Mathematics

Do Negative and Imaginary Numbers Exist?

Numbers for the greater part of history have been viewed alternately as concepts and as quantities. Now, this raises problems about many types of numbers, which include negative numbers and imaginary numbers, because these cannot be viewed as quantities although there are compelling theories that can treat them logically as concepts. In what way are these concepts real when they cannot be represented in the real world, now presents a problem of mathematical realism, which remains unresolved today. This post discusses the issues of realism in mathematics outlining how there are two ways of counting, only one of which involves negative numbers, while the other doesn’t. Why both schemes are real and yet only one appears in the real world is then discussed, along with the meaning of complex numbers, and why physical theories using these numbers are incomplete.

Russell’s Conception of Numbers

Let me begin discussing the problem by illustrating Bertrand Russell’s theory of numbers. The number 5 for Russell is a concept, and concepts can be depicted by sets. For example, 5 is the set of all sets that comprise 5 objects. There are some fundamental issues with this notion of numbers, such as the fact that to determine if a set has 5 members, you must have the notion of 5 even before you can construct the set of all sets that have 5 members. Therefore, to define the number 5, you must presuppose the existence of 5.

Some mathematicians have argued that this precondition can be obviated if we can suppose that there is a proto sequence of numbers defined in some a priori manner to which we can map the other sets in a one-to-one fashion. For example, we can say that the number 0 is represented by the empty set {}. Then the number 1 can be represented by the set that has one member – e.g. {{}}. Then the number 2 will be defined by a set that has two members, which are 0 and 1 such as {{}, {{}}}. In this way you can construct the entire sequence of numbers from zero to infinity, zero being depicted as the empty set.

These methods, however, miss an important issue, which is that we still require the ability to distinguish an object from other objects. In the above scheme, for example, we must have the ability to distinguish the boundaries of a set by the two braces { and }. To do this, you must parse the sequence of braces and match the opening and closing braces and count them—the counting is depicted by the braces and comma separator. So, the necessity to know that a set has a certain number of members doesn’t go away; it is replaced by the counting of commas and braces, which needs the concept of numbers in the first place.

The right answer to this problem is that we must suppose the existence of some numbers as a concept, and then use it to construct other numbers. We will see toward the end of this post that these presupposed numbers must be 0 and 1, and the other numbers are to be conceived as positive and negative components of 0, and positive fractions of 1.

The Problem of Negative Numbers

Even if we suppose that the Russell’s scheme of defining numbers is unproblematic, we are still left with the issue of negative numbers. To define the negative numbers in the same ways as positive numbers, we would have to say that the number -5 is the set of all sets that have -5 members! But you can never actually find a set that has -5 members. You can at best find a set that has 0 members, which is the empty or the null set. Therefore, even if the definition of positive numbers were unproblematic, that of negative numbers is.

The problem is that we think of numbers as concepts. And these concepts must have some physical instantiation—e.g. in the size of a set. You cannot conceive a size that is -5.

Historically, negative numbers became important when people were dealing with money. You could borrow money, and then you had a debt. How do you represent money that is owed? Accountants invented the use of negative numbers to represent debt. But factually debt is a moral notion: you expect that someone will pay the debt in the future, but now the money is physically with the person who hasn’t yet paid it. Therefore, we are not talking about the money itself, but the direction in which it is going to flow in the future.

Demystifying Negative Numbers Using Directions

One way to generalize this idea is to always associate a number with a direction. For example, in geometry we can speak of a straight line with a zero in the center and speak about positive and negative directions. Negative numbers are now understood by the fact that a number is not just quantity but also a direction, quite akin to vectors. If this idea is extended to a multidimensional space, direction itself becomes a quantity. For example, in a two-dimensional space, you can have two numbers—a quantity and a direction. This indeed forms the basis of some well-known coordinate systems of counting.

