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Posts tagged with books

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).




On the afternoon of the Nobel announcement, Nash said that he had won for game theory and that he felt that game theory was like string theory: a subject of great intrinsic intellectual interest that the world wishes to imagine can be of some utility.


He said it with enough scepticism in his voice to make it funny.

"A Beautiful Mind: A Biography of John Forbes Nash, Jr." (1998)

(Source: fisher.osu.edu)




Leonardo da Vinci’s ability to embrace uncertainty, ambiguity, and paradox was a critical characteristic of his genius. —J Michael Gelb
Say you want to use a mathematical metaphor, but you don’t want to be really precise. Here are some ways to do that:
Tack a +ε onto the end of an equation.
Use bounds (“I expect to make less than a trillion dollars over my lifetime and more than $0.”)
Speak about a general class without specifying which member of the class you’re talking about. (The members all share some property like, being feminists, without necessarily having other properties like, being women or being angry.)
Use fuzzy logic (the ∈ membership relation gets a percent attached to it: “I 30%-belong-to the class of feminists | vegetarians | successful people.”).
Use a specific probability distribution like Gaussian, Cauchy, Weibull.
Use a tempered distribution a.k.a. a Schwartz function.
Tempered distributions are my favourite way of thinking mathematically imprecisely.
Tempered distributions have exact upper and lower bounds but an inexact mean and variance. T.D.’s also shoot down very fast (like exp{−x²} the gaussian) which makes them tractable.
For example I can talk about the temperature in the room (there is not just one temperature since there are several moles of air molecules in the room), the position of a quantum particle, my fuzzy inclusion in the set of vegetarians, my confidence level in a business forecast, ….. with a definite, imprecise meaning.
Classroom mathematics usually involves precise formulas but the level of generality achieved by 20th century mathematicians allows us to talk about a cobordism between two things without knowing everything precisely about them.
It’s funny; the more advanced and general the mathematics, the more casual it can become. Like stingy stickler things that build up to a chummy, whatever-it’s-all-good.
 
Our knowledge of the world is not only piecemeal, but also vague and imprecise. To link mathematics to our conceptions of the real world, therefore, requires imprecision.
I want the option of thinking about my life, commerce, the natural world, art, and ideas using manifolds, metrics, functors, topological connections, lattices, orthogonality, linear spans, categories, geometry, and any other metaphor, if I wish.

Leonardo da Vinci’s ability to embrace uncertainty, ambiguity, and paradox was a critical characteristic of his genius. —J Michael Gelb

Say you want to use a mathematical metaphor, but you don’t want to be really precise. Here are some ways to do that:

  • Tack a onto the end of an equation.
  • Use bounds (“I expect to make less than a trillion dollars over my lifetime and more than $0.”)
  • Speak about a general class without specifying which member of the class you’re talking about. (The members all share some property like, being feminists, without necessarily having other properties like, being women or being angry.)
  • Use fuzzy logic (the  membership relation gets a percent attached to it: “I 30%-belong-to the class of feminists | vegetarians | successful people.”).
  • Use a specific probability distribution like Gaussian, Cauchy, Weibull.
  • Use a tempered distribution a.k.a. a Schwartz function.

Tempered distributions are my favourite way of thinking mathematically imprecisely.

Tempered distributions have exact upper and lower bounds but an inexact mean and variance. T.D.’s also shoot down very fast (like exp{−x²} the gaussian) which makes them tractable.

For example I can talk about the temperature in the room (there is not just one temperature since there are several moles of air molecules in the room), the position of a quantum particle, my fuzzy inclusion in the set of vegetarians, my confidence level in a business forecast, ….. with a definite, imprecise meaning.

Classroom mathematics usually involves precise formulas but the level of generality achieved by 20th century mathematicians allows us to talk about a cobordism between two things without knowing everything precisely about them.

