The world is in turmoil for want of sound reasoning. Economics and the environment are but two of many areas of human endeavour badly betrayed through a failed combination of physical and information science and the rule of law. Logic is the fabric of pure mathematics as the foundation of applied mathematics on which all science is based from the physical through biological and medical to the social sciences. However the symbolic logic of today seems of scarce more use than the syllogisms of Aristotle as observed by Francis Bacon nearly 400 years ago: The logic now in use serves rather to fix and give stability to the errors which have their foundation in commonly received notions than to help the search after truth. So it does more harm than good [Novum Organon Aphorism XII, 1620]

Heather, Michael

Rossiter, Nick

English

Northumbria Research Link

Northumbria Research Link
Citation: Heather, Michael and Rossiter, Nick (2009) A new process foundation for the applied topos.  
In: The 88th Peripatetic Seminar on Sheaves and Logic, Centre for Mathematical Sciences, 3-5 April  
2009, University of Cambridge, Cambridge. 
URL: 
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88th Peripatetic Seminar on Sheaves and Logic 
Centre for Mathematical Sciences University of Cambridge 
3
rd
- 5
th
 April 2009 
 
 
A NEW PROCESS FOUNDATION FOR THE APPLIED TOPOS 
 
60
th
 birthday tribute to the pioneering work of Martin Hyland and Peter 
Johnstone in the tradition of Cambridge mathematics 
 
Michael Heather & Nick Rossiter  
School of Computing, Engineering and Information Sciences, Northumbria University, 
NE1 8ST, UK 
michael.heather@cantab.net nick.rossiter@unn.ac.uk,  
WWW home page: http://computing.unn.ac.uk/staff/CGNR1/ 
  
EXTENDED ABSTRACT 
 
The world is in turmoil for want of sound reasoning. Economics and the environment are but 
two of many areas of human endeavour badly betrayed through a failed combination of 
physical and information science and the rule of law. Logic is the fabric of pure mathematics 
as the foundation of applied mathematics on which all science is based from the physical 
through biological and medical to the social sciences. However the symbolic logic of today 
seems of scarce more use than the syllogisms of Aristotle as observed by Francis Bacon 
nearly 400 years ago:   
 
The logic now in use serves rather to fix and give stability to the errors which have 
their foundation in commonly received notions than to help the search after truth. So 
it does more harm than good [Novum Organon  Aphorism XII, 1620]. 
 
The well known phrase of ‘new foundations’ was used by Quine for new axioms of set theory 
but it echoes a long tradition at Cambridge for foundation principles from Bacon’s call for a 
new version of Aristotle’s organon, through Newton’s Principia to the Principia 
Mathematica of Whitehead and Russell. However these  are but conspicuous nodes in a 
continuum of mathematical advancement at Cambridge. The separate painstaking work of 
Hyland and Johnstone on the in-depth study of the topos should surely be seen in that 
tradition when its significance is fully recognised. The historical development of the 
argument is summarised in the table below. 
 
The world’s present problems lie in in the logic of open interactive global systems --- this is 
the structure of the topos. In the light of current problems thrown up by the real world, the 
outcome of the work of the last thirty years shows clearly the need for new foundations. The 
topos so far has been constructed in a category theory based in a set theory of closed world 
axioms. Martin Hyland and Peter Johnstone have faithfully expounded the topos in the  
context of the twentieth century mathematics of proof theory on the one hand with the 
‘effective topos’ and within model theory on the other with sheaves, relations, allegories, 
sketches as well as in extensions of set theory like topology, homotopy, cohomology and 
differential geometry.  
 
An example of a topos structure is the tranching in the securatization of subprime mortgages. 
Based on first order models these were considered safer because of the internal spread of risk: 
 
As do the physicists who promote nuclear energy, those bankers who promote 
securitisation and the originate-and-distribute model stress that these financial 
innovations can benefit society. Indeed, some of the bankers who do the promoting 
are trained physicists and the models that they use to make their case borrow from 
science some very advanced mathematics. [Securitisation and the Originate and 
Distribute Model: Does it have a future? Speech by Thomas Huertas, Director, 
Banking Sector, FSA Euro 50 Group, London 21 April 2008] 
 
Unfortunately this ‘very advanced mathematics’ is not advanced enough. For it needs the 
internal structure of a preorder with a subobject classifier that can be either the initial object 
(false) or the terminal object (true) according to the extraneous ambient context. This is just 
an example of the well known ‘Schrödinger cat’ in quantum mechanics. Another example is 
the cause of climate cange which is likewise undecidable at the  first order level of statistical 
modelling. 
 
Another important area calling for more rigorous category theory is information technologies 
and communications including the development of the quantum computer where the current 
theories of the different interpretations of quantum mechanics are first order models and need 
the higher order construction of the topos. There are very many examples of problems with 
failed information systems. Whatever the design criteria and the project management issues 
there is always the underlying conceptual problem of the mapping of the real world on to the 
von Neumann architecture of current computers.  The architypal situation is to be found in 
Codd’s relational database model where there is always a need to loose information in the 
process of normalization of data. It is not usually possible to know in advance the status of 
this lost information. The problem arises directly out of the theoretical basis of set theory. 
 
