About Us
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Hypothesis is a New York City based company that creates games and stories to invite imaginative thinking about unsolved problems.
Hypothesis brings together artists, educators, mathematicians, scientists, software developers and other visionaries and doers. We aim to transmit knowledge between people with elite mathematical information and curious people working beyond prescribed culture, for the benefit of both parties. By working with small groups of friends who come from a variety of different perspectives, we accomplish this more fluidly than is possible for rigid institutions.
Testimonials
Some things folks have said about our director’s related work:
“To say I enjoyed this program is an understatement. It was genuinely incredible. I feel like it taught me to think in a whole new and different way.”
— Caleb, student (11th grade) from Expedition given through Columbia University
“It’s definitely worth it. Even if you think you won’t be that interested the concepts might surprise you.”
— Student (9th grade) from Expedition given through BEAM
“We got to learn about concepts and ways of thinking that we wouldn’t learn anywhere else.”
— Yuki, student (grado 8) from Quinn’s Center for Talented Youth course
“Joe is able to see the world from the perspective of those with little to no mathematical comfort. He can break down the basics and present them in ways that struggling students can understand and use.”
— Alex Ott, Associate Dean, Bronx Community College
“It’s hard to make things like ... the fundamental theorem of algebra fun for a 14-year-old in the middle of his or her summer vacation, but Joe can and has. He is an exceptional educator and I cannot recommend him highly enough.”
— Joshu Fisher, Program Manager, Center for Talented Youth
© 2020 Hypothesis LLC
Newsletters
Boletín
Past Newsletters:
Boletínes Anteriores:
- 2025 January 12 — The Myth of AI
- 2024 October 31 — Haberdashers and Hauntings
- 2024 June 13 — Software Engineers Wanted
- 2024 January 18 — The Difficulties
- 2023 October 9 — Progress
- 2022 September 8 — The Banishment of Dagon
- 2021 September 5 — Solar Bound
- 2021 May 23 — Aliens in the News
- 2021 April 11 — The Shapes of the Numerals
- 2021 February 22 — The Moon & Our Hands
2025 January 12 — The Myth of AI
On Friday February 21 from 7:00 to 8:30pm, I will present Banisher at the Every Thing Goes Book Cafe, located at 208 Bay Street in Staten Island, NY 10301. I will show some of the hand-made sets, teach how to play the game, and invite feedback about the game’s psychospiritual theme. There will be a $5 suggested donation.
A common interest held by myself and the organizer of the above mentioned event is a humanistic push-back against artificial general intelligence. Following are some of my thoughts on this topic, formulated before having made this serendipitous connection.
There is no such thing as artificial intelligence. Intelligence can only emanate from consciousness, and consciousness cannot embed in a discrete medium because consciousness requires a continuum. Consider the analogy of sand on a beach: with enough particles, the beach takes on dynamic properties that are better explained in terms of the forces acting upon it, but at no point does the sand acquire an individuality.
The myth of artificial intelligence is a result of corporate influence over an internet that has vast information, and the use of that influence to weaponize paranoia. The application of this technology to large language models and generation of images results in an emulation of human creativity which is necessarily incapable of actual creativity. Synthetic “intelligence” is not taking over; rather, tasks that used to be performed by human beings are being assigned by human beings to automation. You have a role in choosing whether or not to automate tasks, and whether or not to be affected by automation. You are responsible for how you direct your attention.
If it seems like actions taken by an automaton are directed by a conscious agency, i.e., that an “AI” has passed your Turing test, the cause is one of the following: either targeted marketing has successfully compartmentalized your personality, or you are experiencing synchronicities while interfacing with an electronic device. Regarding the latter (lesser understood) possibility, synchronicities occur on or off electronic devices and with or without prosaic explanations. They are typically seen more in the places where you look for them. If you are interested in maintaining a personality impervious to corporate control, and feel susceptible to being manipulated by “artificial intelligence”, I recommend looking for and then interacting with synchronicity using methods whereby automation is not among the possible explanations, and keeping track of your findings by some personal and private means.
Have fun. Keep thinking. Stay curious.
2024 October 31 — Haberdashers and Hauntings
For Halloween, I bring you musings about mathematical models for visits from the spirits world, but first some updates.
A software engineer for Gardens of Danu has been contracted. This was another fortunate outcome of connections made during my attempt to unionize MoMath. Maybe more people should try that. Anyway, the vision continues to coalesce.
I’m up to eight completed handmade Banisher sets with eleven more to go. I’ve made no progress toward commodification and have doubts about whether a mass-producible version is a good idea. While the original goal was to interest arithmaphobic people in group theory, the game has been more successful with interesting arithmaphilic people in its psychospiritual properties. Much of the value comes from the sets being handmade. Meanwhile, my non-Hypothesis work has been relevant to Hypothesis’ goals while keeping my financial situation stable.
Now let’s get to the spooky stuff.
Haberdashers
The Haberdasher puzzle is about dissecting a shape into smaller pieces and building another from the pieces. It’s a well exposited topic with recreational value that also motivates modern research in topology. I encourage you to research and explore this as well as anything following for which you’d like more explanation; I’ll gloss over details that are easy to find.
Here I focus on something I haven’t seen elsewhere: an analogy between one-dimensional dissection puzzles and quantum theory as an entry point for mathematical modeling of the paranormal. Unlike much modeling for spookiness, this concept does not involve higher-dimensional geometry nor extrasensory perception (perhaps topics for another time).
Commensurability
We’ll need the mathematical concept of commensurability (there are others), and it will be most useful to focus on positive numbers thought of as lengths of line segments. For one length to be commensurable to another means that they are each a finite multiple of some common length. For example, 4 and 6 are commensurable to one another because they’re both multiples of 2.
For natural numbers, this idea is trivial because any pair of natural numbers is a multiple of 1. If we add fractions to the mix, nothing changes: we can always find a least common divisor of a pair of rational numbers. For example, 2/3 and 5 are both multiples of 1/3. The idea gets interesting, and got very interesting historically, with the consideration of lengths as measurements of shapes in space.
