On The Nature Of Physicality And Mathematical Descriptors
Posted: Thu Jan 01, 2015 12:27 am
Take a moment to consider objects as sets of properties. Objects have properties, sets have members, so there is a parallel here (even if properties are not fundamental, they can still be objects in a set or be had by objects) We may make additional sets out of these sets by asking what properties they have as members and making a logical construction to determine the members of the new set. For example, if we say Hp =def has p and P =def the set of all properties, then we can say a set is defined by the statement: ((Hp∨Hq)∧(¬Hx))∧(p,q,x∈P). If this is the case, then so long as objects have property p or q and do not have property x, they are included in this set. Now, consider that objects do in fact have properties, and do not need to be sets for this method to work. Therefore, we have found a logical way of dividing up objects into sets.
Let us consider how we might go about finding a logical construction that defines what it is to be physical; what is the set of all physical objects? For example, say that physical objects are those that have location. This is useful, but now we have to consider the much more subtle problem of what it is to have location. To have location usually to be in a space. Generally, locations are specified by coordinates that specify points in a space. It might be tempting to identify a location with a point. However, points have locations; they are not locations themselves. Further, other objects such as shapes and extensions have locations too. Thus, location is a property of these sorts of objects, and is not identical with any of them. Let us return to the concept of a space. The minimum requirement for something to be a space is that it is a set of objects with added structure. This added structure (mathematical structure) involves adding mathematical objects such as measures, groups, fields, topologies, metrics, orders, equivalence relations and categories, to name a few. Since sets can contain any kind of object, what this boils down to is that a "physical" space (a space that contains exclusively physical objects, but may not actually have the property of being physical) is a structured set of physical objects. Thus, to have a location is to be a member of a structured set of exclusively physical objects. Therefore, a thing having a location in a space-time manifold, for example, is not sufficient to warrant categorizing something as physical.
Let us take a step back now and look at objects that are physical. We need to search for any commonalities among them. Then, we need to locate commonalities that are unique to physical objects. What are some physical things? Tables and chairs are physical. Particles are physical. Points are not physical, nor are shapes. What do chairs and particles have that shapes and points lack? They have energy. Energy is, in effect, the ability to do work. Work, in essence is change. Therefore, the thing that differentiates particles and chairs from points, shapes, even physical space, is that they change without losing their identities. Perhaps the essence of physicality is change with identity.
It is a controversial claim that only physical things change. What about mental objects, for example? Beliefs seem to change over time. However, this may be an illusion. It may be that mental objects are eternal and are simply instantiated at different times. For example, when we think of a number, it is a mental object in time, but it does not change over time. The same could be said of the belief that it is sunny. The belief itself does not change when a person changes their mind about that belief. The belief is replaced by another belief. Hence, beliefs can exist in time, but do not change. This is consistent with the multiple realizability hypothesis, which earns it bonus points.
What about propositions? Do they change? I think not. Even propositions that can be true at one moment and not another do not actually change. This is because they include, contextually, an implicit timestamp. In other words, “it is raining” actually means “it is raining at such and such a place in space-time”. This proposition can never be made to be false if it is true and vise versa and hence, never gains or loses a property. All propositions, when considered in their entirety do not gain or lose properties. Further propositions may not even exist in time because they refer to a location in space-time and particular conditions at the locale just as coordinates refer to a point in a mathematical space. The coordinates are location-less. The same is true for propositions that refer to temporal phenomena - they do not actually have a location in space-time; they only refer to such a location.
It seems that only physical objects have the ability to change without losing their identities. Let us consider how this might happen. Objects have properties and they change when they gain or lose properties. Normally, when an object gains or loses a property, it is said to no longer be the same object. The implications of this way of looking at numerical identity was a source of great controversy in ancient times. The Greek philosopher Zeno developed an argument that involved the following premises: (1) objects cannot survive gaining or losing properties; (2) physical objects lose and gain a property every time they move. Conclusion: motion is impossible; change is an illusion because it is a contradiction, for it involves something coming into and going out of existence at the same time. One popular way to resolve this paradox is to simply say that spatial location is not a property that plays into determining the identity of an object. A less ad hoc version of this hypothesis is that relational properties can be gained or lost by physical objects without destroying their identities. However, this way of looking at things appears to fail. However, there are particles that seem to have all the same properties (bosons) in a system, except that they are in different locations. It seems clear that these objects, though they are very similar - qualitatively identical - they are not numerically identical.
