Unlimited quantities

It’s a rainy day in Maine, and my son is working on projects from Richard Scarry’s Best Rainy Day Book Ever, which consists of about 200 pages of coloring, cutouts, drawing exercises, games, and the like.  He’s having a grand old time with me on the porch as I type this essay.  I’m interrupted periodically by questions about color, for which I am singularly unqualified; he asks what color I think various things should be, but fortunately, he usually rejects my suggestions.

It’s been several weeks since I’ve seen my son, the last time being for a quick trip for him to say his goodbyes to the dog, and it’s good to have him back.  The house was getting too quiet for me.  Part of the joy of being a parent is watching your child’s mind grow and expand and change.  Inevitably you compare what your child is going through to your own experience – and it makes me understand why my mom enjoys being a grandmother so much, because she doesn’t just compare her son to herself, but further triangulates against what I was like growing up.  She has three sets of comparisons – her/grandson, her/son, and son/grandson – and that gives her a plane shape to compose.  My comparisons are inevitably linear, one-dimensional; hers are now two-dimensional and accordingly more complex and rich.  She was also an elementary school teacher, back when she was in the convent and in the years shortly after, and so she has the comparisons from all of her schoolchildren.  To me, the dimensionality of her universe of childhood expands again with the additional of that non-familial vector, and thus becomes infinitely richer than anything I could conceive of as a simple parent of just one child.

That, really, is a good window into how my mind works.  When you ask me about a comparison, my mind starts mapping out a space in which “information” consists of sets of vectors.  It’s not a real space, and it’s not really a set of vectors, but the mental image helps me set up a system of logical (or, if numbers are involved, actual mathematical) functional statements, which can then be tested for (logical) truth or falsehood, and (logical) completeness.  This largely happens in the background; my conscious thinking is often consciously redirected away from the logical processing of information about the world.  For example, I intentionally try to read non-linear fiction (James Joyce, say, or Don Quixote) when I have a particularly daunting puzzle in my life or in my work.  Distracting my conscious mind, and allowing my subconscious logical processing to be as uninhibited as possible, is a really good way of allowing the logical processing to complete.

Troubling for a lot of people who are in my life, this works for my understanding of human relationships as well.  It’s not that I consciously think of human relations as sets of functions interacting with one another in a kind of simultaneous solution of differential equations… but it’s not that far off.  And it’s not that I quantify or quantize human relations – but that’s only because when I think of quantification, I don’t think of rigid quantities.  I think of continuously integrated functions, of cumulative distributions which interact with other potential cumulative distributions, with incomplete descriptions of correlations which reveal spaces – spaces being unbounded n-spaces, meaning spaces of higher dimensions which aren’t ever fully described but still can be “described” verbally as non-unioned sets which share certain descriptive elements but without enough overlap to explain outcomes fully or even satisfactorily.  This is how I think of people, or creatures, or societies, coming together – even in love, even in the inexpressible dimensions of attraction, and desire, and care, and sacrifice, and beauty.  I see them in terms of a foundational set theory of n-spaces, of integrable and differentiable functions, with too much mutual unintelligibility but each being bounded in their own way and describable in their own terms as closed.

I’m sure this is gibberish, and indeed while I understand what I mean, I know that what I’m saying is complete garbage.  So let me try this from another direction.  My conception of the universe is as follows:

• There is a functionally infinite (although bounded) virtual lattice which describes every potential point in the universe, everywhere.  Each point of the lattice is separated from each other point by a distance equal to the Planck unit of distance (the smallest distance expressible in our current understanding of quantum physics).  To give a sense of scale, that’s about 10²⁶¹ lattice points in the space described by a proton, so we’re talking a really fine lattice.
• Each point on the lattice is a point in space at which an expression of quantum physical potential (which is to say the instantiation of energy, howsoever it may exist) can exist for a given time, that time being expressed as a Planck unit of time (the smallest theoretical length of time).
• There exists a function which describes the state of the lattice – that is, the population of each node in the lattice at each moment in time – and describes the transition possibilities from the prior moment in time and to the next moment in time.  This function may or may not be discrete, mathematically speaking, but it does exist.
• The integral of this function over time, from 0 to infinity, describes the cumulative potential function of the universe.

