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Tuesday, April 10, 2012

A Di-Lemma

A reader with the monometopian name of OneBrow wrote in comment to an earlier post: 
As traditionally presented, the argument from motion rarely lists all the assumptions at the beginning. Assumptions such as "motion must be externally activated from potential" and "motion can not be in an infinite chain" get worked into the argument, often by declaring them to be obvious (the very notion of being an assumption). By then end of the argument, there are a fairly good-sized number of assumptions (each distinct argument having its distinct subset).
Now, in mathematics, a lemma is a proven statement used as a step in the proof of a bigger theorem.  There is no formal difference between a lemma and a theorem -- both follow the same rules of proof -- but there is a material difference.  The main interest centers on the matter of the theorem, while the matter of the lemma is of little interest in itself at the moment.  In the same manner a carpenter may use a hammer in the construction of a cabinet without any imputation that the hammer is simply assumed, and not itself constructed.

For example, in the proof that two continuous functions from X into a Hausdorff space Y are equal if they coincide on a dense subset of X makes use the lemma that if A is dense in X and Q is open in X then A∩Q is dense in Q.  [Example: the rational numbers are dense in the real numbers, so the rational numbers in the interval (0,1) are dense in that interval.]  Now, the proof of the lemma is lengthy enough that to insert it into the midst of the proof of the theorem would constitute a digression.  But even disregarding length, the proof of the lemma is "another topic" relative to the purpose of the theorem.


With this in mind, let us consider the Unobrower's complaint wrt the Proof from Motion.  When life hands you a lemma, make lemma-nade.  We have been handed two lemmas, which puts us in a di-lemma.

Definition.  The argument from motion does not proceed from local motion only, that is "change of location."  The word κινεσις (kinesis) used by Aristotle or motus used by Aquinas is usually translated as "motion" but more precisely means "change."  To refute Parmenides and Zeno, who held that change was in illusion, Aristotle developed the notion of Potency and Act, and defined change as a "motion" from potency to act.  A green apple changes to a red apple if first of all it was potentially red and then was moved to being actually red.  The motion from potency to act was called kinesis, usages preserved to this day in potential energy and kinetic energy.  

Why an apple turns red
Lemma 1. Whatever is being changed is being changed by another.  That is, nothing can move itself from potency to act.  The reason is simple.  Whatever exists potentially does not (yet) exist actually.  That which does not (yet) exist cannot do diddly-squat, so in particular that which is not-X cannot make itself X.  (Ex nihilo nihil fit.)  Thus, anything that is being changed is being changed by another.  (Btw, note the present imperfect tense, which corresponds to the Latin original.) 
Example, the apple is moved to become red by anthocyanin, which is activated by light in the regions of 3,600 to 4,500 Å.  (This is light within the near-ultraviolet, violet, blue and green regions of the spectrum.  Whatever absorbs these colors will appear red.)  Without light, apples become yellow as they ripen, even if the apple would otherwise normally turns red.  Thus the light possesses redness actually (in an eminent sense) and moves the apple to redness.  
In a similar manner, we see that the dog walks across the room because the local motion of its parts (the legs) move the whole.  (And the legs are moved by the muscles, etc.)   Thus the legs possess local motion actually (in a formal sense) and move the dog to local motion.   
For a more general discussion of this Lemma, especially as regards the usual counterarguments, see the pdf: "Whatever is Changing is Being Changed by Something Else’: A Reappraisal of Premise One of the First Way," David S. Oderberg

An essentially-ordered series.
This cannot regress infinitely.
Lemma 2. An essentially-ordered series cannot regress infinitely.  Mr. Brow's version -- "motion can not be in an infinite chain" -- is incorrect.  In fact, Aquinas cheerfully admits that an accidentally-ordered series might well regress without limit.  (He believed the world had a beginning, which would put a cap on any series; but he lived long before general relativity and the big bang theory.)   

