Axiom of Choice

“Choice and Accountability.” We hear the phrase frequently and if you are a veteran of the LDS Young Women’s program, you’ve probably heard it a lot. The two go together, since you can’t be accountable for a decision, unless you have true alternatives to choose from, and your choice is not completely determined by circumstance outside yourself. I’m compressing and there are fine points here, but that’s how most would see it I imagine.

In mathematics, the Axiom of Choice is best stated in formal terms, but it would take a while to get you up to speed on it, so let me be vague. It says: if you have any collection of sets, none of which are empty, it’s possible to pick one thing from each. Sounds innocent and obvious, right? One consequence is the Banach-Tarski theorem, (1924) or Banach-Tarski Paradox. This is a really odd thing but also really fun: A solid three dimensional sphere can be cut into a finite number a pieces in a judicious way and those pieces reassembled into two solid spheres identical to the first. Or: one sphere spawns two, each of the same volume as the original. So the 4th century Christian-philosophers were right! Ex nihilo creation?

Alfred Tarski - argued against the axiom of choice like a theologian - fun but ineffective


The Banach-Tarski Paradox is not a logical paradox, but it gives one pause.[1] The power to choose may have unforeseen consequences.

I think it is fair to say that in Mormonism, we believe in the power of responsible choice. Indeed, if you believe the stories, war was fought in Heaven over the issue of exercising that power. I think that if we drill down far enough, most Saints would agree that Mormonism requires that some form of free-will (FW) for human beings exists. What’s the potential for paradox here? Yeah. That old black magic of exhaustive foreknowledge (EF). But in this case it’s worse. Banach-Tarski only looks like a logical paradox, because of our modern physics-inspired views about conservation. When Augustine encountered the real paradox of FW vs. EF, he chose EF. Now lots of paper has been cut and fitted into lots of books to try and weasel here. But Augustine was a very smart guy, and I think he saw the right of it. Pelagians out, EF gets the win (not without some Augustinian angst of course). Assuming this is right for a moment, we should observe that modern Christianity owes its commitment to EF (such as it is) to Augustine.[2]

There is a thread of EF that runs through Mormonism and I think we can say that we may owe some of this to Augustine, by way of the Puritans. Related to this is the question of agency. Granting agency is a big theme in Mormonism. But it seems unlikely that this term, used in this way, is equivalent to free will. As Truman Madsen said once, free will is the driver, agency is the car. Road blocks effect where the car may go, but not necessarily where the driver would go. A sort of brain-body thing.[3]

Here’s the question: discursively, how committed are we to EF? Joseph Smith can be construed as being on the side of EF – “I believe that God foreknew every thing”[4] but I think a little contextualization serves us well here. Smith was speaking only, I think, of certain outcomes in the Garden of Eden. Room to weasel there. But what about definite commitments to EF? Well you won’t find the hard definition in Mormon Discourse. But perhaps we can find something definite enough to talk about.[5]

First, it’s clear that Brigham Young took a pretty hard stand against omniscience. As a practical matter he just found it hard to believe that there would not always be things to learn. Mormon deification plays into this and Young is using the same analogical reasoning that gave rise to the Heavenly Mother ideas: we shall never cease to learn, so neither does God.

We shall never cease to learn, unless we apostatize from the religion
of Jesus Christ. Then we shall cease to increase, and will continue to
decrease and decompose, until we return to our native element. Can you
understand that? It is a subject worthy the attention of the eminent
divines of Christendom, and they may search into it until they are tired,
and still know comparatively little about it, while I preach it to you in
a few words.

Young makes clear elsewhere that this decomposition is the fate of the sons of perdition and the devils of hell. But again, the important point is that he thinks we may never cease to learn, ergo, God never ceases to learn (his conclusion, not mine). BY is teaching this in a very definite fashion, counter to Orson Pratt.[Compare his of JD 4:126-7] Pratt subscribed to omniscience and he and Young went at it over the issue. So the score is 1-1. It’s safe to say that at the time of this discussion the rest of the Church authorities were on Young’s side, or at least silent about it.

Casual uses of “omniscient” probably should not count. But I think Ezra Taft Benson can be counted as being in favor of EF. See his talk 4 October 1987.

Elder Bruce R. McConkie was pretty much in favor of EF. You can find this any number of places, but take for instance his Mormon Doctrine. Just look up omniscience.

