Comte’s first volume of his positive philosophy ends with a study of mechanics. Although mechanics is for Comte even more concrete than geometry, it still falls within mathematics. Indeed, Comte is against all readings of mechanics, where some elements of analysis are interpreted as real forces, although they would be nothing but means for making calculations. What forces are real can be decided only by observation and experience. Furthermore, Comte adds, mechanics does not investigate what is the nature of these forces, but merely the movements caused by them.
The abstract nature of mechanics, characteristic to mathematics in general, Comte notes, is seen in the fact that mechanical calculations are simplified by assuming bodies to be passive or inert, although in reality they are in many ways active. This idealisation of bodies as inert in mechanics, Comte explains, is not to be confused with inertia in the sense expressed in one of the basic laws of mechanics - the fact that bodies tend to move in straight lines and retain their state of movement. Together with the other two basic laws of mechanics - one being Newton’s law of action and reaction, the other being Galilei’s discovery that forces are independent of one another and can thus be composed with the parallelogram law - the law of inertia is, according to Comte, based on observation, not on any a priori deduction. Particularly, Comte adds, the law of inertia cannot be deduced from the law of sufficient reason.
Comte divides mechanics, expectedly, into statics dealing with instantaneous forces and uniform movement or equilibrium arising from them and dynamics dealing with continuous forces and varied movement arising from them. Within both statics and dynamics, he then differentiates a part examining solid bodies from a more complex part examining fluids. An important problem in this classification in Comte’s opinion concerns the relative status of statics and dynamics. Statics is clearly the older discipline, studied by ancient mathematicians long before dynamical questions. But as Comte has said earlier, the historical order of disciplines does not necessarily correspond to the order of the disciplines in a completed science. Indeed, when dynamics was finally introduced at the start of the Modern Age, statics was regarded as a mere abstract limit case of dynamics.
Yet, as Comte’s favourite mathematician, Lagrange, had argued, the notion of virtual displacement - an application of his calculus of variations to mechanics - could be used to reduce all of dynamics to statics. In effect, Comte is referring to the central idea of the so-called d’Alembert principle that all apparently dynamical systems can be regarded as being in equilibrium. Comte further links Lagrange’s idea with Poinsot’s notion of a force couple, which Comte thinks is a modification of a notion of force from translation to rotation.
Just like with other parts of mathematics, Comte is especially interested in the applications of mechanics. Thus, he points out that statics is used for finding mass centres of bodies, while in dynamics we are trying to calculate movement of a particle from forces affecting it, or the other way around, to find forces creating a known movement. We need not go in great detail to theorems that Comte lists as consequences of the three basic mechanical principles. I will just point out that Comte speaks against interpreting Maupertuis’ principle of least action in a theological or metaphysical manner suggesting that bodies would somehow choose to move in accordance with the principle.
maanantai 21. helmikuuta 2022
lauantai 22. tammikuuta 2022
Auguste Comte: Course of positive philosophy 1 - Geometry
While many philosophers had considered it an important problem to put geometry on secure foundations, Comte finds such attempts mere unfounded metaphysics. For him, geometry is simply a natural science with an empirical basis. Of course, it is the most abstract natural science, dealing only with static spatial properties of things, in abstraction from any movement. Still, Comte feels no need to prove the basic axioms of geometry, since he can just assume them as bare facts. On the other hand, he also feels no need to consider the possibility of other geometries with other axioms, since experience appears to agree with the ordinary Euclidean geometry.
Comte supposes that this empirical science has a very practical purpose, namely, that of measuring spatial features of things. Of course, he adds, not all measuring is geometry, for instance, if we fill an oddly shaped container with water and then measure the volume of the water, this is still not geometry. Instead, geometry, like all mathematics, is an art of finding out quantities indirectly, through calculations.
When we measure spatial features of things, Comte continues, we can measure all the dimensions of it or then only some of them. When a geometer speaks of planes or lines, he adds, they are considering just such abstractions, that is, they are ignoring some of the dimensions the thing has. Thus, lines in reality always are wide and thick, we just concentrate on their length. We can even ignore all the dimensions of a thing and consider only its position in relation to other things - this is the origin of the notion of point.
While the essence of geometry is indirect measuring of spatial features of things, a geometer must assume some ways to directly measure these spatial features. This implicitly assumed form of measuring, Comte suggests, is the measurement of straight lines or lengths by comparing them with a length of some other thing, like a ruler. Other geometrical figures (curves, areas and volumes) are then to be measured with the help of these straight lines.
Comte admits that geometry is full of other things beyond mere measuring, that is, full of propositions about spatial properties of things. Still, he insists, even these properties are ultimately studied, because they could help with measuring (perhaps in some more concrete science). Because one cannot know beforehand what properties help with measurements, Comte advocates studying as many such properties as possible.
Comte mentions the old distinction between synthetic and analytic methods in geometry, but seems to have no idea what these terms meant originally and what their difference was supposed to be. Instead, Comte suggests that the distinction is just a roundabout way to distinguish ancient from modern geometry. Ancient geometry, Comte suggests, was mainly dealing with concrete and individual figures, taking one type of figure and finding all its characteristics. Because of the uniqueness of the chosen figure, none of these characteristics could be assumed to hold for other entities. Modern geometry, on the other hand, deals with abstract geometrical problems, which can then be applied to many different contexts.
Comte has little patience with ancient geometry, and he especially dislikes the common habit of starting to teach geometry from works like Euclid’s Elements - as he has noticed earlier, history of a discipline is usually not the most convenient way to teach it. Comte is especially critical of the proofs of early propositions in Elements, where Euclid simply places figures on top of each other, to show their similarity - this is just as futile as an attempt to prove the parallel axiom would be.
Although Comte is very critical of ancient geometry, he does admit that Greek mathematicians did make some progress. Particularly, they perfected the study of the simplest kind of figures, namely straight lines and polygons and polyhedras. Furthermore, while Comte ridicules the Greek use of diagrams as proofs of proposition, he does suggest it to be a sort of precursor for modern projective geometry. Another field developed by ancient geometers to perfection, Comte concludes, is trigonometry, where they used certain lines (sines,tangents etc.) to represent angles and their relations and so simplified calculations.
Comte associates the birth of modern geometry with Descartes. While it had been long known that geometric forms could be described in terms of spatial situations of their limiting points etc., it was the invention of Descartes, Comte says, to reduce the talk of situations to talk of lengths and other magnitudes through the notion of a coordinate system. In effect, Comte concludes, Descartes was able to transform at first sight qualitative properties like geometric figures into quantitative properties.
The outcome of the Cartesian transformation of geometry, Comte explains, was that now in two-dimensional geometry lines could be expressed by equations and equations by lines (in three-dimensional case, Comte adds, equations express surfaces, while lines are expressed by pairs of equations). These equations do not just characterise some random properties of the lines, Comte notes, but explicate how they could be generated. He adds that when a geometer is looking for an equation to describe a line, there is no need to choose any specific coordinate system - sometimes the searched for equation might be easier to describe e.g. in terms of polar coordinates, which are especially convenient when describing rotations. Then again, he admits, when finding lines to describe equations, it is best to pick the rectilinear coordinate system, which is the most natural for us to decipher.
