Transcendental Territory

transcendent mindLast time we considered the possibility that human consciousness somehow supervenes on the physical brain, that it only emerges under specific physical conditions. Perhaps, like laser light and microwaves, it requires the right equipment.

We also touched on how Church-Turing implies that, if human consciousness can be implemented with software, then the mind is necessarily an algorithm — an abstract mathematical object. But the human mind is presumed to be a natural physical object (or at least to emerge from one).

This time we’ll consider the effect of transcendence on all this.

This is not the religious kind of transcendence, this is the mathematical kind. A special property that π and e and many other numbers have.[1]

We start by considering three Yin-Yang pairs with regard to numbers.

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Yin-Yang-1The first is the finite versus the (countably) infinite.

On the one side, precise numbers that match their quantities. “Dave donated a dozen dimes!” “Twenty-two teens turned twenty.” “Only ate one waffle.”

On the other side, a row of three dots (…) or a lazy eight (∞). “The road goes on forever.” “My curiosity is endless.” “Close your eyes and count to infinity.”

But the thing about countable things is that they’re countable.

More to the point, computers can count things really good.[2]

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Yin-Yang-2So the second Yin-Yang pair is the countable versus the uncountableCantorville — the discrete versus the continuous.

On the one side, everything from the first pair, the countable things, even infinite ones. Here numbers are cardinals; they stand for quantities of discrete objects.

On the other side, the real numbers, the smooth and continuous. These numbers are magnitudes; they stand for points along a number continuum. They are a different kind of number.[3]

Calculating with real numbers offers some challenges, especially with regard to chaos. Calculation necessarily rounds off real numbers, so there is a loss in absolute precision.

We’re only at the second level, and already calculation is in trouble. To the extent calculation involves discrete symbols (i.e. digital calculation), we can’t calculate with real number values, only their approximations.

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Yin-Yang-3The final Yin-Yang pair is real numbers versus the transcendental numbers.

Again, on the one side, everything from so far (including the complex numbers). Sane numbers tamed with algebra.

On the other side, wild mysterious numbers with some vaguely magical properties.

Firstly, their decimal expressions never form any repeating patterns (as with rational numbers).

Secondly, there is no algebraic expression that specifies their value (as with ordinary real numbers).

That latter property results in the “game” demonstrated in this YouTube video. It’s way worth watching (seriously, please do watch it, at least the first half):

I knew about algebraic roots, but I never realized they implied the game here.[4] It’s a neat way to look at it, and it got me thinking about transcendent numbers. (Which invokes Euler’s Identity, hence the Beautiful Math post.)

One can definitely make a case that God invented the integers, that the countable correlates with physical reality whereas the real numbers (let alone the transcendentals) are abstract inventions.

The problem is that God presumably invented circles, too, and little old π is one of those things you notice if you look at circles. It’s just the ratio between the diameter and circumference.[5]

There is something magical about the transcendentals that sets them apart. The name is certainly evocative.

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The question I want to ask is whether consciousness could be, in some sense, transcendental (and just to reiterate, I don’t mean that in the spiritual sense).

If so, does that present a problem with regard to an algorithmic theory of mind?

(The presumption being that transcendental calculation is somehow a problem, so there really are two thesisesthesisispoints to demonstrate here.[6])

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There is a Yin-Yang situation regarding numbers like π. On the one hand, no perfect circles exist, so π never actually occurs anywhere physically. On the other hand, its true value underlies every circle and sinuous process!

circlesLook at it this way: All the inaccurate real-world instances that involve π are inaccurate in their own way.

The average of all those inaccuracies converges on the true value (proof it does lurk beneath all physical circles).

While no individual circle is transcendental, circles certainly are.

So why might a human brain be a transcendental process?

The answer partly may lie in the sheer complexity and scale of the brain.

Not only are the parts complex, there are hundreds of trillions of them! Perhaps transcendence emerges from the multitude just as it does with circles.

synapse photoSynapses are hugely complicated in their own right (and amazing). Is it possible that a full model of a synapse is complex enough to be subject to chaos?

That’s not at all a stretch.

