Prev: geometry precisely defining ellipsis and how infinity is in the midsection #427 Correcting Math
Next: Accounting for Governmental and Nonprofit Entities, 15th Edition Earl Wilson McGraw Hill Test bank is available at affordable prices. Email me at allsolutionmanuals11[at]gmail.com if you need to buy this. All emails will be answered ASAP.
From: Marshall on 12 Apr 2010 02:08
On Apr 11, 7:51 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote:
> Marshall wrote:
> > On Apr 11, 5:09 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote:
> >> Marshall wrote:
> >>> On Apr 11, 3:54 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote:
> >>>>> I reject your claim that intuition, of whatever kind, plays
> >>>>> any part in our thinking about the naturals.
> >>>> So is the naturals collectively a finite syntactical notion *to you*,
> >>>> since you'd reject the idea they're an intuition notion?
>
> >> Can you share with us your resounding, firm, answer to that simple
> >> question?
>
> > You mean, again? You want me to say "yes" again? Okay:
>
> > Yes.
>
> So your resounding answer is "yes" that the notion of the naturals
> is based on an intuition but is a notion of syntactical formal system.

Not a "formal system" in this sense:

http://en.wikipedia.org/wiki/Formal_system

The naturals and their operators are a model, not a theory.


> How would you know though such a formal system is syntactical
> consistent?

The question of consistency is asked of theories, not of
models. I guess you are not really clear about the distinction.
Since the naturals and their operators are a model,
it makes no sense to ask about their consistency.


> >>> Other than that, and as much as I hesitate to accede to any
> >>> formula of yours given how unreliably you use terminology,
> >>> my answer is "yes." We can completely capture enough
> >>> about the naturals to uniquely characterize them up to
> >>> isomorphism. For example, we have many syntactic
> >>> representations of natural numbers available to us that
> >>> can represent any natural up to resource limits, and we
> >>> have simple algorithms on those representations that
> >>> can compute successor, addition, etc. of any naturals
> >>> up to resource limits. These things are entirely
> >>> mechanical, and free of any vague or intuitive aspects.
> >> I'm sorry: "up to resource limits" isn't an intuition or technical
> >> notion of the natural numbers. So you don't seem to understand
> >> the notion of the natural numbers, despite your having claimed
> >> otherwise.
>
> > Of course it's not an intuition. I've already made it clear that
> > I'm not talking about that. But it certainly *is* a technical
> > notion when discussing any "entirely mechanical" implementation
> > of an algorithm.
>
> "Entirely mechanical" is not a technical definition, just in case
> you're not aware.

It's a term; I never claimed it was the definition of anything.
I guess this is another area where you don't know very
much. I will merely note your lack of familiarity with the
technical terminology around resource limits and around
the mechanical execution of algorithms and move on.


> If the naturals are of syntactical notion then
> they must be described by a formal system.

Is that your intuition?


> So far you keep saying
> it's syntactical but giving no hint what such natural-number formal
> system be!

You haven't been asking about "formal systems". I'm
just describing the natural numbers, not any formal
system. I've said on other occasions that you would
really do well to learn the difference, but oh well.


> > I am pretty sure a marginally competent ten year old
> > can explain what a base ten natural number is (whether
> > or not his school uses those exact terms) and can
> > explain how to find the successor, how to add two
> > naturals, multiply, etc.
>
> But that's what we'd call _intuition_

You are of course free to call it whatever you want.
I do not call it that however. Has my repeated rejection
of the term "intuition" really gone unnoticed by you,
or are you just teasing?


> and the 10-year old
> would most likely never claim he could use such an intuition
> to prove the consistency of PA, right?

Sure. But we were talking about the natural numbers, not PA.


> But AK would do
> so and you'd do the same or at least support such a claim.

I am a poor judge of what AK would say. For myself, I am
unclear enough about what you mean when you say "intuition"
to be able to answer you. My best guess would be: would
we consider the hunch of a ten year old to be a proof that
PA is consistent? Speaking for myself, no. If you meant
something else, please clarify. If you do decide to clarify,
please recall that PA is not the natural numbers, that PA is
a theory and the naturals+their operators are a model,
and that consistency is a question that only applies to
theories. Also recall that the fact that all the axioms of
PA are true for the naturals is a model-theoretic proof
that PA is consistent.


> Oh I learnt the naturals numbers the same way mathematicians
> would learn: it's an intuitive notion!

The above sentence does not signify anything clearly enough
for me either to agree nor to disagree.

Every time I engage with you in any least substantive
way, I end up feeling like I have wasted my time.
Not sure why I do it. I really would be better served
just to laugh at your silly ideas and be done with it.


