Table Of ContentSTUDIES IN LOGIC
AND
THE FOUNDATIONS OF MATHEMATICS
VOLUME 80
Editors
H. J. KEISLER, Madison
A. MOSTOWSKI, Warszawa
A. ROBINSON, New Haven
P. SUPPES, Stanford
A. S. TROELSTRA, Amsterdam
Advisory Editorial Board
Y. BAR-HILLEL, Jeritsalem
K. L. DE BOUVEYE, S U ~ ~tiUar a
H. HERMES, Freihurg i. Br.
J. HINTIKKA. Helsinki
J. C. SHEPHERDSON, Bristol
E. P. SPECKER, Ziirich
NORTH-HOLLAND PUBLISHING COMPANY -AMSTERDAM . OXFORD
AMERICAN ELSEVIER PUBLISHING COMPANY, 1NC.-NEW YORK
LOGIC COLLOQUIUM '73
PROCEEDINGS OF THE LOGIC COLLOQUIUM
BRISTOL. JULY 1973
-~
Edited by
H. E. ROSE
Lecturer in Mathematics, Bristol
and
J. C. SHEPHERDSON
Professor of Pure Mathematics, Bristol
1975
NORTH-HOLLAND PUBLISHING COMPANY- AMSTERDAM . OXFORD
AMERICAN ELSEVIER PUBLISHING COMPANY, 1NC.-NEW YORK
@ NORTH-HOLLAND PUBLISHING COMPANY-197s
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0 7204 2282 5
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PREFACE
This volume contains most of the proceedings of a conference on
mathematical logic held under the auspices of the Association for Symbolic
Logic in July 1973 in Bristol, England. The main areas of interest of the
conference were: Philosophy of mathematics, metamathematics of
algebra, proof theory, category theory and theory of computation. There
were fourteen invited papers, thirty contributed papers and a further
thirteen contributed papers presented by title only. Abstracts of most of
these papers appear in the Journal of Symbolic Logic for 1974. The
speakers and titles of the invited papers were:
P. Bernays, Mathematics as a domain of theoretical science and also of
mental experience
M. Dummett, Philosophical foundations of intuitionistic logic
A. Robinson, Concerning progress in the philosophy of mathematics
G. Muller, Set theory as a ‘frame’ for mathematics
W. W. Boone, An algebraic characterization of groups with solvable
word problem
A. Kino, Introduction to proof theory
P. Martin-LGf, On the formalization and proof theoretical analysis of
intuitionistic mathematics
S. Maclane, Internal logic in toposes and other categories
F. W. Lawuere, Coherent topos: algebraic geometry =geometric logic
D. Scott, Continuous functions and computability
C. Strachey, Some unsolved problems in the theory of computation
R. Milner, Processes, a mathematical model of computing agents
C. C. Elgot, Monadic computation and iterative algebraic theories
E. Engefer,O n the structure of algorithmic problems
In the following pages, ten of the first eleven papers are expanded and
developed versions of the corresponding lectures given at the conference;
Kino’s introductory lecture has been replaced by the paper by Kino and
viii PREFACE
Myhill listed below. The remaining twelve contributed papers are
concerned with classical set theory, recursive function theory,
metamathematics of algebra, proof theory, model theory and combinatory
logic.
We should like to take this opportunity, on behalf of all participants at
the conference, to thank the following organizations for financial support:
International Union for the History and Philosophy of Science, Interna-
tional Computers Ltd, British Academy, British Council and University of
Bristol.
It was with deep regret that, during the final preparations for this book,
we heard of the death of Abraham Robinson. As an editor of the ‘Studies in
Logic’ series and as a friend he gave us much help and encouragement in his
usual urbane and efficient manner both during the conference and while
working on this volume. His death is a great loss to the communities of
mathematical logicians and philosophers of mathematics alike.
April 1974 H. E. Rose
J. C. Shepherdson
MATHEMATICS AS A DOMAIN OF THEORETICAL
SCIENCE AND OF MENTAL EXPERIENCE
Paul BERNAYS
Zurich, Switzerland
Ladies and Gentlemen!
