Mathematics, Treachery and the Magician
In the brain, the neural basis of mathematical thought is quite distinct from that of language processing, although there is some overlap. So when people are presented with something to understand that contains both language-based and mathematics–based material, they tend to approach it through one path or the other. But this doesn't mean that the way language is used in mathematical discussion itself, becomes irrelevant. In fact, it becomes remarkably relevant in set theory when we start to talk about "infinite sets". Without getting bogged down in mathematics symbols, we are going to look at an example issue.
The point here is not simply mathematical critique. It's about the mind. Because mathematics is an activity of the mind, and the mind arises through the brain. Mathematics itself is not a method of understanding that is devoid of psychology. And if we want to understand the brain and the nature of the mind, then it is going to become necessary to understand the psychology of mathematics.
Now there is a backdrop to all this, in physics. Because where physics is at, right now, in the understanding of quantum decoherence, is a place that leaves most commonplace presumptions about life, the world, and even experience as a human being, without a foundation. A full explanation of that would require another article, however, the point to import here is that what is becoming understood in physics - specifically in the science of quantum decoherence - is about the nature of objectivity itself. The commonplace position taken on this is that the material world in which we live is objective because it is separate from us, even though we are understood to have evolved within it. This position is becoming progressively untenable. The crux of the matter is that quantum decoherence, especially now in its extension quantum Darwinism, shows us that what makes the world objective is not that it is fundamentally separate from us - the observers and experiencers of it - but rather, what makes it "objective" is the principle of redundancy.
Extremely briefly, and somewhat simplistically put, if a sufficient number of things or people agree in some way on something, then there is redundancy of agreement. And when redundancy of agreement reaches a point where the whole field is nothing but redundancy, then whatever it is that is agreed on, becomes "objective". That, as I said, is far from a perfect explanation because it is highly condensed and simplified, but it should suffice here. In physics, the way in which the very objectivity of the material world itself in which we live comes about, is through just such a redundancy of what physicists call "pointer states". Quantum Darwinism is already a good understanding of how this redundancy is the way quantum coherence becomes the objectivity of the classical world, the world we all know as the material world in which we live. The world in which the human species evolved. But the principle that objectivity or stability arises from redundancy extends beyond physics, into all kinds of classical situations, into neuroscience, and into psychology.
It even extends into modern mathematics. And to say that, to say that numbers and mathematical structures are just built on the principle of redundancy, in itself, sounds like treachery to the mathematical ideology that says numbers and mathematical structures exist transcendentally in some Platonic or Pythagorean heaven. It's not really treachery to philosophy or theology, though. It doesn't betray Plato or Ficino, because it's not something that is said in the registers in which they are speaking, in the first place. If you want to say that Number is above the beings, and yet arises together with the essence from the One, then I would certainly have no argument with that. But that, is philosophy or theology, and not modern mathematics. And here, we are talking about modern mathematics. That's not all, though. Because we're also talking psychology.
In modern mathematics Cantor's reasoning about the nature of infinity is now widely accepted, although it was widely rejected when it was first proposed. That metamorphosis from rejection to wide acceptance is in itself a demonstration of the redundancy phenomenon. The taking of the multiple cardinalities of infinities to be "objectively true" when once the idea was laughed at, is based on two things. Firstly, it is based on reaching an agreement in the way infinities are conceived. Secondly, it is based on stability that is achieved through redundancy of agreement.
Now this doesn't mean that Cantorian theory (as strengthened up by modern mathematics), is "wrong". What it means, in the context of the redundancy phenomenon is that what is "objectively true", whether it is a mathematical understanding or the material world itself, is something whose "objectivity" is not based on it being separate from and independent of the intelligence that finds it so. Why? Because the intelligence that finds it so evolved. It evolved in a classical world whose objectivity is based on redundancy, and not "separation". And the redundancy principle goes right through from quantum coherence up into the cognitive processes of human beings.
Nothing in science or mathematics is ever established as "true" except by redundant agreement. Just as in quantum Darwinism the very objectivity of the material world itself is only a stabilisation of agreement between redundant pointer states.
