Re: Why does Cantor a target for cranks?

In article <1166433380.992217.225570@xxxxxxxxxxxxxxxxxxxxxxxxxxx>,
"Albrecht" <albstorz@xxxxxx> wrote:

Virgil schrieb:

In article <1166185380.308915.39920@xxxxxxxxxxxxxxxxxxxxxxxxxxx>,
"Albrecht" <albstorz@xxxxxx> wrote:

Virgil schrieb:

In article <1166099791.719017.67260@xxxxxxxxxxxxxxxxxxxxxxxxxxx>,
"Albrecht" <albstorz@xxxxxx> wrote:

I don't know if Cantor is a target for cranks. In my opinion Cantor's
ideas are targets for questions because multiple causes.
1st: Cantor's ideas contradict not only the opinions of nearly all
great mathematicians before Cantor but also the opinions of nearly
all
great philosophers before Cantor.

That's to be expected of most new ideas.

No. The most new ideas augment the former ideas.

Little new ideas may be merely minor extensions of old ideas, but big
new ideas are, ore tend to be viewed as, revolutionary. Consider, for
example, relativity and quantum theory.

Einstein didn't blow away Newton. Heisenberg didn't expel Maxwell. The
revolutions in physics and sciences mostly wasn't in the sense of a
total negation of the former paradigm. Your examples are examples which
show that revolutions in science extend the former ideas but not
dismiss them. The revolutionary aspect was the opening of new fields of
thinking - not the changes in old fields of thinking.






2nd: Many mathematicians and nearly all mathematical laymen seems to
believe that the set of reals is greater than the set of naturals in
an
absolute sense.

In the Cantor sense, perhaps. Cantor defines an order relation on set
"sizes", which he calls cardinalities, by saying:
Card(A) <= Card(B) means there is an injective function from A to B.
One can easily verify that these "sizes" are ordered by the stated
definition, so that one has a proper order relation on cardinalities.

If one wants a concrete model, on can define in ZFC or NBG, the
cardinal
of a set to be the smallest, by inclusion, of all ordinals which biject
with the given set.

All this is well known. You need to affirm your own knowledge? But
that's not the subject of my 2nd thesis. You seems to belong to the
mathematical laymen I talked about. Without use of the axiom of
Infinity it is totally nonsensical to say that there are more objects
of any kind than natural numbers.

It is not nonsensical, it is merely unjustified without axioms to
justify it. But it is equally unjustified to claim that in the
non-physical world of mathematics there cannot be infinitely many
objects without having some sort of axiom to that effect.

That's wrong. Are you able to understand that there are things
thinkable which are nonsensical. How would you distinguish between
senseful things and nonsensical things? How want you to know whether
the unicorns in the math world are green or yellow striped?
Has to have an axiomatic system infinite many axioms to be definite?
You know only that for sure, what your axioms state (and their
syllogism) and axioms only make sense if they are well founded.

But how does one "well found" them without making assumptions about what
may properly be assumed?

Those who would require axiom systems to conform to their notions of
physical reality may very well have incorrect notions of physical
reality.


My point is that, at least in in pure mathematics, one cannot conclude
anything at all without SOME assumptions. And what you get depends on
what you assume.

What is bad with the assumption that math unicorns are yellow striped?

I do not know anything either bad or good about it, but I would not
chose to impose it on any mathematical axiom system.

What is nice with the assumption that infinite aggregates exist?

Together with suitable other assumptions, it leads to interesting
theorems.

How do you distinguish between good and bad assumptions in a scientific
way?

Science has nothing to do with it. Axiom systems do not fit into test
tubes.




They can't see that the consequences of ZF only hold in
the framework of ZF.

And the conseqeunces of other axiom systems likewise only hold within
those systems. So what?

Axiom systems are not the world.

They are the world of pure mathematics, as you cannot draw any
conclusions from nothing.

There is no "pure math" in this absolut sense. Pure math depends on the
real world too. Pure math bases on things like identity, causality,
equality, and so on. All this things are aspects of the real world.

On the contrary, things like identity, causality and equality are our
interpretations imposed on the real world.

That may be what you see, but I do not see anything mathematical being
valid outside of some axiom system. With NO assumptions at all, one
cannot deduce anything.

