UD Creationists and Proof

A reader sent me a link to a comment on one of my least favorite major creationist websites, Uncommon Descent (No link, I refuse to link to UD). It’s dumb enough that it really deserves a good mocking.

Barry Arrington, June 10, 2016 at 2:45 pm

daveS:
“That 2 + 3 = 5 is true by definition can be verified in a purely mechanical, absolutely certain way.”

This may be counter intuitive to you dave, but your statement is false. There is no way to verify that statement. It is either accepted as self-evidently true, or not. Think about it. What more basic steps of reasoning would you employ to verify the equation? That’s right; there are none. You can say the same thing in different ways such as || + ||| = ||||| or “a set with a cardinality of two added to a set with cardinality of three results in a set with a cardinality of five.” But they all amount to the same statement.

That is another feature of a self-evident truth. It does not depend upon (indeed cannot be) “verified” (as you say) by a process of “precept upon precept” reasoning. As WJM has been trying to tell you, a self-evident truth is, by definition, a truth that is accepted because rejection would be upon pain of patent absurdity.

2+3=5 cannot be verified. It is accepted as self-evidently true because any denial would come at the price of affirming an absurdity.

It’s absolutely possible to verify the statement “2 + 3 = 5”. It’s also absolutely possible to prove that statement. In fact, both of those are more than possible: they’re downright easy, provided you accept the standard definitions of arithmetic. And frankly, only a total idiot who has absolutely no concept of what verification or proof mean would ever claim otherwise.

We’ll start with verification. What does that mean?

Verification is the process of testing a hypothesis to determine if it correctly predicts the outcome. Here’s how you verify that 2+3=5:

  1. Get two pennies, and put them in a pile.
  2. Get three pennies, and put them in a pile.
  3. Put the pile of 2 pennies on top of the pile of 3 pennies.
  4. Count the resulting pile of pennies.
  5. If there are 5 pennies, then you have verified that 2+3=5.

Verification isn’t perfect. It’s the result of a single test that confirms what you expect. But verification is repeatable: you can repeat that experiment as many times as you want, and you’ll always get the same result: the resulting pile will always have 5 pennies.

Proof is something different. Proof is a process of using a formal system to demonstrate that within that formal system, a given statement necessarily follows from a set of premises. If the formal system has a valid model, and you accept the premises, then the proof shows that the conclusion must be true.

In formal terms, a proof operates within a formal system called a logic. The logic consists of:

  1. A collection of rules (called syntax rules or formation rules) that define how to construct a valid statement are in the logical language.
  2. )

  3. A collection of rules (called inference rules) that define how to use true statements to determine other true statements.
  4. A collection of foundational true statements called axioms.

Note that “validity”, as mentioned in the syntax rules, is a very different thing from “truth”. Validity means that the statement has the correct structural form. A statement can be valid, and yet be completely meaningless. “The moon is made of green cheese” is a valid sentence, which can easily be rendered in valid logical form, but it’s not true. The classic example of a meaningless statement is “Colorless green ideas sleep furiously”, which is syntactically valid, but utterly meaningless.

Most of the time, when we’re talking about logic and proofs, we’re using a system of logic called first order predicate logic, and a foundational system of axioms called ZFC set theory. Built on those, we define numbers using a collection of definitions called Peano arithmetic.

In Peano arithmetic, we define the natural numbers (that is, the set of non-negative integers) by defining 0 (the cardinality of the empty set), and then defining the other natural numbers using the successor function. In this system, the number zero can be written as z; one is s(z) (the successor of z); two is the successor of 1: s(1) = s(s(z)). And so on.

