Mathematics – How to Determine If a Theory of Everything Is Necessarily Unique?

mathematicstheory-of-everything

There is a lot of interesting debate over whether a "theory of everything" (ToE) is allowed to exist in the mathematical sense, see Does Gödel preclude a workable ToE?, Final Theory in Physics: a mathematical existence of proof? and Arguments Against a Theory of Everything. Since as far as I can tell this is still an open question, let's assume for now that formulating a ToE is possible. I'm curious about what can be said about the uniqueness of such a theory.

Now the hard part, where I'm pretty sure I'm about to back myself into a logical corner. To clarify what I mean by unique: physical theories are formulated mathematically. It is possible to test whether two mathematical formalisms are equivalent (right? see bullet point 3 below). If so, it is possible to test the mathematical equivalence of various theories. From the Bayesian point of view, any theory which predicts a set of observables is equally valid, but the degree of belief in a given theory is modulated by observations of the observables and their associated errors. So now consider the set of all possible formulations of theories predicting the set of all observables – within this set live subsets of mathematically equivalent formulations. The number of these subsets is the number of unique ToEs. Now the question becomes how many subsets are there?

Possibilities:

It can be proven that if a ToE exists, it is necessarily unique ($1$ subset).
It can be proven that if a ToE exists, it is necessarily not unique ($>1$ subset).
It can be proven that it is impossible to say anything about the uniqueness of a ToE, should it exist (it is impossible to test mathematical equivalence of theories).
We don't know if we can say anything about the uniqueness of a ToE, should it exist.

So this is really asking about the ensemble of closed mathematical systems (physical theories) of an arbitrary (infinite?) number of variables (observables). This is honestly a pure math question, but here strongly physically motivated.

I suspect the answer is probably the fourth bullet, but surely there has been some research on the topic? Hopefully someone familiar with the ToE literature can shed some light on the question.

Best Answer

I will expand my comment above into an answer:

If you search for a TOE that is a mathematical theory, it has to be at least a logic theory, i.e. you need to define the symbols and statements, and the inference rules to write new (true) sentences from the axioms. Obviously you would need to add mathematical structures by means of additional axioms, symbols, etc. to obtain a sufficient predictive power to answer physically relevant questions.

Then, given two theories, you have the following logical definition of equivalence: let $A$ and $B$ be two theories. Then $A$ is equivalent to $B$ if: for every statement $a$ of both $A$ and $B$, $a$ is provable in $A$ $\Leftrightarrow$ $a$ is provable in $B$.

Obviously you may be able to write statements in $A$ that are not in $B$ or vice-versa, if the objects and symbols of $A$ and $B$ are not the same. But let's suppose (for simplicity) that the symbols of $A$ and $B$ coincide, as the rules of inference, and they only differ for the objects (in the sense that $A$ may contain more objects than $B$ or vice versa) and axioms.

In this context, ZFC and Bernays-Godel set theories are equivalent, when considering statements about sets, even if the axioms are different and the Bernays-Godel theory defines classes as mathematical objects, while ZFC does not.

Let's start to talk about physics, and TOE, following the discussion in the comments. It has been said that two TOEs must differ only in non-physical statements, since they have to be TOEs after all, and thus explain every physical observation in the same way. I agree, and from now on let's consider only theories in which the physical statements are true.

Let $A$ be a TOE. Let $a$ be an axiom that is independent of the axioms of $A$ (that means, roughly speaking, that there are statements undecidable in $A$, that are decidable in $A+a$, but all statements true in $A$ are still true in $A+a$). First of all, such an $a$ exists by Godel's theorem, as there is always an undecidable statement, given a logical theory. Also, $a$ is unphysical, since $A$ is a TOE. Finally, $A$ and $A+a$ are inequivalent (in the sense above), and are TOEs.

One example is, in my opinion, the generalized continuum hypothesis (GCH): without entering into details, it has been shown with the theory of forcing that it is independent of the axioms of ZFC set theory. Thus $ZFC$, $ZFC+GCH$ and $ZFC+\overline{GCH}$ (ZFC plus the negation of GCH) are all inequivalent theories that contain $ZFC$. It is very likely that a TOE must contain set theory, e.g. ZFC. Let $A$ be such a TOE. Also, it is very likely that $GCH$ is not a physically relevant axiom (at least it is not for our present knowledge). Then $A$ and $A+GCH$ would be inequivalent TOEs, then a TOE is not unique.

