Proof and perception
IV
M. Otte
Institut für Didaktik der Mathematik
Bielefeld, Germany
All mathematical knowledge we might have seems to fall
into two categories.
1. Knowledge that certain mathematical claims
follow from certain other mathematical claims, or bodies
of claims.
2. Knowledge of the consistency of certain mathematical
claims, or bodies of claims.
This view leads to the impression that mathematical
knowledge is simply logical knowledge. Today, however, our
logical investigations rely on mathematics and because of
this, logic appears as a chapter of mathematics. Therefore
in the end, mathematics appears as discipline which is
supposed to justify itself. Such a "naturalism" is rather
unusual. The core of this circularity seems to come down to
the following. Ever since Euclid and until the 19th century,
as a rule, axiomatics was used as an instrument for dealing
with the complexities of some realm of intended applications
(think of Euclid's axiomatization of geometry and of
Newton's axiomatization of mechanics). Ever since Hamilton
and Boole, and particularly after Hilbert and Gödel,
axiomatics began to be used as an instrument for dealing
with the complexities of mathematics itself. It became a
means of mathematizing or modeling mathematical theories
themselves!
While the layman, as a rule, is instructed that
absolute "what-is"-questions make no sense in the realm of a
scientific treatment of objective phenomena, and that man
must rather be content with the structural representations
of nature in mathematical language, in the case of
mathematical knowledge the relational point of view is
constantly challenged by another opposite position which
tries to give explicit definitions of every mathematical
term. In this manner attempts are made to justify the
abstract mathematical language through another less abstract
language. A typical illustration of this attitude can be
found in Bertrand Russell's Introduction to Mathematical
Philosophy (Routledge London 1998/1st ed. 1919).
While Russell emphasizes the concepts of
"relation" and "relational structure" with great care
because this is conductive to an illustration of the general
principle "that what matters in mathematics, and to a very
great extent in physical science, is not the intrinsic
nature of our terms, but the logical nature of their
interrelations", for "we know much more about the form of
nature than about the matter", he takes much trouble in
explaining what the term "number" means.
According to the previous principle, what we
should be asking for is not "what" numbers "are," but only
what the forms used to describe the relationships between
them are. This is, indeed, the core of the pragmatic maxim
which stands behind Peano and Hilbert's axiomatic approach
to arithmetic and according to which, ultimately,
mathematical concepts must be defined through the operative
possibilities opened up by the forms used to represent these
concepts, that is, via the differences in behavior made
possible or triggered by them. This can be done without any
reference whatever to any absolute arithmetical ontology.
Yet, as was said already, an axiomatized theory is not taken
as an abstract symbolic game carried out on meaningless
signs, but is rather developed in connection with certain
sets of possible applications or intended interpretations.
Within the historical development of abstract group theory,
for example, a theory presented in axiomatic terms from the
very beginning, group representations of various kinds (as
linear transformations or permutations, for instance) have
played an essential role.
Still, to try to secure the applicability of an
axiomtized theory in an a priori fashion by providing final
interpretations of the undefined terms is completely
wrongheaded. The question, in particular, what numbers are,
in the sense of first looking for objects with which numbers
can be identified, is ill posed. Not what numbers are is the
question, but rather to which purpose numbers are
constructed, and which function within the process of
cognition they fulfill. Numbers do not exist outside the
theory of arithmetic (as points and lines do not exist
outside axiomatized geometry; here lies the difference
between Euclid's and Hilbert's conception of an axiomatized
theory). Hence it is problematical to try, as Russell did,
to first obtain some arbitrary objects, say in the frame of
set theory, in order to pass these off subsequently as the
numbers. Russell claims that Peano´s axiomatic
characterization of arithmetic is insufficient if "we want
our numbers not merely to verify mathematical formulae but
to apply in the right way to common objects". To this one
could reply by pointing out first of all, that theories are
applied as structured wholes and not in a piecemeal fashion,
and second that it is impossible to specify before hand all
the possible applications of a theory. Indeed, the
applications are often uncovered in time, alongside the
elaboration of the abstract theory itself; as Lebesgue once
remarked: arithmetic applies when it applies.
