith in the Creator of the universe and the human mind.

    Respondeo:

    As you know, my main point was that God is called omnipotent because He can do all things that are possible absolutely:

    which is the second way of saying a thing is possible. For a thing is said to be possible or impossible absolutely, according to the relation in which the very terms stand to one another, possible if the predicate is not incompatible with the subject, as that Socrates sits; and absolutely impossible when the predicate is altogether incompatible with the subject, as, for instance, that a man is a donkey.

    It must, however, be remembered that since every agent produces an effect like itself, to each active power there corresponds a thing possible as its proper object according to the nature of that act on which its active power is founded; for instance, the power of giving warmth is related as to its proper object to the being capable of being warmed. The divine existence, however, upon which the nature of power in God is founded, is infinite, and is not limited to any genus of being; but possesses within itself the perfection of all being. Whence, whatsoever has or can have the nature of being, is numbered among the absolutely possible things, in respect of which God is called omnipotent. Now nothing is opposed to the idea of being except non-being. Therefore, that which implies being and non-being at the same time is repugnant to the idea of an absolutely possible thing, within the scope of the divine omnipotence. For such cannot come under the divine omnipotence, not because of any defect in the power of God, but because it has not the nature of a feasible or possible thing. Therefore, everything that does not imply a contradiction in terms, is numbered amongst those possible things, in respect of which God is called omnipotent: whereas whatever implies contradiction does not come within the scope of divine omnipotence, because it cannot have the aspect of possibility. Hence it is better to say that such things cannot be done, than that God cannot do them. Nor is this contrary to the word of the angel, saying: "No word shall be impossible with God." For whatever implies a contradiction cannot be a word, because no intellect can possibly conceive such a thing.
    If in the 21st Century you have demonstrated that a triangle of more than 180 degrees indeed has being, then God can draw it. Not being God, I could not do that back in the 1200s. The error was mine, not God's.

    Now perhaps you begin to see why I said all my work was like so much straw.

    Best,
    Tom

    P.S. Thanks for the invitation to "post medium," but compared with the bliss of seeing God face to face, even Goldenskate palls. But I look forward to meeting you in person.

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    I should be studying!
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    While I certainly agree with Spun Silver's points, I wonder if the original premise of Euclid was based upon the assumption that the earth was flat, or at least that the triangle was to be drawn on a flat plane.

    Just a novice butting into the conversation. I can see my way out of it, if requested.

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    Rooting for the divas with Kwanford Spun Silver's Avatar
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    Oh Alsace - welcome to the hot seat of a debate with our own Euclid. Can you take over the driving on the numbers? Thanks! :chorus:

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    I like pie. Tonichelle's Avatar
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    numbers... ewwwwwwwwwwwwwwwwwwwwwwwwww

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    The Zamboni Rocks!!! sillylionlove's Avatar
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    When I was in college one of the dorms was called Aquinas Hall. It was on the far end of campus and was by far the oldest and nastiest dorm. I think the 4 years that I attend college I was in that dorm 2 times!!

    How's that for some numbers!!


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    Custom Title Mathman's Avatar
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    Quote Originally Posted by Dee4707 View Post
    Math, your a genius at math but I think you need to go to the circus!!!
    Step this way, folks! You ain’t seen nothin’ yet.

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    Custom Title Mathman's Avatar
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    Quote Originally Posted by Alsace View Post
    While I certainly agree with Spun Silver's points, I wonder if the original premise of Euclid was based upon the assumption that the earth was flat, or at least that the triangle was to be drawn on a flat plane.
    Yes, that’s right. On a flat plane the angles of a triangle add up to 180 degrees. On a surface of positive curvature, like a sphere, they add up to more than 180 degrees, and on a surface of negative curvature, like a hyperboloid they add up to less than 180 degrees.

    So the question is, in the actual 3-dimensional universe that we live it, which of these possibilities is correct? If we drew a triangle in space with one vertex at the Earth, another in the Andromeda Galaxy and the third in the center of the Coma Cluster, what would the angles be?

    This is not a question that can be answered (as both Euclid and Thomas Aquinas incorrectly believed) by pure reason. We have to draw the triangle and see. Fortunately, there actually are experiments (one is the experiment addressed to Toni below) which are equivalent to drawing large triangles in space.

