Quatsch mit Soße.
Zum Jahreswechsel mal, außer der Reihe, einfach nur Klamauk:
Mathematik wird ja gerne benutzt, um allem möglichen esoterischen Unsinn (von Astrologie bis Wirtschaftswissenschaften) einen wissenschaftlichen Anstrich zu verleihen. Vor allem die Chaostheorie wird da oft ins Feld geführt.
Und selbst die Topologie ist vor solchen Vereinnahmungen nicht sicher, wie ich vor einiger Zeit aus einer älteren Arbeit von Alan Sokal gelernt habe, in der unter anderem politische, philosophische und gesellschaftliche Implikationen der Topologie analysiert wurden.
Im Kapitel Differential Topology and Homology geht es um die Rolle der Topologie in den Sozialwissenschaften:
Differential Topology and Homology
Unbeknownst to most outsiders, theoretical physics underwent a significant transformation — albeit not yet a true Kuhnian paradigm shift — in the 1970’s and 80’s: the traditional tools of mathematical physics (real and complex analysis), which deal with the space-time manifold only locally, were supplemented by topological approaches (more precisely, methods from differential topology) that account for the global (holistic) structure of the universe. This trend was seen in the analysis of anomalies in gauge theories; in the theory of vortex-mediated phase transitions; and in string and superstring theories. Numerous books and review articles on “topology for physicists” were published during these years.
At about the same time, in the social and psychological sciences Jacques Lacan pointed out the key role played by differential topology:
This diagram [the Möbius strip] can be considered the basis of a sort of essential inscription at the origin, in the knot which constitutes the subject. This goes much further than you may think at first, because you can search for the sort of surface able to receive such inscriptions. You can perhaps see that the sphere, that old symbol for totality, is unsuitable. A torus, a Klein bottle, a cross-cut surface, are able to receive such a cut. And this diversity is very important as it explains many things about the structure of mental disease. If one can symbolize the subject by this fundamental cut, in the same way one can show that a cut on a torus corresponds to the neurotic subject, and on a cross-cut surface to another sort of mental disease.
As Althusser rightly commented, “Lacan finally gives Freud’s thinking the scientific concepts that it requires”. More recently, Lacan’s topologie du sujet has been applied fruitfully to cinema criticism and to the psychoanalysis of AIDS. In mathematical terms, Lacan is here pointing out that the first homology group of the sphere is trivial, while those of the other surfaces are profound; and this homology is linked with the connectedness or disconnectedness of the surface after one or more cuts. Furthermore, as Lacan suspected, there is an intimate connection between the external structure of the physical world and its inner psychological representation qua knot theory: this hypothesis has recently been confirmed by Witten’s derivation of knot invariants (in particular the Jones polynomial) from three-dimensional Chern-Simons quantum field theory.
Analogous topological structures arise in quantum gravity, but inasmuch as the manifolds involved are multidimensional rather than two-dimensional, higher homology groups play a role as well. These multidimensional manifolds are no longer amenable to visualization in conventional three-dimensional Cartesian space: for example, the projective space RP3 , which arises from the ordinary 3-sphere by identification of antipodes, would require a Euclidean embedding space of dimension at least 5. Nevertheless, the higher homology groups can be perceived, at least approximately, via a suitable multidimensional (nonlinear) logic.
Noch lustiger finde ich den Abschnitt über Mannigfaltigkeiten Manifold Theory: (W)holes and Boundaries. (“Whole” ist wohl seine Übersetzung des französischen Worts “ensemble”, auf deutsch: “Menge”.)
Manifold Theory: (W)holes and Boundaries
Luce Irigaray, in her famous article “Is the Subject of Science Sexed?”, pointed out that
the mathematical sciences, in the theory of wholes [théorie des ensembles], concern themselves with closed and open spaces … They concern themselves very little with the question of the partially open, with wholes that are not clearly delineated [ensembles flous], with any analysis of the problem of borders [bords] …
In 1982, when Irigaray’s essay first appeared, this was an incisive criticism: differential topology has traditionally privileged the study of what are known technically as “manifolds without boundary”. However, in the past decade, under the impetus of the feminist critique, some mathematicians have given renewed attention to the theory of “manifolds with boundary” [Fr. variétés à bord]. Perhaps not coincidentally, it is precisely these manifolds that arise in the new physics of conformal field theory, superstring theory and quantum gravity.
In string theory, the quantum-mechanical amplitude for the interaction of n closed or open strings is represented by a functional integral (basically, a sum) over fields living on a two-dimensional manifold with boundary. gif In quantum gravity, we may expect that a similar representation will hold, except that the two-dimensional manifold with boundary will be replaced by a multidimensional one. Unfortunately, multidimensionality goes against the grain of conventional linear mathematical thought, and despite a recent broadening of attitudes (notably associated with the study of multidimensional nonlinear phenomena in chaos theory), the theory of multidimensional manifolds with boundary remains somewhat underdeveloped. Nevertheless, physicists’ work on the functional-integral approach to quantum gravity continues apace, and this work is likely to stimulate the attention of mathematicians.
As Irigaray anticipated, an important question in all of these theories is: Can the boundary be transgressed (crossed), and if so, what happens then? Technically this is known as the problem of “boundary conditions”. At a purely mathematical level, the most salient aspect of boundary conditions is the great diversity of possibilities: for example, “free b.c.” (no obstacle to crossing), “reflecting b.c.” (specular reflection as in a mirror), “periodic b.c.” (re-entrance in another part of the manifold), and “antiperiodic b.c.” (re-entrance with 180o twist). The question posed by physicists is: Of all these conceivable boundary conditions, which ones actually occur in the representation of quantum gravity? Or perhaps, do all of them occur simultaneously and on an equal footing, as suggested by the complementarity principle?
At this point my summary of developments in physics must stop, for the simple reason that the answers to these questions — if indeed they have univocal answers — are not yet known. In the remainder of this essay, I propose to take as my starting point those features of the theory of quantum gravity which are relatively well established (at least by the standards of conventional science), and attempt to draw out their philosophical and political implications.
PS: Die Geschichte der Sokal-Affäre ist ja sicher allgemein bekannt: die Arbeit Transgressing the Boundaries: Towards a Transformative Hermeneutics of Quantum Gravity wurde 1996 in der Fachzeitschrift Social Text veröffentlicht.
Bemerkenswerter als der Hoax ist, daß Sokals Theorien geringfügige Modifikationen “echter” sozialwissenschaftlicher Theorien sind. Die oben erwähnten “Anwendungen” der Topologie in den Sozialwissenschaften gab es also wirklich!
Vielleicht wäre es mal ein Thema auch für diese Reihe, über solche Anwendungen der Topologie zu schreiben. Es hat da ja bestimmt auch nach 1996 noch neue Entwicklungen gegeben. (Nur um auch einen Vorsatz fürs neue Jahr zu haben. Selbst wenn dann wohl sowieso nichts draus wird. Hinweise auf lustige Anwendungen der Topologie nehme ich aber jedenfalls gerne entgegen…)