Chapter 18 B. Quantity (Die QuantitaBt) Ⅲ. Degree (Grad)

§103 The limit is identical with the totality of the quantification itself.As multiple, the limit itself is the quantity (or extensive quantity) of extension, but as a simple stipulation, the limit itself is the quantity (or deep quantity) or degree of connotation. [Explanation] The difference between continuous quantity and separate quantity is different from denotative quantity and connotative quantity. This difference lies in the fact that the former relates to general quantity, while the latter relates to the limit of quantity or the specificity of quantity itself.The quantity of extension and the quantity of connotation are also not two different quantities, one of which never contains the other's stipulation; any quantity of extension is also a quantity of connotation, and any quantity of connotation is also a quantity of extension.

Note: The quantity or degree of connotation is, by its nature, different from the quantity or quantification of denotation.Therefore, as often happens, someone who does not recognize this distinction, and equates the two forms of magnitude without taking it into consideration, must point out that this is not permissible.In physics, there is no distinction between the two. For example, when physics explains the difference in specific gravity, it says that if one object has twice the specific gravity of another object, the material molecules contained in the same space ( or atoms) will be twice as numerous as the other object.The same is true with respect to the specific gravity of heat and light, if a greater or lesser number of particles (or molecules) of heat and light is used to account for different degrees of temperature or brightness.Physicists who adopt this interpretation, when their claims are denounced as unfounded, no doubt often defend themselves by saying that they do not attempt to explain the (notoriously unknowable) "freedom" behind those phenomena. things], they use the above terms purely for reasons of convenience.The so-called more convenient means that it is easier to calculate; but it is difficult for us to understand why the intrinsic quantity is not as easy to calculate as the extensional quantity, since it also has its definite number.If the purpose is purely for convenience, then it is most convenient not to calculate or think at all.Moreover, it is sufficient to object to the apology of the physicists just mentioned, that an explanation of their kind goes in any case beyond the sphere of perception and experience, and touches that of metaphysics and speculation, which they sometimes declare It is frivolous or even dangerous speculation.It has certainly been seen in experience that if two purses full of money, one of which weighs twice as much as the other, must be due to the fact that one purse contains two hundred dollars and the other only Contains one hundred dollars.We can see these coins and feel them with our senses.Conversely, atoms and molecules and the like are outside the sphere of sensory perception, and only the mind can decide whether they are acceptable or meaningful.But (as mentioned above in the footnote to §98) the abstract intellect fixes in the form of atoms the moment of the plurality contained in the concept of being-for-itself, and insists on it as a last principle.The same abstract intellect, in the present problem, contradicts the naive intuition and the real and concrete thinking. It recognizes the quantity of extension as the only form of quantity, and does not recognize its specific stipulations for the quantity of connotation. Unreliable assumptions try to attribute the quantity of connotation to the quantity of extension in a crude way.

Among the many criticisms of modern philosophy, one of the more frequently heard charges is that modern philosophy reduces everything to the same.Hence modern philosophy has been dubbed the same philosophy.But the discussion presented here consists in pointing out that philosophy alone insists on a distinction between conceptually and empirically different things, whereas those who call themselves empiricists elevate abstract identity to the highest principle of cognition.Therefore, only their narrowly empiricist philosophy can most properly be called identity philosophy.Furthermore, it is quite correct to say that there is no purely extensive quantity, nor a purely connotative quantity, just as there is no simple continuous quantity, nor a simple separate quantity, and that these two determinations of quantity It is not two independent quantities that are opposed to each other.Every intensional quantity is also extensional, and conversely, every extensional quantity is also intensional.For example, a certain degree of temperature is a connotative quantity to which there is an entirely simple feeling.If we look at the thermometer, we can see that this degree of temperature corresponds to a certain expansion of the mercury column.The amount of this extension varies with temperature or the amount of extension at the same time.The same is the case in the spiritual world: a character with a greater content acts on a wider range than a character with a lesser content.

§104 In the degree, the concept of quantification is set up.Quantitative is a self-neutral and simple quantity, but in this way, the stipulation that makes quantity quantitative is completely outside of it, in other quantities.This is the contradiction in which the self-existing, neutral limit is absolute externality, and infinite quantitative progress is posited. ——This is a process of directly changing from immediacy to its opposite, into indirectness (that is, beyond the quantification just set up), and vice versa, it is also a process of directly changing from indirectness to its opposite, into indirectness. direct process.

[Explanation] Numbers are thoughts, but as a thought that exists completely outside itself.Since number is thought, it does not belong to intuition, but is a thought whose determination is the externality of intuition. —Thus not only can the quantity be increased or decreased to infinity, but the quantity itself, by virtue of its conception, constantly exceeds itself outwardly.The progress of infinite quantity is nothing but the meaningless repetition of the same contradiction, which is quantification in general, and degree when its determination is brought into play.It would be superfluous to speak of the contradictions of this infinitely progressive form.

