1. Michael P. Frank mpf@cise.ufl.edu
A Mathematical Theory of Existence
Michael P. Frank
CISE Department
University of Florida
Hi, I’m Michael Frank, and I’m one of the newer faculty members in the CISE department here at UF. I only got my “Doctor of Philosophy” degree a couple of years ago, but I’ve been somewhat of an amateur philosopher my whole life. Today, I’m going to present to you some of my own personal philosophy regarding some the biggest philosophical questions, which I have developed over a period of many years. Similar ideas have also been discussed recently in books by various authors [cf. Tipler’s Physics of Immortality, Kurzweil’s The Age of Spiritual Machines], but I’m not yet aware of anyone who takes the quite the same overall philosophical approach as myself.
I asked Rob if I could talk here today because I think that a lot of these ideas may resonate well with this particular group of hopefully open-minded individuals. But, you may be surprised where some of this leads.
The title of this talk is “A Mathematical Theory of Existence,” but don’t be scared off: This talk isn’t really going to contain any mathematics per se. Rather, it is a philosophical theory that rests, on its foundation, on some basic concepts from the philosphy of mathematics and computer science.
A talk presented to UF’s Atheist/Agnostic Student Association (AASA) Wednesday, November 7, 2001

2. Overview of Talk The big questions
The Mathematical Existence Postulate
Implications of the theory:
Existence of the Universe
The success of Ockham’s razor
What sorts of deities are possible
A meaningful goal for civilization
Discussion
Here’s an overview of the talk. First, we’re going to list some of the big philosophical questions, really the biggest we know of, which we are going to try to address in this philosophy.
Next I’m going to introduce the core principle of this philosophy, which is something I call the “Mathematical Existence Postulate.” The whole philosophy depends on it critically, so I’m going to outline the arguments of why we should think this postulate is correct.
Next, I’m going to talk about some of the logical implications of this postulate for philosophy, when one takes into account current scientific knowledge. We’ll address the implications for these issues (refer to bullets) and others.
Finally, I’ll be happy to stay around for discussion after the talk. We can also discuss various points as they arise during the course of the talk. Feel free to interrupt with questions.

3. The Big Questions Why is there something, rather than nothing?
Why is it that we can comprehend the universe through science?
Is there a God or gods?
If so, what are its/their characteristics?
What is our fate after death?
What is humankind’s destiny?
OK, so here are some of the really big questions we’re interested in answering. The third and fourth ones hit at one of the main interests of this organization.
Amazingly, starting from extremely simple premises and some deduction informed by results in mathematics, science, and computer science, we can form rationally justified answers to all of these questions. There are even indications that further mathematical & technological advances may enable the study of these questions to be turned into a quantitative science.
Let’s start with the first of these questions, which is perhaps the simplest, most direct and compelling. Some have claimed it is forever beyond the scope of science. But I will argue otherwise.

4. Why is there something? Traditional form: Modern variants:
“Who created the world?”
Modern variants:
“What caused the Big Bang?”
“What was there before the Big Bang?”
“Why does the abstract universe described by physical law really exist physically?”
All these questions are ill-conceived!
We’ll see why...
Of course, historically this question has been around since the dawn of humanity. When our primitive forebears first started making tools and fashioning crude implements, they must have quickly noticed that good artifacts required an intelligent maker, and so if the world was an artifact, it must have a maker - who was he? Where was he? This was probably the earliest motivation for the conception of a God or gods.
Of course, nowadays people like to think of ourselves as being much more sophisticated. Modern cocktail-party philosophy is more likely to phrase this same question in other terms that the questioner thinks are more scientific. But I will argue that all these variations on this question are totally ill-conceived, that they “just don’t get it,” philosophically speaking, and that they are all completely swept aside in a more mathematically mature view of philosophy.
This entirely mathematically based philosophy rests most crucially on a single premise, which I’ll now present and justify

5. The Mathematical Existence Postulate (MEP)
Logical truths & mathematical structures exist a priori, independently of anything else.
Postulated to be true in “reality”:
What does this mean?
Why should we believe it?
What can we deduce from it?
Answers to many big questions!
OK, here it is, and it’s important enough that I’ll give it a name so I can refer to it later. For those who don’t know, a postulate is a proposed statement (a hypothesis or premise) that purports to state an objective fact about reality. In physics, postulates are the statements that say that some abstract mathematical rule actually describes the real world. Here, we are saying something a little bit different. We are saying that these things - logical truths & mathematical structures (terms which I’ll define in a moment) have an independent, objective existence as mathematical entities. A priori means “in the beginning,” not in the sense of in the beginning time-wise, but rather, as a conceptual starting point, a foundational principle.
The basic idea here is not to dissimilar to Plato’s forms, but whereas he had forms for every idea or concept, even vague or ill-defined ones, we are only making a claim about mathematical objects, which by their nature are very precisely defined.
I’ll explain what this posulate is saying in a little more detail (such as what I mean by terms such as “truths,” “structures,” and “exists”), and then I’ll try to give you a little justification for why you (like many mathematicians) should agree that it’s true, or at least that you should grant, for the sake of argument, that it’s reasonably plausible.
Then I’ll talk about the conclusions that it seems you can draw from this principle, at least if you throw some pretty solid results from math, science, and CS into the mix. Amazingly, it seems that this simple principle leads to interesting and compelling answers to many of the big questions, questions that are often claimed to be beyond the bounds of rational thought.

