The result of the previous chapter was that the limitation of magnitudes was seen to depend on the qualitative contents measured by the magnitudes, or as Fichte also puts it, quality was shown as the source of quantitative. Thus, we are not anymore dealing with the empty continuity of counting formal, indifferent units into infinity nor with a positive or negative relation of magnitudes measured by this mutual relation. Instead, Fichte emphasises, the content forms the foundation of limits and relations of a magnitude that is derived from this content.
The first form of this magnitude, Fichte says, is an extensive magnitude. Often by an extensive magnitude, Fichte admits, is meant simply any limited quantity in general. Yet, he is here using the term in a special meaning, where the content is recognised as the foundation of this limitation: the content alone extends itself into a determined quantity against other contents and does not just receive a limit from outside. Thus, the content is by its own nature extended and limited from others.
Extensive magnitude is limitation, Fichte clarifies, but only with the distinction that it is the qualitative determination that limits itself and divides into a manifold of quantitatively distinguishable parts. There are thus two aspects, outward and inward, in any extensive magnitude. Outwardly, Fichte begins, extension is related to something else, since limitation implies in general an opposition. Here the opposition is not anymore formal opposition of different amounts, but a determined opposition, where the content itself determines its own extension opposed to other similar extensions (you might picture here a sort of force field). Just like earlier, we might speak of the relations that the numeric expressions of these extensions have to one another.
Inwardly, Fichte adds, extension is an infinitely divisible or distinguishable manifold sharing a continuous, unified quality. Here we meet again the opposition of continuous and discrete magnitude, but now finally balanced. Thus, Fichte notes, we have a similarly continuing extension, and if we formally ignored the content that determines the limits of this extension, it would seem to be able to continue in infinity. Such a return to earlier stages of quantity, Fichte explains, is caused by the one-sidedness of the concept of continuity. Conversely, the extension as a magnitude in general is an internally distinguishable, discrete manifold. Just like with continuity, taking this discreteness one-sidedly would lead us to an infinity – this time to infinite divisibility. Yet, Fichte points out, both continuity and discreteness are by themselves untrue and presuppose one another or their common unity. This unity, he says, is just the extensive magnitude or the determined content quantifying itself and giving itself its own extension and thus showing itself as both a continuous quantity and a discrete plurality of parts.
Fichte notes that an extensive magnitude appears to also contain the opposed conceptual moment of intensity. He begins his argument from the notion of extension as a limited and determined continuous quantity that rejects from its extension everything opposing its determination. In other words, Fichte suggests, the outward limitation of the determination of the extensive magnitude is necessarily connected with a positive self-assertion of this determination within these limits. Now, at the level of quantity, this self-assertion must also have a measure, which is then its force or intensity, in opposition to mere extension or outward limitation.
The intensive magnitude, Fichte continues, is the simple, undivided determination, not yet as a pure quality, but as a quantitative measure. In other words, it is still a magnitude, but not a manifold of parts (that would be an extension). It is more like the grade of the content, designating its inner energy or its strength and weakness. The grade, Fichte suggests, is a simple measure of a simple content, but it can be designated through the general expression of all quantities, that is, through number. Thus, the increase or decrease of intensity can be expressed through determined numbers, but these numbers do not designate any sum of individuals or no amount of grades. Instead, Fichte says, a numerically expressed grade shows a certain position in a scale of grades.
It shouldn't be surprising that Fichte manages to again introduce here the opposition of continuous and discrete. While intensity remains same as to its content, he says, it can be thought as continuously strengthening or weakening in infinity, whereby the grade of intensity seems indifferent. In other words, the content itself should remain the same, no matter how great or small of an intensity it has.
In addition to the indifferent continuum of strengthening or weakening in infinity, Fichte says, the intensity also contains the aspect of discreteness as a quantitative determination. In other words, an intensity has a determined grade and is locked within certain limits of strengthening and weakening, but only in a series of other grades that give the continuum of strengthening and weakening internal distinctions that reproduce the moment of discreteness. The measure of intensity, Fichte thinks, is only in relation to others, and a determined intensity can be called great or small and strong or weak only when it is compared to other similar intensities, and so the same grade can appear at the same time great and small according to different comparisons and relations. Grade, Fichte concludes, is thus the highest mediation of the oppositions of quantity.