Using Peano arithmetic, addition is defined recursively:

  1. For any number x, x + 0 = x.
  2. For any number numbers x and y: s(x)+y=x+s(y).

So, using peano arithmetic, here’s how we can prove that 2+3=5:

  1. In Peano arithemetic form, 2+3 means s(s(z)) + s(s(s(z))).
  2. From rule 2 of addition, we can infer that s(s(z)) + s(s(s(z))) is the same as s(z) + s(s(s(s(z)))). (In numerical syntax, 2+3 is the same as 1+4.)
  3. Using rule 2 of addition again, we can infer that s(z) + s(s(s(s(z)))) = z + s(s(s(s(s(z))))) (1+4=0+5); and so, by transitivity, that 2+3=0+5.
  4. Using rule 1 of addition, we can then infer that 0+5=5; and so, by transitivity, 2+3=5.

You can get around this by pointing out that it’s certainly not a proof from first principles. But I’d argue that if you’re talking about the statement “2+3=5” in the terms of the quoted discussion, that you’re clearly already living in the world of FOPL with some axioms that support peano arithmetic: if you weren’t, then the statement “2+3=5” wouldn’t have any meaning at all. For you to be able to argue that it’s true but unprovable, you must be living in a world in which arithmetic works, and that means that the statement is both verifiable and provable.

If you want to play games and argue about axioms, then I’ll point at the Principia Mathematica. The Principia was an ultimately misguided effort to put mathematics on a perfect, sound foundation. It started with a minimal form of predicate logic and a tiny set of inarguably true axioms, and attempted to derive all of mathematics from nothing but those absolute, unquestionable first principles. It took them a ton of work, but using that foundation, you can derive all of number theory – and that’s what they did. It took them 378 pages of dense logic, but they ultimately build a rock-solid model of the natural numbers, and used that to demonstrate the validity of Peano arithmetic, and then in turn used that to prove, once and for all, that 1+1=2. Using the same proof technique, you can show from first principles, that 2+3=5.

But in a world in which we don’t play semantic games, and we accept the basic principle of Peano arithmetic as a given, it’s a simple proof. It’s a simple proof that can be found in almost any textbook on foundational mathematics or logic. But note how Arrington responds to it: by playing word-games, rephrasing the question in a couple of different ways to show off how much he knows, while completely avoiding the point.

What does it take to conclude that you can’t verify or prove something like 2+3=5? Profound, utter ignorance. Anyone who’s spent any time learning math should know better.

But over at Uncommon Descent? They don’t think they need to actually learn stuff. They don’t need to refer to ungodly things like textbook. They’ve got their God, and that’s all they need to know.

To be clear, I’m not anti-religion. I’m a religious Jew. Uncommon Descent and their rubbish don’t annoy me because I have a grudge against theism. I have a grudge against ignorance. And UD is a huge promoter of arrogant, dishonest ignorance.

Living with Social Anxiety

I’ve been thinking about writing this for a long time. Probably a few years by now. I think it’s probably about time to out myself.

I’ve mentioned in the past that I’ve dealt with mental illness. But I’ve never gone in depth about it. There are a lot of reasons for that, but the biggest one is just that it’s really frightening to reveal that kind of personal information.

I’ve had trouble with two related things. I’ve written before about the fact that I’m being treated for depression. What I haven’t talked about is the fact that I’ve got very severe social anxiety.

To me, the depression isn’t such a big deal. I’m not saying that depression isn’t serious. I’m not even saying that in my case, my depression wasn’t/isn’t serious. But for me, depression is easily treated. I’m one of the lucky people who respond really well to medication.

Back when I first realized that something was wrong, and I realized that what I was feeling (or, more accurately, what I was not feeling) was depression, I went to see a doctor. On my first visit with him, he wrote me a prescription for Zoloft. Two weeks later, I started feeling better; between 5 and 6 weeks after starting to take the medication, I was pretty much recovered from that episode of depression.

I wouldn’t say that I’ve ever totally recovered. There’s always a lingering residue of depression, which I’m constantly struggling against. It doesn’t go away, but it’s manageable. As long as I’m aware of it, it doesn’t have a huge impact on my life.

On the other hand, social anxiety. For me, that’s a really big deal. That’s the thing that shapes (and warps) my entire life. And as hard as it is to talk about something like depression, talking about SA is much harder.

