Metamathematics.

[spe...@gmail.com]

Is metamathematics mathematics or is it beyond mathematics?

[Julio Di Egidio]

On Saturday, 7 March 2020 13:20:15 UTC+1, spe...@gmail.com wrote:

> Is metamathematics mathematics or is it beyond mathematics?

It is mathematics. And meta-meta-mathematics is still mathematics...

At the boundary of mathematics we rather find the philosophy of mathematics.

Julio

[Dan Christensen]

On Saturday, March 7, 2020 at 7:20:15 AM UTC-5, spe...@gmail.com wrote:
> Is metamathematics mathematics or is it beyond mathematics?

Why does it matter?


Dan

[FredJeffries]

On Saturday, March 7, 2020 at 4:20:15 AM UTC-8, spe...@gmail.com wrote:
> Is metamathematics mathematics or is it beyond mathematics?

Like many (most?) other words, that depends on who is using it and what they are trying to say.

Wikipedia says 'Metamathematics is the study of mathematics itself using mathematical methods', so that would (it seems to me) make it a branch of mathematics: kind of like using a microscope to study microscopes.
https://en.wikipedia.org/wiki/Metamathematics

On the other hand, Wolfram MathWorld, after saying 'Metamathematics is another word for proof theory' goes on 'The branch of logic dealing with the study of the combination and application of mathematical symbols is also sometimes called metamathematics or metalogic' (but, of course, that study would probably be classified as 'mathematics')
https://mathworld.wolfram.com/Metamathematics.html

Andrej Bauer has some (to me) interesting answers to people looking for unshakable 'foundations' for mathematics:

'When logicians speak of "foundations" of mathematics, they may give the impression that they are "building the cathedral" starting from its foundation. But it is much better to view what they are doing as a study of how the cathedral is built and how we can improve it.'
https://math.stackexchange.com/questions/912366/models-of-set-theory#912501

'It is far more fruitful to think of logic and model theory as just another branch of mathematics, namely the one that studies mathematical methods and mathematical activity with mathematical tools. They follow the usual pattern of "mathematizing" their object of interest'

'The success of set theory has lead many to believe that it provides an unshakeable foundation for mathematics. It does not, at least not the mystical kind that some would like to have. It provides a unifying language and framework for mathematicians.'
https://mathoverflow.net/questions/23060/set-theory-and-model-theory/23077#23077

[Jim Burns]

On 3/7/2020 7:20 AM, spe...@gmail.com wrote:

> Is metamathematics mathematics or is it beyond mathematics?

Since metamathematics _uses_ mathematics to _study_ mathematics
(the most famous example being Godel using formal arithmetic
to study arithmetic and extensions of arithmetic),
metamathematics is mathematics.

I think you might be hinting at how it is that the prefix "meta"
comes from the Greek for "after" or "beyond", which doesn't seem
to be what metamathematics actually _is_ Maybe it would make
more sense _in Greek_ to call such studies endomathematics, or
maybe something else. However, this is how we _actually use_
"metamathematics".

I've heard that the earliest use of "meta" in (approximately)
this way was in the title of Aristotle's _Metaphysics_ , where
it meant, pretty much, "essays by Aristotle that are being placed
_on the shelf_ after Aristotle's Physics". So, even then, I think
"meta" was not used how someone might say it "should" be used.

[Khong Dong]

On Saturday, 7 March 2020 09:53:12 UTC-7, Jim Burns wrote:
> On 3/7/2020 7:20 AM, spe...@gmail.com wrote:
>
> > Is metamathematics mathematics or is it beyond mathematics?
>
> Since metamathematics _uses_ mathematics to _study_ mathematics
> (the most famous example being Godel using formal arithmetic
> to study arithmetic and extensions of arithmetic),
> metamathematics is mathematics.

What a crackpot utterance! Where _exactly_ did _you_ learn Goedel was
"using formal arithmetic to study arithmetic and extensions of arithmetic"?

By many respected professors and PhD's, GIT is a piece of _informal_
reasoning - using knowledge of *informal arithmetic*. Good grief!