Sets
In terms of the emerging understanding of objectivity as arising from redundancy, the concept of the set is not the discovery of an objective thing independent of the intelligence being used by mathematicians. On the contrary, it is a combination of the manifest nature of the world in which we live, stabilised through the principle of redundancy, and the psychology of the intelligence itself, that has evolved in this world. In other words the concept of a set is a psychological concept, and to fully understand it, it is necessary to understand the evolutionary psychology as well, not just the math. And the deep point is that the psychology behind the math is not understandable just through math. To come to understand it and spread the understanding requires language as well.
Let me give you a concrete example. Even today Cantor's argument for whether or not an infinite set is countable or uncountable relies on the "one to one correspondence" argument. We don't need to go into exactly what that is here, but it is based on rigorous mathematical conceptualisation. Those who have studied mathematics will probably know exactly what I am talking about. Its rigorous definition makes it look as though we can leave the psychology of mathematics out of the picture. So what we're going to do here, is to show just how much psychology is involved. And we are going to do it in a way that is as transparent as possible, as informal as possible, and as accessible as possible, without obscuring things behind mathematical symbols or modern psychology jargon.
In what follows, this should all be very familiar ground to anyone who has studied mathematics. But we are going to broaden out the path.
So Cantor's theory of infinite sets involves "sets". To begin with what we are labelling a "set" is understood at the relatively informal level as a "well defined collection of distinct objects". Now that is a clear concept in the mind described by words. But words are treacherous. And for it to have a meaning that is agreed on there must be a redundancy of minds that agree with each other on what it means. (Not unlike a pointer states agreeing with each other in quantum Darwinism). In actual fact, in immediately understanding what is being said here, a mode of cognition is being used in which the whole question of what constitutes "distinct" and what constitutes "objects", has been sidestepped. And underlying this, is the assumption that what we are looking at here is something that is "objective", otherwise we wouldn't be talking about "objects". But the whole question of what makes something "objective" and what that means, is also not even on the table. We are not going to get into all that, but it needs to be pointed out.
Having created this idea of a "set", we can now move forward and then conceive a thing, an object in the mind, called an "infinite set". A set in which the "number" of elements is infinite.
Here, we have a somewhat fuzzy concept of what we mean by "infinite set", despite more careful definitions of a "set". The reason it is "fuzzy" is that according to the definition of what a "set" is, we encounter physically verifiable "sets" in the material world, all the time. It is not difficult to empirically verify the concept. But when it comes to the "infinite set" we are in the realm of a mental extension of what we empirically encounter in the material world itself.
Psychologically, what we have done is to form a small apperception mass out of two distinct concepts. The concept of the set, and the concept of an infinity, or "an infinite number of". Whether or not this is legitimate is questionable and some people would say that this is a category error. However, we are not going to go there, because what we are looking at, is psychology.
In conceiving the "infinite set" we have envisaged infinitely many distinct elements that we can confine to a "closure", like drawing a line around them or putting them in a bag or a box, and we have called that "the set". It is something that is taken to be "complete". And we have said that there are infinitely many elements inside the box, so in intuitive terms unless the box is infinitely large the elements have to be infinitely small. In mathematics, we might try to get around any conceptual difficulties associated with that "infinitely small" by introducing another word, "infinitesimal", and supposing it to mean something essentially different from "infinitely small". But in any case, we can now carry that combination of concepts into what is basically a "closure" with "infinitely many" objects inside, conceived as a single object in the mind that is given the label "infinite set".
The question then arises as to whether this "infinite set" is "countable" or "uncountable". Of course what we are actually referring to here is whether the elements in the set are "countable" or "uncountable". These are words, and words are treacherous, so in the mathematical theory they are carefully defined in order to avoid any ambiguity. However, therein lies the beginning of something very interesting indeed. Much more interesting than the mathematical theory.
Here now, in words, is how the mathematical theory then proceeds:
We start with one "infinite set" which is the set of "natural numbers". Now "natural numbers" are literally the numbers we commonly use to count with. In the number base 10, they are 1, 2, 3, 4, 5, 6… and so on. We will then use this set as the "tool" with which to test whether or not another "infinite set" is to be designated "countable" or "uncountable".
What we have thrown into the language / concept mix so far that will turn out to be psycholinguistically potent, are the words "countable", "uncountable", and the "natural numbers" - which is important because they happen to be the numbers we commonly count with.