These are not the first cause of the
Waldstein Sonata
In an essentially-ordered series, each changer receives its power to change from the concurrent operation of a logically prior changer.  Logically prior, not temporally prior.  That is, the changers are operating right here, right now, not at some remote time in the past.  For example: suppose you hear Beethoven's Waldstein Sonata.  In simplified terms:
  • the auditory cortex in your brain is moved by
  • the auditory nerve, which is moved by
  • the cochlea, which is moved by
  • the bones of the middle ear, which are moved by
  • the tympanum, which is moved by
  • compression waves funneled by the pinnae and ear canal, which are moved by
  • the vibrations of the strings in the piano, which are moved by
  • the hammers, which are moved by
  • the keys, which are moved by
  • the pianist's fingers, which are moved by
  • (skipping some intermediaries) an act of will of the pianist
An accidentally-ordered series.
This might regress infinitely.
The thing to notice is that none of these movers possess the power to move unless it is being concurrently moved by a prior mover.  None of the intermediate changers has the power in itself to produce the Waldstein.  They are called "instrumental causes" or "secondary causes."  The piano is an instrumental cause (lulz) since if the pianist stops playing, the effect, the Waldstein, will not happen.   

Now, suppose the series were to regress indefinitely.  Then there would be no first changer.  But then none of the subsequent changers in the series would have any power whatsoever and there would be no change in the world.  But there is change, therefore there must be a first mover for any essentially-ordered series.

This is also addressed somewhat in the Odenberg paper linked above. 

Someone once must have come up with
this without having to be taught it
by another
Scotus made an argument that even accidentally-ordered series imply a first mover.  Paraphrased, it runs as follows: Suppose you are teaching some Little Darlings™ the Pythagorean theorem.  You are a teacher.  But where did you learn the Pythagorean theorem?  Someone else taught you.  You are a taught teacher.  But your teacher also learned the theorem in this manner.  Indeed, everyone you have ever met who teaches the Pythagorean theorem is a taught teacher.  You can even imagine that this proceeds with infinite regress into the past.  Everyone who has ever lived has been a taught teacher of the Pythagorean theorem.  But reason indicates that there must be an untaught teacher, because the content of the theorem could not have come from an endless chain of teachers simply passing on what they have been told.  Someone had to come up with it.  We might say the same about emails being forwarded endlessly since the infinite past.  Even if the series regresses infinitely, the content of the email cannot come from an endless series of forwarding.  Somehow or other there must be an author of the email, an unsent sender who sent the email without having received it from another. Similarly, the unchanged changer. 

Mr. Brow concluded by stating "By then end of the argument, there are a fairly good-sized number of assumptions."   A fairly good-sized number?  But it is not clear that there are more than the two mentioned, the Major and the minor premise, which are not assumptions but lemmas proven independently.  (And how many assumptions lie behind the topological theorem mentioned above?)

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Da man.
For interested parties, here is the digest version given by Thomas in Summa theologica.  Note that the two things Mr. Brow believes were stated as mere assumptions "declared to be obvious" have been highlighted in bold-face.
The first and more manifest way is the argument from motion. It is certain, and evident to our senses, that in the world some things are in motion. Now whatever is in motion is put in motion by another, for nothing can be in motion except it is in potentiality to that towards which it is in motion; whereas a thing moves inasmuch as it is in act. For motion is nothing else than the reduction of something from potentiality to actuality. But nothing can be reduced from potentiality to actuality, except by something in a state of actuality. Thus that which is actually hot, as fire, makes wood, which is potentially hot, to be actually hot, and thereby moves and changes it. Now it is not possible that the same thing should be at once in actuality and potentiality in the same respect, but only in different respects. For what is actually hot cannot simultaneously be potentially hot; but it is simultaneously potentially  cold. It is therefore impossible that in the same respect and in the same way a thing should be both mover and moved, i.e. that it should move itself. Therefore, whatever is in motion must be put in motion by another. If that by which it is put in motion be itself put in motion, then this also must needs be put in motion by another, and that by another again. But this cannot go on to infinity, because then there would be no first mover, and, consequently, no other mover; seeing that subsequent movers move only inasmuch as they are put in motion by the first mover; as the staff moves only because it is put in motion by the hand. Therefore it is necessary to arrive at a first mover, put in motion by no other; and this everyone understands to be God. 
The only thing I see missing is an explicit definition of essentially-ordered series, which is implicit in the underlined part.  You can translate that as subsequent movers [pianos] move only inasmuch as they are put in motion by the first mover [pianist].  He mentions in another article that accidentally-ordered series might regress infinitely; but offhand I do not recall where he did this.  Remember, he wrote this for graduate students in theology.  They would already have mastered basic metaphysics and physics for their master degree.  The same is true of my math books.  Many of them do not bother to rehash undergraduate math.  E.g., Atiyah's Introduction to Commutative Algebra starts with the assumption that you have already studied abstract algebra.

