Must you believe in irrational numbers to go to heaven?

What is truth? The more I study it, the less I seem to understand it. How do we know it when we see it? How do we hold on to it? Is it stable? Are the truths of the present moment, the truths of the next as well? How can we tell?

Take this: the proof that the square root of ‘2’ is irrational. A real number. A number that if you wanted to write it down, you would never in all eternity finish. It is endless in its extent, but here are the first 10,000 numbers to get you started:

and so on and on . . .

Now look at this proof that the square root of ‘2’ is an irrational number.

Were you convinced? I followed it, but when I get to the end I cannot hold it all in my mind very long after. It seems to slip away. How did that go again? I have to trust my memory. I’m done reading it now. Do I still believe it? Did I miss something? Is it still true if I can’t remember the details? I could reconstruct it, I suppose, but then, when I turn away again for just a moment, it’s suddenly gone again and I have to trust that I did it right, or the person who wrote it did. Faith has entered in. Faith in the past. Yet, if you asked me if I thought that the square root of two is an irrational number, I would say, ‘you betcha.’ If you asked for some probability on how certain I was, I would say 100%. If you pressed me to lay down a wager, I would put a house up against a dollar bet.

Yet a lot of what goes into my sense that this is the truth is held in in places I just don’t think about at any given time. It relies on much other knowledge that I just don’t keep in front of me: Definitions of number; square root; what marks on the paper mean; what the words used in the proof mean, e.g., what ‘the’ and ‘is’ mean; what ‘odd’ and ‘even’ numbers point to; the context for my trust in rationality itself; my memory of what elements a proof must have; what it should leave out (is it relevant that it be written in black on white?); to what the symbols refer; how the order of presentation makes the proof coherent . . . Get the idea? I could go on for pages and pages. Most of this depends on brain structures that we barely have conscious access to. It depends on memory, which cognitive scientists now know is mostly a construction and is very loosely connected with what happens and matches reality only in broadest brush strokes. What does this imply for the proof given above?

A man at a meeting I recently attended stood up and said he did not believe in real numbers (numbers like the square root of ‘2’). An eminent historian of science had just given a lecture in which real numbers appeared and in the question and answer session this man stood and declared that there was no evidence for real numbers existing and they should be left out of physics. A philosopher next to me whispered, “He’s crazy.” “Yeah,” I said. I also knew he was crazy because he had cornered me in an opening social and I could tell he was wacky after a short conversation. After the session, several of us said, “How about that crazy guy?” (being in Canada, we said of course, “How ‘aboot’ that crazy guy?”)

He did not find the proof convincing for some reason, it would seem. Something, that formed the background assumptions necessary for structuring the proof had broken down in this man. We all referred to him as ‘crazy.’

We use ‘truth’ a lot in our religious discourse. But much of what we mean, relies on a memory of spiritual events. It is embedded in a matrix of other beliefs and framings about our world, personal history, personality, language, cognitive structures for which we have only limited access, and a host of other beliefs about the world.

This is not a bad thing. It is grounded in whatever we mean by ‘truth.’ But we seem sometimes to assume that by ‘the truth’ we mean something simple and accessible. Something that forms a recipe for which we all would agree has these x elements.

I don’t really have a profound point here. This man just set me thinking about ‘truth’ as such and I wondered: What do you mean by ‘truth?’

(And when is not believing some line of evidence worthy of being dubbed crazy?)


  1. Thomas Parkin says:

    I think you should rejoice because now you are primed to really love reading Borges.

    Also, truth is a knowledge of things as they were, as they are and as they will be. Our inability to capture or maintain that knowledge notwithstanding. The fact that at any given time the best we are doing is approximating doesn’t make it any less so. :) ~

  2. The real question (so to speak!) is whether or not you must believe that while there are just as many rational numbers as integers, there are many more real numbers than integers.

  3. Something tells me SteveP is already familiar with Borges.

    This reminds me of a point I used to try and make in my ethics class. The reason that we want our understood truths to be objective and their understood truths to be subjective is because it allows us to completely ignore them. You can ignore crazy people (unless they have knives). If I dispute a subjective principle (“those people are nice” “no they are not”), then I have a different opinion, which is just fine. If I dispute an objective principle (“the sky isn’t blue, it is orange with squiggly green lines in it”), then I am crazy. And, as stated earlier, once you have turned your opponent into a crazy person, you can ignore them (as ya’ll did with the crazy guy in the meeting). It is very powerful rhetoric.

  4. “I think you should rejoice because now you are primed to really love reading Borges.”

    Could I love him anymore than I do? No.

    Cynthia, there are about 46 more reals than integers, right? (I saw a picture Cantor’s triangle and I counted them and that’s how how many more reals there were.)

    John, I always turn my opponents into crazy persons, regardless of the issue.

  5. Mark D. says:

    What do you mean by ‘truth?’

    Truth is the way things really are. There is no such thing as subjective truth, or rather anyone who uses the term “truth” to refer to something subjective is essentially free riding on a term that by convention means exactly the opposite, namely convergence with reality. “Subjective truth” is an oxymoron.

