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Ultranumerics

Metamath

Welcome to Metamath. As the name suggests, metamath, here, is about what is beyond mathematics itself. The way we approach this, is through what we call here, Object Theory. Probably the best way to introduce this is from the Preface to the book Objects and Structures.

The Object Theory we refer to here is a schema of thought, rather than a thesis. It is essentially a suggestion of somewhere similar that we may need to explore as we move towards new horizons in the scientific knowledge of our own mind and the nature of our own intelligence.

Where we are now, in these times, we have already realised that mathematical "proofs" are relative things, rather than absolutes, and that some things must remain forever uncertain or undecidable. This knowledge was achieved through intellectual reasoning itself. Through the use of the human mind. But now, there is another dimension to this story. We can no longer be confident that mathematical reasoning is the study of something separate from the processes of mind that are being used to do the reasoning.

"Traditionally" mathematicians have loved to build towers of logic and reasoning, certain mathematical structures, in other words, "proofs", without ever considering the structures in the psychology and functioning of the minds doing the reasoning, independently of the act of reasoning itself. It is as if we have had a religious faith in the idea that we been discovering truths about something that is independent of us. Independent of the mind we are being, and the nature of our own intelligence. The methods used since the ancient Greeks and before, have succeeded in relating to the way things are, and behave, in the apparently timeless world of numbers and mathematical structures. But is this world of numbers and mathematical structures really fundamental to the phenomena of our world?

The position of modern neuroscience is that there is nothing we can think, comprehend, understand, perceive, or conceive, that is not a construct of brain function. In short, the implicit position is that the human mind, indeed the human intellect, and the mathematical intellect, is a construct of brain function. So brain function is a context even for our understanding of mathematical reasoning. It means that we no longer truly have the luxury of considering the psychology of our mental processes as a subject separate from the way we do mathematics.

Details about brain function are beyond our scope here, but it is well known that we do have a natural cognitive bias in the way we understand things. Psychologists speak of the apperception mass, and we can consider that to inevitably arise from the principle of neuroplasticity in brain function. We have to learn new things through the medium of what we have already learned.

We have seen this particularly in the difficulties we had in coming to terms with the facts of quantum mechanics. Our pre-conditioning led us to resist, for example, the idea that something can be in two places at once, or that until it is observed or measured, it doesn't exist except as a probability wave.

There are certain things that we must appreciate, once we know that what we are looking at, or contemplating, is in fact a construct of brain function. The first is that our brains and minds do not exist in isolation. On the contrary, they exist literally, in networks and overall as a network. That means that it is possible for whole networks of networks of minds to come to a consensus on something, that is literally a network consensus. In that, whatever structures and processes of mind and thought are underlying the reasoning taking place in any individual mind, it will be related in some way to what is inherent in the network as a whole. In a sense, we can even speak of "network mind".

Neuroplasticity in brain function plays a key role in learning, and in mathematical learning especially, teaching also plays a key role. It is through these mechanisms of teaching and learning, together with the principle of neuroplasticity, that "network mind" in the form of "schools" of understanding, can come into existence.

Not all mathematical theorising is empirically testable in the way that successful theories of physics have been. The history of set theory in its application to the concept of infinite sets is sufficient demonstration of that. So if empirical testing is not available, the question then arises, what exactly is it, that mathematicians are looking for, that is other than, or better than, just a matter of consensus? What is it that constitutes even just what is aspired to, or believed by some to be "truth" or "proof" in mathematics?

The position we have currently come to, collectively, is that we must show that there is consistency of theories and results, and structures of understanding, with the axioms upon which they are built. We now know that there are always axioms, and there are always things that must remain undecidable.

For our purposes here, where we are putting mathematics in the context of the modern neuroscience, we are going to be talking about something that we define as "genuinely objective". We are going to define this notion of genuinely objective as something that is not dependent on an individual's mind and brain function, but more importantly, not dependent on a consensus in networks of individual minds and brain functions.

This idea that there is something that is "genuinely objective” is by no means an assertion that there is anything that we can ever know about that is other than a construct of brain function. Rather, it is simply saying that what is "genuinely objective" is not dependent on the way any particular individual mind or networks of mind, or brain or networks of brains, is working or functioning.

In this way, the principle of the brain itself, the same principle that nature instantiates in all our brains, the principle through which we have our experience of being, and experience of mind, and thought, and indeed, mathematical reasoning and understanding, that principle, itself, we can take to be "genuinely objective".

Intoduction

What even is mathematics?

What even is mathematics anyway? Why isn't mathematics an art?

All our understanding of mathematics is something that exists in human brain function. Everything we know about it, and understand about it, only exists as a construct of human brain function. If we think we are experiencing something other than a construct of brain function, something other than what we are being - because what we are being is a construct of brain function - then we don't really know what mathematics is, or what it really means.

It is our prerogative to know what it is. When we do, there may perhaps be a metamath that is more like an art. An art in which we represent what numbers are, in a way transcendental to how they are compelled to behave, in relation to each other.

There is nothing in anything that we understand as mathematics, or in anything we understand in science, that is not a construct of brain function. But also, all of this is about structures of relations between things that are taken to be distinct. Without things that are distinct, or that at least appear to be distinct, or are treated as though they are distinct, there is nothing to understand. So ultimately what we are looking at in objective thought and understanding are structures of relations between objects that are taken to be distinct.

Structures

Structures of relations between things that are taken to be distinct

What is this?

It is saying that there can only be a genuine multiplicity of things, if those things are distinct from each other. This applies to everything. From everyday objects, to quantum particles, to people, to mathematical structures, to the things we call numbers. Always, what we are looking at, is a structure of relations between things that appear to be distinct.

There is another way we can say this: Anything is only distinct from anything else, by virtue of the structure of which it is a part. Ultimately, the only thing that makes things distinct, is structure. The structure can only be understood objectively, in terms of what it creates. Everything that we study objectively, in science, and even in pure mathematics, is structures, and all these structures, are part of the overall structure.

In objective thinking, such as in mathematics and science, everything we think about is an object. Otherwise we wouldn't be thinking about it objectively. That's what objective thinking and understanding, is. It is thinking and understanding about objects, related in structures. Of course, all this requires the brain, and the intelligence we are being through the brain. So actually, every object that we can think about in this way, is an object of thought. An object in the mind. Some of these objects represent material objects, others don't. And material objects themselves, are objects. But they are not the only kind of objects, because anything we think of, if we think of it as if it is separate from the mind we are thinking with, then it is an object. Even if it is not really separate from the mind we are thinking with. This is the position as far as Object Theory is concerned.

So then, if we are talking about objective thought, and understanding, what we are talking about is structures of relations between objects that appear to be distinct.

The Nature of Structures

  • An object is anything considered to be an object, in object oriented thinking.
  • If two objects are distinct, then there must be a relation between them.

The relation between, say two objects A and B, we can represent with a double-headed arrow (or two arrows) as:
Stacks Image 125
  • A relation itself can be considered as an object.
  • Objects connected by relations are structures. Structures are essentially networks of objects with relations between the objects.
  • Any structure can be considered as an object.
  • Any object can be taken to be a structure, even if we do not know what that structure might be.
In actuality, structures are likely to be complex networks of objects, the objects being the nodes or vertices, and the relations being the edges of the network.

Numbers as Processes

A key feature of Object Theory as explored in Objects and Structures is the treatment of numbers as processes. Specifically, objects called infinite iteration processes or IIPs.

Why would a number be an infinite iteration process? Essentially, it is because any number is an object, in the schema of thought of Object Theory. In Object Theory, any object can also be considered as an infinite iteration process. The reason for this is explained in the Premises of Object Theory.