You miss totally the point. Surely we need always assumptions to
deduce something. But there are different kinds of assumptions. I
think, this is the centre of your problem. We may have the assumption
that the augmentation of a defined quantum with another defined quantum
leads to a defined quantum. The addition is based on this assumption.
We can prove this assumption in a finite number of cases and we can't
see any causes why this assumption should not hold in any of infinite
many cases. We call assumptions like this plausible assumptions. We may
use assumptions like this to build up axiomes.

It has been a plausible assumption for several millennia that there are
more than any finite number of points on a line.

Yes. And nobody disagree with this idea. But it makes no sense to state
that the line consists of infinite many points since this statement
leads to contradicting consequences.

If one were to insist that lines and points had any physical existence,
here might be contradicting consequences, but I am not aware of any
contradictions that arise from geometry in the abstract by assuming that
a line contains infinitely many points.

The sentence that there are more than any finite number of points on a
line isn't identical with the sentence that there are infinite many
points on the line (in the sense: actual infinite many points build up
a line). Herein lies a great difference. And the fault of Cantor' s
idea and ZFC.

The idea that there must be infinitely many points in a line has been
obvious since Euclid.


Cantor uses an assumption which isn't plausible and which isn't
provable for any case:

"Infinite discret aggregations exist".

Assumptions are assumed because unprovable.
It is equally unprovable to assume Cantor's assumptions wrong.

You use the assumption that it is false without being able to prove your
assumption. So you are doing the very thing you are criticizing.

I don't use the assumption that the sentence is false. I use the
assumption that the assumptions should be plausible.

Plausible to whom? I find the assumption that an endless sequence must
have an end to be implausible.


I am not aware that anyone has said that all mathematics MUST be done
all in one axiom system, though I can see that it might be a great
advantage if it were ever shown to be achievable.

I'm sure: This will be the death of all math if it will be (seemingly)
achievable.

Only in the sense of a Phoenix's death, from whose ashes it will be
reborn more beautiful than ever.



4.: Nobody knows if ZF is inconsistent or not. Therefore deductions
of
ZF must be proved very well and in some cases like e.g. the
Banach-Tarski-Paradoxon, math must be adapted to avoid antinomies.

No one can prove the consistency of any system sufficiently complex as
to allow the construction of standard arithmetic, so systems not known
to be inconsistent is as good as it gets.

Why are you unable to see that your argument isn't useful.

I think it supremely useful in showing that certain forms of logical
perfection are unachievable.

You think only in terms of aximatic systems.

Without assumptions, one has no mathematics at all. You too make
assumptions, including the assumption that all mathematics of any worth
must somehow mirror your version of "reality".

That last is an assumption that many mathematicians do not choose to be
constrained by.




We live in a
real world. All we know is not sure. But there are things which are
more sure than other things.

The axiomatic method does not in any way dispute what you have just
said.

The axiomatic method needs inputs. The axioms. This inputs stem from
the real world from real brains from real peoples.
Are you blind to be unable to see the point? If someone uses an
axiomatic system he has to reason the axioms he use. What are your
criterias to choose the one axiomes and to dismiss the others?

We choose to investigate the systems which interest us.
Just as those who play games choose which set of rules by which they
wish to play.


It is clear that the truth of what is derived from an axiomatic system
is ultimately dependent on the truth of its axioms, and that the truth
of its axioms can never be assured unequivocally.


Why do you try to suppress this truth to
defend the axiomatic method?

Why do you attack a method that has been a bastion of logic for
millennia?

What do you mean? I don't attack axiomatic method. I attack the faith
that axiomatic method is the only one method and I attack the
arbitrariness to choose axiomes in axiomatic method.

Which means that you are attacking the axiomatic method.

The axiomatic method is only the requirement that one make explicit all
ones assumptions over and above basic first order logic.
Within that method one has perfect freedom to choose whatever axioms one
wishes by whatever criteria one wishes.



Is Euclid your version of a mathematical anti-Christ?

Nonsense.





There are much more causes. Maybe, I'd listed the most important one
why many deep thinking people criticise Cantor's work and the
consequences of Cantor's ideas.

Do they equally criticize Goedel's work?

Do you?

No.

Me to.

Best regards
Albrecht S. Storz
.

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