Marshall
From: Marshall on 12 Apr 2010 02:13
On Apr 11, 10:43 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote:
> Marshall wrote:
> > On Apr 11, 9:55 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote:
> >> To be fair the "standard theorists" and I don't seem to fight on
> >> the relativity of mathematical truth in general. They know
> >> ~(1+1=0) is true or false, relative to what kind of arithmetic
> >> we've chosen as the underlying one (e.g. arithmetic modulo 2 or
> >> arithmetic modulo 3).
>
> > In other words, what a sentence means is "relative" to
> > what meaning we choose for the symbols it is composed of.
>
> NO.

No? You don't think the meaning of a sentence depends
on the meaning of the symbols it contains? Really?


> Semantics and truth are 2 _different notions_!

Is there any relationship between them at all?


> And I'm taliking about "the relativity of mathematical truth".

Yes you are. That's what's so funny.


Marshall
From: Alan Smaill on 12 Apr 2010 07:12
Nam Nguyen <namducnguyen(a)shaw.ca> writes:

> Alan Smaill wrote:
>> Nam Nguyen <namducnguyen(a)shaw.ca> writes:
>>
>>> To claim some formula being
>>> absolutely true (or false) is to destroy the notion of such symmetry.
>>
>> Do you agree that symmetry is only
>> broken if the cases "P" and "not P" are treated differently?
>
> Of course I'd not agree. Even in relativity, given an appropriate context,
> P and ~P must necessarily be treated differently because of LEM.

Now you've really lost me.
LEM *is* symmetric itself, isn't it?

> The Principle addresses a different issue: P or ~P can't be uniformly
> treated as true or false in _all_ contexts. That's all the Principle of
> Symmetry would stipulate, and all I've really said (as above).

So how do you know it is relevant to the particular context at hand,
that of assuming "not P" hypothetically (ie for sake of argument), deriving
a contradiction, and concluding "P". And also in the context where "P"
and "not P" are switched systematically in such an argument?


--
Alan Smaill
From: Nam Nguyen on 13 Apr 2010 00:18
Marshall wrote:
> On Apr 11, 7:51 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote:
>> Marshall wrote:
>>> On Apr 11, 5:09 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote:
>>>> Marshall wrote:
>>>>> On Apr 11, 3:54 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote:
>>>>>>> I reject your claim that intuition, of whatever kind, plays
>>>>>>> any part in our thinking about the naturals.
>>>>>> So is the naturals collectively a finite syntactical notion *to you*,
>>>>>> since you'd reject the idea they're an intuition notion?
>>>> Can you share with us your resounding, firm, answer to that simple
>>>> question?
>>> You mean, again? You want me to say "yes" again? Okay:
>>> Yes.
>> So your resounding answer is "yes" that the notion of the naturals
>> is based on an intuition but is a notion of syntactical formal system.
>
> Not a "formal system" in this sense:
>
> http://en.wikipedia.org/wiki/Formal_system
>
> The naturals and their operators are a model, not a theory.

Not only you're clueless as to exactly what the natural numbers be,
you also don't use the technical terms correctly.

First of all, if the naturals are collectively a model of a language
then you should have not resoundingly said "yes" that they'd be "a finite
syntactical notion", which refer to syntactical symbols, formulas, axioms,
axiom-system, rules of inference: but not to interpretation, or truth values
as when we talk about model truths.

Secondly model would *require* intuition: if there's no intuition there's no
interpretation, no truth values, hence no model! _You've contradicted yourself_
by first rejecting intuition "plays any part in our thinking about the naturals"
and now stating "The naturals and their operators are a model".

>> How would you know though such a formal system is syntactical
>> consistent?
>
> The question of consistency is asked of theories, not of
> models. I guess you are not really clear about the distinction.

I wasn't asking you about the general consistency. I was asking you
about *your contradictory claims* about the natural numbers. My hope
here was that my question would help you to realize how wrong you
were in rejecting intuition "plays any part in our thinking about
the naturals".

> Since the naturals and their operators are a model,
> it makes no sense to ask about their consistency.

Again it was about one of your contradictory claims that the naturals
are of syntactical notion, which is something that we'd *only* refer
to when talking about formal systems. Your statements flip-flop back
and forth between the naturals being model theoretical and syntactical.

Being model theoretical and being syntactical are 2 different (and opposing)
notions. Can you now make up your mind and tell us once for all, which
ways the natural numbers are: model theoretical, or syntactical?

>
>> and the 10-year old
>> would most likely never claim he could use such an intuition
>> to prove the consistency of PA, right?
>
> Sure. But we were talking about the natural numbers, not PA.

You're wrong. My conversation here started when I responded to
AK's statement:

>> Anyone can intuit whatever they want. I said nothing about such
>> matters, I merely noted I have, like any number of logic students
>> over the decades, produced a proof of the consistency of PA in the
>> course of my studies. Intuition has no more and no less to do with
>> it than with any proof in mathematics.

So AK did prove the consistency of PA. And his passage here seemed to
deny PA's consistency proof would have to do anything with intuition
[of the naturals used in the consistency proof of PA]. And that's why
I later protested:

Nam:

>> Then, it also looks like a poor choice of using the _intuition_ about
>> the naturals as a foundation of reasoning, as the school of thought AK,
>> TF seem to have subscribed to, would suggest.

