Re: What is more primary than numbers?

[Bruce Kellett]

On Mon, Dec 17, 2018 at 5:59 PM Jason Resch  wrote:

> On Mon, Dec 17, 2018 at 12:03 AM Bruce Kellett 
> wrote:
>
>> On Mon, Dec 17, 2018 at 4:30 PM Jason Resch  wrote:
>>
>>> On Sun, Dec 16, 2018 at 9:39 PM Bruce Kellett 
>>> wrote:
>>>
 On Mon, Dec 17, 2018 at 1:50 PM Jason Resch 
 wrote:

> On Sun, Dec 16, 2018 at 7:21 PM Bruce Kellett 
> wrote:
>
>
>> Are you claiming that there is an objective arithmetical realm that
>> is independent of any set of axioms?
>>
>
> Yes. This is partly why Gödel's result was so shocking, and so
> important.
>
>
>> And our axiomatisations are attempts to provide a theory of this
>> realm? In which case any particular set of axioms might not be true of
>> "real" mathematics?
>>
>
> It will be either incomplete or inconsistent.
>
>
>
>> Sorry, but that is silly. The realm of integers is completely defined
>> by a set of simple axioms -- there is no arithmetic "reality" beyond 
>> this.
>>
>>
> The integers can be defined, but no axiomatic system can prove
> everything that happens to be true about them.  This fact is not commonly
> known and appreciated outside of some esoteric branches of mathematics, 
> but
> it is the case.
>

 All that this means is that theorems do not encapsulate all "truth".

>>>
>>> Where does truth come from, if not the formalism of the axioms?
>>>
>>
>> You are equivocating on the notion of "truth". You seem to be claiming
>> that "truth" is encapsulated in the axioms, and yet the axioms and the
>> given rules of inference do not encapsulate all "truth".
>>
>> I think I worded that badly.  What I mean is given that truth does not
> come from axioms (since they cannot encapsulate all of it), then where does
> it come from?  Does it have an independent, uncaused, transcendent
> existence?
>

I don't know what that would mean. I don't think the truth of arithmetical
statements comes from some underlying consistent model in which the axioms
are "true". How do you determine the truth of the Godel sentence in some
axiomatic system? Only by going to some more general system, not by
reference to some underlying model.


Do you agree that arithmetical truth has an existence independent of the
>>> axiomatic system?
>>>
>>
Since truth does not equal 'theorem of the system', there is a sense in
which this is true. But it does not mean that the truth of any
syntactically correct statement is independent of any axiom set.


>
>> I agree that there are true statements in arithmetic that are not
>> theorems in any particular axiomatic system. This does not mean that
>> arithmetic has an existence beyond its definition in terms of some set of
>> axioms. You cannot go from "true" to "exists", where "exists" means
>> something more than the existential quantifier over some set. Confusing the
>> existential quantifier with an ontology is a common mistake among some
>> classes of mathematicians.
>>
>
> I agree, let us ignore "exists" for now as I think it is distracting from
> the current question of whether "true statements are true" (independent of
> thinking about them, defining them, uttering them, etc.).
>

True statements are true by definition!



> What I am curious to know is how how many of these statements you agree
> with:
>
> "2+2 = 4" was true:
> 1. Before I was born
> 2. Before humans formalized axioms and found a proof of it
> 3. Before there were humans
> 4. Before there was any conscious life in this universe
> 5. As soon as there were 4 physical things to count
> 6. Before the big bang / before there were 4 physical things
>

"2+2=4" is a tautology, true because of the meanings of the terms involved.
So its truth is not independent of the formulation of the question and the
definition of the terms involved.

Bruce

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Re: What is more primary than numbers?

[Jason Resch]

On Mon, Dec 17, 2018 at 12:05 AM Brent Meeker  wrote:

>
>
> On 12/16/2018 9:36 PM, Jason Resch wrote:
>
>
>
> On Sun, Dec 16, 2018 at 10:22 PM Brent Meeker 
> wrote:
>
>>
>>
>> On 12/16/2018 4:39 PM, Jason Resch wrote:
>>
>>
>>
>> On Sun, Dec 16, 2018 at 5:53 PM Brent Meeker 
>> wrote:
>>
>>>
>>>
>>> On 12/16/2018 1:56 PM, Jason Resch wrote:
>>>
>>>
>>>
>>> On Sun, Dec 16, 2018 at 3:28 PM Brent Meeker 
>>> wrote:
>>>


 On 12/15/2018 10:24 PM, Jason Resch wrote:



 On Sat, Dec 15, 2018 at 11:35 PM Brent Meeker 
 wrote:

>
>
> On 12/15/2018 6:07 PM, Jason Resch wrote:
>
>
>
> On Sat, Dec 15, 2018 at 7:57 PM Brent Meeker 
> wrote:
>
>>
>>
>> On 12/15/2018 5:42 PM, Jason Resch wrote:
>>
>> hh, but diophantine equations only need integers, addition, and
>>> multiplication, and can define any computable function. Therefore the
>>> question of whether or not some diophantine equation has a solution can 
>>> be
>>> made equivalent to the question of whether some Turing machine halts.  
>>> So
>>> you face this problem of getting at all the truth once you can define
>>> integers, addition and multiplication.
>>>
>>>
>>> There's no surprise that you can't get at all true statements about
>>> a system  that is defined to be infinite.
>>>
>>
>> But you can always prove more true statements with a better system of
>> axioms.  So clearly the axioms are not the driving force behind truth.
>>
>>
>> And you can prove more false statements with a "better" system of
>> axioms...which was my original point.  So axioms are not a "force behind
>> truth"; they are a force behind what is provable.
>>
>>
> There are objectively better systems which prove nothing false, but
> allow you to prove more things than weaker systems of axioms.
>
>
> By that criterion an inconsistent system is the objectively best of
> all.
>
>
 The problem with an inconsistent system is that it does prove things
 that are false i.e. "not true".


> However we can never prove that the system doesn't prove anything
> false (within the theory itself).
>
>
> You're confusing mathematically consistency with not proving something
> false.
>

  They're related. A system that is inconsistent can prove a statement
 as well as its converse. Therefore it is proving things that are false.


 But a system that is consistent can also prove a statement that is
 false:

 axiom 1: Trump is a genius.
 axiom 2: Trump is stable.

 theorem: Trump is a stable genius.

>>>
>>> So how is this different from flawed physical theories?
>>>
>>>
>>> The difference is that mathematicians can't test their theories.
>>>
>>
>> Sure they can:  A set of axioms predicts a Diophantine equation has no
>> solutions.  We happen to find it does have a solution.  We can reject that
>> set of axioms.
>>
>>
>> Then the axioms must have also included enough to include Diophantine
>> equations (e.g. PA) so you have added axioms making the system inconsistent
>> and every proposition is a theorem.  The only test of the theory was that
>> it is inconsistent.
>>
>
> There is also soundness  which I
> think more accurately reflects my example above.
>
>
> "...a system is sound when all of its theorems are tautologies."  Which is
> to say it is true that the theorem follows from the axioms.  Not that it is
> true simpliciter.
>

How about this:

Arithmetic soundness[edit

]
If *T* is a theory whose objects of discourse can be interpreted as natural
numbers , we say *T* is
*arithmetically
sound* if all theorems of *T* are actually true about the standard
mathematical integers. For further information, see ω-consistent theory
.


Jason

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Re: What is more primary than numbers?

[Jason Resch]

On Mon, Dec 17, 2018 at 12:03 AM Bruce Kellett 
wrote:

> On Mon, Dec 17, 2018 at 4:30 PM Jason Resch  wrote:
>
>> On Sun, Dec 16, 2018 at 9:39 PM Bruce Kellett 
>> wrote:
>>
>>> On Mon, Dec 17, 2018 at 1:50 PM Jason Resch 
>>> wrote:
>>>
 On Sun, Dec 16, 2018 at 7:21 PM Bruce Kellett 
 wrote:


> Are you claiming that there is an objective arithmetical realm that is
> independent of any set of axioms?
>

 Yes. This is partly why Gödel's result was so shocking, and so
 important.


> And our axiomatisations are attempts to provide a theory of this
> realm? In which case any particular set of axioms might not be true of
> "real" mathematics?
>

 It will be either incomplete or inconsistent.



> Sorry, but that is silly. The realm of integers is completely defined
> by a set of simple axioms -- there is no arithmetic "reality" beyond this.
>
>
 The integers can be defined, but no axiomatic system can prove
 everything that happens to be true about them.  This fact is not commonly
 known and appreciated outside of some esoteric branches of mathematics, but
 it is the case.

>>>
>>> All that this means is that theorems do not encapsulate all "truth".
>>>
>>
>> Where does truth come from, if not the formalism of the axioms?
>>
>
> You are equivocating on the notion of "truth". You seem to be claiming
> that "truth" is encapsulated in the axioms, and yet the axioms and the
> given rules of inference do not encapsulate all "truth".
>
> I think I worded that badly.  What I mean is given that truth does not
come from axioms (since they cannot encapsulate all of it), then where does
it come from?  Does it have an independent, uncaused, transcendent
existence?


> Do you agree that arithmetical truth has an existence independent of the
>> axiomatic system?
>>
>
> I agree that there are true statements in arithmetic that are not theorems
> in any particular axiomatic system. This does not mean that arithmetic has
> an existence beyond its definition in terms of some set of axioms. You
> cannot go from "true" to "exists", where "exists" means something more than
> the existential quantifier over some set. Confusing the existential
> quantifier with an ontology is a common mistake among some classes of
> mathematicians.
>

I agree, let us ignore "exists" for now as I think it is distracting from
the current question of whether "true statements are true" (independent of
thinking about them, defining them, uttering them, etc.).

What I am curious to know is how how many of these statements you agree
with:

"2+2 = 4" was true:
1. Before I was born
2. Before humans formalized axioms and found a proof of it
3. Before there were humans
4. Before there was any conscious life in this universe
5. As soon as there were 4 physical things to count
6. Before the big bang / before there were 4 physical things


>
> There are syntactically correct statements in the system that are not
>>> theorems, and neither are their negation theorems.
>>>
>>
>> Yes.
>>
>>
>>> Godel's theorem merely shows that some of these statements may be true
>>> in a more general system.
>>>
>>
>> So isn't this like scientific theories attempting to better describe the
>> physical world, with ever more general and more powerful theories?
>>
>
> Except that physics is not an axiomatic system, and does not confuse
> theorems with truth. It is not useful to classify physical theories as
> 'true' or 'false',
>

Isn't this what professors do with physics tests? Ask there students to
prove something or determine what some physical law says should happen?
Then they grade an item as wrong if the answer given was "false" under the
working theory.


> even though this is often done in mistaken homage to Popper. The
> descriptions of the phenomena that physical theories give are either
> consistent with the data or not -- even adequate descriptions are not
> necessarily "true" in any sense.
>
>
Would you liken consistency with the data to soundness in a system of
axioms?


>
>
>> That does not mean that the integers are not completely defined by some
>>> simple axioms. It means no more than that 'truth' and 'theorem' are not
>>> synonyms.
>>>
>>>
>> I agree with this.
>>
>
> Good.
>
>
Jason

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Re: What is more primary than numbers?

[Jason Resch]

On Mon, Dec 17, 2018 at 12:00 AM Brent Meeker  wrote:

>
>
> On 12/16/2018 9:30 PM, Jason Resch wrote:
>
>
>
> On Sun, Dec 16, 2018 at 9:39 PM Bruce Kellett 
> wrote:
>
>> On Mon, Dec 17, 2018 at 1:50 PM Jason Resch  wrote:
>>
>>> On Sun, Dec 16, 2018 at 7:21 PM Bruce Kellett 
>>> wrote:
>>>
>>>
 Are you claiming that there is an objective arithmetical realm that is
 independent of any set of axioms?

>>>
>>> Yes. This is partly why Gödel's result was so shocking, and so important.
>>>
>>>
 And our axiomatisations are attempts to provide a theory of this realm?
 In which case any particular set of axioms might not be true of "real"
 mathematics?

>>>
>>> It will be either incomplete or inconsistent.
>>>
>>>
>>>
 Sorry, but that is silly. The realm of integers is completely defined
 by a set of simple axioms -- there is no arithmetic "reality" beyond this.


>>> The integers can be defined, but no axiomatic system can prove
>>> everything that happens to be true about them.  This fact is not commonly
>>> known and appreciated outside of some esoteric branches of mathematics, but
>>> it is the case.
>>>
>>
>> All that this means is that theorems do not encapsulate all "truth".
>>
>
> Where does truth come from, if not the formalism of the axioms?  Do you
> agree that arithmetical truth has an existence independent of the axiomatic
> system?
>
>
> No.  You are assuming that arithmetic exists apart from axioms that define
> it.
>

I am saying truth about the integers exists independently of any system of
axioms that are capable of defining the integers.


> There are true things about arithmetic that are not provable *within
> arithmetic*.
>

It's unclear what you mean by "within arithmetic".


> But that is not the same as being independent of the axioms.  Some axioms
> are necessary to define what is meant by arithmetic.
>

You need to define what you're talking about before you can talk about it.
But in any case, the axioms don't define arithmetical truth, which is my
only point.

If they don't, then formalism, nominalism, fictionalism, etc. all fall, and
what is left is platonism.

Jason

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Re: What is more primary than numbers?

[Brent Meeker]

On 12/16/2018 9:42 PM, Jason Resch wrote:



On Sun, Dec 16, 2018 at 10:27 PM Brent Meeker > wrote:




On 12/16/2018 4:43 PM, Jason Resch wrote:



On Sun, Dec 16, 2018 at 6:02 PM Brent Meeker
mailto:meeke...@verizon.net>> wrote:



On 12/16/2018 2:04 PM, Jason Resch wrote:



On Sun, Dec 16, 2018 at 4:01 PM Bruce Kellett
mailto:bhkellet...@gmail.com>> wrote:

On Mon, Dec 17, 2018 at 8:56 AM Jason Resch
mailto:jasonre...@gmail.com>> wrote:

On Sun, Dec 16, 2018 at 3:28 PM Brent Meeker
mailto:meeke...@verizon.net>>
wrote:


But a system that is consistent can also prove a
statement that is false:

axiom 1: Trump is a genius.
axiom 2: Trump is stable.

theorem: Trump is a stable genius.


So how is this different from flawed physical theories?


Physical theories do not claim to prove theorems - they
are not systems of axioms and theorems. Attempts to
recast physics in this form have always failed.


Physical theories claim to describe models of reality.  You
can have a fully consistent physical theory that
nevertheless fails to accurately describe the physical
world, or is an incomplete description of the physical
world.  Likewise, you can have an axiomatic system that is
consistent, but fails to accurately describe the integers,
or is less complete than we would like.


But it still has theorems.  And no matter what the theory is,
even if it describes the integers (another mathematical
abstraction), it will fail to describe other things.

ISTM that the usefulness of mathematics is that it's
identical with its theories...it's not intended to describe
something else.


A useful set of axioms (a mathematical theory, if you will) will
accurately describe arithmetical truth.  E.g., it will provide us
the means to determine the behavior of a large number of Turing
machines, or whether or not a given equation has a solution.  The
world of mathematical truth is what we are trying to describe. 
We want to know whether there is a biggest twin prime or not, for
example.  There either is or isn't a biggest twin prime.  Our
theories will either succeed or fail to include such truths as
theorems.


This is begging the question. You taking one piece of mathematics,
arithmetic, and using it as a theory describing another piece of
mathematics, Turing machines. And then you're calling a successful
description "true". But all you're showing is that one contains
the other.


I'm not following here.

Theorems are not "truths" except in the conditional sense that it
is true that they follow from the axioms and the rules of inference.


I agree a theorem is not the same as a truth. Truth is independent of 
some statement being provable in some system.


OK.

Truth is objective.  If a system of axioms is sound and consistent, 
then a theorem in that system is a truth.


No, c.f. Donald Trump.

But we can never be sure that system is sound and consistent (just 
like we can never know if our physical theories reflect the physical 
reality they attempt to capture).


But sometimes we can be sure that our theory does not reflect reality, 
even if it is sound and consistent.


Brent



Jason
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Re: What is more primary than numbers?

[Brent Meeker]

On 12/16/2018 9:36 PM, Jason Resch wrote:



On Sun, Dec 16, 2018 at 10:22 PM Brent Meeker > wrote:




On 12/16/2018 4:39 PM, Jason Resch wrote:



On Sun, Dec 16, 2018 at 5:53 PM Brent Meeker
mailto:meeke...@verizon.net>> wrote:



On 12/16/2018 1:56 PM, Jason Resch wrote:



On Sun, Dec 16, 2018 at 3:28 PM Brent Meeker
mailto:meeke...@verizon.net>> wrote:



On 12/15/2018 10:24 PM, Jason Resch wrote:



On Sat, Dec 15, 2018 at 11:35 PM Brent Meeker
mailto:meeke...@verizon.net>> wrote:



On 12/15/2018 6:07 PM, Jason Resch wrote:



On Sat, Dec 15, 2018 at 7:57 PM Brent Meeker
mailto:meeke...@verizon.net>> wrote:



On 12/15/2018 5:42 PM, Jason Resch wrote:



hh, but diophantine equations only need
integers, addition, and multiplication,
and can define any computable function.
Therefore the question of whether or not
some diophantine equation has a solution
can be made equivalent to the question
of whether some Turing machine halts. 
So you face this problem of getting at
all the truth once you can define
integers, addition and multiplication.


There's no surprise that you can't get at
all true statements about a system  that
is defined to be infinite.