[Khong Dong]

On Saturday, 7 March 2020 10:34:04 UTC-7, Khong Dong wrote:
> On Saturday, 7 March 2020 09:53:12 UTC-7, Jim Burns wrote:
> > On 3/7/2020 7:20 AM, spe...@gmail.com wrote:
> >
> > > Is metamathematics mathematics or is it beyond mathematics?
> >
> > Since metamathematics _uses_ mathematics to _study_ mathematics
> > (the most famous example being Godel using formal arithmetic
> > to study arithmetic and extensions of arithmetic),
> > metamathematics is mathematics.
>
> What a crackpot utterance! Where _exactly_ did _you_ learn Goedel was
> "using formal arithmetic to study arithmetic and extensions of arithmetic"?
>
> By many respected professors and PhD's, GIT is a piece of _informal_
> reasoning - using knowledge of *informal arithmetic*. Good grief!
>
>
> >
> > I think you might be hinting at how it is that the prefix "meta"
> > comes from the Greek for "after" or "beyond", which doesn't seem
> > to be what metamathematics actually _is_ Maybe it would make
> > more sense _in Greek_ to call such studies endomathematics, or
> > maybe something else. However, this is how we _actually use_
> > "metamathematics".

Maybe you, Jim Burns, should learn a bit of English to understand that
mathematicians/logicians in the right mind would take "meta" in
"metamathematics" to mean "above/beyond". Then again, a crank as you are
would not hesitate to call "apple" "orange" - just like you've thought "meta"
wouldn't mean "above/beyond" (in the context of "metamathematics").

Only a crank of sci.logic would think "meta" would mean "something else" (your
words) and not "above/beyond"!

> > I've heard that the earliest use of "meta" in (approximately)
> > this way was in the title of Aristotle's _Metaphysics_ , where
> > it meant, pretty much, "essays by Aristotle that are being placed
> > _on the shelf_ after Aristotle's Physics". So, even then, I think
> > "meta" was not used how someone might say it "should" be used.

Who cares about a crank like Jim Burns deciding how/what the word "meta"
"should be used".

[Khong Dong]

On Saturday, 7 March 2020 06:01:04 UTC-7, Julio Di Egidio wrote:
> On Saturday, 7 March 2020 13:20:15 UTC+1, spe...@gmail.com wrote:
>
> > Is metamathematics mathematics or is it beyond mathematics?
>
> It is mathematics. And meta-meta-mathematics is still mathematics...

What a delusion. Mathematicians/logicians didn't/don't just attach the prefix
"meta" just so that it means NOTHING!

[Khong Dong]

On Saturday, 7 March 2020 05:20:15 UTC-7, spe...@gmail.com wrote:
> Is metamathematics mathematics or is it beyond mathematics?

Metamathematics is an introspection on how we'd do mathematical reasoning -
inference - in arriving at conclusions, hence it is above and beyond mathematics.

[Julio Di Egidio]

It does mean something, just you I have no idea what it means. Or anything
else you are mentioning, for that matter. Really, the meaning of words...

Julio

[Khong Dong]

On Saturday, 7 March 2020 10:58:40 UTC-7, Julio Di Egidio wrote:
> On Saturday, 7 March 2020 18:50:54 UTC+1, Khong Dong wrote:
> > On Saturday, 7 March 2020 06:01:04 UTC-7, Julio Di Egidio wrote:
> > > On Saturday, 7 March 2020 13:20:15 UTC+1, spe...@gmail.com wrote:
> > >
> > > > Is metamathematics mathematics or is it beyond mathematics?
> > >
> > > It is mathematics. And meta-meta-mathematics is still mathematics...
> >
> > What a delusion. Mathematicians/logicians didn't/don't just attach the prefix
> > "meta" just so that it means NOTHING!
>
> It does mean something, just you I have no idea what it means.

It's you who has no clue here. In mathematical reasoning, (software) data
modeling, ..., "meta MUST" necessarily mean "above/beyond". Get real.

> Or anything
> else you are mentioning, for that matter. Really, the meaning of words...

You're in delusion, as mentioned.

[Julio Di Egidio]

On Saturday, 7 March 2020 19:02:57 UTC+1, Khong Dong wrote:
> On Saturday, 7 March 2020 10:58:40 UTC-7, Julio Di Egidio wrote:
> > On Saturday, 7 March 2020 18:50:54 UTC+1, Khong Dong wrote:
> > > On Saturday, 7 March 2020 06:01:04 UTC-7, Julio Di Egidio wrote:
> > > > On Saturday, 7 March 2020 13:20:15 UTC+1, spe...@gmail.com wrote:
> > > >
> > > > > Is metamathematics mathematics or is it beyond mathematics?
> > > >
> > > > It is mathematics. And meta-meta-mathematics is still mathematics...
> > >
> > > What a delusion. Mathematicians/logicians didn't/don't just attach the prefix
> > > "meta" just so that it means NOTHING!
> >
> > It does mean something, just you I have no idea what it means.
>
> It's you who has no clue here. In mathematical reasoning, (software) data
> modeling, ..., "meta MUST" necessarily mean "above/beyond". Get real.

