## The Mathematical Universe - by Max Tegmark 2007

This document contains comments about the article "The Mathematical Universe" - by M.Tegmark 2007. For a copy select: http://arxiv.org/abs/0704.0646.

The document is of phylosophical nature and is very interesting because he discusses (free interpretation) the rules to describe the physical reality. At the same time it also raises many questions.
The document starts with the sentence:
I explore physics implications of the External Reality Hypothesis (ERH) that there exists an external physical reality completely independent of us humans.
The issue is when you want to describe the physical reality you should do that based on observations (as much as possible) independent of the human point of view. For example in order to describe the colour red of light you should use the words photons and frequency. See also: Reflections
Next he writes
I argue that with a sufficiently broad definition of mathematics, it implies the Mathematical Universe Hypothesis (MUH) that our physical world is an abstract mathematical structure.
IMO the physical reality can not be a mathematical structure. What we want is to describe (understand) the present state and the processes involved in the evolution of the Universe. In order to do that we will use mathematics.

### I. INTRODUCTION

The idea that our universe is in some sense mathematical goes back at least to the Pythagoreans, and has been extensively discussed in the literature. Galileo Galilei stated that the Universe is a grand book written in the language of mathematics.
The point is that only a small part of all physical processes can be acurately described by some form of mathematics. Most of what is happening can only be described in a very general sense, based on approximations.

### A. The External Reality Hypothesis

There exists an external physical reality completely independent of us humans.
Such a physical reality does not exist. There exists only a physical reality and we humans are part of it. As only explained above: You should try to describe this reality independent of human point of view.
Indeed, adherents of the Copenhagen interpretation of quantum mechanics may reject the ERH on the grounds that there is no reality without observation.
What is missing is a clear explanation what the link is between the first part of this sentence and the final part.
The issue is that there exists an reality and that you need observations in order to describe this reality. You do not need the Copenhagen interpretation, nor quantum mechanics nor ERH to accept/understand this.
Our most successful physics theories to date are generally regarded as descriptions of merely limited aspects of the external reality.
That is true and that is all what we will ever have.
In contrast, the holy grail of theoretical physics is to find a complete description of it, referred to as a "Theory of Everything",
All what we is describe the physical reality as accurate as possible. We will never find a complete description. It sounds like claiming: all people are beautifull.
Put differently, such a description must be expressible in a form that is devoid of human "baggage", e.g "particle" and "observation"
There is nothing wrong with the words particle and observation in order to describe the physical reality.

### B. Reducing the baggage allowance

All these theories have two components: mathematical equations and "baggage", words that explain how they are connected to what we humans observe and intuitively understand.
The so called "baggage" words are important because they describe the objects involved. The word intuition should not be used to describe the reality.
See Figure 1:
However, these cases where the arrows are well understood form a minority. Deriving biology from chemistry or psychology from biology appears unfeasible in practice.
That is correct. But what is the point. The concept of a family tree to link different theories is rather adhoc.
The starting point of understanding our physical reality is chemistry, which includes nuclear physics, electromagnetime and astrophysics. Chemical reactions are the starting points of the changes in nature.
However, could it ever be possible to give a description of the external reality involving no baggage? If so, our description of entities in the external reality and relations between them would have to be completely abstract.
A mathematical structure is precisely this: abstract entities with relations between them.
You can not describe the physical reality only with mathematics. Again you must desribe the objects involved.
Familiar examples include the integers and the real numbers.
You should describe the physical reality only by postive integers.

### C. Implications for a Mathematical Universe

2. Something that has a baggage-free description is precisely a mathematical structure.
Mathematical structures pure sang have no physical meaning.

### D. Description versus equivalence

If a future physics textbook contains the Theory of Everything (TOE), then its equations are the complete description of the mathematical structure that is the external physical reality.
The problem with the word TOE is that there exist no general accepted definition. My definition is as good as any one else. My prediction will be that a book that contains a TOE has no predictive power.

### E. Evidence for the MUH

This predictive power of the mathematical universe idea was expressed by Dirac in 1931: "The most powerful method of advance that can be suggested at present is to employ all the resources of pure mathematics in attempts to perfect and generalize the mathematical formalism that forms the existing basis of theoretical physics, and after each success in this direction, to try to interpret the new mathematical features in terms of physical entities
Dirac makes a clear distinction between theoretical physics and physical entities.
I know of no other compelling explanation for this trend than that the physical world really is completely mathematical.
What Dirac writes is not in agreement with this claim.

### III. PHYSICS FROM SCRATCH

Suppose we were given mathematical equations that completely describe the physical world, including us, but with no hints about how to interpret them.
Those mathematical equations don't exists so what is the point. If you start with this proposition (thought experiment) which is not realistic all what follows is not realistic.