If so, that means each synapse is just a little unpredictable (mathematically).

The synapse knows what it’s doing, but for us to determine that precisely may be effectively impossible (like the three-body problem or weather prediction).

The network of the brain is also highly complex and vast. Neurons all operate in parallel and talk to each other in variable frequency pulse trains. It’s even easier to imagine that chaos plays a role here. (It’s harder to imagine it wouldn’t!)

So it’s possible the parts transcend calculation and even more possible the whole network does.

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Z80The obvious question is: A CPU has many billions of transistors, why can’t that multitude be transcendent?

Another form of the question is: A software model can model trillions or quadrillions of (virtual) parts; why isn’t that multitude transcendent?

The answer is that, potentially, it could if those parts, or the network of those parts, had the same indeterminacy as does the operation of the human brain.

But so far computer technology works very hard to remove all indeterminacy from all levels of computer operation! It’s considered noise that degrades the system.

There is research into the idea of introducing noise or uncertainty into algorithms, and it’s possible that may bear fruit some day.[7] (It’s not the same as “fuzzy logic” which is just logic over a value range.)

DivideAs it stands now, algorithms are fully deterministic at all levels. There is nothing that’s allowed to transcend.

Keep in mind that if hardware is the only possible source of indeterminacy or transcendence (as is the case in the physical world brains inhabit), then conscious is not algorithmic.

It can’t be if it supervenes on hardware!

In order for consciousness to be strictly algorithmic, any indeterminacy or transcendence must come from the software steps. And as we’ve seen, those amount to: Input numbers; Do math on numbers; Output numbers.

Where is the transcendence?

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signI’ve been wondering if Turing’s Halting Problem or Gödel’s Incompleteness Theorems might play any role in this. It’s possible to read their conclusions as addressing transcendental territory.[8]

In the Turing case, no algorithm can transcend the algorithmic context such that it can solve the halting problem.

In the Gödel case, no axiomatic arithmetic system can transcend its context such that all true statements in the system can be proved in the system.

Either way there’s chaos theory telling us that some calculable systems are so sensitive to input conditions that any rounding off of real numbers degrades the calculation.

This all seems to suggest (to me, anyway) that real world processes, while wildly mathematically “inaccurate” on their own account, converge on mathematically ineffable transcendence given sufficiently large numbers.

Think of it as actually doing quadrillions of steps in an infinite mathematical series. How close to its real transcendent value would π be after 500 trillion steps?


[1] This all started out as a flight of fancy, but the more I think about it, the more it fits.

[2] Even if it takes them forever, just like it would you.

[3] Transfinite mathematics involves multiple levels of infinity, but I think the countable versus uncountable one is the foundation. (I’m not convinced the others exist meaningfully.)

[4] Which, if you didn’t watch the video, is that you can reduce any algebraic expression to zero using only addition (and subtraction), multiplication (and division), or exponentiation.

(The video at the bottom has some extra bits they didn’t include in the main video. It’s the same one linked to at the end of the main one.)

[5] It’s when you look closely at π that you realize how weird it is. See the Pi Day post for how far down the rabbit hole that goes!

[6] AKA: “theses” 🙂

[7] One problem is that calculating random numbers is nearly impossible. It takes some real world source (semiconductor noise is a good one) for true randomness.

(The difficulty of calculating random numbers is just another illustration of the limits of discrete math and algorithmic processes.)

[8] Cantor is clearly addressing the countable-uncountable divide, and it’s possible Turing and Gödel are as well.

About Wyrd Smythe

The canonical fool on the hill watching the sunset and the rotation of the planet and thinking what he imagines are large thoughts. View all posts by Wyrd Smythe

27 responses to “Transcendental Territory

  • SelfAwarePatterns

    I always knew that Pi and e were trouble, from the minute I laid eyes on them. 🙂

    Carl Sagan reportedly had a message from a deistic god buried extremely far into the digits of Pi in his book Contact (which sounds much better than the movie).

