Modern Pythagorean

[wij]

Pythagorean: Any number can be infinitely approached by ratio p/q. There is
no REAL number (e.g. the diagonal of a square) that cannot be approached by
ratio number.

The limit theory perfectly says THE LIMIT of lim(x->c) f(x) is L, and
jumps to (logically invalid) conclusion f(c)=L (EQUAL).
lim(x->1) x= 1 ... yes x can be 1
lim(x->1) 1/(x-1)=? ... no x cannot be 1
lim(x->∞) n/x=0 ... whatever, the magic of infinity emerges. (no x in
ℝ can make 1/x=2/x=0, ∞ is also said not in ℝ)

PS. I believe TM (program,algorithm) should/would be the foundation of math.
These are for programmers.

[Ben Bacarisse]

wij <wyni...@gmail.com> writes:

> Pythagorean: Any number can be infinitely approached by ratio p/q. There is
> no REAL number (e.g. the diagonal of a square) that cannot be approached by
> ratio number.
>
> The limit theory perfectly says THE LIMIT of lim(x->c) f(x) is L, and
> jumps to (logically invalid) conclusion f(c)=L (EQUAL).

No. Take a look at any good text on the subject and you will find that
is explicitly /not/ a conclusion (though it is a possibility). f need
not even be defined at C. You don't seem to have even a basic
understanding of limits.

> lim(x->1) x= 1 ... yes x can be 1

Yup.

> lim(x->1) 1/(x-1)=? ... no x cannot be 1

Indeed. f need not even be defined at c for lim(x->c) f(x) to be
defined, though in this case, neither is defined.

> lim(x->∞) n/x=0 ... whatever, the magic of infinity emerges. (no x in
> ℝ can make 1/x=2/x=0, ∞ is also said not in ℝ)

You seem to think that lim(x->∞) is defined in the same way as
lim(x->c). It's unfortunate that the notation suggests they are the
same, but most students don't get confused. Do you know how lim(x->∞)
f(x) is defined?

> PS. I believe TM (program,algorithm) should/would be the foundation of math.
> These are for programmers.

But I am sure you won't do the work needed to develop a theory (and then
theorems) based on an algorithmic foundation to mathematics -- something
I'd like to see, by the way.

--
Ben.

[Richard Damon]

On 8/29/22 12:46 AM, wij wrote:
> Pythagorean: Any number can be infinitely approached by ratio p/q. There is
> no REAL number (e.g. the diagonal of a square) that cannot be approached by
> ratio number.
>
> The limit theory perfectly says THE LIMIT of lim(x->c) f(x) is L, and
> jumps to (logically invalid) conclusion f(c)=L (EQUAL).

Nope, as Ben says, this is not a true statement of limit theory.

In fact, there is a special name for functions that have the property
that f(c) = lim(x->c) f(x), that is they are continuous functions.

> lim(x->1) x= 1 ... yes x can be 1
> lim(x->1) 1/(x-1)=? ... no x cannot be 1
> lim(x->∞) n/x=0 ... whatever, the magic of infinity emerges. (no x in
> ℝ can make 1/x=2/x=0, ∞ is also said not in ℝ)
>
> PS. I believe TM (program,algorithm) should/would be the foundation of math.
> These are for programmers.

The problem with making the TM the foundation of math is that the TM can
only perform "computable" operations, and not everything in math is
actually computable.

[Skep Dick]

On Monday, 29 August 2022 at 13:49:22 UTC+2, richar...@gmail.com wrote:
> On 8/29/22 12:46 AM, wij wrote:
> In fact, there is a special name for functions that have the property
> that f(c) = lim(x->c) f(x), that is they are continuous functions.

HAHAHAHAHAHAHAHAHA😂😂😂😂😂😂😂😂😂😂😂😂😂😂😂😂

So you are overtly admitting that lim(x->∞) f(x) is NOT continuous.

> The problem with making the TM the foundation of math is that the TM can
> only perform "computable" operations, and not everything in math is
> actually computable.

Tadaaaaa! https://ncatlab.org/nlab/show/function+realizability

In this topos any x in R is represented as a continuous data stream, and functions on infinite/continuous data-streams are computable/realizable operatons in the usual sense of "realizability" https://ncatlab.org/nlab/show/realizability

In its finite form you would even recognize it as the stream-processing paradigm. https://en.wikipedia.org/wiki/Stream_processing

[Skep Dick]

On Monday, 29 August 2022 at 13:49:22 UTC+2, richar...@gmail.com wrote:

> The problem with making the TM the foundation of math is that the TM can
> only perform "computable" operations, and not everything in math is
> actually computable.

Also, this needs to be spelled out for you. Nobody is trying to make **Turing Machines** the foundation of math.

This is comp.theory - we are not married to any particular model of computation. We explore ALL models of computation.

Especially the ones which challenge the "standard" definitions.

[Richard Damon]

Yes, I am not saying that we can't talk about "other" computation forms,
where they are appropriate.

The problem with you is that when someone IS talking about Turing
Machine, or a specific branch of logic, you ignore the fact that they
are talking within a topic space an just spew irrelevent garbage.

