Hereditary Mechanics and Boltzmann

8/12/2019 Hereditary Mechanics and Boltzmann

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BOLTZMANN'S NACHWIRKUNG ANDOLTZMANN S NACHWIRKUNG ANDHEREDITARY MECHANICSEREDITARY MECHANICS

Maria Grazia IANNIELLOUniversit di Salerno

Giorgio ISRAELUniversit di Roma "La Sapienza"

The origins of modern viscoelasticity theories (and, more generally, of

hereditary systems theories) are usually traced back to the work of

Ludwig Boltzmann and Vito Volterra. This indication poses a first task to

the historian: going beyond a generic reference in order to analyze

Boltzmann's and Volterra's contributions and to place them within the

scientific context of their times. However, as one faces this study manyproblems arise, so forcing to extend the analysis to a much more complex

framework than a mere description.

Historiography on hereditary theories is extremely poor, being

limited to some papers written by Volterra. In his first writings on these

topics,1Volterra appropriately mentioned other contributions preceding his

own work which was considered by him, at least in part, as a developmentof Wiechert's, Meyer's and especially Boltzmann's work.2However, in his

following papers (especially in his historical writings3) these references

1 Volterra V. 1909 c,d.2 Boltzmann L. 1874, Meyer O.E.1874, Wiechert E. 1893.3 Volterra V. 1912 b,c.


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M.G. Ianniello, G. Israel Boltzmann's Nachwirkung and hereditary mechanics

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disappeared: to those readers who are not involved in a careful search for its

origins, the mathematical theory of hereditary phenomena appears mainly

as an achievement of Volterra.

However, a superficial analysis of Boltzmann's and Volterra's works

show that this omission is rather unfair: the core of the theory is already

present in Boltzmann's contribution. Therefore, after completing the job of

reconstructing the historical genesis of the theory, the whole story could be

set aside as one of the many cases in which the priority in a discovery has

not been fully acknowledged.

However, a more careful reading of the two scientists' contribution

shows that there are differences in their approaches suggesting an

explanation of the progressive disappearance of the reference to Boltzmann

in Volterra's last works.

First of all, we can point out a difference in the methodological

approach of Boltzmann and Volterra. Following Boltzmann's views, themathematical analysis of a specific problem is based on and justified by the

experimental analysis; calculations must be developed up to the point of

making possible an immediate comparison with the experiment.4 This

experimental concern is much less intense in Volterra's work. For instance,

the specific problem of elastische Nachwirkung which is the starting point

of Boltzmann's contribution, is seen by Volterra more as an empiricalreference than as an experimental context, in order to lay the foundations

of a general theory of hereditary phenomena which Boltzmann's

elastische Nachwirkung theory should be part of.

4 Boltzmann L. 1874, p.619.


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The difference between an empirical and an experimental approach is

crucial in order to fully understand the main trends of classical mathematical

physics. Following the first approach, in order to build a physico-

mathematical theory, we need only, as a starting point, a set of clearly stated

empirical observations; being then completely free of any care of

experimental verification of the mathematical results. This is the point of

view of the abstract empiricism of the French physico-mathematical

tradition deriving from D'Alembert's and Lagrange's work, which in turn is

rooted in the way how Newtonian mechanics was transplanted to the

continent and in particular to France. Here the empirical approach was

conjugated with the cartesian abstract geometric approach and transformed

in a philosophical reference to Condillac's sensism. One should not

underestimate the influence of this approach on the history of mathematical

physics of countries like France, Italy and England and on the foundation of

scientific branches like analytical mechanics. On the other side, we cannotspeak of a clearly stated theory of an experimental approach before the

work of Joseph Fourier: here we can find the first time a clearly stated

conception of the relationships between mathematical laws and experimental

validation.

So, when speaking of Boltzmann and Volterra, we are facing two

different scientific conceptions springing from two different traditions ofclassical mathematical physics. We observed that this difference is quite

evident from the physical side of the question, which was considered by

Boltzmann from an experimental point of view, while Volterra conceived it

as an empirical support to the mathematical structure of the theory.

Apparently less evident is the difference on the mathematical side, since the


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equations obtained by Boltzmann in 1874 are nothing but the integro-

differential equations which Volterra considered later as the methodological

core of his hereditary theories. However, a more accurate analysis allows to

detect a significant difference, perhaps the most significant one. In order to

explain this aspect we shall give a very short account of the main aspects of

the results of Boltzmann and Volterra.5

Boltzmann introduced the first time the concept of elastische

Nachwirkung in a paper of 1874.6In fact this concept was well known and

studied in the context of an experimental scientific literature and in particular

in some papers of Weber, Kolrausch and, later, of Meyer.7 These papers

deal with the problems of the torsion of a wire and of the strain of an elastic

horizontal bar: the central hypothesis is that the deformation of the

mechanical system is a function not only of the stress (i.e. of the forces)

acting in the moment of the experiment but also of all the stresses acting on

the system in the all of his past. Boltzmann takes this hypothesis (which isthe core of the concept of elastische Nachwirkung) as a starting point but

blames the papers of Weber, Kolrausch and Meyer for their lack of rigor.