We are now led to a compelling picture in which a negative number is in fact two things—a magnitude and a direction. Similarly, even a positive number must be defined by a magnitude and direction. Direction itself could then be defined by multiple quantities depending on the dimensionality of the space in which we define the number. For example, the spherical coordinate system uses two angles and one linear quantity.

Numbers and Geometry

This leads to the idea that number cannot be understood without geometry. I mean, if we plainly and simply just talk about positive numbers, then potentially we could think of numbers without a direction and hence a geometry. But the moment we introduce negative numbers, we have introduced a direction and hence the notion of space. This space could be as simple as a straight line, which can then be expanded into multiple dimensions.

Clearly, if we are going to locate points in space, then we need an origin which represents the zero. Following this zero, we must also select a coordinate system which in the simplest case is constituted of which chosen direction is called positive vs. negative. For instance, you could call the right side of the zero as the direction of positive numbers. This further reinforces the idea that we cannot think of negative numbers without geometry, although we could potentially think of positive numbers as pure abstract concepts.

This is a somewhat dissatisfying. After all, we want to think of positive numbers as concepts, because it is seems that there is an idea 5 which appears in collections of 5 objects. There are potentially infinite collections of 5 things, but there is only one idea called 5. The only way we can solve this problem is by saying that we must reconceive the space in which we count both positive and negative numbers as a semantic space, or a space of concepts. In this space, both +5 and -5 are ideas, rather than quantities. By this approach, we could say that numbers are concepts and yet not defined without a space.

Cardinal and Ordinal Numbers

However, now we would have to distinguish between the concept 5 and the instances of 5, quite like we distinguish between the concept dog and the instance of the idea of dog. In the idea space, the number 5 is a single entity, which is expanded into 5 entities in the instance space. Let’s call this instance space the individual space or the space of things. The idea 5 is the universal idea of fiveness in the semantic space which incarnates in the individual space as five distinct things. However, the idea of -5 never incarnates in the individual space.

You could say that the ordinal -5 exists—if you think of numbers as concepts existing in a semantic space—but the cardinal -5 doesn’t exist because you can never find a collection of -5 objects. Nevertheless, you can label objects as -1, -2, -3, -4, and -5 ordinally.

For example, when we say that the electron has a negative charge and the positron has a positive charge, we cannot say that the electron is missing a charge. We must rather say that the electron has a charge, which is signified by an ordinal negative number. That ordinal number represents a concept, and the electron therefore has a property denoted by that concept. It is not the absence of charge but the presence of the opposite charge.

The inevitable conclusion is that we will always have a positive number of particles, but we can have these particles with positive or negative charge. That negative charge is not a cardinal, but an ordinal—i.e. a conceptual property—instantiated from a concept space into the individual space. Even if we say that a particle is -5 meters away, we are superimposing a conceptual space on the individual space by defining an origin on the individual space just like we had defined the origin of the conceptual space as zero. The origin of the conceptual space is absolute—represented by the concept zero—but the superimposition of this conceptual space on the individual space is completely arbitrary.

In short, if we were counting objects, we can only count as 1, 2, 3, etc. But if we were describing properties of these objects, then we can count in both positive and negative directions. Negative numbers are conceptually real, but individually unreal. You can never have a set that has -1 members, but you can have an electron with -1 charge.

The Problem of Negative Multiplication

All children are taught: the multiplication of two negative numbers creates a positive number. However, the reasons for this rule are never explained. Suppose you are asked to multiply -2 x -2. If you treat these as quantities or sets of individual members, then there can never be a set with -2 members, and the operation makes no sense. However, if you treat this operation geometrically, then -2 is two things—a quantity and a direction. 2 x 2 makes perfect sense as quantity. And (-) x (-) makes sense as inversion of direction. This minus sign can be treated logically as the NOT operation, which means that NOT of NOT cancels each other. The trick is to realize that we are in a space; where are dealing not with a single entity called a negative number, but with a number and a direction.