You are talking nonsense, the exact meaning of anything is context dependent,
and the exact meaning of "meta" in the theoretical context is *about*, e.g.
meta-mathematics is mathematics about mathematics. Goedel's Incompleteness
Theorem indeed being a prototypical example.

You try and get real... (EOD.)

Julio

[Khong Dong]

Well, you know a competent poster would know, *without saying*, the word
"meta" also means "about" in addition to "above/beyond/...": all of which is
context sensitive (nobody - except Jim Burns - would deny this)!

> Goedel's Incompleteness
> Theorem indeed being a prototypical example.

Right. So perhaps you could teach Jim Burns that: he's apparently ignorant on
the meaning of the word "metamathematics".

[Jim Burns]

On 3/7/2020 12:34 PM, Khong Dong wrote:
> On Saturday, 7 March 2020 09:53:12 UTC-7,
> Jim Burns wrote:
>> On 3/7/2020 7:20 AM, spe...@gmail.com wrote:

>>> Is metamathematics mathematics or is it beyond mathematics?
>>
>> Since metamathematics _uses_ mathematics to _study_ mathematics
>> (the most famous example being Godel using formal arithmetic
>> to study arithmetic and extensions of arithmetic),
>> metamathematics is mathematics.
>
> What a crackpot utterance! Where _exactly_ did _you_ learn
> Goedel was "using formal arithmetic to study arithmetic and
> extensions of arithmetic"?

Among the places this is mentioned is the second paragraph of
Goedel's 1931 paper, "On Formally Undecidable Propositions of
Principia Mathematica and Related Systems I".
|
| Before going into details, we shall first indicate the main
| lines of the proof, naturally without laying claims to exactness.
| [...] For metamathematical purposes it is naturally immaterial
| what objects are taken as basic signs, and we propose to use
| natural numbers for them. [...] Metamathematical concepts and
| propositions thereby become concepts and propositions concerning
| natural numbers, or series of them, and therefore at least
| partially expressible in the symbols of the system PM itself.
| [...]
|
[Trans. Bernard Meltzer (1963)]

> By many respected professors and PhD's, GIT is a piece of
> _informal_ reasoning - using knowledge of *informal arithmetic*.
> Good grief!

My use of the word "formal" here is intended to convey descriptions
of arithmetic that do not stop with individual sums and products
and so on: more than 2+2=4, 7*11*13=1001, and so on.
Essentially, I include in "formal arithmetic" variables that
range over _all_ natural numbers and axioms that describe
_all_ the sums, _all_ the products.

You (NN) seem to have some fundamental need to disagree with me
no matter what I say. But maybe you will agree that Goedel uses
variables.

[Peter Percival]

Khong Dong wrote:
> On Saturday, 7 March 2020 09:53:12 UTC-7, Jim Burns wrote:
>> On 3/7/2020 7:20 AM, spe...@gmail.com wrote:
>>
>>> Is metamathematics mathematics or is it beyond mathematics?
>>
>> Since metamathematics _uses_ mathematics to _study_ mathematics
>> (the most famous example being Godel using formal arithmetic
>> to study arithmetic and extensions of arithmetic),
>> metamathematics is mathematics.
>
> What a crackpot utterance! Where _exactly_ did _you_ learn Goedel was
> "using formal arithmetic to study arithmetic and extensions of arithmetic"?
>
> By many respected professors and PhD's, GIT is a piece of _informal_
> reasoning - using knowledge of *informal arithmetic*. Good grief!

Has it become impossible for you to contribute to this newsgroup without
ill-tempered ranting?

[Khong Dong]

On Saturday, 7 March 2020 11:51:06 UTC-7, Peter Percival wrote:
> Khong Dong wrote:
> > On Saturday, 7 March 2020 09:53:12 UTC-7, Jim Burns wrote:
> >> On 3/7/2020 7:20 AM, spe...@gmail.com wrote:
> >>
> >>> Is metamathematics mathematics or is it beyond mathematics?
> >>
> >> Since metamathematics _uses_ mathematics to _study_ mathematics
> >> (the most famous example being Godel using formal arithmetic
> >> to study arithmetic and extensions of arithmetic),
> >> metamathematics is mathematics.
> >
> > What a crackpot utterance! Where _exactly_ did _you_ learn Goedel was
> > "using formal arithmetic to study arithmetic and extensions of arithmetic"?
> >
> > By many respected professors and PhD's, GIT is a piece of _informal_
> > reasoning - using knowledge of *informal arithmetic*. Good grief!
>
> Has it become impossible for you to contribute to this newsgroup without
> ill-tempered ranting?

