Mathematics

The Reality of Rational and Irrational Numbers

In the previous post we talked about the problem of mathematical realism of negative and complex numbers; the issue was that you can construct these numbers logically and conceptually, but you will never find them in the real world. The problem of irrational numbers is the opposite: you can easily find irrational numbers such as √2, π, and e in the real world, but they appear to have infinite irreducible complexity. Similarly, many rational numbers such as 1/3 have infinite digits in them. Unlike negative numbers which are fully understood but never seen, rational and irrational numbers are seen but never fully understood. In what sense are these numbers then conceptually real?

The Circle Example

We can describe the problem of the reality of the irrational numbers by drawing a circle. It takes a small amount of time to draw a circle using a protractor and pencil, but the circumference of the circle is described by 2πr in which π is an irrational number. Irrational numbers have an infinite fractional part, in which the digits don’t repeat. This leads to the question: Why does a geometrically finite operation result in a number that has infinite digits? Mathematicians will argue that the number itself is not infinite, which is true. However, an irrational number contains infinite information. If we looked at the same problem informationally then finite information is needed to represent the geometrical task, but the result of that finite information contains infinite information.

Something that contains infinite information cannot be grasped or understood, because even though the physical entity appears finite, the concept is infinitely complex.

The Example of Dividing a Line

All rationals such as 1/3 = 0.3333 .. also have infinite digits but the sequence repeats. Rationals can be geometrically constructed by scaling operations. For example, to trisect a line, draw a second line comprised of equal segments that intersects it at the start; then join the end of the second line with the first, and draw parallel lines at the points where the segments in the second line begin; wherever the parallel lines intersect the first line divides the line into three parts. Effectively, we have taken a line in which the segments are of well-known length (e.g. unit length) and scaled it to another line of unknown length. Any arbitrary fraction can be constructed by this basic geometrical procedure.

However, we are still left with the paradox of how an operation like trisecting a line produces a number that has infinite digits in it? The operation is clearly finite; so, how could it create infinite informational complexity? Why is it that an operation that has a clear end results in a number that hasn’t any end? The problem of infinite digits presents itself in the case of both rational and irrational numbers, and it needs an answer.

What is a Scaling Operation?

There are two intuitive ways to think about a scaling operation. First, we are trying to define or describe one concept in terms of another. Suppose you see a rope and think of it as a snake. The parts of the rope would now be described in terms of the parts of the snake. So, you divide the rope into parts that resemble the form of a snake. For instance, some part of the rope would be called the ‘head’ and the other is called ‘tail’. If the rope is big, then the snake would also be big; if the rope is small, the snake will appear to be small. The rope here is the numerator and the snake is the denominator: we are trying to divide the rope and produce its parts such that they resemble the parts of the snake.

Second, we can think of the scaling operation as a representation of one concept inside another; these representations occur when we know the world because then we encode the external world in our minds. Now, you can think of the numerator as the mind which is divided into parts by the presence of the external object as the denominator. Our mind—whatever it was before the perception—is absorbed in the thought of the external object and it therefore takes the form of that object. Each mind can see the same thing differently, so we must keep the original mind and its modification in the composite picture.

These two intuitive ways of thinking about a fraction are different applications of the same idea. In the first case, an external object is being divided by the concept from our mind. And in the second case, the external idea is dividing our mind. This notion about division becomes much clearer if we represent numbers as a tree structure because then the two entities are higher and lower nodes in the tree. The higher node is the dividend and the lower node is the divisor, but they exist simultaneously as two separate entities.

If, however, we collapse the tree hierarchy and view the hierarchy of nodes as a single node, then we arrive at a fractional part of the whole. The problem of rational numbers is that we cannot always exactly represent an idea in the mind, or perfectly see one idea in terms of another. In such cases, the ‘fraction’ results in infinite digits—if we try to collapse the hierarchy. But, factually, we don’t have to collapse the hierarchy; we can represent a fraction by expressions such as 1/3 or 7/24. Notably, all rational numbers can be represented as the ratio of two numbers, so there isn’t a problem of understanding how they are created, although there is a problem about the outcome of the process.

Common examples of such divisions are conceptual idealizations and approximations. For example, we can think of a planet or a billiard ball in terms of a point particle. Clearly, the ball or the planet is not a point, so the idealization must discard some features. Conversely, we might want to think of a rock as a chair, and some part of the ideal chair may be missing. Similarly, in dividing two numbers we may be left with a remainder, and we can either approximate the fraction or continue the division process indefinitely. When the process continues indefinitely, we have two concepts that are mutually irreducible.