6. What’s a “Logical Truth?”
A true conditional statement: that
if such-and-such axioms & rules of inference are held to be true,
then such-and-such consequence necessarily follows logically.
Many logical truths can be proven mechanistically (by algorithms)...
But, there are also infinitely many true but unprovable statements (Gödel).
Now, let me define the terms in the MEP a little more clearly. What do I mean by a logical truth? I mean a precisely defined mathematical truth. Well, a truth in mathematics is always a truth within some particular domain of reasoning that has been set up by giving some precise definitions, axioms, and rules of inference. So, any such truth is really of a conditional form, that is, it says that if one makes that particular set of definitions, and adopts those particular axioms and rules, then such-and-such statement does not fail to be satisfied within the domain of discourse so defined.
The fact that truths are conditional statements is important, because it means that the truth of the statement does not depend on whether the domain in question “exists” or is “real” or not (whatever these mean). A logical truth is true, regardless, because it just says that if the described context were to exist, as described, then it would be logically compelled to have such-and-such property. This conditional status is what gives truths an “a priori” status that is independent of whatever else you accept in your ontology.
Now, many truths can actually be proven via a mechanical process of applying inference rules repeatedly until one derives the truth from the axioms. For a long time, mathematicians hoped that any statement could eventually be proved or disproved. But, Kurt Gödel (pronounced “girdle”) showed that this was not so, that in any reasonably expressive domain of mathematical discourse, there are an infinite number of true statements within that domain that cannot be proven from the set of axioms provided. No finite or even infinite-computable set of axioms can suffice. In other words, there will always be facts (about any mathematical system) that, although they may be objectively true (not undetermined), they cannot be proven. An example might be Goldbach’s conjecture that every even positive integer is the sum of two primes. It may just happen to be true, but have no proof.

7. What’s a “Mathematical structure”?
Any entity that can be well-described within some logical/mathematical axiom system. Examples:
Discrete structures: Strings, graphs, groups, models of computation, mathematical theories
Continuous structures: Functions, fields, fractal sets, topological spaces, etc.
Mathematicians speak of a structure “existing” if its properties are consistent within the axiom system it’s defined in.
Anyway, Gödel was kind of an tangent, which it turns out is not really too relevant to our discussion, so let’s go on… The statement of the MEP earlier also referred to mathematical structures. These are simply the objects of mathatical descriptions and statements. Any entity that you can give a precise, definitive description of in mathematical terms, and where its definition is self-consistent, is said to “exist” in mathematics, in the domain of discourse of a particular set of definitions & axioms. In mathematics we encounter both discrete and continuous structures, although as an interesting aside, it is conceivable that it could someday be proven that some or all of the continuous structures aren’t self-consistent, and don’t actually exist. (I have a sketch of a potential proof that the famous continuous structure R, the set of all real numbers, does not exist, if anyone is interested in fleshing it out.)

8. Ontology: Picture #1
An infinity of (disconnected) mathematical structures of different sorts...
OK, so this is the particular picture of ontology (the study of what exists) that we have come up with so far. There is an infinity of mathematical objects (structures) that we can define, and they exist, with all of their logically true properties. The simple picture is that they exist independently, separately from each other; later we’ll refine this.
In this particular I have suggestively represented the different structures as geometric objects with different shapes and colors. Of course, geometric entities are but one of many kinds of mathematical objects. The colors and shapes portrayed here are not intended to be meaningful, but just to suggest that different mathematical objects have different true properties.
The positions, orientations, and sizes portrayed are even less meaningful; the objects of math are not (in general) actually together in a space with spatial relationships with each other. The sizes shown are arbitrary, and the locations and orientations were chosen so that the shapes don’t overlap, so as to suggest that all the objects are separate and independent of each other.

9. Complex Structures can have Simple Descriptions
Benoit Mandelbröt’s fractal set:
John Conway’s Game of Life:
Now, the next important point I want to make is that a mathematical object is more than just a collection of the statements that you use to define it. An object with an extremely simple and concise definition may have an infinitely rich, complex structure of logically-true properties. Let’s see a couple of examples.
Benoit Mandelbröt (ö is pronounced “ur”, but I’m going to just say “Mandelbrot”), an applied mathematician (check this) at IBM Research discovered (I think it was in the early 1970s) that a certain extremely simple rule generates a structure of enormous beauty and infinite depth, detail and complexity. This object is now called the Mandelbröt set. The rule is simply that for any point c on the complex plane, you start with z=0 and repeatedly square z and add c, and see whether the result goes to infinity or not. Counting the time until the sequence of numbers exceeds a certain size permits coloring the plane using an artistically chosen pallette. The choice of colors is arbitrary, but the shapes are inherent to the structure itself.
As another example, mathematician John Conway discovered that the following simple rule (“The Game of Life”), applied on a grid of binary cells, generates an enormously complex behavior. A cell is on if, at the previous time step, it had exactly 3 “on” neighbors, or was “on” itself and had exactly 2 “on” neighbors. Applying this rule to even very small & simple intial patterns generates complex & growing patterns with clusters of cells moving around, growing, generating other patterns, etc. It was eventually proved that this system is Turing universal, which means that any computation of any kind of structure can be embedded within it. For example, you could create a simple life pattern that generates (gradually, over infinite time) all provable truths of mathematics.
Infinite depth + detail
Universal computational capability

10. Why should we believe these things exist a priori?
They don’t depend on human language:
The same underlying truths & structures can be described in any (Turing-complete) language.
They don’t depend on human thought:
The same truths & structures could be discovered by any intelligent entity.
(Crucial) They don’t depend on any being, God, or universe existing:
A valid conditional “if this, then that” is true regardless of whether anyone exists to think about it! It could not be false, no matter what.
OK, now I want to convince you of the truth of the MEP, or at least, make you inclined to consider it to be a reasonably plausible hypothesis. I claim that these math objects, numbers and fields and Mandelbrot sets and Life histories, have an independent existence. Here is why.
First, it’s clear that these objects don’t depend in any way on the accidents of human language, because mathematicians & computer scientists have learned that all sufficiently powerful languages are equivalent in terms of what they can express, and so the very same underlying truths can be expressed and derived, and the same structures defined and described, in any language. Our particular language may have influenced which structures mathematicians have chosen to define and explore the properties of so far, but, using any language, one can methodically explore structures in such a way that any structure will eventually be explored. The order of exploration may depend on circumstances such as language, but the total collection of all entities (truths & structures) does not.
Similarly, these things don’t even depend on human beings existing at all, because a hypothetical alien across the galaxy could (if he has a Turing-complete language capability) explore the exact same structures, and would do so if he were exploring all structures methodically.
Finally, and this is perhaps the most crucial point, the truth of these truths and the existence of these structures does not depend on any intelligent beings, Gods, or even universes existing, because remember, the truths are just conditionals that say, if such-and-such, then so-and-so; even if nothing exists, it’s still true that if such-and-such definitions were defined, then consequence so-and-so would follow. And as for the structures, even if nothing is “real” or exists physically, a definition of some object still logically implies its properties.

