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?)

+ + +

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.