The intensity, Fichte notes, on the one hand, limits itself outwardly in relation to other intensities by having a different grade than them. On the other hand, it also limits itself internally by allowing the determination of its content remain the same within the limits of strengthening and weakening. This means, Fichte suggests, that intensity must give itself an equally determined extension, which leads us to the mediation between extension and intension. This mediation, in Fichte’s opinion, is a particular application of the more general principle that qualities must also have a quantitative determination.
The result of Fichte’s investigation has been that extensive and intensive magnitudes are reciprocally conditions of one another. Since both thus by themselves move to the other, their truth is only their unity, which is then also the truth of the third stage of quantity. This unity of both, Fichte suggests, is the quantitative determination appearing from qualitative content that gives itself an appropriate and specific quantity with determined intensity in itself and in determined extension toward other self-quantifying qualities. In relation to itself, such self-quantifying quality can be increased or decreased, but this changeability remains locked within determined limits. Beyond these limits, the content itself changes, and thus this qualitative determination can be thought only within a series of such determinations and limited by them. At the same time, Fichte notes, the concept of quantitative is thus raised to a new and higher meaning, since it receives a qualitative sense.
The mere formal continuity of infinite increase and decrease of magnitude, Fichte thinks, has received a limit and a deeper meaning by becoming an expression of the content. On the other hand, the content has its specific, fundamental character in a series of magnitudes, within which it can grow or diminish, but beyond which it will immediately become something else. Mere quantitative difference has thus become qualitative, and continuity of quantitative increasing and decreasing is thereby limited: what is just quantitatively changed in an unremarkable fashion suddenly leaps over the limit of its changeability and becomes qualitatively different, breaking the series of quantitative graduality. Quantitative in general, Fichte suggests, is absorbed by qualitative and assimilated as a mere moment in it. Quantity is nothing in itself, but only the expression of qualitative and even in its formal limitation it shows itself as dependent and subsumed to the content that places in these quantitative limits qualitative distinctions.
Fichte concludes with a summary of the development of the concept of quantity. Its journey began with the notion of this (Dieses) that was formally, but not really distinguished from other these. This first concept of quantity developed into the notion of a limited quantity or magnitude, which was expressed or determined as a countable number, as a measure, and finally, as a determined grade of extension and intensity. Through this development of quantity, the notion of qualitative determination protruded more and more forcefully: something has to be qualitatively determined or has to have a content, in order to be quantified. In other words, Fichte states, quantity presupposes in general the quality as determining and governing it. Transition into quality, he adds, is, like every dialectical step forward, also a return to more essential or a conscious highlighting of an implicit presupposition. Quality is therefore not deduced from quantity, since this would mean deriving more full and real from the empty and abstract. Instead, Fichte insists, the new concept appears, when something hidden in the earlier conceptual context is raised to consciousness by showing how the earlier concept in its purity and isolation leads to the next concept. Because of this isolation, Fichte ends the chapter, the new thought is at first still the negation of the previous, until the following conceptual level mediates both and collects them in a common higher expression.
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lauantai 31. elokuuta 2024
sunnuntai 25. elokuuta 2024
Immanuel Hermann Fichte: Outline of a system of philosophy. Second division: Ontology – Measure
Last time, Fichte had reached the concept of allness, by which he meant a certain determined collection of ones. Such all, he continues, is then externally limited: it is these ones we are talking about, not those others. Internally the ones collected together are still left undetermined, but they are still distinguished from other ones through an external limitation within the whole field of quantity. Indeed, the essential character of such an external limitation, Fichte says, is that the limiting distinction of quantities does not arise from any internal qualities. Just like all quantities, this quantitative limitation can also be described as continuous, and then it could be called, Fichte suggests, a measure. This measure, he insists, should be taken as a determination of something: measure as the determination gives something its limitation, but the limitation is here understood just externally or quantitatively.
Measure, Fichte begins, is at first completely indifferent toward any qualitative distinctions of what is measured. In other words, the measured content is in general just continuous and undistinguished. Fichte's examples of such a measure are an hour of time and a quarter of a foot, where it is indifferent what period of time or length is measured and the distinction of this time and length from others seems completely arbitrary. Measure is thus an arbitrary quantitative limit for something that remains similar or continuous, because any possible qualitative distinctions have been ignored. Measure, Fichte continues, is one of the most comprehensive, but also most abstract determinations of thinking, because it leaves the measured content undetermined. It is a comprehensive determination, he explains, because having a limited magnitude is an essential condition for being something. Thus, just like everything is numerical in the sense of being at least one, everything has also essentially a limit or measure and having this measure makes something into this. At this stage, Fichte notes, all determinations appear completely quantitative, thus, here all qualitative remains still beyond.
