One of the principle arguments both for our confidence in the application of our physical theories to unobservable situations¹ and the reality of the postulated objects is that our physical theories are particularly simple. The background idea is that when we approximate a function by fitting points or some other general method we expect to get a complex unwieldy object back thus the simplicity of our physical theories shows they aren’t just good approximations based on lots of data points but somehow really get at what is happening. However, I’m skeptical that our intuitions about simplicity are correct. In particular I worry that our idea of what’s simple is deeply influenced by what we find useful. To explain further let me offer an example.

Suppose you are given a box that lets you dial in any² number between 0 and 1 and returns some output value between 0 and 1 within some experimental error³. If after trying many values you derive a polynomial with 25 coefficients that lets you very closely approximate the average result⁴ for a given input you probably wouldn’t think you’d hit on anything deep about the operation of the box. In fact you’d probably guess that greater precision (averaging over more tests) would reveal subtle distinctions between your approximate function and the true value. On the other hand if after the same number of tests it appears that sin(x) is an equally good approximation you might think this was the true function and expect this to be born out by further experiments. You might even make hypothesises about the box’s mechanism on this basis.

My worry is that those theories we take to be simple and elegant really aren’t simple at all. For instance is it really the case that sin(x) is a simpler function than some 25 term polynomial with integer coefficients between 1 and 10? The obvious way to answer this is to ask how many symbols it takes to define each function but this answer depends on what we take to be our primitive terms. To put the point more formally the Kolmogorov complexity of a string depends on our choice of a universal prefix-free machine. However, it’s reasonable to think that so long as we pick one system to represent out theories in and stick with it then it will function as a useful measure of a theories complexity⁵.

However, in practice we never really fix one system and insist on writing all our theories in terms of it. When people discovered that the sin function was frequently useful in describing physical systems they stuck it into their toolkit. They didn’t stick with whatever previous system they had been using and include the definition of sin(x) in all of their theories. Yet if our idea of what a simple theory is changes in response to what seems to make good predictions we no longer have a good argument for the truth of our theories. If it had turned out that a parametrized solution to the equation y^3+x*y=x^2 had been widely useful in physical theories instead of solutions to x^2+y^2=1 then it would probably have been those functions rather than cos(x) and sin(x) that we regarded as elementary functions.

I don’t doubt that evolution has endowed us with a notion of simplicity that works well in everyday macroscopic scenarios. What I’m skeptical of is the claim that the abstract mathematical theories that underlie particle physics and cosmology are really especially simple. Certainly it’s true that they can be expressed in a form that strikes us as elegant and appears simple but they only do so by making use of many layers of abstraction. I’m not so sure that if we examined the mathematical framework for quantum mechanics written out as a formal statement in PA it would still strike us as particularly simple.

In short I’m worried that we underestimate the power of additional layers of abstraction. Sure, the mathematical concepts used in modern physics are the result of a series of definitions and abstractions each one of which strikes us as simple and elegant but the essential question is whether alternative theories giving similar agreement with the data would admit a similar chain of definitions. Given that no real work (to my knowledge) has been done about the additional complexity each layer of abstraction brings to a theory what reason do we really have to be confident about the simplicity of physics?

I’m listening to an interview on KQED’s forum (local NPR station’s call in show) with science journalist Timothy Ferris. Apparently he just wrote a new book about amateur astronomy which I don’t doubt is well researched and accurate but as people called in he apparently felt the need to opine on time travel and quantum mechanics ’scientific’ matters and I was appalled. Since he’s also written a book called “The Whole Shebang” his misleading answers can’t be explained as mere failure to research. However, I’m inclined to think that in this case the fault lies with the physicists themselves (either for doing bad philosophy or using mislead metaphors.

It started with someone bringing up the Fermi paradox (why haven’t advanced alien civilizations contacted us yet). The host then steered the question towards whether this was an argument against time travel as well (failure to see time travelers). Timothy Ferris replied that he didn’t find it very compelling because he expects time will be lack a deck of cards so that if you go back in time of forward in time you end up in one of many possible pasts or many possible futures. While he admitted it was just his expectation he clearly conveyed the sense that it was a possibility that experts would take seriously.

I happen to think the very idea of something being time travel requires that we go back into the past not merely enter some universe that looks like the past.¹ However, let’s set this point aside. I suspect the journalist was referencing some approaches to quantum mechanics that go by the name of sum over histories or multiple histories. Possibly he meant to refer to many minds or many worlds theories. The problem is that traveling to an ‘alternate’ past doesn’t even make sense in any of them. Supposing it was true and even meaningful that we have multiple histories in this QM sense we would have multiple presents as well. What the hell would it even mean for a person, who is really a superposition, to visit one component of a prior superposition? Pure many worlds theories only really make sense² if we understand them as collapsing down to a many mind’s theory and it certainly isn’t clear what it would mean for a mind that rides atop the superposition to time travel by itself, certainly not in the sense of some dude from the future appearing.

