They all have at least one thing in common: each time we extend our reach in terms of scope or depth, we find new ways of seeing things which we saw before and/or see new things which were before not seen. There are many more of these various patterns which I may write about in the future. Even the idea of a binary relationship is being transcended into finitary versions (which I use in my work). We are beginning to ask questions which require answers which transcend relationship properties such as causality, equivalence, and inference in all of their forms. Then we found other questions which led to boundary, multi-valued, fuzzy, and fractal logics, among a few others I haven’t mentioned yet.Įven our validity claims are evolving. Logic was, for example, limited to quite sophisticated methods that all were restricted to a boolean context. The framing of our answers to mathematical questions is also evolving. This pattern continues in other ways: Holors, for example, which extend and include mathematical entities such as Complex numbers, scalars, vectors, matrices, tensors, Quaternions, and other hypercomplex numbers, yet are still capable of providing a different algebra which is consistent with real algebra. We also know there are abstractions and/or extensions of Complex numbers where the ‘air starts to get thin’ and mathematical properties start to ‘fade away’: Quaternions, Octonians, Sedenions,… This pattern continues in other ways, as well. They have both been found, because the domain of questioning exceeded the range of answers being sought within the properties each of those numbers. There are also numbers which lay outside of R which we call Imaginary numbers. We all know there are numbers which lay outside of Q which we call Irrational numbers. I’ll introduce you to a pattern you may have already noticed in the current stage of our mathematical endeavour. We need to extend our examination of mathematical knowledge, both in depth and in scope, out farther and in deeper than numbers (sets and categories as well – even more below) have yet done. It is my firm belief that mathematics will overcome this current situation and eventually be quite able to examine these kinds of questions in a much more expansive and deeper way. I am not criticising mathematics in any way. Mathematics has been, and always shall be my ‘first love’, and it has provided for me all of these years. These constants, identities, and variations being referred to in this post, and others like it, all lay embedded in a far deeper substrate than current mathematics has yet explored. The reason why it is so hard to prove is actually very easy to answer. Why is it so hard to prove that e+pi or e*pi is irrational/rational?
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