Believers in eternal truth often point to mathematics as a model of a realm with timeless truths. Mathematicians explore this realm with their minds and discover truths that exist outside of time, in the same way that we discover the laws of physics by experiment. But mathematics is not only self-consistent, but also plays a central role in formulating fundamental laws of physics, which the physics Nobel laureate Eugene Wigner once referred to as the “unreasonable success of mathematics in physics”. One way to explain this “success” within the dominant metaphysical paradigm of the timeless multiverse is to suppose that physical reality is mathematical, i.e. we are creatures within the timeless Platonic realm. The cosmologist Max Tegmark calls this the mathematical universe hypothesis. A slightly less provocative approach is to posit that since the laws of physics can be represented mathematically, not only is their essential truth outside of time, but there is in the Platonic realm a mathematical object, a solution to the equations of the final theory, that is “isomorphic” in every respect to the history of the universe. That is, any truth about the universe can be mapped into a theorem about the corresponding mathematical object. If nothing exists or is true outside of time, then this description is void. However, if mathematics is not the description of a different timeless realm of reality, what is it? What are the theorems of mathematics about if numbers, formulas and curves do not exist outside of our world?
Let us consider a game of chess. It was invented at a particular time, before which there is no reason to speak of any truths of chess. But once the game was invented, a long list of facts became demonstrable. These are provable from the rules and can be called the theorems of chess. These facts are objective in that any two minds that reason logically from the same rules will reach the same conclusions about whether a conjectured theorem is true or not. Platonists would say that chess always existed timelessly in an infinite space of mathematically describable games. By such an assertion, we do not achieve anything except a feeling of doing something elevated. Further, we have to explain how we finite beings embedded in time can gain knowledge about this timeless realm. It is much simpler to think that at the moment the game was invented, a large set of facts become objectively demonstrable, as a consequence of the invention of the game. There is no need to think of the facts as eternally existing truths, which are suddenly discoverable. Instead we can say they are objective facts that are evoked into existence by the invention of the game of chess. The bulk of mathematics can be treated the same way, even if the subjects of mathematics such as numbers and geometry are inspired by our most fundamental observations of nature. Mathematics is no less objective, useful or true for being evoked by and dependent on discoveries of living minds in the process of exploring the time-bound universe.
The Mandelbrot Set is often cited as a mathematical object with an independent existence of its own. Mandelbrot Set is produced by a remarkably simple mathematical formula – a few lines of code (f(z) = z2+c) describing a recursive feed-back loop – but can be used to produce beautiful colored computer plots. It is possible to endlessly zoom in to the set revealing ever more beautiful structures which never seem to repeat themselves. Penrose called it “not an invention of the human mind: it was a discovery”. It was just out there. On the other hand, fractals – geometrical shapes found through out Nature – are self-similar because how far you zoom into them; they still resemble the original structure. Some people use these factors to plead that mathematics and not evolution is the sole factor in designing Nature. They miss the deep inner meaning of these, which will be described later while describing the structure of the Universe.
The opposing view reflects the ideas of Kant regarding the innate categories of thought whereby all our experience is ordered by our minds. Kant pointed out the difference between the internal mental models we build of the external world and the real objects that we know through our sense organs. The views of Kant have many similarities with that of Bohr. The Consciousness of Kant is described as intelligence by Bohr. The sense organs of Kant are described as measuring devices by Bohr. Kant’s mental models are Bohr’s quantum mechanical models. This view of mathematics stresses more on “mathematical modeling” than mathematical rules or axioms. In this view, the so-called constants of Nature that arise as theoretically determined constants of proportionality in our mathematical equations, are solely artifacts of the particular mathematical representation we have chosen to use for explaining different natural phenomena. For example, we use G as the Gravitational constant because of our inclination to express the gravitational interaction in a particular way. This view is misleading as the large number of the so-called constants of Nature points to some underlying reality behind it. We will discuss this point later.
The debate over the definition of “physical reality” led to the notion that it should be external to the observer – an observer-independent objective reality. The statistical formulation of the laws of atomic and sub-atomic physics has added a new dimension to the problem. In quantum mechanics, the experimental arrangements are treated in classical terms, whereas the observed objects are treated in probabilistic terms. In this way, the measuring apparatus and the observer are effectively joined into one complex system which has no distinct, well defined parts, and the measuring apparatus does not have to be described as an isolated physical entity.
As Max Tegmark in his External Reality Hypothesis puts it: If we assume that reality exists independently of humans, then for a description to be complete, it must also be well-defined according to non-human entities that lack any understanding of human concepts like “particle”, “observation”, etc. A description of objects in this external reality and the relations between them would have to be completely abstract, forcing any words or symbols to be mere labels with no preconceived meanings what-so-ever. To understand the concept, you have to distinguish between two ways of viewing reality. The first is from outside, like the overview of a physicist studying its mathematical structure – a bird’s eye view. The second way is the inside view of an observer living in the structure – the view of a frog in the well.
Though Tegmark’s view is nearer the truth (it will be discussed later), it has been contested by others on the ground of contradicting logical consistency. Tegmark relies on a quote of David Hilbert: “Mathematical existence is merely freedom from contradiction”. This implies that mathematical structures simply do not exist unless they are logically consistent. They cite the Russell’s paradox (discussed in detail in later pages) and other paradoxes – such as the Zermelo-Frankel set theory that avoids the Russell’s paradox – to point out that mathematics on its own does not lead to a sensible universe. We seem to need to apply constraints in order to obtain consistent physical reality from mathematics. Unrestricted axioms lead to Russell’s paradox.
