Nothing that can be called thinking is formalized or formalizable; nor can it be likened to a mechanical process (Church's hypothesis). Rather, thinking sets into motion human imagination and passion.Having already written extensively on the imagination,' I will limit myself here to outlining its basic structure. At the two opposite poles of knowledge, as well as in its center, lies the creative power of the human being, that is, radical imagination. It is thanks to the imagination that the world is (...) presented in this form and not in some other: it is imagination that creates axioms, postulates, and the fundamental patterns that subtend the structures of knowledge; finally, it is imagination that both furnishes the hypothetical models and idea-images of knowledge, and makes possible their potential development and/or insight into them. This imagination, however, both in itself and in its essential operations – which include its social forms, experienced on this level as the creation of an anonymous collective force – is neither formalized nor formalizable. Obviously, the imagination contains – just as does everything else that exists – a totalizing, identity-bearing dimension (which for brevity's sake we have elsewhere called ensidiqt,ce)2; but this dimension of the imagination is not the essential one, neither in its operations nor its results, no more than the arithmetic relations between tones are the essential element of a Bach fugue. (shrink)

"Full of vitality as well as profundity, and resonating with something I can only term friendship, these meditations / memoirs belong to the great tradition of metaphysical prose, alongside the works of Nietzsche, Shklovsky, Kierkegaard,...

The existence of a partnership between knowledge and armies - and, connected with it, between knowledge and wars, conquests, and the entire apparatus of empires - has been affirmed since the time of Xenophon. The troops clear a path that the scholars follow, and an increase of knowledge is a side effect of the incursion. The great linguistic discoveries of the eighteenth century - that is, the Zend and Sanskrit languages - would have been impossible without the expansion of the (...) French and British empires into Asia; and Bonaparte, in his foray into Egypt, was accompanied by a large contingent of scholars. After the uniformed troops march in, official scholarly missions follow. For example, in 1849, Italian archives were opened to French scholars as a result of the French occupation of Rome. Renan's access to Italian libraries, monasteries, and manuscript collections was a result of the presence of French troops in Italy. This was a period of preliminary analyses, of the publication of catalogues, and of amazement at the wealth of available documents. In brief, it was a period of successful compilation, of rapid and exhilarating erudition. It was a time when everything rang true. In 1860, it was once more Renan who set off, on the margins of the French intervention in Syria and Lebanon, in search of epigraphical materials. In fact, all of nineteenth-century French, German, and English archaeology follows in the wake of the colonial expansion of the various empires. (shrink)

When the world reveals a part of its beauty, what should our reaction be? How can we respond adequately? Is not our initial reaction one of a “discrepancy between our impressions and their habitual expression?” It is this question that Proust poses in one of the crucial passages early on in his masterpiece. Describing his walks along Méséglise's Way, and “the humble discoveries” he made there, the narrator details for us the overwhelming, decisive impression made on him by a shaft (...) of sunlight:After an hour of rain and wind, against which I had struggled cheerfully, as I came to the edge of the Montjouvain pond, beside a little hut with a tiled roof in which M. Vinteuil's gardener kept his tools, the sun had just reappeared, and its golden rays, washed clean by the shower, glittered anew in the sky, on the trees, on the wall of the hut and the still wet tiles of the roof, on the ridge of which a hen was strutting. The wind tugged at the wild grass growing from cracks in the wall and at the hen's downy feathers, which floated out horizontally to their full extent with the unresisting submissiveness of light and lifeless things. The tiled roof cast upon the pond, translucent again in the sunlight, a dappled pink reflection which I had never observed before. And, seeing upon the water, and on the surface of the wall, a pallid smile responding to the smiling sky, I cried out aloud in my enthusiasm, brandishing my furled umbrella: “Gosh, gosh, gosh, gosh!” But at the same time I felt that I was in duty bound not to content myself with these unilluminating words, but to endeavor to see more clearly into the sources of my rapture. (shrink)

A multiplicity of circumstances - including geographic and political isolation, and differences of social organization and customs - has led different groups of people to develop mathematical knowledge independently of each other. Yet history has shown us again and again that by some necessity these separate groups often encounter the same problems. The solutions they propose, however, are often different. This suggests a series of questions. First of all: what is the relationship between the solutions? Is one solution an alternative (...) to the other, and if so, how to account for their differences? Or, on the contrary, are the solutions complementary? Both of these configurations can be found in the history of mathematics: in the following article we will investigate two elementary examples of these phenomena and trace the historical circumstances under which different solutions have come into contact. (shrink)