Trans | Internet-Zeitschrift für Kulturwissenschaften | 15. Nr. | August 2004 | |

HerausgeberIn | Editor | Éditeur: Josephine Papst (Graz) Buch: Das Verbindende der Kulturen | Book: The Unifying Aspects of Cultures | Livre: Les points communs des cultures |

Bernhard Lauth (Department of Philosophy, Logic and Theory of Science, University of Munich, Germany)

The subject of this paper is the role of transtheoretical structures in scientific theories. But what exactly are transtheoretical structures? The answer to this question is quite simple: basically, transtheoretical structures are mathematical structures which can occur in different areas of scientific research and therefore recur in a variety of otherwise more or less unrelated theories.

In the natural sciences such structures have been known for a very long time. Prominent examples are vector spaces and vector fields. Vector spaces play a crucial role in almost all areas of classical and non-classical physics, from Newtonian mechanics to Einstein's special and general relativity.

A related example is topology. Topological structures have first emerged in a geometrical context and serve to describe certain relationships between geometrical objects that remain invariant under continuous transformations and distortions. Today, topological and metrical structures occur in nearly all areas of pure and applied mathematics, in set theory, in astronomy, in molecular biology, in the cognitive sciences and in many other areas of scientific research.

This does not come as surprise, since basic topological concepts (like the concepts of open neighbourhood and continuous transformation) are very general by their definition, indeed, much more general than most other geometrical concepts. More general structures have a wider range of possible applications and, accordingly, a higher probability of transtheoretical occurrences.(1) Sometimes, however, we hit upon rather specific structures, which nevertheless may have a surprisingly wide range of empirical applications.

An important example is provided by deterministic models and deterministic systems. Because of their significance for the following considerations we will take a closer look at the underlying definitions.

A deterministic system is defined by its possible *states*
and its possible state *transitions*. More precisely: deterministic
laws regulate the possible state transitions of a physical system
in such a way that the state of the system at any time *t *>
0 is completely and uniquely determined by the initial state of
the system at time *t* = 0.

Take the example of classical mechanics. The physical state
of a mechanical system with *n* degrees of freedom, say,
is completely determined by its canonical (position and momentum)
coordinates, hence by the position of the system in a 2*n*-dimensional
vector space (the phase space of the system, which should not
be confused with the ordinary three-dimensional space). Possible
state transitions are determined by the so-called Hamiltonian
equations. These form a system of 2*n* first order differential
equations (one for each canonical coordinate) which entail unique
solutions for each possible set of initial conditions. Thus, if
we know the state of the system at some arbitrarily chosen time
*t = *0, we can - at least in principle - predict the states
of the system at all later times *t* > 0.

The following table lists in its left column the basic concepts which are relevant to all kinds of deterministic processes. In the right column these concepts are illustrated by appropriate examples from classical mechanics.

Let **Z **denote the collection of all possible states of
the system and let denote
the actual state of the system at time *t*. The family of
states is called a deterministic
process, iff there exists a uniquely determined state *z´*
**Z **for all times *t, t´
* *T *with *t
< t´ *and all *z*
**Z** such that = *z´
*holds for all , where
denotes the collection of all
physically possible state transitions (paths) : .

Actually, *all *classical systems are deterministic in
the above sense. The reason is that state transitions in classical
physics are always determined by appropriate differential equations
which entail unique solutions for all possible initial conditions(2). Even quantum physics has
a deterministic component, since the state transitions of a quantum
system are determined by the well-known Schrödinger equation(3). Unfortunately, solutions
of this equation yield only statistical predictions about the
(probabilities of) possible outcomes of physical measurements
and observations, namely in accordance with Heisenberg's uncertainty
relation.

Transtheoretical structures do not only occur in the exact natural sciences, they also emerge in the social sciences and humanities, as is obvious from appropriate examples. I will sketch only three examples here, which should suffice to illustrate my thesis.

