This post discusses how points in a conceptual space are defined differently than in a physical space. The difference is that a physical space defines locations in relation to an origin, whereas a conceptual space defines locations in relation to a boundary. In a physical space, points are constructed through absolute proximity to a single origin. In a conceptual space, points are constructed through their relative distance to two endpoints. Many changes in distance and order arise due to this difference. These insights are useful in order to describe a different kind of geometry of space and time.

Object vs. Meaning – What is the Difference?

In classical physics, the universe could have a single isolated particle. Meanings are unlike such isolated particles, because they are defined in relation to other meanings. For example, you cannot define “red” all itself; you have to define “red” in relation to “green” and “blue”. Similarly, you cannot define “hot” by itself; you have to define it in opposition to “cold”. Every concept we employ to describe the material world involves oppositions or distinctions, due to which the world is said to be comprised of duality in Sāńkhya.

Owing to this fact, there is a fundamental difference in the descriptions of space in Sāńkhya. For instance, in physical space, we choose a property such as temperature and arbitrarily designate some point as 0 0C. We then sequence objects on this scale in the order of their increasing temperature. Such a space, of course, has no logical beginning or end. You can have temperatures as high as you like, and as low as you can imagine (practically, however, there might be physical limitations in this scale).

A conceptual space instead is defined by extremes—e.g. hot and cold. Within these extremes are positions such as lukewarm, tepid, warm, cool, cooler, etc. A boundary comprises and defines the opposite locations, which represent opposite meanings. We might find this unintuitive because since the time of Euclid space has been defined as a set of dimensions centered on an origin. But Euclid’s approach isn’t the only way to define geometry. We can also define positions in relation to a boundary, rather than an origin. In fact, when we define positions in relation to a boundary, we also define positions in relation to the origin, although quite differently than how they are defined at present.

The Meaning of Center and Boundary

The boundary, as we have seen, represents logical extremes—e.g. hot vs. cold. The center, on the other hand, is the position “mid-way” between these extremes. It represents the idea “warm”, which is both hot and cold, and yet neither hot nor cold.

In Sāńkhya, these positions can be distinguished using the three modes of nature. For example, “cold” is tamo-guna, “hot” is rajo-guna, and “temperature” is sattva-guna. The mode of sattva-guna precedes those of rajo-guna and tamo-guna, and the latter two modes are oppositions created from and within sattva-guna. Just like the numbers +1 and -1 can be created from the number 0, similarly, a state devoid of duality produces a state of duality. The center represents the number zero, while the locations on the boundary will represent +1 and -1. The numbers +1 and -1 are opposites of each other, but zero is not the opposite of anything. It is the ideal number, because it is free of duality.

In this approach, we don’t discard the center just because we have the extremes. In fact, the extremes are created from the center, which is a more abstract and therefore higher concept. However, we do not create the content in the space from the origin outwards until we reach an extreme. Rather, we create an extreme from the origin, and once the limits of the space are defined, the content is already limited by the boundaries.

In current geometry, we pick an origin and then define a point infinitely close to the origin, followed by another point infinitely close to the previous point, and so on, until we have constructed every point in space. As it turns out, this process is infinite, which means that we cannot know the origin of space because any point can be the origin and we cannot know the ends of the space because it stretches infinitely in all directions. In conceptual geometry, we define three entities—the origin, and the two extremes—and thereafter every other point is defined by the combination of the three entities. In other words, we begin with the three modes of nature, and construct all other positions using the modes.

The Relation to Zeno’s Paradox

The difference between semantic and physical approaches lies at the root of Zeno’s paradoxes in which Zeno presumes a destination to be reached, and defines successive locations in relation to that destination. If the destination is 1, then intermediate locations are 1/2, 1/4, 1/8, etc. The problem is that if you begin from +1 then you can never reach 0 by incrementally dividing the edges. This is clearly understood if we begin in hot and cold and try to arrive at temperature. We know that if hot and cold are properties, then there is also a higher property called ‘temperature’. But this higher property is actually not at the same “level” as the previous properties. Zeno’s approach has similarities to conceptual space because we start by defining the origin and the end, and then construct the intermediate points. This is unintuitive if we think in terms of physical geometry where we construct points from origin outward. In this construction, we do not think about a goal, because we suppose that the goal would be automatically reached after all the intermediate points are created. In semantic space, we start with the goal and then create all the intermediate steps as the process or procedure required to reach the destination.

In a semantic geometry, you start by defining the origin and the two extreme positions, and then construct everything through a combination of these three positions. The combination of these three positions then constitutes the “mixing” of the three modes of nature. In a sense, we do not construct all the intermediate points between origin and boundary before we reach the boundary. Rather, we set the origin and the boundary, and then define all the intermediate steps that are needed to reach it. Zeno’s method also constructs space as a binary tree of locations, rather than a set of linear points. These are based on two radically different approaches to the idea of space and distance.

