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Peirceverance
Peirceverance
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I think that theThe suggested solution you suggest resolves Zeno's paradox! It is very similar to Whitehead's argument presented in Part I of Process and Reality and presupposed in the Theory of Extension in Part IV. If you recall from Aims of Education he holds that a line is a "moving point" while a "plane is a moving line." Along with von Neumann's work on mathematical rings, Whitehead relies on a continuous, point-free projective geometry of non-finite dimension. Whitehead's "extensa" or extensive structures are logically and metaphysically prior to time and space as topological relational systems (mathematically speaking, they may be seen as spatial, but this pertains to "spatial reasoning"). This allows for the openness in which "universal relativity" (not to be confused with Einstein's as only a single application) and "atomicity" are compatible. He writes:

I think thisTherefore, Zeno's paradox is a definitive claim and is consistent with what you include with some slight variation. These distinctions may seem petty but they are crucial for the methodology of a post-Kantian speculative cosmology."mathematical fallacy" as Whitehead explains:

In his ‘Achilles and the Tortoise’ Zeno produces an invalid argument depending on ignorance of the theory of infinite convergent numerical series. Eliminating the irrelevant details of the race and of motion—details which have endeared the paradox to the literature of all ages—consider the first half-second as one act of becoming, the next quarter-second as another such act, the next eighth-second as yet another, and so on in-definitely. Zeno then illegitimately assumes this infinite series of acts of becoming can never be exhausted. But there is no need to assume that an infinite series of acts of becoming, with a first act, and each act with an immediate successor,† is inexhaustible in the process of becoming. Simple arithmetic assures us that the series just indicated will be exhausted in the period of one second. The way is then open for the intervention of a new act of becoming which lies beyond the whole series. Thus this paradox of Zeno is based upon a mathematical fallacy. The modification of the ‘Arrow’ paradox, stated above, brings out the principle that every act of becoming must have an immediate successor, if we admit that something becomes. For otherwise we cannot point out what creature becomes as we enter upon the second in question. But we cannot, in the absence of some additional premise, infer that every act of becoming must have had an immediate predecessor.The conclusion is that in every act of becoming there is the becoming of something with temporal extension; but that the act itself is not extensive, in the sense that it is divisible into earlier and later acts of becoming which correspond to the extensive divisibility of what has become (PR, 69, emphasis added).

I think that the solution you suggest resolves Zeno's paradox! It is very similar to Whitehead's argument presented in Part I of Process and Reality and presupposed in the Theory of Extension in Part IV. If you recall from Aims of Education he holds that a line is a "moving point" while a "plane is a moving line." Along with von Neumann's work on mathematical rings, Whitehead relies on a continuous, point-free projective geometry of non-finite dimension. Whitehead's "extensa" or extensive structures are logically and metaphysically prior to time and space as topological relational systems (mathematically speaking, they may be seen as spatial, but this pertains to "spatial reasoning"). This allows for the openness in which "universal relativity" (not to be confused with Einstein's as only a single application) and "atomicity" are compatible. He writes:

I think this is a definitive claim and is consistent with what you include with some slight variation. These distinctions may seem petty but they are crucial for the methodology of a post-Kantian speculative cosmology.

The suggested solution is very similar to Whitehead's argument presented in Part I of Process and Reality and presupposed in the Theory of Extension in Part IV. If you recall from Aims of Education he holds that a line is a "moving point" while a "plane is a moving line." Along with von Neumann's work on mathematical rings, Whitehead relies on a continuous, point-free projective geometry of non-finite dimension. Whitehead's "extensa" or extensive structures are logically and metaphysically prior to time and space as topological relational systems (mathematically speaking, they may be seen as spatial, but this pertains to "spatial reasoning"). This allows for the openness in which "universal relativity" (not to be confused with Einstein's as only a single application) and "atomicity" are compatible. He writes:

Therefore, Zeno's paradox is a "mathematical fallacy" as Whitehead explains:

In his ‘Achilles and the Tortoise’ Zeno produces an invalid argument depending on ignorance of the theory of infinite convergent numerical series. Eliminating the irrelevant details of the race and of motion—details which have endeared the paradox to the literature of all ages—consider the first half-second as one act of becoming, the next quarter-second as another such act, the next eighth-second as yet another, and so on in-definitely. Zeno then illegitimately assumes this infinite series of acts of becoming can never be exhausted. But there is no need to assume that an infinite series of acts of becoming, with a first act, and each act with an immediate successor,† is inexhaustible in the process of becoming. Simple arithmetic assures us that the series just indicated will be exhausted in the period of one second. The way is then open for the intervention of a new act of becoming which lies beyond the whole series. Thus this paradox of Zeno is based upon a mathematical fallacy. The modification of the ‘Arrow’ paradox, stated above, brings out the principle that every act of becoming must have an immediate successor, if we admit that something becomes. For otherwise we cannot point out what creature becomes as we enter upon the second in question. But we cannot, in the absence of some additional premise, infer that every act of becoming must have had an immediate predecessor.The conclusion is that in every act of becoming there is the becoming of something with temporal extension; but that the act itself is not extensive, in the sense that it is divisible into earlier and later acts of becoming which correspond to the extensive divisibility of what has become (PR, 69, emphasis added).

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Peirceverance
Peirceverance
  • 2.2k
  • 13
  • 22

I think that the solution you suggest resolves Zeno's paradox! It is very similar to Whitehead's argument presented in Part I of Process and Reality and presupposed in the Theory of Extension in Part IV. If you recall from Aims of Education he holds that a line is a "moving point" while a "plane is a moving line." Along with von Neumann's work on mathematical rings, Whitehead relies on a continuous, point-free projective geometry of non-finite dimension. Whitehead's "extensa" or extensive structures are logically and metaphysically prior to time and space as topological relational systems (mathematically speaking, they may be seen as spatial, but this pertains to "spatial reasoning"). This allows for the openness in which "universal relativity" (not to be confused with Einstein's as only a single application) and "atomicity" are compatible. He writes:

The creatures are atomic [i.e., undivided]. In the present cosmic epoch there is a creation of continuity. Perhaps such creation is an ultimate metaphysical truth holding of all cosmic epochs, but this does not seem to be a necessary conclusion. The more likely opinion is that extensive continuity is a special condition arising from the society of creatures which constitute our immediate epoch (PR, 35-6).

This compatibility allows for the "becoming of continuity" and not a "continuity of becoming" as is assumed in Zeno's paradox. Under the conditions of our epoch

The extensive continuity of the physical universe has usually been construed to mean that there is a continuity of becoming. But if we admit that ‘something becomes,’ [in the physical sense,] it is easy, by employing Zeno’s method, to prove that there can be no continuity of becoming. [Where we are speaking exclusively of the way that actual entities, considered physically rather than logically, divide the extensive continuum there is a becoming of continuity, but no continuity of becoming. [Under those assumptions, the actual occasions are the creatures which become, and [to that same extent, in the physical sense only] they constitute a continuously extensive world. In other words, [physically speaking,] extensiveness becomes, but ‘becoming’ is not itself extensive (PR, 35).

I think this is a definitive claim and is consistent with what you include with some slight variation. These distinctions may seem petty but they are crucial for the methodology of a post-Kantian speculative cosmology.