Fichte reminds us that all previous quantitative determinations have shared an indifference toward content and thus did not just limit things, but also immediately cancelled this limit. In other words, all quantities could be increased or decreased. Now, Fichte notes, same is true also of measure: every given measure could again be cancelled or made more or less, just because there is no internal limitation through a specific content. When such an abstractly thought measure is increased or decreased without any limit, or more precisely, by cancelling every given limit, Fichte says, it becomes its own opposite or measureless. Thus, it becomes more and more evident that the concept of measure must be complemented with some internal determination.
In the superficial increasing or decreasing of measure, Fichte thinks, is still found the contradiction that a relation to the content is ignored. Measure is cancelled through such external increasing or decreasing and reduces to more abstract determination of magnitude in general. On the other hand, determination of content would limit the increasing and decreasing and make measure more stable. This means, Fichte explains, that only a specific something should have a determined measure that cancels all increasing and decreasing. So, in the concept of measure, we for the first time have determined and not abstract magnitudes.
A determined measure cannot then be arbitrarily changed without any limit. This does not mean that it cannot be changed at all, Fichte explains, but only that it cannot be increased or decreased without changing the measured content. Change is still an essential moment of this concept, because it remains a quantity and the positing and cancelling of limits forms the character of quantitative. Compared with the previous levels, here the change of measure is linked to something lying beyond merely quantitative or to the content. This content, Fichte thinks, itself cancels its measure and so finds through its own internal progress another measure. In other words, change of measure is only the quantitative expression of this internal development. Indeed, Fichte points out, since quantity in itself is meaningless, the highest and most developed forms of quantity are mere expressions of something else.
By introducing determination to quantity, Fichte remarks, we encounter again the familiar relation of opposition of determined against its other and here especially against otherwise determined measures: every thesis of a determination is inevitably combined to an antithesis. This opposition means, Fichte explains, firstly, that the determined measure is determined only as a negation against another determined measure beyond it, and similarly this other measure is just a negation of the first. Secondly, as a variable measure it is also a negation of itself, that is, it has a tendency to change into a different measure and thus in a sense oppose itself. Indeed, Fichte points out, the change of measure shows both forms of opposition: a changing measure is opposed to both itself and to other determined measures.
What could distinguish a measure from another – or in case of a changing measure, from itself – is the content, Fichte says, but we still abstract here from it. Thus, we have only a quantitative difference that can appear merely by comparing quantities with one another or by putting them in relation. Hence, Fichte argues, measure is determined only in a synthetical relation to other, equally determined measures. This synthesis behind the determined measures is the common sphere of measures comparable to one another. With this comparison of determined measures, Fichte suggests, returns the distinction or the discreteness of magnitudes, because the synthetic element or the measure for comparing and so distinguishing measures is once again number (Zahl), or at this stage, amount (Anzahl). This amount, he explains, is the same measure, but now understood as a discrete collection of units: determinations and changes of measures can be compared to one another only according to numeric relations.
The basis of amount, Fichte continues, is one or unit that is used as a common yardstick for all measures, and the different ones or units are collected into the amount. This sounds like what happened in collecting ones into a number, but Fichte finds a difference: for abstract numbers, the content is completely indifferent or anything that could be just abstractly distinguished could also be numbered. Amount, on the other hand, is expressly connected to the content, that is, its ones or units are all expressly similar to one another: only things that can be regarded as similar can be collected into an amount. Thus, Fichte emphasises, just like the predominantly continuous determined measure contains a moment of discreteness, similarly the predominantly discrete amount contains a moment of continuity – because the units comprehended in the amount are similar, we can regard the amount also as continuous or as determined measure, just like a determined measure can be thought or measured only as numbered or as amount. Determined measure and amount are thus only one another complementing expressions of the same limited magnitude.
