I suppose a world *without law at all* would be one in which people habitually defect on the Prisoner's Dilemma. Even when it's the *least* 'true' a PD can be and still be a PD (so, iterated, with reputational incentives, all that stuff). There are no Schelling points, and thus no coördination: Nash equilibria and Moloch for all.

The "rule of law", perhaps, is a list of particular properties of the law (collection of Schelling points): that they contain no proper nouns, for example (the same law binds the King), that they do not ...

45y

In this hypothetical world where there is a shared understanding that something
like the Fugitive Slave Law is unjust and ought not to be enforced, but for some
reason the people sharing this understanding, can't be bothered to rewrite the
law, what's going on? Why are there written laws at all in that world? What's
their role in coordination?
I'm not saying correspondence between written and enforced law is identical with
justice; I'm saying that if there are written laws, and they do not correspond
to the law as enforced, then this is symptomatic of a lack of rule of law. In
particular, it implies that the written law is meant to misinform, most likely
in order to prevent the true norms from becoming common knowledge, which only
makes sense if the enforcers intend to apply official standards unevenly in
order to benefit from the information asymmetries they set up.

One cause might be that some of the underlying drivers of performance (of which IQ is but one) correlate with race. For a non-IQ-related example, if your interview process for a basketball team includes a jumping test, this will have a "disparate impact" because on average blacks jump better than whites. Therefore, even if you can demonstrate that you use the jump test because the regression analysis showed it was a good predictor of performance, the usual suspects will scream "algorithmic bias" and now even if you prove (possibly in ...

Curiously, from a civilisational perspective it doesn't matter whether you invest the money or just stuff it in your mattress; either way you're creating capital relative to the alternative, which is to spend it now on some form of consumption. (Note that, in this view, charitable giving is a consumption good rather than a capital good. In practice, of course, it's more complicated, because Africans with mosquito nets are more likely to generate endogenous economic growth than Africans with malaria. But to first order, saving African live...

15y

So, my understanding of the economic theory is that burying the money under a
mattress should create capital that grows at the rate of GDP growth, whereas
investing it should create capital that grows at the (faster) rate of the stock
market. However, this doesn't answer the question I was trying to address in
this post, which is how much should we trust the economic theory? My intuition
is that the better causal models we can come up with to explain why the theory
works, the more we should trust it, as long as those causal models don't predict
that the theory is likely to break down when you apply it to reality. I don't
really see anything that I would call a causal model in your comment.

I don't know if I count as part of the "movement", but I can't agree on these demands, because they all assume that notions such as "public policy" and "government" are valid and legitimate.

Suppose we were to turn them round and write them as negative demands; 5, 6, and 8 all reduce to freedom of contract. 7 is covered by "no crimes, only torts" since the concept of a "victimless tort" is obviously meaningless. 1 and 2 are fundamentally just questions of governance procedure, and become a lot le...

I think you're misusing the notion of "rule of law", possibly because of the Jewish-upbringing factors you mention. Economist Don Boudreaux would argue, after Hayek, that legislation and law are different things. My version of this, heavily influenced by David D. Friedman, is that *law* is really a collection of Schelling points, and *legislation* (the statute book) is one way (of many) in which new Schelling points can be created. This is far more obvious to someone who was brought up under the common-law system — and the US's legal tra...

45y

What would it look like not to have the rule of law, on this model?

most of us probably won't be able to find much of a trunk build that we can agree on

I think you're wrong as a question of fact, but I love the way you've expressed yourself.

It's more like a non-monotonic DVCS; we may all have divergent head states, but almost every commit you have is replicated in millions of other people's thought caches.

Also, I don't think the system needs to be Byzantine fault tolerant; indeed we may do well to leave out authentication and error correction in exchange for a higher raw data rate, relying on Release Early Release Often...

On the other hand, if you’re Dr. Evil and you’re in your moon base preparing to fire your giant laser at Washington, DC when you get a phone call from Austin “Omega” Powers

So, does this mean ata is going to write an Austin Powers: Superrational Man of Mysterious Answers fanfic?

How exactly are abstract, non-physical objects -- laws of nature, living in their "transcendent aerie" -- supposed to interact with physical stuff? What is the mechanism by which the constraint is applied? Could the laws of nature have been different, so that they forced electrons to attract one another?

I feel I should link to my post The Apparent Reality of Physics *right now*. To summarise: both the "descriptions" and "rules" views are wrong as they suppose there is something to be described or ruled. The (to me, obvious) dissolution is to state that a Universe *is* its rules.

There is a further subtlety here. As I discussed in "Syntacticism", in Gödel's theorems number theory is in fact talking about "number theory", and we apply a metatheory to prove that "number theory is "number theory"", and think we've proved that number theory is "number theory". The answer I came to was to conclude that number theory isn't talking about anything (ie. *ascription of semantics to mathematics does not reflect any underlying reality*), it's just a set of symbols and rules for manipulating sam...