Another pervasive example to be found over a wide field of applications is that of turbulence 
This has a long history of study by Cambridge applied mathematicians following the work of 
G I Taylor in the middle of the last century. However all attempts to produce an analytical 
solution to the problem have failed. Chaos theory is a first order ‘make do’ that again can 
lead to even dangerous consequences when risk analysis is needed in hazardous 
environments. 
 
Despite the later work on the axioms of set theory throughout the twentieth century, 
nevertheless the classical foundation for category theory still rests ultimately on Whitehead 
and Russell’s principia mathematica. The best candidate to provide the new foundations 
which accords with the naturaleness of (and as defined in ) category theory seems to be the 
process philosophy of Whitehead’s later period. There metaphysics replaces the model and 
assumptions can be discarded to give a reliable theory needed for the applied mathematics of 
the 21
st
 century. 
 
 
 
 
 
MATHEMATICAL 
DEVELOPMENT 
SIGNIFICANCE CONSEQUENCES CATEGORY 
THEORY 
Port Royal Logic (1662)  different possible 
logical 
views/meanings   
closed world assumption 
(CWA) only holds 
locally 
Exact/coexact 
adjointness 
Frege’s  predicate logic 
(1879) 
meaning as 
arithmetic 
Not universal because of 
CWA 
co-limits 
Whitehead & Russell’s 
(1910) 
principia mathematica  
axioms of set theory 
give rise to paradox 
pm repudiated explicitly 
by BR 
implicitly by ANW by 
conduct 
 
category of sets   
a model restricted 
by the axioms of 
set theory 
Gödel’s doctoral thesis 
(1930)  
first order (boolean) 
semantics complete 
20
th
 century science 
holds to first order 
Query eg methods for 
economic/climate  
models, LHC, etc 
validates current 
CT development  
to first order 
Gödel’s metamathematical 
5
th
 theorem (1930) 
predicates realizable 
as coded by  number 
logic of mathematics is 
not the logic of physics 
justifies number 
in CT by 
assumption 
Gödel’s 6
th
 theorem (1930) axiomatic systems of 
number/sets are 
undecidable 
Hilbert’s programme/ 
Entscheidungsproblem  
undecidable 
CT needs higher 
order validation 
Church conjecture (1932) effective 
calculability of 
number 
scientific method 
relying on 
measurement only 
valid to first order 
The ‘effective 
topos’ is a 
model 
Church-Turing Thesis 
(1936) 
effective 
computability of 
number 
computer with von 
Neumann architecture 
is a calculator not a 
logic machine 
 CT in computer 
science 
restricted to 
categorification 
Church-Turing-Deutsch 
Principle (1985) 
effective quantum 
computability in 
parallel 
quantum computer 
needs full quantum 
theory beyond first 
order model  
CT in quantum 
mechanics 
restricted to 
categorification 
Whitehead’s Process & 
Reality (1929) 
Universe as non-
staionary process 
Process is metaphysics 
while set theory is a 
model 
 arrow as 
process; 
 physical 
Universe as a 
topos; 
 CT as 
metaphysics 
instead of 
categorification of 
sets 
 


