When mathematical truths and their applicability to the physical world are discussed, there's a certain kind of flawed (in my opinion) thinking that is often employed, and it comes out in sentences like "rocks behave isomorphically to integers, while clouds don't", or "two apples plus two apples is four apples, unless someone stole one from the bowl", and so on. I'll try to explain why I consider such reasoning flawed, and what are the more suitable descriptions of what's going on with the apples and the rocks and the clouds and the drops of water and such.

Two mistakes combine together and create a shared state of confusion; the mistakes themselves are almost independent and I think they stand or fall separately.

1. The first mistake is conflating counting things and bringing things together spatially. One usually goes with the other because that's how we learn to count, by looking at groups of things located closely together - but it doesn't have to be this way. When we say "take 2 apples and add 2 more apples, now count - you have 4 apples", we automatically imagine the two additional apples being brought in and placed next to the two original ones, but that's just a mental crutch you can easily do without. Let me show you: suppose I ask you to consider two sheep in England and two more sheep in Australia - how many sheep are there together? You see the English sheep in your mind's eye, there they are standing, and you count one, two. Now please resist the temptation to teleport the two Australian sheep and place them next in sequence. Instead, just fly with your mind's eye all the way to Australia in a split second and home on that field - there they are - and continue counting: three, four. There, you just counted 2+2 sheep without bringing them together in space, in real life or your imagination. There's nothing to it and you do it all the time - if someone asks you to count the number of chairs in a large and busy-looking living room, you don't bring them together in your mind, you just gaze-travel over the room and count them off one by one.

Now consider the implication of that to clouds or other such objects. Suppose someone tells you "well, apples obey the law of arithmetics, but one drop of water plus another drop of water equals one larger drop of water, not two" - it should be clear that they are naively conflating spatial movement and adding/counting. Adding two apples and two apples is not a spatial operation of bringing them together, it's a mental act of viewing them as one whole collection that can be counted. It's just that bringing them spatially together, in reality or your imagimation, is the easiest way to carry that "viewing" out. For drops of water or clouds, the spatial operation becomes a distraction, so just resist it. One cloud plus one cloud most certainly equals two clouds: just count them in your mind, here's one and there's two, maybe drifting next to each other, or maybe they're on separate continents. You don't have to merge them - nothing about "+" says you do.

2. The second mistake is a sort of a map-territory confusion where we naively grant the territory the power of holding discrete objects for our needs. It may be helpful to realize and to remember that at least on our macro scale of reality, on the scale of things we perceive with our senses, discrete, separate objects are a feature of the map, not the territory; they exist in your mind, not the reality. In the reality, there's just a lot of atoms everywhere (I'm simplifying; to be more thorough, think of elementary particles and many-worlds if you feel like it, and the virtual vacuum particles and so on) with no "natural" way of separating them into different conglomerates.

Normally, this really isn't an important point to insist on, and there's no harm whatsoever in just thinking of reality as being made up of objects like apples and clouds and whatnot. But imagine now an argument over an issue like "Was 2+2=4 before humans were around to invent that equation? I believe that long before humans, on an Earth devoid of life, when two rocks rolled onto a beach with three rocks already on it, there were five rocks altogether on the beach". What's really happening in that story? Well, there's the spatial confusion dealt with above, when we find it easiest to imagine the two rocks to be added rolling into the scene. But even before that, zoom in on those three rocks on the beach. What are they? What business do you have saying there's a "rock" there? There's a bunch of atoms of various kinds, and lots of other atoms (of air, sand, etc.) around and next to those, with somewhat different densities and behavior, but no definite boundary between any of them. On the nanoscale, there's constant exchange of atoms everywhere. To say "this is a discrete object, a rock" you need to be able to somehow ignore this inherently fuzzy boundary, maybe by picking a scale and smoothing out all the fine details below it - in human experience, the crudeness of our senses does that job for us, but our senses aren't there in that picture. The reality doesn't know or care about "rocks" - there's just atoms.

This isn't to say, I find it important to note, that the existence of three rocks on that beach is somehow a "subjective" claim. Imagine that there are alien nanobots everywhere on that Earth, recording everything faithfully on the nanoscale and storing the data somewhere for someone to discover; a billion years later, you come along, parse the data, reconstruct the scene - and of course you'll then recognize that there were three rocks on the beach. The fact that there are three rocks on that beach is an objective fact of nature; it's just that the meaning of that statement relies on the procedure of discretization being carried out, on someone or something defining what we consider a single discrete object and how we isolate it; and nature won't do that for you. You do that with your brain. That's why counting - and addition - are inherently mental acts carried out on mental constructs; on the map, not the territory.

All this is not to say that math is "invented" rather than "discovered"; I think my analysis is silent on that, and both platonism and formalism remain possible. It's just that an example of five rocks on the beach before humans are around is not helpful in resolving that question - without human-style discretization you can't meaningfully say it was five rocks there, rather than just a bunch of atoms. It may still be true - I certainly believe so - that the laws that govern the behavior of discrete objects, though they are normally mental entities, are universal and 2+2 was 4 before humans were around and in any alien mentality.

(It remains to acknowledge that reality may be discrete on the micro level - spacetime itself may be discrete, we can certainly speak of single photons, etc. This is irrelevant on the level of our everyday perceptions, however - the level of apples, rocks, clouds and so on).