Boltzmann considers an elastic parallelepiped whose edges are parallel

to the coordinate axes and are subjected to a uniform deformation in the

direction of the axes (Fig.1). Following the classical treatment of Clebsch

and Lam8

it is possible to calculate the forces acting on the unitaryboundary surface and hence the equations of motion and equilibrium. In

fact, if !, ", #are the deformations of the unitary lenghts on the axes x, y,

5 One could find a more detailed account in a forthcoming extended version of this paper.6 Boltzmann L. 1874.7 Weber W. 1835, Kolrausch F. 1864, 1866, Meyer O.E. 1874.8 See Lam G. 1852.


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z, it easily seen that the force acting on the unitary boundary surface

Fig.1

orthogonal to the x-axis is

[1] N1 = $( !+"+ #) + 2 !

where $and are the constants of Lam. The expression of the forces

acting on the other faces is quite similar.

Then by assuming the classical approach as his starting point

Boltzmann shows how he intends to modify it:

The forces acting on the boundary surfaces of the parallelepiped at a given instantdo not depend only on the deformations of the body at that time but also on the previous

deformations; nevertheless under the hypothesis that, for a given deformation, the fartherthe instant when it took place the smaller the effect it produces; that is the force requiredto produce any deformation is smaller if a deformation previously happened in the samedirection. I want to define the circumstance when a deformation which took placeprevioulsy reduces the force required to produce a deformation in the same direction as


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the decreasing in force caused by that previous deformation.9

In this way, Boltzmann introduces the concept of heredity in a form

which is essentially the same that will be adopted later by Volterra;

furthermore he states what Volterra will consider the fundamental postulate

of the elastic hereditary action, namely that the elastische Nachwirkung

tends to zero when time tends to infinity.

Boltzmann then introduces the hypothesis that if, starting at any instant,

the elastic body undergoes a deformation !(%) in a time interval d%, the

decreasing in force provoked by this deformation on the force acting at the

time t, is proportional to d%, !(%)and to a function of the time t -%= &

before which the deformation happened. It then follows, at least for not too

large deformations, what Boltzmann calls the superposition principle ,

that is the assumption that the decreasing in force produced by a given

deformation, which took place at a past instant of time, doesn't depend onthe states through which the body went in the meantime. Volterra will

make this hypothesis again, in essentially the same form, and call it

principle of invariability of the heredity .

Now Boltzmann is in position to deduce the equations of motion for an

isotropic body undergoing elastische Nachwirkung. First he writes again

the expression of the forces Ni. In order to do that it is enough to modifyequation [1] assuming that the deformations are no longer constant (and

therefore given by the numbers!, ", #) but that they are functions of time

!(t), "(t), #(t). He keeps the assumption that, at any instant the forces act in

9 Boltzmann L. 1874, p. 620-1.


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a spatially uniform way in the three directions of the axes. The first and

second term should be modified by two terms representing the decreasing in

force caused by the elastische Nachwirkung as well as all previous forces.

In these two terms the time dependence of the decreasing of force will be

represented by two functions '(t),((t). One can then write for N1:

[2] N1=$[!(t)+"(t)+#(t)] + 2!(t)

)*

0

"

d&'(&)[!(t-&)+"(t-&)+#(t-&)]) 2)*

0

"

d&((&)!(t-&) .

Likewise for N2 and N3. Now, generalizing Clebsch's treatment and

using notations similar to Lam, Boltzmann writes the equations of motion

for the case of the component parallel to the x-axis (where T1 is the

tangential component):

[3]

+,-,.

N1 = $(t)+2du(t)dx )*0

"

d&[((&)/(t-&)+2((&)du(t-&)dx ]

T1 = dv(t)dz +

dw(t)dy

)*

0

"

d&((&)[dv(t-&)dz +dw(t-&)

dy ]

where u(t), v(t), w(t) are the displacements parallel to the three axes.

This is the general mathematical approach to the problem,which allows,

for any given case, to write the corresponding equation of motion, whichwill be given by an integro-differential equation. The functions '(t),((t) will

be determined, for every particular case, by the experimental analysis. For

the sake of brevity, we point out only the main lines of Boltzmann's

treatment of the problem.

First he considers the case of a cilindric wire with length and radius R,


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hanging vertically from a fixed end (Fig.2). The axis is supposed to be

oriented like the x-axis and its origin to be coinciding with the upper end of

the wire; the lower end is free and loaded with a weight having a very great

moment of inertia K, so that we can assume that on the wire acts at any

time a uniform torsion.