I have studied a bit of logic just for fun, so I may be wrong...If someone thinks so and can correct me is welcome ;-)

Incorporating Comments

There were comments which I will incorporate into this answer. It seems that comments are only supposed to be temporary, and some of these observations I think are useful.

Hilbert's program was an attempt to establish that set theoretic mathematics is consistent using only finitary means. There is an interpretation of Gödel's theorem that goes like this:

Gödel showed that no system can prove its own consistency
Set theory proves the consistency of Peano Arithmetic
Therefore Gödel kills Hilbert's program of proving the consistency of set theory using arithmetic.

This interpretation is false, and does not reflect Hilbert's point of view, in my opinion. Hilbert left the definition of "finitary" open. I think this was because he wasn't sure exactly what should be admitted as finitary, although I think he was pretty sure of what should not be admitted as finitary:

No real numbers, no analysis, no arbitrary subsets of $\Bbb Z$. Only axioms and statements expressible in the language of Peano Arithmetic.
No structure which you cannot realize explicitly and constructively, like an integer. So no uncountable ordinals, for example.

Unlike his followers, he did not say that "finitary" means "provable in Peano Arithmetic", or "provable in primitive recursive Arithmetic", because I don't think he believed this was strong enough. Hilbert had experience with transfinite induction, and its power, and I think that he, unlike others who followed him in his program, was ready to accept that transfinite induction proves more theorems than just ordinary Peano induction.

What he was not willing to accept was axioms based on a metaphysics of set existence. Things like the Powerset axiom and the Axiom of choice. These two axioms produce systems which not only violate intuition, but are further not obviously grounded in experience, so that the axioms cannot be verified by intuition.

Those that followed Hilbert interpreted finitary as "provable in Peano Arithmetic" or a weaker fragment, like PRA. Given this interpretation, Gödel's theorem kills Hilbert's program. But this interpretation is crazy, given what we know now.

Hilbert wrote a book on the foundations of mathematics after Gödel's theorem, and I wish it were translated into English, because I don't read German. I am guessing that he says in there what I am about to say here.

What Finitary Means

The definition of finitary is completely obvious today, after 1936. A finitary statement is a true statement about computable objects, things that can be represented on a computer. This is equivalent to saying that a finitary statement is a proposition about integers which can be expressed (not necessarily proved) in the language of Peano Arithmetic.

This includes integers, finite graphs, text strings, symbolic manipulations, basically, anything that Mathematica handles, and it includes ordinals too. You can represent the ordinals up to $\epsilon_0$, for example, using a text string encoding of their Cantor Normal form.

The ordinals which can be fully represented by a computer are limited by the Church-Kleene ordinal, which I will call $\Omega$. This ordinal is relatively small in traditional set theory, because it is a countable ordinal, which is easily exceeded by $\omega_1$ (the first uncountable ordinal), $\omega_\Omega$ (the Church-Kleene-th uncountable ordinal), and the ordinal of a huge cardinal. But it is important to understand that all the computational representations of ordinals are always less than this.

So when you are doing finitary mathematics, it means that you are talking about objects you can represent on a machine, you should be restricting yourself to ordinals less than Church-Kleene. The following argues that this is no restriction at all, since the Church-Kleene ordinal can establish the consistency of any system.

Ordinal Religion

Gödel's theorem is best interpreted as follows: Given any (consistent, omega-consistent) axiomatic system, you can make it stronger by adding the axiom "consis(S)". There are several ways of making the system stronger, and some of them are not simply related to this extension, but consider this one.

Given any system and a computable ordinal, you can iterate the process of strengthening up to a the ordinal. So there is a map from ordinals to consistency strength. This implies the following:

Natural theories are linearly ordered by consistency strength.
Natural theories are well-founded (there is no infinite descending chain of theories $A_k$ such that $A_k$ proves the consistency of $A_{k+1}$ for all k).
Natural theories approach the Church Kleene ordinal in strength, but never reach it.

It is natural to assume the following:

Given a sequence of ordinals which approaches the Church-Kleene ordinal, the theories corresponding to this ordinal will prove every theorem of Arithmetic, including the consistency of arbitrarily strong consistent theories.

Further, the consistency proofs are often carried out in constructive logic just as well, so really:

Every theorem that can be proven, in the limit of Church-Kleene ordinal, gets a constructive proof.

This is not a contradiction with Gödel's theorem, because generating an ordinal sequence which approaches $\Omega$ cannot be done algorithmically, it cannot be done on a computer. Further, any finite location is not really philosophically much closer to Church-Kleene than where you started, because there is always infinitely more structure left undescribed.

