Do Gödel’s Theorems Impact Negatively on Physics?

he precise relation between Kurt Gödel’s incompleteness theorems and physics has often been discussed by physicists and philosophers. (It’s usually the first incompleteness theorem that’s deemed relevant in this respect.)

So here’s an example (from John M. Myers and F. Hadi Madjid) of what can be taken to be a very tangential (or simply weak) connection between Gödel’s first incompleteness theorem and (quantum) physics:

“We show how Gödel’s first incompleteness theorem has an analog in quantum theory… to do with the set of explanations of given evidence. We prove that the set of explanations of given evidence is uncountably infinite, thereby showing how contact between theory and experiment depends on activity beyond computation and measurement.”

And in the following the science journalist Davide Castelvecchi tells us that physicists have “tried” to apply Gödel’s theorems to “concrete problems”:

“Since the 1990s, theoretical physicists have tried to embody Turing’s work in idealized models of physical phenomena. But ‘the undecidable questions that they spawned did not directly correspond to concrete problems that physicists are interested in’, says Markus Müller, a theoretical physicist at Western University in London, Canada.”

Finally, John D. Barrow (who’ll be discussed later) takes a circumspect position on this issue. He writes:

“We introduce some early considerations of physical and mathematical impossibility as preludes to Gödel’s incompleteness theorems… We argue that there is no reason to expect Gödel incompleteness to handicap the search for a description of the laws of Nature, but we do expect it to limit what we can predict about the outcomes of those laws, and examples are given.”

The Basic Argument

Take these two statements:

1) Mathematical systems contain unprovable statements. 2) Physics uses and depends on mathematics.

Then, from 1) and 2) above, we can construct the following (simple) form of the general argument:

ia) If physics utilizes and depends on mathematics, ib) and Gödel’s theorems apply to mathematical systems, ii) then Gödel’s theorems must also apply to physics.

Thus physical theories (or even a/the Theory of Everything) must either be complete and inconsistent or consistent and incomplete. Either way, physics loses… Or does it?

The British-American theoretical and mathematical physicist Freeman Dyson did see a strong link between Gödel incompleteness and physics. Firstly he explained Gödel incompleteness in this way:

“[N]o finite set of axioms and rules of inference can ever encompass the whole of mathematics; given any set of axioms, we can find meaningful mathematical questions which the axioms leave unanswered.”

Dyson then applied his explanation of Gödel incompleteness to “the physical world”:

"I hope that an analogous situation exists in the physical world. If my view of the future is correct, it means that the world of physics and astronomy is also inexhaustible; no matter how far we go into the future, there will always be new things happening, new information coming in, new worlds to explore, a constantly expanding domain of life, consciousness, and memory.”

There is indeed a sense in which the word “incompleteness” can be applied to problems in physics. But is this Gödel incompleteness? Not really. The passage above (from Dyson) should really be read as a statement of scientific incompletability, not Gödel incompleteness. (In philosophy there’s also a relevant distinction made between incompletability and insolubilia.) In other words, Dyson’s words aren’t about a Gödelian lack of proof within a system — or even within all systems. They’re about (to use Dyson’s own word) the “inexhaustible” nature of “physics and astronomy”. Yes; the words “inexhaustible” and “incomplete” are near-synonyms; though this still isn’t a reference to Gödel incompleteness. In other words, Gödel incompleteness and scientific incompletability are two very different things.

Another related kind of incompleteness in physics simply applies to the situation in which new observations can’t be accounted for by older theories. Thus the older theories must be incomplete and yet still contain some — or even much — truth. Again, this has no direct or strong connection to Gödel incompleteness.

Besides all that, Freeman Dyson himself admitted that the connections between Gödel’s theorems and physics he highlighted only amount to an “analogous” (i.e., not a logical) link. (See my related piece on Stephen Hawking in which he too uses the derivative word “analogy”: ‘Deflating Gödelised Physics: With Stephen Hawking’.)

The English cosmologist, theoretical physicist and mathematician John D. Barrow also put the case against a thoroughly Gödelised physics in the following:

“[I]t is by no means obvious that Gödel places any straightforward limit upon the overall scope of physics to understand the nature of the Universe just because physics makes use of mathematics.”

Here we can highlight the word “understand”. To put what I believe is Barrow’s position in a simple statement:

Scientific understanding doesn’t require Gödel completeness.

