The n-Category Café

Daivd wrote:

what if the modal logic that emerged with it, and its extension to dynamic/temporal logic, is merely a useful tool in computer science?

We have Aristotle trying to formalise concepts like necessity and people like Avicenna and scholastic philosophers pushing the ideas further (partly motivated by the ontological argument, I believe). I get the impression that mathematicians looked down on modal logic and then suddenly a philosopher, Kripke, makes a connection with the far-out idea of many worlds and the whole thing becomes respectable mathematics. And then, as if by magic, through a Curry-Howard-like isomorphism, modal logics like S4 turn out to be exactly what you need to formally describe things like partially evaluating computer programs while you compile them. This is an amazing journey for a concept to take and it really shows how you can’t tell, a priori, where an idea might lead, or even what you should be reading to get good ideas for use in a particular field.

David asks:

So are there any indications as to what gave Zeno’s argument its value?

You will know more about the history of Zeno’s paradox than I do. All I meant to say is that Zeno’s argument is an interesting, albeit wrong, argument, which is thought provoking. It requires quite a bit of sophistication (well, we teach it in school today, but it took mankind a while to figure it out) to resolve it:

as we know, the resolution is:

- the concept that Zeno was lacking was that of a convering series (an infinite converging sum).

- the wrong assumption that Zeno made was that every series (namely of time intervals spent by Achilles chasing the tortoise) with non-vanishing entries has to diverge.

This is wrong. I don’t blame it on Zeno, but I point it out as an example that there are plenty of arguments which may sound sophisticated and may sound right, but are based on hidden false assumptions and indeed wrong.

And that science/math happens only when we stick to those arguments where we have reasonable control over the assumptions and the meaning of the terms involved.

There are, and of course have been in history, plenty of interesting open questions which one might long to answer in a scientific manner, but which are simply out of reach for exact science.

Another famous such paradox is Fermi’s paradox (which is actually closely related to brainimania).

Or speculations about consciousness. I can imagine that one day in the far future, mankind understands the phenomenon of consciousness on a much deeper, scientific, level than we do today, possibly figuring out how it is related to quantum cosmology (as some contemporary cosmologists suggest). But we are certainly not there yet and can’t reason about it within exact science.

Which is not to say that it is not fun, and even useful, to wonder about this and exchange arguments about it while, say, sitting around a camp fire. But not in articles that come with the appearance of scientific papers.

And are these indications lacking in Boltzmann brain discussions?

I find the Boltzmann brain argument itself weird. I do see some value, in the above campfire-sense, in the general discussion about the meaning and relevance of life-in-the-cosmos. It can certainly be fun and thought-provoking. But for it to come closer to science, even to speculative science, the proponents should put more effort into trying to scrutinize their arguments. And maybe those of their leading figures.

David wrote:

So are there any indications as to what gave Zeno’s argument its value? And are these indications lacking in Boltzmann brain discussions?

The value of a paradox or puzzle depends to some extent on how intelligently one responds to it. If the modern theory of the continuum was a response to Zeno’s paradox — I don’t know if it really was — then we’d have to say this paradox was very useful. But, we’d also have to admit there was a lot of ‘added value’ in the intelligent response to the puzzle.

The Boltzmann brain puzzle could prove useful if it led to intelligent reflections about the thermodynamics of life, the arrow of time, etc. But it seems often people use it to carry out some variant of the following argument:

“If the universe lasts forever, it will suffer a heat death and eventually the number of brains arising from vacuum fluctuations — so-called ‘Boltzmann brains’ — will exceed the number of brains that arise the usual way, through evolution. So, if the universe lasts forever, it’s most likely that I am a Boltzmann brain. But that’s absurd! Therefore, the universe must not last forever.”

This is silly, because if the argument leading up to But that’s absurd! were really ironclad, we couldn’t just declare it absurd out of some feeling of emotional revulsion, and then conclude that the premise — the long-lived universe — must be false.

After all, lots of physics arguments do lead to conclusions that seem absurd: nothing moves faster than light, every particle has an antiparticle that can annihilate it, etc. But when these arguments are solid, the absurd conclusions have an interesting way of being true.

