Here's an interesting lecture on philosophical problems with modern mathematics. What kind of world is it where all mathematics students aren't shown the construction of the Stern-Brocot Tree and its connection with Continued fractions and Farey sequences?
The problem, as Aristotle pointed out quite clearly over a couple of thousand years ago, is that we shouldn't expect to be able to prove anything about a whole from logically prior parts. That's not how logic works. See On Indivisible Lines: page one:
Aristotle wrote something once, somewhere, which was something along the lines of:
I will add that I once actually programmed such a construction as described above using the "Higher Order Functors" in Moscow ML. I do not have any copy of the code, however. I suspect that there quite a few copies around, because I distributed multiple USB memory sticks containing backups of all my work in mid April 2011. See Catholic Church and 2009 Coup D'Etat in Honduras. I suspect the Mormons of having had as much to do with this as any others. See Abuse Helplines Used to Cover Up Abuse and The Mormons, the FBI, the CIA and Soviet Spies. See also Whitney Webb on US Government Ties to Organised Crime and then you will understand why Justice is so elusive in today's world: see Lori on the Fake Economy.
The idea of starting at a whole and deriving constructions as a process of synthesizing it from elements produced by dividing the whole is inherent in Islamic geometry. See this post: Geometry in Islamic Art, in particular, this video, from 43 seconds:
The problem, as Aristotle pointed out quite clearly over a couple of thousand years ago, is that we shouldn't expect to be able to prove anything about a whole from logically prior parts. That's not how logic works. See On Indivisible Lines: page one:
Aristotle wrote something once, somewhere, which was something along the lines of:
What is it that pluralises the one and makes it many?And his answer was something along the lines of:
That which pluralises the one and makes it many is error and the nature of error.So, if you want a sound logical basis for mathematics, maybe look at Gauss' error function, and a co-construction of some sort of abstract space and its dual, for example, a normed vector space, whose dual is some sort of Fourier transform, and look for way of building in the steps of that construction by encoding computations into elements of the space. If such computations include iterative proof searches, for example, then you could still hope to produce a coherent whole without needing to know in advance all such questions such as "Is the Riemann hypothesis true", or "Is PA2 consistent". This is what I tried to express here: On Tarski's Definition of Truth "Convention-T". For further discussion, see this post: Logic, in particular this brief note on constructivism, intuitionism and foundations which was the result of an attempt to engage someone in a rational discussion on this subject. The attempt was a failure, I hardly need to add, ...
I will add that I once actually programmed such a construction as described above using the "Higher Order Functors" in Moscow ML. I do not have any copy of the code, however. I suspect that there quite a few copies around, because I distributed multiple USB memory sticks containing backups of all my work in mid April 2011. See Catholic Church and 2009 Coup D'Etat in Honduras. I suspect the Mormons of having had as much to do with this as any others. See Abuse Helplines Used to Cover Up Abuse and The Mormons, the FBI, the CIA and Soviet Spies. See also Whitney Webb on US Government Ties to Organised Crime and then you will understand why Justice is so elusive in today's world: see Lori on the Fake Economy.
The idea of starting at a whole and deriving constructions as a process of synthesizing it from elements produced by dividing the whole is inherent in Islamic geometry. See this post: Geometry in Islamic Art, in particular, this video, from 43 seconds:
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