Also, numbers other than conceptual constructs such as the so-called "transfinite" numbers or "hyperreal" numbers are synonymous with some counting system. Such counting systems are infinite iteration processes. The process is iterative and there is no set limit to the process - it does not in itself halt, and is infinite. So a rational number is either a "halting point" of an IIP or can still be regarded as an unhalted IIP - for example the number "5" is also the number "5.0000…" where the zeros after the point extend infinitely.

By treating numbers as IIPs infinities can be thought about in a powerful new way.

Fractals

Stacks Image 156
In Objects and Structures the Mandelbrot and Julia sets are examined from the point of view of Object Theory, using empirical computer evidence, which leads to both a different perspective on the relation between the complex numbers and the continuum, and hence between the reals and the continuum. The nature and behaviour of the fractal geometry itself can be seen as a salient manifestation of this relation between complex numbers and the continuum. This relation is essentially the same as that between the reals and the continuum.

Mandelbrot & Julia

Julia Sets

The Mandelbrot and Julia sets are created using an infinite iteration. There are just two numbers used in the iteration: z, and c. It is often explained that the Mandelbrot set consists of all the c values that in the iteration will result in connected Julia sets, whilst c values outside of the Mandelbrot set, result in Julia sets that are disconnected.

This is literally true, but it rather glosses over the deep meaning of these objects. So here, we are going to gradually show the deeper context - a context way beyond questions concerning just the Mandelbrot and Julia sets themselves. To do so we'll begin at the beginning, and bear in mind, this is not a lesson in mathematics. We are in the domain of metamathematics.

Julia sets are so called because they are mathematical objects that have been understood and described through set theory. They are understood to be a specific set of "points" or numbers. These "points" or numbers, are points or numbers on the complex plane. The complex plane is a two-dimensional representation of the infinity of complex numbers. It is conceived as a two-dimensional plane because complex numbers have two parts.

A complex number is a number that has a "real" part, which is essentially an ordinary number, and an "imaginary" part, that represents quantities that we don't find existing on their own as actual quantities we can measure, in the everyday world. They are, however quantities that exist in mathematics, without which, mathematics and many of its descriptions of natural phenomena would be incomplete.

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Object Theory Context

So there are a number of ideas, concepts, or objects, here, to begin with, before we go any further. The main objects are "points", "numbers", "imaginary quantities", and "set". These are all part of one of the current mathematical approaches to these objects.

In terms of the psychology here, the mathematical understanding we are using is inherited from an entire network of minds, that extends back in time, and is evolving. This is true also of other mathematical approaches to these objects.

The network of relations between these conceptual objects, such as "sets" and "numbers", is a structure in the view of Object Theory, and is now a feature of the collective nature of our understanding. In terms of modern neuroscience it is also a feature of the collective way in which human brains and minds are operating, as a network. Which is itself, a structure.

The Object Theory position on this is that the fact that something may be objectively true in mathematics does not mean that it is something independent of brain function and the way in which thought arises through it. The objectivity in it is simply a manifestation of the part of the principle by which brain function gives rise to our intelligence and experience, that does not depend on any individual brain organ. Hence, we find this aspect of our thought to be the manifestation of something "objective".

The fact that this objectivity happens, is not surprising, because the principle of the brain does not come into existence in the first place, through the evolution of a single, isolated brain. The evolution of the brain is a matter of networks and multiple instances of the evolving principle.

In Object Theory there is a difference between what is objective, because it is handled by object-oriented thought, and what is genuinely objective, in that it is not dependent on any particular network of brains or minds.

For example, is a "point" or"number" something that is genuinely objective? Even if it is, is the concept of a straight line consisting of an "infinite number of points" genuinely objective? What about the concept of a "set"?

All these concepts are objects in object-oriented thinking. Anything we treat as objective, or as an object of thought, Object Theory allows.
The Geometric Objects
There are infinitely many Julia sets, and any Julia set is either completely connected, or completely disconnected. In terms of their fractal geometry connected Julia sets are one continuous object, even though some of them (still infinitely many) exist as as a geometric object composed of infinitely many distinctly identifiable geometric objects, connected only by single points. An example is below:
Stacks Image 179
Look carefully and you will see that there is basically one "shape" that is repeated. It is actually repeated infinitely many times, at different sizes, and in different positions. We cannot see that in just the one image, but in "live" computer imagery we can zoom in on any part and find more and more of the same thing, ad infinitum. There are plenty of demonstrations on the web. In this particular example of a Julia set, the repeats are attached to each other through single points.

Below is another example, in which the distinct objects are still only connected by single points, even though in the image there appears to be a larger area of connection. Computer images can be misleading because the image itself depends upon the number of iterations used in the algorithm creating the image.

These connection points or synapses can often appear as quite large areas when there are insufficient iterations. Increasing the number of iterations sufficiently - essentially increasing the power of the computing - results in the reduction of these areas to the representation of a point.

The larger, graded areas in green, represent different rates of escape. We will talk about what this means later.
Stacks Image 185
Other Julia sets are indeed connected by larger areas:
Stacks Image 191
There is a specific rule governing whether or not the connections between the distinctly identifiable objects are a single point or not.
Other Julia sets look somewhat more simple:
Stacks Image 200
There are also infinitely many Julia sets that are completely disconnected. In disconnected Julia sets all the distinctly identifiable objects are "points" and they are only distinctly identifiable by their locations on the complex plane, in other words, that they are distinct numbers. Julia referred to these sets as "dust", and they are often referred to as Cantor dust, Fatou dust, or "fractal dust". An example of a disconnected Julia set is below:
Stacks Image 206
In these objects what we are looking at in the computer imagery is rates of escape, rather than continuous areas of objects constituting the set. We will talk about rates of escape shortly. These objects are all manifestations of the principle of chaos, which is not randomness. So these differences are not random. All the Julia sets are related to each other in an orderly though non-linear way, through another mathematical object, which is the Mandelbrot set. This is probably the most famous fractal, geometric object:
Stacks Image 212

The Mandelbrot set as a geometric object

Some Important Background
So we can say that the Julia sets, connected (C) and disconnected (D) are related as objects to the object of the Mandelbrot set M. In Object Theory notation we might write this as:
Stacks Image 223
The way this relation is usually understood is to say that the Mandelbrot set is the set of all connected Julia sets, or a "map" of the connected Julia sets. However, there is more to it, than that.

All the Julia sets and the Mandelbrot set are objects resulting from the infinite iteration equation usually expressed as:
Stacks Image 229
This means that we start with some value for z, and c, which are both complex numbers, and then change z on the left into to the expression on the right, which then becomes a new value for z, whereupon we go through the process again. We can do this ad infinitum. This procedure is known as an infinite recurrence, or infinite iteration. In Object Theory it is an infinite iteration process, or IIP.
The study of the behaviour of numbers on the complex plane, in this way, is a part of complex dynamics. In this field of study it is well known that under iteration of this kind, z follows a path on the complex plane known as an orbit. This has something conceptually in common with the way astronomical bodies might orbit another astronomical body, but in general, most orbits in complex dynamics are not stable in the way that, say, the orbit of the moon around the Earth is (at least for the time being). Rather, we often find orbits in which the iterating value of z "falls" in some kind of spiral towards some point, known as an attractor. Orbits can also occur along a straight line, if they are on the real axis of the complex plane.

However, other orbits do the opposite. They spiral outwards away from the origin (the centre) of the plane and keep moving outwards infinitely. These orbit are said to escape to infinity. Another term used is that the values of z diverge.

