
Why Philosophers Should Care About Computational Complexity - sweis
http://www.scottaaronson.com/blog/?p=735
======
bumbledraven
_A given Turing machine M either accepts, rejects, or runs forever (when
started on a blank tape)... [W]hich one it does is an objective fact,
independent of our formal axiomatic theories, the laws of physics, the biology
of the human brain, cultural conventions, etc._ (p. 43)

Nice way to put it. As Franzen points out in "Inexhaustibility", although non-
logicians sometimes find it puzzling, the above is related to what
mathematicians mean when they say that a statement is _true_ without further
qualification. For instance, take Gödel's first incompleteness theorem. It
states that if _F_ is any consistent formal system capable of proving
statements about whether or not arbitrary Turing machines halt, then there are
_true_ statements which cannot be proved within _F_. (Here, _consistent_ means
that it's not possible within _F_ to prove both a statement " _S_ " and its
negation "not _S_ ".)

In the same sense, its _true_ (as Aaronson wrote in "Logicians on Safari")
that "there’s a finite (and not unimaginably-large) set of boxes, such that if
we knew how to pack those boxes into the trunk of your car, then we’d also
know a proof of the Riemann Hypothesis."

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strlen
They already do. Incompleteness Theorem ("some logical statements are true but
are unprovable", e.g., "this program will terminate" for some arbitrary
programs a.k.a. the Halting Problem) is one of the most significant
achievements in logic.

Here's philosophers' take on Kurt Godel:

<http://plato.stanford.edu/entries/goedel/>

I am, however, curious as to how other problems in computational complexity
figure in philosophy, e.g., P vs. NP completeness. Wish I had the time to read
the essay.

~~~
jerf
I haven't read the paper, but here's one example I've seen. Assume for the
moment that the universe is closed, complete (there is nothing but our
universe), and deterministic under the hood. Obviously, nobody in the universe
has free will, right? Because it's all closed and all outcomes are
predetermined.

But maybe that's not a useful definition of free will. The conventional, fuzzy
definition of free will has an omniscient narrator in it; "I do not have free
will if the omniscient narrator knows in advance everything I will do" is a
reasonable expansion of the conventional idea. But in my hypothesized
universe, there _is_ no omniscient narrator. There are only various entities
with various degrees of computational power, with a varying but rather
quantifiable amount of information that any entity can obtain about any other.

Suppose it can be demonstrated that it is computationally infeasible for any
entity due to being too complex to ever predict the actions of any human-sized
(or greater) other entity, even if granted all the remaining resources in the
universe to compute the actions. Or suppose it is demonstrated to be quite
easy. Either way, that would be an interesting philosophical contribution, no?

(I'm just sketching the idea here. I'm not trying to defend it or attack it.
Oh, and one of the foundations of philosophy is that there is never One True
Definition; all loaded terms in this post are ultimately ill-defined, and I've
avoided the distraction of even beginning to nail them down on purpose.)

~~~
ionfish
Your suggestion confuses an epistemic limitation with a metaphysical one. Just
because no one knows what we're going to do next doesn't mean there isn't a
fact of the matter about it.

While the idea of God looking down on us from on high is a compelling story,
if you take God away but leave determinism it doesn't do anything to resolve
the problem: our actions are still pre-determined and cannot be changed.

Of course, whether or not that makes us _free_ is a further question. A
compatibilist [1] would claim that we are.

[1] <http://plato.stanford.edu/entries/compatibilism/>

~~~
jerf
"Confuses" is one way to read it. "Shines a spotlight on the intersection of"
would be a somewhat friendlier way to read it. Is a fact that can't be known,
even in theory, even a sensibly a "fact"? I deny your implicit claim that that
isn't an interesting question because the answer is transparently obvious.

Epistemology becomes much richer when fed with mathematical proofs of
statements like this, and some older snap answers at least have to be
reconsidered.

~~~
ionfish
I don't think it's by any means _uninteresting_ , and I agree that complexity
theory has a role to play in enriching epistemology. That's why I said
elsewhere in this discussion that I think this kind of work is important.

However, it's not in any way obvious that there are metaphysical implications
to the limits this sort of result would place on knowledge. Facts are (for
present purposes) simply to be identified with physical states of affairs. If
some physical state of affairs is unknowable (and "System S will be in state R
at time T" certainly _looks_ like a physical state of affairs), does that mean
it does not (or will not) actually obtain? I have a hard time seeing how one
could make that argument, although of course that shouldn't stop you, if you
think you have a good angle!

To address the particular example under discussion: if you take free will to
consist of the impossibility of predicting an agent's actions and conclude
that because this prediction is impossible, we are free, you still need to
provide a further argument that your definition of free will is the correct
one.

