yori

> You cannot work backward by assuming the field axioms (which are unfortunately named because they are not axioms at all but properties) to derive the definition o the field operators.

Of course, when we say they are field axioms we mean those properties hold true for the elements of the field. If you see those properties, they talk about distributivity over the elements of the field and additive inverses of the elements also belong to the field, so the distributivity automatically applies to additive inverses too.

After that with a little algebra, "product of additive inverses of two elements is equal to the product of the two elements" comes out as a result (not a definition).

Of course, by "product" we mean whatever * represents. It is not necessarily the multiplication operator we see in numbers.

> If we'd chosen some other definition of multiplication, a lot of the "intuitive" properties of multiplication that hold over the natural numbers (such as the distributivity of multiplication over addition and subtraction) would no longer be true over the integers.

This is backward reasoning. The chosen definition of multiplication is not to keep things "intuitive". If you start with the field axioms, the chosen definition of multiplication is pretty much dictated by the axioms. If you choose another definition of multiplication, you would end with contradictions like 1 = 0 and such nonsense! And mathematicians abhor contradictions!

"Product of additive inverses of two elements is equal to the product of the two elements" is dictated by the field axioms in all fields.

Not every arithmetic property needs to be proved from Peano axioms. One can but it is tedious and unnecessary. A much better starting point is the set of field axioms where the distributivity property is already available as an axiom.

The assumptions made in the article are perfectly fine as per field axioms. Granted it would have been nicer if distributivity over addition was used instead of distributivity over subtraction. But it is not a big leap to derive distributivity over substraction from field axioms by distributing multiplication over a positive number and the additive inverse of another positive number.

Wherever you see an assumption made about negative number, just mentally replace it with additive inverse of a positive number and you would be fine.

Like mentioned in another comment on this thread, the assumptions are well known field axioms. They form a good starting point.

And why start with Peano axioms? They seem like a bad starting point because it would take pages upon pages of proof and it won't easily extend to other algebraic structures like rings and fields.

> There is no simple way to solve this equation. I had to resort to using the Lambert W function to solve this.

How do you know when to use Lambert W function?

What your parent comment meant was that it is not possible to learn well from typical blog posts like this that try to condense the subject into a 3000 word article. Of course, if you copy-paste the content of a book into a blog post, then parent comment's point no longer applies.

Off-topic question: I see a lot of new posts coming from .cc websites. What is the appeal that .cc provides? Is it that the content is released under Creative Commons? That's not the case here. What kind of messaging or symbolism or underlying meaning one is aiming for when they are going for a .cc domain?

Example code on the website:

  10 MODE 2
  20 COLOUR RND(7)
  30 PRINT "HELLO WORLD"
  40 GOTO 20

Happy to see "colour" spelled the English way. :-)

Yes, I didn't get the usage of the term "LARP" in this context too. It comes off as rude and condescending to both LARPers and email users. I am not sure the term fits appropriately in this context. All LARPers are aware that they are role playing in a game, not the real world, and they do not pretend otherwise.

> Withdrawn mostly from Reddit in favor of early 2000’s style forums that I pay money for and HN

What are the names of these early 2000's style forums? I would like to join them too.

"There are only two hard things in Computer Science: cache invalidation and naming things."

What's the deal with Hagoromo chalk? Why is it a romantic gesture?