1. The first example is taken from theoretical linguistics. Grammatical structures in natural and formal languages can be mapped by appropriate Chomsky grammars. These grammars correspond to certain types of languages. The best known examples are recursively decidable and recursively enumerable languages, primitive recursive languages, context free versus context sensitive languages and regular languages. Basically, these languages form the so-called Chomsky hierarchy. The following diagram(4) shows the derivation of an English sentence from an unrestricted (type-0) grammar.

The corresponding mathematical structures do not only occur in (appropriate fragments of) the natural languages; the same structures also occur in the computer sciences. The reason is that there exists a one-to-one correspondence between languages in the Chomsky hierarchy and certain types of automata. For example, recursively enumerable languages correspond to Turing machines, regular languages to finite automata and context-free languages to pushdown automata respectively(5).

2. My second example is taken from economics. Theoretical models of economic growth are typically deterministic in the above sense, i.e. they describe the changes of macroeconomic quantities, like the GNP, public and private consumption, the level of exports and imports, etc. in terms of mathematical equations which yield unique solutions for each possible set of initial values of these quantities.

This example might give rise to certain objections against the possibility of deterministic models for social phenomena, since economic predictions hardly ever correspond to economic reality. The following table matches the actual economic growth in Germany from 1975 - 1994 against the predictions of an expert council (the so called Sachverständigenrat)(6). One can easily see that actual and predicted values differ more or less significantly for most of these years.

The notorious unreliability of economic predictions does not
provide much support for a deterministic view of economic phenomena.
Nevertheless, economic predictions are still better than random
predictions or astrological horoscopes. This is obvious from the
positive correlation (*r = *0.64) between actual and predicted
values.

The following diagram exhibits the same data in a different format. Ideally (i.e. if all predicted values were exactly correct) all data points would be located on a 45 line through the origin of the coordinate system. In reality, the data are scattered in a more or less irregular way around the 45 line. This scattering reflects the difference between actual and predicted growth.

3. My last example is taken from the cognitive sciences. Connectionist theories of cognition have attempted to provide mathematical models for the cognitive functions of the human brain. Such models (mainly, neural networks) have first emerged in the context of artificial intelligence research, but they have found applications in cognitive psychology and neurobiology as well. The basic idea of these models is that the cognitive functions of the human brain (like pattern recognition, language processing, motor control, logical and mathematical thinking, etc.) can be explained by the parallel distributed processing of information in a large number of interconnected processing units.

Artificial networks are composed of a finite number of interconnected
microprocessors, whereas natural networks consist of large numbers
of biological neurons along with their synaptic connections.
The actual state of a single neuron is mathematically represented
by the so-called activation function. In the simplest case this
is a binary function that can take on only two values (e.g. 1
= This diagram(8) shows
a simple neural circuit as it might occur in the processing of
visual information. It consists of a small number of input-units
(retina cells, for example) which obtain information from the
outside world. Behind this input layer there is a second layer
of so called "hidden units" which are all connected
to the same output-unit. Excitatory synapses are represented
by Y's, inhibitory synapses by T's in this diagram. One can easily
see that all neurons in the centre of the receptive field are
connected to the "target neuron" This pattern is typical of the perception of contrasting colours or forms, such as the perception of a bright spot on a dark background or vice versa. |

The second diagram(9)
shows a network of linear threshold units which corresponds to
the *exclusive or* (XOR) in mathematical logic. The output
unit responds iff either the left or the right input unit is active.

The above examples clearly show that transtheoretical structures do not only emerge in the natural sciences but also in the social sciences and humanities. But examples themselves do not provide an answer to more basic questions, e.g.: under which conditions can mathematical structures from the natural sciences be transferred to and applied within the social sciences? More specifically: under which conditions can social phenomena be described by deterministic models? And to what extent can human behaviour be explained and predicted by such models?

