https://www.artima.com/shop/modern_mathematics may be of interest to people too.
Practical Foundations of Mathematics
11–20 of 24 posts
Re: Practical Foundations of Mathematics
#12Looking at the table of contents, this seems like a really odd title. In what sense is this practical and a foundation for mathematics?
It's "practical" for the academic. Everyone else should just get a discrete math book and work through the first few chapters.
Re: Practical Foundations of Mathematics
#13>The logical calculus is easier to execute than any of the techniques of mathematics itself, yet only in 1934 did Gerhard Gentzen set it out in a natural way. Even now, mathematics students are expected to learn complicated (epsilon, delta)-proois in analysis with no help in understanding the logical structure of the arguments. Examiners fully deserve the garbage that they get in return. With an opening like that how…
Anything public you can share yet? I’ve been interested in how much LLMs “understand” symbolism.
Re: Practical Foundations of Mathematics
#14Practical in the same sense as André Weil's book on number theory is basic :)
Re: Practical Foundations of Mathematics
#15A someone interested in programming language theory, this book looks very interesting. Unfortunately the HTML is incomplete and very hard to read, and the book itself is quite expensive.
I noticed this.
When reding about denotations the cent sign appears. What do you suppose it actually should be ?
Re: Practical Foundations of Mathematics
#16>The logical calculus is easier to execute than any of the techniques of mathematics itself, yet only in 1934 did Gerhard Gentzen set it out in a natural way. Even now, mathematics students are expected to learn complicated (epsilon, delta)-proois in analysis with no help in understanding the logical structure of the arguments. Examiners fully deserve the garbage that they get in return. With an opening like that how…
Anything public you can share yet? I’ve been interested in how much LLMs “understand” symbolism.
I am interested in this problem as well. Please share any notes.
I am attempting to create parameterized "logic" problems (similar to the zebra puzzle) which cannot be solved by LLMs even when they are trained on it, or even when they "reason" on it.
Meanwhile this approach is even simpler, where it is demonstrated that LLMs cannot recognize 3 state DFAs. https://arxiv.org/pdf/2501.02825
Re: Practical Foundations of Mathematics
#17With llms and theorem provers doing the formalism, what is left is intuition - love the intuitionist focus in this book, also tying into what Bessis said in [1]
I intend to read this book in depth.
Author: please fix the html rendering.
[1] https://www.amazon.com/Mathematica-Secret-World-Intuition-Cu...
Re: Practical Foundations of Mathematics
#18Earlier quoted context omitted.
The practical application of these foundations is the field of programming language theory. At least that is the application I know of. Whether programming language theory is practical or not depends, I think, on your attitude to programming languages. For example, if you think C is a masterpiece of clean and elegant design I suspect you won't have much time for programming language theory. If you think C is riddled…
You do understand how a mathematical Theory (a set of true propositions made by relating operations i.e. from axioms and inference rules) and a Model (a set of elements with all the operations defined and where all the propositions are true) are related, i presume? PLT is not one theory but a set of theories from which you are free to pick and choose any subset you would like to model in your language's abstract mach…
C arguably doesn't actually have formal semantics, the struggles of the standard committee trying to figure out the pointer provenance being one obvious example.
Re: Practical Foundations of Mathematics
#19For folks looking - The pdf you can find/buy has the math correctly rendered
https://www.amazon.com/Practical-Foundations-Mathematics-Cam...
Re: Practical Foundations of Mathematics
#20A someone interested in programming language theory, this book looks very interesting. Unfortunately the HTML is incomplete and very hard to read, and the book itself is quite expensive.
Have you read TAPL? I bought the physical copy and found it 100% worth it. Really easy to work through, even with a layperson's understanding of math