Posts tagged: math
Klint Finley: What does it mean to be a (or, rather THE) “mathpunk”?
Tom Henderson: Ha! Okay. When I was maybe 20 years old, my high school girlfriend was telling me about a punk band called “Green Dave.” I told her that I found punk to be totally unimpressive, because it was a musical genre that, near as I could tell, was founded upon not knowing how to play your instrument.
She set me straight. The point of punk, she said, was that ANYone could get the experience of being in a band, of performing in front of peers, of expressing yourself, without there being a prerequisite to participate.
This blew my mind, and it was that conversation that turned me from a nascent douchebag into a self-aware poser.
Later, a girlfriend who had honest-to-god Southern California punk credibility — this was the time that The Offspring was getting radio play so, what, she was probably most deep in the hardcore scene? — got me interested in the music, and explained to me that punks could be astronomers or Shakespeare devotees with no clash. (Pardon the pun.)
So, these things are tucked into my brain. Later, I move to Portland. I move to Portland with the extensive plan of “take math classes until head blows up, or degree achieved.”
This is the first serious long-term plan I’ve ever had. I figure, Shit, I’m a guy with long term plans now? I need to re-roll my character sheet. I start with appearance (self-aware poser), and ramp up the mathematical angle, to cobble together a philosophy of punk rock mathematics.
It is this:
1) People use the average Joe’s poor mathematics as a way to control, exploit, and numerically fuck him over.
2) Mathematics is the subject in which, regardless of what the authorities tell you is true, you can verify every last iota of truth, with a minimum of equipment.
Therefore, if you are concerned with the empowerment of everyday people, and you believe that it’s probably a good idea to be skeptical of authority you could do worse than to develop your skills at being able to talk math in such a way that anyone can ask questions, can express curiosity, can imagine applying it in the most weird-ass off-the-wall ways possible.
This does not entirely mesh well with the actual practice of learning mathematics, because that is mostly time spent alone or in small groups being very very confused almost all the time, but it’s still the bullseye I keep in mind.
You know, it dovetails with the improv comedy thing… In improv, I’m guided entirely by audience reaction. It’s possible to improvise toward interest in a mathematical discussion in roughly the same way.
Full Story: Technoccult: The Philosophy of Punk Rock Mathematics – Technoccult interviews Tom Henderson
Klint Finley
For all intents and purposes, “the fifth line” was a code for asking whether one was Jewish or not. (People of other nationalities, like Tatars and Armenians, against whom there were prejudices and persecution—though not nearly on the same scale as against the Jews—were also picked up this way.) My “fifth line” said that I was Russian, but my last name—which was my father’s last name, and clearly sounded Jewish—gave me away.
Even if I hadn’t been using my father’s last name, my Jewish origin would have been picked up by the admissions committee anyway, because the application form specifically asked for the full names of both parents. Those full names included patronymic names, that is, the first names of the grandparents of the applicant. My father’s patronymic name was Joseph, clearly Jewish, so this was another way to find out (if his last name weren’t so obviously Jewish). The system was set up in such a way that it would pick up those who were at least one-quarter Jewish and everyone of those was classified as a Jew, much like it was in Nazi Germany.
Having established that by this definition I was a Jew, the woman said:
“Do you know that Jews are not accepted to Moscow University?”
“What do you mean?”
“What I mean is that you shouldn’t even bother to apply. Don’t waste your time. They won’t let you in.”
I didn’t know what to say.
“Is that why you sent me this letter?”
“Yes. I’m just trying to help you.”
Full Story: The New Criterion: The Fifth Problem: Math & Anti-Semitism In The Soviet Union
Tom Henderson, author of the forthcoming book Punk Mathematics, will keynote EsoZone Portland 2011 on November 18th at p:ear. Admission is free. Tom’s talk is tentatively titled “Time, Space, and the Self are Illusions - So Do ‘You’ Wanna Go ‘Out’ with ‘Me’ ‘Tonight’?” He’ll cover:
Tom has a masters in mathematics from Portland State University. According to the Kickstarter page for his book:
Punk Mathematics will be a series of mathematical stories. It is written for readers who are interested in having their minds expanded by the strange metaphors and implications of mathematics, even if they’re not always on friendly terms with equations. Better living through probability; the fractal dimension of cities and cancers; using orders of magnitude to detect bullshit; free will and quantum economics; and the mathematics of cooperation in a networked world on the brink of a No Future collapse.