Any possible answer to these questions is closely connected to philosophical questions about the human mind. An immediate and unrestricted transfer of mathematical structures from the natural to the social sciences would correspond to a naturalistic view of the human mind. According to this view the human mind should be considered as a natural outcome of biological evolution and man himself as an integral part of physical nature. This means (among other things) that human beings are subject to the same physical, chemical and biological laws like all other phenomena in nature. Under this premise, deterministic models and structures can be transferred immediately and necessarily from physics or biology to the human mind, since mental processes themselves are (composed of) physical and/or biochemical processes.

Moreover, a naturalistic conception of the mind entails that we can infer anthropological determinism more or less directly from physical determinism. By "anthropological determinism" I understand the thesis that human thinking and acting is subject to deterministic laws. The logical connection between these assumptions should become clear from the following diagram:

(1)

Physical determinism:All physical processes are subject to deterministic laws.

(2)Naturalism:All human actions / mental operations, etc. are physical processes.

(3)Anthropological determinism:All human actions / mental operations, etc. are subject to deterministic laws.

It is important to understand that neither naturalism nor physical
determinism *alone* entails anthropological determinism.
It is the *combination* of both principles which makes the
above conclusion inevitable.

Thus, if one wishes to avoid the conclusion of the argument (anthropological determinism), one must reject at least one of the above premises. There are indeed many objections that have been raised against one or the other premise of this argument. I will consider the case of physical determinism first.

The most obvious objection against physical determinism is derived from quantum physics. The conclusiveness of this objection, however, hinges on two conditions that have remained a matter of debate till today. First, quantum physical indeterminism refers to microscopic (atomic or subatomic) processes where the magnitudes of physical effects are small with respect to Planck's constant. On the other hand, the propagation of neural activities in the human brain is a comparatively macroscopic process which can be described - at least approximately - by appropriate differential equations and hence by deterministic models(10).

Quantum physical processes ("quantum jumps") *can*
have macroscopic effects, however, as has been illustrated by
the famous thought experiment with Schrödinger's cat. Nevertheless,
at the present state of brain research there seems to be no decisive
proof for the relevance of quantum physics as far as neural activities
in the human brain are concerned.

Secondly, even if a significant impact of quantum physical effects onto the neural processing of information could be proved, this would not amount to a definite refutation of anthropological determinism. The statistical nature of quantum physical predictions admits of different explanations. Some authors (like John von Neumann, J. Jauch and C. Piron) have attempted to prove that the mathematical formalism of quantum physics cannot be embedded into a logically consistent deterministic theory of microscopic phenomena. If they were right, Heisenberg's uncertainty relation would not just reflect our ignorance, but rather exhibit a genuine indeterminism of the physical micro world itself.

On the other hand, David Bohm has argued already 50 years ago
that quantum physics *can* actually be interpreted as a deterministic
theory in a more or less classical sense. Unfortunately, Bohm's
interpretation hinges on certain assumptions (the existence of
"hidden variables") which cannot be verified by independent
empirical tests. If Bohm is right, Heisenberg's uncertainty relation
would reflect our incomplete knowledge of the relevant (micro-)
physical conditions.

Quite a different and more important critique of anthropological determinism does not refer to physical considerations but rather rejects the premises of the naturalistic philosophy of mind which equates human thinking and feeling with physical or biochemical processes in the human brain. Obviously, I cannot even roughly discuss the relevant theories and arguments in this paper. Instead, I will restrict myself to three final remarks.

(1) First of all, one should note that the naturalistic theory
of mind is indeed a sufficient, though in no way necessary condition
for the applicability of mathematical structures from the natural
sciences in the realm of human thinking and acting. In fact, it
suffices to assume that mental processes are in some sense *isomorphic*
to the corresponding brain processes(11).

This is obvious from the fact that mathematical structures, by their very definition, remain invariant under isomorphisms (i.e. isomorphic structures share the same mathematical properties). Thus, in a certain sense, transtheoretical determinism is compatible even with a classical Cartesian dualist conception of the mind.