Determined measure and amount, Fichte reminds the reader, are to be thought as determined only in relation to other equally determined measures or in a system of measures and numbers. Thus, he argues, limited magnitude is not like it appeared at first: it cannot be arbitrarily increased or decreased, because every limitation expresses a determined relation to other magnitudes, which measure or quantitatively determine one another in a reciprocal relation and only thus gain their limit or determination. Such a relation to a system of other measures, Fichte thinks, is infinite, but not anymore in the superficial sense that every posited limit can be cancelled or that every division of a quantity can be divided further. Instead, he calls it a positive infinity that lies in the relation of numbers in general: just like this is this only as opposed to an infinitely different this, similarly every determined magnitude is determined only as opposed to every other magnitude.
Fichte has argued that it is the relation between magnitudes that determines these magnitudes and makes them more than just abstractions. This means, he suggests, that determinations that the magnitudes have in this relation to one another should remain the same, although the numeric expressions of the magnitudes and their determinations would change. Fichte is evidently speaking here of the mathematical notion of a variable being a function of another variable: the function should remain the same, no matter what numeric expressions the variables should have. In an uncareful fashion he suggests that all these relations or functions are reducible to six basic calculations that form three pairs of oppositions: in the first pair, magnitudes are taken as collections of units that are either combined (in addition) or separated (in subtraction), in the second pair, either a new magnitude is assembled by using a given magnitude as a unit for the amount corresponding to another magnitude (in multiplication) or such an assembled magnitude is disassembled (in division), and finally, in the third pair, either one and the same number is taken as both a unit and an amount of multiplication (in exponentiation) or the original number for the result of such exponentiation is searched for (in taking roots).
Every determined magnitude could now be called a quantitative thesis, Fichte says, in the sense that these magnitudes as mutually measuring one another are at first placed in a positive relation with one another, expressed through some equation like x + y = 5 or 2x = y (Fichte explicitly speaks only of additions and subtractions or magnitudes having a determined result and of cases where one magnitude is multiplied with a number to form the other magnitude). Now, every thesis should lead to a further antithesis, by which Fichte here means cases where the two magnitudes are in a converse relation of the sort xy = a, where the increase of x leads to y decreasing and vice versa. Indeed, he points out, the position and negation are immediately connected, when we note that there are three magnitudes that we are speaking of: in equation xy = a, y is in a direct, “positive” relation to a (when y increases, a increases also), but in a converse, “negative” relation to x.
No thesis and antithesis without a synthesis, Fichte is eager to point out. Indeed, the synthesis should already be implicitly contained in the antithesis and thus needs only to be made explicit for the consciousness. This happens, Fichte suggests, when we make a magnitude have both a positive and negative relation to itself: in other words, when in the equation xy = a we assume that x = y. In effect, we are now moving to equations involving exponentiation, which as the phase of synthesis should then be the highest expression of numeric relations. What Fichte sees in exponentiation – and here he is closely following Hegel – is a case where a variable magnitude that despite being increased or decreased still retains a more qualitative relation to another variable magnitude (one could think here of quantities of different dimensions). We thus now enter the final stage in Fichte’s study of quantities, which is concerned of magnitudes and their variability as governed not just by their quantitative relations to other magnitudes, but by their own internal determination.
Measure, Fichte begins, is at first completely indifferent toward any qualitative distinctions of what is measured. In other words, the measured content is in general just continuous and undistinguished. Fichte's examples of such a measure are an hour of time and a quarter of a foot, where it is indifferent what period of time or length is measured and the distinction of this time and length from others seems completely arbitrary. Measure is thus an arbitrary quantitative limit for something that remains similar or continuous, because any possible qualitative distinctions have been ignored. Measure, Fichte continues, is one of the most comprehensive, but also most abstract determinations of thinking, because it leaves the measured content undetermined. It is a comprehensive determination, he explains, because having a limited magnitude is an essential condition for being something. Thus, just like everything is numerical in the sense of being at least one, everything has also essentially a limit or measure and having this measure makes something into this. At this stage, Fichte notes, all determinations appear completely quantitative, thus, here all qualitative remains still beyond.
Fichte reminds us that all previous quantitative determinations have shared an indifference toward content and thus did not just limit things, but also immediately cancelled this limit. In other words, all quantities could be increased or decreased. Now, Fichte notes, same is true also of measure: every given measure could again be cancelled or made more or less, just because there is no internal limitation through a specific content. When such an abstractly thought measure is increased or decreased without any limit, or more precisely, by cancelling every given limit, Fichte says, it becomes its own opposite or measureless. Thus, it becomes more and more evident that the concept of measure must be complemented with some internal determination.
