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A
second comment about mathematics in physics is that in
the empirical sciences equations and inequalities
express universal logical quantification, which is
changed to particular logical quantification when any
of their constituent variables are given numerical
values by measurement, or by calculation with the
equation whose initial conditions have been satisfied
by measurement. Each
measurement performance is an individual measurement
instance, which is not the same as an individual
entity. Different
measurement performances will likely result in different
measurement values, either due to measurement errors
in the execution of the measurement procedures
described in the statements of an experimental design,
or due to different initial conditions in the ranges
of the variables.
An equation is not given particular logical
quantification merely by associated inequalities
expressing limits on the range of possible values of
its variables, as in the indeterminacy relations in
quantum theory. Such
conditions may severely restrict the range of
applicable values.
But there is always a universal claim made by
the equation, because there are an indefinitely large
number of possible measurement performances in
repeatable experiments.
Also equations are not given particular
quantification, when only their parameters such as
constant coefficients are given specific values.
Furthermore no mathematically expressed theory
need be a more "general theory" relative to
any other theory, in order to be a logically universal
statement. In
the case of quantum physics the microphysical quantum
theory need not also be a macrophysical theory, in
order to be a universal theory, just as it need not be
a biological or a sociological theory, in order to be
a logically universal expression.
Logical reductionism is not a condition for
logical universality.
Therefore equations can express universality
like substantive statements in logic, and they may be
included in semantical description lists to add to the
semantics of a meaning complex associated with a term
and to reduce the meaning’s vagueness, if the term
is a mathematical variable.
To summarize this alternative to the wholistic
view: the meanings associated with descriptive terms
are not simple wholes, but are complexes having
component parts, which in turn are meaning complexes
associated with other descriptive terms to which the
former are related in universal affirmations believed
to be true. The
composite meaning complex is exhibited in a semantical
description, which consists of a list of one or
several universal affirmations having a common subject
term with its component parts exhibited as predicates.
The universal affirmations may also include
mathematical expressions with descriptive terms
appearing as variables.
Vagueness is a property of concepts, and exists
to the extent that descriptive terms cannot be related
to one another in universal statements believed to be
true. Vagueness
limits the size of a semantical description list, and
thus also limits the propagation of semantical change
through the web of beliefs.
Consider next the relation between the language
of observation and the language of theory, the second
basic assumption in the doctrine of closed-off
theories. The
word "theory" is still used conventionally
to refer to Newton's "theory" of gravitation,
to Einstein's "theory" of relativity, and to
the quantum "theory", even though the
physics profession had decided many years ago either
to accept or to reject these expressions as physical
explanations. In
this conventional usage the term "theory"
does not have the same meaning as it did when these
expressions were firstly advanced for testing as
proposed explanations of problematic phenomena. When they were firstly proposed, these expressions
represented statements that had a much more
hypothetical status in the judgment of the profession
than they do today, and they were typically topics of
controversy. There
is, therefore, an ambiguity between "theory"
understood as an accepted or rejected explanation, and
"theory" understood as a tentative proposal
submitted for empirical testing.
Unlike the former archival understanding, which
Hanson calls "almanac" science, the latter
Pragmatist understanding of "theory", which
Hanson calls “research science”, pertains to the
function of language at the frontier of the
development of a science, where the function under
consideration is empirical testing.
Only this latter understanding is strategic in
the Pragmatist philosophy of science, even though the
former meaning and still conventional usage may occur
in its expository discourse.
From this functional or Pragmatist view
theories may be defined as universal statements that
are proposed for testing, and explanations are former
theories that have been tested and not falsified.
Theory that is tested and not falsified by a
competent test ceases to be a theory and is given the
status of an explanation, even though there may be
other tested and nonfalsified former theories
addressing the same problem also having the status of
explanations accepted by some scientists in the same
profession. Some
scientists are uncomfortable with this pluralism, but
the contemporary Pragmatist philosophers recognize
such pluralism as historically characteristic of
science.
