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p.10 The 'paradox' (Whitehead 1925 p.32) of modern science, or of the modern scientific mentality, is the way in which highly abstract mathematically formulated theories have been so effectively applied to practical affairs.
p.11 Like Duhem, he [Bachelard] sees the history of science, or rather the history of a concept or of a theory, as crucial in establishing and understanding the relation between theoretical and experimental domains.
p.12 Bachelard insists, scientific thought has to break with common sense, and this transition amounts to an epistemological rupture. However, common sense has a tenacious hold on our thought... Common-sense thought deals in images, scientific thought in concepts.
p.18 In aiming at knowledge of reality, our very general conception of the structure of that reality is revealed in the standards by which knowledge claims are assessed, in Bachelard's terminology, in the epistemological values revealed in the evaluation of such claims and in the sort of information that is taken to be important in advancing knowledge in the area concerned.
p.18 Determination of the object of knowledge goes via epistemological values, i.e. via the ways in which it is presumed to be possible to have knowledge of it. This approach is summed up in Bachelard's remark (NES p.143), 'Tell me how you are sought, I will tell you what you are.'
p.19 Science has a philosophical component. If this is so, then scientific change can require philosophic change.
p.25 there is no process of critical reflection which can be both rational and genuinely innovative.
p.30 Descartes' procedure is reductive... He presumes that there are simple starting points for knowledge (epistemological foundations) and that these are found in our clear and distinct perception of simple ideas... Any other idea must be reduced to, or be seen in terms of, such simple ideas if knowledge concerning its objects is to be possible.
p.32 Contemporary science thus works without the presupposition of any simple unrevisable 'givens', whether at the theoretical or at the empirical level.
p.55 It is with dream worlds and dream images, as revealing the structures of the unconscious, that Bachelard associates alchemy... It is for this reason the antithesis of science and belongs with poetry. It is a voyage of the subject into itself... but not in the direction of the world of material objects. It is a voyage that needs to be taken, but not under the illusion that its end will be objective (scientific) knowledge.
p.64 The conditions for creative science are not wholly separable from those of creative artistic work.
p.64 In proposing a new theory or a new set of experimental techniques it is necessary for an individual, or group of individuals, to think outside the accepted norms, to break with them. But for scientific development this cannot be just an arbitrary break; it has to be rationally, critically motivated. It has to arise out of a questioning of the correctness or legitimacy of accepted experimental or theoretical practices... This requires scientists themselves to regard the framework within which they are working as open, as having a potential for development, rather than as fixed, final and closed and therefore the framework within which all future research must be envisaged as proceeding. It is just such an attitude on the part of scientists which Bachelard claims to characterise contemporary science.
p.66-67 Conceptual innovation requires creativity on the part of the subject, who asserts his autonomy in thinking beyond accepted norms. But if this conceptual innovation is to be even a candidate for cognitive advance, the subject has not merely to be able to communicate his critically constructive thought, he has to be able to convince others that there are reasons for taking it seriously.
p.75 Axiomatic systems are mathematically constructed psychological robots (RA p.25). But reasoning is absent from the functioning of robots.
[JLJ - Reasoning might become present in the results of a diagnostic test, as in, "we believe such and such a position, because it is indicated in the results of the well-constructed and relevant diagnostic test". You can make a robot execute a diagnostic test, then perform one action or another based on the results. The robot might then appear to behave intelligently, based on clever programmer-interpretations of richly-detailed signs or cues present in the environment, but the robot did not construct the diagnostic test, modify it to keep it relevant over time or challenge, nor understand what what it does does.]
p.77 Even thought within a formal system, thought according to rules, requires more than mechanical rule-following. Calculation is in the first place used to reach a result, but to be sure of the correctness of the result one must be able to go back and check the calculation. The calculation and its result have to be evaluated; if correct the calculation is seen to prove its result.
p.77 Logical formulation in mathematics is necessary for rigour, but it reflects only one aspect of mathematical thought. It omits the less tidy processes of mathematical discovery... These are the processes which have to precede axiomatisation.
p.77 Axiomatisation is only possible by reflection on a preceding activity. It is a reordering of past thought (a rethinking, never a first thinking) in the interests of rigour. This is why, for Bachelard, rationalism is something which has no beginning.
p.78 [Russell] The number 2 is not purely mental, but is an entity which may be thought of. Whatever can be thought of has being, and its being is a precondition, not a result, of it being thought of.
