What is the principle of the continuum

synecheia - the Aristotelian concept of the continuum

 

- the context as the substance of size

 

 

Development of the continuum principle with Aristotle (after Dehn, Waschkies and Seidel)

The concept of connection (synecheia, lat. continuum, literally what has grown together, what has grown together) has a key role for Aristotle. Like the concept of size, the concept of connection goes far beyond mathematics, even if, thanks to Cantor and others, the mathematical meaning of the potential and actual infinite of the continuum was again intensely discussed. Originally, context meant the coherence of the world as a whole. Heidegger and Benjamin speak in a similar way of the "rulers" or "rulers".

With this term Aristotle found his position in the natural philosophy tradition. He was the first to distinguish nature (physis) of their material (hyle), using it as an expression for the substance hyle literally chooses the wood, the plant, that is, a biological expression with inner liveliness. The overgrown (synecheia) shows the property of wood on which Aristotle was located. The following comment is intended to show how relationship and size relate to one another, such as material and form. However, while Aristotle used the concept of greatness in the figure (scheme) a mathematical independence (chorizein) (Phys. II.2, 193b31), he did not introduce a comparable term to mathematically make the context independent. In this comment, the thesis should be developed that this is the dimension (slide topic) is.

The predecessors of Aristotle show what power is connected with context. The Phythagoreans are probably closest to the origins. For Philolaos the hearth fire (hestia) the center, the cohesion (synoch) and the measure of nature (metron physeos) manufactured and secured. (»Philolaos says there is a fire in the middle around the center, which he hears (hestia) of the universe and house of Zeus and mother of gods and altar and cohesion (synoch) and measure of nature ((metron physeos) «, Diels Kranz, Fragments of the Pre-Socratics, 44 A 16). With Hestia, Philolaos refers to the original world of gods. The hearth goddess Hestia is not simply responsible for the unity of the world, but with the hearth fire for the context, the warming and the invigorating. When it protects the hearth fire, it secures the foundations of life. At the same time, Hestia stands for the greatest crisis in Greek history. She had left Olympus when conditions there had radically changed with the arrival of Dionysus. If Philolaos nevertheless refers to Hestia, he wants to go back with his idea of ​​cohesion to an original world order that was deeply shaken by the new circumstances after Hestia's departure. When Hestia left Olympus, the original connection and with it the protection of the world threatened to perish. If Philolaos takes up her name again, he would like to revive this world order in her name with his natural philosophy. He could stand here in the tradition of the Orphic traditions, which he wants to continue in his own way.

In Parmenides there is little evidence of such a basic attitude, but he too uses the term connection. In the fragment 8 of the didactic piece on nature that he has handed down, it says, "It never was, never will be, since it is now, all at once, the one, continuous" (oude pot 'ên oud' estai, epei nun estin homou pan, hen, suneches; Source). The coherent creates the inner unity of the One, that is, that of the unity, through which it becomes one.

Heraclitus no longer mentions the connection directly in fragment 30 (syneches), but only speaks of dimensions (metra). It goes back all the more clearly to the meaning of fire. However, he no longer calls on the name of Hestia, but instead speaks of the "everlasting fire (pure aeizôon): »This world order (cosmon tonde), the same for all beings, created neither gods nor men, but the eternally living fire was and is and will be, glowing according to measure (metra) and extinguishing according to measure «(source).

Aristotle absorbs all of this. With the context he sees a principle emerging from physics, which for him takes the place that the number has in Plato. While for Plato the number is the material of which mathematics is made, for Aristotle it is the context. Instead of contrasting the idea of ​​number with perception of nature, as Plato does, he wants to show how the principle of connection can be obtained from an understanding of nature.