Of course, there’s a good chance that the argument leading up to But that’s absurd! is not actually solid. So, people should find a hole in it.

In short: one would need to examine the whole puzzle in a disciplined way to get something really interesting out of it. I haven’t seen people doing that. On the other hand, I haven’t looked very hard — since I have more interesting things to think about.

You say that the sole observation we have is that some number of class 2 observers exist - however, this is not the case. I additionally have the observation that I am a class 2 observer.

Consider the following situation - the dictator has just decided to round up 100 citizens (including me) and shoot some of them for fun. However, to make it exciting, he has announced that he won’t kill all of them - he’ll flip a coin, and if it comes up heads then he’ll have 90 of the citizens (chosen randomly from this group) shot while the rest are fired at with blanks, but if it comes up tails, he’ll have only 10 shot while the rest are fired at with blanks. Now the day of the execution comes, and the prisoners are taken before the firing squad and fired at. After the bullets and blanks are fired, I realize that I’ve survived, and I can see a few of the other prisoners who have also survived (it’s a large field that we’re in, so I can’t tell what’s gone on beyond the couple people closest to me).

If you were right that the only information we have is the number of class 2 observers, then the only information I have is that at least a few people have survived, so I have no evidence whether the coin came up heads or tails. Yet intuitively, it seems that I have somewhat strong evidence that the coin came up tails (because otherwise I would probably be dead).

Kenny, in your example you get to experience a pre-execution state, in which you are waiting around to be randomly allocated into one of the states victim or survivor, with either $p_v=0.9$ and $p_s=0.1$ or vice versa. But in my example, I have no experience of a pre-life disembodied existence in which I was waiting around to be allocated at random into the body of a class 1 or class 2 observer.

We could make your example less violent, and repeatable, by changing it to a paintball game with either 90% red paint and 10% blue, or vice versa. Then (just by witnessing your own shooting, and no others) you could certainly make money over time by betting that the majority paint colour was the one that you were hit with.

But short of reincarnation in multiple universes, what’s the equivalent in cosmology? Maybe we could assume that there is a multiverse, consisting of many universes obeying theory A and many universes obeying theory B. Then if someone states explicitly that what they want is a strategy that – if adopted by all observers in the multiverse – will maximise the number of observers across the multiverse who correctly guess the theory that applies to their universe … well, then I can’t deny that a good strategy for doing that would be to guess the theory that implies you belong to the majority class.

But is such a strategy really what cosmology is, or ought to be? Why is the weight 1 per observer? Maybe it would be more “scientific” to weight observers by their lifespans? Or the number of “mental events” they experience? I’m being facetious, but my point is, why should we care about this pan-multiverse weighted average of correct guesses at all? What if someone comes up with a theory that we think is very unlikely, but if it were true it would multiply the number of human-like observers in the universe to which it applies by $10^1000000$? Should we say “yes, we’ll act as if we have a good reason to believe that this theory applies to our universe” because, even if we’re wrong, we’ve followed a strategy that – if also followed by everyone else in the multiverse – will lead to a vast number of human-like observers being correct, overall?

Clearly, the latter is much more probable as a result of a random fluctuation.

Sorry to say so, but it is these “clearly”-arguments which worry me.

In the above entry I gave an argument in which I came to the opposite conclusion. In fact, I feel “clearly” the conclusion you come to is weird.

Do we have a means to clarify the situation using solid reasoning?

Greg and Urs:I think you’re making great progress! Greg raised what seemed to me a fatal objection, and Urs escaped it with Houdini-esque ease, and then Greg agreed.

The other issue Greg raises — whether we can draw sensible conclusions from assuming we are ‘typical observers’ — seems really difficult and frustrating. For one thing, we have to decide who counts as an observer. That might be so difficult that it vitiates the whole argument.

Does a fruit fly count as an observer? If so, should I be shocked that I’m not a fruit fly, since there are so many more of those?

Well, presumably I wouldn’t be blogging now if I were a fruit fly — so given that fact about myself, it’s not so improbable that I’m human.

But if I get to use facts like that, it’s also not improbable that I’m a left-handed expert on $n$-categories whose cousin is a folk singer.