    On determinism, neuroscientist Michael Gazzaniga, in his book ‘Who’s In Charge?’ points out that there is substantial evidence is that the brain is mostly deterministic. He notes that this makes sense when you think about, evolutionarily, what the brain is for, which is to make movement decisions for an organism based on sensory inputs. Rampant indeterminism would destroy any evolutionary advantage in that function.

    Is the brain *fully* deterministic? No one knows. In truth, due to chaos theory dynamics, it may never be known. The question is whether a fully deterministic system could approximate its workings. Again, no one knows for sure, but the fact that the brain is at least mostly deterministic gives me hope. Ultimately, the only way we’ll know for sure is if someone succeeds, or after the brain has thoroughly been mapped and understood, fails anyway.

    • Wyrd Smythe

      “Carl Sagan reportedly had a message from a deistic god buried extremely far into the digits of Pi in his book Contact (which sounds much better than the movie).”

      Yes, a raster pattern of a circle buried deep in the digits of pi. When Ellie meets the aliens they tell her that even more complex messages are buried in other transcendental numbers.

      I wrote about this last Pi Day in Here Today; Pi Tomorrow and quoted the relevant passage from the book. (I like both the book and the movie.)

      The funny thing is that Sagan was right. Sort of. Transcendental numbers can have the numerical quality of being “normal.” Pi has shown to be normal! That means, somewhere in the string, every possible finite sequence occurs. So there definitely is a raster pattern of a circle in the digits of pi.

      There’s also every GIF, JPEG, PNG, and every other image format, image ever created or potential. Also all the images of just random gibberish. And every novel, magazine or other form of printed material ever. In every language. In every variation of typos and whatnot. And every audio file. And so on.

      {{I recently went and grabbed a 10-million digit file of π so I could play around with digit distributions. I just started mucking about with it, but check this out:

      0:   999440 (0.099944, +0.000056)
      1:   999333 (0.099933, +0.000067)
      2:  1000306 (0.100031, -0.000031)
      3:   999964 (0.099996, +0.000004)
      4:  1001093 (0.100109, -0.000109)
      5:  1000466 (0.100047, -0.000047)
      6:   999337 (0.099934, +0.000066)
      7:  1000207 (0.100021, -0.000021)
      8:   999814 (0.099981, +0.000019)
      9:  1000040 (0.100004, -0.000004)

      This is a digit frequency histogram. The first number is the digit. The second is the number of times it appeared in the 10-mega digit string. The third number is the percentage; we’d expect 10% if π is normal (and that’s what we got). The fourth number is the difference from 10% — not much! It’d be interesting to get a lot more digits and see if the differences approach zero.}}

      “Rampant indeterminism would destroy any evolutionary advantage in that function.”

      Makes sense. (A world where that wasn’t true sounds like something Greg Egan would write. He likes to turn things on their ear. He has one, which I’ve not read, where SR works the opposite. The faster you go, the more the external world slows down. A civilization is threatened with a supernova, so they send out a fast ship of scientists to make a long loop through space (that takes the scientists generations) so they can solve the problem and return “shortly after they left” comparatively speaking.)

      Absolutely agree with your last paragraph.

      The point of this post, really, is that question about whether the brain is fully deterministic. My guess is that it’s not, although that would seem to require quantum behavior. Chaotic behavior is deterministic, but utterly unpredictable. Chaos destroys calculation, so it’s really hard for software to be chaotic, but physical systems can be.

      (As I pointed out, we work really hard to keep it out of computers!)

      While we disagree about the likelihood hard AI, perhaps you can at least appreciate why I think it’s such a big leap from what we know is possible.

      This post, and the previous one, are the point I’ve been headed to all along.

      • SelfAwarePatterns

        Sometimes I wonder, if the stuff of spacetime is quantum in nature, as periodically gets pondered in science articles, whether that means we’d eventually hit the end of Pi digits. Or if this is just a case where our mathematics, built upon observed patterns at the level of reality we live in, is just different than the fundamental layers.

        I can appreciate incredulity toward hard AI, and I’ve written myself several times that I think we’ll have to understand human minds in order to accomplish it. (We don’t need that understanding to have very intelligent systems, just for ones we’d consider “conscious”.) But to me the possibility logically follows from what we currently know.

