Most of the book is criticism. On page one we get "Traditional retirement planning has failed." It all feels mostly like a sales tactic: His approach is the one that works!

He spends a lot of time criticizing Monte Carlo methods. His explanation of the approach in general is so bad it must be deliberately so. Unsavory rhetoric.

There is a reasonable argument buried in his hate for MC methods: He
thinks they don't go far enough in considering more extreme possible
futures, while their confidence intervals make them *seem* all-knowing
when they're not. They generally won't model Social Security
disappearing altogether, for example, but they'll still allocate 100%
of probability, as if there's a 0% chance of these sorts of "extreme"
futures.

So his solution is a sort of manual Monte Carlo, putting lots of weird things into (his, of course) calculators.

He also correctly, I think, focuses on the human reality of living just one life, not a probability distribution of lives, so what we really need is to eliminate extreme downside risk as much as possible.

How much of the retirement planning literature is just confusion about inflation? Possibly a lot, I think.

It turns out his solution is mostly from the skinny-FIRE school of thought: Spend less! Then there's the Robert Kiyosaki part: Cash flow! Real estate!

I object to telling people to become landlords like it's easy, good, and risk-free. The general idea of not dipping into your capital seems fine though. But now are we just back to the original methods he despises, only worse off because we're imagining the capital is safe (and even grows with inflation)? I think yes.

There are some good vibes about enjoying your life, and maybe his recommendation to consider a longevity annuity is reasonable, but for the most part this book is more an advertisement than a resource.

"Every act of creation is first an act of destruction." (page 21)

I think this is supposed to be Picasso? But his usage is unattributed? Sort of weird, given how much he likes to include well-known quotes otherwise.

(Regarding long-term increases in life expectancy) "There is some debate about this statistic with some people claiming the gains came from reductions in infant mortality." (page 46)

This is not a debate! See, for example, Life expectancy is historically misleading .

investment returns over longer time periods of seven years or more are not random. (page 69)

This is part of his idea that he can time the market based on P/E. Valuations aren't nothing, but his takes are a bit much.

"let’s apply conservative estimates by cutting my pension and social security in half so that my combined income sources provide just $15,000 per year instead of $50,000." (page 129)

How does this make sense?

"Different tax rates would only be marginally meaningful if your income fell dramatically after retirement. Are you planning on poverty or financial security?)" (page 142)

It's sort of amazing that he would say this. Your effective tax rate could easily be cut in half at retirement if you're earning a bunch and saving most of it before retirement.

"“Today’s scientists have substituted mathematics for experiments, and they wander off through equation after equation, and eventually build a structure which has no relation to reality.”-Nikola Tesla, inventor" (page 149)

I do like this quote.

"Complicated math is usually more symptomatic of covering up ignorance than expressing wisdom." (page 192)

Also not terrible.

"Retirement planning done right is really about life planning, not calculating magic numbers." (page 151)

"Every dollar bill you spend in your twenties is like spending a hundred dollar bill in retirement." (page 179)

What? Really?

This is assuming 10% annual return over 45 years? And it's nominal? (With inflation, more like $20 purchasing power?)

I guess he's just trying to be dramatic, but I would rather he not be dramatic with numbers.

"The main thing to note is how every other model relies on history to be your guide. This cash flow-based model is the only approach that requires no historical analysis because it is entirely based on numbers fully known in the present. This is important because the past is not the future." (page 199)

Tresidder paints "cash flow" methods as immune from risk, and generally somehow automatically protected from inflation. This isn't the case!

"Lifestyle planning isn’t really a separate retirement model per se." (page 208)

(The book is structured and presented as if it is.)

]]>“There is only one success – to be able to spend your life in your own way.” (Christopher Morley)

Graeber had a popular article and this is the book version. As with Rushkoff's, the titular concept is mostly a hook. I'm more interested in the critique of work generally.

The first part of the book, most directly on "bullshit jobs" and related surveys and interviews, isn't very good, compared to Graeber's Debt. He wants to have it both ways, for one thing: We must trust the people who say their jobs are bullshit, and at the same time not trust people who say their jobs aren't.

Then there's the definition of a bullshit job, which is never done
quite consistently. I think it's supposed to be jobs that don't make
any difference (if they aren't done, nothing changes). But this bleeds
in at least two directions: Jobs where we may not *like* the
difference they make, and jobs that *wouldn't* be necessary if things
were set up differently.

So Graeber repeatedly criticizes Obama for using job preservation as a reason to not completely upend the health care system. If the current employees were all sent home, it would be bad: people wouldn't get insurance reimbursement, etc. The current jobs are only (or primarily) "bullshit" to Graeber because he thinks a single payer (or similar) system could take care of people with fewer employees.

And while I know part of Graeber's point is to critique the Puritan work ethic and "middle class thinking" in general, many of his malcontent subjects just aren't very sympathetic.

Graeber did an academic thesis on "life cycle service," the historical practice in northern Europe in which a young person would spend seven or so years as an apprentice or servant, with some other family, collecting some pay until they could set off on their own. I thought this was interesting in relation to the modern idea of FIRE (Financial Independence, Retire Early) in which a person similarly works for others until they have enough money to be independent and then does whatever they want.

Graeber talks a lot about "spiritual violence," and while this may be worse when a job feels pointless (see also Sayre's Law) I don't think it's unique to "bullshit jobs" but common to any job that has you doing what you really don't want to do. Hence the Morley quote about success as being "able to spend your life in your own way." People really like self-determination.