And lest you think that fruit flies are too stupid to count as ‘observers’, don’t forget that there could be far-future observers who think we’re too stupid to count as observers.

Wasn’t the whole point of decoherence the premise that we could now safely remove ‘the human observer’ from the equation and we don’t have to worry about this sort of thing.

I don’t think these anthropic arguments are related to quantum mechanical issues of what counts as an observer or a measurement. Although the specific cosmological theories like inflation rely on QM, the actual anthropic issues can be factored out and discussed in a purely classical context.

I suspect that, if your view of probability is closer to, e.g., Fuchs and Peres than to Doering and Isham, many of these myth-flavoured questions are simply ill-posed.

Clearly many of these questions are ill-posed, which is part of the reason why the above entry starts with the words “Here is something that disturbs me.”

But I am not following what you are suggestiong Doering and Isham’s “views” have to do with this.

In the article that you point to they wish to identitfy a topos whose internal truth values may be identified with elements in $[0,1]$, such as to be able to say “a probability is the same as a truth value in this topos”. For whatever that’s worth (I haven’t studied their article in detail), this at least involves a desire for formal reasoning.

I find the disturbing aspect of “these myth-flavoured questions” is the lack of interest in any formal reasoning. Hence in science. It’s all about good fireplace stories, instead.

I could have been clearer. Doering and Isham are concerned that what they call an “instrumentalist” position is no good for doing cosmology (pp. 2-3); Fuchs and Peres disagree. (For a statement by Peres alone, see arXiv:quant-ph/9711003, p. 9.) I bet the actual topos-theoretic work can be divorced from these motivational issues, though.

I could have been clearer. Doering and Isham are concerned that what they call an “instrumentalist” position is no good for doing cosmology (pp. 2-3); Fuchs and Peres disagree.

Oh, I see, I had skipped over the intro.

I bet the actual topos-theoretic work can be divorced from these motivational issues, though.

Yes, I think so, too.

I have recently said something related elsewhere: this kind of investigation might benefit from having 1. less chat and 2. more theorems.

To be convincing in these matters, it may help to let more proven facts speak more for themselves. So I guess we agree.

An argument like, “Your promising-looking calculations are problematic because they’re sensitive to details of trans-Planckian physics” (cf. Brandenberger, arXiv:1103.2271) looks to me much more scientifically fruitful than one about how many Boltzmann brains we think are flying about spacetime.

I’ve been trying to understand a bit of cosmology, because — well, honestly — some of the “campfire” discussions conjure up the bleakest scenarios I can imagine, and it’s hard not to think about them. Sometimes, I feel like a character in a story co-authored by Greg Egan and Jorge Luis Borges, fretting over not just the Eternal Return but the Eternal Return with variations … not exactly the most productive frame of mind for thinking about stochastic Petri nets, Rényi entropy or anything else for which we could hope to make progress!

The more I learn, though, the more I wonder if I should rather be unhappy over the scientific process, instead of the conclusions. Take, for example, Linde and Noorbala’s recent article (arXiv:1006.2170) on the “measure problem” in eternally inflationary cosmology. As they say,

The main problem here is that in an eternally inflating universe the total volume occupied by all, even absolutely rare types of the “universes,” is indefinitely large. Therefore comparison of different types of vacua involves comparison of infinities. As emphasized already in the first papers on the probability measure in eternal inflation [6–8], such a comparison is inherently ambiguous and depends on the choice of the cutoff, which is required to regularize the infinities.

A bit later, when discussing candidate measures:

Out of all of these measures, the original proper time cutoff measure $P(\phi,t)$ is the simplest. However, this measure suffers from the youngness problem [14], which was especially clearly formulated in [15]: This measure exponentially rewards parts of the universe staying as long as possible at the highest values of energy density. As a result, this measure exponentially favors life appearing in the parts of the universe with an extremely large temperature, which contradicts the observational data.

Again, we’re back to assuming typicality. And then on we go with the Boltzmann-brain business … We have an “inherently ambiguous” situation, and the means people have taken to address it make no sense to me. Every “solution” looks like an elaborate contrivance which does nothing but reproduce the starting point.