It’s funny; the more advanced and general the mathematics, the more casual it can become. Like stingy stickler things that build up to a chummy, whatever-it’s-all-good.

 

Our knowledge of the world is not only piecemeal, but also vague and imprecise. To link mathematics to our conceptions of the real world, therefore, requires imprecision.

I want the option of thinking about my life, commerce, the natural world, art, and ideas using manifolds, metrics, functors, topological connections, lattices, orthogonality, linear spans, categories, geometry, and any other metaphor, if I wish.




I’ve been asked variants of this question a number of times. So here’s a thorough answer for everyone.

 

First, some well-written books on topics that interest me:

My interests are like economics / psychology / philosophy / probability / trying-to-understand-the-universe, so that’s what I gravitate toward.

 

Second, some personalisation advice: Read anything mathematical that talks about something you’re interested in.

  • If you are into weather / fluids, check out MIT OCW’s courses from the atmospheric science department (free pdf’s).
  • If you’re into the philosophy of quantum mechanics, try Itamar Pitowsky's book or the Stanford Encyclopedia of Philosophy (it will get into hilbert spaces soon enough).
  • I haven’t read it, but Doug Hofstadter’s Gödel Escher Bach seems to have inspired a lot of people.
  • Are you into puzzles? computer graphics? metaphysics? linguistics? liquids? animals? music theorymorality? bargaining theory? economic growth? interpersonal relationships? politics? the function of cells? systems theory? how the body moves? There are mathematical takes on all of those.

I’ve seen speech-pathology students with bad-mathematics syndrome take off like a fish in water when they do maths that relates to what they know—building sawtooth waves, just intonation, EQ’s. You know, ear stuff.

I recommend starting from what you know and using mathematics as a bridge to other things. (“Hey, I didn’t realise audio engineering gets me trigonometry for free! And from there I can go on to land surveying.”)

Third, get used to reading  s l o w l y. I’m naturally a slow reader so this wasn’t a big adjustment for me. But people who are used to breezing through a paperback in an afternoon are often dismayed when they can’t do the same with mathematics. Expect mathematics books to take 10, 20, 30 times as long per page—or more—as regular English reading. You’re expected to re-read passages, turn back to refer to earlier definitions, grab pen & paper and play with a few examples yourself, and possibly cross-refer to other books / Wikipedias.

In fact, the speed/depth tradeoff is part of what makes mathematics pleasurable. After 2-3 years of reading news I’m typically left with a shallow, jerky, shiftless sensation. After 2-3 years of reading mathematics, I literally see the world differently—in a good way. I feel I’ve developed something worthwhile that makes my mind a more interesting place to be in, rather than a jumble of chattering trifles. So don’t worry if you only get through 2-3 pages at a sitting. Not every page will take 20 minutes to comprehend, but some single pages need to slosh around in your brain for days or months. Don’t worry about it. That’s normal. Go slow; go deep.

 

Finally: I try to write for who didn’t do the whole calc 1-2-3-4, ode, real analysis thing. In an ideal world, my writing would come out well enough that an artist or writer with no mathematical confidence could parse it and be inspired with a new thought-shape.

That’s for the original posts.I also try to make people aware of things that are well-known in my circles but not well-known in general. For example, did you know that some guy was actually able to define complexity?

There are three ways to explore this blog:




scottthought asked: Any advice on where to start exploring math for someone who didn't take/ study/ or pay attention to much math beyond high school.

I’ve been asked variants of this question a number of times. So here’s a thorough answer for everyone: Where should I start reading mathematics if I never got far with it in school?

 

First, some well-written books on topics that interest me:

My interests are like economics / philosophy / probability / trying-to-understand-the-universe, so that’s what I gravitate toward.

 

Second, some personalisation advice: Read anything mathematical that talks about something you’re interested in.