Graeber does connect at least a little bit with Black's Abolition of Work and the idea given book form in Fully Automated Luxury Communism. He spends some time on work as caring work rather than factory production. And he also comes out in support of UBI.

(Oh right - I read No more work back in 2018...)

]]>A smattering of his ideas as I remember them:

- The standard economics story of money (as arising because barter is too hard) is silly because people hardly ever barter in the first place, except sometimes with strangers.
- The standard economics story of money starting with coinage, then
later developing debt instruments (see also Dalio, "money as
starting from metal, then notes exchangeable for metal, then fiat
money") is silly because ideas and methods of debt come
*first*. - The moralism of "always paying your debts" in culture and language.
- No single theory of money (Chartalism, etc.) is right on its own because "money has no essence" and money is just one piece of a big evolving world. (Similarly, Graeber has no single thesis on debt.)
- "Baseline communism" refers to the natural "from each according to his ability, to each according to his needs" that we have in real human relations, like feeding a child or asking for someone to hand you something.
- Debt as two equals agreeing to no longer be equal: Now
*you*owe*me*. (In the extreme: debt peonage.) - The difference of kind that comes from quantification: Not just owing me a favor, but owing me $47.39. The relation to dehumanization, as when slavery puts a price on a person, which is only possible by removing or denying their complex real (unquantifiable) value. (Reminds me of The Tyranny of Metrics.)
- The necessity of violence to make people pay back debts, as when military power stands behind the IMF.
- Occasional hints of primitivism or anarcho-primitivism.
- And really, shouldn't we get rid of capitalism?

Dalio has a bunch of signals and a framework of big cycles organizing history. It's a little bit psychohistory and a little bit cycle-finding. For Dalio, the biggest cycle is the debt cycle, followed by cycles of internal disorder and external disorder. He describes money as starting from metal, then notes exchangeable for metal, then fiat money.

For Dalio, the step after fiat money is collapse. He cares a lot about whether a currency is The Reserve Currency. Combining his signals, he gets charts that show the rise and fall of the Dutch guilder, the British pound, and (soon) the US dollar. China is ascendant, never mind that their currency has been fiat longer than the dollar.

Dalio says he's identifying timeless principles of cause and effect. Is he right? Historical causality is hard to test, but there are at least a few problems with Dalio's claims.

For most of history, the idea of a reserve currency hasn't really made sense. The modern monetary system is in this light novel, not timeless. Dalio has a narrow understanding of money compared to the anthroplogical view in Debt: The First 5,000 Years, for example.

It's hard to find falsifiable predictions in Dalio's science. He
admits to not having good tools for predicting *when* any given thing
might happen. In this sense, his whole system is basically noticing
that things have fallen apart in the past and concluding that things
could fall apart in the future. It isn't even wrong.

Looking at the charts, I'm reminded of a common tension in data work: Is data leading to theory, or is theory leading to data? Data can be found to support most theories; what was the process here? It's rather easy to imagine that Dalio's employees are skilled in finding what he thinks is there.

Not that Dalio hasn't looked at history and so on. The Patriot movement in the Netherlands does kind of feel like it rhymes with MAGA energy. He knows about triangular debt in China. He's even concerned about inequality, though mostly because it can lead to wealth-destroying crises.

Does Dalio have the courage of his convictions? Perhaps not. He frequently comes off as defensive, and digresses substantially into his concerns about public scrutiny of the prominent. The historical theory he seems to hold most instinctively is a great man theory in which he and the people he talks to are great; he sounds like Trump talking about how he knows the best people.

If you think your ideas are timeless, you may be right about what happens next, but you probably won't predict things that have never happened before. So Dalio is limited to predicting rearrangements of the world order, not substantially new things. If Marx thought much change was inevitable, Dalio thinks real change is fundamentally impossible, that recurrence is the nature of society.

So Dalio picks China as the new king of the hill.

I'm curious about many of the things Dalio talks about. What causes what in the world? How do we operate in reality? What lessons can we learn from history? I have to think Dalio believes he's sincere about his program. At the same time, reporting suggests his ability to follow his own advice is as constrained as his ability to imagine a better world. For all the lip service to evolution, Dalio doesn't seem to be its exemplar.

]]>For a new idea, there was by definition no prior explanation. It is generally quite difficult to come up with new useful ideas, and often takes many years. Only very few people have any understanding of the new idea at first.

Someone, maybe the discoverer, tries to communicate the new idea. This may not be a great explanation. It may be cloaked in specialist language, heavy with background preconceptions, hidden from broader communities.

The trap of a poor explanation is that it can still be good enough for the most active, prepared, diligent seekers. Some will realize it isn't a good explanation, but understand anyway and move on. Others will fight through the poor explanation and conclude, with a kind of Stockholm syndrome, that the explanation is canonical. The poor explanation starts to be repeated to students.

This cycle can get harder to break as it becomes common knowledge that a subject is difficult, even if the difficulty is not innate but due to the poor explanation. It becomes hard to differentiate between removing unneeded complication and "dumbing down" the subject because people have become attached to the way they learned it. Maybe the subject is now a "filter class" that people take pride in having overcome.

An explanation should not be a filter but a fortifier, lifting up the listener from ignorance to understanding. Things should not be oversimplified, but should be made as simple as possible, so that as many people as possible can profit, as quickly as possible, from sharing in mastery—and contribute to the next ideas.

