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"Mathematico-Cognition: Towards a Universal Ontology"
By Marc Geddes
Version 6
This version: 7th Sep, 2006
Auckland, New Zealand
Abstract:
'The skeleton out-line for a new metaphysics is presented. The
argument against reductive materialism is based on Mathematical
Platonism. The theory presented is a variant of Many-Aspect Monism, or
'Fundamental Property Pluralism'. The framework is aiming to be
universal in scope - a logical scaffolding capable of integrating all
general classes of knowledge under a single explanatory umbrella. It
is proposed that reality manifests itself as 3 different fundamental
knowledge domains - Physical, Volitional and Mathematical/Cognitive.
It is proposed that each domain has associated with it its own
definition of 'causality'. The radical new idea proposed is that
mathematical entities are not static, but can change their state by
moving through 'mathematical time'in 'configuration space'.
Reality itself is postulated to have a two-level structure reflecting
the difference between objects (concrete things with definite locations
in physical space and time) and classes (abstract things which are
universal in scope).
"Mathematico-Cognition: Towards a Universal Ontology"
Mathematical Platonism
Mathematical Platonism is the idea that mathematical concepts have
objective reality. The basic position is that human mathematicians are
engaged in *discovery* of mathematical facts that exist *out there* in
reality. Mathematical facts are not created by humans, but are things
which exist external to human society and are discovered. Mathematical
entities are patterns, or abstractions derived from concrete facts.
Mathematical Platonism is the idea that these abstractions have a real
existence external to the human mind.
Since mathematical Platonism is central to the theory presented here,
an over-view of the arguments in favor of Platonism will first be
given.
First, why should we believe in the objective existence of mathematical
entities? Surely, some will argue, mathematical entities are really
just abstract fictions (or invented languages) we use for describing
what are really material processes. This position is known as
nominalism.
However, there's an argument known as *The argument from
Indispensability*. Certain mathematical theories (for instance
analysis) are indispensable for modern physics. Physics uses
quantifiers which range over domains that include mathematical entities
not in space and time. Thus, the argument goes; since we have to accept
our best scientific theories of the world, we should accept that the
entities referred to in our theories really exist.
Now one could try to remove the references to mathematical entities in
scientific theories. For instance the philosopher Hartry Field (1980)
has proposed this - he suggested trying to remove talk of real numbers
in Newton's theory of gravity and replacing numbers with space-time
points and regions. But if one tries to do this, one finds that the
theories become enormously unwieldy - mathematical entities such as
numbers are just so *useful* in science. If there are entities in our
theories which it is very useful to refer to, this provides some
pragmatic grounds for believing in their existence. The argument at
work here is Occam's razor: in science the general rule of thumb is
that simple explanations are favored over more complex ones. Since in
science references to mathematical entities simplify scientific
theories, the simplest explanation is that these mathematical entities
really exist.
The physicist David Deutsch in his book 'The Fabric Of Reality', uses
the principle to establish 'Criterion for reality'. The idea is that we
should regard as real those postulated entities which, if we tried to
replace them with something else would complicate our explanations.
Deutsch's principle was this:
'If according to the simplest explanation, an entity is complex and
autonomous, then that entity is real.' ('The Fabric Of Reality', Pg 91)
As Detusch points out, mathematical entities do appear to match the
criteria for reality: 'Abstract entities that are complex and
autonomous exist objectively and are part of the fabric of reality.
There exist logically necessary truths about these entities, and these
comprise the subject-matter of mathematics.'
Professor of mathematics Roger Penrose also neatly makes the point that
mathematicians strongly feel they are engaged in discovery, not
creation, and that mathematical entities appear to have complex,
autonomous structure not put there by humans:
'The Mandelbrot set provides a striking example. It's wonderfully
elaborate structure was not the invention of any one person, nor was it
the design of a team of mathematicians. Benoit Mandelbrot himself, the
Polish-American mathematician who first studied the set, had no real
prior conception of the fantastic elaboration inherent in
it...Moreover, the complete details of the complication of the
structure of Mandelbrot's set cannot really be fully comprehended by
any one of us, nor can it be fully revealed by any computer. It would
seem that this structure is not just a part of our minds, but it has a
reality of its own.'