Conventional bivalent logic is assumed to be based on the principle that every proposition takes exactly one of two truth values: “true” or “false”. This is a wrong conclusion based on European tradition as in the ancient times students were advised to: observe, listen (to teachings of others), analyze and test with practical experiments before accepting anything as true. Till it is conclusively proved or disproved, it was “undecided”. The so-called discovery of multi-valued logic is nothing new. If we extend the modern logic then why stop at ternary truth values: it could be four or more-valued logic. But then what are they? We will discuss later.
Though Euclid with his Axioms appears to be a Formalist, his Axioms were abstracted from the real physical world. But the focus of attention of modern Formalists is upon the relations between entities and the rules governing them, rather than the question of whether the objects being manipulated have any intrinsic meaning. The connection between the Natural world and the structure of mathematics is totally irrelevant to them. Thus, when they thought that the Euclidean geometry is not applicable to curved surfaces, they had no hesitation in accepting the view that the sum of the three angles of a triangle need not be equal to 1800. It could be more or less depending upon the curvature. This is a wholly misguided view. The lines or the sides drawn on a curved surface are not straight lines. Hence the Axioms of Euclid are not violated, but are wrongly applied. Riemannian geometry, which led to the chain of non-Euclidean geometry, was developed out of his interest in trying to solve the problems of distortion of metal sheets when they were heated. Einstein used this idea to suggest curvature of space-time without precisely defining space or time or spece-time. But such curvature is a temporary phenomenon due to the application of heat energy. The moment the external heat energy is removed, the metal plate is restored to its original position and Euclidean geometry is applicable. If gravity changes the curvature of space, then it should be like the external energy that distorts the metal plate. Then who applies gravity to mass or what is the mechanism by which gravity is applied to mass. If no external agency is needed and it acts perpetually, then all mass should be changing perpetually, which is contrary to observation. This has been discussed elaborately in latter pages.
Once the notion of the minimum distance scale was firmly established, questions were raised about infinity and irrational numbers. Feynman raised doubts about the relevance of infinitely small scales as follows: “It always bothers me that, according to the laws as we understand them today, it takes a computing machine an infinite number of logical operations to figure out what goes on in no matter how tiny a region of space, and no matter how tiny a region of time. How can all that be going on in that tiny space”. Paul Davies asserted: “the use of differential equations assumes the continuity of space-time on arbitrarily small scales.
The frequent appearance of π implies that their numerical values may be computed to arbitrary precision by an infinite sequence of operations. Many physicists tacitly accept these mathematical idealizations and treat the laws of physics as implementable in some abstract and perfect Platonic realm. Another school of thought, represented most notably by Wheeler and Landauer, stresses that real calculations involve physical objects, such as computers, and take place in the real physical universe, with its specific available resources. In short, information is physical. That being so, it follows that there will be fundamental physical limitations to what may be calculated in the real world”. Thus, Intuitionism or Constructivism divides mathematical structures into “physically relevant” and “physically irrelevant”. It says that mathematics should only include statements which can be deduced by a finite sequence of step-by-step constructions starting from the natural numbers. Thus, according to this view, infinity and irrational numbers cannot be part of mathematics.
Infinity is qualitatively different from even the largest number. Finite numbers, however large, obey the laws of arithmetic. We can add, multiply and divide them, and put different numbers unambiguously in order of size. But infinity is the same as a part of itself, and the mathematics of other numbers is not applicable to it. Often the term “Hilbert’s hotel” is used as a metaphor to describe infinity. Suppose a hotel is full and each guest wants to bring a colleague who would need another room. This would be a nightmare for the management, who could not double the size of the hotel instantly. In an infinite hotel, though, there is no problem. The guest from room 1 goes into room 2, the guest in room 2 into room 4, and so on. All the odd-numbered rooms are then free for new guests. This is a wrong analogy. The numbers are divided into two categories based on whether there is similar perception or not. If after the perception of one object there is further similar perception, they are many, which can range from 2,3,4,…..n depending upon the sequence of perceptions? If there is no similar perception after the perception of one object, then it is one. In the case of Infinity, neither of the above conditions applies. However, Infinity is more like the number ‘one’ – without a similar – except for one characteristic. While one object has a finite dimension, infinity has infinite dimensions. The perception of higher numbers is generated by repetition of ‘one’ that many number of times, but the perception of infinity is ever incomplete.
Since interaction requires a perceptible change anywhere in the system under examination or measurement, normal interactions are not applicable in the case of infinity. For example, space and time in their absolute terms are infinite. Space and time cannot be measured, as they are not directly perceptible through our sense organs, but are deemed to be perceived. Actually what we measure as space is the interval between objects or points on objects. These intervals are mental constructs and have no physical existence other than the objects, which are used to describe space through alternative symbolism. Similarly, what we measure as time is the interval between events. Space and time do not and cannot interact with each other or with other objects or events as no mathematics is possible between infinities. Our measurements of an arbitrary segment of space or time (which are really the intervals) do not affect space or time in any way. We have explained the quantum phenomena with real numbers derived from fundamental principles and correlated them to the macro world. The quantities like π and φ etc have other significances, which will be discussed later.