]]>A major metaphor in the book is "phase transitions," like solid to liquid. Bahcall offers "The Innovation Equation" (details in HBR article) for \( M \), "a certain size at which human groups shift from embracing radical ideas to quashing them."

\[ M = \frac{E S^{2} F}{G} \]

The only right hand term that isn't unitless is \( S \), "management span," which is the average number of direct reports that managers have. Since \( S \) is squared, the result is in units of people squared, which I don't think makes sense. With his training as a physicist, it's a little surprising that Bahcall would do this. (Am I missing something?)

Also surprising is that there's no constant term. (For example, gravity isn't mass times mass divided by distance squared; there's a constant that we have to get from data.) Bahcall plugs in \( E = 50\% \), meaning compensation is 50% equity, which seems rare, then management span of 6, and \( F = 1 \) meaning equal returns to project work and politics, finishing with \( G = 12\% \) meaning 12% compensation increase at promotion. Ignoring the units, this gives 150, and he claims a connection with Dunbar's number, saying "That's interesting."

I have to think Bahcall doesn't really take this equation seriously; he must intend it as an illustration for his narrative. Still, it offends my sensibilities because it seems like math-washing: presenting something that looks "fancy" to make it seem more legitimate. In this case, the evidence isn't really anything more than suggestive.

One issue is boundaries: Where does an organization begin and end? DARPA is considered an example of an innovative organization. Do we consider just the DARPA PMs? Also their contractors? The whole Department of Defense?

I think a much more promising attempt might be made by switching from the organization to the individual view: At what point does an individual change their behavior, based on the local structure of their organization and incentives? This would also be more coherent with Bahcall's ideas about having different parts of an organization working on different goals (loonshots vs. "franchise" work). How do you get loonshots? Support people who work on them.

A deeper critique is that Bahcall's "loonshot" forces here seem more
about *doing your job* than about working on crazy innovative ideas
specifically. "Project work" is not necessarily loonshot work.

This "Goals Gone Wild" paper is pretty fun, and reminds me of The Tyranny of Metrics. The paper includes this warning label:

Also referenced (a lot) is Vannevar Bush's 1945 Science: The Endless Frontier, which reminds me of The Beginning of Infinity.

“The graveyard of unexplained experiments, as Land would soon show, is a great place to find a False Fail.” (page 76)

“Some companies are the equivalent of an innovation landfill,” wrote one senior Apple executive, who helped lure some of PARC’s best engineers to Apple. “They are garbage dumps where great ideas go to die. At PARC, the key development people kept leaving because they never saw their products get into the market.” (page 115)

This is an interesting idea: What's out there that's known but ignored, currently, when it shouldn't be ignored? Bahcall uses the example of radar's foundations being known but ignored by the Navy for nearly 20 years. What fun things are out there, already discovered but not yet exposed to sunlight?

“While the individual man is an insoluble puzzle, in the aggregate he becomes a mathematical certainty.” (Sherlock Holmes, in The Sign of the Four, quoted page 117)

Seems like everybody wants to do psychohistory!

]]>“... a teacher has to be adjusted to fit the mind of each boy and girl it teaches ... each kid has to be taught differently.” (Asimov, The Fun They Had, 1951)

The title of this book refers to LLM text generation and Khan's call for “educated bravery” in meeting new technologies, but it also refers to Brave New World, a distopia. In his introduction, Khan mentions Diamond Age, Ender's Game, and The Fun They Had as fiction that “has gone on to inspire very real innovation.” In Diamond Age, three “AI” books are really humans behind the scenes, and the actual AI books train a child army. In Ender's Game, the educational technology serves the goal of genocide. And in Asimov's The Fun They Had, the joke is that AI education would make children long for the current world of schooling. I'm fascinated by the same fiction as Khan. I also think of the Star Trek post-scarcity world (1, 2) as an inspiration. I think Khan is right that LLMs present opportunities, but they also face persistent challenges of education.

There is a kind of “AI exceptionalism” that imagines generative AI as immune to problems faced elsewhere. For example, Khan criticizes Google for having ads. But will commercial LLMs remain disinterested? Khan Academy doesn't offer its Khanmigo AI tutor for free to students because of the cost of AI inference, and it's hard to imagine that LLMs are permanently immune to the attentions of advertisers. And while Khan speaks glowingly of Khanmigo, there is already at least one example of a failed AI ed-bot effort.

More generally, Khan would have education and LLMs as somehow beyond
having a point of view, absolutely impartial. Researchers have worked
hard on “alignment” that creates this illusion, and certainly a
Socratic approach can have value in education. But fundamentally there
*is no* impartial view of the world, and the effect in aiming for one
is to allow the inane while preventing progress.

Khan wants to engage students by “incorporating their interests” in ways that are superficial and misleading. One way to fail in this is to imagine content as irredeemably rigid, and try to apply a false veneer. Another way is to imagine there is no content in particular worth focusing on.

Khan gives one example of imagining a soccer coach modeling goals
scored with a polynomial, and then asking for the degree of this
polynomial and the leading coefficient. This is a weak trick at best,
and doesn't at all get to *why* anyone would care about the question.
Applications can be motivating, but this one is so artificial that it
seems more likely to damage credibility.

