This came up in an email thread, but for the life of me I can’t find it. Since this is the second email (thread) from a colleague I have lost in as many weeks, I’m beginning to suspect my new email client (Thunderbird) is eating them.
Not that you care about that, but because I’ll be rebutting this colleague’s argument, I can’t do so in his own words. (“Why don’t you email him, Briggs?” Because I can’t remember who it was.)
Anyway, the impetus was yet another anti-“white supremacist” article claiming whites were too good at math, so that we should re-define math to be what people want it to be. Like letting students hand in Tik Tok videos instead of proofs, you racists.
I naturally made a stinging witticism in answer to this, along the lines of if math is invented, then everything nitwits and the evil scream about “white supremacist” math is true. Anything goes.
My colleague thought he was disagreeing with this, because it is beyond obviously absurd to let the kiddies make up their own “math”, while also insisting that math was indeed invented, not discovered.
He thought his position was saved by saying math had to align to the natural world (or just “world”), the material stuff that makes up the universe. That still sounds like a failure for “math is invention”, because we discover facts about the world, we don’t invent them.
Sure, our discoveries are tinged with invention, because every fact and observations fits into a theory or scheme somehow—how else do we know what facts to look for and classify them?—but that doesn’t make the facts wrong. It makes them conditional. And it makes some more uncertain than others. But that’s about it, logically speaking. We are always aiming at the truth of the world in science—or we used to, before it became another branch of poiltics.
If the world is only known up to a degree, and not perfectly, and math aligns with what we know of the world, then it seems math would have to change every time we learn a new or refine an old fact about the world. Yet this does not happen. Math remains constant.
This is because, of course, math begins with propositions that are true conditional on the rock-solid argument “This axiom has to be true”. It proceeds from there, building step by step, expanding using, it is hoped and almost always turns out to be so, solid links in the chain.
Still, this did not satisfy my colleague, who pointed to Euclidean and non-Euclidean geometries. One was thought to fit the world, but the other fit it better. Therefore, the one that did not fit was made up. Invented.
If that is so, then it should be able to prove the not-fitting geometry false. Which can’t be done by pointing to any of its theorems, or even its axioms. It works only by requiring the rule “If not fit world, then false”.
What is the proof of that rule? Well, there is none. It is a desire at best.
It a false desire, too. For here is the simple proof that the proposition “Math beyond that which fits the world is invented” is false.
In math, we have the simple proof (using “successor functions”) that the natural numbers run 1, 2, 3, … and so on, all the way up. Never mind about infinity for a moment.
In the world, we have some number of objects, which can be counted as long as one defines a way to count them. Suppose this method of counting exists. Use it to number all the objects in the universe.
It will stop at some number U, for universe. This U cannot be infinite, for if it were, then the universe would be filled with stuff such that nothing could happen. There aren’t, to use one example I read one, an infinite number of basketballs. If there were, that’s all that we would see, given we could “see” anything when we’d be basketballs.
Infinity is not just a large number. Besides, you can’t, as you’ll see in a moment, invoke infinity until you first show there are an infinite number of objects.
“What about multiverses and the like, Briggs?”
Show me one. They are all, at this point, only math. And they may even be a clear case of math inventing the natural world! (Or attempting to.)
Anyway, we have U. But since U is tops, we cannot have U + 1. That number wouldn’t exist. It wouldn’t make answer sense to talk of U + 1 other than as invention. All math involving numbers larger than U would have to be tossed as unworthy, useful only as puzzles, like other inventions.
It’s easy to see that this is nuts. Because math isn’t invented. It is discovered. And it doesn’t have to be that it fits only the natural world. There is also the immaterial world, in which discussions of infinity and sizes of infinities and the like make sense.
Next step is to make this into a Tik Tok.
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This is one article (saved from 2019) about Mathematics education being “racist.” Over the last decade, names and examples in elementary and middle school word problems, and subsequently mandated tests, were becoming so “politically correct” that students were stumbling over the pronunciations and scenarios instead of arriving at the solutions. I actually taught students to substitute just the first letters for the names and broadly cross out all unnecessary scenarios or storylines. This improved scores of all students considerably.
https://www.hoover.org/research/seattle-schools-propose-teach-math-education-racist-will-california-be-far-behindseattle
Can’t you download your email????
I have no problem with letting math be what people want it to be. After all, who needs bridges that don’t collapse, who needs medication with any kind of measurements attached, who needs their money in their bank accounts and who needs those accounts to balance? Why should your paycheck reflect what you earned and how many hours you worked? Random tax withholding is good, too. And welfare checks sent out based on a random number generator. Life would be so exciting that way. Really, you racists, give it up.
