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.

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.

II. THE MATHEMATICAL UNIVERSE HYPOTHESIS

A. The External Reality Hypothesis

There exists an external physical reality completely independent of us humans.

Indeed, adherents of the Copenhagen interpretation of quantum mechanics may reject the ERH on the grounds that there is no reality without observation.

Our most successful physics theories to date are generally regarded as descriptions of merely limited aspects of the external reality.

In contrast, the holy grail of theoretical physics is to find a complete description of it, referred to as a "Theory of Everything",

Put differently, such a description must be expressible in a form that is devoid of human "baggage", e.g "particle" and "observation"

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.

However, these cases where the arrows are well understood form a minority. Deriving biology from chemistry or psychology from biology appears unfeasible in practice.

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.

Familiar examples include the integers and the real numbers.

C. Implications for a Mathematical Universe

2. Something that has a baggage-free description is precisely a mathematical structure.

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.

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

I know of no other compelling explanation for this trend than that the physical world really is completely mathematical.

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.

IV. IMPLICATIONS FOR SYMMETRY, INITIAL CONDITIONS, RANDOMNESS AND PHYSICAL CONSTANTS

A. Symmetry

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.

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.

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.

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.

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.

Because of gravity, clumpier distributions are exponentially more likely.

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.

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.

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.

A. Are we simulated?

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.

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.

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 concept of dark matter should be handled very critically because it depends on visibility which is a human quality.
• The concept of light should also be handled with care. Common understanding is that the speed of light is constant, but the fact that we cannot observe a blackhole makes this tricky. To "solve" this issue by introducing a local reference frame IMO only moves the problem forward, but does not solve the issue.
• The similar problem exist with temperature related to the Cosmic Microwave Background Radiation. This radiation are photons which is measured as a frequency. That means we have an average temperature (frequency) and a fluctuation.
• Schrodingers Cat (which is discussed at page 14) also depends about human observations capabilities and should be considered very critically.
  The physics involved about Schrodingers Cat completely depends about the half-life of a radio active element. You need a Geiger Counter to establish the half-life. When the half-life is 1 minute and when your experiment releases a poison when the element decays, you know for sure that the cat will be dead if you open the box after 1 hour. So what is the point. Anyway this is no prove that there are parallel universes which the article starts you to believe.
• The Schrodinger Cat experiment in a simpler form looks like this:
  You ask a person to go in a room which contains 16 chairs and you ask him to sit down on one chair. He goes in and you close the door.
  Next you make the following proposition:

  "Before I open the door and I look in the room the person is a superposition of 16 states: he sits on chair 1 and he sits on chair 2 and he sits on chair 3 etc.".

  IMO such proposition does not make sense.
  Within the context of Schrodingers Cat, the the state of the cat (Before the obeserver opens the room) is defined as: The Cat is both live and dead.

  For a more detailed discussion about the same issue goto: The Chemical Universe

The Mathematical Universe - This does not make sense

• 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?

For more about the CMB radiation read this: Friedmann's equation - Question 13
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