The present paper expounds the logic notation proposed by Augustus De Morgan in 1850 from within the original context of De Morgan’s account of syllogistic logic and his approach to quantification. The notational system of 1850 is shown to be a flexible tool to state inferences, to prove their validity and to derive formulæ of the respective system by ‘blind’ application of transformation rules. These pertain to the swapping of operator signs, which are of inverse ‘character’ in a two-fold sense: (...) The signs both represent inverse operations and exemplify oppositeness due to their very shapes. Thus, while De Morgan’s initial choice of symbol shapes is due to certain diagrammatic features, his notational system of 1850 leads to an account of logical operations as effected by manipulation of symbol tokens. It is concluded that as an instantiation of what De Morgan terms a ‘calculus of opposite relations’, his notation of 1850 may also be conceived of as a ‘graphicism’ affording for the working of a ‘symbolic machine’ on ‘de-semantification’. (shrink)

More than any other aspect of the Second Scientific Revolution, the remarkable revitalization or British mathematics and mathematical physics during the first half of the nineteenth century is perhaps the most deserving of the name. While the newly constituted sciences of biology and geology were undergoing their first revolution, as it were, the reform of British mathematics was truly and self-consciously the story of a second coming of age. ‘Discovered by Fermat, cocinnated and rendered analytical by Newton, and enriched by (...) Leibniz with a powerful and comprehensive notation’, wrote the young John Herschel and Charles Babbage of the calculus in 1813, ‘as if the soil of this country [was] unfavourable to its cultivation, it soon drooped and almost faded into neglect; and we now have to re-import the exotic, with nearly a century of foreign improvement, and to render it once more indigenous among us’. (shrink)

The “centre-periphery” relationship historically structured scientific exchanges between metropolis and province, between the fount of empire and its outposts. But the exchange, if regarded merely as a one-way flow of scientific information, ignores both the politics of knowledge and the nature of its appropriation. Arguably, imperial structures do not entirely determine scientific practices and the exchange of knowledge. Several factors neutralise the over-determining influence of politics—and possibly also the normative values of science—on scientific practice.In examining these four examples of Indian (...) scientists in encounters with their peers at the centre, exceptional scientists are seen in a social context where the epistemology of science supposedly describes its practice. Imperialism imposes practices and patronage, which moderate the exchange of scientific knowledge. But, at Level Two, the politics of knowledge and the patterns of patronage within it mediate exchanges between the centre and the periphery.The first step in reconfiguring exchanges between centre and periphery —in this case, between Europe and India during the period 1850 to 1930— is to recognise the relation between the acquisition of resources and the maintenance of legitimacy and identity.67 Political life is not confined to the core of political institutions.68 Second, in examining science as practised in the colonies, it is necessary to see stages of scientific institutions, whose development structures the exchange.From the encounter of Ramchandra and De Morgan, it is evident that the centre-periphery framework should be separated from the models of transmission embedded within it. The notion of “translation” helps to suggest that scientists bring personal motives and meanings to each encounter. Ramchandra, for example, sought a novel method of teaching Indians calculus, while De Morgan's interest lay in finding a place for algebra in a liberal education.The hierarchy inherent in the centre-periphery framework compels the conclusion that, at Level Two, the autodidact outside the institutions of science must have his work presented to scientists at the centre by authoritative figures from the centre. This is not mainly a question of imperialism, but rather of patronage. The peripheral scientist could not be granted direct entry into the collegial circle until his efforts at the periphery could be translated into the language and concerns of the central community. Ramanujan's enigmatic formulas were translated into the language of analysis by Hardy, which enabled the creation of a field to which Hardy was committed. Scientists from the periphery who were already part of the circle by virtue of their training, were not necessarily subject to the same degree of attestation as other scientists from the periphery. P.C. Ray, with his DSc from Edinburgh, and his position at Calcutta University, had less difficulty in winning the trust of colleagues at the centre, even when he returned to India. On the contrary, remaining at the periphery, he moved from a context of patronage to a sphere of competition. In addition, Ray's collegiality, even at Level Two, was more comprehensive, and connected him with Level One.Eventually, the professional Indian science graduate found collegiality within the international community of scientists. Saha's self-imposed progressive nationalism constrained his identification with the centre and made him a potential competitor instead. Once having achieved eminence in the world of science, C.V. Raman and Saha shifted their work to journals of physics published in India in order to further the cause of physics research in their own country.69To go beyond the limitations of the centre-periphery model, it is necessary not merely to examine exchanges between scientists functioning in a “shared epistemological universe”,70 but also to recognise the part played by institutions, the experience of colonialism, and the forms of patronage characterising both colonialism and science. Put another way, although there is relative epistemological autonomy within the disciplinary research communities of science, the interplay between knowledge and power structures this exchange.The scientific links between colonial India and Britain at the turn of the century were mediated by structures which prefigured change. Does structure determine all? If it does, we are left with an Orientalist reconstruction of the docile native, and a passive cultural medium into which science percolates. But this neglects the role of scientists in creating new structures within which they worked. A middle position—one more sensitive to the exigencies of colonial scientific life—would be one where the participants are seen not as the dupes of “structure nor the potentates of action”, but as occupying a ground between the two.71. (shrink)

Although it is clear that Sir William Rowan Hamilton supported a Kantian account of algebra, I argue that there is an important sense in which Hamilton's philosophy of mathematics can be situated in the Newtonian tradition. Drawing from both Niccolo Guicciardini's (2009) and Stephen Gaukroger's (2010) readings of the Newton–Leibniz controversy over the calculus, I aim to show that the very epistemic ideals that underpin Newton's argument for the superiority of geometry over algebra also motivate Hamilton's philosophy of algebra. Namely, (...) Hamilton's defense of algebra, like Newton's defense of geometry, is driven by the claim that a mathematical science must have a proper object and thus a basis in truth. In particular, Hamilton aims to show that algebra is not a mere language, or tool, or a mere “art”; instead, he argues, algebra is a bona fide mathematical science, like geometry, because its methods also provide true and accurate insight into a genuine subject matter, namely, the pure form of temporal intuition. (shrink)

In 1792 Dugald Stewart published Elements of the philosophy of the human mind. In its section on abstraction he declared himself to be a nominalist. Although a few scholars have made brief reference to this position, no sustained attention has been given to the central role that it played within Stewart’s early philosophy of mind. It is therefore the purpose of this essay to unpack Stewart’s nominalism and the intellectual context that fostered it. In the first three sections I aver (...) that his nominalism emanated from his belief that objects of the mind—qualities, ideas and words—were signs that bore no necessary relation to the external objects that they were meant to represent. More specifically, it was these signs that were arranged into systems of thought by the ‘operations of the mind’. The next three sections suggest that his treatment of words as signs most probably originated in his views on language and medicine and that his nominalistic philosophy of mind could also be extended to systems that sought to classify the natural world. I conclude by suggesting several avenues of enquiry that could be pursued by future scholars interested in excavating Stewart’s thought. (shrink)