11. What sorts of questions can we attack using the MEP?
Questions of ontology:
What exists? (And why?)
Questions of epistemology:
What can we know about the universe?
Why does science work so well?
Questions of theology: Is there a God? Etc.
Questions of eschatology:
What’s in store for us, our civilization, & the universe? (Some goals, at least.)
Maybe that was enough to convince you of the MEP, and maybe not. But the more mathematics you encounter, the more you get a visceral feel that these things are there, that they’re real, that you’re exploring them, not creating them.
Anyway, for now, let’s just presume that the MEP is true, and look at the questions we can address, using it in the background. It seems that all these questions and more can be addressed, although for some of them, we will need to rely on some results of science and computer science.
Ontology is the study of what exists, and why.
Epistimology is the study of knowledge, of what we can know about things, and why. In particular, we’d like to understand why the universe is, seemingly so fortunately, amenable to scientific explanation and understanding.
Of particular interest to this audience (Atheist & Agnostic Students Association) are the questions of theology: Is there a God or gods, and if so, what are their abilities and properties?
Eschatology is the study of “the end”, of death and the end of the world and/or the universe. In this MEP-based philosophy, we can say some things about death, and about some goals and potential futures for civilization and for life in the universe.

12. What Exists?
By MEP: At least, any self-consistent mathematical structure exists.
Physics suggests our universe is exactly described by a quantum wavefunction
uniquely determined by certain differential equations & boundary conditions
but, w. many details TBD
This wavefunction is a consistent math structure, which must exist by the MEP.
First, ontology. The MEP tells us directly that any self-consistent math structure exists, at least in the mathematical sense.
The overwhelming evidence of the enormous body of experimental and theoretical work that is modern physics teaches us that our universe (and its entire past & future history) is probably exactly and completely described by a mathematical object of a certain type called a “quantum wavefunction.” Although physics is not complete and the exact form of the laws defining this wavefunction is not yet fully known, we know enough to guess that this object is probably uniquely determined by a set of differential equations (if you don’t know what these are, just think “equations”) and boundary conditions, such as the initial state. It even looks as if the initial state can have a small, finite description. Suppose for a moment that we are on the right track, and the entire wavefunction is defined by an initial-state description plus a few equations.
Now, here’s the interesting thing. This wavefunction is a consistent mathematical object, and so by the MEP, it must exist, in this a priori way that is independent of anything. And yet, this object contains every detail of our universe within it. This is very interesting.
Now, after learning that the universe is seemingly just the solution to an equation, the next question asked by the average physicist (who in my opinion is still philosophically somewhat naïve) is this...

13. Why does the Universe of physics exist, physically?
Answer: The question is ill-conceived!
Consider a given complex structure that contains intelligent entities within it:
Nothing about the entities’ thoughts or behavior can depend at all on whether their structure “exists physically” or not!
There can be no evidence for “physical” existence of one’s own universe - The very concept is a chimera!
He thinks, “OK, this wavefunction exists as a mathematical object. But, why does it exist physically? Why is it real & substantial? Why does the universe it describes surround us, and its Earth solidly support us, as opposed to being just an insubstantial, imaginary abstraction which doesn’t do anything? Why does anything exist (physically) at all?”
I argue that here, the physicist (and others who wonder this) are going way off-base, missing the whole point, making an unjustified leap. The entire question rests on an invalid, unjustified, but normally unrecognized preconception. They are all missing what to me is (now) an obvious, simple, self-evident truth…
To see the light, consider this: When we ask “Why does the universe exist physically?”, consider just for a moment, what is our evidence that it does? Before laughing, think of this: If science is right, then every detail of us and our experiences are there in the wavefunction describing our universe. Mathematical structures can contain (all the details of) intelligent entities.
But now, consider this: If this is correct, then nothing we observe, nothing about our thoughts and behavior can possibly depend on whether the universe exists physically or not. Consider if the universe were just a math object that didn’t exist. Yet, it still contains every detail of our thoughts and experiences. That means, no feature of our thoughts and experiences can possibly be construed as evidence for our universe’s physical existence. Furthermore, no conceivable observation or experience could ever constitute evidence of our universe’s physical existence. The very concept is a chimera! Physical existence is inherently unverifiable! It makes no difference at all to anything! The concept is meaningless! The answer to the question, “Why does the universe (or anything) exist at all, as something more than just a math object?” is just:
IT DOESN’T!!!!!!!
More precisely, I should say: Since there can never be evidence for any other level of existence besides mathematical existence, then by Occam’s razor (prefer simpler explanations) we should do away with physical existence and adopt the simpler model that there is only one kind of existence, the mathematical one. This is a strong argument, but not a rigorous proof, that no other kind of existence “exists.”

14. What is the Universe?
Our Universe is just (no more than) a very complex mathematical structure that happens to have a fairly simple underlying description (laws of physics).
It exists for one simple reason only: Any such structure logically must exist,
assuming the MEP is true.
Later, we’ll get to why it does have a simple underlying description.
Whoa, we don’t exist? What happened to, “I think, therefore I am?” But wait, remember, I am only saying that things don’t exist other than as mathematical entities. So, anything that exists, exists because it is a mathematical entity, and it exists in the mathematical sense.
So, if you ask what is the universe? We should say it is (not just “it is described by”) some complex mathematical structure. Physics teaches us that all the apparently complexity of this structure really just boils down to the logical consequences of a very simple underlying description, namely the laws of physics. However, even if there are bad mistakes in physics, or new phenomena are discovered, that would only be evidence that we have the wrong mathematical structure. But it would not be evidence that the universe is not a mathematical structure. In fact, no observation could ever constitute evidence that the universe is not just a mathematical object - some structure will always be consistent with all the evidence (though it may be a complex, contrived one).
The philosophy that the universe is some mathematical object is thus unfalsifiable. This would be a strike against its status as a physical theory, if not for the fact that the MEP makes all math objects exist anyway, and so the identification of our universe with a suitable math object is always simpler than the alternative where there is “something more,” some extra ingredient of “realness” not in the math. I take Occam’s razor as prior to the falsifiability requirement for evaluating physical theories. Also, note that identifying the universe with any particular math object is falsifiable (since such identification makes specific predictions).
One wonders why it turns out that our universe has a simple underlying description rather than a complex one. We’ll address this in a moment.
As for why the universe exists, this is for one simple reason only: Any self-consistent mathematical object logically must exist, based on the MEP.