Although Comte at first appears to say that all lines correspond to an equation and all equations correspond to a line, he is well aware that analytic geometry of his time has imperfections in this department. Firstly, he notes, discontinuous lines cannot be expressed so well in terms of a single equation. Furthermore, he continues, equations with more than three variables or equations with imaginary solutions have no proper geometric model.
Like in the case of abstract mathematics, Comte is mostly interested in the uses geometry could be put to and the problems that could be solved by its help. Solving some of these problems relies on simple algebraic means, such as when we try to find the number of points that are required for determining the course of a curve. Still, most of these problems, Comte notes, rely on the help of differential and integral calculus. Differentiation is useful not just for finding tangents, but also for describing e.g. the curvature of curves. Then again, Comte concludes, integration is the most useful tool, because it helps us to fulfill the true task of geometry, that of measuring lengths, areas and volumes.
Comte supposes that this empirical science has a very practical purpose, namely, that of measuring spatial features of things. Of course, he adds, not all measuring is geometry, for instance, if we fill an oddly shaped container with water and then measure the volume of the water, this is still not geometry. Instead, geometry, like all mathematics, is an art of finding out quantities indirectly, through calculations.
When we measure spatial features of things, Comte continues, we can measure all the dimensions of it or then only some of them. When a geometer speaks of planes or lines, he adds, they are considering just such abstractions, that is, they are ignoring some of the dimensions the thing has. Thus, lines in reality always are wide and thick, we just concentrate on their length. We can even ignore all the dimensions of a thing and consider only its position in relation to other things - this is the origin of the notion of point.
While the essence of geometry is indirect measuring of spatial features of things, a geometer must assume some ways to directly measure these spatial features. This implicitly assumed form of measuring, Comte suggests, is the measurement of straight lines or lengths by comparing them with a length of some other thing, like a ruler. Other geometrical figures (curves, areas and volumes) are then to be measured with the help of these straight lines.
Comte admits that geometry is full of other things beyond mere measuring, that is, full of propositions about spatial properties of things. Still, he insists, even these properties are ultimately studied, because they could help with measuring (perhaps in some more concrete science). Because one cannot know beforehand what properties help with measurements, Comte advocates studying as many such properties as possible.
Comte mentions the old distinction between synthetic and analytic methods in geometry, but seems to have no idea what these terms meant originally and what their difference was supposed to be. Instead, Comte suggests that the distinction is just a roundabout way to distinguish ancient from modern geometry. Ancient geometry, Comte suggests, was mainly dealing with concrete and individual figures, taking one type of figure and finding all its characteristics. Because of the uniqueness of the chosen figure, none of these characteristics could be assumed to hold for other entities. Modern geometry, on the other hand, deals with abstract geometrical problems, which can then be applied to many different contexts.
Comte has little patience with ancient geometry, and he especially dislikes the common habit of starting to teach geometry from works like Euclid’s Elements - as he has noticed earlier, history of a discipline is usually not the most convenient way to teach it. Comte is especially critical of the proofs of early propositions in Elements, where Euclid simply places figures on top of each other, to show their similarity - this is just as futile as an attempt to prove the parallel axiom would be.
Although Comte is very critical of ancient geometry, he does admit that Greek mathematicians did make some progress. Particularly, they perfected the study of the simplest kind of figures, namely straight lines and polygons and polyhedras. Furthermore, while Comte ridicules the Greek use of diagrams as proofs of proposition, he does suggest it to be a sort of precursor for modern projective geometry. Another field developed by ancient geometers to perfection, Comte concludes, is trigonometry, where they used certain lines (sines,tangents etc.) to represent angles and their relations and so simplified calculations.
Comte associates the birth of modern geometry with Descartes. While it had been long known that geometric forms could be described in terms of spatial situations of their limiting points etc., it was the invention of Descartes, Comte says, to reduce the talk of situations to talk of lengths and other magnitudes through the notion of a coordinate system. In effect, Comte concludes, Descartes was able to transform at first sight qualitative properties like geometric figures into quantitative properties.
The outcome of the Cartesian transformation of geometry, Comte explains, was that now in two-dimensional geometry lines could be expressed by equations and equations by lines (in three-dimensional case, Comte adds, equations express surfaces, while lines are expressed by pairs of equations). These equations do not just characterise some random properties of the lines, Comte notes, but explicate how they could be generated. He adds that when a geometer is looking for an equation to describe a line, there is no need to choose any specific coordinate system - sometimes the searched for equation might be easier to describe e.g. in terms of polar coordinates, which are especially convenient when describing rotations. Then again, he admits, when finding lines to describe equations, it is best to pick the rectilinear coordinate system, which is the most natural for us to decipher.
Although Comte at first appears to say that all lines correspond to an equation and all equations correspond to a line, he is well aware that analytic geometry of his time has imperfections in this department. Firstly, he notes, discontinuous lines cannot be expressed so well in terms of a single equation. Furthermore, he continues, equations with more than three variables or equations with imaginary solutions have no proper geometric model.
Like in the case of abstract mathematics, Comte is mostly interested in the uses geometry could be put to and the problems that could be solved by its help. Solving some of these problems relies on simple algebraic means, such as when we try to find the number of points that are required for determining the course of a curve. Still, most of these problems, Comte notes, rely on the help of differential and integral calculus. Differentiation is useful not just for finding tangents, but also for describing e.g. the curvature of curves. Then again, Comte concludes, integration is the most useful tool, because it helps us to fulfill the true task of geometry, that of measuring lengths, areas and volumes.
sunnuntai 9. tammikuuta 2022
Auguste Comte: Course of positive philosophy 1 - Mathematical analysis
As the second part of abstract mathematics Comte distinguishes what he calls transcendent analytics, which he characterises as a mathematics of indirect functions. Comte’s point appears to be that while in algebra one is interested of and manipulates functions and quantities given in the task, in the transcendental analytics - in effect, infinitesimal calculus - one introduces and searches for auxiliary quantities and functions, which are related to the original quantities and functions in a more distant manner.
Although Comte does mention the method of exhaustion, known already to Euclid, as a precursor of infinitesimal calculus, he singles out Leibniz and Newton as its first true inventors. Leibnizian method is based on the notion of infinitesimals or infinitely small magnitudes, like infinitely small parts of a curve. These infinitesimals are then used as a way to express certain functions and quantities (e.g. tangents) with the help of relations of these infinitesimals. After some algebraic manipulation of these relations, the infinitesimals can then be discarded in the end, leaving only the searched for function described in terms of ordinary quantities. This method is in Comte’s eyes full of misleading metaphysics - what are these supposed infinitely small magnitudes?
Instead, Comte has to rely on Lagrangian understanding of infinitesimal calculus. This is remarkable, because here Comte agrees with another philosopher wanting to provide an account of all sciences, namely, Hegel. Lagrange’s idea was to define the result of both Leibnizian and Newtonian method - what Lagrange calls the derivative function - in terms of the so-called Taylor series. Mathematician Brook Taylor had shown that many functions could be expressed as an infinite sum consisting of factors made out of the derivative, its derivative etc. In effect, Lagrange reversed this process - he just assumed that a function could be expressed as a Taylor series and then picked the derivative as included in the first factor of the series. Further derivatives could then be just picked from the further factors of the series.