I don't believe it, but it sounds like it should be testable, and if it hasn't been tested I'd be somewhat interested in doing so. I believe there are standard methods of comparing legibility or readability of two versions of a text (although, IIRC, they tend to show no statistically significant difference between perfect typesetting and text that would make a typographer scream).

You're probably not the only one bothered by the colour scheme, though; historically, every colour scheme I've used on the various iterations of my website has bothered many people. The previous one was bright green on black :S

0[anonymous]11y

(In case it wasn't clear, I wasn't serious about the
speakers-of-languages-with-several-words-for-blue thing.)

That's interesting, because I would see a difference. Given the choice, I'd test it on the barren rock. However, I can't justify that, nor am I sure how much benefit I'd have to derive to be willing to blow up Eta Kudzunae.

Agree, and think your changes alter the question I was trying to ask, which is, not whether destroying Xenokudzu Planet would be absolutely unacceptable (as a rule, most things aren't), but whether we'd *need a sufficiently good reason*.

which has choked out all other life

I think the LCPW for you here is to suppose that this planet is only capable of supporting this xenokudzu, and no other kind of life. (Maybe the xenokudzu is plasma helices, and the 'planet' is actually a sun, and suppose for the sake of argument that that environment can't support sent...

011y

So, if I were building a planet-destroying superlaser (for, um, mining I guess)
I wouldn't see any particular difference between testing it on Kudzu World or
the barren rock next door.

Well, my source is Dr Bursill-Hall's History of Mathematics lectures at Cambridge; I presume his source is 'the literature'. Sorry I can't give you a better source than that.

5[anonymous]11y

Can anyone confirm this? Preferably with citation?

Hmm. I do understand that, but I still don't think it's relevant. I don't try to argue that Premise 1 is true (except in a throwaway parenthetical which I am considering retracting), rather I'm arguing that Premise 2 is true, and that consequently Premise 1 implies the conclusion ("transposons have ethical value") which in turn implies various things ranging from the disconcerting to the absurd. In fact I believed Premise 1 (albeit without great examination) until I learned about transposons, and now I doubt it (though I haven't rejected it so...

111y

I think this scenario is a little difficult to visualize- an entire biosphere we
can't derive a benefit from, even for sheer curiosity's sake? So, applying the
LCPW: the planet has been invaded by a single species of xenokudzu, which has
choked out all other life but is thriving merrily on its own (maybe it's an
ecocidal bioweapon or something). Would it be acceptable to destroy that planet?
I'd say yes. Agree / disagree / think my changes alter the question?

211y

Interestingly, the gut reaction I had to destroying plant planet was "NO! We
could learn so much!". But then I think I value interesting information, not
life.

My ethics were influenced a nonzero amount by reading Orson Scott Card. More to the point, OSC provided terminology which I felt was both useful and likely to be understood by my audience.

I now think that my use of the word "must" in the above-quoted passage was a mistake.

Your comment is a very good argument against a position - but unfortunately not the position I hold. I may have poorly expressed my meaning; it's not strictly the definition of the English word 'life' that I care about, but rather the exploration of my utility function, and whether my preferences are consistent and coherent, or whether they make an arbitrary distinction between "life with moral status" (people, chimps, and kittens) and "life without moral status" (cockroaches, E. coli, and transposons).

Can you suggest a good way for me to explain this in the article itself?

911y

At the very least, you should reconsider the syllogism at the heart of your
article:
1. All life has ethical value.
2. Transposons are life.
Therefore, transposons have ethical value.
We can substitute in your tentative definition of life:
1. All "self-replicating structures with a genotype which determines their
phenotype and is susceptible to mutation and selection" have ethical value.
2. Transposons are self-replicating structures with a genotype which determine
their phenotype and are susceptible to mutation and selection.
Therefore, transposons have ethical value.
Premise 2 is an empirical claim. Premise 1 is a moral claim that is strictly
stronger than the conclusion, and you do not justify it at all.
If you have moral intuitions or moral arguments for the first premise, then
perhaps you should write about those instead. And your arguments ought to make
sense without using the word "life". If your argument is along the lines of
"well, humans and chimpanzees have ethical value, and they're both
self-replicating structures with genomes etc., so it only makes sense that
transposons have ethical value too", that's not good enough. You'd have to say
why being a self-replicating structure with a genome etc. is the reason why
humans and chimpanzees have ethical value. If humans and chimpanzees have
ethical value because of some other feature, then perhaps transposons don't
share that feature and they don't have ethical value after all.

Sorry to have to tell you this, but Pythagoras of Samos probably didn't even exist. More generally, essentially everything you're likely to have read about the Pythagoreans (except for some of their wacky cultish beliefs about chickens) is false, especially the stuff about irrationals. The Pythagoreans were an orphic cult, who (to the best of our knowledge) had no effect whatsoever on mainstream Greek mathematics or philosophy.

311y

Source?