    If anyone makes the claim “this is true”, the audience inevitably takes them to mean that the assertion exemplifies a high degree of fidelity to reality. No one assumes (by default) that what they mean is “this is my opinion”.

  6. Mark D. says:

    Or worse, this is an assertion whose relationship with reality is unknown, unknowable, indeterminate, or meaningless, and further more this assertion isn’t even consistent with itself or anything else I claim to be the case…

  7. In my mind, truth simply is. It is the fundamental nature of things.

    That being said, I don’t know that I know ANY truth. There are subjects about which I am more certain that I am “closer” to the truth and other subjects where I am further away and don’t know anything. I’m also willing to accept that anything I think is right may be shown to be wrong tomorrow.

    In a religious context, this is why I have a problem with a “testimony” and “I know…” phraseology. I honestly don’t “know” if the LDS Church is “true”. I don’t “know” that there is a God. I don’t know really anything. I feel strongly that there is likely a God. I hope the Church is true after having devoted 40+ years of my life to it. But who really “knows” what the truth is?

  8. Mark D, I beg to differ. All truth is in only in subjectivity. Truth is not what is out there, rather, “Truth is the knowledge . . .” Truth is a value of knowledge, not a thing out there. So all truth is in fact subjective. We want our beliefs to be true means that they line up with the way things really are. To be alined with underlying reality (we call this correspondence theory of truth) or to be in some sense non-contratictory (coherence theory of truth), is something we value in our beliefs, and in the end all truth is subjective. Without a knowing subject the notion of truth has no meaning.

  9. I don’t know SteveP and/or Mark. You have lost me on this.
    I don’t know if all truth is subjetive. I was at a baseball game last week, watching from behind the backstop. The pitcher clearly throw a ball__it was called a strike. Which is the truth? For the game, it’s a strike and forever it will be__but is it the “truth”?
    “What is Truth?”. I think the only question Jesus could not answer.

  10. I was told there wasn’t going to be any math.

  11. Greg, there is always math. In fact, to get into heaven, it is well known you will be given a pencil and paper and a problem that begins, “If a train leaves station y, at time t . . .

  12. Thomas Parkin says:

    I’ve noted that the truth is subjective / truth is objective types almost always breakdown as liberal / conservative respectively (I mean in an overarching way, not merely politically). Don’t know why that is, but it pretty clearly is. Possibly the liberal type imagines while the conservative type preserves and chafes at imagination gnawing away at what he hopes to preserve.

    Anyway … I’m not certain why we can’t just say that reality / truth is objective but that we ourselves see subjectively and we can never escape ourselves. Though something is gained in the attempt. I honestly don’t believe in taking any subjective view seriously where an objective advance has not been seriously made – otherwise we just have this world where every person’s subjective response is the ultimate word. I think there is no way around seeing the world as transcending ourselves.

    I personally think that a life where you can’t say “the waterfall is beautiful” except to really mean “I find the waterfall beautiful” is not infinitely enriched by our personality and imagination, but hemmed in by artificial limitations that _are_ entirely subjective.


    I mention Borges only because I’m currently in the middle of my second love affair with him. Before he was mostly fun – now he is, like, the whole world. ~

  13. Has no one ever seen Rashomon?

  14. Aaron Brown says:

    I’m pretty sure the 9,995th digit in the square root of 2 is a “4”, not a “3”. Just sayin’.

  15. I knew who wrote this post the moment I read the title.

    Now I’ll read the post.

  16. Yes.

    At least if you define heaven as being exalted. Something about needing to know the truth of all things… :)

    As for irrational numbers, there are *many* more irrational numbers then rational so, it behooves us to learn to love them. (As rational numbers are countable and irrational ones are not.)

  17. What Thomas said in #12.

    I believe in objective truth; I just don’t believe in my ability to understand much, if any, of it.

    I prefer Nephi’s definition in 1 Nephi 1:3:

    “And I know that the record which I make is true; and I make it with mine own hand; and I make it according to my knowledge.”

    The best I know is truth to me – even as I reserve the right to adjust that truth in the future.

  18. Oh, forgot to say: really fun to read SteveP!

  19. StevenP: “Without a knowing subject the notion of truth has no meaning.”

    True, but without something showing itself to a subject (who, by the way, may not need to be explicitly knowing), there is equally no truth.

    “Subjective” and “objective” are entirely inadequate adjectives for talking about truth.

  20. Jim, no disagreement. We need something like sobjective I suppose.

  21. Thomas, I agree with your assessment of both waterfalls and Borges.

  22. Joseph it’s not true the irrational numbers are uncountable. I counted them. It took me a while, but since I had eternity before this life, it was worth it. Exhausting but worth it. (PS, unlike most people I existed in a Beth-two infinity before this one. I don’t know how I wandered into this tiny little infinity.)

  23. The Right Trousers says:

    I like “sobjective.” I don’t care what it means, I just like it.

