Can't break laws of physics AND Godel's incompleteness theorem

[Rami Rustom]

On Sat, Sep 8, 2012 at 5:16 PM, Josh Jordan <therealj...@gmail.com> wrote:
> This is a lightly edited transcript of David Deutsch's appearance on "On
> Point" with Tom Ashbrook on August 8, 2011. The audio is available at
> http://onpoint.wbur.org/2011/08/18/david-deutsch

...

> Vijay: Hi, Tom. I think that Professor Deutsch's thesis is deeply misguided
> and badly written, and here's why. His basic idea is that, since we can
> reason, we can do anything (aside from violating the limits placed by
> physics). But we also know, from computer science and mathematics, that
> there are many things that are beyond reason. These are called independence
> results or incompleteness results. There are things that we cannot know, and
> these lead us to deep issues and philosophical issues the context of
> computer science and mathematics. Professor Deutsch is most likely aware of
> these things, and still his basic thesis is that because we can reason, we
> can understand anything and therefore we can do anything. And secondly,
> technically, the statement that, "aside from the limits placed by physics
> and incompleteness results, everything else is possible" is essentially a
> truism. So what exactly is he trying to say? What is new here?
>
> Ashbrook: Vijay, stand by. David Deutsch, what do you say?
>
> Deutsch: Okay, I'll deal with the first question first. In mathematics in
> can be proved that the overwhelming majority of mathematical truths cannot
> be proved, and indeed cannot be known. So the question is, how is that
> compatible with the idea that we can do anything? The short answer is this:
> if a mathematician is interested in a certain problem, let's say to do with
> prime numbers, then one way that will lead to the bottle of champagne being
> opened is if this mathematician discovers a proof that this thing is true or
> a proof that it is false. But another way that you could get exactly the
> same success in human terms would be to prove that it is unprovable. This is
> as much a reason for writing a mathematics paper and opening the bottle of
> champagne as proving that's true or proving that's false. And if you can't
> prove that it's unprovable, then maybe the next best thing is that you
> conjecture that it's unprovable. And then you write a paper about what would
> be the consequences if it were, and another paper about what would be the
> consequences if it weren't, and therefore you get twice the number of papers
> just because the thing you were working on is unprovable. So, in the human
> sense, mathematics provides no barrier to progress, even though, as a matter
> of logic, there are things that we can't know. But they're not things that
> matter ultimately to humans. Now, to answer the second question, "what have
> I said that's new?" In a sense, you're quite right, it's almost a trivial
> consequence of regarding the scientific worldview as true, but listen to the
> other commentators! They are saying that gaining control of the universe is
> (a) impossible and (b) wrong, and I am saying that the scientific worldview
> is incompatible with those ancient ideas of limitation.

The caller said that these 2 ideas conflict:

- We can do anything except break the laws of physics.

- Some math problems are not calculable. (This is from Godel's
incompleteness theorem.)

So he's saying that *calculating a math problem* is part of the set of
*do anything*.

DD's response was that these non-calculable math problems do not
provide a barrier to progress. But that is irrelevant. Not having a
barrier to progress =/= do anything.

If that is problematic, then does this solve it?

- We can do any physical things except break the laws of physics.

- We can do any epistemic things except break the laws of epistemology.

So I'm saying that Godel's incompleteness theorem is a law of
epistemology. Another one is the idea that we can not know which of
our conjectural truths is an objective truth.

-- Rami Rustom
http://ramirustom.blogspot.com

[Brett Hall]

I don't understand your shorthand there. I think David Deutsch is correct in saying that (to paraphase and run the risk of getting it wrong) Godel's incompleteness theorem provides no barrier to infinite progress. If you are saying that's wrong...you might have to provide something more substantive than "that's irrelevant".

Not having a barrier to progress does *not* mean you can do *anything*. It means you can do anything not prohibited by the laws of physics. Godel's theorems just say that some things in maths are true, but not provably so (among other things). But so what? There's an infinite number of undecidable propositions. And provable things. And disprovable things. And physically possible things. What do you think the caller's point was?

>
> If that is problematic,

It's not. DD answered it.

> then does this solve it?
>
> - We can do any physical things except break the laws of physics.

That solves nothing, imo. Circularity seems to be a theme lately. Those two things (we can do any physical things) & (we can't break the laws of physics) mean the same thing. Neither statement contributes anything to the other. What you are physically capable of accomplishing is what the laws of physics say is possible. Breaking the laws makes no physical sense. If you accept the notion of physical law...then you accept the physical impossibility of breaking them. They are not laws otherwise. No big deal here.

>
> - We can do any epistemic things except break the laws of epistemology.

What, exactly, does that mean? Don't you think that's circular? Compare it to: You can go anywhere in this room except outside of it. Or...you can legally do anything at all that is not forbidden by the law. Does appending "not forbidden by the law" contribute about as much additional content as "break the laws of epistemology" in your statement, do you think?

>
> So I'm saying that Godel's incompleteness theorem is a law of
> epistemology.

Ok. I think that's kinda well known in the sense that we know it places limits of what can be proved in maths. Insofar as that's relevant to knowledge creation, then yeah, you're correct. But that's not a revelation. I'm not "having a go" - maybe you are just thinking out loud (fair enough...I do a lot of that here too, it seems).

> Another one is the idea that we can not know which of
> our conjectural truths is an objective truth.

I don't think that follows. I think we know what the difference between objective and subjective is. It's not that problematic except in "fringe" cases of interest to philosophers.


Brett

[Rami Rustom]

I'm saying that "no barrier to infinite progress" does not explain
that: There are some things that can't be done which are within the
laws of physics.

> Not having a barrier to progress does *not* mean you can do *anything*. It means you can do anything not prohibited by the laws of physics. Godel's theorems just say that some things in maths are true, but not provably so (among other things).

Godel's incompleteness theorem says that some problems are not solvable.

> But so what? There's an infinite number of undecidable propositions. And provable things. And disprovable things. And physically possible things. What do you think the caller's point was?

That there exists some things that we can't do and they are within the
laws of physics.

>
>>
>> If that is problematic,
>
> It's not. DD answered it.
>
>
>> then does this solve it?
>>
>> - We can do any physical things except break the laws of physics.
>
> That solves nothing, imo. Circularity seems to be a theme lately.

I don't see how its circular. Say I want to put accelerate beyond the
speed of light. This is a physical act. But I can't do it because of
laws of physics.

> Those two things (we can do any physical things) & (we can't break the laws of physics) mean the same thing. Neither statement contributes anything to the other. What you are physically capable of accomplishing is what the laws of physics say is possible.

Moving is a physical act. By introducing a component to that like *at
speed X*, the act of moving at speed X is still a physical act.

> Breaking the laws makes no physical sense. If you accept the notion of physical law...then you accept the physical impossibility of breaking them. They are not laws otherwise. No big deal here.
>
>>
>> - We can do any epistemic things except break the laws of epistemology.
>
> What, exactly, does that mean? Don't you think that's circular? Compare it to: You can go anywhere in this room except outside of it. Or...you can legally do anything at all that is not forbidden by the law. Does appending "not forbidden by the law" contribute about as much additional content as "break the laws of epistemology" in your statement, do you think?
>
>>
>> So I'm saying that Godel's incompleteness theorem is a law of
>> epistemology.
>
> Ok. I think that's kinda well known in the sense that we know it places limits of what can be proved in maths. Insofar as that's relevant to knowledge creation, then yeah, you're correct. But that's not a revelation. I'm not "having a go" - maybe you are just thinking out loud (fair enough...I do a lot of that here too, it seems).
>
>> Another one is the idea that we can not know which of
>> our conjectural truths is an objective truth.
>
> I don't think that follows. I think we know what the difference between objective and subjective is. It's not that problematic except in "fringe" cases of interest to philosophers.