Fig.2

We pose u = 0 , v = xz/(t) , w =

xy/(t)

, where /(t) isthe angle by wich the lower section of the wire is rotated at the instant t

because of the torsion. Then we calculate the moment of all elastic forces

acting on the lower end to obtain the equation of motion (where D is the

moment of the torsion to which the wire is submitted at time t):


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[4] D Kd2/(t)

dt2 =#R4

2 [/(t) )*

0

"

d&'(&)/(t-&) ]

The determination of the form of the function ((t) is experimentally

possible for any specific case. Boltzmann then considers some special

circumstances, which lead him to obtain a particular form for the function

((t). In some cases that allows to obtain particular integrals of equation [4],

or at least some conclusions about the asimptotic behaviour of the solutions.

We can't linger on these developments: we limit ourselves to recall that theydeal essentially with five cases: 1) the wire at the initial time never was

submitted to torsion; 2) the wire undergoes a constant torsion by an angle #

only in the time interval (- %/2,%/2); 3) the wire is not submitted to any

torsion before time t=0, then undergoes a constant moment of torsion; 4)

the wire was rotated by an angle c" in the time interval (-", 0) after

which it is not submitted to any force, and therefore rotates at time t by anangle /' because of the elastische Nachwirkung; 5) the wire, not submitted

to torsion in the time interval (-", -%/2) experienced a torsion between (-%/2,

%/2), after which no force acts on its lower end.

This first part of the article ends with the comparison of the results

obtained in these cases with the experimental results given in several papers

by Kohlrausch, Neesen and Streintz.

In the second part of the article Boltzmann obtains the general form of

the elastic forces in the case when the functions '(t),((t) have the same

form. We have in this case:


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[5]

+,,-,

,. N1 = $/(t)+)*0

"

[/(t)-/(t-&)]F(&)d&& +

+2)*

0

"

[du(t)dx du(t-&)

dx ]f(&)d&&

T1=)*

0

"

[dv(t)dz +dw(t)

dy dv(t-&)

dz dw(t-&)

dy ]F(&)d&&

where u(t), v(t), w(t) are the displacements parallel to the axes.

Concerning the two functions F(&) and f(&) it is enough to know that

for an appropriate &they have a constant value, while when &is very great

they tend to 0 so that the following integrals converge:

)0*

1

"

f(&)d&& e )

0*

1

"

F(&)d&& .

We have therefore that /(t) =du(t)

dx +

dv(t)

dy +

dw(t)

dz . On a

surface element df normal to the x-axis act the forces N1df, T3df, T2df

parallel to the three axes x, y, z. Likewise for the other axes. Using Lam's

equations of motion we have:

[6]

+,-,.1d

2u(t)dt2

=dN1dx +

dT3dy +

dT2dz + X

1d2v(t)

dt2

=

dT3

dx

+

dN2

dy

+

dT1

dz

+

Y

1d2w(t)

dt2 =

dT2dx +

dT1dy +

dN3dz + Z

where is the density and X, Y, Z the accelerating forces acting on the

body from the inside.

These equations are again applied to the case of the wire of Fig.2.


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Apart from the deformation along the x-axis experienced by the wire

because of the load who only superposes to the oscillatory motion, the

displacements u,v,w take the values:

u = 0 ; v = zx/(t)

; v = yx/(t)

We have finally the equation of motion:

[7] D K

d2/(t)

dt2 =#R4

2 )0*

0

"

[/(t)

-

/(t-&)]

f(&)d&&

Boltzmann considers this equation in some special cases. His analysis

mostly employs approximations or Fourier series developments. The article

ends with an experimental appendix in which Boltzmann reports on some

torsion experiments and makes a first comparison between measured values

and theoretical values deduced from his relations.

We have seen therefore that Boltzmann analysis is based on the

classical mathematical elasticity theory and his approach consists in

modifying its structure in order to include the new experimental results in

the framework of a good agreement between the mathematical solution of

the problem and these results.

We already observed that such a link between mathematical analysis of

the problem and experimental verification is characteristic of theexperimental tradition la Fourier. It should however be pointed out

another feature of Boltzmann's approach, i.e. the absence of any interest for

the specific mathematical side. Are there general methods in order to

solve an integro-differential equation?, Could we determine the class of

admissible solutions or demonstrate a theorem of existence and


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uniqueness? Boltzmann is not interested by questions of this kind, which

have however their root also in the above mentioned tradition la Fourier,

as Poincar pointed out. Even if Boltzmann is to be considered the first

who introduced the use of integro-differential equations we must not

forget that he formulated another famous integro-differential equation, the

transport equation, arising in the determination of the distribution of

particles of an ideal gas in an enclosure of which there is an external force F

he never tackled the above mentioned problems, limiting himself to the

use of techniques like the Fourier series developments.

