Kurt Gödel heralds the Age of
Mathematical Enlightenment

The Demise of Number and the Axiomatic
method

Michael Heather and Nick Rossiter

University of Northumbria at Newcastle NE1
8ST

__michael.heather@cantab.net__

http://www.computing.unn.ac.uk/staff/CGNR1

Mathematics over the last three hundred
years has had great success in modelling phenomena of the physical world with
in turn geometry, algebra, differential calculus, set theory, tensors, etc,
often with new mathematics to support new physics. However, this success
sometimes obscures that there are also spectacular failures. The world of real
number, Boolean logic and Hilbert spaces cannot deal adequately with
turbulence, quantum theory consistent with relativity, normalized data in
information systems, consciousness studies or natural language and
intelligence.

A current example is particle condensation
where the *U(1) x SU(2) x SU(3)* Standard Model of elementary particles seems to have outgrown its origins in
group and gauge theory. For the classical symmetry of Landau cannot explain
fractional quantum Hall states at zero temperature. Topological orders can go
further but these only apply to states with a finite energy gap. For gapless
quantum states it is necessary to resort to orders of quantum theory. Ying Xiao-Gang Wen observed at the end of the 20^{th} Century that the
Standard Model of elementary particles in condensate phase in a vacuum is as
primitive and as inadequate as the atomic theory was for the nucleus at the end
of the 19^{th} Century.

Exact formal theory is still awaited for
most of the problems touching human affairs: economics, politics and government,
domestic and international law, the environment, global and interoperable
systems, biology and medicine. Why is there no theory of cancer?

An organism as a biological system must
surely still obey the laws of physics but it is a microlevel physics somewhere
between classical and quantum physics. The problem for applications and design
is that formal theories at the nanoscale level are to date very idealised for
example the Maxwell-Boltzmann gas distribution. These idealisations may well be
sufficiently close to reality for the closed independent systems traditionally
the subject of interest in classical physics. However, it now appears that
biology is concerned with the interoperation of a number of comparable open
systems and the behaviour of the organism is some meta-closure of the
interoperation. Classical mathematics has been successfully used to model
mono-systems. Reliance on set theory imposes a local limitation which is
inadequate for non-local conditions as found in interoperability.

The conclusions
from the work of Kurt Gödel is that the enlightenment of rational scientific
method that began to emerge seriously in the 17^{th} and 18^{th}
centuries still has some way to go even today in the 21^{st} century.
There are the three components of experimental, observational and the
theoretical and the latter is still 'incomplete':

__Experiment__

__ __

ALCHEMY → CHEMISTRY/PHYSICS

__Observation__

__ __

ASTROLOGY → ASTRONOMY

__Formal
Theory__

MATHEMATICS → MATHEMATICS

The subject of
mathematics is continuously evolving usually driven by science and technology.
The current position may be summed up:--

●
Rational
foundation of theory in 19^{th} and 20^{th} centuries rests on
informal intuition expressed as number or axioms

●
Validation
sought in Hilbert's programme

●
Hilbert's
programme has been *Popper--falsified*
by Gödel's results

●
completeness
only of first order predicate logic

●
unattainable
truth

●
number
requires the axiom of choice

●
*Number* needs Plato's ideals:

●
rejected in
Aristotle's reality

●
even
disbelieved by Plato himself later in the *Timaeus*

●
*Axioms* are impredicative and therefore have only
local validity:

●
Euclid
himself couldn't rely on the completeness of his own axioms even for geometry
(eg congruent triangles)

●
set theory
can't be founded on a complete axiomatic system

●
as a tool of
logical positivism axioms are what Bacon would call *idola theatri*

●
mathematics
relying on the intuition of number and axioms rests on superstition just as
much as did alchemy or astrology

●
enlightenment
is a gradual process even for the enlightened:

●
Newton
continued to devote much of his time to alchemy

●
Gödel still
continued with his work in set theory

__Naturality versus idealism__

The crunch point seems to come when moving
from ideal local systems to real non-local context-sensitive systems [Compare
the example above of the Maxwell-Boltzman gas distribution.] It is the
characteristic of naturality which has to be captured for most of today's
pressing problems. Properties of naturality not easily and coherently
represented formally in naďve or axiomatic set theory are:

●
variancy

●
formal
intuition and constructivism

●
process

●
freeness and
co-freeness

●
limits and
co-limits

●
emergence

●
preorders

●
adjointness

Glimpses of these can be found in topology,
studies of the French Bourbaki group, category theory, neo-logicism, universal and paraconsistent logic, the
process of Alfred North Whitehead and the assumption free studies for the last 25 years of the Cambridge
Alternative Natural Philosophy Association (ANPA). However these are all still
very much held back by the constraints of 19^{th} and 20^{th}
century mathematics

World problems are crying out for
mathematicians who appreciate the significance of the work of Karl Gödel