Hmm, infinitary logic looks interesting (I'll read right through it later, but I'm not entirely sure it covers what I'm trying to do). As for Platonism, mathematical realism, and Tegmark, before discussing these things I'd like to check whether you've read http://lesswrong.com/r/discussion/lw/7r9/syntacticism/ setting out my position on the ontological status of mathematics, and http://lesswrong.com/lw/7rj/the_apparent_reality_of_physics/ on my version of Tegmark-like ideas? I'd rather not repeat all that bit by bit in conversation.

011y

Have you looked into Univalent foundations
[http://www.math.ias.edu/~vladimir/Site3/Univalent_Foundations.html] at all?
There was an interesting talk on it [video.ias.edu/univalent/voevodsky] a while
ago and it seems as though it might be relevant to your pursuits.
I've read your post on Syntacticism and some of your replies to comments. I'm
currently looking at the follow up piece (The Apparent Reality of Physics).

The computer program 'holds the belief that' this way-powerful system exists; while it can't implement arbitrary transfinite proofs (because it doesn't have access to hypercomputation), it can still modify its own source code without losing a meta each time: it can *prove* its new source code will increase utility over its old, without its new source code losing proof-power (as would happen if it only 'believed' PA+n; after n provably-correct rewrites it would only believe PA, and not PA+1. Once you get down to just PA, you have a What The Tortoise Said To ...

411y

I might be misunderstanding you, but it looks like you're just describing
fragments of infinitary logic
[http://plato.stanford.edu/entries/logic-infinitary/] which has a pretty solid
amount of research behind it. Barwise actually developed a model theory for it,
you can find a (slightly old) overview here
[http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.pl/1235417263]
(in part C).
Infinitary logic admits precisely what you're talking about. For instance; it
models sentences with universal and existential quantifiers in N using
conjunctions and disjunctions (respectively) over the index ω.
As far as
I don't think the results of Gödel, Löb and Tarski are necessary to conclude
that Platonism is at least pretty questionable. I don't know where the
"mathematics can't really prove things" bit is coming from - we can prove things
in mathematics, and I've never really seen people claim otherwise. Are you
implicitly attributing something like Absolute Truth to proofs?
Anyway, I've been thinking about a naturalistic account of anti-realism
[http://lesswrong.com/lw/8y3/seq_rerun_beautiful_math/5i0x] for a little while.
I'm not convinced it's fruitful, but Platonism seems totally incomprehensible to
me anymore. I can't see a nice way for it to mesh with what I know about brains
and how we form concepts - and the accounts I read can't give any comprehensible
account of what exactly the Platonic realm is supposed to be, nor how we access
it. It looks like a nice sounding fiction with a big black box called the
Platonic realm that everything we can't really explain gets stuck into. Nor can
I give any kind of real final account of my view of mathematics, because it
would at least require some new neuroscience (which I mention in that link).
I will say that I don't think that mathematics has any special connection with
The Real Truth, and I also think that it's a terrible mistake to think that this
is a problem. Truth is evidence based, we figur

Can you explain more formally what you mean by "proves that it itself exists"?

The fundamental principle of Syntacticism is that the derivations of a formal system are fully determined by the axioms and inference rules of that formal system. By proving that the ordinal kappa is a coherent concept, I prove that PA+kappa is too; thus the derivations of PA+kappa are fully determined and exist-in-Tegmark-space.

Actually it's not PA+kappa that's 'reflectively consistent'; it's *an AI which uses PA+kappa as the basis of its trust in mathematics* that's...

Well, I'm not exactly an expert either (though next term at uni I'm taking a course on Logic and Set Theory, which will help), but I'm pretty sure this isn't the same thing as proof-theoretic ordinals.

You see, proofs in formal systems are generally considered to be constrained to have finite length. What I'm trying to talk about here is the construction of metasyntaxes in which, if A1, A2, ... are valid derivations (indexed in a natural and canonical way by the finite ordinals), then Aw is a valid derivation for ordinals w smaller than some given ordinal....

111y

Ok, I think I get it. You're talking about proofs of transfinite lengths, and
using hypercomputation to check those, right? This seems way powerful, e.g. it
pins down all statements in the arithmetical hierarchy (a statement at level N
requires an N-dimensional hyper...quadrant... of proofs) and much more besides.
A finite computer program probably can't use such a system directly. Do you have
any ideas how to express the notion that the program should "trust" such a
system? All ideas that come to my mind seem equivalent to extending PA with more
and more powerful induction postulates (or equivalently, adding more and more
layers of Con()), which just amounts to increasing the proof-theoretic ordinal
as in my first comment.

I think that's a very good summary indeed, in particular that the "*unique non-ambiguous set of derivations*" is what imbues the syntax with '*reality*'.

Symbols are indeed not defined, but the only means we have of duck-typing symbols is to do so symbolically (a symbol S is an object supporting an equality operator = with other symbols). You mention Lisp; the best mental model of symbols is Lisp gensyms (which, again, are objects supporting only one operator, equality).

conses of conses are indeed a common model of strings, but I'm not sure whether t...

Now I see why TDT has been causing me unease - you're spot on that the 5-and-10 problem is Löbbish, but what's more important to me is that TDT in general tries to be reflective. Indeed, Eliezer on decision theory seems to be all about reflective consistency, and to me reflective consistency looks a lot like PA+Self.