Later Khan imagines restructuring a class's study of World War 2 through a focus on baseball, in a way that seems to suggest that baseball was just as important an aspect of World War 2 as anything else. I could see a research project on this narrow subject for an interested student, but I also think that the bigger features of WW2 should not be deprioritized in order to pander to interests.

One theme here is that part of education is learning what's interesting about things you weren't previously interested in. You shouldn't come out of a class with only the same interests that you went in with.

On history specifically, there's probably a lot more to discuss. Khan offers that “it is impossible to engage in calculus if you don’t know Algebra 2 well, but one can imagine engaging in college-level history even without a strong foundation in history from high school.” I don't know whether all history teachers would agree. To me, to the extent that this is true, it suggests a problem with history as a discipline at least as it is taught. I think we should ask more of history, make more connections, draw more conclusions, have more of a sense of both the theory and application of history. I think Khan is imagining history as if, were it math, in high school students would learn one table of equations and then in college learn a second table of equations, never learning what they mean or how to use them. Perhaps because lessons learned from history can have a political interpretation, history is kept in schools with its hands tied? And so we repeat the errors of the past.

In attempting to be impartial, existing curricula can become dogma. Khan is so focused on how to teach everyone Pre-Algebra, Algebra 1, and Algebra 2, that there doesn't seem to be room for re-thinking our choices in what we teach. The “how” of education is important, but I think the “what” and “why” are keystones, and at least as in need of attention in light of technological advancement.

One pet peeve of mine is titles that are not conceptually coherent. This is the case for Pre-Algebra, Algebra 1, and Algebra 2. These are not good titles for courses. They reflect a bulk-frozen structure that we should be embarrassed to offer students. Imagine if a student submitted an outline so lousy, where the top-level bullets were "first stuff," "second chunk," and "last bits." We should expect more clarity of thought. No student should be given a grab bag of things to learn without better organizing concepts and rationale, and that starts at the level of course titles.

I suspect much of what is taught in schools could be improved
(dropped, modified, added to). This is not necessarily easy to do
(New Math?) but there have been improvements over time and I
expect they can be extended. It has become rare for students to
memorize and recite Against Idleness and Mischief, for example,
while calculus, which used to be exclusively a university topic, is
now encountered by many in high school. Improving *what* is taught
should get more attention.

It seems like AI may move some classroom setups in directions that feel old-fashioned: More in-person writing and tests, more focus on spoken communication. Perhaps it will seem more obvious that teachers are a specific type of manager, which has always been the case though not always widely acknowledged. The toolkit expands.

]]>In Linear Algebra, a function is linear if \( f(u+v) = f(u) + f(v) \) and \( f(cu) = cf(u) \). This is a powerful constraint. The common equation of a line, \( y = f(x) = mx + b \), is not linear by this definition when \( b \neq 0 \). In the situation of one numeric input and output, linear functions are multiplication by a constant coefficient, \( y = f(x) = mx \).

If there are two inputs and one output, linear functions are multiplications followed by addition, \( y = f(x_1, x_2) = m_1 x_1 + m_2 x_2 \). This is the dot product of the vector \( \vec{x} = (x_1, x_2) \) and the coefficient vector \( \vec{m} = (m_1, m_2) \). The result is a linear combination of the inputs.

If there are two inputs and two outputs, each output is a linear combination of the inputs, \( (y_1, y_2) = f(x_1, x_2) = (m_1 x_1 + m_2 x_2, m_3 x_1 + m_4 x_2) \), also written \( \vec{y} = M \vec{x} \).

Applying a second linear transform \( N \) to the output of \( M \), the new outputs \( y_1 \) and \( y_2 \) are:

\[ n_1 (m_1 x_1 + m_2 x_2) + n_2 (m_3 x_1 + m_4 x_2), \\ n_3 (m_1 x_1 + m_2 x_2) + n_4 (m_3 x_1 + m_4 x_2) \]

Re-grouping shows that applying these two linear transforms one after the other is the same as applying just one linear transform with different coefficients:

\[ (n_1 m_1 + n_2 m_3) x_1 + (n_1 m_2 + n_2 m_4) x_2, \\ (n_3 m_1 + n_4 m_3) x_1 + (n_3 m_2 + n_4 m_4) x_2 \]

If \( N M = T \), then \( t_1 \) is the dot product of \( (n_1, n_2) \) and \( (m_1, m_3) \), and so on. The coefficients can be arranged so that each pair that appears is adjacent horizontally or vertically.

\[ \begin{bmatrix} n_1 & n_2 \\ n_3 & n_4 \end{bmatrix} \begin{bmatrix} m_1 & m_2 \\ m_3 & m_4 \end{bmatrix} = \begin{bmatrix} n_1 m_1 + n_2 m_3 & n_1 m_2 + n_2 m_4 \\ n_3 m_1 + n_4 m_3 & n_3 m_2 + n_4 m_4 \end{bmatrix} \]

There is a choice here: The coefficients are written in English reading order, left to right and top to bottom. They could just as well be placed top to bottom and right to left, for example. This choice is arbitrary, but the nature of matrix multiplication is not.

This is a terse version of what I think could be a good way to introduce Linear Algebra and matrix multiplication. It is a more traditionally symbolic version of the flow metaphor for matrix multiplication, with more focus on linearity as a driving part of the meaning of Linear Algebra.

]]>The equation on the cover is the closed form solution for the \( n \)th Fibonacci number, derived using linear algebra in an exercise on page 174, just one of the book's treasures.

