Chapter 15 volume thirteen

Chapter One We have stated the noumenon and matter of sensible things in the "Materials" treatise, and later discussed noumenon with actualized existence.Now, the question for our research is: Is there any unchanging and eternal substance besides the sensible substance, and if there is such a substance, what kind of substance it is.We should take into account the opinions of each school. If there is a mistake in his sincere statement, we should seek to avoid the same flaws. If our intentions are not without similarities with those of the other schools and can be confirmed by each other, then we can also have no regrets. In terms of one's own arguments; people want to introduce the old and bring forth the new, so as to make their way known to the present, and wish to benefit from what the ancients have said.

There are two opinions on this question: or mathematical objects - such as numbers, lines, etc. - For the ontology; or that the meaning is the ontology.Because (1) some people think that Italian formulas and mathematical numbers belong to two different levels, (2) some people think that the two have the same nature, and (3) others think that only the mathematical ontology is the ontology, we must first study Whether mathematical objects exist, and if they exist, how they exist is studied. Whether these are actually ideas, whether they can be the principles and noumenon of ready-made things, and other characteristics are temporarily ignored.Later, we will conduct general discussions on Italian forms according to the general requirements; many arguments are already familiar to everyone in our discussions outside the courtyard, and most of our research here should focus on whether the noumenon and principles of existing things are numbers or not. The issue of Italian and Italian is definitely clarified; after discussing Italian, this remains the third topic.

If mathematical objects exist, they must exist, as some say, in sensible objects, or outside sensible things (this has also been said); and if neither exist, then they must either Whether they do not exist, or they exist with special meaning.So our topic is not their existence, but how they exist. Chapter Two To say that "mathematical objects exist independently of sensible things" is an artificial doctrine, which we have said in the discussion of difficult problems, is actually impossible.We have shown that it is impossible for two entities to occupy the same space, and by the same argument that other potentialities and qualities can only be contained in sensible things, and cannot be openly alone.We have already said this.According to this theory, it is also obvious that no substance can be separated; because the division of a substance must be in the surface, the surface must be in the line, and the line must be in the point. Followed by inseparable.What difference does it make if these are sensible objects, or if they are not sensible objects themselves, but participate in them?The result is the same; if the sensible objects are differentiated, the objects participating in it must also be differentiated; otherwise, the sensible reality cannot be differentiated and becomes an independent mathematical reality.

But, again, such a reality cannot exist independently.If, besides sensible solids, there are solids separate from them and preceding them, there must be other separate planes besides planes, and so do points and lines; and this makes sense.However, if these exist, there must be more separated surface line points besides the surface line points of the mathematical solid. (Since simples must precede composites, just as sensible solids precede sensible solids, so, by the same argument, free-standing surfaces must precede fixed solids. These surface lines will therefore be what those thinkers would call Another set of surface lines than the mathematical surface lines on the quasi-mathematical solid; the surface lines on the mathematical solid are with this solid, and the other set will exist before the mathematical solid surface.) Then, according to the same argument, In addition to these innate surface lines, there must be line points prior to them;

Beyond these antecedent points, there are points that precede them, and beyond the point that precedes and precedes, there is nothing else.Now (1) here the area is rather absurd; for we have incurred another set of solids besides the sensible ones; three sets of planes,— There is one set apart from the perceivable three-dimensional, one set on the mathematical three-dimensional body, and one set free from the mathematical three-dimensionality; four sets of lines, and five sets of points.So which set should mathematics study?Certainly not the point of the upper line that exists on a fixed solid body; for learning always precedes things. (ii) The same reasoning applies to numbers; besides every set of points there may be another set of units, besides every set of existing things there may be another set of sensible numbers, and besides sensible numbers another set of A set of ideal numbers; according to the continuous gain, there will be endless number systems of different levels.

Furthermore, how does this answer the difficult questions we have listed before?For astronomical objects, like geometric objects, will exist independently of sensible things; But how can a universe and its parts—or any other thing in motion—exist independently of what is?Similarly, optical (spectacle) and acoustic (music) objects must also have their own independent existence; this means that there must be sound and light other than the individual sound and light and shadow that can be seen and heard.Obviously, then, the same should be true for other senses, and each of the other sense objects has its own independent set; how can it be so for one sense but not for another?But if this is the case, there will still be animals which can exist elsewhere, since there are also sensations.

Also, the development of a certain universal theorem of mathematics has surpassed these ontology.Here we will have another set of intermediate noumenon besides Italian form and interbody—this noumenon is neither number, nor point, nor space measurement, nor time.If this is impossible, it is evident that none of the previously established realities apart from sensible things can exist. If one can admit the existence of mathematical objects as such independent realities, this leads, generally speaking, to conclusions contrary to truth and custom.These, if they exist, must precede the sensible measures of space, but in fact they must follow; for the unfinished measures of space precede in the process of creation, but follow in the noumenal order, Just as inanimate things should come after living things.

Also, when will the mathematical measurement become one, and where will it be unified?In our perceivable world, things are one by the soul, or by a part of the soul, or by other rational things; .But mathematical things are originally distinguishable measurements, so what is the reason for them to hold together and become one? Again, the way in which mathematical objects are created proves our thesis to be true.Measurement first creates length, then width, and finally depth, thus completing the creation process.If what is subsequent to the creative process should be prior to the noumenal order, then the original will be prior to the surface and the line.

In this way, the body is also more complete because the body can become a living thing.Conversely, how can a line or a surface come alive?Such assumptions are beyond our sensory capacity. Again, the three-dimensional is a kind of ontology; because this can already be called "complete".But how can a thread be called a substance?Lines can neither be regarded as form or appearance, like the soul, nor matter, like solid bodies; for we have no experience of putting lines or planes or points together to make anything; if these are a kind of substance , then we see things made up of them.

