Every elementary school student learns how to add, subtract, multiply and divide. Well, some people object to teaching this sort of thing, but let’s suppose it’s true. The rules for addition, etc. are well known, and seem natural. Induction is a little different and most students have trouble grasping it, even though its basis is quite simple: every subset of the counting numbers has a least member. For example, {10, 5, 150, 47, 3}. Obviously, the least member here is 3.

The simple rules of arithmetic seem obvious but the counting numbers (or natural numbers), 1, 2, 3, etc. are a rich system capable is describing deep and complex statements.

Using the standard principles of logic, one can prove theorems about the counting numbers. One assumes that within arithmetic, truth and provability are the same. But it turns out that unless we are willing to let our faith in arithmetic come crashing down, it is a fact that there are “true” statements within arithmetic which cannot be proved.[1] This is one of Gödel’s “incompleteness” theorems and quite a number of 20th century thinkers found it both fascinating and unsettling. Recently, in an interview reported in Nature, Roger Penrose spoke about some well-known inferences based on these results. In answer to the question, “What does mathematics have to say about consciousness?” Penrose answered:

In my 1989 book The Emperor’s New Mind, I said that computers will not achieve any conscious understanding. Gödel’s theorem tells us that mathematical insights fall outside any formal procedure, so understanding is not a computational process. Something else is going on. I have reason to believe it may involve the limits of quantum mechanics. Microtubules [tiny structures in cells] are the best candidate in the brain for where this might happen, as they are so small, but quantum mechanics would have to work on a huge scale to operate there.[2]

Penrose does mathematical physics and is a brilliant guy. But I wonder just a bit about his interpretation of Gödel. In any case, Gödel’s theorem has been the subject of controversy since it was published in 1931 and Penrose’s remark reminded me of what another physicist said to my graduating class in college. “Truth is not always obvious.” Ok, sounds laughable here, but his comment was in regard to relativity, its discovery and acceptance. He went on to suggest that every day experience, or the blur of the daily news cycle may lead us to draw conclusions that are unsound in the deeper broader world. It takes skull sweat, not surface interpretation or the pablum of superficial consistency to find out what’s going on. Even then, you may fail. The problems we were likely to encounter as new graduates would have solutions which would not be solvable by means of the same old Newtonian mechanics. Of course he was using metaphor.

The title of the post comes in here. From my teenage years I’ve been a motorcycle rider. I’ve always enjoyed it but there are unintuitive things about it that can get you into trouble. One of these is countersteering. At speed, especially high speed, you can’t turn a bike effectively just by leaning into a curve. Let’s say you’re on your way to school, late, and your passing a stream of slowpoke cars, accelerating past 100mph. You need to get out of the oncoming traffic lane. Try leaning into it then and you will die. The way to turn rapidly at speed is to turn the handlebars the opposite way you wish to go. Zip -you’re back in the lane, getting ready for the intersection ahead.[3]

Two statements attributed to Joseph Smith reflect these ideas for me.

1. It is my meditation all the day & more than my meat & drink to know how I shall make the saints of God to comprehend the visions that roll like an overflowing surge, before my mind.

2. the things of God are of deep import; and time, and experience, and careful and ponderous and solemn thoughts can only find them out.

Perhaps Penrose is right that consciousness will never emerge from silicon. It is a puzzling problem. One thing Joseph was definite about. We are eternal. We’re going to stick around. Hopefully, that’s a good thing. On the other hand, to the extent that Gödel’s results may reflect the nature of the human mind, I look forward seeing where it might lead.

[1] A very nice accessible example is Goodstein’s “Theorem.”

[2] See the December 23, 2010 issue. There is a huge literature on the subject of consciousness and consciousness as an emergent property to which our own SteveP has made contribution. A lot of the quantum stuff was written about in the 1990s. But there is a continuing interest. There is a lot of literature here that relies on various bits of interpretive writing/speculation about what the models *may* say than on the models themselves. It’s interesting, but I wonder about its ultimate value or meaning. You can still get Penrose’s book on Amazon, and three of his other books as well.

[3] Riding something with more than 100 hp, you better learn to countersteer. And yes, it creates a vicious lean, much quicker than your body weight ever can.


  1. Sorry to be off topic but I just realized for the first time that WVS was my partial differential equations teacher at BYU. Small world! Oh, and keep up the good blogging.

  2. As for Godel, I would think I would interpret it that any mathematically based system of logic, including much of physics, will find that it cannot prove all things that are true. This does seem to put a limit on to what extent we can use science/math to “prove” the truth of all things that are true.

