LessWrong is where I learned about Bitcoin, several years ago, and my greatest regret is that I did not investigate it more as soon as possible, that people here did not yell at me louder that it was important, and to go take a look at it.  In that spirit, I will do so now.

 

First of all, several caveats:

* You should not go blindly buying anything that you do not understand.  If you don't know about Bitcoin, you should start by reading about its history, read Satoshi's whitepaper, etc.  I will assume that hte rest of the readers who continue reading this have a decent idea of what Bitcoin is.

* Under absolutely no circumstances should you invest money into Bitcoin that you cannot afford to lose.  "Risk money" only!  That means that if you were to lose 100% of you money, it would not particularly damage your life.  Do not spend money that you will need within the next several years, or ever.  You might in fact want to mentally write off the entire thing as a 100% loss from the start, if that helps.

* Even more strongly, under absolutely no circumstances whatsoever will you borrow money in order to buy Bitcoins, such as using margin, credit card loans, using your student loan, etc.  This is very much similar to taking out a loan, going to a casino and betting it all on black on the roulette wheel.  You would either get very lucky or potentially ruin your life.  Its not worth it, this is reality, and there are no laws of the universe preventing you from losing.

* This post is not "investment advice".

* I own Bitcoins, which makes me biased.  You should update to reflect that I am going to present a pro-Bitcoin case.

 

So why is this potentially a time to buy Bitcoins?  One could think of markets like a pendulum, where price swings from one extreme to another over time, with a very high price corresponding to over-enthusiasm, and a very low price corresponding to despair.  As Warren Buffett said, Mr. Market is like a manic depressive.  One day he walks into your office and is exuberant, and offers to buy your stocks at a high price.  Another day he is depressed and will sell them for a fraction of that. 

The root cause of this phenomenon is confirmation bias.  When things are going well, and the fundamentals of a stock or commodity are strong, the price is driven up, and this results in a positive feedback loop.  Investors receive confirmation of their belief that things are going good from the price increase, confirming their bias.  The process repeats and builds upon itself during a bull market, until it reaches a point of euphoria, in which bad news is completely ignored or disbelieved in.

The same process happens in reverse during a price decline, or bear market.  Investors receive the feedback that the price is going down => things are bad, and good news is ignored and disbelieved.  Both of these processes run away for a while until they reach enough of an extreme that the "smart money" (most well informed and intelligent agents in the system) realizes that the process has gone too far and switches sides. 

 

Bitcoin at this point is certainly somewhere in the despair side of the pendulum.  I don't want to imply in any way that it is not possible for it to go lower.  Picking a bottom is probably the most difficult thing to do in markets, especially before it happens, and everyone who has claimed that Bitcoin was at a bottom for the past year has been repeatedly proven wrong.  (In fact, I feel a tremendous amount of fear in sticking my neck out to create this post, well aware that I could look like a complete idiot weeks or months or years from now and utterly destroy my reputation, yet I will continue anyway).

 

First of all, lets look at the fundamentals of Bitcoin.  On one hand, things are going well. 

 

Use of Bitcoin (network effect):

One measurement of Bitcoin's value is the strenght of its network effect.  By Metcalfe's law, the value of a network is proporitonal to the square of the number of nodes in the network. 

http://en.wikipedia.org/wiki/Metcalfe%27s_law

Over the long term, Bitcoin's price has generally followed this law (though with wild swings to both the upside and downside as the pendulum swings). 

In terms of network effect, Bitcoin is doing well.

 

Bitcoin transactions are hitting all time highs:  (28 day average of number of transactions).

https://blockchain.info/charts/n-transactions-excluding-popular?timespan=2year&showDataPoints=false&daysAverageString=28&show_header=true&scale=0&address=

 

Number of Bitcoin addresses are hitting all time highs:

https://blockchain.info/charts/n-unique-addresses?timespan=2year&showDataPoints=false&daysAverageString=28&show_header=true&scale=0&address=

 

Merchant adoption continues to hit new highs:

BitPay/Coinbase continue to report 10% monthly growth in the number of merchants that accept Bitcoin.