Peter Percival, as a crank of sci.logic, should know posters typically
don't refer to TF's own characterization of the cranks he made some dialog
with as "ranting". Cranks, especially hypocrite ones, deserve to be insulted,
Peter Percival should have known.

[Khong Dong]

Goedel did think it'd matter, for his Incompleteness effort, apparently.

[spe...@gmail.com]

So mathematics studies themselves. But nothing can contain itself!

[Dan Christensen]

Mathematicians need to talk things out when they get to an impasse. Even publish scholarly papers about it. Ultimately, of course, some kind of formalism (with axioms, rules of inference, etc.) would be necessary for any result in mathematics to have real legitimacy. At some point, the hand waving and speculation has to stop.


Dan

[Julio Di Egidio]

You too just have no idea what you are talking about.

No, meta-mathematics is not the same as informal mathematics. Indeed, again,
see GIT for example.

And no, informal mathematics is not the same as vague mathematics. In fact,
informal rather founds the formal!

And no, philosophy is not hand-waving either. Just shut up.

HTH,

Julio

[Peter Percival]

Hardly. A lot of mathematics was done before axioms were thought up.

[Peter Percival]

Julio Di Egidio wrote:
> On Sunday, 8 March 2020 15:05:17 UTC+1, Dan Christensen wrote:
>> On Sunday, March 8, 2020 at 5:39:04 AM UTC-4, spe...@gmail.com wrote:
>>> On Saturday, 7 March 2020 14:01:04 UTC+1, Julio Di Egidio wrote:
>>>> On Saturday, 7 March 2020 13:20:15 UTC+1, spe...@gmail.com wrote:
>>>>
>>>>> Is metamathematics mathematics or is it beyond mathematics?
>>>>
>>>> It is mathematics. And meta-meta-mathematics is still mathematics...
>>>>
>>>> At the boundary of mathematics we rather find the philosophy of mathematics.
>>>>
>>>> Julio
>>>
>>> So mathematics studies themselves. But nothing can contain itself!
>>
>> Mathematicians need to talk things out when they get to an impasse. Even publish scholarly papers about it. Ultimately, of course, some kind of formalism (with axioms, rules of inference, etc.) would be necessary for any result in mathematics to have real legitimacy. At some point, the hand waving and speculation has to stop.
>
> You too just have no idea what you are talking about.
>
> No, meta-mathematics is not the same as informal mathematics. Indeed, again,
> see GIT for example.
>
> And no, informal mathematics is not the same as vague mathematics. In fact,
> informal rather founds the formal!

Quite so. It must do. How can axioms be chosen without a prior notion
of what is correct mathematics?

[Julio Di Egidio]

On Sunday, 8 March 2020 16:23:33 UTC+1, Peter Percival wrote:

> Julio Di Egidio wrote:

> > And no, informal mathematics is not the same as vague mathematics. In fact,
> > informal rather founds the formal!
>
> Quite so. It must do. How can axioms be chosen without a prior notion
> of what is correct mathematics?

Poor analogy: correctness is among the notions that can be formalised.

How can a mathematician choose axioms without a meaning in mind...

Julio

[FredJeffries]

Sure it can:
The idea of the totality of all abstract ideas is an abstract idea
The catalog of all catalogs is a catalog
The database of all databases is a database
The census of all censuses is a census
...

But, anyhow, metamathematics does not 'contain' mathematics. It is just another mathematical tool. Some mathematical tools, like cardinal numbers, can be used to count thinks. Some mathematical tools can be used to measure things. Some mathematical tools can be used to examine fluid flows or the properties of sub-atomic particles.

Metamathematics is merely the branch of mathematics that examines what mathematicians do.

Like using a microscope to examine a microscope. Or a doctor examining another doctor or himself. Or using your human eye to study human eyes...

[Khong Dong]

Or, according to you apparently, using the set of all set to study some
set theoretical properties of a general set.

Great mathematical education system we're having today!

[spe...@gmail.com]

Not, it can´t.
The set of all is not a set.
The universe is not in the universe.
English is not english:"which is the less number than can´t be expressed with less than twenty words in english?"
There not exist the concept "concept".

