The following is a somewhat extended version of a reply to some questions that I sent today to an interested reader. I thought this description would be relevant and useful even to others, and hence I decided to post it.

The current doctrine of numbers is physical and by physicalI mean that if you add 1 to 1, then you get 2. There is, however, another way to think of numbers as concepts.

For example, suppose you know Newton’s three laws of motion, and then I tell you again about the 3rd law. This doesn’t add to your knowledge, because you already knew of the three laws. Thus, in this case 3 + 1 = 3. The nature of conceptual addition is that it increases upon addition if only if the meaning wasn’t previously present. If you already know of Newton’s three laws, and I again tell you about these three laws, then the result of that addition will be 3 + 3 = 3, because knowledge has not increased. Again, concepts don’t increase if we add the same concept.

So, there are two ways of thinking about numbers. In the present conception of numbers, we think about them as physical things, and when you add two physical things you always get a greater number of things. But in the conceptual addition, you can sometimes get two things (e.g., if you add new knowledge), exactly the same thing (e.g., if you add the same knowledge), and even nothing (e.g., if the new concept negates the previous concept). Thus, the simple operation of ‘addition’ becomes unpredictable if we consider numbers as concepts. From this problem, however, we can open the doors to a new kind of understanding that predicts how knowledge grows.

For example, knowledge doesn’t always grow linearly, even if you add new ideas. Rather, knowledge has a structure, and ideas get added, reformulated, some ideas are removed, and some are moved (or given a new kind of place in the system of ideas), and then these ideas are expressed without a reduction. This complicated dynamic of the nature of ideas, namely, addition, removal, reformulation, movement, and expression, constitutes a new science.

These five processes of the dynamic of knowledge are called the five prāna, or “life force”. By their presence, every system of knowledge becomes a ‘living organism’. Thus, we are living organisms, because we add, remove, reformulate, reorganize, and express ideas. The addition is called prāna, the removal apāna, reformulation samāna, reorganization vyāna, and expression udāna. This is the dynamic of knowledge that occurs in every knowledge system—our individual bodies, the society of knowing beings, and further stages of complexity.

Now, to relate this problem to the nature of God, let us look at the 1st verse from Iśopaniśad:

“That is complete, this is complete. From the complete comes the complete. And after the complete is removed from the complete, the balance is complete”.

It is impossible to understand this verse if we think physically. But it is possible to understand it if we treat the complete conceptually. In this case, God must be understood as ‘knowledge’. From this ‘knowledge’, many varieties of knowledge, which are parts of knowledge, expand. But after all this knowledge has expanded, God hasn’t become ignorant! Thus, even if you remove the complete from the complete, the balance is still complete. Why? Because the complete that you removed was knowledge, and the source that created that knowledge was knowledge.

Thus, we can describe the paradoxical relation between science and religion by the simple operation of number addition: 1 – 1 = 0 in science, and 1 – 1 = 1 in (Vedic) religion. But the solution to this paradox, by treating reality conceptually, opens the door to their reconciliation.

To understand this resolution, let us see how conceptual addition exists in set theory in a primitive form as Union of sets, where a Union of two sets equals to the same set, if these two sets are identical. Even if they are not identical, the union of sets only produces a set that combines the unique elements, not the duplication of elements that already existed.

For example, let’s consider two sets A and B, such that A := {P, Q, R} and B := {Q, R, S}. Then the union of A and B which is denoted as A U B := {P, Q, R, S}. This is how knowledge addition works; we add new knowledge only when the knowledge was previously missing. If the knowledge was previously existing, we don’t say that anything new has been acquired.

However, the conceptual addition is not extended to conceptual subtraction in set theory. The set theory subtraction is called ‘Difference’, and it removes elements. Thus, for example, if A := {P, Q, R}, and B := {Q, R}, then the difference A – B := {P}. This is like saying that if I taught you some scientific theory, then I must have become ignorant in the theory that I previously taught you. As ridiculous as that sounds, that is precisely what set theory Difference leads us to.

Now, lots of problems arise due to the mistaken idea about ‘Difference’ in set theory, because then we stop treating the sets as conceptual entities. We rather treat them as physical entities. The correct set theory would be where ‘Difference’ is also just like Union — you remove some elements from a set, and they are still there, just like by giving knowledge, knowledge is not lost.