For more on Tom, you can follow him on Twitter, read the Technoccult interview with him or listen to this interview on the Acme Science podcast Strongly Connected Components.
EsoZone Portland 2011 will take place over the course of November 18th and 19th. It will include a few pre-scheduled presentations, workshops and performances along with ample free space for ad-hoc “unconference” sessions in the style of BarCamp or Bird of a Feather.
Watch this space for more announcements.
There’s a new interview with Tom Henderson (aka Mathpunk) on the podcast Strongly Connected Components. Tom talks about numeracy, his teaching style and whatever happened to Math for Primates.
Strongly Connected Components: Tom Henderson
My interview with Tom is here.
After I posted that article about technical analysis a couple people commented that it reminded them of the film Pi, about a renegade mathematician somehow using Pi to search for patters in the stock market with a homemade supercomputer in his crummy Manhatten apartment.
Technical analysis was probably the inspiration for the stock market portion of the film, but did you know that the part about renegade mathematicians building supercomputers in their living rooms to calculate Pi is actually based on a true story? Aronofsky almost certainly took the inspiration from this 1992 New Yorker story:
Gregory Volfovich Chudnovsky recently built a supercomputer in his apartment from mail-order parts. Gregory Chudnovsky is a number theorist. His apartment is situated near the top floor of a run-down building on the West Side of Manhattan, in a neighborhood near Columbia University. Not long ago, a human corpse was found dumped at the end of the block. The world’s most powerful supercomputers include the Cray Y-MP C90, the Thinking Machines CM-5, the Hitachi S-820/80, the nCube, the Fujitsu parallel machine, the Kendall Square Research parallel machine, the NEC SX-3, the Touchstone Delta, and Gregory Chudnovsky’s apartment. The apartment seems to be a kind of container for the supercomputer at least as much as it is a container for people.
Gregory Chudnovsky’s partner in the design and construction of the supercomputer was his older brother, David Volfovich Chudnovsky, who is also a mathematician, and who lives five blocks away from Gregory. The Chudnovsky brothers call their machine m zero. It occupies the former living room of Gregory’s apartment, and its tentacles reach into other rooms. The brothers claim that m zero is a “true, general-purpose supercomputer,” and that it is as fast and powerful as a somewhat older Cray Y-MP, but it is not as fast as the latest of the Y-MP machines, the C90, an advanced supercomputer made by Cray Research. A Cray Y-MP C90 costs more than thirty million dollars. It is a black monolith, seven feet tall and eight feet across, in the shape of a squat cylinder, and is cooled by liquid freon. So far, the brothers have spent around seventy thousand dollars on parts for their supercomputer, and much of the money has come out of their wives’ pockets. […]
Pi is by no means the only unexplored number in the Chudnovskys’ inventory, but it is one that interests them very much. They wonder whether the digits contain a hidden rule, an as yet unseen architecture, close to the mind of God. A subtle and fantastic order may appear in the digits of pi way out there somewhere; no one knows. No one has ever proved, for example, that pi does not turn into nothing but nines and zeros, spattered to infinity in some peculiar arrangement. If we were to explore the digits of pi far enough, they might resolve into a breathtaking numerical pattern, as knotty as “The Book of Kells,” and it might mean something. It might be a small but interesting message from God, hidden in the crypt of the circle, awaiting notice by a mathematician. On the other hand, the digits of pi may ramble forever in a hideous cacophony, which is a kind of absolute perfection to a mathematician like Gregory Chudnovsky. Pi looks “monstrous” to him. “We know absolutely nothing about pi,” he declared from his bed. “What the hell does it mean? The definition of pi is really very simple—it’s just the ratio of the circumference to the diameter—but the complexity of the sequence it spits out in digits is really unbelievable. We have a sequence of digits that looks like gibberish.”
Since the publication of that story, the Brothers Chudnovsky have apparently turned their attentions to applying their expertise in supercomputing to other domains. Richard Preston, author of the original piece, wrote a follow-up for the New Yorker in 2005.
You can learn more about them on this NOVA page.
