The Infinite. Part 1: The Finite.

This series examines, from a somewhat naive point of view, the meaning of “infinite” in a number of contexts. Joseph Smith delves deeply into the infinite, and in particular in funeral sermons, even though he does not engage it with rigor. (Parts of this series appeared elsewhere.)

There are several ways Joseph Smith finds need for the infinite. Some of these became or were immediately controversial but most of them have not been carefully examined outside very narrow wedges of literature.

It may be easier to grapple with the infinite by first thinking a bit about its negation: the finite. As a descriptor, “finite” can be an equivalent to being fixed or determined. More often it suggests the idea of boundedness, being limited. Theologically it might refer to having an existence which is conditional, or, it might be used to suggest something is subject to limitation of some kind.

Much of classical theology has centered around the postulate that God is unlimited in “every” way. For one thing, such an assumption seemed to be the only way God could be distinguished from other beings. That idea leads naturally to the notion that God created everything else, from nothing.[1] The postulate had and has many difficult consequences which remain puzzling to the thoughtful. Dealing with them has resulted in a very large literature and has included the option of simply shrugging the shoulders and giving up.

Time and its counterpart, timelessness, are related to the notions of finite and infinite in informal discourse. While the timeless is important in classical theology, it plays a marginal role in Mormonism.

Measurement is clearly related to at least some meanings of finite and measurement involves numbers in some form. For the sophisticated, the so-called counting numbers, 1, 2, 3, 4, . . . form the basis of measurement. Any collection of objects would be declared finite if it can be “counted” using some sufficiently large collection of numbers, 1, 2, 3, . . . K. (Number of people on the earth ~ 6.5 billion. Number of planets in solar system 8 – I guess).

So far, I have hidden an important principle in the discussion. It may seem natural to most of you, but historically it is critical to the landscape of counting and theology. That principle is the so-called “law of excluded middle.” Briefly, it requires that every statement, where the assignment of truth value makes sense, is either “true” or “false.” False being the equivalent of “not true.” There is no other possible assignment to be made. “Middle” values, are altogether excluded.[2] In the case of finite and infinite, infinite is simply the negation of finite. That which is not finite, is infinite. Indeed, that will be our working definition of infinite for the moment.

To be continued. (Part 2 is here.
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[1] This does not mean that creatio ex nihilo was first proposed as a distancing strategy between God and “not God.” To be sure, it came to be used in that way but its initial use was in a more practical vein perhaps.

[2] Determining the meaning of “true” is a different adventure.

Comments

  1. Ah, the great problem of the law of excluded middle in all this…

  2. My thought is increasingly moving towards the idea that the infinite, in any sort of theological sense, is a myth, and that everything is bounded in one way or another for agency and, you know, pretty much anything, to exist. But I’m interested in reading more perspectives on the matter!

  3. Which leads to a great discussion of transfinite.

  4. Perhaps you could tell us about the Riemann sphere and Penrose-Carter diagrams.

  5. Love this stuff, WVS.

  6. Interesting. I’m thinking this could go in some interesting directions. Perhaps the most interesting one for me is the difference between the theoretical acknowledgment and definition of infinity (not finite) and the practical reality of whether not infinity even exists.

    Clearly infinity as a concept is powerful in allowing us do some neat tricks (i.e. prove stuff) with mathematics, but as a practical reality I think it comes up a bit short (i.e. we have to accept it as an axiom). When the practical and theoretical are juxtaposed in this way it seems we can only make an argument as to the veracity of infinity based on its usefulness. But that doesn’t make it “true.”

  7. jmb275, the debate about the infinite stems partly from our ability to perceive reality. We only see so far, hear so much, touch what is within arms length. Infinity is a matter of faith, like faith in God. Indeed, the two are closely connected.

  8. We most commonly deal with infinity when dealing with an infinite collection of infinitely small members. Someone’s declaration on a chalkboard in an office of fluid dynamics graduate students has remained lodged in my mind: “Nature is a continuum!” I think the writer meant something like “Atoms, shmatoms.”

  9. To me it is apparent that our theology is the theology of the infinite. We sing about it in hymns, we read about it in our authoritative scriptures, we think about it. This OP would not make much sense in any other theological context. The most original theological idea is that we participate in these infinities in a real way whereas in most other theologies the infinite only applies to God.

    We have no idea what the implications of the infinite are because this life is the life of the finite. Limited time, limited space, limited resources, limited love.

    I have tried the regression to infinity. Start with what we have now and double it. Try to imagine what that means. Then double it again and re-imagine, then redouble and re-imagine. Somewhere beyond three or four doublings we will find something which feels a little infinite to our constrained minds.

    But the ontological implications of living in multiple dimensions of the infinite are not imaginable or, obviously, quantifiable. This life is the opposition to infinity, which, I believe, we needed to understand in order to progress. In the inverse argument, we, living in the infinities, could not really imagine the finite.