We shall now tell something about Volterra's contribution.

It would not be correct to present Volterra's interest for the

problematics of systems with memory as a simple prosecution of

Boltzmann's researches on the elastische Nachwirkung. Many others

hints concurred to awake the interest of the italian scientist for that

argument. The connection between the concept of function of lines(introduced by Volterra in 188710) and that one of an infinite value-

dependent process is quite clear, as in the case of an elastic body submitted,

in its past history, to an infinite number of stresses. It would be reasonable

at least for chronological reasons to suppose that Volterra had

previously know Boltzmann's work on the elastische Nachwirkung; he

did not cite however that class of phenomena, as an empirical justificationfor his theory of functions of lines. This is true until 1909. Thus it seems

well-founded to say that Volterra was aware of the existence of many classes

of phenomena whose behaviour was definitely influenced by its past and

10 Volterra V. 1887.


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that his conceptions of the theory of functions of lines had been widely

influenced by these phenomena; but the theory of the elastische

Nachwirkung did not pay any relevant role.

On the other hand Volterra gave fundamental contributions to the

theory of elasticity, one of his preferred fields of research. In fact in 1907 he

wrote one of his most important memoirs on the theory of elasticity where

he connected the concept of functions of lines with that of elastic

hysteresis.

In 1907 E. Picard had already written about phenomena with

memory in an article which appeared in the first volume of the journal

Scientia ,11 thus well known in the italian scientific milieu. In this paper

Picard attempted to reconstruct, in front of a general crisis of the foundation

of classical mathematical physics, a coherent frame for mechanics. Here

Picard developed that program of elastic defense of classical which a few

years later Poincare' effectively defined with his metaphore on the necessityof deforming the frames of classical science in order to preserve its

essential core. Among these frame deformations, according to Picard it

was necessary to abandon the strictly non-hereditary determinism la

Laplace. In the hereditary cases observed Picard it will be perhaps

necessary to give up differential equations in favor of functional equations.

Clearly in his reference to the functional equations he explicitely alluded toVolterra's mathematical theories. The deep link, both scientific and human

between Volterra and Picard is well known. The correspondence between

the two scientists shows how in 1909 they had reached the idea of giving

11 Picard E. 1907.


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an explicite name to the functional equations describing those mechanical

phenomena previously christened as hereditary by Picard himself. The

new definition of integro-differential equations was introduced by Volterra

for the first time in the scientific literature in a paper of 1909.12

In another work that followed soon afterwards,13Volterra applied the

integro-differential equations technique to the study of magnetic hysteresis

correcting Hertz's equations by integral terms. In this work appeared the

condition called by Volterra the condition of closed loop (later known as

closed cycle) that is equivalent in the special case of periodic hereditary

coefficients to the superposition principle of Boltzmann.

The way how hereditary theories and integro-differential equations are

introduced by Volterra thus appears largely influenced by a general scientific

program, although inspired by specific applicative themes. The general and

philosophical nature of this approach is clearly seen even in the effort to

construct a mathematical coherent theory as general as possible.14Here weclearly see the difference with Boltzmann's approach.

Reference to Boltzmann's work appeared for the first time in a work

of that same year. The elastische Nachwirkung phenomena are quoted in

the wider class of hereditary phenomena in the sense of Picard.15Volterra's

starting point was a general philosophical program quite similar to Picard's

views, i.e. the need to weaken the principle of determinism in view ofstudying the new class of phenomena.

12 Volterra V. 1909a (in Opere Matematiche, p.269)..13 Volterra V. 1909b.14 E' bene [] osservare [] che il problema della risoluzione delle equazioni integro-differenziali

costituisce in generale un problema essenzialmente distinto dai problemi delle equazioni differenziali ada quelli ordinarii delle equazioni integrali. (Ibidem, in Opere Matematiche, p. 275).

15 Volterra V. 1909c (in Opere Matematiche, p.288).


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Mr. Painlev, in the interesting chapter on mechanics of his book:De la mthode

dans les Sciences,16states that, in a certain sense, problems on heredity are but apparent

and that a more perfect knowledge of the constitution of bodies could dispose of them,through their reduction to non-hereditary form; but whatever opinion one might have on

the subject, as a matter of fact at the present moment it is necessary to take them into

account.

The equations governing some of these problems have been known for a long time.