But you can always prove more true statements
with a better system of axioms.  So clearly
the axioms are not the driving force behind
truth.



And you can prove more false statements with a
"better" system of axioms...which was my
original point.  So axioms are not a "force
behind truth"; they are a force behind what is
provable.


There are objectively better systems which prove
nothing false, but allow you to prove more things
than weaker systems of axioms.


By that criterion an inconsistent system is the
objectively best of all.


The problem with an inconsistent system is that it does
prove things that are false i.e. "not true".


However we can never prove that the system doesn't
prove anything false (within the theory itself).


You're confusing mathematically consistency with
not proving something false.


 They're related. A system that is inconsistent can
prove a statement as well as its converse. Therefore it
is proving things that are false.


But a system that is consistent can also prove a
statement that is false:

axiom 1: Trump is a genius.
axiom 2: Trump is stable.

theorem: Trump is a stable genius.


So how is this different from flawed physical theories?


The difference is that mathematicians can't test their theories.


Sure they can:  A set of axioms predicts a Diophantine equation
has no solutions.  We happen to find it does have a solution.  We
can reject that set of axioms.


Then the axioms must have also included enough to include
Diophantine equations (e.g. PA) so you have added axioms making
the system inconsistent and every proposition is a theorem.  The
only test of the theory was that it is inconsistent.


There is also soundness 
 which I think more 
accurately reflects my example above.


"...a system is sound when all of its theorems are tautologies." Which 
is to say it is true that the theorem follows from the axioms.  Not that 
it is true simpliciter.


Brent

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Re: What is more primary than numbers?

[Bruce Kellett]

On Mon, Dec 17, 2018 at 4:30 PM Jason Resch  wrote:

> On Sun, Dec 16, 2018 at 9:39 PM Bruce Kellett 
> wrote:
>
>> On Mon, Dec 17, 2018 at 1:50 PM Jason Resch  wrote:
>>
>>> On Sun, Dec 16, 2018 at 7:21 PM Bruce Kellett 
>>> wrote:
>>>
>>>
 Are you claiming that there is an objective arithmetical realm that is
 independent of any set of axioms?

>>>
>>> Yes. This is partly why Gödel's result was so shocking, and so important.
>>>
>>>
 And our axiomatisations are attempts to provide a theory of this realm?
 In which case any particular set of axioms might not be true of "real"
 mathematics?

>>>
>>> It will be either incomplete or inconsistent.
>>>
>>>
>>>
 Sorry, but that is silly. The realm of integers is completely defined
 by a set of simple axioms -- there is no arithmetic "reality" beyond this.


>>> The integers can be defined, but no axiomatic system can prove
>>> everything that happens to be true about them.  This fact is not commonly
>>> known and appreciated outside of some esoteric branches of mathematics, but
>>> it is the case.
>>>
>>
>> All that this means is that theorems do not encapsulate all "truth".
>>
>
> Where does truth come from, if not the formalism of the axioms?
>

You are equivocating on the notion of "truth". You seem to be claiming that
"truth" is encapsulated in the axioms, and yet the axioms and the given
rules of inference do not encapsulate all "truth".

Do you agree that arithmetical truth has an existence independent of the
> axiomatic system?
>

I agree that there are true statements in arithmetic that are not theorems
in any particular axiomatic system. This does not mean that arithmetic has
an existence beyond its definition in terms of some set of axioms. You
cannot go from "true" to "exists", where "exists" means something more than
the existential quantifier over some set. Confusing the existential
quantifier with an ontology is a common mistake among some classes of
mathematicians.

There are syntactically correct statements in the system that are not
>> theorems, and neither are their negation theorems.
>>
>
> Yes.
>
>
>> Godel's theorem merely shows that some of these statements may be true in
>> a more general system.
>>
>
> So isn't this like scientific theories attempting to better describe the
> physical world, with ever more general and more powerful theories?
>

Except that physics is not an axiomatic system, and does not confuse
theorems with truth. It is not useful to classify physical theories as
'true' or 'false', even though this is often done in mistaken homage to
Popper. The descriptions of the phenomena that physical theories give are
either consistent with the data or not -- even adequate descriptions are
not necessarily "true" in any sense.



> That does not mean that the integers are not completely defined by some
>> simple axioms. It means no more than that 'truth' and 'theorem' are not
>> synonyms.
>>
>>
> I agree with this.
>

Good.

Bruce

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Re: What is more primary than numbers?

[Brent Meeker]

On 12/16/2018 9:30 PM, Jason Resch wrote:



On Sun, Dec 16, 2018 at 9:39 PM Bruce Kellett > wrote:


On Mon, Dec 17, 2018 at 1:50 PM Jason Resch mailto:jasonre...@gmail.com>> wrote:

On Sun, Dec 16, 2018 at 7:21 PM Bruce Kellett
mailto:bhkellet...@gmail.com>> wrote:

Are you claiming that there is an objective arithmetical
realm that is independent of any set of axioms?


Yes. This is partly why Gödel's result was so shocking, and so
important.

And our axiomatisations are attempts to provide a theory
of this realm? In which case any particular set of axioms
might not be true of "real" mathematics?


It will be either incomplete or inconsistent.

Sorry, but that is silly. The realm of integers is
completely defined by a set of simple axioms -- there is
no arithmetic "reality" beyond this.


The integers can be defined, but no axiomatic system can prove
everything that happens to be true about them.  This fact is
not commonly known and appreciated outside of some esoteric
branches of mathematics, but it is the case.


All that this means is that theorems do not encapsulate all "truth".


Where does truth come from, if not the formalism of the axioms?  Do 
you agree that arithmetical truth has an existence independent of the 
axiomatic system?


No.  You are assuming that arithmetic exists apart from axioms that 
define it.  There are true things about arithmetic that are not provable 
/within arithmetic/.  But that is not the same as being independent of 
the axioms.  Some axioms are necessary to define what is meant by 
arithmetic.


Brent


There are syntactically correct statements in the system that are
not theorems, and neither are their negation theorems.


Yes.

Godel's theorem merely shows that some of these statements may be
true in a more general system.


So isn't this like scientific theories attempting to better describe 
the physical world, with ever more general and more powerful theories?


That does not mean that the integers are not completely defined by
some simple axioms. It means no more than that 'truth' and
'theorem' are not synonyms.


I agree with this.


Jason
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Re: What is more primary than numbers?

[Jason Resch]

On Sun, Dec 16, 2018 at 10:27 PM Brent Meeker  wrote:

>
>
> On 12/16/2018 4:43 PM, Jason Resch wrote:
>
>
>
> On Sun, Dec 16, 2018 at 6:02 PM Brent Meeker  wrote:
>
>>
>>
>> On 12/16/2018 2:04 PM, Jason Resch wrote:
>>
>>
>>
>> On Sun, Dec 16, 2018 at 4:01 PM Bruce Kellett 
>> wrote:
>>
>>> On Mon, Dec 17, 2018 at 8:56 AM Jason Resch 
>>> wrote:
>>>
 On Sun, Dec 16, 2018 at 3:28 PM Brent Meeker 
 wrote:

>
> But a system that is consistent can also prove a statement that is
> false:
>
> axiom 1: Trump is a genius.
> axiom 2: Trump is stable.
>
> theorem: Trump is a stable genius.
>

 So how is this different from flawed physical theories?

>>>
>>> Physical theories do not claim to prove theorems - they are not systems
>>> of axioms and theorems. Attempts to recast physics in this form have always
>>> failed.
>>>
>>>
>> Physical theories claim to describe models of reality.  You can have a
>> fully consistent physical theory that nevertheless fails to accurately
>> describe the physical world, or is an incomplete description of the
>> physical world.  Likewise, you can have an axiomatic system that is
>> consistent, but fails to accurately describe the integers, or is less
>> complete than we would like.
>>
>>
>> But it still has theorems.  And no matter what the theory is, even if it
>> describes the integers (another mathematical abstraction), it will fail to
>> describe other things.
>>
>> ISTM that the usefulness of mathematics is that it's identical with its
>> theories...it's not intended to describe something else.
>>
>
> A useful set of axioms (a mathematical theory, if you will) will
> accurately describe arithmetical truth.  E.g., it will provide us the means
> to determine the behavior of a large number of Turing machines, or whether
> or not a given equation has a solution.  The world of mathematical truth is
> what we are trying to describe.  We want to know whether there is a biggest
> twin prime or not, for example.  There either is or isn't a biggest twin
> prime.  Our theories will either succeed or fail to include such truths as
> theorems.
>
>
> This is begging the question. You taking one piece of mathematics,
> arithmetic, and using it as a theory describing another piece of
> mathematics, Turing machines.  And then you're calling a successful
> description "true". But all you're showing is that one contains the
> other.
>

I'm not following here.


> Theorems are not "truths" except in the conditional sense that it is true
> that they follow from the axioms and the rules of inference.
>

I agree a theorem is not the same as a truth. Truth is independent of some
statement being provable in some system. Truth is objective.  If a system
of axioms is sound and consistent, then a theorem in that system is a
truth. But we can never be sure that system is sound and consistent (just
like we can never know if our physical theories reflect the physical
reality they attempt to capture).

Jason

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Re: What is more primary than numbers?

[Jason Resch]

On Sun, Dec 16, 2018 at 10:22 PM Brent Meeker  wrote:

>
>
> On 12/16/2018 4:39 PM, Jason Resch wrote:
>
>
>
> On Sun, Dec 16, 2018 at 5:53 PM Brent Meeker  wrote:
>
>>
>>
>> On 12/16/2018 1:56 PM, Jason Resch wrote:
>>
>>
>>
>> On Sun, Dec 16, 2018 at 3:28 PM Brent Meeker 
>> wrote:
>>
>>>
>>>
>>> On 12/15/2018 10:24 PM, Jason Resch wrote:
>>>
>>>
>>>
>>> On Sat, Dec 15, 2018 at 11:35 PM Brent Meeker 
>>> wrote:
>>>


 On 12/15/2018 6:07 PM, Jason Resch wrote:



 On Sat, Dec 15, 2018 at 7:57 PM Brent Meeker 
 wrote:

>
>
> On 12/15/2018 5:42 PM, Jason Resch wrote:
>
> hh, but diophantine equations only need integers, addition, and
>> multiplication, and can define any computable function. Therefore the
>> question of whether or not some diophantine equation has a solution can 
>> be
>> made equivalent to the question of whether some Turing machine halts.  So
>> you face this problem of getting at all the truth once you can define
>> integers, addition and multiplication.
>>
>>
>> There's no surprise that you can't get at all true statements about a
>> system  that is defined to be infinite.
>>
>
> But you can always prove more true statements with a better system of
> axioms.  So clearly the axioms are not the driving force behind truth.
>
>
> And you can prove more false statements with a "better" system of
> axioms...which was my original point.  So axioms are not a "force behind
> truth"; they are a force behind what is provable.
>
>
 There are objectively better systems which prove nothing false, but
 allow you to prove more things than weaker systems of axioms.


 By that criterion an inconsistent system is the objectively best of all.


>>> The problem with an inconsistent system is that it does prove things
>>> that are false i.e. "not true".
>>>
>>>
 However we can never prove that the system doesn't prove anything false
 (within the theory itself).


 You're confusing mathematically consistency with not proving something
 false.

>>>
>>>  They're related. A system that is inconsistent can prove a statement as
>>> well as its converse. Therefore it is proving things that are false.
>>>
>>>
>>> But a system that is consistent can also prove a statement that is false:
>>>
>>> axiom 1: Trump is a genius.
>>> axiom 2: Trump is stable.
>>>
>>> theorem: Trump is a stable genius.
>>>
>>
>> So how is this different from flawed physical theories?
>>
>>
>> The difference is that mathematicians can't test their theories.
>>
>
> Sure they can:  A set of axioms predicts a Diophantine equation has no
> solutions.  We happen to find it does have a solution.  We can reject that
> set of axioms.
>
>
> Then the axioms must have also included enough to include Diophantine
> equations (e.g. PA) so you have added axioms making the system inconsistent
> and every proposition is a theorem.  The only test of the theory was that
> it is inconsistent.
>

There is also soundness  which I
think more accurately reflects my example above.

Jason

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Re: What is more primary than numbers?

[agrayson2000]

On Monday, December 17, 2018 at 1:21:15 AM UTC, Bruce wrote:
>
> On Mon, Dec 17, 2018 at 11:36 AM Jason Resch  > wrote:
>
>> On Sun, Dec 16, 2018 at 4:14 PM Bruce Kellett > > wrote:
>>
>>> On Mon, Dec 17, 2018 at 9:04 AM Jason Resch >> > wrote:
>>>
 On Sun, Dec 16, 2018 at 4:01 PM Bruce Kellett >>> > wrote:

> On Mon, Dec 17, 2018 at 8:56 AM Jason Resch  > wrote:
>
>> On Sun, Dec 16, 2018 at 3:28 PM Brent Meeker > > wrote:
>>
>>>
>>> But a system that is consistent can also prove a statement that is 
>>> false:
>>>
>>> axiom 1: Trump is a genius.
>>> axiom 2: Trump is stable.
>>>
>>> theorem: Trump is a stable genius.
>>>
>>
>> So how is this different from flawed physical theories?
>>
>
> Physical theories do not claim to prove theorems - they are not 
> systems of axioms and theorems. Attempts to recast physics in this form 
> have always failed.
>
>
 Physical theories claim to describe models of reality.

>>>
>>> Physical theories are models of reality -- using the word "model" in the 
>>> physicists sense.
>>>  
>>>
 You can have a fully consistent physical theory that nevertheless fails 
 to accurately describe the physical world,

>>>
>>> Like Brent's example of an axiomatic description of Trump..
>>>  
>>>
 or is an incomplete description of the physical world.  Likewise, you 
 can have an axiomatic system that is consistent, but fails to accurately 
 describe the integers, or is less complete than we would like.

>>>
>>> Axiomatic system are always going to fail to capture everything we would 
>>> like to capture about any domain. That is why attempted axiomatisation of 
>>> physics have been rather unsuccessful.
>>>  
>>>
 It is a completely analogous situation. If you hold the physical 
 reality is real because we can study it objectively and refine our 
 understanding of it through observations,

>>>
>>> That is not "why" I hold the physical world to be real. I take the 
>>> physical world to be real because that is the definition of reality.
>>>
>>
>> There is no evidence that physics reality marks the end of our ability to 
>> explain anything deeper.
>>
>
> And there is no evidence that any deeper explanation is possible.
>

*A deeper explanation is certainly possible. I don't see why you reject it 
out of hand. OTOH, the issue at hand is whether arithmetic is that deeper 
explanation. Doubtful IMO. AG*

Let's face it, you could make such a claim about any theory -- there is no 
> evidence that there is not some deeper explanation -- unless, that is, your 
> theory does not account for all the facts. Physics itself is not a theory. 
> We have theories about physical phenomena that are more or less successful, 
> but the theories are not the physical reality.
>  
>
>>  
>>
>>> then the same would hold for the mathematical reality.

>>>
>>> No, mathematical "reality" (note the scare quotes) is a derived realm, 
>>> entirely dependent on the set of axioms chosen in any instance. So it is 
>>> not in any way analogous to physics.
>>>
>>>
>> Did you miss my earlier posts to Brent on this?  The integers and their 
>> relations are not modeled by any axiomatic system, they transcend the 
>> axioms and therefore we must conclude have a reality independent from our 
>> attempts to model them.
>>
>
> It is interesting, then, that Bruno is very proud of the fact that 
> arithmetic depends only on a small set of axioms, or even just on the 
> properties of a pair of combinators. Are you claiming that there is an 
> objective arithmetical realm that is independent of any set of axioms? And 
> our axiomatisations are attempts to provide a theory of this realm? In 
> which case any particular set of axioms might not be true of "real" 
> mathematics?
>
> Sorry, but that is silly. The realm of integers is completely defined by a 
> set of simple axioms -- there is no arithmetic "reality" beyond this.
>
> Bruce
>

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Re: What is more primary than numbers?

[Jason Resch]

On Sun, Dec 16, 2018 at 9:39 PM Bruce Kellett  wrote:

> On Mon, Dec 17, 2018 at 1:50 PM Jason Resch  wrote:
>
>> On Sun, Dec 16, 2018 at 7:21 PM Bruce Kellett 
>> wrote:
>>
>>
>>> Are you claiming that there is an objective arithmetical realm that is
>>> independent of any set of axioms?
>>>
>>
>> Yes. This is partly why Gödel's result was so shocking, and so important.
>>
>>
>>> And our axiomatisations are attempts to provide a theory of this realm?
>>> In which case any particular set of axioms might not be true of "real"
>>> mathematics?
>>>
>>
>> It will be either incomplete or inconsistent.
>>
>>
>>
>>> Sorry, but that is silly. The realm of integers is completely defined by
>>> a set of simple axioms -- there is no arithmetic "reality" beyond this.
>>>
>>>
>> The integers can be defined, but no axiomatic system can prove everything
>> that happens to be true about them.  This fact is not commonly known and
>> appreciated outside of some esoteric branches of mathematics, but it is the
>> case.
>>
>
> All that this means is that theorems do not encapsulate all "truth".
>

Where does truth come from, if not the formalism of the axioms?  Do you
agree that arithmetical truth has an existence independent of the axiomatic
system?


> There are syntactically correct statements in the system that are not
> theorems, and neither are their negation theorems.
>

Yes.


> Godel's theorem merely shows that some of these statements may be true in
> a more general system.
>

So isn't this like scientific theories attempting to better describe the
physical world, with ever more general and more powerful theories?