James E. Talmage, the Nautilus of the modern Mormon correlated doctrine fleet, played with omniscience a bit. For example, p. 77 of the first edition of his Articles of Faith. Current edition, p. 75. (Talmage did not believe in any sort of “backward” eternal existence of the human soul which may play into this.)

The atonement wrought by Jesus Christ is a necessary sequence of the transgression of Adam; and, as the infinite foreknowledge of God made clear to Him the one even before Adam was place on earth

Keeping score is fun, but probably not very useful since what we really need to understand here is not whether this or that person believes in omniscience, the supercategory of EF, but how they reconcile it with FW. By the time you do the work to get down to that point, most people are just smiling and shrugging their shoulders. I mentioned (see [2]) that the escape trajectory from this paradox is to weaken either FW or EF. Let’s examine just two of these. First, limited foreknowledge. In this case, God predicts the future in fine detail but it is *possible* for surprise to happen as events unfold (but see the note on general relativity in [1]). Agents *may* modify the picture since choice cannot be predicted with logical certainty. The second alternative: one is frozen in choice sequences by one’s previous choices, not by an outside being (God). Self-determinism might fit in with Mormonism, even with Joseph Smith’s idea of uncreate selves. An infinite history of choices determines present choice and presumably God could know that history. Some kind of thing like this would be required to hold on to “Choice and Accountability.”[6]

B. H. Roberts found his reconciliation with a God who is temporally omniscient. That is, at this moment, God knows everything that it is possible to know at this moment. Roberts eschewed EF.[7] Dumping FW in favor of self-determnism probably matches Neal Maxwell’s position, but it is unclear if Maxwell believed in EF.

It is interesting that the tension we see between FW and EF is rather like that found in the current political discussions over national security and personal freedom. Want to be safe, or free. Some of both is what we will get, but how much of each?

I think that Mormons who subscribe to the limited foreknowledge idea are precisely those troubled by the negative effect EF has on “Choice and Accountability,” clearly a cardinal principle of Mormonism. What’s more important, EF, C and A, LFW? I’m not sure about LFW in all its glory, but I have all kinds of trouble with EF. So count me in the limited foreknowledge camp. I’m avoiding all the middle knowledge stuff, etc. feel free to bring it up if you want to. I’m quite comfortable giving up EF. For me it is balanced by personal eternity though I’m not necessarily a compatibilist possibly like Maxwell. I think that position is *incompatible* with C and A. And C and A rules in Mormonism. Your milage may vary.

———————————
[1] You may believe that some slight-of-hand is present here, but not so. Tarski felt the axiom of choice was too broad and would lead to paradoxical things. But the Banach-Tarski paradox is not one of them. This pleads for an answer to the question: can I take my regulation softball and cut it up and make two regulation softballs? Sure. But you will need a special saw. <grin>. The Axiom of Choice has a somewhat checkered history as implied by Tarski. In 1940, Godel showed that it is consistent with the rest of mathematics, in the sense that it does not introduce logical contradiction. (It’s also independent, but that’s another story.) Incidentally, near the end of the same decade, Gödel showed that the general relativity theory essentially entails no FW – but think, bald guy on Fringe.

[2] Keep in mind, that EF is a technical condition. Two ways have emerged to deal with the question of shoving accountability and foreknowledge together. Essentially they amount to weakening one or the other. Mormons are fundamentally committed, I think, to humans having free will in a strong form. Otherwise “present” accountability is hard to swallow. There is many an “unless” here, but in a practical way some sacrifice must be made to avoid paradox. (Dennis Potter has given a short “proof” of this.) There are all kinds of subtexts here, like the nature of time, God and man. As an aside, in my experience there are plenty of Mormons who buy into the old Puritan notion of Providence. But yet they want something like FW. You can’t go to Disneyworld and Disneyland at the same time folks. Are we talking Libertarian Free Will here? I expect that random decision-making has to exist as a *possibility* even if not in a given individual, usually.

[3] See Madsen, Timeless Questions – Gospel Insights. Lecture 4. Bookcraft, 1998.

[4] See his speech of Feb. 5, 1840.

[5] At this point we could bring up the Lectures on Faith. They are highly revered in some circles and pretty clearly lean to omniscience. In fact they press the issue as required to generate true faith. This is good old Protestant talk. But it is difficult to know if it means EF.

[6] I admit that most people would find the latter distasteful perhaps. There is some reason to believe that EF is not a very useful state of affairs, and for the pragmatist Mormon that could be significant. Prayer is what I’m thinking of here, an issue I don’t really address above, but one I think is important. Asking God how to proceed in some life decision becomes problematic in EF. If the future is malleable then petitionary prayer makes much more sense (but Providential interpretations of prayer abound in Mormon discourse).