Having instantiated in the mind the object called the "infinite set" of the natural numbers, we then move on to the question of whether or not another "infinite set" is "countable" or "uncountable". And because the naturals are the numbers we use to count with, we use the "infinite set" of natural numbers, as a tool to test whether or not our second set is "countable".
We have not actually defined what we mean by "counting" or "count" as a verb, or "count" as a noun standing for a quantity. But we are already using the words "countable" and "uncountable". Nonetheless, the inference or suggestion that what we are talking about is the process of counting, is unmistakable. The use of the words is not purely accidental.
Language
The words "countable" and "uncountable" already each have dual meanings in ordinary language. These are their common meanings. The first common meaning means that it is possible to apply the process of counting. The second common meaning is that it is possible to complete the process of counting. In the theory, the mathematically defined meaning of the word "countable" is different, and we will come to that in due course. So coupled with the mathematically defined meanings, each word has at least three different meanings.
Despite the fact that the meaning of the word "countable" is about to be rigorously mathematically defined, we are now in the realm of multiple meanings of words. Elsewhere in mathematics, words like "integral" or "differentiation" or "multiplication" or "function" do not, in general, have multiple meanings. Neither do words like "electron" or "gravity" in physics. Despite that each may be described in different ways. So there is no room for confusion, or conflation between meanings. When it comes to the word "countable", however, we are on different territory, despite the way the theory defines the word. And we will now see why that is important.
Situations in which language can have more than one meaning, is often exploited by professional illusionists to create what is known in the profession as "dual reality". The technique is a form of psycholinguistics. The essence of the performance technique is misdirection utilising the fact that words can have more than one meaning. The technique is capable of splitting an audience into two parts, one part understanding in one way, and the other part understanding in another way.
In the theatre of the theory of infinite cardinals, the first significant move is the definition of the word "countable". It is not that in making this move in mathematics we are deliberately performing or setting up a "dual reality" as a performing illusionist might do, but what we are doing, as the theory develops, as we shall see, is conflating three different meanings of the same word, and conflating meanings of words is one technique that can be used by psychological illusionists in setting up "dual reality". So whether we have realised it or not, even in the mathematics, we are now in the realm of psychology.
The definition of "countable" is presented as whether or not the second infinite set can be placed in one-to-one correspondence with the set of natural numbers. And "one-to-one correspondence" is formally and rigorously defined. If there is one-to-one correspondence, then the second set is said to be "countable". Otherwise it is said to be "uncountable".
Using suggestion as a tool of misdirection is a standard technique used by illusionists. In the laying out of the theory the first use of suggestion is in the use of the word "countable" and hence its opposite "uncountable". It suggests that the first and second common meanings of these words is going to continue to apply in what is subsequently done. But in fact this is not the case.
The completion of the misdirection comes with the presentation of the rigorous definition of the word "countable" based on the principle of one-to-one correspondence with the natural numbers. Being a rigorous definition presented in mathematics, the definition now carries authority. The introduction of a fact of undisputed authority by drawing attention to it is also a standard part of misdirection technique.
That authority can now be used to undermine the first and second common meanings of the word "countable" and its opposite "uncountable", even though the words and their common meanings are still in play. This paves the way for psychological manipulation to take place by misappropriating the authority of the definition. It is now possible to use the authority of the definition to undermine what are otherwise perfectly reliable cognitive structures.
The suggestion has been that although the authoritative definition is what decides whether a set is "countable" or "uncountable", the first and second common meanings of the words still stand. They are not now explicitly excluded from the theory.
It is of course possible to apply the process of counting (the first common meaning of the word "countable") to any infinite set, regardless of the one-to-one correspondence definition. And it is not possible to complete counting any infinite set (the second common meaning of the word "countable"), also regardless of the one-to-one correspondence definition. It is impossible simply because it is an infinite set and the process of counting it is therefore unending or open. It never reaches closure.
In fact, the same thing is true of the set of all natural numbers. We can start counting them, but we cannot finish. And yet this set is designated by the rigorous definition as "countable" on the basis of the one-to-one correspondence principle.