    Here at BYU, the math department has signs up all over that say “Truth is Power.” I want to paint over them all. These uppity mathematicians (and I count myself among them) think they have a stranglehold on Truth. Not so. They only have a strong claim to consistency.

    Contemporary mathematics is made from a system called Zermelo-Fraenkel set theory, in which every last object, including the number one, is actually a set. A set! (One is {{}}.) Real numbers are really infinite sets of infinite sequences of rational numbers, which of course are also sets. Must I believe that distances are infinite sets now? I understand how the empty set and sets of sets can be made to represent mathematical objects, but it would be absurd to be absolutely sure that this is how reality is. We only use ZF set theory because it has nice properties.

    It also appeals nicely to intuition – if you’re already crazy. Go read the axioms sometime.

    ZF set theory is usually augmented with the axiom of choice. Explained in English, it seems very intuitive. But if you assume it, you can do crazy things like break a solid ball into pieces and reassemble two identical balls of the same size from the pieces. (This is called the “Banach-Tarski paradox.”) Is this truth, or an artifact of the system?

    I think mathematicians sometimes get confused between consistency and truth because the propositional logic that most mathematics is based on features “truth values” called “true” and “false.” You could rename those suckers to “qutox” and “13” if you like, retaining “true” and “false” for philosophy, and it would change nothing about mathematics. I’d rather rename them to “consistent” and “inconsistent,” though.

    And perhaps the rest of the world (SteveP included) get confused because nobody ever explained that mathematics is actually nothing but a toy. A useful toy, to be sure. A consistent toy. A toy that can be made to approximate reality really well in many circumstances that we care about. But a toy nonetheless. What else could it be, if real numbers are really sets and you can always get two balls for the price of one?

  24. The Right Trousers says:

    Ah, never mind about SteveP being confused. I just re-read. Sorry about that, Steve.

  25. Numbers are like the outcome of quantum events, they do not exist without an observer. Without intelligence there is no such thing as 2^.5 or e or pi.

    Nature does perfectly well without knowing these things.

  26. The Right Trousers says:

    Unless we live in a matrix. Then it possibly uses these things to simulate our very existence.

    The Gospel does not rule this out, AFAIK. :P

  27. Good grief, then we have to know math to get out of the matrix!

  28. Mark D. says:

    Steve P: Truth is not what is out there, rather, “Truth is the knowledge . . .” Truth is a value of knowledge, not a thing out there.

    I didn’t say truth was “out there”, I said it was the way things really were, or alternatively “fidelity to reality”. The scripture says: “And truth is knowledge of things as they are, and as they were, and as they are to come” (D&C 93:24)

    That is a somewhat different way of putting it. The salient point about knowledge is that knowledge that isn’t of (or caused by) things as they really are isn’t knowledge at all. Imagination, speculation, belief, supposition but not knowledge.

    Or more generally speaking you cannot know anything about real world object A unless your belief about or experience of A is causally derived from events pertaining to A, whether routed through some divine conduit or otherwise. Otherwise you have no justifiable reason to conclude that A exists.

    If A isn’t even real, “knowledge” of A isn’t about the way things really are, i.e. not “truth” in the D&C 93:24 sense of the term. Likewise an assertion about properties of A that are entirely other than and underivable from the properties that A has in actual fact isn’t “knowledge” either. It is falsehood, perhaps an unknown falsehood but falsity nonetheless.

  29. 23 The Right Trousers,

    I read the ZF axioms. They didn’t tell me what a set is, only how it behaves. Perhaps you can enlighten us?

    Meanwhile, I suggest that talk of what Truth is without proposing an algorithm for verifying when something is actually true is equally vain…

  30. SteveP,

    “Joseph it’s not true the irrational numbers are uncountable.”

    What are you talking about? Yes they are:

    The real numbers are an uncountable set and the rational numbers ware countable subset and so the difference between the two (the irrational numbers) remains uncountable.

    *And yes, this means even if you have all eternity you could never count them all.* All you could ever do is describe them by using set theory or something equivalent. You could never list them even given all eternity. Hence the name uncountable.

    Disprove this statement SteveP and you will be one of the most famous mathematicians that walk the earth. (But it isn’t happening.)

  31. SteveP, please resist the temptation to ruin the parody by explaining it.

    And Joseph Smidt, please resist the temptation to link to Wikipedia when you are waxing didactic. Last time I followed a link there I was up all night! :)

  32. Dan, SteveP,

    Sorry, I was out of line again. And fine, if we want to imagine that uncountable sets can somehow how be counted given all eternity than *maybe* its possible. But I’m not so sure.

  33. When God travels faster than the speed of light, what is His mass?

    Can God factor the product of two really, really big prime numbers? How long does it take? Does the size of the prime factors matter?

    Can God simultaneously determine both the position and velocity of an electron?

    Does God’s DNA have echoes of the same ancient viruses that happened to infect the population that human DNA has?

    Are the answers to these questions the same for every exalted being, including yourself in due course?

    I was never a very good Sunday School student. :- )

  34. Actually, #32 is also a good one.

    Can God find room for the real numbers in the Hilbert Hotel?