Objective truth and conjectural truth are terms that Popper used.
Objective truth means the actual truth. Conjectural truth are our
ideas that we currently don't have criticisms of. So, any of our
conjectural truths may be flawed, but we don't know which ones. And
some of our conjectural truths may be not flawed, which means they are
objective truths, but we don't know which ones.

So an epistemic thing that we can do is create conjectural knowledge.
And an epistemic thing we can't do is justify our knowledge (which in
Popper's terms means that a person can know that a conjectural truth
is an objective truth). Another epistemic thing we can't do is create
knowledge via induction.

[Brett Hall]

Oh, ok. For example?

It can't be "prove a theorem true that's actually not provably true" because that wouldn't be "within the laws of physics", would it?

>
>
>> Not having a barrier to progress does *not* mean you can do *anything*. It means you can do anything not prohibited by the laws of physics. Godel's theorems just say that some things in maths are true, but not provably so (among other things).
>
> Godel's incompleteness theorem says that some problems are not solvable.

I do not think that is true either. The *problem* is: is this theorem true, false or not provably true or false? Proving that it is not provably true (or false) is a *solution*.

>
>
>> But so what? There's an infinite number of undecidable propositions. And provable things. And disprovable things. And physically possible things. What do you think the caller's point was?
>
> That there exists some things that we can't do and they are within the
> laws of physics.

You'll have to be really specific. Like what, for example?

>
>
>>
>>>
>>> If that is problematic,
>>
>> It's not. DD answered it.
>>
>>
>>> then does this solve it?
>>>
>>> - We can do any physical things except break the laws of physics.
>>
>> That solves nothing, imo. Circularity seems to be a theme lately.
>
> I don't see how its circular. Say I want to put accelerate beyond the
> speed of light. This is a physical act. But I can't do it because of
> laws of physics.

Yeah. So "accelerating beyond the speed of light" whatever that is, isn't *physical*. I suppose you could call it imaginary - but it's not physical because it's not physically possible. Just because you can imagine it, doesn't endow it with some sort of physical status, does it?

>
>
>> Those two things (we can do any physical things) & (we can't break the laws of physics) mean the same thing. Neither statement contributes anything to the other. What you are physically capable of accomplishing is what the laws of physics say is possible.
>
> Moving is a physical act. By introducing a component to that like *at
> speed X*, the act of moving at speed X is still a physical act.

No, for the reason I just explained.

>
>
>> Breaking the laws makes no physical sense. If you accept the notion of physical law...then you accept the physical impossibility of breaking them. They are not laws otherwise. No big deal here.
>>
>>>
>>> - We can do any epistemic things except break the laws of epistemology.
>>
>> What, exactly, does that mean? Don't you think that's circular? Compare it to: You can go anywhere in this room except outside of it. Or...you can legally do anything at all that is not forbidden by the law. Does appending "not forbidden by the law" contribute about as much additional content as "break the laws of epistemology" in your statement, do you think?
>>
>>>
>>> So I'm saying that Godel's incompleteness theorem is a law of
>>> epistemology.
>>
>> Ok. I think that's kinda well known in the sense that we know it places limits of what can be proved in maths. Insofar as that's relevant to knowledge creation, then yeah, you're correct. But that's not a revelation. I'm not "having a go" - maybe you are just thinking out loud (fair enough...I do a lot of that here too, it seems).
>>
>>> Another one is the idea that we can not know which of
>>> our conjectural truths is an objective truth.
>>
>> I don't think that follows. I think we know what the difference between objective and subjective is. It's not that problematic except in "fringe" cases of interest to philosophers.
>
> Objective truth and conjectural truth are terms that Popper used.
> Objective truth means the actual truth. Conjectural truth are our
> ideas that we currently don't have criticisms of. So, any of our
> conjectural truths may be flawed, but we don't know which ones. And
> some of our conjectural truths may be not flawed, which means they are
> objective truths, but we don't know which ones.

Yeah, okay. The "ontological truth" - or objective truth - whatever - in that sense is unobtainable. In another program didn't DD say that too? The presenter said and he agreed that "the final grok is unobtainable" - have you heard this interview? Anyways - that seems to be what you're talking about. I haven't seen "objective" used the way you're pushing it here though. Our knowledge of the laws of physics is objective. It's also conjectural. But all knowledge is conjectural. Science is objective. So is maths and so forth.

There is subjective knowledge too though. Do you know honey is sweet? That "honey tastes sweet" is objective knowledge about your subjective state.

>
> So an epistemic thing that we can do is create conjectural knowledge.
> And an epistemic thing we can't do is justify our knowledge (which in
> Popper's terms means that a person can know that a conjectural truth
> is an objective truth). Another epistemic thing we can't do is create
> knowledge via induction.

Okay. Fine. There's probably clearer ways of saying that. Namely take out the word "epistemic" every time you have used it there and it all means *exactly* the same thing...without this need to clarify some, superfluous (as it turns out) words.

Brett.

[Rami Rustom]

Like solving certain categories of math problems (this is the subject
of Godel's Incompleteness theorem).

> It can't be "prove a theorem true that's actually not provably true" because that wouldn't be "within the laws of physics", would it?

Well I don't think that Godel's Incompleteness theorem is a law of
physics. I'd call it a law of epistemology.

>
>
>>
>>
>>> Not having a barrier to progress does *not* mean you can do *anything*. It means you can do anything not prohibited by the laws of physics. Godel's theorems just say that some things in maths are true, but not provably so (among other things).
>>
>> Godel's incompleteness theorem says that some problems are not solvable.
>
> I do not think that is true either. The *problem* is: is this theorem true, false or not provably true or false? Proving that it is not provably true (or false) is a *solution*.

Thats not the problem I'm talking about nor is it the subject of
Godel's Incompleteness theorem. The theorem is about a category of
math problems that are unsolvable.

>>
>>
>>> But so what? There's an infinite number of undecidable propositions. And provable things. And disprovable things. And physically possible things. What do you think the caller's point was?
>>
>> That there exists some things that we can't do and they are within the
>> laws of physics.
>
> You'll have to be really specific. Like what, for example?

Like solving certain categories of math problems.

>>
>>
>>>
>>>>
>>>> If that is problematic,
>>>
>>> It's not. DD answered it.
>>>
>>>
>>>> then does this solve it?
>>>>
>>>> - We can do any physical things except break the laws of physics.
>>>
>>> That solves nothing, imo. Circularity seems to be a theme lately.
>>
>> I don't see how its circular. Say I want to put accelerate beyond the
>> speed of light. This is a physical act. But I can't do it because of
>> laws of physics.
>
> Yeah. So "accelerating beyond the speed of light" whatever that is, isn't *physical*. I suppose you could call it imaginary - but it's not physical because it's not physically possible. Just because you can imagine it, doesn't endow it with some sort of physical status, does it?

I don't know. I think its a semantics problem. And semantics is not
what I'm discussing. I'm discussing the idea, the conflict that the
listener pointed out. Either there really is a conflict or not. I
don't see how DD's answer shows that there is no conflict.

>>>> Another one is the idea that we can not know which of
>>>> our conjectural truths is an objective truth.
>>>
>>> I don't think that follows. I think we know what the difference between objective and subjective is. It's not that problematic except in "fringe" cases of interest to philosophers.
>>
>> Objective truth and conjectural truth are terms that Popper used.
>> Objective truth means the actual truth. Conjectural truth are our
>> ideas that we currently don't have criticisms of. So, any of our
>> conjectural truths may be flawed, but we don't know which ones. And
>> some of our conjectural truths may be not flawed, which means they are
>> objective truths, but we don't know which ones.
>
> Yeah, okay. The "ontological truth" - or objective truth - whatever - in that sense is unobtainable. In another program didn't DD say that too?