While Comte’s take on calculus of differences is heavily inspired by Lagrange, clearly his own invention is to connect it with Fourier’s notion of periodic functions: idea seems to be that the periodicity of functions is a generalisation of a regularity in the finite differences of a function. Fourier’s notion serves also as a good bridge to more concrete mathematics, as Comte ponders the possibility that Fourier’s work of applying periodic functions to thermal physics might form a new part of concrete mathematics. Yet, Comte doesn’t develop this suggestion further, so I’ll instead consider next time his more developed take on another part of concrete mathematics, that is, geometry.
Although Comte does mention the method of exhaustion, known already to Euclid, as a precursor of infinitesimal calculus, he singles out Leibniz and Newton as its first true inventors. Leibnizian method is based on the notion of infinitesimals or infinitely small magnitudes, like infinitely small parts of a curve. These infinitesimals are then used as a way to express certain functions and quantities (e.g. tangents) with the help of relations of these infinitesimals. After some algebraic manipulation of these relations, the infinitesimals can then be discarded in the end, leaving only the searched for function described in terms of ordinary quantities. This method is in Comte’s eyes full of misleading metaphysics - what are these supposed infinitely small magnitudes?
Newtonian method agrees in its results with Leibnizian method, but Comte finds it much more viable. The basis of Newton's method is the notion of a limit - for instance, we can understand tangent of a curve as a limit of a series of secants of the same curve, where the series is formed by letting the endpoints of the secants approach one another. Despite the increased validity Newton’s method, Comte finds it otherwise more cumbersome to use than Leibnizian method. A common problem in both, Comte suggests, is that they both distinguish analysis too severely from algebra, because the main concepts of both methods - infinitesimals and limits - are not something that can be expressed in algebraic terms. Of course, a more algebraic explication of the notion of limit was in the air - Cauchy had already used the famous epsilon-delta -proofs to show what a limit of something was. Yet, Cauchy's proofs were not widely known and they were not put in form of a clear definition before Bolzano, so it is no surprise Comte still is unaware of the algebraic explication of limit.
Instead, Comte has to rely on Lagrangian understanding of infinitesimal calculus. This is remarkable, because here Comte agrees with another philosopher wanting to provide an account of all sciences, namely, Hegel. Lagrange’s idea was to define the result of both Leibnizian and Newtonian method - what Lagrange calls the derivative function - in terms of the so-called Taylor series. Mathematician Brook Taylor had shown that many functions could be expressed as an infinite sum consisting of factors made out of the derivative, its derivative etc. In effect, Lagrange reversed this process - he just assumed that a function could be expressed as a Taylor series and then picked the derivative as included in the first factor of the series. Further derivatives could then be just picked from the further factors of the series.
Lagrangian method was at least as cumbersome to use as Newtonian. Furthermore, it created an added problem that one has to prove why the factors of Taylor series had the important properties derivatives we supposed to have. For instance, Lagrange had to still show that a line determined by the derivative at a certain point is tangent to the curve at that same point, because it is the line closest to the curve determined by the original function. Despite this added difficulty, Comte says, Lagrangian notion has the benefit that it makes analysis into a mere new modification of algebra, that is, an algebra of Taylor series.
With this general idea of what analysis is, Comte can easily divide analysis into two parts: differential calculus, which tries to find derivative from an original function, and integration, which goes the opposite way, trying to find the original function related to a derivative. Comte notes that the division is not strict, since many problems require the use of both methods. Both differentiation and integration, Comte continues, can then be further divided into differentiation or integration of explicit functions - in other words, problems, where a function is given and its derivative or a function, to which it is derivative, is asked to be provided - and differentiation and integration of implicit functions - in other words, equations, the solving of which requires the use of differentiation or integration.
Comte notes other ways to classify problems of differentiation and integration. For instance, differentiation can involve formulas of only one variable or of many variables (in the so-called partial differentiation). In relation to integration, on the other hand, Comte notes that integration over integration forms a task of far greater complexity than mere simple integration. He also points out that this is not true of differentiation, because finding derivative of a derivative is as simple as finding the first derivative. This difference, in effect, makes integration a much more complex study than differentiation, Comte concludes.
With this general idea of what analysis is, Comte can easily divide analysis into two parts: differential calculus, which tries to find derivative from an original function, and integration, which goes the opposite way, trying to find the original function related to a derivative. Comte notes that the division is not strict, since many problems require the use of both methods. Both differentiation and integration, Comte continues, can then be further divided into differentiation or integration of explicit functions - in other words, problems, where a function is given and its derivative or a function, to which it is derivative, is asked to be provided - and differentiation and integration of implicit functions - in other words, equations, the solving of which requires the use of differentiation or integration.
Comte notes other ways to classify problems of differentiation and integration. For instance, differentiation can involve formulas of only one variable or of many variables (in the so-called partial differentiation). In relation to integration, on the other hand, Comte notes that integration over integration forms a task of far greater complexity than mere simple integration. He also points out that this is not true of differentiation, because finding derivative of a derivative is as simple as finding the first derivative. This difference, in effect, makes integration a much more complex study than differentiation, Comte concludes.
The basis of all these more or less complex problems, Comte insists, should be the differentiation and integration of the ten basic algebraic functions introduced in his study of algebra.. Even here, Comte points out, differentiation is more complete than integration, where we often have no other way to solve even very simple problems, but to give an approximation of its result through numerical methods.
For Comte, it is important not just to structurise methods of mathematical analysis, but to explain what they are useful for or where they can be applied. The applications he suggests are not surprising - differentiation can be used e.g. to find minimal and maximal points of a curve, while integration can be used in determining quantities of areas. More important is the pragmatic principle that even such an abstract field as mathematics should provide some benefit for society.
Following Lagrange again, Comte includes within transcendent analysis also a sort of generalisation of the problem of finding maximal and minimal points of a curve. The earliest example of this general sort, Comte notes, is Newton’s problem of finding what function determines a solid, which would experience minimum resistance when moving through a homogenous fluid in constant velocity. Another early example pointed by Comte is Bernoulli’s problem that if a body moves from a higher point to a lower point, not directly under it, what curve between the points would describe the fastest descent. Such problems involve minimising or maximising certain integrals by choosing a suitable function as a derivative. Lagrange had the idea that just like in differential calculus we consider ever smaller variations of variables, we could also consider small variations of functions. This similarity of method is enough, Lagrange and Comte conclude, to take this calculus of variations as just another modification of analysis.
In contrast, Comte does not want to include within the field of analysis the so-called calculus of differences, invented by the forementioned Taylor. Taylor’s idea was to generalise differentiation and integration. While regular infinitesimal calculus was based on the notion of infinitesimally small differences, Taylor’s calculus would study finite differences: for instance, instead of finding functions to describe tangents of a curve, it would try to find general form for functions of secants of the same curve. In his calculus, Taylor developed many analogues to ordinary differentiation and integration. Indeed, Taylorian calculus can be used to approximate integrals, which cannot be calculated in any other manner, as Comte himself also notes.
Lagrange’s opinion, which Comte evidently accepts, was that Taylorian calculus was not truly in par with infinitesimal calculus, but a part of regular algebra. More particularly, Comte understands it to be a modification of the theory of series. Thus, Comte notes that Taylorian analogue of differentiation corresponds to finding a rule governing a series of numbers or functions, while Taylorian analogue of integration corresponds to calculating a sum of a series. Somewhat ironically, because Comte did not have the modern explication of limit in his use, he didn't know that differentiation and integration can be presented in terms of limits for series of functions, which would thus reduce the difference between Taylorian and infinitesimal calculus.