Yes, you can talk all about whatever alternate form you want that fit
within the confines of computer theory, but you need to talk about them
in the topics that relate to them, and not in the topics that don't.


All you have shown is that you just don't understand the underpinning of
logic where you first establish SOME sort of ground work and work from
there. Yes, you can establish all different sets of ground work, or as I
call them "First Principles", but once established, you work in them.

Your lack of framework means nothing you say can have meaning.

[Richard Damon]

On 8/29/22 8:39 AM, Skep Dick wrote:
> On Monday, 29 August 2022 at 13:49:22 UTC+2, richar...@gmail.com wrote:
>> On 8/29/22 12:46 AM, wij wrote:
>> In fact, there is a special name for functions that have the property
>> that f(c) = lim(x->c) f(x), that is they are continuous functions.
> HAHAHAHAHAHAHAHAHA😂😂😂😂😂😂😂😂😂😂😂😂😂😂😂😂
>
> So you are overtly admitting that lim(x->∞) f(x) is NOT continuous.

You seem to not understand what continuous means.

"Limits" are not continuous, "Functions" are, and lim(x->∞) f(x) isn't a
function (there is no parametric symbol to be a function of).

Function F is said to be continuous at point c if f(c) = lim(x->c) f(x),
and continuous over an interval if it is continuous at all points in
that interval.

Since ∞ isn't a real number, we never evaluate f(x) at ∞ so we can't
talk about it being continuous there. At best we can talk about it being
continuous on an interval that goes to infinity, like [0,∞)

Note, the ∞ limit is open, not closed, as there IS no closed interval to
infinitiy available in The Reals.

>
>> The problem with making the TM the foundation of math is that the TM can
>> only perform "computable" operations, and not everything in math is
>> actually computable.
> Tadaaaaa! https://ncatlab.org/nlab/show/function+realizability
>
> In this topos any x in R is represented as a continuous data stream, and functions on infinite/continuous data-streams are computable/realizable operatons in the usual sense of "realizability" https://ncatlab.org/nlab/show/realizability
>
> In its finite form you would even recognize it as the stream-processing paradigm. https://en.wikipedia.org/wiki/Stream_processing

I will admit that I don't fully understand this, but I don't see
anything in there that actually talks about working in R, or
representing things as continuous data streams.

I do see mentions of the limitations of this sort of work, and even some
mentions of it being in opposition to some of the classical theories,
which says that it can't just be melded into those theories.


I get the impression that you are just throwing the spagetti at the wall
and seeing what sticks, all the while enjoying the mess you are making.

You made the mess, you get to clean it up.

[Ben Bacarisse]

Richard Damon <Ric...@Damon-Family.org> writes:

> On 8/29/22 8:39 AM, Skep Dick wrote:
>> On Monday, 29 August 2022 at 13:49:22 UTC+2, richar...@gmail.com wrote:
>>> On 8/29/22 12:46 AM, wij wrote:
>>> In fact, there is a special name for functions that have the property
>>> that f(c) = lim(x->c) f(x), that is they are continuous functions.
>> HAHAHAHAHAHAHAHAHA😂😂😂😂😂😂😂😂😂😂😂😂😂😂😂😂
>> So you are overtly admitting that lim(x->∞) f(x) is NOT continuous.
>
> You seem to not understand what continuous means.

Indeed. SD's remark has nothing to do with what you wrote about
continuity.

> lim(x->∞) f(x) isn't a function (there is no parametric symbol to be a
> function of).

In a very real sense it is, and a computer algebra system will make that
explicit in the notation (as in, say, limit[(λx.1/x), +inf]), but when a
mathematician is making a general remark about limits (say when defining
them) there is also an implicit parameter. That's traditionally done
using an unbound variable: "Let f be any function from R to R, then
lim_{x->c} f(x) is defined as...".

--
Ben.

[Richard Damon]

Yes, I suppose you could treat f and/or c as an implicit unbound variables.

[Skep Dick]

Parametrising. Binding. What’s the difference?

It still leaves open the question: why can you bind both Reals and inf to c?

What kind of domain contains both of those objects?

[Richard Damon]

The Type Union of Reals and Infintes.

Note, everything that uses that Type is defined polymorphically to use
different but related definitions for the two types.

If you system can't handle that type, it is defective for handling that
operation.

[Skep Dick]

But that union is just infinity?

> Note, everything that uses that Type is defined polymorphically to use
> different but related definitions for the two types.

Yes. Infinity.

> If you system can't handle that type, it is defective for handling that
> operation.

My system? It is your system…

You are handling infinites. I told you that all along!

[Richard Damon]

No, because that omits the Reals. Infinity isn't the whole number line,
just a concept that sort of represents the very end of that line.

>
>> Note, everything that uses that Type is defined polymorphically to use
>> different but related definitions for the two types.
> Yes. Infinity.
>
>> If you system can't handle that type, it is defective for handling that
>> operation.
> My system? It is your system…
>
> You are handling infinites. I told you that all along!

But not for arithmetic. In The Reals, "Infinity" is just a special
placehold that triggers special definitions. It maps conceptually to the
value at the end of the unending number line, but that isn't actually
its definition.