In short, the "countable" defined by the definition has nothing to do with whether or not counting can be applied to a set, or whether or not the counting can be completed. It is simply a misdirection that plays on the conflation of meanings of words and the obscuration of what is going on through the principle of authority.
This will become very important at the next stage, which is the stage at which the idea of cardinality or the "size" of the infinite set, is introduced.
That's the psychological import of what is going on, at the level of understanding that most illusionists would be working at. But we can go deeper.
Let us consider that our second set to be tested for whether or not it is "countable", is indeed a set of numbers.
At the root of the misdirection here is the fact that in using the one-to-one correspondence test the objecthood of the elements of the set being tested is being conflated with the meanings of the number symbols that are these objects.
The meaning of each number symbol that is each element of a set, is only relevant in the set (the natural numbers) being used to assess whether or not its partner set is deemed "countable", precisely because its elements are the components of "counting". To treat the objects in the set being counted/tested according to the meaning of the object, is to step outside the definition of a set.
It is the same with counting the "size" of any arbitrary set, whether its elements are numbers or not. The symbolic meaning of the element in the set that is being counted is irrelevant. That rule is not broken just by the fact that the elements in the set being counted happen to be numbers.
So the first sleight of mind here, the first misdirection, is using the word "countable" and pointedly associating it with the natural numbers because they are the ones we commonly count with. Then, by arguing that the natural numbers are to be called "countable" because they are in one-to-one correspondence with themselves, we have constructed a rigorous mathematical definition that simply confirms the obvious and trivial fact that the naturals are "countable" according to the first common meaning of the word, by using themselves to count with. We have taken a trivial truth and misdirected attention away from it onto an authoritative but superfluous definition based on one-to-one correspondence, which can then be later misapplied.
We have then laid the way open to easy subsequently conflation of the first and second common meanings of the word "countable". The process exactly mimics a standard procedure in the professional technique of psychological illusionism.
The next step is to then to illegitimately apply the rigorous definition of "countable" that is perfectly consistent with the first common meaning of the word "countable" when it is applied to the naturals, to the relation between another set and the naturals - an illicit move. It's not that the bijection test (one-to-one correspondence) necessarily fails due to the illicitness, rather, sometimes it fails, sometimes it does not, depending on the set in question. But it has nothing to do with whether or not the second set is "countable" according to either of the common meanings of the word "countable".
If you never introduce the language concept of "countable" to begin with, and substitute "X" for "countable" and "Y" for "uncountable" in the theory, you will find that when comparing the naturals with the reals, according to the one-to-one correspondence test, the naturals must be labelled "X" and the reals must be labelled "Y". But there is nothing that suggests any further meaning in those labels that points to the idea that the set labelled "Y" is somehow "bigger" or contains more elements than the set labelled "X", except when a misdirection of the mind is involved. The same would be true had we used binary to count with, rather than the natural numbers. The whole idea of "difference in cardinality" is tied to the interplay of language with mathematics. And once a principle has been established such as difference in cardinality, and established through redundancy, then it is possible to build much larger structures on that principle. Which also then become "objectively true" based on redundancy.
Now the remarkable thing is that this doesn't make it "wrong". What it means is that you cannot isolate mathematical reasoning from the intelligence, and what becomes "reasoned" as "true" is not the discovery of an object separate from the intelligence. Rather, it is a discovery of a feature concerning the intelligence. And despite that the brain processes different areas of cognition in different ways or areas of the brain, the different processes are always entangled at some level. And even a psychological test will not necessarily be able to probe down to the level at which they are entangled. Our intelligence is evolutionary, and our mathematical intelligence cannot be separated from the brain, the breathing, the heart, or consciousness as we know it.
A classic example of misdirection in stage illusionism is the trick of the juggler and the gorilla. The audience is invited to detect a particular move taking place in the act of juggling, and in doing so, they completely miss the entry and exit of someone dressed as a gorilla crossing the stage behind the juggler. Other variations of this trick exist. But when it comes to psychologically induced illusions of cognition things can get MUCH more sophisticated, as modern professional illusionists know very well. One of the outcomes of development in the professional art of illusion making is the principle of building an illusion based on misdirection that occurs so much earlier than the main display or climax that its role in what is happening is never even suspected.
