One of the most intriguing aspects of mathematics is that it is universally applicable to every single field of science. Not a single empirical field, and not a single scientific theory, can do without mathematical tools. Mathematics not only helps with making correct predictions but also allows the results of scientific research to be demonstrated in an elegant, succinct way. The origins of mathematics and the secret behind its success have intrigued thinkers of all eras. In this article, we shall review the main ideas of the Philosophy of Mathematics and elaborate on the stance of the Kabbalah of Information on this matter.
Let us begin with a brief overview.
1 - The main ideas of the Philosophy of Mathematics
Currently, there are many theories within the overall Philosophy of Mathematics, but in general terms, they can be divided into two main groups: Platonism and Nominalism.
The essential idea of Platonism is that mathematical principles already existed in nature and were discovered by humanity, while Nominalism believes that they are the product of the human mind. Let us consider this in more detail.
1.1 Platonism: the key principles
1 - Existence: mathematical entities exist.2 - Abstraction: mathematical entities are abstract.3 - Independence: the existence of mathematical entities is independent of human existence.
The notion of abstraction also implies that mathematical entities exist outside of time or space. This means that Platonism assumes that there is another plane of reality, separate from our Universe.
1 - Mathematical entities, structures, and relations do not exist at all.2 - Another interpretation: mathematical entities, structures, and relations do not exist as abstractions.
1.3 The indispensability argument
During the latter half of the 20th century, two philosophers, Quine and Putnam, argued that mathematics is an indispensable part of science. They did this as follows:
1 - We ought to have ontological commitment to all entities that are indispensable to scientific theories (i.e., consider them part of reality).2 - Mathematical entities are indispensable to scientific theories.3 - Therefore, we ought to have ontological commitment to mathematical entities (and thereby consider them to be part of reality).
The history of science has many examples of mathematics succeeding in unique ways. In 1846, while analyzing the orbit of Uranus, French mathematician Urbain Le Verrier predicted the existence of another planet, unknown at the time. It was later discovered and named Neptune. And in 1915, Albert Einstein, having reviewed some solutions to the equations of his own General Theory of Relativity, predicted the presence of gravitational waves in space-time, which would not be experimentally discovered until the LIGO project in 2016. In turn, British physicist Paul Dirac solved the quantum field equations in 1928, which allowed him to predict the existence of the positron (antielectron) ahead of its experimental discovery in 1932. In the 1960s, Peter Higgs used mathematical calculations to predict the existence of the Higgs boson, which was actually discovered 50 years later during experiments at the Large Hadron Collider.
There are countless other examples as well.
Each theory (Platonism and Nominalism) has its upsides and downsides, and there are arguments pro and con for both. The theory of Platonism faces the following counter-arguments:
1 - The epistemological argument: if mathematical entities are abstract, that is, if they exist outside of time and space, then how can we know them?2 - The ontological argument: what is the nature of abstract mathematical entities?3 - The metaphysical argument: numbers are not independent mathematical entities, but instead belong to a certain (higher or lower) order.4 - How do abstract mathematical entities function in time and space?
Nominalism, for its own part, is incapable of explaining what makes mathematics so successful and indispensable for science or how to apply mathematical semantics (information content) in science, nor can it shed light on the literal understanding of mathematical theories.
2 - The stance of the Kabbalah of Information
2.1 General principle
We cannot invent a single concept that would not exist within the information space of Creation. Otherwise, we would have surpassed the Almighty, which is absurd. This is precisely why Ecclesiastes tells us that “there is nothing new under the sun.”
This is how we can explain it: one of the main ideas of the Kabbalah of Information is the definition of Creation as a single information space. This space consists of concepts, the distance between which is determined by the difference in their information content (or their interpretation).
Creation as an information space is divided into realms that differ in terms of the amount and complexity of their respective concepts, as well as in terms of the number of informational dimensions. As we move from one realm to the next, information becomes compressed, and concepts are simplified. Therefore, any concept of a given realm is formed through the transformation of the more complex concepts from the preceding realm.
It follows from the above that all the concepts of our reality are projections of other, more complex concepts resulting from the hard and fast rules of compression and transformation. This idea can be described as the Law of Correspondence.
It bears noting that all of the concepts in our reality stem from the transformation of more complex concepts which are subject to uniform rules. This mechanism is consistent with the Category Theory that exists in mathematics.
2.2 The mathematical nature of Creation
We can find the following evidence proving the mathematical nature of Creation:
1 - Sefer Yetzirah (1:1) tells us, “With 32 mystical paths of Wisdom He created His universe.”