A new process foundation for the applied topos

Northumbria University Research Portal

88th Peripatetic Seminar on Sheaves and Logic Centre for Mathematical Sciences University of Cambridge 3rd- 5th April 2009   A NEW PROCESS FOUNDATION FOR THE APPLIED TOPOS  60th birthday tribute to the pioneering work of Martin Hyland and Peter Johnstone in the tradition of Cambridge mathematics  Michael Heather & Nick Rossiter  School of Computing, Engineering and Information Sciences, Northumbria University, NE1 8ST, UK michael.heather@cantab.net nick.rossiter@unn.ac.uk,  WWW home page: http://computing.unn.ac.uk/staff/CGNR1/   EXTENDED ABSTRACT  The world is in turmoil for want of sound reasoning. Economics and the environment are but two of many areas of human endeavour badly betrayed through a failed combination of physical and information science and the rule of law. Logic is the fabric of pure mathematics as the foundation of applied mathematics on which all science is based from the physical through biological and medical to the social sciences. However the symbolic logic of today seems of scarce more use than the syllogisms of Aristotle as observed by Francis Bacon nearly 400 years ago:    The logic now in use serves rather to fix and give stability to the errors which have their foundation in commonly received notions than to help the search after truth. So it does more harm than good [Novum Organon  Aphorism XII, 1620].  The well known phrase of ‘new foundations’ was used by Quine for new axioms of set theory but it echoes a long tradition at Cambridge for foundation principles from Bacon’s call for a new version of Aristotle’s organon, through Newton’s Principia to the Principia Mathematica of Whitehead and Russell. However these  are but conspicuous nodes in a continuum of mathematical advancement at Cambridge. The separate painstaking work of Hyland and Johnstone on the in-depth study of the topos should surely be seen in that tradition when its significance is fully recognised. The historical development of the argument is summarised in the table below.  The world’s present problems lie in in the logic of open interactive global systems --- this is the structure of the topos. In the light of current problems thrown up by the real world, the outcome of the work of the last thirty years shows clearly the need for new foundations. The topos so far has been constructed in a category theory based in a set theory of closed world axioms. Martin Hyland and Peter Johnstone have faithfully expounded the topos in the  context of the twentieth century mathematics of proof theory on the one hand with the ‘effective topos’ and within model theory on the other with sheaves, relations, allegories, sketches as well as in extensions of set theory like topology, homotopy, cohomology and differential geometry.   An example of a topos structure is the tranching in the securatization of subprime mortgages. Based on first order models these were considered safer because of the internal spread of risk:  As do the physicists who promote nuclear energy, those bankers who promote securitisation and the originate-and-distribute model stress that these financial innovations can benefit society. Indeed, some of the bankers who do the promoting are trained physicists and the models that they use to make their case borrow from science some very advanced mathematics. [Securitisation and the Originate and Distribute Model: Does it have a future? Speech by Thomas Huertas, Director, Banking Sector, FSA Euro 50 Group, London 21 April 2008]  Unfortunately this ‘very advanced mathematics’ is not advanced enough. For it needs the internal structure of a preorder with a subobject classifier that can be either the initial object (false) or the terminal object (true) according to the extraneous ambient context. This is just an example of the well known ‘Schrödinger cat’ in quantum mechanics. Another example is the cause of climate cange which is likewise undecidable at the  first order level of statistical modelling.  Another important area calling for more rigorous category theory is information technologies and communications including the development of the quantum computer where the current theories of the different interpretations of quantum mechanics are first order models and need the higher order construction of the topos. There are very many examples of problems with failed information systems. Whatever the design criteria and the project management issues there is always the underlying conceptual problem of the mapping of the real world on to the von Neumann architecture of current computers.  The architypal situation is to be found in Codd’s relational database model where there is always a need to loose information in the process of normalization of data. It is not usually possible to know in advance the status of this lost information. The problem arises directly out of the theoretical basis of set theory.  Another pervasive example to be found over a wide field of applications is that of turbulence This has a long history of study by Cambridge applied mathematicians following the work of G I Taylor in the middle of the last century. However all attempts to produce an analytical solution to the problem have failed. Chaos theory is a first order ‘make do’ that again can lead to even dangerous consequences when risk analysis is needed in hazardous environments.  Despite the later work on the axioms of set theory throughout the twentieth century, nevertheless the classical foundation for category theory still rests ultimately on Whitehead and Russell’s principia mathematica. The best candidate to provide the new foundations which accords with the naturaleness of (and as defined in ) category theory seems to be the process philosophy of Whitehead’s later period. There metaphysics replaces the model and assumptions can be discarded to give a reliable theory needed for the applied mathematics of the 21st century.      MATHEMATICAL DEVELOPMENT SIGNIFICANCE CONSEQUENCES CATEGORY THEORY Port Royal Logic (1662)  different possible logical views/meanings   closed world assumption (CWA) only holds locally Exact/coexact adjointness Frege’s  predicate logic (1879) meaning as arithmetic Not universal because of CWA co-limits Whitehead & Russell’s (1910) principia mathematica  axioms of set theory give rise to paradox pm repudiated explicitly by BR implicitly by ANW by conduct  category of sets   a model restricted by the axioms of set theory Gödel’s doctoral thesis (1930)  first order (boolean) semantics complete 20th century science holds to first order Query eg methods for economic/climate  models, LHC, etc validates current CT development  to first order Gödel’s metamathematical 5th theorem (1930) predicates realizable as coded by  number logic of mathematics is not the logic of physics justifies number in CT by assumption Gödel’s 6th theorem (1930) axiomatic systems of number/sets are undecidable Hilbert’s programme/ Entscheidungsproblem  undecidable CT needs higher order validation Church conjecture (1932) effective calculability of number scientific method relying on measurement only valid to first order The ‘effective topos’ is a model Church-Turing Thesis (1936) effective computability of number computer with von Neumann architecture is a calculator not a logic machine  CT in computer science restricted to categorification Church-Turing-Deutsch Principle (1985) effective quantum computability in parallel quantum computer needs full quantum theory beyond first order model  CT in quantum mechanics restricted to categorification Whitehead’s Process & Reality (1929) Universe as non-staionary process Process is metaphysics while set theory is a model  arrow as process;  physical Universe as a topos;  CT as metaphysics instead of categorification of sets  

http://nrl.northumbria.ac.uk/id/eprint/413/1/Heather_A%20New%20Process%20Foundation%20for%20the%20Applied%20Topos.pdf

A new process foundation for the applied topos

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