On Buffon Machines and Numbers

The well-know needle experiment of Buffon can be regarded as an analog (i.e., continuous) device that stochastically "computes" the number 2/pi ~ 0.63661, which is the experiment's probability of success. Generalizing the experiment and simplifying the computational framework, we consider probability distributions, which can be produced perfectly, from a discrete source of unbiased coin flips. We describe and analyse a few simple Buffon machines that generate geometric, Poisson, and logarithmic-series distributions. We provide human-accessible Buffon machines, which require a dozen coin flips or less, on average, and produce experiments whose probabilities of success are expressible in terms of numbers such as, exp(-1), log 2, sqrt(3), cos(1/4), aeta(5). Generally, we develop a collection of constructions based on simple probabilistic mechanisms that enable one to design Buffon experiments involving compositions of exponentials and logarithms, polylogarithms, direct and inverse trigonometric functions, algebraic and hypergeometric functions, as well as functions defined by integrals, such as the Gaussian error function.Comment: Largely revised version with references and figures added. 12 pages. In ACM-SIAM Symposium on Discrete Algorithms (SODA'2011

Natural history and temporalization: reflections on Buffon's Natural history

This article presents a rereading of Buffon´s Natural History in the light of the concepts of temporal reversibility and irreversibility. The goal is to determine to what extent Buffon introduces a transformationist concept of natural forms in this work. To that effect, the main points of classical natural history and the doctrine of preformed germs are analyzed. Subsequently, Buffon´s use of the temporal variable is considered. This examination shows that despite his rejection of the theory of preformationism and the scholastic classification system, Buffon continued to use categories based on a reversible temporal matrix.Presenta una relectura de la Historia natural de Buffon a la luz de los conceptos de reversibilidad e irreversibilidad temporal. El objetivo es determinar hasta qué punto Buffon introduce en dicha obra una concepción transformista de las formas naturales. A tales efectos, se analizan los puntos principales de la historia natural clásica y de la doctrina de los gérmenes preformados. Posteriormente, se considera el uso de la variable temporal que realizaba Buffon. Se demuestra, a partir de este examen, que pese a su rechazo de la teoría preformista y del sistema escolar de clasificación, Buffon continúa utilizando categorías que remiten a una matriz temporal de carácter reversible.Fil: Galfione, María Verónica. Universidad Nacional de Cordoba. Facultad de Lenguas. Centro de Inv. En Cs. del Lenguaje de la Facultad de Lenguas; Argentina

In search for a perfect shape of polyhedra: Buffon transformation

For an arbitrary polygon consider a new one by joining the centres of consecutive edges. Iteration of this procedure leads to a shape which is affine equivalent to a regular polygon. This regularisation effect is usually ascribed to Count Buffon (1707-1788). We discuss a natural analogue of this procedure for 3-dimensional polyhedra, which leads to a new notion of affine $B$-regular polyhedra. The main result is the proof of existence of star-shaped affine $B$-regular polyhedra with prescribed combinatorial structure, under partial symmetry and simpliciality assumptions. The proof is based on deep results from spectral graph theory due to Colin de Verdiere and Lovasz.Comment: Slightly revised version with added example of pentakis dodecahedro

History without time : Buffon's Natural History, as a non-mathematical physique

International audienceWhile "natural history" is practically synonymous with the name of Buffon, the term itself has been otherwise overlooked by historians of science. This essay attempts to address this omission by investigating the meanings of "physique," "natural philosophy," and "history," among other terms, with the purpose of understanding Buffon's actual objectives. It also shows that Buffon never claimed to be a Newtonian and should not be considered as such; the goal is to provide a historical analysis that resituates Buffon's thought within his own era. This is done, primarily, by eschewing the often-studied question of time in Buffon. Instead, this study examines the nontemporal meanings of the word "history" within the naturalist's theory and method. The title of his Natural History is examined both as an indicator of the kind of science that Buffon was hoping to achieve and as a source of great misinterpretation among his peers. Unlike Buffon, many of his contemporaries actually envisioned the study of nature from a Baconian perspective where history was restricted to the mere collection of facts and where philosophy, which was the implicit and ultimate goal of studying nature, was seen, at least for the present, as unrealizable. Buffon confronts this tendency insofar as his Histoire naturelle claims to be the real physique that, along with describing nature, also sought to identify general laws and provide clear insight into what true knowledge of nature is or should be. According to Buffon, history (both natural and civil) is not analogous to mathematics; it is a nonmathematical method whose scope encompasses both nature and society. This methodological stance gives rise to the "physicization" of certain moral concepts--a gesture that was interpreted by his contemporaries as Epicurean and atheist. In addition, Buffon reduces a number of metaphysically tainted historical concepts (e.g., antediluvian monuments) to objects of physical analysis, thereby confronting the very foundation of natural theology. In Buffon, as this essay makes clear, natural history is paving the way for a new physique (science of natural beings), independent from mathematics and from God, that treats naturalia in a philosophical and "historical" manner that is not necessarily "temporal.