This is also my conception of divinity, or God – although I really hate using personifications of mathematical functions.  I don’t even like it when in math you’re expected to refer to different things with proper names, like the Couchy distribution or the Lorenz wave transformation function.  It’s not that I don’t like names – history is lousy with them, and I’m usually quite competent at remembering names in that context – but I don’t see why they have any particular place in mathematics.

Anyway, when my father or my religiously or mystically inclined friends start talking about metaphysics, my mind always returns to this world of partial differential equations.  I try to translate their always verbal conceptions of divinity, or god, or God, or the gods, or what have you, into different expressions of what can exist at each point in the cosmic lattice, and different functional expressions of how each state of the lattice transitions to the next state.  It’s usually not possible, but the exercise is helpful, and it also insulates me somewhat from the need to have a conversation about belief, or the consequences of a given expression of belief.  To me, the description of the universe as an enormous set of states – 10⁴⁵ of them each second, with each state consisting a theoretically infinite 3-dimensional space with 10³⁰⁰ individual lattice points inside of each cubic centimeter – is about right.  That means that for every cubic centimeter of space in our experienced universe, in each second, there are 10^13,500 potential states – but even that isn’t truly expressive of the potential variety, because I don’t claim to understand the number of potential dimensions of state existence at each node, at each point in Planck time.  The current model describes roughly 11 dimensions of potential differentiation, each of which has anywhere from 2 to 6 distinct states.  That means that there is potentially an additional call it 2¹¹ or more potential states at each of those nodes.

That means at any given instant, given our current understanding of energy states, there are 10^13,500 points of instantiation per cubic centimeter, which can experience 2¹¹ states. That gives a potential expression of 2¹¹ (roughly 2000 potential energy expression states) times 10¹⁰⁰ states per cubic centimeter, or 2×10^40,000 potential realizable states per cubic centimeter.  Not all of them are possible given the functional possibility of moving from one time point to the next, but in the absolute, any of them could exist.  And if the space of the universe is, say, roughly 100 billion light years wide in all three dimensions, and a light year is roughly 10²⁵ centimeters long, and therefore the universe has roughly 10¹⁰⁸ cubic centimeters, then the universe has approximately 2×10^4,320,000 possible states at any instant. And each state can be differently expressed 10⁴⁵ times each second.

Now it’s impossible to go from any one random state to another random state – we observe, rather, that there is a kind of path dependency, where the starting state defines the set of potential next states which are able to come into being, albeit recognizing that there is a probability at work which allows for variance, even if the functions which govern how states move from time t_n to time t_n+1.  So we observe that there is a set of functions that govern how each realized state moves to the next of an almost (but not quite) infinite set of potential states, subject to some probabilistic likelihood of randomness outside of the functions.  The probabilistic variance, in a quantum sense, vanishes in the fully integrated state over time t₀ (or more likely, time t_-∞) to time t_∞, but remains relevant in understanding instantaneous shifts in state.

The trick, as it were, is to describe that set of functions – and understand their metaphysical or theological meaning – that then allow you to fully describe the state and its behavior (ie., the way in which transitions to future states are possible and the ways in which they are not) – then you’ll know the meaning of the universe or, as a recent essay on the topic described it, you’ll have a God function.  We also, of course, have to have a “plug” – which if we understand the functions, would consist of knowing the starting state of the universe at any point – not at the Big Bang, or at the end, really – which point it is that we actually completely describe is irrelevant.  We just need to have a complete knowledge of the state solution of the universe at any given instant, and a complete knowledge of the set of functions which govern transitions from one instant to the next, and an understanding of the probability density functions which govern instantaneous randomness.  Piece of cake.

Mathematically, however, it will be impossible to ever determine such a function, as it’s been proven by Godel in his famous incompleteness theorm, which states that for any system (and what I described above is a complete system), there exists at least one proposition or potential state which can simultaneously be true (possible) and false (impossible) under the observable dynamics of the system itself.  Put another way, it means that since we are wholly within the system of the universe, we’ll never be able to derive a complete set of proofs for the system itself.  It will always be a mystery.  This is enormously comforting to me.