So try to make them precede by definition.This still does not mean that everything that is prior to definition should be prior to ontology.For all things to be prior to other things in substance, their ability to exist independently should exceed other things after they are separated from other things; <formulas> are combined by their definition <formulas>; these two properties do not have to coincide.Attributes such as a "moving" or a "white", if not separated from the ontology, "white" will be prior to "white" in definition, but will be behind in ontology.

Because the attribute of "whiteness" can only exist with the composite entity I refer to as "white people", and cannot exist independently of it.So it is understood that things obtained by abstraction cannot be prior, and things obtained by adding a decisive noun may not be behind; when we say "white people", we add a decisive noun <person> to "white". This, then, sufficiently shows that mathematical objects are not higher substances than substances, that they are, as reality, only prior in definition, and not prior to sensible things, nor can they exist independently anywhere.But since these do not exist both inside and outside the sensible things, it is clear that they should not exist at all, or only exist in a special sense; "existence" originally has many meanings.So they don't exist in full. Chapter three Just as the general propositions of mathematics do not study those measurements and numbers that are extended out of reality and are considered to be independent objects, the two researches are still measurements and numbers, but these measurements and numbers are no longer as the objects that are quantitative and measurable. Distinguishing primordials, obviously, may also have certain propositions and demonstrations of sensible measure, which are not concerned with the sensibility of primordials, but with some other quality.There are many propositions which are devoted to the motion of things, whatever they may be in themselves, and whatever their accidental properties, it is not necessary here to separate pre-motion from sensible things, or Another movement is established in sensible things, and in this way treats things as substances in movement, either as surfaces, or as lines, either as distinguishable, or as indistinguishable with a place, or only Differentiate objects, but do not create another first-class movable object, which also establishes some propositions and obtains a lot of knowledge.So, since it can be said that it is entirely true that not only separable things exist, but also inseparable things (such as motion), it can be said that mathematical objects that mathematicians assign certain properties should also exist at all.And this can also be said unconditionally, that no other science is better than that each studies its subject as such—regardless of its contingent nature, (for example, medicine with health as its subject, if it has things about health and patients) is " It doesn’t ask whether it’s white or not, it just cares about its health,) each academic discipline only cares about its own theme—the study of health will study the part that can be regarded as the theory of health, and the study of human beings to study that part of things that can be regarded as anthropomorphic—and so is geometry; if its subject happens to be sensible things, though geometry is not studied for their sensibility, mathematics is not for that reason considered Misunderstood the science of sensible things.On the other hand, the study of things which are separate from sensible things is not to be misunderstood. Many characteristics are seen in things, often due to their own attributes; for example, animals have a special endowment to distinguish between male and female; (there is no "female" and "male" that can exist apart from animals) length or face What is seen in things is an attribute rather than a class.In like manner, the simpler and prior to the definition of things we study, the more precise our knowledge, that is to say, is simpler.Therefore, in abstract science, those separated from the measurement of space should be more accurate than those mixed with the measurement of space, and those separated from movement should be more accurate than those mixed with movement; The movement method is more precise because it is the simplest movement; and among the basic movement methods, the uniform, uniform and constant speed movement is the simplest. The same principle can also be applied to optics (painting) and acoustics (music); These two sciences do not study their objects as sight and sound, but as numbers and lines; but numbers and lines are precisely the special endowments of light and sound.The same is true for the study of mechanics. Therefore, if we separate the attributes of things from each other and study them separately, some people draw a line on the ground that is not a foot long and use it as a foot (foot) standard, we do this It is no more wrong than those; for the errors in it are not included in the presuppositions. Every problem is best considered in this way—as the mathematicians and geometers