Finally, it must be said that Russell's project
cannot be accomplished. Indeed, numbers are neither objects
nor simply properties of objects. That numbers cannot be
concrete or set-theoretical objects was shown by Paul
Benacerraf by providing different set theoretical
interpretations of the number concept. Benacerraf's argument
is basically quite simple: "It wa pointed out ... that any
system of objects, whether sets or not, that forms a
recursive progression is adequate. But this is odd, for any
recursive set can be arranged in a recursive progression. So
what matters, really, is not any condition on the objects
(that is, on the set) but rather a condition on the relation
under which they form a progression. To put the point
differently - and this is the crux of the matter - that any
recursive sequence whatever would do suggests that what is
important is not the individuality of each element but
structure which they jointly exhibit." ((cf. P.Benacerraf,
What numbers could not be, in: Benacerraf/Putnam (eds.),
Philosophy of Mathematics, Cambridge Univ. Press 1965, 290).
Numbers are no sets, but can be identified in a variety of
ways with sets. This causes no problem, because what is
important is not the individuality of each element, but the
arithmetical structure. Numbers, by the way, cannot be
identified with whatever concepts either, for this would
amount to, establishing the identity of these concepts in
the usual manner by means of the axiom of extensionality,
only one further set theoretical reduction of the number
concept. Russell, when having discovered the set theoretical
paradoxes wished not only to reduce number theory to set
theory but wanted also to interprete the latter in
conceptual terms, in terms of propositional functions. We
return to this below.
The structuralistic philosophy of science
(Quine, Sneed, Moulines) generally accepts that the identity
of a concept or of a whole theory depends among other things
on certain pragmatic criteria by which the possible
applications of a theory and the values of the bound
variables are identified. But mathematics being justified
through mathematical practice ceases to be paradoxical as
soon as it is understood in terms of a theoretical practice
on systems of signs. This is relational in a double sense:
the objects of which a system S of signs are made are
defined only in relation to the system as a whole, and
reference relationships between S and another system of
entities S' (in terms of other systems of signs or of
sensible objects) are no longer invoked to explain S through
S' or the other way around in an absolute way, but rather as
functional relationships in an almost mathematical sense,
and these relationships are not, as it were, inherent in the
individual objects of the system or in the systems
themselves but they depend on the intervention of the human
subject as a mediating agent. Without morphisms there is no
group theory, and without monotonic functions there is no
arithmetic.
Pragmatism understands reality as a process
rather than as something statically given, and therefore
searches for fundamental epistemic processes, basing our
cognition essentially on two irreducible mechanisms: on the
development of language (Quine), and on perception (Peirce).
While we may talk about certain things like "society" or
"France", we are unable to perceive them. Other things
closer to our corporeal existence we do perceive. As
Gödel once said: "We perceive objects and understand
concepts". And he considered, contrary to Russell, sets as
quasi-spatial objects, not concepts, commenting on Russell's
argumentation against this extensional view of mathematics
that "these arguments could, if anything, at most prove that
the null class and the unit classes (as distinct from their
only element) are fictions (introduced to simplify the
calculus like the points at infinity in geometry), not that
all classes are fictions" (K. Gödel, Russel's
Mathematical Logic, in Schillpp (ed)., The Philosophy of
Bertrand Russell, Open Court, La Salle 1971, 141). Leibniz
already had made this latter point quite clear in relation
to the ontological status of the infinitesimals of
(non-standard) analysis: these may be fictions established
by an infinite process, but the relations leading to them
are real and perceivable. Cabri Geometre offers plenty of
opportunities to observe relational facts, using something
like Leibniz' continuity principle.
All mathematical objects arise from such
perceptual contexts. What we perceive are relational sign
structures. The notion of perception is to be understood in
the sense of semiotics or of model theory, the
representation taking place quite possibly in analogy with
the medium of our own bodily movements.