We humans are very much in that position now. Quantum Darwinism has already set the stage for our rumbling of the original misdirection. We just need a little help. And who is the magician? The answer to that involves the language game again. Because the magician's name is "nature", and that's a word with multiple different meanings.
The point here is not simply mathematical critique. It's about the mind. Because mathematics is an activity of the mind, and the mind arises through the brain. Mathematics itself is not a method of understanding that is devoid of psychology. And if we want to understand the brain and the nature of the mind, then it is going to become necessary to understand the psychology of mathematics.
Now there is a backdrop to all this, in physics. Because where physics is at, right now, in the understanding of quantum decoherence, is a place that leaves most commonplace presumptions about life, the world, and even experience as a human being, without a foundation. A full explanation of that would require another article, however, the point to import here is that what is becoming understood in physics - specifically in the science of quantum decoherence - is about the nature of objectivity itself. The commonplace position taken on this is that the material world in which we live is objective because it is separate from us, even though we are understood to have evolved within it. This position is becoming progressively untenable. The crux of the matter is that quantum decoherence, especially now in its extension quantum Darwinism, shows us that what makes the world objective is not that it is fundamentally separate from us - the observers and experiencers of it - but rather, what makes it "objective" is the principle of redundancy.
Extremely briefly, and somewhat simplistically put, if a sufficient number of things or people agree in some way on something, then there is redundancy of agreement. And when redundancy of agreement reaches a point where the whole field is nothing but redundancy, then whatever it is that is agreed on, becomes "objective". That, as I said, is far from a perfect explanation because it is highly condensed and simplified, but it should suffice here. In physics, the way in which the very objectivity of the material world itself in which we live comes about, is through just such a redundancy of what physicists call "pointer states". Quantum Darwinism is already a good understanding of how this redundancy is the way quantum coherence becomes the objectivity of the classical world, the world we all know as the material world in which we live. The world in which the human species evolved. But the principle that objectivity or stability arises from redundancy extends beyond physics, into all kinds of classical situations, into neuroscience, and into psychology.
It even extends into modern mathematics. And to say that, to say that numbers and mathematical structures are just built on the principle of redundancy, in itself, sounds like treachery to the mathematical ideology that says numbers and mathematical structures exist transcendentally in some Platonic or Pythagorean heaven. It's not really treachery to philosophy or theology, though. It doesn't betray Plato or Ficino, because it's not something that is said in the registers in which they are speaking, in the first place. If you want to say that Number is above the beings, and yet arises together with the essence from the One, then I would certainly have no argument with that. But that, is philosophy or theology, and not modern mathematics. And here, we are talking about modern mathematics. That's not all, though. Because we're also talking psychology.
In modern mathematics Cantor's reasoning about the nature of infinity is now widely accepted, although it was widely rejected when it was first proposed. That metamorphosis from rejection to wide acceptance is in itself a demonstration of the redundancy phenomenon. The taking of the multiple cardinalities of infinities to be "objectively true" when once the idea was laughed at, is based on two things. Firstly, it is based on reaching an agreement in the way infinities are conceived. Secondly, it is based on stability that is achieved through redundancy of agreement.
Now this doesn't mean that Cantorian theory (as strengthened up by modern mathematics), is "wrong". What it means, in the context of the redundancy phenomenon is that what is "objectively true", whether it is a mathematical understanding or the material world itself, is something whose "objectivity" is not based on it being separate from and independent of the intelligence that finds it so. Why? Because the intelligence that finds it so evolved. It evolved in a classical world whose objectivity is based on redundancy, and not "separation". And the redundancy principle goes right through from quantum coherence up into the cognitive processes of human beings.
Nothing in science or mathematics is ever established as "true" except by redundant agreement. Just as in quantum Darwinism the very objectivity of the material world itself is only a stabilisation of agreement between redundant pointer states.
Sets
In terms of the emerging understanding of objectivity as arising from redundancy, the concept of the set is not the discovery of an objective thing independent of the intelligence being used by mathematicians. On the contrary, it is a combination of the manifest nature of the world in which we live, stabilised through the principle of redundancy, and the psychology of the intelligence itself, that has evolved in this world. In other words the concept of a set is a psychological concept, and to fully understand it, it is necessary to understand the evolutionary psychology as well, not just the math. And the deep point is that the psychology behind the math is not understandable just through math. To come to understand it and spread the understanding requires language as well.