  • If you are into weather / fluids, check out MIT OCW’s courses from the atmospheric science department (free pdf’s).
  • If you’re into the philosophy of quantum mechanics, try Itamar Pitowsky's book or the Stanford Encyclopedia of Philosophy (it will get into hilbert spaces soon enough).
  • I haven’t read it, but Doug Hofstadter’s Gödel Escher Bach seems to have inspired a lot of people.
  • Are you into puzzles? computer graphics? metaphysics? linguistics? liquids? animals? music theory? morality? interpersonal relationships? the function of cells? systems theory? how the body moves? There are mathematical takes on all of those.
  • Scott, I can see from your blog that you’re a Rubyist. Google “Dominic Verity category theory part 2”, he explains category theory to Haskellers who like functional programming. You might also want to explore Wikipedia / google on “discrete math”? Posets and graph theory are programming related. Just guessing here.

I’ve seen speech-pathology students with bad-mathematics syndrome take off like a fish in water when they do mathematics that relates to what they know—building sawtooth waves, just intonation, EQ’s. You know, ear stuff.

I recommend starting from what you know and using mathematics as a bridge to other things. (“Hey, I didn’t realise audio engineering gets me trigonometry for free!”)

 

Finally: I try to aim this blog at people who didn’t do the whole calc 1-2-3-4, ode, real analysis thing. In an ideal world, my writing would come out well enough that an artist or writer with no mathematical confidence could parse it and be inspired with a new thought-shape.

That’s for the original posts, and I also try to make people aware of things that are well-known in my circles but not well-known in general. For example, did you know that some guy was actually able to define complexity?

There are three ways to explore this blog:




Fuzzy Logic
Not everything is so simple as true or false. Even declarative statements may evaluate outside {0,1}. So let’s introduce the kind-of: truth ∈ [0,1].
Examples of non-binary declarative statements:
Shooting trap, my bullet nicked the clay pigeon but didn’t smash it. I 30%-hit the mark.
I’m not exactly a vegetarian. I purposely eat ⅔ of my meals without meat, but — like yogini Sadie Nardini — I feel weak if I go 100% vegetarian. So I’m ⅔ contributing to the social cause of non-animal-eating, and I’m a ⅔ vegetarian.
I’m sixteen years old. Am I a child, or an adult? Well, I don’t have a career or a mortgage, but I do have a serious boyfriend. This one is going to be hard to assign a single number as a percentage.
 
So that’s the motivation for Fuzzy Logic. It sounds compelling. But the academic field of fuzzy logic seems to have achieved not-very-much, although there are practical applications. Hopefully it’s just not-very-much-yet (Steven Vickers and Ulrich Höhle have two interesting-looking papers I want to read).
I see three problems which a Sensible Fuzzy Logic must overcome:
Implication. Classical logic (“the propositional calculus”) uses a screwed up version of “If A, then B”. It equates “if” to “Either not A, or else B is true, or else both.”Fuzzy logic inherits this problem — but also lacks one clear, convincing “t-norm”, which is the fuzzy logic word for fuzzy implication.   Can you come up with a sensible rule for how this should work?:      
A implies B, and A is 70% true. How true is B?
Furthermore, should there be different numbers attached to “implies” ? Should we have “strongly implies” and “weakly implies” or “strongly implies if Antecedent is above 70% and does not imply at all otherwise” ?
 You can see where I’m going here. There is an ℵ2 of choices for the number of possible curves / distributions which could be used to define “A implies B”.
Too specific. Fuzzy logic uses real numbers, which include transcendental numbers, which are crazy. Bart Kosko’s book explains FL with familiar two-digit percentages, which are for the most part intuitive. So I can accept that something might be 79% true — but what does it mean for something to be π/4 % true? Or e^e^π^e / 22222222222 % true?We’re encumbering the theory with all of these unneeded, unintuitive numbers.
One-dimensional.  For all of the space, breadth, depth, and spaceship adventures contained in the interval [0,1], it’s still quite limited in terms of the directions it can go. That is [0,1] comprises a total order with an implied norm. Again, why assume distance exists and why assume unidimensionality, if you don’t actually mean to. There are alternatives.Unidimensionality excludes survey answers like                 
N/A
I don’t know
Sort of
Yes and no
It’s hard to say
I’m in a delicate superposition