Do not memorize the formula for the product of two complex numbers—you can always rederive it by recalling that \( i^2 = -1 \) and then using the usual rules of arithmetic (as given by 1.3). (page 2)

I love seeing advice like this in textbooks; sort of a didactic principle of emphasizing what's fundamental and what's a consequence... On the other hand you don't want to re-derive everything every time, but there's some optimization of what gets a spot in memory...

When we think of an element of \( \mathbb{R}^2 \) as an arrow, we refer to it as a vector.

Whenever we use pictures in \( \mathbb{R}^2 \) or use the somewhat vague language of points and vectors, remember that these are just aids to our understanding, not substitutes for the actual mathematics that we will develop. (page 8)

Also good pedagogy here, I think: making a distinction between the thing itself and a mental model or metaphor for it.

Mathematical models of the economy can have thousands of variables, say \( x_1, ..., x_{5000} \), which means that we must work in \( \mathbb{R}^{5000} \). Such a space cannot be dealt with geometrically. However, the algebraic approach works well. Thus our subject is called

linear algebra. (page 8)

I appreciate the attempt to justify the name, but I feel like this one isn't completely satisfying... Good aspect: Gets at the centrality of multi-dimensionality. Less good aspect: Doesn't really justify the "linear" part.

We could define a multiplication in \( \mathbb{F}^n \) in a similar fashion, starting with two elements of \( \mathbb{F}^n \) and getting another element of \( \mathbb{F}^n \) by multiplying corresponding coordinates. Experience shows that this definition is not useful for our purposes. Another type of multiplication, called scalar multiplication, will be central to our subject. (page 9)

I feel like the appeal to experience is not a satisfying justification for matrix multiplication. I like the flow metaphor, and I think even better can be done than my exposition there...

In general, a vector space is an abstract entity whose elements might be lists, functions, or weird objects. (page 14)

Fun example of being both mathematically rigorous and friendly in communication.

I at once gave up my former occupations, set down natural history and all its progeny as a deformed and abortive creation, and entertained the greatest disdain for a would-be science which could never even step within the threshold of real knowledge. In this mood I betook myself to the mathematics and the branches of study appertaining to that science as being built upon secure foundations, and so worthy of my consideration.

—Frankenstein, Mary Wollstonecraft Shelley (page 50)

Axler has inserted some fun quotes at the ends of sections here and there; this is one I liked.

We will frequently use the powerful fundamental theorem of linear maps, which states that the dimension of the domain of a linear map equals the dimension of the subspace that gets sent to 0 plus the dimension of the range. (page 51)

Thus the linear functions of high school algebra are not the same as linear maps in the context of linear algebra. (page 56)

He's using zero-to-zero to show this here. I think this is a really helpful thing to emphasize immediately.

Consider linear maps \( T: U \to V \) and \(S: V \to W \). The composition \( ST \) is a linear map from \( U \) to \( W \). Does \( M(ST) \) equal \( M(S)M(T) \)? This question does not yet make sense because we have not defined the product of two matrices. We will choose a definition of matrix multiplication that forces this question to have a positive answer. (page 72)

Here is at least some justification of matrix multiplication. I think it comes a little late, since matrices were introduced already, and really they sort of co-design...

Note that we define the product of two matrices only when the number of columns of the first matrix equals the number of rows of the second matrix. (73)

Couldn't this be made to feel less arbitrary?

The fundamental theorem of algebra is an existence theorem. Its proof does not lead to a method for finding zeros. The quadratic formula gives the zeros explicitly for polynomials of degree 2. Similar but more complicated formulas exist for polynomials of degree 3 and 4. No such formulas exist for polynomials of degree 5 and above. (page 124)

Axler does a lot to connect these algebra ideas and linear algebra, which is pretty cool.

It's the Abel–Ruffini theorem that shows there aren't formulas for degree five and up. Is this mentioned when teaching the quadratic formula? Would be cool to at least show the crazy equations for degree three and four...

A later example:

This exercise shows that every monic polynomial is the minimal polynomial of some operator. Hence a formula or an algorithm that could produce exact eigenvalues for each operator on each \( \mathbb{F}^n \) could then produce exact zeros for each polynomial [by 5.27(a)]. Thus there is no such formula or algorithm. However, efficient numeric methods exist for obtaining very good approximations for the eigenvalues of an operator. (page 152)

Now we begin our investigation of operators, which are linear maps from a vector space to itself. Their study constitutes the most important part of linear algebra. (page 132)

The word eigenvalue is half-German, half-English. The German prefix eigen means “own” in the sense of characterizing an intrinsic property. (page 134)

I like this kind of explanation.

The main reason that a richer theory exists for operators (which map a vector space into itself) than for more general linear maps is that operators can be raised to powers. (page 137)

A central goal of linear algebra is to show that given an operator \( T \) on a finite-dimensional vector space \( V \), there exists a basis of \( V \) with respect to which \( T \) has a reasonably simple matrix. (page 154)

You may recall from a previous course that every matrix of numbers can be changed to a matrix in what is called row echelon form. If one begins with a square matrix, the matrix in row echelon form will be an upper-triangular matrix. Do not confuse this upper-triangular matrix with the upper-triangular matrix of an operator with respect to some basis whose existence is proclaimed by 5.47 (if \( \mathbb{F} = \mathbb{C} \))—there is no connection between these upper-triangular matrices. (page 160)

The Euclidean algorithm for polynomials (look it up) can quickly determine the greatest common divisor of two polynomials, without requiring any information about the zeros of the polynomials. (page 173)

I'm just amused by the "look it up".

