Table Of ContentTOPICS IN THE FOUNDATION
OF STATISTICS
Edited by
Bas C. van Fraassen
Department ofP hilosophy,
Princeton University
Reprinted from
Foundations of Science
Volume 1, No.1, 1995/96
Springer-Science+Business Media, B.V.
A C.I.P. Catalogue record for this book is available from the Library of Congress
ISBN 978-90-481-4792-2 ISBN 978-94-015-8816-4 (eBook)
DOI 10.1007/978-94-015-8816-4
Printed on acid-free paper
Prepared with pennission of Oficyna Akademicka
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© 1997 Springer Science+Business Media Dordrecht
Originally published by Kluwer Academic Publishers in 1997
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CONTENTS
FOUNDATIONS OF SCIENCE - DISCUSSION: 5-18
Bas van Fraassen A Philosophical Approach to
Foundations of Science
Patrie Suppes A Pluralistic View of Foundations of Science
Arne Collen The Foundation of Science
THE FOUNDATION OF STATISTICS:
David Freedman Some Issues in the Foundation
of Statistics 19-39
Comments by James Berger, E. L. Lehmann,
Paul Holland, Clifford Clogg, Neil Henry 41-67
David Freedman Rejoinder 69-83
Diedrik Aerts Applications of Quantum Statistics in
and Sven Aerts Psychological Studies of Decision Proc~sses 85-97
Maria Carla Operationism, Probability and Quantum
Galavotti Mechanics 99-118
Paul Humphreys Computational Empiricism 119-130
VARIA:
Joseph Agassi Blame Not the Laws of Nature 131-154
***
Biographical Notes 155-157
On AFOS (some history of the initiative
and some recent developments) 159-160
Joseph Agassi Summary of AFOS Workshop, 1994 161-166
Foundations of Science
1 (1995/96), 5-18
FOUNDATIONS OF SCIENCE - DISCUSSION
Bas van Fraassen
Department of Philosophy
Princeton University
Princeton, N J 08544, USA
A PHILOSOPHICAL APPROACH TO FOUNDATIONS OF SCIENCE
Abstract. Foundational research focuses on the theory, but theories ar'! to be related
also to other theories, experiments, facts in their domains, data, and to their uses in
applications, whether of prediction, control, or explanation. A theory is to be identified
through its class of models, but not so narrowly as to disallow these roles. The language
of science is to be studied separately, with special reference to the relations listed above,
and to the consequent need for resources other than for theoretical description. Peculiar to
the foundational level are questions of completeness (specifically in the representation of
measurement), and of interpretation (a topic beset with confusions of truth and evidence,
and with inappropriate metalinguistic abstraction).
In foundational research, focus is on the theory, that is, on the prod
uct of theoretical activity in science, as opposed to, for example, on the
processes of theory testing, choice, evaluation, confirmation, or historic and
other contextual factors that may playa role in such theorizing.
Nevertheless, such research pertains not only to the structure of the
theory itself, but also to its relations to other theories, to the facts in the
domain of the theory, to experiment, to data obtained from experiment and
observation, and to its use in applications, whether of prediction, control, or
explanation.
1. What is a theory? Here I am in agreement WIth Patrick Suppes, that
we must distinguish the theory from its formulations. Thus a set of axioms
or theorems cannot be the theory; the theory is something which admits
B. C. van Fraassen (ed.), Topics in the Foundation of Statistics
© Springer Science+Business Media Dordrecht 1997
6 Bas van Fraassen
of many alternative formulations, which may indeed be given in alternative
languages with different vocabularies and even different logical resources. For
this reason it is is best to identify a theory through a class of mathematical
structures, its models, which can be described in those various ways.
At the same time, however, a theory gives information about the world,
and may be believed or doubted. Therefore, although a theory may be
identified through a class of models, and may even (in accordance with logical
and mathematical practice) be identified with that class, there is clearly more
to it. For a class is not the sort of thing that gives information or may be
believed or doubted.
Historically, this point has sometimes been made by saying that scien
tific theories are not "uninterpreted objects" such as are studied in pure
mathematics. That is not a good choice of words, it seems to me, and may
be misleading. It suggests that a scientific theory could be instead a com
posite, something abstract together with something else that interprets that
abstract thing. But then what is that something else? Is it too something
abstract, like a function? If so we have the same problem all over again - we
have replaced one abstract thing by two. On the other hand if it is suggested
that a function needs no interpretation, though it is an abstract thing, then
why did the first abstract thing need interpretation? If on the third hand the
something else is not an abstract thing at all, but something concrete, we
shall have to study it (scientifically, in the way all concrete things are to be
studied) - and how shall we do that, if not by constructing a mathematical
model of it?
Therefore I prefer to say that a theory can be identified through its class
of models. These models are the main subject of study for foundational
research, taken in its strictest sense now, pertaining to that theory. But the
theory states that these models include a correct representation of what there
is (or of what the theory is about). This statement (by which I do not mean
any linguistic formulation, but rather what all formulations of the theory
"say") is subject to assent or doubt. It is also subject to interpretation prior
to any assent or doubt. To this role of interpretation I shall return below.