        Now, it’s possible that something we *don’t* currently know will prevent it, but until / unless we encounter that something, I think regarding it as impossible is unjustified. But I’m an empiricist, so I fully admit that we won’t know for sure until either someone accomplishes it, or demonstrates that it’s impossible in principle.

      • Wyrd Smythe

        “Sometimes I wonder, if the stuff of spacetime is quantum in nature, as periodically gets pondered in science articles, whether that means we’d eventually hit the end of Pi digits.”

        Mathematically speaking, no. Pi goes on forever. It’s possible to actually derive this from the properties of a circle, which is why the ancient Greeks knew something was very weird about π. (As the guy in the video mentions, people died over this stuff! It was that weird and offensive to consider.)

        In any physical world we can imagine, Planck level is a limit, so, yeah, there is some ultimate precision of π on that basis.

        {We believe it’s impossible to inspect the world below the Planck level. It takes energy to look at small things (hence CERN), and if you use enough energy to look as small as sub-Plank, that much energy in that small a space creates a black hole and whisks away anything you could see. Kinda defeats the purpose! 😀 }

        {{I think I mentioned my hope (wish) that spacetime by Einstein-smooth. It seems definite that matter-energy are lumpy, but I hold out a faint hope spacetime isn’t. I know a theoretical physicist who calls that hope “idiotic.” He’s probably right, but the jury is still out for the moment. 🙂 }}

        “But I’m an empiricist, so I fully admit that we won’t know for sure until either someone accomplishes it, or demonstrates that it’s impossible in principle.”

        Very much likewise!

        From where I sit, there seem strong (but not definitive) arguments against software AI, and I’ve tried to lay those out in these posts.

        Equally, from where I sit, I see little that argues against a physical (non-biological) network that replicates the brain’s structure.

        Really, the whole point is the gap I see in those two. We have a long, long way to go to establish clear connections between calculation and consciousness.

  • Wyrd Smythe

    An interesting aspect occurred to me about how computers work very hard to remove possible sources of transcendence. They are engineered to treat one entire voltage range as logical one and another voltage range as logical zero. There’s usually some forbidden territory between where the system will treat it arbitrarily.

    But as part of their processing, computers throw away vast amounts of tiny “irrelevant noise” that would degrade their behavior. It would make them inaccurate.

    At the very least, this is hugely different from how human brains work. Neurons communicate analog signals with timed pulses, and analog systems are capable of transcendent behavior, especially ones with 500 trillion “moving” parts.

  • Steve Morris

    Interesting post, Wyrd. Lots of IFs though. The brain as a chaotic system? Possibly. No reason to assume that it is, or that this rules out modelling its behaviour though.

    Are you familiar with the use of cellular automata to model turbulence in fluids? Simple rules can give rise to complex non-linear behaviour even in digital systems, i.e. digital computers can model chaotic behaviour of non-linear systems despite the fact that they are not transcendental.

    Engineers are smart. Unless the science says categorically “no” I wouldn’t rule anything out.

    • Wyrd Smythe

      “The brain as a chaotic system? Possibly. No reason to assume that it is, or that this rules out modelling its behaviour though.”

      Given that the brain is a complex analog physical system with lots of parts, I think the odds of it being chaotic are closer to “probably” than “possibly” but it does remain to be seen.

      “[D]igital computers can model chaotic behaviour of non-linear systems despite the fact that they are not transcendental.”

      Indeed. The Mandelbrot is an example of a simple chaotic system easily modeled with an algorithm. You can zoom in on sections of the Mandelbrot to any precision you’re willing to calculate.

      What’s not clear is how you can model chaotic analog systems with infinite precision, and chaos theory (as I understand it; it’s always possible I don’t) says a digital model will always diverge. Improved precision just delays that divergence.

      Fundamentally, there is an inescapable difference between analog and digital. The latter can get awfully close, but never quite there. (My premise is that matters.) At some point you’re down to Planck level, and it’s all “digital.” But now Heisenberg is a problem. 🙂

      “Unless the science says categorically “no” I wouldn’t rule anything out.”