Let me give you a concrete example. Even today Cantor's argument for whether or not an infinite set is countable or uncountable relies on the "one to one correspondence" argument. We don't need to go into exactly what that is here, but it is based on rigorous mathematical conceptualisation. Those who have studied mathematics will probably know exactly what I am talking about. Its rigorous definition makes it look as though we can leave the psychology of mathematics out of the picture. So what we're going to do here, is to show just how much psychology is involved. And we are going to do it in a way that is as transparent as possible, as informal as possible, and as accessible as possible, without obscuring things behind mathematical symbols or modern psychology jargon.
In what follows, this should all be very familiar ground to anyone who has studied mathematics. But we are going to broaden out the path.
So Cantor's theory of infinite sets involves "sets". To begin with what we are labelling a "set" is understood at the relatively informal level as a "well defined collection of distinct objects". Now that is a clear concept in the mind described by words. But words are treacherous. And for it to have a meaning that is agreed on there must be a redundancy of minds that agree with each other on what it means. (Not unlike a pointer states agreeing with each other in quantum Darwinism). In actual fact, in immediately understanding what is being said here, a mode of cognition is being used in which the whole question of what constitutes "distinct" and what constitutes "objects", has been sidestepped. And underlying this, is the assumption that what we are looking at here is something that is "objective", otherwise we wouldn't be talking about "objects". But the whole question of what makes something "objective" and what that means, is also not even on the table. We are not going to get into all that, but it needs to be pointed out.
Having created this idea of a "set", we can now move forward and then conceive a thing, an object in the mind, called an "infinite set". A set in which the "number" of elements is infinite.
Here, we have a somewhat fuzzy concept of what we mean by "infinite set", despite more careful definitions of a "set". The reason it is "fuzzy" is that according to the definition of what a "set" is, we encounter physically verifiable "sets" in the material world, all the time. It is not difficult to empirically verify the concept. But when it comes to the "infinite set" we are in the realm of a mental extension of what we empirically encounter in the material world itself.
Psychologically, what we have done is to form a small apperception mass out of two distinct concepts. The concept of the set, and the concept of an infinity, or "an infinite number of". Whether or not this is legitimate is questionable and some people would say that this is a category error. However, we are not going to go there, because what we are looking at, is psychology.
In conceiving the "infinite set" we have envisaged infinitely many distinct elements that we can confine to a "closure", like drawing a line around them or putting them in a bag or a box, and we have called that "the set". It is something that is taken to be "complete". And we have said that there are infinitely many elements inside the box, so in intuitive terms unless the box is infinitely large the elements have to be infinitely small. In mathematics, we might try to get around any conceptual difficulties associated with that "infinitely small" by introducing another word, "infinitesimal", and supposing it to mean something essentially different from "infinitely small". But in any case, we can now carry that combination of concepts into what is basically a "closure" with "infinitely many" objects inside, conceived as a single object in the mind that is given the label "infinite set".
The question then arises as to whether this "infinite set" is "countable" or "uncountable". Of course what we are actually referring to here is whether the elements in the set are "countable" or "uncountable". These are words, and words are treacherous, so in the mathematical theory they are carefully defined in order to avoid any ambiguity. However, therein lies the beginning of something very interesting indeed. Much more interesting than the mathematical theory.
Here now, in words, is how the mathematical theory then proceeds:
We start with one "infinite set" which is the set of "natural numbers". Now "natural numbers" are literally the numbers we commonly use to count with. In the number base 10, they are 1, 2, 3, 4, 5, 6… and so on. We will then use this set as the "tool" with which to test whether or not another "infinite set" is to be designated "countable" or "uncountable".
What we have thrown into the language / concept mix so far that will turn out to be psycholinguistically potent, are the words "countable", "uncountable", and the "natural numbers" - which is important because they happen to be the numbers we commonly count with.
Having instantiated in the mind the object called the "infinite set" of the natural numbers, we then move on to the question of whether or not another "infinite set" is "countable" or "uncountable". And because the naturals are the numbers we use to count with, we use the "infinite set" of natural numbers, as a tool to test whether or not our second set is "countable".