, — or rather it maps effectively different answers onto the same number.  Sometimes things are both good and bad;
sometimes they are neither good nor bad;
sometimes things are not up for evaluation;
sometimes a generalised function (distribution) expresses the membership better than a single number;
sometimes the ideas are topologically related or order related but not necessarily distance related;
sometimes an incomplete lattice might be best. 

 
So those are my gripes with fuzzy logic. At the same time, Kosko’s book was my introduction to an interesting, new way of thinking. It definitely set my mind spinning. For the logical mind that wants a rigorous framework for understanding ambiguity, vagueness, and gray areas, fuzzy logic is a good start.

Fuzzy Logic

Not everything is so simple as true or false. Even declarative statements may evaluate outside {0,1}. So let’s introduce the kind-of: truth ∈ [0,1].

Examples of non-binary declarative statements:

  • Shooting trap, my bullet nicked the clay pigeon but didn’t smash it. I 30%-hit the mark.
  • I’m not exactly a vegetarian. I purposely eat  of my meals without meat, but — like yogini Sadie Nardini I feel weak if I go 100% vegetarian. So I’m  contributing to the social cause of non-animal-eating, and I’m a ⅔ vegetarian.
  • I’m sixteen years old. Am I a child, or an adult? Well, I don’t have a career or a mortgage, but I do have a serious boyfriend. This one is going to be hard to assign a single number as a percentage.
 

So that’s the motivation for Fuzzy Logic. It sounds compelling. But the academic field of fuzzy logic seems to have achieved not-very-much, although there are practical applications. Hopefully it’s just not-very-much-yet (Steven Vickers and Ulrich Höhle have two interesting-looking papers I want to read).

I see three problems which a Sensible Fuzzy Logic must overcome:

  1. Implication. Classical logic (“the propositional calculus”) uses a screwed up version of “If A, then B”. It equates “if” to “Either not A, or else B is true, or else both.”

    Fuzzy logic inherits this problem — but also lacks one clear, convincing “t-norm”, which is the fuzzy logic word for fuzzy implication.  Can you come up with a sensible rule for how this should work?:
    • A implies B, and A is 70% true. How true is B?
    • Furthermore, should there be different numbers attached to “implies” ? Should we have “strongly implies” and “weakly implies” or “strongly implies if Antecedent is above 70% and does not imply at all otherwise” ?


    You can see where I’m going here. There is an 2 of choices for the number of possible curves / distributions which could be used to define “A implies B”.

  2. Too specific. Fuzzy logic uses real numbers, which include transcendental numbers, which are crazy. Bart Kosko’s book explains FL with familiar two-digit percentages, which are for the most part intuitive. So I can accept that something might be 79% true — but what does it mean for something to be π/4 % true? Or e^e^π^e / 22222222222 % true?

    We’re encumbering the theory with all of these unneeded, unintuitive numbers.

  3. One-dimensional.  For all of the space, breadth, depth, and spaceship adventures contained in the interval [0,1], it’s still quite limited in terms of the directions it can go. That is [0,1] comprises a total order with an implied norm. Again, why assume distance exists and why assume unidimensionality, if you don’t actually mean to. There are alternatives.

    Unidimensionality excludes survey answers like
    • N/A
    • I don’t know
    • Sort of
    • Yes and no
    • It’s hard to say
    • I’m in a delicate superposition
    , — or rather it maps effectively different answers onto the same number.  

    • Sometimes things are both good and bad;
    • sometimes they are neither good nor bad;
    • sometimes things are not up for evaluation;
    • sometimes a generalised function (distribution) expresses the membership better than a single number;
    • sometimes the ideas are topologically related or order related but not necessarily distance related;
    • sometimes an incomplete lattice might be best. 
 