As the Pythagoreans discovered, a square’s diagonal is incommensurable with its side length. For example, the square root of 2 is incommensurable with 1. Thus it’s impossible to write down certain distances as ratios. This challenged the Pythagorean belief that physical reality is a mere shadow of a numerical Platonic ideal. The discovery was occulted and legend has it that someone was murdered for divulging it.
About 2500 years later, a rigorous theory for conceiving of non-commensurable quantities as lengths was a major project for the European intellectual community, leading to further upheavals of long-held beliefs, including proof that certain questions about infinitesimals are independent of mathematics. Perhaps the community was obsessed with having continuity in the 1-dimensional number line. Perhaps this headed mathematics down a rabbit hole of counterintuitive abstraction.
Regardless, the outcome of the work is fascinating. We have a rigorous method for measuring infinite sets, and some are greater than others. There are infinitely many lengths which are pairwise incommensurable to one another, for instance square roots of prime numbers. Among the transcendental numbers, there’s an even greater infinitude of commensurability classes. From exploring the algebra of how these strange numbers interact, we have ring theory and field theory.
But does this have anything to do with physical reality?
Quantum Physics
Quantum physicists tell us there’s a set of tiny units, including ones for distance and time, such that at a more granular scale, classical mechanics ceases to apply. At a distance less than the Planck length, quantum gravity takes precedence and things that otherwise seem like particles instead seem like waves. At a duration less than the Planck time, linear continuity breaks down and there’s no consensus on the nature of causality. The best we can do with this empirically is use probability to manage our ignorance.
Our physical bodies, and hence our physical apparatuses for perceiving the material world, are subject to the same conditions. Whatever happens at the quantum scale, it synthesizes into patterns that we understand in everyday life as objects occupying space and moving through time on a continuum. This understanding applies as long as we don’t transgress the quantum lower bounds.
I’m not an expert on quantum physics, but I’ve thought about what one can do mathematically on a naive model and how that relates to some math developed contemporaneously with the foundations for the physics.
Hauntings
Let’s leave aside the question of whether or not we perceive quantum information by some extrasensory mechanism, and explore the idea of information coming from a discrete set of perceived infinitesimal moments. Let’s start with a simplified model where there’s one person observing moments of time at regular intervals of a fixed length which I’ll call T, and let’s imagine sensory information originating as waves.
Since there are infinitely many numbers less than T, there are infinitely many moments in between every pair of moments observed, making ample room for imperceptible phenomena.
A wave that’s at its crest at every moment of the person’s perception would look like a steady flow of energy. Maybe the person sees a table but when they’re not looking the table fades out of physical reality. A wave that’s always at equilibrium at the moments of perception would be imperceptible; now we’ve got supernatural furniture.
More generally, a wave with wavelength commensurable to T would appear periodic to the person. (This is like including the rational numbers.) The person wouldn’t see the wave’s actual behavior, and there’s a variety of possible observed behavior depending on the relationship between the wavelength and T, but the observed effect of the wave would follow some pattern, making it possible for the person to predict and comprehend it. I’m thinking of waves like this as creating the person’s natural world.
Consider a wave with period incommensurable to T. (This is like including the transcendental numbers.) Even though the wave is periodic, the person would perceive it as a patternless sequence of states. Moreover, depending on the period, it could take any apparent sequence of chaotic states relative to the moments of perception. There’s an uncountable infinitude of ways for this to happen. I’m thinking of waves like this as creating the person’s supernatural world.
In between the person’s moments perception, the waves would interact, including interactions at the quantum scale. I’m thinking of this as where different manifestations of realities could transfer information to one another, providing a mechanism for communication between the natural and supernatural.
Further Speculation
The algebra of this is provided by German ring theory. I’m not saying this was done to model alternative realities suggested by the foundational quantum discoveries, but it was developed contemporaneously with those discoveries. If one wants to know what discrete patterns can be formed from interacting commensurability classes, one will find it there.
I was taught German ring theory by the grumpy French-American professor who later became a friend, John Loustau. He wrote on the board in the traditional German gothic script and interrupted his lectures with nonsequitous pearls of wisdom. I used to stay up all night doing his assignments and it felt like the shattered pieces of my mind were given a place where they could reassemble. The math gave structure to what had in the past been lunacy. When the veil is thin, I appreciate the visits from what he became after death.
To bring our thought experiment closer to a model for our physical world, one would need the set of sources to be moving, one would need the waves to interact as the observer varies the intervals between their moments of perception, and one would need a means of conceptualizing discrete information as a continuum. These things too have already been worked out via the theories, respectively, of dynamical systems, Fourier analysis, and numerical methods. If we want to think of the physical world as elastic rather than rigid, we have topology.
Optional assignment for readers interested in digging deeper: compare and contrast the definitions, and the motivations for the definitions, of the reduced Planck’s constant and the Dehn invariant. If you respond with your thoughts on this, please let me know if I have your permission to share them in a future newsletter.
Have fun. Keep thinking. Stay curious.
2024 June 13 — Software Engineers Wanted
Hypothesis will be looking to contract one or more software engineers later this summer to work on the video game Gardens of Danu. If this is of potential interest to you, email hello [at] hypothesisnyc.org and I will plan to send you a detailed job description and application toward the end of this month.
Thanks to everyone who read and passed around the last newsletter. Thanks to Jeff, Jon, Tracy and Sylvia for the responses. Thanks to John Voight (not the actor, the mathematician who wrote the bible on quaternion algebras, with open access) for the donation, which helped with morale in addition to the financial value.
This time, I’ll share the latest on Gardens of Danu (plus some tangential follow-ups to what I said last time) in hopes of attracting attention from software engineers who might be a good match for that project.
A motivation for Gardens of Danu is to head in the opposite direction from using artificial intelligence for artwork: making things that are only possible by the work of talented and creative human beings. The characters are real handmade dolls, the music is recorded from real instruments, the backgrounds are real paintings, and the other visual elements are objects from real life that are photographed, collaged and animated. The story unfolds in the form of text-based dialogues that are far more clever, nuanced and sophisticated than anything possible with large language models.