15. Generic definition of a “universe”.
Any mathematical structure containing a natural time-like dimension.
Examples:
The structure described by physics.
Successive iterations of the Mandelbröt set calculation.
The game of life, starting from any initial state.
The execution history of any algorithm.
Now, we know what our universe is, but for purposes of later discussion, let’s define “a universe” in a more generic sense which will let us speak of “other universes.”
For our purposes, let’s say that a “universe” is any mathematical structure containing a natural time-like dimension. A time-like dimension is a way of carving a structure into “slices,” together with a linear ordering of the slices, such that the state (or detailed description) of a slice is always determined by the state of the previous and/or next slice. (Even nondeterministic universes obey this definition if the slices are taken as probability distributions over states.)
With this definition, the universe described by physics (including quantum mechanics & general relativity) is probably a universe (although depending on how the unification of QM and GR works out, the definition of universe may have to be loosened a little).
The history of sucessive generations of the game of life is a universe.
The calculation of the Mandelbrot set can be viewed as a universe consisting of a two-dimensional surface wiggling through a four-dimensional space, with a discrete time dimension.
If one takes any algorithm (computer program), and considers its total execution history starting from any (definable) initial state, that’s a universe.

16. Embedding of Structures
A mathematical structure can be embedded (represented, encoded) within another structure.
E.g., A Mandelbrot set in a computer
Multiple, recursive embeddings allowed:
Example: A universe in a Turing machine in a Life game in a computer program in a computer circuit in a universe…
There may even be infinitely many embedded structures, & infinite levels of embedding...
Now, here’s an important concept. As mathematicians and computer scientists well know, one mathematical structure can be embedded (by which I mean represented, encoded, described, contained, etc.) within another structure. For example, we can embed the structure of number theory within the structure of set theory. We can represent a Game of Life state or a Mandelbrot set (at least, a discrete approximation) within a computer program.
Furthermore, you can have many structures embedded in one, and one structure may be found embedded in many others. You can have recursive embeddings; for example, a universal wavefunction being simulated in a Turing machine emulated on a Game-of-Life board being represented in a computer program running on a computer logic circuit embedded in a wavefunction universe, etc.
It’s even possible to describe infinite numbers of structures embedded within other structures, and infinitely many levels of nesting of one structures within another, or structures embedded within themselves, …
These realizations lead us to some important later conclusions…

17. Ontology: Picture #2
An infinity of mathematical structures, each embedded in many others; many are universes...
This realization about embeddings lets us elaborate our earlier ontological picture a bit. An infinity of different mathematical objects exist, but moreover, each object exists an infinity of times in an infinity of different ways, found in exact copies embedded one or more times within an infinity of other objects. So, for example, the orange oval on the upper left exists on its own (independently of other structures), but we can also find a copy of it embedded within the greenish-yellow square towards the upper right, and another copy in the dark purple triangle at bottom center. Each copy of the orange oval contains two separate copies of the blue oval, which also exists by itself, and there’s also a copy of the blue oval in the greenish-yellow square but separate from the orange oval. The greenish-yellow square (including all its contents) can also be found within the flesh-colored circle. As an example of an infinite recursive embedding, the magenta oval contains the blue triangle, plus another copy of itself, which contains another blue triangle and another copy of the magenta oval and all its contents, ad infinitum.
Also possible are approximate representations. For example, the flesh-colored circle contains an approximate copy of the red oval, which differs from the original in that it does not contain the nested copy of itself. (This approximate copy is technically a different object.)

18. Physical Existence: The only sensible definition
Any sentient being says that something “physically exists” if the thing happens to be within the same mathematical structure (universe) as the being itself.
By definition then, our universe physically exists for us, and other universes do not, except insofar as we can incorporate their structure within parts of our universe.
By the way, although earlier I said that physical existence is a meaningless concept the way people normally think of it (as a special kind of existence that is somehow stronger than mathematical existence), I’m now going to show that there is one definition of physical existence that is useful and meaningful. But it’s a relative definition, not an absolute one.
Normally, we say that something exists physically if we can interact with it in some way. So, let’s just define physical existence as follows: From the perspective of a particular sentient being (intelligent entity), something physically exists if it happens to be contained within the same mathematical structure (in our case, our universe) as the being itself.
So, from our perspective, our universe and all its contents exist physically for us, and other universes and math objects do not, except insofar as we can embed them or an approximation of them within parts of our universe (for example in a computer simulation).
However, there are other universes with other intelligent beings in them, and from the perspective of those beings, their universe and its contents exist physically, and ours does not.
You can think of it as somewhat of a democratic or equal-opportunity perspective on physical existence. A departure from the Universe-centrism of the past. Our Universe is just one of many universes, all of which have an equal claim to existence, and all of which seem equally real to their inhabitants.

