Smaller and smaller

1. If you take an infinite line and cut it into an infinite number of pieces, you are faced with a paradox:

2. How can you fit an infinite number of pieces into a line which stretches to infinity? What will be the size of each infinitesimal piece?

3. Georg Cantor, who invented set theory, argued that there are different kind of infinities. Take for example the set of integer numbers (1,2,3,…) and the set of real numbers (1, 1.1, 1.2,…2, 2.1, 2.2…,…). Both sets are infinite but the set of real numbers has to be bigger, since there is an infinite amount of real numbers between two integers.

4. Therefore the problem is how we define the size of a part. In other worlds infinitesimal quantities have to come in different sizes. There can’t be an absolute infinitesimal quantity of reference. No matter how small an infinitesimal might be, there will always be another infinitesimal smaller.

5. This is also true in physics. Initially the atom had been treated as an indivisible and fundamental quantity (atom= that which cannot be divided). Later on the nucleus (protons and neutrons) of the atom as well as the electrons (orbiting the atom) were discovered. Finally it was found that protons and neutrons consist of quarks. Nowadays quarks are considered indivisible. But nobody dares anymore say that quarks may not be divided in the future into smaller constituents. Therefore the process of dividing matter into smaller and smaller parts may never end.    

6. A similar problem is depicted in the previous painting. But instead of ‘atoms’ the universe of the artist consists of lizards, growing smaller and smaller as space is tessellated into smaller and smaller parts.

7. But could there be something in the universe representing the smallest possible thing? Apparently the universe itself has to be the biggest thing ever to have existed. Again here the problem is not only that the universe is expanding but that there could also be other universes in a larger construction, the multiverse. But let’s ignore for a moment those additional elements and consider the universe the biggest possible thing.

8. Going back in time, we can imagine the universe shrinking on and on, until it reaches a state of singularity as it is called. This same singularity is considered to be a point of infinite density in spacetime where the universe came from before the Big Bang (in fact even before spacetime formed).

9. Here we face the same problem of infinity we faced before. How much ‘infinite’ was the singularity? How ‘rigid’ can be an object of infinitesimal size but which contains an infinite amount of energy? In order to quantify this problem we will suppose that the whole universe is a black hole, so that the singularity is still there, at the beginning of space and time, while the event horizon of this black hole is the horizon of the observable universe. In fact there is a formula in physics which estimates the radius of a black hole, and it is called the Schwarzschild radius R_S: 

where G is Newton’s gravitational constant, and c is the speed of light.

10. If we replace in this formula the mass M_S of a black hole with that of the observable universe we take as result the radius of the observable universe! [1]

11. Besides the meaning of this coincidence, we can also ask ourselves which could be the smallest possible object in the universe. To answer this question we might use another formula in physics, which gives the Compton wavelength λ of a particle:

where h is Planck constant, m is the mass of the particle, and c is the speed of light.

12. If we replace in this formula the mass of a particle with that of the observable universe we take a size (wavelength) about equal to 10^-95 m. Incidentally this size is much smaller than Planck length, which is considered the smallest possible size according to quantum mechanics.

13. While again such a size depends on the mass of the object, so that we could assume an even larger mass (for example including dark matter), we could ultimately agree on some infinitesimal quantity, based on some calculation or another, to represent a ‘sufficiently infinitesimal’ quantity according to which we could divide anything into the smallest possible parts.

14. Here however we will move away from the quantitative problem of infinity and infinitesimals. To give an example, let’s think about the universe and the atom at the same time. While we bring both these objects, opposed to each other, into mind, what size do they really occupy within our mind? Let’s imagine two ‘circles,’ or ‘spheres,’ the latter one (the atom) consisting of ‘small black and grey grains’ (protons and neutrons) at the center, and ‘tiny bright dots’ on the circumference (the electrons), and the former one (the universe) consisting of ‘faint distant spots’ (the stars), and ‘milky spiral structures’ (the galaxies) in its volume. Are the stars ‘brighter’ or ‘bigger’ than the electrons in our mind? Is a ‘black hole’ in the universe more real than a ‘dark spot’ in our thought? Is a singularity anything more than a mathematical construction of our own imagination? Are the lizards depicted in the artist’s painting more iconic than the virtual images of true lizards in our mind?

15. No matter what the physical definition or mathematical description of the mind might be, the size or weight of objects is a problem of the senses (how big we see or how heavy we feel the objects), which we communicate with the mathematical or the common language. While the universe must be ‘infinitely massive and vast,’ the atom must be ‘infinitesimally light and small.’ But isn’t the physical properties of what we try to conceptualize, or the parameters of our own conceptualization, the true problem we are faced with?

[1]: [https://archive.org/details/CrossingTheBrachistochrone_201804]

9/9/2018

Picture: Smaller and smaller, M.C. Escher