Reading all of this reminded me of a story I read earlier in the week about someone who claims to have “cracked the code” in Plato’s writings:
The hidden codes show that Plato anticipated the Scientific Revolution 2,000 years before Isaac Newton, discovering its most important idea – the book of nature is written in the language of mathematics. […]
However Plato did not design his secret patterns purely for pleasure – it was for his own safety. Plato’s ideas were a dangerous threat to Greek religion. He said that mathematical laws and not the gods controlled the universe. Plato’s own teacher had been executed for heresy. Secrecy was normal in ancient times, especially for esoteric and religious knowledge, but for Plato it was a matter of life and death. Encoding his ideas in secret patterns was the only way to be safe.
Manchester University: Science historian cracks “the Plato code”
(via Social Physicist)
Tom Henderson, aka Mathpunk on Twitter, is a mathematics lecturer at Portland State University and an improve comedian with the group The Light Finger Five. He edits mathpunk.net and is co-host of the podcast Math for Primates (with scientist and professional weightlifter graduate student and competitive weigh lifter Nick Horton). He received the Pandora Award (Bronze) from Chris DiBona, Open Source Program Manager for Google, for his participation in the game Superstruct.
Klint Finley: What does it mean to be a (or, rather THE) “mathpunk”?
Tom Henderson: Ha! Okay. When I was maybe 20 years old, my high school girlfriend was telling me about a punk band called “Green Dave.” I told her that I found punk to be totally unimpressive, because it was a musical genre that, near as I could tell, was founded upon not knowing how to play your instrument.
She set me straight. The point of punk, she said, was that ANYone could get the experience of being in a band, of performing in front of peers, of expressing yourself, without there being a prerequisite to participate.
This blew my mind, and it was that conversation that turned me from a nascent douchebag into a self-aware poser.
Later, a girlfriend who had honest-to-god Southern California punk credibility — this was the time that The Offspring was getting radio play so, what, she was probably most deep in the hardcore scene? — got me interested in the music, and explained to me that punks could be astronomers or Shakespeare devotees with no clash. (Pardon the pun.)
So, these things are tucked into my brain. Later, I move to Portland. I move to Portland with the extensive plan of “take math classes until head blows up, or degree achieved.”
This is the first serious long-term plan I’ve ever had. I figure, Shit, I’m a guy with long term plans now? I need to re-roll my character sheet. I start with appearance (self-aware poser), and ramp up the mathematical angle, to cobble together a philosophy of punk rock mathematics.
It is this:
1) People use the average Joe’s poor mathematics as a way to control, exploit, and numerically fuck him over.
2) Mathematics is the subject in which, regardless of what the authorities tell you is true, you can verify every last iota of truth, with a minimum of equipment.
Therefore, if you are concerned with the empowerment of everyday people, and you believe that it’s probably a good idea to be skeptical of authority you could do worse than to develop your skills at being able to talk math in such a way that anyone can ask questions, can express curiosity, can imagine applying it in the most weird-ass off-the-wall ways possible.
This does not entirely mesh well with the actual practice of learning mathematics, because that is mostly time spent alone or in small groups being very very confused almost all the time, but it’s still the bullseye I keep in mind.
You know, it dovetails with the improv comedy thing… In improv, I’m guided entirely by audience reaction. It’s possible to improvise toward interest in a mathematical discussion in roughly the same way.
In a nutshell, what is the problem with math education in the US?
I have no idea. Let me instead describe the attitude that students have that is problematic, and you can reconstruct what must be wrong with it from that angle.
"Show me the steps."
Many students want teachers to “show me the steps.”
They want a sequence of steps that they can perform that will give them an answer. This is not unreasonable; they know that their performance on exams, and therefore their performance on the All-Seeing Grade Point Average, is largely determined by being able to Do The Steps.
But “The Steps” are cargo cult mathematics.
The Steps are seeing the sorts of symbols that count as “right”, and trying to replicate that dance of steps. It turns out that the easiest thing in the world is to look at a student’s work, and tell the difference between “Knows what’s going on, made mistakes and dozed off” vs. “Can memorize steps, has no idea what’s going on.”
Now, the way that I explain mathematics, it sort of looks like I’m torturing the poor bastards. I handwave. I refer to certain groupings of symbols as “Alphabet soup” and write it down as a wild scribble with one or two symbols around it.
Because I’m trying to avoid showing The Steps and instead show them enough of The Idea that they can reconstruct what the steps MUST be.
Many students want to know the formulas, so that they can float them on top of their short-term memory, ace the exam, and then skim them off. Why do they want to know that?