19. Dispelling some more creation-related questions
“Who created the universe?”
As you can see, no “who” is needed; all valid structures must exist logically.
“What happened before the Big Bang?”
The question is ill-conceived because time itself (and “before” and “after”) only has meaning within specific universes.
The existence of mathematical structures (including universes) is itself timeless.
Let’s now return to some of the earlier creation-related questions, and see what makes them so ludicrous.
The first point is fairly self-explantory. All self-consistent structures must exist, so the universe requires no creator. However, notice that there is no requirement for all consistent structures to exist within our universe; so, if we want a copy of some structure to exist within our universe, then we generally do have to create that.
People who know just enough about modern cosmology to know there was a Big Bang often ask, “Well, what happened before the Big Bang?” This question is wrongly conceived, because the very concept of time only has meaning within specific universes. Mathematical structures themselves are inherently timeless; they just exist. “Eternal” would be the wrong word here, because it suggests the passage of infinite time. Outside of specific structures, there is no time. Further there is no relation between the time dimension in one universe and that in another.
One may think, “Well, any structure must have its details computed, and the order of computation gives it a time dimension.” But a given structure can in general be computed in many ways; you don’t necessarily have to compute it in the same order as its time dimension. One way to compute what happens in quantum mechanics is to work backwards & forwards through time repeatedly. Another way is to start with an approximate representation of all time, and successively make it more detailed. Einstein’s relativity teaches us that our universe doesn’t even have a unique time dimension; there are many, equally valid, directions (at different relative “angles” to each other) in which the time dimension can be drawn.
Similarly, there are many orders in which the collection of all computable math structures could be computed. Presumably, none is particularly special. All the structures just exist, they do not exist in any particular order in time.

20. Why does science work? A fundamental principle of science:
The enormous success of physics shows that the universe is highly structured & obeys relatively simple underlying laws.
Simple theories work in other fields also.
A fundamental principle of science:
“Ockham’s Razor” (paraphrased): The simplest explanation is most likely to be the right one. - Allows us to predict things.
Can we explain why this is true?
A fundamental principle of reasoning, called Occam’s Razor after Sir William of Ockham, who was the first to state something like it, is that simpler explanations of an observation or phenomenon are more likely a priori to be correct than are more complex ones. This principle is often quantified in computer-based automated reasoning systems, where it is sometimes called the minimum description length (MDL) principle. The prior probability of a hypothesis is sometimes taken to be 2-n where n is the length of the shortest description of the theory in bits, in an appropriate description system (more on this later). Note that more complex theories are not only more likely, but exponentially more unlikely, as they get longer. This particular assignment of probabilities to theories is correct if one imagines that correct theories are chosen at random from random infinite bit-strings. A particular string s of length n will appear at a random position in a random string with probability 2-n. The intuition here is that the more complex theory requires a larger number of independent assumptions to be correct. Independent probabilities multiply, so, the more assumptions, the less likely the whole theory (exponentially so with the number of assumptions).
Now, the interesting thing is that empirically, the universe actually seems to obey this logic. Time and time again, the scientific methodology of finding the simplest theoretical model that (with quantitative precision) explains observed phenomena has paid off with previously unexpected predictions that turned out to nevertheless be correct. This has seemingly held true has science has progressed downward to the innermost depths of physics.
When one does not know where the universe comes from, this seems mysterious. If the universe was created by God, why did he create one that follows simple rules, rather than a system so baroque and complex as to be utterly inexplicable? Indeed, the ancient theological imaginings of heaven, hell, angels, demons etc. seem to fall more towards the baroque than the simple; if God were like us, the universe would be similarly complex and imaginative. Why the simplicity found by physics?

21. Why is the razor surprising?
If all structures exist, there’s a version of our universe where tomorrow, the force of gravity just goes away!
Atmophere dissipates, the sun explodes…
What basis do we have for thinking we’re not in that universe?
Since it exists, some poor folks are in it.
Why are we blessed with stability?
Why not some complex/chaotic physics?
Now, even with the MEP explaining why the universe exists (all math objects exist), it might at first seem surprising that it is simple. After all, if all structures exist, then a version of the universe that contained a few “amendments” to the laws of physics must also exist. For example, one could define a version of physics in which everything is normal until a certain time after the big bang, say tomorrow, at which point spacetime suddenly flattens out (the force of gravity disappears). The Earth’s atmosphere would explode outwards into space at a thousand MPH or so, carrying any loose surface objects along with it. Other surface objects drift away from the surface more slowly. The Earth itself, if it does not explode, would probably bulge out rapidly, perhaps fragmenting into a number irregularly-shaped chunks. The sun would definitely explode right away, incinerating any survivors with million-degree plasma within eight minutes.
This version of physics is consistent with all physical observations to date; it’s just slightly more complicated. But if our universe exists and that one exists, how do we know we’re not in that one? After all, in the universe that’s exactly like ours aside from that amendment, some poor slobs (exactly like us) are there. Many less drastic variations of the universe are also conceivable, such as random objects appearing at random places for no apparent explanation. So, why are we blessed with a stable, simple universe? Why aren’t we in some more complex, chaotic variation?

22. Definitions of Complexity
The complexity of a structure can be fairly well-defined as follows:
The length of the shortest computer program that can generate that structure.
Other definitions have been proposed.
This particular definition is known as Kolmogorov complexity.
Can be shown to be fairly insensitive to one’s choice of programming language.
The minimum description length principle gives us a hint. One can quantify the complexity of a structure as the length of the shortest computer program that can generate the structure; this is called its Kolmogorov complexity or K-complexity. It can be shown that the K-complexity is fairly stable across programming languages; in any “reasonable” programming language, you can write a fairly short interpreter for any other reasonable language; “unreasonable” languages are those that don’t have short interpreters in most languages. So, the quantity is a fairly objective one, though it is still a little bit imprecise.
There are other, similar measures of complexity, which we will not go into at present.

