Two Plus Two Equals Four Until You Redefine Addition

Like everything else, the Babylon Bee had fun with this:

In a mathematics lesson delivered to her kindergarten class Tuesday, local teacher and closed-minded bigot Becky Delatorre reportedly insisted that two plus two equals four, all the time, to the exclusion of all other numbers, no matter how anyone feels about it.

Well…we turn to the famous Russian mathematician Israel Gelfand’s Lectures on Linear Algebra (Dover Books on Mathematics), who at the start makes this definition:

Breaking it down, in the italics he makes a definition of what a vector space is.  At the core of that definition is what linear algebra (which itself is at the core of numerical methods, computer simulations, etc.) is all about: everything that happens is basically scalar addition (which is what the kindergarten teacher in the Bee was trying to teacher her bratty students) and scalar multiplication, and lots of it for large models.

That definition made, Gelfand sets forth eight (8) rules for these two operations to follow in order to be valid.  At this point, Gelfand puts in the kicker:

It is not an oversight on our part that we have not specified how elements of R are to be added or multiplied by numbers.  Any definitions of these operations are acceptable as long as the axioms listed above are satisfied.  Whenever this is the case we are dealing with an instance of a vector space.

What he is saying is this: for a valid vector space, we can redefine addition and multiplication as long as it meets the eight rules!  An example of how that works is here.

This is an interesting twist in mathematics that, mercifully, doesn’t have much practical application.  But I suppose it’s possible to shut up (or put to sleep, either result works) a class of unruly kindergartners with the eight rules.  And having endured stuff like this makes attacking Bill Clinton’s Eucharistic Theology even more fun.

Better stick with 2 + 2 = 4


We’re Stuck at the Other End of Science, Too

Sabine Hossenfelder laments that what should be the “cutting edge” of science isn’t cutting it these days:

Very plausibly, the main reason why we haven’t made progress is that we’re not doing the right thing. We’re looking in the wrong places. We are letting ourselves be guided by the wrong principles. It’s about time that we rethink this because, clearly, it’s not working. One of the things that I’ve spent a lot of time thinking about is what would be good principles to look at. Interestingly, in high-energy particle physics and also in cosmology, people pay a lot of attention to aesthetic criteria that they use to select theories they think are promising. And we know that paying attention to beauty is not very scientific. It’s certainly a human desire, but it’s questionable whether it will bring us anywhere.

To be honest, her article struck a chord with me at the “trailing edge” of science, geotechnical engineering.  I’ve gotten the feeling that we’re stuck in neutral in many ways: we’re doing a great many technological tweaks, but we’re not really moving the ball down the field the way we should.  Some of this is due to our regulatory environment, some due to the way our research is funded (which isn’t as different from Hossenfelder’s as we’d care to admit) but ultimately, as is the case in her field, it’s the way the community looks at the problem.  I lamented some of this last year in this post regarding my own specialty, pile dynamics:

Numerical methods and computer power have both vastly improved since Smith’s day. So is it possible to see another paradigm shift in the way we perform forward and inverse pile dynamics? The answer is “yes,” but there are two main obstacles to seeing that dream become a reality.

The first is the nature of our research system. As noted above, Smith’s achievement was done in a large organisation with considerable resources and the means to make them a reality. It was also a long-term effort. Today the piecemeal nature of our research grant system and the organisational disconnect among between universities, contractors and owners incentivises tweaking existing technology and techniques rather than taking bolder, riskier steps with the possible consequence of a dead-end result and a disappointed grant source.

The second is the nature of our standard, code and legal system. Getting the wave equation accepted in the transportation building community, for example, was an extended process that took longer than developing the program in the first place. Geotechnical engineering is a traditionally conservative branch of the profession. Its conservatism is buttressed by our code and standard system (which is also slow-moving) and the punishment meted out by our legal system when things go wrong, even when the mistake was well-intentioned. Getting a replacement will doubtless be a similar extended process.

My guess is that this problem extends to other fields of science and engineering as well.  If we want to make progress, we need to address these issues directly.

He Was Right, They Do Flunk Clock

Some British schools are taking analogue clocks out of schools:

Schools are removing analogue clocks from examination halls because teenagers are unable to tell the time, a head teachers’ union has said.

Teachers are now installing digital devices after pupils sitting their GCSE and A-level exams complained that they were struggling to read the correct time on an analogue clock.

In 1971, when I was in prep school, I obtained for my dorm room one of the earliest types of digital clocks, namely one with those “flip cards” which changed to display the time digitally.  A friend looked at the clock and said, “Your little brother will flunk clock.”  He was partially wrong because I was the little (or younger) brother in my family.  But he was right about one thing: when people have digital clocks, they will forget how to read an analogue one.