[FredJeffries]

'The Category of Categories as a Foundation for Mathematics'

https://link.springer.com/chapter/10.1007/978-3-642-99902-4_1

Lawvere F.W. (1966) The Category of Categories as a Foundation for Mathematics. In: Eilenberg S., Harrison D.K., MacLane S., Röhrl H. (eds) Proceedings of the Conference on Categorical Algebra. Springer, Berlin, Heidelberg

[FredJeffries]

Fine. I have been trolled. EOD

[Jim Burns]

On 3/8/2020 5:39 AM, spe...@gmail.com wrote:
> On Saturday, 7 March 2020 14:01:04 UTC+1,
> Julio Di Egidio wrote:
>> On Saturday, 7 March 2020 13:20:15 UTC+1,
>> spe...@gmail.com wrote:

>>> Is metamathematics mathematics or is it beyond mathematics?
>>
>> It is mathematics. And meta-meta-mathematics is
>> still mathematics...
>> At the boundary of mathematics we rather find the
>> philosophy of mathematics.

> So mathematics studies themselves.

Mathematics can be used to study mathematics. Why not?

Goedel wasn't the first to use mathematics to study mathematics.

Descartes used numbers and equations to study geometry. Maybe
we take the connection for granted now, but they're very
different. And, it led to a proof that trisecting an arbitrary
angle with compass and straightedge is impossible. Using
mathematics to study mathematics gives us insight that we might
not have otherwise.

> But nothing can contain itself!

If you try to make that statement precise, I think you might find
yourself doing metamathematics.

Also, if you try to make that statement precise, I think that it
will turn out that it is true if you mean it in certain ways but
it is false if you mean it in certain other ways.

In the most-often-used set theories, ZFC for example, there is no
set of all sets. However, one can construct _in ZFC_ a set V and
membership relation 'e' (the set of all (x,y) in VxV such that
x e y) that satisfy the axioms of ZFC.
Does ZFC "contain" ZFC?
It depends upon what you mean by that.

Godel arithmetizes the reasoning behind arithmetic.
Maybe that means that arithmetic can contain arithmetic.

He uses that arithmetization to prove that, for arithmetic and
for any extension, there are unprovable sentences.
Maybe that means that arithmetic cannot contain arithmetic.

[Peter Percival]

Where has Fred Jeffries said that he thinks there is set of all sets?
Do you deny that one may look at a microscope with a microscope, or that
a doctor may examine himself?

[Me]

On Sunday, March 8, 2020 at 5:23:39 PM UTC+1, spe...@gmail.com wrote:

> The set of all is not a set.
> The universe is not in the universe.

In NF and NFU both is the case:

1. U := {x : x = x}

is a set

2. U e U .

Hint: https://en.wikipedia.org/wiki/Universal_set

Moreover there are set theories
(differing form NF and NFU; with no universal set) where there are sets A such that A e A.

See: https://en.wikipedia.org/wiki/Non-well-founded_set_theory

Lit.: https://www.amazon.com/-/de/dp/1575860082/ref=sr_1_fkmr0_1?__mk_de_DE=%C3%85M%C3%85%C5%BD%C3%95%C3%91&keywords=barwise+jon+and+larry+moss&qid=1583693240&sr=8-1-fkmr0

The subject of non-wellfounded sets came to prominence with the 1988 publication of Peter Aczel's book on the subject. Since then, a number of researchers in widely differing fields have used non-wellfounded sets (also called "hypersets") in modeling many types of circular phenomena. The application areas range from knowledge representation and theoretical economics to the semantics of natural language and programming languages. Vicious Circles offers an introduction to this fascinating and timely topic. Written as a book to learn from, theoretical points are always illustrated by examples from the applications and by exercises whose solutions are also presented. The text is suitable for use in a classroom, seminar, or for individual study. In addition to presenting the basic material on hypersets and their applications, this volume thoroughly develops the mathematics behind solving systems of set equations, greatest fixed points, coinduction, and corecursion. Much of this material has not appeared before. The application chapters also contain new material on modal logic and new explorations of paradoxes from semantics and game theory."

[graham...@gmail.com]

On Saturday, March 7, 2020 at 10:20:15 PM UTC+10, spe...@gmail.com wrote:
> Is metamathematics mathematics or is it beyond mathematics?

my take.......


ABOUT Mathematics often refers to the study of FORMULAS PER SE


FORMULA ::- FORMULA or FORMULA
FORMULA ::- FORMULA and FORMULA
...


So WFF are the beginning and end of meta-mathematics, and they lead to practically nothing... philosophy of logic.


Much better to study REDUCTION FORMULA

Given the FOUNDATION of Horn Clauses (a&b&..z)->f



A and B -> A or B
A and ~B -> A or B
~A and B -> A or B
~A and ~B -> ~(A or B)


See how much more PRACTICAL actually reducing OR is than a simple derelict WFF ?


That is why I think

UNIFY( function(args) , function(args) )


*IS* metamathematics, logic, and eventually mathematics.



If you cannot PREDICATE(IT)
It is NOT MATHEMATICS!