In 'Shadows of the Mind', Penrose goes on to make the very telling
point that mathematical theories often turn out to be useful for
science in a manner which goes far far beyond what the math was
originally used for. The example is given of the mathematics of
Einstein's general theory of relativity:
'In the early years after Einstein's theory was put forward, there were
only a few effects that supported it and the increase in precision over
Newton's scheme was marginal. However, now, nearly 80 years after the
theory was first produced, its overall precision has grown to something
like *ten million times* greater' ('Shadows of The Mind', Pg 415).
This is not at all what we would expect if the math was just an
invention of the human mind. As Penrose points out: 'Einstein was not
just 'noticing patterns' in the behavior of physical objects. He was
uncovering a profound mathematical substructure that was already hidden
in the very workings of the world.'
Accepting all this, a philosopher might concede that mathematical
entities have objective existence, but try to identity them with the
material world. However there are complications. It appears that there
exist perfectly good mathematical facts which be cannot be directly
identified with material facts in any simple way. A striking example of
this transfinite numbers and infinite sets- here references are clearly
made to infinite entities yet all available evidence would indicate
that all material entities are finite. Nor can infinite sets be argued
away as fictions - they are perfectly precise and logical mathematics,
having the same 'reality' as any other results in mathematics. Greg
Cantor developed a rigorous treatment of transfinite numbers and later
Abraham Robinson and John Conway did the same for infinitesimals.
Semantic considerations provide even more evidence for believing in the
existence of abstract entities. 'The Fregean argument' is based on the
idea that only in the context of a sentence does a word have meaning.
If a certain expression functions as a singular term in a sentence, and
the sentence is true, the sentence cannot be meaningful unless there is
an actual real singular entity to which the term is referring. For
instance if '2' functions as a singular term in a true sentence,
there must be a real entity '2' to the terms refers.
It is not clear whether mathematical Platonism conflicts with
materialism in the weak sense. Physicalism or materialism in the weak
sense, is here defined as simply the idea that everything has physical
properties associated with it. It is possible that all mathematical
properties have physical properties associated with them. However
mathematical Platonism appears to cast serious doubt on *reductive*
materialism, the idea that all mathematical properties can be
completely reduced to - or explained in terms of - physical
properties. It appears instead that physicalism is not an exclusive
metaphysics, or even a complete one. Mathematical properties appear to
be just as real as physical properties and further it is doubtful that
such properties can be identified with physical properties in any
simple way.
Functionalism
"Functionalism is a theory in the philosophy of mind that thinks of
mental states rather as we think of patterns. A pattern - say a
six-pointed star - can be made out of anything...The thing that makes
the pattern a star and not a circle or a crescent is the mutual
relation of its constituent parts, not the material out of which those
parts are made." (Consciousness: Guide to the Debates)
The assumption is that mental states are constituted in *computations*
and that there is no esoteric non-computational physics involved.
There's good evidence for this, based on the fact that all known laws
of physics are computational in nature.
Non-reductive Physicalism.
Non-reductive physicalism is the idea that mental concepts are
constituted in (associated with) physical processes, but descriptions
of mental concepts cannot be completely converted into descriptions of
material processes by precise laws.
A popular alternative philosophical position known as 'Eliminative
Materialism' - the idea that mental experiences are really just
illusions or mis-representations of material processes is based on weak
premises and non sequiturs
'Eliminative materialism' (the idea that 'qualia' are illusions or
misrepresentations of what are really entirely material processes) is
based on arguments by philosophers Paul and Patricia Churchland and
Daniel Dennett (in fact the position traces back to earlier arguments
by philosophers Paul Feyerabend and Quine). These arguments run
roughly as follows: (a) Qualia are simply abstract (or theoretical
entities) and (b) should be replaced by the objective scientific
viewpoint. But the argument undermines itself. One can agree that
Qualia are 'theoretical abstractions' and also agree that the correct
view-point requires an objective scientific account, but the conclusion
that Qualia are fictions doesn't follow from (a) and (b) at all!
There are examples of abstract entities (mathematical concepts) that
many (Platonists) take to be objectivity real, yet clearly don't
directly fit into the causal networks of the brain at all. As Kripke
showed (1972) such entities don't require causal contact. They can
be referenced through meaningful descriptions. Nor is causal contact
required to obtain evidence of such entities. Evidence for the
existence of something is based on the explanatory power of the
postulated entities for our theories of the world.
Conceding that the most accurate view-point of something is the
objective scientific view-point does not establish that Qualia are
illusions either. Although humans only know Qualia subjectively,
Qualia themselves could be a part of objective science. Suppose that in
order to achieve an accurate model of the behaviour of volitional
agents one needs to introduce mental concepts into one's explanations
right from the start - i.e. suppose this is an *in principle*
requirement? Then one would have to conclude that some mental concepts
are just as fundamental and real as physical concepts and the ontology
of objective science would have to be broadened to include these mental
concepts.