The Role of Prime Numbers in Irreducibility

According to the fundamental theorem of arithmetic, every natural number can be expressed as a multiple of prime numbers. Hence, both numerator and denominator can be expressed as prime number factors. Some of these factors may cancel out, and ultimately every fraction is the attempt to see one set of primes in terms of other primes.

Each of these primes is mutually irreducible, which means they are concepts which cannot be expressed in terms of other concepts. Since the primes are infinite, there are infinite mutually irreducible concepts. How these mutually irreducible primes are constructed from something more fundamental is a separate issue that I will not cover here. There are mathematical conjectures such as the Reimann Hypothesis and the Goldbach’s Conjecture which try to reduce all primes to a certain series and the sum of other primes, but the true nature of primes remains the biggest unsolved problem in mathematics today.

The real issue is that we still don’t understand how irreducible concepts can be constructed from some fundamental concepts. Without this construction, each irreducible concept will become fundamental and there would be infinite such fundamental ideas. Ultimately, it would entail that the world is infinitely unknowable because we cannot grasp infinite irreducible concepts, and we cannot construct them from fundamental ideas. So, the construction of primes is a very fundamental problem with huge ramifications. Note the inherent contradiction in this problem: the primes are mutually irreducible, and yet we want a method by which we can reduce them to something more fundamental. The only prime to which every other prime can be reduced is 1, so we need to find a way to construct all primes from 1, and then every other number from these primes.

Given that there are mutually irreducible numbers represented by primes, we can say that the reason for infinite digits in a rational number owes to the attempt to interpret one prime in terms of another and given the mutual irreducibility this is impossible. This impossibility is represented by the infinite series of digits in most rationals.

Of course, not every rational has infinite digits. For example, 1/2 is 0.5 and it has a finite number of digits. We can say that it is possible to interpret some concepts in terms of others, or they are mutually reducible. Those with infinite digits are mutually irreducible. This analysis in terms of concepts gives us a grasp of why some rationals have infinite digits while others don’t. We can also see that this problem arises if we try to collapse the tree hierarchy, because otherwise there is a succinct representation. Finally, the problem is demystified when we treat numbers as concepts. We could also invert this idea and say that all concepts are numbers, and some concepts are mutually irreducible.

The Problem of Irrational Numbers

Once we reduce the understanding of all rationals to those of primes, with division defined as the interpretation of one concept in terms of other concepts, we are still left with the problem of irrationals which cannot be expressed as fractions. While natural numbers, fractions, and irrational numbers are all infinite, Georg Cantor showed that the ‘number’ of real numbers (real numbers include natural numbers, fractions, and irrational numbers) is exponentially higher than the ‘number’ of natural numbers. If the count of natural numbers is N, then the count of real numbers is 2N. Since the count of natural and rationals is N (Cantor showed they have the same count), irrationals are dominant in 2N.

The number 2N has a neat interpretation—if there is a set with N objects, then there is a set called the power set which has 2N members. A power set is the set of all subsets of a set. For example, consider a set of three members {A, B, C}. This set has 8 possible subsets which can be represented as a power set {{A}, {B}, {C}, {A, B}, {B, C}, {C, A}, (A, B, C), {}}, which include the original set as well as a null set. This result can be used to understand the nature of real numbers if we say that each real is a member of the power set.

Now, if A, B, and C are viewed as concepts, then a member of a power set is a relation between these concepts. Generally, relations can be ordered which means we consider A → B and B → A as two separate ordered relations. The power set doesn’t deal in this order, but in simply identifying the members that are being related. Effectively, a power set is all the ways in which you can slice and dice a collection of individual entities.

The fact that real numbers have an order therefore means that some collections are logically prior to others. So, there is an order in between the power set members, but there is no order inside the members of the power set. Alternately, to map the members of the power set to real numbers, we would need to order these individual subsets. With this interpretation, we can see that real numbers are not like natural numbers or rationals at all. However, if we treat the natural numbers as concepts, and rationals as interpretation of one concept in terms of another concept, then the real numbers are groupings of concepts.

Real Numbers and the Nature of Time

Suppose N represents all the possible facts (Cantor had showed that the ‘count’ of rationals equals the count of natural numbers, so these facts include all the interpretive or perceptual events, which means all possible facts include facts about observers). A subset of all these facts become real at a given time, and each moment in time therefore represents a member of the power set. These members are now ordered as moments in time. Thus, if we treat all the facts as possibilities which become reality in time, and a subset of all that is eternally possible becomes real at any moment, then there is a neat interpretation of real numbers as moments in time that convert the possibility into reality.