Aristotle quotes Zeno well on this point: "It is the same to say something once and to say it forever." Note 1: If we regard the general definition of quantity in mathematics as stated above (§99) as something that can be increased or decreased, no one can deny the correctness of the view on which this definition is based, but the problem It still depends on how we understand this kind of things that can be increased or decreased.It would be unsatisfactory if our answer to this question were simply to have recourse to experience, since, except in experience, we have only representations of quantities, not ideas, which would only be shown as a possibility ( increase or decrease), while we would have no real insight into the necessity of quantitative change.On the contrary, in the process of logical development, quantity is not only recognized as a stage of the thinking process which determines itself, but the fact also shows that the concept of quantity contains a necessity beyond itself. Therefore, what we have here The increase or decrease of the amount discussed is not only possible, but inevitable.

Note 2: The infinite progress of quantity is always insisted upon by the reflective understanding, and used to discuss the question of infinity.But for this form of infinite progress, what we said earlier in the discussion of qualitative infinite progress applies just as well.We have said that such infinite progress does not express true infinity, but only bad infinity.It never goes beyond mere ought, and so remains practically limited.This quantitative form of infinite progress is what Spinoza rightly called infinity imaginationis, which is only an imagination.Many poets, such as Harrell and Kropstock, often use this representation to vividly describe the infinity of nature, and even describe the infinity of God himself.For example, we find Harrell in a famous poem about the infinity of God, saying:

We pile up gigantic numbers, mountain after mountain, ten thousand after ten thousand, worlds I pile worlds, time I add time, when I look up at you from the dreadful peak,— With dazed eyes: The power of all numbers, multiplied thousands of times, is still far away from a part of you. Here we first encounter quantity, especially number, which constantly surpasses itself, which Kant described as "terrifying".In fact, the real horror lies in the boredom of forever setting boundaries and always exceeding them without making any progress.The poet mentioned above, after his description of the infinity of badness, adds the epilogue:

I get rid of their entanglement, and you appear in front of me whole. This means that true infinity cannot be regarded as something purely beyond finite things, and that we must give up progressus in infinitum if we want to gain awareness of true infinity. Attachment 3: As you all know, Pythagoras once thought about numbers philosophically, and he believed that numbers are the fundamental principle of all things.Such a view may at first appear to ordinary consciousness to be a complete paradox, even nonsense. The question of what is number then arises.To answer this question, we must first remember that the task of philosophy as a whole is to trace things back to thought, and to definite thought.But number is undoubtedly a thought, and the thought closest to sense-things, or rather, number is the thought of sense-things themselves, insofar as we understand them as being distinct and plural.Thus we find the first step to metaphysics in our attempt to explain the universe as numbers.In the history of philosophy, everyone knows that Pythagoras stood between the Ionian philosophers and the Eleatic philosophers.The former, as pointed out by Aristotle, still stays in the theory that the essence of things is matter (JBFη), while the latter, especially Parmenides, has progressed to "existence" as "form". ’, so it is the principle of Pythagorean philosophy that forms, as it were, a bridge between sensuous and supersensible things.

From this we can know why some people think that Pythagoras's theory of number recognition as the essence of things is obviously going too far.They admit that we can count things, but they argue that there is more to things than counting.Saying that things have something more than number, of course anyone can admit that things are not just numbers, but the problem is only how to understand what this thing more than number is.Ordinary sense-consciousness, according to its own point of view, does not hesitate to point to the perceptual side of the senses in order to answer the question posed here, thus saying: things are not only countable, but also visible, smellable, palpable, etc. Wait.In modern language, their criticism of Pythagoras' philosophy can be boiled down to one point, that is, his theory is too idealistic.But in the light of what we have just said about the place of Pythagorean philosophy in history, the opposite is in fact the case.We must admit that things are not only numbers, but this should be understood as the idea of ​​simple numbers is not enough to fully express the concept or specific essence of things.Therefore, rather than saying that Pythagoras' philosophy on number has gone too far, it is rather to say that his philosophy has not gone far enough, and it was not until the Eleatic school further reached the philosophy of pure thinking.