You’re going to need to set it to music, add some rockin’ graphics, and maybe then you can make Tik Tok. Or add a kitten or cute puppies. Maybe infinite puppies from a mulitverse!!!!
I will note I think that math is both defined and discovered. First, you define numbers. Then arithmetic and geometry are discovered. Algebra starts in on the defined/made up and by the time you get to calculus, which Newton INVENTED, it ceases to be discovered and is definately made up.
Oh, and PARENTS WANT THIS so stop complaining. Parents hate their kids and send them to indoctrination centers, throwing snit fits when those centers closed for Covid. Parenting means loving your government day care centers and indoctrination centers.
Bill Cosby once famously asked: “What’s a cubit?”
ergo
Math is racist, patriarchist, misogynist and rape culture
The use of basketballs to explain infinity used to be racist until it was “discovered” that white men can’t jump.
So yes, Math of and by itself is not intrinsically racist
but once it’s brought in “as a language” to help define the material or immaterial world it becomes racist
Math as a language is key to its racism
RT’s comment is corollary to this concept
The English language is racist … aks anyone
I see what RT sees. When my students have no idea how to pronounce the names of the characters in the word problems because they are so multi-cultural, it keeps us from getting to the math. I like the idea of substituting first letters.
As far as the rest of your argument, my guess is that the general response will be something along the lines of “Shut up, they argued”. You can’t have a logical argument to prove that things aren’t illogical. It fits the same pattern as “white people are racist so any white person who says they aren’t racist must be racist”.
I am reminded of an old rocket science story: Somehow the mainframe was glitching, so whitecoats in-charge decided that manual calculations could be undertaken in the interests of project efficiency, much to the chagrin of IBM execs. But, this meant the services of actual mathematicians would be required. Those same mathematicians, many of whom IBM had recently rendered ‘redundant’ through ‘advanced’ technology. As it turned out, most had already returned to Russia or Poland where they could still be gainfully engaged with ‘hands-on’ practical work. Pleas were issued. Specialists were found. The Day was saved. The U.S. Government even now continues purchasing Russian built rocket engines. They are strangely – reliable. This has something to do with mathematics being revealed and then unpacked and then applied – not just imagined through the creative might of free thinkers.
Re: Sheri … Newton INVENTED calculus
Newton or Leibniz “invented” or “discovered” Calculus?
Newton and Leibniz each “invented” a “language” for calculus.
But calculus concepts had been building and advancing through the ages.
https://www.thegreatcoursesdaily.com/invented-calculus-newton-leibniz/
From the piece: Newton’s teacher, Isaac Barrow, said “the fundamental theorem of calculus” was present in his [Barrow’s] writings but somehow he [Barrow] didn’t realize the significance of it nor highlight it. … [Barrow’s] pupil [Newton] presumably learned things from him. Fermat invented [discovered] some of the early concepts associated with calculus: finding derivatives and finding the maxima and minima of equations. Many other mathematicians contributed to both the development of the derivative and the development of the integral.
So again, it’s down to language (and intrinsically racism).
The “racial” or “cultural” difference between Newton and Leibniz supposedly worked against England
It became a huge mess, that, incidentally, led to the retardation of British mathematics for the next century because they didn’t take advantage of the developments of calculus that took place in continental Europe.
Thanks for taking us down this rabbit hole, Sheri
Physicist Eugene Wigner (1902-1995) addressed these issues in his famous article, “On the Unreasonable Effectiveness of Mathematics.”
He and I disagree about the definition of mathematics, because I think (more consistently with his article) that mathematics investigates the fundamental structure of reality. But he and I agree with you that mathematics is discovered, not invented: that’s why it is so “unreasonably effective.”
Particularly notable is the fact that mathematical discoveries of past centuries (for example, some of Euler’s work) have helped us understand recent discoveries in quantum physics. If mathematics did not reflect some deep structure of reality, it would be quite a miracle. Anyway, here’s a link to Wigner’s article:
https://www.dartmouth.edu/~matc/MathDrama/reading/Wigner.html
I love math … yes, I’m a racist (word problems were my favorite)
I once read that Isaac Asimov (in one of his four or five hundred volumes) wanted to be a Mathematician. He got a “B” in his second Calculus course and gave up on that idea.