Thus the pseudoinverse provides what is called a best fit to the equation above. (page 222)

I thought a bit about how linear regression is solved. The thing that's usually written down, \( (X^T X)^{-1} X^T y \), is really crummy computationally if \( X \) isn't well-behaved, because of the inverse...

It seems like nobody actually implements regression this way. In R,
`lm`

is done via QR decomposition, anyway.

The pseudoinverse seems to work too, but doesn't always give the same solution, when multiple solutions are possible. I suspect it isn't used in practice because SVD is more work than QR. There may be more reasons.

Recall that an operator on \( V \) is called diagonalizable if the operator has a diagonal matrix with respect to some basis of \( V \). Recall also that this happens if and only if there is a basis of \( V \) consisting of eigenvectors of the operator (see 5.55). (page 243)

7.54 eigenvalues of unitary operators have absolute value 1 (page 262)

Makes me want to see some quantum examples...

8.50 trace of matrix of operator does not depend on basis (page 327)

How cool is this? Could it come earlier?

Our definition of the determinant leads to the following magical proof that the determinant is multiplicative. (page 357)

We're having fun!

Vandermonde matrices have important applications in polynomial interpolation, the discrete Fourier transform, and other areas of mathematics. (page 366)

Makes me want to read an applied book.

]]>I find that in my own elementary lectures, I have, for pedagogical reasons, pushed determinants more and more into the background. Too often I have had the experience that, while the students acquired facility with the formulas, which are so useful in abbreviating long expressions, they often failed to gain familiarity with their meaning, and skill in manipulation prevented the student from going into all the details of the subject and so gaining a mastery.

—Elementary Mathematics from an Advanced Standpoint: Geometry, Felix Klein (page 369)

I really like Rushkoff in some respects. I wish his Team Human podcast would make really nice basketball jerseys and sweatshirts. (I consider myself on Team Human quite independent of whatever Rushkoff says; I think it's a nice idea.)

I don't think escapism is unique to the ultra-rich. Every time I put on headphones, in some ways I'm trying to find safety in isolation. Not only the wealthy are preppers. Everyone could do well to reflect on how interdependent we all really are.

There's a bit of a failure of imagination in not exploring in the direction of Atlas Shrugged. Perhaps a Marxist can't imagine the proletariat being replaced by technology. I still think that in the near term "There will continue to be more real problems with not having enough people to work than there are with having too many people put out of work by technology," but there are big questions about what's going to happen with all the people not working conventional jobs (of which there are many already today).

I don't think Rushkoff has a good argument here. But he says so *many*
things that the book is quite thought-provoking regardless. He
characterizes QAnon as a kind of internet addiction. And you get
to hear about the many parties he's been invited to. Lucky you.

He [Richard Dawkins] couldn’t acknowledge that his own commitment to scientism is based on something passional—something more like

faithin an empiricist universe. In other words, his insistence on living in an evidence-only universe isn’t based on evidence at all. It’s an assumption. It’s part of a system of meaning, developed by a community of people over time. It just happens to be a meaning system that ignores meaning itself. Worse, by rejecting the validity of any other meaning system, it is prone to instilling in its adherents a sense of superiority over others. Those who strive for meaning are mere “moralists.” (page 34)

I agree with parts of this critique; science should recognize its foundation and scope.

Initially the market economy, introduced just after the Crusades in the late Middle Ages, benefited the former peasants of feudalism. This was a lateral, peer-to-peer economy, not a hierarchical one. Local farmers and bakers didn’t generally aspire to be “rich” so much as to sustain themselves. Their currencies were optimized for trade; they were less a way of saving or hoarding money than facilitating the exchange of goods between people. It worked so well that Europe saw its greatest period of economic growth to this day—measured in the prosperity of the common folk. Towns got so wealthy that they invested in building cathedrals to spur future pilgrimages and tourism. People worked less, ate more, and grew taller than ever before—and in some cases, ever since. (page 40)

I don't think this is the dominant historical view, but this time does seem to be Rushkoff's "make economics great again" period. It's a rather bold claim, saying that everybody used to be better off, and even taller than now!

The citation here is the unpublished (but available online) Of Human Wealth. The authors (Lietar and Belgin) call the period 1050-1290 "the first European renaissance." On page 92 they have this graph in classic Excel style:

On page 91, the "ever since" is explained:

In a study of the skeletons of bodies in the same geographical area (the City of London), informative findings emerged. The women of the 10-12th century were on average taller than in any other period in recorded history, at least in London. During that period, the average female Londoner was a whopping 7 centimeters taller than her counterpart in Victorian times and 1 centimeter taller than the average woman of London today! (Of Human Wealth, page 91)

The source for all this is London Bodies, the book companion to a 1998 exhibit at the Museum of London. The graph in Of Human Wealth seems to be based on a table on the last page:

It isn't clear that this is an authoritative source for drawing conclusions about trends in height over time. It also isn't clear that the table is even consistent with the rest of its own book, and further it seems that Of Human Wealth is rather abusing what there is in London Bodies anyway.