In the linear approach to spatial distance, points are labeled sequentially. In the hierarchical approach however, there is another dimension (hierarchy) which we don’t see unless we treat the locations conceptually. In this approach, eventually (if we create all the points) the points can be numbered linearly. But it is not necessary that all the points have been created. If, in fact, we consider the order in which we have to create the points, then 1/2 comes before 1/4. In a sense, if space is created in this way, then you can move without traversing all the intermediate points. It would seem magical, because while a physical motion theorist is thinking that he has to go from 1/32 to 1/16 to 1/8 to 1/4 to 1/2, for the semantic motion theorist 1/2 comes before 1/32 or 1/16. So he can directly jump to 1/2, which the physical motion theorist would consider a “non-local” phenomenon.

In physical ordering, 1/4 comes before 1/2. In semantic ordering, 1/2 comes before 1/4. Once we change the order, we change the idea of distance between points. Now, what you consider physically close may not be semantically close at all. For example, you might stand in front of a prison, but you won’t necessarily get inside a particular prison cell. It may seem physically close, but you cannot reach that location if you haven’t been judged as a criminal. There will similarly be possibilities which scientists call “wormholes” whereby you can travel to great physical distances in very short periods of time, because the semantic distance is quite small. The notion of semantic space therefore opens up a new way of thinking which is empirically verifiable through experiments.

Two Notions of Zero

I noted above that the numbers +1 and -1 can be created from 0. Similarly, the numbers +1 and -1 can be added to create 0. It is now worth noting that in these two instances we are talking about two kinds of zeros—ordinal and cardinal. An ordinal number represents order, such as first, second, third, etc. A cardinal number represents quantity such as one, two, three, etc. The number zero can both be a cardinal or an ordinal.

The cardinal number 0 is obtained by adding +1 and -1. The individual numbers +1 and -1 exist inside this 0, but if we view this quantity “externally” we would not see the opposites +1 and -1 inside it. The fact is that 0 could also be the sum of +1/2 and -1/2 and if we see this quantity externally, we cannot distinguish between the 0 that adds +1 and -1 vs. the 0 that adds +1/2 and -1/2. We can conceive of infinite different zeros which are identical cardinally, but have different compositions in terms of opposites. The situation is analogous to a person who has a certain amount of money in his wallet but does not know the exact denominations in which the money is stored. The money could be, for example, stored in smaller or larger denominations, or combinations thereof.

The ordinal number 0 is one that precedes both +1 and -1. Just as 1/2 precedes 1/4 even though 1/4 is smaller than 1/2, similarly, 0 seems larger than -1 but 0 actually precedes -1. The reason is that both +1 and -1 are created from zero. In the same way, we can also conceive a 0 from which the pair +1/2 and -1/2 are created. This means that 0 is prior to -1/2 even though 0 is larger than -1/2. We can now see that there are infinite types of cardinal zeros which precede +1 and -1, +1/2 and -1/2, etc. However, there is only one ordinal zero because relative to the ordering, the position of 0 is fixed.

In semantic thinking, the ordinal zero represents “temperature”, which is prior to both “hot” and “cold”. Conversely, the cardinal zero represents “warm” which is the combination of “hot” and “cold”. Owing to this, the cardinal zero (“warm”) is both hot and cold, while the ordinal zero (“temperature” is neither hot nor cold. These two numbers are not the same, even though we could call them both zero. This creates problems in classical Aristotelian logic which is not expected to break the principles of non-contradiction and mutual-exclusion—i.e. allowing categories such as “both” and “neither”.

Non-Contradiction and Mutual-Exclusion

Assume that there are two contradictory propositions P and not-P. Aristotelian logic says that only one of P or not-P can be true, implying that {both P and not-P} cannot be true. This is called the principles of Non-Contradiction (NE). Aristotelian logic also says that at least one of P or not-P must be true, implying that {neither P nor not-P} cannot be true. This is called the principle of Mutual Exclusion (ME). If we allow “both” (NC) and “neither” (ME) we are breaking the fundamental principles of Aristotelian logic.

And yet, all the problems in modern Set Theory (and thereby Number Theory) arise because Aristotelian logic doesn’t know how to deal with “both” and “neither”. Consider the examples shown in the figure below which illustrate Set Theory’s problems.

Suppose you have a class of people called “friends”, and by negating this class you can obtain another class called “non-friends”. The sum of these two classes appears to constitute the “universe” from your standpoint (the world is either my friend or a non-friend). And yet, this “universe” doesn’t contain you because you are neither your friend nor a non-friend to yourself. This breaks the principle of Mutual-Exclusion.

There are similarly numerous cases—which we call “paradoxes”—in which we encounter statements that are both true and false. The most common of all is the Liar’s Paradox in which a person says: “I am a liar”. The problem is that if the speaker is indeed a liar then by his admission, he must be lying and therefore he becomes truthful. Conversely, if he is telling the truth, then by his claim, he must be lying. This violates Non-Contradiction.

The problem of set theory is that it thinks about what is inside and outside the set but it doesn’t think what is on the boundary between inside and outside. Yes, there is indeed a material entity called a boundary. In fact, there are two of them!

