Vivian wrote:When you call the universe "mathematics", are you talking arithmetic with equations for proof, etc.? Or are you talking geometry? I know the universe is full of geometry - all kinds, everywhere. From what I saw, Tegmark was illustrating the geometry of the universe. That is easy to see - although, admittedly, I did not go all the way. I do not understand what you mean by "the universe is mathematics". Can you explain exactly what you mean by the universe being pure mathematics? Or, is it too deep a subject? Don't fear saying so. I know my limits.
Ok.. where to start.. while keeping it simple.
Discovery vs. Invention:
Let's take the value PI. It is created by an Equation and in decimal reads 3.141592653589...etc.
It is the ratio of the circumference of a circle vs. the diameter of a circle and, as far as we have calculated it out, appears to be an infinitely long number.
Did we Invent that Equation or Discover it? Suppose an Alien race invented it before us? Suppose another Alien race invented it before them and so on and so on. Ultimately, one must conclude than none of them actually invented it. It was Discovered. So for any race there can be the first to discover it, but none of them actually invented it. So if each race discovered it, it's logical that it pre-existed the discovery by any intelligent race. So when did it first qualify as existing? The first race to discover it? Logically, it's always existed, awaiting some race to discover it.
This would apply to all Math. All Math Existed at the very beginning of Time, waiting to be discovered. And if every Equation existed since time began, then so did all their respective solutions. The Equations and Solutions are Mathematical Truths.. and are therefore Timeless.
The Absolute Void:
In my search for understanding our Reality, I run into a lot of Obstacles. What is Time? Did Time have a beginning? What was happening before Time began? What is Matter and Energy? How Small is Small? How Big is big? Is the space between stuff empty or full of something?
For example: What is Matter? We already know that the more we look into ever increasing smallness, that everything we find is made of still smaller stuff. Where does it end? How small is smallest? Is there some ultimate basic particle or stuff at the smallest scales? What's it made of? Where did it come from and how long has it existed?
These such questions led me to take a stand that rids us of all these questions: The Absolute Void.
In the absolute Void there is no Time, no Energy, no Matter, no such thing as Distance, Size or Dimensions, no Scale, no Change. Any physical term that describes Existence is absent there. It is the only true place that actually Exists. Don't think of a huge empty space.. because that now implies size and distance. It has neither.
Now that I have an absolute Nothing.. can we get a Universe from it? YES! Because the one thing we can't exclude is Truth, because Truth has no Physical Properties. As stated in my opening.. Mathematical Truths are Timeless and Truths don't occupy any Space. They simply Exist. And that's all we really need.
If we go back to Pi (3.14159...etc) we may notice something.. it has sequence. If you were a smart observant Number located in that string of numbers then you might notice that there are other numbers that preceded you and followed you. In those that follow, the string of numbers may extend to an infinite length (the Future) but if looking towards the beginning, it had a definitive beginning as the Number 3. So for that little simple Universe of Pi, the sequence had a beginning and thus Time (from its perspective) had a beginning, even if it has no end in the other direction (future).
That same would apply to any Equation that produces a growing Sequence. It had a first state or beginning. Time is Sequence.
Of course Pi is a simple little one dimensional Universe that has no choice to be other than it is. But we are not limited to just one Equation. We are limited by an Infinite number of Equations. I should also mention that Math is not limited to numbers alone. For example, suppose we had grid where the top is numbered by digits of Pi and the left side is numbered by digits of Pi forwarded by 3 places. Now for every box in the Grid put a True or False if the Top number is greater or lesser/equal than the Side number. You will get an initial arbitrary pattern of True and False to fill in the grid.
Now put an empty grid on top of that 2D grid and look at the True/False value under that layer and the neighbors of a specific grid square. Make up your own rules on how the neighbors from around and under a grid square will influence the True of False status of each new blank layer added. Some rules you pick may cause the pattern to fill in or empty out. Some will produce an endless variety of T/F patterns. After enough layers are added, patterns may begin to emerge that have interesting aspects. Patterns may Oscillate or Move or may create Waves of patterns. This is called a Cellular Automaton and is just one of many branches of Mathematics.
We are adding new blank layers on top of each previous layer to allow change, otherwise known as Time. This Automaton can be expressed as an Equation embedded with changes and growing towards its future. As it grows, even more Macro patterns may emerge, some being very persistent. These persistent moving patterns could be called the Geometry of Matter. The rules you made up could be called Energy. Some of these can become very spectacular, with patterns that behave as Particles exchanging waves of information with other Particles with patterns emerging with the behavior of Quarks and almost anything, eventually Atoms may even emerge. In this case as all below, everything is nothing more than interactive information following a set of basic Rules you made up creating a Pattern Geometry.
At this time we must realize that this is a Solution for an Automaton based on a set of rules. Thus we must also realize that such a solution Exists for every possible set of Rules. It goes without saying, but I'll say it anyway, that the vast majority of such Mathematical Universes will never amount to anything. Just plain old Junk. But some will be awesome and highly complex, even with simple rules for the relationships of the Basic Cells.
How much space does one of these Automatons take up? None! They are composed of Information only and take up no more space than Pi. Each has a beginning and that's where its concept of Time having a beginning is derived from. Its size is determined by it own definitions and can grow to Infinity, while still never taking up any Real Space.. because in the Absolute Void.. Real Space, Size and Distance doesn't exist. So since they take up no Real Space, there is room for an Infinite number of Universes, each with its own set of rules.
There is no arbitrary limit on their complexity. Any intelligent life that forms in such a Universe would notice that everything is stepped, because it is made of Cells in all directions. It is growing constantly because it needs new blank layers for the patterned Geometries to propagate onto. Depending on the Equation, it can have any number of Dimensions, such as we have at least 4 dimensions in this Automaton we Exist inside of. From inside our Automaton, we have a beginning of Time and a minimum Cell Size called the Planck Length. This Minimum Cell size establishes a Scale for all things in the Larger Scales. That way all Atoms are of compatible and controlled size.
Ok, I'm running out of time now.. I may add more later.. I hope this gives Vivian a grasp of what a Mathematical Universe entails.