This approach can be considered successful if the coherent can be understood as the matter of greatness, whereby Aristotle expressly refers to the concept of greatness, which was newly introduced since Eudoxus (megethos) relates. In doing so, he has also found an answer to atomism, which in the end could never decide whether the emptiness or the full is the substance of greatness. Modern science follows atomism and as a rule understands the atoms as the substance of size, but then leans back towards the ordered space and speaks, for example, of a structural realism that gives the spatial structures ontological weight. In a figurative sense, modern mathematics sees the numbers and the simplest geometric figures as their atoms and thus as the substance of mathematical magnitude, in the 20th century the set and its elements. This direction will be fundamentally questioned in the following with Aristotle.

However, Aristotle did not provide a complete representation of the concepts of connection, size and between. The term context (synecheia, Continuum) is used in the various books of the physics continuously developed (Phys. III, V, VI and VIII). In other places such as the Celestial science it is assumed. Here - with Dehn and Waschkies - three different definitions of the continuum can be demonstrated, whereby Waschkies assumes that the books of the physics are arranged according to their origin and build on one another.

To understand the context, Aristotle developed an early form of topology. He differentiates between things that are separate, successive and finally grown together. Only with them does he see a connection. However, this approach has not yet been systematically merged with his ideas about the order of the numbers and the positions of the figures.

Since Aristotle does not yet systematically differentiate between context and its mathematical independence, other advanced ideas remain unconnected with him. (This can of course also be due to the fact that large parts of his work have been lost.) By differentiating the order of the numbers and the position of the figures, Aristotle was apparently looking for an approach from which the crisis that had broken out in Greek mathematics could be answered : Sizes and numbers are incompatible. For Aristotle, their incompatibility is due to the fact that the order of the numbers and the position of the figures must be systematically differentiated from one another.

Their relationship is still open today and, with Cantor's continuum hypothesis, led to the fundamental crisis of modern mathematics, in which the Greek fundamental crisis of mathematics is repeated on a new level. In the following, after a short repetition of the first definition of the continuum as the indivisible, the order of the numbers and the position of the figures will be explained in order to then show how the principle of the coherent was introduced into this problem area. The order of the numbers and the position of the figures are understood as an approach in which way a mathematical independence can be found in the context, which corresponds to the relationship between size and figure. - That should be in further comments on the in-between (metaxy), the principles of mathematics and the concept of time.

(I) The continuum is infinitely divisible

The unlimited divisibility was recognized as the distinguishing feature of size from the natural number. If this property is considered in and of itself and emphasized as a principle in its own right, then this results in the first definition: The continuum (syneches) is the infinitely divisible. What is indefinitely divisible must be a continuum. For Max Dehn this is the analytical concept of the continuum.

This concept of the continuum is used in Phys. III addressed. It is presupposed there and not justified further. Just as suddenly it is assumed that the understanding of size gained with Eudoxos is known. These are the places any unbiased reader must stumble upon. Seidel is absolutely right when, for him, size suddenly "thuds into" the train of thought (Seidel, p. 209). Aristotle's formulations show that there is little more than a sketch to order the train of thought.

The movement »seems to be in the realm of the Related to belong, in the term 'connected', however, appears (the determination) 'unlimited'; if one determines 'connected', it happens incidentally that one often uses the term 'unlimited', because 'infinitely divisible' - that is just 'connected' "(Phys., III.1, 200b).

“The meaning of 'unlimited' is not the same in application to (space)size, change and time- as if this were a single natural object (physis) - but here the subordinate (acholondein) stated in accordance with the above factually, e.g. change (is an unlimited process), because the (spatial)sizein which locomotion, property change and growth take place (this does so indefinitely); the time then it is because of the change "(Phys. III.7, 207b, cf. also Phys. III.4, 202b, Phys. III.6,206a and again in retrospect Phys., IV.11, 219a).

The size can be divided indefinitely. It is continuous. Because the size changes continuously (and not in discrete quantum leaps), the movement is also continuous. And because the movement is continuous, it is also the time in which the movement occurs.

Note: Here, Aristotle has already sketched out the basic outline of the train of thought, which Hegel also used in his Science of logic time develops step by step out of size, or perhaps better said the concept of time is constructed. See the logic studies on Hegel.