So $\Omega$ knows all and proves all, but you can never fully comprehend it. You can only get closer by a series of approximations which you can never precisely specify, and which are always somehow infinitely inadequate.

You can believe that this is not true, that there are statements that remain undecidable no matter how close you get to Church-Kleene, and I don't know how to convince you otherwise, other than by pointing to longstanding conjectures that could have been absolutely independent, but fell to sufficiently powerful methods. To believe that a sufficiently strong formal system resolves all questions of arithmetic is an article of faith, explicitly articulated by Paul Cohen in Set Theory and the Continuum Hypothesis. I believe it, but I cannot prove it.

Ordinal Analysis

So given any theory, like ZF, one expects that there is a computable ordinal which can prove its consistency. How close have we come to doing this?

We know how to prove the consistency of Peano Arithmetic--- this can be done in PA, in PRA, or in Heyting Arithmetic (constructive Peano Arithmetic), using only the axiom

Every countdown from $\epsilon_0$ terminates.

This means that the proof theoretic ordinal of Peano Arithmetic is $\epsilon_0$. That tells you that Peano arithmetic is consistent, because it is manifestly obvious that $\epsilon_0$ is an ordinal, so all its countdowns terminate.

There are constructive set theories whose proof-theoretic ordinal is similarly well understood, see here: "Ordinal analysis: Theories with larger proof theoretic ordinals".

To go further requires an advance in our systems of ordinal notation, but there is no limitation of principle to establishing the consistency of set theories as strong as ZF by computable ordinals which can be comprehended.

Doing so would complete Hilbert's program--- it would removes any need for an ontology of infinite sets in doing mathematics. You can disbelieve in the set of all real numbers, and still accept the consistency of ZF, or of inaccessible cardinals (using a bigger ordinal), and so on up the chain of theories.

Other interpretations

Not everyone agrees with the sentiments above. Some people view the undecidable propositions like those provided by Gödel's theorem as somehow having a random truth value, which is not determined by anything at all, so that they are absolutely undecidable. This makes mathematics fundamentally random at its foundation. This point of view is often advocated by Chaitin. In this point of view, undecidability is a fundamental limitation to what we can know about mathematics, and so bears a resemblence to a popular misinterpretation of Heisenberg's uncertainty principle, which considers it a limitation on what we can know about a particle's simultaneous position and momentum (as if these were hidden variables).

I believe that Gödel's theorem bears absolutely no resemblence to this misinterpretation of Heisenberg's uncertainty principle. The preferred interpretation of Gödel's theorem is that every sentence of Peano Arithmetic is still true or false, not random, and it should be provable in a strong enough reflection of Peano Arithmetic. Gödel's theorem is no obstacle to us knowing the answer to every question of mathematics eventually.

Hilbert's program is alive and well, because it seems that countable ordinals less than $\Omega$ resolve every mathematical question. This means that if some statement is unresolvable in ZFC, it can be settled by adding a suitable chain of axioms of the form "ZFC is consistent", "ZFC+consis(ZFC) is consistent" and so on, transfinitely iterated up to a countable computable ordinal, or similarly starting with PA, or PRA, or Heyting arithmetic (perhaps by iterating up the theory ladder using a different step-size, like adding transfinite induction to the limit of all provably well-ordered ordinals in the theory).

Gödel's theorem does not establish undecidability, only undecidability relative to a fixed axiomatization, and this procedure produces a new axiom which should be added to strengthen the system. This is an essential ingredient in ordinal analysis, and ordinal analysis is just Hilbert's program as it is called today. Generally, everyone gets this wrong except the handful of remaining people in the German school of ordinal analysis. But this is one of those things that can be fixed by shouting loud enough.

Torkel Franzén

There are books about Gödel's theorem which are more nuanced, but which I think still get it not quite right. Greg P says, regarding Torkel Franzén:

I thought that Franzen's book avoided the whole 'Goedel's theorem was the death of the Hilbert program' thing. In any case he was not so simplistic and from reading it one would only say that the program was 'transformed' in the sense that people won't limit themselves to finitary reasoning. As far as the stuff you are talking about, John Stillwell's book "Roads to Infinity" is better. But Franzen's book is good for issues such as BCS's question (does Godel's theorem resemble the uncertainty principle).

Finitary means computational, and a consistency proof just needs an ordinal of sufficient complexity.

Greg P responded:

The issue is then what 'finitary' is. I guess I assumed it excluded things like transfinite induction. But it looks like you call that finitary. What is an example of non-finitary reasoning then?