The difference between these two sources of
cognition has been clarified by Russell's theory of
description. Russell distinguishes between acquaintance with
a thing and knowledge about it. It is the second kind, which
is stated by a denoting phrase; the first comes by an
indication of the thing. Thus acquaintance comes from
perception, knowledge through conceptual thought. The
Kantian distinction between synthetic and analytic knowledge
reappears at this point. In synthetic knowledge an object is
directly' given, in analytic knowledge it is
indirectly presented, namely by means of some properties or
relations. A brief look at the history of axiomatics
suggests that the shift from Euclid to Hilbert represented a
change from a synthetic to an analytic understanding of
mathematics, from a paradigm of evidence to a paradigm of
consistency. This is in a certain sense correct, yet it only
tells half the story. In order to get rid once and for all
of the evidence paradigm in mathematics, Hilbert was led
sometimes to say that existence means consistency. But as
outlined by Hilbert himself, finite consistency proofs are
an attempt to justify the possible through the actual and
therefore depend on concrete, actual sign manipulations.
Thus, consistency proofs still rely on existence, on
indexical signs, that is, ultimately, on knowledge by
acquaintance.
Knowledge by acquaintance is important in
mathematics because mathematics is an activity and all its
objects are either constructed or are to be understood at
least in relation to a system of means of mathematical
construction.
One might object that what is at stake in pure
mathematics and with respect to its proof procedures is
identity rather than existence. Now with respect the claims
of type 1 above identity is assumed to begin with, but with
respect to the second type it has to be established.
Identity of functions (of propositional functions in
particular as well as of concepts) is commonly established
by means of the axiom of extensionality: F=G if F(x)=G(x)
for all x. But keeping completely silent about the nature of
the universe of discourse to which the quantified variables
belong makes no sense. From a logical point of view already
there exist type-theoretical restrictions on the bound
variables. Logical considerations aside, it remains true
that our mathematical theories do not refer to a pre-given
and pre-structured world as such. Therefore, identity
statements make sense only in relation to certain practices
and in contexts where there exist individuating conditions,
like they are provided, for instance, in the classroom by
the blackboard or the computer screen. Proof and Perception
part II (Otte, 1998) gives a
good illustration as to what this means.
One might claim, as Russell did at times, that
mathematics is essentially intensional, just being a part of
language and conceptual logic and that mathematical activity
consists in operating with concepts or predicates. Letting
aside all logical problems (we do after all have only
countably many linguistic expressions), we believe that
adopting a certain theory or language implies not only
certain modes of expression, but also a conception of the
world, or an ontology. The objects to which our bound
variables refer thus are not just given. Every access to the
world is theory-laden and every theory is characterized by
its own system of dissecting the world as well as by the
type of meanings ascribed to the objects. On the other hand
a theory in general does not fully determine its ontology,
as the example of the different number concepts shows.
Finally it is a fact that the human subject is capable to
single out objects of the external world before she masters
language and theory by a combination of perception and
activity. Thus the question arises of how perception or
intuition on one hand and language and communication on the
other work together. We certainly cannot treat this problem
exhaustively here. Suffice if to say that mathematical
theory always has to employ perception.
What we perceive are, on the one hand,
particular things such as points on the computer screen or
on the blackboard. We ascertain the identity of the latter
by ostentation, representing them by indices. All the
letters we use in geometry and algebra are such indices for
singular things. This kind of perception has an important
role in the transition from an intensional to an extensional
point of view.
Even if we believe that consistency were
sufficient for existence we shall have to secure unambiguous
relations of reference. Kant, however, believing that things
have essential properties, claims that a different kind of
perception is necessary in addition and concludes from this
that existential assertions of a more elaborate kind are
essential to deal with questions of consistency. Kant's
argument is that a contradiction essentially comes from
imaging an object and making various assumptions about the
postulated object that are inconsistent with the nature of
that object. But a contradiction could never arise if we
rejected the object. And as conceptual thinking and logic
can never categorically assert the existence of anything,
existence in the end depends on intuition and perception.