  10. re#2, over the last decade since my mission, I’ve been moving to a similar perspective about everything being bounded. However recently life changed for me and I decided it was best to get a new degree (the economy and all that, and I see the hand of the Lord moving me here).

    Right now I’m just finishing up a course on mathematical proofs and concepts. Recently we covered the concepts of finite and infinite and as I understood it it was one of those spiritual/theological eureka moments for me and my perspective has completely shifted.

    First I learned that (as was mentioned above) finite is anything that you can count using a set of natural numbers {1, 2, 3, …, n} and infinite it anything that was not finite (i.e. no matter how big the number, n, infinity will always be bigger) which was basically the idea of infinity that I had before.

    Next however I learned that there are different orders of infinity and that some are “relatively small” known as countably infinite sets, and that others were what my prof called “big” known as un-countably infinite (which is any infinite set that is not countably infinite).

    Over the two weeks that we covered these concepts I took the opportunity study and to further clarify these ideas with the prof (who has no religious belief as far as I can tell). I felt like I was being instructed by the Lord about his very real nature. Its difficult to explain in a only few paragraphs the things I’ve learned and the ideas that I’ve felt, but it has brought to me a strong testimony of the reality of the actually infinite and divine nature of our Father in Heaven. (which testimony has been strongly reinforced by my scripture study (especially in the the Book of Abraham and the Doctrine & Covenants) since then.

    Really looking forward to the next post in this series.

  11. Rio, glad to know you’ve encountered Aleph numbers. Neat stuff there.

  12. BTW, consider what adding a dimension does to finiteness.

    E.g. a 2-D finite shape (a square) becomes a 3-D shape (a box, solid or not). It is, from the 2-D perspective, now infinite.

  13. Discussions of the infinite are much more interesting when informed by a knowledge of modern analysis.

    Georg Cantor used diagonalization (among other techniques) to prove the notion of the uncountable. If one accepts the concept of an infinite collection, one is logically forced to accept the construction of a “larger” infinite collection, ad infinitum.

    In treating the infinite, one has to be extremely careful. The whole foundation of calculus relies on carefully constructing the notion of a “limit” of a sequence, and defining when such a limit exists. The Cauchy criterion helps us handle the infinite without ever actually reaching such a thing as “infinity”.

    Gödel, in dealing with questions raised by Cantor, among others, demonstrated that any countable collection of beliefs or axioms about the natural numbers cannot prove every truth about said numbers, and that such a collection can’t be used to prove its own consistency.

    Moving on, it has since been shown that there are classes of problems or questions that can be asked that are “undecidable”. No formal method can be used to definitively answer the question for all inputs.

    There are very important open questions about infinite collections of problems, such as the P versus NP problem, that we may believe we have the answer to but it may not be provable, and what’s worse, may not be possible to prove that it’s unprovable.

    IMHO, it is very naive to talk about the “infinite” in relation to God without accepting what has happened to the concept of the infinite over the past century or so. In relation to space or time, the “infinite” we talk about tends to be the countably infinite. If I only had a countably infinite number of “time”, “space”, or “thought” units to work in, it turns out that it would be impossible for me to compute even a fractional portion of what I might deem “all truth”. Are we unintentionally limiting God that way, by only giving Him a countably infinite amount of resources with which to work? But how can our linear minds conceive of anything more than countably infinite time? And if we managed somehow to have uncountably infinite space (somehow still counted by God), how would one manage to visit it all in countably infinite time?

    What about the possible implications of turing completeness and computational irreducibility? We have very strong reasons to believe that some questions can only be answered with a long computation that there is no shortcut for. If we posed such questions to God, does He know the answer immediately? Does he freeze time and then compute the answer? Can he break RSA encryption (never mind Shor’s algorithm — find something similar if you’re picky) at a moment’s notice? Does P = NP in his universe of knowledge?

    I can’t accept the idea of “infinite” as the opposite of “finite”. In set theory, I’d say the equivalent of “opposite” is “complement”, and thus the opposite of something depends on your universe of discourse. The opposite of finite is only infinite if your universe of discourse is infinite. And it’s only countably infinite if your universe of discourse is countably infinite…

    =D I hope to hear more interesting thoughts!

  14. Relative to talking about God I think Mormons put sufficient restrictions on God and reject the idea he is the source of all being such that we really can’t have the same discussion. How one talks about infinity (and what kind of infinity) really depends upon the subject matter at hand to which ones mathematics is being applied.

  15. it's a series of tubes says:

    We have very strong reasons to believe that some questions can only be answered with a long computation that there is no shortcut for. If we posed such questions to God, does He know the answer immediately?

    Though he approaches issues from a secular humanist / transhumanist perspective, you might find Ray Kurzweil’s “The Age of Spiritual Machines” interesting reading – particularly towards the end re the concept of the universe as a great computational engine.

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