Re: Godel and Kant, and 'incompleteness'

John Jones schrieb:
The precision of mathematics is dependent on the more general
precision we employ for object definitions in the physical world.
Mathematics could not be more precise than this.

Actually the physical world and mathematics are to different
things. Mathmatics builds models of the physical worlds. And
guess what, most of the time they are wrong. The problem is
that when mathematics cannot find a closed solution, numerics
has to be applied. i.e. a bunch of fortran routines. But these
routines do make a discretization of time space. And thus there
is an error in the computation, besides the usual error of the
mesurement of the initial side conditions.

On the other hand closed solutions are often based of idealized
assumptions on side conditions. Things like a straight line
and so one. Closed solutions are then overly precise, and can
lead to wrong predictions, because the reality might behave
totally different. The same holds to formal logic. Its an
idealized model of a particular form of reasoning. The reasoning
for a given case, might be much more complex. And writing
it down in a formal way, might be too tedious.

I don't know what you mean by "object definitions" in the
"physical world". I think formal logic has a very high degree
of precision not only because it works with idealized models.
A further reason for its precious is the it uses mathematical
definition and proof techniques. These techniques I cannot
relate to "object definitions" in the "physical world". These
techniques are rather cultural achievement of our society.
Mathematical proofs and also proofs in mathematical logic
are not only sound waves emitted by the author of the proof.

They are speech acts. These speech acts perform an assertion
of the premisses and conclusion of the proof. And these speech
acts function because 50% of the university education of any
mathematicians has the only goal, to assure this communication.
So that when he leaves university, he can communicate with his
colleagues. The communication standards are directed towards
an unambiguous transmission of proofs. This is also archived
by using definitional and symbolic devices.

I think as soon as a transmission is ambiguous, and the
ambiguity cannot be resolved, the precision is lost. I don't
see that the same precision is found in any other communication
form other than mathematics. I must admit its very hard work
also to follow a mathematical text. But one can assume that
it contains mathematical assertions, and thus one is not
so much lost on its way. On the other hand arbitrary texts
are often less self contained, less precise and have sometimes
even problems to express assertions.

That other texts have sometimes problems to express assertions
has to do with the other 50% of the unversity education.
Mathematics not only provides a method to communicate savely,
it also provides some basic abstractions that can be used
during communication. These basic abstractions, lets call them
modelling brics, can be used in the fabric of the model that
is to be discussed. These modelling brics are again idealized
and have a broad range of applications.

When these modelling brics are recalled a whole set of
standard proofs pops to the mathematical mind. Example of
modelling brics are numbers, sets, etc.. If we look at the
current landscape we will see that each science has his
preferred modelling brics. In psychology you will see
utility functions based on linear or non linear combinations,
in economics you will see convex functions, etc.. But there
will be also a great deal of natural language.

What could be fruitful in the interaction between mathematical
logic and the other sciences, would be the introduction
of new modelling brics in each science. And/or the improvement
of modelling brics by the needs of each science. But this
is a very long way. But maybe software will play a special
role in this transfer. Computer algebra has already changed
the landscape how some math is done, and what can be done by
the individual or the team.

There are at least three different meanings of the word 'rational'. A
person can be precise in going about a murder, but we do not say that
the murder is necessarily rational. The idea of 'Mental illness'
> brings in a third meaning of the word rational based on social
expectations and behavioural taboos.

Yes sire.


.

Relevant Pages

  • Re: Response to Karen and to Willem on recursive proofs
    ... Because programming is APPLIED mathematics. ... than two places of precision. ... It appears that computer scientists with heroic exceptions like ... To sniff like a maiden aunt, "there is no excuse" is to be in denial ...
    (comp.programming)
  • Re: Applications of Mathematics
    ... precision using floating-point arithmetic. ... scientific mathematics. ... for determining mathematical nonsense? ... Toni Lassila, who I realize now was probably mocking ...
    (sci.math)
  • Re: its just simple questions..please answer these.
    ... >I've got some problems in doing mathematics. ... >the right is a fraction. ... >expressed with the precision of the least precise number. ... When the root of an approximate number is found, ...
    (sci.math)
  • its just simple questions..please answer these.
    ... I've got some problems in doing mathematics. ... the right is a fraction. ... expressed with the precision of the least precise number. ... When the root of an approximate number is found, ...
    (sci.math)
Loading