Threshold & Discontinuity
“Continuity becomes a problem when you try to figure out what it means to be “between.” [1]
Where and how does distinctiveness arise out of simplicity (homogeneity)? The idea of “threshold” allows a form of answer to this question to begin . Our perception depends upon distinction. The distinctions created by acts of perception necessarily recognize (because they depend upon) both sides of a threshold. Afterall, a threshold is constituted by the marking off of surrounding contents from each other by watersheds of difference. This marking off is a kind of metaphysical cut in which reality is both held together and separated in a single act of recognition. It is based upon the existence of discontinuity within what is, and in turn, defines what the existence of discontinuity demands; given that the being of what is possesses the character of being whole.
Modernism has been struggling with this problem through its history. Seurat attempted to paint the discontinuity of impression. Each daub of paint standing with isolated independence, while at the same time acting as a threshold joining neighbouring daubs together into a stable-tending complex. In a similar manner, Father Gregor (Johann Mendel) found genetic discontinuity from within the statistical regularity of the Law of Segregation. He showed that the traits or “factors” paired in successive generations of peas do not shade into one another as Darwin had claimed they would. Instead they continue to appear or not appear independently; “segregated” from each other. There is no homogeneous mixing of hereditary material; no smooth continuity. What there is however, is genetic implication in spite of the disjunction of its transferrence.
This kind of action was most potently described in the 19th century by the mathematician Richard Dedekind. The Dedekind Cut is a mathematical definition of irrational numbers which separates the digital from the continuous in arithmetic.[2] Neither whole numbers (0, 1, 2, 3 …) nor rational ratios are continuous. There always exist more of the same between them. Irrational numbers also exist between. However, these have the further difficulty that they are impossible to distinguish positively. How can the sequence of real numbers extend continuosly? Without “space” between succeeding numbers, (if you can always find more numbers between any two in the sequence) it would seem that no two numbers can define locations which are truly side by side.[3] The genius of Dedekind was the realization that this difficulty defined the very nature of reality. Each arithmetic separation was a threshold which both cut and maintained continuity. By itself it promised an account of smooth change within a discontinuously experienced multiplicity of difference.
Likewise, change is also a conundrum of the discontinuous. Without continuity, how can change come about between entities? What about entropy? It describes a process of change: between singularity and multiplicity, between the central and the peripheral. Max Plank’s so called “elementary quantum of action” describes a distinct limit to the size (smallness) of step in the increase or decrease of energy. The quantum (energy “atom”) proves that energy does not change in a smooth or continuous manner. It itself is proved by its necessity as a constant in the equation satisfying Boltzmann’s conception of entropy as the sum of logarithms of probabilities; the probabilities under consideration being different combinations of distinct bits and pieces. In other words, entropy cannot be described as a continuity either. Instead it is a process described by quantum changes which themselves result in a continuity of directedness . . . a directedness seemingly aimed at producing an ever devolving distinctiveness from the once homogeneous singularity of physical existence.
[1] William R. Everdell, The First Moderns: Profiles in the Origins of Twentieth-Century Thought (Chicago: The University of Chicago Press, 1997): 33.
[2]His theoretical cut states that if a function (number field) can be cut at any point or number so that all remaining values are either greater or smaller than that point/number, and that same point can belong equally to either set, then the interval is continuous by definition. It is not necessary to examine numbers themselves to determine if they are between or next to other numbers.
[3]This language seems to make numbers out to be more than they are - as if they had breadth and were not mere abstract locators of infinitely small points. Numbers, as points, are by definition cuts in reality.
“Continuity becomes a problem when you try to figure out what it means to be “between.” [1]
Where and how does distinctiveness arise out of simplicity (homogeneity)? The idea of “threshold” allows a form of answer to this question to begin . Our perception depends upon distinction. The distinctions created by acts of perception necessarily recognize (because they depend upon) both sides of a threshold. Afterall, a threshold is constituted by the marking off of surrounding contents from each other by watersheds of difference. This marking off is a kind of metaphysical cut in which reality is both held together and separated in a single act of recognition. It is based upon the existence of discontinuity within what is, and in turn, defines what the existence of discontinuity demands; given that the being of what is possesses the character of being whole.
Modernism has been struggling with this problem through its history. Seurat attempted to paint the discontinuity of impression. Each daub of paint standing with isolated independence, while at the same time acting as a threshold joining neighbouring daubs together into a stable-tending complex. In a similar manner, Father Gregor (Johann Mendel) found genetic discontinuity from within the statistical regularity of the Law of Segregation. He showed that the traits or “factors” paired in successive generations of peas do not shade into one another as Darwin had claimed they would. Instead they continue to appear or not appear independently; “segregated” from each other. There is no homogeneous mixing of hereditary material; no smooth continuity. What there is however, is genetic implication in spite of the disjunction of its transferrence.
This kind of action was most potently described in the 19th century by the mathematician Richard Dedekind. The Dedekind Cut is a mathematical definition of irrational numbers which separates the digital from the continuous in arithmetic.[2] Neither whole numbers (0, 1, 2, 3 …) nor rational ratios are continuous. There always exist more of the same between them. Irrational numbers also exist between. However, these have the further difficulty that they are impossible to distinguish positively. How can the sequence of real numbers extend continuosly? Without “space” between succeeding numbers, (if you can always find more numbers between any two in the sequence) it would seem that no two numbers can define locations which are truly side by side.[3] The genius of Dedekind was the realization that this difficulty defined the very nature of reality. Each arithmetic separation was a threshold which both cut and maintained continuity. By itself it promised an account of smooth change within a discontinuously experienced multiplicity of difference.
Likewise, change is also a conundrum of the discontinuous. Without continuity, how can change come about between entities? What about entropy? It describes a process of change: between singularity and multiplicity, between the central and the peripheral. Max Plank’s so called “elementary quantum of action” describes a distinct limit to the size (smallness) of step in the increase or decrease of energy. The quantum (energy “atom”) proves that energy does not change in a smooth or continuous manner. It itself is proved by its necessity as a constant in the equation satisfying Boltzmann’s conception of entropy as the sum of logarithms of probabilities; the probabilities under consideration being different combinations of distinct bits and pieces. In other words, entropy cannot be described as a continuity either. Instead it is a process described by quantum changes which themselves result in a continuity of directedness . . . a directedness seemingly aimed at producing an ever devolving distinctiveness from the once homogeneous singularity of physical existence.
[1] William R. Everdell, The First Moderns: Profiles in the Origins of Twentieth-Century Thought (Chicago: The University of Chicago Press, 1997): 33.
[2]His theoretical cut states that if a function (number field) can be cut at any point or number so that all remaining values are either greater or smaller than that point/number, and that same point can belong equally to either set, then the interval is continuous by definition. It is not necessary to examine numbers themselves to determine if they are between or next to other numbers.
[3]This language seems to make numbers out to be more than they are - as if they had breadth and were not mere abstract locators of infinitely small points. Numbers, as points, are by definition cuts in reality.
posted by haro d' aires at 4:55 a.m.
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