Do Gödel’s Theorems Impact Negatively on Physics?

 

The 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:

“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?

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Both Consciousness & Quantum Mechanics are Sexy… So Let’s Join Them Together!

 There are some people who drop the words “quantum mechanics” into almost all scientific and philosophical conversations. More relevantly, the words (to cite just two examples) “quantum dualism” (i.e., in the Cartesian — not Niels Bohr’s complementarity — sense) and “quantum coherence” also feature strongly in debates about consciousness.

Quantum physics is sexy and exotic. It’s also counter-intuitive. Hence the appeal.

It’s here that I’ll rely primarily on the science writer John Horgan’s interviews with various scientists and philosophers in his excellent book The End of Science.

Quantum mechanics is so sexy that it even explains free will and, therefore, mind-body dualism. Or at least the Australian neurophysiologist and philosopher John Eccles (who won the Noble Prize in 1963) believed so. In concrete terms, Eccles (who died in 1997 — a year or so after Horgan’s interview) claimed that the mind (or self) must

“exert its influence over the brain by ‘deciding’ which neuron will fire and which will not”.

Yet, despite all that neuroscience (or physical theory), John Eccles still conceded: “We have no proof of any of this.” So perhaps what was truly motivating Eccles (as John Horgan suggests) is his aversion to what he called “cheap materialism”. Indeed Eccles classed himself as a “religious person”. Moreover, he added:

“[T]he very nature of the mind is the same as the nature of life. It’s a divine creation.”

So whereas as Eccles believed that quantum mechanics explains free will, so the mathematical physicist Roger Penrose believes that it explains consciousness itself. In fact the two theories tie together (if in a fairly vague way).

For Eccles, quantum indeterminism (or, in his case, the superposition of physical states at the synapses of a neurone) is explained by the “action” of the self or mind. With Penrose, the self is kept out of the picture; though quantum indeterminism is kept in. More specifically:

“Penrose conjectured that microtubules perform non-deterministic, quasi-quantum computations that somehow give rise to consciousness.”

But what of the scientific sceptics who play down quantum mechanics — at least when it comes to consciousness and mind? John Horgan had this to say on the matter:

“There is one issue on which Crick, Edelman, and indeed almost all neuroscientists agree: the properties of the mind do not depend in any crucial way on quantum mechanics.”

Note here that it’s neuroscientists who “agree” on this; not philosophers, psychologists, physicists and others. Yet it’s neuroscientists who prod and probe into the the brain. So, at least on the surface, it would seem that it’s neuroscientists who should know what they’re talking about. Sure, philosophical and conceptual issues will impinge on what they claim; though surely they’re the best people to ask when it come to this question:

Do the mind and consciousness depend in any crucial way on quantum mechanics?

So how could, say, a philosopher or a psychologist answer that question?

Yet the obvious point here is that neuroscientists don’t prod and probe at the quantum level. Indeed many (i.e., not all!) neuroscientists may be more or less illiterate when it comes to quantum mechanics. So no wonder many of them ignore the facts and realities of quantum mechanics; or, at the very least, play them down.

Having said all that, Horgan does tell us that the molecular biologist and neuroscientist Francis Crick did believe that

“some neural equivalent of Heisenberg’s uncertainty principle might restrict our ability to trace the brain’s activity in minute detail”.

Sure; that was a very minor admission from Crick.

In a certain — even in a strong — sense it’s certainly the case that the brain itself (if not consciousness) does depend on quantum mechanics. Yet that’s simply because every physical object and physical event depends on quantum mechanics. That is, the micro-constituents of all physical objects and events are ultimately made up of particles, forces and fields that have no meaning outside of the theoretical ambit of quantum mechanics.

So this is a good place to take a look at Daniel Dennett’s quantum car.

Daniel Dennett’s Quantum Car

Daniel Dennett driving his quantum car.