So the arguments of Eliminative materialists are weaker than generally
thought.
Different levels of Causality
Brain processes are enacting things which are *mathematical* in nature
- 'algorithms' (See 'Functionalism'). Mathematical entities are
abstracted patterns. But abstracted patterns themselves (like
'algorithms') don't exist directly inside physical causal networks,
only particular instances of them do. This is clear by pointing to the
fact that many different brains could enact the *same* computation
(algorithm) - the philosophical term is that the algorithm is 'multiply
realizable'. So the particular physical processes in the brain can't
be *identical* to the mathematical entity (the algorithm) itself.
It was an argument similar to this that led to the demise of the
original 'Identity Theory' of mind (a theory which attempted to
identity mental states with physical processes). Again, the trouble is
that many different brain states could be associated with the *same*
algorithm (or have the same mental states) which shows that physical
processes cannot be identified with mathematical entities in any simple
way. The weaker 'Token Identity' theories concede this, but still
attempt to equate mental states with physical processes. Couldn't one
simply say that there's some general high-level properties of physical
matter which can be equated with the algorithm, and hence dispense with
ghostly mathematical entities? The reason one can't really say this
boils down to Occam's razor and inference to the best explanation
again. Attempting to replace the concept of 'algorithm' with some high
level properties of physical matter is results in descriptions that are
enormously complex and unwieldy. And therefore such an arbitrary scheme
should be rejected, for reasons explained earlier. Inference to the
best explanation requires that we accept that mathematical entities
such as 'algorithms' really do have an objective existence above and
beyond a particular instantiation in material processes.
If we are prepared to grant objective reality to Abstracted patterns
(See 'Mathematical Platonism') one way to make sense of this is to
generalize the notion of causality to include *abstract* kinds of
causality. So long as postulated metaphysical entities (like for
instance 'Qualia') are defined as being at least *partially* inside the
network of physical causality, there'll be observable consequences and
scientific evidence for the existence of these entities can be
gathered. But this does *not* mean that metaphysical entities like
Qualia fit completely inside physical causal networks. Take the
analogy of a three-dimensional object (say a cube) passing through a
2-dimensional plane which we'll call Flatland. Just like a cube passing
through a 2-d plane has part of itself intersecting the plane, a Quale
could be *part* of physical causality without being entirely reducible
to physical reality.
Extra time dimensions
The combination of the above arguments can be used to argue for extra
time dimensions.
The view taken here is that everything in reality can be defined as
part of an 'event' (a cause and effect relationship). Causality is
central to modern science. Philosopher Donald Davidson revived the
notion of 'Events' as a fundamental category of ontology.
So *Time* is here being defined as 'Causality' - or ordered cause and
effect relations between things. Then *Time dimension* is defined to
mean a particular linear ordering of causal events. If as suggested,
there's more than one definition of causality, then this can be
interpreted as evidence for the existence of extra time dimensions.
*Multiple time dimensions* simply mean that there's more than one valid
way to define cause and effect.
There's nothing mystical about the notion of extra time dimensions. A
'time dimension' is simply a co-ordinate system for marking off events.
On the macroscopic scale there appear to be 3 dimensions of space and
1 of time. For instance to locate an event (say meeting a friend), one
has to give 4-ordinates. 3 involve space: Distance left-right,
distance forward-back and distance up-down. One involves time - for
instance 10'clock. This is the classical physics Einsteinian conception
of a 4-dimensional space-time (3 spatial dimensional, 1 time
dimension).
There have been string theorists exploring the possibility of extra
time dimensions. It may be that this idea could resolve some of the
paradoxes and confusion surrounding quantum mechanics. In quantum
physics, things behave as if they occupy more than one state at once.
Counter-factual states - 'what if's' or things that *could* have
happened strangely interfere with what *did* ?happen.? No one is quite
sure what this means. But string theorists Edward Witten and Cumrun
Vafa believe that string theory could resolve these puzzles. Cumrun
Vafa has proposed a 12-dimensional version of string theory (his
'F-variation') with two dimensions of time and 10 of space.
Cosmologists Stephen Hawking and James Hartle proposed the notion of
'Imaginary Time'. . In this theory, the time dimension can behave as if
it's just another dimension of space. The theory was originally
intended to deal only with extreme situations such as the beginning of
the universe, where the mystery of the origin of time could be resolved
by supposing that ordinary time changes into 'Imaginary Time'.
