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\section{\usemenu{slacpub7205::context::slacpub7205006}{A Model for Scientific Investigation}}\label{section::slacpub7205006}
I restrict our formalism so that it can serve as an abstract
model for physical measurement in the following way.
We assume that we encounter entities one at a
time, save an entity so encountered, compare it with the next entity
encountered, decide whether they are the same or different, and record
the result. If they are the same, we record a ``0'' and if they
are different we record a ``1''. The first of the two entities
encountered is then discarded and the second saved, ready to be
compared to the next entity encountered. The recursive pursuit
of this investigation will clearly produce an ordered string of ``0''
's and ``1'' 's, which we can treat as a bitstring. We further assume
that this record  which is our abstract version of a {\it laboratory
notebook}  can be duplicated, communicated other investigators, treated
as the input tape for a Turing machine, cut into segments which
can be duplicated, combined and compared using our bitstring
operations, the results recorded, and so on.
Our second assumption
is that if we cut this tape into segments of length $N$ and determine
how many such segments have
the Hamming measure $a$, the probability we will find
the integer $a$, given $N$ will approach $2^{N}{N!\over a!(Na)!}$
in the sense of the {\it law of large numbers}. Without further tests all
such strings characterized by the two integers $a\leq N$ will be
called {\it indistinguishable}. It should be obvious that I make this
postulate in order to be able to, eventually, derive the Planck
black body spectrum from my theory. Remember that Planck's formula has
stood up to all experimental tests for 97 years, a remarkable
achievement in twentieth century physics! We have recently learned that
in fact it also represents
to remarkable precision the cosmic background radiation at $2.73 ^oK$.
For those who want to know how and why
the quantum revolution started with the discovery of Planck's formula, rather
than just myths about what happened, I strongly recommend Kuhn's last major
work\cite{41}.
Any further structure coming out of our investigation is to
be found using the familiar operation of {\it discrimination} ${\bf
a}\oplus {\bf b}$ between two strings ${\bf a}$, ${\bf b}$ of equal length,
by {\it concatenation} ${\bf a}\Vert {\bf b}$ (which doubles
the string length for equal length strings), and by taking the {\it Dirac inner
product} ${\bf a}\cdot {\bf b}$ which takes two strings out of the category of
{\it bitstrings} and replaces them
by a positive integer. This third operation is {\it also} how we
determine the Hamming
measure of a single string: ${\bf a}\cdot {\bf a}\equiv a$. It will become
our abstract version of {\it quantum measurement}, which we interpret
as the determination of a {\it cardinal}.
Clearly the category change between ``bitstring'' and ``integer'' is needed
if we are to have a theory of {\it quantitative measurement}.
I take this to be the hallmark of {\it physics} as a science.
The category change produced by taking the inner product
allows us to relate two strings which combine
by discrimination to the integer equation:
\begin{equation}
2{\bf a}\cdot {\bf b}= a+b({\bf a\oplus b})\cdot ({\bf a\oplus b})
\end{equation}
If it is taken as axiomatic that (a) we can know the Hamming
measure of a bit string and (b) that this implies that
we can know the Hamming measure of the discriminant between two
bitstrings, then this {\it basic bitstring theorem}
seems very natural.
Once we start combining bitstrings and recording their Hamming
measures, and in particular writing down sequential records of these integers,
the analysis clearly becomes {\it context sensitive}. It is our
abstract model for a {\it historical record}.
The underlying philosophy
is the assumption that in appropriate units {\it any} physical
measurement can be abstractly represented by a positive integer
with an uncertainty of $\pm {1\over 2}$. If we were using real numbers,
this would be expressed by saying that the value of the physical
quantity represented by $n\pm {1\over 2}$ has a 50\% chance of
lying in the interval between $n{1\over 2}$ and $n+{1\over 2}$.
But in discrete physics, such a statement is {\it meaningless}
in the sense used by operationalists. Clearly, part of our conceptual
problem is to develop a language describing the uncertainty
in the measurement of integers
which does {\it not} require us to construct the real numbers.
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