Another possible solution to the problem of physical numerical identity is continuity. The continuity hypothesis is that if an object is continuous with itself, then it is identical with itself. However, there are many contradictions in this account. For example, if an object gradually fissions into two pieces that are identical to each other and the original object in every way except for their spatial locations, which one is the “real” one? A more relevant example of a contradiction in the continuity account comes from quantum mechanics. In the quantum world, motion is discontinuous. If we only had one particle of a certain kind, then this would not be a problem, because we could infer that the object in one location at one time is the same as the object with the same properties in another location at a different time. However, what of the many cases in which there are multiple indistinguishable particles? How could we conceivably determine which particle is numerically identical with which particle at a different time? One response to this is to say that there is simply an identity property which physical objects share with their selves at other moments. In other words, there is a fact of the matter which object is which, even if we cannot find that out in reality. However, quantum statistics rules this out. To understand why, consider two indistinguishable particles which can each be in one of two boxes. We would think that there are four possible configurations. One in which both particles are in one box, one in which they are in the other box, one in which they are both in separate boxes, and one in which the particles are switched. Now, even if we couldn’t distinguish between these particles, the frequency of these occurrences, if the particles were randomly distributed, would be as follows: one fourth of the time they would be in one box, one fourth of the time they would be in the other box and half the time, they would both be in separate boxes (though we could not tell which was which). However, in reality, when indistinguishable quantum particles are distributed into two boxes, they both occupy one box one third of the time, they occupy the other box one third of the time and they are evenly distributed one third of the time. This implies that the even distribution of the particles is one state and one state only. In other words, there is no fact of the matter about which particles are which.
What are the implications of this strange finding? For one thing, this implies that lone particles do not actually change over time. If they "change" locations, they are no longer the same particle that they were before. What of larger physical systems? Well, if a system’s identity is determined by its members (particles), then it too cannot maintain an identity over time. This seems to be a big blow to the hypothesis that physicality is defined by identity and change. It is. However, something does change. For one, the universe changes. If it did not, there would be no illusion of change. Or would there be?
Let us consider a phenomenological account of change. It is possible that there is only one moment that exists. We infer that there is change, and time, because we have the sensational experience of change. However, as we explored earlier, mental and abstract objects may not actually change. If they do not change, then it is possible that this sensation of change is actually a static mental object, just like the beliefs and other kinds of experiences. What of memories? Why would we have memories if there is not a past? Well, memories must, even in the context of time, exist at the "present" (or only) moment. We experience the past by the juxtaposition of various miniature "worlds" of experience. Memories, sensations, emotions, predictions and abstract thoughts all exist together in the mind. Therefore, our memories are like projections onto a plane. Our experience of time resembles the image of a mirror facing a mirror. There is information about different experiences contained within a larger image and so forth. Therefore, let us question the supposition that there is something that changes at all. A belief can "change" by being replaced with another belief, but how do we know that a belief has changed? We know this because the old belief is juxtaposed with the new belief, one in memory and the other in more recent experience. Therefore, it is conceivable that the change that occurs as a result of the replacement of one thing with another (which we have shown happens in the physical, the mental and the abstract worlds) is as much an illusion as the maintenance of identity over time.
The conclusion that there is no change does not imply that there is no time, nor does it imply that there is no physical world. Time could be a dimension of a manifold, or of our experience. The physical world may be distinguished by some unique property other than location or change, which we have eliminated. There is another common account of the physical. This account is that physical objects are those objects that we can sense. However, there appears to be no clear boundary between sensory experiences and imagined ones. Illusions and hallucinations dispatch this account rather quickly.