> That does not mean that the integers are not completely defined by some
> simple axioms. It means no more than that 'truth' and 'theorem' are not
> synonyms.
>
>
I agree with this.


Jason

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Re: What is more primary than numbers?

[Brent Meeker]

On 12/16/2018 4:43 PM, Jason Resch wrote:



On Sun, Dec 16, 2018 at 6:02 PM Brent Meeker > wrote:




On 12/16/2018 2:04 PM, Jason Resch wrote:



On Sun, Dec 16, 2018 at 4:01 PM Bruce Kellett
mailto:bhkellet...@gmail.com>> wrote:

On Mon, Dec 17, 2018 at 8:56 AM Jason Resch
mailto:jasonre...@gmail.com>> wrote:

On Sun, Dec 16, 2018 at 3:28 PM Brent Meeker
mailto:meeke...@verizon.net>> wrote:


But a system that is consistent can also prove a
statement that is false:

axiom 1: Trump is a genius.
axiom 2: Trump is stable.

theorem: Trump is a stable genius.


So how is this different from flawed physical theories?


Physical theories do not claim to prove theorems - they are
not systems of axioms and theorems. Attempts to recast
physics in this form have always failed.


Physical theories claim to describe models of reality.  You can
have a fully consistent physical theory that nevertheless fails
to accurately describe the physical world, or is an incomplete
description of the physical world.  Likewise, you can have an
axiomatic system that is consistent, but fails to accurately
describe the integers, or is less complete than we would like.


But it still has theorems.  And no matter what the theory is, even
if it describes the integers (another mathematical abstraction),
it will fail to describe other things.

ISTM that the usefulness of mathematics is that it's identical
with its theories...it's not intended to describe something else.


A useful set of axioms (a mathematical theory, if you will) will 
accurately describe arithmetical truth.  E.g., it will provide us the 
means to determine the behavior of a large number of Turing machines, 
or whether or not a given equation has a solution.  The world of 
mathematical truth is what we are trying to describe.  We want to know 
whether there is a biggest twin prime or not, for example.  There 
either is or isn't a biggest twin prime.  Our theories will either 
succeed or fail to include such truths as theorems.


This is begging the question. You taking one piece of mathematics, 
arithmetic, and using it as a theory describing another piece of 
mathematics, Turing machines.  And then you're calling a successful 
description "true". But all you're showing is that one contains the 
other.   Theorems are not "truths" except in the conditional sense that 
it is true that they follow from the axioms and the rules of inference.


Brent




Jason
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Re: What is more primary than numbers?

[Brent Meeker]

On 12/16/2018 4:39 PM, Jason Resch wrote:



On Sun, Dec 16, 2018 at 5:53 PM Brent Meeker > wrote:




On 12/16/2018 1:56 PM, Jason Resch wrote:



On Sun, Dec 16, 2018 at 3:28 PM Brent Meeker
mailto:meeke...@verizon.net>> wrote:



On 12/15/2018 10:24 PM, Jason Resch wrote:



On Sat, Dec 15, 2018 at 11:35 PM Brent Meeker
mailto:meeke...@verizon.net>> wrote:



On 12/15/2018 6:07 PM, Jason Resch wrote:



On Sat, Dec 15, 2018 at 7:57 PM Brent Meeker
mailto:meeke...@verizon.net>> wrote:



On 12/15/2018 5:42 PM, Jason Resch wrote:



hh, but diophantine equations only need
integers, addition, and multiplication, and
can define any computable function. Therefore
the question of whether or not some
diophantine equation has a solution can be
made equivalent to the question of whether
some Turing machine halts.  So you face this
problem of getting at all the truth once you
can define integers, addition and multiplication.


There's no surprise that you can't get at all
true statements about a system  that is
defined to be infinite.


But you can always prove more true statements with
a better system of axioms.  So clearly the axioms
are not the driving force behind truth.



And you can prove more false statements with a
"better" system of axioms...which was my original
point.  So axioms are not a "force behind truth";
they are a force behind what is provable.


There are objectively better systems which prove
nothing false, but allow you to prove more things than
weaker systems of axioms.


By that criterion an inconsistent system is the
objectively best of all.


The problem with an inconsistent system is that it does
prove things that are false i.e. "not true".


However we can never prove that the system doesn't
prove anything false (within the theory itself).


You're confusing mathematically consistency with not
proving something false.


 They're related. A system that is inconsistent can prove a
statement as well as its converse. Therefore it is proving
things that are false.


But a system that is consistent can also prove a statement
that is false:

axiom 1: Trump is a genius.
axiom 2: Trump is stable.

theorem: Trump is a stable genius.


So how is this different from flawed physical theories?


The difference is that mathematicians can't test their theories.


Sure they can:  A set of axioms predicts a Diophantine equation has no 
solutions.  We happen to find it does have a solution.  We can reject 
that set of axioms.


Then the axioms must have also included enough to include Diophantine 
equations (e.g. PA) so you have added axioms making the system 
inconsistent and every proposition is a theorem.  The only test of the 
theory was that it is inconsistent.


Brent

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Re: What is more primary than numbers?

[Bruce Kellett]

On Mon, Dec 17, 2018 at 1:50 PM Jason Resch  wrote:

> On Sun, Dec 16, 2018 at 7:21 PM Bruce Kellett 
> wrote:
>
>
>> Are you claiming that there is an objective arithmetical realm that is
>> independent of any set of axioms?
>>
>
> Yes. This is partly why Gödel's result was so shocking, and so important.
>
>
>> And our axiomatisations are attempts to provide a theory of this realm?
>> In which case any particular set of axioms might not be true of "real"
>> mathematics?
>>
>
> It will be either incomplete or inconsistent.
>
>
>
>> Sorry, but that is silly. The realm of integers is completely defined by
>> a set of simple axioms -- there is no arithmetic "reality" beyond this.
>>
>>
> The integers can be defined, but no axiomatic system can prove everything
> that happens to be true about them.  This fact is not commonly known and
> appreciated outside of some esoteric branches of mathematics, but it is the
> case.
>

All that this means is that theorems do not encapsulate all "truth". There
are syntactically correct statements in the system that are not theorems,
and neither are their negation theorems. Godel's theorem merely shows that
some of these statements may be true in a more general system. That does
not mean that the integers are not completely defined by some simple
axioms. It means no more than that 'truth' and 'theorem' are not synonyms.

Bruce



> For example:
> https://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorems
>
> "*Gödel's incompleteness theorems* are two theorems
>  of mathematical logic
>  that demonstrate the
> inherent limitations of every formal axiomatic system
>  capable of modelling
> basic arithmetic . These
> results, published by Kurt Gödel
>  in 1931, are important
> both in mathematical logic and in the philosophy of mathematics
> . The theorems
> are widely, but not universally, interpreted as showing that Hilbert's
> program  to find a
> complete and consistent set of axioms
>  for all mathematics
>  is impossible."
>
>
> And
> https://en.wikipedia.org/wiki/Halting_problem#G%C3%B6del's_incompleteness_theorems
>
> "Since we know that there cannot be such an algorithm, it follows that
> the assumption that there is a consistent and complete axiomatization of
> all true first-order logic statements about natural numbers must be false."
>
>
> Jason
>

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Re: What is more primary than numbers?

[Jason Resch]

On Sun, Dec 16, 2018 at 7:21 PM Bruce Kellett  wrote:

> On Mon, Dec 17, 2018 at 11:36 AM Jason Resch  wrote:
>
>> On Sun, Dec 16, 2018 at 4:14 PM Bruce Kellett 
>> wrote:
>>
>>> On Mon, Dec 17, 2018 at 9:04 AM Jason Resch 
>>> wrote:
>>>
 On Sun, Dec 16, 2018 at 4:01 PM Bruce Kellett 
 wrote:

> On Mon, Dec 17, 2018 at 8:56 AM Jason Resch 
> wrote:
>
>> On Sun, Dec 16, 2018 at 3:28 PM Brent Meeker 
>> wrote:
>>
>>>
>>> But a system that is consistent can also prove a statement that is
>>> false:
>>>
>>> axiom 1: Trump is a genius.
>>> axiom 2: Trump is stable.
>>>
>>> theorem: Trump is a stable genius.
>>>
>>
>> So how is this different from flawed physical theories?
>>
>
> Physical theories do not claim to prove theorems - they are not
> systems of axioms and theorems. Attempts to recast physics in this form
> have always failed.
>
>
 Physical theories claim to describe models of reality.

>>>
>>> Physical theories are models of reality -- using the word "model" in the
>>> physicists sense.
>>>
>>>
 You can have a fully consistent physical theory that nevertheless fails
 to accurately describe the physical world,

>>>
>>> Like Brent's example of an axiomatic description of Trump..
>>>
>>>
 or is an incomplete description of the physical world.  Likewise, you
 can have an axiomatic system that is consistent, but fails to accurately
 describe the integers, or is less complete than we would like.

>>>
>>> Axiomatic system are always going to fail to capture everything we would
>>> like to capture about any domain. That is why attempted axiomatisation of
>>> physics have been rather unsuccessful.
>>>
>>>
 It is a completely analogous situation. If you hold the physical
 reality is real because we can study it objectively and refine our
 understanding of it through observations,

>>>
>>> That is not "why" I hold the physical world to be real. I take the
>>> physical world to be real because that is the definition of reality.
>>>
>>
>> There is no evidence that physics reality marks the end of our ability to
>> explain anything deeper.
>>
>
> And there is no evidence that any deeper explanation is possible. Let's
> face it, you could make such a claim about any theory -- there is no
> evidence that there is not some deeper explanation -- unless, that is, your
> theory does not account for all the facts. Physics itself is not a theory.
> We have theories about physical phenomena that are more or less successful,
> but the theories are not the physical reality.
>

Admittedly then your believe that physics is not derivative from anything
more fundamental is a quasi religious belief--it's held without any
evidence for or against (in your view).

On the other hand, there is evidence that physics is derived from more
fundamental structures.  But you reject them. Why?


>
>
>>
>>
>>> then the same would hold for the mathematical reality.

>>>
>>> No, mathematical "reality" (note the scare quotes) is a derived realm,
>>> entirely dependent on the set of axioms chosen in any instance. So it is
>>> not in any way analogous to physics.
>>>
>>>
>> Did you miss my earlier posts to Brent on this?  The integers and their
>> relations are not modeled by any axiomatic system, they transcend the
>> axioms and therefore we must conclude have a reality independent from our
>> attempts to model them.
>>
>
> It is interesting, then, that Bruno is very proud of the fact that
> arithmetic depends only on a small set of axioms, or even just on the
> properties of a pair of combinators.
>

A simple set of axioms allows us to define the Integers as well as
computation, but those axioms can only scratch the surface regarding all
the truth about the integers and their relations.


> Are you claiming that there is an objective arithmetical realm that is
> independent of any set of axioms?
>

Yes. This is partly why Gödel's result was so shocking, and so important.


> And our axiomatisations are attempts to provide a theory of this realm? In
> which case any particular set of axioms might not be true of "real"
> mathematics?
>

It will be either incomplete or inconsistent.



>
> Sorry, but that is silly. The realm of integers is completely defined by a
> set of simple axioms -- there is no arithmetic "reality" beyond this.
>
>
The integers can be defined, but no axiomatic system can prove everything
that happens to be true about them.  This fact is not commonly known and
appreciated outside of some esoteric branches of mathematics, but it is the
case.

For example:
https://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorems

"*Gödel's incompleteness theorems* are two theorems
 of mathematical logic
 that demonstrate the
inherent limitations of every formal axiomatic system

Re: What is more primary than numbers?

[Bruce Kellett]

On Mon, Dec 17, 2018 at 11:36 AM Jason Resch  wrote:

> On Sun, Dec 16, 2018 at 4:14 PM Bruce Kellett 
> wrote:
>
>> On Mon, Dec 17, 2018 at 9:04 AM Jason Resch  wrote:
>>
>>> On Sun, Dec 16, 2018 at 4:01 PM Bruce Kellett 
>>> wrote:
>>>
 On Mon, Dec 17, 2018 at 8:56 AM Jason Resch 
 wrote:

> On Sun, Dec 16, 2018 at 3:28 PM Brent Meeker 
> wrote:
>
>>
>> But a system that is consistent can also prove a statement that is
>> false:
>>
>> axiom 1: Trump is a genius.
>> axiom 2: Trump is stable.
>>
>> theorem: Trump is a stable genius.
>>
>
> So how is this different from flawed physical theories?
>

 Physical theories do not claim to prove theorems - they are not systems
 of axioms and theorems. Attempts to recast physics in this form have always
 failed.


>>> Physical theories claim to describe models of reality.
>>>
>>
>> Physical theories are models of reality -- using the word "model" in the
>> physicists sense.
>>
>>
>>> You can have a fully consistent physical theory that nevertheless fails
>>> to accurately describe the physical world,
>>>
>>
>> Like Brent's example of an axiomatic description of Trump..
>>
>>
>>> or is an incomplete description of the physical world.  Likewise, you
>>> can have an axiomatic system that is consistent, but fails to accurately
>>> describe the integers, or is less complete than we would like.
>>>
>>
>> Axiomatic system are always going to fail to capture everything we would
>> like to capture about any domain. That is why attempted axiomatisation of
>> physics have been rather unsuccessful.
>>
>>
>>> It is a completely analogous situation. If you hold the physical reality
>>> is real because we can study it objectively and refine our understanding of
>>> it through observations,
>>>
>>
>> That is not "why" I hold the physical world to be real. I take the
>> physical world to be real because that is the definition of reality.
>>
>
> There is no evidence that physics reality marks the end of our ability to
> explain anything deeper.
>

And there is no evidence that any deeper explanation is possible. Let's
face it, you could make such a claim about any theory -- there is no
evidence that there is not some deeper explanation -- unless, that is, your
theory does not account for all the facts. Physics itself is not a theory.
We have theories about physical phenomena that are more or less successful,
but the theories are not the physical reality.


>
>
>> then the same would hold for the mathematical reality.
>>>
>>
>> No, mathematical "reality" (note the scare quotes) is a derived realm,
>> entirely dependent on the set of axioms chosen in any instance. So it is
>> not in any way analogous to physics.
>>
>>
> Did you miss my earlier posts to Brent on this?  The integers and their
> relations are not modeled by any axiomatic system, they transcend the
> axioms and therefore we must conclude have a reality independent from our
> attempts to model them.
>

It is interesting, then, that Bruno is very proud of the fact that
arithmetic depends only on a small set of axioms, or even just on the
properties of a pair of combinators. Are you claiming that there is an
objective arithmetical realm that is independent of any set of axioms? And
our axiomatisations are attempts to provide a theory of this realm? In
which case any particular set of axioms might not be true of "real"
mathematics?

Sorry, but that is silly. The realm of integers is completely defined by a
set of simple axioms -- there is no arithmetic "reality" beyond this.

Bruce

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Re: What is more primary than numbers?

[agrayson2000]

On Sunday, December 16, 2018 at 10:01:18 PM UTC, Bruce wrote:
>
> On Mon, Dec 17, 2018 at 8:56 AM Jason Resch  > wrote:
>
>> On Sun, Dec 16, 2018 at 3:28 PM Brent Meeker > > wrote:
>>
>>>
>>> But a system that is consistent can also prove a statement that is false:
>>>
>>> axiom 1: Trump is a genius.
>>> axiom 2: Trump is stable.
>>>
>>> theorem: Trump is a stable genius.
>>>
>>
>> So how is this different from flawed physical theories?
>>
>
> Physical theories do not claim to prove theorems - they are not systems of 
> axioms and theorems. Attempts to recast physics in this form have always 
> failed.
>
> Bruce
>

QM can be interpreted as a system of axioms or postulates, and the HUP can 
be interpreted as a theorem, or consequence of those axioms or postulates. 
AG 

>
>  
>

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Re: What is more primary than numbers?

[Jason Resch]

On Sun, Dec 16, 2018 at 6:02 PM Brent Meeker  wrote:

>
>
> On 12/16/2018 2:04 PM, Jason Resch wrote:
>
>
>
> On Sun, Dec 16, 2018 at 4:01 PM Bruce Kellett 
> wrote:
>
>> On Mon, Dec 17, 2018 at 8:56 AM Jason Resch  wrote:
>>
>>> On Sun, Dec 16, 2018 at 3:28 PM Brent Meeker 
>>> wrote:
>>>

 But a system that is consistent can also prove a statement that is
 false:

 axiom 1: Trump is a genius.
 axiom 2: Trump is stable.

 theorem: Trump is a stable genius.

>>>
>>> So how is this different from flawed physical theories?
>>>
>>
>> Physical theories do not claim to prove theorems - they are not systems
>> of axioms and theorems. Attempts to recast physics in this form have always
>> failed.
>>
>>
> Physical theories claim to describe models of reality.  You can have a
> fully consistent physical theory that nevertheless fails to accurately
> describe the physical world, or is an incomplete description of the
> physical world.  Likewise, you can have an axiomatic system that is
> consistent, but fails to accurately describe the integers, or is less
> complete than we would like.
>
>
> But it still has theorems.  And no matter what the theory is, even if it
> describes the integers (another mathematical abstraction), it will fail to
> describe other things.
>
> ISTM that the usefulness of mathematics is that it's identical with its
> theories...it's not intended to describe something else.
>

A useful set of axioms (a mathematical theory, if you will) will accurately
describe arithmetical truth.  E.g., it will provide us the means to
determine the behavior of a large number of Turing machines, or whether or
not a given equation has a solution.  The world of mathematical truth is
what we are trying to describe.  We want to know whether there is a biggest
twin prime or not, for example.  There either is or isn't a biggest twin
prime.  Our theories will either succeed or fail to include such truths as
theorems.