p.80 'Number is only a moment of numeration and all numeration is a method of thought. One could also say that number is a synthesis of acts' (CA p.174). Numbers and arithmetic have a wholly subjective origin and correspond to an internal experience: the experience of a rationally active subject. The conditions of rigour are, Bachelard says, inextricably linked to those of voluntary action. 'If we want to know with a maximum of rigour, we must organize acts, substitute totally the constructed for the given' (CA p.174). The bare structure of constructive thought is that of the step-by-step 'construction' of a number in counting. [JLJ - Nope. The bare construction of constructive thought is the execution of the 'scheme' - modified over time, impressed into us and which we believe in out of necessity - which tells us how to 'go on'. Now such schemes usually involve the recognition of and enumeration of certain cues, but 'counting' itself is and must be part of a scheme of understanding, of some kind.]
p.80 Construction as a rational process, which can yield knowledge in so far as one has a conception of and hence knows the method of construction, is a discrete and linearly ordered process, one whose formal archetype is to be found in the construction of the natural number sequence. Grasp of a method of construction does not carry with it an ability to visualise or imagine the product. Taken by itself it yields only an intellectual, non-experiential conception of the entity constructed.
p.82-83 Bachelard sees the demand for rigour as initially arising out of the recognition of error... It is only to the extent that mathematical objects are constructs that certain knowledge of their properties is possible... the construction goes on in the development of organised mathematical theories; in the process of agreeing on precise definitions finding adequate sets of axioms, and standardising proof procedures. In other words, it takes place in the process of formalisation, of imposing standards of rigour... it is not a matter of free, unconstrained construction, for it is a methodical reconstruction of something which was given... The direction of future development is not determined although it is constrained. [JLJ - sounds like Foucault's discursive formation]
p.84 It is reflection on rational activity which forms the basis for abstraction.
p.88 What is it to believe in the reality of something? Bachlard's answer (NES p.34) is that it is essentially to be convinced that there is more to it than is evident from what is immediately given, to believe that there is objective knowledge to be gained.
p.89 This 'quasi reality' of mathematical objects emerges extrinsically. As mere products of definition they have no independent reality.
p.113 Russell's work... it is the relations which are of primary concern, not only in geometry but, in his view, in natural science generally.
This illustrates a shift within mathematics from thinking in terms of objects as primary and then dealing with their relations, to treating relations, relational structures and their interrelations directly. Structures, in their turn are characterised in terms of their interrelations, in terms of the groups of transformations under which they are invariant.
p.120 It is characteristic of modern science to insist on the importance of detailed and precise experiments as well as on working toward very general abstract theories
p.120-121 'In point of fact, scientific objectivity is only possible if one has first broken with the immediate object... Far from marvelling at the object, objective thought must treat it ironically' (PF p.1).
p.124 An object of knowledge always bears the marks of thought. The object of scientific knowledge is thought of, and hence constituted, as objectively as possible; it is constituted by the network of its causal and theoretical relations, not by its relation to the subject.
p.124 Initial institution of material objects or natural phenomena as objects of scientific study can thus only proceed in the light of some vague, schematic (theoretical) conception of their nature.
p.125-126 one would say that the object of scientific knowledge has to be located in a problematic. The object is always an object of interest, one for which the process of objectivation has not been achieved, an object 'which does not refer purely and simply to past knowledge encrusted on a name' (RA p.55). As he says, the position of the scientific object, that which it is actually instructive to study empirically, is much more complex than either that of the immediate object of ostension or that of an object of discourse (the referent of a name), and it is much more 'engaged' in a cognitive sense. To grasp the object of knowledge one has to have some idea of the methods by which one can come to have more knowledge of it, and hence also of what it would be to have such knowledge (the form it would take, the questions it would answer). It is the surrounding problematic which supplies this, so the problematic is an antecedent to all experience which is to be instructive. Such a problematic is initially founded on a specific doubt, a doubt specified by the object of knowledge. This questioning doubt will not at first be precise. It is part of the way in which progress is made that it becomes more precise; progress in part consists in giving more precise articulations to the questions to be answered and to the methods suitable for answering them.
p.126 Scientific progress is the product of a continual dialogue between reason and experience (pp.47-48), it is a product of the interference of theoretical and empirical conceptions of the items under investigations.
p.129 Bachelard... sees contemporary science as committed to a non-idealist philosophy... It is committed (without proof) to a transcendent reality (the thing-in-itself) and this enters with the idea of the history of science as revealing a sequence of approximations. The extent to which we can think of science as making objective progress and as legitimately taking objective knowledge as its goal is therefore intimately bound up with the extent to which history can be seen on the model of a continuable sequence of approximations.
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