But having established the possibility of something by
intuition we cannot just reject it no more than we could
establish it just by postulation. Our intuitions do not
necessarily obey conceptual postulation. Kant gives the
following example: Postulating a triangle and then saying
that it does not have three angles is a contradiction but
"the proposition above-mentioned does not enounce that three
angles necessarily exist, but, upon the condition that a
triangle exists, three angles must necessarily exist in it"
(Kant, B 622). There is, however, never a contradiction if
there were no triangle, there is nothing there about which
we have made contradictory assumptions. Triangles have
however a possible existence within our intuition and proofs
of consistency thus provide real knowledge. Kant wants
mathematics first of all to be applicable to the real world
and therefore emphasizes the importance of an intuition that
reaches beyond mere ostentation. "It is a commonplace of
newer Kant scholarship that he already knew about
non-euclidean geometry and that in fact its very possibility
reinforces Kants doctrine that euclidean geometry is
synthetic a priori because only its concepts are
constructible in intuition" (J. Webb, Tracking
Contradictions in Geometry, in: J. Hintikka (ed) Essays on
the Development of the Foundations of Mathematics Kluwer
1995).
The argument of post-Kantian pure mathematics
suggests an asymmetry, an asymmetry as it appears in Frege's
analysis of A=B in terms of sense and reference (see
Proof and Perception II). The
reference, according to Frege, being an object, is defined
quite precisely; but the sense is not so defined. All that
is really provided is a criterion for when the senses of two
sentences are the same or are different. The lack of
symmetry is due to the difference between existence and
identity or object and concept. Concepts or ideas might have
an identity, but do not exist in the same manner as
particular objects exist. According to Kant, however, they
have a possible existence in intuition.
Taking into account our observations with
respect to the application of the axiom of extensionality,
we might claim, as Kant did, that our world is always
conceptually structured to begin with. In his recent Mind
and World (Harvard UP 1994), J. McDowell has claimed that in
order to resolve the problems of modern analytical theories
of truth, we have, with Kant, to recognize that concepts and
intuitions, understanding and sensibility must be integrated
even in the mere intake of experiental content
characteristic of sense perception. Even when making
perceptual judgments we have to employ conceptual ideas. We
thus realize that the distinction between ideas and objects
is only a relative one, and that all we deal with in our
cognitive activity or in our perception are signs rather
than objects. In this way the difference between the two
types of knowledge is present already in our perceptions.
And it is represented there by two different types of sign,
namely by indices and icons.
An image of a unicorn is an icon which does not
contain any existence claim, but gives only an image, or, if
you wish, a description of this beast. The same is true of a
general triangle or of an axiomatic system, etc. But if in
the course of a geometrical proof I say "triangle A" is
congruent to "triangle B", then I will have to indicate
these different triangles and my representation will change
character. As has been said before an icon is very often
required to resolve confusion. An index is always necessary
if I want to solve a problem of reference. An index,
however, does not tell me anything about the object
indicated. Kant was convinced that all mathematical
judgments are intuitive. He distinguishes, however, between
pure intuition and empirical intuition. In empirical
intuition, only particulars are given to us. It is the kind
of intuition we have just spoken of. It is very meaningful
in the context of the Cabri-Géomètre as it
leads us to new hypotheses via the perception of fixed
points and other invariants. We do not obtain mathematical
proofs, however, in this way.
A mathematical proof is tied to a comprehension
and thus to a generalization, and this is where another kind
of intuition takes effect, that is the perception of ideas
or of general concepts. In Part II, for instance, the
concept of the center of gravity has led to a proof, which
yields a solution of the problem quoted for all polygons.
That the midpoint M remained fixed had to be understood as
an indication of the invariance of the center of gravity of
the whole constellation. The center of gravity is not an
empirical thing, but it is immediately accessible to our
experience, and thus to our intuition. The idea of number,
which Russell so absolutely wanted to see fixed, also
becomes perceptible to us through its effects within the
system of our activities. We really can perceive the
effects, which our ways of using the number concept have,
and algebra is nothing but a means of representing these.
Although we cannot provide a non-circular definition of the
term 'number', we shall certainly recognize a number if we
meet one.
As has been said, such intuitions of ideas are
necessary to obtain the mathematical knowledge of the first
type in our above classification. Observing a mere fact does
not yield a proof yet. A proof requires an interpretative
idea, and that means a generalized representation. Such an
idea has no reality beyond leading to the ordering and
clarification of a confusing situation comparable, say, to
that of the person shown in the picture in part
I is confronted with. With regard to Nicolas Balacheff
sending me this image together with the request to write
something about "Proof and Perception", this is the moment
where I may indeed consider my task completed. This fourth
part "Proof and Perception IV" thus is the last.
©
Michael Otte
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