Thus I will mention those regarding the subsequent elasticity17 which Boltzmann18

established in the isotropic case starting from empirical concepts and that Wiechert later

rediscovered under a new perspective.19

Up to these very last times, however, an analysis was lacking for the general study of

these equations, allowing to treat them in a complete way. I will shortly point out why it is

so. The problems of non-hereditary mechanics and mathematical physics, because of their

very nature, depend on ordinary or partial differential equations; as it is well-known, the

initial data are givenby the arbitrary constants or the arbitrary functions arising from the

integration of these equations. On the contrary, in order to deal with the problems of

hereditary mathematical physics, the analysis of differential equations is no longer

sufficient. Since the present state of the system depends on its previous history, and this is

detailed by the values taken by certains parameters during a given time interval, it is clearly

necessary to take into account quantities depending on all the values of these parameters

regarded as functions of time.20

It is not possible to describe here even roughly (and it would be

somewhat off our point) Volterra's contribution on the general theory of

integro-differential equations and hereditary processes. We shall only recall

that Volterra specific starting point is, like Boltzmann, the problem of the

torsion of a wire. If &is the angle of torsion and M the torsion moment,

16 Painlev P. 1909.17 This is Volterra's translation of elastische Nachwirkung. In subsequent papers he shall make use of

the term lasticit residuelle.18 The reference is to Boltzmann L. 1874.19 The reference is to Wiechert E. 1893.20 Volterra V. 1909c (in Opere Matematiche, p.288-9).


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according to Hooke's law, we have:

[8] & = KM (where K is a constant).

The hereditary approach implies the following modification of Hooke's

law:

[9] & = KM + '

where ' is depending from all the values taken by M in the time

interval (-", 0). Then Volterra develops 'in a series analogous toTaylor

series:

[10] ' = )*

t0

tM(%)'(t;%) d% +

12)*

t0

tM(%1)M(%2)'(t;%1;%2) + ...

so obtaining, with a first order approximation:

[11] &(t) = KM(t) + )*

t0

t M(%)'(t;%) d%

The dynamical equation is obtained in a way different from the one

followed by Boltzmann. Making use of D'Alembert's principle, Volterra

replaces the torsion M with the difference M $

2&

$t2 (where is a constant) . The equation is similar to Boltzmann's

equation:

[12] &(t) = K [M(t) $2&(t)

$t2] + )0

*

t0

t

[M(%)$2&(%)

$%2]'(t;%)d%

Let us now discuss another aspect of the differences between the


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approaches of the two scientists. While Boltzmann only emphasizes the

modification of the mathematical form of the forces acting on a body when

not only the deformations of the body in the given moment but also the

effect of the deformations which previously affected it are taken into

account, Volterra emphasizes a different implication of the consideration of

the past stresses. In Volterra's views the consideration of the past stresses

implies that the mechanical evolution of the system is no longer determined

by its initial state but by all its past as well. It is quite obvious that there is

no formal difference between the two points of view. But the different

interpretation of the passage from the ordinary differential equation of

motion to an integro-differential equation which according to the first

approach is seen as a modification of the components of force and according

to the second as the substitution of the vector of numbers giving the initial

conditions with a vector of integrals (i.e. with a functional) has an

important conceptual consequence. In the latter case attention is focused onthe abandonment of the deterministic principle of mechanical phenomena as

a consequence of the fact that we are taking account of the past stresses: the

evolution of the mechanical system is no longer and solely determined by its

initial state (a vector of numbers) but by the whole past of the system

(described as a set of integrals). Volterra emphasizes this aspect more and

more markedly; and, in connection with the growht of this emphasis, thereference to the elastische Nachwirkung disappears from his writings.

This theme, on the other hand, is totally absent in Boltzmann's work.

The deep implications of hereditary theories on the deterministic

view of mechanical phenomena is confirmed by a dispute which took place

around 1910 on the mechanics of heredity, opposing the orthodox


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determinist Painlev to Volterra and Picard, who were upholding a less strict

version of determinism.21Painlev had thrown his dards against hereditary

theories (clearly having as goals, Picard and Volterra). The argument

developed by Painlev against hereditary theories was inspired by an

orthodox laplacian conception: the evolution of a mechanical system is

strictly determined by its initial state, and appeal to the knowledge of its past

history can only be useful in those cases in which technics is not powerful

enough to determine the initial state of the system with sufficient

approximation. There are, however, no essentially hereditary processes,

since the principle of causality in the form expressed by laplacian

determinism has a universal validity. And Painlev concluded in

extremely drastic and polemical way, observing that the conception

following which, in order to predict the future of a system it is necessary to

know all his past, is the very negation of science.

Volterra's answer was not as sharp, and it shows many similarities tothe soft attitude exhibited by Poincar on this sort of questions. While in the

brief mention given in his 1908 paper, Volterra gave an instrumental

justification of the use of the hereditary point of view, in his answer to

Painlev, written in 1912, he goes a step forward, declaring himself ready, if

necessary, to modify the frames of classical reductionism so as to make it

capable of explaining new classes of phenomena. Therefore, even if notwilling to destroy the principle of determinism, he cannot at the same time

accept the escape of considering hereditary theories as a mere technical

device to be dispended as soon as possible. On the contrary, he stresses

21 For a more detailed account, seeIsrael G. 1984.


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again their descriptive and explanatory value, by making a comparison with

the role given by Newton to the concept of action at distance.

