A Mathematical Theory of Existence cise.ufl/~mpf/philosophy

1. A Mathematical Theory of Existencehttp://www.cise.ufl.edu/~mpf/philosophy.ppt Michael P. Frankmpf@cise.ufl.edu CISE Department University of Florida 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

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?

4. Why is there something? • Traditional form: • “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...

5. The Mathematical Existence Postulate (MEP) Logical truths & mathematical structures exist a priori, independently of anything else. • What does this mean? • Why should we believe it? • What can we deduce from it? • Answers to many big questions! Postulatedto be truein “reality”:

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).

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.

8. Ontology: Picture #1 • An infinity of (disconnected) mathematical structures of different sorts...

9. Benoit Mandelbröt’s fractal set: John Conway’s Game of Life: Complex Structures can have Simple Descriptions 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.

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.)

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.

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!

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.

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.

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...

17. Ontology: Picture #2 • An infinity of mathematical structures, each embedded in many others; many are universes...

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.

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.

20. Why does science work? • 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?

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?

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.

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.

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!

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.

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.

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!

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.

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!

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).

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.

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/0011122)everything-list@eskimo.com

39. What if this line of thought doesn’t work out? • If the current effort fails, and further, if we can mathematically prove that noa 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).