Posted on 11/30/2004 6:21:11 PM PST by betty boop
Actually, it's slightly amusing in the context of this thread to note that there plainly is one case in which the church (albeit the Anglican Church) attacked science and the church was right: a pamphlet attacking Edmund Halley (who used calculus to predict the return of the eponymous comet and was a notorious 'freethinker') written by Bishop Berkeley contains (in mocking tones) a proof of the absurdity of infinitesimals in the context of classical logic.
N = qd + r for 0 <= r <= N-1
Now if d = 0, r must lie outside the range for remainders (r = N). We can liberalize the definition, and instead ask for solutions to the equation with minimal r, in which case one finds that any number can be q, so the problem has too many solutions to define an operation.
At the level of generality one uses in calculus, division by a 'static' zero still makes no sense because division now is the same as multiplication by the reciprocal (the reciprocal of a being defined independently from division as the solution to ax = 1). But division by 'dynamic' zeros can sometimes be made sense of. In an old approach to real and complex functions usually called by the German name Funktiontheorie, 'removable singularities'--those where the lack of definition of a function can be removed to give a point of continuity by definining the value at the point to be the limit at the point--are always 'removed', so x/x = 1 as functions even though when x = 0 one has 0/0. Of course 5x/x = 5, while x/(x^2) = 1/x, which still makes no sense for x = 0.
Sure you get rid of base-dependence, but if one treats 'terminating' decimal, binary, ternary, . . . expansions as ending in a repeated string of zeros, the set of numbers with repeating place-value expansions is base independent--the rationals--and the really intersting break, between numbers which admit place-value expansions generated by a Turing machine and those which don't is the same as the break between numbers which admit continued fractions expansions generated by a Turing machine and those which don't.
I love Bertarnd Russel's definition for the number one. What is it? Twenty pages long? Or is it fifty?
Yes, I was being a little loose with the language. Division by zero is undefined under the real numbers because it is not a real expression.
When I learned trigonometry, the instructor always wrote the symbol for infinity when a problem involved division by zero rather that saying it was undefined. I always liked his approach.
I worded it badly. I meant to say that most transcendentals have no simple pattern in their continued fraction expansion (or decimal expansion also.) Of course some transcendentals such as e have "interesting" continued fraction expansions.
My overall point was that periodicity of expansions applies to rationals in decimal notation but with continued fractions, one gets the quadratic irrationalities. There isn't a good regular expansion for cubics (Jacobi doesn't always work) or for much else. It's funny to me anyway.
The algebraics are strange in some ways too, by Roth's theorem. None of them have good rational approximations. Likewise there is a (controversial because some reviewers think the paper is wrong) paper by (I think) von der Poorten claiming that no finite state machine can generate the decimal (or binary) expansion of an algebraic irrationality.
Tell me, truly, does anybody really think these transcendentals (pace Doctor Stochastic and The_Reader_David) are some kind of generative sisters of Kant's Transcendental Ego? (No doubt Kant was a dropout mathematician and worded it badly!)
I wasn't suggesting that, Matchett. My point is that God knows who His elect are, but we don't. Then there is the matter of falling into a situation where we may place doctrine ahead of our duty to love God with our whole heart and soul and mind and strength, and our neighbor as ourself (which would be to flirt with idolatry). The commandment does not say, "Love thy neighbor who is elect of God." It simply says: "Love thy neighbor." I think God wants us to love our neighbor as God loves each of us, or at least as much as this is possible for us.
One other point: God saves whom He wills. We don't know much about that, either. To say otherwise seemingly implies we wish to place a limit on God. And this will not do! FWIW.
Does this mean that the singularity itself is "dense?" If the physical theories do not function, how do we know this? By logical inference from an entirely hypothetical reversal of its expansion?
Thanks for the lovely "Greeting Card," Eastbound!
As the densities get really big (lots of stuff packed into a small space), the methods of current physical theory no longer converge to an answer. It's not just that the errors get large (they do) but one fails to get any result at all.
This implies that new theories are needed and that experiments to select among the theories are also needed.
It seems that the implication here that we need new theories is premised on the hypothesis that the universal expansion is factually a time-reversable process. But on what basis can we say that this is a valid statement, rather than an inference drawn from a (uncorroborated and uncorroboratable) thought experiment?
Do you see weakness in the theory of the Big Bang?
There's no weakness in the big bang, at least the BB explains more of the observed phenomena than any competing theory. The problem is that the earliest moments (using the terms loosely) of the BB are not described well by current theory (nor by any other; they all have similar problems). All other theories do much worse; in fact, so far they all fail. (There's really only one, the steady state, and it doesn't work.)
I think the problem is that it's probably impossible to apply the physical laws when space, time, and matter do not yet exist. It may be that science cannot tell us much about the "early universe" -- i.e., what's going on in the Planck era -- in principle, because its method cannot find anything to "work with," so to speak.
Pretty weird, huh?
"A man's gotta know his limitations."
-- Clint Eastwood as Dirty Harry
I’d like to take the approach of simplifying the issue for Lurkers to the thread.
Infinity of space/time is a necessary pre-condition to atheism (metaphysical naturalism). This is because a disbelief in God or someone or something “beyond” seeks to explain away the extraordinary unlikelihood of the physical constants, the beginning of real time and the unreasonableness of math in this universe as happenstance, i.e. that anything that can happen, has (the plentitude argument).