Firstly may I apologise for the fact that the abstract of my talk only
partly corresponds to what I have to say. I considered the general theory
of knowledge in more detail than I will do today. So allow me to deviate
from that abstract, and in particular to begin in an other way.
Ferdinand Gonseth often speaks of the problem of beginning (problbme
du commencement). Let me begin, at least provisionally, with some
words about positivism. Many contemporary philosophers hold the
opinion that the appropriate view in philosophy of science is positivism.
However, the majority of scientists do not take a positivistic view, and
further it cannot be maintained that the results of science are clearly in
favour of that view. Positivism appears to many people to be a reasonable
attitude of an enlightened man because it suggests a radical opposition to
intellectual authority. However, from an antagonism to confessional
doctrines it passes to a general antagonism to any kind of belief, whether
it is our instinctive belief in the existence of exterior objects, or a more
abstract belief in a frame of objectivity which we are induced to adopt in
science.
The antagonism to the belief in external things and beings is of course
not practicable in real life but it seems to some philosophers that it can be
maintained as a philosophical attitude. As you know, the doctrine has
been set up that natural science deals properly only with our sensations.
Perhaps it is not possible to refute directly this view, yet it can hardly be
claimed that we get from it a satisfactory understanding of physical
science. The relations which are stated in physics, i.e. the relations which
constitute the contents of the physical laws, are not (apart from very
2 PAUL BERNAYS
special exceptions) between sensations but rather between physical
entities. Only indirectly do we get information from them concerning our
sensations. Still, it might seem that positivism has a better justification
with respect to mathematics: as a device for abstaining from any
Hypostasierung or, as Willard V. Quine calls it, ontological commitment.
An element of positivism is contained both in the intuitionistic view of
mathematics and in the view denoted as ‘formalism’, which arose from
Hilhert’s proof theory. I think one main point of Hilbert’s proof theory can
be accepted without adopting the philosophy of formalism. Indeed this
main point is the idea expressed by Hilbert in his talk ‘Axiomatisches
Denken’: to make mathematical proof as such an object of mathematical
investigation, a possibility resulting from the method of logical formaliza-
tion. Yet it was certainly not Hilbert’s intention that mathematics should
consist only of proof theory (though some of his statements describing his
attitude to metamathematics might suggest this view). Also intuitionistic
mathematics can be accepted without the adoption of the positivistic views
concurrent with it.
After this first consideration let us begin again, let us go back to the
traditional view of mathematics, which is that mathematics is the science
of quantities. This view certainly originated from the work of Euclid,
where geometry was based on a theory of quantities whose axioms were
stated in the KOLUO~i~v vo~ai( common notions). This system of axioms, as
it seems, was regarded as expressing evident facts, whereas the other
axioms were taken as postulates. Yet the application of the concept of
quantity to the geometrical entities was taken as unproblematic. It is one
of the characteristics of the newer axiomatizations of geometry, since
Pasch, that the privileged status of the concept of quantity has been
abolished. For every kind of geometrical entity quantitative characters
become established separately by axioms. In particular for linear quantity
one introduces axioms of betweenness or of order. The emphasis on the
concept of quantity is related to the traditional opposition of quality and
quantity. The inadequacy of this opposition may be seen by noting that
linear quantities are measured by observing betweenness. Reflecting on
this we come to replace the old opposition by that one of quality and
structure; and, in fact, mathematics may be regarded as the science of
structures.
This characterization has still to be specified more precisely. The
objectivity of mathematics has to be distinguished from the objectivity of
MATHEMATICS AS A DOMAIN OF THEORETICAL SCIENCE 3
physics, and also from the objectivity of phenomenal description. On the
one hand this means that structures are considered as such, that is,
independent of their occurrence in real objects. Thus the objectivity of
mathematics is a kind of phenomenological objectivity and may be
explained by the analogy with works of art, in particular musical
works-which indeed cannot be identified with their various perfor-
mances. On the other hand the structures we consider in mathematics
have nothing to do with the complications peculiar to sensual phenomena
and are therefore more easily conceptualised. We may express this by
saying that the structures considered in mathematics are idealized struc -
tures. The difference between concrete structures and idealized structures
has been stressed especially by Stephan Korner.
