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«(in: AUSEJO Elena & HORMIGON Mariano (Editors) (1995) Paradigms and Mathematics. Madrid, Siglo XXI de España Editores, pp. 325-335.) If we do not ...»

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Free University of Berlin (Germany)

(in: AUSEJO Elena & HORMIGON Mariano (Editors) (1995) Paradigms and

Mathematics. Madrid, Siglo XXI de España Editores, pp. 325-335.)

If we do not come up to more, I would be glad to transmit the following two things: firstly

there are not only the mathematical sciences which meet with problems on the occasion of

their attempt to relate their history with the kuhnian pattern of interpretation, because the same thing is happening and even in a more severe way with the social sciences and particularly with psychology. Secondly it is just mathematics which due to their very special role in the framework of psychology are contributing heavily to this distortion with regard to the kuhnian system of reference.

Speaking about the history of psychology always turns out to be a delicate thing. For there does neither exist conformity between the representatives of this matter as to the beginning of this history nor as to the subjects that it should include.

It happens to be even more controversial to speak about the part which mathematics are playing within this history. For the most influential protagonists of contemporary psychology take mathematization as an intrinsic tendency of all the sciences and considering mathematics as the last resort of truth, there does not remain any margin for a critical investigation relative to this role and its changes. To such a degree that for these people talking about mathematics in psychology is the same thing as talking about progress and so that's that.

Nevertheless we will now risk a glance at the history of psychology and the part that mathematics are playing there.

The first contributions to modern psychology go back to the 19th century and just to scientists working in the field of classical natural sciences and well acquainted with mathematics. Among this kind of men is ranking the physicist G.Th. Fechner (1801-1987) and the physiologist E.H. Weber (1795-1878), both founders of the so called psychophysics. Its programme consisted in establishing regular relationships between physical scales (e.g. the weight of an object) and psychical scales (e.g. the just noticeable difference between weights which is necessary to distinguish them).

It is only natural that in the context of this kind of study the question of the utility of mathematical functions suited to model the curves which the experiments yielded pretty soon rose. So that already at the beginning of modern psychology the people made use of mathematics in a really amazing manner. Let us take the famous Law of Fechner which held a logarithmic relationship between physical variables (the independent variable, e.g. the volume of an acoustic stimulus) and the intensity of the subjective sensation (the dependent variable, e.g. the acoustic reception of the same stimulus). By the way, it’s simple to derive this relationship on the basis of the already mentioned just noticeable differences by means of the differential or difference calculus, respectively. The following graph shall serve to illustrate this kind of psychophysical laws.

Eckart Leiser (1) For a rather long time psychology followed this course accumulating this type of regular relationships which characterize psychical mechanisms, going more and more beyond the sensorial functions which formed the specific scope of psychophysics. Another prominent representative who marks this more extensive programme in which the original psychophysics ended up was W. Wundt (1832-1920). His laboratory, founded in 1859, is taken in general to be the origin of modern psychology in its proper sense. Wundt, philosopher, carried through a systematic generalization of the project of a forthcoming scientific psychology. According to him it should devote itself to the explication of psychical phenomena basing it on a set of primary elements which by the way of combining themselves end up in more complex arrangements held together by some principles of association. Due to this background the historians of psychology used to name this project the psychology of elements or psychology of association.

What matters in the present context is that questions of this character almost automatically give rise to mathematical concepts. For example one problem to deal with was what is the function like which could describe the relationship between the number of elements (say senseless syllables) and the number of presentations of their corresponding arrangements necessary to obtain stable associations? To get a more vivid idea we can imagine the

following pairs of senseless syllables:

yer-fis dak-mun per-xol hen-sof get-tok which will be presented as many times as we need to come to a correct reproduction.

Or, once we have reached such stable associations with regard to a given arrangement:

what is the curve like which could describe their decay taking place in the course of time?

Later on this kind of problem led to the investigations of H. Ebbinghaus (1850-1909), a disciple of Wundt, who was intimately engaged in the study of the mechanisms of memory.

The following graph shows one of these laws of memory.

Eckart Leiser (2) As a matter of fact this sort of experiment already requires more than one person and something like statistical techniques. For given the poorly standardized conditions of the experimental setting and the imponderabilities of the subjective dynamics on the part of the investigated person as to the results we always will remain with a considerable fluctuation.

But the above mentioned scientists maintained that fundamentally and under perfect conditions one can do with a single subject to obtain all the data proper for generalizations relative to the human psyche. Preferably it should be the scientist himself as the best trained person in inspecting his own psychical apparatus. Clinging to this pure-case-concept one can only attribute a marginal relevance to the individual differences, while the real thing that counts is the general structure of the human psychical organization. The object of psychology according to this comprehension is to elucidate the essential characteristics of this general structure by means of theoretical models in which mathematics at the most can play an auxiliary part.

This view of psychology persisted for a rather long time and brought forth other schools like the school of Würzburg (to whose more famous representatives K. Bühler and 0. Külpe belonged). At the same time, this means at the beginning of the century, this concept of psychology spread from Europe to North America where, under the protagonism of E.B.

Titchener (1867-1927), it adopted the name structural psychology.