I think DD does use Popper's terminology of conjectural truth and
objective truth.

> The presenter said and he agreed that "the final grok is unobtainable" - have you heard this interview? Anyways - that seems to be what you're talking about. I haven't seen "objective" used the way you're pushing it here though. Our knowledge of the laws of physics is objective.
> It's also conjectural. But all knowledge is conjectural. Science is objective. So is maths and so forth.
>
> There is subjective knowledge too though. Do you know honey is sweet? That "honey tastes sweet" is objective knowledge about your subjective state.

Yes.

>
>>
>> So an epistemic thing that we can do is create conjectural knowledge.
>> And an epistemic thing we can't do is justify our knowledge (which in
>> Popper's terms means that a person can know that a conjectural truth
>> is an objective truth). Another epistemic thing we can't do is create
>> knowledge via induction.
>
> Okay. Fine. There's probably clearer ways of saying that. Namely take out the word "epistemic" every time you have used it there and it all means *exactly* the same thing...without this need to clarify some, superfluous (as it turns out) words.

If you do that, then you how would you resolve the contradiction
between these two ideas:

- its not possible to solve certain categories of math problems

- we can do anything within the laws of physics

[Brett Hall]

Godel's incompleteness theorems are probably second only to quantum theory in the amount of woo that sometimes floats around them. 

Yet, interesting (amazing even) as they seemed to me when I first encountered them in some uni subject called "Logic and Computability" I never thought they could cache out anything truly strange. They were just another theorem in maths. See for example DD's comments at the end of this message:

http://groups.yahoo.com/group/Fabric-of-Reality/message/18263

I know you aren't spouting woo here, but it's a similar extrapolation of Godel that isn't warranted. It seems to me that you don't need Godel to make your point. You could just use...well any bit of mathematics you wanted and be just as wrong. Let's try Pythagoras;

Would you say: Pythagoras' theorem places *physical limits* on what we can do with triangles. I would love to be able to construct a right angled triangle where the square on the hypotenuse is not equal to the sum of the squares on the two shorter sides. Indeed, I can imagine an infinite number of triads that violate that law right now...but pesky mathematics gets in the way. This *prevents me from making infinite progress with triangles* because I am restricted to only some smaller subset of real, mathematically possible triangles.

There are statements that are true, but not provably so. This is one of the incompleteness theorems, more or less. It gets the most woo attention. Anyways I look at it like this, for the purpose of this argument:

You want to make infinite progress. So you're on a road which branches into two. Both lead off to "infinite progress" but one of the branches is blocked by Godel's theorems. Effectively you can see an infinite amount off into the distance but...everytime you take a step on that path you end up back at the same juncture. You want to go down that road, trying to prove true things that the proof says you can't and so you are delivered back to where you began. This path to providing proofs for all those true statements where it can be shown, there are no proofs to demonstrate they are true, or false, is a path to nowhere. To you it just *seems* to go somewhere. That's an illusion though. And it's not a barrier to infinite progress at all because...

The other road - not barred - leads to infinite progress too. And, tellingly *it branches* an infinite number of times too and you can choose any of these branches. When you take a step along those roads, they branch and branch again. Some are blocked by physical law, or mathematical proof or philosophical absurdity, but who cares? You just discover the branches that aren't and continue to make infinite progress. You are never *ultimately* blocked. There are always alternative routes. Indeed they are the only ones rightly deserving of the name "route to infinite progress". The others are just dead ends.

It can't be "prove a theorem true that's actually not provably true" because that wouldn't be "within the laws of physics", would it?

Well I don't think that Godel's Incompleteness theorem is a law of
physics. I'd call it a law of epistemology.

DD would know the most about this, on the planet, probably. Physics and computability are related in a deep way that he has shown. Godel's theorem is related to computability so places limits on what can be computed. But don't take my word for it; ask the hive mind:

http://en.wikipedia.org/wiki/Gödel's_incompleteness_theorems#Relationship_with_computability

Not having a barrier to progress does *not* mean you can do *anything*. It means you can do anything not prohibited by the laws of physics. Godel's theorems just say that some things in maths are true, but not provably so (among other things).

Godel's incompleteness theorem says that some problems are not solvable.

I do not think that is true either. The *problem* is: is this theorem true, false or not provably true or false? Proving that it is not provably true (or false) is a *solution*.

Thats not the problem I'm talking about nor is it the subject of
Godel's Incompleteness theorem. The theorem is about a category of
math problems that are unsolvable.

Ok. This is where we have to be careful and where the theorem starts to get endowed with qualities that it just does not possess. I don't think it quite does this. Loosely speaking maybe - but isn't the *solution* to a certain category (as you put it) of math problems just that *there is no procedure for proving them true or false*. But that is solvable. The assertion that "its unsolvable" is the solution. As DD points out - that's not a barrier to progress because now you can write more papers. Conjecture: what if it's true and write that paper. Conjecture it's false and write that paper. Conjecture that it's undecidable and write that paper. Yeah, you are prevented from ever having a proof either way...but you are prevented from consistently constructing 4 sided triangles too. So what? Doesn't matter. Not for infinite progress.

But so what? There's an infinite number of undecidable propositions. And provable things. And disprovable things. And physically possible things. What do you think the caller's point was?

That there exists some things that we can't do and they are within the

laws of physics.

You'll have to be really specific. Like what, for example?

Like solving certain categories of math problems.

"Within the laws of physics" surely cannot include those things shown to be mathematically impossible. Right?

If that is problematic,

It's not. DD answered it.

then does this solve it?

- We can do any physical things except break the laws of physics.

That solves nothing, imo. Circularity seems to be a theme lately.

I don't see how its circular. Say I want to put accelerate beyond the

speed of light. This is a physical act. But I can't do it because of

laws of physics.

Yeah. So "accelerating beyond the speed of light" whatever that is, isn't *physical*. I suppose you could call it imaginary - but it's not physical because it's not physically possible. Just because you can imagine it, doesn't endow it with some sort of physical status, does it?

I don't know. I think its a semantics problem. And semantics is not
what I'm discussing.

Well, you say that...and yet it's all about what "physical" or "epistemic" means. You're confused by these words, so if you stop using them...I reckon your problem is solved...at least as far as some of what you write is concerned. Again, take this sentence of yours:

We can do any physical things except break the laws of physics.

Just take out the word "physical". It then becomes much better. It says:

We can do anything except break the laws of physics.

After all the verb "do" implies some sort of physical action, right? What else could it mean? What other sort of things can be "done"?

I'm discussing the idea, the conflict that the
listener pointed out. Either there really is a conflict or not.

No conflict.

I
don't see how DD's answer shows that there is no conflict.

It does. Interesting though. This then becomes a problem of communicating in different ways. DD answer convinced me. The same words didn't convince you.  Not sure what to do about that...

Another one is the idea that we can not know which of

our conjectural truths is an objective truth.

I don't think that follows. I think we know what the difference between objective and subjective is. It's not that problematic except in "fringe" cases of interest to philosophers.

Objective truth and conjectural truth are terms that Popper used.

Objective truth means the actual truth. Conjectural truth are our

ideas that we currently don't have criticisms of. So, any of our

conjectural truths may be flawed, but we don't know which ones. And

some of our conjectural truths may be not flawed, which means they are

objective truths, but we don't know which ones.

Yeah, okay. The "ontological truth" - or objective truth - whatever - in that sense is unobtainable. In another program didn't DD say that too?

I think DD does use Popper's terminology of conjectural truth and
objective truth.