In the superficial increasing or decreasing of measure, Fichte thinks, is still found the contradiction that a relation to the content is ignored. Measure is cancelled through such external increasing or decreasing and reduces to more abstract determination of magnitude in general. On the other hand, determination of content would limit the increasing and decreasing and make measure more stable. This means, Fichte explains, that only a specific something should have a determined measure that cancels all increasing and decreasing. So, in the concept of measure, we for the first time have determined and not abstract magnitudes.
A determined measure cannot then be arbitrarily changed without any limit. This does not mean that it cannot be changed at all, Fichte explains, but only that it cannot be increased or decreased without changing the measured content. Change is still an essential moment of this concept, because it remains a quantity and the positing and cancelling of limits forms the character of quantitative. Compared with the previous levels, here the change of measure is linked to something lying beyond merely quantitative or to the content. This content, Fichte thinks, itself cancels its measure and so finds through its own internal progress another measure. In other words, change of measure is only the quantitative expression of this internal development. Indeed, Fichte points out, since quantity in itself is meaningless, the highest and most developed forms of quantity are mere expressions of something else.
By introducing determination to quantity, Fichte remarks, we encounter again the familiar relation of opposition of determined against its other and here especially against otherwise determined measures: every thesis of a determination is inevitably combined to an antithesis. This opposition means, Fichte explains, firstly, that the determined measure is determined only as a negation against another determined measure beyond it, and similarly this other measure is just a negation of the first. Secondly, as a variable measure it is also a negation of itself, that is, it has a tendency to change into a different measure and thus in a sense oppose itself. Indeed, Fichte points out, the change of measure shows both forms of opposition: a changing measure is opposed to both itself and to other determined measures.
What could distinguish a measure from another – or in case of a changing measure, from itself – is the content, Fichte says, but we still abstract here from it. Thus, we have only a quantitative difference that can appear merely by comparing quantities with one another or by putting them in relation. Hence, Fichte argues, measure is determined only in a synthetical relation to other, equally determined measures. This synthesis behind the determined measures is the common sphere of measures comparable to one another. With this comparison of determined measures, Fichte suggests, returns the distinction or the discreteness of magnitudes, because the synthetic element or the measure for comparing and so distinguishing measures is once again number (Zahl), or at this stage, amount (Anzahl). This amount, he explains, is the same measure, but now understood as a discrete collection of units: determinations and changes of measures can be compared to one another only according to numeric relations.
The basis of amount, Fichte continues, is one or unit that is used as a common yardstick for all measures, and the different ones or units are collected into the amount. This sounds like what happened in collecting ones into a number, but Fichte finds a difference: for abstract numbers, the content is completely indifferent or anything that could be just abstractly distinguished could also be numbered. Amount, on the other hand, is expressly connected to the content, that is, its ones or units are all expressly similar to one another: only things that can be regarded as similar can be collected into an amount. Thus, Fichte emphasises, just like the predominantly continuous determined measure contains a moment of discreteness, similarly the predominantly discrete amount contains a moment of continuity – because the units comprehended in the amount are similar, we can regard the amount also as continuous or as determined measure, just like a determined measure can be thought or measured only as numbered or as amount. Determined measure and amount are thus only one another complementing expressions of the same limited magnitude.
Determined measure and amount, Fichte reminds the reader, are to be thought as determined only in relation to other equally determined measures or in a system of measures and numbers. Thus, he argues, limited magnitude is not like it appeared at first: it cannot be arbitrarily increased or decreased, because every limitation expresses a determined relation to other magnitudes, which measure or quantitatively determine one another in a reciprocal relation and only thus gain their limit or determination. Such a relation to a system of other measures, Fichte thinks, is infinite, but not anymore in the superficial sense that every posited limit can be cancelled or that every division of a quantity can be divided further. Instead, he calls it a positive infinity that lies in the relation of numbers in general: just like this is this only as opposed to an infinitely different this, similarly every determined magnitude is determined only as opposed to every other magnitude.
