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The book is a historical investigation of the problem of infinity in Greek ontology and physics more specifically, the problem of the infinite size of the world and of.
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Rather the mind is able to consider the perceived object without some of its properties, such as being perceived, being made of sand, marble, bronze, etc. However, this is analogous to the logical manipulation of definitions, by considering terms with or without certain additions. Hence, Aristotle will sometimes call the material object, the mathematical object by adding on.

As a convenience, the mind conceives of this as if the object were just that. On this view, abstraction is no more and no less psychological than inference. Conceptually, we might think of the process as the mind rearranging the ontological structure of the object.

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As a substantial artifact, what-it-is, the sand box has certain properties essentially. The figure drawn may be incidental to what-it-is, i. In treating the object as the figure drawn, being made of sand is incidental to it. This is the concept which does most of the work for Aristotle. In his discussions of precision, Aristotle states that those sciences which have more properties removed are more precise.

Arithemetic, about units, is more precise than geometry, since a point is a unit having position. A science of kinematics geometry of moving magnitudes where all motion is uniform motions is more precise than a science that includes non-uniform motions in addition, and a science of non-moving magnitudes geometry is more precise than one with moving magnitudes. Aristotle solves the separability problem with a kind of fictionalism. The language and practice of mathematicians is legitimate because we are able to conceive of perceptible magnitudes in ways that they are not.

The only basic realities for Aristotle remain substances, however we are to conceive them. A primary characteristic of substances is that they are separate.

Michael Forster: "to apeiron" in the Cosmology of Anaximander

It is a subject in our science in our discourse in the science. The mental and logical mechanism by which we accomplish this is the core of Aristotle's strategy in diffusing platonisms. The English adverb is normally followed by a noun phrase. In the case where we examine or study an object X qua Y or X in-the-respect-that X is Y , we study the consequences that follow from something's being an Y.

In other words, Y determines the logical space of what we study. If X is a bronze triangle a perceptible magnitude , to study X qua bronze will be to examine bronze and the properties that accrue to something that is bronze. To study X qua triangle is to study the properties that accrue to a triangle. Unless it follows from something's being a triangle that it must be bronze, the property of being bronze will not appear in one's examination. For example, Aristotle says De anima iii.

We can study a perceptible triangle qua triangle because it is a triangle. For convenience, we can call this principle qua -realism. We begin with perceptible magnitudes. These are volumes, surfaces, edges, and corners. They change in position and size. They are made of some material and are the quantities of substances and their interactions.

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The volumes, surfaces, and edges have shape. Times and corners do not. Different sciences treat different perceptible magnitudes qua different things. Moreover, since there are many perceptible magnitudes, there will be enough, qua line, to prove any theorem that involves lines.

The plurality problem is trivially solved. The science will speak of Y. This too will not interfere with mathematical practice and so will not violate non-revisionism. In Metaphysics vi. We can now characterize the way in which mathematical objects are eternal and lack change. Namely, generation and change are not among the predicates studied by geometry or arithmetic. Hence, it is correct to say that qua lines, perceptible lines lack generation, destruction, and change with appropriate provisos for kinematics and mathematical astronomy.

Whether the precision problem is also solved and how it is solved is more controversial. On the ancient and medieval interpretation, the problem of precision is solved by allowing mental representations to be as precise as one chooses. The contemporary interpretation of considering Aristotle's mathematical objects as physical object treated in a special way has a more difficult task.


Aristotle and Mathematics

There are five ways in which Aristotle may attempt to solve the precision problem. Our difficulty is that while Aristotle raises the problem of precision, he does not explicitly explain his solution to it.

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  8. Perceptible magnitudes have perceptible matter. A bronze sphere is a perceptible magnitude. For solving the plurality problem, Aristotle needs to have many triangles with the same form. Since perceptible matter is not part of the object considered in abstraction or removal , he needs to have a notion of matter which is the matter of the object: bronze sphere MINUS bronze perceptible matter. Since this object must be a composite individual to distinguish it from other individuals with the same form, it will have matter.

    He calls such matter intelligible or mathematical matter.

    A brief synopsis of the notion of "Infinite" (Apeiron) in early Greek philosophy

    Aristotle has at least four different conceptions of intelligible matter in the middle books of the Metaphysics , Physics iv, and De anima i:. Since Aristotle's concern in discussing 4 is with the nature of the parts of definitions and not with questions of extended matter, it is unclear whether the non-definitional parts are potential extended parts or merely forms of extended parts, although the former seems more plausible. Ancillary to his discussions of being qua being and theology Metaphysics vi.

    Aristotle reports Posterior Analytics. Metaphysics xiii.

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    The discovery of universal proofs is usually associated with Eudoxus' theory of proportion. For Aristotle this creates a problem since a science concerns a genus or kind, but also there seems to be no kind comprising number and magnitude. Aristotle seems more reticent, describing the proofs as concerning lines, etc. He seems to identify such a super-mathematics Metaphysics vi. Another possibility is that the common science has theorems which apply by analogy to the different mathematical kinds.

    Elsewhere Metaphysics xi. If so and if this is by Aristotle, it would correspond to the general theory of proportions as it comes down to us. One may well wonder if scholars have been led astray by a hyperbole about universal proofs. Ironically, extant Greek mathematics shows no traces of an Aristotelian universal mathematics. The theory of ratio for magnitudes in Euclid, Elements v is completely separate from the treatment of ratio for number in Elements vii and parts of viii, none of which appeals to v, even though almost all of the proofs of v could apply straightforwardly to numbers.

    For example, Euclid provides separate definitions of proportion v def. Compare the rule above alternando ,which is proved at v. In Plato's Academy, some philosophers suggested that lines are composed of indivisible magnitude, whether a finite number a line of indivisible lines or a infinite number a line of infinite points. Aristotle builds a theory of continuity and infinite divisibility of geometrical objects. Aristotle denies both conceptions. Yet, he needs to give an account of continuous magnitudes that is also free from paradoxes that these theories attempted to avoid.

    The elements of his account may be found principally in Physics iv. Aristotle's account pertains to perceptible magnitudes. However, it is clear that he understands this to apply to magnitudes in mathematics as well. Aristotle has many objections to thinking of a line as composed of actual points likewise, a plane of lines, etc.

    To say that a line is comprised of an infinity of potential points is no more than to say that a line may be divided with a line-cutter, with the mind, etc.

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    The continuity of a line consists in the fact that any actualized point within the line will hold together the line segments on each side. Otherwise, it makes no sense to speak of a potential point actually holding two potential lines together. Suppose I have a line AB and cut it at C. The lines AC and CB are distinct. Is C one point or two? This merely means that we can treat it once or twice or as many times as we choose.

    Note that Aristotle says the same thing about a continuous proportion. Greek mathematicians tend to conceive of number arithmos as a plurality of units. Their conception involves:. For Aristotle and his contemporaries there are several fundamental problems in understanding number and arithmetic:. Aristotle presents three Academic solutions to these problems. Units are comparable if they can be counted together such as the ten cows in the field. Units are not comparable, if it is conceptually impossible to count them together a less intuitive notion. Greeks used an Egyptian system of fractions.

    Additionally, Greeks used systems of measure, as we do, with units of measure being divided up into more refined units of measure. This feature of measure may be reflected in Plato's observation Rep. Since Aristotle esp.