A possible route to a solution (to the Löb problem Eliezer discusses in "Yudkowsky (2011a)") that I'd like to propose is as follows: we know how to construct P+1, P+2, ... P+w, etc. (forall Q, Q+1 = Q u {forall S, [](Q|-S)|-S}). We ...

011y

Are you referring to something like ordinal analysis
[http://en.wikipedia.org/wiki/Ordinal_analysis]? Can I drop the mention of
cardinals and define kappa as the smallest ordinal such that the proof-theoretic
ordinal of PA+kappa is kappa? Sorry if these are stupid questions, I don't know
very much about the topic.
Also I don't completely understand why such a system could be called
reflectively consistent. Can you explain more formally what you mean by "proves
that it itself exists"?

Oh, I'm willing to admit variously infinite numbers of applications of the rules... that's why transfinite induction doesn't bother me in the slightest.

But, my objection to the existence of abstract points is: what's the definition of a point? It's defined by *what it does*, by duck-typing. For instance, a point in R² is an ordered pair of reals. Now, you could say "an ordered pair (x,y) is the set {x,{x,y}}", but that's silly, that's not what an ordered pair *is*, it's just a construction that exhibits the required behaviour: namely, a constructo...

011y

These might be stupid questions, but I’m encouraged by a recent post to ask
them:
Doesn’t that apply to syntactic methods, too? It was my understanding that the
symbols, strings and transformation rules don’t quite have a definition except
for duck typing, i.e. “symbols are things that can be recognized as identical or
distinct from each other”. (In fact, in at least one of the courses I took the
teacher explicitly said something like “symbols are not defined”, though I don’t
know if that is “common terminology” or just him or her being not quite sure how
to explain their “abstractness”.)
And the phrase about ordered pairs applies just as well to ordered strings in
syntax, doesn’t it? Isn’t the most common model of “strings” the Lisp-like
pair-of-symbol-and-pair-of-symbol-and...?
--------------------------------------------------------------------------------
Oh, wait a minute. Perhaps I got it. Is the following a fair summary of your
attitude?
We can only reason rigorously by syntactic methods (at least, it’s the best we
have). To reason about the “real world” we must model it syntactically, use
syntactic methods for reasoning (produce allowed derivations), then “translate
back” the conclusions to “real world” terms. The modelling part can be done in
many ways—we can translate the properties of what we model in many ways—but a
certain syntactic system has a unique non-ambiguous set of derivations,
therefore the things we model from the “real world” are not quite real, only the
syntax is.

The locus exists, as a mathematical object (it's the *string* "{x \in R²: |x|=r}", not the *set* {x \in R² : |x|=r}). The "circle" on the other hand is a collection of points. You can apply syntactic (ie. mathematical) operators to a mathematical object; you can't apply syntactic operators to a collection of points. It is syntactic systems and their productions (ie. mathematical systems and their strings) which exist.

011y

Hmm. I’m not quite sure I understand why abstract symbols, strings and
manipulations of those must exist in the a sense in which abstract points, sets
of points and manipulations of those don’t, nor am I quite sure why exactly one
can’t do “syntactic” operations with points and sets rather than symbols.
In my mind cellular automatons look very much like “syntactic manipulation of
strings of symbols” right now, and I can’t quite tell why points etc. shouldn’t
look the same, other than being continuous. And I’m pretty sure there’s someone
out there doing (meta-)math using languages with variously infinite numbers of
symbols arranged in variously infinite strings and manipulated by variously
infinite syntactic rule sets applied a variously infinite number of times... In
fact, rather than being convenient for different applications, I can’t quite
tell what existence-relevant differences there are between those. Or in what way
rule-based manipulations strings of symbols are “syntactic” and rule-based
manipulations of sets of points aren’t—except for the fact that one is easy to
implement by humans. In other words, how is compass and straightedge
construction not syntactical?
(In terms of the tree-falling-in-the-forest problem, I’m not arguing about what
sounds are, I’m just listing why I don’t understand what you mean by sound, in
our case “existence”.)
[ETA. By “variously infinite” above I meant “infinite, with various
cardinalities”. For the benefit of any future readers, note that I don’t know
much about those other than very basic distinctions between countable and
uncountable.]

I have also (disappointingly/validatingly) thought of this and then read Tegmark. (It's even more disappointing/validating than that, though, since as well as Tegmark, you appear to have invented Syntacticism. You even have all my arguments, like subverting the simulation hypothesis and talking about 'closure'). However, I have one more thing to add, which *may* answer the problem of regularity. That one thing is what I call the 'causality manifold': Obviously by simulating a universe we have no causal effect upon it (if we are assuming the mathematical ...

Yes, it still works, because of the way the subjective probability flow on Tegmark-space works. (Think of it like PageRank, and remember that the s.p. flows from the simulated to the simulator)

It is technically possible that the differences between how much the two Universes simulate each other can, when combined with differences in how much they are simulated by other Universes, can cause the coupling between the two not to be strong enough to override some other couplings, with the result that the s.p. expectation of "giving Omega the $100" is...