do, by separating the inseparable.Man as a person is an indistinguishable thing; arithmetic considers the properties of the man as an indistinguishable but countable thing.The geometer treats this person neither as a person nor as an indistinguishable thing, but as a solid.For obviously, even if he sometimes becomes non-indistinguishable, whatever is due to him (solidity) must belong to him, in addition to these properties (indistinguishability and humanity).The geometers, then, would be right in saying that he is a solid; it is true that they speak of existing things, and that the subject of which they speak actually exists; , and the existence of matter. Also, since the good and the beautiful are different (the good is always predominated in action, while the beautiful is also to be seen in inactive things), those who think that the sciences of mathematics and physics have nothing to do with either beauty or goodness are wrong.For Mathematics has much to say about beauty and goodness, and has done much to prove them; It cannot be said that mathematics and science do not involve beauty and goodness.The principal forms of beauty are "order, symmetry, and definiteness," to which only the sciences of mathematics and science are superior.And since these (such as order and clarity) are evidently the cause of many things, it is natural that the sciences of mathematics must also study causal principles of this kind which have beauty as their cause.We will discuss these issues in more detail elsewhere. Chapter Four Much has been said about mathematical objects; we have shown that mathematical objects exist, and in what sense they exist, in what sense they are prior, and in what sense they are not prior.Now, when we come to Italian forms, we should first consider the Italian theory itself, never getting involved with the nature of numbers, but concentrating on the original meanings as conceived by the founders of the Italian theory.The proponents of the Italian theory are introduced to the Italian because of the pursuit of the truth of things. They accepted the teachings of Heraclitus and described all sensible things as "perpetually passing away", so if cognition or thought The need for an object can only be sought in some other eternal reality than sensible things.Since all things are like flowing water without a moment's rest, it is impossible to desire to know something about it.At that time, Socrates concentrated on the analysis and debate of ethics and morality, and he first raised the problem of the general definition of ethical virtues.The early naturalist Democritus only made some superficial definitions of heat and cold in physics, and only occasionally touched on the problem of definition; as for the Pythagoreans, they had studied a few things before—— Like chance, morality, or marriage—definitions, they connect these things in numbers. This is natural. Socrates devoted himself to synthetic dialectics. He took "what is this" as the starting point of all theories (synthetic theory), and then explored how things are; because until this period, people did not have such an argument. It is not necessary to speculate on the opposites based on noumenal knowledge, and to inquire whether the opposites belong to the same science; Two great things can be attributed to Socrates—inductive speculation and general definitions, both of which concern the foundation of all learning.But Socrates did not separate the generality or the definition from the thing, but they (the Italianists) separated it and made it independent, and this is what they call the Italian.By roughly the same argument, this of course leads to the conclusion that everything generally said has a form, which is almost as if a person who wants to count things and finds that there are too few things to count, makes them increase on purpose. , and then count.In fact, the general formula is more than the individual sensible things, but when they seek the cause of things, they go beyond the things and go to the general formula.For a certain thing, there must be another entity with the same name that is separated from the ontology, (the same is true for other groups, there must be a "to unify the many" (general formula)), regardless of whether these "many" are temporal or supernatural things . Again, none of the methods used to prove the existence of general formulas is convincing; for some arguments do not necessarily lead to such conclusions, and some lead to general formulas in things that we usually think of as having no general formulas.According to this principle, how many academic disciplines all things belong to, there will be as many types of general formulas; according to this argument of "unifying many with one", even if it is negated ("nothing" or "non-being"), it will also have its general formulas; According to the principle that the thought of a thing does not perish when it perishes, we shall again have the general form of perishable things; for we retain the image of perishable things.In some rather clever debates, some have brought those things which are not independent classes into the sense of "relation", and others