Monday, June 14, 2010

Order Underpins Everything

I discovered the work of Kevin H. Knuth, and took a dive into his papers and this recent talk given at the Perimeter Institute. The theme of his research is that a simple ordering relation among elements is more fundamental than, and can be used to derive, more familiar theories. The talk is entitled “The Role of Order in Natural Law”, and was part of a workshop on the topic of laws of nature.

In older papers, such as “Deriving Laws from Ordering Relations”, and “The Origin of Probability and Entropy”, Knuth shows how a partially ordered set, or lattice, can give rise to probability theory and information theory. There are broader applications, too, which follow from his demonstration of how order leads to algebra, and , through assigning numbers to elements, can further be used to derive measure theory; this is the link to order being revealed as foundational to other areas of science and mathematics. The derivation of probability theory, which is the first half of the talk, is easy to follow and interesting.

In a new paper, co-authored by colleague Newshaw Bahreyni, Knuth constructs a causal set (a bit different partially ordered set compared to the lattice) and undertakes to derive special relativity. The steps here are also pretty straightforward; a causal set is one where events are ordered according to a relation where some events have the potential to be informed of or influenced by, other events, but the relationship is not reciprocal. Completely ordered chains of events are selected to be “observers”, and events on chains are quantified by simply assigning integers. Then the relation of off-chain events to the earliest potentially informed events on 2 observer-chains gives rise to time-like and space-like coordinates. The Minkowski metric follows from this analysis, and Knuth and Bahreyni also derive Lorentz transformations.

Knuth also thinks that ordered pairs of numbers also could play a role in the foundations of quantum mechanics. He is a co-author of this paper by Philip Goyal (who also gave a talk at the same workshop), which looks to derive the complex quantum amplitudes from ordered pairs of real numbers. This looks very interesting as well.

From the conclusion of the causal set paper: “Given the belief that physical law represents underlying order, one would expect that given this underlying order, one ought to be able to derive the most fundamental aspects of physical law.”

Knuth’s program , while not yet tackling gravity, shares the spirit of many of the quantum gravity programs I’ve reviewed which explore the idea that the space-time of general relativity itself emerges from a more fundamental network of causally-linked elements. Causal sets themselves have played a featured role in the research program spearheaded by Rafael Sorkin (see old post here). Progress has been made in this program over a number of fronts, although incorporating quantum mechanics explicitly has evidently been a challenge. The arxiv has recent update papers on this causal set program by Sorkin and Joe Henson.

UPDATE 18 June 2010: I finished listening to Goyal's talk, and I am excited by his program. If even the foundations of QM end up being derived from 1) the ordering of events, 2) the logic that derives operationally from this order, and 3)an assumption about the limits on the accessibility of degrees of freedom (the essence of complementarity), this would be a great achievement. Events (which Goyal interestingly thinks about as Aristotelian actualizations) would be fundamental: "The quantum formalism is not explicitly about space and matter" -- these would be secondary notions; and time enters only in the minimal (but crucial) sense of ordering.


Allen said...

Hey, have you heard of Quentin Meillassoux?

Here's a fairly brief paper that introduces his ideas.

I'm not so much taken with his "problem of ancestrality", but I find his ideas about Kant, correlationism, and facticity to be very interesting.

I'd be interested to hear your thoughts!

Steve said...

Thanks. I will check that out.

Steve said...

On the first reading, I find the paper deep and difficult, but interesting and suggestive. I'll re-read and look for some of his other writings.

Allen said...

It is a bit difficult to parse. It really gets going in section 3, "The Principle of Factialty".

I thought this was the central paragraph of the paper:

"I call 'facticity' the absence of reason for any reality; in other words, the impossibility of providing an ultimate ground for the existence of any being. We can only attain conditional necessity, never absolute necessity. If definite causes and physical laws are posited, then we can claim that a determined effect must follow. But we shall never find a ground for these laws and causes, except eventually other ungrounded causes and laws: there is no ultimate cause, nor ultimate law, that is to say, a cause or a law including the ground of its own existence. But this facticity is also proper to thought. The Cartesian Cogito clearly shows this point: what is necessary, in the Cogito, is a conditional necessity: if I think, then I must be. But it is not an absolute necessity: it is not necessary that I should think. From the inside of the subjective correlation, I accede to my own facticity, and so to the facticity of the world correlated with my subjective access to it. I do it by attaining the lack of an ultimate reason, of a causa sui, able to ground my existence."

I read his "After Finitude", and thought it was quite good and fairly readable. The first chapter on ancestrality was slow, but from chapter 2 onwards it was very good.

So far I don't buy his argument for speculative materialism, BUT I think he highlights an important point about facticity, which I hadn't seen made anywhere else.

But, as he says in the final paragraph, there's more to come:

"Now, my project consists of a problem which I don’t resolve in After Finitude, but which I hope to resolve in the future: it is a very difficult problem, one that I can’t rigorously set out here, but that I can sum up in this simple question: Would it be possible to derive, to draw from the principle of factiality, the ability of the natural sciences to know, by way of mathematical discourse, reality in itself, by which I mean our world, the factual world as it is actually produced by Hyperchaos, and which exists independently of our subjectivity? To answer this very difficult problem is a condition of a real resolution of the problem of ancestrality, and this constitutes the theoretical finality of my present work."

Steve said...

The idea that there is no necessary ground for the contingent, but something like a supercontingency or hyperchaos, was the part I thought was deep and thought provoking, although I don't yet know if it helps or if I can buy it.

I have to say that the modern continental style of philosophy isn't my favorite (surely the language issue is part of it). I've never been able to really get into any philosophers since Merleau-Ponty. But if you thought it was fairly readable I may take a shot here and get the book.

Allen said...

The argument he uses to arrive at the possibility of hyperchaos seems very compelling to me.

If there is no reason for the laws that we have, then it seems plausible that those laws could change for no reason.

Even if there is some sort of "super-law" that enforces the consistent application of the physical laws, then what enforces the consistent application of the super-law?

If there is no reason that the world is *this* way instead of some other way, then it seems plausible that the world could, for no reason whatsoever, spontaneously become some other way, or even disappear altogether. But even if you deduce a reason, what is the reason for the reason? Can there be an explanation that explains both itself and everything that follows from it?

One thing that does strike me is that hyperchaos as he describes it would require *two* things. First, the power to create and destroy. And second, a dimension of time along which things come into existence and are destroyed.

But the existence of this time dimension would also seem to be contingent...wouldn’t it? I lean towards some sort of static, timeless variation of his idea, where things aren’t created and destroyed but rather just exist, eternally and without reason.

On continental philosophy, I’ve also found it rather difficult to decipher, but I really liked "After Finitude."

Based on that positive experience, I’m currently reading "A Thing of This World: A History of Continental Anti-Realism," by Lee Braver. From it’s product description: "At a time when the analytic/continental split dominates contemporary philosophy, this ambitious work offers a careful and clear-minded way to bridge that divide."

The first two chapters, "Defining Realism" and "Kant’s Revolution", were very good. The third chapter on Hegel was comprehensible, but dragged a bit. I’m currently on the fourth chapter, about Nietzsche, and it’s been much more enjoyable.

It’s been interesting to see Hegel and Kant’s writings contrasted with quotes from Bertrand Russell, Hilary Putnam, Donald Davidson, and other analytical philosophers. So far so good I think.