In an empirical test the semantics of the
vocabulary in all the relevant discourse is controlled
by a strategic decision that is antecedent to the
performance of the test.
This is the decision as to what statements are
presumed for testing and what statements are proposed
for testing. The
former language is the explicit statements of test
design together with usually many tacit assumptions,
and the latter is the explicit statements of the
theory. This decision is entirely pragmatic, since it is not based on
the syntactical or the semantical characteristics of
language, but rather is based on the use or function
of the language, namely empirical testing.
The test design statements are those that by
prior decision and agreement among cognizant members
of the profession have the status of definitions.
These statements are presumed to be true
regardless of the outcome of the test, and serve to
identify the subject of investigation throughout the
test. The
theory is the language that by prior decision and
agreement among the cognizant members of the
profession has the less certain status of a
hypothesis. The
hypothesis is believed to be true to the extent that
it is considered worthy of testing, although the
proponent and his entourage of cheering advocates may
be quite firmly convinced. But if the test outcome is a falsification, then it is the
statements of theory and not the statements of test
design, that are judged to have been falsified.
However, a falsification may lead some
interested scientists, such as the theory's proponent
and advocates, to reconsider the beliefs underlying
the test design, even while admitting that the test
was executed in accordance with its design.
This role reversal between test design and
theory may result in productive research.
When the falsified theory is made a test design
statement characterizing the problematic phenomenon,
the problem has become reconceptualized.
As Conant discovered to his dismay, the history
of science is replete with such prejudicial responses
to scientific evidence that have been productive and
strategic to the advancement of basic science in
historically important episodes.
The decision distinguishing test design and
theory language made prior to the experiment may but
need not result in identifying mathematical equations
as the statements of theory and of identifying
colloquial discourse or substantive language as the
statements of test design.
The decision is not based on syntactical
characteristics of the language, and the test design
statements often include mathematically expressed
statements together with statements in substantive
language describing the measured phenomenon, the
measurement procedures, and the design and operation
of the measurement apparatus.
Even more relevantly the decision is not based
on semantical criteria, as advocates of the
naturalistic philosophy of the semantics of language
believe. The
decision is not based on any purportedly inherent
distinction between observation and theory, whether or
not, as in the case of quantum theory, the observation
concepts are called "classical" or
"macroscopic", and the theoretical concepts
are called "quantum" or
"microscopic". The distinction between statements of test design and
statements of theory is neither syntactical nor
semantical; it is distinctively and entirely
pragmatic.
Consider the language of an empirical test
before the test is executed.
In order for the test design statements to
characterize evidence independently of the theory
proposed for testing, the test design statements and
the theory statements must be logically independent.
Neither set of statements may be merely a
transformation of the other, and the test design
statements may neither deductively imply nor
contradict the theory or any of its alternatives.
Furthermore the statements of the theory are
too hypothetical to function as definitions, except
perhaps for the proponent and other advocates of the
theory, who may believe in the theory as strongly as
they believe in the truth of the test design
statements. But
for all those critical researchers for whom the test
is a decision procedure, the semantical consequence of
the logical independence and hypothetical status of a
theory relative to the universal statements of test
design, is that each of the terms common to both the
test design statements and theory statements have
their semantics defined in relation to the meanings of
the other terms in the test design statements, such
that they characterize the subject matter of the
experiment, but not defined in relation to the
meanings of the other terms of the theory.
In other words the theory statements are not
included in the same semantical description list as
the test design statements, even though both sets of
statements are mutually consistent and contain the
same common subject term.
The meaning of each term common to the test
design and theory statements is therefore vague with
respect to the meanings of the other terms of the
theory. And
on the artifactual thesis of the semantics of language
the observation language in turn is merely the test
design statements with their logical quantification
changed from universal to particular, to enable their
use to describe the particular ongoing or historical
experiment performance.