Jason

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Re: What is more primary than numbers?

[Jason Resch]

On Sun, Dec 16, 2018 at 5:53 PM Brent Meeker  wrote:

>
>
> On 12/16/2018 1:56 PM, Jason Resch wrote:
>
>
>
> On Sun, Dec 16, 2018 at 3:28 PM Brent Meeker  wrote:
>
>>
>>
>> On 12/15/2018 10:24 PM, Jason Resch wrote:
>>
>>
>>
>> On Sat, Dec 15, 2018 at 11:35 PM Brent Meeker 
>> wrote:
>>
>>>
>>>
>>> On 12/15/2018 6:07 PM, Jason Resch wrote:
>>>
>>>
>>>
>>> On Sat, Dec 15, 2018 at 7:57 PM Brent Meeker 
>>> wrote:
>>>


 On 12/15/2018 5:42 PM, Jason Resch wrote:

 hh, but diophantine equations only need integers, addition, and
> multiplication, and can define any computable function. Therefore the
> question of whether or not some diophantine equation has a solution can be
> made equivalent to the question of whether some Turing machine halts.  So
> you face this problem of getting at all the truth once you can define
> integers, addition and multiplication.
>
>
> There's no surprise that you can't get at all true statements about a
> system  that is defined to be infinite.
>

 But you can always prove more true statements with a better system of
 axioms.  So clearly the axioms are not the driving force behind truth.


 And you can prove more false statements with a "better" system of
 axioms...which was my original point.  So axioms are not a "force behind
 truth"; they are a force behind what is provable.


>>> There are objectively better systems which prove nothing false, but
>>> allow you to prove more things than weaker systems of axioms.
>>>
>>>
>>> By that criterion an inconsistent system is the objectively best of all.
>>>
>>>
>> The problem with an inconsistent system is that it does prove things that
>> are false i.e. "not true".
>>
>>
>>> However we can never prove that the system doesn't prove anything false
>>> (within the theory itself).
>>>
>>>
>>> You're confusing mathematically consistency with not proving something
>>> false.
>>>
>>
>>  They're related. A system that is inconsistent can prove a statement as
>> well as its converse. Therefore it is proving things that are false.
>>
>>
>> But a system that is consistent can also prove a statement that is false:
>>
>> axiom 1: Trump is a genius.
>> axiom 2: Trump is stable.
>>
>> theorem: Trump is a stable genius.
>>
>
> So how is this different from flawed physical theories?
>
>
> The difference is that mathematicians can't test their theories.
>

Sure they can:  A set of axioms predicts a Diophantine equation has no
solutions.  We happen to find it does have a solution.  We can reject that
set of axioms.

Jason

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Re: What is more primary than numbers?

[Jason Resch]

On Sun, Dec 16, 2018 at 4:14 PM Bruce Kellett  wrote:

> On Mon, Dec 17, 2018 at 9:04 AM Jason Resch  wrote:
>
>> On Sun, Dec 16, 2018 at 4:01 PM Bruce Kellett 
>> wrote:
>>
>>> On Mon, Dec 17, 2018 at 8:56 AM Jason Resch 
>>> wrote:
>>>
 On Sun, Dec 16, 2018 at 3:28 PM Brent Meeker 
 wrote:

>
> But a system that is consistent can also prove a statement that is
> false:
>
> axiom 1: Trump is a genius.
> axiom 2: Trump is stable.
>
> theorem: Trump is a stable genius.
>

 So how is this different from flawed physical theories?

>>>
>>> Physical theories do not claim to prove theorems - they are not systems
>>> of axioms and theorems. Attempts to recast physics in this form have always
>>> failed.
>>>
>>>
>> Physical theories claim to describe models of reality.
>>
>
> Physical theories are models of reality -- using the word "model" in the
> physicists sense.
>
>
>> You can have a fully consistent physical theory that nevertheless fails
>> to accurately describe the physical world,
>>
>
> Like Brent's example of an axiomatic description of Trump..
>
>
>> or is an incomplete description of the physical world.  Likewise, you can
>> have an axiomatic system that is consistent, but fails to accurately
>> describe the integers, or is less complete than we would like.
>>
>
> Axiomatic system are always going to fail to capture everything we would
> like to capture about any domain. That is why attempted axiomatisation of
> physics have been rather unsuccessful.
>
>
>> It is a completely analogous situation. If you hold the physical reality
>> is real because we can study it objectively and refine our understanding of
>> it through observations,
>>
>
> That is not "why" I hold the physical world to be real. I take the
> physical world to be real because that is the definition of reality.
>

There is no evidence that physics reality marks the end of our ability to
explain anything deeper.


>
>
>> then the same would hold for the mathematical reality.
>>
>
> No, mathematical "reality" (note the scare quotes) is a derived realm,
> entirely dependent on the set of axioms chosen in any instance. So it is
> not in any way analogous to physics.
>
>
Did you miss my earlier posts to Brent on this?  The integers and their
relations are not modeled by any axiomatic system, they transcend the
axioms and therefore we must conclude have a reality independent from our
attempts to model them.

Jason

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Re: What is more primary than numbers?

[Brent Meeker]

On 12/16/2018 2:04 PM, Jason Resch wrote:



On Sun, Dec 16, 2018 at 4:01 PM Bruce Kellett > wrote:


On Mon, Dec 17, 2018 at 8:56 AM Jason Resch mailto:jasonre...@gmail.com>> wrote:

On Sun, Dec 16, 2018 at 3:28 PM Brent Meeker
mailto:meeke...@verizon.net>> wrote:


But a system that is consistent can also prove a statement
that is false:

axiom 1: Trump is a genius.
axiom 2: Trump is stable.

theorem: Trump is a stable genius.


So how is this different from flawed physical theories?


Physical theories do not claim to prove theorems - they are not
systems of axioms and theorems. Attempts to recast physics in this
form have always failed.


Physical theories claim to describe models of reality. You can have a 
fully consistent physical theory that nevertheless fails to accurately 
describe the physical world, or is an incomplete description of the 
physical world.  Likewise, you can have an axiomatic system that is 
consistent, but fails to accurately describe the integers, or is less 
complete than we would like.


But it still has theorems.  And no matter what the theory is, even if it 
describes the integers (another mathematical abstraction), it will fail 
to describe other things.


ISTM that the usefulness of mathematics is that it's identical with its 
theories...it's not intended to describe something else.


Brentent

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Re: What is more primary than numbers?

[Brent Meeker]

On 12/16/2018 1:56 PM, Jason Resch wrote:



On Sun, Dec 16, 2018 at 3:28 PM Brent Meeker > wrote:




On 12/15/2018 10:24 PM, Jason Resch wrote:



On Sat, Dec 15, 2018 at 11:35 PM Brent Meeker
mailto:meeke...@verizon.net>> wrote:



On 12/15/2018 6:07 PM, Jason Resch wrote:



On Sat, Dec 15, 2018 at 7:57 PM Brent Meeker
mailto:meeke...@verizon.net>> wrote:



On 12/15/2018 5:42 PM, Jason Resch wrote:



hh, but diophantine equations only need integers,
addition, and multiplication, and can define any
computable function. Therefore the question of
whether or not some diophantine equation has a
solution can be made equivalent to the question of
whether some Turing machine halts.  So you face
this problem of getting at all the truth once you
can define integers, addition and multiplication.


There's no surprise that you can't get at all true
statements about a system  that is defined to be
infinite.


But you can always prove more true statements with a
better system of axioms.  So clearly the axioms are not
the driving force behind truth.



And you can prove more false statements with a "better"
system of axioms...which was my original point. So
axioms are not a "force behind truth"; they are a force
behind what is provable.


There are objectively better systems which prove nothing
false, but allow you to prove more things than weaker
systems of axioms.


By that criterion an inconsistent system is the objectively
best of all.


The problem with an inconsistent system is that it does prove
things that are false i.e. "not true".


However we can never prove that the system doesn't prove
anything false (within the theory itself).


You're confusing mathematically consistency with not proving
something false.


 They're related. A system that is inconsistent can prove a
statement as well as its converse. Therefore it is proving things
that are false.


But a system that is consistent can also prove a statement that is
false:

axiom 1: Trump is a genius.
axiom 2: Trump is stable.

theorem: Trump is a stable genius.


So how is this different from flawed physical theories?


The difference is that mathematicians can't test their theories.

Brent

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Re: What is more primary than numbers?

[agrayson2000]

On Sunday, December 16, 2018 at 10:53:18 PM UTC, agrays...@gmail.com wrote:
>
>
>
> On Sunday, December 16, 2018 at 10:01:55 PM UTC, Jason wrote:
>>
>>
>>
>> On Sun, Dec 16, 2018 at 3:39 PM  wrote:
>>
>>>
>>>
>>> On Sunday, December 16, 2018 at 8:58:33 PM UTC, Jason wrote:



 On Sun, Dec 16, 2018 at 2:14 PM  wrote:

>
>
> On Sunday, December 16, 2018 at 2:11:06 AM UTC, Jason wrote:
>>
>>
>>
>> On Sat, Dec 15, 2018 at 8:06 PM  wrote:
>>
>>>
>>>
>>> On Sunday, December 16, 2018 at 1:41:08 AM UTC, Jason wrote:



 On Sat, Dec 15, 2018 at 7:28 PM  wrote:

>
>
> On Saturday, December 15, 2018 at 11:04:55 PM UTC, Jason wrote:
>>
>>
>>
>> On Saturday, December 15, 2018,  wrote:
>>
>>>
>>>
>>> On Saturday, December 15, 2018 at 9:28:32 PM UTC, Brent wrote:



 On 12/15/2018 7:43 AM, Jason Resch wrote:



 On Sat, Dec 15, 2018 at 1:09 AM Brent Meeker <
 meek...@verizon.net> wrote:

>
>
> On 12/14/2018 7:31 PM, Jason Resch wrote:
>
> On Fri, Dec 14, 2018 at 8:43 PM Brent Meeker <
> meek...@verizon.net> wrote:
>
>> Yes, you create a whole theology around not all truths are 
>> provable.  But you ignore that what is false is also provable.  
>> Provable is 
>> only relative to axioms.
>>
>>
> 1. Do you agree a Turing machine will either halt or not?
>
> 2. Do you agree that no finite set of axioms has the power to 
> prove whether or not any given Turing machine will halt or not?
>
>
> 3. What does this tell us about the relationship between 
> truth, proofs, and axioms?
>
>
> What do you think it tells us.  Does it tell us that a false 
> axiom will not allow proof of a false proposition?
>
  
 It tells us mathematical truth is objective and doesn't come 
 from axioms. Axioms are like physical theories, we can test them 
 and refute 
 them if they lead to predictions that are demonstrably false. 
 E.g., if they 
 predict a Turing machine will not halt, but it does, then we can 
 reject 
 that axiom as an incorrect theory of mathematical truth.  
 Similarly, we 
 might find axioms that allow us to prove more things than some 
 weaker set 
 of axioms, thereby building a better theory, but we have no 
 mechanical way 
 of doing this. In that way it is like doing science, and requires 
 trial and 
 error, comparing our theories with our observations, etc.


 Fine, except you've had to quailfy it as "mathematical truth", 
 meaning that it is relative to the axioms defining the Turning 
 machine.  
 Remember a Turing machine isn't a real device.

>>>
>>> This seems to be the core problem with Bruno's proposal or model 
>>> of reality; how does an imaginary device produce the illusion of 
>>> matter 
>>> (and space and time)? AG 
>>>

 Brent

>>> -- 
>>
>>
>> The solution us easy. Don't assume they're only imaginary.
>>
>
> *If they're responsible for the existence of the matter and 
> spacetime illusion, then they aren't composed of matter and don't 
> exist in 
> spacetime. So, the only alternative is that they exist in our 
> imagination; 
> hence, they're imaginary. QED. AG *
>
>>
>>
 Imaginary mean exists only in imagination.

 Simple counter example to your proof: If this universe is a 
 simulation run on a computer by an advanced alien species, you would 
 conclude that computer and alien species is imaginary on the basis 
 that it 
 can't be located in spacetime.  But clearly this computer and alien 
 civilization does not exist only in our heads, for if they didn't we 
 wouldn't have heads with which to imagine them.

>>>
>>> *If you insist on asserting something, anything, exists, but not in 
>>> spacetime, you have a huge burden of proof since it's impossible to 
>>> prove 
>>> your assertion by any empirical test. So, you're not dealing with a 
>>> scientific hypothesis, since it can't be falsified. AG *
>>>
>

Re: What is more primary than numbers?

[agrayson2000]

On Sunday, December 16, 2018 at 10:01:55 PM UTC, Jason wrote:
>
>
>
> On Sun, Dec 16, 2018 at 3:39 PM > wrote:
>
>>
>>
>> On Sunday, December 16, 2018 at 8:58:33 PM UTC, Jason wrote:
>>>
>>>
>>>
>>> On Sun, Dec 16, 2018 at 2:14 PM  wrote:
>>>


 On Sunday, December 16, 2018 at 2:11:06 AM UTC, Jason wrote:
>
>
>
> On Sat, Dec 15, 2018 at 8:06 PM  wrote:
>
>>
>>
>> On Sunday, December 16, 2018 at 1:41:08 AM UTC, Jason wrote:
>>>
>>>
>>>
>>> On Sat, Dec 15, 2018 at 7:28 PM  wrote:
>>>


 On Saturday, December 15, 2018 at 11:04:55 PM UTC, Jason wrote:
>
>
>
> On Saturday, December 15, 2018,  wrote:
>
>>
>>
>> On Saturday, December 15, 2018 at 9:28:32 PM UTC, Brent wrote:
>>>
>>>
>>>
>>> On 12/15/2018 7:43 AM, Jason Resch wrote:
>>>
>>>
>>>
>>> On Sat, Dec 15, 2018 at 1:09 AM Brent Meeker <
>>> meek...@verizon.net> wrote:
>>>


 On 12/14/2018 7:31 PM, Jason Resch wrote:

 On Fri, Dec 14, 2018 at 8:43 PM Brent Meeker <
 meek...@verizon.net> wrote:

> Yes, you create a whole theology around not all truths are 
> provable.  But you ignore that what is false is also provable.  
> Provable is 
> only relative to axioms.
>
>
 1. Do you agree a Turing machine will either halt or not?

 2. Do you agree that no finite set of axioms has the power to 
 prove whether or not any given Turing machine will halt or not?


 3. What does this tell us about the relationship between truth, 
 proofs, and axioms?


 What do you think it tells us.  Does it tell us that a false 
 axiom will not allow proof of a false proposition?

>>>  
>>> It tells us mathematical truth is objective and doesn't come 
>>> from axioms. Axioms are like physical theories, we can test them 
>>> and refute 
>>> them if they lead to predictions that are demonstrably false. E.g., 
>>> if they 
>>> predict a Turing machine will not halt, but it does, then we can 
>>> reject 
>>> that axiom as an incorrect theory of mathematical truth.  
>>> Similarly, we 
>>> might find axioms that allow us to prove more things than some 
>>> weaker set 
>>> of axioms, thereby building a better theory, but we have no 
>>> mechanical way 
>>> of doing this. In that way it is like doing science, and requires 
>>> trial and 
>>> error, comparing our theories with our observations, etc.
>>>
>>>
>>> Fine, except you've had to quailfy it as "mathematical truth", 
>>> meaning that it is relative to the axioms defining the Turning 
>>> machine.  
>>> Remember a Turing machine isn't a real device.
>>>
>>
>> This seems to be the core problem with Bruno's proposal or model 
>> of reality; how does an imaginary device produce the illusion of 
>> matter 
>> (and space and time)? AG 
>>
>>>
>>> Brent
>>>
>> -- 
>
>
> The solution us easy. Don't assume they're only imaginary.
>

 *If they're responsible for the existence of the matter and 
 spacetime illusion, then they aren't composed of matter and don't 
 exist in 
 spacetime. So, the only alternative is that they exist in our 
 imagination; 
 hence, they're imaginary. QED. AG *

>
>
>>> Imaginary mean exists only in imagination.
>>>
>>> Simple counter example to your proof: If this universe is a 
>>> simulation run on a computer by an advanced alien species, you would 
>>> conclude that computer and alien species is imaginary on the basis that 
>>> it 
>>> can't be located in spacetime.  But clearly this computer and alien 
>>> civilization does not exist only in our heads, for if they didn't we 
>>> wouldn't have heads with which to imagine them.
>>>
>>
>> *If you insist on asserting something, anything, exists, but not in 
>> spacetime, you have a huge burden of proof since it's impossible to 
>> prove 
>> your assertion by any empirical test. So, you're not dealing with a 
>> scientific hypothesis, since it can't be falsified. AG *
>>
>>>
>>>
> It can be falsified. I think you missed the posts I wrote in response 
> to John.  The basic idea is this:
>
> Theories predict certain observations.  We can check for those 
> observatio

Re: What is more primary than numbers?