What else might be unique to physical objects? Let us take a step back and consider why we identified change and identity as the key players in physicality. Energy. It seems that all physical objects have energy, and it is the driver of change. Therefore, if we consider the property of having energy to be the common property that is unique to all physical objects, we might be able to save our hypothesis without modifying the theory in desperate ways. What does it mean to have energy?
The word energy is derived from the Ancient Greek word energeia (ἐνέργεια) which translates to "activity" or "operation". In other words, change. The law of conservation of energy is understood to be the result of its translational symmetry with space-time. That is, that the laws of physics do not change over time or from location to location. It seems that to say a system has energy is simply to say that it changes. However, let us examine the definition of energy more carefully.
Energeia can mean "activity" or "operation". Take the word "operation" literally, and we are back in business. Energy can mean, "to operate". Operators are mathematical, unchanging objects. Perhaps energy is a kind of operator that translates into activity or change. Is this simply a case of equivocation? I think not; energetic transformations happen via operators. Perhaps energy is a blanket term for some structure that exists in the physical world in addition to the traditional space-time account.
Minkowski space is one kind of structure in the physical world. However, General Relativity shows that there is additional structure in space. Mass bends space-time. However, General Relativity does not really account for the exact locations of the various masses in space-time. This seems like an unsatisfactory account of the structure of a physical space. After all, a space is a set with added structure. In this case, a set of physical objects. Should the structure not be complete? That structure ought to be statable in its entirety in mathematical terms. Might it be possible to describe the entire physical world as a space, or nested spaces? Could it even be possible to describe the entire universe, including phenomenological structures in this way? This is an exciting prospect. However, even if it is possible, I would bet that the descriptors of truths in the universe are as numerous and enigmatic as mathematical truths are. That is, there are an infinite number of them and they all must be proven in their own unique way. Gödel’s Incompleteness Theorem shows that no set of axioms can account for all mathematical truths in any given system. If this is the case, it is likely that, if the universe is describable in mathematical spatial terms, it is infinitely complex and no finite set of axioms can describe it.
Nonetheless, the concept of describing both the physical world and the mental world as spaces is promising. Perhaps we will develop ever more detailed models and spaces that describe the universe until the realms of the physical and the mental are no longer distinct. That is the main conclusion of my rumination. That physicality, at base, appears to have no real distinguishing features. Perhaps the reason for this is because the descriptors of physical reality can also be applied to other realms of experience. Perhaps it is even an accident that we discovered so many physical truths before understanding the structure of consciousness.
Thanks for reading!
Cheers,
Oliver
Let us consider how we might go about finding a logical construction that defines what it is to be physical; what is the set of all physical objects? For example, say that physical objects are those that have location. This is useful, but now we have to consider the much more subtle problem of what it is to have location. To have location usually to be in a space. Generally, locations are specified by coordinates that specify points in a space. It might be tempting to identify a location with a point. However, points have locations; they are not locations themselves. Further, other objects such as shapes and extensions have locations too. Thus, location is a property of these sorts of objects, and is not identical with any of them. Let us return to the concept of a space. The minimum requirement for something to be a space is that it is a set of objects with added structure. This added structure (mathematical structure) involves adding mathematical objects such as measures, groups, fields, topologies, metrics, orders, equivalence relations and categories, to name a few. Since sets can contain any kind of object, what this boils down to is that a "physical" space (a space that contains exclusively physical objects, but may not actually have the property of being physical) is a structured set of physical objects. Thus, to have a location is to be a member of a structured set of exclusively physical objects. Therefore, a thing having a location in a space-time manifold, for example, is not sufficient to warrant categorizing something as physical.
Let us take a step back now and look at objects that are physical. We need to search for any commonalities among them. Then, we need to locate commonalities that are unique to physical objects. What are some physical things? Tables and chairs are physical. Particles are physical. Points are not physical, nor are shapes. What do chairs and particles have that shapes and points lack? They have energy. Energy is, in effect, the ability to do work. Work, in essence is change. Therefore, the thing that differentiates particles and chairs from points, shapes, even physical space, is that they change without losing their identities. Perhaps the essence of physicality is change with identity.