The American philosopher Daniel Dennett made these points:

“Most biologists think that quantum effects all just cancel out in the brain, that there’s no reason to think they’re harnessed in any way. Of course they’re there; quantum effects are there in your car, your watch, and your computer. But most things — most macroscopic objects — are, as it were, oblivious to quantum effects. They don’t amplify them; they don’t hinge on them. Roger [Penrose] thinks that the brain somehow exploits quantum effects.”

In other words:

Sure; there are quantum happenings in the brain as a whole or in neurons. Then again, there are quantum happenings in your car, watch and television.

It may be true that in order for Dennett’s car to be a car, it doesn’t depend on the quantum effects which are occurring inside it. However, why should that also be true of the brain and its relation to mind or to consciousness? The nature and functioning of a car (or watch) is very different to the reality and functioning of the brain and its relation to consciousness. A car is (to use Dennett’s word) “oblivious” to the quantum effects inside — though only if it is treated qua car! However, it’s of course the case that a car can also be analysed as a medium of quantum effects; though not, again, qua car.

Then again, it is strictly true that a car — even qua car — doesn’t depend on quantum effects/events/conditions? Surely it does so in the simple sense that if there were no quantum effects/events/conditions, then there would be no car either. And, yes, it’s true that this also applies to literally all other physical objects — including biological and artificial objects.

Of course we’ll now need to know exactly why “quantum effects” don’t transfer to the brain as a whole. Alternatively, why aren’t quantum effects (to use Dennett’s words) “amplified” and “exploited” by the brain? More specifically, we’ll need to know why such things don’t cause (or bring about) mental phenomena or consciousness. In other words:

Why is there such a sharp dividing line between Dennett’s quantum effects in the brain (or in neurons) and consciousness itself?

Surely there can’t be such a neat and tiny cut-off point (an — as it were - Heisenberg cut) between these two worlds. Then again, it’s not logically absurd to argue that there is indeed such a clear cut-off point.

Large-Scale Quantum Effects

Yet, despite all the above, the mathematical physicist Roger Penrose argues (as do many biologists and physicists) that quantum effects/events/conditions do indeed have an effect on the large scale. He makes that point plain in the following passage:

“The very existence of solid bodies, the strengths and physical properties of materials, the nature of chemistry, the colours of substances, the phenomena of freezing and boiling, the reliability of inheritance — these, and many other familiar properties, require the quantum theory for their explanations.”

It can now be argued that even though these “solid bodies” and “materials” do “require the quantum theory for their explanations”; that doesn’t also automatically mean that such quantum effects are in any way substantive. It simply means that quantum mechanics is a part of the whole picture (as in the case of Dennett’s car). So, in the sense of supplying a complete picture of such bodies and materials — then, yes, of course quantum theory will be required.

Yet Dennett himself does accept that quantum events/effects/conditions/etc. influence (or affect) the large scale. For example, he says that quantum mechanics is

“stunningly successful at predicting and explaining many phenomena, including everyday phenomena such as the reflection and refraction of lights, and the operation of the proteins in our retinas that permit us to see”.

Of course it may still be the case that because

quantum mechanics can (to use Penrose’s words) “predict and explain” such things as (to use Dennett’s words) “the reflection and refraction of lights”

that this doesn’t also mean that

quantum mechanics fully accounts for these things.

…. Then again, surely it does mean that!

And if all this is true of the aforementioned light and protein molecules, then why can’t it also be true of the brain and consciousness? Of course the parallels between

quantum mechanics and the reflection and refraction of light

and

quantum mechanics and the mind (or consciousness)

may not be parallel (or equivalent) in every respect. However, surely that wouldn’t matter too much in this case. What matters is whether or not quantum mechanics is having an impact on the brain and therefore on the mind (or consciousness). It doesn’t need to be the case that quantum mechanics does so in precisely the same way in which it impacts on (to use Penrose’s words again)

“solid bodies, the strengths and physical properties of materials, the nature of chemistry, the colours of substances, the phenomena of freezing and boiling, the reliability of inheritance”.

