‘Reverse #Mathematics’ Illuminates Why Hard Problems Are Hard #Metamathematics #pigeonhole

https://www.quantamagazine.org/reverse-mathematics-illuminates-why-hard-problems-are-hard-20251201/

The #metamathematics - isn't it #metalogical?!?

- a setting of the #ontology of #mathematics or #metalogics -

Now that I have tried to take a look behind or in front of physics in the form of #metaphysics, a fundamental consideration of #mathematics as #metamathematics is now due. In this context, I will of course also address the well-known #fundamental #crisis of mathematics.

More at: https://philosophies.de/index.php/2023/05/24/die-metamathematik/

The #metamathematics - is metalogical after all?!?

- a setting of the ontology of mathematics or #metalogics -

The basis of my essay is mainly the paper by #Matthias #Neuber "Mathematik und Ontologie - Der lange Weg zum 'strukturellen Formalismus'" in which he precisely depicts the foundational crisis of mathematics (which persists until today) and derives possible ways out of it (not only in the mathematical sense).

More at: https://philosophies.de/index.php/2023/05/24/die-metamathematik/

The #metamathematics - is metalogical after all?!?

- a setting of the ontology of mathematics or #metalogics -

After I have now tried to take a look behind or in front of physics in the form of "metaphysics", now a basic consideration of mathematics as "metamathematics" would be due. In this context I will of course also deal with the well-known "fundamental crisis of mathematics". So, there is probably enough to fathom ;-).

More at: https://philosophies.de/index.php/2023/05/24/die-metamathematik/

The #metamathematics - is metalogical after all?!?

- a setting of the ontology of mathematics or #metalogics -

The Intro

After I have now tried to take a look behind or in front of physics in the form of "metaphysics", now a basic consideration of mathematics as "metamathematics" would be due. In this context I will of course also deal with the well-known "fundamental crisis of mathematics". So, there is probably enough to fathom ;-).

More at: https://philosophies.de/index.php/2023/05/24/die-metamathematik/

The #metamathematics - is metalogical after all?!?

- a setting of the ontology of mathematics or #metalogics -

The Intro

"Two times three makes four
Widdewiddewitt and three makes nine
I make the world
Widdewidde as I like it!"
("Hej Pippi Langstrumpf" song by Henning Wehland)

Pippi Langstrumpf is, of course, much freer in this respect than the rest of the "mathematical world". In the following essay I would like to get to the bottom of the "logic of mathematics" in the form of the "metalogics" once more exactly. After I have now tried to take a look behind or in front of physics in the form of "metaphysics", now a basic consideration of mathematics as "metamathematics" would be due. In this context I will of course also deal with the well-known "fundamental crisis of mathematics". So, there is probably enough to fathom ;-).

More at: https://philosophies.de/index.php/2023/05/24/die-metamathematik/

ACM Communications 4/23 paper by Bentkamp et al
"Today, even if most mathematicians do not consciously use logic on a daily basis, they can feel reassured to know that their definitions and arguments can be encoded in it" p82

Nope. Nope. And Nope.
#formalLogic
#MetaMathematics
#ComputerScience

So, I think I need some help from #mathstodon and the wider community with understanding of #computability and #recursiveenumeration.

If you go to

http://jdh.hamkins.org/alan-turing-on-computable-numbers/

there is a wonderful article by Dr. Joel David Hamkins, a mathematician whose work I deeply admire.

However, if you scroll down to the comments, you will notice a comment from Nathan Harvey (that’s me!) contesting some of the claims of the article. In particular, Dr. Hamkins makes the claim that with Turing’s original definition of computable numbers and functions, addition is not a computable function. He appears to view computable functions as consumers of the output of the programs that represent the reals, not as consumers of the programs themselves, and I give an example where the analysis changes and make reference to Turing’s definition.

But! I can be wrong here. Dr. Hamkins is the real stuff. I just keep coming back, after spending time to consider his points and trying to reframe them to ensure I understand, thinking that my point wasn’t refuted or even really addressed. And as I read the responses, I fail to see any comments about my example or my point about the difference between consuming outputs of the computation versus the actual program, … and I keep thinking the point is getting missed. But that’s dangerous territory that can lead one to crankdom and obstinate ignorance. I don’t want to do that to myself.

So if there are any mathematicians who enjoy the area of computable functions and want to give it a quick read, I would appreciate any comments on my point. Even if it’s just a comment “No, Nate, you’re wrong and deeply misguided” with no further explanation. After one or two of those from other mathematicians, I’ll take the L and shrink off to read more books on the topic.

And if you don’t know the answer but have followers who work in that or related areas of math, a boost would be appreciated. This is an area of math that is deeply interesting to me and I thought I understood it well, but self-taught people are known to go off the rails.

Some hashtags to meet the right eyes:
#math #mathematics #metamathematics #constructivism #Turing #formallanguages

Some attags, not to get their direct response (unless interested themselves in doing so), but if they find the discussion respectful and the topic interesting, a boost might benefit the discussion:
@ProfKinyon @MartinEscardo @BartoszMilewski