[Philip Thrift]

On Sunday, December 16, 2018 at 3:38:59 PM UTC-6, agrays...@gmail.com wrote:
>
>
>
> On Sunday, December 16, 2018 at 8:58:33 PM UTC, Jason wrote:
>>
>>
>>
>> On Sun, Dec 16, 2018 at 2:14 PM  wrote:
>>
>>>
>>>
>>> On Sunday, December 16, 2018 at 2:11:06 AM UTC, Jason wrote:



 On Sat, Dec 15, 2018 at 8:06 PM  wrote:

>
>
> On Sunday, December 16, 2018 at 1:41:08 AM UTC, Jason wrote:
>>
>>
>>
>> On Sat, Dec 15, 2018 at 7:28 PM  wrote:
>>
>>>
>>>
>>> On Saturday, December 15, 2018 at 11:04:55 PM UTC, Jason wrote:



 On Saturday, December 15, 2018,  wrote:

>
>
> On Saturday, December 15, 2018 at 9:28:32 PM UTC, Brent wrote:
>>
>>
>>
>> On 12/15/2018 7:43 AM, Jason Resch wrote:
>>
>>
>>
>> On Sat, Dec 15, 2018 at 1:09 AM Brent Meeker  
>> wrote:
>>
>>>
>>>
>>> On 12/14/2018 7:31 PM, Jason Resch wrote:
>>>
>>> On Fri, Dec 14, 2018 at 8:43 PM Brent Meeker <
>>> meek...@verizon.net> wrote:
>>>
 Yes, you create a whole theology around not all truths are 
 provable.  But you ignore that what is false is also provable.  
 Provable is 
 only relative to axioms.


>>> 1. Do you agree a Turing machine will either halt or not?
>>>
>>> 2. Do you agree that no finite set of axioms has the power to 
>>> prove whether or not any given Turing machine will halt or not?
>>>
>>>
>>> 3. What does this tell us about the relationship between truth, 
>>> proofs, and axioms?
>>>
>>>
>>> What do you think it tells us.  Does it tell us that a false 
>>> axiom will not allow proof of a false proposition?
>>>
>>  
>> It tells us mathematical truth is objective and doesn't come from 
>> axioms. Axioms are like physical theories, we can test them and 
>> refute them 
>> if they lead to predictions that are demonstrably false. E.g., if 
>> they 
>> predict a Turing machine will not halt, but it does, then we can 
>> reject 
>> that axiom as an incorrect theory of mathematical truth.  Similarly, 
>> we 
>> might find axioms that allow us to prove more things than some 
>> weaker set 
>> of axioms, thereby building a better theory, but we have no 
>> mechanical way 
>> of doing this. In that way it is like doing science, and requires 
>> trial and 
>> error, comparing our theories with our observations, etc.
>>
>>
>> Fine, except you've had to quailfy it as "mathematical truth", 
>> meaning that it is relative to the axioms defining the Turning 
>> machine.  
>> Remember a Turing machine isn't a real device.
>>
>
> This seems to be the core problem with Bruno's proposal or model 
> of reality; how does an imaginary device produce the illusion of 
> matter 
> (and space and time)? AG 
>
>>
>> Brent
>>
> -- 


 The solution us easy. Don't assume they're only imaginary.

>>>
>>> *If they're responsible for the existence of the matter and 
>>> spacetime illusion, then they aren't composed of matter and don't exist 
>>> in 
>>> spacetime. So, the only alternative is that they exist in our 
>>> imagination; 
>>> hence, they're imaginary. QED. AG *
>>>


>> Imaginary mean exists only in imagination.
>>
>> Simple counter example to your proof: If this universe is a 
>> simulation run on a computer by an advanced alien species, you would 
>> conclude that computer and alien species is imaginary on the basis that 
>> it 
>> can't be located in spacetime.  But clearly this computer and alien 
>> civilization does not exist only in our heads, for if they didn't we 
>> wouldn't have heads with which to imagine them.
>>
>
> *If you insist on asserting something, anything, exists, but not in 
> spacetime, you have a huge burden of proof since it's impossible to prove 
> your assertion by any empirical test. So, you're not dealing with a 
> scientific hypothesis, since it can't be falsified. AG *
>
>>
>>
 It can be falsified. I think you missed the posts I wrote in response 
 to John.  The basic idea is this:

 Theories predict certain observations.  We can check for those 
 observations.  If we find them, the theory has passed a test. If we don't 
 find them we keep looking. If we find observations that contradict the 
 predictions of the theory, then we reject that theory and 

Re: What is more primary than numbers?

[Philip Thrift]

Keep in mind that what an "abstraction" is means different things to 
mathematical nominalists, constructivists, platonists.

- pt

On Sunday, December 16, 2018 at 3:37:44 PM UTC-6, Brent wrote:
>
> Numbers are an abstraction and generalization from counting.   But 
> counting takes seeing some things a similar enough to be counted, yet not 
> identical.  I can count the dogs in my yard because what's a dog and what's 
> not seems clear. But it's hard to count trees in my yard:  Is that a bush 
> or a tree?  Is that sprout a tree, or does it have to grow up first?
>
> Brent
>
> On 12/16/2018 1:29 AM, 'scerir' via Everything List wrote:
>
> A *numerus* (literally: "number"*i*) was the term used for a unit of the 
> Roman 
> army .. In the Imperial Roman 
> army  (30 BC – 284 
> AD), it referred to units of barbarian 
>  allies who were not integrated 
> into the regular army structure of legions 
>  and auxilia 
> . 
>
> I'm inclined to think that numbers - for there obiectivity - need a good 
> "counter" (somebody or somethink). 
>
> 'I raised just this objection with the (extreme) ultrafinitist Yessenin 
> Volpin during a lecture of his. He asked me to be more specific. I then 
> proceeded to start with 2^1 and asked him whether this is "real" or 
> something to that effect. He virtually immediately said yes. Then I asked 
> about 2^2, and he again said yes, but with a perceptible delay. Then 2^3, 
> and yes, but with more delay. This continued for a couple of more times, 
> till it was obvious how he was handling this objection. Sure, he was 
> prepared to always answer yes, but he was going to take 2^100 times as long 
> to answer yes to 2^100 then he would to answering 2^1. There is no way that 
> I could get very far with this.' -Harvey M. Friedman
>
> Dunno if in each every part of this universe there is a good  "counter". 
> Maybe universe itself, as a whole, is a "counter"?. 
>
>  'Paper in white the floor of the room, and rule it off in one-foot 
> squares. Down on one's hands and knees, write in the first square a set of 
> equations conceived as able to govern the physics of the universe. Think 
> more overnight. Next day put a better set of equations into square two. 
> Invite one's most respected colleagues to contribute to other squares. At 
> the end of these labors, one has worked oneself out into the doorway. Stand 
> up, look back on all those equations, some perhaps more hopeful than 
> others, raise one's finger commandingly, and give the order "*Fly*!" Not 
> one of those equations will put on wings, take off, or fly. *Yet the 
> universe "flies"*.(Wheeler on page 1208 of *Gravitation*)
>
>  
>
>
>

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Re: What is more primary than numbers?

[Bruce Kellett]

On Mon, Dec 17, 2018 at 9:04 AM Jason Resch  wrote:

> On Sun, Dec 16, 2018 at 4:01 PM Bruce Kellett 
> wrote:
>
>> On Mon, Dec 17, 2018 at 8:56 AM Jason Resch  wrote:
>>
>>> On Sun, Dec 16, 2018 at 3:28 PM Brent Meeker 
>>> wrote:
>>>

 But a system that is consistent can also prove a statement that is
 false:

 axiom 1: Trump is a genius.
 axiom 2: Trump is stable.

 theorem: Trump is a stable genius.

>>>
>>> So how is this different from flawed physical theories?
>>>
>>
>> Physical theories do not claim to prove theorems - they are not systems
>> of axioms and theorems. Attempts to recast physics in this form have always
>> failed.
>>
>>
> Physical theories claim to describe models of reality.
>

Physical theories are models of reality -- using the word "model" in the
physicists sense.


> You can have a fully consistent physical theory that nevertheless fails to
> accurately describe the physical world,
>

Like Brent's example of an axiomatic description of Trump..


> or is an incomplete description of the physical world.  Likewise, you can
> have an axiomatic system that is consistent, but fails to accurately
> describe the integers, or is less complete than we would like.
>

Axiomatic system are always going to fail to capture everything we would
like to capture about any domain. That is why attempted axiomatisation of
physics have been rather unsuccessful.


> It is a completely analogous situation. If you hold the physical reality
> is real because we can study it objectively and refine our understanding of
> it through observations,
>

That is not "why" I hold the physical world to be real. I take the physical
world to be real because that is the definition of reality.


> then the same would hold for the mathematical reality.
>

No, mathematical "reality" (note the scare quotes) is a derived realm,
entirely dependent on the set of axioms chosen in any instance. So it is
not in any way analogous to physics.

Bruce

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Re: What is more primary than numbers?

[Jason Resch]

On Sun, Dec 16, 2018 at 4:01 PM Bruce Kellett  wrote:

> On Mon, Dec 17, 2018 at 8:56 AM Jason Resch  wrote:
>
>> On Sun, Dec 16, 2018 at 3:28 PM Brent Meeker 
>> wrote:
>>
>>>
>>> But a system that is consistent can also prove a statement that is false:
>>>
>>> axiom 1: Trump is a genius.
>>> axiom 2: Trump is stable.
>>>
>>> theorem: Trump is a stable genius.
>>>
>>
>> So how is this different from flawed physical theories?
>>
>
> Physical theories do not claim to prove theorems - they are not systems of
> axioms and theorems. Attempts to recast physics in this form have always
> failed.
>
>
Physical theories claim to describe models of reality.  You can have a
fully consistent physical theory that nevertheless fails to accurately
describe the physical world, or is an incomplete description of the
physical world.  Likewise, you can have an axiomatic system that is
consistent, but fails to accurately describe the integers, or is less
complete than we would like.

It is a completely analogous situation. If you hold the physical reality is
real because we can study it objectively and refine our understanding of it
through observations, then the same would hold for the mathematical reality.

Jason

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Re: What is more primary than numbers?

[Jason Resch]

On Sun, Dec 16, 2018 at 3:39 PM  wrote:

>
>
> On Sunday, December 16, 2018 at 8:58:33 PM UTC, Jason wrote:
>>
>>
>>
>> On Sun, Dec 16, 2018 at 2:14 PM  wrote:
>>
>>>
>>>
>>> On Sunday, December 16, 2018 at 2:11:06 AM UTC, Jason wrote:



 On Sat, Dec 15, 2018 at 8:06 PM  wrote:

>
>
> On Sunday, December 16, 2018 at 1:41:08 AM UTC, Jason wrote:
>>
>>
>>
>> On Sat, Dec 15, 2018 at 7:28 PM  wrote:
>>
>>>
>>>
>>> On Saturday, December 15, 2018 at 11:04:55 PM UTC, Jason wrote:



 On Saturday, December 15, 2018,  wrote:

>
>
> On Saturday, December 15, 2018 at 9:28:32 PM UTC, Brent wrote:
>>
>>
>>
>> On 12/15/2018 7:43 AM, Jason Resch wrote:
>>
>>
>>
>> On Sat, Dec 15, 2018 at 1:09 AM Brent Meeker 
>> wrote:
>>
>>>
>>>
>>> On 12/14/2018 7:31 PM, Jason Resch wrote:
>>>
>>> On Fri, Dec 14, 2018 at 8:43 PM Brent Meeker <
>>> meek...@verizon.net> wrote:
>>>
 Yes, you create a whole theology around not all truths are
 provable.  But you ignore that what is false is also provable.  
 Provable is
 only relative to axioms.


>>> 1. Do you agree a Turing machine will either halt or not?
>>>
>>> 2. Do you agree that no finite set of axioms has the power to
>>> prove whether or not any given Turing machine will halt or not?
>>>
>>>
>>> 3. What does this tell us about the relationship between truth,
>>> proofs, and axioms?
>>>
>>>
>>> What do you think it tells us.  Does it tell us that a false
>>> axiom will not allow proof of a false proposition?
>>>
>>
>> It tells us mathematical truth is objective and doesn't come from
>> axioms. Axioms are like physical theories, we can test them and 
>> refute them
>> if they lead to predictions that are demonstrably false. E.g., if 
>> they
>> predict a Turing machine will not halt, but it does, then we can 
>> reject
>> that axiom as an incorrect theory of mathematical truth.  Similarly, 
>> we
>> might find axioms that allow us to prove more things than some 
>> weaker set
>> of axioms, thereby building a better theory, but we have no 
>> mechanical way
>> of doing this. In that way it is like doing science, and requires 
>> trial and
>> error, comparing our theories with our observations, etc.
>>
>>
>> Fine, except you've had to quailfy it as "mathematical truth",
>> meaning that it is relative to the axioms defining the Turning 
>> machine.
>> Remember a Turing machine isn't a real device.
>>
>
> This seems to be the core problem with Bruno's proposal or model
> of reality; how does an imaginary device produce the illusion of 
> matter
> (and space and time)? AG
>
>>
>> Brent
>>
> --


 The solution us easy. Don't assume they're only imaginary.

>>>
>>> *If they're responsible for the existence of the matter and
>>> spacetime illusion, then they aren't composed of matter and don't exist 
>>> in
>>> spacetime. So, the only alternative is that they exist in our 
>>> imagination;
>>> hence, they're imaginary. QED. AG *
>>>


>> Imaginary mean exists only in imagination.
>>
>> Simple counter example to your proof: If this universe is a
>> simulation run on a computer by an advanced alien species, you would
>> conclude that computer and alien species is imaginary on the basis that 
>> it
>> can't be located in spacetime.  But clearly this computer and alien
>> civilization does not exist only in our heads, for if they didn't we
>> wouldn't have heads with which to imagine them.
>>
>
> *If you insist on asserting something, anything, exists, but not in
> spacetime, you have a huge burden of proof since it's impossible to prove
> your assertion by any empirical test. So, you're not dealing with a
> scientific hypothesis, since it can't be falsified. AG *
>
>>
>>
 It can be falsified. I think you missed the posts I wrote in response
 to John.  The basic idea is this:

 Theories predict certain observations.  We can check for those
 observations.  If we find them, the theory has passed a test. If we don't
 find them we keep looking. If we find observations that contradict the
 predictions of the theory, then we reject that theory and look for
 something better.

>>>
>>> *As I previously wrote, I could o

Re: What is more primary than numbers?

[Bruce Kellett]

On Mon, Dec 17, 2018 at 8:56 AM Jason Resch  wrote:

> On Sun, Dec 16, 2018 at 3:28 PM Brent Meeker  wrote:
>
>>
>> But a system that is consistent can also prove a statement that is false:
>>
>> axiom 1: Trump is a genius.
>> axiom 2: Trump is stable.
>>
>> theorem: Trump is a stable genius.
>>
>
> So how is this different from flawed physical theories?
>

Physical theories do not claim to prove theorems - they are not systems of
axioms and theorems. Attempts to recast physics in this form have always
failed.

Bruce

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Re: What is more primary than numbers?

[Jason Resch]

On Sun, Dec 16, 2018 at 3:28 PM Brent Meeker  wrote:

>
>
> On 12/15/2018 10:24 PM, Jason Resch wrote:
>
>
>
> On Sat, Dec 15, 2018 at 11:35 PM Brent Meeker 
> wrote:
>
>>
>>
>> On 12/15/2018 6:07 PM, Jason Resch wrote:
>>
>>
>>
>> On Sat, Dec 15, 2018 at 7:57 PM Brent Meeker 
>> wrote:
>>
>>>
>>>
>>> On 12/15/2018 5:42 PM, Jason Resch wrote:
>>>
>>> hh, but diophantine equations only need integers, addition, and
 multiplication, and can define any computable function. Therefore the
 question of whether or not some diophantine equation has a solution can be
 made equivalent to the question of whether some Turing machine halts.  So
 you face this problem of getting at all the truth once you can define
 integers, addition and multiplication.


 There's no surprise that you can't get at all true statements about a
 system  that is defined to be infinite.

>>>
>>> But you can always prove more true statements with a better system of
>>> axioms.  So clearly the axioms are not the driving force behind truth.
>>>
>>>
>>> And you can prove more false statements with a "better" system of
>>> axioms...which was my original point.  So axioms are not a "force behind
>>> truth"; they are a force behind what is provable.
>>>
>>>
>> There are objectively better systems which prove nothing false, but allow
>> you to prove more things than weaker systems of axioms.
>>
>>
>> By that criterion an inconsistent system is the objectively best of all.
>>
>>
> The problem with an inconsistent system is that it does prove things that
> are false i.e. "not true".
>
>
>> However we can never prove that the system doesn't prove anything false
>> (within the theory itself).
>>
>>
>> You're confusing mathematically consistency with not proving something
>> false.
>>
>
>  They're related. A system that is inconsistent can prove a statement as
> well as its converse. Therefore it is proving things that are false.
>
>
> But a system that is consistent can also prove a statement that is false:
>
> axiom 1: Trump is a genius.
> axiom 2: Trump is stable.
>
> theorem: Trump is a stable genius.
>

So how is this different from flawed physical theories?

Jason

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Re: What is more primary than numbers?

