Time and Hard Sums

I’ve been thinking recently that maybe there’s an elegant way of describing Time simply as difficulty. Mathematicians now have a charmingly naive term, “hardness,” for describing the relative knottiness of a calculation problem. If asked to give an explanation to a layperson, they will often say something like: “Well, suppose you had a perfect computer. A hard problem takes it more time to solve than an easy one.” If you have a problem like the “traveling salesman” puzzle–given n cities, how do you figure out the shortest route by which he can visit all of them–it’s really easy if you have three or even five cities, but if you have a few hundred, no computer in the universe, using the fastest theoretical algorithm, could solve it in less than a billion years, by which time the salesman would long ago have moldered into atoms. These algorithms are called “NP hard.” There are even more difficult problems still, ones that ask, for instance, whether it is possible to prove whether or not a given problem has an algorithm to solve it at all–i.e. problems that can only be solved by the emergence of as-yet-undiscovered further problems that will require unimaginable algorithms of the future if they are to be solved.

If we describe the universe as a computational system–and the fact that all science expresses its conclusions in numerical terms strongly suggests that science “votes with its feet” for that hypothesis–then we can see all entities in the universe as the workings of problem-solving algorithms. The ones that are easily solved have been solved already and have stopped, and constitute the eternal constants of physics that are true at every instant through all time, like the instantaneous coexistence of the probabilistic quantum world described by David Bohm. The ones that are a bit harder but still solvable are the deterministic processes in nature that Newtonian science describes. The ones left over constitute the whole world of change and becoming, ranging from chemical reactions through self-cloning living organisms to ourselves, arranged in a nice pyramid of emergent temporal features as described by the great philosopher J. T. Fraser.

If time is a river, there would be nothing to indicate the passage of time unless some parts of the river were flowing faster than other parts. “Hardness” gives us a nice way of measuring which ones are faster and which ones are slower.

Here’s a nice thought-experiment to prove this idea. The increase of entropy (thermal disorder) is usually recognized by physicists as a reliable marker of the passage of time. Thermal disorder is what we call heat. If I am right, a large amount of local computation should be the same thing as a large amount of local time; and a large amount of time should be correlated with an increase of heat. If you are using a laptop, and it’s actually on your lap, you can feel the heat of computation generating time on your thighs.

By Frederick Turner

Professor, poet, lecturer, black belt, and more.

33 replies on “Time and Hard Sums”

“…a large amount of local computation should be the same thing as a large amount of local time;”

Would you put that in poetry please? Wherever heat is found is found enlarging time?

If you do these time/space/purpose experiments for “the fun of it” you are, according to Amelia Earhart speaking current culture’s values. But anyone who reads science fiction religiously is right in sync with the times!

The point is that the mathematics of algorithmic theory and proof theory can now quite exactly quantify the amount of difficulty in a given procedure. And I believe the equations for that quantification are isomorphic with the equations for calculating the increase of entropy in a thermodynamic system, and possibly also with the equations in information theory for calculating the amount of information loss in transmitting a message (or the amount of free energy that would have to be spent in boosting the signal to compensate for the loss).

To put it in the poetry of traditional proverbs: Time is money; where there’s muck, there’s money. Or Shakepeare, in As You Like It:

Time travels in divers paces with
divers persons. I’ll tell you who Time ambles
withal, who Time trots withal, who Time gallops
withal and who he stands still withal.

ORLANDO: I prithee, who doth he trot withal?

ROSALIND: Marry, he trots hard with a young maid between the
contract of her marriage and the day it is
solemnized: if the interim be but a se’nnight,
Time’s pace is so hard that it seems the length of
seven year.

ORLANDO: Who ambles Time withal?

ROSALIND: With a priest that lacks Latin and a rich man that
hath not the gout, for the one sleeps easily because
he cannot study, and the other lives merrily because
he feels no pain, the one lacking the burden of lean
and wasteful learning, the other knowing no burden
of heavy tedious penury; these Time ambles withal.

ORLANDO: Who doth he gallop withal?

ROSALIND: With a thief to the gallows, for though he go as
softly as foot can fall, he thinks himself too soon there.

ORLANDO: Who stays it still withal?

ROSALIND: With lawyers in the vacation, for they sleep between term and term and then they perceive not how Time moves.

Stuart Kaufman argues that time’s arrow comes about with the increasing nonergodicity of the universe. He says that history occurs when there are more choices in the future than there were in the past. If you haven’t read “Reinventing the Sacred,” you should. Then there’s my book, which is now out, btw.

It is very clear what you are saying. My trouble with “difficulty” was my associating it with “trouble,” as trouble occurs when things get heated up and time goes away (expands) if only in the sense that consciousness of the present moment occupies the whole of time on a pin head so to speak. “Difficulty” connotes, if not denotes, trouble, which seems to preclude the other realms of heat–the heat of passion, or love, excitement, anger, sadness and other timeless emotions without difficulty. Shakespeare gives us the most familiar ones in spades. Does an epiphany, such as the one you experienced once out walking, finding yourself surrounded by Eden-like blossoms (which you never found again) for instance, generate heat? I think so–but not difficulty. Nor would you call the heated consummation of young lovers difficult, but rather effortless.

Do you mean by isomorphic analogous–or coterminous?

An after-thought: Or maybe “entanglement” –as in God the Father (the One) is an example of easily solved, and stopped; God the Son (the Two), a bit harder but still solvable; but it takes God the Spirit (ironically) for things to become entangled, heat up and produce something new–new time maybe. What did Newton say about the Trinity?

I love the stuff about the trinity, which is an amazing generator of insight.

Isomorphic means that the relations of the parts are the same, though the parts may be different, thus one behaves the same as the other under change.

I think you made my point–“timeless” experiences,epiphanic or chairotic moments, are perhaps indeed devoid of difficulty. But it’s only through time–difficulty–that time is conquered. If Keats were to become as unconscious as the nightingale, he wouldn’t be Keats, wouldn’t do the difficult work of being a poet. “To thy high requiem become a sod.”

Yes, I did make your point, didn’t I–a deeply ironic one, that only time conquers time.

I am intrigued still (pardon the circuitous route and detour from the subject here) by the resonance between two definitions of ‘isomorphic,’ which one dictionary defines as “having similar appearance but genetically different,” and you define as “the relations of the parts are the same, though the parts may be different, thus one behaves the same as the other under change.” It occurs to me that metaphors which involve the passage of time (“dawn to dusk” and “birth to death,” both expressing the abstraction of beginning to end) are isomorphic in relation, i.e. interchangeable of parts against the background of change (Unchanging Change)–maybe “change” is another word for time?

But again I’d define change as difference, and difference as the amount of difficulty getting computationally from one to the other (or resolving one in terms of the other, or restating one in the other’s terms). This nicely describes what physicists call the history function in elementary particle physics, ie the sum of all the possible pathways from one state of the particle to the next.

maaaaaaaybe. I happened to read a nifty arlctie on a christian perspective on time by dorothy sayers this morning: Strong Meat. I read it from the book, letters to a diminished church. She argues against the struggle against aging and putting faith in progress.

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