The paths of Wisdom, in this case, are 22 letters of the Hebrew alphabet and 10 numbers.
2 - The transition from one information realm to another is similar to a phase transition in a number of ways: the number of degrees of order increases, and the number of dimensions and degrees of freedom decreases, as does complexity and information entropy (uncertainty).
3 - In my other article, What Is Everything Made Of?, I wrote that the outlook of the Kabbalah of Information is close to the ideas of structural realism. The ontological structural realism approach can be briefly described as follows:there are no substances, just structures and relations between them. A significant number of eminent scientists have shared the ideas of structural realism.
I mentioned above that the process of transformation of concepts within the information space of Creation can be described in terms of the mathematical Category Theory. In this context, transformations may differ from each other, but there are certain objective, invariant states.
Quantum physicist Max Born said, “Invariants are the concepts of which science speaks in the same way as ordinary language speaks of “things...” He further added that reality is always a certain kind of structural invariance. Max Born believed that the notion of invariance is vital for the concept of relations that make up reality, not only in physics, but in all aspects of our world.
British scientist Arthur Eddington wrote, “What sort of thing is it that I know? The answer is structure.”The outstanding quantum physicist Hermann Weil and the brilliant mathematician Henri Poincaré shared a similar stance.
In the information space of Creation, the realms of Atziluth, Beriah, Yetzirah, and Asiyah have an invariant structure: each of them encompasses ten Sefirot, which emerged after the transformation of the Sefirot system in the realm of Atziluth. In other words, the structure of the Tree of Sefirot is the invariant of Creation.
The book Eitz Chaim tells us that “three realms – Beriah, Yetzirah, and Asiyah – are the seals of the realm of Atziluth.”
On top of that, each Sefirah contains the entire Tree of Sefirot, and so forth. This reaffirms that the Tree of Sefirot is the invariant of Creation.
4 - The Law of Likeness, which I talked about in previous articles (“And G-d said, Let us make man in our image, after our likeness,” from Bereishit; and “They shall make an ark of acacia wood, two and a half cubits its length, a cubit and a half its width, and a cubit and a half its height,” from Shemot, Parashat Terumah), stipulates that the distance between concepts in the information space is determined by the difference in their information content (interpretation). This further proves the mathematical nature of Creation and makes it possible to find connections between the concepts from our world and other realms by operating with dimensions and mathematical relationships.
5 - Equally important is the Law of Dimensionality, which can be phrased as follows: no structure (for example, a soul) can exist in an information space where the number of dimensions is lower than the number of dimensions of this structure (a soul).
All of the ideas listed above are reaffirmed by the essential teachings of Kabbalah and will be discussed in more detail in a separate article.
6 - Kabbalah mentions the concept of the Measuring Rod.
In this context, it’s a certain process that the Almighty used for determining the dimensions of all objects in our reality.
The fact that trillions of elementary particles (like electrons) are completely identical to one another in all aspects, save for their position in space and time, prompted physicist and Nobel Prize winner Richard Feynman to famously say, “They are all the same electron!”
The academic community was not receptive towards this idea, but the Kabbalah of Information can rephrase Richard Feynman’s statement as ‘they are all the same concept of the Electron.’
2.3 - The Kabbalah of Information and the Philosophy of Mathematics
The stance of the Kabbalah of Information is somewhat similar to Platonism, but with a number of important nuances:
1 - Mathematical entities exist.
2 - Mathematical entities are not just abstractions: they exist as concepts within the information space of Creation.
3 - Are mathematical entities (concepts) independent of human existence? Formally, it may seem that they are.
But if we bear in mind that Man is the highest, ultimate goal of Creation, we can conclude that all elements of Creation are, without exception, inextricably linked with human existence.
These statements allow us to respond to the arguments against the theory of Platonism.
1 - What is the nature of abstract mathematical entities?
The answer is: mathematical entities are informational concepts of Creation.
2 - How can we obtain knowledge on abstract mathematical entities that exist outside of time and space?
The answer is: mathematical entities are concepts that exist within the same holistic information space (Creation) as humans (including both their bodies and their souls).
3 - How can abstract mathematical entities that exist outside of time and space find such successful application in our world, which does exist within time and space?
The answer is: all of Creation is a single information space.
To purchase Eduard Shyfrin’s book ‘From Infinity to Man: The Fundamental Ideas of Kabbalah Within the Framework of Information Theory and Quantum Physics’ please click here. To purchase Eduard Shyfrin’s book ‘Travels with Sushi in the Land of the Mind’ please click here.