[7] See his Mormon Doctrine of Deity. Compare Kent Robson, “Omnipotence, Omnipresence and Omniscience in Mormon Theology.” Sunstone 10/4 (1983): 21.

Comments

  1. I deal with the paradox similarly to the way I deal with the theodicy–as to the theodicy I don’t believe in the omnis–rather I believe that God is very powerful and very good. Similarly, I believe that God is very knowledgeable. Whether that means EF of not, I don’t know, but I sort of doubt it.

  2. Eric Russell says:

    I think that agency is indeed LFW, even if some leaders sometimes misuse the word. I’d be interested in Madsen’s grounds for his car analogy. As for EF, I don’t think giving it up is such a terrible thing and I don’t think anything but absolute literalism demands it.

  3. Thomas Parkin says:

    I completely reject EF. God has a pretty good idea where things are going – partly because things have powerful tendencies to go one way. But, in the details there is all kinds of wiggle room and unexpected turns. His words never fail not because He knows the precise end from the beginning, but because His influence is such that He is able to bring things to happen. An example of this: when Jesus is asked why He is doing this and that, He replies, “because it was prophesied that I would.”

    I like the old idea of foreordination. Look we’re going to set you apart to do this and that kind of thing, but should things wind up going some other way for you, just know that there will be plans B,C,D, or whatever we need to have.

  4. WVS,
    I give you compatiblism:

    http://en.wikipedia.org/wiki/Compatibilism

    Substitute determinism with EF–
    Augustine withstanding, I just don’t see any convincing reason that free-will and EF are mutually exclusive. If God is atemporal, then all things can be before him in space and time–yet for temporal humans choice is still there.

  5. I don’t see that they are mutually exclusive either. God knows the past yet I had complete Free Will then. So him “knowing” something does not negate free will. If I believe that God isn’t bound to time the same way that we are, it means he can “remember” the future just like he remembers the past.

  6. mmiles, if you go with Mormon materialism, God has a material body. Hence he is not atemporal.

  7. Thomas Parkin says:

    When, many years ago, I first encountered the determinism crowd I was rather shaken by it. So, I tried several experiments on myself. I tried to make simple choices,- like placing a coin in both hands then choosing which coin to drop, or choosing which of several books to take off a shelf,- then observe myself closely while making that choice. (Yes, indeed, I was just that odd.)

    What I discovered was that at the moment of choice something would happen, something would say ‘this book’ or ‘that book.’ But where that speaker came from I could not determine. It wasn’t my consciousness calling for the voice, let alone determining the voice, as far as I could tell. The speaking voice didn’t even necessarily sound like the voice of my consciousness. Something was choosing, or seemed to be choosing, but while it came from my subconscious – or, rather, because it came from my subconscious – I could not determine what it was that was choosing, or why.

    This shook me further. While I sat on it I came to feel that I (??) must decide against freedom as I had thought of it up to that point. Either my choices were being determined and I was simply observing the result of some infinitely complex equation running through even the simplest things, or something was free and was choosing but if that something was me, my will to this or my will to that, my freedom was centered in some part of me that my consciousness was not in control of.

    Where are we, I of course thought, with free agency then? I wasn’t concerned with the idea of responsibility, since whether I had consciously chosen my actions or not I had still been the agent of actions and was responsible for them.

    That idea of freedom, certainly the ‘illusion’ my freedom, stayed with me, though, and slowly I came to see my old observations in a new way. In the very act of observing the choosing there came an elongation of the process, in other words an intervention. I came to decide this: somewhere in that field where the voices that are produced by my unconscious process enters into my consciousness there is a weighing up by something that I recognize as _me_. It may happen quickly, and it may not even be a product of consciousness, but there is a sifting and deciding and I am, in some measure, in charge of it. Somewhere on that field there emerges an uncaused effect, a first cause, magic, my choice. The more I observed now, the more I believed I saw it, that it was not an illusion. Where the two or more ideas are held in stasis, something I recognize as me enters in and chooses. I have replayed those experiments many times, now, and the more I do it the more I sense myself intervening in the process. In a way, this may be nothing more than a slow extension of my awareness of, turning a flashlight on the borders of, my unconscious. But it is surely me that is doing it.