23. Implications of Simplicity
One expects that: The simpler a given description D of a structure S,
the more likely that D will happen to appear within a “typical” structure S’,
and the more likely S itself will be found (at least partially) embedded within S’.
Therefore we presume the following:
Structures having simple descriptions are vastly more common throughout the “multiverse” of all consistent structures.
So, now what does complexity and simplicity have to do with the MEP philosophy? We mentioned earlier that structures can appear embedded in other structures. However, if a structure S has some simple description (program) D (that is, S has low K-complexity), then one expects that it is more likely that D may be found (by accident, or on purpose) within some other random, “typical” structure S’, and therefore that it is more likely that D will be elaborated (and S thereby embedded) by some mechanism within S’. In other words, the structures with simple underlying descriptions, are more frequently found embedded in other structures throughout the “multiverse” of all structures, similarly to how a short bit-string is expected to be found more often in a long random bit-string.
This argument is informal, since we have not quantified what constitutes a typical structure. In a sense, we are defining typical structures recursively, as those which occur most often embedded within other typical structures. The definition seems circular, but it may be that the constraint of self-consistency causes the definition to “bottom out” at a unique fixed point, though this is not yet clear. Moreover, it’s not yet clear how to evaluate which structures occur “most often” when some structures may contain infinitely many substructures. However, these questions are amenable to mathematical study, and answers may be forthcoming. At the moment, it is an open question whether this intuition of simpler structures being “more common” can be made more precise and justified objectively. But for now, let’s suppose this works out.

24. Expectations for Our Universe
Now consider the set of all intelligent entities (throughout all copies of all universes) capable of investigating their universe.
One would expect that most of these would find themselves in universes having the simplest laws that can support such beings, since those universes (we expect) are much more common. (More on this later.)
So, as such beings, we must statistically expect Ockham’s razor to work!
Now, if the “frequency of occurrence” of structures can be quantified based on their K-complexity, then consider the following. Consider a “random” intelligent entity capable of questioning the nature of his universe, selected from all intelligent entities throughout all embedded copies of all universes. Now, since the universes with simple underlying descriptions are so much more common than more complex universes, and since even simply-defined universes can contain huge numbers of distinct intelligent beings (e.g., a simple program can enumerate all possible variations of nondeterministic quantum events in our universe’s history, generating enormous numbers of variations of ourselves), one expects that of the random intelligent entities, “most” would find themselves in one of those simple universes.
So, each of us, after observing of ourselves that we are intelligent beings, should conclude that our statistical expectation a priori should be that we are in one of the simplest universes (if not the simplest) that contains large numbers of such beings in its full elaboration. So, we should expect physics to be as simple as it can be while still retaining the ability to spontaneously evolve large numbers of intelligent beings. Since our knowledge of physics is not yet complete, we do not yet know exactly how simple it will turn out to be.
It is interesting to note that many aspects of physics seem optimized for producing intelligent life, and lots of it. Quantum theory (in its many-worlds interpretation) generates enormous numbers of variations on the universe, which explore all different combinations of small random events, so that life can arise somewhere even if it involves some unlikely molecular accidents at some point along the way. The sheer size and ongoing expansion of the universe gives lots of space for life to fill. Analyses have shown that if any of the constants of physics had even been slightly different from what they are, the universe would be drastically different, and life probably could not have formed at all.

25. Theological Questions
Q: Does a God or gods exist?
A: Every god that is (or is part of) a consistent mathematical structure exists (in our mathematical sense).
If there is a precise & self-consistent mathematical description of what you mean by God, then your God exists (as a math object).
Q: Can a God do anything it wants?
A: It can do anything you define it to do within the context of the structure it’s defined in.
Now, let’s move on to the topic which is possibly of most interest to AASA, namely the big theological questions. Is there a God?
Well, here’s something interesting. If there is some precisely definable, self-consistent mathematical structure (or sub-structure) that exactly characterizes something that you would recognize as being what you mean by God, then that God exists, and is that structure. I believe that mathematical existence is the only kind of existence - anything else seems superfluous - so if your God is not a math structure, or at least some fuzzily-defined entity that can be found embedded within some math structure, then I don’t believe your God exists.
Most traditional religious people would not, I expect, want to believe that whatever they mean by God could ever be encompassed in a math structure. So, most such traditional Gods don’t exist. However, if you can accept a somewhat less traditional definition (which we will see in a moment), then we will see that infinitely many such gods (under the new definition) exist.
One thing commonly proposed as a defining characteristic of God is omnipotence - the ability to do anything. Can a God that is in part of a math structure do anything? Well, it can do anything your definition enables it to do, within the context of the structure within which it’s defined. It can’t do anything outside of that structure! Only a limited kind of omnipotence is logically possible.

26. Limits & Capabilities of Gods
Q: What can any God or demigod not do (no matter how you define it)?
A: It cannot affect structures that have no interaction with its universe, or cause any structures to exist or not to exist mathematically.
Q: What are some interesting things a suitably-defined God might be able to do?
Simulate any desired universes in complete detail
Assuming the God has at least Turing-machine power.
Influence, by informing or inspiring, other entities that are observing its behavior
Entities who are simulating, or part of, its universe.
In particular, here are some things that any God that actually exists (as a math object, as opposed to the fictional Gods that don’t exist because they don’t have a precise and self-consistent definition) can’t do: They can’t cause any structure to exist or not to exist mathematically, because existence is determined by pure immutable logic. The structures that do exist exist independently, they don’t need a God to make them, and the ones that don’t exist can’t exist because the properties used to define them aren’t logically consistent; some of the defining properties can’t logically be true of any object satisfying the other defining properties. So God, as opposed to being the creator of everything, is completely impotent when it comes to creating anything, in the absolute sense! Of course, if a god is a sentient being in a universe, he can form copies of things within his universe, just as you or I can make things (embedded representations of pure-formal objects) within our universe.
So, any god is totally impotent in that one respect, but what interesting powers could be had by our putative deity? If an entity has at least Turing-machine power, then he can simulate any desired universes in complete and utter detail. Further, he can influence other entitites that are observing his behavior - either because they are in the same universe as him and interact with him directly, or because they are simulating his universe and observing his behavior, or because he created some entities embedded within his universe, perhaps in a computer simulation, and is interacting with them. (I don’t mean to be sexist when I say “him,” one could just as well use “her” “them” or “it.”)

