"Mechanical picture of the chemical affinity of monovalent similar atoms.

I have once previously treated the problem of the dissociation of gases, on the basis of the most general possible assumptions, which of course I had to specialize at the end. Since here I prefer perspicuousness to generality, I shall make special assumptions that are as simple as possible. The reader must not misunderstand the following and perhaps believe that I hold the opinion that chemical attraction acts precisely according to the laws of the force assumed here. These laws are rather to be considered the simplest, most perspicuous possible picture of forces which have a certain similarity to chemical forces, and hence in the present case can be substituted for them with a certain degree of approximation." - Boltzmann (1895), Lectures on Gas Theory, trans. S. Brush, University of California Press, 1964, p. 376.

"There is no question that for the calculation of natural processes the mere equations, without their foundation, are sufficient; likewise, empirically confirmed equations have a higher degree of certainty than the hypotheses used in deriving them. But on the other hand, it appears to me that the mechanical basis is necessary to illustrate the abstract equations, in the same way that geometrical constructions illuminate algebraic relations. Just as the latter are not made superfluous by mere algebra, so I believe that one cannot completely dispense with the intuitive representation of the laws valid for the action of macroscopic masses provided by molecular dynamics, even if he doubts the possibility of knowledge of the latter, or indeed the existence of the molecules. A clear understanding is just as important for knowledge as the establishment of results by laws and formulas." - Boltzmann (1895), Lectures on Gas Theory, trans. S. Brush, University of California Press, 1964, p. 407.

"Die Differentialgleichungen der mathematisch-physikalischen Phänomenologie sind offenbar nichts als Regeln fur die Bildung und Verbindung von Zahlen und geometrischen Begriffen, diese aber sind wieder nichts anderes als Gedankenbilder, aus denen die Erscheinungen vorhergesagt werden können. Genau dasselbe gilt auch von den Vorstellungen der Atomistik, sodass ich in dieser Beziehung nicht den mindesten Unterschied zu erkennen vermag. Ueberhaupt scheint mir von einem umfassenden Thatsachengebiete nieinals eine directe Beschreibung, stets nur ein Gedankenbild möglich. Man dad daher nicht mit Ostwald sagen, du sollst dir kein Bild machen, sondern nur, du sollst in dasselbe moglichst wenig willkürliches aufnehmen." - Boltzmann (1897), Ueber die Unentbehrlichkeit der Atomistik in der Naturwissenschaft. *Annalen der Physik und Chemie* 296 (2), 231-247.

"The qualitative laws of natural phenomena and their quantitative relations under very simple circumstances, for example the conditions of equilibrium of a heavy parallelipiped of edges in the ratio 1:2:3, can of course be pictured in the mind without starting from a very large finite number of elements. However, as soon as one wants to specify the quantitative laws for complicated conditions one always must start from differential equations, that is first imagine a large finite number of points in the manifold, in short one must think atomistically, and this is not altered by the fact that afterwards we can increase the number of imagined points and so come arbitrarily close to the continuum without ever reaching it.

However this may be, there is a special attraction today in treating mechanics, the most perspicuous scientific discipline, by means of a method that is the very opposite of the modern one and in laying down very special mental pictures from the outset. To begin with the reader may be unable to overcome the feeling that we are merely playing with mental pictures and losing sight of reality. Unperturbed by this we shall first of all try to build up the edifice of ideas as clearly and consistently as possible. If it then agrees with reality, the arbitrary features in the fundamental ideas will thereby have been excused. Indeed we wanted only a picture of atoms and by being clearly aware of this, we do not run the danger of trusting the picture more than reality and becoming blind to the latter." - Boltzmann, Lectures on the Principles of Mechanics, as translated in *Theoretical Physics and Philosophical Problems*, D. Reidel Publishing Company, Dordrecht, Holland (1974), p. 228.

"Advisable as it is to separate the factual from the hypothetical and never to multiply the latter beyond need, I believe that without any hypothetical features one could never go beyond an unsimplified memory mark for each separate phenomenon. All simplifications of memory pictures, all laying hold of law-like features, all rules for summarizing complicated phenomena and predetermining them according to simple prescriptions, rest on the use of pictures drawn from other kinds of simple phenomena and acts of the will.

People have put forward as ideal the mere setting up of partial differential equations and prediction of phenomena from them. However, they too are nothing more than rules for constructing alien mental pictures, namely of series of numbers. Partial differential equations require the construction of collections of numbers representing a manifold of several dimensions. If we remember the meaning of their symbolism they are nothing more than the demand to imagine very many points of such manifolds… and, using certain rules, constantly to derive from them new points of the manifold, to imagine, as it were, a progressive movement of the points in the manifold.

