Date: Mon, 03 Aug 1998 23:47:26 +0300 From: Yair Mahalalel <yairm-AT-tabs.co.il> Subject: Re: Points and the metaphysics of geometry LC wrote: > Points, the way most people employ the term, is _truly_ a metaphysical > notion devoid of concept or of function. Lines do not have to exist > because of an infinitesimally divisible nature of the line into points > that would leave gaps per force exposed as one's resolution of the line > would increase. Nor is the line composed of an infinity of points and > gaps, or even absent points. It is the line which must be thought of > independently of any points, since the line is the reality of the > continuous, and the mathematical point truly a metaphysical nothing. The mathematical point is a definition. It is defined by different means in different mathematical systems. For example, in Cartesian geometry a point can be an ordered set of numbers that determine its location in a predefined coordinate space. It can conversely be defined by the intersection of N subspaces (N being the dimension of the space), each with dimension N-1, such as the intersection of two straight lines on a surface or the intersection of three flat planes in a three dimensional space. Since it is merely a definition in a closed logical/mathematical system it cannot be a metaphysical anything - It does not impose anything upon reality, it doesn't even mean that such a point exists. As for the line, we do not have to think about it at all, with or without gaps. The word has many uses (my line of thought contains many gaps, I'm sure, in any resolution :)), even in mathematics. If we'd like to return to Cartesian geometry, a (straight) line is usually related a linear function, and this set of functions has been proved to comply with yet another mathematical definition termed "continuous". And the proof involves non discrete set of discrete series of numbers, or points. All this does not contradict the statement that lines do not have to exist /because/ of the points. You are more than welcome to devise a mathematical system which defines lines and continuity without any use of points (or knots!), but I can hardly see any benefit in a geometrical system where lines cannot have ends and cannot intersect with anything. > Further even, points do not exist as unidimensional (or worse still, > dimensionless) elements. Agreed. It escapes my imagination how a mathematical point can exist, not to say measured or sensed. By the way, a unidimensional object is usually termed a line. > Points are not numbers and numbers points. This is of course true in the pure literal meanings of the words. Many mathematical systems, however, use numbers to specify the location of a geometrical point, or use the term point to denote a dimensionless subspace of another space. > Numbers mean nothing unless they have dimensionality. Numbers have no synthetic meaning. They are a pure "unreal" abstraction that have no place other than Plato's warehouse of lines, triangles, cats, objects and subjects. Yet pure numbers are far from being meaningless, they are useful crutches. The meaningless statement 2+2=4 is a powerful meaningless abstraction that can be applied to a variety of meaningless countable objects and accomplish a variety of meaningless objectives. > As Reich was fond > of saying, there is no zero in natural research. The only possible > concept of a point is as the intersection of two or more lines - such > points are called knots - and the whole problem of Euclidean Geometry > and Cartesianism stems from confusing point-knots with dimensionless > points in order to construct a line; point-knots are not elements of a > line, but the topological product of the synthesis of lines. It is of > these point-knots that one can say that a line passes between them or in > the gaps between them, and not _through_ the points _and_ their gaps. > Only the _concept_ of molar line subordinates the line to the point. What is this obsession with lines? Why is a dimensionless point such an enemy of the public while the unidimensional line is the savior of the day? Do you find the existence of a mathematical line more comprehensible than a mathematical point? Why is our experience of finite three dimensional discrete objects more likely to accept objects with zero length and zero width than objects that also have zero height? Mathematical abstractions are just that, do you also want to discard infinities because they cannot be verified on the basis of empirical data? As for the mathematical line - it is *not* molar! It is not constituted by an infinity of countable discrete points. It is constituted by a non countable set of infinite converging series. Is the simple difference between Aleph 0 and Aleph 1 so difficult to grasp?! > Question: does a striated space exist other than through and in the > mind, as representation or as plane of transcendence? Answer: No. Question: Does the continuous one exist as such? > > Lambda C > Yair. --- from list nietzsche-AT-lists.village.virginia.edu ---
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