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 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
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.
Next he writes
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 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.
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.
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
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.
There exists an external physical reality completely independent of us humans.
What is missing is a clear explanation what the link is between the first part of this sentence and the final part.
Indeed, adherents of the Copenhagen interpretation of quantum mechanics may reject the ERH on the grounds that there is no reality without observation.
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.
That is true and that is all what we will ever have.
Our most successful physics theories to date are generally regarded as descriptions of merely limited aspects of the external reality.
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.
- In contrast, the holy grail of theoretical physics is to find a complete description
of it, referred to as a "Theory of Everything",
There is nothing wrong with the words particle and observation in order to describe the physical reality.
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
The so called "baggage" words are important because they describe the objects involved. The word intuition should not be used to describe the reality.
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.
See Figure 1:
That is correct. But what is the point. The concept of a family tree to link different theories is rather adhoc.
However, these cases where the arrows are well understood form a minority. Deriving biology from
chemistry or psychology from biology appears unfeasible in practice.
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.
You can not describe the physical reality only with mathematics. Again you must desribe the objects involved.
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 should describe the physical reality only by postive integers.
Familiar examples include the integers and the real numbers.
C. Implications for a Mathematical Universe
Mathematical structures pure sang have no physical meaning.
2. Something that has a baggage-free description is precisely a mathematical structure.
D. Description versus equivalence
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.
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
Dirac makes a clear distinction between theoretical physics and physical entities.
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
What Dirac writes is not in agreement with this claim.
I know of no other compelling explanation for this trend than that the physical world really is completely mathematical.
III. PHYSICS FROM SCRATCH
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.
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
B. Initial conditions
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.
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.
Using this methodology you can make testable predictions. Without those you cannot.
1. How the MUH banishes them
IMO (sorry again) this does not make sense.
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
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.
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.
Let us contunue reading.
Why would you do that? How many massive particles requires the state of one massive particle?
- 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.
The concept of Galaxies and Galaxy Clusters show that the Universe is not simple.
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.
Immediately followed by:
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.
Because of gravity, clumpier distributions are exponentially more likely.
5. How complex is our world?
This does not make sense because many words and concepts are not clearly defined.
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
They belong to pure speculation.
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
Concepts like "some form of computer simulation" are extremely vaque and should not be used.
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?
A simple: No.
VII. THE COMPUTABLE UNIVERSE HYPOTHESIS
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.
In this section, we will explore the consequences of augmenting the Mathematical Universe Hypothesis with a
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.
Computable Universe Hypothesis (CUH):
The mathematical structure that is our external physical reality is defined by computable functions.
B. The MUH and the Philosophy of science
The second part probably is correct. I strongly doubt the first part
If the MUH is correct, then a number of hotly debated
issues in the philosophy of science are cast in a different
The second part probably is correct. I strongly doubt the first part
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 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.
- 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.
- Schrödingers Cat (which is discussed at page 14) also depends about human observations capabilities and should be considered very critically.
The physics involved about Schrödingers 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 Schrödinger 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 Schrödingers 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
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
For more about the CMB radiation read this: Friedmann's equation - Question 13
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