The test design statements similarly supply the
vocabulary that describes the observed test outcome,
even if the outcome contradicts the claims of the
tested theory, thus falsifying the theory.
Consider next the language of the empirical
test after the test is executed.
When the test is executed, a falsifying test
outcome produces no semantical change except for the
proponents and advocates of the theory, who had been
convinced of the theory’s truth, and then choose to
reconsider their belief in the theory due to the test
outcome. But
a nonfalsifying outcome produces a semantical change,
especially for the critics of the theory for whom the
test is a decision procedure.
After the test the theory no longer has the
hypothetical status that it formerly had merely as a
proposal, but assumes the status of an explanation,
which is neither more nor less contingent than other
accepted universal empirical statements including
the test design statements.
The semantical outcome is that both the test
design statements and the theory statements (now
elevated to the status of an explanation) are
semantical rules exhibiting the composition of the
meanings of the univocal terms common to both sets of
statements. Those component parts defined by the test design statements
remain unchanged.
But the semantical descriptions for these terms
now include not only the test design statements but
also the statements constituting the tested and
nonfalsified theory.
These theory statements are additional
information learned from the test outcome that
resolves some of the vagueness in the vocabulary terms
common to both the theory and the test design
statements. In
summary: the descriptive terms common to both test
design and theory statements have part of their
semantics defined by the test design statements
throughout the test, both before, during, and after
the test is executed. And these common terms have part of their semantics augmented
and thus defined by the statements of the tested and
nonfalsified theory added after the test.
In Heisenberg's doctrine of closed-off theories
the naturalistic philosophy of language requires
retention of the Newtonian concepts for observation in
the context of the quantum theory.
But the resulting equivocation is unnecessary,
if it is remembered that the Newtonian concepts are
never involved, since the Newtonian theory is a
falsified microphysical theory.
It is sufficient to use a less precise
vocabulary that Heisenberg calls "everyday"
words used by physicists in order to describe the
experimental set up, which is a macrophysical
phenomenon. The
meanings of these everyday concepts are vague about
the fundamental constitution of matter.
After the quantum theory was recognized as
experimentally adequate, the vagueness in these
everyday concepts was resolved by the equations
constituting the statements of the quantum theory,
because the quantum theory is the tested and
nonfalsified theory, which after the test became a
semantical rule contributing meaning parts to the
complex meanings of these univocal terms.
For this resolution of vagueness to occur it is
not necessary for the Newtonian macrophysical laws to
be made logical extensions of the quantum theory by
logical reduction procedures, because the Newtonian
theory is falsified as a microphysical theory.
Nor is it necessary for the Newtonian
macrophysical laws to be replaced by a macrophysical
theory that is an extension of the quantum laws.
The univocal semantical thesis neither implies
nor requires Hugh Everett's "many worlds"
interpretation (which furthermore is in principle
empirically untestable), nor does it imply or require
any other reductionist development of a
macrophysical quantum theory, i.e. a macrophysical
theory which is deductively or reductively a logical
extension of the microphysical quantum theory.
It is sufficient merely that the scientist
realize that the nonfalsifying test outcome has made
the quantum theory and not classical physics an
empirically warranted microphysical theory.
Heisenberg's doctrine of closed-off theories is
incorrect, and Einstein's semantical thesis expressed
in his admonition to Heisenberg is correct, because
the vocabulary used for observation after the quantum
theory's acceptance is a univocal vocabulary with
meaning parts from the quantum theory.
The descriptive terms in the equations of the
quantum theory contribute to, and thereby resolve some
of the vagueness in, the meaning complex associated
with the descriptive terms used for observation.
Thus the quantum theory decides what the
scientist observes in the Wilson cloud chamber. The macrophysical description is not in contradiction to the
microphysical quantum theory including the
indeterminacy relations.
The quantum concepts are included in the
univocal meaning complexes associated with the
observation description.
The Newtonian concepts were never included,
because the macrophysical description never affirmed a
Newtonian microphysical theory.
