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







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 20th 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 19th 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 17th and 18th centuries still has some way to go even today in the 21st century. There are the three components of experimental, observational and the theoretical and the latter is still 'incomplete':










Formal Theory




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 19th and 20th 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:



       formal intuition and constructivism


       freeness and co-freeness

       limits and co-limits





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 19th and 20th century mathematics


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