Calculus was my downfall as well. I wonder now if it was simply a culture/language problem? NOT
Pretty sure that Indian Mathematician Srinivasa Ramanujan CANCELs any concept of racial/cultural divides in Math
“Pretty sure that Indian Mathematician Srinivasa Ramanujan CANCELs any concept of racial/cultural divides in Math.”
That Indian math dude is an Indo-European Aryan. So he’s Hitler. That’s why he’s good at Hitler Math.
“Pretty sure that Indian Mathematician Srinivasa Ramanujan CANCELs any concept of racial/cultural divides in Math”
Unless, of course, he was of one of the Y-DNA R1 haplogroups that mark many Europeans, Persians, and Northern Indians.
Of course, “invent” and “discover” mean the same thing, anyway.
Columbus invented the West Indies
In Math, you “discover” a concept and then you have to “invent” a means to communicate the concept
Columbus “discovered” a land and then had to “invent” a means to communicate it – “West Indies” worked well
well..
“If math is discovered rather than invented, then ET, when we meet him, will turn out to use the same math.”
Nice global statement – but wrong: some math is discovered, some is invented.
The invented falls, I suspect, into two categories (ignoring notations) : error and convenience. The error stuff generally gets corrected as the error is discovered (e.g. I don’t think there really are 1200+ discernibly different subclasses of prime numbers); the convenience stuff gets discarded when no longer useful. The obvious examples there are Calculus and Logs – both effective and useful inventions for describing the world within the computational limitations of the time, but cumbersome and unnecessary today.
In contrast, continuous fraction arithmetic has largely disappeared from use but is more probably (I think?) a discovery than an invention because it looks pretty fundamental and offers much greater accuracy in computation than anything else we know – and if Enterprise is going to navigate between stars it will need +100 digit accuracy.
The one I want to ask ET about, however, is multiplication because it looks an invented convenience to me..
“There aren’t, to use one example I read once, an infinite number of basketballs. If there were, that’s all that we would see, given we could “see” anything when we’d be basketballs.”
Not necessarily. If there’s an infinite amount of space for the infinite amount of basketballs (and there would have to be), then the space between any given two basketballs could in theory be large enough that we might never even see one, let alone more than one.
Infinitely large space between the basketballs? You could calculate how much space per basketball by dividing … oh! Maybe not.
Is the infinity of space a larger infinity than the infinity of basketballs?
Math is an Muslim invention because the Indians discovered the number zero. How is this racist? Indians and Muslims cannot be racist. It is racist to think that white men invented math. white men culturally appropriated math.
Hey! The Arabs got their eponymous numerals, including zero, from the Hindoos. Which makes maths Aryan, again, and therefore racist.
Stephen J and Simon Platt … getting into Set Theory
See: https://www.encyclopedia.com/science/encyclopedias-almanacs-transcripts-and-maps/set-theory-and-sizes-infinity
Yes, as you say (and there would have to be), if the basketball is an official NBA basketball, you would need an infinite space to fit them all. That said…
Mathematicians generally regarded infinity as a way of speaking about a limit that could never be reached. They balked at the idea of considering it in more concrete terms, such as an infinitely large group of items.
Georg Cantor found ways to work with infinite sets, which many believed could not exist. He further alarmed his contemporaries by demonstrating that while all infinite sets are indefinitely large, some are nonetheless larger than others.
So if you can define your “set of space” such that your “set of basketballs” do not entirely “fill” the space, you could probably define your “set of basketballs” to “overfill” your “set of space”.
As Simon Platt suggests “space” suggests some sort of dimensional component (length, breadth, depth), and as he realized, it cannot have quantifiable dimension. Yet an “official” NBA Basketball by definition has Dimensional components.
Since both the basketball and space would have otherwise quantifiable Dimensional components, without any other rule about their existence, they would always push against their dimensional component such that any given space would contain a basketball and anyy given basketball would be contained by space
That’s my conjecture, I’ll leave the proof to the reader
“Is the infinity of space a larger infinity than the infinity of basketballs?”
Those infinities are small potatoes next to the infinity of Hitler.
Sander van der Wal
This more or less agrees with your take
https://www.history.com/news/who-invented-the-zero
The Indians were the first to use the zero to actually mean null or “nothing” whereas Babylonians and Mayans used it as a place holder whereas Indians used it as a place holder and a value. With the Indians also showing that a number subtracted from itself is zero.
The Muslims incorporated zero into their Algebra.
It is racist to think that white men invented math. white men culturally appropriated math.
All of our points exactly.