That’s confused and I was annoyed that he said it with such apparent authority but what really got my goat was when he talked about how interesting it would be if we ended up with quantum computers since we couldn’t explain their processing power with just one universe and would have to say that they use other universes to do their computations. This is just a lie that is being pushed on the public. The fundamental laws of nature could just offer us an oracle that computer anything we wanted as fast as we want. For all we know there is some special experiment we can do that reveals the true bits of 0′ (the set of the halting problem). Worse this is certainly not anything scientists have or can test. It is purely unjustified bad philosophical speculation that misleads the public.

I’m not sure whether to be mad at the people who promote this crap or applaud the physicists for great PR. Maybe we should just adopt this for math. Push the whole confusion about Godel’s theorem a bit more and try selling the Banach-Tarski paradox as a proof that “space is an illusion.”

Bit about quantum computer

While there may be more sophisticated versions of Bayesianism as I understand the theory it treats ambiguous gambles the same way it treats risky gambles. That is even in situations where one does not know the odds offered one is still modelled as having some prior probability. For instance if someone was to ask my to make a fair $5 bet one whether or not I would pull our a red ball when I reached into their hat Bayesianism would have me assign a prior probability distribution to the ratio of red to non-red balls in the hat, even if I had never seen this person before in my life nor been given any clue about the ratio.

However, it appears some recent research suggests that different brain areas are implicated in making risky decisions and ambiguous decisions. In fact it even appears that risk-aversion and ambiguity-aversion can vary separately between individuals. Does this doom Bayesianism?

I don’t have a clue. This just begs the question of what the hell theories like Bayesianism are supposed to accomplish. If it is a claim about rationality then this result is irrelevant. If it is about a ‘good-enough’ approximation to actual scientific decision making then this could just be pushing the goodness of the approximation. If it aims to be some kind of idealization of actual decision making given arbitrary time and generally agreed upon errors ruled out then yes this conclusion could be very damning.

It is very difficult to underestimate the importance of precision and naming. Each of these different notions of Bayesianism really should be given differing names.

So here comes another film inspired philosophical puzzle. Well really a less sophisticated version of this problem occurred to me a long time ago while reading Permutation City by my favorite sci-fi author Greg Egan and seeing someone last week totally consumed with a virtual world reminded me about it but everyone knows what I’m talking about when I mention the matrix. However, some people may not be familiar with the Sleeping Beauty problem so I will provide a short summary in the post body.

The point I want to illustrate is that whether or not this is the actual world or a matrix style simulation is an instance of the sleeping beauty problem. In particular the same reasoning that supports the two-thirds solution to the sleeping beauty problems seems to guarantee we are in a repetitive simulation. I don’t know if this puzzle has appeared in the literature yet (it seems kinda obvious to me so I have a hard time believing it hasn’t) but it is new to me (no I haven’t read all the papers I link so it could be there). Since I find the reasoning supporting the two-thirds solution compelling I expect some other solution will ultimately be forthcoming but I expect that solution will be more interesting than the problem itself.

More concretely it certainly seems possible that we are actually in a virtual world like the matrix. It remains possible even if we suppose that our lives are in fact arbitrarily long (the ‘real’ world has conquered aging) and every time we die in the virtual world our memories are wiped and the simulation is restarted. While we might think ‘a priori’ such a world is unlikely since it seems possible we should assign it some non-zero probability. However, since this matrix world repeats the experience we are having infinitely many times it would seem that the same reasoning which allows us to support the two-thirds solution in the sleeping beauty problem requires that we find the matrix world arbitrarily more probable than our own world, i.e., the probability of the matrix world conditionalized on matrix world or real world is one (or arbitrarily close to one). Of course the virtual world aspect isn’t central to the problem. We could repeat the same argument in favor of Nietchzie’s world of eternal recurrence or any situation which involves making us experience our lives so far arbitrarily many times. (more…)

Now that I’m back from Europe and my life is settling down again I will hopefully make more frequent posts. Although I still need to figure out how to teach an introductory philosophy course about Phil of science but including Leibniz as well as Kuhn.

Anyway I was having a conversation with Sharon today when she said (more or less), “If my dad had been home schooled that would explain his personality.” While intuitively I immediately knew what she meant, though a precise analysis is somewhat complicated, the sentence itself seemed deeply troublesome. In particular it seemed to come out necessarily false, or at least self-defeating like a Moorean sentence, on a possible world’s account of counterfactuals.

Why? Well I’m not entirely sure. My intuition seems to be based on a principle like: an explanation of some effect must necessarily cite things which ’cause’ the explanadum. For example the fact that Mary threw a rock towards my bedroom window at 3pm does not explain why my window glass is broken if it was actually smashed at noon by John. This holds despite the fact that throwing a rock towards a window is quite definitely the type of act which causes window glass to break. For a more detailed discussion read below. (more…)