      Absolutely! (Although, honestly, I have finally ruled out all possibility that I’ll marry Lucy Liu or Lisa Edelstein.)

  • Steve Morris

    MIke, “Sometimes I wonder, if the stuff of spacetime is quantum in nature, as periodically gets pondered in science articles, whether that means we’d eventually hit the end of Pi digits.”
    I think that if that happened, then the circumference of a circle would become scale-dependent at the Planck scale. It’s rather like the way the length of a coastline is scale-dependent and the question “how long is the coastline of this island?” is meaningless.
    In any case, pi has a definite value:
    https://en.wikipedia.org/wiki/Leibniz_formula_for_%CF%80

  • Steve Morris

    Wyrd, intriguing thought if the mind really isn’t a Turing machine. Does that imply that it is capable of doing things that a Turing machine can’t, such as solve the halting problem?

    • Wyrd Smythe

      I’ve wondered that, and I think it might imply that, yes. There’s a potential tie-in to Gödel as well in that Incompleteness suggests there are intuitive truths impossible to prove formally. It’s possible a mind can intuit halting — intuition has been shown to be surprising accurate, at least in some cases.

      (Wielding Gödel philosophically is a questionable. It applies strictly to axiomatic arithmetic systems, but if mind is algorithmic then it’s mathematical, so Gödel might have some bearing.)

      The Traveling Salesman is thought to be a problem in NP and thus intractable to digital calculation (it’d doable, the algorithm is known, it just takes longer than the age of the universe). Yet it seems that bees may solve the problem naturally.

      The brain may turn out to be a kind of analog computer (think of the dynamic motion equations you solve playing, say, raquetball), but like most analog computers doesn’t work by crunching numbers.

      • Steve Morris

        Interesting thought. I actually have a half-finished sci-fi novel that’s a mash-up of religion, an ancient secret society, and and some Turing-related ideas. It’s like Dan Brown meets The Matrix! Your series of posts relates directly to the underlying premise of the novel, although in my novel I have turned a lot of ideas inside out and upside down. I think it’s still scientifically rigorous, though. May even finish writing it one day.

      • Wyrd Smythe

        Cool, go for it! (Does that mean I’ll get passing mention in the credits? 🙂 )

      • Steve Morris

        I could name one of the characters Wyrd 🙂

      • Wyrd Smythe

        Or even just Smythe would do. [Back in my Special Relativity series, I floated an idea about how FTL “ansibles” might work (without obviously violating Einstein… there might still be something lurking… it’s still not clear to me that FTL communication between points in the same reference frame has to be ruled out… I understand it’s supposed to be ruled out, but I’m not sure I understand exactly why.) Anyway, if anyone wanted to use that idea, I just asked they call it “Smythe Waves”… 🙂 ]

  • Disagreeable Me (@Disagreeable_I)

    Hi Wyrd,

    As stated on Self Aware Patterns, I’m not really seeing an argument here.

    There are transcendental numbers, where transcendental just means that they cannot be written as an algebraic expression.

    There are uncomputable numbers, which are a superset of the transcendental numbers (not including e and Pi by the way) which are just numbers for which there exists no algorithm to enumerate the digits to whatever precision we might desire. Any number with randomly chosen digits is such a number. It’s tricky to really define what specific uncomputable numbers might be, because a definition might be tantamount to an algorithm. One example is the so-called Chaitin’s constant (more a family of constants), which seems to have a clear definition but the definition is not particularly useful because using it to find the value of the constant would require being able to solve the halting problem.

    So, yes, there are some pretty weird numbers. I wouldn’t go so far as to call them even vaguely magical.

    I’m not seeing the connection to the mind and consciousness, though. It seems to me that you’re conflating transcendental in the sense of non-algebraic with transcendental in the sense of, I don’t know, generic mysteriousness, mysticism, pre-rational intuition and that sort of stuff. That’s a clear equivocation to me. These are entirely different uses of the term and it is not at all legitimate to draw the inferences you are making in my view. You might as well conclude from the existence of irrational numbers that we are all crazy.