Under my syntacticist cosmology, which is a kind of Tegmarkian/Almondian crossover (with measure flowing along the seemingly 'backward' causal relations), the answer becomes trivially "yes, give Omega the $100" because counterfactual-me exists. In fact, since this-Omega simulates counterfactual-me and counterfactual-Omega simulates this-me, the (backwards) flow of measure ensures that the subjective probabilities of finding myself in real-me and counterfactual-me must be fairly close together; consequently this remains my decision even in the Al...

011y

Does syntacticism work if you know Omega likes simulating poor you, and each
simulated rich you is counterbalanced by many simulated poor yous? Or only in
special cases like you mentioned?

Indeed. Circles are merely a map-tool geometers use to understand the underlying territory of Euclidean geometry, which is precisely real vector spaces (which can be studied axiomatically without ever using the word 'circle'). So, circles don't exist, but {x \in R² : |x|=r} does. (Plane geometry is one *model* of the formal system)

011y

And how exactly would you define the word “circle” other than {X \in R² :
|x|=r}?
(In other words, if a geometric locus of points in a plane equidistant to a
certain point exists, but circles don’t, the two are different; what is then the
latter?)

It doesn't seem odd at all, we have an expectation of the calculator, and if it fails to fulfill that expectation then we start to doubt that it is, in fact, what we thought it was (a working calculator).

Except that if you examine the workings of a calculator that *does* agree with us, you're much much less likely to find a wiring fault (that is, that it's implementing a different algorithm).

...if (a) [a reasonable human would agree implements arithmetic] and (b) [which disagrees with us on whether 2+2 equals 4] both hold, then (c) [The human decides she v

011y

Well, this seems a bit unclear. We are operating under the assumption that the
set up looks very similar to a correct set up, close enough to fool a reasonable
expert. So while the previous fault would cause some consternation and force the
expert to lower his priors for "this is a working calculator", it doesn't follow
that he wouldn't make the appropriate adjustment and then (upon seeing nothing
else wrong with it) decide that it is likely that it should resume working
correctly.
Yes, it would be true, but what exactly is it that 'is true'? The human brain is
a tangle of probabilistic algorithms playing various functional roles; it is
"intuitively obvious" that there should be a Solomonoff-irreducible (up to some
constant) program that can be implemented (given sufficient background knowledge
of all of the components involved; Boolean circuits implemented on some
substrate in such and such a way that "computes" "arithmetic operations on
integers" (really it is doing some fancy electrical acrobatics, to be later
interpreted first into a form we can perceive as an output, such as a sequence
of pixels on some manner of screen arranged in a way to resemble the numerical
output we want etc.) and that this is a physical fact about the universe (that
we can have things arranged in such a way lead to such an outcome).
It is not obvious that we should then reverse the matter and claim that we ought
to project a computational Platonism on to reality any more than logical
positivist philosophers should have felt justified in doing that with
mathematics and predicate logic a hundred years ago.
It is clear to me that we can perceive 'computational' patterns in top level
phenomena such as the output of calculators or mental computations and that we
can and have devised a framework for organizing the functional role of these
processes (in terms of algorithmic information theory/computational
complexity/computability theory) in a way that allows us to reason generally
about th

I don't understand the meaning of the word "symbols" in the abstract, without a brain to interpret them with and map them onto reality.

Think in terms of LISP gensyms - objects which themselves support only one operation, ==. The only thing we can say about (rg45t) is that it's the same as (rg45t) but not the same as (2qox), whereas we think we know what (forall) means (in the game of set theory) - in fact the only reason (forall) has a meaning is because some of our symbol-manipulating rules mention it.

I think ec429 “sides” with the first intuition, and you tend more towards the second. I just noticed I am confused.

No, I'd say nearer the second - the *mathematical expression* of the world of P2 "exists" indifferently of us, and has just as much "existence" as we do. Rocks and trees and leptons, and their equivalents in P2-world, however, don't "exist"; only their corresponding 'pieces of math' flowing through the equations can be said to "exist".

011y

I don’t quite get what you mean, then. If the various “pieces of math” describe
no more and no less than exactly the rocks and trees and leptons, how can one
distinguish between the two?
Would you say the math of “x^2 + y^2 = r^2” exists but circles don’t?

What is clear to me is that when we set up a physical system (such as a Von Neumann machine, or a human who has been 'set up' by being educated and then asked a certain question) in a certain way, some part of the future state of that system is (say with 99.999% likelihood) recognizable to us as output (perhaps certain patterns of light resonate with us as "the correct answer")

But note that there are also patterns of light which we would interpret as "the wrong answer". If arithmetic is implementation-dependent, isn't it a bit *odd* t...