For Comte, it is important not just to structurise methods of mathematical analysis, but to explain what they are useful for or where they can be applied. The applications he suggests are not surprising - differentiation can be used e.g. to find minimal and maximal points of a curve, while integration can be used in determining quantities of areas. More important is the pragmatic principle that even such an abstract field as mathematics should provide some benefit for society.
Following Lagrange again, Comte includes within transcendent analysis also a sort of generalisation of the problem of finding maximal and minimal points of a curve. The earliest example of this general sort, Comte notes, is Newton’s problem of finding what function determines a solid, which would experience minimum resistance when moving through a homogenous fluid in constant velocity. Another early example pointed by Comte is Bernoulli’s problem that if a body moves from a higher point to a lower point, not directly under it, what curve between the points would describe the fastest descent. Such problems involve minimising or maximising certain integrals by choosing a suitable function as a derivative. Lagrange had the idea that just like in differential calculus we consider ever smaller variations of variables, we could also consider small variations of functions. This similarity of method is enough, Lagrange and Comte conclude, to take this calculus of variations as just another modification of analysis.
In contrast, Comte does not want to include within the field of analysis the so-called calculus of differences, invented by the forementioned Taylor. Taylor’s idea was to generalise differentiation and integration. While regular infinitesimal calculus was based on the notion of infinitesimally small differences, Taylor’s calculus would study finite differences: for instance, instead of finding functions to describe tangents of a curve, it would try to find general form for functions of secants of the same curve. In his calculus, Taylor developed many analogues to ordinary differentiation and integration. Indeed, Taylorian calculus can be used to approximate integrals, which cannot be calculated in any other manner, as Comte himself also notes.
Lagrange’s opinion, which Comte evidently accepts, was that Taylorian calculus was not truly in par with infinitesimal calculus, but a part of regular algebra. More particularly, Comte understands it to be a modification of the theory of series. Thus, Comte notes that Taylorian analogue of differentiation corresponds to finding a rule governing a series of numbers or functions, while Taylorian analogue of integration corresponds to calculating a sum of a series. Somewhat ironically, because Comte did not have the modern explication of limit in his use, he didn't know that differentiation and integration can be presented in terms of limits for series of functions, which would thus reduce the difference between Taylorian and infinitesimal calculus.
While Comte’s take on calculus of differences is heavily inspired by Lagrange, clearly his own invention is to connect it with Fourier’s notion of periodic functions: idea seems to be that the periodicity of functions is a generalisation of a regularity in the finite differences of a function. Fourier’s notion serves also as a good bridge to more concrete mathematics, as Comte ponders the possibility that Fourier’s work of applying periodic functions to thermal physics might form a new part of concrete mathematics. Yet, Comte doesn’t develop this suggestion further, so I’ll instead consider next time his more developed take on another part of concrete mathematics, that is, geometry.
keskiviikko 22. joulukuuta 2021
Auguste Comte: Course of positive philosophy 1 - Solving equations
In a way that sounds rather old-fashioned these days, Comte defines mathematics as a science of measuring magnitudes. Of course, back in his days, mathematics was mostly about numbers, so the definition makes more sense. Furthermore, Comte instantly qualifies his statement, by noting that immediate measuring of, say, length with a ruler or temperature with thermometer is not yet mathematics. Instead, mathematics is all about indirect methods of measuring unknown magnitudes through their relations to others, known magnitudes. In other words, mathematics has to do with solving equations between magnitudes.
Comte is convinced that mathematics is a universal science that applies in principle to anything. His conviction means that everything should be in principle quantifiable. Comte mentions that this is in direct opposition to Kant’s table of categories, where qualities are kept strictly separate from quantities (this seems a rather peculiar way to understand Kant’s division of categories, but I’ll let it pass now). Even organic and social phenomena should be quantifiable, although their complexity might prevent us ever giving a full quantification of them, Comte hastens to add.
Comte divides mathematics into two parts. One part or concrete mathematics deals with measuring magnitudes in empirical and phenomenal matters, such as geometry and mechanics - I shall leave these disciplines to a later post. The other part deals with measuring magnitudes in abstract fashion, or as Comte puts it, with the logic of mathematics. More precisely, the topic of abstract mathematics consists of, Comte says, equations between abstract functions. By function Comte means some type of dependency, for instance, a sum of two magnitudes is their function, because the sum is dependent on what the magnitudes are. Now, abstract function is a dependency that can be understood only on the basis of bare magnitudes or numbers - sum of two magnitudes is the same, no matter whether the magnitudes are units of length, time, mass etc. On the contrary, concrete function expresses a dependency, understanding of which requires something more than mere numbers, such as geometric or mechanical properties.
Comte’s definition might still leave it unclear what to actually include in the abstract functions. He points out that we can at least enumerate some examples of pairs of simple abstract functions (pairs, because they consist of a function and its inverse). He adds that we can then know that any complex function that could be constructed from these simple functions is also an abstract function. These pairs would at least include, Comte recounts, addition and its inverse or subtraction, multiplication and division, raising to a power and roots, and exponential and logarithmic functions.
A more intricate question is whether to include among abstract simple functions also sinus and inverse sinus. The question is difficult, because if sinus is taken as a simple, unanalysable function, then it seems far from numerical, since it receives its meaning from a certain geometrical context (e.g. a unit circle). Then again, sinus can be defined in a purely numerical fashion, but then it is not anymore a simple function. Yet, just because of this dual nature Comte accepts the pair among simple functions, and instantly notes that other functions might also deserve to be included for the same reason, for instance, Jacobi’s theta-function.
Comte divides abstract mathematics into two disciplines, corresponding to two stages of solving equations. Firstly, one transforms or resolves functions given in the equation into other, more easily solvable forms - this is the task of algebra. Secondly, one finds the values of these easier functions - this is the task of arithmetic. Comte notes that this notion of arithmetic is more extensive than what is usually meant by it, since it includes also e.g. the use of logarithmic tables. He also points out that arithmetic is in a sense just a special case of algebra, since finding a value for a certain formula just means turning it into form (10^n)a + (10^(n-1))b + (10^(n-2))c + (10^(n-3))d + …, where a, b, c, d ... etc. are natural numbers smaller than 10. We might thus say that abstract mathematics is nothing but algebra.
Comte divides algebra further into a study of what he calls indirect functions - transcendent analysis or infinitesimal calculus - and study of direct functions - algebra in the proper sense. I shall concentrate in the rest of this post on the latter, leaving infinitesimals for the next one. Well, there’s not that much of philosophical interest in what Comte still has to say about algebra. He notes that the current state of algebra was far from complete, since general solutions had been found only for polynomial equations up to fourth degree of complexity - he was apparently unaware of the recently discovered fact that such general solutions could not be given for more complex polynomials. Still, this supposed incomplete state of algebra gives him an opportunity to mention that numerical methods of solving equations form a second part of algebra.
More interesting is Comte’s idea that because algebra abstracts from all the conditions for the meaningfulness of functions and equations, it instead aims for being as general as possible. Thus, in algebra all the functions or operations are defined so that they always will have results, no matter whether these results can be interpreted meaningfully. Hence, the notion of number is extended, first, to negative numbers smaller than zero, because all subtractions should produce some results, and finally even to seemingly impossible imaginary numbers, which are roots of negative numbers.