    You’re also drawing in other ideas which I don’t see as particularly related to transcendental numbers, like chaos and complexity.

    You may have something, I don’t know, but you haven’t really joined the dots for me and so I’m afraid it comes across to me like a mish-mash of disparate ideas that don’t really belong together. Perhaps you’re aiming more for poetry than argument and I’m missing the point.

    • Wyrd Smythe

      Hello DM-

      Welcome to my blog… 🙂

      “It seems to me that you’re conflating transcendental in the sense of non-algebraic with transcendental in the sense of, I don’t know, generic mysteriousness, mysticism, pre-rational intuition and that sort of stuff.”

      Yet I said, “This is not the religious kind of transcendence, this is the mathematical kind.” And later, “(and just to reiterate, I don’t mean that in the spiritual sense)”

      I’m talking strictly about the impossibility of calculating with such numbers in any system that processes discrete symbols.

      “There are transcendental numbers, where transcendental just means that they cannot be written as an algebraic expression.”

      Exactly so. They require infinite series which is the same as saying infinite calculation. How can you calculate with numbers that go on forever? You can’t.

      “I wouldn’t go so far as to call them even vaguely magical.”

      Okay, your call. I’m not the one that named them “transcendental” and though.. The mathematicians who discovered them, and who were very excited by them, did. (Did you note the enthusiasm Mr. Pampena showed in the video over these numbers?) That they’re so weird is what makes them a little magical (obviously a metaphor, since none of us believes in magic).

      (As I’ve said, it seems more like an ungrounded, unfounded, unsupported by any shred of evidence belief that “information processing” will give rise to self-awareness is the belief in the magic of numbers.)

      “These are entirely different uses of the term and it is not at all legitimate to draw the inferences you are making in my view.”

      Except that I’m not using it that sense at all, so you’ve apparently missed what I am getting at. In a word: incalculability.

      There are things you simply cannot calculate with a system that processes discrete symbols. The analog and discrete worlds are different. The break from countable to uncountable numbers is bad enough. Chaos enters the picture at that point. The break between algebraic numbers and non-algebraic numbers is even starker. Now we’re dealing with numbers we can’t even write down!

      “You’re also drawing in other ideas which I don’t see as particularly related to transcendental numbers, like chaos and complexity.”

      Yes. Other aspects that also support my argument and make this one stronger.

      “[Y]ou haven’t really joined the dots for me”

      Fair enough. We see the world differently, so naturally we have a different sense of it. 🙂

      • Disagreeable Me (@Disagreeable_I)

        Hi Wyrd,

        You say you’re not using it in the religious sense, but the leap you’re making seems to belie that. But, OK, you say this is about calculability so let’s continue…

        > I’m talking strictly about the impossibility of calculating with such numbers

        Don’t you run into the same problem with your everyday rational numbers? Most of these cannot be exactly represented in binary. The best you can do is to model the fraction explicitly (e.g. recording numerators and denominators) and defer actually rendering this into a single value. But you can do that with pi and e also. You can record your values in terms of pi or e.

        So, again, I don’t think the existence of transcendental numbers has any relevance here.

        > How can you calculate with numbers that go on forever? You can’t.

        You can do so by getting answers to whatever precision you require, as well as having an algorithm that will continue to fetch more precise digits as you need them. I think where we differ is that I don’t think you ever really need infinite precision for any purpose. Having an algorithm to get as many digits you want is in all cases sufficient.

        > The analog and discrete worlds are different.

        Not that different in this respect! You couldn’t calculate with this stuff with analog systems either, because analog systems are inherently imprecise. Due to the impossibility of measuring any quantity to infinite precision, you’ll get no more accurate a reproduction with an analog computation than you will with a discrete one. Indeed you’ll probably be less accurate. Try to find a precise value of pi with a compass and a tape measure and you’ll see what I mean — you will get much better results with a discrete algorithm.

        I would also hazard that due to quantum uncertainty it is incorrect to assume that there actually is a precise value to measure, in many cases.

        > Now we’re dealing with numbers we can’t even write down!