[agrayson2000]

On Sunday, December 16, 2018 at 8:58:33 PM UTC, Jason wrote:
>
>
>
> On Sun, Dec 16, 2018 at 2:14 PM > wrote:
>
>>
>>
>> On Sunday, December 16, 2018 at 2:11:06 AM UTC, Jason wrote:
>>>
>>>
>>>
>>> On Sat, Dec 15, 2018 at 8:06 PM  wrote:
>>>


 On Sunday, December 16, 2018 at 1:41:08 AM UTC, Jason wrote:
>
>
>
> On Sat, Dec 15, 2018 at 7:28 PM  wrote:
>
>>
>>
>> On Saturday, December 15, 2018 at 11:04:55 PM UTC, Jason wrote:
>>>
>>>
>>>
>>> On Saturday, December 15, 2018,  wrote:
>>>


 On Saturday, December 15, 2018 at 9:28:32 PM UTC, Brent wrote:
>
>
>
> On 12/15/2018 7:43 AM, Jason Resch wrote:
>
>
>
> On Sat, Dec 15, 2018 at 1:09 AM Brent Meeker  
> wrote:
>
>>
>>
>> On 12/14/2018 7:31 PM, Jason Resch wrote:
>>
>> On Fri, Dec 14, 2018 at 8:43 PM Brent Meeker  
>> wrote:
>>
>>> Yes, you create a whole theology around not all truths are 
>>> provable.  But you ignore that what is false is also provable.  
>>> Provable is 
>>> only relative to axioms.
>>>
>>>
>> 1. Do you agree a Turing machine will either halt or not?
>>
>> 2. Do you agree that no finite set of axioms has the power to 
>> prove whether or not any given Turing machine will halt or not?
>>
>>
>> 3. What does this tell us about the relationship between truth, 
>> proofs, and axioms?
>>
>>
>> What do you think it tells us.  Does it tell us that a false 
>> axiom will not allow proof of a false proposition?
>>
>  
> It tells us mathematical truth is objective and doesn't come from 
> axioms. Axioms are like physical theories, we can test them and 
> refute them 
> if they lead to predictions that are demonstrably false. E.g., if 
> they 
> predict a Turing machine will not halt, but it does, then we can 
> reject 
> that axiom as an incorrect theory of mathematical truth.  Similarly, 
> we 
> might find axioms that allow us to prove more things than some weaker 
> set 
> of axioms, thereby building a better theory, but we have no 
> mechanical way 
> of doing this. In that way it is like doing science, and requires 
> trial and 
> error, comparing our theories with our observations, etc.
>
>
> Fine, except you've had to quailfy it as "mathematical truth", 
> meaning that it is relative to the axioms defining the Turning 
> machine.  
> Remember a Turing machine isn't a real device.
>

 This seems to be the core problem with Bruno's proposal or model of 
 reality; how does an imaginary device produce the illusion of matter 
 (and 
 space and time)? AG 

>
> Brent
>
 -- 
>>>
>>>
>>> The solution us easy. Don't assume they're only imaginary.
>>>
>>
>> *If they're responsible for the existence of the matter and spacetime 
>> illusion, then they aren't composed of matter and don't exist in 
>> spacetime. 
>> So, the only alternative is that they exist in our imagination; hence, 
>> they're imaginary. QED. AG *
>>
>>>
>>>
> Imaginary mean exists only in imagination.
>
> Simple counter example to your proof: If this universe is a simulation 
> run on a computer by an advanced alien species, you would conclude that 
> computer and alien species is imaginary on the basis that it can't be 
> located in spacetime.  But clearly this computer and alien civilization 
> does not exist only in our heads, for if they didn't we wouldn't have 
> heads 
> with which to imagine them.
>

 *If you insist on asserting something, anything, exists, but not in 
 spacetime, you have a huge burden of proof since it's impossible to prove 
 your assertion by any empirical test. So, you're not dealing with a 
 scientific hypothesis, since it can't be falsified. AG *

>
>
>>> It can be falsified. I think you missed the posts I wrote in response to 
>>> John.  The basic idea is this:
>>>
>>> Theories predict certain observations.  We can check for those 
>>> observations.  If we find them, the theory has passed a test. If we don't 
>>> find them we keep looking. If we find observations that contradict the 
>>> predictions of the theory, then we reject that theory and look for 
>>> something better.
>>>
>>
>> *As I previously wrote, I could offer some information about the 
>> predictions of modern physics; not only what they are, and how they're 
>> tested, but how they came about. I wouldn't have refer to some paper

Re: What is more primary than numbers?

[Brent Meeker]

Numbers are an abstraction and generalization from counting.   But 
counting takes seeing some things a similar enough to be counted, yet 
not identical.  I can count the dogs in my yard because what's a dog and 
what's not seems clear. But it's hard to count trees in my yard:  Is 
that a bush or a tree?  Is that sprout a tree, or does it have to grow 
up first?


Brent

On 12/16/2018 1:29 AM, 'scerir' via Everything List wrote:


A /numerus/ (literally: "number"/i/) was the term used for a unit of 
the Roman army .. In the 
Imperial Roman army 
 (30 BC – 284 AD), 
it referred to units of barbarian 
 allies who were not 
integrated into the regular army structure of legions 
 and auxilia 
.


I'm inclined to think that numbers - for there obiectivity - need a 
good "counter" (somebody or somethink).


'I raised just this objection with the (extreme) ultrafinitist 
Yessenin Volpin during a lecture of his. He asked me to be more 
specific. I then proceeded to start with 2^1 and asked him whether 
this is "real" or something to that effect. He virtually immediately 
said yes. Then I asked about 2^2, and he again said yes, but with a 
perceptible delay. Then 2^3, and yes, but with more delay. This 
continued for a couple of more times, till it was obvious how he was 
handling this objection. Sure, he was prepared to always answer yes, 
but he was going to take 2^100 times as long to answer yes to 2^100 
then he would to answering 2^1. There is no way that I could get very 
far with this.' -Harvey M. Friedman


Dunno if in each every part of this universe there is a good 
"counter". Maybe universe itself, as a whole, is a "counter"?.


 'Paper in white the floor of the room, and rule it off in one-foot 
squares. Down on one's hands and knees, write in the first square a 
set of equations conceived as able to govern the physics of the 
universe. Think more overnight. Next day put a better set of equations 
into square two. Invite one's most respected colleagues to contribute 
to other squares. At the end of these labors, one has worked oneself 
out into the doorway. Stand up, look back on all those equations, some 
perhaps more hopeful than others, raise one's finger commandingly, and 
give the order "*Fly*!" Not one of those equations will put on wings, 
take off, or fly. *Yet the universe "flies"*.(Wheeler on page 1208 of 
_Gravitation_)


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Re: What is more primary than numbers?

[Brent Meeker]

On 12/15/2018 10:24 PM, Jason Resch wrote:



On Sat, Dec 15, 2018 at 11:35 PM Brent Meeker > wrote:




On 12/15/2018 6:07 PM, Jason Resch wrote:



On Sat, Dec 15, 2018 at 7:57 PM Brent Meeker
mailto:meeke...@verizon.net>> wrote:



On 12/15/2018 5:42 PM, Jason Resch wrote:



hh, but diophantine equations only need integers,
addition, and multiplication, and can define any
computable function. Therefore the question of whether
or not some diophantine equation has a solution can be
made equivalent to the question of whether some Turing
machine halts. So you face this problem of getting at
all the truth once you can define integers, addition
and multiplication.


There's no surprise that you can't get at all true
statements about a system  that is defined to be infinite.


But you can always prove more true statements with a better
system of axioms. So clearly the axioms are not the driving
force behind truth.



And you can prove more false statements with a "better"
system of axioms...which was my original point.  So axioms
are not a "force behind truth"; they are a force behind what
is provable.


There are objectively better systems which prove nothing false,
but allow you to prove more things than weaker systems of axioms.


By that criterion an inconsistent system is the objectively best
of all.


The problem with an inconsistent system is that it does prove things 
that are false i.e. "not true".



However we can never prove that the system doesn't prove anything
false (within the theory itself).


You're confusing mathematically consistency with not proving
something false.


 They're related. A system that is inconsistent can prove a statement 
as well as its converse. Therefore it is proving things that are false.


But a system that is consistent can also prove a statement that is false:

axiom 1: Trump is a genius.
axiom 2: Trump is stable.

theorem: Trump is a stable genius.

Brent

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Re: What is more primary than numbers?

[Jason Resch]

On Sun, Dec 16, 2018 at 2:14 PM  wrote:

>
>
> On Sunday, December 16, 2018 at 2:11:06 AM UTC, Jason wrote:
>>
>>
>>
>> On Sat, Dec 15, 2018 at 8:06 PM  wrote:
>>
>>>
>>>
>>> On Sunday, December 16, 2018 at 1:41:08 AM UTC, Jason wrote:



 On Sat, Dec 15, 2018 at 7:28 PM  wrote:

>
>
> On Saturday, December 15, 2018 at 11:04:55 PM UTC, Jason wrote:
>>
>>
>>
>> On Saturday, December 15, 2018,  wrote:
>>
>>>
>>>
>>> On Saturday, December 15, 2018 at 9:28:32 PM UTC, Brent wrote:



 On 12/15/2018 7:43 AM, Jason Resch wrote:



 On Sat, Dec 15, 2018 at 1:09 AM Brent Meeker 
 wrote:

>
>
> On 12/14/2018 7:31 PM, Jason Resch wrote:
>
> On Fri, Dec 14, 2018 at 8:43 PM Brent Meeker 
> wrote:
>
>> Yes, you create a whole theology around not all truths are
>> provable.  But you ignore that what is false is also provable.  
>> Provable is
>> only relative to axioms.
>>
>>
> 1. Do you agree a Turing machine will either halt or not?
>
> 2. Do you agree that no finite set of axioms has the power to
> prove whether or not any given Turing machine will halt or not?
>
>
> 3. What does this tell us about the relationship between truth,
> proofs, and axioms?
>
>
> What do you think it tells us.  Does it tell us that a false axiom
> will not allow proof of a false proposition?
>

 It tells us mathematical truth is objective and doesn't come from
 axioms. Axioms are like physical theories, we can test them and refute 
 them
 if they lead to predictions that are demonstrably false. E.g., if they
 predict a Turing machine will not halt, but it does, then we can reject
 that axiom as an incorrect theory of mathematical truth.  Similarly, we
 might find axioms that allow us to prove more things than some weaker 
 set
 of axioms, thereby building a better theory, but we have no mechanical 
 way
 of doing this. In that way it is like doing science, and requires 
 trial and
 error, comparing our theories with our observations, etc.


 Fine, except you've had to quailfy it as "mathematical truth",
 meaning that it is relative to the axioms defining the Turning machine.
 Remember a Turing machine isn't a real device.

>>>
>>> This seems to be the core problem with Bruno's proposal or model of
>>> reality; how does an imaginary device produce the illusion of matter 
>>> (and
>>> space and time)? AG
>>>

 Brent

>>> --
>>
>>
>> The solution us easy. Don't assume they're only imaginary.
>>
>
> *If they're responsible for the existence of the matter and spacetime
> illusion, then they aren't composed of matter and don't exist in 
> spacetime.
> So, the only alternative is that they exist in our imagination; hence,
> they're imaginary. QED. AG *
>
>>
>>
 Imaginary mean exists only in imagination.

 Simple counter example to your proof: If this universe is a simulation
 run on a computer by an advanced alien species, you would conclude that
 computer and alien species is imaginary on the basis that it can't be
 located in spacetime.  But clearly this computer and alien civilization
 does not exist only in our heads, for if they didn't we wouldn't have heads
 with which to imagine them.

>>>
>>> *If you insist on asserting something, anything, exists, but not in
>>> spacetime, you have a huge burden of proof since it's impossible to prove
>>> your assertion by any empirical test. So, you're not dealing with a
>>> scientific hypothesis, since it can't be falsified. AG *
>>>


>> It can be falsified. I think you missed the posts I wrote in response to
>> John.  The basic idea is this:
>>
>> Theories predict certain observations.  We can check for those
>> observations.  If we find them, the theory has passed a test. If we don't
>> find them we keep looking. If we find observations that contradict the
>> predictions of the theory, then we reject that theory and look for
>> something better.
>>
>
> *As I previously wrote, I could offer some information about the
> predictions of modern physics; not only what they are, and how they're
> tested, but how they came about. I wouldn't have refer to some paper. I
> haven't seen any plausibility arguments concerning predictions of
> arithmetic being the cause of the alleged illusion of matter and spacetime.*
>

That's not surprising, as you have said numerous times, you refuse to read
the papers.


> * Not one such argument as far as I can recall. None of t

Re: What is more primary than numbers?

[agrayson2000]

On Sunday, December 16, 2018 at 9:29:47 AM UTC, scerir wrote:
>
> A *numerus* (literally: "number"*i*) was the term used for a unit of the 
> Roman 
> army .. In the Imperial Roman 
> army  (30 BC – 284 
> AD), it referred to units of barbarian 
>  allies who were not integrated 
> into the regular army structure of legions 
>  and auxilia 
> . 
>

> I'm inclined to think that numbers - for there obiectivity - need a good 
> "counter" (somebody or somethink). 
>
 

> 'I raised just this objection with the (extreme) ultrafinitist Yessenin 
> Volpin during a lecture of his. He asked me to be more specific. I then 
> proceeded to start with 2^1 and asked him whether this is "real" or 
> something to that effect. He virtually immediately said yes. Then I asked 
> about 2^2, and he again said yes, but with a perceptible delay. Then 2^3, 
> and yes, but with more delay. This continued for a couple of more times, 
> till it was obvious how he was handling this objection. Sure, he was 
> prepared to always answer yes, but he was going to take 2^100 times as long 
> to answer yes to 2^100 then he would to answering 2^1. There is no way that 
> I could get very far with this.' -Harvey M. Friedman
>
 

> Dunno if in each every part of this universe there is a good  "counter". 
> Maybe universe itself, as a whole, is a "counter"?. 
>

*That would require a frame-independent clock in a non-material Platonic 
reality;  the ultimate Bird's Eye frame and possibly an observer as well. 
AG*

 'Paper in white the floor of the room, and rule it off in one-foot 
> squares. Down on one's hands and knees, write in the first square a set of 
> equations conceived as able to govern the physics of the universe. Think 
> more overnight. Next day put a better set of equations into square two. 
> Invite one's most respected colleagues to contribute to other squares. At 
> the end of these labors, one has worked oneself out into the doorway. Stand 
> up, look back on all those equations, some perhaps more hopeful than 
> others, raise one's finger commandingly, and give the order "*Fly*!" Not 
> one of those equations will put on wings, take off, or fly. *Yet the 
> universe "flies"*.(Wheeler on page 1208 of *Gravitation*)
>
>  
>

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Re: What is more primary than numbers?

[agrayson2000]

On Sunday, December 16, 2018 at 2:11:06 AM UTC, Jason wrote:
>
>
>
> On Sat, Dec 15, 2018 at 8:06 PM > wrote:
>
>>
>>
>> On Sunday, December 16, 2018 at 1:41:08 AM UTC, Jason wrote:
>>>
>>>
>>>
>>> On Sat, Dec 15, 2018 at 7:28 PM  wrote:
>>>


 On Saturday, December 15, 2018 at 11:04:55 PM UTC, Jason wrote:
>
>
>
> On Saturday, December 15, 2018,  wrote:
>
>>
>>
>> On Saturday, December 15, 2018 at 9:28:32 PM UTC, Brent wrote:
>>>
>>>
>>>
>>> On 12/15/2018 7:43 AM, Jason Resch wrote:
>>>
>>>
>>>
>>> On Sat, Dec 15, 2018 at 1:09 AM Brent Meeker  
>>> wrote:
>>>


 On 12/14/2018 7:31 PM, Jason Resch wrote:

 On Fri, Dec 14, 2018 at 8:43 PM Brent Meeker  
 wrote:

> Yes, you create a whole theology around not all truths are 
> provable.  But you ignore that what is false is also provable.  
> Provable is 
> only relative to axioms.
>
>
 1. Do you agree a Turing machine will either halt or not?

 2. Do you agree that no finite set of axioms has the power to prove 
 whether or not any given Turing machine will halt or not?


 3. What does this tell us about the relationship between truth, 
 proofs, and axioms?


 What do you think it tells us.  Does it tell us that a false axiom 
 will not allow proof of a false proposition?

