Concept and Form:
The Cahiers pour l'Analyse
and Contemporary French Thought

This project is funded by an Arts and Humanities Research Council (AHRC) research grant and is supported by the Centre for Research in Modern European Philosophy (CRMEP) and Kingston University's Faculty of Arts and Social Sciences.

Jean le Rond D’Alembert (1717-1783) played an important role in post-Newtonian debates about the nature and status of the sciences, in particular mathematics and mechanics. For twelve years he served as co-editor with Denis Diderot of the quintessential Enlightenment text, the Encyclopédie. ‘D’Alembert’s philosophy’, as Paul F. Johnson explains, ‘is characterized by an abiding commitment to the clarity and precision which attends mathematical abstraction. He believed that in its essence the natural order is internally structured by laws whose operation can be articulated under the principles of geometry. All natural phenomena are to be explained under the terms of those basic mathematical principles that govern the scientific domain in which they are located (chemistry or astronomy for example), and all scientific domains could be brought ultimately to perfect consistency and systematic order within a comprehensive theory. The events and processes which constitute the natural order reflect the reality of the mathematical structure which underlies them.’¹

The article reproduced here consists of the entirety of the entry on ‘Elements of the Sciences’ in the Encyclopédie (volume V, 1755). D’Alembert defends the general validity of clear and systematic scientific method, regardless of its domain of application. He argues that the elements of a science are best understood as ‘the propositions or general truths that provide a basis for other [propositions]’ and thus serve as the seed or ‘germ that need only be developed, in order to know the objects of that science in more detail’ (CpA 9.16:208).

The concept of ‘element’ itself connotes a relation of part to whole (as in ‘elements of a whole’): d’Alembert begins by asking how this relation might then apply to science ‘as a whole’, thus indirectly posing the problem of a doctrine of science. For a divine or omniscient mind, the ‘elements of all the sciences would be reduced to a single principle, whose principal consequences would be the elements of each particular science.’ Human minds have a more limited perspective. ‘However ordered the arrangement of our propositions, however precise the observation of our deductions, there will of necessity always be some gaps [vies]; not all the propositions will hold together directly, and they will form, so to speak, different and disunited groups’ (CpA 9.16:207).

Very quickly the relevance of d’Alembert’s framework to that of the Cahiers becomes clear. D’Alembert thinks of the discourse of science as a determinant chain, a series of propositions which serve as either principles or consequences for other propositions. Crucially, the elementary links in this chain can be identified even if the chain itself is partially ‘discontinuous’:

From all we have said it should be obvious that we should only tackle the elements of a science when the propositions that constitute it are no longer isolated and independent from one another, and when we can identify principal propositions, propositions from which others follow as their consequences. How are we to distinguish these primary propositions? In the following way: If the propositions that form the whole [l’ensemble] of a science do not follow immediately one after the other, we will identify the places where the chain is broken. The propositions that form the head of each part of the chain are those that should be treated as elements (CpA 9.16:209).²

A proposition will be treated as ‘principal’ i.e. ‘elementary’ on several conditions: if it is relatively distinct from others, if it is the source of other propositions (that can be treated as its consequences), and if it is not implied (or at least not fully implied) by other propositions (CpA 9.16:209). Some isolated propositions (propositions that are neither ‘the principle nor the consequence of another’) should also be included among the elements, ‘for the elements of a science must contain at least the germ of all the truths that form the object of that science; consequently, the omission of a single isolated truth would render the elements imperfect’ (CpA 9.16:209). The sequence that constitutes the elements of a science is thus no less ‘perfect’ for being discontinuous, i.e. for containing propositions that do not determine any others and yet remain true.

The general goal is to establish this fundamental chain or set of elementary propositions, which determine the development of a particular science, in the clearest and simplest terms. D’Alembert adopts a broadly empiricist (or rational-empiricist) approach here. Since a science can only draw its true principles from ‘observation’, so ‘the metaphysics of each science can only consist of the general consequences that result from observation, considered from the widest possible point of view.’ We should ensure then that the ‘metaphysics of the propositions’ which shape a science is limited to the ‘clear and precise exposition of the general and philosophical truths upon which the principles of [that] science are established’. Philosophy deals only in ‘facts or chimera’, and whatever is true is by the same token clear and available to all, without obscurity or convolution. ‘The truth is simple, and wants to be treated as it is’ (CpA 9.16:210). Any science worthy of the name should adopt, as its founding principles, ‘simple facts, clearly observed and confirmed facts: in Physics, the observation of the universe; in Geometry, the principal properties of extension; in Mechanics, the impenetrability of bodies; in Metaphysics and in Morals the study of our soul and its affections, and so on’ (CpA 9.16:210).