Aristotle, however, does not justify why he used the subordination (acholondein) forms. For it can also be argued conversely that the movement is continuous and therefore the variable that changes in and with the movement must also be continuous. It can be said that time is continuous, and hence the movement that follows it. Obviously these are the first sketches, the underlying principle (ark) to recognize.

Whatever the arrangement, it is the infinitely divisible of the coherent, in which size, movement and time have something in common. Size, movement and time are continuous.

The statement "the quantity differs from the number because it is infinitely divisible" is now rephrased as "the quantity differs from the number because it is continuous and not discrete". With this, no new knowledge has been gained in terms of content, but a first definition for the continuum has been found and an approach to bring these terms into an internal order that needs to be explained in more detail.

Order (taxis) and order of numbers

The order of the numbers (one, two, three, ...) seems to be a matter of course. It forms the pattern of a series in an almost prototypical manner. However, the understanding proves to be premature when it must be seen, apparently against all reason, that the order of the numbers cannot be illustrated geometrically. The mind blocks three insights: (i) Numbers are not sizes and they are not size. There is no point in asking what size the number 7 is. The 7 is neither high, wide, colored, salty or rough, it is neither a surface nor a sphere. Numbers have no dimension. This knowledge was not first represented by Frege, but it is implicit in Aristotle. Numbers only exist in counting, within the process of counting. (ii) Conversely, therefore, there is nothing that has a number as a property. For example, there is no point in saying that something is size 7. The size always relates to one dimension. It can be measured that a house is 7 m high or 7 m wide, that it has a floor area of ​​7 square meters, or an enclosed space of 7 cubic meters, but it is not "the size 7". (iii) Although each number is 1 larger than its predecessor, there is no space between two adjacent numbers. One number immediately follows the other; there is no further natural number between two neighboring natural numbers. (Only for the sizes is that there is always at least one other size between any two sizes.) If an attempt is made to represent the numbers geometrically, e.g. as points on a number line, the distance would have to be shown on the one hand, and it should not contain anything on the other.

But nobody can escape the geometric illustration, because otherwise a cycle threatens: Numbers are necessary to describe what numbers are. The one has a double meaning: the one is (i) the first number, the one, and it is (ii) the distance between two adjacent numbers. Each number is one larger than its predecessor. The same one is used as a number and to describe a property of numbers. In order to avoid this ambiguity, there is inevitably the geometric intuition to represent the numbers by points and their distances by lengths between the points. This is neither an optical illusion nor an illusion or dream, but the imagination of the soul is principally led astray in relation to the numbers.

In order to avoid geometric intuition, one can try to differentiate the two meanings by color, as Seidel suggested. Each number has a size value and two numbers are separated from each other by a space. For example, the numbers 12 and 5 are spaced 7 apart, or the numbers 1 and 2 are spaced 1. The spacing between two numbers cannot be measured because neither the numbers have a size nor are there any other numbers between two neighboring numbers, it is him can only be calculated as the difference.

Nobody can deny the discomfort if it has to be accepted that and how the numbers evade all spontaneous attempts to form images and a spatial perception. Only those who have gone through this crisis of spontaneous perception will be able to find mathematics. Numbers and their laws are in principle imperceptible, but every perception follows the laws of numbers when it counts objects and compares counted objects. Not even in a dream can the elementary rules of numbers be violated. Man is unable to imagine anything that contradicts the numbers. If, however, he encounters something like the length of the diagonal compared to the outside, then the imagination is completely in crisis. This length can be "seen", but it cannot be counted, although otherwise it is true that humans cannot see anything that they cannot count.

Location (thesis) and order of the figures

Before going into this in detail and examining what is special about what has grown together, the concept of sequence must be clarified. In general, it applies to Aristotle that two geometric figures of the same genus, for example two points, two lines or two surfaces, are separated from one another if a geometric figure of another genus lies between them, for example a line between two separate points, one Area between two separate lines, a space between two separate areas.

