Last 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.
We start by considering three Yin-Yang pairs with regard to numbers.
The 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.
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.
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.
The 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. 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.
There is something magical about the transcendentals that sets them apart. The name is certainly evocative.
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
thesises… thesisis… points to demonstrate here.)
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!
Look 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.
Synapses 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.
The 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. (It’s not the same as “fuzzy logic” which is just logic over a value range.)
As 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?
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?
 This all started out as a flight of fancy, but the more I think about it, the more it fits.
 Even if it takes them forever, just like it would you.
 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.)
 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.)
 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!
 AKA: “theses” 🙂
 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.)
 Cantor is clearly addressing the countable-uncountable divide, and it’s possible Turing and Gödel are as well.