Effectively, a real number is not describing an individual object, but a collection of objects. This collection is not reducible to the members, but is an object, although this object cannot be placed in the same space as the original individual objects. For example, if the original objects are points then the collection of objects is not a point in the same space. However, if we force the idea that the collection is in the same space as the original objects, then we end up with irrational numbers—with infinite digits in them. This entails the flaw in thinking that the collection of things is also a thing of the same type as the original things.

To truly think about the collection of things, we need another type of ‘dimension’. That ‘dimension’ is time. Unlike space which constitutes all the things, the subsets of these things are not a thing in the same space, but something in another dimension. Thus, all the facts in the universe at a given time are individual points in space. However, the subsets of these facts put together is also an individual entity represented by a real number. So, it is possible to treat real numbers as points in some ‘space’, but it is not the same space.

Real numbers arise out of a flaw in thinking in which a collection of things is of the same type as those things. If you think of the collection as a boundary that segregates things into sets, then those boundaries are not in the same space. However, if you want to force the idea that the boundary is in the same space, then you can get an irrational number.

Sets as Mathematical Functions

A set can always be represented by a function. For example, the unit circle is represented by the function x2 + y2 = 1. When we map sets of objects to real numbers, we can effectively say that we are numbering functions. The fact that real numbers are uncountable means that if the natural numbers and fractions are countable then the number of functions defined over these numbers are uncountable. If the numbers exist in some ‘space’ then the functions are in a different ‘space’ (this ‘space’ as we have said above is time).

This leads us to a new understanding of time in which it instantiates different functions at different moments which manifests different possibilities into reality. Hence, time cannot be homogeneous—i.e. a single function. Similarly, every function has a domain where it is valid, which means that the laws cannot be valid everywhere. An idea foundational to modern science is that laws of nature—i.e. the functions that describe the world—remain unchanged over time and space. If time is identified as functions that operate on a domain, then the selection of the function will change the laws over time and space.

We can now distinguish between Eternal Time and passing time. The Eternal Time is all the functions which exist eternally as mathematical objects. The passing time is that these functions are selected one by one and applied to different domains. The collection of Eternal and passing time can now be called Causal Time which creates a new world by selecting a possibility. However, this possibility is governed by the choice of a function and a domain, therefore, it is not like the parameterized time used in modern science.

The Meaning of Infinite Series

A standard artifact of modern mathematics is the idea that numbers can be expressed as a series, which are basically infinite functions. For example, the number π can be expressed as the series 4 – 4/3 + 4/5 – 4/7 + 4/9 – 4/11 + … One could write this series by defining a recursive function f(n) = 4/n – 4/n+2 + f(n+3). The recursion here represents a hierarchy or a self-repeating pattern. The number π is the function f(n) and it is a finite algebraic expression. You input the number 1 into the expression, and you get the output π.

Therefore, if functions or algebraic expressions were real, then π would be real, because it is nothing but the algebraic expression f(n) above which naturally leads to π. Therefore, instead of talking about whether π is real, we can also ask if f(n) is real. Notably, f(n) is not of infinite complexity, and hence it is not an infinite object. However, because of recursion you can compute this function to infinite accuracy, or to just a finite accuracy.

If we think of a tree topology, then recursion is simply expanding the tree to more branches and leaves, and this expansion represents greater and greater accuracy of π. However, even if you terminate this recursion arbitrarily, you will get some approximate value of π. This is an important idea because we now identify π not with a numerical value but with a function which produces a value. The value may have infinite digits, but the function is finite.

This idea can be related to some very fundamental results in modern mathematics such as the sum of all natural numbers or 1 + 2 + 3 + 4 + 5 … = -1/12. Clearly, under normal circumstances if you add positive numbers, you will never get a negative number, let alone a negative fraction. How then do we understand such mathematical results? One way to resolve this problem is to define a recursive function g(n) = n + g(n+1) and in the ‘space’ of all functions this function will be location -1/12. By this approach, we clearly distinguish between the number -1/12 which represents a function rather than a value. And yet, you can substitute the entire series by this function and the function by a number. Obviously, some sleight of hand is involved in these tricks which means that this is not to be done under all circumstances, but in some cases, this would be a very valid answer.

The trick here is semantics. Functions such as f(n) and g(n) are shorthand for expressions. To map these expressions to numbers one must conceive a semantic space. The points or locations in this space are functions and because the locations can be mapped to numbers the functions can be mapped to numbers. The numbers applied to these functions are valid in that semantic space and not in the space of values. But if we conceive a semantic space in which expressions are mapped to numbers, the above results will be true.