The synthesis of these elements is like a surrealist cadavre exquis drawing: each artist has creative freedom but only sees what they need to in order to do their part, and I assemble the pieces. Most of the parts are already finished, and it’s going to be awesomely weird.
The puzzle part of the game is both accessible and mathematically provocative. It’s based on a recent invention by some game theorists (TBA) at NYU, and is full of unsolved math problems. A motivation for inventing it was to find something easy to learn but with a game tree too complicated to be solved efficiently by AI. It lent itself to giving each non-player character a fixed heuristic which you can figure out how to beat once you figure out what it is. That part was playtested extensively by young students from my outreach courses at BEAM and Columbia. Then this past semester, a student (TBA) at College of Staten Island did an honors thesis with me on the puzzle game. His results are implementable to increase the difficulty for the later game stages, making for a dynamic range of challenges. So far we have almost all the NPC heuristics coded but not the latest contribution from my thesis student.
The themes of Gardens of Danu relate to conspiracy theories. Since the last newsletter included some conspiratorial thinking about mathematical information, I’ll differentiate that from what’s done in the game.
In The Difficulties, I described a pattern whereby mathematical information is limited and controlled in the USA. That’s my current working theory that explains why this country has such an internationally substandard knowledge of math among its general public, while also being home to the most competitive math doctoral programs and research jobs in the world. My theory helps me avoid spinning my wheels in a system that doesn’t seem designed to spread mathematical knowledge, and helps me find effective means of divulgation. I don’t think my theory is outlandish. I laid it out last time, so here I’ll just make one more comment in its defense: it’s much easier to rip people off if they can’t do the math, so whoever is doing the ripping off is unlikely to consider the system broken.
The conspiracy theories in Gardens of Danu, on the other hand, are outlandish. They are insane and playful, not necessarily intended to be believed, yet meticulously researched. This is a response to the commodification of information during our age of competing complicated narratives. We provide humor about things that can otherwise be confusing and scary.
Each character in the game has its own bizarre personality and tells you about some equally bizarre theory, complete with references in case you want to learn more. We want to inoculate the player with so much fringe information that they cannot take any of it at face value without applying some critical thinking and coming up with their own opinions. It’s a crash course in independent thinking so as not to be used by a predatory ideology. We cover topics such as extraterrestrial and ultraterrestrial life, breakaway technology, psyops, disinformation, drugs, gods, magic, death, and horseshoe crabs.
As far as marketing and whatnot, the target audience is... well, myself. It’s art, not a money-making gimmick, and my responsibility is to the vision. I can’t guarantee that anyone other than the creators will appreciate it, but suspect that some people will love it, and hope that it will open new ways of being curious about math and science. The fact that there exists no category of commercial product with which to compare it is a crucial motivation for creating it. If you’re going to work on this project, you should be down with this philosophy.
Feel free to write back to me with any questions or comments. If there’s something related to math, education or art that you’d like to hear my perspective on, don’t hesitate to ask. In the meantime...
Have fun. Keep thinking. Stay curious.
2024 January 18 — The Difficulties
When working on one of these newsletters, I usually start by writing about difficulties with math education that explain my motivation for starting Hypothesis, then scrap what I wrote, deciding that I should instead focus on my creative processes. This time, I talk about the difficulties. I hope to provide perspective on some of the lesser-known obstacles to the Hypothesis mission for the benefit of other people struggling with those obstacles. In part, this is a response to current discourse about manipulation of information. I’ll fill in a piece I have not seen explained elsewhere: how differences between math and science lead to different patterns in how the knowledge from those areas is controlled. The more central issues I’ll address are the ongoing pervasive dislike of math among the public in the United States, and the frustration faced by the educators who try to improve that situation. My main point is that the education system, rather than being broken, is functioning as intended as a gatekeeping device to limit and control those who acquire mathematical interest and ability. I’ll explain this, and suggest ways of dealing with that problem as a student and as an educator.
Since some of what I’ll say has an anti-academic flavor, I balance that with some pro-academic observations. It’s because I have an academic job that I’m free to speak my mind without concern for the entrepreneurial value of my newsletter. The reason I have advanced mathematical abilities is because of my academic training, and I’m not aware of a current effective alternative training method. I think that our educational infrastructure has value despite its problems, and should be renovated with a coherent view of the challenges.
I’m operating on the premise that there are parties who can and do control information, including what’s transmitted through mainstream education, to manipulate the population. If you’re lacking in background on that, I recommend the “Propaganda” track from Propaganda and Control of the Public Mind, by Noam Chomsky (1997) and if you don’t mind it being a little dirty, I recommend the “Dumb Americans” track from Life is Worth Losing, by George Carlin (2006). For a friendly intro to this mode of thought in general, I recommend the “Conspiracy theory” track from Robert Anton Wilson Explains Everything (Or Old Bob Exposes His Ignorance), by Sounds True (2001). I don’t think that everyone in a position of power is intentionally manipulating the population, but I do think that anyone with influence over the system has an obligation to be proactive rather than negligent, to help correct the situation.
I don’t worry about who exactly I’m up against; I work in defiance of them without fear. Here are some tools that allow me to do this. Trying to control people is an inherently paranoid act, so you can safely assume that anyone trying to control you is more afraid of you than you are of them, thus you can focus on what needs to be done while they go insane trying to figure out how anyone could be compassionate. People who try to control others tend not to get along with one another, making their internal cooperation ineffective. Provided you’re maintaining your networks appropriately, anyone trying to hunt you down ultimately just encounters their worst nightmare: their own projected fear. This works regardless of who the controlling party is.
Now as we approach the topic of how mathematical information is controlled in the United States, let’s start with the difference between math and science, which is well known to people working in those fields but not well known publicly.