This rules out the case of a sequence, because e.g. two lines form a sequence if they lie on a common straight line. In this special case, they are not separated from one another by an area, but by a third segment. And accordingly, two surfaces can lie next to each other on a common base. Then they are separated from one another by a third area and not by a room.

Figure 1: Layers of lines and areas

(1a) Lines: Infinitely long lines separated from one another must run parallel, otherwise they would intersect. Between them lies a surface.The distance is the same everywhere. - Finely long lines (stretches) do not have to run parallel. When they are inclined towards each other, there will be two opposite points where they are closest to each other. - Finitely long lines can run on a common infinitely long line. Their distance is given by the distance between the opposite end points.

(1b) Areas: Infinitely large areas must be parallel to each other. Between them lies a space, the height of which is their distance. - Finely large surfaces (e.g. circles) can be inclined to one another. - Finitely large areas (e.g. circles) are in sequence if their distances are finitely long lines that lie on a common infinitely long line.

These examples show that, in the case of the sequence, figures are separated from one another by the distance between their boundary points.

(II) The continuum is what has grown together

The definition of size by Eudoxus, Euclid and their generalization by Hilbert can suggest that the coherent is already given when points can be arranged closely - "without interruption", without gaps - one after the other. Aristotle clearly contradicts and distinguishes for better clarity: Something can (a) "separate", choris, "following in series", ephexes be arranged, (b) it can be directly "adjacent", echomenos so that the individual elements "touch", haptesthai (with the root hapto, touch, grasp), until it is (c) "coherent", "grown together", syneches or "at the same time" in a basic understanding that is important for the theory of time, ama is.

Aristotle's way of portraying it suggests that what is grown together emerges from the sequence like a border crossing. About the concepts considered by Aristotle ephexes, echomenos, haptesthai and syneches To be able to differentiate more clearly, one has to consider first the special case of the numbers, which do not have a size of their own, then the particularity of the sequence and finally the merged.

Figure 2: Numbers, Atoms, and Points

(2a) Natural numbers have no inner distance (they do not have a size, but are size values) and are separated from each other by an outer distance.

(2b) Atoms have an inner indivisible distance (they have a size and are "full" inside) and can be separated by an outer distance.

(2c) Atoms can be ordered linearly and moved so close to one another that they touch at boundary points, but do not penetrate or intersect. They are then close to each other.

(2d) Overgrown context (using the example of a red organic cell structure) (compactness)

Here a real game of thought unfolds, which lives from numerous multiple meanings and the systematic differentiation of numbers, sizes and magnitudes, points, internal and external distances as well as vanishing quantities (the "between"). Only with the concept of the coalesced does Aristotle gain the special meaning of the coherent, which is far more than just that which is infinitely divisible.

(2b) If figures (lines, surfaces or bodies) are considered, there is both an inner distance between the outer points of the figure that are furthest apart from one another (which describe an in-between within the figure) and an outer distance between the various figures. They are differentiated here by black and gray double arrows. If these two types of distances are not clearly distinguished, this leads to a number of contradictions. The inner distances are the size of the figures, the outer distances the size of their distance from one another.

(2b) can result from (2a) if every number is replaced by a figure. In the simplest case, all inner distances (diameter) and all outer distances are the same.

(2c) When the outer gaps disappear, the different figures touch each other. They "stick together", like Heidegger very vividly the word echomenos has translated. As is so often the case, Aristotle chooses a very complex formulation in order to meet the facts in the greatest generality.

»Of (something) that follows next (I speak) when there is nothing of the same genus between that which comes after the beginning separately according to the situation or naturally or on the basis of some other (rule) and that which it is supposed to follow next. By this I mean, for example, that (between) a line (and its immediate successor) neither a line nor several lines can be shown (and that between two immediately adjacent units) neither one unit nor several units (may lie) "(Phys. V .3 226b34-227a3, cited and translated Waschkies, p. 226f).