I got a few extra whereas’s (not infinite)
The second whereas should have been the start of a new thought:
The Indians DID use it as a place holder but also as having a value
Interesting!!
And found this: https://www.britannica.com/science/infinity-mathematics
God bless, C-Marie
Query:
If a tree falls in the forest, and there was no scientist around to hear it, see it, or measure it, did it happen?
Response:
It depends on the consequences that any particular tree’s falling may or may not have on any current government policy or cultural sensitivity restrictions that the scientist is allowed to operate within.
—
Speaking of Isaac Newton, there’s a good presentation here by Dr. E Michael Jones about how Alchemy and the desire to create gold from lesser metals influenced his research and therefore ultimately led to his position at the mint wherein alchemy would eventually morph into usurious capitalism where gold and profits and debt could always be multiplied into infinity thanks to the abstractions of mathematics! A lecture worthy of your time:
Newton and the Exploitation of Science
https://www.bitchute.com/video/dnzU2vSA8hDz/
Does Hilbert’s infinite room hotel paradox disposes of the notion of seeing nothing but basketballs if space is also infinite? Let’d divide space into numbered cubes and put each basketball (also numbered) into its corresponding cube. Now let’s move multiply the number on each basketball by 2 and move it into the correspondingly numbered cube…voila: an empty cube beside each basketball. Or this why Hilbert didn’t believe infinity was “real?”
Loved the Querry and Response, Johnno!
Copy and pasted on to others … hope that was ok!
God bless, C-Marie
Sorry, John – I was just being facetious.
No worries Simon, I figured as much
“Space is big. You just won’t believe how vastly, hugely, mind-bogglingly big it is. I mean, you may think it’s a long way down the road to the chemist’s, but that’s just peanuts [or basketballs] to space.” ? Douglas Adams, The Hitchhiker’s Guide to the Galaxy
Listening to it today … I just realized how reminiscent Zaphod Beeblebrox is of Donald Trump
BobK
Way cool
Making room for 1 guest
Making room for an infinite bus of guests
https://medium.com/i-math/hilberts-infinite-hotel-paradox-ca388533f05
Choice! A rally cry, a dog whistle, for the debased woke folk and their fellow travelers. However, will I (he) have a choice to take the vaccine or not? Will I (he) have the choice of a specific shot like the traditional J&J or will the “health” overlords force me (he) to take the new (experimental) mRNA injection? My body my choice canned, canceled, ignored, vilified, withdrawn, terminated, and trashed. Also, while we are in the “health” realm, let us open the southern border. Walk right in, Corona-doom concerns canceled. Meanwhile, try that strut at the airport. Consistent health measures find no home here (canceled).
Re: Paul Murphy “If math is discovered rather than invented, then ET, when we meet him, will turn out to use the same math.”
Reminds me of an Isaac Asimov short story (since he already showed up on this thread) of a group of explorers from Earth landing on Mars and discovering evidence of a long-gone Martian civilization – including what they eventually realized was a Martian university. And the Rosetta Stone for interpreting the Martian writings they came across was a chart they found in one of the ‘campus’ buildings, with a half-dozen or so rows, and columns with varying lengths, ultimately recognized as a periodic table.
So is chemical periodicity something that is discovered or invented, or something that includes elements of both? And if I expect my students to understand that an element that has 3 protons has substantial chemical similarity to one that has 11 protons (as well as a handful of others) does that make me racist?
It’s not about word problems; it’s about White Supremacy. Or so claims the Bill & Melinda Gates Foundation which funds an org called “A Pathway to Equitable Math Instruction”. The Oregon Dept. of Education has adopted the program. They say:
We need to prepare students to “play the game” in order to “change the game.”This means that we must support students from non-dominate communities’ access to math and math achievement…
Not sure what “non-dominant communities” are, but they are not White, or Asian for that matter. They may be Latinx, whatever that is.
From the Pathway website:
https://equitablemath.org/
A Pathway to Equitable Math Instruction is an integrated approach to mathematics that centers Black, Latinx, and Multilingual students in grades 6-8, addresses barriers to math equity, and aligns instruction to grade-level priority standards. The Pathway offers guidance and resources for educators to use now as they plan their curriculum, while also offering opportunities for ongoing self-reflection as they seek to develop an anti-racist math practice. The toolkit “strides” serve as multiple on-ramps for educators as they navigate the individual and collective journey from equity to anti-racism.
Btw, Oregon public school students are below the national average in math proficiency. Expect scores to drop even further.