Prominent companies that began accepting Bitcoin in the past year include: Dell, Overstock, Paypal, Microsoft, etc.

 

On the other hand, due to the sustained price decline, many Btcoin businesses that started up in the past two years with venture capital funding have shut down.  This is more of an effect of the price decline than a cause however.  In the past month especially there has been a number of bearish news stories, such as Bitpay laying off employees, exchanges Vault of Satoshi and CEX.io deciding to shut down, exchange Bitstamp being hacked and shut down for 3 days, but ultimately is back up without losing customer funds, etc.

 

The cost to mine a Bitcoin is commonly seen as one indicator of price.   Note that the cost to mine a Bitcoin does not directly determine the *value* or usefulness of a Bitcoin.   I do not believe in the labor theory of value: http://en.wikipedia.org/wiki/Labor_theory_of_value

However, there is a stabilizing effect in commodities, in which over time, the price of an item will often converge towards the cost to produce it due to market forces. 

 

If a Bitcoin is being priced at a value much greater than the cost (in mining equipment and electricity) to create it, people will invest in mining equipment.  This results in increased 'difficulty' of mining and drives down the amount of Bitcoin that you can create with a particular piece of mining equipment.  (The amount of Bitcoins created is a fixed amount per unit of time, and thus the more mining equipment that exists, the less Bitcoin each miner will get).

If Bitcoin is being priced at a value below the cost to create it, people will stop investing in mining equipment.  This may be a signal that the price is getting too low, and could rise.

 

Historically, the one period of time where Bitcoin was priced significantly below the cost to produce it was in late 2011.  It was noted on LessWrong.  The price has not currently fallen to quite the same extent as it did back then (which may indicate that it has further to fall), however the current price relative to the mining cost indicates we are very much in the bearish side of the pendulum.

 

It is difficult to calculate an exact cost to mine a Bitcoin, because this depends on the exact hardware used, your cost of electricity, and a prediction of the future difficulty adjustments that will occur.  However, we can make estimates with websites such as http://www.vnbitcoin.org/bitcoincalculator.php

According to this site, every available Bitcoin miner will never give you back as much money as it cost, factoring in the hardware cost and electricity cost.   Upcoming more efficient miners which have not yet released yet are estimated to pay off in about a year, if difficulty grows extremely slowly, and that is for upcoming technology which has not yet even been released. 

 

There are two important breakpoints when discussing Bitcoin mining profitability.  The first is the point at which your return is enough that it pays for both the electricity and the hardware.  The second is the point at which you make more than your electricity costs, but cannot recover the hardware cost.

 

For example, lets say Alice pays $1000 on Bitcoin mining equipment.  Every day, this mining equipment can return $10 worth of Bitcoin, but it costs $5 of electricity to run.  Her gain for the day is $5, and it would take 200 days at this rate before the mining equipment paid for itself.  Once she has made the decision to purchase the mining equipment, the money spent on the miner is a sunk cost.  The money spent on electricity is not a sunk cost, she continues to have the decision every day of whether or not to run her mining equipment.  The optimal decision is to continue to run the miner as long as it returns more than the electricity cost. 

Over time, the payout she will receive from this hardware will decline, as the difficulty of mining Bitcoin increases.  Eventually, her payout will decline below the electricity cost, and she should shut the miner down.  At this point, if her total gain from running the equipment was higher than the hardware cost, it was a good investment.  If it did not recoup its cost, then it was worse than simply spending the money buying Bitcoin on an exchange in the first place.

 

This process creates a feedback into the market price of Bitcoins.  Imagine that Bitcoin investors have two choices, either they can buy Bitcoins (the commodity which has already been produced by others), or they can buy miners, and produce Bitcoins for themself.   If the Bitcoin price falls sufficiently that mining equipment will not recover its costs over time, investment money that would have gone into miners instead goes into Bitcoin, helping to support the price.  As you can see from mining cost calculators, we have passed this point already.  (In fact, we passed it months ago already).