-111y

I did note that, maybe not explicitly but it isn't really something that anyone
would expect another person not to consider.
It doesn't seem odd at all, we have an expectation of the calculator, and if it
fails to fulfill that expectation then we start to doubt that it is, in fact,
what we thought it was (a working calculator). This refocuses the issue on us
and the mechanics of how we compress information; we expected information 'X' at
time t, but instead received 'Y' and decide that something is wrong with out
model (and then aim to fix it by figuring out if it is indeed a wiring problem
or a bit-flip or a bug in the programming of the calculator or some
electromagnetic interference).
No. But why is this? Because if (a) [a reasonable human would agree implements
arithmetic] and (b) [which disagrees with us on whether 2+2 equals 4] both hold,
then (c) [The human decides she ve was mistaken and needs to fix the machine].
If the human can alter the machine so as to make it agree with 2+2 = 4, then and
only then will the human feel justified in asserting that it implements
arithmetic.
The implementation is decidedly correct only if it demonstrates itself to be
correct. Only if it fulfills our expectations of it. With a calculator, we are
looking for something that allows us to extend our ability to infer things about
the world. If I know that a car has a mass of 1000 kilograms and a speed of 200
kilometers for hour, then I can determine whether it will be able to topple a
wall given that I have some number that encoded the amount of force it can
withstand. I compute the output and compare it to the data for the wall.
I tend to think it depends on a human-like brain that has been trained to
interpret '2', '+' and '4' in a certain way, so I don't readily agree with your
claim here.
I'll look over it, but given what you say here I'm not confident that it won't
be an attempt at a resurrection of Platonism.

When you look at the statement 2+2=4 you think some form of "hey, that's true". When I look at the statement, I also think some form of "hey, that's true". We can then talk and both come to our own unique conclusion that the other person agrees with us.

I think your argument involves reflection somewhere. The desk calculator agrees that 2+2=4, and it's not reflective. Putting two pebbles next to two pebbles also agrees.

Look at the discussion under this comment; I maintain that cognitive agents converge, even if their only common con...

011y

Agreement with statements such as 2+2=4 is not a function that desk calculators
perform. It is not the function performed when you place two pebbles next to two
pebbles.
Agreement is an evaluation performed by your mind from its unique position in
the universe.
The conclusion that convergence has occurred must be made from a context of
evaluation. You make observations and derive a conclusion of convergence from
them. Convergence is a state of your map, not a state of the territory.
Mathematical realism appears to confuse the map for the territory -- as does
scientific realism, as does physical realism.
When I refer to physical reality or existence I am only referring to a
convenient level of abstraction. Space, time, electrons, arithmetic, these all
are interpretations formed from different contexts of evaluation. We form
networks of maps to describe our universe, but these maps are not the territory.
Gottlob Frege coined the term context principle
[http://en.wikipedia.org/wiki/Context_principle] in his Foundations of
Arithmetic, 1884 (translated). He stated it as "We must never try to define the
meaning of a word in isolation, but only as it is used in the context of a
proposition."
I am saying that we must never try to identify meaning or existence in
isolation, but only as they are formed by a context of evaluation.
When you state:
I look for the context of evaluation that produces this result -- and I
recognize that the pebbles and agreement are states formed within your mind as
you interact with the universe. To believe that these states exist in the
universe you are interacting with is a mind projection fallacy.

But you don't have to have unlimited resources, you just have to have X large but finite amount of resources, and you don't know how big X is.

Of course, in order to *prove* that your resources are sufficient to find the proof, without simply going ahead and trying to find the proof, you would need those resources to be unlimited - because you don't know how big X is. But you still know it's finite. "Feasibly computable" is not the same thing as "computable". "In principle" is, in principle, well defined. "In practice&qu...

It's P(I will find a proof in time t) that is asking for the probability of a definite event. It's not that evaluating this number at large t is so problematic, it's that it doesn't capture what people usually mean by "provable in principle."

Suppose that a proof is a finite sequence of symbols from a finite alphabet (which supposition seems reasonable, at least to me). Suppose that you can determine whether a given sequence constitutes a proof, in finite time (not necessarily bounded). Then construct an ordering on sequences (can be done, it...

0[anonymous]11y

Finite amount of time, yes. Feasible amount of time, no, unless P = NP. When I
said [http://lesswrong.com/lw/7r9/syntacticism/4vwg] that you were considering
agents with unlimited resources, this is what I meant--agents for whom "in
principle" is not different from "in practice." There are no such agents under
the sun.

One can certainly compute the digits of pi, so that since (as non-intuitionists insist anyway) either the $n$th digit is even, or it is odd, we must have P($n$th digit is even) > P(axioms) or P($n$ digit is odd) > P(axioms).

I don't think that's valid - even if I *know* (P=1) that there is a fact-of-the-matter about whether the *n*th digit is even, if I don't have any information causally determined by whether the *n*th digit is even then I assign P(even) = P(odd) = ½. If I instead only believe with P=P(axioms) that a fact-of-the-matter exists, then I a...

I am still arguing with you because I think your misstep poisons more than you have yet realized, not to get on your nerves.

I wasn't suggesting you were trying to get on my nerves. I just think we're talking past each other.

