Posts tagged with assumptions

1. The monad, of which we will speak here, is nothing else than a simple substance, which goes to make up compounds; by simple, we mean without parts.

2. There must be simple substances because there are compound substances; for the compound is nothing else than a collection or aggregatum of simple substances.

Gottfried W. Leibniz, Monadology

 

Crazy how a “father” of calculus was so illogical in his seminal work of 1714.

  • The existence of compound things does not imply the existence of partless atoms.
  • He asserts, doesn’t prove, that a compound is “nothing more than" a collection of simple substances. (atoms)







I’ve collected a few tidbits about non-wellfoundedness on isomorphismes:

  • the opposite of the idea of “indivisible atoms" at the "bottom" of everything
  • turtles all the way down
  • (infinite regress is OK)
  • a > b > c > a
  • (so the two options I can think of for non-wellfounded sets are either an infinite straight line or a circle—which biject by stereographic projection)

as well as examples of irreducible things:

  • if you take away one Borromean ring
    http://upload.wikimedia.org/wikipedia/commons/thumb/5/5a/Borromean_Rings_Illusion.png/774px-Borromean_Rings_Illusion.png
    then the whole is no longer interlinked
  • Twisted products in K-theory are different to straight products.
    https://upload.wikimedia.org/wikipedia/commons/thumb/6/6e/M%C3%B6biusStripAsSquare.svg/1000px-M%C3%B6biusStripAsSquare.svg.png
    A Möbius band is different to a wedding band.
    image

    Yet 100% of the difference is in how two 1-D lines are put together. The parts in the recipe are the same, it’s the way they’re combined (twisted or straight product) that makes the difference.

So Leibnitz’s assertions are not only unsupported, but wrong. (Markov, causality, St Anselm’s argument, conservation of mass, etc. in Monadology 4, 5, 22, 44, 45.)

tl,dr: Leibniz, like Spinoza, uses the word “therefore” to mean “and here’s another thing I’m assuming”.




I have a colleague whose wife appears to be remarkably materialistic.

He feels that he makes a good living so that she can enjoy fancy handbags and days at the spa; he wants kids but she isn’t convinced that they can afford them. (He makes [an] upper middle-class income; she works in the low-paying fashion industry.) Whenever he talks about her, he inevitably refers to “how women are” or “how New York women are” — his wife is not unusual in any way, she’s just like every woman. Also, did you know that women always blame their bad behavior on their hormones (PMS!) but men are not able to do so?

In case it’s not clear, this man is not some poor misunderstood statistical genius with a gift for generalization; he is simply a cretin.

Anonymous commenter on the Falkenblog

(Source: falkenblog.blogspot.com)




The seeds of my dissent from economic orthodoxy were pretty much sown for me by my 1st professor on the 1st day of my 1st economics class.

This prof had gone to a great personal trouble to begin our exposure to the dismal science with a very down-to-earth and super-important lesson. She went so far as to spend her own cash on some things from the store, of varying cost, and gave us all at the beginning of the class random items. Some people got candy, some got socks, one or two got things of greater value.

This was a masterful teaching stroke, by someone who cared deeply about her subject and teaching it to newbies: she would have us all participate in voluntary trade within the classroom and end up than we started. Gains from trade—the fundamental point about economics—are really “the only thing we know about welfare”. Sure, some people start off with more—more wealth, more smarts, better looks, genes that will make them grow taller so they can reach the mayonnaise jars from the back—but hey, at least we can make all of them better off and not hurt anyone by allowing them to trade freely.

Right?

We each reported, on a scale of 1 to 10, how satisfied we were with the Stuff we had been randomly given at the beginning of the class, and the prof wrote these scores down on the board. Then we were asked to stand up, walk about the room, and see if anyone would voluntarily exchange Stuff with us. Multiple transactions were allowed, even encouraged—and after a few minutes of cluelessly blitzing with each other, the trading day was closed and we resumed our seats.

The prof asked our scores again, fully expecting that ∀i in the class, utility before utility after.

But one girl reported a lower score.

Instead of taking this as evidence against her belief that transactions are always mutually beneficial—a cornerstone of normative economic theory—the prof instead scolded the girl. "Well, what’d ya do that for?!”

By the way, this was not a prof who prepended test questions with the phrase “According to the theory we learned in class,” which means I still dispute that I got that one about the lobstermongers right! (Since it asked about “What would happen” not “what the theory says would happen”.)

At the time I thought the outburst a bit rude and over the years to come I remembered the episode. (well, obviously) I still think of it as a microcosm of certain intellectual misdeeds by economists. The framework is too important to hold onto; if anyone undermines then you get angry and yell at them! It’s a plausibility war, after all.

Not too far off from real comments by economists: But if you took away the mutually-beneficial assumption, then you’d have no theory at all! (Regardless of whether nullset is the only true theory we have.)

The assumptions about what goes on in transactions are so appealing that even when you see them violated in front of your eyes, they’re still so implausible and—hey—what about all this stuff I learned about indifference curves? If I saw so many graphs with them not overlapping or going backwards, then that has to be the truth, because maths!