have led to "third person". Generally speaking, the arguments of general formulas annihilate things whose existence should be of more concern to those who believe in general formulas than the existence of the general formulas; One, not two <undetermined two> as the first, will be the relative number prior to the number, and more prior to the absolute number. ——In addition, there are other conclusions. People follow the development of Italian thinking, and they will inevitably conflict with the principles they hold earlier. Again, according to the assumptions from which we build up the forms, there should be general forms not only of noumena, but of many other things; The science of noumenal things; thousands of similar problems will follow.) But according to the claims of general formulas and the requirements of examples, if they can be participated, there should only be noumenal meanings, because their being participated It is not participating in attributes, but participating in the indefinable ontology. (To illustrate what I mean, for example, if a thing participates in the "absolute multiple", it also participates in the "eternal multiple", but this is incidental; for the multiple can only be "eternal" in its nature.) So The general formula will be the ontology.But the same noun refers to individual entities as well as to entities in the Italian world. (If not, then what is the true meaning of the noumenon in the so-called "one unites many" Italian world outside of individual things?) If the Italian form and the individual things participating in the Italian form have the same form, they will be There must be certain common characteristics. ("2" is the same in the destructible "2"s, or in the eternal "2", why is it not the same in the "absolute 2" (this 2) and "individual 2"?) But if they do not have the same form, they only have the same name, as if people call Gallia "man" and a piece of wood "man" without paying attention to the commonality between the two. However, if we assume elsewhere that ordinary definitions apply to general formulas, such as the definition of "plane circle" and other parts apply to "this circle" (Italian circle) and wait to add "what is this actually "<What makes this formula a formula>, we must ask whether this makes no sense at all.This addition would add to which element of the original definition?Complementary to "center" or "plane" or other parts of the definition?For all the elements of how to be (in Italian man) are Italian, such as "animal" and "two feet."Also, the meaning of "plane" is mentioned here, and "being a meaning" must conform to the meaning of being a family and genus, and being a family and genus is a certain property common to all varieties. Chapter five Finally, you can discuss this question, what effect does the general formula have on the sensible things in the world (whether they are eternal or arising and passing away at any time).For they can neither move nor change things.Nor do they contribute to cognition (for they are not substances of these things, in which they would have to exist), and if they do not exist in the individual things in which they participate, they may be regarded as The reason, such as "white" enters the composition of things, so that a white thing can become white <whiteness>.But this argument, first employed by Anaxagoras, and later by Eudoxus in his answer to the doubt, and by some others, is easily broken; defensible objections. Also, when it is said that all things evolve "from" the general formula, this "from" cannot be the ordinary meaning of the word.The general formula is a model, and other things participate in it, which is nothing more than poetic metaphor and fiction.Take a look at the Italian style, what exactly is it producing?There is no Italian model for things to copy, things will exist and become, whether there is a Socrates or not, a person like Socrates will always appear.Even if Socrates is transcendent and eternal, there will be such people in the world.The same thing can have several models, so it must also have several general formulas; for example, "animal", "two legs" and "human" are all general formulas of human beings.And the general formula is not only a model of sensible things, but also of the general formula itself, just as the genera, which are the families to which the varieties belong, become the genera to which the genera belong, so that the same thing will again be the family of the genera. The blueprint is a copy again. Also, it seems impossible to separate the noumenon from the location of the noumenon; then, since the form is the noumenon of things, how can it be independent from things? In Phaedo the problem is stated thus - the general formula is the cause of "is" (what is) and "will be" (becoming thing); but the general formula exists except for something else which moves , things that participate in the general formula do not come into being; yet many other things (such as a house or a ring) that they say have no general formula also come into being.Obviously, then, the cause of the above-mentioned things can also be the cause of the existence ("is") and the