As bad as people react to depression, the reaction to social anxiety is worse. Depression is commonly viewed as more weakness than illness. But social anxiety is treated as a joke.

It’s no joke. For those of us who deal with it, it’s a huge source of pain. It’s had a huge effect on my life. But I’ve always been afraid to talk about it. The thing is, I think that things like this are important to talk about. Our society has a huge stigma against mental illness. I really believe that needs to change. And the only way that it will change is when we stop treating it as something to be ashamed of, or something that needs to stay hidden. And that means that I’ve got to be willing to talk about it.

Social anxiety is part of who I am, and I can’t escape that. But I can talk about what it is. And I can, publicly, say to kids who are in the same situation that I was in 30 years ago: Yes, being like this sucks. But despite in, you can live a good life. You can find friends who’ll care about you, find a partner who’ll love you, build a successful career, and thrive. Even if your SA never goes away, even if there’s always some pain because of it, it doesn’t have to rule your life. You can still be happy.

The first thing I need to do is to explain just what SA is. But I need to be very clear here: like anything else that involves peoples’ inner perceptions, I can only talk about what it’s like for me. Different people experience things differently, so what it’s like for me might be totally different from what it’s like for someone else. I don’t mean to in any way cast any shade on anyone else: their feelings and perceptions may be different from mine, but they’re just as valid. This is just my experience.

So. What is social anxiety?

It’s very difficult to explain. The best I can do is to say that it’s the absolute knowledge that I’m freak, combined with a terror of what will happen when anyone finds out. I know, on a deep physical level that if people figure out who/what I am, that they’ll hate me – and worse, that they’ll actively turn on me, attack me, harm me.

It’s not true. I know perfectly well that it’s not true. I can feel like this even with my closest friends – people who I know will always support me, who would never do anything to hurt me. But deep down, on a level below conscious thought, I know it. It doesn’t matter that intellectually I’m aware that it’s not true, because my physical reaction in social situations is based on what my subconscious knows.

So every time I walk into a room full of people, every time I walk into a store, every time I pick up the phone, every time I walk over to a coworker to ask a question, that’s what I’m feeling. That fear, that need to escape, that certainty that I’m going to mess up, and that when I do, I’m going to be ostracized or worse.

What makes it worse is the fact that the way I behave because of the social anxiety increases the odds that other people will think I’m strange – and when people see me that way, it increases the stress that I feel. When you’re putting a substantial part of your effort and concentration into squashing down the feeling of panic, you’re not paying full attention to the people you’re interacting with. At best, you come off as distant, inattentive, and rude. At worst, you’re seen reacting in odd ways, because you’ve missed some important social cue.

It’s not a small thing. Humans are social creatures. We need contact with other people. We can’t live without it. But my interactions are always colored by this fear. I have to fight against it every day, in everything I do. It colors every interaction I have with every person I encounter. It’s there, all the time.

When people talk about social anxiety, they mostly talk about it as being something like excessive shyness. I hope that this descriptions helps make it clear that that’s not what it’s really about.

Where’d this craziness come from?

For me, it’s really a kind of PTSD, or so a doctor who specializes in SA told me. I feel really guilty saying that, because to me, PTSD is something serious, and I have a hard time putting myself into the same basket as people who’ve gone through real trauma. But in medical terms, that’s what’s happening.

I’ve written about my past a little bit before. I had a rough childhood. Most of the time when you hear that, you think of family trouble, which couldn’t be farther from the truth for me. I had a really wonderful family. My parents and my siblings were/are great. But in school, I was the victim of abuse. I was a very small kid. I’m fairly tall now (around 5’11”) when I started high school, I wasn’t quite 5 feet tall. At the beginning of my junior year, I was still just 5’1″. So, I was short, skinny, hyperactive geeky kid. That’s pretty much the formula for getting picked on.

But I didn’t just get picked on. I got beaten up an a regular basis. I don’t say that lightly. I’m not talking about small stuff. The small abuses would have been bad enough, but that’s not what happened to me. This was serious physical abuse. To give one example, in gym class one day during my senior year, I had someone tackle me to the ground; then grab my little finger, say “I wonder what it would feel like if I broke this?”, and then snap it.