Comte is convinced that mathematics is a universal science that applies in principle to anything. His conviction means that everything should be in principle quantifiable. Comte mentions that this is in direct opposition to Kant’s table of categories, where qualities are kept strictly separate from quantities (this seems a rather peculiar way to understand Kant’s division of categories, but I’ll let it pass now). Even organic and social phenomena should be quantifiable, although their complexity might prevent us ever giving a full quantification of them, Comte hastens to add.
Comte divides mathematics into two parts. One part or concrete mathematics deals with measuring magnitudes in empirical and phenomenal matters, such as geometry and mechanics - I shall leave these disciplines to a later post. The other part deals with measuring magnitudes in abstract fashion, or as Comte puts it, with the logic of mathematics. More precisely, the topic of abstract mathematics consists of, Comte says, equations between abstract functions. By function Comte means some type of dependency, for instance, a sum of two magnitudes is their function, because the sum is dependent on what the magnitudes are. Now, abstract function is a dependency that can be understood only on the basis of bare magnitudes or numbers - sum of two magnitudes is the same, no matter whether the magnitudes are units of length, time, mass etc. On the contrary, concrete function expresses a dependency, understanding of which requires something more than mere numbers, such as geometric or mechanical properties.
Comte’s definition might still leave it unclear what to actually include in the abstract functions. He points out that we can at least enumerate some examples of pairs of simple abstract functions (pairs, because they consist of a function and its inverse). He adds that we can then know that any complex function that could be constructed from these simple functions is also an abstract function. These pairs would at least include, Comte recounts, addition and its inverse or subtraction, multiplication and division, raising to a power and roots, and exponential and logarithmic functions.
A more intricate question is whether to include among abstract simple functions also sinus and inverse sinus. The question is difficult, because if sinus is taken as a simple, unanalysable function, then it seems far from numerical, since it receives its meaning from a certain geometrical context (e.g. a unit circle). Then again, sinus can be defined in a purely numerical fashion, but then it is not anymore a simple function. Yet, just because of this dual nature Comte accepts the pair among simple functions, and instantly notes that other functions might also deserve to be included for the same reason, for instance, Jacobi’s theta-function.
Comte divides abstract mathematics into two disciplines, corresponding to two stages of solving equations. Firstly, one transforms or resolves functions given in the equation into other, more easily solvable forms - this is the task of algebra. Secondly, one finds the values of these easier functions - this is the task of arithmetic. Comte notes that this notion of arithmetic is more extensive than what is usually meant by it, since it includes also e.g. the use of logarithmic tables. He also points out that arithmetic is in a sense just a special case of algebra, since finding a value for a certain formula just means turning it into form (10^n)a + (10^(n-1))b + (10^(n-2))c + (10^(n-3))d + …, where a, b, c, d ... etc. are natural numbers smaller than 10. We might thus say that abstract mathematics is nothing but algebra.
Comte divides algebra further into a study of what he calls indirect functions - transcendent analysis or infinitesimal calculus - and study of direct functions - algebra in the proper sense. I shall concentrate in the rest of this post on the latter, leaving infinitesimals for the next one. Well, there’s not that much of philosophical interest in what Comte still has to say about algebra. He notes that the current state of algebra was far from complete, since general solutions had been found only for polynomial equations up to fourth degree of complexity - he was apparently unaware of the recently discovered fact that such general solutions could not be given for more complex polynomials. Still, this supposed incomplete state of algebra gives him an opportunity to mention that numerical methods of solving equations form a second part of algebra.
More interesting is Comte’s idea that because algebra abstracts from all the conditions for the meaningfulness of functions and equations, it instead aims for being as general as possible. Thus, in algebra all the functions or operations are defined so that they always will have results, no matter whether these results can be interpreted meaningfully. Hence, the notion of number is extended, first, to negative numbers smaller than zero, because all subtractions should produce some results, and finally even to seemingly impossible imaginary numbers, which are roots of negative numbers.
tiistai 14. joulukuuta 2021
Auguste Comte: Course of positive philosophy 1 - Classification of sciences
Classification of disciplines has been a staple of philosophy since the time of Aristotle, and Comte’s studies make no exception. Of course, there have been plenty of classifications presented as the correct one, so it is reasonable to ask what is so special in Comte’s. Comte himself has a clear answer: earlier classifications derive from a time when all fields of science had not reached the status of positivism.
Before introducing his classification of sciences, Comte considers the question of what he should actually be classifying. He first distinguishes theoretical sciences from practical arts and delineates between these two extremes the field of engineering, which aims at applying results of theoretical sciences to practical questions. He notes that arts and engineering are essentially dependent on theoretical sciences. Indeed, he adds, one art can depend on various sciences, for instance, agriculture requires theoretical knowledge of plants, of chemicals and even of sun, moon and stars. Thus, he concludes that the basic classification should be made at the level of theoretical sciences, not on the level of their practical applications.
Another distinction Comte makes is that between general or abstract sciences and particular or concrete sciences. Abstract sciences, he explains, deal with what is possible, for instance, according to known physical and chemical laws. Concrete sciences then deal with actual instances of such laws: examples include natural history and mineralogy. Comte also points out that just like practical arts depend on theoretical sciences, concrete sciences depend on abstract sciences, being their specifications. Thus, the disciplines good for classification are abstract theoretical sciences.
Comte goes on to speak about the method one should use in the classification. He points out that while we strive for what could be called a natural classification, we can approach such classification only through various artificial classifications. Indeed, he notes, we often begin classification of a new science historically, that is, by noting new ideas and discoveries in the order in which they were found. The more a science is developed, Comte notes, less and less possible it becomes to use the historical approach, because the number of theorems involved becomes too unwieldy. Historical classification is then replaced by a dogmatic approach, where the ideas in question should form a systematic whole. This dogmatic approach abbreviates the historical approach. In the particular case of classifying all sciences, there is the further link that the most abstract disciplines, from which the dogmatic approach begins, are also historically the earliest to reach a more complete stage.
Comte’s classification is meant to be a basis for a complete reform of the educational system. His idea is simple: if we can organise general theoretical sciences into a hierarchical system, in accordance with the dogmatic approach, we then know the ideal order of science education - the education must always start with the most abstract science and move towards more concrete ones. That way, researches working with problems of the more concrete sort would have the necessary tools for understanding more abstract field, on which the more concrete questions depend.
The actual classification Comte suggests seems somewhat problematic - this is just to be expected, since the development of science has been explosive in the last two centuries. Comte’s main dividing line between sciences of inorganic and organic nature seems acceptable, but subdivisions of the two feel less successful. For instance, Comte divides study of inorganic nature into study of stellar phenomena or astronomy and study of terrestrial phenomena, which he then divides into study of more mechanical phenomena or physics and chemistry. Considering that Comte clearly states that zoology and botany are not divisions of the abstract study of organic nature, being more like concrete applications of the study of organisms for two actual species of organisms, one might protest that astronomy is also just application of the same physical laws into stellar objects. Indeed, this is even more evident nowadays, when we know that chemistry could also be used to describe elements of the stellar objects.
Comte’s division of the study of organisms is also problematic. He suggests dividing this whole into a study of individual organisms or physiology and a study of interactions of organisms or sociology. One has to wonder if this is just a circumspect way to distinguish study of humans from study of plants and animals, trying to avoid the same criticism that Comte himself leveled against distinguishing zoology and botany, or if he truly will accept study of all populations of organisms as application of sociology. Even if the latter would be true, it is still doubtful whether we really can meaningfully separate study of an individual organism from study of the interactions of organisms in the same species.