Then, there are some other orbits that are more interesting in their behaviour. They follow an orbiting path for many, many iterations, perhaps thousands, or more, that seems to neither fall towards and attractor, nor escape to infinity, at least for the time being. The orbit does orbit around an attractor, or attractors, and often the behaviour of the orbit is exotic. In systems of analysis or investigation into this behaviour we can theoretically place such an orbit either into a set of escaping orbits, or non-escaping orbits. However, for an arbitrary example of such an orbit it is not possible to definitively state whether or not the orbit escapes, unless we find a point in the orbit that breaks free of a circle of radius 2 around the origin. It is possible to prove that when this happens, escape will occur.

The images below (generated using Fractally) show some orbits for the iteration
Stacks Image 238
starting with z = 0, but for different values of c. Click on any image to see a larger picture.
  • Orbit with 3-period
    Orbit with 3-period
  • Orbit with 3-period and convoluting spiral arms
    Orbit with 3-period and convoluting spiral arms
  • Orbit with 3-period and and 4 attractors
    Orbit with 3-period and and 4 attractors
  • Orbit with 3-period and very convoluted spiral arms
    Orbit with 3-period and very convoluted spiral arms
  • Orbit with 3-period beginning to become exotic
    Orbit with 3-period beginning to become exotic
  • Exotic orbit with 3-period, still bounded
    Exotic orbit with 3-period, still bounded
  • Orbit with 3-period beginning to be attracted to infinity but still bounded
    Orbit with 3-period beginning to be attracted to infinity but still bounded
  • Orbit with 3-period and 4 attractors, beginning to be attracted to infinity also
    Orbit with 3-period and 4 attractors, beginning to be attracted to infinity also
  • Orbit with 3-period where c is approaching exit from the Mandelbrot set
    Orbit with 3-period where c is approaching exit from the Mandelbrot set
  • Orbit with 3-period very close to escaping to infinity but still bounded
    Orbit with 3-period very close to escaping to infinity but still bounded
  • Orbit into an attractor outside the Mandelbrot set
    Orbit into an attractor outside the Mandelbrot set
  • Orbit that breaks free of the circle of radius 2, and begins to escape to infinity
    Orbit that breaks free of the circle of radius 2, and begins to escape to infinity
  • Exotic bounded orbit
    Exotic bounded orbit
  • Exotic bounded orbit
    Exotic bounded orbit
  • Exotic bounded orbit
    Exotic bounded orbit
  • Exotic bounded orbit
    Exotic bounded orbit
  • Exotic bounded orbit
    Exotic bounded orbit
  • Exotic bounded orbit
    Exotic bounded orbit
  • Exotic bounded orbit
    Exotic bounded orbit
We have the benefit of computer imagery, but Julia didn't. Only later, when Mandelbrot started to investigate this behaviour, were early computers available.
Julia investigated the behavior of the iteration for different initial values z0, of z. (c being a constant). It turns out that some initial values z0 result in z escaping to infinity under the iteration, whilst others do not. Whether or not a given z0 will result in z escaping to infinity under iteration, becomes a central question. It is a question that is answered by the Mandelbrot set, but at the time Julia was working, there was no knowledge of the Mandelbrot set.
If the initial value of z, z0, is set at zero, then the orbit begins at c, and then proceeds with an iteration containing only one parameter, in the same form as:
Stacks Image 279
except that z = c in this iteration. This is simply because:
Stacks Image 285
For other z0 values, we could ask, "for a given starting value z0, what are the c values for which z0 does not escape to infinity? What are the values for which it does? We could then try some different values for z0, one at a time, and for each one test to see which c values result in z0 escaping, or not escaping. We would start to get some results like the following, in which the illuminated areas are the c values for which our iteration of z from the chosen z0 does escape. (Credit: John Whitehouse, eddaardvark.co.uk, 2021)
  • z0 = 1.1, 0.1i
    z0 = 1.1, 0.1i
  • z0 = 0.5, 0.5i
    z0 = 0.5, 0.5i
  • z0 = 0.5, 0.4i
    z0 = 0.5, 0.4i
  • z0 = 0.4, 0.3i
    z0 = 0.4, 0.3i
  • z0 = 0.1, i
    z0 = 0.1, i
  • z0 = -0.9, 0.91 zoomed in
    z0 = -0.9, 0.91 zoomed in
  • z0 = -0.9, 0.9i zoomed in more, on upper object
    z0 = -0.9, 0.9i zoomed in more, on upper object
We can see already that what we are looking at here, is various parts of this complete object:
Stacks Image 308
This is the Mandelbrot Set shown as a geometric object. This object in its complete state is all the values of c for which the iteration of z does not escape to infinity, provided that we set z0 = 0. The illuminated area surrounding it but outside it is then the values of c for which the iteration does escape to infinity.

But to get the complete object we do need to set z0 = 0.

Now if for any fixed value c, we make iterations for z0 values all over the complex plane, then for that value of c, if we put into one set all the z0 values whose orbits do not escape to infinity under the iteration, then we will get a Julia set that is connected if c is within the Mandelbrot set, and disconnected if c is outside the Mandelbrot set.

This seems fairly straightforward, but it glosses over some deep, underlying issues.
Going deeper
Remember above we said that by putting z0 = 0, we are then working with an iteration that is essentially:
Stacks Image 320
This means we can simply pick a point z, and put it into the iteration above. Depending on where z is, there are three things that can happen, given that this is an infinite iteration that we can only iterate finitely, and given that the behavior we are looking at is non-linear. Non-linear in this context means that there is not an equation that we can use, shortcutting step by step calculation, by which we can just calculate where z will be, after n iterations, no matter how large n is.

In fact, this is somewhat of an understatement. The non-linear behaviour of an exotic orbit means that whether or not the orbit will escape to infinity is super sensitive to z0 or the starting value of the iteration. It turns out, not just super sensitive, but infinitely sensitive. Because z0 or the starting value is itself, a number that needs to be expressed with infinite precision.

The three things that can happen by iterating finitely are:

  • The orbit can be found to fall into an attractor.
  • The orbit can be found to escape the circle of radius 2 around the origin, whereupon we know it will eventually escape to infinity if it is not already doing so, because we can mathematically show this to be true.
  • We can be uncertain about whether or not the orbit will escape to infinity.

In the last case, there is no such thing as certainty, because we cannot iterate infinitely. An "uncertain" orbit is an orbit that orbits around an attractor, or attractors, for very large numbers of iterations, and must always lie conceptually somewhere "between" the orbits that escape, and the orbits that do not (orbits that remain bounded). If one set is the orbits that escape, and another set is the orbits that do not, then the uncertain orbits cannot be said to belong to one set or the other, except by an approximation.

The approximation exists in computed imagery by regarding the orbit as bounded if it has not escaped within a given number of iterations. This is usually called the "escape time algorithm", but it's not really about time. It is about number of iterations.

The same principles are true if instead of using the iteration equation above, we use our chosen point as the value of c in the iteration:
Stacks Image 326
and then iterate for values of z0 all over the complex plane. When we do this, the z0 values that give orbits that do not escape to infinity, create the Julia set for that value of c. The Julia set is often defined in this way.

As we have already discussed, values of c outside the Mandelbrot set will result in an orbit that escapes to infinity when we iterate with that value of c, for z0 = 0. However, if we iterate for z0 all over the complex plane, with that value of c in the iteration, then theoretically, not all orbits escape to infinity. Those that do not, however, are for z0 values that are isolated points surrounded by other points with iterations that do escape to infinity. These isolated points whose orbits remain bounded, together constitute a disconnected Julia set and are referred to as dust.

In contrast, when c is in the Mandelbrot set, then iterating for z0 values all over the complex plane, we find that all the orbits that do not escape begin with z0 values that are within the Mandelbrot set, whilst all those that do escape are for z0 values outside the Mandelbrot set. This is true for all points c within the Mandelbrot set. These are the connected Julia sets.