generation ("will be") of things with meaning, which they say, and things can be independent of The general formula depends on these reasons for its existence.With regard to Italian, it is possible to bring together many objections of this kind, or in a more abstract and precise point of view. Chapter Six Now that we have dealt with the problems of form, it is time to consider again those who maintain that number is a separable substance and a consequence of the first cause of things.If number is a reality, according to some people's claims, its essence is only number and nothing else, and then there should be <such various number systems>, (a) number can be (sub) first, second , one after the other is that each number is of a different kind—such numbers are without exception, each of which is incommensurable, or (ugly) they are sequential numbers without exception. , and any number, like what they call the number of mathematics (arithmetic), can communicate with any other number; in the number of mathematics, the units of each number are not different from each other.Or (yin) some of the units can be interlinked, some can not; for example, 2, assuming that the first order is after 1, so the order is 3, and the rest, the units in each number can be interlinked, for example, the first The units in the first 2 are interoperable, and so are the units in the first 3 and the rest of the numbers; but the units in the "absolute 2" (Ben 2) cannot communicate with the units in the Absolute 3 (Ben 3) The units are interoperable, and the numbers in the rest of the sequence are also similar. Mathematical numbers are counted like this - 1, 2 (which consists of another 1 followed by the previous 1), and 3 (which consists of another 1 followed by the first two 1s), the remainders are similar; The numbers in the formula are counted in this way—after 1, there is a distinct 2, which does not include the previous number, and the following 3 does not include the previous 2, and the remainder is similar.Or so, (B) one kind of numbers is like the one we first explained, the other is like the one described by mathematicians, and the last one we talk about should be the third kind. Again, the various systems of numbers must be either separable from things, or inseparable from objects of vision, (but this is not the way we have thought before, but only in the sense that objects of vision are contained in them composed of numbers)——either one kind is like this, the other kind is not as good, or all kinds are like this or neither is like this. These must be the only ways in which column numbers are possible.The Sunny School regards one as the origin of all things, the essence of all things, and the elements of all things, and the series of numbers are all composed of one and other things. The number systems they describe do not belong to the above-mentioned categories; it is just that all the numbers in them are not interoperable. The kind of number system that no one has claimed yet.This should be reasonable; besides the above-mentioned possible ways, there should be no other number systems.Some people say that there are two types of number systems, and that the successive numbers are different in kind and are the same as the Italian forms, while the numbers in mathematics are different from the Italian forms and different from the sensible things, and that the two types of number systems can be separated by the sensible things ; others say that only mathematical numbers exist, and that numbers, apart from sensible things, are the origin of realities.The Pythagoreans also believed that the number system was only of the category of mathematical numbers; but they believed that numbers are not separated from sensible things, and sensible things are composed of numbers.They constituted the whole universe out of numbers, and the numbers they used were not abstract units; they assumed that numbers had spatial dimensions.But how the first 1 can constitute a measure, this they do not seem to be able to explain. Another thinker said that only the numbers of general formulas, that is, number systems of the first kind, exist, and others said that the numbers of general formulas are the numbers of mathematics, and the two are the same. Lines, surfaces, and volumes are similar.There are those who mean that things as mathematical objects are different from their meanings; and among those who hold the contrary opinion, there are those who speak of mathematical objects only mathematically—those who do not count meanings, nor say that meanings exist; Others do not speak of mathematical objects in a mathematical way. They say that not every measure of space can be distinguished as measure, nor can two units be taken arbitrarily to make 2. People, except the Pythagoreans, think that numbers consist of abstract units; but, as mentioned above, they think of numbers as measures.It should have been stated how many ways there are without omission; all these claims are false, and some of them are more fanciful than others. Chapter seven So let us first study whether the units can be connected, and if so, which method should be adopted