There’s one very conspicuous absence in Comte’s classification - mathematics. This is simply because mathematics plays a very special role for Comte. In a sense it is a part of the classification, the most abstract science there is. In another sense it is for Comte a general methodology for all concrete sciences. Because of this role, mathematics is tightly linked to things it is used for, although it also has a part that is pure of all applications - but this is a discussion I’ll be entering later.
Before introducing his classification of sciences, Comte considers the question of what he should actually be classifying. He first distinguishes theoretical sciences from practical arts and delineates between these two extremes the field of engineering, which aims at applying results of theoretical sciences to practical questions. He notes that arts and engineering are essentially dependent on theoretical sciences. Indeed, he adds, one art can depend on various sciences, for instance, agriculture requires theoretical knowledge of plants, of chemicals and even of sun, moon and stars. Thus, he concludes that the basic classification should be made at the level of theoretical sciences, not on the level of their practical applications.
Another distinction Comte makes is that between general or abstract sciences and particular or concrete sciences. Abstract sciences, he explains, deal with what is possible, for instance, according to known physical and chemical laws. Concrete sciences then deal with actual instances of such laws: examples include natural history and mineralogy. Comte also points out that just like practical arts depend on theoretical sciences, concrete sciences depend on abstract sciences, being their specifications. Thus, the disciplines good for classification are abstract theoretical sciences.
Comte goes on to speak about the method one should use in the classification. He points out that while we strive for what could be called a natural classification, we can approach such classification only through various artificial classifications. Indeed, he notes, we often begin classification of a new science historically, that is, by noting new ideas and discoveries in the order in which they were found. The more a science is developed, Comte notes, less and less possible it becomes to use the historical approach, because the number of theorems involved becomes too unwieldy. Historical classification is then replaced by a dogmatic approach, where the ideas in question should form a systematic whole. This dogmatic approach abbreviates the historical approach. In the particular case of classifying all sciences, there is the further link that the most abstract disciplines, from which the dogmatic approach begins, are also historically the earliest to reach a more complete stage.
Comte’s classification is meant to be a basis for a complete reform of the educational system. His idea is simple: if we can organise general theoretical sciences into a hierarchical system, in accordance with the dogmatic approach, we then know the ideal order of science education - the education must always start with the most abstract science and move towards more concrete ones. That way, researches working with problems of the more concrete sort would have the necessary tools for understanding more abstract field, on which the more concrete questions depend.
The actual classification Comte suggests seems somewhat problematic - this is just to be expected, since the development of science has been explosive in the last two centuries. Comte’s main dividing line between sciences of inorganic and organic nature seems acceptable, but subdivisions of the two feel less successful. For instance, Comte divides study of inorganic nature into study of stellar phenomena or astronomy and study of terrestrial phenomena, which he then divides into study of more mechanical phenomena or physics and chemistry. Considering that Comte clearly states that zoology and botany are not divisions of the abstract study of organic nature, being more like concrete applications of the study of organisms for two actual species of organisms, one might protest that astronomy is also just application of the same physical laws into stellar objects. Indeed, this is even more evident nowadays, when we know that chemistry could also be used to describe elements of the stellar objects.
Comte’s division of the study of organisms is also problematic. He suggests dividing this whole into a study of individual organisms or physiology and a study of interactions of organisms or sociology. One has to wonder if this is just a circumspect way to distinguish study of humans from study of plants and animals, trying to avoid the same criticism that Comte himself leveled against distinguishing zoology and botany, or if he truly will accept study of all populations of organisms as application of sociology. Even if the latter would be true, it is still doubtful whether we really can meaningfully separate study of an individual organism from study of the interactions of organisms in the same species.
There’s one very conspicuous absence in Comte’s classification - mathematics. This is simply because mathematics plays a very special role for Comte. In a sense it is a part of the classification, the most abstract science there is. In another sense it is for Comte a general methodology for all concrete sciences. Because of this role, mathematics is tightly linked to things it is used for, although it also has a part that is pure of all applications - but this is a discussion I’ll be entering later.
torstai 25. marraskuuta 2021
Auguste Comte: Course of positive philosophy 1 (1830) - Toward the age of positivism
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| (1798-1857) |
Of course, there are clear signs of Comte being a child of his time, especially in his idea of the three stages of human thought: theological, metaphysical and positive. Similar notions of progressive stages of humanity, leading from crude religious thoughts to scientific outlook had been a staple of French thought. One need mention just Saint-Simon, whose work indeed had an influence on Comtee. Compared to the systems of his predecessors, who might have distinguished a dozen stages, Comte’s version seems much more streamlined, which shows his attempt to transform such crude historical schemes into a real law of human progression.
A more important novelty is Comte’s idea that the three stages are not really distinct, but more like abstractions from a concrete continuum. Thus, he notes that we really cannot determine a specific spot where e.g. the positive stage began, because it has progressed in different manners in different sciences, and while physical sciences have already managed to eradicate theological and metaphysical notions, they still abound in human sciences.
While it is quite easy to understand what Comte means by the theological stage, where everything is explained by actions of divinities, the notion of metaphysical stage is not so simple to understand. Metaphysics should supposedly replace gods with abstract forces that act as explanatory causes. This description could fit a number of theories, for instance, Neoplatonic hierarchies of abstractions, but when we see Comte suggesting that metaphysics ultimately strives toward unifying all these causes in the notion of Nature, it appears that he is especially referring to various materialist philosophies that try to explain phenomena through some ultimate group of material existents.
Now, an obvious question such historical schemes suggest is whether they are meant to be just a very general description of past events or whether they imply that such progression has been necessary. Comte at least seems to take the latter route. He states that the three stages can be found even in the development of individual human beings, at least in the sense that we all must begin as theologians, searching for purposeful actors behind everything we experience. Indeed, Comte says, at the dawn of humanity such behaviour was quite rational, since there simply was not enough information to tell how e.g. the stars moved.
Even if the beginning of human history is necessarily theological, it seems to require more justification to state that it is necessary to move from this stage toward the so-called positive stage. Comte does not delineate any argument for this stance, at least in the first chapter of his Course, but he appears to have the idea that when we gather more and more information about the phenomena around us, we firstly notice that we really cannot find the ultimate causes of them, and secondly, also notice that we actually need no such explanations. In other words, moving toward the positive stage means rejecting all theological assumptions, but not by assuming other, materialistic assumptions. Thus, the positive stage is one of agnosticism and skepticism about ultimate causes, that is, in it we merely describe the laws or regularities of phenomena, but do not try to explain them.
The final justification of this historical scheme should apparently be given in the science of human societies, the foundation of which is one task Comte sets for himself in the course. This scientific study of humanity should replace psychology of his times, which he thinks to be still filled with theological and metaphysical assumptions. Comte especially criticises the use of self-observation as a method of psychology. In physical sciences, he notes, we have already learned that human observation might fake us, suggesting e.g. sun to be a much smaller object than it actually is. Why should we assume that observation of our own actions would be more trustworthy?