Thus if we go to the bottom of it, Maxwell's electromagnetic equations in their Hertzian form likewise contain hypothetical features added to experience, which are fashioned, as always, by transferring the laws we have observed in finite bodies to fictitious elements of our own making. These equations, like all partial differential equations of mathematical physics, …are likewise only inexact schematic pictures for definite areas of fact, even though the pictures are pieced together from elements that are somewhat different from the atoms to which we are accustomed. The justification of these equations Hertz seeks only after the event in agreement with experience, just as we should with atomist pictures." - Boltzmann, Lectures on the Principles of Mechanics, as translated in *Theoretical Physics and Philosophical Problems*, pp. 225-6, D. Reidel Publishing Company, Dordrecht, Holland (1974).

"The cause why the above pictures are clear is obvious: they are prescriptions for thinking spatial circumstances that everybody can easily and palpably represent for himself in approximation, by means of ruler and pencil or wooden sticks and knitting needles, and which are so well known that their mere idea usually is sufficiently clear even without drawing. A minimum of ideas is employed. The transition from a few individual imaginable points to very many is achieved through general rules. The more we can represent by means of these simple pictures, which we can in any case at present not dispense with in the representation of certain phenomena, the more comprehensible nature must seem to us." - Boltzmann, Lectures on the Principles of Mechanics, as translated in *Theoretical Physics and Philosophical Problems*, D. Reidel Publishing Company, Dordrecht, Holland (1974), p. 230.

"Let us now further develop our picture by assuming certain fictitious laws for the way these material points change place with time." - Boltzmann, Lectures on the Principles of Mechanics, as translated in *Theoretical Physics and Philosophical Problems*, D. Reidel Publishing Company, Dordrecht, Holland (1974), p. 230.

"For my feeling there is still a certain lack of clarity in the differential coefficients with respect to time. Except for the few cases where one can find an analytic function that has exactly the prescribed differential coefficients with respect to time, then in order to set up a numerical picture one will always have to imagine time as divided into a finite number of parts before one proceeds to the limit. Perhaps our formulae are only very closely approximate expressions for average values that can be constructed from much finer elements and are not strictly speaking differentiable." - Boltzmann, Lectures on the Principles of Mechanics, as translated in *Theoretical Physics and Philosophical Problems*, D. Reidel Publishing Company, Dordrecht, Holland (1974), p. 243-4.

"We have deliberately gone rather far away from reality, in order to obtain as precise and clear a picture as possible, that is, one free from vague concepts but offering the most definite indications for the purpose of calculations, so that in every definite case the result to be expected can be unambiguously and securely predetermined to any degree of approximation. THe requirement that the picture should be thus unambiguous seems to me to be what Hertz understands by the requirement that the picture should coincide with the laws of thought; for I cannot really imagine any other law of thought than that our pictures should be clearly and unambiguously imaginable and that from them results always agreeing with experience can be derived as readily as possible. Nor am I in the least of the opinion that anything, say geometrical images, can be derived from the laws of thought alone." - Boltzmann, Lectures on the Principles of Mechanics, as translated in *Theoretical Physics and Philosophical Problems*, D. Reidel Publishing Company, Dordrecht, Holland (1974), p. 251.

"instead of asking how things are really constituted I should like to ask more modestly by means of what pictures our experience is currently most simply and unambiguously represented." - Boltzmann, Lectures on the Principles of Mechanics, as translated in *Theoretical Physics and Philosophical Problems*, D. Reidel Publishing Company, Dordrecht, Holland (1974), p. 254.

"When I say that mechanical pictures might be able to illuminate such obscurities, I do not mean by this that the position and motion of material points in space is something whose simplest elements are completely explicable. On the contrary, to explain the ultimate elements of our cognition is altogether impossible" - Boltzmann, Lectures on the Principles of Mechanics, as translated in *Theoretical Physics and Philosophical Problems*, D. Reidel Publishing Company, Dordrecht, Holland (1974), p. 257.

"we have evaded this difficulty [of formulating the law of inertia without introducing an absolute space] by never speaking of anything real or existing, but replacing matter by mere mental pictures, namely material points, without worrying whether this might not also be done successfully in other ways…

No one can stop us forming mental pictures as we wish, nor therefore including in them a co-ordinate system… over and above the material points. After the event we call these mental pictures true only because they are useful in predicting future phenomena" - Boltzmann, Lectures on the Principles of Mechanics, as translated in *Theoretical Physics and Philosophical Problems*, D. Reidel Publishing Company, Dordrecht, Holland (1974), p. 261.

"To form a picture of what in a given case we should expect, further complicated activities of the will (constructions, calculations) may be required. The picture can be so comprehensive that we may use it under the most varied conditions to construct a successful solution. If we experiment with the pictures themselves, calling to mind by volitions their common features and their differences, while seeking to construct a successful solution in cases that differ from observed ones, then we may be said to speculate. The result will have to be tested by experience, as with simple conjectures." - Boltzmann, On the Question of the Objective Existence of Processes in Inanimate Nature, as translated in *Theoretical Physics and Philosophical Problems*, p. 59, D. Reidel Publishing Company, Dordrecht, Holland (1974).