In addition, even if there is no thing itself, there will be the state of things and general natural phenomena, and their stipulations are mainly based on specific numbers and the relationship between numbers.The combination of the difference of sounds and the harmony of tones has the specificity of numbers.Everyone knows that it is said that Pythagoras recognized number as the essence of things because of the enlightenment he got from observing the phenomenon of tones.Although it is extremely important for scientific research to trace the phenomenon of tones back to the specific numbers on which they are based, it must not be allowed to regard the determinations of thought as mere determinations of numbers.It is true that people initially had a tendency to associate the most general rules of thought with the most basic numbers, so it is said that the first is pure and direct thought, the second represents the difference and indirectness of thought, and the third is the unity of the two.But this connection is entirely external, and the numbers themselves have no properties sufficient to represent these particular thoughts.The further one takes this method of conceiving, the more arbitrary will be the association of particular numbers with particular ideas.For example, people can think that 4 is the combination of 1 and 3, and also the combination of these two kinds of number thinking, but 4 can also be said to be twice as much as 2.Similarly, 9 is not only the square of 3, but also the sum of 8 and 1, 7 and 2, and so on.To think that a certain number or a certain figure is of extraordinarily important importance, as has been the practice of many secret societies lately, is certainly a pastime on the one hand, but on the other hand it is also a sign of intellectual weakness.People can certainly say that behind these numbers and figures, there are deep meanings that can arouse our many thoughts.But in philosophy the question is not what we can think, but what we actually think.The true element of thought is not in arbitrarily chosen symbols, but has only to be sought in thought itself. §105 Quantity is external to itself in its determinate nature of being-for-itself, and this external existence of it constitutes its quality.Quantity, in its external being, is itself and is itself related to itself.In quantification, externality (that is, quantity) and being-for-itself (that is, quality) are united.Quantities thus established within themselves are the proportions of quantities—this determinateness being a direct quantity, the exponent of proportion, as an intermediary process, i.e., the connection of a certain quantity with another, forming a proportionality. Two ways.At the same time, these two aspects of proportion are not calculated according to their direct (numerical) values, but their (numerical) values ​​only exist in this proportional relationship. Note: The infinite progress of quantity seems at first to be number continually surpassing itself. But on closer inspection, quantity is shown to return to itself in the course of this progression.Because from the point of view of thought, the meaning contained in the infinite progress of quantity is generally only the process of determining numbers by numbers, and this process of determining numbers by numbers results in the proportion of quantities.For example, take 2:4 as an example, here we have two numbers, what we are looking for is not their direct values, but only the mutual relationship between these two numbers. But the connection between these two items (the exponent of the ratio) is itself a number, and the difference between this number and the two items in the ratio is that as soon as the number (ie, the index) changes, the ratio of the two items changes accordingly, and vice versa. , although the two terms are changed, their proportions are not affected, and as long as the index remains unchanged, the proportions of the two terms remain unchanged.We can therefore substitute 3:6 for 2:4 without changing the ratio, since the exponent 2 remains the same in both cases. §106 The two terms of proportion are still directly quantitative, and qualitative and quantitative determinations are still external to each other.But as far as the truth of quality and quantity is concerned: Quantity itself is related to itself in its externality, or the quantity that exists for itself is united with the quantity that is neutral to determination—so The quantity is the scale (Maβ). Note: Through the dialectical movement of each link of quantity examined above, it is proved that quantity returns to quality.We see that the concept of quantity is originally a sublated quality, that is to say, a quality that is not identical with "existence", and has nothing to do with "existence", and is only an external regulation.This conception of quantity is, as has been said, the basis of the ordinary mathematical definition of quantity, that is, of the conception of a quantity as something that can be increased or decreased.At first glance, this definition seems to mean that quantity is only something that can be changed in general (because being able to increase or decrease is just another way of saying quantity), so it may make the relationship between quantity and definite existence (the second stage of quality, as far as it is concerned) In essence, it can also be recognized as a variable) without difference.It may therefore be added to the content of the definition of quantity that in quantity we have a changeable thing which, though changed, remains the same.The concept of quantity thus involves an inherent contradiction.And this contradiction constitutes the dialectics of quantity.However, the result of the dialectics of quantity is not simply to return to quality, as if it is a concept that regards quality as true and quantity as false, but progresses to the unity and truth of both quality and quantity, and to the concept of quality. amount, or scale. Here we can also say that when we observe the objective world, we use the category of quantity.In fact the object we have in mind of this observation is always the acquisition of knowledge of scale.This point is often implied even in our daily language. When we want to know the nature and relationship of the quantity of things, we call it measure (Messen).For example, when we measure the lengths of different strings in vibration, it is with regard to knowing the qualitative difference in the pitch corresponding to the length of the string, which is caused by the vibration of each string.Likewise, in chemistry we try to ascertain the quantities of the various substances used in combination, and thereby find the dimensions governing these compounds, that is to say, to know those quantities which give rise to specific qualities.Again, in statistics, the numbers used in research are important only because of the qualitative consequences conditioned by those numbers.On the other hand, if it is just a collection of figures without the guiding ideas mentioned here, then it can be justifiably regarded as frivolous stuff, which can satisfy neither theoretical interest nor practical requirements.