When the ordinal is not computable, if it is bigger than the Church-Kleene ordinal, then it is infinitary. If you use the set of all reals, or the powerset of $\Bbb Z$ as a set with discrete elements, that's infinitary. Ordinals which can be represented on a computer are finitary, and this is the point of view that I believe Hilbert pushes in the Grundlagen, but it's not translated.

[Physics] Final theory in Physics: a mathematical existence proof

This is an interesting questionning, rather than question, as I am not sure what is the question stated, because too many issues are questionned or questionnable

I think it is different, but very much related to a previous question regarding the physical provability of the Church-Turing thesis stating that any computing device that can be built will compute no more than what is computable by a Turing machine.

One issue with the Church-Turing thesis is that the concept of proof in an axiomatic theory is fundamentally the same as the concept of a computer program, i.e., a Turing Machine. This is not in the sense that a program can churn out theorems and proofs, as in Ron Maimon's presentation of Gödel's proof, but because a program can be "read" as a proof of its specification ("Given a value x such that P(x) holds, there is a result y such that Q(x,y) holds.") and conversely a proof may be read as a program that actually computes whatever is stated by the theorem. This presentation is of course a very simplified version of a result by Curry and Howard (1980) which is still researched.

Hence one major issue with the possible limitations of calculability, whether there are such physical limitations or not, and whether we can prove or not the existence of such limitations, is that mathematical proofs are directly concerned by the same limitations. One crucial aspect of such limitations, which I will address below is the denumerable nature of intellectual processes.

We can assume, we must assume, that our way of doing mathematics, including physical theories, is consistant. This is really (to me at least) a physical view of it: we do find inconsistencies, usually called paradoxes, but there are resolved (have been so far) like experimental inconsistencies in physics, by refinements of the théories and evolution of concepts to overcome the difficulty and state the problems appropriately.

Assuming mathematics is essentially consistent is essential, because whatever we can prove should not be questionned by future extensions, if any are physically possible, of the concepts of calculability or provability.

Now some results make deep assumptions that are not always obvious to interpret. In the case of Gödel incompleteness result, one major aspect is that logical formulae, theorems and proofs can be encoded as integers. This means that our logical deduction systems are fundamentally denumerable entities (like Turing machines). If it turned out that a breakthrough in physics allowed us to deal effectively with non-denumerable systems, then results relying on this denumerability would be in question. This is precisely the case for Gödel incompleteness result, as stated (possibly it could come back in another form).

I addressed this denumerability aspect in my answer to the question on the physics provability of Church-Turing thesis. At the time this answer was entirely based on my informal understanding of these issues. Intending to improve a bit on my answer, I looked for some litterature, and the topic is currently actively researched. While my knowledge of this literature remains more than shallow, it seems that my intuition was correct, that a proper handling of the fundamentally discrete (or denumerable) character of calculability, and provability, is essential in the current state of the art, for deriving Church-Turing thesis from the laws of physics, and that continuity or real numbers are a major issue.

One approach I have looked at (I am limited to papers in open access on the web) relies on assuming a specific property of the physical world, presented as dual of the limitation on the speed of light and information, which is a limitation on the space density of information, both limitations together ensuring density limitation in space-time. The translation of this new law in physical terms can actually be subtle to account for various existing physical laws. This apparently excludes unregulated use of real numbers.

If this limited density law is actually verified, I think it would also mean that Gödel theorem is also a consequence of physical laws.

Whether such a limited information density law is actually verified is another matter. If it is not, then doors remain open for an extension of the concepts of calculability and provability.

In such a case, assuming that our mathematics are otherwise consistent, all provable results would remain provable, but we might be able to prove new theorems that were true but not provable in the classical denumerable setting.

So the answer to the question whether a theory of Everything would evade Gödel's incompleteness theorem is very much dependent on what Everything is, since it actually determines the context in which calculability or provability have to be defined. Would a theory of Everything include a law limiting information density.

Note that even with Gödel's result being valid, there could be a possibility of a theory of Everything, in which all true facts regarding the universe would indeed be true. It is just that you would not be able to prove it (so that the universe would keep some mystery for us to wonder in starry nights). enter image description here

On the other hand, there could be no such theory of everything. But, what would the ultimate definition of a theory if physics allowed us to question the discrete nature of the language in which they are expressed ?

For the rest, other than taking 42 as ultimate answer, I can only suggest leaving the matrix to get the Truth about our world, or reading Simulacron-3.