The presenter said and he agreed that "the final grok is unobtainable" - have you heard this interview? Anyways - that seems to be what you're talking about. I haven't seen "objective" used the way you're pushing it here though. Our knowledge of the laws of physics is objective.

It's also conjectural. But all knowledge is conjectural. Science is objective. So is maths and so forth.

There is subjective knowledge too though. Do you know honey is sweet? That "honey tastes sweet" is objective knowledge about your subjective state.

Yes.

So an epistemic thing that we can do is create conjectural knowledge.

And an epistemic thing we can't do is justify our knowledge (which in

Popper's terms means that a person can know that a conjectural truth

is an objective truth). Another epistemic thing we can't do is create

knowledge via induction.

Okay. Fine. There's probably clearer ways of saying that. Namely take out the word "epistemic" every time you have used it there and it all means *exactly* the same thing...without this need to clarify some, superfluous (as it turns out) words.

If you do that, then you how would you resolve the contradiction

There is no contradiction. The problem is yours, internally. It's not that there is a contradiction in reality. It's that you think there is one, where there isn't.

between these two ideas:

- its not possible to solve certain categories of math problems

We are confused about the use of the verb "solve". I think one possible solution is: there is no proof of this theorem. Discovering *that* makes the problem soluble. And most importantly, is no barrier to infinite progress for the reasons I've explained in this post.

- we can do anything within the laws of physics

We sure can.

I see no contradiction.

:)

Brett.

[Josh Jordan]

On Sat, Oct 13, 2012 at 6:11 AM, Rami Rustom <rom...@gmail.com> wrote:
> So I'm saying that Godel's incompleteness theorem is a law of
> epistemology.

According to BoI (p 186), Goedel's Theorem is a law of physics:

"[T]here is nothing mathematically special about the undecidable
questions, the non-computable functions, the unprovable propositions.
They are distinguished by physics only. Different physical laws would
make different things infinite, different things computable, different
truths – both mathematical and scientific – knowable. It is only the
laws of physics that determine which abstract entities and
relationships are modelled by physical objects such as mathematicians’
brains, computers and sheets of paper."

[Elliot Temple]

On Oct 13, 2012, at 6:50 AM, Rami Rustom <rom...@gmail.com> wrote:

> On Sat, Oct 13, 2012 at 6:02 AM, Brett Hall <brha...@hotmail.com> wrote:
>>
>> On 13/10/2012, at 21:12, "Rami Rustom" <rom...@gmail.com> wrote:
>>
>>> The caller said that these 2 ideas conflict:
>>>
>>> - We can do anything except break the laws of physics.
>>>
>>> - Some math problems are not calculable. (This is from Godel's
>>> incompleteness theorem.)
>>>
>>> So he's saying that *calculating a math problem* is part of the set of
>>> *do anything*.
>>>
>>> DD's response was that these non-calculable math problems do not
>>> provide a barrier to progress. But that is irrelevant. Not having a
>>> barrier to progress =/= do anything.
>>
>> I don't understand your shorthand there. I think David Deutsch is correct in saying that (to paraphase and run the risk of getting it wrong) Godel's incompleteness theorem provides no barrier to infinite progress. If you are saying that's wrong...you might have to provide something more substantive than "that's irrelevant".
>
> I'm saying that "no barrier to infinite progress" does not explain
> that: There are some things that can't be done which are within the
> laws of physics.

Like Josh posted about: Proving is a physical act. It's within the laws of physics.

> Another epistemic thing we can't do is create knowledge via induction.

Because "create knowledge via induction" does not refer to anything the laws of physics allow one to do. It doesn't refer to any possible physical process. (Often, I think it does not refer to any impossible physical process either. It's just too vague to map to any set of instructions/actions for how a person could do it.)


-- Elliot Temple
http://fallibleideas.com/

[jon_o...@trendmicro.com]

Hi,

On Monday, 15 October 2012 4:36 PM, Josh Jordan writes

> According to BoI (p 186), Goedel's Theorem is a law of physics:

I am not exactly sure how Goedel's Theorem is a law of physics.
You can start with some very basic assumptions about arithmetic (Peano's axioms http://en.wikipedia.org/wiki/Peano_axioms ).
Basically they say
1. zero is a natural number
2. equality of natural numbers work like we expect it to (reflexive, symmetric, transitive, closed)
3. each successor to a natural number is also a natural number. If x is a natural number then x+1 is also a natural number.
From there you can get to Goedel's incompleteness.
I don't see much physics in Peano's axioms.
So Goedel's theorem would appear to be present in almost any universe (any with arithmetic).


BoI says on page 186

"[T]here is nothing mathematically special about the undecidable questions, the non-computable functions, the unprovable propositions.

They are distinguished by physics only. Different physical laws would make different things infinite, different things computable, different truths - both mathematical and scientific - knowable. It is only the laws of physics that determine which abstract entities and relationships are modelled by physical objects such as mathematicians'

brains, computers and sheets of paper."

I interpret this paragraph from David a bit differently.
It is would appear to be alluding to Oracles - which can change the hierarchy of undecidable problems / languages.
http://en.wikipedia.org/wiki/Oracle_machine
So we could imagine a universe (or even this one) that some process (a quantum effect, some decay of particles, whatever) solves a problem which is undecidable by Turing Machines in some finite time.
You attach this apparatus to Turning Machines and call it an oracle - and that lets you solve those problems in one step.
This can result in some previously undecidable problems becoming decidable, but in general you still have other undecidable problems.

In this sense, the laws of physics could affect what is decidable / undecidable in a given universe.

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[Richard Fine]

On 15 Oct 2012, at 08:00, "jon_o...@trendmicro.com"

<jon_o...@trendmicro.com> wrote:

> Hi,
>
> On Monday, 15 October 2012 4:36 PM, Josh Jordan writes
>> According to BoI (p 186), Goedel's Theorem is a law of physics:
>
> I am not exactly sure how Goedel's Theorem is a law of physics.
> You can start with some very basic assumptions about arithmetic (Peano's axioms http://en.wikipedia.org/wiki/Peano_axioms ).
> Basically they say
> 1. zero is a natural number
> 2. equality of natural numbers work like we expect it to (reflexive, symmetric, transitive, closed)
> 3. each successor to a natural number is also a natural number. If x is a natural number then x+1 is also a natural number.
> From there you can get to Goedel's incompleteness.

"Get to" how? Using natural deduction, right? But isn't natural
deduction a part of physics?

- Richard

[David Deutsch]

On 15 Oct 2012, at 08:00, jon_o...@trendmicro.com wrote:

> Hi,
>
> On Monday, 15 October 2012 4:36 PM, Josh Jordan writes
>> According to BoI (p 186), Goedel's Theorem is a law of physics:
>
> I am not exactly sure how Goedel's Theorem is a law of physics.
> You can start with some very basic

The thing is, the difference between 'very basic' and 'very complicated' is purely a consequence of the laws of physics, just like the difference between computable and non-computable.

> assumptions about arithmetic (Peano's axioms http://en.wikipedia.org/wiki/Peano_axioms ).
> Basically they say
> 1. zero is a natural number
> 2. equality of natural numbers work like we expect it to (reflexive, symmetric, transitive, closed)
> 3. each successor to a natural number is also a natural number. If x is a natural number then x+1 is also a natural number.
> From there you can get to Goedel's incompleteness.
> I don't see much physics in Peano's axioms.
> So Goedel's theorem would appear to be present in almost any universe (any with arithmetic).
>
>
> BoI says on page 186
> "[T]here is nothing mathematically special about the undecidable questions, the non-computable functions, the unprovable propositions.
> They are distinguished by physics only. Different physical laws would make different things infinite, different things computable, different truths - both mathematical and scientific - knowable. It is only the laws of physics that determine which abstract entities and relationships are modelled by physical objects such as mathematicians'
> brains, computers and sheets of paper."
>
> I interpret this paragraph from David a bit differently.
> It is would appear to be alluding to Oracles - which can change the hierarchy of undecidable problems / languages.
> http://en.wikipedia.org/wiki/Oracle_machine
> So we could imagine a universe (or even this one) that some process (a quantum effect, some decay of particles, whatever) solves a problem which is undecidable by Turing Machines in some finite time.
> You attach this apparatus to Turning Machines and call it an oracle - and that lets you solve those problems in one step.
> This can result in some previously undecidable problems becoming decidable, but in general you still have other undecidable problems.
>
> In this sense, the laws of physics could affect what is decidable / undecidable in a given universe.

