In this series of blog posts, I will summarize and quote parts of some books that deal with the problems of reductionism.
The first book that draw my attention, was the famous book "Beyond reductionism", New perspectives in the life sciences, which is a report from the Alpbach Symposium held in 1968. Participants were, among others, Arthur Koestler [holons!], J.R. Smythies, Ludwig von Bertalanffy [systems theory], Paul Weiss, Jean Piaget[developmental psychology], Paul Maclean [three functional brains idea], W.H. Thorpe, Viktor Frankl. The book breathes a fresh atmosphere. It is a lively report of the thoughts and research of many great scientists, especially from the biology, psychology and linguistic departments. One can also see how current academia have diverged from this line of thought, in favor of a more reductionist paradigm (as evidenced by molecular biology, and especially neuropsychology). There are many problems facing the reductionist paradigm. I will deal with some of these in this series of postings.
The first article concerns the lecture of Jean Piaget and Bärbel Inhelder (University of Geneva), titled “The gaps in empiricism”. I will quote from a section of this long lecture. First there is an intro on behaviorism, which I will skip. Then, an enlightening part on empiricism and mathematics follows (all emphasis is mine):
Empiricism and mathematics
“In so far as empiricism seeks to limit knowledge to that of observable features, the problem it has failed to solve is the existence of mathematics, and this problem becomes particularly acute when it comes to explaining psychologically how the subject discovers or constructs logico-mathematical structures.”
“Classical empiricism, as argued by Herbert Spencer for example, considered that we derive mathematical concepts by means of abstraction from physical objects: certain Soviet schools of thought share this view, though it is in fact not consistent with the theory of dialectics. In contrast to this attitude, contemporary logical empiricism has well understood the difference between physics, on the one hand, and logic and mathematics, on the other, but instead of seeking a possible common source of knowledge in these respective fields it has maintained that there are two entirely different sources. It has thus aimed at reducing physical knowledge to experience alone (the root of synthetic judgments) and logico-mathematical knowledge to language alone (whose general syntactic and semantic features pertain to analytical judgments).”
“This view poses several problems. Firstly, from the linguistic point of view, while Bloomfield's positivism (and even earlier Watson's behaviorism) aimed at reducing all thought and, in particular, logic to a mere product of language, Chomsky's transformational structuralism reverts to the rationalist tradition of grammar and logic (in doing this, as we have just seen, he exaggerates to the point where he regards basic structures as innate). In the second place, the great logician Quine was able to show the impossibility of defending a radical dualism of analytic and synthetic judgments (this "dogma" of logical empiricism, as Quine amusingly termed it). Moreover, a collective study by our Centre for Genetic Epistemology has been able to verify Quine's objections experimentally by finding numerous intermediaries between the analytic and synthetic poles. Finally, psychogenetically, it is obvious that the roots of the logico-mathematical structures must go far deeper than language and must extend to the general coordination of actions found at the elementary behavior levels, and even to sensory-motor intelligence; sensory-motor schemes already include order of movements, embedding of a sub-scheme into a total scheme and establishing correspondences. The basic arguments of logical empiricism are thus shown today to be refuted in all the linguistic, logical and psychological areas where one might have hoped to prove them.”
“As far as the connections between logico-mathematical structures and physical reality are concerned, the situation seems just as clear. It became clarified through experimental analyses of the nature of experience itself. While empiricists aimed at reducing everything to experience, and were thus obliged to explain what they meant by "experience", they have simply forgotten to prove their interpretation experimentally. In other words, we have been given no systematic experimental study on what experience actually is.”
“From our prolonged and careful studies of the development of experience and of the roles which it plays in both physical and logico-mathematical knowledge, the following facts emerge.”
“It is perfectly true that logico-mathematical knowledge begins with a phase in which the child needs experience because it cannot reason along deductive lines. There is an epistemological parallel: Egyptian geometry was based on land-measuring, which paved the way for the empirical discovery of the relationship between the sides of a right-angled triangle with sides of 3, 4 and 5 units, which constitutes a special case of Pythagoras' theorem. Similarly, the child at the preoperatory level (before 7-8 years) needs to make sure by actually handling objects that 3+2=2+3 or that A=C if A=B and B = C (when he cannot see A and C together).”
“But logico-mathematical experience which precedes deductive elaboration is not of the same type as physical experience. The latter bears directly on, and obtains its information from, objects as such by means of abstraction—"direct" abstraction which consists of retaining the interesting properties of the object in question by separating them from others which are ignored. For example, if one side of a rubber ball is coated with flour, the child discovers fairly early on that the further the ball drops in height the more it flattens out when it hits the ground (as indicated by the mark on the floor). He also discovers at a later age (10-11) that the more this ball flattens out the higher it bounces up; a younger child thinks it is the other way round. This is therefore a physical experience because it leads to knowledge which is derived from the objects themselves.”
“By contrast, in the case of logico-mathematical experience, the child also acts on the objects, but the knowledge which he gains from the experience is not derived from these objects: it is derived from the action bearing on the objects, which is not the same thing at all. In order to find out that 3+2=2+3, he needs to introduce a certain order into the objects he is handling (pebbles, marbles, etc.), putting down first three and then two or first two and then three. He needs to put these objects together in different ways—2, 3 or 5. What he discovers is that the total remains the same whatever the order; in other words, that the product of the action of bringing together is independent of the action of ordering. If there is in fact (at this level) an experimental discovery, it is not relevant to the properties of the objects. Here the discovery stems from the subject's actions and manipulations and this is why later, when these actions are interiorized into operations (interiorized reversible actions belonging to a structure), handling becomes superfluous and the subject can combine these operations by means of a purely deductive procedure and he knows that there is no risk of them being proved wrong by contradictory physical experiences. Thus the actual properties of the objects are not relevant to such logical mathematical discoveries. By contrast, it is just these properties which are relevant when—as in one of our recent experiments—the child is asked questions about how the behaviour of pebbles (which stay where you put them) differs from that of drops of water.”
“The method of abstraction peculiar to logico-mathematical structures is therefore different from that in elementary physical experiences. The former can be called a "reflective abstraction", because, when the child slowly progresses from material actions to interiorized operations (by "superior" we mean both "more complex" and "chronologically later") the results of the abstractions carried out on an inferior level are reflected on a superior one. This term is also appropriate because the structures of the inferior level will be reorganized on the next one since the child can now reflect on his own thought processes. At the same time this reflection enriches the structures that are already present. For example, primitive societies and children are already aware of the one-to-one correspondence, but it needed Cantor to discover the general operations of establishing relationships by means of "reflective abstraction" and he needed a second reflective abstraction in order to establish a relationship between the series 1,2,3 ... and 2, 4, 6 ... and thus to discover transfinite numbers.”
“In this light we understand why mathematics, which at its outset has been shown to stem from the general coordination of actions of handling (and thus from neurological coordinations and, if we go even further back, from organic self-regulations), succeeds in constantly engendering new constructions. These constructions must of necessity have a certain form. In other words, mathematical thought builds structures which are quite different from the simple verbal tautologies in which logical empiricism would have us believe.”