Now, while number theory is based on set theory, the concept of number addition is not like set Union, because 1 + 1 is not 1. It is rather 2. Therefore, the idea of number addition is different than the idea of set Union. However, the idea of number subtraction is exactly like the idea of set Difference. For example, 1 – 1 = 0, and if you subtract a set from itself, you get a null set.

So, even though set theory starts by using a novel idea of Union, because it is incompatible with the idea of Difference, the definition of sets has to be modified to be simply a collection of physical things. In this new conception of sets, if you add two sets of 3 and 5 cars, you must get a set of 8 cars. You cannot say that the set of 8 cars is simply one thing – i.e., cars. Because if you said that, then you would be led to a paradox: Are there 8 different things, or just 1 thing?

Of course, the fact is that both kinds of statements are true, although in different ways. You must say that there are 8 individual cars. And you must say that there is just one type of thing—i.e., car. But we cannot do this in sets because it requires the notion of modalities (these are called the ‘modes of nature’ in Vedic philosophy). In the individual modality, you must say that there are 8 cars, and in the universal modality you must say that there is only one type of thing—i.e., car.

The mode in which we see that one type of thing is called sattva-guna, and the mode in which we see eight different things is called tamo-guna. Then there is also a mode called rajo-guna in which we create contrasts between things by comparison – e.g., cars are defined by their distinction to trucks, so by a car we also mean something that it is not a truck (but the distinction to a horse can be contextually applied or removed, depending on the context). This third mode creates contextuality in the study of concepts, where something is not other things, but what it is not only depends on the context (a car can also be towed by a horse, in which case, it now becomes a ‘carriage’, not a ‘car’).

So, there is just one reality, but, based on different modes we must be able to say that there is only one thing, and that there are eight things, and then some unspecified number of things based on the context. These modes are like “perspectives” on the same reality. In one perspective, you see eight individual things, and in another perspective, you just see one thing. Then by relating to another group of things, you assign it additional attributes. So, by changing the perspective, the same thing is just a single entity, or it is many individual entities, or few different types of entities. These are statements about the exact same reality, and therefore, they are simultaneously true. However, without modes, they lead to a contradiction: How can we say that the same thing is just one thing, and then say that it is eight different things, and then say something else purely based on the context in which the truth is presented?

So, the modes of nature, or modalities, constitute a radically different way of viewing reality in which reality is known completely through many perspectives. Vedic philosophy identifies three fundamental perspectives – universal, individual, and contextual – and these combine in innumerable ways to create infinite perspectives. Due to these infinite perspectives, reality is just One, and yet it is simultaneously known in infinite ways through infinite perspectives.

We cannot reconcile these perspectives because they are indeed contradictory. However, each of these perspectives is true. This leads to the paradox that truth must be self-contradictory. This principle of contradiction is called Achintya-Bheda-Abheda Tattva, where two things are simultaneously different from each other, and yet they are the descriptions of same thing, and because we cannot logically reconcile them, therefore, reality is said to be ‘inconceivable’.

There is, however, a solution to this inconceivability, and that is the notion of reality and its perspectives. And that means that we must reconceive reality in terms of many modes. This modal conception of reality constitutes a step forward from the Achintya doctrine, and can be called the Chintya doctrine. This is the essence of the Vedanta Sutra commentary entitled “Conceiving the Inconceivable” that was published recently. This commentary demonstrates how reality is described in contradictory ways from different perspectives—called modes.

As noted above, in addition to this perspective approach, we also require a conceptual approach, so that we can say that by removing the complete from the complete, the balance is still complete. And these two ideas can be reconciled if we say that the original ‘complete’ is the ‘Reality’ and the removed complete is the infinite ‘Perspectives’ on that reality. The original ‘Reality’ – the supremely beautiful Lord Kṛṣṇa – manifests infinite perspectives, but by manifesting this knowledge, He doesn’t become ignorant. So, even when the complete is removed from the complete, the balance is still complete, and that is because the complete is knowledge. If this was some physical entity being removed from another physical entity, the removal would reduce the thing it is removed from, and that will lead to reductionism.

The key point is this: The basis on which we understand God is also the basis on which we can understand concepts, and the basis on which we understand concepts is also the basis on which we can solve the paradoxes of mathematics. The first is religion, the second is philosophy, and the third is science. These are not fundamentally different subjects, if properly understood.

If this provokes your interest, then you can read “Gödel’s Mistake” and “Conceiving the Inconceivable”. The former is the description of the problem and its solution in mathematics, and the latter is the discussion of the same problem and its solution in Vedanta philosophy.