27. Gods: A Proposed Definition
Let a god be any substructure (in any universe) that has these properties:
Sentient (has a creative intelligence)
Immortal & Infinitely Intelligent:
Has as many unique thoughts as desired
This requires unlimited computational power:
Access to as much storage as desired
Can perform as many computational operations as it chooses to, for any desired computation.
A god can simulate any or all universes!
So, let me propose what I think would be a good definition of the word “god,” since the traditional definition is too fuzzy to be useful. Usually God is conceived as a sentient, thinking being with a creative intelligence. We can handle that, because we know that such beings (for example, ourselves) can be part of mathematical structures.
Further, people usually define God as being Immortal and Infinitely Intelligent. Either of these I would say is simply a way of saying that the being in question can entertain as many unique thoughts as it desires, and can compute whatever it wishes to compute. This obviously requires unlimited computational power, that is, access to as much information storage space as needed (since any with bounded store we will eventually just repeat previous states, no more unique thoughts) and ability to perform an unlimited number of computational operations.
Given this definition, we can deduce that there exist an infinite number of gods, so defined, assuming only that some universe (not necessarily ours, but, more on this later) provides a lush enough environment that intelligent life can not only evolve within it, but, once arisen, maintain itself indefinitely.
Armed with these powers, a god can indeed simulate all universes (and describe all other mathematical structures), or, can choose to selectively explore only a subset of universes.

28. Demigods: A Proposed Definition
A “demigod” is a mortal entity having some god-like capabilities:
Sentient
Can do any (simple enough) computation.
Can at least begin to simulate universes that are sufficiently simple.
Except that the number of computational operations may be limited, without choice.
We are weak demigods, but improving.
Just for fun, let’s define “demigod”. A demigod is like a god (sentient) except that it is not immortal (infinitely intelligent) and can only perform a number of computations that is not of its own choosing. A demigod can simulate sufficiently simple universes for a short time or in an approximate way. Under this definition, we humans are weak demigods, but advancing to higher levels of demigodhood as our computer power increases.

29. Gods & Our Universe
Q: Does a God or gods control or influence events in our universe?
A: There is no evidence for this, and a universe without a God has a simpler description, so probably no (i.e. the vast majority of universes like ours likely have no God manipulating them).
But, more on this later...
Q: Is a God or gods observing events in our universe (without interfering)?
A: Based on our definitions, infinitely many gods & demigods may watch exact copies of our universe!
Now, we’ve talked about properties gods may or may not have in general. Let’s now talk about the possible relationships between gods and our own universe.
Of course, one big question that has been on people’s minds since the dawn of human history is whether gods are influencing our universe. Most ancients were convinced that nearly everything was the gods’ doing, but now we know from the experience of science that any phenomenon that has been methodically studied has yielded to a physicalistic explanation. Current best-available physical theory apparently does an excellent job of explaining everything that we can observe methodically: galaxies, stars, planets, chemistry, life, etc. The only “unexplained” things are the alleged fringe phenomena such as psychics, ghosts, UFOs, angels, demons, magic, miracles, etc. that have never yielded to methodical characterization, not for any lack of dedicated enthusiasts who wish to study them, but rather most probably because there is nothing there to be studied other than fables and hoaxes. (One remark though: Aliens armed with nanotechnology are seemingly not beyond the bounds of physics, and could go a long way towards explaining all these phenomena, if one assumed the aliens have a perverse desire to keep their existence a secret, and to mess with people’s heads.)
So, assuming that physics really does explain all phenomena, the question of whether a God is pulling the strings for us must be probably not. The Godless universe is simpler, so presumably more common; most universes like ours probably don’t have an extra God structure tacked onto them. However, this really depends on whether our physics is more commonly found embedded within mindless structures that are just methodically spewing out all the universes, or within computer simulations operated by intelligent gods (or powerful demigods). More on this later.
Here’s a really interesting point that logically follows from the MEP: There must be infinitely many gods choosing to observe any detail of our universe, by looking at the exact mirror-image copies of it that are embedded within their limitless computer simulations. If you care about their opinion of you, you may want to be good. You can talk to them if you like.

30. Are Miracles Possible?
Q: Could a god that is observing our universe interfere with it?
A: They could do whatever they wanted with the copy they are running, but that would not affect other copies.
Interestingly, it seems that we can’t know for sure whether we happen to be in a god-controlled copy (unless we see an otherwise-inexplicable miracle).
You may ask, what is the good of praying to the infinitely many gods that are watching you unless they can do something to help? Can they do something?
Well, of course they can do whatever they want with the copy they are running - temporarily bending our laws of physics (but not theirs) in some way that helps you out, perhaps - but the rub is that whatever a god does with his own copy of the universe won’t have a whit of an effect on the infinity of other copies that he’s not running.
The relevant thing is the probability that our particular instance of our universe is one that is being watched by a god, that he pays attention to your particular plea, and decides for whatever reason to humor it. It seems that we can’t know for sure whether we happen to be in a god-controlled copy of our universe, or, even if we are, if the god is even paying attention. But there seems scant real evidence that anything has been done in the past to bend the laws of physics for us. But who knows, maybe the laws of physics used to get broken for us all the time, but now that we know the laws, they don’t get broken much any more because our particular god wants to keep his existence a secret from us, for whatever reason.
However, it seems more likely that human fabrication and exaggeration can explain all the miracle stories, and that we are in one of the many godless instances of our physical system that would be elaborated within any of the infinity of mathematical structures that have an automatic Turing-complete capability of enumerating all possible structures within them.
But, who knows, perhaps in many gods’ universes, it is quickly discovered that our universe is particularly interesting for some reason, and perhaps it is a favorite pastime of the gods, throughout their infinite lives, to create copies of our universe which they fiddle with and manipulate in various ways. In which case it might not be unlikely for ours to be one of the many variations.

