Structures and Morphisms
Mathematical Structures from Bounded Infinity
More speculative explorations in metamathematics, on the subject of numbers, quantities and infinities.
See also Objects and Structures
Scientific and technological progress relies on our understanding of the objective aspect of the world in which we live.
This objective aspect is the aspect of the world that is its structures and behaviours that do not depend on anyone's individual mind, beliefs, or opinions, or, indeed, on those of any group of people. This is why we call it "objective".
A concomitant scientific fact is that what we actually experience as our world is a construct of brain function. Each one of us experiences a subjective experience of the world, each in our own way. However, the world that we are having our subjective experience of, is nonetheless still a construct of human brain function in general. The idea that there is something separate from brain function that constitutes human consciousness, or human self, that is observing and experiencing the world through brain function, is not tenable in the face of modern neuroscience. Rather, the entire experience of self, mind and world, is a construct of brain function. Specifically, in each one of us, a construct happening through the same brain organ.
The implications of this are not something that everyone can follow. It is much, much easier to presume that there is a "real world" that corresponds to the material world we experience being in, that is other than a construct of brain function. In other words, it has nothing to do with us other than that we happen to have evolved within it.
Post naivety accepts the basic modern neuroscientific facts about the brain and our experience of mind, self, and world, at face value. It accepts the scientific evidence as it stands.
The world in which we live still has an objective facet that is the aspect of it that science is coming to understand. And "objective" still means that this aspect of the world does not depend on anyone's individual brain function. Nor does it depend on networks of beliefs, or scientific paradigms. Rather, networks of belief, or scientific paradigms, are our way of coming to understand the nature of the objective facet of the world.
Without this way of understanding there would be no science or technology of the kind that follows from scientific knowledge proper.
The real purpose of technology is not to make more efficient war or to increase our advantages over others, but rather, to be a benefit to humanity by assisting in overcoming our evolutionary roots.
Our evolutionary roots are not a "golden age" in the past. Rather, the whole principle of natural evolution is based on contention - the survival of the fittest. Even "harmony" as it arises in nature, as perfect adaptation to the environment, is "harmony" in the overall context of contention. At present, our science and technology exists in the same context.
So understanding the objective facet of the world in which we live is an advantage. It is not a spurious, random, spiritually irrelevant thing, that happens to be happening.
From the point of view of naivety, we think of our world as composed of things that are as we experience them through our senses, or things whose existence kind of correlates to that. But science has progressively shown that the fundamental nature of our world is not like this, at all.
Object and structures is way of conceiving the objective aspect of the world, in relation to the fact that it is a construct of brain function. It doesn't take the world to be "real" in the sense that it is not a construct of brain function, or has nothing to do with us other than that we happen to have evolved within it. In other words, it doesn't take the world to be separate from us. It respects the fact that the intelligibility of the world, for us, is entirely a matter of the nature of the intelligence we are being. It respects the fact that this is a feature of evolution, and specifically, the evolution of evolutionary intelligence. There are higher truths that may be articulated, but it is not directly concerned with them. Rather, it is concerned with comprehending the objective facet of the world in which we live, in the way that science understands it, and will continue to do so.
Seeing things in terms of objects and structures is probably rather too stratospherically abstract for most people. However, it is approach that can be explored. Its exploration would no doubt go much further than I have been able to take it.
The basic observation is that all our scientific and mathematical understanding is always in terms of structures of relations between distinct objects. And so the question arises, what exactly is it that makes objects distinct from each other, in the first place? Obviously, they are different structures. In material terms, they are different structures of structures of structures, and ultimately different structures of atoms, and then finally different structures of quantum correlations. These correlations themselves are expressed using natural numbers. Natural numbers are themselves distinct objects.
In fact, what we refer to as natural numbers is none other than just our way of "labelling" an infinity of distinct objects. This is not an infinity of "actual things" that we might be able to scientifically encounter, but rather, it's just an abstract infinity of distinct objects. And so it is necessary to distinguish between our way of "labelling" such an infinity, and the infinity of distinct objects itself, which is actually a tacit concept in our mind, as soon as we deal with these "labels", that we call the natural numbers.
Of course, the "natural numbers" that we usually deal with, are not an absolute thing, but rather, are just numbers to the base 10. We could use any base we like, such as hexadecimal or binary, and the numbers would still be "labels" for the infinity of distinct objects. Ultimately, we could use a unary base or an infinite number base. The infinite number base would simply be using infinitely many unique "labels" to label the infinity of distinct objects. These "labels" would be none other than an infinity of distinct objects. But they would be an infinity of distinct object of a particular kind, rather than the abstract infinity of distinct objects.
In the end, "labels" are essentially "representations" of objects, but are also objects in their own right. In science, all our equations and representations of things like quantum correlations, are all "representations". All "representations" and the things that they represent, are structures of relations between distinct objects. And in the final analysis, ordered numbers are "representations" of infinite iteration processes.
When we look closely at objects and distinctions between objects, we find these infinite iterations. And somewhere behind the appearance of our world, are structures of infinite iterations.
We see tantalising clues to this, in the orbit of electrons around nuclei, which really, is just a mathematical structure. The electron "shell", as we know, actually consists of "subshells". What we are looking at, is patterning in structures of relations between objects that appear to be, or are taken to be, distinct. We take the object that we call "expansion" (such as the expansion of the universe) to be distinct from the object that we call "orbit" (such as the orbit of the electron around the atom, of the orbit of the Earth around the Sun or the orbit of stars around a black hole at the centre of a galaxy).
