Where ever one looks, things mostly seem to be in lumps or clumps of matter. We live on a lump of matter, one of a number of lumps of matter orbiting an even bigger lump of matter. We look into the sky when the bigger lump of matter is conveniently on the other side of our lump of matter and we see evidence of other lumps of matter similar to the lump of matter that our lump of matter orbits.
We see stars, in short, which poetically speaking float in a void empty of matter. We can see that these stars are not evenly distributed and that they gather together in clumps which we call galaxies. Actually stars seem to clump together in smaller clumps such as the Local Cluster of a dozen or so stars, and most galaxies have arms or other features that show structure at all levels.
The galaxies, which we can see between the much closer stars of our own galaxy, also appear to be clustered together in clumps, and the clumps seem to be clumped together. Of course, the ultimate clump is the Universe itself, but at all levels the Universe appears to have structure, to be organised, to be formed of lumps and clumps, variously shaped into loops, whorls, sheets, arms, rings, bubbles, and so on.
OK, but in the other direction, towards the smaller rather than the larger, our planet has various systems, weather, orogenic, natural, social and evolutionary. All sorts of systems at all levels, from global scope to the scope of the smallest element.
In other personal worlds, below the level our interactions with our families, we have all the systems that make up our own bodies. The system that circulates our blood, the system that processes our food, the system that maintains our multiple systems in a state homeostasis.
That is, not a steady state, but a state where all the individual systems self-adjust so that the larger system does not descend into a state of chaos, leading to a disruption of the larger whole. Death.
By and large most systems in our environment are made up of molecules, which are in turn made up of atoms. Atoms are a convenient stopping point on the scale from very large to very small. They are pretty “well defined”, in that they are a very strong concept.
Atoms are rarely found solo. They are sociable critters. They form relationships with other atoms, but some atoms are more sociable than others, forming multiple bonds with other atoms. Some are more promiscuous than others, changing partners frequently.
These relationships are called molecules, and range from simple to complex, containing from two or three atoms, to millions of atoms. The really large molecules can be broken down to smaller sub-molecules which are linked repeatedly to make up the complex molecules.
To rise higher up the scale for a moment, these molecules, large and small are organised into cells, which are essentially factories for making identical or nearly identical copies of themselves. The differences are necessary to make cells into muscles or organs and other functional features, and cells that make bones and sinews and other structural parts of a body.
As I said, atoms are a convenient stopping point. Every atom of an element is identical at least in its base state. It may lose or gain electrons in a “relationship” or molecule, but basically it is the same as any other element of the same sort.
Each atom consists of a nucleus and surrounding electrons, a model which some people liken to a solar system. There are similarities, but there are also differences (which I won’t go into in this post). The nucleus consists a mix of protons and neutrons. While the number neutrons may vary, they don’t significantly affect the chemical properties of the atom, which makes all atoms of an element effectively the same.
Each component of an atom is made up of smaller particles called “elementary” particles, although they may not be fundamentally elementary. At this level we reach the blurry level of quantum physics where a particle has an imprecise definition and an imprecise location in macroscopic terms.
Having travelled from the largest to the smallest, I’m now going to talk mathematics. I’ll link back to physics at the end.
We are all familiar with counting. One, two, three and so on. These concepts are the atoms of the mathematical world. They can be built up into complex structures, much like atoms can be built into molecules, organelles, cells, tissues and organs. (The analogy is far from perfect. I can think of several ways that it breaks down).
Below the “atomic” level of the integers is the “elementary” level of the rational numbers, what most people would recognise as fractions. Interestingly between any two rational numbers, you can find other rational numbers. These are very roughly equivalent to the elementary particles. Very roughly.
One might think that these would exhaust the list of types of numbers, but below (in a sense) the rational numbers is the level of the real numbers. While many of the real numbers are also rational numbers, the majority of the real numbers ate not rational numbers.
The level of the real numbers is also known as the level of the continuum. A continuum implies a line has no gaps, as in a line drawn with a pencil. If the line is made up of dots, no matter how small, it doesn’t represent a continuum.
A line made up of atoms is not a continuum, nor is a line of elementary particles. While scientists have found ever more fundamental particles, the line has apparently ended with quarks. Quantum physics seems to indicate that nature, at the lowest level, is discrete, or, to loop back to the start of this post, lumpy. There doesn’t seem to be a level of the continuum in nature.
That leaves us with two options. Either there is no level of the continuum in nature and nature is fundamentally lumpy, or the apparent indication of quantum physics that nature is lumpy is wrong.
It’s hard to believe that a lumpy universe would permit the concept of the continuum. If the nature of things is discrete, it’s hard to see how one could consider a smooth continuous thing. It’s like considering chess, which fundamentally defines a discontinuous world, where a playing piece is in a particular square and a square contains a playing piece or not.
It’s a weak argument, but the fact that we can conceive the concept of a continuum hints that the universe may be fundamentally continuous, in spite of quantum physics’ indications that it is not continuous.