We must also note here the importance and propriety of Suppes' call
to make mathematics rather than meta-mathematics the tool of philosophy
of science with respect to foundational discussions. The class of models is
a class of mathematical structures, described in any way mathematics al
lows. Mathematical description has its limits of course, in that mathematics
describes its subject matter only up to isomorphism.
2. Logical positivism and logical empiricism began, somewhat unfortu-
A Philosophical Approach to Foundations of Science. 7
nat ely, by assimilating the relation between a theory and its domain to the
relation of theory to evidence. Both are important, and to some extent they
overlap. Those relations of theory to experiment that fall under such epis
temic headings as confirmation and testing I do not count as belonging to
the area of foundations.
However, when we consider the relation of the theory to fact (to its do
main, to the world, however we want to phrase this), it is important to ask
whether the theory itself models the processes of measurement or experi
ment, or whether those are described in terms of some other theory. This
becomes crucial for those theories in physics that are potentially universal
theories, theories of everything, such as quantum mechanics and general rel
ativity. When the adequacy of the theory requires it to have models which
correctly represent certain phenomena, and this adequacy can be ascertained
only through certain types of processes, we face a potential consistency prob
lem. Does the theory represent not only those phenomena but also those
processes which give experimental access to those phenomena? If so, do
the two sorts of representations 'mesh'? This is the form of the measure
ment problem in quantum mechanics, but may crop up in any foundational
discussion, in some form or other.
3. Uses of a theory comprise prediction and explanation. Both of these
are topics that tend to lead us from foundations into problems concerning
the language of science. Prediction requires the user to relate himself to some
model or models of the theory, in a way that is analogous to someone who
specifies his own location on a map, or gives the co-ordinates of his location,
or specifies the co-ordinate system which is 'rigidly attached' to him. The
linguistic resources to do this do not belong to the purely theoretical language
which is sufficient for the presentation of the theory, but to the language of
applied science. Since the description of the models is mathematical, this
co-ordination with the theory required for an application may involve more
than specifying the user's own co-ordinates. For example, he may wish to use
a certain formal theory to predict thermal phenomena, while the equations
of that theory describe equally well e. g. gas diffusion.
It seems to me that Toulmin's analogy between a theory and a map is
quite apt here, even if he used it differently. A circuit diagram might, by
coincidence, share some structure with a roadmap or railway map, or a map
of one city might share some structure with that of another. In that case
I can use one in place of the other, by ignoring its origin, paying attention
only to the relevant structure, and locating myself on it in a different way.
There is a certain limitation of language that is evident here, where theory
8 Bas van Fraassen
and practice meet. For describing the act of self-location on the map is very
different from doing it, and until it is done, conditions for the use of the map
are not met.
The use of a theory in explanation may require even more, at least if
accounts of explanation in terms of the logic of questions are correct. Issues
of completeness for theories - such as the much discussed completeness of
quantum mechanics issue - relate solely to the expressive power of the the
oreticallanguage, i. e. the language in which the models and the domain
of the theory are described. Use of the theoretical description as explana
tion, however, may require attention to very specific interests and concerns
characterizing the context of inquiry. The very same description may be
able to play the role of explanation in one context and not in another - so
explanatory completeness does not follow from expressive completeness.
4. Some of the problems touched on under the above headings do not
pertain to foundations of the sciences at all, although they belong to philoso
phy of science or in some cases to science. In addition, some of the problems
indicated were left dangling, so to speak. I will here add comments only
about one. Among the relations of a theory to its domain there are those
which determine whether or not the theory is true. Prior to the question
of truth, however, are questions of interpretation and of possibility. The
question of truth does not arise except relative to an interpretation. On a
purely theoretical level we can investigate which interpretations the theory
admits, and thereby answer the question: how could the world (domain,
fact) possibly be the way this theory says it is? That it is not easy to answer
this question in general is clearly due to two factors. The first is the diffi
culty of developing any interpretation at all for sufficiently complex theories
presented in ma.thematical form. The second is the further demand that
the question be answered through an interpretation on which the theory is
compatible with other accepted parts of science. In the case of quantum me
chanics, for example, we have found that it admits no interpretations which
cohere with certain traditional assumptions about causality, while on the
other hand, interpretations that imply the existence of superluminal signals
are not generally considered satisfactory either.
In conclusion I would like to draw attention to ongoing work to elaborate
on the conception of foundational research in the sense of section (1) above,
which received its original impetus from the work of Suppes. Some history
is provided by way of introduction in Fred Suppe (Suppe, 1989); a recent
contribution to be noted is (da Costa and French, 1990).
References
Da Costa, N.C.A. and French, S. (1990), The model-theoretic approach in
the philosophy of science, Philosophy of Science 57, 248-265.
Suppe, F. (1989), The Semantic Conception of Theories and Scientific Re
alism. Urbana: University of lllinois Press.