In his "Remarks on the Origin of the
Relations of Uncertainty" in a memorial volume
dedicated to him titled The
Uncertainty Principle and Foundations of Quantum Mechanics
(1977), which was in press at the time of his death in
1976, Heisenberg concludes the brief four-page article
by saying that there have been attempts to replace the
traditional language with its classical concepts by
a new language which should be better adapted to the
mathematical formalism of quantum theory.
But he adds that during the preceding fifty
years, physicists have preferred to use the
traditional language in describing their experiments
with the precaution that the limitations given by the
uncertainty relations should "always be kept in
mind". He concludes that a "more precise"
language has not been developed and in fact it is not
needed, since there seems to be general agreement
about the conclusions and predictions drawn from any
given experiment in the field.
Regrettably Heisenberg never repudiated his
doctrine of closed-off theories.
But contrary to his doctrine of closed-off
theories, Heisenberg’s statement that the
contemporary physicist must keep quantum effects
"in mind" when the physicist is describing
macrophysical objects, even while not explicitly
accounting for quantum effects that are experimentally
undetectable in the circumstances, is prima
facie evidence of a semantical change in the
univocal macrophysical vocabulary used to describe
experiments due to the development of quantum theory.
In other words a “more precise” language
with a less vague semantics has in fact evolved.
This semantical evolution consists in the fact
that the concepts employed for description contain
component parts from the quantum theory. That is how the limitations of the uncertainty relations are
“always kept in mind”: they have become built into
the semantics of those terms, even when those terms
are used to describe observations.
Heisenberg’s semantical theory of
equivocation is the result of the acceptance of the
naturalistic philosophy of the semantics of language
together with the assumption that meanings are simple,
indivisible wholes. However, all such views are
untenable, because they imply what can only be called
"double think".
The equivocation thesis demands that the modern
physicist indulge in a contrived cognitive duplicity
with himself, a pretext of simultaneously both knowing
and not knowing the modern quantum theory.
But concepts are not known like physical
objects to which one may simply close one’s eyes;
they are
knowledge. Scientists never did in practice carry on the kind of
cognitive duplicity that the equivocation semantical
theses require, and since the ascendancy of the
contemporary Pragmatism, philosophers no longer expect
that they should.
Heisenberg might have obtained greater utility
from his insightful idea of “everyday” concepts,
had he rejected Bohr’s philosophy of observation
language, and realized that neither these everyday
concepts nor the Newtonian concepts nor any other
concepts are inherently observational.
In the Pragmatist perspective “everyday”
concepts are distinctive only because they are vague
in a very strategic fashion: they are the concepts
used in test design statements, and are vague relative
to the concepts in the theories proposed for testing
prior to execution of the test and prior to the
production of a nonfalsifying test outcome.
In the case of the quantum theory experiments,
everyday concepts are vague because they are not
defined by the axiomatic systems of either the
Newtonian or the quantum theories or of any other
proposed microphysical theory prior to the execution
of the tests.
In “On the Methods of Theoretical Physics”
in Ideas and
Opinions (1933) Einstein said that if you want to
find out anything from the theoretical physicists
about the methods they use, stick closely to one
principle: don’t listen to their words, but rather
fix your attention on their deeds.
This might be construed as good advice for
anyone attempting to understand Heisenberg’s
philosophy of science.
The philosophy of science that Heisenberg
practiced as a result of Einstein's influence and
chronicled in his autobiographical works, is
historically more important than the one he expounded
as a result of Bohr's influence and set forth as his
doctrine of closed-off theories.
In the philosophy he practiced, he anticipated
the contemporary Pragmatist philosophy of science by
at least a quarter of a century.
Because contemporary Pragmatism is based on an
artifactual view of the semantics of language, it
implies the interdependence of belief and semantics.
On the naturalistic view of semantics, the
truth of empirical statements is dependent on meanings
which in turn are determined independently by the
natural processes of perception which are thought to
capture some presumed or preferred ontology.
