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Topic: mathematical infinite as a matter of method
Replies: 25   Last Post: May 4, 2013 11:24 PM

 Messages: [ Previous | Next ]
 fom Posts: 1,968 Registered: 12/4/12
Re: mathematical infinite as a matter of method
Posted: May 4, 2013 9:59 AM

On 5/4/2013 1:21 AM, Graham Cooper wrote:
> On May 4, 2:17 pm, fom <fomJ...@nyms.net> wrote:
>> On 5/3/2013 8:20 PM, Graham Cooper wrote:
>>
>>
>>

>>> You don't just 'choose' a foundational theory, none of ZFC holds up
>>> to Induction, when it's derived proofs are more closely examined
>>> there is a lot of convoluted circular reasoning...

>>
>> of the late nineteenth and twentieth century went to great pains
>> to reject epistemology. They are simply rediscovering it within
>> their formalisms while continuing along that path of rejection.

>
>
> Nonsense, Plato wouldn't be so unkind...
>
> the incompleteness theorem is just a note from mum that they couldn't
> do their homework...
>
> ...the homework told them so...
>
> Q1 You can't solve this!
>

>>
>> There are not many choices...
>>
>> http://en.wikipedia.org/wiki/M%C3%BCnchhausen_Trilemma
>>
>> At least I know why I make the choices I make.
>>
>> AxAy(xcy <-> (Az(ycz -> xcz) /\ Ez(xcz /\ -ycz)))
>>
>> AxAy(xey <-> (Az(ycz -> xez) /\ Ez(xez /\ -ycz)))
>>
>> Unfortunately, they would do you no good either.
>>

>
> Well with ordinary subset
>
> xcy <-> Aa aex->aey
>
> is sufficient to eliminate quantifiers.
>
> Aa ...
> and
> { a | ... }
>
> both mean " ALL A such that ... "
>
> When you see a 'C'
> in your equations there is a hidden ALL()
> ranging over all the elements

There is nothing "hidden" about it.

When a mathematician begins a proof, the statement
of the premises are to be considered as true. Thus,
proofs do not begin with statements containing free
variables.

Meanings are hidden when one confuses the
product of logical analysis -- namely a language
signature of undefined symbols -- with the
stipulation that a theory is foundational. In
the latter case, the theory ought to indicate
some means by which the primitive relations are
recognized independent of stipulation.

There is little difference between purport and
authorial intention. But, mathematical logic is
supposed to frown upon intention. Rather, the
goal of logical analysis is to discern an
objectively recognizable underlying logical form.

>
> Quantified Logic and Set Theory are equivalent theories, you don't
> need the syntax of both.

When one actually uses pencil and paper,
one knows what constitutes transformation
rules in a proof and what constitutes
statements to which those transformation
rules are applied.

And, for the record, your statement only
holds in the sense of "set theory" obtained
relative to the Russellian view. Cantor
rejected the "extension of a concept"
interpretation and attributed Russell's

>
> ------------
>
> AxAy(xcy <-> (Az(ycz -> xcz) /\ Ez(xcz /\ -ycz)))
>
>
> RE: ~ycz
>
> which just means x is PROPER subset of y
>
> I use not(ss(..,..))
>
> in a different way, to remove an ALL() quantifier.
>
> ~An p(n) <-> En ~p(n)
>
> Proof By Counter-Example
>
> ----------------
>
> e.g p(n) <-> x MOD 2 = 0
>
> ~An p(n) == not all numbers are even
>
> .........
>
> PROOFBYCOUNTEREXAMPLE.PRO
>
> nat(0).
> nat(s(X)) :- nat(X).
> even(0).
> even(s(s(X))) :- even(X).
> odd(s(0)).
> odd(s(s(X))) :- odd(X).
>
> e(A,nats) :- nat(A).
> e(A,evens) :- even(A).
> e(A,odds) :- odd(A).
>
> e(A, not(evens)) :- e(A, odds).
> e(A, not(odds)) :- e(A, evens).
>
> intersects(S1,S2) :- e(A,S1),e(A,S2).
> not(ss( S1 , S2 )) :- intersects( S1, not(S2) ).
> ********************
>
>
> This tiny program will work out that
>
> ?- not(ss(nats,evens)) .

>> YES
>
>
> i.e ~nats c evens
>
> by actually testing 0, finding it's both nat & even,
> testing s(0), and finding not s(0) e evens
>
> Are all natural numbers even -> NO!
>
> (this is early work since I didn't have to program ALL() yet!)
>
> -------------
>
> This kind of CATEGORICAL NEGATION
>
> if it's ODD its not EVEN
> if it's BLUE it's not RED
>
> is used in human logic every day!
>
> e(A, not(evens)) :- e(A, odds).
> e(A, not(odds)) :- e(A, evens).
>
> -------------
>
> PROLOG SET THEORY is the paradigm that early logicians could only
> glimpse a small part of the program.
>

Some, perhaps. Frege changed his mind.

"The more I have thought the matter
over, the more convinced I have become
that arithmetic and geometry have
developed on the same basis -- a
geometrical one in fact -- so that
mathematics in its entirety is
really geometry"

Negation is eliminable.

Date Subject Author
4/21/13 fom
4/21/13 Virgil
5/2/13 Hercules ofZeus
5/2/13 fom
5/2/13 Virgil
5/3/13 Graham Cooper
5/3/13 fom
5/3/13 Brian Q. Hutchings
5/3/13 Graham Cooper
5/3/13 fom
5/3/13 Graham Cooper
5/3/13 fom
5/3/13 fom
5/4/13 Graham Cooper
5/3/13 Graham Cooper
5/3/13 fom
5/3/13 Graham Cooper
5/4/13 fom
5/4/13 Graham Cooper
5/4/13 fom
5/4/13 Graham Cooper
5/4/13 fom
5/4/13 Graham Cooper
5/4/13 fom
5/4/13 Graham Cooper
5/3/13 Graham Cooper