    Not sure though how I would apply it to conscience.

  3. It’s nonsense to say that Godel means computer intelligence is impossible, though. I like Penrose and that book The Emporer’s New Mind was really interesting, however he got that point completely wrong. A brain doesn’t work anything like a computational system such as arithmetic, of course. There’s much more going on at a lot higher level than what we might imagine as the “machine language” level of the brain. Put it this way: computer designers go to a whole lot of trouble to guarantee that we will get a predictable outcome from the same inputs inside a computer. The laws of physics are taken advantage of in such a way that we can almost always predict exactly what will occur. In other words, we could do a hand-crank version of the program running on the given machine and know what the answer would be, exactly what the machine would do, though a rare glitch might come from a stray cosmic ray changing a 0 to a 1 or vice versa.

    Contrast this with the way the brain works, which is very much undetermined, wildly different from brain to brain, deeply dependent on analog values like the exact level of electrolytes in the region of the axon of a given neuron, and so on. The butterfly effect, in which macro scale structures (for instance, thoughts) are highly dependent on miniscule difference in initial conditions, is in full force. The brain is extremely nonlinear in trillions of different dimensions. There’s really no predictability of what will happen in any given brain. The emergent properties themselves (thoughts, insights, perceptions) are selected over billions of years for maximum survival value. There is not a whole lot of overlap between the way brains work, and our current computers, in fact. A brain isn’t at all mappable to a consistent formal system of mathematical calculations and proof, such as Euclidean geometry, or propositional calculus, or what have you.

    As such, Godelian arguments about formal systems don’t really apply to thinking at all. When we do learn how to program an intelligent machine, it will be by carefully mimicking the processes that we discover going on in human brains, as we learn more about them (which we’re a long way from understanding well enough to replicate). So anything that applies to organic brains will apply to human-designed ones as well, and there will be no Godel incompleteness issues. The idea is very silly and comes from lack of understanding of how brains work.

    In essence a brain is just a biological machine that is manufactured by a zygote using a program written in DNA. The laws of physics, chemistry, etc. are what apply, are what make it go. Anything applicable to one such machine can theoretically be true of a functional duplicate assembled by other means out of other materials.

  4. Thanks for another fine post, WVS — there must be some productive interface between math and religion, and I am sure you will find it and blog about it. It’s worth pondering why math has been used so productively in science but not at all in religion.

  5. #1. Whahahahahahahaha. My clutches extend through the universe.

    Tatiana: it is a complex issue. Thanks for taking the time to respond.

    Dave: I think mathematics finds much subtle expression in religion but formal application is difficult for several reasons. One is the imprecision of discourse in religion and another is that most people feel they have an intuitive grasp on religious discourse (which they probably don’t have). A mathematical approach to religion (I’ve seen a few transfinite notions applied) would find most people unprepared to participate and religion is nothing if not participatory. Maybe.

  6. WVS, I don’t comment often on BCC, but I really have to say that I always love your posts, and this one is no exception.

  7. I wish we heard more about “skull sweat” in religious contexts (not least because it’s just a delightful turn of phrase).

  8. Senile Old Fart says:

    A course in partial differential equations (but taught by someone other than WVS) convinced me to seek a career field that did not require complex computations.

  9. Thanks, Kathleen.

  10. “Turning rapidly at speed.”

    This is the way all two wheel vehicles turn at any speed. You first turn the handle bars counter to the turn to tip the bike over into the turn, then quickly turn them back to catch the turn. I taught every one of my children how to ride bikes this way. It saved so many crashes and bloody knees. I wish my father had known that.

  11. My guess is that the brain is mostly classical chaos theory at work with a small admixture of quantum indeterminacy. A computer will never be chaotic. I think that programming does not permit that. You are not supposed to be able to create a logic state which another computer, bigger and better, cannot figure out and predict. In chaos theory there are states whose antecedents are too small to measure and are therefor utterly unpredictable. If even a little bit of quantum indeterminacy is added to that mix there is no possible way to compute or predict the outcome.

    So imagine a computer whose logic states respond to and are determined by non-measurable fluctuations which occurred hours, even years, ago.

    So if God’s brain is 10^6 larger than ours with a commensurately larger number of connections, each sensitive to classical thermal noise and a little quantum tunneling, this is utterly beyond the entire universe to predict.

    I have loved these and similar quotes from JS. I wish the Church were a place where we could discuss the mind of God. But “to comprehend the visions that roll like an overflowing surge, before [JS]’s mind” is much to dangerous to the unity of the Church and would threaten its very survival.