  35. The Right Trousers says:

    Whoa there, #30! SteveP says he came from Beth-two infinity. I suppose that means his time dimension was extremely dense. Like, super-uncountable. I guess he could map an uncountable set onto what seemed to him to be discrete events, or whatever the equivalent is in that sort of existence. That, my friend, is head-asploding groovy.

    It’s not exactly “counting” of course (in the sense of mapping to the naturals), but I think he was joking. Maybe.

    #29: With an axiomatic object, all you can do is describe behavior. Other things can be “constructed” from these axiomatic objects, and that’s about as close as mathematics gets to defining what something “is.”

    We intuit sets as bags of items, but a bag of items is what a logician might call “only a model” (of sets). And thus they are rather silly.

  36. Actually SteveP, you should watch out. Legend has it the first person to demonstrate that the square root of 2 lost his life over it. (I’d make a link, but Dan would like me to refrain.)

    Those crazy Pythagoreans. So to this day I try to stay away from demonstrating this fact. You have to be careful. This is like missionaries going over a body of water. :)

    And hopefully I can join you in your beth-two universe when I die.

  37. #12 Thomas, if I met you in a playground, would you push me on a swing? That’s in no way a come on but I think I love you. In a completely acceptable way, of course. Everything you wrote is exactly what I would have written except for the part about Borges because I’m Borges ignorant and except for the part where you said you’re not sure why it is that liberals lean toward the notion of truth being subjective while conservatives lean toward objective because I think I know why but I can’t think of a way to say it that won’t insult someone. So, I won’t. Plus, I’m under-slept and keep getting myself into trouble that way.

  38. Mark D. says:

    Can God factor the product of two really, really big prime numbers? How long does it take? Does the size of the prime factors matter?

    A quantum computer can do this in polynomial time. God no doubt has superior technology.

  39. Mike S,#7
    Dallin H. Oaks gave a talk on testimony that made a lot of sense. I think your point that truth just is is true :)

    I think a lot of times when we each feel pretty strongly about something, it’s because we have a bit of the truth, somewhere behind what we feel. We usually muck it up and extrapolate the hell out of that grain of truth and end up talking past each other, as we each fight over the mountains of philosophies of men, built upon a grain of pure truth, which sometimes we don’t even recognize anymore. I think of this everytime the Prop 8 battles come up, as both sides are filled with strong convictions based on some truths, to something as simple as a discussion in priesthood meeting where two commenters just keep talking past each other trying to one-up each other with their own points.

    But long story short, regarding truth and knowing, I think Oaks gave a very good explanation. You know you love your wife, and if you said you loved her, you’d know you would be telling the truth. I have no problems with “know” now because his talk made it pretty clear to me how a knowledge of the gospel is entirely individual. I can’t prove it to you. I can’t convince you of it, by telling you over and over again that I know it. But I can tell you things that did help me come to that personal knowledge that you can also have.

    This post is a really good start, but like the author also feels doesn’t really go anywhere (that’s ok). But I really am looking forward to the day I can see a smart guy of faith like the author her put together a nice argument which synthesize all the uncertainties, probabilities, social constructs, etc. with our faith. I’m probably not saying my meaning right… but I like things like this that make you think, doubt, and can build faith at the same time. Someone just needs to put it all together in a concise, clear way on a silver platter for us.

    That’s my take anyway.

  40. Good one, SteveP.
    We translate “I know the Church is true” into various languages quite directly: Se que la iglesia es verdadera, etc. I wonder how those from other cultures (heck, even cultures within the USA–like Arkansas) would really express what a “testimony” is if we (whoever “we” might be) didn’t supply the vocabulary. Maybe: “This feels good.” “This shines nicely in my head.” “This falls on me well.” I wonder if our translators have ever made up a word for TRUE in a culture where the word didn’t actually exist.

  41. Dave P. says:

    I’ll stick firm with what I know to be true: i^2 = -1

  42. fun post, interesting ideas.

  43. 39 Chris,

    I am still believe that it does depend on what the meaning of is is.

    On June 8, 1978, the meaning of the color black changed, and what was true (and you knew it to be true because a Prophet had said so) was not true. One day true, the next day false.

    This (so I am told) is qualitatively different from the earth being flat (because it wasn’t, we just thought it was), because God had never said it was flat. But what of Jericho? Did not God stop the sun (without ripping the earth apart)? Practicing Christians far more devout than I “know” that He did, because God said so.

    But what the heck do I know? Even Socrates knew only one thing. Or so we hear from Plato anyway. And he would know what his master said, wouldn’t he?

  44. Truth is the embodiment of what is. Knowledge is the personal set of truths (elements of Truth) that one knows. I have knowledge of many things, and therefore I know things which are true, but I reserve the right to alter what is in my set of truths as I increase the number of things that I know.

    This in no way will stop me from recklessly stating things that I know to be true, even while acknowledging that what I know to be true right now will most likely have altered in some way by tomorrow (if not sooner).