"A proof exists" is a much murkier statement and it is much more difficult to discuss its probability.

As a first approximation, what's wrong with "\lim_{t -> \infty} P(I can find a proof in time t)"?

Also, I don't see why the prior has to be oracular; what's wrong with, say, P(the 3^^^3th decimal digit of p...

0[anonymous]11y

I slipped. It's P(I will find a proof in time t) that is asking for the
probability of a definite event. It's not that evaluating this number at large t
is so problematic, it's that it doesn't capture what people usually mean by
"provable in principle."
If A is logically implied by B, then P(A) >= P(B), or else you are committing a
version of the conjunction fallacy. One can certainly compute the digits of pi,
so that since (as non-intuitionists insist anyway) either the $n$th digit is
even, or it is odd, we must have P(nth digit is even) > P(axioms) or P(n digit
is odd) > P(axioms). P becomes an oracle by testing, for each assertion x,
whether P(x) > P(axioms). There might be ways out of this but they require you
to think about feasibility.

lengthiness is not expected to be the only obstacle to finding a proof

True; stick a ceteris paribus in there somewhere.

You are trying to reason about reality from the point of view of a hypothetical entity that has infinite resources.

Not so; I am reasoning about reality in terms of what it is theoretically possible we might conclude with finite resources. It is just that enumerating the collection of things it is theoretically possible we might conclude with finite resources requires infinite resources (and may not be possible even then). Fortunat...

Paul Almond

To Minds, Substrate, Measure and Value Part 2: Extra Information About Substrate Dependence I make his Objection 9 and am not satisfied with his answer to it. I believe there is a directed graph (possibly cyclic) of mathematical structures containing simulations of other mathematical structures (where the causal relation proceeds from the simulated to the simulator), and I suspect that if we treat this graph as a Markov chain and find its invariant distribution, that this might then give us a statistical measure of the probability of being i...

011y

Thanks for following up on Almond. Your statements align well with my intuition,
but I admit heavy confusion on the topic.

But then, how do you determine whether information exists-in-the-universe at all? Does the number 2 exist-in-the-universe? (I can pick up 2 pebbles, so I'm guessing 'yes'.) Does the number 3^^^3 exist-in-the-universe? Does the number N = total count of particles in the universe exist-in-the-universe? (I'm guessing 'yes', because it's represented by the universe.) Does N+1 exist-in-the-universe? (After all, I can consider {particles in the universe} union {{particles in the universe}}, with cardinality N+1) If you allow encodings other than unary, le...

I am *aware* it can be very small. The only sense in which I claimed otherwise was by a poor choice of wording. The use I made of the claim that "Agents implementing the same deduction rules and starting from the same axioms tend to converge on the same set of theorems" was to argue for the proposition that there is a fact-of-the-matter about which theorems are provable in a given system. You accept that my finding a proof causes you to update P(you can find a proof) upwards by a strictly positive amount - from which I infer that you accept that...

0[anonymous]11y

I am still arguing with you because I think your misstep poisons more than you
have yet realized, not to get on your nerves.
No. "I can find a proof in time t" is a definite event whose probability maybe
can be measured (with difficulty!). "A proof exists" is a much murkier statement
and it is much more difficult to discuss its probability. (For instance it is
not possible to have a consistent probability distribution over assertions like
this without assigning P(proof exists) = 0 or P(proof exists) = 1. Such a
consistent prior is an oracle!)

Your conclusion on truth is a physical state in your mind, generated by physical processes. The existence of a metaphysical truth is not required for you to come to that conclusion.

I think a meta- has gone missing here: I can't be *certain* that others tend to reach the same truth (rather than funny hats), and I can't be *certain* that 2+2=4. I can't even be *certain* that there is a fact-of-the-matter about whether 2+2=4. But it seems damned likely, given Occamian priors, that there is a fact-of-the-matter about whether 2+2=4 (and, inasmuch as a reflective...

011y

Restating my claim in terms of sheep: The identification of a sheep is a state
change within a context of evaluation that implements sheep recognition. So a
sheep exists in that context.
Physical reality however does not recognize sheep; it recognizes and responds to
physical reality stuff. Sheep don't exist within physical reality.
"Sheep" is at a different meta-level than the chain of physical inference that
led to that classification.
"Truth" is at a different meta-level than the chain of physical inference that
lead to that classification. There is no requirement that "truth" is in the set
of stuff that has meaning within the territory.
When you look at the statement 2+2=4 you think some form of "hey, that's true".
When I look at the statement, I also think some form of "hey, that's true". We
can then talk and both come to our own unique conclusion that the other person
agrees with us. This process does not require a metaphysical arithmetic; it only
requires a common context.
For example we both have a proximal existence within the physical universe, we
have a communication channel, we both understand English, and we both understand
basic arithmetic. These types of common contexts allow us make some very
practical and reasonable assumptions about what the other person means.
Common contexts allow us to agree on the consequences of arithmetic.
The short summary is that meaning/existence is formed by contexts of evaluation,
and common contexts allow us to communicate. These processes explain your
observations and operate entirely within the physical universe. The concept of
metaphysical existence is not needed.

