diffyqs
Reals-as-Oracles
| diffyqs | Reals-as-Oracles | |
|---|---|---|
| 2 | 1 | |
| 53 | 1 | |
| - | - | |
| 8.8 | 6.9 | |
| 2 days ago | 15 days ago | |
| TeX | TeX | |
| GNU General Public License v3.0 or later | - |
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diffyqs
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Getting ready for next year specialization
253 uses clp3, and the textbook they recommend for 256 is https://www.jirka.org/diffyqs/
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Braille Readers
For LaTeX, I would first advise looking at the following section from the differential equations textbook I used as an example of what to expect. - Here is what is typed - Here is the readable output If learning that seems appealing, then I would recommend looking at the documentation on Overleaf, which provides a comprehensive introduction to the basics of the system. Note, though, that learning it can be a process, and I have no idea how well braille conversion of the source would go.
Reals-as-Oracles
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Math Limitations
I think having a language that helps understand those limitations is a useful achievement. Much of mathematics does have that. A notable exception is the definition of real numbers. They are usually presented as a string of infinite decimals, or a converging sequence, or a set of numbers less than something. All of those notions obscure the basic limitation of knowing the real number and give a veneer of similarity to rational number. Rational numbers are numbers that we can have in our hand while irrational numbers are ones which we can never have. It is important to have a setup that respects that difference.
This is what motivated me to come up with a new definition of real numbers, namely, they are objects (I call them oracles) that answer Yes or No when asked if the number ought to be between two given rational numbers. Abstracting out what properties such an object should have, one can come up with a space of these oracles, define an arithmetic, and prove that they satisfy the axioms of real numbers.
For details: https://github.com/jostylr/Reals-as-Oracles/
In many ways, this is giving a definitional support to the use of interval analysis which is, of course, a very practical concern. It also brings our some cool stuff about mediants and continued fractions (nothing new about that, but nicely motivated).
It also fits in with the adjacent post about busy beaver numbers and its conclusion about knowing a number is in an interval.
What are some alternatives?
ra - Basic Analysis, undergraduate real analysis textbook
ml-pen-and-paper-exercises - Pen and paper exercises in machine learning
maths_formulae - A LaTeX File with Maths Formulae from High School Maths and Common Core Engineering
LaTeX-examples - Examples for the usage of LaTeX
LiX - Compose papers, dissertations, textbooks, novels, newspapers etc. quicker using abstractions to the standard LaTeX document classes - or create your own classes easily.
maths_book - Planning for an entire maths LaTeX book
the_statistics_handbook - the statistics handbook open source repository
internal-methods - Notes on how to use the internal language of toposes in algebraic geometry