Yes but in view of the fact that I pointed out above, "in this sense ... could affect" actually means "totally determines".

(BTW 'in a given universe' would be better phrased as 'under given laws of physics'.)

-- David Deutsch

[Rami Rustom]

Ok so all unsolvable (abstract) problems are unsolvable because of the
limits of the laws of physics.

So the idea that knowledge cannot be created by induction is a limit
by some law of physics. Maybe its the laws of physics that make our
brains work.

And the idea that knowledge is created via conjecture and criticism is
a fact of some laws of physics, the ones that our brains follow.

And the idea that we are fallible, and that any one of our ideas can
be mistaken, and that we can't know which of our ideas is an objective
truth, are facts because of the limits of laws of physics.

So that means that epistemology emerges from physics. Right?

(Side note: This contradicts Bruno's comp idea. He says that physics
emerges from arithmetic.)

-- Rami

[jon_o...@trendmicro.com]

On Monday, 15 October 2012 8:31 PM, Richard Fine writes:

>> From there you can get to Goedel's incompleteness.
>
> "Get to" how? Using natural deduction, right? But isn't natural deduction a part of physics?

The "get to" is a combination of simple rules and the axiom system.
There are (at least 3) interesting sets of undecidable problems

(A) the undecidable problems derived from arithmetic
(B) the undecidable problems where the laws of physics provides an oracle for lookups.
These provides a way of punching holes in set (A) and makes these problems decidable for a particular laws of physics.
(C) the fundamental undecidable problems (like the halting-problem)
These problems are problems where the laws of physics cannot provide a lookup mechanism.

I think it would be a serious stretch to consider sets (A) and set (C) as being "physics".
They are either pure mathematics or computation theory.
There is no mention of either (i) the laws of physics or (ii) observation in either of these areas.

Work on (B) would definitely be in "physics".
Currently there are no known undecidable problems (of type (A) according to arithmetic) where the laws of physics provides an oracle.
For example, quantum computing does not change any of the problems in (A).
It would be wonderfully exciting if we do determine areas where the laws of physics solve a problem which was previously considered undecidable.

[Brett Hall]

On 16/10/2012, at 7:21, "jon_o...@trendmicro.com" <jon_o...@trendmicro.com> wrote:

> On Monday, 15 October 2012 8:31 PM, Richard Fine writes:
>
>>> From there you can get to Goedel's incompleteness.
>>
>> "Get to" how? Using natural deduction, right? But isn't natural deduction a part of physics?
>
> The "get to" is a combination of simple rules and the axiom system.
> There are (at least 3) interesting sets of undecidable problems
>
> (A) the undecidable problems derived from arithmetic
> (B) the undecidable problems where the laws of physics provides an oracle for lookups.
> These provides a way of punching holes in set (A) and makes these problems decidable for a particular laws of physics.
> (C) the fundamental undecidable problems (like the halting-problem)
> These problems are problems where the laws of physics cannot provide a lookup mechanism.
>
> I think it would be a serious stretch to consider sets (A) and set (C) as being "physics".
> They are either pure mathematics or computation theory.

Not meaning to be facetious, but have you read FoR or BoI? Some good stuff in there about the supposed elevation of pure mathematics above scientific theories and philosophy and the implicit hierarchy most people assume exists. Good stuff about the nature of pure mathematics.

"Necessary truth is the subject matter of mathematics, not the reward we get for doing it." A proof, even of an incompleteness theorem, is still just a physical process.

All proofs are.

Each of A, B and C above is a question/problem about some limit on what can be computed or calculated or proved or whatever. Whatever verb you want to use, it's a physical process you are talking about.

Physics determines what can and cannot be shown in mathematics. You can never rule out glitches, errors, misconceptions, etc, etc. I can't remember if it was Feynman (I think it was) and I certainly can't remember the quote exactly but:

Even if you do mathematics with pen and paper you are assuming you know how pens, ink and paper work. You only think you know any of that because of what you think you know about the laws of physics. And we know those laws permit all sorts of things. Quantum theory permits (mandates) that somewhere each time a "2" is written it morphs to a "3". However unlikely, sometimes it happens and because it can't be ruled out you don't have access to a realm of Platonic certainty where you do pure mathematics without needing to be concerned about what the laws of physics are and are not.

So to say that proofs can happen *independent* of physics or our knowledge of those laws is to assume access to a Platonic realm for *supernatural* reasons.

Computation theory requires a computer. Computers are physical things.

Pure mathematics also requires a computer to do its proofs with. Even if that computer is a human brain.

Brett.

[Richard Fine]

On 15 Oct 2012, at 21:21, "jon_o...@trendmicro.com"

<jon_o...@trendmicro.com> wrote:

> On Monday, 15 October 2012 8:31 PM, Richard Fine writes:
>
>>> From there you can get to Goedel's incompleteness.
>>
>> "Get to" how? Using natural deduction, right? But isn't natural deduction a part of physics?
>
> The "get to" is a combination of simple rules and the axiom system.
> There are (at least 3) interesting sets of undecidable problems
>
> (A) the undecidable problems derived from arithmetic
> (B) the undecidable problems where the laws of physics provides an oracle for lookups.
> These provides a way of punching holes in set (A) and makes these problems decidable for a particular laws of physics.
> (C) the fundamental undecidable problems (like the halting-problem)
> These problems are problems where the laws of physics cannot provide a lookup mechanism.
>
> I think it would be a serious stretch to consider sets (A) and set (C) as being "physics".
> They are either pure mathematics or computation theory.
> There is no mention of either (i) the laws of physics or (ii) observation in either of these areas.

Suppose I have a pair of rocks in a box. I take another two rocks and
add them to the first two. Now I have four rocks in the box.

It's a physical reality that carrying out the process of "adding" on
these groups of rocks produces a consistent result. In fact, taking
*any* box containing two rocks, and putting two more rocks in it, will
result in a box containing four rocks. That is a state-transition that
will happen anywhere in the universe. It's not possible - barring
other interfering processes - to get any other result.

Isn't that a law of physics?

It's a very parochial law, of course - overly specific (and thus not
very useful). But when we abstract away the details that are not
relevant to what's happening - what we end up with is arithmetic.

And I don't see that abstracting away the details makes it any less a
law of physics; when we say "metal conducts electricity" we don't
worry about having abstracted away the shape of the metal, or that
there are small impurities in it, or that it weighs a particular
amount.

Most, if not all, of mathematics and computation theory is rooted in
physical reality like this. So I don't see why we can't call results
within them laws of physics - at least in the sense that they are
immutable, inviolable properties of the universe.

(That they may not be the subject of study by people who call
themselves "physicists" is neither here nor there).

- Richard

[jon_o...@trendmicro.com]

Hi Brett

On Tuesday, 16 October 2012 9:16 AM, Brett Hall wrote:
> but have you read FoR or BoI?

I have read BoI and not read FoR.
I totally agree with the central thesis of BoI that finding good explanations, and those explanations are real is the basis of science.
It is very refreshing to see it stated so clearly.