Math is metaphysical. Apart from using pencil and paper or other technology to organize one’s work, to participate in math is to witness objective truths about the structure of ultimate reality by the power of thought. For example, take the statement “1 + 2 = 3”. One need only know the meaning of the notation to verify the statement, and can perform that verification mentally. A mathematical theory is a more interesting set of truths which, similarly, depends upon rigor rather than evidence. Such a theory transcends the material world and exists in the mental plane where causality is independent of spacetime and where epistemological certainty is possible.
Scientific truths don’t exist insofar as having observable objective certainty because they’re empirical rather than abstract. A scientific prediction can be shown to be statistically viable by evidence, but no amount of evidence rules out that it might be contradicted in the future. For example, if I put one stone in my left hand and two in my right hand, I expect to be holding three stones, but I can’t rule out the possibility that I accidentally drop them, or that one of them breaks, or any variety of things that have nothing to do with the mathematical fact that 1 + 2 = 3. Scientific theories are ideas that explain what we’ve observed well enough to make predictions. These theories are part of our process of understanding reality, and often need to be improved or replaced as new discoveries are made.
I’ll next explain how science and math are controlled differently in the United States because of the different natures of their truths. I’ll then describe how this relates to the insipidness of mainstream U.S. math curricula.
Scientific theories are susceptible to being commodified and manipulated because of their lack of objective certainty. With enough money and influence, one can obtain and publicize data that supports a theory fitting one’s agenda. Having more scientists from diverse perspectives makes for a larger pool of data from which to select. Thus, we see a lot of funding to encourage more people from diverse perspectives to become scientists. They go on to develop theories that broaden the market of information. A scientist who follows the money has a shot at a lucrative but ethically compromised career, whereas one who follows an internal moral code has a shot at a fulfilling and worthwhile life but will most likely struggle for funding.
Mathematical theories, on the other hand, because they are objectively true, cannot be changed by any amount of money or influence. So instead of manipulating mathematical information itself, money and influence are used to decide who can understand the information. This is controlled via curricular choices for school math as well as children’s books, museums, popular science expositions, et cetera. As school curriculum is the most pervasive in that list, I’ll focus on that.
Our education system presents math as an increasingly abstract subject and with no apparent purpose: an accumulation of complicated puzzles that one must solve for no reason but passing the classes. This culminates in calculus presented as a series of tedious computations rather than as the elegant theory of measuring curved and moving phenomena. Math as rigorous thinking in the mental plane is not revealed until one takes courses for which calculus is a prerequisite, even though there are more accessible entry points to the topics of those courses. Courses past calculus are only taken by people continuing in math, which generally does not include science majors. This excludes the vast majority of the population from accessing the info, and selects a small group of students with a specific skill who will access it, while also ensuring the terminus of each of those two types of students. Let’s look at the typical path in each case.
The common case is the student who doesn’t excel at math in school. Unfortunately, they’re unlikely to pursue any further advanced engagement with the subject. This is not necessarily due to any shortcoming in their potential mathematical ability; rather, it’s due to their having been presented with the material in a demoralizing manner. They come away with the perception that math is boring, intimidating, dehumanizing, ridiculous, or some combination of those things. This works across all cultural and socioeconomic categories. Even successful scientists typically experience discomfort with working in the rigorous language of pure mathematics.
Let’s now consider the rarer case: the student who excels at math in school. The token example of this is a person who diligently performs computations without concern for the motivation. After succeeding at this through the calculus sequence, they’re pipelined into competition for prestige. These students tend to also be socioeconomically advantaged and have less experience with challenges beyond academic ones, making them easier to manipulate using academically defined status as the reward. The ones who become mathematicians work in abstraction disconnected from scientific application. Their research results are then reported in a way that’s only accessible to other experts of the same niche. The experts are compartmentalized such that the scientific application of their work is unknown to them, a macrocosm of the strategy used in The Manhattan Project. There are exceptions to this and it’s a nuanced topic in its own right which I’m treating in broad strokes, but that’s the general trend. A “successful” mathematician must trust the system and disregard the work of others. Moreover, the next generation of professors come from among these people, despite their usually having received no pedagogical training. A successful research career is sufficient for a professor to maintain a university teaching position, even for a professor who habitually alienates students of service courses, and for this to happen is typical. Laziness about pedagogy is built into the culture. Thus poor teaching compounds the effect of insipid curricula, and enhances the overall effect.
The solution on which I wish to focus is: how to create interaction between real math (that is, math as metaphysics) and the general population, while avoiding the control system described above. I’ll speak to some of the roles that students and educators might play in this.
To those who are currently struggling with math in school, it may be tempting to walk away from the subject all together, but the fact is: what they’re teaching really does unlock access to myriad things. Some of it will be useful to you and some of it won’t, but it’s up to you to supplement your school experience with a real education guided by your personal enjoyment, interests and curiosity. You can and should interact with the information. You can and should participate in mathematics. Here is something that might help if you’re facing the problem of an inaccessible teacher: Calculus for the Practical Man, by J. E. Thompson.
To the math PhDs who didn’t have much experience outside academia, I gave you a hard time in this newsletter, but the fact that you got a doctorate in arguably the most challenging and unpopular topic in the country means you are resilient, hard-working, and passionate about your interests. What often happens when mathematicians discover the importance of better education is that they focus on innovative ways to present the math that they like as though they’ll be welcomed with open arms. When that doesn’t happen, and they encounter obstacles to funding and access to audiences, they get whiney. Not only is that naive, but there are already many tried and true alternative ways of teaching math that invite enthusiastic participation; they’re used in enrichment programs for wealthy children, plus a few scholarship recipients, who are already bound for academic inculcation. (They’re also used in other countries.) The type of innovations we need are ones that have broad accessibility and dissemination built into the design so that they’ll spread regardless of the dominance hierarchy. To create such a thing requires a lot of trial and error. The good news is that, if you’re a mathematician, you are hardcore enough to work on immensely difficult problems. This problem might just be difficult enough to pique your interest. Personally, I enjoy working on it.
