Gap in understanding of Logical Pinpointing

6 Incorrect 12 November 2012 05:33PM

I had originally found myself very confused by how the second order axiom of induction restricted PA to a single model, leading to this discussion where I thought doing so would violate the incompleteness theorem.

What I misunderstood is that while the axiom schema of induction effectively quantifies over properties definable in PA, the SOL version quantifies over ALL properties, including those you can't even define in PA.

This seems like a really subtle point that wasn't obvious to me from the article, knowing FOL but not SOL. I actually realized my mistake when I saw that Eliezer was representing properties as sets in the visual diagram.

Anyway, I thought others might benefit by learning from my mistake. Also, I'd like to point out that this definition of the natural numbers yields no procedure that lets you produce theorems of PA as second-order logic has no computable complete definition of provability. Just use the axiom schema of induction instead of the second order axiom of induction and you will be able to produce theorems though.

What mathematics to learn

6 Incorrect 23 November 2011 06:14PM

There is, of course, Kahn Academy for fundamentals. We have already had a discussion on How to learn math.

What resources exist detailing which mathematics to learn in what order? What resources exist that explain the utility of different mathematical subfields for the purpose of directing studies?

Cognitive Style Tends To Predict Religious Conviction (psychcentral.com)

10 Incorrect 23 September 2011 06:28PM

http://psychcentral.com/news/2011/09/21/cognitive-style-tends-to-predict-religious-conviction/29646.html

Participants who gave intuitive answers to all three problems [that required reflective thinking rather than intuitive] were one and a half times as likely to report they were convinced of God’s existence as those who answered all of the questions correctly.

Importantly, researchers discovered the association between thinking styles and religious beliefs were not tied to the participants’ thinking ability or IQ.

participants who wrote about a successful intuitive experience were more likely to report they were convinced of God’s existence than those who wrote about a successful reflective experience.

I think this is the source but I can't be sure:

http://www.apa.org/pubs/journals/releases/xge-ofp-shenhav.pdf

http://lesswrong.com/lw/7o4/atheism_autism_spectrum/4vbc

Reddit /r/psychology discussion

How can humans make precommitments?

6 Incorrect 15 September 2011 01:19AM

How can you precommit to something where the commitment is carried out only after you know your commitment strategy has failed?

It would seem to make it impossible to commit to blackmail when the action of blackmail has negative utility. How can you possibly convince your rational future self to carry out a commitment they know will not work?

You could attempt to adopt a strategy of always following your commitments. From your current perspective this is useful but once you have learned your strategy has failed, what's to prevent you from just disregarding the strategy?

If a commitment strategy will fail you don't want to make the commitment but if you will not follow the commitment even when the strategy fails then you never made the commitment in the first place.

For example, in nuclear war why would you ever retaliate? Once you know your strategy of nuclear deterrence has failed, shooting back will only cause more civilian casualties.

I'm not saying commitments aren't useful, I'm just not sure how you can make them. How do you prevent your future self from reasoning their way out of them?

I apologize if reading this makes it harder for any of you to make precommitments. I'm hoping someone has a better solution than simply tricking your future self.