Idealization is in particular present in the conception of geometrical
figures, and in this case it can be observed that idealization does not only
occur in science but is already instinctively applied when we look at
visual structures and form concepts about them.
There is a general relation of the study of idealized structures to
theoretical science in the following sense: all theoretical description in
natural science is schematic (as Gonseth calls it), in particular it is
restricted to a certain scale of quantity (Grossenordnung); and the
schemata used in theoretical science are just the idealized structures we
consider in mathematics, though not all idealized structures come to be
applied in theoretical physics.
I have stressed the difference between mathematical objectivity, physi-
cal objectivity and the objectivity of sensual description. We have
however to admit that there are methodical analogies between mathema-
tics and empirical science: there is a kind of experience in mathematics, to
which especially George P6lya points in his investigations on
mathematical heuristics. There is also an analogy with theoretical physics
in so far as the general programme is not stable but changes essentially
with the development of the science. This means that iri mathematics not
only the questions to be solved within the particular theories but also the
suitable theoretical setting up of the whole of mathematics constitutes a
problem.
Let us recall some newer developments in this respect. At the beginning
of our century the theory of algebraic functions was the central domain; it
included function theory, algebra, algebraic geometry, theory of Riemann
surfaces and topology. The privileged role of this frame was abandoned
4 PAUL BERNAYS
about the time of Hilbert. Then came the idea of taking axiomatic set
theory as the general frame of mathematics, and now this position of set
theory is being challenged. One has various embracing theories, and
axiomatic set theory itself is extended by model theory, where set
theoretic concepts are used independently from the axiomatization. The
modified situation is especially clear in the general theory of mappings
which is called the theory of categories.
From this aspect of mathematics we can also deal with the criticism of
classical mathematics advanced by the intuitionists. Thus Heyting argues
that it is too naive to ask questions like ‘Does there really exist a
well-ordering of the continuum?’ But the question need not be put in this
way. The question rather may be asked: ‘Is it suitable to adopt the strong
extrapolation of classical analysis as it is formulated in the conceptions of
set theory, in particular in the theory of transfinite cardinals?’ At the
present time no deciding experience has been found to answer this
question. Mathematicians have different opinions about the suitability of
stronger or only weaker methods of idealizations.
At all events the fact that our mathematical idealizations are so
successful is a striking mental experience and by no means something
trivial.
THE PHILOSOPHICAL BASIS OF
INTUITIONISTIC LOGIC
Michael DUMMETT
University of Oxford, U.K .
The question with which I am here concerned is: What plausible
rationale can there be for repudiating, within mathematical reasoning, the
canons of classical logic in favour of those of intuitionistic logic? I am,
thus, not concerned with justifications of intuitionistic mathematics from
an eclectic point of view, that is, from one which would admit intuitionis-
tic mathematics as a legitimate and interesting form of mathematics
alongside classical mathematics: I am concerned only with the standpoint
of the intuitionists themselves, namely that c!assical mathematics emp-
loys forms of reasoning which are not valid on any legitimate construal of
mathematical statements (save, occasionally, by accident, as it were,
under a quite unintended reinterpretation). Nor am I concerned with
exegesis of the writings of Brouwer or of Heyting: the question is what
forms of justification of intuitionistic mathematics will stand up, not what
particular writers, however eminent, had in mind. And, finally, I am
concerned only with the most fundamental feature of intuitionistic
mathematics, its underlying logic, and not with the other respects (such as
the theory of free choice sequences) in which it differs from classical
mathematics. It will therefore be possible to conduct the discussion
wholly at the level of elementary number theory. Since we are, in effect,
solely concerned with the logical constants-with the sentential operators
and the first-order quantifiers-our interest lies only with the most general
features of the notion of a mathematical construction, although it will be
seen that we need to consider these in a somewhat delicate way.
Any justification for adopting one logic rather than another as the logic
for mathematics must turn on questions of meaning. It would be impos-