Patrick Suppes
Stanford University
Stanford, CA 943005, USA
A PLURALISTIC VIEW OF FOUNDATIONS OF SCIENCE
Before writing this I read Bas van Fraassen's statement. I find little I
can disagree with in what he says. Consequently, what I would like to do
is address several clusters of concepts and issues that are often somewhat
neglected in discussions of the philosophical foundations of science.
I. Epistemology of Experiments
Bas rightly emphasizes the importance of going beyond models of theories
in analyzing the content of science. What I want to stress is how far the whole
activity of experimentation, including its reporting, is from the standard
theoretical model of science, which Bas and I substantially agree about.
The first point is that experimental reports, much more than theoreti
cal papers, are not generally understandable. They are like detailed sports
reporting, intelligible only to the initiated. There is, of course, the insis
tence of journals that reports be as brief as possible, and certainly there
is no requirement they be generally intelligible. However, I think there is
a deeper epistemological reason, which also applies to sports reporting. It
is not possible to describe in ordinary language, even augmented by some
technical terms, the backhand stroke of a tennis player with any accuracy.
Similarly, it is not possible to describe in language the many activities of an
experimentalist. This applies to his actions setting up and running experi
ments, but also to his perceptions in digesting the results in their first "raw"
References
Da Costa, N.C.A. and French, S. (1990), The model-theoretic approach in
the philosophy of science, Philosophy of Science 57, 248-265.
Suppe, F. (1989), The Semantic Conception of Theories and Scientific Re
alism. Urbana: University of lllinois Press.
Patrick Suppes
Stanford University
Stanford, CA 943005, USA
A PLURALISTIC VIEW OF FOUNDATIONS OF SCIENCE
Before writing this I read Bas van Fraassen's statement. I find little I
can disagree with in what he says. Consequently, what I would like to do
is address several clusters of concepts and issues that are often somewhat
neglected in discussions of the philosophical foundations of science.
I. Epistemology of Experiments
Bas rightly emphasizes the importance of going beyond models of theories
in analyzing the content of science. What I want to stress is how far the whole
activity of experimentation, including its reporting, is from the standard
theoretical model of science, which Bas and I substantially agree about.
The first point is that experimental reports, much more than theoreti
cal papers, are not generally understandable. They are like detailed sports
reporting, intelligible only to the initiated. There is, of course, the insis
tence of journals that reports be as brief as possible, and certainly there
is no requirement they be generally intelligible. However, I think there is
a deeper epistemological reason, which also applies to sports reporting. It
is not possible to describe in ordinary language, even augmented by some
technical terms, the backhand stroke of a tennis player with any accuracy.
Similarly, it is not possible to describe in language the many activities of an
experimentalist. This applies to his actions setting up and running experi
ments, but also to his perceptions in digesting the results in their first "raw"
B. C. van Fraassen (ed.), Topics in the Foundation of Statistics
© Springer Science+Business Media Dordrecht 1997
10 Patrick Suppes
form. The written account can only hint at the main features of what is done
or what is observed.
The central epistemological point of these remarks is that this radical
incompleteness of descriptions of experiments is not a mark of bad science,
but is an essential, unremovable feature of almost all science. The drastic
descriptive limitation of what we have to say about experiments is, in my
view, a fundamental limitation of our scientific knowledge, possible or actual.
Moreover, this radical incompleteness of the experimental reporting leads
to more appeals to authority in experimental work than in theoretical work.
It is common to hear, in every part of experimental science: "Well, we
know those results are right and can be trusted because we know X and his
lab." Of course, there is the answer that experiments can be repeated by
others, and this is the great empirical check against being overwhelmed by
authority. But it is still an important point, which can be easily amplified
by various historical examples, that in many respects theoretical results can
be evaluated for error much easier and more directly than experimental
work. To bring this up to current science, computational experiments and
simulations need to be included as well.
Part of the epistemology of experiments recognized by everyone is the pres
ence of experimental error, but the theory of error has not crept into the
philosophical foundations of science, but remains on the ground floor of ac
tual experimental work, with only an occasional philosophical nod to its
importance. Yet error is a central concept of a proper general epistemology,
and, on the other hand, has a long technical history of theoretical develop
ment, at least since the early work of Simpson in the 18th century. Its con
ceptual place in science, however, is, in my view, still not fully recognized.
Let me give just one personal example. I have spent many years working on
the foundations of measurement, and I recently went to a gathering of the
measurement theorists' "clan" in Kiel, Germany. It was generally agreed
that a really proper inclusion and analysis of error in foundational theories
of measurement is the number one general problem.
Finally, I emphasize, as I have before, the 'hierarchy of models used in
the analysis of experimental data. Modern statistics has developed within
the set-theoretical view of mathematics, as can easily be seen by perusing
the pages of The Annals of Statistics and other leading journals. The many
levels of data reduction usually needed to get to detailed statistical analysis
is an epistemological problem, not a statistical problem as such, and needs
more philosophical analysis, with close attention to the varying practices
in different parts of science. Skepticism about explicit objectivity holding
"all the way down" is certainly one of the reasons for the recent increase