The first boundary is subtle matter which exists before gross matter exists (e.g. temperature), and since the gross material distinctions (e.g. hot vs. cold) are created from this subtle matter, this boundary is neither of those distinctions. It is not a violation of logic. It is rather recognition of a new category from which opposites are mutually created. The second boundary is gross material which is created after gross material dualities are created, and it lies right in the middle of these oppositions—e.g. warm. The boundary simply says that it is equidistant from both the extremes or in the “middle” of the extremes of hot and cold. You cannot define the center unless you define where the extremes are. As a result, the boundary is both of the extremes combined.

The Boundary as a Real Object

In current set theory, objects are on either side of a boundary but not on the boundary. But in the above examples, the “self” is on the boundary. For instance, if we describe the world as the sets “my friends” and “my non-friends” then “me” is logically prior to both these sets (and falls in the “neither” category) and not included in these sets. Therefore, it must lie on the boundary where the boundary denotes that which is neither of the sets. Similarly, I might be sometimes a liar and sometimes truthful. While I can classify my statements into two categories (truths and lies) I am in both the classes because I spoke both truth and lies, and the boundary denotes that which is in both the opposites.

The peculiar property of the boundary is that it is in touch with opposites, and can be considered both the opposites. Similarly, because it is only demarcating the extremes of these opposites, it can be regarded as being neither of them. Set theory and Aristotelian logic are incomplete because they don’t deal with boundaries which can constitute the logical categories “both” and “neither”. In a flat space, we cannot distinguish between when a boundary represents “both” and when it represents “neither”. However, in a hierarchical space, the “both” boundary is at the “same level” as the opposites, while the “neither” boundary is at a “higher level” as a logically more abstract concept. Owing to this property of hierarchical space, “both” and “neither” need not be confused or colluded.

Square of Oppositions vs. Chatuskoti

Aristotelian logic defines a Square of Oppositions that respects Non-Contradiction and Mutual-Exclusion. In this logical system, “both” and “neither” are disallowed.

The Sāńkhya approach also constructs a square, which is called Chatuskoti in which “both” and “neither” are allowed, and “neither” is a very important category. The below figure provides a simplified illustration of this idea in which subtle matter called sattva-guna creates gross matter comprised of rajo-guna and tamo-guna (which are opposites). Thereafter, rajo-guna and tamo-guna combine to produce more variety, constructing a space of duality with many positions in between the extremes.

Based on the above picture (which has been deliberately conceived with numbers rather than truth values) we can now relate the top and bottom categories (“both” and “neither”) to the two kinds of zeros we discussed above. There is a zero which is neither +1 nor -1, and it represents sattva-guna which precedes rajo-guna and tamo-guna. There is another zero which follows rajo-guna and tamo-guna and combines them.

If we think of +1 and -1 as the logical extremes of the gross visible material space, then there is something that is logically prior to this space, which does not exist in this space, and is yet the origin of this space. This “origin” beyond the visible space is the first zero which we called the “ordinal zero”. Similarly, once we have the logical extremes, then they can be combined to create a range of possible values that lie in between +1 and -1. The zero resulting from the combination of +1 and -1 is within the visible material space, and therefore appears as a “quantity”—we called it the “cardinal zero”.

Cardinals and Ordinals Revisited

In physical space the quantity 0 precedes the quantity 1 because 0 is less than 1. In conceptual space, the quantity 0 is less than quantity 1, but 0 follows 1 because it is created from the boundary. There is, however, another zero (the ordinal zero) which is logically prior to both 1 and -1. Again, it seems that -1 is less than 0, so we must begin counting from -1. But that’s not how we count. We have to start counting from 0, and we have two directions in which we can count—namely, +1 and -1—because 0 precedes +1 and -1 and both logical extremes are produced from 0 at once.

We have already seen how 1/4 is quantitatively lower than 1/2 but in terms of order, 1/2 comes before 1/4. Now we have also seen how the order between 0 and 1 is not quite what we think in physical space—ordinal zero precedes 1 and cardinal zero follows 1. Owing to all these factors, we now have a serious discrepancy between two kinds of spaces—cardinal numbers and ordinal numbers. The order in which the world is created using concepts is not the order in which we perceive the world sensually.

The world of quantities or cardinal numbers is what we perceive through our senses, and it gives us a definition of distance in which 1/4 is less than 1/2 and therefore 1/4 precedes 1/2. The world of ordinals too (when treated as concepts) creates another space—the conceptual space—in which quantities are not important, just the order is important. This is the world that we don’t see by the senses, but we can conceive with the mind, employing a theory about nature as comprised of three modes or logical states—0, +1, -1.

Modern science is based on cardinality or quantities. There is, however, another science based on ordinals or concepts. These two sciences are like chalk and cheese: (1) the logic they employ is different, (2) they conceive numbers as concepts rather than quantities, (3) the order between numbers is not driven by their quantitative values, and (4) notions such as distance are different because the order of positions in space changes.

A description of the world based on conceptual spaces, the three modes of nature, ordinality instead of cardinality, and a logic that admits “both” and “neither” into our discourse is not impossible, nor is it unscientific, or illogical. But it is not how we think of “science” at present. To understand this science, we have to begin with changes to logic, then changes to numbers, then changes to space, and then changes to matter. This can seem daunting to many, but hopefully this post helps us see how this is possible.