The task of creating a social science is in Comte’s eyes intricately linked to a second task, that of forming a system out of all individual sciences. Only through social science, Comte says, can we recognise logic, that is, scientific methodology, and so it helps us to understand relations between all sciences. Comte notes that creation of such a system or hierarchy of sciences satisfies our need for unification, evident in the earlier replacement of polytheism with monotheism and of pluralism of forces with monistic Nature. Comte speculates that this unification has its limits and that we can probably never reduce laws of different sciences into one law. Still, he notes, there’s at least a homogenous method combining all sciences into a unity.
In addition to these two tasks, Comte suggests several benefits his undertaking might have. Firstly, delineating the relations between scientific disciplines might suggest fruitful interdisciplinary studies. Comte mentions as examples Cartesian application of algebra into geometry and recent studies in organic chemistry. Another benefit lies in the reorganisation of education, where mere literary studies could be replaced by a curriculum designed around the system of sciences and beginning with their general methodology.
The final benefit Comte suggests links his undertaking again to the contemporary discussions in French philosophy. He notes that theological and metaphysical thinkers had disputed about the best possible form of governance: it is likely that he is referring here, on the one hand, to Catholic conservatives siding with absolute monarchy, and on the other hand, various materialist leaning thinkers speaking for more republican or at least constitutional state. Comte appears to be suggesting that this debate cannot be solved through philosophical disputations, but only through scientific description of how human societies work and how they could be organised most effectively - a task for the social sciences.
torstai 21. lokakuuta 2021
Louis-Gabriel-Ambroise vicomte de Bonald: Philosophical demonstration of the constitutive principle of society (1830)
In his book Démonstration Philosophique du Principe Constitutif de la Société, one of the leading stars of French conservative school of philosophy, de Bonald, returns to questions of political philosophy. Still, he also has something to say in the introduction of the work about different philosophical schools, especially as they had appeared in contemporary France. In a previous book, Recherches philosophiques, de Bonald had distinguished two separate schools of philosophy - Platonists, who lean toward innate ideas as the source of knowledge and uphold spiritualism and theism, and Aristotelians, who lean toward sensations as the source of knowledge and uphold materialism and atheism.
Now, de Bonald, distinguishes also a third school, eclecticism, represented by thinkers such as Maine de Biran, who were not part of the empirical tradition of Condillac, but who also were not attached to the very Catholic inspired school of de Bonald. De Bonald’s quick judgement is that such thinkers are simply inconsistent and that there really are only two possible philosophical positions to choose from - and in the end, only Catholicism is actually true.
In the book itself, de Bonald attempts to give a new justification to his political theory. He begins with a very traditional account of family: family consists of three roles, father, mother and child. Family as such is always monogamous, de Bonald says, because polygamous family would mean just combining many families together under same father (note how de Bonald conveniently forgets the possibility of polyandry). He goes even so far as to suggest that serial monogamy, based on the possibility of divorce, is just polygamy in disguise, although here the different families succeed one another and do not overlap in time.
De Bonald’s idea of family is not just heteronormative, but also patriarchal, as he insists that all the power in the family should reside with the father. Power of the father is absolute and independent of the mother and the child and divides into two parts: power to judge what is good for the family and power to combat any obstacles against the good of the family. Despite the power of family being concentrated to the father, the purpose of the family, de Bonald notes, is to take care that human species will continue through its individuals and especially children.
Between the power of father and the service of children lies the role of mother, who works as a sort of minister for the father, de Bonald tells. In a remarkably insulting statement de Bonald reveals that mother as a mediating element of the family resembles both men and children - being servant to one and controlling the other - and could thus be called a manchild.
Families tend to reproduce and thus multiply, and the aim of a state, de Bonald states, is to guarantee the continual regeneration of family life, just like families guarantee the continual regeneration of human individuals. Thus, he concludes, states should be like big families. If states have grown from a single family, this happens quite naturally, de Bonald insists, by central power remaining always in the hereditary line of succession. Then again, even in a case where a group of unrelated individuals and families combine into a state, there is usually some heroic person who acts as the central node in bringing all together, de Bonald assures the reader. He is explicitly criticising the idea of a social contract made in a state of a nature, where a group of unorganised individuals could invent a political structure to guide them.
De Bonald’s ideal of a state is thus monarchic, as we know from his previous writings. Central power must always be unified, absolute and independent of everything else, he says and adds that a king should have the final authority in deciding the ownership of the land and soil of the state. Just like in an ideal formation of a state, the monarchy should be perpetuated through heredity.
An ideal state, de Bonald continues, shouldn’t be just a two-rung hierarchy, with nothing mediating between the monarch and the subjects, like in the Ottoman empire, the favourite example of despotism for early modern thinkers. Instead, de Bonald argues that like domestic society or family had to have a mediating position of woman, political society or state should have a mediating position of the nobility. Like woman in the family, de Bonald notes, nobility should share the nature of extremes, being subject to the monarch, but being also like little kings, having absolute authority in their own piece of land.
While it seems that there is no place for the opinion of the subjects to be heard in Bonaldian state, he does allow a position for them in the form of General Estates, although it has only a consultative role in the state affairs. De Bonald dilutes this concession by noting that, as was the tradition, one third of the Estates was to be filled by nobility, who represented the political society, while the second third of the Estates was then supposed to represent church or the religious society. In his earlier works, de Bonald had thought that the final or the Third Estate was superfluous to the proceedings, but he now finds a justification for their inclusion: they represent the domestic societies or the ordinary families.
Just like de Bonald preferred monogamous family over polygamous, he also prefers monocratic state over polycratic or democratic state. In a democracy, he insists, all social roles are confused, everyone being both a ruler and a subject at the same time. He is certain that democracy works only in small communities, like Swiss cantons, or then in such backward and almost savage countries like America.
Unlike with the case of family, de Bonald notes that there are various middle positions between the ideal monarchy and democracy. Main one of them is aristocracy, which de Bonald calls also acephalous or headless monarchy. This need of a central power, he notes, often forces aristocracies to elect a figurative monarch, which still lacks the status of absolute ruler, because of its dependency on the nobility. Another type of middle position is provided by English representative monarchy, which de Bonald interprets as an unstable combination of three states (monarchy, aristocracy and democracy), which can only work in such an isolated country.
We already mentioned that beside domestic and political societies de Bonald speaks also of religious society. The place of the monarch or father lies naturally with God, whom de Bonald assumes everyone should be aware of through innate ideas and who accounts for the preservation of the whole world. Analogically to the domestic and the political society, the natural form of religious society is monotheistic, while the respective perverted form is polytheism, which de Bonald calls either idolatry (in a natural state) or paganism (in an organised state).
The problem with forming religious society, de Bonald notes, is the mediation between God and humans, since the gulf between the two extremes is infinitely wide. This problem is especially strong with the question of sacrifice, de Bonald thinks. For him, sacrifice is an essential element in all societies and particularly for all mediators, who according to de Bonald should sacrifice themselves in order to pay for the care that the rulers provide for their subjects. Thus, de Bonald insists, nobles are meant to put their lives on stake for their country. In an even more drastic and patriarchal fashion, de Bonald thinks that women should yield their whole life to the service of their husbands.
In a religious society, the role of mediator has at first been in the hands of fathers of the families, who have shown their gratitude for God through sacrifices - de Bonald refers to the story of Abraham, who was willing to sacrifice even his own son for God, who then accepted animals as substitutes. When families combined into states, this role of mediators was taken up by a priestly class, who continued the tradition of animal sacrifices. Still, the gulf remained, and de Bonald concludes, it could be bridged only by someone partaking both of the nature of God and human and sacrificing himself for the whole humanity - here’s a justification for main tenets of Christianity.
