Gödel and Einstein. |

[This introduction is cut and pasted from part 1. Skip if you've read it before.]

This piece, of course, isn't about deflating Kurt Gödel's metamathematics or even deflating his own comments about physics. It's about deflating other people's applications of Gödel's theorems to physics.

Indeed Gödel himself wasn't too keen on applying his findings to physics – especially to quantum physics. According to John D. Barrow:

Indeed Gödel himself wasn't too keen on applying his findings to physics – especially to quantum physics. According to John D. Barrow:

“Godel was not minded to draw any strong conclusions for physics from his incompleteness theorems. He made no connections with the Uncertainty Principle of quantum mechanics....”

More broadly, Gödel's theorems may not have the massive and important applications to physics which some philosophers and scientists believe they do have.

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Firstly, we have two statements:

1)
Mathematical systems contain unprovable statements. (Or: Gödel's
theorems apply to all mathematics.)

2)
Physics uses mathematics.

This
is the argument:

ia)
If physics utilises mathematical systems,

ib)
and Gödel's discoveries apply to such mathematical systems,

ii)
then Gödel's discoveries also apply to physics.

Thus
a physical theory is either complete and inconsistent or consistent
and not complete. Either way, physics/science looses...
Or does it?

Freeman
Dyson, for one, did see a strong link between Gödel incompleteness
and physics. He
wrote:

“...
no 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.
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.”

The
above should be classed, however, as scientific

*incompletability*, not scientific*insolubility*. In other words, it not about a Gödelian lack of proof within a system - or even within all systems. It's about the “inexhaustible” (to use Dyson's own word) nature of “physics and astronomy”. Yes, the words “inexhaustible” and “incomplete” are near-synonyms; though this still isn't Gödel incompleteness.
Besides
which, Dyson himself says that the link between Gödel and physics only amounts to an “analogous” - not a logical - link.

On
the other hand, John D. Barrow puts the case against a thoroughly
Gödelised physics in
the following:

“....
it 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. The
mathematics that Nature makes use of may be smaller and simpler than
is needed for incompleteness and undecidable to rear their heads.”

Here
we can highlight the word “understand”. To put what I believe is
Barrow's position: scientific understanding doesn't require Gödel
completeness. It 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 mean that we can describe and explain a physical theory (or nature itself) without the notion of completeness having any substantive affect on that description or explanation. Perhaps that's because the completeness would only be relevant to the mathematics required to understand or describe a physical theory (or nature).
On
that last point. Barrow also makes the technical point that even
taking into consideration the necessary and vital role maths plays in
physics, it may still be the case that the (as Barrow puts it) “mathematics Nature makes use of may be smaller and simpler than is
needed for incompleteness and undecidable to rear their heads”.
That's a technical point that I - as neither a mathematician nor a
physicist - find hard to comment upon. It's clearly, nonetheless, a
statement that Gödel incompleteness doesn't apply to all maths. And
that mathematical remainder may be all that's required for
mathematical physics.

Thus,
to sum up. Does it follow that Gödel incompleteness is automatically
a negative conclusion for physics? Doesn't it all depend on a whole
host of other factors? As just stated, physics may not require the
entirety of mathematics. Moreover, it may require only those parts of
maths that aren't affected by Gödel's theorems. And even if Gödel's
theorems do somehow affect the maths employed in physics, that may
not be to the detriment of physics in any substantive way.

Succinctly
in three statements:

1)
Physics may not (or does not) utilise the entirety of mathematics.

2)
Physics may utilise only the parts of mathematics which aren't
affected by Gödel's theorems.

3)
Physics may survive (or not be substantively touched) even if it is affected by Gödel's theorems.

To
put all that another way. Only certain aspects of mathematics are
applied to physical reality. Those aspects are decidable or
computable.

The following is an alternative argument which is nonetheless related:

i)
Physics doesn't need (or have) strict proofs.

ii)
Gödel's theorems are primarily about proof.

iii)
Therefore the most important aspect of Gödel's theorems may not be
directly applicable to physics.

More
specifically, one formulation of Gödel incompleteness to physics
doesn't seem to work. Say that the (weak) argument is 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.

As
stated, physics doesn't - strictly speaking - have proofs. (Though
the mathematics included it it may need proofs.) Neither does it

*require*proofs. Secondly and relatedly,*discovering*everything isn't the same thing as*proving*everything. And what sort of claim or aim is it anyway to “discover everything”?
Tangentially,
there is a sense in which the word “incompleteness” can indeed be
applied to problems in physics. Is this Gödel incompleteness? Not
really.

One
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. Again, this has no direct
connection to Gödel incompleteness.

**Physical Laws as Axioms?**

Perhaps
taking the laws of physics as axioms is at the root of the problem.
After all, if one takes physical laws as axioms; then, somewhere
along the line, there may be Gödel incompleteness or inconsistency.

Yet
physical laws both are and are not axioms. For one, they aren't
self-evident or intuitively acceptable. One reason for that is that
physical laws are things which couldn't - even in principle - by
intuitively obvious because intuitions can't apply (in any strict
sense) to laws which govern things which lie outside experience (at
the cosmological or the quantum scale, for example). Added to that,
if physical laws are axioms, and what we derive from laws are
theorems, then what about the unpredictable consequences or
predictions which we derive from our axiomatic laws?

Take
a purely formal logical deduction or argument. In such a thing, we
move “from an incontestable premises to an acceptable
conclusion via an impeccable rule of inference”... Can all that -
in any way whatsoever - be applied to axiomatic physical laws and
their theorems? Indeed is it correct to use the word “axiom” at
all in physics?

In
addition, can any law of physics ever be as simple and as pure as an
axiom in a logical or mathematical system?

There
is another direct consequence of this way of thinking.

Gödel's
theorems require that the axioms of a theory or 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 physical laws bear a strong resemblance to the theorems
which are a derived from the axioms of a logical or mathematical
system? In other words, do we have entailment (or strict deduction)
from physical axioms/laws to physical theorems? Do we have either
metaphysical or logical entailment or deduction when it comes to
axioms/laws and the theorems derived from them?

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