To apply the metaphysical methodology of mathematics to the logic and form of process in natural philosophy requires a metaphysics above modelling, a methodology more than method and a mathematics beyond the set based topics of arithmetic, algebra, geometry and topology. At the start of the twentieth century Alfred North Whitehead together with his former student Bertrand Russell was able to expound the form and logic of the mathematics of his day by the extensive treatment of axioms and theorems. The technical quality of this work found world acclaim and became the foundation for the advancement of science by the application of models still with us today

Heather, Michael

Rossiter, Nick

English

Northumbria Research Link

Northumbria Research Link
Citation:  Heather,  Michael  and  Rossiter,  Nick  (2009)  Process  categories:  the  metaphysics, 
methodology  &  mathematics,  philosophy  of  nature  and  process  philosophy.  In:  The  7th  annual 
meeting of Whitehead Metaphysical Society, 8-9 May 2009, Krakow and the University of Silesia, 
Poland. 
URL: 
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Philosophy of Nature and Process Philosophy 
Seventh annual meeting organized by The Whitehead Metaphysical Society  
with the Philosophy of Nature and Natural Science Section of Polish Philosophical Society,  
 Ancient and Medieval Philosophy Department of the Pontifical Academy of Theology in 
Krakow and the University of Silesia, Poland   
 8-9 May 2009 
 
PROCESS CATEGORIES: 
THE METAPHYSICS, METHODOLOGY & 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 
 
To apply the metaphysical methodology of mathematics to the logic and form of process in 
natural philosophy requires a metaphysics above modelling, a methodology  more than 
method and a mathematics beyond the set based topics of arithmetic, algebra, geometry and 
topology.  
 
At the start of the twentieth century Alfred North Whitehead together with his former student 
Bertrand Russell was able to expound the form and logic of the mathematics of his day by the 
extensive treatment of axioms and theorems.  The technical quality of this work found world 
acclaim and became the foundation for the advancement of science by the application of 
models still with us today. 
 
Rather curiously the authors themselves were less impressed with their own work. Russell in 
the later editions of Principia Mathematica effectively renounced much of the foundation on 
which it is built. Whitehead recalls that when Frege‟s attention was first drawn to Russell‟s 
paradox,  Frege‟s letter to Russell in response began,  “Alas, arithmetic totters!”  At the time 
they were in the midst of writing  the Principia Mathematica and this event led Russell to 
introduce his notion of types. Whitehead concludes:  
Thus the number “three” as applied to entities of one type has a different 
meaning to the number “three” as applied to entities of another type. For 
example, if we are considering two types, there are two different meanings of 
the number “three”. (Essays in Science and Philosophy, Rider 1948, page 79).   
 
Thus while never showing quite the same explicit dissatisfaction as Russell, Whitehead  
seems only to stand by the earlier work where generally it is correct in some context. The 
significance for today is that any context of valid application is always extremely narrow, 
being confined to some retricted locality. The post-modern world of global interconnectivity, 
interoperation and open systems is well beyond any narrow validity. Degeneracy in the 
meaning of number is commonly ignored: the tottering nature of arithmetic hardly 
recognised. In physics „ measurement‟ as number is still paramount in the scientific method 
despite Gödel‟s proof that any system dependent that concept is undecidable.  A recent 
example where this has become apparent is the academic discipline of international 
economics. It is found wanting where it relies on a method of axioms and theorems.  
 
Whitehead himself escalated his thoughts to a higher conceptual level with his metaphysics 
of process. However while he had a rich source to drawn on from the work of Grassmann, 
Frege and Hilbert to develop the form and logic of set theory in his Principia Mathematic, 
sadly he had no corresponding source to draw on for a formal presentation of  Process & 
Realiy. This seems to have been no great problem for Whitehead but it may be for us. The 
history of thought well demonstrates that scientific ideas can only be widely disseminated, 
understood  and applied when expressible in mathematical form.  
 
A mathematician of less calibre than Whitehead might have been tempted to pursue some 
formal primative representation of process but he made no attempt at this. Nevertheless since 
his time there has been advanced the subject of category theory  relying on the concept of the 
arrow. More from historical reasons that subject has grown out of set theory and topology and 
therefore is based in the axioms of set theory. This is really rather restrictive because the 
underlying permanency inherent in the concept of the set is anti-process thinking. The 
application of categories in both pure and applied mathematics tends to proceed by way of 
the model approach from which Whitehead was able to escape into metaphysics. Metaphysics 
presupposes a top down approach and to treat that bottom-up, as in finitary category theory, 
proves rather awkward for it is contravariant. 
 
A top-down approach begins with the Universe as process. The highest structure in category 
theory is the topos which can be identified as  an arrow, itself containing distinguishable 
identity arrows. That is distinguishable in the manner that every entity in the Universe is 
different but related to every other entity. This identification of the Universe as process seems 
really self-evident in the sense that everything is process. This concept  does not view the 
Universe as a container where things go on. Rather the Universe is just  „goings on‟. This  
does not even need the concept of entity or „thing‟. Following Aristotle, types can be defined 
as categories. This is in effect only a label for a recursive process. Within the topos, arrows 
are components relating other identity arrows. Each identity arrow may itself be composed of 
arrows that relate other identity arrows.  
 