That was, pretty much, my life every day from 5th grade until I graduated high school. Everything I did became a reason to abuse me. If I answered a teachers question in class? That was a reason to beat me: I’m making them look bad. If I didn’t answer a question in class, that was a reason to beat me: I should be satisfying the teacher so that they don’t have to.

It wasn’t limited to school. My house was vandalized. The gas lines on our grill were cut. A swastika was burned into the street in front of my house. We had so many mailboxes destroyed that we literally build a detachable mount for the mailbox, and brought it in to the house every night. Then in retribution for depriving the assholes of the privilege of smashing our mailbox, they set the wooden mailbox post on fire.

Hearing this, you’d probably ask “Where was the principal/administration when all of this was going on?”. The answer? They didn’t really give a damn. The principal was an ex-nun, who believed that you shouldn’t punish children. If one children hits another, you shouldn’t tell them that hitting is wrong. You should sit them down and talk to them about “safe hands”, and what you need to do for your hands to be safe.

After the finger-breaking incident, my parents really freaked out, and went in to see the principal and assistant principal. Their reaction was to be furious at my parents. The AP literally shouted at my father, saying “What do you want, a god-damned armed guard to follow your kid around?”. (To which, I think, the response should have been “Fuck yeah. If you’re doing such a shit job protecting your students that the only way to stop them from having their bones broken for fun is to hire armed guards to follow them around, then you should damn well do that.”) Unfortunately, my parents didn’t believe in lawsuits; they wouldn’t sue the school, and they just didn’t have the money to move me to a private school. So I got to suffer.

(Even now, I would dearly love to find that principal… I’d really like to explain to her exactly what a god-damned idiot she is, and how ashamed she should be of the horrible job she did. A principal’s number one job is making sure that the school is a safe place for children to learn. She failed, horribly, at that – and, as far as I could tell, never felt the slightest bit of guilt over all of the things she allowed to happen in her school.)

So now, it’s literally 30 years since I got out of high school. But it’s very hard to get past the things that were pounded into you during your childhood. The eight years of daily abuse – from the time I was 10 years old until I turned 18 – basically rewired my personality.

The effects of that are what made me the way I am.

How does social anxiety really affect my daily life?

Socially, it’s almost crippling. I don’t have much of a social life. I’ve got a small group of very close friends who I don’t get to see nearly enough of, and I have a very hard time meeting new people. Even with people that I’ve known for a long time, I’m just not comfortable. Sometimes I really need social contact, but most of the time, I’d rather be alone in some quiet place, where I don’t need to worry about what other people think. I’d really like to be able to socialize more – in particular, there are a lot of people that I’ve met through this blog that I think of as friends, who I’d love to meet in person, but I never do. Even when I have the change, I usually manage to muck it up. (Because I always believe that people are looking for some reason to reject me, I see rejection in places where it doesn’t exist.)

Professionally, it’s been up and down. It definitely has held me back somewhat. In any job where you need to promote yourself, someone with SA is in deep trouble.

At one point, I even lost a job because of it. I didn’t get fired, but that’s only because I quit when it became obvious that that’s what was coming, and there was no point sticking around waiting for it. My manager at the time found out I was getting treated for SA. From the moment he found out, he stopped trusting anything I said about anything. To make matters worse, at the time, he was in trouble for a project that was literally 2 years overdue, and he needed a scapegoat. The “crazy” guy was the obvious target.

As an example of what I mean: one of the times he accused me of incompetence involved actors, which is a programming model that I used in my PhD dissertation. Actors are a model of concurrent computation in which everything is asynchronous. There are no visible locks – just a collection of active objects which can asynchronously send and receive messages. (I wrote a post about actors, with my own implementation of a really silly actor-based programming language here.)