We have not actually defined what we mean by "counting" or "count" as a verb, or "count" as a noun standing for a quantity. But we are already using the words "countable" and "uncountable". Nonetheless, the inference or suggestion that what we are talking about is the process of counting, is unmistakable. The use of the words is not purely accidental.
Language
The words "countable" and "uncountable" already each have dual meanings in ordinary language. These are their common meanings. The first common meaning means that it is possible to apply the process of counting. The second common meaning is that it is possible to complete the process of counting. In the theory, the mathematically defined meaning of the word "countable" is different, and we will come to that in due course. So coupled with the mathematically defined meanings, each word has at least three different meanings.
Despite the fact that the meaning of the word "countable" is about to be rigorously mathematically defined, we are now in the realm of multiple meanings of words. Elsewhere in mathematics, words like "integral" or "differentiation" or "multiplication" or "function" do not, in general, have multiple meanings. Neither do words like "electron" or "gravity" in physics. Despite that each may be described in different ways. So there is no room for confusion, or conflation between meanings. When it comes to the word "countable", however, we are on different territory, despite the way the theory defines the word. And we will now see why that is important.
Situations in which language can have more than one meaning, is often exploited by professional illusionists to create what is known in the profession as "dual reality". The technique is a form of psycholinguistics. The essence of the performance technique is misdirection utilising the fact that words can have more than one meaning. The technique is capable of splitting an audience into two parts, one part understanding in one way, and the other part understanding in another way.
In the theatre of the theory of infinite cardinals, the first significant move is the definition of the word "countable". It is not that in making this move in mathematics we are deliberately performing or setting up a "dual reality" as a performing illusionist might do, but what we are doing, as the theory develops, as we shall see, is conflating three different meanings of the same word, and conflating meanings of words is one technique that can be used by psychological illusionists in setting up "dual reality". So whether we have realised it or not, even in the mathematics, we are now in the realm of psychology.
The definition of "countable" is presented as whether or not the second infinite set can be placed in one-to-one correspondence with the set of natural numbers. And "one-to-one correspondence" is formally and rigorously defined. If there is one-to-one correspondence, then the second set is said to be "countable". Otherwise it is said to be "uncountable".
Using suggestion as a tool of misdirection is a standard technique used by illusionists. In the laying out of the theory the first use of suggestion is in the use of the word "countable" and hence its opposite "uncountable". It suggests that the first and second common meanings of these words is going to continue to apply in what is subsequently done. But in fact this is not the case.
The completion of the misdirection comes with the presentation of the rigorous definition of the word "countable" based on the principle of one-to-one correspondence with the natural numbers. Being a rigorous definition presented in mathematics, the definition now carries authority. The introduction of a fact of undisputed authority by drawing attention to it is also a standard part of misdirection technique.
That authority can now be used to undermine the first and second common meanings of the word "countable" and its opposite "uncountable", even though the words and their common meanings are still in play. This paves the way for psychological manipulation to take place by misappropriating the authority of the definition. It is now possible to use the authority of the definition to undermine what are otherwise perfectly reliable cognitive structures.
The suggestion has been that although the authoritative definition is what decides whether a set is "countable" or "uncountable", the first and second common meanings of the words still stand. They are not now explicitly excluded from the theory.
It is of course possible to apply the process of counting (the first common meaning of the word "countable") to any infinite set, regardless of the one-to-one correspondence definition. And it is not possible to complete counting any infinite set (the second common meaning of the word "countable"), also regardless of the one-to-one correspondence definition. It is impossible simply because it is an infinite set and the process of counting it is therefore unending or open. It never reaches closure.
In fact, the same thing is true of the set of all natural numbers. We can start counting them, but we cannot finish. And yet this set is designated by the rigorous definition as "countable" on the basis of the one-to-one correspondence principle.
In short, the "countable" defined by the definition has nothing to do with whether or not counting can be applied to a set, or whether or not the counting can be completed. It is simply a misdirection that plays on the conflation of meanings of words and the obscuration of what is going on through the principle of authority.