 

The second breakpoint is when the Bitcoin price falls so low that it falls below the electricity cost of running mining equipment.  We have passed this point for many of the less efficient ways to mine.  For example, Cointerra recently shut down its cloud mining pool because it was losing money.  We have not yet passed this point for more recent and efficient miners, but we are getting fairly close to it. Crossing this point has occurred once in Bitcoin's history, in late 2011 when the price bottomed out near $2, before giving birth to the massive bull run of 2012-2013 in which the price rose by a factor of 500.

 

Market Sentiment: 

I was not active in Bitcoin back in 2011, so I cannot compare the present time to the sentiment at the November 2011 bottom.  However, sentiment currently is the worst that I have seen by a significant margin. Again, this does not mean that things could not get much, much worse before they get better!  After all, sentiment has been growing worse for months now as the price declines, and everyone who predicted that it was as bad as it could get and the price could not possibly go below $X has been wrong.  We are in a feedback loop which is strongly pumping bearishness into all market participants, and that feedback loop can continue and has continued for quite a while.

 

A look at market indicators tells us that Bitcoin is very, very oversold, almost historically oversold.  Again, this does not mean that it could not get worse before it gets better. 

 

As I write this, the price of Bitcoin is $230.  For perspective, this is down over 80% from the all time high of $1163 in November 2013.  It is still higher than the roughly $100 level it spent most of mid 2013 at.

* The average price of a Bitcoin since the last time it moved is $314.

https://www.reddit.com/r/BitcoinMarkets/comments/2ez90b/and_the_average_bitcoin_cost_basis_is/

The current price is a multiple of .73 of this price.  This is very low historically, but not the lowest it has ever ben.  THe lowest was about .39 in late 2011. 

 

* Short interest (the number of Bitcoins that were borrowed and sold, and must be rebought later) hit all time highs this week, according to data on the exchange Bitfinex, at more than 25000 Bitcoins sold short:

http://www.bfxdata.com/swaphistory/totals.php

 

* Weekly RSI (relative strength index), an indicator which tells if a stock or commodity is 'overbought' or 'oversold' relative to its history, just hit its lowest value ever.

 

Many indicators are telling us that Bitcoin is at or near historical levels in terms of the depth of this bear market.  In percentage terms, the price decline is surpassed only by the November 2011 low.  In terms of length, the current decline is more than twice as long as the previous longest bear market.

 

To summarize: At the present time, the market is pricing in a significant probability that Bitcoin is dying.

But there are some indicators (such as # of transactions) which say it is not dying.  Maybe it continues down into oblivion, and the remaining fundamentals which looked bullish turn downwards and never recover.  Remember that this is reality, and anything can happen, and nothing will save you.

 

 

Given all of this, we now have a choice.  People have often compared Bitcoin to making a bet in which you have a 50% chance of losing everything, and a 50% chance of making multiples (far more than 2x) of what you started with. 

There are times when the payout on that bet is much lower, when everyone is euphoric and has been convinced by the positive feedback loop that they will win.  And there are times when the payout on that bet is much higher, when everyone else is extremely fearful and is convinced it will not pay off. 

 

This is a time to be good rationalists, and investigate a possible opportunity, comparing the present situation to historical examples, and making an informed decision.   Either Bitcoin has begun the process of dying, and this decline will continue in stages until it hits zero (or some incredibly low value that is essentially the same for our purposes), or it will live.  Based on the new all time high being hit in number of transactions, and ways to spend Bitcoin, I think there is at least a reasonable chance it will live.  Enough of a chance that it is worth taking some money that you can 100% afford to lose, and making a bet.  A rational gamble that there is a decent probability that it will survive, at a time when a large number of others are betting that it will fail.

 

And then once you do that, try your hardest to mentally write it off as a complete loss, like you had blown the money on a vacation or a consumer good, and now it is gone, and then wait a long time.

 

 

Eliezer's writing style of A->B, then A, then B, though generally clear, results in a large amount of redundancy.