To establish a first reference to the kuhnian model of scientific development perhaps we could speak of a pre-paradigmatic phase of psychology whose typical feature was an assembly of individual researchers inspired by their respective idiosyncrasies and at most united by invoking a concept of scholarship derived from the classical sciences. Far from any pretension to enter into questions of obligatory norms to be observed in the theoretical and practical work of psychology they managed to coexist peacefully in their respective ivory towers.

But shortly afterwards there was a radical break in the evolution of psychology so far described, a break which was initiated by the functional manifesto of North American psychologist J.R. Angell (l869-1949) [ANGELL, 1907] and later intensified by another manifesto by J.B. Watson (1878-1958) [WATSON, 1913] in favour of a total abolition of the psychology which existed until then and its substitution for behaviourism.

More concretely: what impact did this break have on psychology in general and the function of mathematics within it? It is not feasible to analyze this problem without going back to some factors outside psychology that formed the historical background of this evolution. In the scientific and philosophical field we must mention the trancendental influence that came from Charles Darwin's ideas and the biological and evolutionist thinking inspired by them. With relation to human psychology it would mean that psychism is nothing more than another result of an ubiquitous adaptation. Going beyond the human species as Eckart Leiser (3) such it would suggest that each individual, far from being a realization of the human being, characterized by a general structure is a different product of its own adaptive processes and more or less accurate according to each case.

However, as far as psychology is concerned Darwinism has another effect: there is not any longer a coherent object left for psychological research and the theories that arise from it, but this is converted into a population of exemplary individuals which comprise a variety of characteristics and levels of adaptation. In this population man and his psychism as the objective of the psychology that existed until then is diminished to the size of a dot, which not only implies the dissolving of the scientific objective but also the corresponding theoretical pretensions. From now on the principal objective of psychology consists in researching the adapting value of those characteristics and in identifying the most adapted of fellow human beings to distinguish them from the less adapted ones. So, Angell's term functionalism refers to this programme of questioning the usefulness of psychic mechanisms as adaptive functions instead of going into its structures and lawful characteristics. And if this weren't enough later on this last type of questioning is denounced as being a useless luxury and moreover a sort of very little scientific obscurantism. What happened was that this programme was not even begun because functionalism came into the shadow of behaviourism, its more radical child. At the same time behaviourism proclaimed the definite renouncing of all theoretical pretensions in favour of total empirism being content with finding data about the observable responses provoked by an observable stimulus considering the psychic processes either non-existent or like a black box which is out of scientific reach.

To establish another reference to the kuhnian concepts of scientific evolution: how to expect the shaping of a paradigmatic basis for psychology faced with a complete renouncement of a theoretical body and therefore, a sufficiently exhaustive delimitation of the object under research? However, there was a hegemony indeed that in the United States for many decades succeeded in excluding any other way of focusing the official argument of psychology. So, the sociological factors subjected in the kuhnian analysis to maintain a hegemony determined the state of things but without the counterbalance of having to offer solutions to both theoretical and practical problems put forward by the object of research. In that sense we can talk about a hegemony without any rational basis. This hegemony continues to the fore in many Latin American countries.

To get back to the second part of our question: what does all this mean to the use of mathematics in psychology? In fact, it was precisely a kind of logical mathematical background which brought about this hegemony, namely a terminology inspired by the logical empirism of the Vienna Circle, the physicalism and the cult around a total abstraction which finds its ideal in mathematics and formal logic. So paradoxically it was ideologically disfigured mathematics which gave rise to this irrational and intangible basis of behaviourist hegemony.

From time to time attempts were made to take these logic-mathematical pretensions seriously. Here it's worth mentioning E.C. Tolman (1886-1959) and his topological model to represent the mental maps of behaviour and more particularly we mention C.L. Hull (1884The latter spent his life desperately designing a mathematical apparatus capable of representing the afore mentioned black box based on the so-called intervening variables in order to reproduce his experimental data obtained from his rats [HULL, 1952]. However, this project as the mathematical apparatus was inflated more and more, ended in a total disaster.

This marks the beginning of the decomposing of the behaviourist hegemony which finalized breaking up into several intricate sects mutually at war.

Eckart Leiser (4) As we once again establish a reference to the kuhnian concepts it was not in the first place due to an internal crisis that this behaviourism broke down as the impossibility of solving research problems out of the reach of a paradigm which until now had been successful, but due to outside intervention. It was the famous linguist Noam Chomsky (born 1928) who, in 1959 took the whole argumentative basis of behaviourism to pieces in a review of one of Skinner's texts, contrasting his position with some elementary facts of linguistics. This caused a general upset in North American psychology which had repercussions in the whole world and was followed by a precarious phase in which behaviourists found themselves devoid of their ideological clothing. Some of the supporters joined ranks with the Chomsky project and the others in search of a substitution to Skinner's abstractionism and his compromise with a logic-mathematical vision of the world fell in with the cybernetic situation of these years under the lemma of cognitionist change.

It was another attempt to forcefully adopt a language outside psychology as it happened formerly to the language of the classical sciences to try to find some watchword of scholarly method. This implied a fundamental inversion of means and ends: the cybernetic machinery does not serve as a useful model to represent some characteristics of the cognitive processes by means of simplification and abstraction and bearing in mind their character of a tool but quite the reverse: the mind is nothing more than a computer and methodical abstraction to arrive at this analogy it is converted into an ontological quality. And thus a fetishization of models is produced. The cybernetic categories substitute a genuine psychological language or epistem in such a way that the mind is seen as a place where


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