'When Are Relations Neither True Nor False? [Logic]

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From: Nam Nguyen on 11 Apr 2010 23:01

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.
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).

From: Nam Nguyen on 11 Apr 2010 23:02

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

I meant "... is NOT based on an intuition but is a notion ..."

>
>>
>>
>>>> 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. If the naturals are of syntactical notion then
> they must be described by a formal system. So far you keep saying
> it's syntactical but giving no hint what such natural-number formal
> system be!
>
>>
>> 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_ and the 10-year old
> would most likely never claim he could use such an intuition
> to prove the consistency of PA, right? But AK would do
> so and you'd do the same or at least support such a claim.
>
>> I could certainly supply you
>> with my own version of that explanation, but if you
>> can't already do it yourself, then I propose you
>> ought to first go learn that. If you can, then there
>> is no point in my retyping it.
>
> Oh I learnt the naturals numbers the same way mathematicians
> would learn: it's an intuitive notion!
>
> It's you who'd go against the mainstream of mathematicians
> and made the following odd notion:
>
> >>>>> I reject your claim that intuition, of whatever kind, plays
> >>>>> any part in our thinking about the naturals.
>
> [What textbook's author would even hint such a bogus an idea?]

From: Nam Nguyen on 12 Apr 2010 00:55

Nam Nguyen wrote:
> 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.
> 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).

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).

The battle here is they'd believe our intuition of the natural numbers
(or equivalently our intuition of the standard model of PA) is an absolute
notion, that everyone "must" intuit them exactly the same way, while
I've been saying that they're wrong in that respect. And they're wrong
by more than one account, but the first one being that the concept of the
natural numbers is an intuition which is subjective (hence relative) to an
individual perception. You might believe GC is true in your perception of
the naturals while I may believe "there are infinitely many counter examples
of GC" is true in my own perception of them, and none of us is _absolutely_
correct or incorrect.

Their belief on the absolute truths of the natural numbers is very much
akin to the belief before that arithmetic syntactical provability be
absolute in the sense that either a formula or its negation written in the
language of arithmetic must be _syntactically provable_ and there be
no middle ground.

Some how they've managed not to pay attention to the history that has only
been fairly recent.

From: Marshall on 12 Apr 2010 01:36

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.

Of course, calling this "the relativity of mathematical truth
in general" is inappropriately grandiose. The only thing this
says about anything at all is:

The meaning of a symbol depends on context.

I'll alert the media.

Marshall

From: Nam Nguyen on 12 Apr 2010 01:43

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. Semantics and truth are 2 _different notions_! And I'm
taliking about "the relativity of mathematical truth".

If you read people's writing carefully.