The "Saxon" numbers seem to correspond to page 56, in the "Saxon Bodies" chapter:

From this period, the eleventh and twelfth centuries, comes our best group of medieval skeletons from a parish churchyard in London. Excavations between 1975 and 1979 at the cemetery of St Nicholas Shambles on Newgate Street produced over 200 burials. The average heights of the people buried in the cemetery were 173cm (5ft 8in.) for men and 157cm (5ft 2in.) for women. (London Bodies, page 56)

Here the value for men matches the table, but the number for women is quite different. Indeed nowhere in the "Saxon Bodies" chapter does the 163cm number appear.

Regardless of whether the number in the table is a typo or not, there are many reasons to be skeptical. People buried at a particular church are not at all guaranteed to be a random sample of the population. Then there's the different methods for the heights: the first six periods use long bone lengths, while the last two use height records. The Victorian period is particularly suspect because it's based exclusively on heights of "habitual criminals," not known to be the most well-nourished of a given population. The mapping that the authors of Of Human Wealth do from London Body's period labels to centuries is also not quite right but hardly even worth mentioning compared to everything else.

I'm sure I don't know enough about history to provide a complete evaluation of this whole "first European renaissance" theory, but from what I can see it doesn't seem very convincing. I'm surprised Rushkoff would even mention it, and the fact he does makes me more skeptical of everything else he says.

]]>- Work really hard.
- Be a perfectionist.
- Micro-manage. (Tony tries to clarify that what he recommends isn't micro-managing; the distinction is something like managing outputs as good vs. process as bad.)
- Push people hard.
- Have interviews done by a small group of people who will work with the new person, not a bunch of people from all over the company.
- Don't have a lot of perks at the office.
- Blame Google for Nest not going how you wanted.

I think the most interesting thing, to me, is the tension between intrinsic motivation vs. external pushing for generating results. Tony seems like a pusher, and he's gotten results. But do bosses have to be so hard-driving?

Fadell comments on how much time he spent on "people problems" (HR or personnel issues) as a leader. This might always be the case, but I also wonder whether particular leadership styles (and related hiring choices) lead to more of this. Fadell says he followed a "no assholes rule" in hiring. Might anosognosia be at play?

I'm not sure I'd like to work with Fadell, but I tend to like his goals and his Build Collective seems to be supporting cool things. (Ends vs. means?) The book is almost a promotional tie-in for this venture operation.

]]>There are times when having someone with bipolar live at home with family is no longer realistic, and the person is not ready to live on their own. In that case, a group home might be an option. (page 78)

It Won’t Work Right Away, and Not Everything Will Work (page 91)

Sometimes someone will go off medication without medical supervision. This is dangerous but understandable. (page 94)

Sometimes You Need to Be a Caregiver (page 122)

No matter how stable someone becomes, there will probably be another mood episode eventually. (page 123)

Know the difference between your need to control things and what will actually work. (page 135)

]]>Repeated mood episodes are part of bipolar disorder: expect setbacks. It’s not a disorder that anyone truly gets over, at least not yet. (page 145)

The Chloralkali process is electrolysis of very salty water. It produces chlroine gas, which will kill you, and sodium hydroxide aka lye, which isn't healthy either and which you can use to pulp wood and make paper. This is very cool, and something you could probably do at home. It's a little dangerous. I wonder if this is part of how we lose education about the material world, by avoiding parts that are both interesting and risky.

Maybe it's just my perspective as a computer guy, but I feel like the work of the physical world is too often ignored and undervalued. Something like Hammerbacher has said, it feels like everyone wants to get rich at a keyboard while the lab benches are empty.

The book in a couple places recommends going to YouTube to see a video, with search terms to use. I feel like I've heard this kind of thing in a few books recently. Is this what books do now? They can't embed video, so they tell you to go find the video yourself on YouTube? I'm not 100% opposed, but this feels a little gross to me.

]]>Here's the graph in Outlive:

What a remarkably horizontal white line! It supports Attia's position, in that he's arguing that we need to invest more in fighting other causes of death (heart disease, cancer, dementia, diabetes) so he's trying to make them seem like big unsolved problems. And certainly, they are big unsolved problems. But the claim that mortality rates haven't been improving is misleading.

Let's first go to the graph in The Rise and Fall of American Growth:

Attia copied this graph fairly exactly, except for adding "top" before
and removing the asterisk after "diseases," which is just as well
since the asterisk doesn't seem to have an explanation in *Growth*
anyway.

Such a horizontal white line! Here, it supports Gordon's position, in that he's arguing that things were getting better in the US from 1870-1970, but things aren't getting better so much any more. So here's a decrease that stops decreasing by 1970.

The biggest problem with this presentation is that the data is not age-adjusted, and as such it's sensitive to the distribution of ages in the population. For example, with low death rates for people in their early years, one way to lower the raw death rate is to arrange for the birth of a lot of babies. The raw death rate doesn't tell us much about anything in particular, and certainly isn't the "real" death rate in any meaningful way. Let's compare with a CDC graph that is age-adjusted:

The green line is the age-adjusted death rate. It was 1,446 in 1950, 1,222.6 in 1970, and 723.6 in 2018. That's a 50% improvement from 1950, and a 41% improvement from 1970.

Adjusting for age is important, because for example, when people don't die young, you get more old people. Baby booms and so on also matter. The population pyramid looks different now than it did in 1900. Compare the 1900 and 2000 shapes as presented in a Stanford visualization:

Both the raw and age-adjusted overall death rates don't tell us everything about what's happening at every age, so let's look at that. Here are death rates for the oldest nine age buckets in data from the Human Mortality Database from 1933 to 2021, shown with a quick graph I made.