In this post, I have attempted to reduce the number of rules needed to remember by half. The numbers are the rules from the original post.

So, without further ado, a good definition for a word:

1. can be shown to be wrong³⁷ and is not the final¹³ authority^{18 19}
2. has strong justifications³³ for the word's existence³² and its particular definition,²⁰ which leave no room for an argument^{17 22}
3. agrees with conventional usage⁴
4. explains what context the word depends on³⁶
5. limits its scope to avoid overlap with other meanings²⁵
6. does not assume that definitions are the best way of giving words semantics¹²
7. directs a complex mental paintbrush³⁵ to paint detailed pictures of the thing you're trying to think about²³
8. is a brain inference aid¹³ that refers to and instructs one on how to find a specific/unique²⁴ similarity cluster²¹ that is apparent from empirical experience^{28 29 30}, the cluster's size being inversely proportional to the word's length³¹
9. is not a binary category^{9 11} and cannot be used for deductive inference²⁷
10. requires observing only¹⁴ a few³ real-world¹ properties that can be easily⁵ verified² and are less abstract⁶ than the word being defined (in particular, the definition cannot be circular¹⁶)
11. is not just a list of random properties^{10 21}
12. contains no negated properties^{10 33}
13. specifies exhaustively all of the correct connotations of the word^{25 26}
14. makes the properties of a random object satisfying the definition be nearly independent³⁴
15. has examples⁶ which satisfy the definition, including the original example(s) that motivated the definition being given¹⁵ and typical/conventional examples⁷
16. tells you which examples are more typical or less typical⁹
17. captures enough characteristics of the examples to identify non-members⁸

And there you go. 17 rules, follow them all and you can't use words wrongly.

Any intuition-dominant thinker who's struggled with math problems or logic-dominant thinker who's struggled with small-talk knows how difficult and hopeless the experience feels like. For a long time I was an intuition thinker, then I developed a logical thinking style and soon it ended up dominating -- granting me the luxury of experiencing both kinds of struggles. I eventually learned to apply the thinking style better optimized for the problem I was facing. Looking back, I realized why I kept sticking to one extreme.

I hypothesize that one-sided thinkers develop biases and tendencies that prevent them from improving their weaker mode of thinking. These biases cause a positive feedback loop that further skews thinking styles in the same direction.

The reasons why one style might be overdeveloped and the other underdeveloped vary greatly. Genes have a strong influence, but environment also plays a large part. A teacher may have inspired you to love learning science at a young age, causing you to foster to a thinking style better for learning science. Or maybe you grew up very physically attractive and found socializing with your peers a lot more rewarding than studying after school, causing you to foster a thinking style better for navigating social situations. Environment can be changed to help develop certain thinking styles, but it should be supplementary to exposing and understanding the biases you already have. Entering an environment that penalizes your thinking style can be uncomfortable, stressful and frustrating without being prepared. (Such a painful experience is part of why these biases cause a positive feedback loop, by making us avoid environments that require the opposite thinking style.)

Despite genetic predisposition and environmental circumstances, there's room for improvement and exposing these biases and learning to account for them is a great first step.

Below is a list of a few biases that worsen our ability to solve a certain class of problems and keep us from improving our underdeveloped thinking style.

---

Intuition-dominant Biases

Overlooking crucial details

Details matter in order to understand technical concepts. Overlooking a word or sentence structure can cause complete misunderstanding -- a common blunder for intuition thinkers.

Intuition is really good at making fairly accurate predictions without complete information, enabling us to navigate the world without having a deep understanding of it. As a result, intuition trains us to experience the feeling we understand something without examining every detail. In most situations, paying close attention to detail is unnecessary and sometimes dangerous. When learning a technical concept, every detail matters and the premature feeling of understanding stops us from examining them.

This bias is one that's more likely to go away once you realize it's there. You often don't know what details you're missing after you've missed them, so merely remembering that you tend to miss important details should prompt you to take closer examinations in the future.