This rather beautiful set of relations between infinite objects was unravelled by Mandelbrot. Many other mathematicians have subsequently shown remarkable properties inherent in the Mandelbrot set, regarding numbers and their relations. Not least, are the fractal properties of the Mandelbrot set that you can easily find demonstrated on the Web, such as the "infinite zooms" into the infinite geometric structure of the set, that you can find on YouTube.

Math and the Brain

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The concept of a "point" is an alternative concept for a "number" when it is put into the context of "dimensions", or "axes" or "the complex plane" or the "Riemann sphere", and so on. It can alternatively exist in without necessarily inferring a specific number, such as when it is in the context of objects such as a "set" or "infinite set" of numbers.

All these concepts are objects in object oriented thought. They come together to make the various areas of mathematical study as networks of "concept structures" that constitute domains of thought, conceptualisation, and understanding, that then become established and accessible, to multiple minds.

These concepts and concept structures, in the context of object oriented thought, are essentially "objects" as defined and understood in the mind. The domain of mind and understanding that they constitute, is then a "structure". As far as Object Theory is concerned it does not matter whether these are only in the mind, or whether they are are regarded as independent of the mind as it arises through brain function.

Any mind arising through the brain function of a brain organ, can be referred to as arising through the principle of the brain. It literally does not do so in isolation, because the brain organ, and the principle of brain function, does not come into existence in isolation, in the first instance. Rather, brain organs, and the principle of brain function, comes into existence through multiplicity, and the principle of causal networks. So what becomes established as knowledge, which is itself a construct of brain function, enabled by the principle of the brain and the way it has evolved and is still evolving, always has involved multiple brains, and network principles.

The same is true for what becomes established as how we experience the objective world. Our experience of the objective world itself, including the objective nature of it, that we experience, also arises as a construct a brain function. The nature of the world's objectivity arises from the part of the nature of the principle of the brain, that is itself "objective", simply in the sense that it does not depend on the particular functioning of any specific brain.

This way of looking at our situation as human beings, is Post Naive Realism or post naive materialism. Its counterpart, outside the sphere of science - and indeed mathematics - is in the relation between our being as human beings, and the brain, which ultimately comes down to a matter of responsibility for the self we are being.

It is in that, that science, and mathematics, and the humanities, come together. The idea of separating the subject of mathematics from subjects that are called "the humanities" on the grounds that it deals with what is "objective" and "rational" is mistaken. More mistaken still, is the idea that mathematics deals with "truth", that is in some way transcendental to the principle of the brain.

Nothing in our object oriented thinking is transcendental to the functioning of the brain. Our desire to see our rational mind as something separable from brain function and the principle of the brain, is not only a part of naive realism, but actually, also contrary to the fundamental facts of modern neuroscience.

Knowledge, and the condition of our mind, and the way it works, is a matter of brain function and the principle of the brain. What gets passed on as knowledge, and passed around, through the network of human minds and brains, either "horizontally" in the present period, or through evolutionary time, as thought, ideas, memes, beliefs, and processes of thinking, and so on, does not get propagated simply through Darwinian principles of evolution. This is not some radical new idea, but is now well known. On the contrary, the brain changes itself, in the present, and changes other brains, both in the present, and across generations, through principles of neuroplasticity, in teaching, and learning.

So what becomes established and understood as mathematical structures, is part of the way the mind we are being, collectively becomes established in the way it is working, conceiving, and comprehending. Mostly, in mathematics this is in compliance with the genuinely objective nature of material phenomena. However, the notion that any of this is about anything that is separate from the principle of the brain, simply arises from the psychological conviction that there is something that is separate from the experience of mind and being that we are experiencing, as it arises through brain function. That notion is naive realism.

In contrast, post naive realism is the understanding that there is nothing we know or experience that is not a construct of brain function, and therefore, there is nothing we know or experience in ordinary or rational understanding that is not secondary to the principle of the brain.

Post naive realism extends even to the object we call "numbers" and the structures of relations between them. The Pythagorean view of numbers likes to uphold the notion that numbers are essentially transcendental to our existence. Some mathematicians may then like to believe that mathematical thought is somehow transcendental.

(Post naive realism does not accept this, and this is not a new or recent view. An attentive reading of Plato will reveal that Plato (or Socrates) did not hold calculation and the study of the existential nature of numbers and geometry, to be of prime importance. Rather, he saw numbers as we encounter them, and geometry as we encounter it, as symbolic, and frequently referred to being as the fundamental).

What we currently understand in mathematics is essentially a particular understanding of the way in which our own mind is working. This is not separate from the way natural phenomena works. Because knowledge and thought structures get propagated through principles of teaching and learning and neuroplasticity, and because the principle of the brain is essentially an evolutionary network affair, this knowledge becomes established in networks of minds and brains.

That is why, the Object Theory approach distinguishes between what is considered objective, and what is genuinely objective, the latter being that which is not dependent on any particular brain or network of brains.

This does not imply that what is objective, but not genuinely objective by that definition, has diminished importance. On the contrary, our object-oriented mind can be regarded as part of the way human brain function is developing, collectively, in the evolution of the principle of the brain.

Infinite Processes

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Informally, a process is generally something you might expect has a beginning, a middle, and an end. Contrary to some definitions a process doesn't have to be stepwise. Pouring a bucket of water into another bucket, is a process. Processes can be continuous, or stepwise.

What about an infinite process? Even if we have never found one existentially, in the world, in mathematics you can still propose such a thing. If it is a process, then you might expect it to have a beginning, a middle, and an end. But if it is infinite, then where would the beginning be, and where would the end be? Suppose it were a process of drawing an infinitely long line. You might begin at the beginning, but you never arrive at the end. Or you might see the drawing of it in the middle, but never the beginning or the end.

You might want to say there are an infinity of points in the area of a circle, and an infinity of points around the circumference of a circle. Would you want to say these are infinities of different sizes, related by the number π ?

As soon as we use the word "infinity", or the word "number" or "numbers", we are using those words to stand for objects of thought. It doesn't matter to Object Theory whether we believe those objects to be separate from the mind, on not. Rather, what we are interested in is the structures of these objects. But when you start from the premise that what you are thinking about is separate from the mind you are using to think about it, which is naive realism, then the way you are thinking, is different.

It is not that we can start thinking about mathematics, with personal mind. Mathematics is impersonal. But that doesn't mean it is separate from the mind.

What is the actual structure of relation between the infinity object and the objects we call numbers? What do you actually mean and understand by a number?

Cantor proposed a relation, which was rejected by his contemporaries, and then subsequently became a mainstream rational belief or conceptualisation system that still persists today. In Object Theory terms, such a system is a structure.

Cantor's "proof" that the infinity of real numbers is larger than the infinity of natural numbers has had the status of received wisdom despite giving rise to the (until recently) open question called the continuum hypothesis. Surprisingly, it is still being repeated and propagated as established fact, even though in 2015 Malliaris and Shelah published proof that these infinities are the same "size", for which they won the Hausdorff Medal in 2017.
The Object Theory Context
Not all mathematical "proofs" relate to what can be measured in natural material phenomena. In the context of post naive realism, even what is accepted as proven in mathematics, and therefore a "matter of fact", is still always problematic in that the achieving of this status is mixed up with the notion that mathematical understanding is about something independent of the way a network of minds happens to be working. This essentially springs from the mind's own preconception that its thought structures and structures of comprehension are focussed on something external to and separate from its own workings.

A "classic" example of this is the continuum hypothesis. It arose out of a structure of understanding that is theoretical only, rather then accessible to empirical evidence - namely, structures of understanding based on Cantorian theory.

Where valid measurement that constitutes empirical verification is concerned, and we have arrived at something that is genuinely objective, then in the context of Object Theory this does not mean we have arrived at the knowledge something independent of brain function, the principle of the brain, and hence, the way the human mind network is working. Rather, "genuinely objective" means that this structure is not dependent on any particular network of minds.