We were working on a scheduling problem for our system. Our team had a meeting to discuss how to implement a particular component of that. After a lot of discussion, we agreed that we should implement it as an actor system. So I wrote a lightweight actors framework on top of our thread library, and implement the whole thing in actors. My coworkers reviewed the code, and accepted it with a lot of enthusiasm. My manager scheduled a private meeting where he accused my mental illness of impairing my judgement, because what kind of idiot would write something like that to be totally asynchronous?

So I left that company. Fortunately, skilled software engineers are in high demand in the NYC area, so finding a new job wasn’t a problem. I’ve had several different jobs since then. SA really hasn’t been a huge problem at any of them, thank goodness. It’s always a bit of a problem because my natural tendency is to try to disappear into the background, so it’s easy for people to not notice the work I’m doing. But I’ve mostly learned how to overcome that. It’s not easy, but I’ve managed.

When job-hunting, after that terrible experience, I learned to be careful to learn a bit about what the work culture of a company is like before I go to work there. I’ve tried to work something into conversations with people at the company after I have an offer, but before I accept it. It gives me a chance to see how they react to it. If I don’t like their reaction, if it seems like there’s a good chance that it’ll cause trouble, I’ll just take a different job someplace where it won’t be a problem. Like I said before, it’s a good time to be a software engineer in NYC – I can afford to turn down offers from companies that I don’t like.

So, yeah. I’m kind of crazy. Writing this is both difficult and terrifying. Posting it is going to be even worse. But I think it’s important to get stuff like this out there.

Despite all of this, I’ve wound up in a good place. I’m married to a lovely woman. I’ve got two smart, happy kids. I’ve got a great job, working with people that I really, genuinely like and enjoy working with, and they seem to like me back. It’s been a long, hard road to get here, but I’m pretty happy where I am.

This has gotten to be quite long, and I’ve been working on it on and off for a couple of months. I think that I’ve got to just let go, and post it as is. Feel free to ask questions about anything that I can clarify, and feel free to share your own stories in the comments. If you want to post something anonymously, feel free to email it to me (markcc@gmail.com), and I’ll post it for you so that theres nothing on the blog that could identify you.

Also note that I’m going to tightly moderate replies to this post. I’m not interested in having my blog turn into a place where jerks can abuse people sharing painful personal stories.

Elon Musk’s Techno-Religion

A couple of people have written to me asking me to say something about Elon Musk’s simulation argument.

Unfortunately, I haven’t been able to find a verbatim quote from Musk about his argument, and I’ve seen a couple of slightly different arguments presented as being what Musk said. So I’m not really going to focus so much on Musk, but instead, just going to try to take the basic simulation argument, and talk about what’s wrong with it from a mathematical perspective.

The argument isn’t really all that new. I’ve found a couple of sources that attribute it to a paper published in 2003. That 2003 paper may have been the first academic publication, and it might have been the first to present the argument in formal terms, but I definitely remember discussing this in one of my philophy classes in college in the late 1980s.

Here’s the argument:

  1. Any advanced technological civilization is going to develop massive computational capabilities.
  2. With immense computational capabilities, they’ll run very detailed simulations of their own ancestors in order to understand where they came from.
  3. Once it is possible to run simulations, they will run many of them to explore how different parameters will affect the simulated universe.
  4. That means that advanced technological civilization will run many simulations of universes where their ancestors evolved.
  5. Therefore the number of simulated universes with intelligent life will be dramatically larger than the number of original non-simulated civilizations.

If you follow that reasoning, then the odds are, for any given form of intelligent life, it’s more likely that they are living in a simulation than in an actual non-simulated universe.

As an argument, it’s pretty much the kind of crap you’d expect from a bunch of half drunk college kids in a middle-of-the-night bullshit session.

Let’s look at a couple of simple problems with it.

The biggest one is a question of size and storage. The heart of this argument is the assumption that for an advanced civilization, nearly infinite computational capability will effectively become free. If you actually try to look at that assumption in detail, it’s not reasonable.