29 comments:

  1. But then none of the subsequent changers in the series would have any power whatsoever and there would be no change in the world.

    I did not follow why this is. Why does there have to be a first mover in the chain in order for any element of the chain to have power to change? Is it because the chain, itself, must be endowed with the power to change, and since it can't endow itself, must be endowed by something else?

    Couldn't one just argue that the power to change came from nowhere, or was always there? But I guess "where does the power to change come from" was the question to begin with, so I haven't answered the question, but just restated it.

    But this cannot go on to infinity, because then there would be no first mover, and, consequently, no other mover; seeing that subsequent movers move only inasmuch as they are put in motion by the first mover; as the staff moves only because it is put in motion by the hand. Therefore it is necessary to arrive at a first mover, put in motion by no other.

    I did not understand this either. It seems the Ox is just saying "there must be a first mover, because if there isn't then there wouldn't be," (i.e., begging the question), but I think I am just misunderstanding.

    Alas and alack, I don't have a graduate degree in metaphysics, so I assume I am lacking the foundation necessary to understand.

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    1. Thinking about it, I think I answered my own question. The power for something to change can't come from nowhere because then it wouldn't exist (ex nihilio, nihil fit), so it must come from somewhere. If I say it came from the thing before it, I also haven't answered the question, because I can just ask where did that get its power to change. If I appeal to an infinite regression, I haven't ever actually answered the question, but just basically said "it was always there", which is as good as "it came from nowhere", which (see above).

      Am I anywhere near the right track?

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    2. A series of toppling dominoes can regress indefinitely because the power of the nth domino to knock down the n+1th domino is there once it has been knocked itself. That it, once n has been knocked, the n-1th domino can disappear from the universe and n would still knock over the n+1th. The same applies to the classic example of A being the parent of B being the parent of C. The ability of B to father C exists regardless whether A has continued to exist or not. So there can be an infinite regress of fathers (or mothers, if you like).

      The same does not apply to a golf swing or the playing of the Waldstein. The piano strings have no power whatsoever to produce the sonata unless concurrently there is a pianist tickling the ivories. If there were no pianist, the keys would not move, the hammers would not strike, the strings would not vibrate, the air would not shimmer with those compression waves we call the Waldstein Sonata.

      Similarly, the golf club has no power whatsoever to strike the ball unless concurrently there is a golfer swinging it. (Even then, alas, some duffers do not connect; but that is a different story.)

      Thus, there must be a first mover in the series, since otherwise the second(ary) movers (instrumental causes) would not move.

      Even for accidentally-ordered series like the dominoes or the fathers, we have the example of the taught teachers of the Pythagorean theorem or the sent senders of the forwarded email. There must have been an untaught teacher; there must have been an unsent sender because the act of teaching or the act of forwarding does not create the content. But there is not requirement in such a series that the first cause continues to exist after that.

      In an essentially-ordered series, the first cause must exist right here, right now because otherwise the motion would not be happening right here, right now. The series drills down in the present; it does not fade back into the past. An infinite series in the present would not have happened yet, since there is no final term in an infinite series. You would wait forever for the golf club to swing.

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    3. I see, but now don't we have to prove that the universe is essentially-ordered and not accidentally-ordered?

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    4. No, only that there are such series. Besides, the universe is not an element, but a set; viz., {X|X exists}

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    5. OFloinn: "A series of toppling dominoes can regress indefinitely because the power of the nth domino to knock down the n+1th domino is there once it has been knocked itself. That it, once n has been knocked, the n-1th domino can disappear from the universe and n would still knock over the n+1th. The same applies to the classic example of A being the parent of B being the parent of C. The ability of B to father C exists regardless whether A has continued to exist or not. So there can be an infinite regress of fathers (or mothers, if you like)."