Probably because, for their entire mathematical careers, math has been a sequence of Steps, and if they get them wrong, they get red pen, bad grades, No No No Look What You Did. Plus, bonus, there is no apparent relevance of these algorithms other than To Get The Answer.
What’s wrong with math education in the US? What’s wrong is, Whatever it is that makes my students uninterested in learning any more math than is required to minimize feeling stupid.
So that we’re clear, lots of my students are totally awakened to the interesting weirdnesses of mathematics. But, it takes some doing, and I can’t do it by myself. Hence the podcasts and the lunatic twitter stream and the plans for TV shows and online games and godknowswhat else.
I’m trying to get across that if you are highly motivating, if you have a high degree of fire and “Fuck yeah!” and “What, that’s impossible, but true!”, you can get students to express interest in theorems named after dead Hungarians.
I’ve always been “bad at math” (and things I see as related: chemistry, physics, mechanics, etc.) Is there any hope for me (for example, have I just had bad math education in the past?), or is it an unchangeable function of how my brain works?
That’s the real question, isn’t it? And I’m totally unqualified to answer it because I’m “good at math.” I tell students that “Math will wait for you until you are ready.”
One of the best Einstein quotes in this regard is the one where he says, “It’s not that I’m so smart; it’s that I stay with the problem longer.”
Well, have you had students who have been able to turn that around? Go from being “bad at math” to being really into it?
Yes.
Let me tell you a theory about math knowledge. A mathematical concept can be expressed in symbols (algebra), in pictures (geometry and diagrams), verbally, and numerically. This is a common theory; my additional spin is that math knowledge also exists as a performative concept. Like, the way that I direct the attention of the students (“If you ignore this alphabet soup for a minute, you can see it’s really just a product of two things…”) Or, the way I will use physicality. Like, the other week, I climbed onto the chair and then onto the desk while I was trying to explain slope.
ANYway, the theory goes that you don’t understand a mathematical concept until you understand it in TWO modalities. I do very well with visual knowledge, so my notes of understanding are full of color and pictures and mindmaps and arrows linking concepts, and I highlight the holy hell out of math books. However, I don’t believe I KNOW a concept until I can explain it verbally, because I can barely understand anything if someone just talks it at me.
First swipe is through my best modality, second swipe is through my worst modality. The whole “learning style” thing may be overstated, but it remains true that getting students to understand things in a variety of modalities seems like the way to go.
Maybe they don’t get the picture. So you ask them many verbal questions. (Questions, not explanations, 99% of the time.)
I don’t know if you saw the article I posted here at Technoccult a few weeks back, but it looks like the whole "learning style" thing is complete bunk.
Which makes sense, I mean who really “learns best” by having someone lecture at them for hours or reading a book with no illustrations anything?
There may be some people who CAN learn that way but I don’t know if anyone really learns best that way.
But yeah, multiple modalities always seems like a good way to go.
Sure, maybe. But as a teacher I have to do something, and those somethings may as well be grouped by, “What things do I need to prepare? Should I work out a lot of pictures, a lot of numerical ‘recall this fact…’, a strong narrative for the problem at hand?”
You know? It’s like… all this shit is imaginary.
Mathematics is like unicorn anatomy. You imagine this thing, and it doesn’t exist, yet it still comes with facts. I know how many legs a unicorn has.
So, if you’re trying to imagine a thing that doesn’t exist you can use multiple modalities like tweezers — “The thing isn’t a picture, but here’s what a picture of it would be like. It’s not a verbal thing, but here’s the best we’ve got.” The real thing is the underlying Platonic concept.
Post-Platonic? Now I wanna go make Xeroxes of Plato with his eyes X’d out. Thanks, punk-rock atemporality!
Whoa, tangent.
Let me expand briefly on one thing I know is wrong, and I hope that a networked learning environment can fix.
Networked learning might — might! — solve the example problem. Students need familiar examples, but what is familiar is different to different students. I’m hoping that we can teach the web to send people the right example.
You’re the co-host of the podcast Math for Primates. What’s the purpose of the podcast, and who is your target audience?
Math for Primates started from the concept that there are certain things that humans are always interested in. Really, they like other humans. That’s the best thing. The internet used to just be a box with text, but once there was a critical mass of social information on it, it was a box with people inside! We love looking into boxes that have people in ‘em!