The difficulty arises because Aristotle wants to describe both the order of touching figures and the order of the numbers with a single formulation. The following statement is understandable for the figures: If circles touch each other, as in this example, circle follows circle and no other genre intervenes. If, on the other hand, the circles are separated by a distance, then circle, distance, circle, distance, ... follow one another. The distance between two circles is not a circle, but a length, i.e. of a different kind.

On the other hand, it is far more difficult to understand that even with units that form a sequence, there is no other genus between successive units. What is meant is that there is no other number between two numbers, as has already been explained with reference to FIG. (2a). The fact that Aristotle does not clearly separate these two cases of consecutive geometrical figures and natural numbers that follow one another is an indication for me that he has not yet clearly separated the principles of size and dimension from one another. (Max Dehn has already suspected this, who, despite all agreeing with Aristotle, objects that "whether the independence of the term 'size' from dimension is recognized is very doubtful", Dehn, p. 20).

After first emphasizing the common features of the successive numbers and figures, Aristotle wants to clarify their difference in the second step. While geometrical figures can touch each other at common edge points (e.g. if a circle touches the next one), no contact between numbers is conceivable. They follow one another closely - there is no number between two neighboring numbers - but they do not touch.

“So if there should be point and unity in the way that they are considered to exist for themselves, then it is not possible that unity and point are the same: namely, the (points) are entitled to contact, the units ( only) sequence, and with the (points) something can lie in between - every line lies between points - with the (units) this necessity is not: there is no 'in the middle' between duality and unity "(Phys. V. 3, 227a).

The more concise formulation in metaphysics is perhaps a little clearer:

“So the point and the one are not identical. Because there is contact for points, but not for ones, but only succession; for points there is a mean (metaxy), but not with ones ”(Metaphysik XI, 12, quoted in Seidel, p. 318).

Nevertheless, Aristotle provokes a misunderstanding with this imprecise - literally wrong - statement. (2c) shows that points do not touch each other here, but figures touch each other at their border points. Seidel immediately hooks: “I have you! The only passage in physics in which Aristotle says that points can touch ”(Seidel, p. 287). As I understand it, Aristotle did not mean that, but Seidel is literally right.

(2d) From the perspective of 20th century mathematics, the really exciting question is whether there is something like a border crossing in (2c) when the inner distances between the circles become smaller and smaller and finally converge to 0. At the limit crossing this would mean that the finite circles in each case merge into the real numbers, which have neither an inner distance nor an outer distance from one another. There can be no smallest unit, because as already seen, the rational numbers and the irrational numbers cannot be reduced to a common unit. A smallest unit would also contradict the assumption that the continuous set of real numbers should be infinitely divisible. Before Aristotle clearly rejects this assumption in his third definition of the continuum, he introduces a positive definition of the continuum instead.

The term synecheia (Context) took Aristotle from medicine. Today synechia means malformations when something fails when organs grow together. The clinical picture of synechia shows what is important to Aristotle: If things are going well, the coalescence creates a connection that creates a high level of resilience to external disturbances.

Originally was synecheswhat's stapled together. This is how the atomists used it when several atoms are stapled together. Proclus also uses it in his Euclid Commentary (In Eucl., Pp. 278.9-9.2, quoted and translated by Waschkies, pp. 382f), who does not follow Aristotle's path here. Aristotle distinguishes the fused from what is stapled together. What has grown together cannot be broken down into the individual elements that are attached, but rather it has become a whole with new properties. For Max Dehn this is the synthetic term of the continuum.

(III) The continuum does not consist of points

However, this is initially only an intuitive understanding that can stimulate further considerations. It is intuitively clear what happens when organs grow together, but what are "overgrown numbers" or "overgrown points"? Aristotle gives no answer to this, but only makes it clear in his final third definition that a continuum cannot be composed of mere dimensionless points.