I'm not prejudging whether or not you could come up with a convincing argument
to that effect—it might well be possible. But to my mind, the point of the
omniscient narrator example is to add determinism to the system: because our
actions are known in advance, they can be known in advance; if they can be
known in advance, they must be determined. I take this to be the implicit
argument when people suggest that we're not free because God knows what we
will do.

Obviously there are different ways in which we can understand the modalities
of time, and I won't try to pretend that this isn't a contentious area.

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gnosis
This is basically aimed at analytic philosophers. Continental philosophers
generally aren't interested in this sort of thing.

~~~
thebooktocome
What evidence do you have for this?

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aheilbut
It's interesting that one of the references Aaronson cites [63] is the pg
essay "How to do philosophy" <http://www.paulgraham.com/philosophy.html>

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ionfish
This looks fascinating, and in yours truly is guaranteed at least one reader,
once I have some time to devote to it. I think there's a lot of room for
interesting work in this area.

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forkandwait
The problem with Godel and friends is that they point to the limits of
rationality. Without a transcendent rationality (and morality) upon which to
discurse, philosophers are out of a job, so to speak. If you use logic to
understand everything, the last thing you want to deal with is somebody
telling you the limits to logic. So they blow him off.

At least I think the above is true for "analytic" philosophers. The
"continentals", on the other hand, have been engaging in an extended _attack_
on transcendent rationality since Nietszche who said, to paraphrase, that
every transcendent metaphysics was an attempt to justify a morality, and every
morality was facilitated a (usually inarticulate) "will to power." (This line
sort of starts with Hegel, who posited that rationality emerged from history
and the development of a collective human spirit, but wasn't "out there"
before that). Godel seems like ammunition for these guys, but I don't think
they have picked him up much. I guess if you are anti-rationalist, the last
thing you read up on is advanced logic, even if it is so advanced it comes to
its own frontier.

~~~
joe_the_user
I believe Alain Badiou might be an exception here.

<http://en.wikipedia.org/wiki/Alain_Badiou>

~~~
mjsergey
The book "The Mathematics of Novelty" seems like a pretty good introduction to
Badiou. As with most non-fiction published by re.press it is creative commons:
[http://re-press.org/books/the-mathematics-of-novelty-badiou%...](http://re-
press.org/books/the-mathematics-of-novelty-badiou%E2%80%99s-minimalist-
metaphysics/).

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yason
I've always considered philosophy as a sort of the mind's struggle for its own
life. The introspective mind somehow can't validate itself without somehow
having the absolute last say about something.

Given this track, a philosophical mind will produce a lot of thoughts, models,
and theories of how anything that already exists _could_ exist in the first
place. That is itself an interesting mind-game that in some cases might even
produce an insight that is applicable to something real, but generally that's
where it should stay.

The mind can't think itself into existence, even if it wants to believe it
can.

If it does the result bears resemblance to as if someone would try to describe
forest but only be allowed to draw straight lines in one color. At best it
could be exemplary line art, at worst it's just ridiculous and childish and
most people will turn the other way.

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p4bl0
It seems nobody has mentionned Gregory Chaitin[1] yet, and in my opinion it's
kind of mandatory when discussing this very topic. So now it's done :-). I
like his work a lot. His not really working on this subject anymore (now he's
developping a new field called metabiology which is also fascinating but not
very relevant here) but he used to work on Algorithmic Information Theory and
he wrote a lot of books and essays (you can find almost everything on his
webpage) on the subject we're discussing here. _The Limits of Mathematics_[2]
might be the most relevant to Hacker News users.

[1] <http://www.cs.umaine.edu/~chaitin/> [2]
<http://news.ycombinator.com/item?id=1725936>

~~~
bumbledraven
_It seems nobody has mentionned Gregory Chaitin[1] yet_

With good reason. The logician Torkel Franzen said it well in "Gödel's
Theorem: An Incomplete Guide" [Chapter 8, p 148]: ``Chaitin's claim to have
discovered "the amazing fact that some mathematical statements are true for no
reason" has no apparent content, but seems rather to be based on the general
associations surrounding the word "random".''

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ez77
Unabridged work posted on the Electronic Colloquium on Computational
Complexity: <http://eccc.hpi-web.de/report/2011/108/>

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draggnar
one interesting perspective on complexity is that of Ashby and his law of
requisite variery.
[http://en.wikipedia.org/wiki/Variety_(cybernetics)#The_Law_o...](http://en.wikipedia.org/wiki/Variety_\(cybernetics\)#The_Law_of_Requisite_Variety)

this work was later manifested in the work of stafford beer creating a control
center in chile during the
70s.(<http://en.wikipedia.org/wiki/Project_Cybersyn>).

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da_dude4242
I think irreducibility is well received in philosophy. Evolutionary Biology
not so much for political reasons. Sure there are are plenty of models that
incorporate emergent phenomena but it's a heated area.