The problem is, we live in a quantum universe. That is, we live in a universe made up of discrete entities. You can take an object, and cut it in half only a finite number of times, before you get to something that can’t be cut into smaller parts. It doesn’t matter how advanced your technology gets; it’s got to be made of the basic particles – and that means that there’s a limit to how small it can get.

Again, it doesn’t matter how advanced your computers get; it’s going to take more than one particle in the real universe to simulate the behavior of a particle. To simulate a universe, you’d need a computer bigger than the universe you want to simulate. There’s really no way around that: you need to maintain state information about every particle in the universe. You need to store information about everything in the universe, and you need to also have some amount of hardware to actually do the simulation with the state information. So even with the most advanced technology that you can possible imagine, you can’t possible to better than one particle in the real universe containing all of the state information about a particle in the simulated universe. If you did, then you’d be guaranteeing that your simulated universe wasn’t realistic, because its particles would have less state than particles in the real universe.

This means that to simulate something in full detail, you effectively need something bigger than the thing you’re simulating.

That might sound silly: we do lots of things with tiny computers. I’ve got an iPad in my computer bag with a couple of hundred books on it: it’s much smaller than the books it simulates, right?

The “in full detail” is the catch. When my iPad simulates a book, it’s not capturing all the detail. It doesn’t simulate the individual pages, much less the individual molecules that make up those pages, the individual atoms that make up those molecules, etc.

But when you’re talking about perfectly simulating a system well enough to make it possible for an intelligent being to be self-aware, you need that kind of detail. We know, from our own observations of ourselves, that the way our cells operates is dependent on incredibly fine-grained sub-molecular interactions. To make our bodies work correctly, you need to simulate things on that level.

You can’t simulate the full detail of a universe bigger that the computer that simulates it. Because the computer is made of the same things as the universe that it’s simulating.

There’s a lot of handwaving you can do about what things you can omit from your model. But at the end of the day, you’re looking at an incredibly massive problem, and you’re stuck with the simple fact that you’re talking, at least, about building a computer that can simulate an entire planet and its environs. And you’re trying to do it in a universe just like the one you’re simulating.

But OK, we don’t actually need to simulate the whole universe, right? I mean, you’re really interested in developing a single species like yourself, so you only care about one planet.

But to make that planet behave absolutely correctly, you need to be able to correctly simulate everything observable from that planet. Its solar system, you need to simulate pretty precisely. The galaxy around it needs less precision, but it still needs a lot of work. Even getting very far away, you’ve got an awful lot of stuff to simulate, because your simulated intelligences, from their little planet, are going to be able to observe an awful lot.

To simulate a planet and its environment with enough precision to get life and intelligence and civilization, and to do it at a reasonable speed, you pretty much need to have a computer bigger than the planet. You can cheat a little bit, and maybe abstract parts of the planet; but you’ve got to do pretty good simulations of lots of stuff outside the planet.

It’s possible, but it’s not particularly useful. Because you need to run that simulation. And since it’s made up of the same particles as the things it’s simulating, it can’t move faster than the universe it simulates. To get useful results, you’d need to build it to be massively parallel. And that means that your computer needs to be even larger – something like a million times bigger.

If technology were to get good enough, you could, in theory, do that. But it’s not going to be something you do a lot of: no matter how advanced technology gets, building a computer that can simulate an entire planet and its people in full detail is going to be a truly massive undertaking. You’re not going to run large numbers of simulations.

You can certainly wave you hands and say that the “real” people live in a universe without the kind of quantum limit that we live with. But if you do, you’re throwing other assumptions out the window. You’re not talking about ancestor simulation any more. And you’re pretending that you can make predictions based on our technology about the technology of people living in a universe with dramatically different properties.

This just doesn’t make any sense. It’s really just techno-religion. It’s based on the belief that technology is going to continue to develop computational capability without limit. That the fundamental structure of the universe won’t limit technology and computation. Essentially, it’s saying that technology is omnipotent. Technology is God, and just as in any other religion, it’s adherents believe that you can’t place any limits on it.

Rubbish.