So those are my gripes with fuzzy logic. At the same time, Kosko’s book was my introduction to an interesting, new way of thinking. It definitely set my mind spinning. For the logical mind that wants a rigorous framework for understanding ambiguity, vagueness, and gray areas, fuzzy logic is a good start.




What constitutes a dramatic situation? What makes for an interesting story?
The most interesting stories, to me, often come from the humblest of places. The history of salt; the manufacturers of parts for electrical sockets, water fountains, sock factories, chair legs, bucket handles, straws, Parmalat-style packaging (box & lid), keycaps, microphone covers, whiteboard backing, broomhandles; import duties & tariffs in 17th-century Britain; the company that makes machines that can print on diapers (flexigraphic printing); traders of wicker, cork, gypsum, shale, spring steel, vinyl, styrofoam peanuts; antimicrobial coatings for deli slicers, baby changing tables, and elevator buttons; a variety of glues and solvents; and so on. The duller the subject, the more I feel there is to uncover.
Like my father, who worked in television, I have little interest in melodrama. Whether it’s the evening news or another form of abrupt exaggeration, we’re not interested. My father’s favourite character in the Bible was instrumental in bringing down the walls of Jericho — but never shared his name.
Anyway, enough family history. Here is a bona fide fascinating story about how the humble box redirected the rivers of money that flow opposite today’s global goods market. Fortunes were made and unmade; cities brought low and raised high.
via shippingandlogistics:

 
The Box: How the Shipping Container Made the World Smaller and the World Economy Bigger charts the historic rise of the intermodal shipping container and how it changed the economic landscape on a major scale.
The New York Times: A revolution that came in a box

Before the container, Mr. Levinson writes: ”It was not routine for shoppers to find Brazilian shoes and Mexican vacuum cleaners in stores in the middle of Kansas. Japanese families did not eat beef from cattle in Wyoming, and French clothing designers did not have their exclusive apparel cut and sewn in Turkey and Vietnam.”
Which is to say: the line from ”containerization” to globalization is a straight one. 

What constitutes a dramatic situation? What makes for an interesting story?

The most interesting stories, to me, often come from the humblest of places. The history of salt; the manufacturers of parts for electrical sockets, water fountains, sock factories, chair legs, bucket handles, straws, Parmalat-style packaging (box & lid), keycaps, microphone covers, whiteboard backing, broomhandles; import duties & tariffs in 17th-century Britain; the company that makes machines that can print on diapers (flexigraphic printing); traders of wicker, cork, gypsum, shale, spring steel, vinyl, styrofoam peanuts; antimicrobial coatings for deli slicers, baby changing tables, and elevator buttons; a variety of glues and solvents; and so on. The duller the subject, the more I feel there is to uncover.

Like my father, who worked in television, I have little interest in melodrama. Whether it’s the evening news or another form of abrupt exaggeration, we’re not interested. My father’s favourite character in the Bible was instrumental in bringing down the walls of Jericho — but never shared his name.

Anyway, enough family history. Here is a bona fide fascinating story about how the humble box redirected the rivers of money that flow opposite today’s global goods market. Fortunes were made and unmade; cities brought low and raised high.

via shippingandlogistics:

The Box: How the Shipping Container Made the World Smaller and the World Economy Bigger charts the historic rise of the intermodal shipping container and how it changed the economic landscape on a major scale.


The New York Times: A revolution that came in a box

Before the container, Mr. Levinson writes: ”It was not routine for shoppers to find Brazilian shoes and Mexican vacuum cleaners in stores in the middle of Kansas. Japanese families did not eat beef from cattle in Wyoming, and French clothing designers did not have their exclusive apparel cut and sewn in Turkey and Vietnam.”

Which is to say: the line from ”containerization” to globalization is a straight one.