285 karma · joined December 17, 2019
And the first amendment is not a definition of freedom of speech. Read it again.
Yes there is.
> For example the nxn diagonal matrices with real entries
That's a model, not the theory.
> I don't know why you are conflating the words "compute" and "define"
This whole discussion is about definition.
Your own comment started with "You first define..."
And no, you are not able to compute arbitrary products of arbitrary reals in this way anyway, if by computation you mean a finitary algorithmic process.
Yes, you are. Cauchy sequences are a model of the real numbers (which are the Dedekind-complete ordered field).
> I'm explaining how to "compute: \pi*\pi using repeated addition"
How will you define π times π as a repeated addition, i.e. something of the form x + x + … + x?
> there is always a subset for which multiplication works like repeated addition
A subset, yes.
[0] https://en.wikipedia.org/wiki/Theory_(mathematical_logic)
Nope.
The question addressed by the article is precisely whether multiplication is repeated addition.
> adding the elements of the Cauchy sequences. (And similarly for multiplication.) This is the definition of addition (multiplication)
Nope.
The concept of a real closed field [0] (and its categorical second-order version, the Dedekind-complete ordered field) stands on its own, without multiplication being defined in terms of repeated addition. It is completely independent of whether you happen to encode the reals as Cauchy sequences, Dedekind cuts, or something else. The (equivalence classes of) Cauchy sequences are not the same thing as the real numbers, even if we sometimes abuse terminology in this way for expediency. The distinction becomes increasingly important as you delve into more exotic algebraic structures.
Another illustrative example is the Hessenberg product [1]. Even the ordinary product cannot really be reduced to "repeated addition", because you have to use the infinitary concept of a limit. And not just your everyday limit [2], but a limit on a proper-class sized domain [3]!
See also the category-theoretic product [4].
[0] https://en.wikipedia.org/wiki/Real_closed_field
[1] https://en.wikipedia.org/wiki/Ordinal_arithmetic#Natural_ope...
[2] https://en.wikipedia.org/wiki/Limit_of_a_sequence
[3] https://en.wikipedia.org/wiki/Order_topology#Ordinal-indexed...
Doesn't work in every context where multiplication is defined.
> It's even a useful thing to do, how else would you define multiplication?
Perhaps something along the lines of: A multiplication is an endomorphism on an additive semigroup.
I wouldn't say multiplication is literally repeated addition. I'd say it reduces to repeated addition when the multiplier is a natural number.
"denying the possibility of change" is the opposite of what they did.
Your response: I think your knee-jerk focus on who the president is (and denying the possibility of change) demonstrates what I was saying: the rhetoric has convinced people that it doesn't matter and it's all some national team sport.
Your (knee-jerk?) characterization is the opposite of what that comment is saying.
I suggest reading chapter 2 (starting at "Abolition of the family!") for the first, and chapter 2.3 and 2.4 for the second.
https://en.wikipedia.org/wiki/Assurance_contract#Dominant_as...
https://link.springer.com/article/10.1023%2FA%3A100495710953...
> Many types of public goods can be produced privately by profit seeking entrepreneurs using a modified form of assurance contract, called a dominant assurance contract. I model the dominant assurance contract as a game and show that the pure strategy equilibrium has agents contributing to the public good as a dominant strategy.
I suggest you read how she reacted to being exposed.
> I would extend that courtesy to Glenn Greenwald too, had he not exhausted it writing shit like this.
Again, what "shit" are you referring to? Why are you so angry that he's reporting on this issue?
> Anyway, here’s the larger point: it’s absolutely ludicrous to claim that Marc Andreeson’s life and reputation would be ruined even if the allegation were true. That’s why this is such trivial bullshit to write about.
So you give the perpetrator a free pass because their victim in this case happens to be a high-profile person who might be able to defend themselves against the slander and all its consequences?
What hope is left for the little guy?
Why are you so angry that it was foiled?
An NYT reporter trying to ruin people's lives and reputations for petty ideological reasons is anything but "trivial bullshit".
And there's nothing ironic about it.
https://news.ycombinator.com/item?id=26070324
Can you explain what you mean by this?
Amazon, Apple, Google, Twitter, Facebook, etc. are multibillion-dollar corporations that control a colossal share of online communications.
Let's not play pretend here.
There are many comments on previous HN threads defending censorship and information restriction, precisely as the GP has described it.
https://news.ycombinator.com/item?id=25693742
https://news.ycombinator.com/item?id=25691631
Let's try again:
If the USSR had such a wonderful society, why did they have to systematically mass murder ordinary people who wanted to leave this wonderful society?
Don't deflect. Don't evade. Answer.