A philosophy rationally based ‘observations and facts’ will avoid all sterile scholastic speculation about the general properties of being or ideas. All legitimate abstraction comes directly from ‘the study of particular beings’ (CpA 9.16:210), and is based on simple, uncontroversial points of principle. The science of mechanics, for instance, should have nothing to do with ‘the obscure and contentious metaphysics of the nature of movement’; while the deluded Zeno (author of the famous paradoxes concerning rest and movement) wastes his time trying to find out if bodies actually move, ‘Archimedes finds the laws of equilibrium, Huygens those of percussion, and Newton those of the system of the world’ (CpA 9.16:211).

Although principles that depend ‘less on the nature of things than on language’ are ‘secondary’ in relation to the ‘primitive truths’ of a science, communication of such truths depends on the clarity of the propositions that convey them. D’Alembert proceeds, then, to reflect on what is involved in the definition and use of the elementary terms used in a science. ‘To define, following the force of the word, is to mark the boundaries and limits of a thing; thus to define a word is to determine and circumscribe its sense or meaning in such a way that one is unable to doubt this given sense, nor extend it, nor restrict it, nor attribute to it the sense of any other term’ (CpA 9.16:211). Terms should be defined by ‘examining the simple [i.e. non-decomposable or non-derivable] ideas they contain’, e.g. our ideas of ‘existence’ or ‘sensation.’ The ‘degree of simplicity of the ideas’ at work in a term is determined not by the operations of the mind but by ‘the greater or lesser simplicity of the object’ to which it refers (CpA 9.16:212). Simple ideas are acquired either through simple sensation (e.g. of colour or warmth) or through abstraction, which ‘is in fact nothing more than the operation by which we consider a particular property in an object without attending to which ones connect with what others to constitute the essence of the object’ (CpA 9.16:212). D’Alembert introduces an additional distinction: the terms used in a science are either ‘vulgar’ i.e. familiar, or ‘scientific’ i.e. technical. Any definition of a vulgar term which contains only a single simple idea would be redundant. The work of definition can then proceed according to the criterion of economy: there is no need, in mechanics, to define familiar terms like time and space, since they contain only a single simple idea, but the notion of movement, although equally familiar, requires definition in order to clarify the two simple ideas it contains (‘that of the space traversed, and that of the time required to traverse it’) (CpA 9.16:213). The shorter a definition the clearer it will be, and the use of technical terms in a science should be kept to a minimum, in keeping with the rule that they must easily be defined by ‘other, more vulgar and more simple terms’, none of which require any additional explanation (CpA 9.16:214).

In partial anticipation of Kant, d’Alembert derides as ‘pointless’ any attempt to determine whether a definition applies to either word or thing.

“

Ignorant as we are of what beings are in themselves, the knowledge of the nature of a thing (at least in regard to us) can only consist in the clear and simplified or de-composed [décomposée] notion, not of the real and absolute principles of this thing, but of those that it appears to us to contain. A definition can only be conceived under this latter point of view: in this case, it will be more than the simple definition of a name, since it will not be limited to explaining the meaning of a word, but it will rather simplify or de-compose its object; and it will also be less than the definition of a thing [in itself], since the true nature of the object, however much we de-compose it, may always remain unknown (CpA 9.16:213).

”

In the last part of his article, d’Alembert ponders a few general questions regarding the treatment of elements.

First of all, how should we order them? Rather than follow the actual process of their empirical discovery or invention, we should adopt a ‘methodical’ approach. An ‘analytic method’ is best suited to mathematics and other sciences whose ‘object is not outside us, and whose progress depends solely on meditation’. Analysis ‘proceeds from composite ideas to abstract ideas, and moves from established consequences to unknown principles.’ A ‘synthetic method’, on the other hand, ‘descends from principles to consequences and from abstract ideas to composite ones’, and is better suited to sciences concerned with objects external to us. ‘In general, the analytic method is more suited to the finding of truths, or for showing how they were found. The synthetic method is more suited to explaining and helping us understand the truths we have found. The one teaches us to struggle against difficulties in going back up to the source; the other places the mind in the source itself, from which it needs only follow an easy course’ (CpA 9.16:215).