And then you decided to join the conversation:

Marshall:

>> I reject your claim that intuition, of whatever kind, plays
>> any part in our thinking about the naturals.

So, Marshall. Let's not forget the conversation here is about
the proof consistency of PA, as well as about intuition-or-not-
intuition of the naturals.

>> But AK would do
>> so and you'd do the same or at least support such a claim.
>
> I am a poor judge of what AK would say.

Then perhaps you should have stayed silent instead of trying
to defend someone whose statement-validities you only have a
"poor" judgment - at best!

> For myself, I am
> unclear enough about what you mean when you say "intuition"
> to be able to answer you.

Since you don't understand a very simple notion such as intuition
[that a 10-year old student would understand], let me explain that
to you.

You yourself claimed that the natural numbers [collectively]
is a model [I'm assuming of the language L(PA)]. But how would you
know that for sure? The answer is you CAN NOT know that for sure,
because you yourself will NEVER be able to list out all the necessary
n-ary relations (sets) required by the definition of a language model.
The only reason left for you to have a feeling that you know the
naturals is a model at all is your INTUITION. Do you understand now
that you were wrong before when you resoundingly claimed:

>> I reject your claim that intuition, of whatever kind, plays
>> any part in our thinking about the naturals.

> My best guess would be: would
> we consider the hunch of a ten year old to be a proof that
> PA is consistent? Speaking for myself, no. If you meant
> something else, please clarify.

Above I already explained to you the simple notion of intuition that a
10-year old would understand!

> If you do decide to clarify,
> please recall that PA is not the natural numbers, that PA is
> a theory and the naturals+their operators are a model,
> and that consistency is a question that only applies to
> theories.

Where did I claim otherwise? My questions to you were meant
for you to realize you didn't know what you were talking about
when rejecting that intuition "plays any part in our thinking
about the naturals". You should read the conversation more
carefully.

> Also recall that the fact that all the axioms of
> PA are true for the naturals is a model-theoretic proof
> that PA is consistent.

Marshall, how do you prove that - WITHOUT INTUITION?
You're clearly clueless about what you are asserting.

>> Oh I learnt the naturals numbers the same way mathematicians
>> would learn: it's an intuitive notion!
>
> The above sentence does not signify anything clearly enough
> for me either to agree nor to disagree.
>
> Every time I engage with you in any least substantive
> way, I end up feeling like I have wasted my time.

> Not sure why I do it. I really would be better served
> just to laugh at your silly ideas and be done with it.

What silly ideas? You meant when I alluded that the notion
of the naturals is a notion of intuition? If so, then you'd
be also laughing of, say, Shoenfield!
From: Nam Nguyen on 13 Apr 2010 00:34
Alan Smaill wrote:
> Nam Nguyen <namducnguyen(a)shaw.ca> writes:
>
>> Alan Smaill wrote:
>>> Nam Nguyen <namducnguyen(a)shaw.ca> writes:
>>>
>>>> To claim some formula being
>>>> absolutely true (or false) is to destroy the notion of such symmetry.
>>> Do you agree that symmetry is only
>>> broken if the cases "P" and "not P" are treated differently?
>> Of course I'd not agree. Even in relativity, given an appropriate context,
>> P and ~P must necessarily be treated differently because of LEM.
>
> Now you've really lost me.
> LEM *is* symmetric itself, isn't it?

I have no idea what you meant here by "LEM *is* symmetric itself".
You have to explain what you meant by that.

The word "symmetry" I've used here is in the context of the Principle
of Symmetry I articulated before with some clear examples and are about
concepts, formulas - but not about LEM].

>
>> The Principle addresses a different issue: P or ~P can't be uniformly
>> treated as true or false in _all_ contexts. That's all the Principle of
>> Symmetry would stipulate, and all I've really said (as above).
>
> So how do you know it is relevant to the particular context at hand,
> that of assuming "not P" hypothetically (ie for sake of argument), deriving
> a contradiction, and concluding "P". And also in the context where "P"
> and "not P" are switched systematically in such an argument?

I'm not sure what you're trying to argue here. My argument only concerns
the alleged "absolute" truths that DB tried to suggest, and in which I counter
claimed any such absolute truth would break the Principle of Symmetry.
That's all I've really argued in this conversation!
First  |  Prev  |  Next  |  Last
Pages: 134 135 136 137 138 139 140 141 142 143 144 145 146 147
Prev: geometry precisely defining ellipsis and how infinity is in the midsection #427 Correcting Math
Next: Accounting for Governmental and Nonprofit Entities, 15th Edition Earl Wilson McGraw Hill Test bank is available at affordable prices. Email me at allsolutionmanuals11[at]gmail.com if you need to buy this. All emails will be answered ASAP.