      'Indefinite' and 'infiinite' are not the same thing. Some series could have as many items as the cube of the number of elementary particle in the universe, and the series will still not be of infinite count. One could "remove from the universe" any number of prior [n-1] items of the series, and the series will still not be of infinite count. Only if the series itself is "(eternally) removed from the universe" of time-and-change (as, for example, the series of natural numbers), can a series be of infinite extent/duration.

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    6. 'Infinity' is a/the number to which one cannot, even in principle, count. 'Infinity' is the answer to the question, "What is n/0?"

      No matter the value of 'n', if n is in principle countable, then [n-1] != [n], and thus [n] != [infinity], and thus the series of which n is the count is not of infinite extent or duration.

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  2. Mr.Flynn, I think Aquinas mentions the possibility of an accidental-ordered series in Question 46 (on the duration of the world) of the first book of the Summa:
    http://www.newadvent.org/summa/1046.htm

    Dr. Feser also cites question 13 (on the existence of God) of the Summa Contra Gentiles:
    http://www2.nd.edu/Departments/Maritain/etext/gc1_13.htm

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  3. "Mr. Brow concluded by stating "By then end of the argument, there are a fairly good-sized number of assumptions." A fairly good-sized number? But it is not clear that there are more than the two mentioned, the Major and the minor premise, which are not assumptions but lemmas proven independently."

    Ah, but you're assuming that there is truth, that we can know at least some truth, that we can reason from known truth to previously unknown truth, and that we can know that we know truth. This is dangerous ground! Well, to the likes of Onebrow.

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  4. But is the Waldstein series really concurrent?

    Eg.
    # the bones of the middle ear, which are moved by
    # the tympanum, which is moved by

    is it not a temporal process?

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    1. You will note the use of the present imperfect. Whatever is changing is being changed by another. As Whitehead pointed out in his Principle of Relativity, experience is in intervals, not in fictitious instantaneous moments.

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    2. Experience is also subjective, not physical nor metaphysical. A metaphysical proof might discuss the reality of experiences, but can not usefully rely on experiences to determine the nature of things.

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  5. <>

    I think this is what shows that God is not a watchmaker god but rather, is needed for the continued existence of the universe. There is nothing that allows the change of potential to action except the act of God's will. Which also introduces the idea of the First Mover being a willed first mover rather than a multiverse which cannot cause the change from potential to act.

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  6. "God as watchmaker" isn't false, it's impomplete.

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    1. Most heresies lack and grow distorted. God as watchmaker lacks and grows distorted into something else.

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    2. *Everything* we say, or can say, about God is incomplete. Necessarily.

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    3. I'd just about bet that you subscribe to 'hylomorphism' (and quite possibly visciously so), and that you *also* subscribe to praying to dead people and to statues of dead people, rather than praying to Christ.

      Now, in their own right, praying to dead people and statues of dead people is *worse* than a heresy, besides being futile. But, further, in light of 'hylomorphism', praying to dead people is equivalent to praying to things which don't even exist.

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    4. Is there any other passive-aggressive snark you'd like to bring into play?

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  7. I appreciate the effort you put into this post. First, I did post some detail about why the effort seems impossible to me. It's not about the individual proofs, specfically.

    If you have been a regular reader of Good Math, Bad Math, you probably recall reading one or another of Chu-Carroll's discussions of Cantor Crankery such as this most recent one. The attempt to prove a proposition along the lines of "the God of Catholicism exists" from a proposition such as "motion happens" is akin to trying to order the reals to align with the counting numbers.

    A handful of more specific criticisms concerning your post:

    "That which is not-X cannot make itself X" is not a special case nor a particular for "That which does not (yet) exist cannot do diddly-squat". It is quite compatible for the non-existent to do diddly-squat and yet for the extant to alter themselves by virtue of their form.

    Unless you claim you can show "Ex nihilo nihil fit" from "motion happens", it's an additional assumption, and not a trivial one. Even then, it does not support the attempted conclusion in that paragraph.