“But if that which is continuous and touching and what follows behaves as we have stated above, namely that those things are continuous, the outermost limits of which are one ('grown together'), but those things are touching, whose outermost limits are locally at the same time and immediately following those between which nothing of the same kind lies between them, it is impossible for something continuous to consist of indivisible, such as a line to consist of points, provided that the line is continuous, but the point is indivisible ”(Phys. VI.1 , 231a, Prantl translation after Seidel, p. 313. Zekl translates: "whose edges form a unit").

The constant does not consist of the discreet. Aristotle develops the definition of the coherent from the understanding of the coherent and no longer from the question of divisibility or size. Dehn and Waschkies emphasize that a new independent definition of the continuum has been found that differs from the earlier ones in the physics differs.

For this purpose, modern mathematics found the concept of compactness in the topology. Just as cell fibers are formed in an organic cell structure, each of which covers an entire area, so for compact quantities there is a subset of finitely many within all overlaps, ie all sets of quantities that together contain ("cover") the quantity There are sets that already cover the set. Aristotle's approach is thus taken seriously. The compact set is not defined by the points it contains, but by the sets that cover it. No matter how small these quantities are, a finite number of them are sufficient. The best illustrative example is the covering of the real numbers by circles. For example, if the number interval from 0 to 1 is covered by any number of circles, there are a finite number of these circles that overlap sufficiently or touch each other and completely cover the interval. At first glance, that seems to be a matter of course. The idea is that this still applies if the individual amounts of the overlap are getting smaller and smaller. However, since they are always quantities, they never become as small as dimensionless points. All the basic theorems of analysis are proved with this property.

The connection as a matter of size (according to Happ, Waschkies and Seidel)

Happ and Waschkies are convinced “that Aristotle does that syneches finally with the hyle the megethi identified ”(Waschkies, p. 368). How can that be justified?

It is worth remembering Seidel's distinction between the two meanings of the number. Seidel generalizes this to the other dimensions.

Figure 3: Dimension and size

(3a) The one-dimensional straight line is divided by points into segments of equal length. Each point is of size 0 (it is not expanded). Each distance is of size 1 measured as the length of the distance. Inside, each section is contiguous, delimited by a point on the outside. If the points are counted for differentiation and designated as ›1, 2, 3, ...‹, the normal representation of the natural numbers results.

(3b) Areas are two-dimensional. Their edge is their shape. The size of the area is measured by its area (in the simplest case with rectangles of height times width), the size of the edge is the length of the outer line.

(3c) Bodies are three-dimensional. Their surface is their shape. The size of the body is measured by its volume and results from the internal distances, for example in the case of a cylinder from height times diameter times 2 times pi. The size of the shape is the area of ​​the surface.

The examples show that the outer border (the edge) is always one dimension smaller than the content. Since both the size of the content and the size of the border can be calculated and result in a number, the question arises whether there is a general formula to calculate the size of the content from the size of the border and vice versa the size of the border from the Calculate size of content. This is what differential and integral calculus do. Seidel can therefore simply say:

»The first derivation of the material is the form. (...) To descend ... to a form manageable for the physicist, we have the work of Isaac Newton, where the surface is the first derivative of the body, the path is the first derivative of the surface and the point is the first derivative of the path «(Seidel, pp. 241, 243).

Here Seidel deliberately skips the mathematical independence of material and form and puts material and form directly into a mathematical relationship: the form is one dimension smaller than the material. The form is the derivative of the substance and the substance is the integral of the form. (We will come back to this later when it comes to the squares of time and Kepler's laws, the relationship of which Seidel rightly sees foreseen in Aristotle.)

This statement, "the first derivation of the material gives the form" hits the real point, but has to be broken down into several sub-statements in order to avoid ambiguities and misunderstandings. Integral and derivative only exist within mathematics, when it is no longer matter and form that are considered, but their mathematical independence. On the basis of this, however, the following statements can be considered, between which there are internal analogies:

(a) In mathematics, the following applies: The derivative of f (x) = xn is f '(x) = n xn-1. From this simple formula, the rules of the differential calculus and its scope are explained step by step in mathematics.

