maths_book
Reals-as-Oracles
| maths_book | Reals-as-Oracles | |
|---|---|---|
| 4 | 1 | |
| 194 | 1 | |
| - | - | |
| 2.9 | 6.9 | |
| 11 months ago | 19 days ago | |
| TeX | TeX | |
| Creative Commons Zero v1.0 Universal | - |
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maths_book
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LaTeX for books?
Yes, I'm working on one these days: https://github.com/pelegs/maths_book
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People seemed to like this at r/asktransgender, so I'm sharing it here too: trans tapir, a figure in a mathematics book I'm currently working on
Hihi. The first chapter of the book is almost 100% done, and the rest is getting written almost daily. You can find the most updated version here: https://github.com/pelegs/maths_book/raw/main/bookmain.pdf (I will keep the book free and open source)
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Illustration in a math book using the trans pride flag for a pun: appropriate?
For context: I use tapirs as a theme throughout the book, e.g. as chapter headings, hence the trans tapir pun. The book is open source and will remain free for everyone.
- Math book for science students written entirely in LaTeX
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
stacks-project - Repository for the Stacks Project
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.
diffyqs - Notes on Diffy Qs, a textbook for differential equations
OpenLogic - An open-source, customizable intermediate logic textbook
internal-methods - Notes on how to use the internal language of toposes in algebraic geometry
maths_formulae - A LaTeX File with Maths Formulae from High School Maths and Common Core Engineering
the_statistics_handbook - the statistics handbook open source repository