De Bonald’s account of the role of sacrifice in religion makes God sound like a mafia boss demanding payment for his protection. Furthermore, like his account of family and state, his account of church or the earthly representative of God and his mediator is quite hierarchical, de Bonald practically endorsing Catholicism.
All attempts to reform Catholic church, de Bonald stated, inevitably lead the church toward the equivalent of democracy, where all individuals by themselves mediate their relation to divinity. Indeed, de Bonald notes, reformist churches often endorse the ideals of democracy and even allow divorce, which he thought to be just another name for polygamy, analogue of democracy in family life. Like with state, de Bonald admits there are various middle positions between Catholicism and full reformism, such as Orthodox, Anglican and Lutheran churches, which at least admit the need for a priest class, even if they do not submit under the authority of Catholic church.
Now, de Bonald, distinguishes also a third school, eclecticism, represented by thinkers such as Maine de Biran, who were not part of the empirical tradition of Condillac, but who also were not attached to the very Catholic inspired school of de Bonald. De Bonald’s quick judgement is that such thinkers are simply inconsistent and that there really are only two possible philosophical positions to choose from - and in the end, only Catholicism is actually true.
In the book itself, de Bonald attempts to give a new justification to his political theory. He begins with a very traditional account of family: family consists of three roles, father, mother and child. Family as such is always monogamous, de Bonald says, because polygamous family would mean just combining many families together under same father (note how de Bonald conveniently forgets the possibility of polyandry). He goes even so far as to suggest that serial monogamy, based on the possibility of divorce, is just polygamy in disguise, although here the different families succeed one another and do not overlap in time.
De Bonald’s idea of family is not just heteronormative, but also patriarchal, as he insists that all the power in the family should reside with the father. Power of the father is absolute and independent of the mother and the child and divides into two parts: power to judge what is good for the family and power to combat any obstacles against the good of the family. Despite the power of family being concentrated to the father, the purpose of the family, de Bonald notes, is to take care that human species will continue through its individuals and especially children.
Between the power of father and the service of children lies the role of mother, who works as a sort of minister for the father, de Bonald tells. In a remarkably insulting statement de Bonald reveals that mother as a mediating element of the family resembles both men and children - being servant to one and controlling the other - and could thus be called a manchild.
Families tend to reproduce and thus multiply, and the aim of a state, de Bonald states, is to guarantee the continual regeneration of family life, just like families guarantee the continual regeneration of human individuals. Thus, he concludes, states should be like big families. If states have grown from a single family, this happens quite naturally, de Bonald insists, by central power remaining always in the hereditary line of succession. Then again, even in a case where a group of unrelated individuals and families combine into a state, there is usually some heroic person who acts as the central node in bringing all together, de Bonald assures the reader. He is explicitly criticising the idea of a social contract made in a state of a nature, where a group of unorganised individuals could invent a political structure to guide them.
De Bonald’s ideal of a state is thus monarchic, as we know from his previous writings. Central power must always be unified, absolute and independent of everything else, he says and adds that a king should have the final authority in deciding the ownership of the land and soil of the state. Just like in an ideal formation of a state, the monarchy should be perpetuated through heredity.
An ideal state, de Bonald continues, shouldn’t be just a two-rung hierarchy, with nothing mediating between the monarch and the subjects, like in the Ottoman empire, the favourite example of despotism for early modern thinkers. Instead, de Bonald argues that like domestic society or family had to have a mediating position of woman, political society or state should have a mediating position of the nobility. Like woman in the family, de Bonald notes, nobility should share the nature of extremes, being subject to the monarch, but being also like little kings, having absolute authority in their own piece of land.
While it seems that there is no place for the opinion of the subjects to be heard in Bonaldian state, he does allow a position for them in the form of General Estates, although it has only a consultative role in the state affairs. De Bonald dilutes this concession by noting that, as was the tradition, one third of the Estates was to be filled by nobility, who represented the political society, while the second third of the Estates was then supposed to represent church or the religious society. In his earlier works, de Bonald had thought that the final or the Third Estate was superfluous to the proceedings, but he now finds a justification for their inclusion: they represent the domestic societies or the ordinary families.
Just like de Bonald preferred monogamous family over polygamous, he also prefers monocratic state over polycratic or democratic state. In a democracy, he insists, all social roles are confused, everyone being both a ruler and a subject at the same time. He is certain that democracy works only in small communities, like Swiss cantons, or then in such backward and almost savage countries like America.
Unlike with the case of family, de Bonald notes that there are various middle positions between the ideal monarchy and democracy. Main one of them is aristocracy, which de Bonald calls also acephalous or headless monarchy. This need of a central power, he notes, often forces aristocracies to elect a figurative monarch, which still lacks the status of absolute ruler, because of its dependency on the nobility. Another type of middle position is provided by English representative monarchy, which de Bonald interprets as an unstable combination of three states (monarchy, aristocracy and democracy), which can only work in such an isolated country.
We already mentioned that beside domestic and political societies de Bonald speaks also of religious society. The place of the monarch or father lies naturally with God, whom de Bonald assumes everyone should be aware of through innate ideas and who accounts for the preservation of the whole world. Analogically to the domestic and the political society, the natural form of religious society is monotheistic, while the respective perverted form is polytheism, which de Bonald calls either idolatry (in a natural state) or paganism (in an organised state).
The problem with forming religious society, de Bonald notes, is the mediation between God and humans, since the gulf between the two extremes is infinitely wide. This problem is especially strong with the question of sacrifice, de Bonald thinks. For him, sacrifice is an essential element in all societies and particularly for all mediators, who according to de Bonald should sacrifice themselves in order to pay for the care that the rulers provide for their subjects. Thus, de Bonald insists, nobles are meant to put their lives on stake for their country. In an even more drastic and patriarchal fashion, de Bonald thinks that women should yield their whole life to the service of their husbands.
In a religious society, the role of mediator has at first been in the hands of fathers of the families, who have shown their gratitude for God through sacrifices - de Bonald refers to the story of Abraham, who was willing to sacrifice even his own son for God, who then accepted animals as substitutes. When families combined into states, this role of mediators was taken up by a priestly class, who continued the tradition of animal sacrifices. Still, the gulf remained, and de Bonald concludes, it could be bridged only by someone partaking both of the nature of God and human and sacrificing himself for the whole humanity - here’s a justification for main tenets of Christianity.
De Bonald’s account of the role of sacrifice in religion makes God sound like a mafia boss demanding payment for his protection. Furthermore, like his account of family and state, his account of church or the earthly representative of God and his mediator is quite hierarchical, de Bonald practically endorsing Catholicism.
All attempts to reform Catholic church, de Bonald stated, inevitably lead the church toward the equivalent of democracy, where all individuals by themselves mediate their relation to divinity. Indeed, de Bonald notes, reformist churches often endorse the ideals of democracy and even allow divorce, which he thought to be just another name for polygamy, analogue of democracy in family life. Like with state, de Bonald admits there are various middle positions between Catholicism and full reformism, such as Orthodox, Anglican and Lutheran churches, which at least admit the need for a priest class, even if they do not submit under the authority of Catholic church.
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