As for those who have already self-incentivized to participate in math, I encourage you to apply your creativity to the possible uses for it. Use math to add rigor to your conversation with the universe. I can tell you as someone who got a late start doing that: it’s never too late.
The next newsletter, I anticipate, will be back to focusing on the particular projects on which Hypothesis is working. However, if there are topics you’d like to know my thoughts about, feel free to respond with suggestions.
Have fun. Keep thinking. Stay curious.
2023 October 9 — Progress
Since the last time I wrote to you, I... started a new job, got divorced, moved, fell in love, and made progress on Hypothesis projects. So I’ll focus on that last part, but this gives you some idea of why you haven’t heard from me in so long. Moving forward, I plan on sending updates about three times per year.
The new job has been a step in the right direction. The people I work with are cool, and I have time for my creative pursuits. It’s been liberating to be able to work on my projects and finance collaborations without the ethical conundrum of everything needing to be commercial.
Nothing, the picture book collaboration with Dennis Ryan, had been stuck behind a case of painter’s block and is now unstuck. Dennis, who in my opinion is the best living illustrator, expects to have the watercolor paintings that will comprise the images for the book finished by this winter. What that means for thepublication date, I’m not really sure because I’ve never done this before but I’ll keep you posted.
After about three years of experimenting with Banisher, and testing it with the 30 play testers listed in the last newsletter, the right combination of game mechanics aligned and that part is now done. The game has a competitive version and acooperative version, is for one to four players, and can be played on eleven different patterns, called “clocks”, of varying complexity. For any algebraists reading this: the clocks are Cayley graphs of non-cyclic groups with two generators of order at most twelve. One need not understand the math to play the game, but the game is challenging and is not for everybody. It’s a strategy game of complete information that involves complicated visualization. People who like that sort of thing tend to like it a lot, and commonly report that it’s unlike anything else out there.
I’ve completed five handmade sets, of which there will be a total of nineteen. The main task remaining is a lot of embroidery. I do that when taking public transportation, so if you happen to be on a subway, bus or ferry in New York City and see a tall guy embroidering yellow and blue floss into a purple grommeted piece of canvas, that’s most likely me.
These handmade sets are not for sale. They’re being lent out and placed in locations that facilitate productive experimentation, but are not commodities. As alluded to in the previous newsletter, the game has personal significance, which I’m still figuring out. My current understanding, which I realize may sound strange, is that the game taps into liminal information by combining left- and right-brain stimulation, making the outcome of a session synchronistically significant. The left brain is primed by the activity of visualizing paths and computing logical outcomes; the right brain is primed by the sense impressions from the colors, shapes and textures of the game components. This needs to be brought together with the correct thematic interpretation, which I think is mostly there but is not finished yet. It has something to do with spirits orchestrating physical events through nonlinear interaction with time. The making and usage of the handmade sets is part of the process of figuring this out and, once it’s figured out, I intend to design an affordable, mass-producible facsimile of my magical creation that’s appropriate for broader distribution.
There’ve been some Banisher play sessions. These are opportunities for folks to play the game and share their observations. Thank you to Eden Morris (of Brooklyn, New York), an artist who studies topology, for hosting a fun and well-attended play session on 2023 May 29. Thank you to the Math Club at the College of Staten Island for letting me present and share the game on 2023 October 3. If you would like to host a play session or find out if you can join one, email me at hello@hypothesisnyc.org. I am willing to travel (within reason) to your house or apartment to show the game to you and your friends.
Due to the help I’ve received with Gardens of Danu, the quality of the work done so far on this video game is extraordinary. We do not have a release date yet (once again, because I’ve never done this before).
The world of the game is as follows. Certain moons of our solar system are populated by creatures called Liralunes. Each Liralune has a collection of objects from Earth that they’ve arranged in a circle they call a garden. This garden magically generates snacks which double as game pieces for a 2-player puzzle game. When visiting a Liralune on their moon, you alternate between talking to them and playing the snacking game against them. You must advance at the snacking game to advance the conversation. You’re trying to find out who the Liralunes are, and what the nature is of non-human intelligent life. Each Liralune gives you a different set of information about this, rooted deeply in beliefs held by actual people, and often conflicting with one another. The further out you get in the solar system, the trippier the information becomes. It’s up to you to decide what’s real.
Thanks to Michael Adams, who worked on this in his spare time for free for three years, we have a graphics template and computer algorithms ready to assign to the puzzle stages. This allows each Liralune to play the puzzle game using a distinct strategy.
For each algorithm that the computer uses, we need a sequence of starting states for the puzzle so that it increments in difficulty. This requires a lot of testing. Thanks to Bridge to Enter Advanced Mathematics, who let me play test a hands-onversion of the game with groups of kids from New York for a 10- week Saturday course, we found starting states for about half of the puzzles. Thanks to Cole from Florida, who reached out looking for a fun project to work on because “high school can be boring some days” (Amen!), we have a few more of those starting states. We’re gradually working through the rest. If you’d like to pitch in on that, send me an email at hello@hypothesisnyc.org.
The audio and video elements are coming together too. Thanks to musician Dylan Sparrow, we have 23 tracks of original music assigned to the various parts of the game, which are perfect forthe epic wacky style. Filmmaker Allen Cordell has beencontracted to make animations. I’ve been a fan of Allen’s work since we went to film school together at SUNY Purchase and his approach here is as awesome as anticipated.
Last but not least, the progress that’s the most above and beyond my expectations is with the Liralune conversations. I’d been planning on writing the dialogue trees myself, then found a fantastic science fiction writer who has a greater knowledge of alien conspiracy theories than I do and who was up for the job. The dialogue trees she’s composing are both comprehensively researched and hilarious, giving each character a unique and bizarre personality. I could see someone playing the game just to read the text.
Thanks to Abhijit, Dennis, Jeff, Sylvia and Victoria for your helpful and thoughtful comments on my last newsletter. Thanks to Victoria pointing out the following: “hypnogogic” refers to states while falling asleep whereas “hypnopompic” refers to states while waking up.