This also means that there can never be a √-1 unless you create a new geometry and step outside the space. This is precisely what imaginary arithmetic does, when it postulates a two-dimensional space instead of a single dimensional line. Multiplication by √-1 preserves the quantity but changes the direction by 900. This confuses most people because they are used to thinking of numbers as quantities without a direction. They haven’t realized that all numbers only exist within a space which must have both quantity and direction.

Positive and negative numbers just support two directions, but imaginary numbers add two more directions to create four directions on a two-dimensional plane, and the combination of real and imaginary numbers create infinite directions. Unlike the multiplication of positive numbers which only changes quantities, and unlike the multiplication by negative numbers which flips the direction by 1800, imaginary multiplication changes direction by 900 at a time. The whole thing would sound much less problematic if we realized that numbers must always have a direction.

The Problem of Complex Numbers

A new problem arises when we think about three-dimensional complex numbers which must have 3 dimensions each of real and imaginary components. Now, we are breaking the commonsense intuition that there are only 3 dimensions of space and requiring that we have a 6-dimensional space. This problem can only be solved by thinking of hierarchical space in which the imaginary space is factually another space outside the real space, and the real number space becomes an object in that space. This object acquires a location and direction in the higher space, which means we are changing the direction and size of one space inside another space while doing complex number operations. We are no longer dealing with just objects; we are dealing with objects that are also spaces.

The problem is that the hierarchy of spaces is potentially infinite, and to accommodate that we would need not just a three-dimensional imaginary space, but another such space to contain the imaginary space, and then an even higher space to contain the previous space, and so on, until we come to the end of the hierarchy. This entails the need for potentially infinite spaces, which makes the mathematics infinitely complicated.

Theories such as quantum mechanics restrict the problem to a single imaginary space but suppose that there are infinite objects in six dimensions. The right way to think about the problem is that there are infinite objects and infinite spaces. The same thing is a space containing the lower object, and an object contained in the higher space.

Since the same thing is both an object and a space—although in different higher and lower contexts—unless we capture the context in the theory, simply using 3 dimensions for the object and 3 dimensions for the space isn’t going to work. This is the mathematical reason why theories such as quantum mechanics become incomplete. They are right in thinking that the quantum object must be described by complex numbers because that object is also a space, but the relation between the objects/spaces is missing. A truly hierarchical geometry is needed to solve this problem in which quantum objects are not in a single space, but successively contained inside a cascading hierarchy of spaces.

Dual Numbering Schemes

Suppose you have a coin with two faces—head and tail. As properties of the coin, you can call the head +1 and the tail -1. But as individual things, you must say that head is first, and tail is second, due to which the coin has two faces rather than zero faces. The problem here is that we see the same world as both universal concepts and individual things. As universal concepts,  the tail can be -1, but as individual things it must be +2.

This problem can only be solved by saying that there are two separate spaces—of universals and individuals. In the space of universals, you can count negative numbers, but in the space of individuals you must only count positive numbers. Unfortunately, our language operates in such a way that we apply the same label to describe both universals and individuals. For example, the label ‘barber’ can denote the universal concept of a person who shaves others, and an individual person who shaves others. We can say that these are two modes in language by which the same symbol is used interchangeably, and through the context we decide which mode language is operating in. So, factually, we cannot get rid of these two modes, but we can distinguish between them.

Matter and Soul—Philosophical Issues

I will now discuss the relation between the foregoing ideas and Vedic philosophy. Let’s begin by outlining the distinction between matter and soul. The distinction is that the soul is the individual, but matter is the universals. The universals can be described by opposites such as tall and short, male and female, black and white, etc. which constitute the duality or opposition of concepts, which is why the material world is duality. We can represent these things by positive and negative numbers. However, the soul—as the individual—can never be described by negative numbers; there can only be a positive number of souls.

When the soul is covered by matter, it applies the dualities of the material world to itself—e.g. becoming man or woman. But put together, man and woman are two individuals rather than zero (which they would be if we added up the opposites of duality). In short, man and woman don’t cancel each other out because there is an underlying soul.