    I have doowloaded the Oderberg article, though, and will read it thoroughly.

    I have seen no good defintion for the interval of change required for a series to be essentially ordered instead of accidentally ordered. The distinction is largely superficial and subjective, from what I can tell. For example, you shine a laser at two targets (liend up so you can hit both aiming for one place). Target A is four light-seconds away, target B is eight. You shine the laser for five seconds. Using the interval definition, the series source-light-A is essentially ordered, while source-light-B is not. I don't see anyting useful that says about either sequence.

    Your choice of an act of will by the pianist is an arbitrary stopping point. The pianists decision is often affected by (depending on the pianist) things like audience reaction, notes on paper, or music he is hearing while playing.

    If somethings have the form to change themselves, their needs to be no other reason for series to begin and end.

    It seems you think I meant that all arguemnts rest on that particualr pair of assumptions. I am not such a fool. You can always choose different assumptions to rest things on. The choice of the particualr assumptions is beside my point. My point is that such assumptions must always exist.

    As for the number of assumptions that lie behind the topological theorem, there are several, and you can choose from different starting assumptions, just like any other formal system. I don't think any individual axiom is required. However, there will be several axioms (or a small number of really powerful axioms), regardless, just like with any proof of God's existence.

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    1. You had claimed that the two lemmas were merely assumed. I showed that even in the precis contained in the Summa theologica that they were argued, contrary to your claim. Naturally, you reject the argument -- That is a foregone conclusion -- but your contention that the argument was a bunch of a priori assumptions is empirically untenable.
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      You appear to believe that I would be astonished by the existence of axioms in topology. I am not. But in mathematics we are generally clear on what are the assumptions and what are the propositions and conclusions. You have proven you can link to Wikipedia; but the point of the illustration was the use of subsidiary proofs, not whether they were based in ZF or not.
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      Your claim that "The attempt to prove a proposition along the lines of 'the God of Catholicism exists' from a proposition such as 'motion happens' is akin to trying to order the reals to align with the counting numbers" is not an argument. For one thing, in what way can the two be "akin"? It is not enough simply to assert an equivalence; you must show it. Secondly, the proof from motion was devised by Aristotle and I am confident that Aristotle was not trying to prove that 'the God of Catholicism' exists for the excellent reason that he had never heard of Catholicism -- Just as you have evidently never heard of Orthodoxy.
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      Your example with the lasers shows that you do not understand "essentially-ordered."
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      Your example with the pianist shows you do not understand "causation."
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    2. I thought using the expression "such as", as opposed to "for example", indicated "axioms of this sort", as opposed to "these exact two axioms". I believe they were argued. Their rejection would have been based on other, equivalent axioms. It can not be otherwise.

      No, I did not think you were in any way unaware fo the axioms. I was being a little cheekish at the end. I apologize for the offense. Although, since you engage in your share of such cheekiness, I was suprised you were offended.

      The two are akin, in that they ignore the basic limitations of the systems involved. Classical/propositonal logic does not have the power to go from lesser contraints to greater contraints, just as there are not enough rationals to form a 1-1 relationship with the reals, regardless of the construction of the argument (in logic)/assignment (of numbers).

      Perhaps I have been misinformed, but my understanding was that Aquinas argued that the Prime Mover, once established, must have all the traits assigned to the Catholic God. Since you seem to feel that the Orthodox God is different, what would you say that difference is?

      Perhaps my examples show that I understand it well enough to understand the flaws and the subjective standards used when claiming that there is some natural, metaphysical notion of "interval". Perhaps not. I have been looking for someone with the knowledge and patience to explain how that concept can be rigorously applied (Dr. Feser and the poster on his site apply it quite casually and subjectively). Is that you?

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    3. my understanding was that Aquinas argued that the Prime Mover, once established, must have all the traits assigned to the Catholic God.

      Sure, that's what he spent the next 500 or so pages establishing. Aristotle went only as far as the God; but he cannot be accused of having some axe to grind or pre-conceived notion he was aching to get to.

      Since you seem to feel that the Orthodox God is different, what would you say that difference is?