Many-aspect Monism (Fundamental Property Dualism)
Many-Aspect monism is a variant of Non-reductive physicalism. The idea
is that there is only one reality (monism) but this reality takes on
the appearance of multiple forms or properties. All of these different
properties are really reflecting (at least approximately) the same
underlying reality. They only *appear* to be different things. It is
here proposed that the notion of some underlying fundamental substance
can be entirely dispensed with. Like the aether before relativity
theory it's not necessary. Many-aspect monism does not require an
underlying substance. There are simply different kinds of fundamental
properties.
The idea has similarities to an earlier theory known as 'Identity
Theory'. That theory originated with Ullin Place (1956 paper, "Is
Consciousness A Brain Process?). It was defended by philosopher Jack
Smart (1959 paper, "Sensations and Brain Processes'. The defense makes
use of a distinction from philosopher Gottlob Frege (1848-1925) between
the sense (Sinn) of an expression and what the expression refers to
(Bedeutang). For instance the terms "morning star" and "evening star"
have different senses - one refers to a bright star seen early in the
day, another to a bright star seen later in the day - but in fact they
both refer to the *same* entity - the planet Venus. Similarly,
material brain processes and consciousness may *seem* to be different
things (we use mental and physical concepts in different senses), but
they are not.
The theory of Monism dates back to Baruch Spinoza (1632-1677), and a
variant known as Dual-aspect Monism was championed by Bertrand Russell
(1872-1970). Russell pointed out that reality consists of two general
types of properties - Intrinsic - or Independent properties of objects
- and Extrinsic - or properties of objects relative to other objects.
Russell proposed to equate Intrinsic properties with Mental properties
and Extrinsic properties with Physical things. So the idea was
everything had these two properties associated with it - mental and
physical. (Hence the expression 'Dual-aspect Monism'). David
Chalmer's has proposed 'Type-F Monism' as an option to resolve
the mind-body problem.
Triple aspect monism
A variant of Many-aspect monism is here being proposed. It is
suggested that reality manifests itself as 3 different fundamental
classes of properties. The idea is that everything in reality has
3-fundamental aspects associated with it. There are, if you like, 3
different valid perspectives through which one could view the whole of
reality (see summaries of 'Many Aspect Monism' above).
It was argued above that there was more than one form of causality and
by implication, more than one time dimension. It is here proposed that
there are three different kinds of causality. If there are really
three different kinds of causality, 'events' along each of the three
time-lines will have their own fundamental metaphysical categories
associated with them.
The Mathematico-Cognition Ontology
The ontology here proposed has 9 primitives:
(1) Physical (PH): Knowledge domain related to concrete material
things and processes. This domain is associated with energy - the
capacity to do work. Physical entities have locations in physical
space.
(2) Volitional (VO): Knowledge domain related to teleological (goal
directed) things and processes. This domain is associated with
volition - the capacity to make choices.
(3) Mathematico-Cognitive (MA): Knowledge domain related to
Mathematical and Cognitive properties. This domain is associated with
abstract knowledge, or knowledge which is universal in scope.
(4) Matrix (MA): Background substance in which continuants reside
(5) Continuant (CO): Entities which maintain stable identities
(6) Occurrent (OC): Processes - things with temporal parts
(7) Independent (IN): Loosely, Intrinsic. Property of the thing in
itself
(8) Relative (RE): Loosely, Extrinsic. Property of the thing
formed from its relationship with other things
(9) Mediating (ME): Property of an object which describes its
ability to mediate between two other objects.
Categories from each set can be combined. A 3*3*3 matrix gives the
possible combinations. There are 27 fundamental categories proposed.
John Sowa's KR ontology was a major influence on the ontology here
proposed and some of the categories here used are close in meaning to
the categories of the KR ontology. In particular all the physical
categories here proposed are similar to the physical categories in the
KR ontology. However some additional primitives are being used here,
and many of the categories for the Volitional and Mathematico-Cognitive
domains are quite dissimiliar to the KR categories - although the
same names are used where the category in question is in fact quite
similar.
Preliminary skeleton descriptions for the 27 fundamental categories are
given below, by identifying the categories which appropriate names
which suggest possible definitions.