In Vedic philosophy, the opposites of matter are differentiated when the soul is added to matter. Without the presence of the soul, the dualities cancel each other out. However, in the presence of the soul, the dualities become two things. Therefore, the counting of individuals must be separate from the counting of material properties. The counting of material properties can use negative numbers, but individuals can only use positive numbers. The scientific novelty here is that matter must be described as opposites or duality. This duality is mutually defined, and the opposites are therefore mutually connected.

However, there is a process by which the soul is said to be ‘liberated’ from matter, which is the scenario in which it stops applying the dualities of matter to itself. It also means that the idea that the soul can be reduced to matter is a flawed idea; fundamentally, they pertain to two different spaces or descriptions—one of duality and the other non-dual.

Creation from Numbers

This so-called non-duality doesn’t mean that there is only one thing. It rather means that there are many things, but they will be described by positive numbers. Matter, on the other hand, can be described by both positive and negative numbers. We can also say that matter is a property of the soul which can be positive or negative. The soul is the object to which that property is attached. Under those properties the soul imagines to have become that positive or negative property, but there is a ‘space’ in which the soul is only a positive number, or an individual counted relative to the first individual called God.

There are further descriptions in Vedic philosophy where the soul is considered a part of God, which means that the soul is a fraction of one. Therefore, when we say that there are multiple souls, what we mean is that the original soul is 1, and the subsequent souls are fractions; however, there are no negative fractions. These positive fractions can also be counted in a hierarchy. Matter on the other hand is also fractions but it can be both positive and negative fractions. These positive and negative fractions can be said to cancel each other when matter is devoid of souls, so matter can be represented by a zero.

Thus, it can be said that the primordial reality is 1 and 0 (God and matter) but their combination expands that reality into all the numbers because 1 has positive fractions and 0 has both positive and negative fractions. That combination creates the infinite set of individuals which are in one sense just fractions of the primordial reality.

The implication for mathematics is that we cannot construct a theory of numbers out of something that doesn’t presuppose any number. We must rather formulate the theory of numbers using two numbers—1 and 0—and create all other numbers from them. The process of this creation is that 1 divides into fractions and 0 splits into positive and negative numbers (which can also include fractions). The positive fractions created from 1 overlap with the positive fractions produced by splitting 0, and this constitutes the fundamental confusion in number theory, but it can be overcome in a new theory.

The Role of Time

There is a sophisticated theory of how everything springs from 1 and 0; the essence of this theory is that the division of 1 into fractions constitutes the free will of God, and the splitting of 0 into opposite parts is the function of time. Thus, God creates the consciousness of the soul by “dividing Himself by Himself” according to the Śrīmad Bhagavatam. Time instead creates the universal set of events independent of any soul, constituting the ‘market’ of give and take without fixing the players in the market—i.e. who gives and who takes. The soul becomes a participant actor in this market. The market automatically expands and contracts, which means the collective history of the world is predictable.

However, since the soul’s participation is a choice, the individual history is subject to the collective marketplace expanding or contracting, but otherwise free (this freedom is further subject to the limits of karma which enables and disables the participation of the soul in certain types of markets). God, time, and matter constitute what Western philosophy calls Being, Becoming, and Nothingness, following Hegel. The Being and Becoming are masculine while Nothingness is feminine. In a subsequent post, I will try to discuss the nature of Becoming in terms of familiar mathematical ideas.

This outline of the number scheme can be used to construct a new type of mathematics in which there is a hierarchy which results in fractions of the whole, then fractions of fractions, and so forth, and the manifest world is produced. Unlike the impersonal and atheistic descriptions of modern science, this would be theistic. However, the usefulness of this scheme goes beyond the theism vs. atheism debate, because the scheme would be useful in solving outstanding problems of modern science, which can be taken to mean that since the science works in explaining matter, its truth is also the evidence for the metaphysical presuppositions that underlie the development of the theory.