in the two methods we have discussed before. ⑦ It is possible that any unit is not connected with any unit, and it is also possible that the units in "Ben 2" and "Ben 3" are not connected. Generally, the units in each Italian number are not connected with other Italian numbers. of each unit.Now (1) if all the units are the same but interlinked, what we get is a mathematical number—there is only one series of numbers, and the Italian formula cannot be such a number. How can "human-like" and "animal-like" or any other meaning be such a number?Each thing has its own meaning, for example, people have "humanity", and animals have "animalism"; but there are infinitely many similar and undifferentiated ones, and any individual 3 must be regarded as "humanity" just like other 3s. ".But the Italian form would not exist at all if it could not be a number.What principle will the Italian style be derived from?The numbers derived from 1 and the undetermined 2 are just the principles and elements of numbers, and meanings cannot be listed as prior or subsequent to numbers.But, (2) If the units are not connected, no number is connected to any number, and such a number cannot be a mathematical number; since mathematical numbers consist of undifferentiated units, this property also proves to be true. .This also cannot be an Italian number.In such a number system, 2 will not be the first number generated by "one and undetermined two", and the other numbers cannot have the series sequence of "2, 3, 4...", because no matter whether it is like the Italian theory or not According to the originator of the Italian formula 2, the units in Italian formula 2 are derived simultaneously from the "unequal" (the "unequal" column is generated when it is balanced) or in other ways,-if one of them is If the other is prior to the other, this will be prior to the 2 combined; if one is prior to the other, the sum of the two will be prior to the other and then to the one. Also, because "Ben 1" is the first, after "Ben 1" there is an individual 1 that precedes the other 1s, and another individual 1 follows the number of units of the previous 1.Now (1) if all the units are the same but interlinked, what we get is a mathematical number—there is only one series of numbers, and the Italian formula cannot be such a number. How can "human-like" and "animal-like" or any other meaning be such a number?Each thing has its own meaning, for example, people have "humanity", and animals have "animalism"; but there are infinitely many similar and undifferentiated ones, and any individual 3 must be regarded as "humanity" just like other 3s. ".But the Italian form would not exist at all if it could not be a number.What principle will the Italian style be derived from?The numbers derived from 1 and the undetermined 2 are just the principles and elements of numbers, and meanings cannot be listed as prior or subsequent to numbers.But, (2) If the units are not connected, no number is connected to any number, and such a number cannot be a mathematical number; since mathematical numbers consist of undifferentiated units, this property also proves to be true. .This also cannot be an Italian number.In such a number system, 2 will not be the first number generated by "one and undetermined two", and the other numbers cannot have the series sequence of "2, 3, 4...", because no matter whether it is like the Italian theory or not According to the originator of the Italian formula 2, the units in Italian formula 2 are derived simultaneously from the "unequal" (the "unequal" column is generated when it is balanced) or in other ways,-if one of them is If the other is prior to the other, this will be prior to the 2 combined; if one is prior to the other, the sum of the two will be prior to the other and then to the one. Also, because "Ben 1" is the first, then after "Ben 1" there is an individual 1 that precedes the other 1s, and another individual 1 is actually the third 1 immediately after the previous 1, Then the two in sequence after the original 1,—then the units must be preceded by the order in which they point; for example, in 2 there is a third unit which precedes the 3, and the fourth and fifth units already exist. In 3, it exists before the two numbers 4 and 5.Although none of these thinkers said that the units are completely disconnected in this way, according to their principles, this is the case, although it is actually impossible.For it is reasonable that if there is a first unit or first 1, the units should be preceded and followed, and if there is a first 2, the 2s should also be preceded and followed. ; after the first, it is reasonable that there must be a second, and if there is a second, there must be a third, and the rest follow in sequence, unit next to the first 1, and it is impossible to say that 2 is the first 2 after the Italian 1), but they produced the first unit or the first 1 without There is a second 1 and a third 1, they made the first 2, but they no longer make the second 2 and the third 2. If all units are not connected, it is also clearly impossible to have "Ben 2" and "Ben 3"; the same is true for other numbers.Because