Fictionalism. Drunken Risibility.

Applied mathematics is often used as a source of support for platonism. How else but by becoming platonists can we make sense of the success of applied mathematics in science? As an answer to this question, the fictionalist empiricist will note that it’s not the case that applied mathematics always works. In several cases, it doesn’t work as initially intended, and it works only when accompanied by suitable empirical interpretations of the mathematical formalism. For example, when Dirac found negative energy solutions to the equation that now bears his name, he tried to devise physically meaningful interpretations of these solutions. His first inclination was to ignore these negative energy solutions as not being physically significant, and he took the solutions to be just an artifact of the mathematics – as is commonly done in similar cases in classical mechanics. Later, however, he identified a physically meaningful interpretation of these negative energy solutions in terms of “holes” in a sea of electrons. But the resulting interpretation was empirically inadequate, since it entailed that protons and electrons had the same mass. Given this difficulty, Dirac rejected that interpretation and formulated another. He interpreted the negative energy solutions in terms of a new particle that had the same mass as the electron but opposite charge. A couple of years after Dirac’s final interpretation was published Carl Anderson detected something that could be interpreted as the particle that Dirac posited. Asked as to whether Anderson was aware of Dirac’s papers, Anderson replied that he knew of the work, but he was so busy with his instruments that, as far as he was concerned, the discovery of the positron was entirely accidental.

The application of mathematics is ultimately a matter of using the vocabulary of mathematical theories to express relations among physical entities. Given that, for the fictionalist empiricist, the truth of the various theories involved – mathematical, physical, biological, and whatnot – is never asserted, no commitment to the existence of the entities that are posited by such theories is forthcoming. But if the theories in question – and, in particular, the mathematical theories – are not taken to be true, how can they be successfully applied? There is no mystery here. First, even in science, false theories can have true consequences. The situation here is analogous to what happens in fiction. Novels can, and often do, provide insightful, illuminating descriptions of phenomena of various kinds – for example, psychological or historical events – that help us understand the events in question in new, unexpected ways, despite the fact that the novels in question are not true. Second, given that mathematical entities are not subject to spatial-temporal constraints, it’s not surprising that they have no active role in applied contexts. Mathematical theories need only provide a framework that, suitably interpreted, can be used to describe the behavior of various types of phenomena – whether the latter are physical, chemical, biological, or whatnot. Having such a descriptive function is clearly compatible with the (interpreted) mathematical framework not being true, as Dirac’s case illustrates so powerfully. After all, as was just noted, one of the interpretations of the mathematical formalism was empirically inadequate.

On the fictionalist empiricist account, mathematical discourse is clearly taken on a par with scientific discourse. There is no change in the semantics. Mathematical and scientific statements are treated in exactly the same way. Both sorts of statements are truth-apt, and are taken as describing (correctly or not) the objects and relations they are about. The only shift here is on the aim of the research. After all, on the fictionalist empiricist proposal, the goal is not truth, but something weaker: empirical adequacy – or truth only with respect to the observable phenomena. However, once again, this goal matters to both science and (applied) mathematics, and the semantic uniformity between the two fields is still preserved. According to the fictionalist empiricist, mathematical discourse is also taken literally. If a mathematical theory states that “There are differentiable functions such that…”, the theory is not going to be reformulated in any way to avoid reference to these functions. The truth of the theory, however, is never asserted. There’s no need for that, given that only the empirical adequacy of the overall theoretical package is required.

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Rhizomatic Extreme-Right.

In the context of the extreme right-wing politics in the contemporary age, groupuscules can be defined as numerically negligible political, frequently meta-political, but never party-political entities formed to pursue palingenetic ideological, organizational or activistic ends with an ultimate goal of overcoming the decadence of the liberal-democratic system. Though, they are fully formed and autonomous, they have small active memberships and minimal, if any public visibility or support, which is now inflating. Yet they acquire enhanced influence and significance through the ease with which they can be associated, even if only in the minds of political extremists, with other group lets which are sufficiently aligned ideologically and tactically to complement each other’s activities in their bid to institute a new type of society. As a result the groupuscule has Janus-headed characteristic of combining organizational autonomy with the ability to create informal linkages with, or reinforce the influence of other such formations. This enables groupuscules, when considered in terms of their aggregate impact on politics and society, to be seen as forming a non-hierarchical, leaderless and centreless, or rather polycentric movement with fluid boundaries and constantly changing components. This groupuscular right has the characteristics of a political and ideological subculture rather than a conventional political party movement, and is perfectly adapted to the task of perpetuating revolutionary extremism in an age of relative political stability.

The outstanding contrast between the groupuscular and party-political organization of the extreme right is that instead of being formed into a tree-like hierarchical organisms it is now rhizomatic. The use of the term was pioneered in the spirit of post-structuralist radicalism by Deleuze and Guattari to help conceptualize the social phenomena to which, metaphorically at least, the attributes of supra-personal organic life-forms can be ascribed, but which are not structured in a coherently hierarchical or systematically interconnected way which would make tree-based or dendroid metaphors appropriate. When applied to groupuscular right the concept of rhizome throws itself into relief its dynamic nature as a polycentric, leaderless movement by stressing that it does not operate like a single organism such as a tree with a tap-root, branch and canopy, and a well-defined beginning and an end. Instead, it behaves like the root-system of some species of grass or tuber, displaying multiple starts and beginnings which intertwine and connect with each other, constantly producing new shoots as others die off in an unpredictable, asymmetrical pattern of growth and decay. If a political network has a rhizomes political structure it means that it forms a cellular, capillary network with ill-defined boundaries and no formal hierarchy or internal organizational structure to give it a unified intelligence. Thanks to its rhizomic structure the groupuscular right no longer emulates a singular living organism, as the slime-mould is so mysteriously capable of doing. Nor is it to be seen as made up of countless tiny, disconnected micro-organisms. Instead, following an internal dynamic which only the most advanced life sciences can model with any clarity, the minute bursts of spontaneous creativity which produce and maintain individual groupuscules constitute nodal points in a force-field or web of radical political energy which fuels the vitality and viability of the organism as a whole. These qualities duplicate the very features of the Internet for making it impossible to shut down or wipe out the information it contains simply by knocking out any one part of it, since there is no mission control to destroy. The groupuscularity of the contemporary extreme right makes it eminently able to survive and grow even if some of the individual organizations which constitute it are banned and their websites closed down.

From Slime Mould to Rhizome