(b) The derivation of a global figure of a function (for example the parabola as figure of the function f (x) = x²) gives the local figure at every single point x0 of this function. Mathematicians say: The whole function y = x² looks like a parabola, but at every single point it looks like a tangent. This can be generalized to: Local properties are the derivatives of global properties.

(c) The following applies within geometry and differential geometry: The derivation of the content of a figure results in the figure's edge. If the content is understood as the expanse of the figure and the edge of the figure, to put it simply, as "the figure", then one can say: The derivation of the expanse results in the figure. At this crucial point, the ambiguity should be noted: "edge of a figure" and "figure" are equated. - Instead of the "edge of a figure", the term "shape of a figure" is sometimes used with a similar ambiguity: "shape" and "shape of a figure" are usually linguistically equated.

(d) If quantities are defined in meta-mathematics as something that is internally connected and limited to the outside, then it can be said: The outer limit is the derivation of the inner connection, or, to put it more simply: the limit is the derivation of the Context.What lies outside the limit is no longer related to what lies within the limit. This is just another formulation of the definition of limit.

To make the confusion perfect, in geometry the size of a figure is usually understood to mean its content (for example, the size of a rectangle is the enclosed content and not the length of its edge), but in the theory of perception and on it Building on psychology and philosophy, the outer edge: An object is as big as I see (perceive) it. Since I see its outside (e.g. the height of a building) and not its inside, the (visible) outer limit is the size. - Therefore it can be said here: The size (the outer visible edge) is the derivation of the connection (that which is not visible inside and outside). Kant describes what is visible from the outside as the "phenomenon" (what appears) and what is not visible from inside as the "thing in itself." In this respect it can be said that the phenomenon is the derivation of the thing in itself.

(e) If finally the size is understood as the form and the interior as the substance, then the result is: The form is the derivative of the substance. Form and substance are principles of physics. If the mathematical concept of derivation is applicable to them, then with derivation, in the truest sense of the word, a mathematical principle of physics has been found, which was Newton's concern.

Aristotle was evidently aware of the ambiguities addressed here, but did not get around to clarifying it. Happ and Waschkies assume that he thought in this direction, but remained very cautious as long as the underlying terms were not yet clarified for him. There is no surviving text in which Aristotle systematically deals with this question, but Waschkies and Happ name two places where it is addressed:

»It could therefore seem that the form and shape of each one is his place, whereby the (spatial) size and the material of this size (e hyle e tou megethous) is limited; that is the (external) demarcation of each "Phys. IV.2, 209b4).

»So some come into doubt about the circle and the triangle, as if it didn't belong to them, through lines and through continuity (syneches), but ... lead everything back to numbers and claim that the concept of the line is that of two "(Met. VII.11 1036b).

This suggests that Aristotle created geometric figures such as the circle and triangle through continuity (syneches) and did not want to trace back to numbers, and that he was expressly of a substance of size (e hyle e tou megethous) and had the vague idea that the outer figure is the shape of this substance.

Happ comments:

“So in every size one distinguishes the determinable from the determinant. Since in our place no other body properties apart from the expansion are considered, this is hyle here pure indefinite expansion. This is raised to certainty by the final sentence of our section, which describes a method of abstraction: (The determinable hyle to name is correct), 'because if one considers a body (e.g.) a sphere) the limitation and its properties (pade) takes away, nothing is left but the hyle '. The result of this subtraction process is undoubtedly not the absolutely attributeless hyle, but the hyle as indeterminate poson, in other words: the hyle of mathematical objects ”(Happ, p. 606f).

"The hyle noete is therefore the unlimited mathematical expansion that is lifted out of the sensually perceptible by thinking and, as something thought, is not bound to the restrictions imposed by the cosmos aisteos according to the view of Aristotle is subject to "(Happ, p. 609).

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