Feel free to write back to me with any questions or comments. If there’s something I haven’t mentioned related to math, education or art that you’d like to hear my perspective on, or if there’s something I did mention that you’d like to hear more about, don’t hesitate to ask. In the meantime...
Have fun. Keep thinking. Stay curious.
2022 September 8 — The Banishment of Dagon
A period of focused work on Banisher began on 2021 August 22. What I thought would be a few weeks extended through September, then through the rest of 2021, then all the way to June of 2022 when I packed away the materials to get ready for my summer teaching gigs.
As the story for the game developed, I became engrossed in studying its spiritual context as a work of non-fiction. Since the game was framed as a banishing ritual to liberate us from an evil god, studying it in this way led into heavy territory, the beginning of which I wrote about in the previous newsletter. I’ll share some more about that, though there’s a lot more that I’m leaving out for now. As always, feel free to respond with any questions, comments, reactions, etc.
The basic idea for Banisher’s story came from a conversation with my friend Raymundo Jackson, a librarian from New Paltz NY, back in 2005 when I was working on a documentary about the Westboro Baptist Church. I was interviewing people asking their views on whether that group’s message of hate was consistent with Christian teachings. Of the many priests, students of theology, cult members, and other people I talked to, Raymundo’s take on it was the most interesting.
Raymundo prefaced his response by making a distinction between a religion and a belief system: that a religion is a rigid dogma imposed upon a population in order to control them, and a belief system is an ever-adapting theory on the nature of reality. His spiritual ideas were part of his personal belief system, loosely based on Taíno teachings, which he invited me to explore, critique, and borrow from as I liked. Incidentally, Raymundo passed away unexpectedly at the age of 50 on 2021 November 8. He was the kind of guy who would make time to have a mind-altering conversation with you no matter what mundane tasks beckoned his attention, and someone I’m grateful to have known.
Raymundo believed there was a problem with all Abrahamic religion, tracing back to the prophet Abraham himself. He referenced the story of how Abraham was told by his god to sacrifice his son Isaac as a test of his faith. As the story goes, just as Abraham was about to do this, he was interrupted by an angel providing him with a ram to sacrifice instead. Raymundo believed that unbeknownst to Abraham or his many descendants and followers, this god was a disguised form of Dagon (enemy of Yahweh), and the fact that Abraham was willing to murder his own son out of rote obedience showed his subjugation to this less compassionate deity. Raymundo described ancient imagery of Dagon as part man and part fish, holding a shield with a six-pointed star, and wearing a cross atop a pope-like hat which was used for impaling, and explained that this imagery secretly retained this significance in its current usage. He described how Dagon’s requirement of blood sacrifice was continuous with the many wars, the murdering and torture of martyrs, the child abuse, and the psychological oppression that have all transpired between various forms of Judaism, Christianity and Islam, which all stem from this common source. According to Raymundo, their god is real, and this god is the problem (but this god might not be Yahweh).
At the time of that conversation, I tended more toward atheism but found Raymundo’s concept useful. I had been raised Catholic in a small town, hated it, and by fourteen was blasting metal and refusing to go to church. At twenty-six, when I had that chat with Raymundo, I shared the sentiment that religion itself is the problem, not so much the individual people. I was compelled by the idea that there really are deities, but that you have to work to recognize them and not be fooled, taking into consideration the character and effects of the worship. I’d been ruminating on it off and on since then.
When I drafted a story for Banisher, I spun Raymundo’s concept together with impressions of my own. The first sentence was “The Sepian Epoch began with a simultaneous psychic communication to all living things from ![[image blocked]](https://hypothesisnyc.org/images/newsletters/denarius.png)
, a deity that identified as the source of all our beliefs, whereupon dreams and reality became one.” would reveal itself to humanity in clear form with new instructions on how to achieve spiritual fulfillment. At first everything would be great, but then anyone skeptical about the new arrangements would be scapegoated by the others. From that point, things would get continuously worse, with the sadistic deity already in control of most of the population, leaving it up to the skeptics to perfect the banishing ritual in hiding.
A turning point in how seriously I took this occurred when discussing it with my mom, who’s still Catholic, but with whom I have a better relationship than I did at fourteen. She’s interested in apparitions of the Virgin Mary. She described a famous series of these that occurred in Garabandal, Spain in the early 1960s. Four young girls reported receiving messages from Mary after a public spectacle where the four stared into the sky in ecstasy. One of the messages was a prophecy that there will be a moment when everyone in the world will be simultaneously confronted with an examination of conscience. This resonated with my story’s opening sentence. A part of another message the girls received was that “Many Cardinals, Bishops and priests are following the road to perdition, and with them they are taking many more souls.” That one caused controversy in the Catholic church especially since the girls’ story has been corroborated by Mother Theresa and Padre Pio, both of whom were later canonized. To me, the message of this apparition evidences a significance of Mary (or whatever she symbolizes) as apart from the god who is maintained by the Catholic bureaucracy. In Raymundo’s model, maybe it’s a call-back to the true Yahweh (or whatever he represents). In any case, it felt like something important I needed to know more about, and also a chance to bond with my mom over a common interest.
We made spontaneous plans to visit Garabandal. We got the tickets, got our itinerary all set, and were about to finalize our Covid paperwork a couple of days before the trip. But I had a series of hypnagogic experiences at my mom’s house that made it impossible for us to go.
One night, I woke up urgently to a feeling of utter despair that is beyond description. It was companied with a stench like rotten frosting. Rather, that’s the closest thing I can think of to compare it to; though it was a distinct and unique smell. It was the stench of wickedness, inescapable and everywhere, yet I knew I was not perceiving it through ordinary perception. I had a sense of enormous troll-like things walking around in the woods, though this was not exactly literal. The Earth had become a living hell where we were trapped and would rot but could not die. All I could do was try to think of ways to prevent anyone else from ever having to experience this. It didn’t matter what anyone had done; no human deserved it. I cried and dug around through scrap objects that I found in a drawer, which took on significance that were too emotional to describe but something like intense nostalgia, and somehow resolved the matter, then went back to sleep.