    So far, all experiments and measurements of this type are compatible with the Euclidean model. However, the discovery in 1998 of “dark energy” throws a monkey wrench into the mix. This is because dark energy mimics negative curvature and we don’t know how it evolved over the course of cosmic history (the expansion of the universe appears to have received a big burst of acceleration about 5 billion years ago, for instance – some 9 billion years after the creation of light.)

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    Custom Title Mathman's Avatar
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    Quote Originally Posted by Tonichelle View Post
    numbers... ewwwwwwwwwwwwwwwwwwwwwwwwww
    Dear Toni – it is my life’s mission to make you like math. So here goes.

    Now…take a deep breath…you can do this…

    What is the size, shape and ultimate fate of the universe? How can we obtain data that bear on such questions?

    Buy a telescope (OK, you need a pretty big one). Count the number of galaxy clusters that you see at a distance of 1 billion light years. Let’s say it is 10,000.

    Now count how many you see at a distance of 2 billion light years. Let’s say it is 40,100.

    (Wait, wait, come back. I am almost to the punch line.)

    Now use algebra to solve this equation:

    4x + 4 = 40,100/10,000.

    Did you get .0025?

    Now take the square root. (There are two square roots; take the negative one.) This is the curvature of the universe, k = negative .05 (the units are inverse billion light years).

    Conclusion: The universe is non-Euclidean (in particular the three angles of a triangle will always add up to less than 180 degrees. It is infinite and will expand forever.

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    Custom Title Mathman's Avatar
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    Quote Originally Posted by Thomas Aquinas
    It would seem that either you have faulty instruments or mathematics has made remarkable advances in the past 750 years. The latter would rather confirm than shake my faith in the Creator of the universe and the human mind.
    Esteemed Doctor,

    Oh, If only you could have held on to this mortal coil for another four centuries or so! You missed a doozie of an intellectual revolution.

    In the European Middle Ages there were two paths to Truth. The first was appeal to authority. (Such-and-such is true because Aristotle says so. Our only duty in later times is to study Aristotle so that we fully understand his arguments.)

    The second path was the application of the formal rules of logic to a set of indisputable first principles.

    Mathematics falls into this second category. The reason that the angles of a triangle must add up to 180 degrees is because (as Euclid shows), this conclusion follows by logical necessity from the five postulates of the Euclidean system.

    The weakness of this point of view is obvious – but in the case of Euclidean geometry, it was not truly taken seriously until the nineteenth century. How do we know whether, in the real world, the basic postulates are right or not? (As we say in modern techno-speak, garbage in, garbage out.)

    But in the 16th century a third way (spearheaded by Galileo, at the time called a heretic) sprung onto the scene, the empirical method. As an alternative to consulting ancient texts and/or cogitating in our cells, we can go forth into the world and find out for ourselves. Do the angles of a triangle add up to 180 degrees? Draw some and see! When we find circumstances where they don’t, we know that God, in his infinite wisdom, saw beyond the postulates of Euclid after all!
    Now perhaps you begin to see why I said all my work was like so much straw.
    In the 20th century a fellow named Einstein taught us to ask, "relative to what?" Your modesty is surpassed only by the clarity of your thought and the constancy of your character.

    In our impossible, irresistible quest to catch a glimpse of that which -- for all our pride of intellect -- we can see only through a glass darkly, none has labored more heroically than you.
    Last edited by Mathman; 06-28-2007 at 08:20 PM.

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    I like pie. Tonichelle's Avatar
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    ok bringing science AND math into the same problem is even worse

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    Rooting for the divas with Kwanford Spun Silver's Avatar
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    Mathman, you are something of an angelic doctor yourself.

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    Da' Spellin' Homegirl Grgranny's Avatar
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    WOW!!!
    This is sure all Greek to me. Can't understand a thing. I am surprised I even passed Algebra and Geometry in high school. I think the teacher, whom I loved, was taking it easy on me. I probably had D minuses. Or is it minus's or minus'?

    OT The teacher was also my piano teacher and knew how dumb I was.
    Although, one time she had me go through a new piece and my classmate was waiting and she told my classmate if she practiced she could be as good as I. One of the only compliments I ever received any time.

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    The Zamboni Rocks!!! sillylionlove's Avatar
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    Yikes...this is way to deep for me. I hate math. I will ways hate math. I have always been horrible at math and will always be horrible at math.

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