31. What about Free Will?
Q: Isn’t the concept of man (or God) as a mathematical structure (or sub-structure) inconsistent with the concept of Free Will?
A: Not at all; this is just jumping to conclusions.
“Claiming a logical inconsistency where none exists is a far graver intellectual error than not chancing to discover some contradiction that does exist.” -me
We are part of the universe. Our own process of deliberate, willful choice is an inextricable part of how the overall structure is determined.
Any simulation must include our decision process.
By the way, some people might protest that

32. Warning: Don't Try This At Home!
Life after Death
Q: What will become of me when I die?
A: You’re probably toast, but if it’s any comfort…
Other entities effectively indistinguishable from you will live on in other universes that are similar enough to ours to contain a “you”, but different enough to let the other you go on.
Cf. “Quantum Immortality Theorem”
Of the scenarios that keep you around, whichever occurs most often in the multiverse will most likely be the one you find yourself in. (Maybe not heaven, but, more on this later.)

33. Fate of the Universe
May depend on choices made by us (& any other) sentients in the universe.
A worthy goal: Maximize the number of computational operations that we can do over all future history.
Is this number finite or infinite?
An open scientific question. (Cf. Dyson ‘79, Krauss & Starkman ‘99, Dyson ‘01).
If infinite, we can create/become a God.

34. A Critical Question
Earlier, we said that the simplest universes appeared most often as substructures of other structures.
But, what if the most common way for a universe to appear as a substructure is for a god to be simulating it?
In that case, it is up to the gods to shape the overall probability distribution of universes!

35. What Gods May Do
In each universe that evolves a God or Gods, those gods have many options.
They can simulate all possible universes, or only selected ones.
They can preferentially simulate universes they like, to increase the probability that intelligent entities will find themselves in such universes, rather than others.
We can only hope that the gods have good taste!

36. What about Heaven?
Note that a god that is simulating our universe could copy our simulated selves into new environments, perhaps improved variations of our universe.
Perhaps the god would choose to copy entities it approved of to environments that were enjoyable for those entities (Heaven-like environments)?
On the other hand, perhaps it wouldn’t.

37. Ethics & Morality
Where do our systems of values, ethics, & morality come from, historically?
They co-evolved with our culture.
Pragmatic tools for a stable society.
Where should we get our values from?
Social sciences can tell us what “works.”
But, our ultimate goals are really up to us.
Someday, perhaps we can simulate other universes’ sentients, & ask their opinion.

38. The Universal Probability Distribution (UPD)
The idea of a structure having a definite “frequency of occurrence” within the metaverse of all structures is critical.
Can these “true” frequencies be objectively quantified by some mathematical analysis?
Researchers are actively working on this.
Outcome is still unclear.
If an answer is found, then the deepest philosophical questions may finally yield to scientific quantification...
Schmidhuber ‘00 (quant-ph/ )

39. What if this line of thought doesn’t work out?
If the current effort fails, and further, if we can mathematically prove that no a priori basis for a UPD can exist, then:
At least, we would finally know that the “true” probabilities of various metaphysical situations are, ultimately, unknowable.
The UPD would be, in a sense, the only inexplicable thing (cf. Adams ‘01)...
But, it effectively determines everything!
How likely any possible occurrence is...

40. The One True God?
If the UPD can’t be objectively quantified, perhaps then the “one, true God” is this:
Some principle, beyond mathematics, that enumerates the countless structures of mathematics in the special ordering that breathes the life of the probabilities that we experience (such as the preference for simple laws) into what would otherwise be a featureless, chaotic expanse, an infinity of bizarre, complex, inscrutable, and equally improbable worlds.

41. Directions for Future Work
Complete the work of physics & find the simplest laws describing the physics of our universe.
Analyze cosmological limits of computation, determine whether creating our own God is feasible.
Engineer our universe (or at least as much as we can grab) into a computer of maximum possible power.
Try to deduce the “true” probability distribution (if it exists) over all structures/universes/gods/sentients.
Come up with nice God algorithms that we may want to implement in our universe. Decide which to implement so as to best nudge the universal probability distribution in the direction that we favor.

42. Conclusion
Simple existence/origins problems can be neatly solved (or at least, swept aside) by the MEP.
The origin of Occam’s razor can be addressed, but not yet in an entirely satisfactory way.
Under the MEP, computational god-like beings do exist, observe universes, & help to shape the universal probability distribution.
A noble purpose for humankind: Become ‘gods,’ and join the quest to shape the UPD in a ‘good’ direction.
Whether we can ever know the overall shape of the UPD is still unclear; it may be forever inscrutable.
But, science is having a crack at it!

43. Appendix: Simple God-like (but non-sentient) Algorithms
Here is perhaps the simplest God-like algorithm:
Enumerate and simulate in parallel all Turing machines on all finite inputs.
Here’s another pretty simple one:
Enumerate, in parallel, all computable sets of logical axioms.
In parallel, for each set, enumerate its logical consequences until an inconsistency is found (then throw that set away).
As an appendix, here are some high-level descriptions of algorithms that are god-like in that they can (and do) generate all possible structures, including ours. These two algorithms are perhaps the simplest universes that contain our universe, and therefore ourselves, as a sub-structure. If our universe indeed has a simple description, then it will appear repeatedly in many ways in the constellation of substructures produced by either of these algorithms. Given the extreme simplicity of these two algorithms, they (and thus our simple universe) will appear relatively often within other dumb algorithms that enumerate structures. Something similar to these algorithms may also be used by gods who are trying to find interesting universes to explore. One suspects that gods will not want to waste their time exploring boring infinities of all possible structures, as these algorithms would do; they will instead seek out the ones that are “interesting”, particularly ones that spontaneously evolve intelligent life, and spend a lot of time fiddling around with variations on those universes. The interesting question is whether universes like ours occur more often as automatically generated sub-structures of simple little structures like these, or as intensely-studied, potentially-manipulated playthings of the gods. If we could answer this question of frequency, it would tell us which kind of universe we are most likely living in. However, one thing we can say is that if our universe does have a god, he sure likes to stay hidden, and he’s probably not going to mess around with things in any really overt way (since he hasn’t much up till now). But you never know, maybe he stepped away from his computer for a short coffee break, and when he gets back he’ll take a look at his monitor and say, “Ack! In the 100 simulated years that passed while I was gone, they discovered nuclear weapons and used them on each other! I was hoping they wouldn’t do that. Oh well, guess it’s time to terminate this particular experiment.”