Therefore while sets are represenatable as number, process is representable as the arrow:  
 
      
An arrow points in some ordered sense. That is it defines an ordering. At the highest level an 
arrow neither begins nor ends but just goes on. Alternatively this may be conceived as 
beginning with itself and ending with itself. In this situation the arrow identifies itself and is a 
„being‟ or an „object‟ which is the term used in finitary category theory.  
 
 
 
 
 
 
 
  
 
 
The arrow may begin on one entity and end on another.  
 
 
 
   
 
 
 
When the two beings are indistinguishable this is an isomorphism and would be given the 
same name. 
 
 
 
 
 
 
 
 
 
Here the unit 1 indicates an identity and the subscript is an arbitrary label. If the beings are 
distinguishable then some other label is also needed.  
 
 
 
 There the left hand being maps on to the right hand being. It would then be appropriate to 
give the arrow the label r (as like a function) it takes the values of the right hand being. 
Alternatively this can be viewed as a typing in the direction of the arrow. Here the left types 
the right. In process there is only one arrow and this arrow is just a subprocess where the left 
and right beings are really just the beginning and ending of the subprocess.  An isomorphism 
is a special case of adjointness There is always an implied reiprocal arrow in the opposite 
direction: 
 
 
 
 
 
 
 
 
This is Whitehead‟s „prehension‟ where a being „feels‟ another.  In finitary category theory 
the term used is „adjointness‟. For an isomorphism each arrow will be the inverse of the 
other. The more general adjointness amounts to an equivalence equipped with two measures 
the unit and co-unit of adjunction. For an isomorphism these are respectively just the initial 
and final process. Otherwise they are values in the preorder between the initial and the  final. 
A being is an intension for which there is any possible extension, adjoint to it.  
 
Category theory is not retricted by the constraints of set membership. It therefore avoids the 
problems of independence and atomicity of elements. It avoids too the undecidability of  
Gödel as well as Russell‟s paradox. It therefore has the the property of recursion to be found 
in the real world 
 
 
 
 
 
 
  
 
 
 
 
 
 
 
 
It is recursive up (or down) to four levels, that is across three layers. There is closure at the 
fourth level because any higher level is an alternative equivalent level. These four levels in 
finitary category theory are known as „topos‟, „natural transformation‟ „functor‟ and 
„category‟. Thus the topos is the identity natural transformation or the top „being‟. This can 
be identified with Whitehead‟s „category of the ultimate‟. It is the  metaphysical universe 
which consists of relations between relations between physical entities. Perhaps more 
correctly it should be termed Meta-meta-physics. 
 
It turns out that the topos has  general properties that might have been predicted from our 
experience of the physical Universe. The logic is intuistionistic—Heyting not Boolean. The 
topos has no intrinsic number system although a natural number object can be defined if 
desired.  It needs no axioms but can be freely generated by induction without assumptions or 
approximations.  
 
This demonstrates the methodological strength of metaphysics over modelling. A model is by 
its nature always an inferior theory whereas the metaphysics is either superior or at least co-
extensive with it. A model is a microcosm within the Universe whereas the Universe is an 
instantiation of the metaphysics. Consequently the mathematics of process categories is co-
extensive with ultimate closure. 
 
 


Process categories: the metaphysics, methodology & mathematics, philosophy of nature and process philosophy