So many problems can be anticipated if we just have the vision to see the consequences…

The Matter of Women Pilots Should Have Been Settled in the 1930’s

The Daily Beast correctly notes the achievement of Southwest pilot Tammy Jo Shults but with the usual caveat:

Just how masterfully Tammie Jo Shults, the pilot of the badly crippled Southwest Airlines Flight 1380, handled the problem of an engine exploding at 30,000 feet is winning admiration from thousands of her fellow pilots—and should finally help to temper the hubris of what has been a notoriously testosterone-charged profession.

That last point was settled a long time ago, as I noted here:

One of the more interesting parts of our story concerns the substantial number of women who flew and took parts in the competitions that Chet coordinated. This is not just a “hindsight” kind of thing; it garnered a good deal of attention at the time as well.

Aviation was still a new technology in the 1930’s, and attitudes about “roles” weren’t as fixed about flying as they were about other activities. Moreover many of the women who flew came from the upper reaches of society, which put the whole role of women in a different light than elsewhere.

When we think of women’s aviation in those times, we usually think of Amelia Earhart.  But there were many other women who achieved great things in the pilot’s seat: as I noted here, in the lead-up to the May 1934 Langley Day air show in Washington, my grandfather hosted “…a reception at the Willard Hotel in honour of Laura Ingalls, the New Jersey “society girl” who had just (22 April 1934) come from setting a world’s record by flying 15,000 miles from the U.S. to South America and back, which included crossing the Andes Mountains.”

As Camille Paglia noted, the 1920’s and 1930’s were indeed the “favourite period” of feminism, with plenty of lessons for us now.

The Core Problem With Liberal Arts Curricula/Degrees

I’ve taken flak for saying this in other contexts, but this comment on a frustrated history PhD’s failure to land a tenure-track position hits the nail on the head:

my own experience has been very similar although now i’m glad that i left the humanities and i believe that the humanities themselves should be completely erased as actual graduate disciplines. so many wasted minds, so much wasted capital for really very little societal value and much much grief to very smart individuals only to keep up a very vague idea that a university teaches you “how to think” or how to “read” or “how to write.”

the irony is that, given the current way in which most writing and reading occurs, taking a humanities course in milton, cervantes, or baudelaire may actually make you a worse writer and reader for today’s environment… but i digress.

i encourage you to think of this as an opportunity and to go outside of the field completely. i myself am now a software engineer although i once did a phd in comparative literature. it is possible to change, and it can be very fulfilling.

Those Unscientific Science Journalists

NPR wants a new one, but…

NPR, which to its credit at least attempts to cover science and health, is looking for a new Science Editor. Unfortunately, actually being trained in science is not required for the job.

Under the qualifications section, the ad says, “Education: Bachelor’s degree or equivalent work experience.” Amazingly, not only is a background in science unnecessary, college itself is optional. Despite such a low bar, whoever gets hired for the job will be responsible for covering “consumer health trends, medicine, public health, biotech and health policy.” Seriously?

It’s fair to say these days that a journalism position is an advocacy position.  But that’s one thing that’s discredited many scientific initiatives: the total lack of people with science training either reporting, advocating, or setting public policy on these kinds of issues.  No where is this more evident than climate change, where the biggest carbon-free solution in the mix–nuclear power–has been shunted aside, even when most of scientific community is good with it.

But that’s the American way, and has been for years.  It’s little wonder that countries such as China, Iran and Russia, where more of the educated population is trained in the sciences, are perceived as such threats.

I Guess YouTube Will Flag the Boring Video, Too

YouTube is doing some strange things these days, and this is yet another:

YouTube announced Friday it will start flagging videos published by organizations that receive government funding.

Viewers will be able to see labels on videos from government-funded outlets above the video’s title on the page.

“News is an important and growing vertical for us and we want to be sure to get it right, helping to grow news and support news publishers on YouTube in a responsible way,” YouTube News senior product manager Geoff Samek said.

I guess that includes this masterpiece, which I use in my Soil Mechanics class:

Watch it for a minute or two and see why I call it the “Boring Video.”  I told my students that labelling it as such was my attempt at “truth in advertising.”

This video was produced at the University of California at Davis with a grant from the Feds.  Like so many documents and other material in this field, it was produced with government funding, and use of this kind of material is widespread amongst the Federal and State agencies charged with civil and military works, and used in the teaching of civil engineering, most of which in this country takes place at state (government) universities.

So I guess that YouTube will, once it figures all this out, label this as “propaganda.”