On Tuesday, 16 October 2012 9:16 AM, Brett Hall wrote:
> Physics determines what can and cannot be shown in mathematics.
> You can never rule out glitches, errors, misconceptions, etc, etc.
> I can't remember if it was Feynman (I think it was) and I certainly can't remember the quote exactly but:

For 99.99% or more of science, the laws of physics impose all sorts of constraints.
But what you are suggesting is taking things too far.
There is an area where the laws of physics say little, observation is not that relevant (beyond the exception below)
and David's arguments about seeking explanations does not add much (beyond the exception below where it says a lot).
This area is decidability.
This area is an amazingly small when compared to the rest of science.

As I said before, there is a caveat / exception.
There is the potential that the laws of physics poke holes in what arithmetic says is decidable / undecidable.
The set where this happens is currently thought to be the empty set (no elements - for example Quantum computing does not change it from being empty).
So saying that this set (currently thought to be empty) dominates what is otherwise huge sets (infinite / uncountable) is a weak argument.


If this discussion group is going to take the view that decidability theory / pure mathematics is suspect - its claims are unjustified,
then this seems inconsistent with Beginning of Infinity.
If baseline pure mathematics should be questioned, then we should also question David's use of the no-go theorems
and other derivations starting from axioms in Chapter 13 on choices and group decision making.

I suggest a different approach.
That approach used by David in the Chapter 13 is a good approach.
Start by using these axioms / theorems as starting points
(which will apply under many different laws of physics - whether those laws apply elsewhere - or whether we have not found additional, relevant laws of physics here).
They are a starting point where we build up explanations of the universe around us based on observation, criticism, etc.

Cheers
Jon Oliver

[Alan Forrester]

On 19 Oct 2012, at 01:19, jon_o...@trendmicro.com wrote:

> Hi Brett
>
> On Tuesday, 16 October 2012 9:16 AM, Brett Hall wrote:
>> but have you read FoR or BoI?
>
> I have read BoI and not read FoR.
> I totally agree with the central thesis of BoI that finding good explanations, and those explanations are real is the basis of science.
> It is very refreshing to see it stated so clearly.

BoI explains that explanations are not just relevant to science, they are relevant for making progress in every field. For example, explanations are important for making progress in moral philosophy as explained on pp 120-121 of BoI.

> On Tuesday, 16 October 2012 9:16 AM, Brett Hall wrote:
>> Physics determines what can and cannot be shown in mathematics.
>> You can never rule out glitches, errors, misconceptions, etc, etc.
>> I can't remember if it was Feynman (I think it was) and I certainly can't remember the quote exactly but:
>
> For 99.99% or more of science, the laws of physics impose all sorts of constraints.
> But what you are suggesting is taking things too far.
> There is an area where the laws of physics say little, observation is not that relevant (beyond the exception below)
> and David's arguments about seeking explanations does not add much (beyond the exception below where it says a lot).
> This area is decidability.
> This area is an amazingly small when compared to the rest of science.
>
> As I said before, there is a caveat / exception.
> There is the potential that the laws of physics poke holes in what arithmetic says is decidable / undecidable.

Arithmetical statements are either true or false and whether they are true or false is not determined by the laws of physics.

Arithmetic doesn't say anything about decidability/undecidability: the laws of physics determine what is decidable/undecidable. On BoI, p. 186, David writes:

"So there is nothing *mathematically* special about the set of decidable questions, the non-computable functions, the unprovable propositions. They are distinguished by physics only."

See BoI Chapter 8 and FoR, Chapter 10 for the relevant arguments. Do you have a criticism of those arguments?

> The set where this happens is currently thought to be the empty set (no elements - for example Quantum computing does not change it from being empty).
> So saying that this set (currently thought to be empty) dominates what is otherwise huge sets (infinite / uncountable) is a weak argument.

No such argument is made in BoI or FoR to the best of my knowledge. Where is this argument made in either book? Page references would be useful and quotes too.

Also, it is interesting that you write about weak arguments, when Chapter 13 of BoI criticises the idea that that arguments can be weighed, especially on pages 340-341. Arguments are either right or wrong, they are not strong or weak.

> If this discussion group is going to take the view that decidability theory / pure mathematics is suspect - its claims are unjustified,
> then this seems inconsistent with Beginning of Infinity.

David attacks the idea of justification in BoI, see the entries in the index for justificationism. In particular, in the terminology section for Chapter 1 the entry for justificationism reads:
"The misconception that knowledge can be genuine or reliable only if it is justified by some source or criterion."

So claiming that mathematics is unjustified is entirely consistent with BoI.

If ideas have to be justified then the idea that the putative source or criterion of justification is correct has to be justified and this leads to an infinite regress.

In addition, if there is such a source, then its success is also inexplicable. If there was such an explanation then the source wouldn't the foundation of that explanation.

We should look for explanations instead and decide between those explanations by considering which ones solve problems instead of looking for justifications.

> If baseline pure mathematics should be questioned, then we should also question David's use of the no-go theorems
> and other derivations starting from axioms in Chapter 13 on choices and group decision making.

No it doesn't because justification is unnecessary and impossible.

> I suggest a different approach.
> That approach used by David in the Chapter 13 is a good approach.
> Start by using these axioms / theorems as starting points (which will apply under many different laws of physics - whether those laws apply elsewhere - or whether we have not found additional, relevant laws of physics here).
> They are a starting point where we build up explanations of the universe around us based on observation, criticism, etc.

To the best of my knowledge, David doesn't use that approach, nor does he say that he uses it. If you think he does, then page references and quotes would be useful.

Alan

[jon_o...@trendmicro.com]

On Saturday, 20 October 2012 4:10 AM Alan Forrester wrote:
>> I totally agree with the central thesis of BoI that finding good explanations, and those explanations are real is the basis of science.

> BoI explains that explanations are not just relevant to science, they are relevant for making progress in every field.

Indeed - what I stated is not inconsistent with that.

On Saturday, 20 October 2012 4:10 AM Alan Forrester wrote:
> Do you have a criticism of those arguments?

Yes. On the basis of your email I have considered long and hard what I learned in computation theory 20 years ago,
and read up some more recent writings. I have re-read Chapter 8 thoroughly.
I will carefully go through my objections and post that to the list.

On Saturday, 20 October 2012 4:10 AM Alan Forrester wrote:
>> The set where this happens is currently thought to be the empty set (no elements - for example Quantum computing does not change it from being empty).
>> So saying that this set (currently thought to be empty) dominates what is otherwise huge sets (infinite / uncountable) is a weak argument.

> No such argument is made in BoI or FoR to the best of my knowledge.
> Where is this argument made in either book? Page references would be useful and quotes too.

Let LUP(maths) be the set of problems which mathematics / computation theory says is undecidable
Let LDP(laws physics) be the set of problems which is in LUP(maths) but which the laws of physics makes decidable
So LDP(laws physics) is the set of problems where the laws of physics is making the difference.

This argument is made of page 186. David writes

"So there is nothing *mathematically* special about the set of decidable questions, the non-computable functions, the unprovable propositions.
They are distinguished by physics only."

So on page 186, BoI is saying that
the set of decidable questions are distinguished by physics only.
But with our current understanding of the laws of physics, LDP(laws physics) is empty.
Quantum computing has not put any problems in this set.
Neither has special relativity or general relativity.
If we had an infinity hotel, then yes LDP(laws physics) would be non empty.

On Saturday, 20 October 2012 4:10 AM Alan Forrester wrote:
> If ideas have to be justified then the idea that the putative source or criterion of justification is correct has to be justified and this leads to an infinite regress.