>>>  
>>> It tells us mathematical truth is objective and doesn't come from 
>>> axioms. Axioms are like physical theories, we can test them and refute 
>>> them 
>>> if they lead to predictions that are demonstrably false. E.g., if they 
>>> predict a Turing machine will not halt, but it does, then we can reject 
>>> that axiom as an incorrect theory of mathematical truth.  Similarly, we 
>>> might find axioms that allow us to prove more things than some weaker 
>>> set 
>>> of axioms, thereby building a better theory, but we have no mechanical 
>>> way 
>>> of doing this. In that way it is like doing science, and requires trial 
>>> and 
>>> error, comparing our theories with our observations, etc.
>>>
>>>
>>> Fine, except you've had to quailfy it as "mathematical truth", 
>>> meaning that it is relative to the axioms defining the Turning machine. 
>>>  
>>> Remember a Turing machine isn't a real device.
>>>
>>
>> This seems to be the core problem with Bruno's proposal or model of 
>> reality; how does an imaginary device produce the illusion of matter 
>> (and 
>> space and time)? AG 
>>
>>>
>>> Brent
>>>
>> -- 
>
>
> The solution us easy. Don't assume they're only imaginary.
>

 *If they're responsible for the existence of the matter and spacetime 
 illusion, then they aren't composed of matter and don't exist in 
 spacetime. 
 So, the only alternative is that they exist in our imagination; hence, 
 they're imaginary. QED. AG *

>
>
>>> Imaginary mean exists only in imagination.
>>>
>>> Simple counter example to your proof: If this universe is a simulation 
>>> run on a computer by an advanced alien species, you would conclude that 
>>> computer and alien species is imaginary on the basis that it can't be 
>>> located in spacetime.  But clearly this computer and alien civilization 
>>> does not exist only in our heads, for if they didn't we wouldn't have heads 
>>> with which to imagine them.
>>>
>>
>> *If you insist on asserting something, anything, exists, but not in 
>> spacetime, you have a huge burden of proof since it's impossible to prove 
>> your assertion by any empirical test. So, you're not dealing with a 
>> scientific hypothesis, since it can't be falsified. AG *
>>
>>>
>>>
> It can be falsified. I think you missed the posts I wrote in response to 
> John.  The basic idea is this:
>
> Theories predict certain observations.  We can check for those 
> observations.  If we find them, the theory has passed a test. If we don't 
> find them we keep looking. If we find observations that contradict the 
> predictions of the theory, then we reject that theory and look for 
> something better.
>

*As I previously wrote, I could offer some information about the 
predictions of modern physics; not only what they are, and how they're 
tested, but how they came about. I wouldn't have refer to some paper. I 
haven't seen any plausibility arguments concerning predictions of 
arithmetic being the cause of the alleged illusion of matter and spacetime. 
Not one such argument as far as I can recall. None of the advocates of this 
theory are able to offer any motivational predictions and their 
plausibility BASED on your Platonic arithmetic theory; not one! AG *

>
> Jason 
>

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Re: Towards Conscious AI Systems (a symposium at the AAAI Stanford Spring Symposium 2019)

[Bruno Marchal]

> On 15 Dec 2018, at 00:12, Philip Thrift  wrote:
> 
> 
> 
> On Friday, December 14, 2018 at 5:00:33 PM UTC-6, Brent wrote:
> 
> 
> On 12/14/2018 2:59 AM, Bruno Marchal wrote: 
> >> On 13 Dec 2018, at 21:24, Brent Meeker > 
> >> wrote: 
> >> 
> >> 
> >> 
> >> On 12/13/2018 3:25 AM, Bruno Marchal wrote: 
>  But that is the same as saying proof=>truth. 
> >>> I don’t think so. It says that []p -> p is not provable, unless p is 
> >>> proved. 
> >> So  []([]p -> p) -> p  or in other words Proof([]p -> p) => (p is true)  
> >> So in this case proof entails truth?? 
> > But “[]([]p -> p) -> p” is not a theorem of G, meaning that "[]([]p -> p) 
> > -> p” is not true in general for any arithmetic p, with [] = Gödel’s 
> > beweisbar. 
> > 
> > The Löb’s formula is []([]p -> p) -> []p, not []([]p -> p) -> p. 
> > 
> > 
> > 
> >> 
> >>> For example []f -> f (consistency) is not provable. It will belong to G* 
> >>> \ G. 
> >>> 
> >>> Another example is that []<>t -> <>t is false, despite <>t being true. In 
> >>> fact <>t -> ~[]<>t. 
> >>> Or <>t -> <>[]f. Consistency implies the consistency of inconsistency. 
> >> I'm not sure how to interpret these formulae.  Are you asserting them for 
> >> every substitution of t by a true proposition (even though "true" is 
> >> undefinable)? 
> > No, only by either the constant propositional “true”, or any obvious truth 
> > you want, like “1 = 1”. 
> > 
> > 
> > 
> > 
> >> Or are you asserting that there is at least one true proposition for which 
> >> []<>t -> <>t is false? 
> > You can read it beweisbar (consistent(“1 = 1”)) -> (consistent (“1=1”), and 
> > indeed that is true, but not provable by the machine too which this 
> > provability and consistency referred to. 
> > 
> > 
> > 
> > 
> >>> 
>  Nothing which is proven can be false, 
> >>> Assuming consistency, which is not provable. 
> >> So consistency is hard to determine.  You just assume it for arithmetic.  
> >> But finding that an axiom is false is common in argument. 
> > Explain this to your tax inspector! 
> 
> I have.  Just because I spent $125,000 on my apartment building doesn't 
> mean it's appraised value must be $125,000 greater. 
> 
> > 
> > If elementary arithmetic is inconsistent, all scientific theories are 
> > false. 
> 
> Not inconsistent, derived from false or inapplicable premises. 
> 
> > 
> > Gödel’s theorem illustrate indirectly the consistency of arithmetic, as no 
> > one has ever been able to prove arithmetic’s consistency in arithmetic, 
> > which confirms its consistency, given that if arithmetic is consistent, it 
> > cannot prove its consistency. 
> 
> But it can be proven in bigger systems. 
> 
> > Gödel’s result does not throw any doubt about arithmetic’s consistency, 
> > quite the contrary. 
> > 
> > Of course, if arithmetic was inconsistent, it would be able to prove 
> > (easily) its consistency. 
> 
> Only if you first found the inconsistency, i.e. proved a contradiction.  
> And even then there might be a question of the rules of inference. 
> 
> Brent 
> 
> 
> 
> 
> I have read in various texts that at some point matter (all there is in the 
> universe) may reach a point of inconsistency: All matter itself would just 
> disintegrate.  That's all, folks!

That is what Black-Holes are in Newton physics, and they appear if any two 
masses are at distance 0 of each other. The *quantum* black hole are attempts 
by nature to survive God’s dividing number by zero!

With Mechanism we have partial control. It is up to us to try not dividing by 
zero and multiplying by the infinite. But we can’t control all bugs (security) 
without losing Turing universality (liberty).

Matter will remain apparent, but its semantic will differ, as it is a mix of 
contingent with what is observable for all universal 
number/machine/combinator/… 

With (indexical, digital) Mechanism, that is mainly the classical 
Church-Turing-Post-Kleene thesis, the physical reality is phi_i independent. It 
should imply the structure on which a measure one exist, may be like Turing 
universal groups (like the unitary groups). With mechanism, the origin of the 
physical laws is ia problem in mathematics, and partial solution already 
obtained compare well with the general data of contemporary physics. Major 
advantage: the Gödel-Löb-Solovay split of G and G* (the truth on “me” and what 
I can justify on “me”) allows to distinguish, in the observable, the 
justifiable and the non justifiable, but also the knowable and non knowable, 
the observable and the non observable, and this is used to distinguish the 
quanta and the qualia in the sensible realm. 

For the universal machine there is a rich corona in between the rational 
(justifiable) and the surrational (true but non provable) and the Lobian 
machine (those knowing that they are universal) are aware of that corona in the 
first person (non justifiable nor even describable) way. 


Bruno





> 
> - pt
> 
> -- 
> You received this message because you are sub

Re: What is more primary than numbers?

[John Clark]

On Sat, Dec 15, 2018 at 11:44 AM Jason Resch  wrote:


> > *Pure numbers may not correspond to point in time and space, but their
> relationships do. *
>

Where and when did 2+2=4 happen? Does that relationship between 2 and 4
ever change?


> *> Doesn't the fact that "John Clark is conscious of every point of time
> in his life, *
>

But I haven't been  conscious of every point of time in my life, only about
2/3 of the time.


> > *and none of those John Clark view points ceases to exist", already
> contradict your idea of change?*
>

Every point along the time axis of my world-line corresponds to a DIFFERENT
arrangement of matter in space. Things change and I live.


> *> "Computer Science is no more about computers than astronomy is about
> telescopes."*
>

Bad analogy. The Andromeda Galaxy would exist without telescopes but
computations would not exist without computers or brains made of something
that can change.


> > *People are convicted without there being any eye witnesses or direct
> evidence all the time.  *
>

Memory can be very unreliable so I don't give much credibility to
eyewitness testimony even if they're honest, I think circumstantial
evidence and the deductions that come from it are far more compelling .


> > *In those cases it requires indirect evidence *
>

Indirect evidence is still evidence, invisible evidence is not. You claim
to have a invisible photo of my fingerprints on the murder weapon, but
that's not good enough to send me to jail, not even in Trump's America, at
least not yet.


> >> it's easy to understand why we can't see beyond the Hubble volume, but
>> it's very hard to understand why we can't detect non-material Turing
>> machines if they exist,
>>
>
> *> Why is that hard to understand?*
>

I can't explain why we can't detect non-material Turing machines if they
exist and are a vital part of reality and I'm pretty sure you can't either
because if you could you'd have done so by now.


> >> and if they are responsible for our consciousness it's even harder to
>> understand why a change in the matter in our brain changes our
>> consciousness and a change in our consciousness changes the matter in our
>> brain.
>>
>
> > T*his is explained well in Markus Muller's paper:*
> https://arxiv.org/pdf/1712.01826.pdf
>

Muller wasn't even trying to explain that, the paper isn't talking about
what we're discussing, he specifically says "*The question of consciousness
is irrelevant for this paper*". Muller also says in the paper "*objective
reality is not assumed on this approach*" and that's fine but he certainly
isn't the first to question realism, Bell told us more that 50 years ago
that things can't be local and realistic. But we were talking about
subjective reality.


> > *Among the predictions he reached assuming only the existence of
> computations:*
> *The Big Bang:  Abby will identify a singular state in the past, where the
> universe was particularly “small” and “simple” in the algorithmic sense. *


As I said as far as consciousness is concerned it doesn't matter if the
argument is sound or not, but as long as we're on the subject, It would
seem to me  it would be even simpler if the previous state of the universe
was more or less the same as it is today, if Fred Hoyle's Steady State
Cosmology had turned out to be correct I have no doubt somebody would be
using virtually identical arguments to say that Solomonoff induction had
predicted the Steady State. After all an algorithm that does not need to
change anything doesn't need to be very long. I'd be a lot more impressed
if he predicted something we didn't already know, something testable and
not obvious.

Another problem is it's not entirely clear what he means by simple, is the
Mandelbrot Set simple? In one way its infinitely complex and yet it can be
generated with just a few lines of code. You mentioned Kolmogorov
complexity, it says the complexity of a mathematical object is the size of
the smallest algorithm that can produce it, but the problem is in general
there is no way to compute the exact size of that smallest algorithm, you
can't even compute a lower bound for it.


> * > Addressing your point with changing matter of the brain being
> correlated to changing experience*


But that's not all, changing experience also corresponds with changing
matter in the brain. logically if X then Y is true and Y then X is also
true then  X=Y.

>  *In particular, her observations do not fundamentally supervene on this
> “physical universe”;*


I don't know what torturous logic you used to reach that conclusion.

>  it is merely a useful tool to predict her future observations.


And that useful tool predicts she will have no future observations.whatsoever
if a bullet is blasted into the physical matter in her brain.

>  *Nonetheless, this universe will seem perfectly real to her,*


And that is what's important if we're still talking about subjectivity.

> *If the measure µ that is computed within her computational univers

Re: Towards Conscious AI Systems (a symposium at the AAAI Stanford Spring Symposium 2019)

[Philip Thrift]

On Sunday, December 16, 2018 at 11:27:50 AM UTC-6, Bruno Marchal wrote:
>
>
> > On 15 Dec 2018, at 00:00, Brent Meeker  > wrote: 
> > 
> > 
> > 
> > On 12/14/2018 2:59 AM, Bruno Marchal wrote: 
> >>> On 13 Dec 2018, at 21:24, Brent Meeker  > wrote: 
> >>> 
> >>> 
> >>> 
> >>> On 12/13/2018 3:25 AM, Bruno Marchal wrote: 
> > But that is the same as saying proof=>truth. 
>  I don’t think so. It says that []p -> p is not provable, unless p is 
> proved. 
> >>> So  []([]p -> p) -> p  or in other words Proof([]p -> p) => (p is 
> true)  So in this case proof entails truth?? 
> >> But “[]([]p -> p) -> p” is not a theorem of G, meaning that "[]([]p -> 
> p) -> p” is not true in general for any arithmetic p, with [] = Gödel’s 
> beweisbar. 
> >> 
> >> The Löb’s formula is []([]p -> p) -> []p, not []([]p -> p) -> p. 
> >> 
> >> 
> >> 
> >>> 
>  For example []f -> f (consistency) is not provable. It will belong to 
> G* \ G. 
>  
>  Another example is that []<>t -> <>t is false, despite <>t being 
> true. In fact <>t -> ~[]<>t. 
>  Or <>t -> <>[]f. Consistency implies the consistency of 
> inconsistency. 
> >>> I'm not sure how to interpret these formulae.  Are you asserting them 
> for every substitution of t by a true proposition (even though "true" is 
> undefinable)? 
> >> No, only by either the constant propositional “true”, or any obvious 
> truth you want, like “1 = 1”. 
> >> 
> >> 
> >> 
> >> 
> >>> Or are you asserting that there is at least one true proposition for 
> which []<>t -> <>t is false? 
> >> You can read it beweisbar (consistent(“1 = 1”)) -> (consistent (“1=1”), 
> and indeed that is true, but not provable by the machine too which this 
> provability and consistency referred to. 
> >> 
> >> 
> >> 
> >> 
>  
> > Nothing which is proven can be false, 
>  Assuming consistency, which is not provable. 
> >>> So consistency is hard to determine.  You just assume it for 
> arithmetic.  But finding that an axiom is false is common in argument. 
> >> Explain this to your tax inspector! 
> > 
> > I have.  Just because I spent $125,000 on my apartment building doesn't 
> mean it's appraised value must be $125,000 greater. 
>
> ? 
>
>
>
> > 
> >> 
> >> If elementary arithmetic is inconsistent, all scientific theories are 
> false. 
> > 
> > Not inconsistent, derived from false or inapplicable premises. 
>
> In classical logic false entails inconsistent. Inapplicable does not mean 
> anything, as the theory’s application are *in* and *about* arithmetic. 
>
> If we are universal machine emulable at some level of description, then it 
> is absolutely undecidable if there is something more than arithmetic, but 
> the observable must obey some laws, so we can test Mechanism. 
>
>
>
>
>
>
> > 
> >> 
> >> Gödel’s theorem illustrate indirectly the consistency of arithmetic, as 
> no one has ever been able to prove arithmetic’s consistency in arithmetic, 
> which confirms its consistency, given that if arithmetic is consistent, it 
> cannot prove its consistency. 
> > 
> > But it can be proven in bigger systems. 
>
> Yes, and Ronsion arithmetic emulates all the bigger systems. No need to 
> assume more than Robinson arithmetic to get the emulation of more rich 
> system of belief, which actually can help the finite things to figure more 
> on themselves. Numbers which introspect themselves, and self-transforms, 
> get soon or later the tentation to believe in the induction axioms, and 
> even in the infinity axioms. 
>
> Before Gödel 1931, the mathematicians thought they could secure the use of 
> the infinities by proving consistent the talk about their descriptions and 
> names, but after Gödel, we understood that we cannot even secure the finite 
> and the numbers with them. The real culprit is that one the system is rich 
> enough to implement a universal machine, like Robinson Arithmetic is, you 
> get an explosion in complexity and uncontrollability. We know now that we 
> understand about nothing on numbers and machine. 
>
>
>
> > 
> >> Gödel’s result does not throw any doubt about arithmetic’s consistency, 
> quite the contrary. 
> >> 
> >> Of course, if arithmetic was inconsistent, it would be able to prove 
> (easily) its consistency. 
> > 
> > Only if you first found the inconsistency, i.e. proved a contradiction. 
>
>
> That is []f, that does not necessarily means arithmetic is inconsistent. 
> The proof could be given by a non standard natural numbers. 
>
> So, at the meta-level, to say that PA is inconsistent means that there is 
> standard number describing a finite proof of f. And in that case, PA would 
> prove any proposition. In classical logic, proving A and proving ~A is 
> equivalent with proving (A & ~A), which []f, interpreted at the meta-level. 
> Now, for the machine, []f is consistent, as the machine cannot prove that 
> []f -> f, which would be her consistency. G* proves <>[]f. 
>
> It is because the domain here is full of ambiguities, that the logic 

Re: Towards Conscious AI Systems (a symposium at the AAAI Stanford Spring Symposium 2019)

[Bruno Marchal]

> On 15 Dec 2018, at 00:00, Brent Meeker  wrote:
> 
> 
> 
> On 12/14/2018 2:59 AM, Bruno Marchal wrote:
>>> On 13 Dec 2018, at 21:24, Brent Meeker  wrote:
>>> 
>>> 
>>> 
>>> On 12/13/2018 3:25 AM, Bruno Marchal wrote:
> But that is the same as saying proof=>truth.
 I don’t think so. It says that []p -> p is not provable, unless p is 
 proved.
>>> So  []([]p -> p) -> p  or in other words Proof([]p -> p) => (p is true)  So 
>>> in this case proof entails truth??
>> But “[]([]p -> p) -> p” is not a theorem of G, meaning that "[]([]p -> p) -> 
>> p” is not true in general for any arithmetic p, with [] = Gödel’s beweisbar.
>> 
>> The Löb’s formula is []([]p -> p) -> []p, not []([]p -> p) -> p.
>> 
>> 
>> 
>>> 
 For example []f -> f (consistency) is not provable. It will belong to G* \ 
 G.
 
 Another example is that []<>t -> <>t is false, despite <>t being true. In 
 fact <>t -> ~[]<>t.
 Or <>t -> <>[]f. Consistency implies the consistency of inconsistency.
>>> I'm not sure how to interpret these formulae.  Are you asserting them for 
>>> every substitution of t by a true proposition (even though "true" is 
>>> undefinable)?
>> No, only by either the constant propositional “true”, or any obvious truth 
>> you want, like “1 = 1”.
>> 
>> 
>> 
>> 
>>> Or are you asserting that there is at least one true proposition for which 
>>> []<>t -> <>t is false?
>> You can read it beweisbar (consistent(“1 = 1”)) -> (consistent (“1=1”), and 
>> indeed that is true, but not provable by the machine too which this 
>> provability and consistency referred to.
>> 
>> 
>> 
>> 
 
> Nothing which is proven can be false,
 Assuming consistency, which is not provable.
>>> So consistency is hard to determine.  You just assume it for arithmetic.  
>>> But finding that an axiom is false is common in argument.
>> Explain this to your tax inspector!
> 
> I have.  Just because I spent $125,000 on my apartment building doesn't mean 
> it's appraised value must be $125,000 greater.