Hermeneutics and Nature

This paper contributes to the on-going research into the ways in which the humanities transformed the natural sciences in the late Eighteenth and early Nineteenth Centuries. By investigating the relationship between hermeneutics -- as developed by Herder -- and natural history, it shows how the methods used for the study of literary and artistic works played a crucial role in the emergence of key natural-scientific fields, including geography and ecology

Against “revolution” and “evolution”

Those standard historiographic themes of “evolution” and “revolution” need replacing. They perpetuate mid-Victorian scientists’ history of science. Historians’ history of science does well to take in the long run from the Greek and Hebrew heritages on, and to work at avoiding misleading anachronism and teleology. As an alternative to the usual “evo-revo” themes, a historiography of origins and species, of cosmologies (including microcosmogonies and macrocosmogonies) and ontologies, is developed here. The advantages of such a historiography are illustrated by looking briefly at a number of transitions the transition from Greek and Hebrew doctrines to their integrations by medieval authors; the transition from the Platonist, Aristotelian, Christian Aquinas to the Newtonian Buffon and to the no less Newtonian Lamarck; the departures the early Darwin made away from Lamarck’s and from Lyell’s views. Issues concerning historical thinking about nature, concerning essentialism and concerning classification are addressed in an attempt to challenge customary stereotypes. Questions about originality and influence are raised, especially concerning Darwin’s “tree of life” scheme. The broader historiography of Darwinian science as a social ideology, and as a “worldview,” is examined and the scope for revisions emphasised. Throughout, graduate students are encouraged to see this topic area not as worked out, but as full of opportunities for fresh contributions

A Quantized Johnson Lindenstrauss Lemma: The Finding of Buffon's Needle

In 1733, Georges-Louis Leclerc, Comte de Buffon in France, set the ground of geometric probability theory by defining an enlightening problem: What is the probability that a needle thrown randomly on a ground made of equispaced parallel strips lies on two of them? In this work, we show that the solution to this problem, and its generalization to $N$ dimensions, allows us to discover a quantized form of the Johnson-Lindenstrauss (JL) Lemma, i.e., one that combines a linear dimensionality reduction procedure with a uniform quantization of precision $\delta>0$. In particular, given a finite set $\mathcal S \subset \mathbb R^N$ of $S$ points and a distortion level $\epsilon>0$, as soon as $M > M_0 = O(\epsilon^{-2} \log S)$, we can (randomly) construct a mapping from $(\mathcal S, \ell_2)$ to $(\delta\mathbb Z^M, \ell_1)$ that approximately preserves the pairwise distances between the points of $\mathcal S$. Interestingly, compared to the common JL Lemma, the mapping is quasi-isometric and we observe both an additive and a multiplicative distortions on the embedded distances. These two distortions, however, decay as $O(\sqrt{(\log S)/M})$ when $M$ increases. Moreover, for coarse quantization, i.e., for high $\delta$ compared to the set radius, the distortion is mainly additive, while for small $\delta$ we tend to a Lipschitz isometric embedding. Finally, we prove the existence of a "nearly" quasi-isometric embedding of $(\mathcal S, \ell_2)$ into $(\delta\mathbb Z^M, \ell_2)$. This one involves a non-linear distortion of the $\ell_2$-distance in $\mathcal S$ that vanishes for distant points in this set. Noticeably, the additive distortion in this case is slower, and decays as $O(\sqrt[4]{(\log S)/M})$.Comment: 27 pages, 2 figures (note: this version corrects a few typos in the abstract

Thou shalt not say "at random" in vain: Bertrand's paradox exposed

We review the well known Bertrand paradoxes, and we first maintain that they do not point to any probabilistic inconsistency, but rather to the risks incurred with a careless use of the locution "at random". We claim then that these paradoxes spring up also in the discussion of the celebrated Buffon's needle problem, and that they are essentially related to the definition of (geometrical) probabilities on "uncountably" infinite sets. A few empirical remarks are finally added to underline the difference between "passive" and "active" randomness, and the prospects of any experimental decisionComment: 17 pages, 4 figures. Added: Appendix A; References 7, 8, 10; Modified: Abstract; Section 4; a few sentences elsewher