To repeat: perhaps the quantum-mechanical explanations of these phenomena are at a different level to actually arguing that quantum-mechanical events actually bring about, cause or even constitute such phenomena. Again, is there a difference between

quantum mechanics being part of the explanation of consciousness (i.e., because quantum happenings occur within neurons, etc.)

and

quantum-mechanical effects/events/states bringing about or constituting consciousness

Surely if quantum mechanics can explain consciousness, then it may be because quantum-mechanical events/effects/conditions also bring about or actually constitute consciousness.

Conclusion: The Sexy Mystery of Consciousness

Kurt Gödel’s theorems are a helpful way to sum things up here.

Gödel’s theorems are also sexy. They’re applied to domains to which they shouldn’t (it can be argued) be applied. That is, theorems which were originally and primarily about mathematical systems are now applied left, right and centre. Nonetheless, since physics depends so heavily on mathematics, then perhaps there is something to the many connections which are made between Gödel’s theorems and everything else under the sun.

As for the sex appeal of quantum mechanics itself, John Horgan writes:

“[Francis] Crick’s partner Christof Koch summed up the quantum-consciousness thesis in a syllogism: Quantum mechanics is mysterious, and consciousness is mysterious. Q.E.D.: quantum mechanics and consciousness must be related.”

More specifically, the obvious question is:

Why do (or would) non-determinism (or acausality), quantum computations, quantum coherence, etc. give rise to consciousness and the “what it is like” to, say, smell a rose?

Is it this argument again? -

i) Consciousness is mysterious.
ii) Quantum mechanics is mysterious.
iii) Therefore consciousness and quantum mechanics simply must be related.

Could it be that some people believe the above simply because there are no other conclusive answers at present? Thus do they also believe this must be answer? Yet if that were the case, then many other outrageous — or simply radical — things may be the answer too. So why is this quantum stuff more feasible than (as the philosopher Patricia Churchland put it) “pixie dust in the synapses” when it comes to explaining consciousness?

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John Dewey’s Naturalist Position on Logic’s Relation to Science

 

One may not be entirely convinced by John Dewey’s naturalist take on logic.

For example, a primary argument against Dewey’s naturalisation of logic — which won’t be pursued here — is that if the (to use Dewey’s own word) “eternal” truths and principles of logic were really disputable and amendable, then Dewey wouldn’t have even been able to argue this case in the first place. Thus the American philosopher must have — and did — assume and use logical principles and truths (e.g., the law of identity, the law of excluded middle and the law of non-contradiction) which he - and others — questioned and claimed to have a non-absolute status. Of course Dewey might well have assumed and used such logical principles, happily admitted that he did so, and yet still rejected their absolute (or eternal) status.

Having said all that, none of these counter-arguments against Dewey’s thoroughgoing logical naturalism will be advanced in this piece.

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The American philosopher John Dewey (1859 — 1952) clearly had a very particular position on logic. In his eyes, logic should be derived from the methods and practices used in science. And because science is always on the move, then Dewey also believed (as expressed in his Essays on Experimental Logic) that logicians shouldn’t see logical principles as

“eternal truths which have been laid down once and for all as supplying a pattern of reasoning to which all inquiry must conform”.

More generally, Dewey expressed his naturalist position on logic (or at least on “thinking”) in this way:

“[T]hinking, or knowledge-getting, is far from being the armchair thing it is often supposed to be. The reason it is not an armchair thing is that it is not an event going on exclusively within the cortex or the cortex and vocal organs. It involves the explorations by which relevant data are procured and the physical analyses by which they are refined and made precise.”

Thus Dewey believed that just as science doesn’t offer us eternal (or absolute) truths, neither should logic. Indeed if one accepts that logic should — and sometimes does — base its principles on the methods and reasonings found in science, then this attitude to logic will evidently make at least some sense.

There’s another point about logic that’s worth making here.

Platonic Logic?