Of course scientific ideas have to be justified.
They have to agree with observation.
They have to be open to criticism, and survive that criticism.
They should be falsifiable - and people should test that ideas.
Etc etc etc.
In summary science should follow the scientific method and provide good explanations.

I suspect / guess that we have different meanings of the word "justified".
If so then I shall avoid using the word "justify" in my emails.

On Saturday, 20 October 2012 4:10 AM Alan Forrester wrote:

>> I suggest a different approach.
>> That approach used by David in the Chapter 13 is a good approach.
>> Start by using these axioms / theorems as starting points (which will apply under many different laws of physics
>> - whether those laws apply elsewhere - or whether we have not found additional, relevant laws of physics here).
>> They are a starting point where we build up explanations of the universe around us based on observation, criticism, etc.


> To the best of my knowledge, David doesn't use that approach, nor does he say that he uses it.
> If you think he does, then page references and quotes would be useful.

Please reread Chapter 13.

On page 336-337, David writes
"This is what Arrow did. He laid down five elementary axioms ...
...
Arrow proved that the axioms that I have listed despite their reasonable appearance, logically inconsistent with each other.
...
It seems to follow that a group of people jointly making decisions is necessarily irrational in one way or another."

In this section, David has clearly followed the approach I suggested.
Note: This is just one of many forms of good explanation - it is certainly not the only form of good explanation.

Cheers
Jon

[Josh Jordan]

On Wed, Oct 24, 2012 at 2:11 AM, jon_o...@trendmicro.com
<jon_o...@trendmicro.com> wrote:
> On Saturday, 20 October 2012 4:10 AM Alan Forrester wrote:
>>> I totally agree with the central thesis of BoI that finding good explanations, and those explanations are real is the basis of science.
>> BoI explains that explanations are not just relevant to science, they are relevant for making progress in every field.
>
> Indeed - what I stated is not inconsistent with that.

No. In the introduction to BoI, David writes, "In this book I shall
argue that all progress, both theoretical and practical, has resulted
from ... the quest for ... good explanations."
So the "central thesis" of BoI is not merely about "the basis of
science". It's about something broader: the cause for "all progress".

> [snip]

> Let LUP(maths) be the set of problems which mathematics / computation theory says is undecidable
> Let LDP(laws physics) be the set of problems which is in LUP(maths) but which the laws of physics makes decidable
> So LDP(laws physics) is the set of problems where the laws of physics is making the difference.

> [snip]

> But with our current understanding of the laws of physics, LDP(laws physics) is empty.

> [snip]

> If we had an infinity hotel, then yes LDP(laws physics) would be non empty.

No. If we had an infinity hotel, then LUP(laws maths) -- in
particular, the definition of "computable" -- would be different.
Therefore, LDP(laws physics) would still be empty.

> On Saturday, 20 October 2012 4:10 AM Alan Forrester wrote:
>> If ideas have to be justified then the idea that the putative source or criterion of justification is correct has to be justified and this leads to an infinite regress.
>
> Of course scientific ideas have to be justified.

No. Justify - "to prove or show to be just, right, or reasonable".
(http://www.merriam-webster.com/dictionary/justified)

So it's not a matter of whether or not ideas (scientific or not)
*have* to be justified, because ideas *cannot* be justified.

> They have to agree with observation.
> They have to be open to criticism, and survive that criticism.
> They should be falsifiable - and people should test that ideas.
> Etc etc etc.
> In summary science should follow the scientific method and provide good explanations.

We cannot be sure that we never will notice serious mistakes even
in theories that
- agree with our observations to date,
- have survived criticism so far,
- have passed our tests,
- etc.

Those theories are therefore not justified.

> [snip]

> On Saturday, 20 October 2012 4:10 AM Alan Forrester wrote:
>
>>> I suggest a different approach.
>>> That approach used by David in the Chapter 13 is a good approach.
>>> Start by using these axioms / theorems as starting points (which will apply under many different laws of physics
>>> - whether those laws apply elsewhere - or whether we have not found additional, relevant laws of physics here).
>>> They are a starting point where we build up explanations of the universe around us based on observation, criticism, etc.
>
>> To the best of my knowledge, David doesn't use that approach, nor does he say that he uses it.
>> If you think he does, then page references and quotes would be useful.
>

> On page 336-337, David writes
> "This is what Arrow did. He laid down five elementary axioms ...
> ...
> Arrow proved that the axioms that I have listed despite their reasonable appearance, logically inconsistent with each other.
> ...
> It seems to follow that a group of people jointly making decisions is necessarily irrational in one way or another."
>
> In this section, David has clearly followed the approach I suggested.

No. The passage above about Arrow's theorem above is a deliberate
example of specious reasoning. Note the word "seems" in the final
sentence. As the book goes on to say, Arrow's Theorem doesn't apply
to creative decision-making processes that involve a search for good
explanations. So although it may *seem* to follow from Arrow's
Theorem that all group decision making is irrational, it doesn't
*actually* follow.

Therefore, that passage is not an example of David actually using
the axiomatic approach. At least, not as a means of drawing
conclusions that he himself accepts.

[Alan Forrester]

What you're quoting is the conclusion, not the argument.

The Turing machine has a tape with squares of finite size. Each square has one of a finite number of distinguishable symbols that can be read by a head with finite memory. That head moves by one square during each step of the evolution of the machine. In BoI, David points out that these requirements are not mandated by the laws of logic. Rather, it could be the case that the head could move by an infinite number of squares in a finite time, as in his suggested way of checking for prime pairs. The idea of the Turing machine is, in substance, a conjecture about the laws of physics that could have been false. The set of computable functions is picked out precisely by the fact that they can be computed by physical objects.

The issue is not whether the actual laws of physics could compute functions other than those computed by the Turing machine. It is whether there is any imaginable set of such laws that could do so.

> On Saturday, 20 October 2012 4:10 AM Alan Forrester wrote:
>> If ideas have to be justified then the idea that the putative source or criterion of justification is correct has to be justified and this leads to an infinite regress.
>
> Of course scientific ideas have to be justified.

The phrase "of course" is bad. It implies that your position is necessarily correct, and so carries a lot of infallibilist baggage.

> They have to agree with observation.
> They have to be open to criticism, and survive that criticism.
> They should be falsifiable - and people should test that ideas.
> Etc etc etc.
> In summary science should follow the scientific method and provide good explanations.
>
> I suspect / guess that we have different meanings of the word "justified".
> If so then I shall avoid using the word "justify" in my emails.



> On Saturday, 20 October 2012 4:10 AM Alan Forrester wrote:
>
>>> I suggest a different approach.
>>> That approach used by David in the Chapter 13 is a good approach.
>>> Start by using these axioms / theorems as starting points (which will apply under many different laws of physics
>>> - whether those laws apply elsewhere - or whether we have not found additional, relevant laws of physics here).
>>> They are a starting point where we build up explanations of the universe around us based on observation, criticism, etc.
>>
>> To the best of my knowledge, David doesn't use that approach, nor does he say that he uses it.
>> If you think he does, then page references and quotes would be useful.
>
> Please reread Chapter 13.
>
> On page 336-337, David writes
> "This is what Arrow did. He laid down five elementary axioms ...
> ...
> Arrow proved that the axioms that I have listed despite their reasonable appearance, logically inconsistent with each other.
> ...
> It seems to follow that a group of people jointly making decisions is necessarily irrational in one way or another."
>
> In this section, David has clearly followed the approach I suggested.
> Note: This is just one of many forms of good explanation - it is certainly not the only form of good explanation.

p. 194 terminology section of chapter 8: "Proof A computation which, given a theory of how the computer on which it runs works, establishes the truth of some abstract proposition."