Expecting solutions to sound a certain way

The Internship has a great example of this bias (and a few others) in action. The movie is about two middle-aged unemployed salesmen (intuition thinkers) trying to land an internship with Google. Part of Google's selection process has the two men participate in several technical challenges. One challenge required the men and their team to find a software bug. In a flash of insight, Vince Vaughn's character, Billy, shouts "Maybe the answer is in the question! Maybe it has something to do with the word bug. A fly!" After enthusiastically making several more word associations, he turns to his team and insists they take him seriously.

Why is it believable to the audience that Billy can be so confident about his answer?

Billy's intuition made an association between the challenge question and riddle-like questions he's heard in the past. When Billy used his intuition to find a solution, his confidence in a riddle-like answer grew. Intuition recklessly uses irrelevant associations as reasons for narrowing down the space of possible solutions to technical problems. When associations pop in your mind, it's a good idea to legitimize those associations with supporting reasons.

Not recognizing precise language

Intuition thinkers are multi-channel learners -- all senses, thoughts and emotions are used to construct a complex database of clustered knowledge to predict and understand the world. With robust information-extracting ability, correct grammar/word-usage is, more often than not, unnecessary for meaningful communication.

Communicating technical concepts in a meaningful way requires precise language. Connotation and subtext are stripped away so words and phrases can purely represent meaningful concepts inside a logical framework. Intuition thinkers communicate with imprecise language, gathering meaning from context to compensate. This makes it hard for them to recognize when to turn off their powerful information extractors.

This bias explains part of why so many intuition thinkers dread math "word problems". Introducing words and phrases rich with meaning and connotation sends their intuition running wild. It's hard for them to find correspondences between words in the problem and variables in the theorems and formulas they've learned.

The noise intuition brings makes it hard to think clearly. It's hard for intuition thinkers to tell whether their automatic associations should be taken seriously. Without a reliable way to discern, wrong interpretations of words go undetected. For example, without any physics background, an intuition thinker may read the statement "Matter can have both wave and particle properties at once" and believe they completely understand it. Unrelated associations of what matter, wave and particle mean, blindly take precedence over technical definitions.

The slightest uncertainty about what a sentence means should raise a red flag. Going back and finding correspondence between each word and how it fits into a technical framework will eliminate any uncertainty.

Believing their level of understanding is deeper than what it is

Intuition works on an unconscious level, making intuition thinkers unaware of how they know what they know. Not surprisingly, their best tool to learn what it means to understand is intuition. The concept "understanding" is a collection of associations from experience. You may have learned that part of understanding something means being able to answer questions on a test with memorized factoids, or knowing what to say to convince people you understand, or just knowing more facts than your friends. These are not good methods for gaining a deep understanding of technical concepts.

When intuition thinkers optimize for understanding, they're really optimizing for a fuzzy idea of what they think understanding means. This often leaves them believing they understand a concept when all they've done is memorize some disconnected facts. Not knowing what it feels like to have deeper understanding, they become conditioned to always expect some amount of surprise. They can feel max understanding with less confidence than logical thinkers when they feel max understanding. This lower confidence disincentivizes intuition thinkers to invest in learning technical concepts, further keeping their logical thinking style underdeveloped.

One way I overcame this tendency was to constantly ask myself "why" questions, like a curious child bothering their parents. The technique helped me uncover what used to be unknown unknowns that made me feel overconfident in my understanding.

---

Logic-dominant Biases

Ignoring information they cannot immediately fit into a framework

Logical thinkers have and use intuition -- problem is they don't feed it enough. They tend to ignore valuable intuition-building information if it doesn't immediately fit into a predictive model they deeply understand. While intuition thinkers don't filter out enough noise, logical thinkers filter too much.

For example, if a logical thinker doesn't have a good framework for understanding human behavior, they're more likely to ignore visual input like body language and fashion, or auditory input like tone of voice and intonation. Human behavior is complicated, there's no framework to date that can make perfectly accurate predictions about it. Intuition can build powerful models despite working with many confounding variables.  