Rumors have been spreading that in The New Normal everyone is above average.
In terms of the space discussion and whether or not elements and mathematics are consistent elsewhere in the universe, the answer is also – It depends on who you ask.
If you believe in God as the Creator, and God is consistent and never-changing, then yes, logic, math and everything else is consistent everywhere.
If you believe in Evolution and Something from Nothing, then you fall into one of two sub-categories:
a) The sort that are still hopeful that consistent processes you are able to observe will make stuff happen and that you just haven’t quite figured it out yet, but someday, man, just someday…
b) The sort that threw in the towel and admitted that every consistent process you observe can’t solve the origins problem… but that’s only in your part of the universe… but maybe the answer lies in some other universe that made ours and math and all fanciful things are something entirely different and magical there, but believe them, when you get there, you’ll finally understand.
In terms of the b)elievers, you could just argue that this could just be Heaven, where God exists, and sucks for them that that God is still consistent and unchanging One and the math is still the same, and they therefore still fail, and nobody gives a rat’s ass about the colour of their skin there.
I cannot do the most simplest algebra but I worked to get a matemathical job – As I always say in my work as an accountant – numbers dont lie, words do
Great logical argument. PS did you mean “to use one example I read once”
I can see why advanced math and advance physics actually belong in philosophy…..whenever infinite is involved, it’s philosophy.
Enjoyed the discussion!
Mr Briggs,
What about imaginary numbers – e.g., i = the square root of -1? Surely they’re invented.
Stephen Hawking (I think) plugged i into the equations for the pointy end of the Big Bang to turn it into a curve such that it could be said that the universe had no beginning which is an idea that a lot of people find soothing. But me, I just can’t see how an imaginary square root of a negative number could actually affect things in the real world. Maybe that’s just another of my personal limitations.
jrm
Not at all … again … concepts (square root of -1) are discovered, the “language” is invented … complex numbers (and functions)
Visualizing the Riemann zeta function and analytic continuation
https://www.youtube.com/watch?v=sD0NjbwqlYw
There are a lot of videos on the Riemann Hypothesis
jrm
Not at all … again … concepts (square root of -1) are discovered, the “language” is invented … complex numbers (and functions)
Visualizing the Riemann zeta function and analytic continuation
https://www.youtube.com/watch?v=sD0NjbwqlYw
There are a lot of videos on the Riemann Hypothesis
Since we’ve been discussing infinity a lot
Riemann’s zeta function gets us the peculiar result that
1+2+3+4+…. converges? to -1/12 (I still don’t get it and I can’t visualize that either)
jrm,
The difficulty you may be having is thinking mathematical objects must have direct observable real-world objects to correspond to. This isn’t so. We nowhere observe, for instance, π, but we all believe it to be what it is.
This is one of the first and maybe one of the best videos on Riemann that I found
https://www.youtube.com/watch?v=zlm1aajH6gY
Speaking of PI, I garnered an appreciation of it because of Spirograph.
It really bothered me that I could never connect subsequent lines to the beginning line. No matter the circle or ring or pencil point I used, I could never connect the last line with the first, even circling numerous times until you almost ran out of ink and started filling the paper with lines, you could see that the lines would not converge. It wasn’t until I started learning about PI that I realized the Spirograph problem.
So observing NOT-PI is almost as good as observing PI.
I may have shared this before but it’s worth sharing again
https://www.khouse.org/articles/1998/158/
PI in the bible within .00001 accuracy!