    So, I decide that we are free, after all. And if free, then the future cannot be predetermined. If there is contingency there is contingency and God is also subject to it. I don’t believe God travels backwards through time, and so I think the phrase ‘remembers’ the future is kind of nonsense. But, if He does, then that contingency that happens in the present must also move backwards through time making not only the future but the past indeterminate. While God may not experience time in precisely the same way we do, He occupies only one point in time and space and in some way experiences the forward flow of events. The past is determined and the future, as Yoda says, is always in motion.

    Groovy.

  8. wondering says:

    Detailed visions of the future in the Book of Mormon, including even small details like people’s names, are perhaps difficult to reconcile with free will.

  9. Thomas Parkin says:

    wondering,

    I don’t think so. All one needs is a general idea of the future, and the details can then be pressed on to match the prophecy, or the prophecy can fail, as per Paul, in which case the details might be scrubbed from the final product. I think prophecy, even when fairly specific (there are very few examples of this), is always a bad indication of determinism … since God’s power in the present can always be substituted for His supposed exhaustive foreknowledge in the past.

    I just think people have a very hard time letting go of the magic wand, all penetrating God. In which case they will, it seems to me, have a difficult time embracing Mormonism.

  10. Eric Russell says:

    AccuScore isn’t bad at simulating games and forecasting scores, but I would never bet against AccuGod. Lack of EF notwithstanding.

  11. A pretty good fictional attempt at blending EF and FW (compatabilism–determinatism vs. causation) is Isaac Asimov’s Foundation series with Hari Seldon “psychohistory” (really psycho future). http://en.wikipedia.org/wiki/Hari_Seldon

    How does quantum mechanics and the Heidegger uncertainty principle fit in the EF?

  12. DavidH, nice one with Asimov. As far as uncertainty goes, the outcome is as yet uncertain. The blending of QM and gravity is part of the puzzle and the way in which it is done makes a difference perhaps in the possibility of EF. And then there is the question of how you consider spirit-body interface in Mormonism. And, if you go the Roberts route, perhaps non-material centers of consciousness get to play.

  13. #11/#12,
    Exactly what I was getting at. Mormon materialism isn’t a barrier.

  14. I’m not sure about LFW in all its glory, but I have all kinds of trouble with EF.

    Bingo. You are a brother.

  15. Bill, this was simply brilliant. Axiom of Choice and FW! Wonderful, and clever combination. I reject EF completely. Compatiblist arguments have always seemed wishful thinking. If from some perspective God sees everything, then there is only one thing to see from some perspective, and FW is an illusion (I think Dennett does a nice job of laying this out in ‘Freedom Evolves’). The EF comes right out of Plotinus, and is not the weeping God of Mormonism, in which God does more than sit in some self-contemplating oneness looking down on existence like a great 4D map of all that happens.

  16. SteveP, thanks.

  17. I have always been troubled by the notion of an omnicient God who knows our every reaction to every situation we are in. Free will seems impossible with these parameters where we could be set up to fail or set up to succeed. The thoughts from you big- brained people give me some interesting insights to consider.

  18. what's this? says:

    If from some perspective God sees everything, then there is only one thing to see from some perspective, and FW is an illusion

    Why must perception = causation?

    Also, #6 – why can’t a physical body be atemporal? One reasonable reading of Alma 40:8 would seem to indicate that the yardstick by which we measure time is simply inapplicable with respect to God.

  19. I like how the original post is careful to distinguish free will and agency. If we’re defining free will as something like the ability to surprise God, than connections between free will and Mormon concepts of agency seem marginal to me. I thought Joseph Smith’s big contribution to the subject was to reject ex nihilo creation in favor of uncreated spirits. If God created everything about everyone’s spirits and environments, in what sense is anyone responsible for any choice, predicable or otherwise? Intelligences that are coeternal with God have instead some part of their decision-making for which God is not responsible. If using that is what agency constitutes, than eternal foreknowledge seems a different issue.

  20. Bill, my brain is tired now. Thanks.

  21. Eric Russell says:

    Brian, the ability to surprise God isn’t a definition of free will, just a logical ramification of it. Free will is freedom from all determinism. Agency, using 2 Ne 2 along with D&C 101:78 as a basis, is the same thing.