So, the concept I pitched to Nick was, “Let’s talk about math from the platform of ‘Math that humans are likely to want to know, because it’s about other humans’”
Social conflict. Sex. Beauty.
It gives us an excuse to talk extensively about game theory. And, game theory is a key place to teach humans mathematics, because we seem to have some optimized “cheat detection” in our brains.
Let me give you an example, it’s something like, uh…
There are four face-down cards on a table. There is a rule: “If the number showing is even, then the back of the card MUST have a vowel.”
Now, given an E, 3, 8, D, what is the smallest number of cards you need to flip over to verify that the rule is being followed?
Maybe I fucked up the puzzle. But, anyway, the answer as I’ve phrased it is NOT E and 3.
You need to make sure that 8 has a vowel on the back, and you need to make sure that D does NOT have an even number on the back.
Everyone gets this wrong, basically. Well, non-mathematicians always do, and I’m pretty sure I got it wrong because I get every answer wrong on the first try. Punk as fuck.
Now, if you ask the same people a logically equivalent question: “You see four people. Two are drinking beer and two are drinking coke. Whose IDs do you have to check?”
No one says you have to check the ID of the coke drinker. Because who cares how old they are? If it’s the same puzzle, but phrased as a problem of possible social cheating, we nail it.
Wow. That’s interesting.
This is interesting to us. We think it’s fascinating that, given just a change of context, people can do logic puzzles more effectively.
So! We believe that if we put the context of mathematics into social situations, and maybe some other human-centered situations (like, we want to talk about group theory, but we will try and make it about “Symmetry” because that’s something that human eyes will pluck right out).
I have to say, that your podcast has made Game Theory seem a lot more approachable to me. I used to think of it as something that was mathematical and scary. And I guess it’s still mathematical, but it seems entirely approachable and not scary.
Precisely!
The thing about math is, you can only answer yes/no questions. How many questions in life are really just yes/no and not “it depends”? Very few. So, the problems that we can attack in mathematics must be very simple indeed.
It’s just that they have a large number of component parts sometimes, because we are trying to build a complex and nuanced model out of stuff that is so simple that it admits a “yes/no” answer, always.
We are talking about putting together an entire mathematical text starting from game theory as the first principles.
Start with relatively simple social games that we can understand. Simplify them until they admit mathematical analysis. Now, introduce the minimal set of tools to solve this problem.
That’d be great. Because what still scares me away game theory is knowing that most texts are still probably going to be incomprehensible to me.
They may well be. Don’t get me wrong, the learning curve is always steep. I tell my students, “You say you’re bad at math, but the truth is, HUMANS are bad at math.”
It takes a lot of quiet reflection to make any of it make sense.
So, our target audience is, humans. But, only humans who are willing to be surprised and confused, and who think that paradox is something to be explored rather than fled.
But not necessarily humans who already have a strong background in math.
Heavens no. We have been referred to as taking the “Beavis and Butt-Head approach to higher mathematics.” And we are very proud of that. This was coming from someone who hope to have on the show soon, with a doctorate in mathematics and a grown-up job and everything.
On to something completely different… Can you tell us a little about superstructing and how you got involved with it?
Deep cleansing breath.
Sure.
Superstructing means to build upon something that is already there, right? To extend a structure, build on top of existing structures.
But, when Jane McGonigal and the Institute for the Future use it, they mean something pretty precise: “Superstructing: A new way of working together, at extreme scales, supported by game platforms and mechanics.”“
"Extreme scale" means that an individual working alone for 5 minutes should be able to contribute to a project. But it also means that in principle you might be taking on some enormous problem space to explore collaboratively. And you’ll need hundreds of person-hours. The game platforms and mechanics provide the support. If you define some huge problem… ok, what do you do?
The designer of a good and superstruct-y game-for-good will have clear missions, things that you can do, ways to compete and cooperate. For points, for gear, for social status, whatever. The idea is that you can use game mechanics to extend human capabilities so that they are able to achieve goals that previously it would take a whole institution to do but you do it in such a way that you can also extend the power of extant institutions with the networked abilities of social primates.
The individuals form a network and get stronger. The institutions get large numbers of humans thinking and sharing and communicating, and get smarter. You’ve superstructed, built on what is there.