  45. I actually agree with #23 Right Trousers

    Is 2+2=4 a true statement? I think it is true because we have defined it to be so. But in some situation it might not be, or rather the implementation might not be. There is always noise in any system. We have defined mathematics to describe those systems, but they are not perfect. Yet the models are “true.” Hence, I am happy to say that math is “true” and 2+2=4 is true within its own context by definition.

    If we liken this to the church, I have no problem saying that the church is “true.” Why? Because in it’s own context we have defined it to be so. But again, there is always noise in the system, and it may or may not perfectly describe objective reality.

    As for whether or not there is objective truth, I dunno. I honestly can see both sides of the story. I hope and long for there to be objective truth, and I hope in Mormonism we are close to it. But I confess I do not know.

  46. Doug Hudson says:

    Does the Church send missionaries to Uqbar?

    Seriously, it occurs to me that Joseph Smith finding and translating the Book of Mormon could come straight out of a Borges story. (I mean that as a compliment.)

  47. #23 TRT says, “mathematics is actually nothing but a toy.”

    Are there no Platonists here to rise up and defend the Kingdom? I’m not sure if I’m one or not, but I can’t think about e^(pi*i) = -1 without wondering how it is that you can raise one transcendental number to another which has been multiplied by an imaginary number and come out with something so wonderfully ordinary.

    Dan, #31 thanks for letting me off the hook for explaining. We Beth-2 folk really struggle with the humor of Aleph-1, linear time.

    Seriously, looking at what Margaret says makes sense. We often see truth as an epistemc value for our knowledge, where we mean something like, our beliefs about the world are right. But more often we are expressing experiences that run to the depth of what we are. When Christ said, “I am the truth.” I don’t think he meant, “I have the truth.” or “There are truths about me you need to know.” (which I think are both right by the way), but I think he meant that literally (and you know me and taking the scriptures literally). We experience Christ. ‘Knowing’ seems like the wrong word here, because it seems to imply some sort of intellectual assent or list of facts we’ve consented to. The trouble is in language, truth means a lot of things. Everything from right knowledge to coherent mathematical structures. In science we mean lining up our beliefs with some ontological reality or, more weakly, a useful description of an unknowable reality. When I think about truths about my relationship with God, none of these things feel right. It’s more like the love that chris #39 mentions which is an experience of being, not a belief about the world.

  48. too. much. math. for. me. My head hurts.

    Impressed at your smarts… want to take my accounting & stats finals for me? Oh wait… that won’t get me into heaven for sure.

    Alas, I’m maintaining that ignorance is bliss and waiting for the next post!

  49. Re #38 (Mark D.)

    A quantum computer can [factor] in polynomial time. God no doubt has superior technology.

    Polynomial time doesn’t help. Once you admit that God needs an algorithm with complexity greater than O(1), you’re cooked. Our everyday intuition doesn’t give big numbers and big sets the respect they deserve. They are an impressive obstacle, even for Someone who is really good at arithmetic. :- )

  50. Dan Weston (43)–

    On June 8, 1978, the meaning of the color black changed, and what was true (and you knew it to be true because a Prophet had said so) was not true. One day true, the next day false.

    I can see the point you’re making, and I think it has validity in many instances for the Church, but I don’t know if this particular instance fits well.

    Prior to the June 1978 revelation on the priesthood, the “truth” was largely just a policy statement: “Blacks are not ordained to the priesthood.” The justification for that policy was actually about as far from an absolute truth as we get–everyone under the sun speculated, taught, opined, and debated countless explanations for why they were not allowed to have the priesthood (including many bad, badder, and badderer explanations), but at the end of the day the only certainty was that we didn’t know for certain why, or whether it would always be that way.

    We still don’t have a company line (as far as I know) explaining definitively why the ban existed or why it was removed when it was removed.

    In other words, on June 8, 1978, the only “certain truth” that changed was a rule–hardly a “truth”–that blacks could be ordained.

  51. I know beyond a shadow of a doubt and with every fiber of my being that this is a seriously groovy post.

    Furthermore, I think I may have just developed a serious crush on The Right Trousers and would like him/her to post more math jargon comments.

    I am greatly comforted that reality is entirely unaffected by my understanding of it. If there is a God, and I don’t believe in God, that doesn’t affect reality. If there is no God, and I do believe in God, that doesn’t affect reality either. And I have no emotional investment in being “right.”

  52. Scott, thank you for responding. I was hoping you would!

    But I think you did miss the point I was making: not what is true, but who believes it is true, when they believed it, and why.

    In my limited experience with the LDS Church, the phrase “is just a policy statement” seems out of place. Many devout members would probably be disturbed to hear that deeply held “truths” are/were “mere policy”. Strangely, having been so made aware of this, these same members seem reluctant to go back and find out how they could have been so mistaken.

    When truth changes, I would hope that God would provide an explanation. When mistaken understanding of truth changes, I would definitely demand that man would provide an explanation.

    “We still don’t have a company line (as far as I know) explaining definitively why the ban existed or why it was removed when it was removed.”

    The essence of epistemology is trying to decide whether you are bothered by the above quote or you are not. I am.