Came across this article, published in 1991 but hardly dated:

David T. Lykken, What's Wrong With Psychology, Anyway? (PDF, 39 pages)

Anyone who's interested in psychology as a science might, I think, find it fascinating. Lots of stuff there about rationality-related failures of academic psychology. Several wonderful anecdotes, of which I'll quote one in full that had me laughing out loud --

In the 1940s and ’50s. there was a torrent of interest and research surrounding the debate between the S-R [Stimulus-Response] reinforcement theorists at Yale and Iowa City and the S-S [Stimulus-Stimulus] expectancy theorists headquartered at Berkeley. As is usual in these affairs, the two sides produced not only differing theoretical interpretations but also different empirical findings from their rat laboratories, differences that ultimately led Marshall Jones to wonder if the researchers in Iowa and California might not be working with genetically different animals. Jones obtained samples of rats from the two colonies: and tested them in the simple runway situation. Sure enough, when running time was plotted against trial number, the two strains showed little overlap in performance. The Iowa rats put their heads down and streaked for the goal box, while the Berkeley animals dawdled, retraced, investigated, appeared to be making “cognitive maps” just as Tolman always said. But by 1965 the torrent of interest in latent-learning had become a backwater and Jones's paper was published obscurely (Jones & Fennel, 1965).

(I came across the reference to the article in the HN discussion about a project, of independent interest, to try and replicate a sample of articles from three reputable journals in psychology in a given year)

Discussion article for the meetup : Tel Aviv, Israel

Discussion article for the meetup : Tel Aviv, Israel

John McCarthy, the inventor of LISP and one of the founders of the study of AI, died earlier this week. McCarthy was actually the person who came up with the phrase "Artificial Intelligence", in 1955. I find it likely that one day, not very soon, the first thinking self-aware machines will study their history and honor McCarthy's memory.

Sustainability of Human Progress is a set of pages jmc worked on mainly in the late 90s and early 2000s, I think, though he continued to update them occasionally later. This work isn't as widely known as it ought to be. It may be of interest to the LW crowd, even though McCarthy's underlying assumptions of how the human progress will proceed differ from those popular here.

"He who refuses to do arithmetic is doomed to talk nonsense." - John McCarthy

Steven Landsburg argued, in an oft-quoted article, that the rational way to donate to charity is to give everything to the charity you consider most effective, rather than diversify; and that this is always true when your contribution is much smaller than the charities' endowments. Besides an informal argument, he provided a mathematical addendum for people who aren't intimidated by partial derivatives. This post will bank on your familiarity with both.

I submit that the math is sloppy and the words don't match the math. This isn't to say that the entire thing must be rejected; on the contrary, an improved set of assumptions will fix the math and make the argument whole. Yet it is useful to understand the assumptions better, whether you want to adopt or reject them. 

And so, consider the math. We assume that our desire is to maximize some utility function U(X, Y, Z), where X, Y and Z are total endowments of three different charities. It's reasonable to assume U is smooth enough so we can take derivatives and apply basic calculus with impunity. We consider our own contributions Δx, Δy and Δz, and form a linear approximation to the updated value U(X+Δx, Y+Δy, Z+Δz). If this approximation is close enough to the true value, the rest of the argument goes through: given that the sum Δx+Δy+Δz is fixed, it's best to put everything into the charity with the largest partial derivative at (X,Y,Z).

The approximation, Landsburg says, is good "assuming that your contributions are small relative to the initial endowments". Here's the thing: why? Suppose Δx/X, Δy/Y and Δz/Z are indeed very small - what then? Why does it follow that the linear approximation works? There's no explanation, and if you think this is because it's immediately obvious - well, it isn't. It may sound plausible, but the math isn't there. We need to go deeper.

Speaking of things that are funny to some and not others, an instructive example is the Orange Head joke. Usually when it's told, the audience is sharply divided into those who think it's hilarious and those who struggle to see what's funny. 

Here's the Orange Head joke:

It's business as usual for a bartender, and one day as he is cleaning his bar when an unusual customer walks in. The man is dressed in an expensive suit, has a beautiful supermodel hanging off each arm, and has a limo parked outside. Furthermore, the man has an orange for a head.

The customer sits down at the bar and orders everyone a drink. He pays for it from a roll of hundreds and manages to get the attention of every woman in the joint, despite having an orange for a head.

The bartender is not a man to pry, but he feels compelled to ask about this man's life.

"Excuse me," says the bartender, "I can't help but notice that you're obviously fabulously wealthy and irresistable to women, but you have an orange for a head. How did that happen?"

So the man told his story.

"A while back, when I was penniless, I was walking along the beach and saw an old lamp, half buried in the sand. I picked it up and gave it a clean, and POOF! out popped a genie. The genie explained that he had been trapped in that lamp for two hundred years, and that he was so grateful to me for freeing him that he would give me three wishes.

"For my first wish I asked for an unlimited fortune. The genie said 'It is done!' and from then on, whenever I needed money, it was there.

"For my second wish I asked for the attention of all the most beautiful women in the world. The genie said it was done, and since then I have been able to get any woman I wanted.

"For my third wish -- and, this is the bit where I kinda fucked up -- I asked for an orange for a head."

 

Do you think it's funny? 

If you search for this joke's key words, you'll see many pages where, after it's told, people react incredulously and ask where the joke was. Others at the same time are laughing their heads off. Here's a blog post that attempts to analyze this, though it doesn't get far.

(I personally think it's hilarious, and easily the best joke I heard last year. When I retold it at my blog, I got many concurring comments, but also comments from people who didn't see anything funny, even after those who did tried to explain what they found in it. Several people went on to convince themselves it's garbled and there must be an "original" version in which the final remark makes sense and is funny - and offered several ideas of how it might go).