How I got involved: I’ve been wrestling with knowledge management for years. I have several linear feet of journals full of mindmaps, but, y’know, you can’t grep dead trees. Back when I was trying to use a file cabinet for knowledge management (ha ha ha ha) I tucked the printout of an NPR transcript (ha ha ha) into a file marked “Ludology”, because I was getting interested in play and games. It included an interview with performance and games researcher, Jane McGonigal. I’m pretty sure that this must have been after ILoveBees, the ARG she designed for the Halo launch.
Anyway, serendipity led me to clean out the hideous file cabinet, I see the Ludology file, I check to see what this McGonigal person is up to, and I find her New Yorker talk.
TOTAL HEAD EXPLODEY
I suddenly felt totally okay about playing EverQuest for three years and stacking up pizza boxes to my sternum, because, hell, there are lots of ways of getting in worse trouble living off Hollywood Blvd when you’re 23.
"Ah ha! I was doing research on early gameplay and networked collaboration! How wise of me!" And, lo and behold, she was starting a new game called Superstruct.
I played the game, drank the Kool-Aid, got the t-shirt. (I’m wearing it now, in fact.)
So what did you do as part of Superstruct?
I wanted to simultaneously win the game, and help it realize the potential I saw in Jane’s Big Idea. The first problem was that the interface was very bad. It would log you out constantly, it was hard to search, it was hard to keep track of what you had done so that you could nurture it.
It was totally gorgeous, the design was beautiful, but the functionality was not what the really hardcore lunatic Super-Empowered Hopeful Individuals (SEHIs) needed in order to shine. Filter failure, basically.
I remember taking a look at it from time to time and giving up within minutes of hitting the site. I couldn’t figure out how the hell to participate.
Right, that’s because when Global Extinction Awareness System released the report, it woke up a swarm of No Future assholes who did their best to disrupt the site. (Possibly government operatives were involved; one storyline got a researcher in jail due to the riots after the report came out. [No one was arrested in real life - .ed])
So, I decided that the interface itself was like our first boss fight. In-game, there was a story line of all the shenanigans that dedicated hackers and griefers can do…
So, obviously, the SEHIs needed to save the project by duplicating efforts on more resilient networks. I did not have the technical skills necessary to do exciting things, so what I did was tried to locate anyone who might help the interface be improved, and do the best social engineering I could manage.
Foundation (who is in Portland) wrote a screen-scraper that would relog in as often as necessary, so he could scrape the site and get interesting information.
He was able to use this tool to send mass messages; I suggested that what we needed was to wake up the SEHIs who were clearly interested but maybe turned off by the site.
So, we identified all SEHIs who had a minimal amount of activity (“Has joined a superstructure”) and sent them a “secret” message. Basically, we told them how bad ass they were (“You are the CORE SEHIs”), and where they could find additional off-site resources.
That’s the thing that I was really proud of. The project wouldn’t be good without lots of active people, and we did what we could to try and maintain excitement and intrigue in the face of a somewhat boring “There are no RSS feeds!” obstacle.
I also delivered an address from Open Source Scientists which people liked a lot. That was fun. I felt like people weren’t bringing their A game, so I basically told everyone, “I’m offering a resource as a prize for you to do something, and I think I will win this game even if I give you that prize.” It was cocky and snarky, and I got to show off my alarmingly large forehead.
What I really wanted to do was some data visualization so that we could reduce redundancies. Lots of people had really great solutions, but some of those solutions were duplicated.
I envisioned hundreds of superstructures circling each other like marine organisms, infecting and eating and mating with each other. Alas, I did not have the skills, nor the data. So as to remedy that I’m teaching myself Python and regular expressions for the next data analysis project that arrives.
Once I know what I don’t know about social network analysis and random graph theory and data mining, I’ll have a clear path toward datamancy: being able to convert information on what people are doing into game-able decision points.
Really, I just want to be able to look at people doing cool interesting things collaboratively, through a lens of computation, make a pretty picture, and strategize from there.
Maybe it will even work!
Sounds good. I think we can call it a wrap unless you have more…
Just one more thing. Tell everyone to go sign up for Evoke!
Will do! I think I’ll give Evoke a shot this time.
From what I can tell, it’s got the secret sauce from Superstruct, packaged in a way that will make a jillion times more clear how to participate. (Jillion being a technical math term.) And, it’s about resilience, entrepreneurship, and helping other primates — it’s what the world needs!
How many zillions in a jillion?
A jillion is a squintillion with a zillion zeroes at the end. Glad I could clarify that.