This will become very important at the next stage, which is the stage at which the idea of cardinality or the "size" of the infinite set, is introduced.
That's the psychological import of what is going on, at the level of understanding that most illusionists would be working at. But we can go deeper.
Let us consider that our second set to be tested for whether or not it is "countable", is indeed a set of numbers.
At the root of the misdirection here is the fact that in using the one-to-one correspondence test the objecthood of the elements of the set being tested is being conflated with the meanings of the number symbols that are these objects.
The meaning of each number symbol that is each element of a set, is only relevant in the set (the natural numbers) being used to assess whether or not its partner set is deemed "countable", precisely because its elements are the components of "counting". To treat the objects in the set being counted/tested according to the meaning of the object, is to step outside the definition of a set.
It is the same with counting the "size" of any arbitrary set, whether its elements are numbers or not. The symbolic meaning of the element in the set that is being counted is irrelevant. That rule is not broken just by the fact that the elements in the set being counted happen to be numbers.
So the first sleight of mind here, the first misdirection, is using the word "countable" and pointedly associating it with the natural numbers because they are the ones we commonly count with. Then, by arguing that the natural numbers are to be called "countable" because they are in one-to-one correspondence with themselves, we have constructed a rigorous mathematical definition that simply confirms the obvious and trivial fact that the naturals are "countable" according to the first common meaning of the word, by using themselves to count with. We have taken a trivial truth and misdirected attention away from it onto an authoritative but superfluous definition based on one-to-one correspondence, which can then be later misapplied.
We have then laid the way open to easy subsequently conflation of the first and second common meanings of the word "countable". The process exactly mimics a standard procedure in the professional technique of psychological illusionism.
The next step is to then to illegitimately apply the rigorous definition of "countable" that is perfectly consistent with the first common meaning of the word "countable" when it is applied to the naturals, to the relation between another set and the naturals - an illicit move. It's not that the bijection test (one-to-one correspondence) necessarily fails due to the illicitness, rather, sometimes it fails, sometimes it does not, depending on the set in question. But it has nothing to do with whether or not the second set is "countable" according to either of the common meanings of the word "countable".
If you never introduce the language concept of "countable" to begin with, and substitute "X" for "countable" and "Y" for "uncountable" in the theory, you will find that when comparing the naturals with the reals, according to the one-to-one correspondence test, the naturals must be labelled "X" and the reals must be labelled "Y". But there is nothing that suggests any further meaning in those labels that points to the idea that the set labelled "Y" is somehow "bigger" or contains more elements than the set labelled "X", except when a misdirection of the mind is involved. The same would be true had we used binary to count with, rather than the natural numbers. The whole idea of "difference in cardinality" is tied to the interplay of language with mathematics. And once a principle has been established such as difference in cardinality, and established through redundancy, then it is possible to build much larger structures on that principle. Which also then become "objectively true" based on redundancy.
Now the remarkable thing is that this doesn't make it "wrong". What it means is that you cannot isolate mathematical reasoning from the intelligence, and what becomes "reasoned" as "true" is not the discovery of an object separate from the intelligence. Rather, it is a discovery of a feature concerning the intelligence. And despite that the brain processes different areas of cognition in different ways or areas of the brain, the different processes are always entangled at some level. And even a psychological test will not necessarily be able to probe down to the level at which they are entangled. Our intelligence is evolutionary, and our mathematical intelligence cannot be separated from the brain, the breathing, the heart, or consciousness as we know it.
A classic example of misdirection in stage illusionism is the trick of the juggler and the gorilla. The audience is invited to detect a particular move taking place in the act of juggling, and in doing so, they completely miss the entry and exit of someone dressed as a gorilla crossing the stage behind the juggler. Other variations of this trick exist. But when it comes to psychologically induced illusions of cognition things can get MUCH more sophisticated, as modern professional illusionists know very well. One of the outcomes of development in the professional art of illusion making is the principle of building an illusion based on misdirection that occurs so much earlier than the main display or climax that its role in what is happening is never even suspected.
We humans are very much in that position now. Quantum Darwinism has already set the stage for our rumbling of the original misdirection. We just need a little help. And who is the magician? The answer to that involves the language game again. Because the magician's name is "nature", and that's a word with multiple different meanings.