Architect of Ideas - Some Thoughts About Calculus and Infinity

This is what happens when people take ideas seriously. Ideas like Toby's one, that a proof in maths is a story, and you are free to go wherever you like with it.

Here's my comment:

Here's a way you can develop your idea of "there's always one more" as a definition of infinity. You could define "infinitely long" as "there''s always one longer" so if you have a line of length N then you can double it and get a line of length 2N, and there's always a longer one. Then you can think of infinitely short as "there's always one shorter" So if you have a line of length N then there's a shorter one of length N/2 which half as long. And then you might think of ratios of lines, and products of lines, which are areas. I think this is a bit closer to the way the ancient Greeks thought about arithmetic, because they were putting everything in terms of geometric constructions. So there is one book of Euclid's Elements which is all about this. It's Book II. Maybe if you mount your phone above you, you could use a mirror on the desk to show you in the same frame as what you're drawing.

11:44 Janet's taking about "That moment when you pass through zero" and relating it to Universes and Black Holes. See Peter Voit and Edward Frenkel on Unification in Physics and Mathematics.

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David Lynch made me a phone holder for my birthday present in 2020: What Is David Working on Today? 6/26/20 - Phone Holder

See Wisteria.

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See Toby's video on "Measurement" by Paul Lockhart.

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My comment there was: She calculates the geometric mean I mentioned above with Algebra! And she gets a fundamental theorem: flower times (flower minus one) is one. Now if you're a madman in the rainforest in Bolivia it's easy to see that you can dualise that to be infinity, which is when flower divided by (flower plus one) is one, because when your flower is infinite then flower plus one is flower.

What might be better would be me for me to try to rewrite that story so that it fits better with Toby's Algebra and Janet's idea of infinity.

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See Turn Up And Write - Plotting.

See Carol Wood on model theory (which is when you seriously get into the semantics of this stuff) in Andrej Bauer's "Five Stages of Accepting Constructive Mathematics":

See Compactness theorem:

In mathematical logic, the compactness theorem states that a set of first-order sentences has a model if and only if every finite subset of it has a model. This theorem is an important tool in model theory, as it provides a useful (but generally not effective) method for constructing models of any set of sentences that is finitely consistent.

Yesterday evening I was listening to piece of music that sounded to me like a good representat ion of all these little creatures scuttling around the gaps in a really huge structure. Here it is:

Valentina Ciniglio · Alf Leyla We Leyla/nel cuore della notte mordo il tempo

Carol Wood was a student of Abraham Robinson.

... [who] became known for his approach of using the methods of mathematical logic to attack problems in analysis and abstract algebra. He "introduced many of the fundamental notions of model theory". Using these methods, he found a way of using formal logic to show that there are self-consistent nonstandard models of the real number system that include infinite and infinitesimal numbers. Others, such as Wilhelmus Luxemburg, showed that the same results could be achieved using ultrafilters, which made Robinson's work more accessible to mathematicians who lacked training in formal logic. Robinson's book Non-standard Analysis was published in 1966. Robinson was strongly interested in the history and philosophy of mathematics, and often remarked that he wanted to get inside the head of Leibniz, the first mathematician to attempt to articulate clearly the concept of infinitesimal numbers.

An Ultrafilter is a concept in order theory where there is a duality between filters and ideals. There an Utrafilter corresponds to a Maximal Ideal. See Norman Wildberger and Daniel Tubbenhauer on Formal and Structural Coincidents.

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As for my book, Standard ML For The Lady Programmer: I thought I was getting there:

... but it's all gone wrong. I got the STeX3 IDE to work once, and I could browse the documentation. It worked quite well, but then when I closed the window and restarted it again it starts the process of "Indexing files" and never finishes. By never, I just mean that after an hour I gave up waiting for it to finish!