It’s enormously comforting to me simply because I know total knowledge is impossible.  I’m comforted by the fact that the system of the universe has enabled the construction of sentience – that is, a recursively reflective intellect which can both observe information, construct theories of how that information came to exist, and in recursion and reflection create doubt about those theories and construct further theories about knowledge construction without bound – such that I can understand that such a system can exist, even if in so doing that sentience becomes aware of its own limitations, becomes aware of the inevitability (not potential, not likelihood, but essentiality) of its own failure to understand the system and, by extension, its place within it.  Even as it attempts to understand, sentience will with perfect certainty – indeed, the only perfect certainty possible at all – fail to gain that understanding.  I can be aware that a functional system exists but I can also prove my own inability to understand, describe, or use it.

It doesn’t mean I can’t develop a kind of understanding of the behavior of us – whether “us” is particles, molecules, cells, creatures, beings, humans, lovers, failed lovers, parents, children, owners, slaves, whatever.  I can interpret all of that, potentially, and because of the magnitude of uncertainty I’m aware of I’m probably better at real estimation than most, in the sense that I interpret but am constantly aware of the instability of my own estimates. I’m as likely to misunderstand or use incomplete or misunderstood or incorrectly measured data and end up with a bad understanding as I am likely to form a correct understanding of those smaller systems, in other words, and while I’m confident of my own interpretive power, I’m humbled by the failure of data and failure of completeness as much as I am forced to act as if I’m making a pretty good estimate of how I think the world works.  I know that I can’t understand all of us at all times, or even at this time, this instantiation that has already passed 10⁴⁵ times for each second it took to type this sentence.  I have to accept my ability to roughly cobble together a good enough understanding of my own small part of that system while still accepting my – and every possible one’s – inability to understand the system as a whole.  And accept that I do.

I was asked by a friend of mine, a teacher, how I came to be numerate, because I told her that being truly numerate was no different than learning a different language, no different than being taught to be bilingual.  I came to understand it by being shown images of infinity at an early age, not true infinity – and except in conceptual spaces I don’t think there is a true infinity, although conceptually bringing infinity into our understanding at an early age has real value.  I was shown notions of the expansiveness of the world of potential.  It’s easy to start with: play games of chance.  Play cards.  Play draughts: an 8×8 two-dimensional lattice, 64 nodes and three potential states (red, black, or unoccupied) at each node during the game.  Each turn is discrete, much like each “turn” of the universe occupies a Planck time state.  Play as many games as you can, with rules which are arbitrary – and then allow for a change in rules and play again, and see how the rules impact the potential for end states of “winning” or “losing” or “drawing”.  Play, but in a defined space which a child’s mind can understand.  And every day, every moment, as soon as one board space or set of rules seems to be understandable, expand the space of the game.  Change the rules.  Make the pieces expand from two colors to three, then four, then eight, then without limit.  Expand the board in size, or add a dimension.  And keep doing it until the player keeps expanding the game space and the rules and the playing pieces on their own.

When the player stops, when they get tired and stop expanding the playing surface, or when they fix the rules and say “I don’t want to keep changing the game”, or when they decide they know what the potential states are for a given piece, then they are, in our parlance, adults.  Then they are dead, really, and we should make room for the next children who we teach again, to play a game – first with a few rules, and just a couple of colors, and a small playing board.  But what happens if the game expands, if the rules can change, if we add another color…

My son, and my mother, and my ex-wife and I played a game last night, building railroads across Europe.  The rules of the game as given to us in the box had already been discarded long ago.  We left behind the time limits, and the requirements to only build one connection at a time.  The game was more interesting than the one in the box, even though it was harder to figure out when someone had won and when someone had lost.  The game lasted far longer than the time described on the box.  I don’t know who won.  We were all playing, and we all built railroads far beyond the boundaries of the map of the game.  We had a lot of fun.  It didn’t matter that it was raining.

Postscript: I’ve updated some simple math errors, and clarified the role of randomness in instantaneous state shifts in my construction of how the universe functions in an edit to this post roughly 18 hours after the original.  Apologies to anyone who was confused by my mistakes or thought that the original text gave the impression that I believed that state shifts occur through a linear, closed-form solvable process.  Oops!