The way we conceive infinities, whether we are a mathematician or not, depends upon how our mind works. Even in formal mathematics, the way infinity is conceived is dependent upon the theoretical structures that have been adopted. Sometimes these are calculative, and the outcome of the calculation can be empirically verified. Sometimes the theories are not calculative in such a way that they can be empirically verified.

Infinities in mathematics are often infinities that are said to "converge", if they are part of a theoretical structure that can lead to numerical calculation, and empirical confirmation. Converging infinities are generally part of a process. They are, as it were, "contained" in something finite. As a process, they proceed towards a finite position or point or number. But even infinities that are not said to converge, in mathematics, are often conceived within something finite. Such as the idea that there are an infinite number of "points" on a straight line, or the circumference of a circle.

Such "points" are related to the concept of a number. They are related to numbers that specify the position of the "points". But then, "position" implies the necessity of something else, relative to which "position" is quoted or measured. We are always looking at structures of concepts, or conceptual objects.

All our conceptions of infinity, are based on other objects of conception, and the relations between those objects.

In the final analysis, all our conceptions of infinity, or, indeed, anything else in object oriented understanding, consists of thought about structures, or networks of relations, between objects of thought. And any thought that we are thinking about something outside the mind, in this respect, is naive realism.

So in Object Theory we just look at the structures themselves, as structures of objects. And then some interesting things emerge.

Infinite Iteration Processes

Numbers, Processes and Infinity
For example, in this approach, a process is an object. And we might represent structures as processes, because any object of thought is a process in brain function and thinking. We do not yet have knowledge of the neural process, but we don't need to for our purposes.

The object called an infinity, conceived in mathematics, is always related to other mathematical objects in a structure that can be understood as a process. Hidden from consciousness, it is a psychological process, even though it is an intellectual one, but hidden psychological processes can become exposed by presuming structures and processes in the first instance, and examining what we are doing, in those terms.

For example, it may be perfectly ordinary to consider a number as a thing, rather than a process, but there is an important way in which all numbers are processes, or halted conditions of processes. In fact, infinite processes.

The number 8, for example, as an isolated number, isn't just an object that has no relation to processes. And that is quite apart from the fact that it relates to the processes of brain function, in which the knowledge and thought of it arises.

All ordinary numbers are related to each other in a structure that is related to the process through which they are created. For example, the number 8 doesn't actually exist, in our thought, with any meaning to it, except in relation to the process for creating natural numbers, or the process of counting with natural numbers. And that process is an infinite process. So even a single number 8, in isolation, is part of an infinite process.

And if you look carefully, you will see that the process of counting, or the process of creating natural numbers, is not just an infinite process, but an infinite iteration process, or IIP.

You might think that a quantity, such as a quantity of length represented by a straight line, or a quantity of velocity, is a thing that is not in itself a process, but if we are to be allowed to measure it, or express it, using numbers, and numbers are infinite iteration processes, or parts thereof, then that quantity certainly has a relation to an infinite iteration process.

And so it is, that in Object Theory, both continuum quantities, and numbers, are thought about in terms of infinite iteration processes or IIPs. And because infinite iteration processes, or IIPs are already infinite, this provides a new way of approaching the concept of infinity, and the way it relates to numbers.
Numbers and Quantites
We can represent the infinite iteration process or IIP of creating (or indeed counting with) the natural numbers to the number-base 10, for example, as:
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Here {N} is not a set. That's a different object, in a different schema of understanding. Here, it is the object to which the infinite iteration process of creating natural numbers - the IIP symbolised - applies, and because an IIP produces the natural numbers, then natural numbers themselves are also an IIP. If an IIP produces an "output" then the "output" is itself an IIP. A "process" does not necessarily have to be the mechanism that produces an output of interest, a mechanism such as, for example, an algorithm - it can also be the output itself because it exists through the process and is dependent on it.

The production of natural numbers (or for that matter the process of counting) is essentially an infinite loop algorithm - a nested IIP, which is an IIP in its own right.

Now consider a finite iteration process, such as the production of natural numbers less than 100 in the same order that natural numbers are produced by
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This iteration process halts when its output equals 99 because that is part of the specification we have just given it. Any object that we think or contemplate with, in object oriented or mathematical thought, is something we ourselves, instantiate. We write the symbol, we specify the meaning, we specify the context. It's not about something separate from the mind we are using. So the specification of this iteration process we are now talking about, states that it halts according to its specification - when its output is 99. However, all the other numbers it produces on the way up, are potential halting conditions of the iteration process, other than its specified halting condition.

There is also an infinite iteration process, for example, for expressing the ratio 123 / 456 as a real number. However, this iteration process does not halt, even in its specified condition, unless we add a condition that it should halt after a specific number of decimal places.

The IIP will produce 0.269736842105263157894 and then infinitely repeat the same string: 736842105263157894. We can already see in this IIP that it actually consists of nested IIPs. For example, the process of infinitely repeating the string 736842105263157894 as a string is itself an IIP. The repeating sequence 736842105263157894 is actually produced from the IIP as a "block" of digits that is then infinitely repeated as a block.

So there is more than one way that we could represent the structure of this IIP, just as there is more than one way we could write an algorithm to achieve a specific task.

There can be many layers of nesting in an IIP. For example, in the number above each one of those digits is selected from a set of 10 possible digits for that position after the decimal point.

So it is also that any real number expressed in decimal, for example, is essentially an infinity of related IIPs, one for each digit position, and each one of those IIPs is a halted condition of an IIP that circulates through the digits 0 to 9.

Now consider an irrational number, such as π. The chosen IIP for producing it (there is more than one) also does not halt (it has infinitely many digits after the decimal point). In the case of π, however, being irrational, there is no secondary, nested IIP that we can find, that repeats a given block of digits. Nevertheless, as an IIP this does not make things more difficult for us.

The fact that its IIP does not halt, does not make its IIP essentially any different to that of a repeating or non-repeating decimal number. Even the number 5, for example, is essentially a halted IIP. Because we could write it with infinitely many zeros after the point, and for that matter, according to some arguments, we could also write 4.999999…. And specify infinitely many 9s after the point.
So real numbers in general, can be treated as IIPs, and for that matter therefore so can imaginary numbers. The Object Theory approach then looks at structures of these objects, that are IIPs - sometimes halted, sometimes unhalting.

Structures are essentially networks of objects in which the edges of the network are the relations between the objects. By engaging this approach it is possible examine structures of relations between standard objects such as numbers and various infinities, and to enter a completely new way of comprehending mathematical structures and relations beyond binding to pre-existing notions about numbers and quantities, and even the nature of the objective world.

One thing this notably concerns is the continuum hypothesis that we mentioned earlier, which was based on the Cantorian argument for "cardinalities" - specifically the of the infinity of the reals and the infinity of the naturals. We now know, as we mentioned, from the work of Malliaris and Shelah, that these two infinities are not two different "sizes".

In the Object Theory approach this should not surprise us at all, because the continuum and the infinity of the reals, are two distinct objects to begin with, unless we can show them to be one and the same object. In Object Theory they are two distinct IIPs, and in fact must be, in order for our understanding of them in mathematics to have proceeded the way it has.
The Continuum Hypothesis
The Continuum Hypothesis is essentially the hypothesis that there is not a set whose cardinality is strictly between that of the infinity of the reals and the infinity of the naturals. This structure of reasoning rests on the concept of transfinite numbers. It rests on the concept of the set and the concept of cardinality. In Object Theory terms, essentially, it is an artefact of a structure (of understanding) that does not exist genuinely objectively.