This polemics gives us hints for reflection. The introduction of integro-

differential equations and of hereditary theories posed a serious problem to

the supporters of orthodox causality. There comes a spontaneous question:

Why did Boltzmann who was the first to introduce the study of

hereditary phenomena and of integro-differential equations, even in other

contexts not catch this critical element?

The implications of the hereditary theories on the principle of

determinism appear clearly if we observe that its mathematical translation is

the theorem of existence and uniqueness of the solutions of a system of

ordinary differential equations. We should therefore devote our attention to

another chapter of history of science, i.e. the history of the above mentioned

theorem a chapter studied until now in a totally unsatisfactory way.

However unsatisfactory are these histories,22it is quite clear that the abovementioned connection could not be clear before the turn of the century. It is

true that the first versions of this theorem go back to 1820 and in particular

to Cauchy's work, but the point of view was always local. We can find a

more precise definition of the conditions for the existence and uniqueness of

the solutions only in Lipschitz' work in 1868. For a local existence theorem

based only on the continuity hypothesis it is necessary to wait for Picard'sand Peano's contributions at the turn of the century, while results on

uniqueness will be the last ones. Finally it should be observed that the first

steps towards the global analysis of the solutions on which rests the

22 The best reference is Dieudonn J. 1978 (Vol. II, Cap.VIII).


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possibility of connecting this theorem with the principle of determinism

through the concept of dynamical system starts only at the end of last

century with the work of Poincar on celestial mechanics. We dispose

therefore of a simple answer to the previously posed question concerning

Boltzmann's attitude. The natural answer is that he could not know in

1874 the more general and global version of the existence and uniqueness

theorem, the only one whereby the theorem could be seen as the

mathematical translation of the principle of determinism.

However, we have here a good example of how the more natural

explanations, arising from the chronological development of mathematical

techniques, can generate misleading conclusions.

In fact the connection between the existence and uniqueness theorem

and the principle of determinism has become commonplace, especially in

contemporary mathematical physics, but it does not have any intrinsic basis.

As Mario Bunge observed, this union only makes sense if one admits thatcausali ty were exhausted by uniform, unique, and continuous

succession23. Only if the empiricist reduction of becoming to uniform,

unique, and continuous succession in time is accepted, then the reduction of

mathematics of "empirical science" to differential equations may follow.24

The above point of view is therefore the reflection of a scientific philosophy

that is founded on a special interpretation of the concept of causality and onthe transformation of mechanical atomism in a sort of geometrical

atomism where the continuous point of view has the upper hand. In fact,

history of science gives a good proof of that. Let us recall the first pages of

23 Bunge M. 1959: 19793, p.74.24Ibidem , p.75.


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theEssai Philosophique des Probabilits by Laplace.25We find here the

most explicit formulation of Newtonian mechanics program even with

regard to the mathematical aspects. Firstly we have an atomistic approach:

every mechanical system is formed by a very high number of particles

whose motion is determined by Newton's law of dynamics. Secondly, the

motion of bodies is subject to a principle of causality, and therefore to a

strict determinism. The program of mechanics consists in determinating the

specific form of the acting forces, in writing the differential equations of

motion for all the elementary components and in integrating them. This can

all be ready in the above mentioned work of Laplace. However the theorem

of existence and uniqueness was not yet known nor was it under the form

that would have permitted this identification before the end of the 19th

century. This strikingly shows the stricly metaphysical nature of that

program: in fact, Laplace stated the above mentioned connection two

centuries before the final formulation of the theorem.26 We shall therefore give two quite different answers to our question

concerning Boltzmann's attitude. First Boltzmann did never give any

substantial importance to the consequences deriving from the

mathematical structures he used. Therefore, the absence of a theorem of

existence and uniqueness in the classical form had no meaning to him as to

the problem, whether the phenomena under investigation were or notessentially ruled by a causal law. Secondly, Boltzmann's mechanism was not

tied to a causal conception based on the deterministic principle typical of the

25 Laplace P.S. de 1825.26 On this topic see G. Israel, "Il determinismo e la teoria delle equazioni differenziali ordinarie.

Un'analisi retrospettiva a partire dalla meccanica ereditaria",preprint.


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laplacian physico-mathematical tradition. Therefore, the problem around

which focused the discussion between Picard, Painlev and Volterra, which

was deeply rooted in the French physico-mathematical tradition, could not

attract his attention.

The above mentioned issue leads us to analyze the core of Boltzmann's

scientific conception on the specific question of the relationships between

physics and mathematics in the analysis of physical phenomena.