Physical Domain
Independent Matrix: ENERGY
Relative Matrix: LOCATION
Mediating Matrix: EXTENSION
Independent Continuant: OBJECT
Relative Continuant: JUNCTURE
Mediating Continuant: STRUCTURE
Independent Occurent: PROCESS
Relative Occurent: PARTICIPATION
Mediating Occurent: SITUATION
Energy -Capacity to do work
Location-Position. Point ordered with respect to other points
Extension-Spatial extent.
Object-Mass (non-kinetic or concentrated) energy
Juncture-Geometric relationship between two objects
Structure-Aggregated objects
Process-Integrated behaviours of objects
Participation-Interaction
Situation-Things which impose constraints on object behaviour
Volitional Domain
Independent Matrix: VOLITION
Relative Matrix: UTILITY
Mediating Matrix: UNITY
Independent Continuant: AGENT
Relative Continuant: ROLE
Mediating Continuant: REASON
Independent Occurent: ACTION
Relative Occurent: ACTIVITY
Mediating Occurent: PURPOSE
Volition - Capacity to make choices
Utility - Preference for a desired outcome relative to other
outcomes
Unity - Co-ordinated choices, or loosely, memes.
Agent - Entity demonstrating goal-directed behaviour
Role - Social status of an agent relative to other agents
Reason - A culture or sub-culture - loosely, ordered groups of
agents.
Action - External behaviour of an agent
Activity - Effect of agent behaviour on other agents
Purpose- Co-ordinated agent behaviour - projects or events
Mathematico-Cognitive
Independent Matrix: INFORMATION
Relative Matrix: STATE
Mediating Matrix: COMPLEXITY
Independent Continuant: FORM
Relative Continuant: PROPOSITION
Mediating Continuant: THEORY
Independent Occurent: MODALITY
Relative Occurent: THOUGHT
Mediating Occurent: DELIBERATION
Information - A variance, or 'difference'.
State - Manifestation of a system relative to other possible
systems.
Complexity - Not ordinary statistical complexity but a measure of
the quality of the information - loosely a measure of knowledge -
or the amount of work it took to produce the information.
Form - Pure mathematical entity - integrated knowledge
represented as a static thing
Proposition - Representation of the relationship between two forms
Theory- Propositions which have been integrated into unitary
explanatory frameworks
Modality Qualia - A reflection on perception
Thought Qualia - A reflection on the link or association
between two perceptions
Deliberation Qualia -A reflection on an integrated set of
perceptions
The Mathematico-Cogntive domain represents the 'Platonic' or abstract
world (See 'Mathematical Platonism'). This is a 'space of possible
states' (see the Matrix categories). This is an abstract 'state
space' (or 'configuration space). The radical idea embedded in
this ontology is that the space of mathematical possibilities is not
static - as standard Platonism requires. Recall that more than one
kind of causality was proposed. The scheme here proposed is that
configuration space has is own kind of 'abstract time' (or
'mathematical time') associated with it. It's the movements of
mathematical continuants through this 'time' that make up the
mathematical occurrents. In other words, it's being suggested that
some mathematical entities are not static - mathematical truth is not
fixed in the way most Platonists believe. Some mathematical entities
can under-go changes of state which are 'movements' through
configuration space.
It is also here postulated that many cognitive properties are identical
to mathematical properties. Thus the label
'Mathematico-Cognition'for this domain. The cognitive properties
deemed identical to mathematical properties are those properties
associated with the perception, integration and reflection on knowledge
which is universal in scope.
The Mathematico-Cognitive domain represents knowledge which is
universal in scope. Mathematical (abstract) entities do not have
locations in physical space or time. This is a major difference to
Volitional and Physical entities. The Physical and Volitional domains
are the 'Object Level', concerned with concrete things. The
Mathematico-Cognitive domain is the 'Class Level', concerned with
abstract definition.
A major possibility will be suggested here ; the possibility that the
mathematical occurrents are identical to Qualia (the phenomenal aspects
of consciousness). These properties are sufficiently subtle and
original that they could one day provide an explanation for the great
puzzle of Qualia. Recall that the mathematical Occurrents represent
the integrated 'movements' of mathematical Continuants through
configuration space. (recall that in this scheme, not all mathematical
truths are fixed). If this idea is correct then Qualia are a sort of
'higher-order causality' - that form of abstract causality which
takes place on the Class level of reality (in configuration space) and
is associated with the perception, integration and reflection on
knowledge. Spoken more poetically, Qualia are 'mathematical
motions'.
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Received on Wed Sep 06 2006 - 03:01:11 PDT
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