It is a controversial claim that only physical things change. What about mental objects, for example? Beliefs seem to change over time. However, this may be an illusion. It may be that mental objects are eternal and are simply instantiated at different times. For example, when we think of a number, it is a mental object in time, but it does not change over time. The same could be said of the belief that it is sunny. The belief itself does not change when a person changes their mind about that belief. The belief is replaced by another belief. Hence, beliefs can exist in time, but do not change. This is consistent with the multiple realizability hypothesis, which earns it bonus points.
What about propositions? Do they change? I think not. Even propositions that can be true at one moment and not another do not actually change. This is because they include, contextually, an implicit timestamp. In other words, “it is raining” actually means “it is raining at such and such a place in space-time”. This proposition can never be made to be false if it is true and vise versa and hence, never gains or loses a property. All propositions, when considered in their entirety do not gain or lose properties. Further propositions may not even exist in time because they refer to a location in space-time and particular conditions at the locale just as coordinates refer to a point in a mathematical space. The coordinates are location-less. The same is true for propositions that refer to temporal phenomena - they do not actually have a location in space-time; they only refer to such a location.
It seems that only physical objects have the ability to change without losing their identities. Let us consider how this might happen. Objects have properties and they change when they gain or lose properties. Normally, when an object gains or loses a property, it is said to no longer be the same object. The implications of this way of looking at numerical identity was a source of great controversy in ancient times. The Greek philosopher Zeno developed an argument that involved the following premises: (1) objects cannot survive gaining or losing properties; (2) physical objects lose and gain a property every time they move. Conclusion: motion is impossible; change is an illusion because it is a contradiction, for it involves something coming into and going out of existence at the same time. One popular way to resolve this paradox is to simply say that spatial location is not a property that plays into determining the identity of an object. A less ad hoc version of this hypothesis is that relational properties can be gained or lost by physical objects without destroying their identities. However, this way of looking at things appears to fail. However, there are particles that seem to have all the same properties (bosons) in a system, except that they are in different locations. It seems clear that these objects, though they are very similar - qualitatively identical - they are not numerically identical.
Another possible solution to the problem of physical numerical identity is continuity. The continuity hypothesis is that if an object is continuous with itself, then it is identical with itself. However, there are many contradictions in this account. For example, if an object gradually fissions into two pieces that are identical to each other and the original object in every way except for their spatial locations, which one is the “real” one? A more relevant example of a contradiction in the continuity account comes from quantum mechanics. In the quantum world, motion is discontinuous. If we only had one particle of a certain kind, then this would not be a problem, because we could infer that the object in one location at one time is the same as the object with the same properties in another location at a different time. However, what of the many cases in which there are multiple indistinguishable particles? How could we conceivably determine which particle is numerically identical with which particle at a different time? One response to this is to say that there is simply an identity property which physical objects share with their selves at other moments. In other words, there is a fact of the matter which object is which, even if we cannot find that out in reality. However, quantum statistics rules this out. To understand why, consider two indistinguishable particles which can each be in one of two boxes. We would think that there are four possible configurations. One in which both particles are in one box, one in which they are in the other box, one in which they are both in separate boxes, and one in which the particles are switched. Now, even if we couldn’t distinguish between these particles, the frequency of these occurrences, if the particles were randomly distributed, would be as follows: one fourth of the time they would be in one box, one fourth of the time they would be in the other box and half the time, they would both be in separate boxes (though we could not tell which was which). However, in reality, when indistinguishable quantum particles are distributed into two boxes, they both occupy one box one third of the time, they occupy the other box one third of the time and they are evenly distributed one third of the time. This implies that the even distribution of the particles is one state and one state only. In other words, there is no fact of the matter about which particles are which.
What are the implications of this strange finding? For one thing, this implies that lone particles do not actually change over time. If they "change" locations, they are no longer the same particle that they were before. What of larger physical systems? Well, if a system’s identity is determined by its members (particles), then it too cannot maintain an identity over time. This seems to be a big blow to the hypothesis that physicality is defined by identity and change. It is. However, something does change. For one, the universe changes. If it did not, there would be no illusion of change. Or would there be?
