The Mathematico-Cognition Reality Theory (MCRT) Version 6.0 -------------------------------------------------------------------------------- "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 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 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 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 Domaiin 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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