        But we can! There’s more than one way to represent a number. Apparently, you’re not very concerned with the inability of a decimal representation to precisely capture a number like one third, perhaps because we can also represent it as 1 over 3. But we can also represent pi and e in ways other than decimal expansion. One way to do so is with infinite sum notation.

        And of course you’ve also got the symbols themselves. Nothing wrong with writing down e as “e”. Conceptually it’s no different than writing 1 as “1”. Would you say we can’t write down 1?

      • Wyrd Smythe

        “Don’t you run into the same problem with your everyday rational numbers?”

        To some extent, yes, of course. As you say they can be represented as “a/b” and it is possible to design machines that work with algebraic symbols.

        So, yes, absolutely! There are limitations with what can be calculated with numbers. That’s the point!

        “But you can do that with pi and e also.”

        No, there is no algebraic formula for pi or e.

        “I think where we differ is that I don’t think you ever really need infinite precision for any purpose.”

        We know that’s a mathematically false assertion.

        “You couldn’t calculate with this stuff with analog systems either, because analog systems are inherently imprecise.”

        In so far as my thesis is that “calculation is limited” I agree, of course. Indeed, you cannot “calculate with this stuff with analog systems either.” That’s the point. Calculation is limited.

        Essentially you’re pointing out that there are problems calculating with non-transcendental numbers, let alone transcendental ones. This is absolutely true.

      • Disagreeable Me (@Disagreeable_I)

        Right, so calculating with all kinds of numbers leads to situations where absolute precision in decimal or binary representations is not possible. Transcendental numbers are not particularly special in this regard, so I don’t see what they have to do with anything.

        Neither do I see what this limitation of decimal/binary representation has to do with consciousness. Consciousness must be robust with respect to small disturbances. It cannot possibly rely on absolutely precise state because absolutely precise state would be disturbed by environmental interactions.

        > We know that’s a mathematically false assertion.

        Well, no, because “purpose” isn’t really a mathematical concept. I’m just saying there is no reason you would ever need an absolutely precise binary or decimal representation of a number. Or at least I can’t think of one. When modelling real systems uncertainty of measurement is of far greater concern than anything to do with transcendental numbers anyway.

      • Wyrd Smythe

        “Transcendental numbers are not particularly special in this regard, so I don’t see what they have to do with anything.”

        It strikes me that they’re a little extra special. They were named transcendental because they seem to go beyond the normal algebraic numbers we use to describe most of reality. And yet we find them lurking everywhere.

        “Neither do I see what this limitation of decimal/binary representation has to do with consciousness.”

        Then you don’t see it. All I can say is that I see the limitations of calculation as being a potential limit with regard to calculation of self-aware consciousness.

        “Consciousness must be robust with respect to small disturbances.”

        Indeed. In fact, it may even supervene on them. Very subtle effects in analog systems can turn out to have subtle effects.

        Nearly all physical systems share a fundamental property of “least action.” Undisturbed soap bubbles are spheres; water seeks a level; light refracts.

        We have a hard time calculating the three-body problem. Calculation gives us approximations that eventually turn out to be wrong (because of chaos). But the physical natural system operates through least action and the whole solar system of myriad objects solves the N-body problem perfectly.

        Nature follows it own physical laws down to the quantum level, so it effectively solves “unsolveable” (i.e. uncalculable) math problems through the agency of physical properties.

        “I’m just saying there is no reason you would ever need an absolutely precise binary or decimal representation of a number.”

        I think we’d love to be able to predict the weather with precision! But it’s not clear that’s possible, even in principle, with a discrete calculation.

        I understand what you’re saying. I’m saying we don’t know, and it’s possible discrete calculation won’t work in calculating mind. (Assuming minds are at least as complex as weather.)

  • Wyrd Smythe

    Note to self: There seems to me no requirement that a Tegmarkian has to also believe in a computational theory of mind. There is plenty that is both mathematical and incalculable, so a belief in an underlying mathematical foundation need not imply mind is software running on some form of Turing Machine.

    A real-time analog network of mathematical relationships can be both purely mathematical and not possible to calculate, even in principle.

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