The way in which this distinction occurs in our observations of the universe, the ways in which all distinctions occur, is connected with the way in which our experiential knowledge of nature arises.
This objective aspect is the aspect of the world that is its structures and behaviours that do not depend on anyone's individual mind, beliefs, or opinions, or, indeed, on those of any group of people. This is why we call it "objective".
A concomitant scientific fact is that what we actually experience as our world is a construct of brain function. Each one of us experiences a subjective experience of the world, each in our own way. However, the world that we are having our subjective experience of, is nonetheless still a construct of human brain function in general. The idea that there is something separate from brain function that constitutes human consciousness, or human self, that is observing and experiencing the world through brain function, is not tenable in the face of modern neuroscience. Rather, the entire experience of self, mind and world, is a construct of brain function. Specifically, in each one of us, a construct happening through the same brain organ.
The implications of this are not something that everyone can follow. It is much, much easier to presume that there is a "real world" that corresponds to the material world we experience being in, that is other than a construct of brain function. In other words, it has nothing to do with us other than that we happen to have evolved within it.
Post naivety accepts the basic modern neuroscientific facts about the brain and our experience of mind, self, and world, at face value. It accepts the scientific evidence as it stands.
The world in which we live still has an objective facet that is the aspect of it that science is coming to understand. And "objective" still means that this aspect of the world does not depend on anyone's individual brain function. Nor does it depend on networks of beliefs, or scientific paradigms. Rather, networks of belief, or scientific paradigms, are our way of coming to understand the nature of the objective facet of the world.
Without this way of understanding there would be no science or technology of the kind that follows from scientific knowledge proper.
The real purpose of technology is not to make more efficient war or to increase our advantages over others, but rather, to be a benefit to humanity by assisting in overcoming our evolutionary roots.
Our evolutionary roots are not a "golden age" in the past. Rather, the whole principle of natural evolution is based on contention - the survival of the fittest. Even "harmony" as it arises in nature, as perfect adaptation to the environment, is "harmony" in the overall context of contention. At present, our science and technology exists in the same context.
So understanding the objective facet of the world in which we live is an advantage. It is not a spurious, random, spiritually irrelevant thing, that happens to be happening.
From the point of view of naivety, we think of our world as composed of things that are as we experience them through our senses, or things whose existence kind of correlates to that. But science has progressively shown that the fundamental nature of our world is not like this, at all.
Object and structures is way of conceiving the objective aspect of the world, in relation to the fact that it is a construct of brain function. It doesn't take the world to be "real" in the sense that it is not a construct of brain function, or has nothing to do with us other than that we happen to have evolved within it. In other words, it doesn't take the world to be separate from us. It respects the fact that the intelligibility of the world, for us, is entirely a matter of the nature of the intelligence we are being. It respects the fact that this is a feature of evolution, and specifically, the evolution of evolutionary intelligence. There are higher truths that may be articulated, but it is not directly concerned with them. Rather, it is concerned with comprehending the objective facet of the world in which we live, in the way that science understands it, and will continue to do so.
Seeing things in terms of objects and structures is probably rather too stratospherically abstract for most people. However, it is approach that can be explored. Its exploration would no doubt go much further than I have been able to take it.
The basic observation is that all our scientific and mathematical understanding is always in terms of structures of relations between distinct objects. And so the question arises, what exactly is it that makes objects distinct from each other, in the first place? Obviously, they are different structures. In material terms, they are different structures of structures of structures, and ultimately different structures of atoms, and then finally different structures of quantum correlations. These correlations themselves are expressed using natural numbers. Natural numbers are themselves distinct objects.
In fact, what we refer to as natural numbers is none other than just our way of "labelling" an infinity of distinct objects. This is not an infinity of "actual things" that we might be able to scientifically encounter, but rather, it's just an abstract infinity of distinct objects. And so it is necessary to distinguish between our way of "labelling" such an infinity, and the infinity of distinct objects itself, which is actually a tacit concept in our mind, as soon as we deal with these "labels", that we call the natural numbers.
Of course, the "natural numbers" that we usually deal with, are not an absolute thing, but rather, are just numbers to the base 10. We could use any base we like, such as hexadecimal or binary, and the numbers would still be "labels" for the infinity of distinct objects. Ultimately, we could use a unary base or an infinite number base. The infinite number base would simply be using infinitely many unique "labels" to label the infinity of distinct objects. These "labels" would be none other than an infinity of distinct objects. But they would be an infinity of distinct object of a particular kind, rather than the abstract infinity of distinct objects.
In the end, "labels" are essentially "representations" of objects, but are also objects in their own right. In science, all our equations and representations of things like quantum correlations, are all "representations". All "representations" and the things that they represent, are structures of relations between distinct objects. And in the final analysis, ordered numbers are "representations" of infinite iteration processes.
When we look closely at objects and distinctions between objects, we find these infinite iterations. And somewhere behind the appearance of our world, are structures of infinite iterations.
We see tantalising clues to this, in the orbit of electrons around nuclei, which really, is just a mathematical structure. The electron "shell", as we know, actually consists of "subshells". What we are looking at, is patterning in structures of relations between objects that appear to be, or are taken to be, distinct. We take the object that we call "expansion" (such as the expansion of the universe) to be distinct from the object that we call "orbit" (such as the orbit of the electron around the atom, of the orbit of the Earth around the Sun or the orbit of stars around a black hole at the centre of a galaxy).
The way in which this distinction occurs in our observations of the universe, the ways in which all distinctions occur, is connected with the way in which our experiential knowledge of nature arises.