no matter whether the units are undifferentiated or each is different, the number must be counted by addition. For example, 2 is added to 1, 3 is added to 2, and 4 is similar.Thus numbers cannot be created from "two" and "one" according to their way of making numbers; (according to addition) 2 becomes a part of 3, and 3 a part of 4, and so on in successive numbers, yet they say that 4 is made of the first A 2 is produced with the undetermined 2,—the product of the two 2s is different from the original 2; otherwise, the original 2 would be a part of 4 with another 2 added.Similarly 2 will consist of "this 1" plus another 1; if so, its other element cannot be "undetermined 2"; 2 creates a given 2 in that way. Also, how can there be other 3s and 2s besides Ben 3 and Ben 2?How are they composed of units preceding and following?All these are absurd fables, neither "Original 2" (the first 2) nor "Ben 3" (absolute 3) can be established.However, if "one and undetermined two" are taken as the elements, then these should all exist.If such results are impossible, it is also impossible to take them as principles of creation. These and similar results, then, must follow if the species of units are different.But (3) if only the units in each number are undifferentiated and communicate with each other, but the units in each number are differentiated and have different types, so the problem still exists.For example, there are ten units in this 10 <Italian 10>, and 10 can be composed of ten 1s, or two 5s.But "this 10" is not composed of any accidental units—the units in the 10 must be different.For, if they were not different, the two 5s that make up 10 would not be different either; but since the two 5s should be different, the units would also be different.However, if they are different, is there no 5 in 10 other than two 5s that are different?If there are no other 5s there, this becomes a paradox; if there are other kinds of 5s, what kind of 10s will the 10 composed of such 5s be?Because in the 10, there is only your own 10, and there is no other 10. According to them, 4 is certainly not composed of any accidental 2's; 他们说那未定之２接受了那已定之２，造成两个２；因为未定之２的性质１５就在使其所受之数成倍。 又，把２脱离其两个单位而当作一实是，把３脱离其三个单位而当作一实是，这怎么才可能？或是由于一个参与在别个之中，象"白人"一样遂成为不同于"白"与"人"（因为白人参与于两者），或是由于一个为别个的差异，象"人"之不同于"动物"和"两脚"一样。 又，有些事物因接触而成一，有些因混和而成一，有些因位置而成一；这些命意均不能应用那组成这２或这３的诸单位，恰象两个人在一起不是使之各解脱其个人而别成为整一事物，各单位之组成列数者意必同然。它们之原为不可区分，于它们作为数而论无关重要；诸点也不可区分，可是一对的点不殊于那两个单点。但，我们也不能忽忘这个后果，跟着还有"先于之２"与"后于之２"，它数亦然。就算４中的两个２是同时的；这些在８之中就得是"先于之２"了，象２创生它们一样，它们创生"本８"中的两４。因此，第一个２若为一意式，这些２也得是某类的意式。同样的道理适用于诸１；因为"第一个２"中的诸１，跟着第一个２创生４而入于本４之中，所以一切１都成意式，而一个意式将是若干意式所组成。所以清楚地，照这样的意式之出于组合，若说有动物的诸意式时，人们将可说动物是诸动物所组成。 总之，分化单位使成不同品种之任何方式均为一荒唐之寓言；我所说寓言的意义，就是为配合一个假设而杜撰的说明。我们所见的一〈单位〉无论在量上和在质上不异于别个一〈单位〉，而数必须是或等或不等——一切数均应如此，而抽象〈单位〉所组成的数更应如此——所以，凡一数若既不大于亦不小于另一数，便应与之相等；但在数上所说的相等，于两事物而言，若品种不异而相等者则谓之相同。倘品种有异，虽"本１０"中之诸２，即便它们相等，也不能不被分化，谁要说它们并不分化，又能提出怎样的理由？ 又，假如每个１加另１为２，从"本２"中来的１和从"本３"中来的１亦将成２。现在（甲）这个２将是相异的１所组成；（乙）这１０个２对于３应属先于抑为后于？似乎这必是先于；因为其中的一个单位与３为同时，另一个则与２为同时。于我们讲来，一般１与１若合在一起就是２，无论事物是否相等或不等，例如这个善一和这个恶一，或是一个人和一匹马，总都是"２"。 假如"本３"为数不大于２，这是可诧异的；假如这是较大，那么清楚地其中必有一个与２相等的数，而这数便应与"本２"不相异。但是，若说有品种相异的第一类数与第二类数这就不可能了。 意式也不能是数。因为在这特点上论，倘真以数为意式，那么主张单位应各不同的人就该是正确的了；这在先曾已讲过。通式是整一的；但"诸１"若不异，"诸２"与"诸３"亦应不异。所以当我们这样计点——"１，２"……他们就必得说这个并不是１个加于前一个数；因为照我们的做法，数就不是从未定之２制成，而一个数也不能成为一个意式；因为这样一个意式将先另一个意式存在着而所有诸通式将成为一个通式的诸部分。这样，由他们的假设来看，他们的推论都是对的，但从全局来看，他们是错的；他们的观念为害匪浅，他们也得承认这种主张本身引致某些疑难，——当我们计点时说"１，２，３"究属是在一个加一个点各数呢，还是在点各个部分呢。但是我们两项都做了；所以从这问题肇致这样重大的分歧，殊为荒唐。 Chapter eight 最好首先决定什么是数的差异，假如一也有差异，则一的差异又是什么。单位的差异必须求之于量或质上；单位在这些上面似乎均有差异。但数作为数论，则在量上各有差异。 假如单位真有量差，则虽是有一样多单位的两数也将有量差。 又在这些具有量差的单位中是那第一单位为较大或较小，抑是第二单位在或增或减？所有这些都是不合理的拟议。它们也不能在质上相异。因为对于诸单位不能系以属性；即便对于列数，质也只能是跟从量而为之系属。又，１与未定之２均不能使数发生质别，因为１本无质而未定之２只有量性；这一实是只具有使事物成为多的性能。假如事实诚不若是，他们该早在论题开始时就有说明，并决定何以单位的差异必须存在，他们既未能先为说明，则他们所谓差异究将何所指呢？ 于是明显地，假如意式是数，诸单位就并非全可相通，在〈前述〉两个方式中也不能说它们全不相通。但其他某些人关于数的议论方式也未为正确。那些不主于意式，也不以意式为某些数列的人，他们认为世上存在有数理对象而列数为现存万物中的基本实是，"本１"又为列数之起点。这是悖解的：照他们的说法，在诸１中有一"原１"〈第一个１〉，却在诸２中并不建立"原２"〈第一个２〉，诸３中也没有"原３"〈第一个３〉。同样的理由应该适用于所有各数。关于数，假使事实正是这样，人们就会得想到惟有数学之数实际存在，而１并非起点（因这样一类的１将异于其它诸１；而２，也将援例存在有第一个２与诸２另作一类，以下顺序各数也相似）。 但，假令１正为万物起点，则关于数理之实义，毋宁以柏拉图之说为近真，"原２"与"原３"便或当为理所必有，而各数亦必互不相通。反之，人苟欲依从此说，则又不能免于吾人上所述若干不符事实之结论。但，两说必据其一，若两不可据，则数便不能脱离于事物而存在。 这也是明显的，这观念的第三翻版最为拙劣——这就是意式之数与数学之数为相同之说。这一说合有两个错误。 （一）数学之数不能是这一类的数，只有持此主张的人杜撰了某些特殊的线索才能纺织起来。（二）主张意式数的人们所面对着的一切后果他也得接受。 毕达哥拉斯学派的数论，较之上述各家较少迷惑，但他们也颇自立异。他们不把数当作独立自在的事物，自然解除了许多疑难的后果；但他们又以实体为列数所成而且实体便是列数，这却是不可能的。这样来说明不可区分的空间量度是不真确的；这类量度无论怎么多怎么少，诸１是没有量度的；一个量度怎能由不可区分物来组成？算术之数终当由抽象诸１来组成。但，这些思想家把数合同于实物；至少他们是把实物当作列数所组成，于是就把数学命题按上去。 于是，数若为一自存的实物，这就必需在前述诸方式中的一式上存在，如果不能在前述的任何一式上存在，数就显然不会具有那样的性质，那些性质是主张数为独立事物的人替它按上去的。 又，是否每个单位都得之于"平衡了的大与小"抑或一个由"小"来另一个由"大"来？（甲）若为后一式，每一事物既不尽备所有的要素，其中各单位也不会没有差异；因为其中有一为大，另一为与大相对反的小。在"本３"中的诸单位又如何安排？其中有一畸另单位。但也许正是这缘由，他们以"本一"为诸奇数中的中间单位。（乙）但两单位若都是平衡了的大与小，那作为整个一件事物的２又怎样由大与小组成？或是如何与其单位相异？又，单位是先于２；因为这消失，２也随之消失。于是１将是一个意式的意式，这在２以前先生成。那么，这从何生成？不是从"未定之２"，因为"未定之２"的作用是在使"倍"。 再者，数必须是无限或是有限（因为这些思想家认为数能独立存在，并就应该在两老中确定其一）。清楚地，这不能是无限；因为无限数是既非奇数又非偶数，而列数生成非奇必偶，非偶必奇。其一法，当１加之于一个偶数时，则生成一个奇数；另一法，当１被２连乘时，就生成２的倍增数； 又一法当２的倍增数，被奇数所乘时就产生其它的偶数。 又，假如每一意式是某些事物的意式，而数为意式，无限数本身将是某事物（或是可感觉事物或是其它事物）的一个意式。可是这个本身就不合理，而照他们的理论也未必可能，至少是照他们的意式安排应为不可能。 