  22. Time is defined by the material in motion. Hence, a material God is not atemporal.

  23. daveonline says:

    Dang, I just wrote out an elegant three part comment, then hit the wrong key and sent it off to another universe, (not this one unfortunately). Let’s see if I can reconstruct it.
    1. We are not atomistic agents who choose (whether freely or deterministically) with consequences only to ourselves. We choose most often with exponential consequences for other. The sins (and rightous choices) of the fathers are passed to the third and fourth generations. With my limited mortal EF I understand that a pregnant mother who drinks or uses cocaine will have agency destroying consequnces for her child. Yet within those limits and expansions, the cool “me” described by Thomas Parkin also acts to expand or contract the range of my agency and those of others around me. This is the nature of the insight given in Abraham when God found himself in a pool of other agents and contemplated how he might act to enlarge the agency of himself and of others.
    2. Eternal learning is not a set of facts to be verified by a multiple choice test, but is due to the hermeneutical nature of our experience creating a meaning unique to the “me” that we are. This occurs both in reaction to external, objective, reality – the loss of a job, winning the lottery, bad health, but also in reaction to seeing that people we are close to and who share the same experience also respond in ways we could never have anticipated. Thus to build on the sphere and TS Eliot. We have had the experience but missed the meaning. We will experience the same sphere of mortality, but our meaning will be unique and create a new sphere. This opportunity for each of our unique mes and sphere of meaning was so essential that a Savior was needed to bring it about. Why? To cross the ontological gap so we could learn to understand and empathize with the unique journey of others and see that in that collaborative process of seeking to appreciate the unique gift and burdens others bear (and not just focus on ours) we might find ourselves back in heavenly collaborative community where one sphere of mortality becomes created anew for each individual, who in sharing that with others creates its own exponential expansion of meaning and intrepretation
    3. I have suspected for a long time that the preexistence was a grand calculus of discussion where we did actually establish and map out the objective events of our lives. Even in this life, the objective fact of getting married and having kids has a level of meaning I could not even fathom 20 years ago. I “knew” that I loved my wife when we were married, but I cannot even comprehend now that I could have really meant it without knowing then the meaning it has for me now when I say that I love her.
    This grand council of group and invididual planning and negotiating likely occured within the context of one on ones’ with God and others where, like Abraham, we bargained for what we wanted/needed or were asked to carry out. The war was a result of a group wanting a guarantee ahead of time that they could be assured a “painless” existence. God, knowing that such was not possible also had to accept asking his own son to accept the greatest pains of all so that our natural interpretive reactions to pain might be overcome and even seen as essential.

    I will leave off with this abbreviated resummarization for now and see if I have time later to fill in more.

  24. I think that HF knows my future as it exists at this moment if I make no major changes in my life. If I use my agency to make a major change, then He will know my future on the new trajectory. As part of either future, He knows that I will have other choices to make, some life changing and some not, and knows what will happen because of each choice but not whether or not I will make the choice.

  25. I don’t know.

  26. Dad,
    I love reading these posts and find them very thought-provoking, but then I wonder….why did we never talk about this stuff in FHE :P

  27. Your mom hoped to protect you.

  28. it's a series of tubes says:

    Time is defined by the material in motion. Hence, a material God is not atemporal.

    Perhaps it’s a semantic point, but would you comment on Revelation 10:6 in light of this? Also, I like the earlier question as to why perception must imply causation – no one seems to have addressed this point, and to me it seems fundamental to the debate at hand.

  29. series of tubes: I think the argument usually goes, if the future is perceived accurately in all ways, then it is fixed (a la Godel). Then free will is an illusion in the sense that at any decision juncture, all choices but one are impossible. The “possibility” of selecting otherwise, does not exist. LFW advocates would see this as a no go. As far as Rev. 10:6 I think the passage is usually interpreted as saying “there will be no more delay.”

  30. Yeah! Someone explained this comic in a way I understand! Thanks WVS!!!

  31. it's a series of tubes says:

    I think the argument usually goes, if the future is perceived accurately in all ways, then it is fixed (ala Godel). Then free will is an illusion in the sense that at any decision juncture, all choices but one are impossible. The “possibility” of selecting otherwise, does not exist.

    Thanks for the summary. I disagree with both the premise and the conclusion, but appreciate the concise way you present it.

    As a physicist, I probably read D&C 88:41 and 67 differently than most; I draw a conclusion that satisfies me with respect to the “all things being before the face of God” issue (reasoning by analogy from how the universe would appear to me as an observer were I to reach the velocity of c).

  32. Kristine N: my pleasure.

  33. #31
    Nicely said. It’s the only thing that makes sense to me in light of scripture, and in light of physics.

Follow

Get every new post delivered to your Inbox.

Join 9,514 other followers