Let us consider a phenomenological account of change. It is possible that there is only one moment that exists. We infer that there is change, and time, because we have the sensational experience of change. However, as we explored earlier, mental and abstract objects may not actually change. If they do not change, then it is possible that this sensation of change is actually a static mental object, just like the beliefs and other kinds of experiences. What of memories? Why would we have memories if there is not a past? Well, memories must, even in the context of time, exist at the "present" (or only) moment. We experience the past by the juxtaposition of various miniature "worlds" of experience. Memories, sensations, emotions, predictions and abstract thoughts all exist together in the mind. Therefore, our memories are like projections onto a plane. Our experience of time resembles the image of a mirror facing a mirror. There is information about different experiences contained within a larger image and so forth. Therefore, let us question the supposition that there is something that changes at all. A belief can "change" by being replaced with another belief, but how do we know that a belief has changed? We know this because the old belief is juxtaposed with the new belief, one in memory and the other in more recent experience. Therefore, it is conceivable that the change that occurs as a result of the replacement of one thing with another (which we have shown happens in the physical, the mental and the abstract worlds) is as much an illusion as the maintenance of identity over time.
The conclusion that there is no change does not imply that there is no time, nor does it imply that there is no physical world. Time could be a dimension of a manifold, or of our experience. The physical world may be distinguished by some unique property other than location or change, which we have eliminated. There is another common account of the physical. This account is that physical objects are those objects that we can sense. However, there appears to be no clear boundary between sensory experiences and imagined ones. Illusions and hallucinations dispatch this account rather quickly.
What else might be unique to physical objects? Let us take a step back and consider why we identified change and identity as the key players in physicality. Energy. It seems that all physical objects have energy, and it is the driver of change. Therefore, if we consider the property of having energy to be the common property that is unique to all physical objects, we might be able to save our hypothesis without modifying the theory in desperate ways. What does it mean to have energy?
The word energy is derived from the Ancient Greek word energeia (ἐνέργεια) which translates to "activity" or "operation". In other words, change. The law of conservation of energy is understood to be the result of its translational symmetry with space-time. That is, that the laws of physics do not change over time or from location to location. It seems that to say a system has energy is simply to say that it changes. However, let us examine the definition of energy more carefully.
Energeia can mean "activity" or "operation". Take the word "operation" literally, and we are back in business. Energy can mean, "to operate". Operators are mathematical, unchanging objects. Perhaps energy is a kind of operator that translates into activity or change. Is this simply a case of equivocation? I think not; energetic transformations happen via operators. Perhaps energy is a blanket term for some structure that exists in the physical world in addition to the traditional space-time account.
Minkowski space is one kind of structure in the physical world. However, General Relativity shows that there is additional structure in space. Mass bends space-time. However, General Relativity does not really account for the exact locations of the various masses in space-time. This seems like an unsatisfactory account of the structure of a physical space. After all, a space is a set with added structure. In this case, a set of physical objects. Should the structure not be complete? That structure ought to be statable in its entirety in mathematical terms. Might it be possible to describe the entire physical world as a space, or nested spaces? Could it even be possible to describe the entire universe, including phenomenological structures in this way? This is an exciting prospect. However, even if it is possible, I would bet that the descriptors of truths in the universe are as numerous and enigmatic as mathematical truths are. That is, there are an infinite number of them and they all must be proven in their own unique way. Gödel’s Incompleteness Theorem shows that no set of axioms can account for all mathematical truths in any given system. If this is the case, it is likely that, if the universe is describable in mathematical spatial terms, it is infinitely complex and no finite set of axioms can describe it.
Nonetheless, the concept of describing both the physical world and the mental world as spaces is promising. Perhaps we will develop ever more detailed models and spaces that describe the universe until the realms of the physical and the mental are no longer distinct. That is the main conclusion of my rumination. That physicality, at base, appears to have no real distinguishing features. Perhaps the reason for this is because the descriptors of physical reality can also be applied to other realms of experience. Perhaps it is even an accident that we discovered so many physical truths before understanding the structure of consciousness.
Thanks for reading!
Cheers,
Oliver