但，数若为有限，则其极限在那里？关于这个，不仅该举出事实，还得说明理由。倘照有些人所说数以１０为终，则通式之为数，也就仅止于１０了；例如３为"人本"，又以何数为"马本"？作为事物之本的若干数列遂终于１０。这必须是在这限度内的一个数，因为只有这些数才是本体，才是意式。可是这些数目很快就用尽了；动物形式的种类着实超过这些数目。同时，这是清楚的，如依此而以意式之"３"为"人本"，其它诸３亦当如兹（在同数内的诸）亦当相似），这样将是无限数的人众；假如每个３均为一个意式，则诸３将悉成"人本"，如其不然，诸３也得是一般人众。又，假如小数为大数的一部分（姑以同数内的诸单位为可相通），于是倘以"本４"为"马"或"白"或其它任何事物的意式，则若人为２时，便当以人为马的一个部分。这也是悖解的，可有１０的意式，而不得有１１与以下各数的意式。又，某些事物碰巧是，或也实际是没有通式的；何以这些没有通式？我们认为通式不是事物之原因。又，说是由１至１０的数系较之本１０更应作为实物与通式，这也悖解。本１０是作为整体而生成的，至于１至１０的数系，则未见其作为整体而生成。他们却先假定了１至１０为一个完整的数系。至少，他们曾在１０限以内创造了好些衍生物——例如虚空，比例，奇数以及类此的其它各项。他们将动静，善恶一类事物列为肇始原理，而将其它事物归之于数。所以他们把奇性合之于１；因为如以３作奇数之本性则５又何如？ 又，对于空间量体及类此的事物，他们都用有定限的数来说明；例如，第一，不可分线，其次２，以及其它；这些都进到１０而终止。 再者，假如数能独立自存，人们可以请问那一数目为先，——１或３或２？假如数是组合的，自当以１为先于，但普遍性与形式若为先于，那么列数便当为先于；因为诸１只是列数的物质材料，而数才是为之作用的形式。在某一涵义上，直角为先于锐角，因为直角有定限，而锐角犹未定，故于定义上为先；在另一涵义上，则锐角为先于，因为锐角是直角部分，直角被区分则成诸锐角。作为物质，则锐角元素与单位为先于；但于形式与由定义所昭示的本体而论，则直角与"物质和形式结合起来的整体"应为先于；因为综合实体虽在生成过程上为后，却是较接近于形式与定义。那么，１安得为起点？他们答复说，因为１是不可区分的；但普遍性与个别性或元素均不可区分。而作为起点则有"始于定义"与"在时间上为始"的分别。那么，１在那一方面为起点？上曾言及，直角可被认为先于锐角，锐角也可说是先于直角，那么直角与锐角均可当作１看。他们使１在两方面都成为起点。 But this is impossible.因为普遍性是由形式或本体以成一，而元素则由物质以成一，或由部分以成一。两者（数与单位）各可为一——实际上两个单位均各潜在（至少，照他们所说不同的数由不同种类的单位组成，亦就是说数不是一堆，而各自一个整体，这就该是这样），而不是完全的实现。他们所以陷入错误的原因是他们同时由数理立场又由普遍定义出发，进行研究，这样（甲）从数理出发，他们以１为点，当作第一原理；因为单位是一个没有位置的点。 （他们象旁的人也曾做过的那样，把最小的部分按装成为事物。）于是"１"成为数的物质要素，同时也就先于２；而在２当作一个整数，当作一个形式时，则１又为后于。然而，（乙）因为他们正在探索普遍性，遂又把"１"表现为列数形式涵义的一个部分。但这些特性不能在同时属之同一事物。 假如"本１"必须是无定位的单元（因为这除了是原理外，并不异于它１），２是可区分的，但１则不可区分，１之于"本１"较之于２将更为相切近，但，１如切近于"本１"，"本１"之于１也将较之于２为相切近；那么２中的各单位必然先于２。然而他们否认这个；至少，他们曾说是２先创生。 又，假如"本２"是一个整体，"本３"也是一个整体，两者合成为２〈两个整体〉。于是，这个"２"所从产生的那两者又当是何物呢？ Chapter Nine 因为列数间不是接触而是串联，例如在２与３中的各单位之间什么都没有，人们可以请问这些于本１是否也如此紧跟着，紧跟着本１的应是２抑或２中的某一个单位。 在后于数的各级事物——线，面，体——也会遭遇相似的迷难。有些人由"大与小"的各品种构制这些，例如由长短制线，由阔狭制面，由深浅制体；那些都是大与小的各个品种。这类几何事物之肇始原理〈第一原理〉，相当于列数之肇始原理，各家所说不同。在这些问题上面，常见有许多不切实的寓言与理当引起的矛盾。（一）若非阔狭也成为长短，几何各级事物便将互相分离。（但阔狭若合于长短，面将合于线，而体合于面；还有角度与图形以及类此诸事物又怎样能解释？）又（二）在数这方面同样的情形也得遭遇；因为"长短"等是量度的诸属性，而量度并不由这些组成，正象线不由"曲直"组成或体不由平滑与粗糙组成一样。 所有这些观点所遇的困难与科属内的品种在论及普遍性时所遇的困难是共通的，例如这参于个别动物之中的是否为"意式动物"抑其它"动物"。假如普遍性不脱离于可感觉事物，这原不会有何困难；若照有些人的主张一与列数皆相分离，困难就不易解决；这所谓"不易"便是"不可能"。因为当我们想到２中之一或一般数目中的一，我们所想的正是意式之一抑或其它的一？ 于是，有些人由这类物质创制几何量体，另有些人由点来创制，——他们认为点不是１而是与１相似的事物—— 也由其它材料如与"１"不同的"众"来创制；这些原理也得遭遇同样严重的困难。因为这些物质若相同，则线，面，体将相同；由同样元素所成事物亦必相同。若说物质不止一样，其一为线之物质，另一为面，又一为体，那么这些物质或为互涵，或不互涵，同样的结果还得产生；因为这样，面就当或含有线或便自己成了线。 再者，数何能由"单与众"组成，他们并未试作解释；可是不管他们作何解释，那些主张"由１与未定之２"来制数的人所面对着的诸驳议，他们也得接受。其一说是由普遍地云谓着的"众"而不由某一特殊的"众"来制数，另一说则由某一特殊的众即第一个众来制数；照后一说，２为第一个众。所以两说实际上并无重要差别，相同的困难跟踪着这些理论——由这些来制数，其方法为如何，搀杂或排列或混和或生殖？以及其它诸问题。在各种疑难之中，人们可以独执这一问题，"假如每一单位为１，１从何来？"当然，并非每个１都是"本１"。于是诸１必须是从"本１"与"众"或众的一部分来。要说单位是出于众多，这不可能，因为这是不可区分的；由众的一部分来制造１也有许多不合理处；因为（甲）每一部分必须是不可区分的（否则所取的这一部分将仍还是众，而这将是可区分的），而"单与众"就不成其为两要素了；因为各个单位不是从"单与众"创生的。（乙）执持这种主张的人不做旁的事，却预拟了另一个数；因为它的不可区分物所组成的众就是一个数。 又，我们必须依照这个理论再研究数是有限抑无限的问题。起初似乎有一个众，其本身为有限，由此"有限之众"与"一"共同创生有限数的诸单位，而另有一个众则是绝对之众，也是无限之众；于是试问用那一类的众多作为与元一配合的要素？人们也可以相似地询问到"点"，那是他们用以创制几何量体的要素。因为这当然不是惟一的一个点；无论如何请他们说明其它各个点各由什么来制成。当然不是由"本点"加上一些距离来制作其它各点。因为数是不可区分之一所组成，但几何量体则不然，所以也不能象由众这个要素的不可区分之诸部分来制成一〈单位〉那样，说要由距离的不可区分之诸部分来制成点。 于是，这些反对意见以及类此的其它意见显明了数与空间量体不能脱离事物而独立。又，关于数论各家立说的分歧，这就是其中必有错误的表征，这些错处引起了混乱。那些认为只有数理对象能脱离可感觉事物而独立的人，看到通式的虚妄与其所引起的困惑，已经放弃了意式之数而转向于数学之数。然而，那些想同时维持通式与数的人假设了这些原理，却看不到数学数存在于意式数之外，他们把意式数在理论上合一于数学数，而实际上则消除了数学数；因为他们所建立的一些特殊的假设，都与一般的数理不符。最初提出通式的人假定数是通式时，也承认有数理对象存在，他是自然地将两者分开的。所以他们都有某些方面是真确的，但全部而论都不免于错误。他们的立论不相符合而相冲突，这就证实了其中必有不是之处。错误就在他们的假设与原理。坏木料总难制成好家具，爱比卡包谟⑥说过，"才出口，人就知道此言有误"。 关于数，我们所提出的问题和所得的结论已足够（那些已信服了的人，可在后更为之详解而益坚其所信，至于尚不信服的人也就再不会有所信服）。关于第一原理与第一原因与元素，那些专谈可感觉本体的各家之说，一部分已在我们的物学著述中说过，一部分也不属于我们现在的研究范围； 但于那些认为在可感觉物体以外，还有其它本体的诸家之说，这必需在讨论过上述各家以后，接着予以考虑。因为有些人说意式与数就是这类〈超感觉〉本体，而这些要素就是实在事物的要素与原理，关于这些我们必须研究他们说了些什么，所说的内容器实义又如何。 那些专主于数而于数又主于数学之数的人，必须在后另论；但是关于那些相信意式的人，大家可以同时观测他们思想的途径和他们所投入的困惑。他们把意式制成为"普遍"，同时又把意式当作可分离的"个别"来处理。这样是不可能的，这曾已为之辩明。那些人既以本体外离于可感觉事物，他们就不得不使那作为普遍的本体又自备有个体的特性。他们想到了可感觉世界的形形色色，尽在消逝之中，惟其普遍理念离异了万物，然后可得保存于人间意识之中。我们先已说过苏格拉底曾用定义〈以求在万变中探取其不变之真理，〉启发了这样的理论，但是他所始创的"普遍"并不与"个别"相分离；在这里他的思想是正确的。结果是已明白的了，若无普遍性则事物必莫得而认取，世上亦无以积累其知识，关于意式只在它脱离事物这一点上，引起驳议。可是，他的继承者却认为若要在流行不息的感觉本体以外建立任何本体，就必需把普遍理念脱出感觉事物而使这些以普遍性为之云谓的本体独立存在，这也就使它们"既成为普遍而又还是个别"。照我们上述的看法，这就是意式论本身的惩结。 Chapter ten 让我们对于相信意式的人提出一个共有的疑难，这一疑难在我们先时列举诸问题时曾已说明。我们若不象个别事物那样假定诸本体为可分离而独立存在，那么我们就消灭了我们自己所意想的"本体"；但，我们若将本体形成为可分离的，则它们的要素与它们的原理该又如何？ 假如诸本体不是普遍而是个别的，（甲）实物与其要素将为数相同，（乙）要素也就不可能得其认识。因为（甲）试使言语中的音节为诸本体，而使它们的字母作为本体的要素；既然诸音节不是形式相同的普遍，不是一个类名，而各自成为一个个体，则βα就只能有一个，其它音节也只能各有一个（又他们〈柏拉图学派〉于每一意式实是也认为各成一个整体）。倘诸音节皆为唯一个体，则组成它们的各部分也将是唯一的；于是α不能超过一个，依据同样的论点，也不能有多数的相同音节存在，而其它诸字母也各只能有一个。然而若说这样是对的，那么字母以外就没有别的了，所有的仅为字母而已。（乙）又，要素也将无从取得其认识，因为它们不是普遍的，而知识却在于认取事物之普遍性。知识必须依凭于实证和定义，这就是知识具有普遍性的说明；若不是每一个三角的诸内角均等于两直角，我们就不作这个"三角的诸内角等于两直角"的论断，若不是"凡人均为动物"，我们也不作这个人是一个动物的论断。但，诸原理若均为普遍，则由此原理所组成的诸本体亦当均为普遍，或是非本体将先于本体； 因为普遍不是一个本体，而要素或原理却是普遍的，要素或原理先于其所主的事物。 当他们正由要素组成意式的同时，又宣称意式脱离那与之形式相同的本体而为一个独立实是，所有这些疑难就自然地跟着发生。 但是，如以言语要素为例，若这并不必需要有一个"本α"与一个"本β"而尽可以有许多α许多β，则由此就可以有无数相似的音节。 依据一切知识悉属普遍之说，事物之诸原理亦当为普遍性而不是各个独立本体，而实际引致了我们上所述各论点中最大困惑者，便是此说，然此说虽则在某一涵义上为不合，在另一涵义上讲还是真实的。"知识"类于动字"知"，具有两项命意，其一为潜能另一为实现。作为潜能，这就是普遍而未定限的物质，所相涉者皆为无所专指的普遍；迨其实现则既为一有定的"这个"，这就只能是"这个"已经确定的个体了。视觉所见各个颜色就是颜色而已，视觉忽然见到了那普遍颜色，这只是出于偶然。文法家所考察的这个个别的α就是一个α而已。假如诸原理必须是普遍的，则由普遍原理所推演的诸事物，例如在论理实证中，亦必为普遍；若然如此，则一切事物将悉无可分离的独立存在〈自性〉——亦即一切均无本体。但明显地，知识之一义为普遍，另一义则非普遍。