  53. #26 TRT has brought up an interesting point. What if this life were nothing but a simulation. It is, like TRT has suggested, not forbidden by the Gospel.

    What happens to truth if this is the case? If this is a simulacrum of a reality then why not pi=3? and e=2.7? I have to admit that IMHO a world based on quantum mechanics really can’t be anything but a simulation. “True reality” can not work in such strange ways.

  54. Oh crap. Race issues. Is this a conspiracy to get me to be more active on BCC? You knew I was reading.
    June 8th is the day after my birthday, and therefore of tremendous importance. In 1955, it was the first full day I lived (on earth).
    I believe some other important things happened (or were at least announced) on that date in later years.

  55. I was a liberal arts major in college. My wife was a math minor, and now a secondary school math teacher. There is a lot about her that I don’t understand. This post does not give me much hope of unraveling some of those mysteries about her. I tend to take terms like “imaginary” and “irrational” numbers at face value. If they are imaginary and irrational, why do I care? She certainly does.

    That’s my subjective truth.

  56. Dan,
    I think I understand you better now. Thanks.

    As a clarification, I used the phrase “just a policy statement” only to emphasize what changed, not the importance of that policy in the hearts and minds of the people. However, I do think that you overstate the likely reaction among LDS people to my word choice–the “deeply held truth” was that blacks couldn’t have the priesthood, but that they would, one day, receive all the blessings that everyone else does.

    As such, although I wasn’t alive in the days leading up to that revelation, I don’t imagine that the concept of a change in that policy was utterly unfathomable for most people–perhaps I’m wrong.

  57. Butch Bowman says:

    I love math, and I often use mathematical models to better understand and teach gospel principles. However, when it comes to understanding what “Truth” is, I tend to favor non-mathematical explanations myself, more along the lines of the comments in #44 (Alex T. Valencic) and #47 (SteveP). On the other hand, I do think this post really illustrates well how mathematical concepts can add to an understanding of what “Truth” is.

    My personal model or theory of what “Truth” is starts with John 14:6: “I am the Way, the Truth, and the Life.” Jesus himself is the Truth. This is possible because of his being God and therefore 1) infinite 2) good and 3) perfect. Thus, he is able to “be” an idea. This understanding or way of thinking about it is helpful to me, because it helps me see that: 1) As a Truth seeker, I should be seeking for Jesus rather than for facts. 2) No earthly institution can contain the infinite and eternal Lord himself, and therefore, there is no earthly repository of all truth. The Church is true, and makes available the fullness of the Gospel, but this does not mean it contains all Truth. You cannot contain the infinite. 3) The absolute Truth or absolute reality has as its core principle a person. Therefore personality is an essential quality of being. 4) As I interact with my environment in living my daily life, God as the Truth is present in every aspect of that interaction, for “by him and through him and of him the worlds are and were created” and “in him we live, and move, and have our being.”

  58. Butch Bowman says:

    And he is the same light that enlightens our eyes and quickens our understandings.

  59. Thomas Parkin says:

    “Does the Church send missionaries to Uqbar?”

    Only in my golden encyclopedia, another like copy of which has never been found.

    Scott, you’re not wrong. I was 12 at the time of the revelation and quite conscious of it. We expected it, and were only to happy to see it. Part of the ‘truth’ we understood was the temporary nature of the circumstance. Other people might have seen it differently, but other people also thought that the Rockies were made by earthquakes when Christ was crucified … and so what are you going to do? Maybe if the Prophet got up in GC and said ‘y’all can stop being ignorant loonies now’ that would do the trick. In fact, he doesn’t really say much about anything, and that does open up this empty space but …

    … as for the church, it can’t win for losing. We all want it to spell things out in detail, and then when it does we rightly complain that they are dogmatic and hierarchical and blah blah blah. I personally like the large areas of speculation, available to anyone who wants to participate without sanction.

    In any case my experience, contra Dan, is that while many people might take the ‘follow the prophet’ mantra farther than I would, few enough, even very conservative members, take church leadership as anything like a sole and unassailable source of truth. Anyone with a smidgen of understanding of this religion knows that we go about this process on our own – that it _is_ a process and that as a search for truth it explicitly requires individual experience and reaction to experience. In fact, I’d say that unless you’ve got a real grasp on this you’ve got no idea what Mormonism is, member or not.

    Natasha *blush* ~

  60. MikeInWeHo says:
  61. Thomas (59),

    Part of the ‘truth’ we understood was the temporary nature of the circumstance.

    Thanks–I think that’s what I was trying to indicate in my first response to Dan Weston above–that in the context of the OP, relating to “eternal” or “absolute” truths, the priesthood ban falls short a bit because the policy was known/assumed/hoped to be not eternal or absolute.

  62. I have often thought that mathematics is a great modeling tool, the ultimate simulation language. That is the way it started and is most often used today. When we count things, for example, we are modeling those things and simulating them in some other medium like paper.