So our knowledge of physics can't be built up from our knowledge of mathematics, since our knowledge of mathematics is dependent on whether we understand the laws of physics.

Alan

[jon_o...@trendmicro.com]

On Thursday, 25 October 2012 6:34 PM Josh Jordan wrote:

>> Let LUP(maths) be the set of problems which mathematics / computation theory says is undecidable
>> Let LDP(laws physics) be the set of problems which is in LUP(maths) but which the laws of physics makes decidable
>> So LDP(laws physics) is the set of problems where the laws of physics is making the difference.
>> [snip]
>> But with our current understanding of the laws of physics, LDP(laws physics) is empty.
>> [snip]
>> If we had an infinity hotel, then yes LDP(laws physics) would be non empty.

> No. If we had an infinity hotel, then LUP(laws maths) -- in particular, the definition of "computable" -- would be different.
> Therefore, LDP(laws physics) would still be empty.

No.
LUP(maths) is the Turing computable problems.
If the laws of physics includes an infinity hotel, then this does not change the definitions of Turing machines
(nor does it change the definitions of Lambda Calculus).
Therefore if we had an infinity hotel, then yes LDP(laws physics) would be non empty.
If the prime-pairs conjecture is not Turing computable, then (as David points out on p185),
then infinity hotel would allow the prime-pairs conjecture to be in LDP(laws physics).

On Thursday, 25 October 2012 6:34 PM Josh Jordan wrote:

>>>> I suggest a different approach.
>>>> That approach used by David in the Chapter 13 is a good approach.
>>>> Start by using these axioms / theorems as starting points (which
>>>> will apply under many different laws of physics
>>>> - whether those laws apply elsewhere - or whether we have not found additional, relevant laws of physics here).
>>>> They are a starting point where we build up explanations of the universe around us based on observation, criticism, etc.
>>
>>> To the best of my knowledge, David doesn't use that approach, nor does he say that he uses it.
>>> If you think he does, then page references and quotes would be useful.
>>
>> On page 336-337, David writes
>> "This is what Arrow did. He laid down five elementary axioms ...
>> ...
>> Arrow proved that the axioms that I have listed despite their reasonable appearance, logically inconsistent with each other.
>> ...
>> It seems to follow that a group of people jointly making decisions is necessarily irrational in one way or another."
>>
>> In this section, David has clearly followed the approach I suggested.

>No. The passage above about Arrow's theorem above is a deliberate example of specious reasoning. Note the word "seems" in the final sentence. As the book goes on to say, >Arrow's Theorem doesn't apply to creative decision-making processes that involve a search for good explanations. So although it may *seem* to follow from Arrow's Theorem that >all group decision making is irrational, it doesn't *actually* follow.
>
>Therefore, that passage is not an example of David actually using the axiomatic approach. At least, not as a means of drawing conclusions that he himself accepts.

I disagree.
David's approach in this Chapter follows a good approach. Starts with the axioms - looks at the theorems based on those axioms.
He then goes on to build up explanations using that as a starting point (p336).
He uses observation to see the relevancy of those axioms (for example on p338 "because in the case of elections the element of persuasion is central to the whole exercise").
He uses observation and criticism to come to conclusions (summarized on p352).

This is a very different approach that starting off with the idea that the pure maths should be ignored.

[Josh Jordan]

On Fri, Oct 26, 2012 at 1:34 AM, jon_o...@trendmicro.com
<jon_o...@trendmicro.com> wrote:
>
> On Thursday, 25 October 2012 6:34 PM Josh Jordan wrote:
>
>>> Let LUP(maths) be the set of problems which mathematics / computation theory says is undecidable
>>> Let LDP(laws physics) be the set of problems which is in LUP(maths) but which the laws of physics makes decidable
>>> So LDP(laws physics) is the set of problems where the laws of physics is making the difference.
>>> [snip]
>>> But with our current understanding of the laws of physics, LDP(laws physics) is empty.
>>> [snip]
>>> If we had an infinity hotel, then yes LDP(laws physics) would be non empty.
>
>> No. If we had an infinity hotel, then LUP(laws maths) -- in particular, the definition of "computable" -- would be different.
>> Therefore, LDP(laws physics) would still be empty.
>
> No.
> LUP(maths) is the Turing computable problems.

Is LUP(maths) defined in a way as to be independent of any particular
laws of physics, or is it meant to refer to the set of problems
defined to be "computable" by mathematicians living under a particular
set of laws of physics? I was taking it as the latter.

> If the laws of physics includes an infinity hotel, then this does not change the definitions of Turing machines
> (nor does it change the definitions of Lambda Calculus).

True, but it points out that there's nothing mathematically privileged
about those definitions, except for the fact that our particular laws
of physics make them natural for us.

> Therefore if we had an infinity hotel, then yes LDP(laws physics) would be non empty.

If a universe had a different laws of physics than ours, then
mathematicians living there would define "computable" differently than
we do, in just such a way as to make LDP(laws physics) empty.

> If the prime-pairs conjecture is not Turing computable, then (as David points out on p185),
> then infinity hotel would allow the prime-pairs conjecture to be in LDP(laws physics).

The definition of "computable" that is natural given the understanding
of physics in our universe would be naturally different than the one
chosen by mathematicians living under different laws of physics. Those
mathematicians could study the consequences of unusual (for them)
definitions of computation, such as Turing machines, and if they did,
they would reach the same conclusions as we have about the power of
Turing machines. But why would they regard those results as having any
bearing on the theory of computation that pertains to their laws of
physics?

[Elliot Temple]

On Oct 18, 2012, at 5:19 PM, jon_o...@trendmicro.com wrote:

> If this discussion group is going to take the view that decidability theory / pure mathematics is suspect - its claims are unjustified,
> then this seems inconsistent with Beginning of Infinity.
> If baseline pure mathematics should be questioned, then we should also question David's use of the no-go theorems
> and other derivations starting from axioms in Chapter 13 on choices and group decision making.

Actually, everything is open to being questioned. All our ideas are fallible. That's part of BoI. None of BoI's ideas are intended to be perfect or go unquestioned.

> I have read BoI and not read FoR.

If you're interested in Godel's incompleteness theorem the fallibility of mathematical proofs, you should *definitely* read FoR, it covers those. It's also great in general with lots of good stuff, of course (e.g. it covers epistemology, induction, MWI, virtual reality, solipsism and time travel.)

You also may enjoy the book Godel, Escher, Bach.

-- Elliot Temple
http://elliottemple.com/

[Elliot Temple]

On Oct 23, 2012, at 11:11 PM, jon_o...@trendmicro.com wrote:

> Of course scientific ideas have to be justified.

Would you be willing to keep an open mind and consider the ideas of David Deutsch and others here?

What you've said is equivalent to, "Of course BoI is false". And you've posted that to the BoI list. What's going on?


If you're willing to consider ideas contrary to those you think you know, one way we could begin is you could post your criticisms of David's arguments in BoI. You say David is completely wrong. What's the argument for that?

> I suspect / guess that we have different meanings of the word "justified".

This is an evasion of David's views. David meant what he said. It's bad to look for excuses to downplay and evade disagreement. It's bad to deny David has the views he says he has and advocates in his books. That's disrespectful and closed minded. This sort of evasion is a strategy that prevents learning from the ideas evaded because one never admits they exist or thinks about them.

David uses "justified" in the normal meaning. I'd guess everyone else here does too, or at least the native or fluent English speakers.

[Elliot Temple]

On Oct 25, 2012, at 10:34 PM, jon_o...@trendmicro.com wrote:

> This is a very different approach that starting off with the idea that the pure maths should be ignored.

No one here is advocating ignoring pure maths. That's a straw man that no one is saying.

-- Elliot Temple
http://curi.us/