On the artifactual view truth and meaning are
mutually determining in empirical statements believed
to be true. Thus
on the artifactual thesis, when Heisenberg used
Einstein's admonition for developing the uncertainty
relations, he had made a prior commitment to the
empirical truth of the equations expressing them, and
he then used their concepts for his theory-dependent
observation of the Wilson cloud chamber tracks made by
the free electron.
Years later Feyerabend will call this
counterinduction even though he seems never to have
recognized its occurrence in Heisenberg's practice.
Had he done so, he might have spoken of the
Galileo-Einstein-Heisenberg tradition, and produced a
philosophy of quantum theory different from than his
relativist philosophy.
Furthermore, given the Pragmatist thesis of
scientific realism, the thesis that empirically
warranted discourse has a semantics describing an
ontology, the artifactual philosophy of the semantics
of language implies the mutual determination of truth
and ontological claims.
In scientific realism, however, the truth is
the empirical warrant earned by testing with no
falsification. Heisenberg
too practiced this, when unlike Bohr, he reported that
he construed the quantum mechanics realistically when
he imitated Einstein’s realism.
He said the decisive step in the development
of special relativity was Einstein's rejection of the
distinction between apparent time and actual time in
the interpretation of the Lorentz transformation equation,
and his taking Lorentz’s apparent time to be
physically real time while rejecting the Newtonian
absolute time as real time.
He said he took the same kind of decisive step,
when he inverted the question of how to pass from an
experimentally given situation to its mathematical
representation, by affirming that only those states
represented as vectors in Hilbert space can occur in
nature and be realized experimentally. This is not a
positivist claim about mere phenomenal appearances; it
is an ontological claim backed by the theory’s
empirical warrant as tested and not falsified. Unlike
Bohm’s hidden variable ontology Heisenberg’s
realist step did not involve creating
“interpretations” by supplementing the quantum
theory with additional characterizations that are not
affirmed by the tested and nonfalsified theory.
Nor is it like Bohm’s informal
characterizations that are separable from the quantum
theory, and which enable no new empirical tests or new
predictions, and therefore offer no empirically
warranted ontological claims.
Similarly for the potentia ontology, if contrary to Heisenberg potentia is taken to refer merely to the fact that position and
momentum measurement instances cannot both occur
simultaneously, such that they may be taken
realistically as a manifestation of reality - rather
than supplementary claims about entities, which cannot
even be referenced in mathematical measurement
language. Initially the quantum theory’s stark and
spartan realism was strange, and “wave” was a
metaphor. But
metaphor is new and unconventional meaning that
convention converts to dead metaphor, i.e. new literal
meaning. Contemporary
realists can let the theory and experiments change the
old literal meanings to new literal meanings, as
continuing experimental research will further enrich
the theory’s descriptive semantics and ontology.
Today’s pragmatic scientific realism is the
thesis that a theory’s ontology - its view of
reality - is described by the semantics of language
that is defined by the context of universal discourse
accepted as empirically true. This language
characterizing manifestations of mind-independent
reality includes the empirically tested and
nonfalsified theory, all its associated test-design
language needed for measurement and/or other
observation, and descriptions of additionally related
experimental findings. Unlike both the Bohm and the
Bohr interpretations this language characterizing
manifestations of reality neither affirms nor denies supplemental ontological claims that
lack of warranting empirical evidence.
Heisenberg’s practice of scientific realism
anticipates Quine’s doctrine of ontological relativity
and also Quine’s rejection of all first philosophy,
namely the use of prior ontological commitments such
as Einstein's commitment to determinism as criteria
for scientific criticism.
Contemporary Pragmatists like Quine reject
ontological criteria in scientific criticism.
They admit only empirical criteria for
scientific criticism, and let the statement of the
empirically tested and nonfalsified theory describe
both the semantics and the ontology of the theory's
domain.
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