Zero and One Grouped Together
Nature is consistent is how she equitably deals with any and all events ever in space-time. How does she enforce the same rules consistently on systems huge and tiny? My research if focused on the idea that we need to get number theory correct for space-time. The correct accounting system is needed for transient events in space-time to appreciate Nature's strange and subtle rules. The target of this research is tensor calculus which is ubiquitous in theoretical physics.
Physics is the study of change. Sir Isaac Newton was the first great mathematical physicist. He not only solved a tremendous range of problem sin physics using both theory and experiment, but he also developed calculus, the mathematical study of change. He conducted a flame war with the independent discover of calculus by Leibniz. The formal study of calculus is know as mathematical analysis. The starting point is a mathematical field. A mathematical field forms a group under addition. The field without zero is also a group under multiplication. The operations are associative. The distributive rule also must work. An identity element exists for both addition and multiplication (one of the rules from group theory).
What comes before a field? What does not change? Two numbers are needed to think about events in space-time. Zero in space-time is here-now. The location is centered on an observer where the clock is in the present moment. One in space-time is there-then. It represents a signal an observer might see in the future.
It is group theory that has brought order to the world of particle physics. Group theory is there for mathematical fields. The first group to start with is the trivial group. As the name suggests, there is only one member in the trivial group. Yet there are an infinite number of ways to represent the trivial group. This study is limited to zero and one. With zero, both the operators of addition and multiplication form the trivial group.
| Group property | Addition + | Multiplication * |
|---|---|---|
| Closure | 0 + 0 = 0 | 0 * 0 = 0 |
| Associative | (0 + 0) + 0 = 0 + (0 + 0) | (0 * 0) * 0 = 0 * (0 * 0) |
| Identity | I = 0: 0 + I = 0 | I = 0: 0 * I = 0 |
| Commutative | 0 + 0 = 0 + 0 | 0 * 0 = 0 * 0 |
| Distributive | 0 * (0 + 0) = 0 * 0 + 0 * 0 |
Zeroes are everywhere. One minor surprise might be the inverse for multiplication. Care was taken to not mention the familiar subtraction and division operations. One reason to avoid these good old friends comes from the needs of the associative property. There will be many situations where:
Only for the trivial group can it be said that the inverse of zero is zero. Dividing by zero remains a bad thing to do.
For an observer, here-now remains here-now. Doesn't here now move forward in space-time? An observer goes from one here-now to the next here-now without paying any fee: 0+0=0.
The trivial group represented by zero and using both the addition and multiplication operator does not represent a mathematical field. The technical definition of a mathematical field stipulates that the addition identity must be different from the multiplication identity. The trivial group remains trivial, unable to change. This may be a great thing: observers continue to observe, never able to close their eyes to the evolving Universe.
Represent the trivial group with unity. This time the addition operator cannot be used because the requirement of closure is lost: 1+1=2. The multiplication operator with unity does form a group.
| Group property | Addition + | Multiplication * |
|---|---|---|
| Closure | 1 + 1 = 1 | 1 * 1 = 1 |
| Associative | (1 + 1) + 1 = 1 + (1 + 1) | (1 * 1) * 1 = 1 * (1 * 1) |
| Identity | I = 1: 1 + I = 1 | I = 1: 1 * I = 1 |
| Commutative | 1 + 1 = 1 + 1 | 1 * 1 = 1 * 1 |
The trivial group can be represented by unity and the multiplication operator. It is different from both representations using zero because it can be used to make a continuous group. Take the set of all positive real numbers. Multiply those by zero and zero is the result. Multiply them by unity, and that forms a continuous group. For example, there is 4 and a fourth whose product is unity.
The mathematician Giuseppe Peano was able to construct the set of natural numbers starting from zero and unity (see the Peano axioms). With the natural (aka counting) numbers in hand, one can then construct the rational numbers using the multiplicative inverse.
Which of the three representations of the trivial group discussed here should one use to do space-time physics? All three. The heart of physics beats from strong logical tensions. Everything in the Universe, great or small, observers the rest of the Universe, and can be observed by the rest of the Universe.
The number zero, here-now, fits the job requirements of a ceaseless observer. The number one, there-then, can produce a signal to be observed later. Zero and one as representing the trivial group is pre-calculus because the trivial group does not allow for change. The fact that the unchangeable trivial group can be represented by both zero and one will have interesting repercussions for the logic of physics.