Every age group is seeing decreasing death rates up until COVID-19 impacts 2020 numbers. I left out younger age bands to avoid crowding, but those also show improvement, especially for the youngest.

For people 85-89, the death rate per 1,000 was 179.87 in 1950, and 154.77 in 1970. It was 103.14 in 2018. That's a 43% improvement since 1950, and 33% improvement from 1970. Even the COVID-19 spike takes us back only to around the same levels as we saw in 2000.

Things have been improving. Improving since 1950, improving since 1970, improving since 2000 apart from COVID-19. It just isn't accurate to suggest that death rates haven't been going down over these periods. There's still room for improvement, but the real gains should be celebrated, not ignored.

The question of whether life expectancies have leveled off or even decreased over shorter, more recent periods is quite distinct from the trend over these longer periods.

I attempted to replicate Gordon's graph using the HSUS data. He cites Ab929-943, which includes ten columns of infectious disease death, by my count. He uses Ab952 for the total death rate, I believe.

It adds only confusion that Gordon's figure is titled "Mortality Rate
*as Percentage of GNP*" (emphasis added). He also cites HSUS series
Bd46 (percentage of GNP spent on health care from 1929 to 1997) and
2012 SAUS table 134 (health care spending). This seems to be a
leftover of an idea from an earlier paper. Gordon's footnote 22 says
that his figure "is an update of McKinlay and McKinlay (1977, figure
3, p. 416)" so let's track that down.

The paper is The Questionable Contribution of Medical Measures to the Decline of Mortality in the United States in the Twentieth Century. It has this Figure 3:

There is no horizontal white line here. The presentation is quite different, arguably better for showing the decrease in death from infectious disease. McKinlay and McKinlay seem to have known to use age-adjusted statistics as well. More relevant to the discussion of GNP, however, their paper has this Figure 2:

The point it's making is that increased spending on health care doesn't seem to be driving a lot of increased reduction in death rate. There's something to this argument, but 50 years later we can also see (in the CDC graph above) that the period roughly 1955-1965 was a fairly flat spot on the death rate graph, with improvement continuing after it.

Gordon mentions the concern that spending on health care isn't delivering matching results in death rate improvement, but it doesn't make it into his figure apart from the title and data citations.

The horizontal white line is basically a coincidence as death has been shifting from younger to older people over time and the population pyramid has been changing shape. This graph is misleading.

]]>My biggest gripe is probably that Harari argues (at length) that animals have consciousness, and that animals are algorithms, and then also claims without a second thought that algorithms implemented via computer have no consciousness.

Harari also makes the fairly common error of scientific exceptionalism: the belief that scientific ways of knowing are fundamentally distinct from all others. For the number of words he releases on related topics, this is frustrating.

Harari does say some reasonable things. I think he's correct in pointing out limitations in our understanding of consciousness. (I think this is such an interesting topic I wrote a short story about it.)

I also like Harari's phrasing around learning history in order to be liberated from it. But throughout the book, the moments of lucidity make the languid and ludicrous majority that much more unbearable.

]]>Easter talks about the Fogg Behavior Model (action requires Motivation, Ability, and a Prompt) and suggests that scrolling on your phone all the time is a bad idea. Okay. He also suggests that audible white noise is caused by EM radiation in air. I struggle to find a reasonable interpretation that makes this true. There's also a good deal of "people used to do this long ago, therefore it's a good idea," which isn't necessarily wrong, but it's too easy to ignore whether there's a good argument beyond the appeal to nature.

The thesis is basically that we should get off the couch, eat healthier, and challenge ourselves. I agree, basically. I'm now listening to Rushkoff's Survival of the Richest, which has some advice which is similar in effect: Focus on the real world rather than disappearing into a technologically mediated isolation pod. Rushkoff gets there by a different path, and from his mouth it sounds like a call to a pro-social future. From Easter, it sounds like rugged individualism for reclaiming masculinity.

We're very lucky to be able to choose our level of difficulty, when we can.

]]>He's a little bit of a tech bro. He talks about how he mellowed and started taking one (1) day off per week. And is he responsible for the popularity of rucking? He used to be very into keto and fasting. He still seems to be a little quick to assume causation. But still: Who doesn't want more healthspan?

- Recommendations
- Exercise
- "Zone 2" sustained cardio
- Intense cardio for improving VO2-max
- Strength (with weights)
- Stability (core, posture, flexitility...)

- Diet
- Get lots of tests done, screen early for cancer, etc.
- Sleep (lots of good)
- Emotional health (good)
- Rapamycin taken once a week
- You can get this from AgeLessRx or HealthSpan, for example.

- Exercise
- Anchor phrases
- Medicine 3.0: Where 1.0 was witchcraft/superstition, 2.0 is scientific "treat the disease" modern medicine, and 3.0 is his style of life-long health improvement to avoid diseases of aging etc. (Attia's practice is called "Early.")
- The Four Horsemen
- Heart disease, stroke, etc.
- Cancer
- Dementia etc.
- Diabetes and friends ("metabolic disfunction") which he says is related to the other three

- The Centenarian Decathlon: Work toward being able to do things when you're really old; overprepare because you will age.

Oh also he talks a little bit about Mendelian randomization, which it turns out is just using genetics as an instrumental variable. Also Bradford Hill criteria for causation. Neat!

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