Like I said, YouTube is doing some strange things these days.  Recently they demonitised “small” YouTube channels (like mine, the pennies rolled in) and frankly I couldn’t figure out what they were trying to accomplish other than getting rid of a large number of accounts that were more hassle to service than they were worth.  The obsession of social media with “propaganda” (and YouTube certainly isn’t alone) is going to kill it for themselves and everyone else.

No, Columbus Wasn’t Worried About Falling Off the Edge of a Flat Earth

I think it’s fair to say that most Boomers (and some who came afterwards) were taught that one reason Columbus sailed west to determine whether the earth was flat.  But this won’t wash, as BizzareVictoria notes:

Everyone knows that in the medieval era, everyone thought the world was flat, and Columbus discovered the Americas in part because he was trying to circumnavigate the globe, to prove it was round, and to end up in India, right?

Except all of this is wrong. 

Eco tells us that people have known the world was spherical since ancient Greece. “Parmenides seems to have guessed its spherical nature, while Pythagoras held that it was spherical for mystical-mathematical reasons [and] subsequent demonstrations of the roundness of the Earth were based on empirical observations: see the texts by Plato and Aristotle. Doubts about sphericity linger in Democritus and Epicurus, and Lucretius denies the existence of the Antipodes, but in general for all of late antiquity, the spherical form of the Earth was no longer debated” (11).

In addition to providing additional backup to this claim, BizzareVictoria goes on to detail how Victorians, frustrated at the opposition of the church to evolution, spread the idea that the medaevals, following Lactantius, thought the earth was flat.

To all that I’d like to add the following:

  1. The Bible does not teach that the earth is flat (cf. Isaiah 40:22.)  There was a great deal of knowledge interchange amongst the civilisations of the Middle East, including the Egyptians, Babylonians, Persians and Greeks, a fact that was better appreciated in ancient times than it is now.
  2. Dante certainly conceived the earth as round, which is closer to Columbus’ time than the Bible.
  3. If there was a central fault in medaeval science, it was an over-reliance on the ancients for scientific fact.  That’s why Galileo butted heads with the schoolmen of his day. In this case, however, that reliance was correct.

As for Lactantius, he wasn’t exactly in the top shelf of Patristic writers, a fact also recognised in medaeval times.  One thing he was dead right on, however, was the rapacity of Late Roman tax collection methods, which doubtless hasn’t endeared him to the bureaucrats.

HT Tim Harding.

Catholic and Christian, the Sweet Combination

Evident in the obituary for a Texas A&M classmate (we both were engineering majors) of mine:

Thomas Craig Kohutek, 63, was called home to be with the Lord January 1, 2018…A faithful servant of the Lord, Tom was an active member of St. Catherine of Siena Catholic Church as well as the Knights of Columbus.

I’ve written before about the revolutionising experience my years at Texas A&M and the effect it had on my life.  That experience was as a Roman Catholic; you can see a video montage of that here.

Somehow I think that directness of commitment has gotten lost between the obsessive churchianity of #straightouttairondale or the loosey-goosey veneer of faith in liberal Catholicism.  To see this and to know some of its roots is refreshing.

But what really tugged at the heartstrings was this, at the end of the obituary:

For this momentary light affliction is producing for us an eternal weight of glory beyond all comparison, as we look not to what is seen but to what is unseen; for what is seen is transitory, but what is unseen is eternal. (2 Cor 4:17)

That sums it up.  So what about you?

Sometimes Science Takes a Detour

And it’s not the detour we’re told happens, either.  Consider this example from J.E. Gordon’s Structures: Or Why Things Don’t Fall Down, during a discussion of Robert Hooke and the development of the theory of elasticity:

In fact, throughout the eighteenth century, remarkably little real progress was made in the study of elasticity.  The reasons for this lack of progress were no doubt complex, but in general it can be said that, while the scientists of the seventeenth century saw their science as interwoven with the progress of technology–a vision of the purpose of science which was then almost new in history–many of the scientists of the eighteenth century thought of themselves as philosophers working on a plane which was altogether superior to the sordid problems of manufacturing and commerce.

It’s worth noting that the eighteenth century is frequently characterised as  “the Age of Reason,” when Europeans (at least, along with their cousins on this side of the Atlantic) began to shed their “superstitions” and move into “the Enlightenment.”  It’s one thing to say that we’re “enlightened” or “guided by reason” but it’s quite another to actually take all of this reason and put it to the solution of present problems.

Today people insist on us “believing in science” or whatever scientific thing they want us to agree with.  The minute we put the question as a subject of belief, we’ve not only missed the whole point; we’ve undermined any claims of our thought processes being guided by reason as well.  And if we simply make our goodness a product of our belief, we may feel better about ourselves, but leave the problems at hand unsolved.