The Measurement System

Every measurement is a ratio, a difference of something that is quantifiable. It is a distinction that tells us about what it is and how it is related to what it is not.

We can, for example, say that the width of a fingernail is one centimeter. A centimeter is based on the value of a meter. A meter is based on the speed of light. The speed of light is based on the length of a second. The second is based on the oscillations of an atomic particle's radiation. The oscillations are based on the second. The speed of light is also based on the meter. Investigate deeply enough and we see that there is no independent measurement, no number that isn't relative to another number. Everything that is big is only big because there are small things. The small things would not be small without the big.

Another thing of note is that the ratios transcend the unit. What does the unit represent if height can be represented by duration, space by time?

The bigger questions: can it be simplified, made more intuitive or practical? Can our measurement system be organized better? Can it become a better tool?

The Mathematics

The problem with using mathematics to model reality is that there are concepts in mathematics which may not physically exist. Infinity is the easiest example. Can we ever observe something infinite? Maybe something can appear to be infinite, but can there ever be proof that it is infinite? Zero is another concept: zero only makes sense when given context (such as when something used to be somewhere that it no longer is), but context itself is something that exists only within the mind. There are irrational numbers, infinite divisible, that cannot be expressed as a ratio. Physically this would imply that infinity exists in the smallest spaces, everywhere.

Can we construct a mathematical structure that best represents reality?

The Autonormalizing Lattice

We're introducing a new semi-constant semi-variable called the kato, represented by the symbol ꝃ. ꝃ is a constant because human beings ultimately decide its value, but it is variable because it is define with the expectation that it will change over time. ꝃ represents the smallest theoretically possible physical numerical value. Today's value will be known as ꝃ_0 and will be set at (10^95)^(-1)_12, which is the inverse of the theoretical total entropy of the universe in base-12.

The kato is meant to be the smallest measurable value in reality where anything lower than it does not have any physical basis. If we want to model the universe with it, its role as the lower cosmic limit must be defined with constraints.

Negative numbers, irrational numbers, and zero cannot exist. To do this, we define the kato-space, called Ꝃ, which looks something like this: for all x in Ꝃ, x = nꝃ where n is a natural number between 1 and ꝃ^(-2). This is a lattice where each point is spaced apart in increments of ꝃ, so it goes all the way from ꝃ, 2ꝃ, 3ꝃ,... all the way up to (10^95)^(2)ꝃ or (10^95). The kato-space contains the smallest measurable value and the largest measurable value, and each value in-between is a multiple of the smallest.

If a smaller value with physical meaning is discovered, the kato is redefined as it. If a bigger value than the inverse kato is found, similarly its inverse will replace the current kato. It may even turn out that the inverse of total entropy is too small to represent anything meaningful, in which case the value of the kato could be increased.

The Discrete-Continuous Interface

We have defined a finite set of numbers that are positioned to exist as the foundation of our mathematical system, but human beings begin at the top to do most of the work. That is to say, for example, the numbers 1 and 2 might not exist in the kato-space, but humanity finds value in the fact that 1 + 1 = 2. Likewise, irrational numbers like e or pi are kato-numbers because they are infinitely divisible and thus not a multiple of ꝃ, yet they have real physical meaning. So we have to round, and the rounding has to be implicit.

For all real numbers greater than ꝃ, if a number is not a multiple of ꝃ, then we assume it to be the nearest number which is a multiple of ꝃ. This means that there will a rounding error of at most (10^95)^(-1)_12 for pi, which is several order of magnitude greater precision than humanity currently finds necessary for scientific applications. If rounding introduces significant errors in real-world applications, a new smaller kato must be discovered and set.

What we've done here is make everything implicitly discrete while making no assertions on the actual state of reality. In our model nothing is truly continuous, but continuous application remains as a useful tool that gives results which are practically perfect but absolutely wrong.

Measurement Construction

The reason a value of entropy was chosen is because it is represented in natural units (nats). Using nats as a base, every existing measurement can be transformed into the same unit, and if everything has the same unit then it is functionally equivalent to having no units. This situation allows for some very nice properties to emerge, but for our purposes here we will focus on the fact that we can structure the whole measurement system as a web of unitless ratios. We would then construct units for human convenience.

Let's start with time as an example. To find the length of a solar year, we find the amount of seconds in a year and then divide it by the Planck time. As both of these numbers have the same units, we are left with a unitless ratio. We get something like 93_12. If we define a second as 1/(10^5)_12 of a day, so that the day has 10,0000 seconds, then we define a constant, the kato-second constant, which, when multiplied with ꝃ, makes the kato-second exactly equal to 1 and the kato-year equal to 10,0000. That value is 93 x 10^90, which is the kato-second:kato ratio. With this, we have defined kato-time and can build a kato-clock.

Repeating this process to define a kato-meter, a kato-hertz, a kato-liter etc. will construct a geometric system of simple ratios that should be cleaner and easier to understand for both the beginner and expert users of these number systems. Mass is energy is momentum is temperature is frequency is time is length is area is information is entropy is curvature is pressure is density, or something like that.

Note: this writing is recognized to be very imprecise, confusing and messy.