    In the modern day, further developments might be seen as an extension of its modeling function. For example, whether a set is countable or not determines the limits of that modeling capability which is just an extension of the great power of mathematics as a modeling language.

    Another example of mathematics as a modeling language is thinking about the orders of infinity. We are trying to model something completely insane by normal means but mathematically it works. We then apply these models to the nature of God and the nature of the universe to try to get a fit. Otherwise this would be beyond comprehension, ie, non-modelable.

    (I know this is simplistic. Mathematics has taken on a life of its own. It can be argued that it has cut itself adrift from the moorings of reality. Mathematical truth, in a technical sense, relates to the accuracy of the proof not necessarily to the accuracy of the model. This truth concerns itself with the language rather than the function of the language.)

    Likewise the Gospel can be thought of as a modeling tool or language for our existence, including eternity. The hard part is that there are no formal methods for establishing the postulates, axioms and theorems, as there are in mathematics, which develop the language. We try to stretch the modeling language of mathematics and philosophy to the Gospel, but these might not necessarily apply. If they applied more perfectly there would be less need for the spirit and revelation. (Discussions like this one are other places where we further develop our Gospel models and language.)

    Truth, in one sense, relates to the accuracy of the models. In an analogy to mathematical modeling, I see modeling tools developing dynamically in the Gospel. As modeling tools develop so do the truths, ie, the accuracy of the model with respect to reality. So much of what we take as absolute and proven truth is not, rather it is cultural and will be overthrown. Much of this poor revelation rests on the limbic systems of the people in charge rather than on some bedrock model of eternity. Much of what we take to be revelation is really a misperception of limbic messages appearing from the unconscious mind.

    The earlier problem of the Church and the Blacks? Bad model. The solution is to get a new and better model. The mistake is to assume that the old model was based on some eternal foundation, it was not. One might argue that it is a perfect example of the confusion of limbic misperception for eternal principle.

    Our view of God, which view is our model, is not static and should not be static. How can a finite mind comprehend the infinite with an unchanging model?

  63. Mark D. says:

    MHH: Our everyday intuition doesn’t give big numbers and big sets the respect they deserve. They are an impressive obstacle, even for Someone who is really good at arithmetic.

    I agree. I don’t think the ability to factorize arbitrarily large numbers is a necessary component of divine power, though.

  64. I’m with Mark D.

    God doesn’t need to factor large numbers. He has a look up table. He made it in the infinite time he had to progress . . .

  65. Ah, I think God uses a divine sort of geometry to nail down all the irrational stuff.

  66. Bruce Rogers says:

    When I took a course in Number Theory at BYU, many years ago, one of the first things that we learned was the proof that the square root of 2 was irrational. I was so impressed with that logic, that I eventuallly majored in math and taught it in HS for two years before continuing my graduate studies toward the PhD. The scriptures tell us that God will “reason with us as one man reasoneth with another”, so I have always tried to reason correctly.

  67. Mathematical truth, in a technical sense, relates to the accuracy of the proof not necessarily to the accuracy of the model.

    I’m a lousy mathematician. Would you mind explaining this one a bit–to me it sounds like a paradox of sorts.

  68. The Right Trousers says:

    #67: I’ll take a stab.

    So you’re brushing your teeth while staring at the 1cm square tiles on the floor of your bathroom. One has been cut in half along the diagonal to make room for the toilet. What’s the length of the cut? (You consider this question because you are either insane or part of a story problem.)

    Most of the time, we would say that “the Pythagorean theorem tells you that its length is sqrt(2).” That’s actually only true if you “model” your problem using Euclidean geometry. This means that you 1) translate things you can measure about your real-life problem into corresponding ideas from geometry (the result is the “model”); and then 2) use the model to solve your problem mathematically. We don’t usually record doing #1, but it’s always implied.

    Here’s my solution to the diagonal cut problem with #1 explicit:

    “I represent distances using real numbers, and bathroom floor tiles using geometric squares. I assume that I can measure distances with perfect accuracy.

    The tiles have 1cm edges, and the diagonal cut across the tile in question is a line segment from one corner to its opposite. This cuts the square in half, with each half being a right triangle. By the Pythagorean theorem, the diagonal cut is sqrt(2) cm.”

    The “model” is a perfect geometric square with real-valued side lengths. How accurate is the model? That depends on the accuracy of the translation from real life. But real life isn’t formal or rigorous enough to be math, so it’s impossible for math to say *anything* about the accuracy of the translation or the model.

    On the other hand, math has a lot to say about the lengths of diagonal cuts of a perfect square. One dude has collected 84 proofs of the Pythagorean theorem online. So paragraph 2’s answer has a lot backing it up. Suppose my model is completely whacked because the tile is actually a parallelogram. Then “the length of the diagonal cut is sqrt(2)” is mathematically true, but “the length of the real-life diagonal cut is sqrt(2) cm” could be very inaccurate.

    FWIW, I think it would be awesome to turn in a story problem answer like this. I’d love to inhabit my 13-year-old body again for 10 minutes just to try it…

  69. One dude has collected 84 proofs of the Pythagorean theorem online. So paragraph 2′s answer has a lot backing it up.

    Reminds me of the joke about the blonde who had sex but wasn’t sure she lost her virginity, so she did it 83 more times to be sure.