(2) Anthropological determinism cannot only be inferred from physical determinism via a naturalistic theory of mind. It is also and more directly implicit in certain theories of cognition. Prominent examples are the above mentioned connectionist models of the mind: if neural networks provide an appropriate model for cognitive processes in the human mind, then these processes are deterministic in an obvious sense, since neural networks are deterministic systems by their very definition: the actual behaviour of the system (its output) is completely and uniquely determined by the actual inputs together with the present state of its neural connections. This is the reason why the behaviour of a neural network can always be simulated and predicted by an ordinary (von-Neumann-) computer.

(3) Finally, I would like to emphasize that the applicability of mathematical structures from physics and biology in the realm of the human mind does not by itself require a deterministic anthropology. Nor does the naturalistic philosophy of mind. This becomes more obvious, when we consider stochastic networks, where the weighted sum of all incoming signals entails the corresponding neural activities with certain probabilities only. This would amount to a non-deterministic network whose behaviour would not be uniquely predictable. Nevertheless, the model would provide us with legitimate transtheoretical structures in the above defined sense.

We thus see that the emergence of transtheoretical structures
in the natural and social sciences is not just an accidental concomitant
of scientific research. It is rather a more or less direct consequence
of certain philosophical conceptions of the human mind. These
conceptions may be hypothetical themselves. But at the same time
the actual emergence of transtheoretical structures provides (meta-)
empirical evidence for the correctness of the underlying ideas.

© Bernhard Lauth (Department of Philosophy, Logic and Theory of Science, University of Munich, Germany)

NOTES

(1) The structuralist philosophy of science claims that scientific theories are best conceived as comprehensive networks of interrelated "theory-elements", where each theory-element is defined by a corresponding core of mathematical models and a variable range of empirical applications.

(2) e.g. Newton's second axiom and the Hamiltonian equations in classical mechanics, Maxwell's equations in classical electrodynamics or Einstein's field equations in general relativity.

(3) In the case of quantum physics, the classical phase space must be replaced by an infinite dimensional Hilbert space.

(4) The
diagram is taken from J. E. Hopcroft, J. D. Ullman; *Introduction
to Automata Theory, Languages and Computation, *Reading Mass.
1979, S. 3

(5) see, for example, J. E. Hopcroft, J. D. Ullman; loc. cit. chapter 9

(6) The
following data are taken from L. Fahrmeir et al.; Statistik: *Der
Weg zur Datenanalyse*, Berlin, 1999.

(7) A more detailed description of neural activity might operate with real valued rather than binary functions

(8) E.
B. Goldstein; *Wahrnehmungspsychologie, *Berlin, 1997, p.
64.

(9) D.
E. Rumelhart et al; *Parallel distributed processing, *Vol.
1, Cambridge Mass., 1986, p. 64.

(10) This
is particularly true in the case of the propagation of electrical
impulses along the axons of a single neuron; see, for example,
R. F. Thompson; *Das Gehirn, *Berlin, 2001, 498 f.

(11) This assumption is crucial for certain "functionalist" conceptions of human cognition.

REFERENCES

L. Fahrmeir et al.; *Statistik: Der Weg zur
Datenanalyse*, Berlin, 1999.

E. B. Goldstein; *Wahrnehmungspsychologie,
*Berlin, 1997.

J. E. Hopcroft, J. D. Ullman; *Introduction
to Automata Theory, Languages and Computation, *Reading Mass.,
1979.

D. E. Rumelhart et al; *Parallel distributed
processing, *Vol. 1, Cambridge Mass., 1986.

R. F. Thompson; *Das Gehirn, *Berlin,
2001.

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For quotation purposes:

Bernhard Lauth (University of Munich, Germany): Transtheoretical structures in the natural and social sciences. In: TRANS. Internet-Zeitschrift für Kulturwissenschaften. No. 15/2003. WWW: http://www.inst.at/trans/15Nr/01_6/lauth15.htm