?



> 
>> 
>> If elementary arithmetic is inconsistent, all scientific theories are false.
> 
> Not inconsistent, derived from false or inapplicable premises.

In classical logic false entails inconsistent. Inapplicable does not mean 
anything, as the theory’s application are *in* and *about* arithmetic. 

If we are universal machine emulable at some level of description, then it is 
absolutely undecidable if there is something more than arithmetic, but the 
observable must obey some laws, so we can test Mechanism.






> 
>> 
>> Gödel’s theorem illustrate indirectly the consistency of arithmetic, as no 
>> one has ever been able to prove arithmetic’s consistency in arithmetic, 
>> which confirms its consistency, given that if arithmetic is consistent, it 
>> cannot prove its consistency.
> 
> But it can be proven in bigger systems.

Yes, and Ronsion arithmetic emulates all the bigger systems. No need to assume 
more than Robinson arithmetic to get the emulation of more rich system of 
belief, which actually can help the finite things to figure more on themselves. 
Numbers which introspect themselves, and self-transforms, get soon or later the 
tentation to believe in the induction axioms, and even in the infinity axioms.

Before Gödel 1931, the mathematicians thought they could secure the use of the 
infinities by proving consistent the talk about their descriptions and names, 
but after Gödel, we understood that we cannot even secure the finite and the 
numbers with them. The real culprit is that one the system is rich enough to 
implement a universal machine, like Robinson Arithmetic is, you get an 
explosion in complexity and uncontrollability. We know now that we understand 
about nothing on numbers and machine.



> 
>> Gödel’s result does not throw any doubt about arithmetic’s consistency, 
>> quite the contrary.
>> 
>> Of course, if arithmetic was inconsistent, it would be able to prove 
>> (easily) its consistency.
> 
> Only if you first found the inconsistency, i.e. proved a contradiction. 


That is []f, that does not necessarily means arithmetic is inconsistent. The 
proof could be given by a non standard natural numbers.

So, at the meta-level, to say that PA is inconsistent means that there is 
standard number describing a finite proof of f. And in that case, PA would 
prove any proposition. In classical logic, proving A and proving ~A is 
equivalent with proving (A & ~A), which []f, interpreted at the meta-level. 
Now, for the machine, []f is consistent, as the machine cannot prove that []f 
-> f, which would be her consistency. G* proves <>[]f.

It is because the domain here is full of ambiguities, that the logic of G and 
G*, which capture the consequence of incompleteness are so useful.

Bruno




> And even then there might be a question of the rules of inference.
> 
> Brent
> 
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> To unsu

Re: Towards Conscious AI Systems (a symposium at the AAAI Stanford Spring Symposium 2019)

[Bruno Marchal]

> On 14 Dec 2018, at 12:01, Philip Thrift  wrote:
> 
> 
> 
> On Friday, December 14, 2018 at 4:49:33 AM UTC-6, Bruno Marchal wrote:
> 
>> On 13 Dec 2018, at 21:05, Brent Meeker > 
>> wrote:
>> 
>> 
>> 
>> On 12/13/2018 3:18 AM, Bruno Marchal wrote:
 Automating Gödel'’s Ontological Proof of God’s Existence ¨ with 
 Higher-order Automated Theorem Provers
 http://page.mi.fu-berlin.de/cbenzmueller/papers/C40.pdf 
 
>>> 
>>> Gödel took the modal logic S5 for its proof, which is the only logic NOT 
>>> available for the machines.
>> 
>> What about S5 makes it not available for machines?
> 
> 
> There are no intensional variant of G leading to S5.
> 
> The axiom “5” is the guilty one (as []p & p obeys S4, and S5 can be defined 
> by S4 + “5”)
> 
> “5” is <>p -> []<>p (the opposite of incompleteness: <>p -> ~[]<>p, but also 
> incompatible in the logic X, Z, etc.).
> 
> Bruno
> 
> 
> 
> 
> 
> How does this relate to the "higher-order theorem provers" that deals with 
> modal systems like S5?
> 
> 
> https://www.ijcai.org/Proceedings/16/Papers/137.pdf 


Rather interesting. The machine makes the right critics! Not quite serious 
about theology though. The definition of God by St-Anselme is a bit too much 
post-529 for the universal machine’s view. But some paragraph, notably the 
requirement of symmetry, gives me the feeling that the machine’s sensibility 
mode ([]p & <>t & p) defined in G1* (with 1 being what they called the collapse 
formula: p -> []p, which does not entail any collapse in arithmetic) might make 
the first person sensibility “believing” in the God of St-Anselme. It would be 
interesting to see what aspect of the ONE would correspond to this.

Bruno



> 
> - pt
> 
> 
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Re: What is more primary than numbers?

[Jason Resch]

On Sunday, December 16, 2018, Philip Thrift  wrote:

>
>
> On Saturday, December 15, 2018 at 7:41:08 PM UTC-6, Jason wrote:
>>
>>
>>
>> On Sat, Dec 15, 2018 at 7:28 PM  wrote:
>>
>>>
>>>
>>> On Saturday, December 15, 2018 at 11:04:55 PM UTC, Jason wrote:



 On Saturday, December 15, 2018,  wrote:

>
>
> On Saturday, December 15, 2018 at 9:28:32 PM UTC, Brent wrote:
>>
>>
>>
>> On 12/15/2018 7:43 AM, Jason Resch wrote:
>>
>>
>>
>> On Sat, Dec 15, 2018 at 1:09 AM Brent Meeker 
>> wrote:
>>
>>>
>>>
>>> On 12/14/2018 7:31 PM, Jason Resch wrote:
>>>
>>> On Fri, Dec 14, 2018 at 8:43 PM Brent Meeker 
>>> wrote:
>>>
 Yes, you create a whole theology around not all truths are
 provable.  But you ignore that what is false is also provable.  
 Provable is
 only relative to axioms.


>>> 1. Do you agree a Turing machine will either halt or not?
>>>
>>> 2. Do you agree that no finite set of axioms has the power to prove
>>> whether or not any given Turing machine will halt or not?
>>>
>>>
>>> 3. What does this tell us about the relationship between truth,
>>> proofs, and axioms?
>>>
>>>
>>> What do you think it tells us.  Does it tell us that a false axiom
>>> will not allow proof of a false proposition?
>>>
>>
>> It tells us mathematical truth is objective and doesn't come from
>> axioms. Axioms are like physical theories, we can test them and refute 
>> them
>> if they lead to predictions that are demonstrably false. E.g., if they
>> predict a Turing machine will not halt, but it does, then we can reject
>> that axiom as an incorrect theory of mathematical truth.  Similarly, we
>> might find axioms that allow us to prove more things than some weaker set
>> of axioms, thereby building a better theory, but we have no mechanical 
>> way
>> of doing this. In that way it is like doing science, and requires trial 
>> and
>> error, comparing our theories with our observations, etc.
>>
>>
>> Fine, except you've had to quailfy it as "mathematical truth",
>> meaning that it is relative to the axioms defining the Turning machine.
>> Remember a Turing machine isn't a real device.
>>
>
> This seems to be the core problem with Bruno's proposal or model of
> reality; how does an imaginary device produce the illusion of matter (and
> space and time)? AG
>
>>
>> Brent
>>
> --


 The solution us easy. Don't assume they're only imaginary.

>>>
>>> *If they're responsible for the existence of the matter and spacetime
>>> illusion, then they aren't composed of matter and don't exist in spacetime.
>>> So, the only alternative is that they exist in our imagination; hence,
>>> they're imaginary. QED. AG *
>>>


>> Imaginary mean exists only in imagination.
>>
>> Simple counter example to your proof: If this universe is a simulation
>> run on a computer by an advanced alien species, you would conclude that
>> computer and alien species is imaginary on the basis that it can't be
>> located in spacetime.  But clearly this computer and alien civilization
>> does not exist only in our heads, for if they didn't we wouldn't have heads
>> with which to imagine them.
>>
>> Jason
>>
>
>
>
>
>
> The simulation hypothesis as you have stated it is an argument for
> materialism.
>
> A simulation on one of our (conventional) computers is a bunch of
> particles moving through CPU and GPU processors and LED pixels. One running
> on some future quantum computer would be using qubit chips.
>
> What is the computer the advanced alien civilization is running "us" on?
> It could be in fact "our" universe as quantum computer [
> https://arxiv.org/abs/1312.4455 ]! Then we would be a bunch of particles
> running on the advanced alien civilization 's quantum computer, just as
> simulations above ore particles running on one of our quantum computers.
>
> - pt
>
>
>
>
You are right, the simulation hypothesis, as generally stated, doesn't
escape materialism.

My example was only to show there can be non-imaginary things that can't be
located in spacetime.

Jason


>
>
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Re: Black holes and computational complexity

[Philip Thrift]

On Friday, December 7, 2018 at 11:45:30 AM UTC-6, Mason Green wrote:
>
> Leonard Susskind thinks there may be a link between the size of a black 
> hole’s interior (which grows with time) and its computational complexity 
> (which does likewise). 
>
> At the end of the article there’s even a suggestion that the expansion of 
> the universe might likewise have a computational origin. 
>
>
> https://www.quantamagazine.org/why-black-hole-interiors-grow-forever-20181206/
>  
>
> -Mason






Black holes are the rabbit holes of theoretical physicists turned computing 
theorists. 

- pt


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Re: What is more primary than numbers?

[Philip Thrift]

On Sunday, December 16, 2018 at 3:29:47 AM UTC-6, scerir wrote:
>
> A *numerus* (literally: "number"*i*) was the term used for a unit of the 
> Roman 
> army .. In the Imperial Roman 
> army  (30 BC – 284 
> AD), it referred to units of barbarian 
>  allies who were not integrated 
> into the regular army structure of legions 
>  and auxilia 
> . 
>
> I'm inclined to think that numbers - for there obiectivity - need a good 
> "counter" (somebody or somethink). 
>
> 'I raised just this objection with the (extreme) ultrafinitist Yessenin 
> Volpin during a lecture of his. He asked me to be more specific. I then 
> proceeded to start with 2^1 and asked him whether this is "real" or 
> something to that effect. He virtually immediately said yes. Then I asked 
> about 2^2, and he again said yes, but with a perceptible delay. Then 2^3, 
> and yes, but with more delay. This continued for a couple of more times, 
> till it was obvious how he was handling this objection. Sure, he was 
> prepared to always answer yes, but he was going to take 2^100 times as long 
> to answer yes to 2^100 then he would to answering 2^1. There is no way that 
> I could get very far with this.' -Harvey M. Friedman
>
> Dunno if in each every part of this universe there is a good  "counter". 
> Maybe universe itself, as a whole, is a "counter"?. 
>
>  'Paper in white the floor of the room, and rule it off in one-foot 
> squares. Down on one's hands and knees, write in the first square a set of 
> equations conceived as able to govern the physics of the universe. Think 
> more overnight. Next day put a better set of equations into square two. 
> Invite one's most respected colleagues to contribute to other squares. At 
> the end of these labors, one has worked oneself out into the doorway. Stand 
> up, look back on all those equations, some perhaps more hopeful than 
> others, raise one's finger commandingly, and give the order "*Fly*!" Not 
> one of those equations will put on wings, take off, or fly. *Yet the 
> universe "flies"*.(Wheeler on page 1208 of *Gravitation*)
>
>  
>


In the finitist theory of Jan Mycielski - recounted in the book* 
Understanding the infinite* by Shaughan Lavine 
[ 
https://books.google.com/books/about/Understanding_the_Infinite.html?id=GvGqRYifGpMC
 
] - the process of 2^n+m would have "gaps" in them: 
...,2^500,2^500+1,...2^501,...  etc.

- pt

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Re: What is more primary than numbers?

['scerir' via Everything List]

A numerus (literally: "number"i) was the term used for a unit of the Roman army 
https://en.wikipedia.org/wiki/Roman_army .. In the Imperial Roman army 
https://en.wikipedia.org/wiki/Imperial_Roman_army (30 BC – 284 AD), it referred 
to units of barbarian https://en.wikipedia.org/wiki/Barbarian allies who were 
not integrated into the regular army structure of legions 
https://en.wikipedia.org/wiki/Roman_legion and auxilia 
https://en.wikipedia.org/wiki/Auxiliaries_(Roman_military) .

I'm inclined to think that numbers - for there obiectivity - need a good 
"counter" (somebody or somethink).

'I raised just this objection with the (extreme) ultrafinitist Yessenin Volpin 
during a lecture of his. He asked me to be more specific. I then proceeded to 
start with 2^1 and asked him whether this is "real" or something to that 
effect. He virtually immediately said yes. Then I asked about 2^2, and he again 
said yes, but with a perceptible delay. Then 2^3, and yes, but with more delay. 
This continued for a couple of more times, till it was obvious how he was 
handling this objection. Sure, he was prepared to always answer yes, but he was 
going to take 2^100 times as long to answer yes to 2^100 then he would to 
answering 2^1. There is no way that I could get very far with this.' -Harvey M. 
Friedman

Dunno if in each every part of this universe there is a good  "counter". Maybe 
universe itself, as a whole, is a "counter"?.

 'Paper in white the floor of the room, and rule it off in one-foot squares. 
Down on one's hands and knees, write in the first square a set of equations 
conceived as able to govern the physics of the universe. Think more overnight. 
Next day put a better set of equations into square two. Invite one's most 
respected colleagues to contribute to other squares. At the end of these 
labors, one has worked oneself out into the doorway. Stand up, look back on all 
those equations, some perhaps more hopeful than others, raise one's finger 
commandingly, and give the order "Fly!" Not one of those equations will put on 
wings, take off, or fly. Yet the universe "flies".(Wheeler on page 1208 of 
Gravitation)

 

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Re: What is more primary than numbers?

[Philip Thrift]

On Saturday, December 15, 2018 at 7:41:08 PM UTC-6, Jason wrote:
>
>
>
> On Sat, Dec 15, 2018 at 7:28 PM > wrote:
>
>>
>>
>> On Saturday, December 15, 2018 at 11:04:55 PM UTC, Jason wrote:
>>>
>>>
>>>
>>> On Saturday, December 15, 2018,  wrote:
>>>


 On Saturday, December 15, 2018 at 9:28:32 PM UTC, Brent wrote:
>
>
>
> On 12/15/2018 7:43 AM, Jason Resch wrote:
>
>
>
> On Sat, Dec 15, 2018 at 1:09 AM Brent Meeker  
> wrote:
>
>>
>>
>> On 12/14/2018 7:31 PM, Jason Resch wrote:
>>
>> On Fri, Dec 14, 2018 at 8:43 PM Brent Meeker  
>> wrote:
>>
>>> Yes, you create a whole theology around not all truths are 
>>> provable.  But you ignore that what is false is also provable.  
>>> Provable is 
>>> only relative to axioms.
>>>
>>>
>> 1. Do you agree a Turing machine will either halt or not?
>>
>> 2. Do you agree that no finite set of axioms has the power to prove 
>> whether or not any given Turing machine will halt or not?
>>
>>
>> 3. What does this tell us about the relationship between truth, 
>> proofs, and axioms?
>>
>>
>> What do you think it tells us.  Does it tell us that a false axiom 
>> will not allow proof of a false proposition?
>>
>  
> It tells us mathematical truth is objective and doesn't come from 
> axioms. Axioms are like physical theories, we can test them and refute 
> them 
> if they lead to predictions that are demonstrably false. E.g., if they 
> predict a Turing machine will not halt, but it does, then we can reject 
> that axiom as an incorrect theory of mathematical truth.  Similarly, we 
> might find axioms that allow us to prove more things than some weaker set 
> of axioms, thereby building a better theory, but we have no mechanical 
> way 
> of doing this. In that way it is like doing science, and requires trial 
> and 
> error, comparing our theories with our observations, etc.
>
>
> Fine, except you've had to quailfy it as "mathematical truth", meaning 
> that it is relative to the axioms defining the Turning machine.  Remember 
> a 
> Turing machine isn't a real device.
>

 This seems to be the core problem with Bruno's proposal or model of 
 reality; how does an imaginary device produce the illusion of matter (and 
 space and time)? AG 

>
> Brent
>
 -- 
>>>
>>>
>>> The solution us easy. Don't assume they're only imaginary.
>>>
>>
>> *If they're responsible for the existence of the matter and spacetime 
>> illusion, then they aren't composed of matter and don't exist in spacetime. 
>> So, the only alternative is that they exist in our imagination; hence, 
>> they're imaginary. QED. AG *
>>
>>>
>>>
> Imaginary mean exists only in imagination.
>
> Simple counter example to your proof: If this universe is a simulation run 
> on a computer by an advanced alien species, you would conclude that 
> computer and alien species is imaginary on the basis that it can't be 
> located in spacetime.  But clearly this computer and alien civilization 
> does not exist only in our heads, for if they didn't we wouldn't have heads 
> with which to imagine them.
>
> Jason
>





The simulation hypothesis as you have stated it is an argument for 
materialism.

A simulation on one of our (conventional) computers is a bunch of particles 
moving through CPU and GPU processors and LED pixels. One running on some 
future quantum computer would be using qubit chips.

What is the computer the advanced alien civilization is running "us" on? It 
could be in fact "our" universe as quantum computer [ 
https://arxiv.org/abs/1312.4455 ]! Then we would be a bunch of particles 
running on the advanced alien civilization 's quantum computer, just as 
simulations above ore particles running on one of our quantum computers.

- pt




 

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