Another night, closer to our planned date of departure, I woke up with a sensation that something huge and heavy was dancing with me, again not a literal sensation, and this time not an unpleasant one. There was a feeling of moving my body from somewhere else like a puppet. As I moved about, I nonchalantly punched a hole in the wall, then decorated it with objects in the room for purposes that again I’m unable to describe and which somehow resolved the matter.
That day, associative ideas were running constantly through my head — too much synchronicity, too much significance. I tried to focus on the Banisher project, but it was spilling over. This wasn’t my first time having experiences like this but it had become increasingly rare, in the past decade or so, that I should fail to contain them in my artwork. (I’ve gained much more control over them ever since I began studying mathematics.)
My mom and I decided it was unwise to take an international trip given my state. Things stabilized for me after I got back home, after making a couple of weird pieces of art, talking to some friends, reading more on spiritual science, and getting my head more into the math and logic aspects of the game development. If you’re interested in spiritual science literature, I recommend starting with How to Know Higher Worlds by Rudolf Steiner. I have more I can say about how that has informed my understanding of this work, which I’ll save for another time.
As of recently, the game is in a place that I feel very good about. The aesthetic design and the mechanics have been evolving as dynamically as (but less dramatically than) the story, with the three informing one another. As of now, with thanks to help from Dennis Ryan and Victoria Manning, I have nineteen handmade boards that fold into cases, and most of the components to fill out those sets. And I think the mechanics of the game are finally done!
I’ll close by thanking all of the play testers so far. So, thank you to: Adam Endres, Adrian Pecarina, Andrew Wellington, Andrew Shih, Ben Blum-Smith, Brian Hargraves, Ben Scaglione, Colin Hargraves, Cuper Vargas, Dan Belina, Dennis Ryan, Donovan Miller, Dylan Sparrow, Ed Bear, Forest Fisher, Gianna Delluomo, Jake Pachman, James Petz, Joshua Kelberman, John Rader, Kim Abrams, Liam Kersting, Marcel Alexander, Molly Miller, Neil Eriyamulla, Rob Roan, Samuel Weisberg, Tracy Pecarina, Victoria Manning, and Wendy Kenneally.
Have fun. Keep thinking. Stay curious.
2021 September 5 — Solar Bound
For the past few weeks, I’ve been immersed in developing my prototype for Banisher, our board game that combines principles from magic and geometric group theory. Most recently, this entailed hand embroidering eleven diagrams in yellow and blue, a four-day endeavor during which my mind was filled with thoughts of what to say in this newsletter. Most of those things, I decided not to say because they were philosophical ramblings in defense of starting a business to accomplish non-commercial objectives, and I feel it’s more productive to focus on the new projects being created.
So, here I’ll do a quick follow-up about that UFO topic from last time and then share some recent developments of our projects. There’s a couple of queries at the end where you might help out if interested. As usual, feel free to respond (if you’re a subscriber receiving this via email) about those or anything else!
As far as the UFO stories in the news, I think that Simon from New Paltz put it best when he said “There is something stranger happening and right here with us (always has been?), and this nuts-and-bolts theory (that UFOs are physical beings/crafts from another planet) is either the only way that we can grasp it, or is being used by those in the know, or by the phenomenon itself, as a deceptive/manipulative way to frame the phenomenon / control the narrative / our perception of it.”
We get bombarded with so much information. It takes discipline to remember that we ourselves are responsible for what we think, especially about the unknown. This is where mathematical training helps. It makes our thinking more rigorous as we consider the information we have, be it logical, intuitive or perceptive. If after that, a phenomenon remains unknown, we can enjoy further study of and play with the mystery rather than falling for irresponsible interpretations (and attacking anyone who disagrees). In the field of pure math, open problems stay open until they are really truly solved; the big questions in life should be no different.
I had intended to focus on Gardens of Danu (our upcoming sci-fi video game) after my summer teaching work was over. My head was swimming with ideas about mythology and extraterrestrials, and Michael (the software developer on the project) had completed code ready to populate with dialogue trees with the NPCs. I made a new draft of my master document of backstory for the game, and wrote some of the dialogue for Silven, the Liralune of Earth’s moon Luna.
Then the excessive screen time caught up with me. After teaching online all summer and spending extra time writing on the laptop, my eyes rebelled with a spell of dryness.
So I switched to the only project that could be done by hand: Banisher. At that point, I had a set of instructions sufficient for remembering how to play (but not for teaching someone to play on their own), and a rough prototype made with foam board, model magic clay, glass pebbles and poker chips. That prototype had been sufficient to work out the game mechanics with my play testers (special shout outs to Victoria from Brooklyn and Brian from Danbury for their awesomely constructive criticism), but it did not convey the intended thematic/emotional effect. That part is very important for Banisher. It’s not intended as merely an eccentric strategy game with weird rules; it’s meant to tap into magical energy from my vision for it and conjure up similar feelings for other people using ancient techniques.
Something I’ve realized about making really weird content is that it’s important to get everything as close to right as you can before pitching it. People have a natural tendency to look for a pre-existing category to lump the idea into and if I don’t like that category, I shouldn’t expect people to read my mind and invent a new category just for me (at least, not without ample magical assistance!). When it comes to content “based on conceptual math”, the battle against indoctrinated negative categories is particularly uphill.
One of the principles of Banisher is a single set of rules that allow for playing on eleven specific diagrams. The diagrams are defined by a set of mathematical conditions but are topological (not geometric) objects, which means there’s some taste that goes into how to draw them. I’d been playing around with the form and aesthetics for these, trying a lot of things. Some of the ideas were informed by dreams. A particularly clear message from a dream occurred in the form of a voice saying “We are of brown, yellow and blue,” along with some new impressions on symbology. Later that day, I discovered a symbol that uniformized the configuration of all the diagrams and the moment I finished drawing it, the mail carrier knocked on my door to deliver a package of brown objects I’d ordered to experiment with as playing pieces. I painted the symbol on my prototype in yellow and blue and added the brown pieces, and had a deep sense of satisfaction with it. The development of Banisher has been rich with synchronous moments like this.