Observing Irrationals in the Real World

We can return to the original question of why it is easy to draw a circle in finite time implying that there is finite information and yet writing this information in the same space as a number involves infinite time. The reason is that the circle is a function which is a finite object in time, but if we want to think of this object as something in space then it has infinite information. The circumference of the circle should in fact be viewed as a collection of points and π represents this collection rather than an individual point.

In terms of length, a circle is the limit of a series of objects from triangle to rectangle, to pentagon, hexagon, heptagon, etc. all the way to an object with infinite sides. This limit converges but it will take us infinite time to measure it. However, we can represent this process of measurement by a recursive function, and this function is finite. Therefore, the reality of π is the finite function rather than the number with infinite digits.

The Importance of Immanence

In an earlier podcast, we discussed the central problem of all concepts as immanence. We noted that the concept ‘color’ is transcendent to all colors and yet immanent in each color. We have now seen two examples of this immanence in this post thus far.

The first type occurs in space in the case of rational numbers when a snake is represented inside a rope. In a simple sense, our observation of the world ‘interferes’ with the world because our ideas become immanent in the external world, and the external world becomes immanent in us. The second type of immanence occurs in time in the case of recursive functions when a function depends upon itself. The past is transcendent to the present, but the future is immanent in the present. Due to this immanence we can say that the future exists in the present but will only become manifest in due time.

As we have seen in the earlier podcast, immanence leads to contradictions. For instance, when a barber shaves himself, there is an immanent barber who is unshaved, and there is a transcendent barber who is shaving this unshaved person. They are the same person and yet there is a difference between the shaver and the shaved connected by shaving. This is the structure of experience and we cannot get behind it, except by removing the experience when the distinction between the shaver, the shaved, and shaving is dissolved. While that distinction exists, there is a logical separation between the shaver, the shaved, and the act of shaving but they are essentially the same person. So, we must speak about a logical separation without separating them physically, and that is the contradiction.

The Metaphysics of Mathematics

The result of the above metaphysics is that we can justify the eternity of mathematical objects called functions, just as we can make the irrational numbers real. We have previously seen how the rational numbers (earlier in this post) and positive and negative integers and complex numbers (in the previous post) are also real.

So, everything in mathematics is real in one sense or another, although how they become real are different. This metaphysics explains why negative numbers are real, but we cannot see negative quantities. Or, why rational numbers are real but not as infinite digits, and why irrational numbers are real, but their infinite digits are only a consequence of the mistaken attempt to place them in the same space as the objects they collect. This metaphysics can be used to justify the realism of mathematics in a new way. But in the process, we must think of numbers as individuals, universals, and functions. In the previous post, we spoke of these as the categories Being, Nothingness, and Becoming.

The Problem of Infinite Divisibility

The realism of rational and irrational numbers in current mathematics entails that space is infinitely divisible, an idea which contradicts atomism. Owing to this fact, if nature is atomic then rational and irrational numbers can never be real. The metaphysics I have described above addresses this issue because all of mathematics can be real without requiring space and time to be infinitely divisible. For instance, real numbers are not in space but as sets they are functions in a different ‘space’ called time. Similarly, rational numbers are not infinite fractions but representations of numbers inside numbers.

Using this metaphysics, all mathematical objects can be discrete, overcoming the problem of infinite divisibility and continuity. However, it also means that the notions of continuity on which modern mathematics, and much of classical physics, depends, are false.

Similarly, it also usurps the idea that mathematical objects are Platonic—i.e. in another ‘world’—because it begs the question of why mathematics is so useful in describing the real world. How can mathematics be ‘reflected’ in this world from another world? How can we describe the real world using numbers if numbers are not in this world?

Mathematical realism is incredibly important to answer these fundamental contradictions (e.g. between infinite divisibility and atomism) and paradoxes (e.g. why mathematics as the science of transcendent objects is useful in describing the real world). And to justify this realism we need a metaphysics that makes mathematical objects this-worldly.

Philosophical Discussion

The cornerstone of this metaphysics is a tripartite distinction between the knower, the known, and the act of knowing. Each of these categories behaves differently, although they are ultimately the same thing in the absence of experience. The creation of experience separates them as three types of entities, but they are not physically separate.

In Vedic philosophy, the state called Brahman is that state devoid of experience in which these distinctions don’t exist. There is hence no knower, known, and knowing. Language doesn’t exist, and its innate structure is not visible. To create language, we must begin with the distinction between the knower, known, and knowing. This distinction is problematic because it is a logical and semantic distinction, not a physical one. As a result, if we cannot solve the problem of experience, everything that follows remains problematic. These problems are reflected in the realism of mathematical entities, so the problem is not just philosophical but has a concrete and logical realization. The philosophy of experience gives us a metaphysics that can be used to understand and resolve these issues.