A positive but minuscule amount.

Right - but if there were no 'fact-of-the-matter' as to whether a proof exists, why should it be non-zero at all?

0[anonymous]11y

But that isn't what either of us said. You mentioned P(you can find a proof). I
am telling you (telling you, modulo standard open problems) that this can be
very small even after another agent has found a proof. This is a standard family
of topics in computer science.

we find it hard to taboo words that are truly about the fundamentals of our universe, such as 'causality' or 'reality' or 'existence' or 'subjective experience'.

I tabooed "exist", above, by what I think it means. *You* think 'existence' is fundamental, but you've not given me enough of a definition for me to understand your arguments that use it as an untabooable word.

words like 'mathematical equations'

I'd say that (or rather 'mathematics') is just 'the orderly manipulations of symbols'. Or, as I prefer to phrase it, 'symbol games'.

...'corr

011y

I understand "symbols" to be a cognitive shorthand for our brains representation
of structures in reality. I don't understand the meaning of the word "symbols"
in the abstract, without a brain to interpret them with and map them onto
reality.
This doesn't really explain anything to me, it just sounds like wisdom.

But why should feasibility matter? Sure, the more steps it takes to prove a proposition, the less likely you are to be able to find a proof. But saying that things are true only by virtue of their proof being feasible... is disturbing, to say the least. If we build a faster computer, do some propositions suddenly *become true*, because we now have the computing power to prove them?

Me saying I have a proof of a theorem should cause you to update P(you can find a proof) upwards. (If it doesn't, I'd be *very* surprised.) Consequently, there is *something* comm...

2[anonymous]11y

Incidentally but importantly, lengthiness is not expected to be the only
obstacle to finding a proof. Cryptography depends on this.
As to why feasibility matters: it's because we have limited resources. You are
trying to reason about reality from the point of view of a hypothetical entity
that has infinite resources. If you wish to convince people to be less skeptical
of infinity (your stated intention), you will have to take feasibility into
account or else make a circular argument.
I am certainly not saying that feasible proofs cause things to be true. Our
previous slow computer and our new fast computer cause exactly the same number
of important things to be true: none at all. That is the formalist position,
anyway.
Not so. If I have P(PA will be shown inconsistent in fewer than m minutes) = p,
then I also have P(I will prove the negation of your theorem in fewer than m+1
minutes) = p. Your ability to prove things doesn't enter into it.

2[anonymous]11y

A positive but minuscule amount. This is how cryptography works! In less than a
minute (aided by my very old laptop), I gave a proof of the following theorem:
the second digit of each of each of the prime factors of n is 6, where
n = 44289087508518650246893852937476857335929624072788480361
It would take you much longer to find a proof (even though the proof is very
short!).
Update!
About feasibility, I might say more later.

Ok, now taboo your uses of "reality" and "preexisted" in the above comment, because I can't conceive of meanings of those words in which your comment makes sense.

011y

The thing about tabooing words, is that we find it easy to taboo words that are
just confused concepts (it's easy to taboo the word 'sound' and refer to
acoustical experience vs acoustic vibrations), and we find it hard to taboo
words that are truly about the fundamentals of our universe, such as 'causality'
or 'reality' or 'existence' or 'subjective experience'.
I find it much easier to taboo the words that you think fundamentals -- words
like 'mathematical equations', namely 'the orderly manipulations of symbols that
human brains can learn to correspond to concepts in the material universe in
order to predict happenings in said material universe'
To put it differently: Why don't you taboo the words "mathematics" and
"equations" first, and see if your argument still makes any sense

Surely can't be exactly what you mean, as exists(our Univese) and ¬exists(everything else) seems coherent if rather unlikely

I would dispute this, on the grounds that my deductions in formal systems come from somewhere that has a causal relation to my brain - the formal system causes me to be more likely to deduce the things which are valid deductions than the things that aren't. So, if I 'exist', I maintain that the formal systems have to 'exist' too, unless you're happy with 'existing' things being causally influenced by 'non-existing' things - in whi...

So if you've ever read Probability Theory, by E.T. Jaynes

I haven't; I probably should.

the position that, in order to make sense when applied to the real world, infinite things have to behave like limits of finite things.

Is this "limits" in the sense of analysis (epsi-delta limits), or is it "limit points" (like ω)? If the former, then that position involves not believing that arithmetic makes sense when applied to the real world. If the latter, then the position doesn't seem different from what most mathematicians believe, beca...

011y

Well, you can say exists, it just seems to be digging you into a hole.

I'm not quite sure how you're defining "causal model" here, but the bit about "get paid to build a factory, which then produces goods, meanwhile you don't consume the goods you were paid" seems causal to me. By not consuming the proceeds of your work, you have caused society to have more capital than otherwise. Heck, the paragraph beginning "But suppose…" is also describing a series of causes and effects, although it glosses over exactly how removing money from circulation drives up the value of money (that&#x... (read more)