Jenny Lorraine Nielsen

This paper reinterprets Abu’l Wafa al-Buzjani’s dissection-construction problems as a major site of exchange between mathematical proof and medieval Islamic artistic practice. Beginning with al-Buzjani’s “tricky triangle,” square-assembly constructions, and dissection proof of the Pythagorean theorem, the paper shows that these puzzles were not recreational curiosities but practical instruments for translating Euclidean reasoning into forms usable by mosaic-makers, architects, and artisans. Al-Buzjani’s interventions corrected approximate workshop methods, especially in the construction of larger squares from smaller ones, while also generating ornamental patterns that entered the visual vocabulary of Islamic architecture. The paper argues that this exchange was reciprocal: artisans needed mathematical precision to prevent accumulated geometric error, while mathematicians found in craft practice a living laboratory for dissection, reconstruction, irrationality, congruence, and visual proof. The “dust board” thus becomes both literal work surface and metaphor for Islamic mathematical culture, where figures could be torn apart, reassembled, proven, beautified, and spiritually charged. The final section argues that medieval Islamic pentagonal and decagonal design practices anticipated the structural logic of Penrose tilings, quasicrystals, and modern theories of quasi-periodic and aperiodic order. While al-Buzjani did not possess a modern formal theory of aperiodicity, his constructional tradition reveals that Islamic geometric art had already discovered, in practical and visual form, patterns of nonperiodic order that modern mathematics would only later name. In this sense, al-Buzjani appears not only as a geometer of construction, but as a figure through whom art pushed mathematics toward discoveries reason had not yet fully formalized. (Winner, AMA Prize for History of Mathematics, 2010)

This dissertation argues that the leading contemporary candidate for a general theory of scientific explanation, New Mechanist Explanation (NME), is unsuitable for the role, and that a Relational Structuralist theory of explanation (RSE), or "New Structuralism," offers the framework the philosophy of science has been in search of. The argument proceeds in four stages. First, I develop a set of motivating criteria (the GAME criteria) for what a general model of scientific explanation must accomplish, and review the historical models of explanation—deductive-nomological, statistical relevance, causal-mechanical, unificationist, structural—in light of those criteria. Second, I take New Mechanism on its own terms and document, through extended case studies across biology, chemistry, ecology, and physics, the substantive areas in which NME fails to capture working scientific explanation. The failures are not isolated edge cases but a coherent pattern: where the explanatory relations are not causal-productive among local parts—where they are structural, topological, statistical, or mathematical-relational—NME cannot do the work, and the working scientific explanations in these areas are not amenable to mechanistic reconstruction. Third, I locate much of NME's difficulty in its inability to ground mathematical explanation. I argue that mathematical explanation is genuine, indispensable explanation, that programs to eliminate it from science have not succeeded and arguably cannot succeed, and that Wigner's puzzle of "the unreasonable effectiveness of mathematics in the natural sciences" receives a natural answer once explanation is read as relational. Fourth, I propose Relational Structural Explanation. RSE rests on three philosophical commitments: the primacy of relation, the ratio reading of relation, and the invariance reading of structure. I develop nine criteria for RSE, demonstrate that RSE meets each of the GAME criteria, work through extended examples (the periodic table, the action potential, the Königsberg bridges, GHZ multipartite entanglement, predator-prey dynamics), and recover the historical models of explanation as sub-classes of RSE corresponding to different relational primitives. I provide a rubric for selecting among sub-classes in particular cases and respond to nine objections to the position. I situate RSE in relation to ontic structural realism, to existing accounts of structural and topological explanation, and to convergent recent work in foundations of physics and metaphysics of science. The dissertation closes by arguing that philosophers of science should adopt RSE as a working general theory of scientific explanation, treating NME as a successful local account of a sub-class of explanations rather than as a candidate for the general role NME has, until now, been advanced to fill. Among the dissertation's specific contributions: (1) NME's defenders face a trapdoor dilemma—either NME retains its parts-and-activities commitments and fails on structural, topological, and field-theoretic cases, or it expands to cover them and ceases to be recognizably mechanistic; (2) the ratio reading of relation, tracing the conceptual ground of explanation through the Latin ratio, the Greek logos, measurement theory, dimensional analysis, Noether's theorem, and topological quantum field theory, dissolves Wigner's puzzle by showing that mathematics is effective in science because both disciplines study the same kind of object, relational structure; (3) the deformation-invariance criterion—that explanatorily robust relations are those surviving continuous deformation of the system—resolves the Newman problem that has challenged structural realism since 1928, by specifying for the first time a principled filter for which structures carry explanatory content; and (4) the four-party entanglement propositions prove, via explicit monogamy-of-entanglement inequalities and the Verstraete–Dehaene–De Moor–Verschelde SLOCC classification, that four parties is the unique threshold at which pigeonhole combinatorics force irreducibly global entanglement structure, demonstrating the framework at its sharpest against mechanism. IF YOU USE THIS CITE THIS.

In recent years, New Mechanism has become one of the most popular and widely discussed philosophical accounts of scientific explanation. Some of its proponents see it as a successor to traditional deductive nomological and statistical approaches to the philosophy of explanation. New Mechanists thus argue for the generality of their approach as a model of scientific explanation. Here we will show that the generality of NME as an account of scientific explanation is restricted. Most significantly, NME is widely recognized to have trouble in accounting for explanation in traditional physics. Yet, in the face of obstacles to NME in the physical sciences, New Mechanists have yet to provide a convincing defense of the generality of their view. In this paper, I (1) summarize when and how NME works, (2) describe briefly what NME is not and can’t be, (3) outline some areas of contemporary scientific explanation where New Mechanism misses the mark; and (4) demonstrate that New Mechanism does not constitute a general theory of scientific explanation but rather covers a restricted scope of explanations.

While science is often presented as a body of knowledge or collection of passively accumulated facts, science should be examined and experienced as a performed process, a human endeavor connected to the ways we access the world. In this white paper, I briefly introduce the practice of science from the perspective of performance theory. By examining science in the context of performance, we may approach certain key questions about science directly. How do scientists perform experiments and practice the scientific method (or, more accurately, scientific methodologies)? How does science rely on particular performer(s) and the particular contexts of those performances? What distinguishes the activity of science from other kinds of human performances? By reframing science as a human performance in the context of human practices and rituals, we more closely approach an accurate and communicable explanation of what the process of “doing science” actually is as something that humans do.

In his famous “It from Bit” essay, John Wheeler contends that the stuff of the physical universe (“it”) arises from information (“bits” – encoded yes or no answers). Wheeler’s question and assumptions are re-examined from a post Aspect experiment perspective. Information is examined and discussed in terms of classical information and “quanglement” (nonlocal state sharing). An argument is made that the universe may arise from (or together with) quanglement but not via classical yes/no information coding.