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S15e. Theoretical physics as a formal model of reality
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Can the meaning of all terms in a physical model be determined
precisely without an infinite regress? I want to show that the
answer is a clear `yes'.
Look at the question `What is a force?' To answer this, one needs
to consider the concepts of force, mass, acceleration, pressure,
stress, recoil, perhaps the gravitational field, etc., in total
a small number of physical items. If we want to define them in reality,
we don't get an infinite chain but a circular definition -- we can only
define one in terms of another, illustrating the concepts by pointing
to situations where we hope everything is obvious.
In practice (i.e., in teaching physics), this works alright since
each of us knows reality already
and only needs enough context to identify the usage of the concepts --
there is essentially only one fit that works, and once the
light goes on, we understand -- or at least the level of
understanding deepens. (Later, when doing high precision measurements,
we may notice that our understanding is not adequate,
and become more careful and sophisticated, and at some advanced s
tage one can probably write a whole book to get definitions
that are really precise...)
But there is another way that is fruitful and neither circular nor
infinite. It is obtained by mimicking how modern logic investigates
its foundations. It assumes that we know at the 'external reality'
level what logic is; then it builds a formal model, a 'formal reality',
in which one can talk about everything one talks in 'real' logic,
but in completely formal terms.
You don't need to know what truth, propositions, etc. are in reality,
but you declare the rules for manipulating
with them -- since this is the heart of the matter.
This is done in exactly the same way as the Greeks declared rules for
manipulating geometric terms. In addition, they had definitions like
'a point is what has no parts'; but in modern geometry, this is
considered to be not a well-defined formal statement
(instead it has the circular character of relating the concept
to reality), and hence is simply dropped from the list of axioms.
So modern geometers define a projective plane by a few simple
statements:
''There are points, there are lines, there is a relation which
tells which points are on which lines, through any two distinct
points there is exactly one line, and any two distinct lines
have exactly one common point.''
That's all, and it is enough to do planar projective geometry with
full clarity and completeness. We do not need to know anything about
the objects to analyze a situation
(unless we want to check it's impact on external reality).
Of course, it is good to have a few more restrictions and concepts to
go really deep, but this is supposed to be just an example.
In the same way, one can discuss _everything_ about
the real logic in the formal model of logic, and reach clarity.
It is my proposal to do this for physics as well.
Actually it has been nearly achieved in classical physics,
and fully achieved in Hamiltonian mechanics.
You start with a phase space and a Hamiltonian which fall from
heaven. (They are motivated by circular arguments, but these arguments
are not part of the theory in the formal sense.)
Having this, you can build a whole world, with atoms,
dynamics, paths, forces, accelerations, stress, etc.
In fact, you can discuss any question about the classical world
in this mathematical frame, without ever needing any undefined term.
Formal reality is define by what is expressible in terms of the
concepts already available, and 'true' reality with its circularity
never enters except as a guide to formulating new concepts and to
discuss their consequences.
This is what I think theoretical physics is about.
It builds a formal model of the world, with a 'formal reality',
in which every important concept from experimental physics has
a well-defined formal meaning, and in which every reasonable
question about the physical world can be posed and investigated.
What can be posed and analyzed in such a framework counts
as understood, and understanding of nature increases by bringing
more and more into such a formal model, until everything about
physical nature is representable.
My vision is that the same is possible and desirable for quantum
physics. For me, realizing this vision is
equivalent to having understood quantum physics.
So I want to have a mathematical quantum model of nature,
in which one can talk about all the things physicists talk about
when they talk about nature in the physical sense. In particular,
there will be concepts like particles, fields, detectors, measurement,
probability, memory, etc. but -- unlike in real nature --
they will have a precise and unambiguous formal definition,
of the same formal quality as force, acceleration, etc. are defined
in Hamiltonian mechanics.
Then we can ask about the "meaning" of each term,
and get a well-defined answer within the formalism,
without infinite regress.