      You misunderstand. Fundamentalists always obsess about the Catholics; hence, "the Catholic God." But the Orthodox Church has the same beliefs, though sometimes expressed differently. So what you want is a term that would include Catholicism, Orthodoxy, and the Peripatetic School of Athens.

      Perhaps my examples show that I understand it well enough to understand the flaws and the subjective standards used when claiming that there is some natural, metaphysical notion of "interval".

      No, they do not. Nor is there anything especially odd about interval. Try the first chapter in Whitehead's The Principle of Relativity with Applications to Physical Science, specifically p.7 et seq.

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    4. "You had claimed that the two lemmas were merely assumed. I showed that even in the precis contained in the Summa theologica that they were argued, contrary to your claim. Naturally, you reject the argument -- That is a foregone conclusion -- but your contention that the argument was a bunch of a priori assumptions is empirically untenable."

      Well, in his defense, as we witnessed just the other day, ol' Onebrow doesn't quite grasp the distinction between 'assumption' and 'conclusion'. Apparently, in his strange little world, 'assumption' can also mean "any conclusion I don't like".

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    5. Thank you for confirming my understanding of Aquinas' efforts.

      I was raised Catholic. The only service I have fond memories of attending was the basement service at St. Francis Xavier, on the campus of St. Louis University. For me, the Catholic notion of God is the prototypical notion of God in Western Christianity, not some strange side idea.

      I'm not saying there is anything odd about the notion of an interval per se. I'm saying that it's use in the separation of essentially ordered and accidentally order series of causes comes across as being subjective.

      Let's take your specific example of someone listening to a piano player. Someone a thousand feet away hears any individual note (say by some tube that allows the sound to carry) about .8 seconds after it starts. Does that make it accidentally ordered? If year, how about at 2,000 or 10,000 feet? If no, how about 500 or 100 feet? Does it depend on the length of the note? I'm interested in a rigorous method of distinction that does not require the subjective perception of the listener; I'm interested in how the answers are determined much more than the answer itself. Metaphysics is a formal process. If you don't have precise definitions, you aren't proving anything.

      While Whitehead deserves a good reading, I don't see passages such as "Events essentially involve this passage" as supporting an Aquinean mindset. I'm curious if there are any particualr notions you interpret as doing so, that I should keep in mind when I read.

      By the way, did you have any thoughts (in agreement ro disagreement) on the article I linkeed to above? Did it clarify for you why I disbelieve such proofs are possible?

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    6. "Events" as opposed to fictional instantaneous moments is not what distinguishes essential from accidental ordering, as the very terms imply. In an essentially-ordered series, the intermediate terms have no causal power on their own. The piano makes no music unless the pianist is playing it. That's why it's called an "instrumental" cause. Similarly, a golf club has no power to cause the ball's flight unless it is being swung by a golfer. It is quite evident that it cannot be "instrumental causes all the way down" any more than it can be "turtles all the way down."

      The use of the present imperfect is intended to break the modernist mental set that insists on "moments" in time and protests that there is an interval between the pressing of the piano key and the hammer striking the chord. You don't need lasers to distant stars to get caught in that cul de sac. Hence, the side reference to Whitehead.

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    7. TheOFloinn,


      So, in the example above, even if the tube is 10,000 feet long and it takes several seconds for sound to travel, the series is essentially ordered (as you understand it), since at no point does a agent with indepenent causal power intervene? This is different from what others have told me, but I can adapt to the language as you are using it. I just want to make sure I understand your terminology.

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    8. Simple: ask yourself this: Can the air in the tube produce the Moonlight Sonata if it is not being moved by something else? Essential comes from esse (to be). It is this dependence of an instrumental cause on a fundamental cause that tells us that even an infinitely large collection of musical instruments cannot make music without musicians; or an infinitely large number of forwarded emails cannot exist unless someone has written the email.

      Time lag, echo, persistence, and such-like are irrelevant. Hence, we say "Whatever IS BEING changed..." and not "Whatever changes..." or "Whatever has changed..."

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  8. Sorry, should have used "rejection/acceptance" in the first paragraph.

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