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Practical Foundations of Mathematics

paultaylor.eu

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Re: Practical Foundations of Mathematics

#21

Post by llm_trw »

memhole wrote: post #13
llm_trw wrote: post #5

>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.

You can email me if you're interested.

This was paid work until recently but the start up pivoted to something else and now I'm wondering what to do with code that takes a few thousand dollars to run every time and is of questionable copyright ownership.

Re: Practical Foundations of Mathematics

#22

Post by jaykru »

This is up there with The Art of Computer Programming for me as a text that has clearly been painstakingly crafted by its author. It shares as well with TAOCP that it is simultaneously so many things: an introduction to programming language semantics, an introduction to category theory, (in later chapters) a reference on pretty sophisticated programming language semantics, and an exploration of the rich connections between all of these topics. I owe huge chunks of my bachelor's thesis [1] to Paul and I eagerly look forward to returning to his text after my retirement from software engineering :)

This book is for you if:

  - You have some exposure to PL semantics (lambda calculus, functional programming) in the operational tradition.
  - You have some background in abstract algebra and/or category theory.
  - You have neither but you're interested and willing to grind.
A word on the title: the contents are practical for doing mathematics; if you're interested in studying lambda calculi and want fancy techniques for writing light, elegant proofs about their properties, this is a great resource. If you're into building systems it may not be as useful to you.

[1] https://github.com/jaykru/thesis/blob/trunk/thesis.pdf

Re: Practical Foundations of Mathematics

#23

Post by rramadass »

Joker_vD wrote: post #18
rramadass wrote: post #10

Earlier quoted context omitted.

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.

C does have formally defined semantics as given by the standard but chose to leave some open-ended to give compiler writers more freedom in the form of a) Unspecified behaviour b) Implementation defined behaviour and c) Undefined behaviour.
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