Less Wrong is a community blog devoted to refining the art of human rationality. Please visit our About page for more information.

Comment author: Thomas 28 April 2017 06:02:55PM *  0 points [-]

You can eliminate paradoxes by declaring operations which lead to them "illegal"

Precisely! You narrow your axiomatic system, hoping that one of contradictors will fall out. They've eliminated self containing sets from the Naive Set Theory and Russell's paradox vanished. Hopefully.

Is i a number? If not, what is it?

It is not "what is it", it is "how does it behave". Whatever behaves as a number, is a number.

Does it have a sign?

If it behaves as it had a sign ... And it does. So yes, it has a sign!

Comment author: Lumifer 28 April 2017 07:37:46PM 0 points [-]

So yes, it has a sign!

Which is what?

Comment author: eternal_neophyte 28 April 2017 06:03:51PM *  0 points [-]

Paul isn't going to cut it

Paul might cut it if you're Thomas Jeffson: https://en.wikipedia.org/wiki/Jefferson_Bible "Paul was the first corrupter of the doctrines of Jesus."

Comment author: Lumifer 28 April 2017 07:37:15PM 0 points [-]

Not for the change of mind from "I completely don't care about humans" to "I'll make my only Son a human (to start with) and let other humans crucify him so that they could wash off the original sin".

Comment author: evand 28 April 2017 06:11:17PM 1 point [-]

What's that? If I don't give into your threat, you'll shoot me in the foot? Well, two can play at that game. If you shoot me in the foot, just watch, I'll shoot my other foot in revenge.

Comment author: Lumifer 28 April 2017 07:35:04PM 1 point [-]

And then I'll bleed on you!

Comment author: Thomas 28 April 2017 04:15:24PM *  0 points [-]

A plausible answer is "you can't do that", but that's also an answer to any paradox whatsoever.

No, this is not true. The only answer we have to paradoxes is - change you axiom system!

How do you feel about imaginary numbers, by the way?

There are no paradoxes there. At least no paradoxes which wouldn't be present in the real numbers already.

SQRT(-1)=i is no paradox whatsoever. People might often say - "oh, well that's paradoxical".

No, it's not. No A & ~ A here. It might be weird to somebody, that you can calculate square roots from negative numbers. I really don't see why, but it might.

Weirdness and paradoxicality are two different things.

I don't mind weirdness, I hate paradoxicality.

Comment author: Lumifer 28 April 2017 04:46:31PM *  0 points [-]

No, this is not true. The only answer we have to paradoxes is - change you axiom system!

You can eliminate paradoxes by declaring operations which lead to them "illegal" (or their outcomes to be undefined).

SQRT(-1)=i is no paradox whatsoever

Is i a number? If not, what is it? Does it have a sign?

Comment author: Lumifer 28 April 2017 04:38:12PM *  1 point [-]

That line of thought seems... misguided. For a quick illustration do s/threat/credible threat/g

Effectively you are trying to estimate The Worst That Could Happen and are telling your AI to discount all outcomes below your estimate.

You will need to trust that estimate A LOT.

Comment author: eternal_neophyte 28 April 2017 03:39:55PM *  0 points [-]

God says "j/k, just kidding"

Either God, Jesus or St. Paul - that all depends entirely on which sect you ask.

Comment author: Lumifer 28 April 2017 03:52:55PM 0 points [-]

Got to be someone from the Holy Trinity -- Paul isn't going to cut it.

Comment author: Thomas 28 April 2017 02:59:18PM 0 points [-]

Not only arithmetic, but an arithmetic with arbitrary large numbers is required for the Godel's Incompleteness Theorem.

You need at least "all integers", that is an infinite set.

Maybe some incompleteness can be proved for the finite sets as well. But it's not known to be so. Nobody proved that yet.

Comment author: Lumifer 28 April 2017 03:36:03PM *  0 points [-]

Finite sets of numbers have obvious problems, well explored in computer science (if you integer has be be expressed in 4 bytes, you can express only a finite set of integers. Or real numbers, for that matter).

Trivially, if your finite set ends at n, what is the sum of (n-1) + (n-2)? A plausible answer is "you can't do that", but that's also an answer to any paradox whatsoever.

How do you feel about imaginary numbers, by the way?

Comment author: Kaj_Sotala 28 April 2017 12:14:30PM 1 point [-]

If you admit that this is an unfair strawman, then why are you bothering to post it?

Comment author: Lumifer 28 April 2017 02:58:53PM 0 points [-]

Caricatures can be highly useful as a pointers at particular failure spots. Caricatures and parodies are, by definition, "unfair strawmen".

Comment author: woodchopper 28 April 2017 11:03:44AM 2 points [-]

I think it's an interesting point about innovation actually being very rare, and I agree. It takes a special combination of things for to happen and that combination doesn't come around much. Britain was extremely innovative a few hundred years ago. In fact, they started the industrial revolution, literally revolutionising humanity. But today they do not strike me as particularly innovative even with that history behind them.

I don't think America's ability to innovate is coming to end all that soon. But even if America continues to prosper, will that mean it continues to innovate? It takes more than prosperity for innovation to happen. It takes a combination of factors that nobody really understands. It takes a particular culture, a particular legal system, and much more.

Comment author: Lumifer 28 April 2017 02:55:57PM 1 point [-]

It takes more than prosperity for innovation to happen. It takes a combination of factors that nobody really understands.

I don't know about that. People have been discussing how does an innovation hub (like Silicon Valley) appear and how one might create one -- that is a difficult problem, partially because starting a virtuous circle is hard.

But general innovation in a society? Lemme throw in some factors off the top of my mind:

  • Low barriers to entry (to experimentation, to starting up businesses, etc.). That includes a permissive legal environment and a light regulatory hand.
  • A properly Darwinian environment where you live or die (quickly) by market success and not by whether you managed to bribe the right bureaucrat.
  • Relatively low stigma attached to failure
  • Sufficient numbers of high-IQ people who are secure enough to take risks
  • Enough money floating around to fund high-risk ventures
  • For basic science, enough money coupled with the willingness to throw it at very high-IQ people and say "Make something interesting with it"
Comment author: Lumifer 28 April 2017 02:43:57PM 2 points [-]

That's a lot of discussion with no data to support it.

Their experiment had null results and probably didn't have enough power to detect anything to start with. I'd classify this under "armchair psychology".

View more: Next