Bayesian probability enables logical thinkers to build predictive models from noisy data without having to use intuition. But even then, the first step of making a Bayesian update is data collection.

Combatting this tendency requires you to pay attention to input you normally ignore. Supplement your broader attentional scope with a researched framework as a guide. Say you want to learn how storytelling works. Start by grabbing resources that teach storytelling and learn the basics. Out in the real-world, pay close attention to sights, sounds, and feelings when someone starts telling a story and try identifying sensory input to the storytelling elements you've learned about. Once the basics are subconsciously picked up by habit, your conscious attention will be freed up to make new and more subtle observations.

Ignoring their emotions

Emotional input is difficult to factor, especially because you're emotional at the time. Logical thinkers are notorious for ignoring this kind of messy data, consequently starving their intuition of emotional data. Being able to "go with your gut feelings" is a major function of intuition that logical thinkers tend to miss out on.

Your gut can predict if you'll get along long-term with a new SO, or what kind of outfit would give you more confidence in your workplace, or if learning tennis in your free time will make you happier, or whether you prefer eating a cheeseburger over tacos for lunch. Logical thinkers don't have enough data collected about their emotions to know what triggers them. They tend to get bogged down and mislead with objective, yet trivial details they manage to factor out. A weak understanding of their own emotions also leads to a weaker understanding of other's emotions. You can become a better empathizer by better understanding yourself.

You could start from scratch and build your own framework, but self-assessment biases will impede productivity. Learning an existing framework is a more realistic solution. You can find resources with some light googling and I'm sure CFAR teaches some good ones too. You can improve your gut feelings too. One way is making sure you're always consciously aware of the circumstances you're in when experiencing an emotion.

Making rules too strict

Logical thinkers build frameworks in order to understand things. When adding a new rule to a framework, there's motivation to make the rule strict. The stricter the rule, the more predictive power, the better the framework. When the domain you're trying to understand has multivariable chaotic phenomena, strict rules are likely to break. The result is something like the current state of macroeconomics: a bunch of logical thinkers preoccupied by elegant models and theories that can only exist when useless in practice.

Following rules that are too strict can have bad consequences. Imagine John the salesperson is learning how to make better first impressions and has built a rough framework so far. John has a rule that smiling always helps make people feel welcomed the first time they meet him. One day he makes a business trip to Russia to meet with a prospective client. The moment he meet his russian client, he flashes a big smile and continues to smile despite negative reactions. After a few hours of talking, his client reveals she felt he wasn't trustworthy at first and almost called off the meeting. Turns out that in Russia smiling to strangers is a sign of insincerity. John's strict rule didn't account for cultural differences, blindsiding him from updating on his clients reaction, putting him in a risky situation.

The desire to hold onto strict rules can make logical thinkers susceptible to confirmation bias too. If John made an exception to his smiling rule, he'd feel less confident about his knowledge of making first impressions, subsequently making him feel bad. He may also have to amend some other rule that relates to the smiling rule, which would further hurt his framework and his feelings.

When feeling the urge to add on a new rule, take note of circumstances in which the evidence for the rule was found in. Add exceptions that limit the rule's predictive power to similar circumstances. Another option is to entertain multiple conflicting rules simultaneously, shifting weight from one to the other after gathering more evidence. 

Followup to: Standard and Nonstandard Numbers

So... last time you claimed that using first-order axioms to rule out the existence of nonstandard numbers - other chains of numbers besides the 'standard' numbers starting at 0 - was forever and truly impossible, even unto a superintelligence, no matter how clever the first-order logic used, even if you came up with an entirely different way of axiomatizing the numbers.

"Right."

How could you, in your finiteness, possibly know that?

"Have you heard of Godel's Incompleteness Theorem?"

