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According to Whitehead’s theory of cosmic epochs and extensive continuum, there are material conditions that drive the more general metaphysical descriptions and principles. We only can account for the factors which characterize our cosmic epoch without overstretching these results to the cosmos in toto.

Unlike many philosophers and scientists, Whitehead views time and space as contingent features of reality which could be otherwise; they are specifications of divisibility accounting for certain respective regions. The extensive continuum is “undivided, yet divisible” in its nature and accords with Whitehead’s bi-metric geometrical framework. But this continuum presupposes a notion of extensiveness or regional togetherness that is non-metrical. Essentially, space and extension are not equivalent.

So temporal and spatial cuts or divisions happen to characterize our cosmic epoch, and this implies that space-time is NOT a necessary ontological modality applicable to the entire assemblage of solar systems. In fact, the author of Principia Mathematica concludes in another epoch it may not be the case that one and one makes two.

Pop culture science continues to advance a hubris which ignores these provisions. They claim to be measuring and accounting for the experience of the universe, in itself. A more modest proposal claims that our laws and discoveries unfold according to the epochal theory of time.

So my question is: how can we apply arithmetic in its pure metaphysical sense without recognizing that the truth of its character depends on the “societies dominating the cosmic epoch in which we live”?

I believe this is not some philosopher splitting hairs but raises a pertinent concern that even mathematics (especially the sciences) in its basic solutions should be open to the pragmatic method of indefinite revisions.

Appreciate your feedback and suggestions—thank you!

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    I'm not familiar with Whitehead's writings on these issues, but it seems like the question you're asking makes independent sense. Can I try to paraphrase the question without reference to Whitehead's specific attitude to it? "If what mathematical facts there are is supposed to depend on facts about the empirically observed world, then is it legitimate to take the logical form of mathematical claims at face value?" Does that sound okay? – Paul Ross Aug 12 '12 at 4:52
  • what would the science look like that accounts for the non-equivalence described by Whitehead? how does ignoring it engender the question " how can we apply arithmetic in its pure metaphysical sense without recognizing the truth of its character depends on the “societies dominating the cosmic epoch in which we live”?"? in what way are they dominating it? do you mean in the sense of statistics coming from staat, from state calculation? – Dr Sister Aug 17 '12 at 15:07
  • Whitehead had the misfortune of developing a philosophy with an eye on physics right before its foundations were upended by quantum mechanics. His metaphysics became obsolete almost at inception, his views of time and space are too tied to classical physics. As for mathematics being open to indefinite revisions, it is not specific to Whitehead at all, it was suggested before him (by Mill, Peirce, etc.) and after (by Quine and most analytic naturalists). But exactly for pragmatic reasons why should we care to "apply arithmetic in its pure metaphysical sense", if that even means anything? – Conifold Aug 26 '18 at 23:32
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The ideas you are entertaining are as current today as they were when written. Whitehead's essential concept of Epochs of history, where basic physics and math principles change epoch to epoch, is the cosmology that is proposed in Smolin and Unger's The Singular Universe and the Reality of Time https://www.amazon.com/Singular-Universe-Reality-Time-Philosophy/dp/1107074061.

Smolin's approach to mathematics is radically nominalist, in that math could have been practically anything, and the math that we find useful is the math that actually matches the fundamental structures of our universe. In earlier and later epochs, any residents of our universe will discover different mathematics laws and principles. https://arxiv.org/pdf/1506.03733.pdf

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