    Seriously, if your reasoning is sound, no proposition can be both true and false, so proving something 84 times does not make it more true than proving it once. In an unsound system, everything is both true and false: the diagonal would be sqrt(2), but also 3.7 or 42.

    As a fellow obsessive, I applaud that online dude, though he is channeling Santayana (as I am with this diatribe of mine!), having succeeded 83 times in proving that “Fanaticism consists in redoubling your effort when you have forgotten your aim.”

    Or as I less cleverly would say: twice proved is once reproved and naught improved.

  70. Georg Cantor already had this fight with Leopold Kronecker. Kronecker definitely won the battle (keeping one of the main geniuses of the 20th century out of a university position) but lost, lost, lost lost, lost the war. I suggest you start using the number “3” ( a perfectly rational number) as a stand-in for pi on all your future building projects.

  71. But perhaps you’ll go to heven, where circles are shaped slightly differently, and two has nn square root.

  72. While it’s true that we use math to model the real world, and the answers are only as good as the models are accurate, there’s also a deeper truth I would like to add.

    The real world is nothing but math. That’s very counterintuitive, but let me tell you the sense in which I mean it.

    The everyday macro world with which we’re all familiar can be fairly well approximated with Newtonian physics, the when you start looking at smaller and smaller parts of it, when you explore the internals of atoms, the approximation totally breaks down. The real world simply isn’t “classical” like this, and if it were, it wouldn’t work. Electrons would spiral inward to combine with their nuclei, and atoms would cease to exist. Everything would implode and chemistry wouldn’t be true anymore. Lots of everyday observations wouldn’t be possible. There would be no colors in an oil slick, laser pointers would stop working, etc. Even plain old light just isn’t classical in behavior. Maxwell’s equations describing exactly how electromagnetism works don’t jibe with classical (Newtonian) physics.

    The speed of light when observed very carefully doesn’t match up with the way waves in a medium behave. Light is a wave (and a particle both) but there’s no medium that is waving, like we’re familiar with. There’s no water or air or rock or ether there on which the wave rides.

    So when we delve deeper to understand reality, we find out that it has this quantum nature, which is a crazy and weird way that it acts, but we have to get used to it, and wrap our minds around it, because that’s how it does act. We know this because QED (quantum electrodynamics) is accurate to experiment to within something like twenty-four decimal places.

    So bear with me now. Go one level deeper and let’s look at the nature of physical law. There is no physical underpinning to these laws. There aren’t any hidden gears and cogs that make the universe behave like this. There’s no mechanical reason for the laws to be true. They just are.

    For instance, take gravity. In the middle ages they thought the planets kept circling round and round because angels were behind them pushing them in their orbits with their wings. Now we have a different theory. Our angels push inward. This is Feynman’s famous joke to describe that we haven’t explained gravity, we’ve just described it. There’s an equation. F = GMm/r^2 Does the moon, for instance, whip out its little calculator each instant and calculate which way it ought to go based on this equation? We don’t think so. We think that space is curved so that when it follows its natural shortest distance between two points, it just falls that way.

    But again, space is curved because there are fields. Particles are essentially knots in the fields. When you slap your hand onto the hard surface of a table and say the word “reality”, the force you feel resisting your hand is just electromagnetic fields. The fields are the underlying reality, I guess you could say, but the fields are just numbers in space. When we go down to the deepest level, the level below which there are demonstrably no more levels, the basic nature of reality is just equations. Physical law is mathematical in nature.

    So you can claim that math only describes reality or models reality, but in actuality, on a deeper level reality is just math, too. I didn’t do this explanation justice. Please read Feynman’s “The Character of Physical Law” for a better explanation. =)

    So any discussion of “ultimate truth” must take this into account.

  73. … and I obviously can’t use the html code for italics correctly. =)

  74. Tatiana, amazing description! I’ve been reading on this idea that it’s math at the bottom and it’s got some good backing (there was also an article in Discover Magazine a bit a go you might find interesting).

  75. I might believe Math to be all true___I just not sure all truth is Math. I need some Chaos in my ” what is Truth” model.

  76. #67 for completeness: The truths of mathematics are relationships between numbers, etc. If mathematics is a modeling language, these truths are all interior to the language, itself.

    In regard to the modeling itself, mathematics is certainly limited. For example, mathematics has a huge difficulty handling discontinuities like shock waves. It also has a hard time with predicting chaotic behavior.

    Mathematics cannot even solve a 3 body problem, for example, an exact, closed solution of the earth, sun and moon. (Practically, the earth’s gravity can be treated as a perturbation on the sun’s gravity to give a pretty good solution.)

  77. Irrational numbers are not real, by definition. Rational numbers are real, irrational, are simply irrational.

  78. Ben,
    Er, no. Irrational numbers are real.

  79. I just want to give a belated thanks for the responses to my question. That statement makes sense now.

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