Of course! Godel's Theorem says that for every consistent mathematical system, there are statements which are true within that system, which can't be proven within the system itself. Godel came up with a way to encode theorems and proofs as numbers, and wrote a purely numerical formula to detect whether a proof obeyed proper logical syntax. The basic trick was to use prime factorization to encode lists; for example, the ordered list <3, 7, 1, 4> could be uniquely encoded as:

2³ * 3⁷ * 5¹ * 7⁴

And since prime factorizations are unique, and prime powers don't mix, you could inspect this single number, 210,039,480, and get the unique ordered list <3, 7, 1, 4> back out. From there, going to an encoding for logical formulas was easy; for example, you could use the 2 prefix for NOT and the 3 prefix for AND and get, for any formulas Φ and Ψ encoded by the numbers #Φ and #Ψ:

¬Φ = 2² * 3^#Φ

Φ ∧ Ψ = 2³ * 3^#Φ * 5^#Ψ

It was then possible, by dint of crazy amounts of work, for Godel to come up with a gigantic formula of Peano Arithmetic [](p, c) meaning, 'P encodes a valid logical proof using first-order Peano axioms of C', from which directly followed the formula []c, meaning, 'There exists a number P such that P encodes a proof of C' or just 'C is provable in Peano arithmetic.'

Godel then put in some further clever work to invent statements which referred to themselves, by having them contain sub-recipes that would reproduce the entire statement when manipulated by another formula.

And then Godel's Statement encodes the statement, 'There does not exist any number P such that P encodes a proof of (this statement) in Peano arithmetic' or in simpler terms 'I am not provable in Peano arithmetic'. If we assume first-order arithmetic is consistent and sound, then no proof of this statement within first-order arithmetic exists, which means the statement is true but can't be proven within the system. That's Godel's Theorem.

"Er... no."

No?

"No. I've heard rumors that Godel's Incompleteness Theorem is horribly misunderstood in your Everett branch. Have you heard of Godel's Completeness Theorem?"

Is that a thing?

"Yes! Godel's Completeness Theorem says that, for any collection of first-order statements, every semantic implication of those statements is syntactically provable within first-order logic. If something is a genuine implication of a collection of first-order statements - if it actually does follow, in the models pinned down by those statements - then you can prove it, within first-order logic, using only the syntactical rules of proof, from those axioms."

Followup to: Logical Pinpointing

"Oh! Hello. Back again?"

Yes, I've got another question. Earlier you said that you had to use second-order logic to define the numbers. But I'm pretty sure I've heard about something called 'first-order Peano arithmetic' which is also supposed to define the natural numbers. Going by the name, I doubt it has any 'second-order' axioms. Honestly, I'm not sure I understand this second-order business at all.

"Well, let's start by examining the following model:"

"This model has three properties that we would expect to be true of the standard numbers - 'Every number has a successor', 'If two numbers have the same successor they are the same number', and '0 is the only number which is not the successor of any number'.  All three of these statements are true in this model, so in that sense it's quite numberlike -"

And yet this model clearly is not the numbers we are looking for, because it's got all these mysterious extra numbers like C and -2*.  That C thing even loops around, which I certainly wouldn't expect any number to do.  And then there's that infinite-in-both-directions chain which isn't corrected to anything else.

"Right, so, the difference between first-order logic and second-order logic is this:  In first-order logic, we can get rid of the ABC - make a statement which rules out any model that has a loop of numbers like that.  But we can't get rid of the infinite chain underneath it.  In second-order logic we can get rid of the extra chain."

Followup to: Mixed Reference: The Great Reductionist Project

Humans need fantasy to be human.

"Tooth fairies? Hogfathers? Little—"

Yes. As practice. You have to start out learning to believe the little lies.

"So we can believe the big ones?"

Yes. Justice. Mercy. Duty. That sort of thing.

"They're not the same at all!"

You think so? Then take the universe and grind it down to the finest powder and sieve it through the finest sieve and then show me one atom of justice, one molecule of mercy.

- Susan and Death, in Hogfather by Terry Pratchett

Suppose three people find a pie - that is, three people exactly simultaneously spot a pie which has been exogenously generated in unclaimed territory. Zaire wants the entire pie; Yancy thinks that 1/3 each is fair; and Xannon thinks that fair would be taking into equal account everyone's ideas about what is "fair".