As with mathematics, even if logic does have eternal principles and truths which somehow exist mind-independently in an abstract realm, it doesn’t at all follow that logicians and mathematicians — any logicians and mathematicians — have unadulterated access to them. Perhaps most logicians and mathematicians simply haven’t discovered (or arrived at) all — or even any — of these eternal principles and truths. So, yes, we may — or can — accept that logic and maths should be beholden to this eternal mind-independent realm. However, it may not be the case that logicians and mathematicians are fully in tune — or in tune at all — with the principles or truths within it. Having said all that, logicians and mathematicians may still have good reasons for believing in this Platonic world’s existence — even if they know little about it.

That said, the use of the word “truth” within a purely logical context is problematic. Many philosophers — including Wittgenstein — have argued that “truth” isn’t the correct word to use when it comes to logical principles, rules, inferences, etc. Instead, thinking in terms of correctness is a better way of looking at these things.

All this means that it would simply be unwise to accept logical principles (or truths) as having some kind of eternal (or absolute) status. Such principles (or truths) may well exist; though logicians may not have access to them in all their fullness. Thus having an absolutist position on logical truths (or principles) may prove to have very negative implications for logic itself and for all those disciplines that (self-consciously) use logic.

Naturalism

It was primarily Dewey’s scepticism about eternal (or absolute) logical truths and principles which made him decide that logicians should base their principles, methods, inferential patterns, etc. on what actually happens in science. That way logicians wouldn’t stick so rigidly to what they believe is the correct (or true) logic. In other words, logic should be fallibilist — just like science. Indeed philosophers like W.V.O. Quine (in the mid-20th century) were fallibilists when it came to both mathematics and logic. Moreover, Quine even became a pragmatist about the Law of Excluded Middle (or at least its applicability) in response to the findings of quantum mechanics and the results of various experiments (see here). In addition, many philosophers believed that statements about the future (i.e., future contingents) can be neither true nor false. Such philosophers, therefore, rejected the Principle of Bivalence. Thus, in this case at the least, logic must include a third value — indeterminate. (See three-valued logic.)

It can now also be argued that certain scientists had rejected the Law of Excluded Middle and the Principle of Bivalence long before most philosophers and logicians had rejected them. And they did so because of science’s relation to the world — or at least science’s relation to “empirical reality”.

On a naturalist (or empiricist) position on logic, the ultimate relation is from the world to logic, not from logic to the world. On the other hand, if logic has no necessary relation to the empirical world (a position that can be called logical Platonism), then logicians may as well stick with their (absolute) principles and truths… for all time. This means that — to such logicians — there may never be any reason to reject these eternal logical principles and truths. Yet what if the world can (as it were) challenge these logical principles and truths? Indeed the world (or at least 20th-century science) has challenged at least some of the assumptions and presuppositions of logic.

Not only is the relation (so Dewey indirectly argued) from the world to logic: the other relation should be from science to logic. In that case, logic seems to take a back seat when it comes to science (or, more usually, physics). And this isn’t so strange if one considers (say) Quine’s additional positions on epistemology, ontology and all the other branches of philosophy. These too, according to Quine, should take a back seat to science. More specifically, Quine argued that there is no— or there shouldn’t be an— a priori epistemology (or an a priori philosophy generally). Therefore there is no fully a priori logic either. So, on a Quinian reading, logic is effectively no different to epistemology in these respects.

And because science’s relation to the world is both more (as it were) direct than logic’s and philosophy’s, then of course the latter (to use Quine’s words) “should defer to science”. Alternatively and to use a term used in the 20th century, logic and philosophy should be naturalised so that they don’t systematically conflict with science and its findings.

So now what about (naturalist) holism?

Most naturalists argue that because single statements, terms, judgements, concepts, etc. aren’t self-sufficient or “atomic”, then neither are the whole disciplines of philosophy and logic. Thus holism - of some description - goes all the way down the line.

To get back to John Dewey’s position on logic and to sum up.

Dewey took a typically pragmatist line on logic. He noted the many successes of science. He therefore believed that whichever logical rules or principles science used when it scored particular successes should also be ones logicians should adopt. And, again, scientists have always adopted new logical methods in their pursuits. Dewey believed that logicians should do so too.

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