We sometimes seek to avert the concept of infinity by using a reasonable substitute. That does not however change the import of infinity.
Take pi for instance. The decimal expansion of it never repeats or terminates. Yet we use reasonable substitutes in our calculations. Zero is another. It alone is neither positive nor negative. No matter how close to zero a number might be, and no matter how effective it might be in getting a job done – it’ll never be zero. Likewise, the inability to describe a universe at Planck length does not substitute for the fact the universe had a beginning.
Returning to the plentitude argument, there is no number of multi-verses which can substitute for infinity of opportunity where from this universe would have to have arisen by happenstance.
Moreover, the beginning of real time (space/time) in this universe means that this universe is finite, not infinite. That there was a beginning is the most theological statement to ever come out of modern science.
Or to put it another way, the issue is geometry. Space/time transforms (relativity, Lorentz transformation). It is expanding from a beginning. All of the fields, waves, energy, etc. exist within space/time. A field in fact exists at all points in space/time. Thus, no space/time – nothing else in this universe.
The fact of a beginning points to something “beyond” which caused it. The alternative materialistic (metaphysically naturalist or atheist) theories suggest that this universe is the effect of a prior material cause. All that accomplishes is to move the goalpost further back to a prior beginning (multi-verse, epyrotic, imaginary time, etc.).
Theologians, philosophers and radical mathematical Platonists - on the other hand – say there is a “beyond” – a being as compared to a becoming (this universe, etc.).
In the case of the radical mathematical Platonists, that “beyond” consists of mathematical structures as existents. Even if true, there remains the issue of causation – where’d they come from? The causation question remains for those who suggest the “beyond” is a “collective consciousness” of physical existence.
This is why the fact of a beginning (finite v infinite) is such a theological statement. Every case points to an uncaused cause, a Creator, God.
The names in credentialed, professional philosophy have--for the most part, most of the time--impeccable reasoning. The majority of students have little reasoning, and the little they have is flawed. We ought to distinguish between real philosophers and those extracurriculars who use some of the form of philosophic process, but without the depth, breadth, and skill of real philosophers.
Is it possible that the Big Bang was a series of mini- implosions? Not implosions as such, but similar to a changing of ice to liquid and gases. In the process, space is created where the ice was, an environment for the liquid and gases to move around. And to reverse the process would merely expand the gases and liquid to its former solid state, using all the space that was created.
Let me try to draw the model:
Supposing the universe was an ice cube. A really big one. How to create space in the middle of an ice cube without melting the whole ice cube? Supposing three lasers, exterior to the cube, were to be aimed at a common intersection within the cube, the combined heat would shrink the molecules at the intersection and we would have a drop of water suspended in space. Suspended because the vacuum created around the drop of water would pull it into a ball-shaped drop of water. Why not a square-shaped drop of water? Because an equal pressure pull from all sides surrounding the drop of water would mould the form into a natural ball.
Maybe that's why planets are round. To remove the presense of heat would cause the drop of water to expand to its former static state, fitting snugly in the middle of the cube.
I created this model while trying to understand what a vacuum was and why planets are round and couldn't come up with anything without visualizing the universe surrounded by something to contain it.
And, of course, the container turns out to be the same as its contents, except for localized changes in form from 'solid' to 'liquid.'
Just another thought experiment. But would time be reversed in the process of re-expansion? I don't think it would be reversed as much as traversed, during the instantaneous process of crystalization.
betty boop: "I wasn't suggesting that, Matchett."
I thought you were since you used the phrase, "Mystical Body ...".
Do you include the totality of the visible church within that phrase?
betty boop: "My point is that God knows who His elect are, but we don't."
True. That's why I'm confused by what you've written and asked you the above question to clarify.
betty boop: "Then there is the matter of falling into a situation where we may place doctrine ahead of our duty to love God with our whole heart and soul and mind and strength, and our neighbor as ourself (which would be to flirt with idolatry). The commandment does not say, "Love thy neighbor who is elect of God." It simply says: "Love thy neighbor." I think God wants us to love our neighbor as God loves each of us, or at least as much as this is possible for us."
Oh, let me reassure you. You needn't worry that I would fall into any such situation as "putting doctrine ahead of our duty to love God and.....". Of course, I can't be responsible for what others choose to do, and in no way would I assume to be their mother.
betty boop: "One other point: God saves whom He wills. We don't know much about that, either. To say otherwise seemingly implies we wish to place a limit on God. And this will not do! FWIW."
I'm aware of that, but thank you for the reminder lesson.
When you don't want to answer a question, though, you really don't need to supply me with wordy justifications.
That's the truth! One imagines this is the reason the statement doesn't sit too well with some scientists as, for instance, Stephen Hawking....
A-G, you wrote:
"Space/time transforms (relativity, Lorentz transformation). It is expanding from a beginning. All of the fields, waves, energy, etc. exist within space/time. A field in fact exists at all points in space/time. Thus, no space/time – nothing else in this universe.... The fact of a beginning points to something “beyond” which caused it.
It needs to be said!!! Thank you for this excellent and informative post, Alamo-Girl.
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