Dear Mohamed,

Yes, I am familiar with your idea about fractal spacetime (i.e., that spacetime may have fractal properties on a quantum scale), which potentially can lead to renormalizable quantum gravity free of short scale troubles. Thank you for mentioning it.

Dear Amrit,

I absolutely loved reading your essay. Thank you for bringing it to my attention. However, infinity you mentioned is not infinity I meant.

In mathematics, there is a principal difference between potential infinity and actual infinity. Potential infinity can be considered as a non-terminating and never complete process of increasing the number, quantity, or size, while actual infinity can be thought about as something that is completed and definite such as the set of all natural numbers.

According to constructive mathematics, the examination of the mathematical objects must be carried out within the framework of the abstraction of potential realizability and with total exclusion of the idea of an actual infinity. So, should we bring ideas of constructivism in physics?

Concept of infinity is linked to concept of zero. Both are inverse of each other and both are abstract and do not have physical existence and therefore cannot be measured. Before the Indian astronomer and mathematician Bramhagupta (597-668AD) invented the concept of zero, the idea did not exist in mathematics. You can think of one apple, 10 apples or 100 apples and so on, but it is not possible to think of zero apples. If you have five apples in a box and if you remove them from box, you are left with zero apples in the box. But the box does not disappear. So zero is related to apples and the location only. Those five apples continue their existence at some other location and in other form. 

All measurable physical quantities has to have boundaries, otherwise you cannot measure them. Infinity cannot have any boundary. If you put a boundary on infinity, it becomes finite and cannot be called infinite. For the same reason there cannot be more than one infinity. If there are two or more infinities then there will be boundaries between them which will make them finite. Therefore infinity is one without a second.

In quantum physics zero point energy has a mass gap. So the particles are not annihilated in this ground state and some finite amount of energy cotinues to exist. In the paper listed below, I have discussed a particle annihilation at Planck energy level without any mass gap. In this state the particle wave completely collapse and the particle motion and its energy disappear. This perfectly motionless state of the energy is associated with an underlying field of the universe which is infinite and one without a second.

Article Periodic quantum gravity and cosmology

Dear Vikram,

I really enjoyed your paper “Periodic quantum gravity and cosmology”. Even though I did not share the main paper’s ideas concerning the universal consciousness and fundamental reducibility of everything to just an energy, I found them thought provoking.

However, regarding the concept of actual infinity I cannot agree with what you’re saying.

You claim, “All measurable physical quantities has to have boundaries, otherwise you cannot measure them.” First, such a statement is logically deficient since there is no way to formulate it unambiguously: No matter where the boundaries limiting physical quantities would be drawn, they would be always ad hoc and so perpetually subject to shifting. Second, the apparent absence of infinite things in the region where all scientific experiments and human experiences happen cannot be an indication of what may happen beyond that region.

Suppose our physical universe was, is and will be spatially infinite; then in such a universe a system with an infinite number of its constituent particles can exist. As a result, a linear superposition of Fock states (that is, states with definite particle number) for the given system would contain states with an infinite number of particles, which theoretically (in principle) could be measurable.

Dear Mohamed,

Upvoting and downvoting someone’s post seems to me a little bit childish. For goodness' sake, dear colleagues, ResearchGate is not Facebook, and a scientific discussion is not a beauty contest!

At the end of the day, the truth in science cannot be reached by a majority of ballots cast.

Your comment "No matter where the boundaries limiting physical quantities would be drawn, they would be always ad hoc and so perpetually subject to shifting," itself suggests that the universe as a whole is indivisible. So the physical measurements related to shifting boundaries are always approximate.

With respect to your second comment, all scientific experiments happen within the region of space-time and are relative and dualistic in nature. Most of the human experience are also dualistic (subject-object experience) in nature. But there exists a non-dualistic human experience in which the boundary between subject and object also disappear. This experience exists beyond the limitations of space-time. And it is in this experience that one realizes the underlying field of the universe which is infinite and one without a second.

A linear superposition of Fock states (that is, states with definite particle number) for the given system would contain states with a very LARGE number of particles, but they cannot be called infinite number of particles because infinity can only be one and not many. Secondly Fock states are with in the limitation of space-time and dualistic.

Dear Vikram,

You brought up here a very interesting (and still open!) problem concerning separation between potential and actual infinity, I want to thank you for this.

But let us concentrate on the question whether infinity is one or many (since this question can demonstrate a counter-intuitive logic behind the concept of actual infinity).

According to the Dedekind definition of infinity, an infinite set of natural numbers (i.e., actual infinity) can be in bijective (one-to-one) correspondence with any proper subset of itself. Consequently, the infinite set of natural numbers can be presented as a direct sum of, say, two infinite subset containing odd and even numbers. In the example of the linear superposition mentioned in the previous post, the quantity of Fibonacci-numbered Fock states is not smaller than total amount of all the Fock states (mathematically speaking, the cardinality of the subset containing the Fibonacci-numbered states is the same as the cardinality of the set of all Fock states). In general, we can consider any finite number of nested infinities in the infinite superposition of the Fock states.