Northumbria University Research Portal

Philosophy of Nature and Process Philosophy Seventh annual meeting organized by The Whitehead Metaphysical Society  with the Philosophy of Nature and Natural Science Section of Polish Philosophical Society,   Ancient and Medieval Philosophy Department of the Pontifical Academy of Theology in Krakow and the University of Silesia, Poland    8-9 May 2009  PROCESS CATEGORIES: THE METAPHYSICS, METHODOLOGY & 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  To apply the metaphysical methodology of mathematics to the logic and form of process in natural philosophy requires a metaphysics above modelling, a methodology  more than method and a mathematics beyond the set based topics of arithmetic, algebra, geometry and topology.   At the start of the twentieth century Alfred North Whitehead together with his former student Bertrand Russell was able to expound the form and logic of the mathematics of his day by the extensive treatment of axioms and theorems.  The technical quality of this work found world acclaim and became the foundation for the advancement of science by the application of models still with us today.  Rather curiously the authors themselves were less impressed with their own work. Russell in the later editions of Principia Mathematica effectively renounced much of the foundation on which it is built. Whitehead recalls that when Frege‟s attention was first drawn to Russell‟s paradox,  Frege‟s letter to Russell in response began,  “Alas, arithmetic totters!”  At the time they were in the midst of writing  the Principia Mathematica and this event led Russell to introduce his notion of types. Whitehead concludes:  Thus the number “three” as applied to entities of one type has a different meaning to the number “three” as applied to entities of another type. For example, if we are considering two types, there are two different meanings of the number “three”. (Essays in Science and Philosophy, Rider 1948, page 79).    Thus while never showing quite the same explicit dissatisfaction as Russell, Whitehead  seems only to stand by the earlier work where generally it is correct in some context. The significance for today is that any context of valid application is always extremely narrow, being confined to some retricted locality. The post-modern world of global interconnectivity, interoperation and open systems is well beyond any narrow validity. Degeneracy in the meaning of number is commonly ignored: the tottering nature of arithmetic hardly recognised. In physics „ measurement‟ as number is still paramount in the scientific method despite Gödel‟s proof that any system dependent that concept is undecidable.  A recent example where this has become apparent is the academic discipline of international economics. It is found wanting where it relies on a method of axioms and theorems.   Whitehead himself escalated his thoughts to a higher conceptual level with his metaphysics of process. However while he had a rich source to drawn on from the work of Grassmann, Frege and Hilbert to develop the form and logic of set theory in his Principia Mathematic, sadly he had no corresponding source to draw on for a formal presentation of  Process & Realiy. This seems to have been no great problem for Whitehead but it may be for us. The history of thought well demonstrates that scientific ideas can only be widely disseminated, understood  and applied when expressible in mathematical form.   A mathematician of less calibre than Whitehead might have been tempted to pursue some formal primative representation of process but he made no attempt at this. Nevertheless since his time there has been advanced the subject of category theory  relying on the concept of the arrow. More from historical reasons that subject has grown out of set theory and topology and therefore is based in the axioms of set theory. This is really rather restrictive because the underlying permanency inherent in the concept of the set is anti-process thinking. The application of categories in both pure and applied mathematics tends to proceed by way of the model approach from which Whitehead was able to escape into metaphysics. Metaphysics presupposes a top down approach and to treat that bottom-up, as in finitary category theory, proves rather awkward for it is contravariant.  A top-down approach begins with the Universe as process. The highest structure in category theory is the topos which can be identified as  an arrow, itself containing distinguishable identity arrows. That is distinguishable in the manner that every entity in the Universe is different but related to every other entity. This identification of the Universe as process seems really self-evident in the sense that everything is process. This concept  does not view the Universe as a container where things go on. Rather the Universe is just  „goings on‟. This  does not even need the concept of entity or „thing‟. Following Aristotle, types can be defined as categories. This is in effect only a label for a recursive process. Within the topos, arrows are components relating other identity arrows. Each identity arrow may itself be composed of arrows that relate other identity arrows.   Therefore while sets are represenatable as number, process is representable as the arrow:         An arrow points in some ordered sense. That is it defines an ordering. At the highest level an arrow neither begins nor ends but just goes on. Alternatively this may be conceived as beginning with itself and ending with itself. In this situation the arrow identifies itself and is a „being‟ or an „object‟ which is the term used in finitary category theory.             The arrow may begin on one entity and end on another.           When the two beings are indistinguishable this is an isomorphism and would be given the same name.          Here the unit 1 indicates an identity and the subscript is an arbitrary label. If the beings are distinguishable then some other label is also needed.      There the left hand being maps on to the right hand being. It would then be appropriate to give the arrow the label r (as like a function) it takes the values of the right hand being. Alternatively this can be viewed as a typing in the direction of the arrow. Here the left types the right. In process there is only one arrow and this arrow is just a subprocess where the left and right beings are really just the beginning and ending of the subprocess.  An isomorphism is a special case of adjointness There is always an implied reiprocal arrow in the opposite direction:         This is Whitehead‟s „prehension‟ where a being „feels‟ another.  In finitary category theory the term used is „adjointness‟. For an isomorphism each arrow will be the inverse of the other. The more general adjointness amounts to an equivalence equipped with two measures the unit and co-unit of adjunction. For an isomorphism these are respectively just the initial and final process. Otherwise they are values in the preorder between the initial and the  final. A being is an intension for which there is any possible extension, adjoint to it.   Category theory is not retricted by the constraints of set membership. It therefore avoids the problems of independence and atomicity of elements. It avoids too the undecidability of  Gödel as well as Russell‟s paradox. It therefore has the the property of recursion to be found in the real world                 It is recursive up (or down) to four levels, that is across three layers. There is closure at the fourth level because any higher level is an alternative equivalent level. These four levels in finitary category theory are known as „topos‟, „natural transformation‟ „functor‟ and „category‟. Thus the topos is the identity natural transformation or the top „being‟. This can be identified with Whitehead‟s „category of the ultimate‟. It is the  metaphysical universe which consists of relations between relations between physical entities. Perhaps more correctly it should be termed Meta-meta-physics.  It turns out that the topos has  general properties that might have been predicted from our experience of the physical Universe. The logic is intuistionistic—Heyting not Boolean. The topos has no intrinsic number system although a natural number object can be defined if desired.  It needs no axioms but can be freely generated by induction without assumptions or approximations.   This demonstrates the methodological strength of metaphysics over modelling. A model is by its nature always an inferior theory whereas the metaphysics is either superior or at least co-extensive with it. A model is a microcosm within the Universe whereas the Universe is an instantiation of the metaphysics. Consequently the mathematics of process categories is co-extensive with ultimate closure.   

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Process categories: the metaphysics, methodology & mathematics, philosophy of nature and process philosophy

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