
Number Theorist Fears All Published Maths Is Wrong - melvinroest
https://www.vice.com/en_uk/article/8xwm54/number-theorist-fears-all-published-maths-is-wrong
======
AnimalMuppet
Title is somewhat misleading. "Number Theorist Fears That No Published Math
Paper Is Certain To Be Correct" would be more like it. The fear isn't that
everything is wrong. The fear is that any given piece _might_ be, and
therefore we can't have real confidence in any of it.

~~~
not_a_cop75
There is a reason for this. Vice, while once ground breaking, has since become
the dumpster diver of the journalism world. And once you give up enough
integrity, it no longer matters where you find your story or what condition
it's in. I wish I could still trust them at face value.

~~~
melvinroest
Thanks for mentioning this, I'll be more critical in sending submissions from
Vice.

------
melvinroest
Related to this, I wish someone had a simple course on this for people who
don’t know that much about (automated) proofs.

I looked it up on HN [1], but couldn’t find any.

[1]
[https://hn.algolia.com/?q=proof+assistant](https://hn.algolia.com/?q=proof+assistant)

~~~
Someone
If you don’t know much about proofs, you first have to learn how incredibly
good mathematicians are at nitpicking (if they want to), and you have to
accept that that is a good thing.

Thing is, if you want to be rigorous, you cannot accept anything as true,
except for the axioms you’re working from, and you want as few and as simple
as possible of those because that decreases the risk that your starting set is
internally inconsistent. You also want every single step in your proof to be
absolutely bloody obvious.

For example, if you want to build math from the ground up, you don’t accept
that integer multiplication or even integer addition exist, but you construct
them. In fact, you don’t even accept that integers exist; you construct them,
for example from natural numbers
([https://en.wikipedia.org/wiki/Integer#Construction](https://en.wikipedia.org/wiki/Integer#Construction))

Starting with the absolute basics also allows you to pick and choose which
axioms to start with, for example by not including the law of the excluded
middle
([https://en.wikipedia.org/wiki/Law_of_excluded_middle#Logicis...](https://en.wikipedia.org/wiki/Law_of_excluded_middle#Logicists_versus_the_intuitionists))
or the parallel postulate
([https://en.wikipedia.org/wiki/Parallel_postulate#History](https://en.wikipedia.org/wiki/Parallel_postulate#History))

For learning how deep that nitpicking can go,
[http://us.metamath.org/index.html](http://us.metamath.org/index.html) (A
proof assistent that is, IMO, simpler to understand than many other ones, but
has really, really long proofs) may be illustrative. See for example their
proof that _2+2=4_.

Once you accept that every detail needs your full attention, I think
[https://leanprover.github.io/theorem_proving_in_lean/](https://leanprover.github.io/theorem_proving_in_lean/)
has a decent introduction.

If you just want an introduction to _a_ theorem prover,
[http://www.cs.ru.nl/~freek/comparison/comparison.pdf](http://www.cs.ru.nl/~freek/comparison/comparison.pdf)
(discusses 17 different ones) also may be useful.