Cohen proved that one cannot prove the nonexistence of an intermediate-sized set between the set of the reals and the set of the naturals, and this implies that an intermediate-sized set might exist, although it cannot be proven. However, in 2015 Malliaris and Shelah showed that the presumption of a difference in cardinality between the reals and the naturals is itself incorrect.

In Objects and Structures we take a different approach, based on foundations divorced from the presumption that a "proof" is of something independent of the way the mind itself happens to be working. Even when the way it is working is the way a whole network of minds is working. Object theory begins with the simpler fact that the continuum and a real number are two distinct objects. And if we try to say that they are not distinct objects, then we have to say why. Only in that way do we reach a genuine understanding within the scheme of thought we are using.

Object Theory, which is ultranumeric, can then be used to show how natural fractal phenomena arising from natural IIPs - specifically, the Julia and Mandelbrot sets - demonstrates empirically the distinction between the continuum and the imaginary numbers. In this context, where there is a distinction, the concept of cardinality or "size" of infinity can be rejected as an artefact of structures of thought in a scheme of understanding that is too limited. The relation between the reals and the continuum is alternatively represented as distinct objects. This object distinction in no way implies, however, that real numbers are incapable of representing continuum quantities.

So the "continuum hypothesis", is in these terms, is essentially empty. It refers to something that is beyond the bounds of the schema of thought that gives rise to it.

Infinite Sets Fallacy

Infinite Sets - an illustration of a fallacy through a card trick

Here is an illustration of how mathematical thought is a consequence of the mind, and the way we are using it, or the way it works. It's not a consequence of some transcendental reality, separate from the mind and intelligence that we are being. Which is why, if we really want to understand the world in which we live, we first need to understand this mind. And in that, is the need to understand the whole of mathematics in terms of the understanding of the operation of our mind.

The Russian mathematician Cantor (1845 - 1918) introduced formally and explicitly the idea of the set in mathematics, and is well known for his work on infinites, and in particular the concept of the infinite set. He also proposed that there are different "sizes" of infinites (cardinalities).

His ideas on sets and infinity were harshly ridiculed in his time, and from 1884 as his mental health deteriorated he was repeatedly hospitalised in a sanatorium. Today, however, his ideas - such as the idea that there are different "sizes" of infinity - seem to be widely accepted or at least still repeated, despite more recent work (Malliaris and Shelah) that contradicts it. How much of this is the realisation of something genuinely objective, and how much is just a matter of network consensus?

A little card trickery
I tell you I am going to place two sets of cards, set A and set B, face down on the table. I tell you they are each infinite sets, and different to each other. Mathematicians do this kind of thing frequently, when they say things like "Let { C } be an infinite set" or "Let { N } be the set of all natural numbers".

I produce a box, and take out one infinite set, and then another box, and take out the other infinite set. I place both infinite sets face down on the table.

I turn over the top two cards, one from each set, to start two new piles, face up, pile C drawn from deck A, and pile D from deck B. We find that the two cards at C and D are different values.

I then begin turning over more cards, from set A, onto the pile C.

Now I explain the rule of the game. The rule is that if we turn over a card from A or B onto its corresponding pile, C or D, and find it is different to the card at the top of the other C or D pile, then we take the next card from the same set A or B that that one came from. As soon as we get a matching pair, we swap to the other set A or B, and continue from there.

We started doing this with deck A, and we are finding that mostly, the cards at C and D are different, so we stay with set A. Every now and then, however, we get a match, and swap to set B. But always, remarkably, when we turn over a card from set B, it matches the card at C, so we have to swap back to drawing from set A.

Clearly, the card sets A and B are not randomly shuffled. They are ordered in some way, and related together. There is a correspondence between them.

At any time, we can count the cards in piles C and D, and always, C will be a larger pile than D. We can actually count them to verify this, because always, they are finite sets. So we conclude that set A must be a larger deck than set B, since there is a clear correspondence between the sets, and the way they are arranged, and we know that both sets are infinite. Therefore, we conclude, there are infinities of different sizes. This seems to us to be a completely logical conclusion, even though both sets are infinite.

So now we attempt to remove the whole of decks A and B so that we can compare their weights and verify their sizes that way. We do this by removing a finite chunk of cards at a time. We manage to remove a chunk of cards but the deck is somehow always still there. We try this with deck B and get the same result. Over and over the same thing happens.

It happens because the decks are infinite.

In fact, we cannot get them off the table, as a whole, unless I use the same magic to put the whole set away back into its box, that I used to take the sets out of the box in the first place, and put them on the table.

So there is no way we can empirically verify that set A is a different size to set B, we just convince ourselves this must be the case.

For any two finite sets of cards it would definitely be the case. No problem there. But we are dealing with infinite sets. They are effectively magic cards. They only came out of the box, onto the table, where we could start manipulating them, in the first place, by means of a trick. What is this magic?

It is the mind itself. The trick began when I said I was going to put two infinite sets on the table. Remember, this is like when mathematicians say thing like "Let { C } be an infinite set" or "Let { N } be the set of all natural numbers". I made it look as though this had nothing to do with the workings of your own mind, when in fact it was exploiting the workings of your mind in order create a psychological illusion.

Getting the cards onto the table in the first place, is where the trick really is. I didn’t really do it. You did. In your mind. Precisely the same thing happens whenever anyone speaks of an infinite set, or an infinite number. Some of the basic ideas behind the theory of infinite sets (Cantorian theory) occur in the same way. They appear convincing, and logical, in a mathematical way, but really, they are psycholinguistic. They depend in the beginning on the subtle use or misuse of language. And then after that, the original language use gets left out of the theorising, which continues on with its own specialist language and symbols.

Then, we can very easily end up with pages of "equation porn" whose sheer complication makes it appear authoritative. And then, as is the way of the world, the best way to confirm authority is with a network consensus. Both are illusory.

In fact, you cannot separate mathematics from language. Both come out of the same principle of the brain, notwithstanding that each corresponds to different brain functions. The brain works as a whole, it is not merely modular. Hence, there is a great deal of mathematics that is actually based on language and the principle of neuroplasticity in the process of learning.


The "Countable" and "Ucountable" Fallacy

Mathematical thought is thought. As such, it is inseparable from brain function, and inseparable from the mechanisms of mind in human beings. That means it is inseparable from human psychology. Psychological processes, even if they are not conscious, can have a major bearing on mathematical thought.

The usual Cantor-style argument is that the real numbers (including non-whole numbers) are “uncountable” whilst the naturals (1,2,3 etc.) are “countable”. This is now, remarkably, it seems, still widely repeated, but this whole idea of “countable” versus “uncountable” relies on a psychological play on the word countable, and the replacement of its normal everyday meaning with the idea of one-to-one correspondence (set bijection). It is in fact nothing to do with understanding mathematics, and rather, a piece of mentalism or psychological illusion, that plays on the psychology of language.

Like many of the ideas behind the theory of infinite sets, it all begins with a misdirection, and psycholinguistics, rather than mathematics.

A popular misdirection employed in transmitting the Cantorian theory concerning "countability" is to present the audience with the simple, but nice and technically sounding idea of bijection. All kinds of misdirection can be covered up if we surround thought-processes with technical sounding language. Again, it is a piece of psychology that relies on the implication that the existence of technical sounding language implies reliability.

Bijection is also called "one-to-one correspondence", and we can see where such correspondence happens between two sets of numbers, if it is there. There are various sets of numbers that you can put side-by-side, and demonstrate this "one-to-one correspondence" or bijection.

We can use this fact to create a psychological illusion before our audience. The misdirection consists essentially of the suggestion that because bijection can exist between the written numbers we use to count with (the natural numbers), and another set of written numbers, that this bijection is necessary in order to count any given set of written numbers.

Allow me to give an example.