Nevermind that people don’t always know what they want, or maybe it’s contradictory or impossible, and even in well-defined classroom experiments they may just, um, do it wrong.

Happy Independence Day. Here’s to hoping you don’t use the independence to shoot yourself in the foot.




Climate Statistics

  • httpness: (studying statistics) Can there be a different standard deviation up and down?
  • isomorphisms: Yes. it's called a semideviation. (Or a quasinorm.) There are a lot of people who argue that semideviations and quasinorms are more natural than standard deviation and norms.
  • httpness: So that's not a normal distribution?
  • isomorphisms: Whatever distribution you're using, there are different measures of dispersion on that -- standard deviation, downside risk / semideviation, interquartile range, kurtosis, etc.
  • httpness: I was just thinking about temperatures. The standard deviation changes depending on the time of year, and the chance of unseasonably warm or cold days changes too.
  • httpness: Here's an example of what I mean. let's say during the summer there _is_ a standard deviation and it's the same up and down. But at another time of year there could be more chance of a very warm day, and at a third time of year there could be more chance of an unseasonably cold day.




As nice as it is to be able to assume normality, … there are problems. The most obvious problem is that we could be wrong.


One … very nice thing … is that, in many situations, … [being wrong] won’t send us immediately to jail without passing “Go.” Under a … broad set of conditions … our assumption [could be wrong, yet we] get away with it. By this I mean that our answer may still be correct even if our assumption is false. This is what we mean when we speak of a [statistic] … being robust.



However, this still leaves at least two problems. In the first place, it is not hard to create reasonable data that violate a normality (or homogeneity of variance) assumption and have “true” answers that are quite different from the answer we would get by making a normality assumption. In other words, we can’t always get away with violating assumptions. Second, there are many situations where even with normality, we don’t know enough about the statistic we are using to draw the appropriate inferences.



One way to look at bootstrap procedures is as procedures for handling data when we are not willing to make assumptions about the parameters of the populations from which we sampled. The most that we are willing to assume (and it is an absolutely critical assumption) is that the data we have are a reasonable representation of the population from which they came. We then resample from the pool of data that we have, and draw inferences about the corresponding population and its parameters.

The second way to look at bootstrap procedures is to think of them as what we use when we don’t know enough.

David Howell

(Source: uvm.edu)




Even though Lil’ Wayne has been a thing for half a decade, I only just now listened to a song of his: Hustler Musik. I like it.

I think this video is juxtaposing different people’s work lives—unemployed responsible guy, cop, drug dealer, stripper.

And check this out at 3:09, 3:17, 3:30 and 3:54 — one of the strippers is reading Tensor Calculus by Synge & Schild.

Also a quantum chemistry book (can’t make out the author).

  • From the main girl’s facial expression at 3:54, I think it isn’t her book. But then again, she  counts her money on it which suggests it is hers.
  • Is the reader currently enrolled in a degree programme? 
  • Both books look in excellent condition — a little too unbent for her to be very far through them. (the softback cover would lift up more if she had made it to chapter 3)
  • How much down-time do you have between dances? I would think there’s some other “duty” or else they would send you home. Maybe working the crowd to sell private dances or trying to get guys to buy drinks.
  • Then again, these dense books are easy to fill up on quickly. When I was working as an artists’ model I would read a bit of maths before I started posing, that way I would have plenty to think about while I stood/sat there.
  • Even though it is a stereotype for a sex worker to say “I’m doing this to put myself through university” — because of the common belief that university is good and valid, much more so than just reading about quantum chemistry because you’re inherently interested in the universe — I don’t think it’s at all unrealistic to show a beautiful woman being into scientific / mathematical erudition. I hate the “attractive people are stupid” stereotype even more than I hate the “nerds rule the world” stereotype. And I actually know a girl who used to dance and at the time had attained an even higher level of mathematical erudition than this girl.
  • A young, attractive girl is much more likely to be able to make good money dancing than by knowing about quantum chemistry. Dancing is also a pick-up job in a way that, for example, working at Fermilab is not. I expect life is freer when you’re doing something like that. Also you don’t have to dance 40-60 hours/week, which leaves plenty of time for intellectual pursuits. I am never surprised to learn that someone with a lot of mathematical erudition is working in a job completely lacking university pre-requisites.
  • I actually have a copy of Synge & Schild — it was recommended supplementary reading in differential geometry class. The writing is good, but for pleasure reading I prefer the diagrams of Solid Shape—a book I’ve extolled in these pages before.
 