Fichte has argued that it is the relation between magnitudes that determines these magnitudes and makes them more than just abstractions. This means, he suggests, that determinations that the magnitudes have in this relation to one another should remain the same, although the numeric expressions of the magnitudes and their determinations would change. Fichte is evidently speaking here of the mathematical notion of a variable being a function of another variable: the function should remain the same, no matter what numeric expressions the variables should have. In an uncareful fashion he suggests that all these relations or functions are reducible to six basic calculations that form three pairs of oppositions: in the first pair, magnitudes are taken as collections of units that are either combined (in addition) or separated (in subtraction), in the second pair, either a new magnitude is assembled by using a given magnitude as a unit for the amount corresponding to another magnitude (in multiplication) or such an assembled magnitude is disassembled (in division), and finally, in the third pair, either one and the same number is taken as both a unit and an amount of multiplication (in exponentiation) or the original number for the result of such exponentiation is searched for (in taking roots).
Every determined magnitude could now be called a quantitative thesis, Fichte says, in the sense that these magnitudes as mutually measuring one another are at first placed in a positive relation with one another, expressed through some equation like x + y = 5 or 2x = y (Fichte explicitly speaks only of additions and subtractions or magnitudes having a determined result and of cases where one magnitude is multiplied with a number to form the other magnitude). Now, every thesis should lead to a further antithesis, by which Fichte here means cases where the two magnitudes are in a converse relation of the sort xy = a, where the increase of x leads to y decreasing and vice versa. Indeed, he points out, the position and negation are immediately connected, when we note that there are three magnitudes that we are speaking of: in equation xy = a, y is in a direct, “positive” relation to a (when y increases, a increases also), but in a converse, “negative” relation to x.
No thesis and antithesis without a synthesis, Fichte is eager to point out. Indeed, the synthesis should already be implicitly contained in the antithesis and thus needs only to be made explicit for the consciousness. This happens, Fichte suggests, when we make a magnitude have both a positive and negative relation to itself: in other words, when in the equation xy = a we assume that x = y. In effect, we are now moving to equations involving exponentiation, which as the phase of synthesis should then be the highest expression of numeric relations. What Fichte sees in exponentiation – and here he is closely following Hegel – is a case where a variable magnitude that despite being increased or decreased still retains a more qualitative relation to another variable magnitude (one could think here of quantities of different dimensions). We thus now enter the final stage in Fichte’s study of quantities, which is concerned of magnitudes and their variability as governed not just by their quantitative relations to other magnitudes, but by their own internal determination.
sunnuntai 16. heinäkuuta 2023
George Peacock: Treatise on Algebra (1830)
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| (1791–1858) |
What Peacock is particularly interested in is the question whether algebra should be somehow based on arithmetics, which is based on concrete groups of objects, like seven apples. Peacock’s answer is a resounding no: arithmetics is far more constrained, allowing e.g. no negative numbers, since you cannot produce a group of, say, minus seven apples, although negative numbers are a common occurrence in algebra.
Not restricted by characteristics of arithmetical numbers, what is to determine what rules are to be assumed in algebra? Peacock makes the bold suggestion that these rules are just assumed: algebra concerns only symbols, and we could in principle choose any rules to govern our calculations in it.
Still, Peacock does not yet reach the modern notion of algebra, where we can have many different systems of calculation with different rules. Instead, he thinks that algebra should be especially the most general system of calculation, where all sorts of calculation are possible. Thus, he accepts not just calculations leading to negative numbers, but also roots of negative numbers, which we nowadays call imaginary numbers. Peacock goes even so far as to suggest that such seemingly nonsensical symbols like 0/0 could be used in a meaningful manner (although he mentions this possibility only in passing, he is referring to certain readings of the infinitesimal calculus).
Peacock still wants that algebra would be of practical use. Here, he suggests that the rules of algebra should at least correspond to the rules of arithmetics at least in those cases where the calculations and their results make arithmetical sense. Then again, he continues, results of algebra could also be used in other mathematical sciences, where some of them might be of relevance. Thus, negative numbers do make sense, for instance, when speaking of debt or of movement to an opposite direction. Even imaginary numbers can have concrete meaning, as describing movement perpendicular to a line.
Algebra becomes then, for Peacock, like a general tool for mathematical problem solving. When a concrete problem is given, it is turned into algebraic symbols: what quantities are known and what are unknown that are supposed to be determined in terms of the known quantities? After doing the algebraic manipulation of the symbols, one must still interpret the results. Often the context of the problem restricts this interpretation, and although algebra would lead to a number of possible results, only some of them might make sense in the context. Sometimes no result suggested by algebra would make sense, and then the problem itself will be impossible.
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.
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