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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
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
…
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Karl Marx: Maybe too Cynical to be Demonized?
Ryan Cooper at The Week is looking for a rehabilitation:
Happy birthday to Karl Marx, who was born 200 years ago on May 5. He was the most astute and influential critic of capitalism in history — and also the most misunderstood.
It is long since time that Marx re-joined the community of ordinary intellectuals, considered as neither the terrifying harbinger of social upheaval, nor a secular pope with the eternally correct description of all human society. He was a genius, but in the end, only another human scholar with a brilliant but incomplete perspective.
But consider this gem from Fritz Raddatz’ Karl Marx: A political biography
:
After a heated argument, first humorous and then serious, as to who should do the chores in the state of the future, the lady of the house asked him: “I cannot picture you in an egalitarian period since your inclinations and habits are thoroughly aristocratic.” “Neither can I, ” Marx replied, “those times must come but we must be gone by then.”
Marx is a complicated figure whose life was neither pleasant nor pretty, as evidenced by this and this. I highly recommend Raddatz’ biography; it more than anything else finally closed the door on me being a socialist or communist.
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We’re Stuck at the Other End of Science, Too
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.
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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…
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Thinking about Mission the Anglican Way
I’m glad to share a new article just published at the blog of The Living Church. I am basically asking why Anglicans have a concrete approach to music, theology, and architecture, but don’t seem to have anything like this when it comes to global mission. Here is the lead: Like most Christians, we Anglicans tend to […]
via “Thinking about Mission the Anglican Way” in *The Living Church* — Duane Alexander Miller’s Blog
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The Only Real Alternative to “Two Kingdoms” Theology is Islam
Some people will complain about anything:
Recently some critics of prominent Trump-supporting Dallas Baptist pastor Robert Jeffress have disapprovingly identified him as a supporter of “Two Kingdoms” theology, an historic Protestant belief about the division of duties between spiritual & earthly rule. Jeffress in public pronouncements has stressed that civil government is called to provide public order, not embody the Sermon on the Mount, on issues like immigration.
Two kingdoms theology’s most expansive expression is Augustine’s City of God, written late in the game for another world power, the Roman Empire (well, the western part.) In those times those in civil authority who “wore the belt” were not allowed to become priests. The system was so corrupt at that stage that Christianity could not see its way clear to fix it (although it made improvements such as getting slavery to dissipate.) Rome collapsed, but the Church, in a different way, laid the foundation for a civilisation that was greater than the one that was there.
I honestly don’t think that the howling social justice warriors who profess and call themselves Christians (and I’ve run into them of late) have really thought through what the New Testament commands us to do vs. what the state should do. The blunt truth is that, like it or not, Christianity has never really set forth a morality for the state, or that the morality of the state should be at unity with that in the church. The religion that has done that is Islam; that may explain in part the affinity that people on the left feel with Islam. Even a secular historian like Ferdinand Lot grasped that truth. Since most of the focus on refugees have been those from the Middle East, it pays to look and see how things have worked out under the various forms of Islam before we unwittingly advocate those things for ourselves.
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If You’re Not Doing It for Jesus, You Shouldn’t Be Doing It
Bethany Jenkins, vice president of forums at the Veritas Forum, which helped to organize the event, reported that Denhollander was asked about her view of the church responding to the issue of sexual abuse. When asked “how do you trust the church to point to justice and truth in these situations?” Denhollander responded “You don’t. You don’t trust the church, you trust Jesus.”
Some Christians are queasy at this statement. But if they are real Evangelicals and not the “corporate” kind, they shouldn’t be. One of the first lessons I learned in the years I worked for the Church of God is that I was doing this for God, not the church and that I, like Denhollander, needed to trust Jesus and not the church. That held me in good stead all the way until the church abolished my department and my position in 2010–and beyond.
Too many Christians practice churchianity rather than Christianity; they equate the church with God and, when the church lets them down, they bail on God. Forms like Christianity like Roman Catholicism, with their high view of the church, set themselves up for that kind of reaction. But those of us who do not have that view of the church have absolutely no business making that equivalence.
Although the #MeToo movement has given Denhollander a larger platform for her message, in many ways she’s swimming against the tide, both in and outside Christianity. But she’s a strong person; we need more like her.
Sailing the Last Voyage with Newton and Pascal
, who at the start makes this definition:
:
Positive Infinity 