WTF is a tensor? I have a much longer post about the topic in my Drafts folder (along with 1150 others), but here’s a quickie preview:

  1. A matrix has two subscripts (row & column); a tensor has three or more subscripts.
  2. Just like the number of rows and columns in a matrix tell you “how many input dimensions” and “how many output dimensions”, tensors can also input/output vectors, matrices, 9-tensors, and so on. A weighted inner product looks like a (0,2) tensor, for example. A matrix looks like a (1,1) tensor, a vector looks like a (0,1) tensor, and a 1-form looks like a (1,0) tensor.
  3. A typical example of a tensor is the stress/strain tensor:

    The piece I have in my drafts folder is talking about foreign exchange rates.
  4. And back to the girl’s pair of textbooks—the two texts do go together. If you think about stress/strain tensors acting on a bridge or something—well if we were talking at a small scale then the forces could be electrical rather than mechanical and operating on a tetrahedron-shaped methane molecule. Tensors are the normal way to combine lots of different forces on different faces of an object.  

To understand tensors, I would recommend looking at the Wikipedia page and Chris Tiee’s essay Covariance, Contravariance, Densities, and All That and maybe also the fourth chapter of MIT OCW’s intro to geophysics lecture notes (that’s on the stress/strain tensors).




Ever since I took too many mathematics classes, I started using the concept of “upper bound” literally. It confuses people.

  • Girlfriend: How long do you think it’ll take you to work out?
  • Me: Oh, I don’t know. I’d say less than five hours.
  • Girlfriend: Five hours?! What are you planning to do there?
  • Me: I didn’t say it would take five hours. I said it would take less than five hours.

Well, it did take me less than five hours. It took me an hour and a half.

 

Same problem with confidence intervals — I give these very literal answers. My former boss told me a story about a “rationality test” she was given, by a statistician or something. First she was asked to guess some fact that only a 5th grader would know, like how many tons the moon weighs or what’s the square mileage of Antarctica. Then the statistician asked her to give 90% confidence bounds. That’s, you’re 90% sure that the value is between these two numbers. Most people fail by saying numbers close to their original guess.

image

My boss just said, "I’m confident that the number is somewhere between zero and a trillion trillion trillion trillion trillion trillion trillion … trillion." Well she was correct! And she was one of the only ones.

You can make confidence bounds as wide as you want and be logically correct. People will look at you in bewilderment when you say things like “I’ll be gone somewhere between two seconds and seven days,” but you will not be a liar.

 

Mathematicians are the only people who go around making statements like “I found out that the answer is greater than 6 and less than 3→3→64→2. I’m pretty sure the answer is 13, though.

(3→3→64→2 is bigger than billions of universes.)

image

It’s not even the difference between certainty-of-proof and casual guesstimation. It’s the difference between giving an upper bound and giving the least upper bound (supremum).

I can say with complete confidence that I will never earn over 10^18737 pounds in my life, no matter how much hyperinflation or life extending medicines lie in the future. I can say the same about 10^18736 pounds. How low am I willing to go with these statements? Ay, there’s the lub.

 

No less of an intellect than Paul Graham swaps upper and lower bounds. In describing how he’s designing a new programming language (arc) with the goal that useful programs should be as short as possible in it, he writes:

That’s part of why I focus on code size. Length is an external constraint. If you start looking at code thinking "what is the lower bound on how long this has to be?" you’re one step from discovering the new operator that will make it that short.

I might be the only person who reads this and is confused. When I hear “lower bound” I think “Nothing is lower than the lower bound. It has to be bigger than the lower bound.” But then he is talking like putting a lower bound on the code means the code is shorter. Zoinks?

And then I’m like, oh. Duh. He means “the lower bound on the upper bound on how long this has to be.” Supremum. Well you could have just said that, Paul. Or I could not think so literally.




Highlights from the episode on Sir Francis Bacon

  • "Male" scientists probe and ravage "female" nature.
  • Christian radicals funded science because of Daniel 12:9. They believed it was Last Days.
  • The New World symbolizes leaving the comfort of the Mediterranean. Aristotelian science = comfort; empirical science = new.
  • Society must “return” from squalid 17th century London to our right and natural state of ease and comfort. In the imagined past food was abundant, disease nonexistent, people happy and well-clothed — and knowledge as in Eden.
  • Science as the path of technological advancement back to Paradise — what Bacon meant by “Knowledge Is Power”

This programme aired on 2 April 2009.

History. The news that’s never too old to be relevant.

(Source: BBC)




There is a prejudice in society against people who wake up late.

Obviously, someone who wakes up at 3pm, eats and gets ready, then works from 5pm until 2am, can get as much done as someone who works from 8:30am until 5:30pm.

And there are even more nocturnal schedules for sure. I used to finish work at 4am or 5am. (EMT’s, I feel ya!)

You would think that a diverse economy like ours could support multiple time-lifestyles, without prejudice. And there are indeed 24-hour Krogers (thankfully) as well as convenience stores. But doctors’ offices, job interviews, government agencies, banks, and gyms are all difficult-ish to access when you get home from work at 5am.

What’s the deal?

Aren’t there profits to be made by keeping your store open later? Plus, drawing customers away from the busiest store hours would reduce stress, mistakes, and costs during peak operating hours. Not to mention, traffic congestion would be reduced if more people commuted at 11am, 2pm, 5pm, 5am, or other times.

Maybe it is the priorities of people with kids forcing the rest of the world to conform to their preferred schedule?

(Source: beckyrussoniello)