Obviously we can biject between two sets of natural numbers:

1 > 1
2 > 2
3 > 3
and son on.

And we can biject between the naturals and the odds, too:

1 > 1
2 > 3
3 > 5
and so on. The same thing applies to bijection with the evens.

However when we do this:

1 > 1
2 > 1.5
3 > 2
4 > 2.6
5 > 2.8
6 > 2.9
7 > 3

and so on, it looks as though the numbers on the left are increasing faster than those on the right, and since the ones on the left are those we are using to count with, we might mis-conclude that if the set on the right is infinite then it is a larger infinity than the set on the left.

Another way of looking at this is to say that between each two natural numbers are infinitely many real numbers. Therefore, we might falsely conclude, the infinity of the reals is larger than the infinity of the naturals.

This is, in fact, a fallacy.

The reason is that it is not necessary to have bijection with the naturals, in order to count with the naturals.

Counting is not a bijection with the naturals, but rather, an iterative process of producing the naturals as "labels" for the objects we are counting, one at a time, on each cycle of counting the distinct objects that are being counted. If those distinct objects are themselves the natural numbers, or the even numbers, for example, then there will of course, be bijection with the naturals being produced by the process of counting, and the real-number objects being counted. But there doesn't have to be.

The implication that there does have to be, is merely a piece of psychological suggestion. It comes from the psycholinguistic blurring of the distinction between different meanings of the word "countable", which we will come to in a moment. It also blurs the distinction between two meanings of the word "number". One meaning of the word "number" is the written symbol or "label" that is generated by the iterative counting process. Another meaning, distinct from that first meaning, is the quantity that the "label" stands for. When we are being informal, we may interchange the word "number" for "quantity". But, in fact, the reason they are two different words, is because they are two different concepts. But like so many of the concepts we use, they can be conflated, depending on the kind of thought processes we are engaged in.

The fact that the "label" or written symbol called a"number", stands for a specific quantity, doesn't mean that the quantity is the same thing as the "label" that stands for it. It is perfectly possible to argue that it is, or that it isn't, using further thought processes. But we are not really understanding what is going on, until we understand that it is about thought processes, and the formation of concepts. The fundamental error is in thinking that what we are thinking about here, is separate from the mind, and the nature of the mind, that is doing the thinking.

In fact, it is all something that is going on in brain function.

Whether we are talking about the natural numbers, or a set of real numbers including non-whole numbers, in both cases, what we are talking about, in its raw form, is a set of distinct objects. And when we are counting distinct objects, we employ an iterative process, that we call "counting". If we want, this iterative process can be an infinite iteration process. An infinite iteration process is not a set. They are two different concepts. Or, actually, in terms of Object Theory, they are two different objects of thought.

Maryanthe Malliaris and Saharon Shelah have in fact now proved within the field of mathematics itself, that the infinities are the same size.

The two kinds of “countable”
"Countable" in the ordinary everyday sense, applied to a set of objects, actually has two meanings. The first, (A), is that it is possible to carry out the act of counting the objects because the situation allows this, whether or not you complete the task. It also means, (B), that it is in principle possible to arrive at an answer by completing the task. This is what countable means before the word is usurped and re-defined as one-to-one correspondence with the natural numbers.

A “real world” example is: The grains of sand in a large static pile on the laboratory table are certainly countable-(A), but probably not countable-(B), if you want to get to lunch on time. But in principle they are countable-(B). The grains of sand on a beach where the tide is breaking are not even countable-(A), not least because they are moving, although we might be able to estimate. Being able to imagine that you can do something in principle, because you can in an imaginary world where you can do anything, doesn’t mean that in principle you can do it. That’s just a play of mind.

An infinity of any objects, if there were such a thing, numbers or otherwise, are by definition uncountable-(B), because you can never complete the task. And again, being able to imagine that you can do something in principle, because you can in an imaginary world where you can do anything, doesn’t mean that in principle you can do it. However, even a supposed infinity of objects would still by definition be countable-(A), just by beginning the task.

The idea of “countability” in Cantorian theory is a piece of mentalism. It starts with defining the word “countable” - a word that already exists in our mind - in terms of one-to-one correspondence with natural numbers, which is mathematically trivial. The problem is that there is an unconscious language trick in play here, of the kind consciously used by mentalists or illusionists to create what is known as “dual reality” (Kenton Knepper). There are two meanings to the word “countable”, the (A) type and the (B) type, and the distinction is critically important in this area. But the distinction between the meanings is glossed over by adopting the mathematical definition.

Defining sets as “countable” or “uncountable” based on one-to-one correspondence with the naturals is therefore not only mathematically trivial but also potentially misleading for the thought processes that are likely to be embarked on. By that definition, an "infinite set" such as the naturals may be called “countable”, when in fact, it is uncountable (B). Furthermore, the implication that there are some objects, whether real numbers or teddy bears, or anything else, that you cannot put in one-to-one correspondence with the natural numbers, and therefore count them, is false.

It is a second piece of mentalism that plays on the word "correspondence". It is yet more psycholinguistics. It deliberately entangles the concept of "correspondence" with specific mathematical relations, and subtly but falsely suggests that disentangled meaning is important in a way that it is not.

The trick here, again, is a language trick not unlike Knepper’s Wonder Words, but in this case disguised through mathematical dressing. The phrase used is “one to one correspondence”. The technical term is by bijection. But the way the word is used in Cantorian theory subtly suggests, like any good mentalist, that it has a meaning and importance beyond what it actually has.

The truth is that you can put any objects you like into “one to one correspondence” with natural numbers, whether they are other natural numbers, teddy bears, or the real numbers. The "one-to-one correspondence" with the natural numbers that is necessary in order to begin counting, does not have to be one in which the objects you are counting have any particular mathematical relation to the numbers you are counting with. And the idea that if that relation does exist, then you can complete counting, is false.

However, you can mislead your audience into believing otherwise, by first subtly suggesting that there is such a thing as an infinite set, whether of natural numbers that you count with, or teddy bears, that exists in the same way as say, an infinite pack of cards. Infinite sets do exist if we want to conceive them, but to imagine they exist like an infinite pack of cards, that you can put in a box, or take out of the box and put on the table, is a misconception. It is a misunderstanding about the nature of infinity.

Dual reality as employed by professional mentalists usually involves a subject on stage understanding words and what is happening differently to how the audience understands it. Mathematical dual reality happens in the mind of the one individual who is the audience who is "understanding" what is going on.

To count, in the everyday way, we put the natural numbers in one-to-one correspondence with the objects we are counting. We do this whether or not these objects are the same or equal to those natural numbers. The illusion pulls the wool over our eyes with regard to what counting means. It creates a whole structure of comprehension that is false, and that mathematicians are particularly vulnerable to, because prior conditioning in how to think about things creates a diversion.

The Cantorian argument is that the real numbers are uncountable because they cannot be put in one-to-one correspondence with the naturals. But in fact, they are every bit as countable (A) as the natural numbers are. And contrary to the psycholinguistics of Cantorian theory, the natural numbers themselves are actually uncountable (B).

The illusion is able to be created in the place due to the inherent tendency to not understand the basic fact about thinking objectively. That fact is that when you think about anything objectively, it merely means you are thinking about objects of thought that you take to be separate from the stuff of your own mind. When we do this, this doesn't mean that these things are separate from the stuff and workings of our own mind. It is just something we work with, a method we must use, in science, as if that were the case. The future of science and mathematics will require that we become more conscious than this. Becoming more conscious in this respect is not about being objective in our thinking - we do that already. It's about becoming transpersonal.


Counting

The point about the “set” of natural numbers is that they are distinct objects. So natural numbers are a structure. That means, that they are not merely a set, but a network.