### B. Initial conditions

The MUH profoundly affects many notions related to initial conditions. The traditional view of these matters is summarized as splitting our quantitative description of the world into two domains, "laws of physics" and "initial conditions". The former we understand and hail as the purview of physics, the latter we lack understanding of and merely take as an input to our calculations.
The physical world (part of) can be described by differential equations, parameters(constants) and initial conditions. Both parts are essential. An example is Newton's Law: F = G*m1*m2/r^2. The parameters are masses m1 and m2 of the objects involved. The initial conditions are positions and velocities involved to describe the initial state of the system described or simulated.
Using this methodology you can make testable predictions. Without those you cannot.

#### 1. How the MUH banishes them

In other words, this would entail a crushing complete defeat of fundamental physical laws. However, contrary to how it may at first appear, it would not constitute a victory for initial conditions in the traditional sense.
IMO (sorry again) this does not make sense.

#### 4. Cosmic complexity

There is an active literature on the complexity of our observable universe and how natural it is. The current consensus is that the initial conditions (shortly before Big Bang nucleosynthesis, say) of this comoving volume of space (our so-called Hubble volume) were both enormously complex and yet surprisingly simple.
Words like complex and simple should be used with care. Something that is complex can not be desribed in a simple way. Either what you describe is not complex or your simple description is not accurate.
To specify the state of its approximate 10^78 massive particles (either classically in terms of positions and velocities, or quantum-mechanically) clearly requires a vast amount of information.
Why would you do that? How many massive particles requires the state of one massive particle?
Yet our universe is strikingly simple in that the matter is nearly uniformly distributed on the largest scales, with density fluctuations only at the 10-5 level.
The concept of Galaxies and Galaxy Clusters show that the Universe is not simple.
Immediately followed by:
Because of gravity, clumpier distributions are exponentially more likely.
Gravity is part of the description of the movement of stars in a galaxy. Gravity can not be used to predict galaxies in the first place.

#### 5. How complex is our world?

If this hypothesis is true and the bird complexity is much lower than the frog complexity, then it implies that the mathematical structure describes some form of multiverse, with the extra frog complexity entering in describing which parallel universe we are in.
This does not make sense because many words and concepts are not clearly defined.

#### V. IMPLICATIONS FOR PARALLEL UNIVERSES

Parallel universes are now all the rage. However, they are as controversial as they are popular, and it is important to ask whether they are within the purview of science, or merely silly speculation.
They belong to pure speculation.

### VI. IMPLICATIONS FOR THE SIMULATION ARGUMENT

Long a staple of science fiction, the idea that our external reality is some form of computer simulation has gained prominence with recent blockbuster movies like The Matrix.
Concepts like "some form of computer simulation" are extremely vaque and should not be used.

A simple: No.

### VII. THE COMPUTABLE UNIVERSE HYPOTHESIS

In this section, we will explore the consequences of augmenting the Mathematical Universe Hypothesis with a second assumption:
Computable Universe Hypothesis (CUH):
The mathematical structure that is our external physical reality is defined by computable functions.
By this we mean that the relations (functions) that define the mathematical structure can all be implemented as computations that are guaranteed to halt after a finite number of steps.
The issue is that the mathematics used to describe the physical reality are differential equations. This fact alone does not support the idea that the Universe is computational.

### VIII. CONCLUSIONS

#### B. The MUH and the Philosophy of science

If the MUH is correct, then a number of hotly debated issues in the philosophy of science are cast in a different light.
The second part probably is correct. I strongly doubt the first part
C. Outlook If the MUH is correct, if offers a fresh perspective on many hotly debated issues at the foundations of physics, mathematics and computer science.
The second part probably is correct. I strongly doubt the first part

### The Mathematical Universe - Reflection

The general question that the article tries to answer is: "What are the laws of nature."
M. Tegmark answers this question by heavily depending on mathematics. IMO this is wrong. What is true that you should consider the laws of nature independent (as much as possible) of human influence.
For Example:

### The Mathematical Universe - This does not make sense

In the above article I use the sentence: "this does not make sense" meaning: I strongly disagree. The Universe, everything around us, is a process in which changes and events take place. Those changes are chemical reactions and movements, rotations. To describe those changes relatif simple methematics is involved. Events like explosions require more complex mathematics but can only be described in more general terms.
• The article contains a lot of mathematics specif see Appendix A. At face value nothing is wrong with this mathematics. The problem is you can not can not study mathematics only and than claim something about any physical problem small or large. That is the wrong way around. The first steps is measuring. The second step combining all the measurements to gether and trying to find relations between these measurements. First relations between the same physical parameters and secondly the relation between different parameters. It is here that mathematics becomes involved. The simpler the better. In fact all what we are this discussing are physical processes, chemical reactions. That means all what we need at small scale is the mathematics to describe these processes. Mainly integer values. At larger scales we need differential equations because concepts like averages and densities become important. Only when you want to describe an exploding star the mathematics becomes realy heavy but strange as it sounds you do not need for example complex numbers.
• The article also discusses issues like parallel worlds and multiverses. See for example Figure 3 page 12 and page 12 in general. I strongly disagree. The major problem is a clear definition. Science depends on experiments on observations. How can you demonstrate that something has happened in a parallel world?

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Created: 23 Januari 2013