¶ Applying the word “discover” to mathematical inquiry imposes false connotations. It obscures how certain principals of a physical (or imaginary) reality represented by mathematical statements or symbols have invaded our awareness. The principal of ? for example was not the invention of a ‘Dupont’ laboratory moonlighter – as was the case with NYLON and KEVLAR. NYLON / KEVLAR were indeed discovered ~ but ? was revealed. “Discovered” seems to have invaded the lexicon of mathematics mainly through the footpaths of 20th century popular science and popular culture. The Nobel Prize is a prime examplar. But, even conceptual math, like ? for example, has no single individual ‘discoverer’. Rather it is better known (or understood) through the words like revealed and perhaps introduced, labeled, or even described. The speed of light is another elucidating case. Although quantitatively deduced through experimentation ~ it was not “discovered” in the colloquial sense of that word. Rather its nature was revealed, quantitatively labeled and symbolized as (c). ¶ The application of mathematical principals to problem solving by engineers (and statisticians) relies on these principal’s accurate representations of a physical universe as that universe has been (so far) revealed to us. If the foundations of mathematics were merely built upon the cryptic notes and proofs of fallible thinkers and their personal “discoveries” over generations, then the topography of scientific endeavor would be like sand dunes in the Sahara. Of course credit must be given where credit is due, but elementary (versus applied) mathematicians (and their work) should not be lexically classed (or lumped in) with the ‘inventor’ of The Tetra Pak* or the ‘discoverer’ of The New World*. One can of course invent a language (say of calculus) but that terminology matrix is a tool whose usefulness depends entirely on being consistently representative of an underlying subject. If this is demonstrated to be useful, then one can say that it has been discovered to hold true as formulae. But, while a language may be invented, it ultimately describes only principals of observation of a reality that most certainly preexisted the logical and terminological signposts of an unfolding eureka moment. Profession Journals of the Mathosphere recently trend towards greater use of ‘reveal’ as a preferable cognate to ‘discover’. Perhaps this because modern academic mathematicians like to represent themselves as humble folks ~ for grant seeking purposes. Or maybe it is out of some perverse inverted vanity in their group.
The first and second (?) are apparitions of the ‘pi’ symbol. The third (?) apparition is an ‘infinity’ symbol [“conceptual math, like ?”]. There is no reckoning with Fonts.
Jan Van Betsuni
I’m okay with revealed vs discovered You could replace revealed for discovered throughout the post and comments.
I think it would especially work with the ET discussion
To Mr Briggs and the other respondents, thank you for your comments. Since I am an old woman who studied medicine, not maths, much that you wrote is opaque to me. Riemann’s zeta function, for instance, is something I’d never heard of before this morning. Having said that, I remain unsure that pi is in the same camp as the square root of negative one. I may not observe pi but I do observe circles. On the other hand, the square root of negative one seems to me to be a logical impossibility since squaring a number, either positive or negative, always produces a positive number. What am I missing?
jrm,
The fault is likely mathematicians calling them “imaginary” numbers. The switch to “complex” was made, but too late!
You see complex numbers in just the same way as you see π, in the sense that they’re “in” things. Or mathematical representations of things. Plus, you never see a circle “visually”; you only see approximations.
You see what a circle is, its essence, mathematically by understanding π, and some other math.
Same thing happens with complex numbers, starting with very humble quadratic equations. In order to solve half of them, complex numbers are needed.
Not only are complex numbers odd, and π, but so are a host of other numbers called “real”. Between any interval you pick, say 0.01 and 0.02, there are an infinite number of numbers, most of which are like π and have digits that go on forever.
There is an excellent book on all this, although it’s not easy going unless you have some math under your belt. James Franklin’s An Aristotelian Realist Philosophy of Mathematics: Mathematics as the Science of Quantity and Structure (link).
jrm & Briggs
I found this link quite interesting
https://www2.clarku.edu/faculty/djoyce/complex/cubic.html
Mathematics reawakened in Western Europe in the 13th century. n the 13th Century
At that time works in mathematics were translated from the Arabic into Latin allowing Western European scholars to learn about the medieval Arabic-language mathematics and the older Greek mathematics, such as Euclid’s Elements. In all this mathematics, only positive numbers were considered to be numbers. Negative numbers were not yet accepted as entities.
So, according to this source, negative numbers themselves were the first leap! In that day, the “revelation” of negative numbers would have been seen as an invention (at least in the West – you can “imagine” people thinking of “negative” values as imaginary as well!)
Briggs and John B()
Ah! Thanks. I begin to understand.
jrm
(Briggs, Sheri, Simon, Sander, Bob K Jan Van, …, everybody!)
Thank you!
Been the most fun and illumination [post and comments] that I’ve experienced in a long time
jrm, et. al.,
If you want to challenge your grey matter, by reviewing math, then Eddie Woo can be your guide. His latest (FEB / MAR 2021) YOUTUBE videos help to demystify imaginary (or) complex numbers for his trigonometry students. This is an exceptionally useful resource for the homeschooling set. Eddie Woo https://www.youtube.com/channel/UCq0EGvLTyy-LLT1oUSO_0FQ . Sometimes I wince (just a little) when he dabbles in statistics – but Mr. Woo is heads and tails above your average HIGH SCHOOL MATH TEACHER.
Here’s a prime example, starting at about the 6 minute mark,
it shows evidence of Pascal’s Triangle throughout the world, time and cultures
https://www.youtube.com/watch?v=gMlf1ELvRzc
(Jan – Thanks for Eddie Woo)