As to the idea of an infinite physical universe, this abstraction (similar to the abstraction of actual mathematical infinity) consists of assumption that process of the cosmic inflation do not terminate in principle. Such an eternally inflated universe can be mentally regarded as an infinite physical object (analogous to the set of all non-negative integers that can be regarded as a mathematical infinite object).

Dear Arkady:

In most of the mathematics, the word infinity is used in the sense of a very very large number and similarly number zero is used in the sense of infinetesimally very small finite number. Therefore in mathematics, both infinity and zero have boundaries. This makes mathematics a relative dualistic science which is not useful in defining the absolute zero and absolute infinity. Absolute zero could be conceived as total absence of any matter (or energy) in the universe. The resulting motionless state of the universe is infinite without any boundary. Mathematics is useless in analyzing this state of the univesre. When the motion takes place, energy is born and then relative dualistic science could be used to analyze the motion.

In absolute infinity there is no duality. It is one without a second. The motion causes apparent dulaity which is caused by the wave-particle dual behavior and  discretization of quanta in quantum mechanics.

Dear Vikram,

I am very sorry to disappoint you but the story about infinity in mathematics is a bit more complicated that you presented here.

Infinity is not a very large number; it is not even an astronomically enormous large number; what is more, it is not a number at all.

According to Encyclopedia of Mathematics, infinity is the concept, which arises in various branches of mathematics such as analytic and geometric theories (where it denotes "improper" or "infinitely-distant" elements) and set theory (where it is used in defying the system of transfinite numbers).

Particularly, in set theory, when dealing with infinite sets such as the set of integers or rational numbers, G. Cantor showed that the cardinality of the infinite set of integers is strictly smaller than that of the infinite set of real numbers. He proposed the continuum hypothesis, which states that the set of real numbers has minimal possible cardinality that is greater than the cardinality of the set of integers. Simply put, Cantor demonstrated that one infinity might be less than other infinity (which, of course, could not be possible if infinity were just a number).

Now, about your sentence “Mathematics is useless in analyzing this state of the universe.” Any analysis (classical or not) implies the use of mathematics as a formal system, that is, the set of axioms and inference rules on which the given analysis is built on. Thus, denying mathematics means denying any analysis whatsoever.

Dear Arkady,

When you read the following article, you will realize that you have not disappointed me. I have quoted following para in sec. 4A of this article.

----quote

Infinity can not be two or more. There can be only one infinity otherwise there will be a boundary between two infinity which will make them finite. Normally in physics, by infinity we mean either spatial or numerical infinity. In light of the equivalence of space-time and energy-vibration continuums, it is possible to introduce a broader concept [not to be confused with Aristotle's concept] which I call potential infinity (omnipotence). By energy approaching infinity, we simply mean the highest possible potency for the form of energy regardless of spatial or numerical considerations, which are but geometrical concepts. Here we are trying to describe something which the vedantins call 'one without a second'. Spatial and numerical concepts help only where there is multiplicity. In other words, all forms of energy with lower potencies have their origin in forms of energy with higher potencies. It

is not a mere coincidence that word Om is considered a sound sphota or sound potency and the particle energies are also measured in terms of potential difference, i.e. electron volts.

-----end quote

Research Philosophy and physics

Dear Vikram,

The crux of our disagreement regarding the concept of infinity lies in the confusion between physics and philosophy (to be exact, metaphysics).

While physical examination of fundamental structures of space, time, and causality involves either mathematics or matters of fact, metaphysics goes beyond these.

I do not argue that metaphysical statements are neither true nor false but meaningless (since, according to the verifiability theory of meaning, a statement is meaningful only if there can be empirical evidence for or against it). I do not say that matters of fact cannot be obtainable independently of experience. I do not state that the metaphysical presuppositions make no contribution to physics predictive success.

What I am trying to set forth here is only that a mathematical structure of any physical theory must be effectively calculable, that is, constructive.

Dear Arkady,

While physical examination of fundamental structures of space, time, and causality involves either mathematics or matters of fact, metaphysics goes beyond these.

mathematical structure of any physical theory must be effectively calculable, that is, constructive.

I do not have any problem with above statements.

according to the verifiability theory of meaning, a statement is meaningful

only if there can be empirical evidence for or against it.

Verifiability theory of meaning is not proposed by a metaphysicist, hence it can apply only to physics, mathematics and other lower sciences. Metaphysicist uses his body-mind complex as an instrument for verification of metaphysical theories and statements. If his instrument is not evolved enough or fine tuned for the purpose, he may not get the empirical evidence for his theories and statements. This evolution and fine tuning of body-mind complex does not happen over night. It may take several births to get there. No such considerations are there for physics and other lower sciences.