Not only Boltzmann believed, as it is well known, in an atomistic

conception of physical phenomena, but this conception was so sharp that it

limited the signification of the use of differential equations in order to avoid

any conclusion about the continuous nature of phenomena. To Boltzmann,

differential equations are nothing but a technical tool that one must use very

cautiously in order to avoid contaminating the analysis of physical

phenomena with a continuous approach. Boltzmann's finitism in

mathematics is far more rooted and definitely different from the pre-intuitionistic finitism of the French functionalist school of which physico-

mathematicians like Volterra, Painlev, Picard e Poincar were followers.

Therefore Boltzmann could not have seen any deterministic implication

deriving from the use of differential equations.

In order to go deeper into the relationships between continuism and

atomism, let us recall something about this connection in the context of theother tradition. Here the resort to a continuous approach was suggested

by the drawbacks of the strict laplacian-newtonian program. The path

followed implied weakening the atomistic assumption and joining it with a

continuist assumption, in a complex but fertile combination: a combination

that it is very well described by the principle called by Volterra the


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principle of the passage from the discontinuous to the continuous.

Let us recall the way how D'Alembert, far before Laplace, obtained the

equation of vibrating strings. He started from Bernouilli representation of a

string like a finite system of masses suspended to a thread without mass,

where to each one he applied Newton's equation. D'Alembert by making

the subdivision (or the mass) tend to zero reintroduced a continuous view of

the string, thus replacing the system of Newton's equations with a partial

differential equation. A similar procedure was followed later by Laplace in

order to obtain the equation of the potential of a sphere on a point outside it.

The body was not more represented by means of a set of real

particles but by an abstract andpurely mathematical atomistic scheme

centered around the notion of geometric point with a mass. This abstract

and purely geometrical atomism lead to a continuous representation of the

phenomena because the elementary unit is not an indivisible element but an

infinitely divisible one. The ambiguous coexistence between the twoapproaches is described very well by Volterra in a paper of 1906.27

In this approach, differential equations have a fundamental role: they

are, as Fourier said, mathematical Analysis and mathematical Analysis is as

extended as Nature.28 The passage from discontinuous to continuous

allows us to develop the Newtonian program even if in a less restricted

sense, by the widening of the range of admissible differential equations. Forinstance, abandoning the strictly mechanical-atomistic interpretation of heat

phenomena seems at this point the only way to save the core of Newton's

program.

27 Volterra V. 1906 (in Opere Matematiche, p.65).28 Fourier J. 1821, p.xxiii.


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Let us now come back to Boltzmann. The fact that Boltzmann wass a

mechanist is almost common knowledge. It is also quite well known that he

was prudent enough not to risk a substantial atomistic interpretation of

phenomena. However, to see Boltzmann's mechanism under the light of the

above mentioned physico-mathematical conception would be completely

misleading. In Boltzmann point of view mechanism is not strictly linked with

determinism. In addition, the continuous representation of phenomena is

only considered nothing more than a mathematical device to be handled

cautiously. The same type of cautiousness if not distrust is shown by him

towards differential equations. This leads to a mechanistic approach that

has little in common with the previous one except for the name. Let's look

at this closer.

Right from the first pages of Boltzmann's Lessons on the Theory of

Gases29, one is impressed by the close link established between the

mechanistic and the atomistic point of view; up to the point of blaming thetrend called by him the continental trend, which considers the

assumption that makes heat a motion of particles [as an assumption that]

will one day be considered false and rejected,30 for having completely

renounced to the mechanistic approach. Boltzmann's approach follows a

different trend opposing the method of mechanical analogy to the one that

he calls the method of purely mathematical formulas and remarking thatexperience teaches us that it is almost exclusively through special

mechanical considerations that it was possible to attain new discoveries.31

29 Boltzmann L. 1895-8.30Ibidem, ed. fr.1987, p.2.31Ibidem, p.3


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Meanwhile, it will be necessary to show prudence concerning the real nature

of phenomena.

We should not be misleaded by this prudence. If Boltzmann accepts

what he calls the more modern standpoint that consists in simply

describing the phenomena32 and admits the well known differential

equations which are related to the internal motions in solid and fluid

bodies,33immediately afterwards refers to the very old idea that bodies

do not fill in a mathematically continuous way the space they occupy but

that they are formed by the lowest particles, molecules, whose smallness

isolatedly makes them imperceptible to our sense organs.34 It should be

observed how by the adverb mathematically, Boltzmann excluded at a

blow any descriptive validity to any kind of schematization based on the so

called passage from discontinuous to continuous. We shall immediately see

how this implies a severe reduction in the descriptive value of differential

equations.To better clarify the restricted boundaries within which Boltzmann

allows the use of differential equations in mathematical physics, we must

refer to his conception of the foundation of mathematics which is radically

finitistic: Boltzmann excludes any consideration of actual infinite and

considers that in Nature each infinite only implies a passage to a limit.35

However this would be an insufficient characterization that would notpermit distinguishing Boltzmann's mathematical thought from the one of

any other intuitionist for instance, of the members of the so-called French

32Ibidem.33Ibidem.34Ibidem.Our italics.35 Boltzmann L. 1877a: in Boltzmann 1909, II, p.167.