Second, should we prefer, in our elaboration of the elements of a science, rigour or facility? For d’Alembert the one implies the other (in anticipation of an epistemological minimalism we find all through the Cahiers), and this applies to all three of the ‘kinds of knowledge’ he recognises: history, the liberal and mechanical arts, and the sciences as such (which have the concerns of ‘pure reasoning’ for their object) (CpA 9.16:216).

Third, how and why should we study such elements? Truly to know the elements of a science is something you must accomplish for yourself. To acquire such knowledge is ‘to gain access to the genius of the inventor, and to put yourself in a position to go further than he did’ (CpA 9.16:217). Elucidation of a suitably ordered set elements orients scientists ‘on the path of discoveries to be made, by presenting to them what has been discovered’, while enabling the rest of us to assess their work and to distinguish true discoveries from false. ‘[F]or anything that cannot be added to the elements of a science as a supplementary form is not worthy of the name discovery’ (CpA 9.16:218).

Notes

1. Paul F. Johnson, ‘D’Alembert, Jean Le Rond’, Routledge Encyclopedia of Philosophy, ed. E. Craig (London: Routledge, 1998). The ‘nature and number of mathematical sciences’, d’Alembert argues in his ‘Preliminary Discourse’ to the Encyclopédie (1751), ‘should not overawe us. It is principally to the simplicity of their object that they owe their certitude. Indeed, one must confess that, since all the parts of mathematics do not have an equally simple aim, so also certainty, which is founded, properly speaking, on necessarily true and self-evident principles, does not belong equally or in the same way to all these parts. Several among them, supported by physical principles (that is, by truths of experience or by simple hypotheses), have, in a manner of speaking, only a certitude of experience or even pure supposition. To be specific, only those that deal with the calculation of magnitudes and with the general properties of extension, that is, Algebra, Geometry, and Mechanics, can be regarded as stamped by the seal of evidence. Indeed, there is a sort of gradation and shading, so to speak, to be observed in the enlightenment which these sciences bestow upon our minds. The broader the object they embrace and the more it is considered in a general and abstract manner, the more also their principles are exempt from obscurities. It is for this reason that Geometry is simpler than Mechanics, and both are less simple than Algebra. This will not be a paradox at all for those who have studied these sciences philosophically. The most abstract notions, those that the common run of men regard as the most inaccessible, are often the ones which bring with them a greater illumination. Our ideas become increasingly obscure as we examine more and more sensible properties in an object’ (D’Alembert, ‘Preliminary Discourse’, http://quod.lib.umich.edu/cgi/t/text/text-idx?c=did;view=text;rgn=main;idno=did2222.0001.083). ↵

2. D’Alembert develops this point in more detail in his ‘Preliminary Discourse’. ‘Viewed without prejudice, [mathematical theorems] are reducible to a rather small number of primary truths. If one examines a succession of geometrical propositions, deduced one from the other so that two neighbouring propositions are immediately contiguous without any interval between them, it will be observed that they are all only the first proposition which is successively and gradually reshaped, so to speak, as it passes from one consequence to the next, but which, nevertheless, has not really been multiplied by this chain of connections; it has merely received different forms. It is almost as if one were trying to express this proposition by means of a language whose nature was being imperceptibly altered, so that the proposition was successively expressed in different ways representing the different states through which the language had passed. Each of these states would be recognized in the one immediately neighbouring it; but in a more remote state we would no longer make it out, although it would still be dependent upon those states which preceded it and designed to transmit the same ideas. Thus, the chain of connection of several geometrical truths can be regarded as more or less different and more or less complicated translations of the same proposition and often of the same hypothesis [...]. It is the same with the physical truths and with the properties of bodies whose connection we perceive. All of these properties gathered together offer us, properly speaking, only a simple and unique piece of knowledge. If others in larger quantity seem detached to us and form different truths, we owe this sorry advantage to the feebleness of our intelligence, and we may say that our abundance in that regard is the effect of our very poverty. Electrical bodies, in which so many curious but seemingly unrelated properties have been discovered, are perhaps in a sense the least known bodies, because they appear to be more known. That power of attracting small particles which they acquire when they are rubbed, and that of producing a violent commotion in animals, are two things for us. They would be a single one if we could reach the primary cause. The universe, if we may be permitted to say so, would only be one fact and one great truth for whoever knew how to embrace it from a single point of view.’ (D’Alembert, ‘Preliminary Discourse’). ↵