They are a graph, in the context of graph theory, because every natural number has a relation (an edge in the network) to every other natural number. The whole idea of them boils down to distinct objects that are related to each other. All our rational thinking and perceiving is built from structures (concept-structures, or structures of brain function if you like), and from the idea of distinct objects.

At the purely abstract level, to create what we call numbers we take a path through this infinite graph of distinct objects with no properties or qualities, a path that never passes through any node twice (a Hamiltonian path), and give unique labels to the objects. But we do it in a way that is related to an iteration process.

Specifically, we use a “label generating machine” (that we have invented) specified by the thing we call a number base, to turn out unique labels (distinct objects), which will also be called called "numbers", through an iteration process. We don't know the collective structure of these "label" objects, but we attach them to the objects of the infinite structure, in the order that they come out of the machine, along the path, thus designating the objects in the infinite structure as "number" objects.

We could assign whatever properties we liked to the objects, if we wanted to, thus identifying them in a way that is distinct from the labels we have just put on them. Otherwise, the network or graph is just the purely abstract infinite structure of distinct objects, then labelled with labels from the natural number label generating machine. Each object is furnished with the label, and the objects then become what we otherwise call the "set" of natural numbers, that all have relations to each other in a fully connected network.

We invented "numbers", through our invention of the labelling machine. They didn’t exist without us, without the intelligence we are being. What exists, as the basis of what we invented as the idea of numbers, is the infinite structure of distinct objects. This structure is fundamental to the way we perceive, conceive, and think - in other words, it is already in the principle through which we have our experience of mind and being, through brain function.

We can use the example of the rotary counter, to illustrate the machine. If we are using base 10, the machine has a potential infinity of number wheels, extending out to the left of the first wheel, each with labels “0” to “9” around their circumference. The machine adds a new wheel each time it needed one. You get the idea. Alternatively, we could use a different label generating machine. We could use a “real number” label generating machine instead. It is the same as the first, except that is has a decimal point to the right of the first wheel, and an infinity of wheels extending off to the right of the decimal point.

Now, if the label is for an irrational number like Pi, the wheels to the right extend to infinity. This could be difficult to handle, so we add another piece of kit to the machine that creates a finite label, like “Pi”, for example, to stand for irrational numbers. We also now say that the edges on the graph between one node on the path and the next adjacent one, is a relation called an “infinitesimal” difference.

In both cases, we are dealing with a structure of distinct objects. This happens in our thought. It is part of our mind, part of the intelligence we are being. In each case, we are saying what it is that makes the objects distinct. The idea of any “set” of distinct objects is part of the way our intelligence is working. It is a fundamental principle to the existence of our intelligence - the principle of distinct objects. All mathematical structures arise from that.

Fundamentally, all structures of distinct objects can be bijected to the objects of the natural number structure.

Whether or not a set of objects is countable is not a matter of bijection, because counting is a process, and not a quantity, or a correspondence, or a bijection, none of which are processes. A process doesn't have to be something that "happens" in time, in the everyday way, although it can be, and ordinary everyday counting is such a process.

Fundamentally, a process is a particular kind of structure in which at least some of the objects themselves - defined by their properties - and the relations between them are conditional on the state of the structure. So a written algorithm is a structure that is not a process, but its running on the computer is.

The object we call "counting" is a such process. It is based on an algorithm that is not a process.

We can colloquially call the algorithm for counting a "counting machine" structure. In order for the structure to become a process, it must have other objects added to its structure.

By definition, all "counting machines" process infinitely if they are allowed to. The structure or algorithm on which they are based, does not in itself result in a process that halts. However, the counting machine may halt when it is applied to objects distinct from itself, which then become part of its structure.

In order to apply it to objects distinct from its original structure, those objects must be identifiable through having some chosen property or properties. When an object is "counted" its relation to the the structure is changed by the structure itself, so that it cannot be counted twice.

The objects are "countable" if the "counting machine" halts when it is applied to the objects in question.

For instance, we could say the property of the object is "apple" in a given orchard. Or we could say that the object is "real number". In the first case, the counting machine will halt. In the second case, it will not.

It is not a question of time, because the counting process doesn't necessarily even have to be one that actually takes place in time. It is simply a question of whether or not the process will halt.

Only if the machine halts is the concept "all the objects" a valid concept.

Genuine Objectivity

In Objects and Structures we talk about "genuine objectivity". Scientific method investigates the objective aspect of nature. Initially, science takes the position that scientists are "independent, objective observers". This tacitly implies Cartesian Duality, that is, that the mind of the scientist is an independent, separate entity, from the world of phenomena that it is studying. Cartesian Duality is so-called because it was proposed by Descartes that mind is a separate "substance" from matter. However, this is not a tenable position to take, in the face of the body of knowledge of modern neuroscience.

Even though science has not yet established how brain function equates to our experience of mind, in any kind of fully developed theory, it is still the case that from the existing body of knowledge and evidence, the idea that our conditional experience of being, as mind and self, arises as a construct of brain function, is not in any serious doubt in the mainstream.

The scientific fact is that what we experience as "self", in the ordinary, everyday way, is the sum total of experience being delivered by brain function. How there arises the one who experiences, as "I", is another question that we do not need to get into here. The fact is that the content of our experience, including the way we think and understand, and the way we express, is all dependent on brain function and the way in which that functioning is working. In the brain, in the "wrong place", a little damage can go along way in disrupting or disabling these features of our experience of being.

It is also a scientific fact that what we experience as the material world, is a construct of brain function. So how is it that science is able to investigate an objective aspect of the world, when the only thing we ever experience as the world, is a construct of brain function?

The answer to that is quite simple. It is that whilst what we experience as the world is a construct of brain function, natural phenomena itself is running on principles that are not dependent on anyone's individual brain function. Another way of putting this, is that in the view of mainstream modern neuroscience, Cartesian Duality that proposes a separation of mind in human beings, and matter, is false. The mind is a construct of material brain function. So we can also say that the objective aspect of the world that science studies, is objective because it does not depend on anyone's individual mind.

Nevertheless, because the only thing we ever experience as the world, is a construct of brain function, and because all our thoughts and understanding are a construct of brain function, this objectivity of the material world is not the same thing as the separation and independence of the material world, from our own brain function and experience of being. Rather, the objectivity of the material world is the aspect of our brain function through which our experience of the material world arises, an aspect that is not dependent on our individual brains.

This is what we would expect, after all, no brain organ evolved in isolation, but rather, all human brain organs are uniquely configured instantiations of the same one principle of the brain, that arises in nature, through which arises our conditional experience of being, as human beings.

The assumption that the world we are experiencing as a result, is something other than a construct of brain function, is in fact a naive misinterpretation of the now known scientific facts. This includes the view that such a world exists, and that the brain is constructing a "predictive model" of it, that is what we actually experience. Rather, what we experience is what we experience, and what we experience, is an experience of a material world, whose objective nature does not depend on our individual experience of it. Nonetheless, this does not mean that there is any such world that we are experiencing, that is separate from our experiencing of it.

The "trick" of nature through which arises human consciousness, of self and world, is far greater than the conceptualisation of it as the evolution of a lifeless material world, in which it happens by chance, that there arises a means for "conscious being" in the form of human beings, to come to know this world.

Material brain function is itself part of the material world, and as we know, the only thing we ever experience as the material world, is a construct of brain function. So what we observe as material brain function, is itself a construct of brain function. This provides a clue to the true nature of the relation between the form of intelligence we are being, and the world we find ourselves occupying.

So in science, including mathematics, genuine objectivity is not a matter of understanding something separate from what we are being. Rather, it is a matter of understanding something that does not depend on individual minds, or even specific networks of minds. This is how we can define what is genuinely objective. Whilst merely being objective, is about using thought structures in which what we are thinking about are relations between objects.

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