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functionalist school, who were yet fully continuist in the field of classical

analysis. The main point is that, for Boltzmann, only mathematical entities

having a physical reality are meaningful. Hence only finite collections of

entities are admissible and the infinite can only be conceived as a limit of

finite collections. However, we should not be misleaded as regards the

meaning of this tendency to a limit: it is not a purely mathematical

procedure. On the contrary, to be fully acceptable it must be justified on the

grounds of procedures which must be concrete, expressed in physical terms

and moreover actually computable. Therefore, the procedures giving rise to

the partial differential equations of classical mathematical physics are only

acceptable to the extent where the procedure of passing from the finite case

to the infinite representation is clearly and concretely defined.

An extremely clear example of this conception of Boltzmann is given

by his justification of Fourier's partial differential equation in heat's theory,

which one can find in his paper Nochmals ber die Atomistik.36

To conclude, for Boltzmann the concepts of differential and integral

calculus freed from any atomistic representation are purely metaphysical, if

by this term we mean (according to a famous definition of Mach) the things

we have forgotten about how we got them.37

In another paper on atomistic,38 Boltzmann contrasts the atomistic

point of view to the approach that he calls phenomenology and that hedefines as the trend to represent limited fields of phenomena by means of

differential equations. He blamed phenomenology for having forgotten

36 Boltzmann L. 1897a: in Boltzmann L. 1905, p.158-9 (fr. transl. in Dugas R. 1959, p.27-8).37Ibidem, p.160.38 Boltzmann L. 1897b.


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the physical roots of the differential equations.

However, perhaps the most radical criticism about the

phenomenology, which shows the strong opposition of Boltzmann to the

mathematical procedure of the passage from discontinuous to continuous,

consisted in the reproach for resorting to every kind of atomistic

representation just simply in order to find a description of phenomena

without any concern about coherence of these different atomistics. Hence

phenomenology leads only to fragmentary and incoherent representations

of phenomena.39

So, if differential equations express a phenomenology but do not

express any essential property of phenomena, there is no sense in looking

at them for the principle of causality. No doubt Boltzmann considered that

mechanical phenomena are subject to what he called the principle of

univocal determination of motions without which these phenomena

would not be a scientific subject but only a curiosity. This principle is part ofa general principle of univocal determination of natural processes.40 But this

principle has little to do with the principle of determinism identified with the

content of the theorem of existence and uniqueness of the solutions of

differential equations. It is enough to say that in hisLessons at the Clark

University of 189941, Boltzmann denied that the initial state of a system

univocally determines its evolution. In fact, this state includes in a close sensethe whole state of the Universe that is never twice the same, and following

Boltzmann, this difficulty could only be overcomed by founding an

39 See once more Boltzmann L. 1897b, p.144-9.40 See Boltzmann L. 1899 (also in Boltzmann L. 1905, pp.253-307).41Ibidem.


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inductive theory based onNahewirkungstheorie (or the theory of contact

actions), according to which motion is determined only by the spacial

elements near the region considered. We could say that Boltzmann

conceives determinism only as a local property.

To conclude, Boltzmann's work on hereditary phenomena is the

starting point of a new branch of mathematical physics, but Boltzmann did

not contributed to the development of the branch in the

phenomenological sense. The reasons for that are perhaps now more

clear.

The case study here examined leads us also to identify the differences

between two different trends in mathematical physics at the end of

Einghteenth century: the mechanistic-atomistic, non deterministic point of

view of Boltzmann and the deterministic-continuist French school to which

Volterra belonged. It also shows how unnecessary and unmotivated are the

relations between determinism and the existence and uniqueness theorem ofthe solutions of ordinary differential equations: two connections which are

again in fashion today in the wake of the study of the expansive dynamics

and the so-called chaotic systems and which mark the recovery (yet

unexpressed) of the deterministic-mechanistic viewpoint coming from

Laplacian tradition. The groundlessness of the above mentioned connection

shows also why some aspects of the present-day debate on the implicationsof the discovery of chaotic systems are meaningless. To those who, like

Prigogine, claim that this discovery leads to the break-down of determinism

from the inside, the laplacian determinist could easily reply that, on the

contrary, this discovery shows the possibility of describing some stochastic

phenomena in the context of the classical deterministic framework. Both are


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wrong: in fact and on this point we should refer to Boltzmann's teaching

mathematical structures cannot have any implication concerning the

substantial nature of phenomena. We see therefore that historical analysis

can also contribute to a less vague use of some conceptual categories which

are of basic importance in the contemporary scientific debate.

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Hereditary Mechanics and Boltzmann

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