All this may simply mean that an understanding of a physical theory — even a full understanding of a physical theory — isn’t affected by Gödel’s theorems. (That, of course, begs the question as to what scientific understanding is.) It may also mean that we can describe (or explain) a physical theory (or nature itself) without the possibility of Gödel incompleteness having any substantive effect on that description or explanation. And perhaps that’s because Gödel incompleteness would only be relevant to the mathematics required to explain or describe Nature — or, more likely, any aspect of Nature.

On that last point.

John Barrow also makes the technical point that even taking into consideration the necessary and vital role mathematics plays in physics, it may still be the case that the

“mathematics Nature makes use of may be smaller and simpler than is needed for incompleteness and [the] undecidable to rear their heads”.

This is clearly a statement that Gödel incompleteness doesn’t apply to all mathematics. And that mathematical remainder may be all that’s required for physics.

Do Gödel’s Theorems Have a Negative Impact on Physics?

So is Gödel incompleteness really a negative conclusion for physics? Well, any answer to that question will obviously depend on a whole host of factors.

As just stated, physics may not require the entirety of mathematics. Moreover, it may require only those parts of mathematics which aren’t affected by Gödel’s theorems. And even if Gödel’s theorems do somehow affect the mathematics employed in physics, then that may still not be (strongly) detrimental to physics as a whole.

To sum all this up in three statements:

1) Physics may not (or does not) utilise the entirety of mathematics. 2) Physics may only utilise those parts of mathematics which aren’t affected by Gödel’s theorems. 3) Physics may survive — or even thrive — even if it is to some degree affected by Gödel’s theorems.

The following is another argument that’s nonetheless related to the three statements above:

i) Physics doesn’t need (or have) strict proofs. ii) Gödel’s first incompleteness theorem is primarily about proof — or about the distinction between proof and truth. iii) Therefore that most important aspect of Gödel’s first theorem may not be directly applicable to physics.

On the other hand, a (weak) Gödelian argument can be put this way:

i) Mathematical systems contain unprovable statements. ii) Physics is based on mathematics. iii) Therefore physics won’t be able to discover “everything that is true”.

Discovering everything isn’t the same thing as proving everything. And what sort of claim (or aim) is it any way to “discover everything”?

Physics doesn’t — strictly speaking — need proofs. However, the mathematics included in physics may well need proof. So do the lack of proofs of mathematical systems glide over to physical theories?

What’s more and as already stated, the fact is that only certain aspects of mathematics are applied to physical reality. And — so it’s argued - those aspects are decidable (or computable).

Physical Laws as Axioms?

Perhaps taking the laws of physics as the exact equivalents of logical and mathematical axioms is at the root of this Gödelian problem. After all, if one takes some given physical laws as axioms, then - somewhere further down the line - there may well be Gödel incompleteness and/or inconsistency.

It’s true that physical laws can be used as axioms or be given an axiomatic status. Indeed just about anything can be used as an axion — any statement, phrase, equation, etc. (For example, the science writer Philip Ball says that the “collapse” of the wave functions is — often tacitly — deemed “axiomatic” in that physicists usually “accept [it] with no questions asked”.)

Yet it’s still the case that physical laws aren’t self-evident or “intuitively acceptable”. One reason for this is that physical laws are things that couldn’t — even in principle — be intuitively obvious because intuitions don’t apply to the laws (even if deemed axiomatic) which generate and govern all the theories and statements of physics. (Particularly, theories or statements about things at the cosmological and quantum scales.) Added to that, if physical laws are axioms, and what we derive from these laws are theorems, then what about the unpredictable consequences (or predictions) which we derive from our axiomatic physical laws?

Moreover, if physical laws were purely and strictly like axioms, then we could move from

“incontestable premises [i.e., axiomatic laws] to an acceptable conclusion [i.e., prediction or theory] via an impeccable rule of inference”.

Yet can that statement (i.e., minus the words in square brackets) be fully applied to physical laws and their resultant theorems? Indeed is it even correct to use the word “axiom” at all in physics? That is, can any law of physics ever be as simple and pure as an axiom in a logical or mathematical system?

There’s another consequence of this Gödelian way of thinking.

Gödel’s theorems require that the axioms of a system be “listable”. Can it be said that all the laws of physics are (or could be) listable? And even if they were listable, would the theorems which we derive from such physical laws bear a strong resemblance to the theorems which are derived from the axioms of a logical or a mathematical system?

In other words, do we have entailment (or strict deduction) from physical axioms (or laws) to physical theorems? And do we have either metaphysical or logical entailment when it comes to physical laws and the predictions, experiments and observations (i.e., the quasi-theorems) which are derived from them?

[I can be found on Twitter here.]