Now it goes a bit deeper, leading to something with which you might help.
My story for Banisher is set in a post-apocalyptic future where the game is a ritual for liberation from an evil deity. The idea is that this deity poses as a savior but in reality feeds sadistically upon our suffering. In the draft, I name that deity by a symbol (to be disclosed at a later time) and not a word. I’d made up this symbol based on my own impressions but then settled for something similar which could be more easily typeset. The closest match comes from an engraving that’s been used in archaic currency, which fits into my vision as the notion that my antagonist controls people by fixation on materialism.
Incidentally, I’d reconnected with an old friend through my teaching work who introduced me to the writings of Rudolph Steiner. Shortly thereafter, I noticed that the symbol I’d chosen occurs in reference to Ahriman, the entity which, according to Steiner, oppresses us by fomenting obsession with materialism. Since then, I’ve been broadening my understanding of Steiner and other elements of Western esotericism as context for my idea. Broadly speaking, I want to strike the right balance between originality (so the game taps a direct source of creativity) and basis in existing work (so that it’s respectful of and relevant to other peoples’ beliefs). The activity around theosophy and anthroposophy have been making the most sense for this lately.
In my latest draft of the story, there’s a secret twelfth diagram which is necessary for completing the banishing ritual. I did this to encourage further exploration of the underlying math, but I did not have anything in particular in mind for the next diagram. Earlier this week, my research led me to the work of the Russian mystic George Gurdjieff. I learned that he taught his pupils to use a 9-sided star called the Enneagram, which he believed to encompass all understanding of the cosmos. The diagram has a similar look to the diagrams from Banisher, though does not follow the same mathematical principle.
There are two more direct synchronous mysteries stemming from Grudjieff’s Enneagram. The first is that a Dutch businessman named Per Striegler created a board game called Life Prison in which the game board is the Enneagram. Apparently it’s meant to teach a psychological profiling technique popular with certain corporations, but I can find no details on how the game is played. The second mystery relates to a book about Gurdjieff published by Solar Bound Press which, based on the description (I haven’t read it yet) may have details on Gurdjieff’s relevance to Banisher that had been eluding me. What makes this synchronous for me is that it was published in New Paltz, NY in 2002: the same year that I moved to New Paltz back when I was an experimental video artist (before my math career) exploring strange spiritual ideas. What makes this a mystery is that I never heard of the Solar Bound Press, and can find no information about them other than that they published this single book.
If chasing down things like that sounds insane to you, hey you’re entitled to your opinion! Regardless, I’d be very interested to know if anyone has or can find further info about the board game Life Prison or about the publishing company Solar Bound Press.
I’ll conclude with some shoutouts. I know we still have a long way to “success”, but real success (without quotation marks) is surviving while doing things that one believes in, and so far we’re successful! My collaborators have been crucial in making this possible. My mom, a retired Spanish teacher, has been helping with the Spanish website translation (any errors you find are very likely mine). Molly Miller (soon to be added to the site) has been helping with art and concepts for Banisher. The scariest new frontier for me in this endeavor has been asking people to help me when I want to pursue something strange, and I’m very glad I did. Also, Bridge to Enter Advanced Mathematics and Columbia SPS have been crucial in giving me work that’s relevant and helps pay the bills while our projects develop.
Oh, and Hypothesis has an Instagram account: @hypothesisnyc . There you’ll find cheeky comments on math and education and business, and some fun stuff about our creative process.
Have fun. Keep thinking. Stay curious.
2021 May 23 — Aliens in the News
When I was a kid, my dad was a politician. We lived in a small town called Carmel in New York State where the most exciting attribute was a lake where you could go swimming. One day, my friend Tom saw a flying saucer over the lake, but I’ll come back to that topic in a bit.
My dad was a County Legislator on the Democratic Party line from 1985 to 1993. This means he was involved in writing and passing laws that required attention at the local level rather than state or federal, and generally supported government regulation of corporations. For example, how do you protect the lake from pollution amidst a growing population? These things weren’t very interesting to me as a kid; my point is that my family often had an inside scoop on local politics. What was interesting was when the local news reported on things my dad had already told me about. I would check the stories to see if they’d gotten them right and often, they hadn’t. Things I’d heard my dad say to reporters on the phone were misquoted. Details would be missing or just plain incorrect. Due to this, I was skeptical of any stories which I had no inside knowledge about — some more so than others. (To get a sense of what sources are reliable, I advise studying the common logical fallacies: arguments that can make it look like something has been proven when it really hasn’t.) As I got older, I came to understand how news reports twist facts to make things entertaining, or to cater to special interest groups. After all, if they can’t keep people watching then they can’t sell advertisements, and those advertisements are not always limited to the designated commercial breaks.
[Aside: I want to clearly disassociate what I’m saying here from Donald Trump’s manipulation of facts where he calls things “fake news”. A nice thing about being an LLC rather than a not-for-profit organization is that I’m free to take sides politically, thus it’s totally legal when I tell you: I think Donald Trump is a narcissist and a national embarrassment. Logical fallacy is his modus operandi. It’s unfortunate that fans of his such as the Q-Anon crowd have given a bad name to media skepticism and to the search for hidden truths.]
Now zoom out to news stories about things that involve the entire state, the entire country, the entire world... If a local newspaper can botch information about a lake that just a few thousand people care about, what happens when the information is far more complicated and involves billions of people? This takes us to our main topic:
How do we react when mainstream news sources like CNN, NBC, and CBS acknowledge the existence of UFO’s? What about when the Pentagon does, too?
I’ve been following the UFO phenomenon for a long time, starting around the time my friend Tom told me he saw a flying saucer over Lake Carmel. I have a set of experiences, investigative data, and opinions about this, but I want to leave those aside for now. Let’s take the recent news reports as a starting point for thought on the topic, and let’s approach it like mathematicians (that is, logically and with an open imagination).
Fact: according to the news sources linked above, the Pentagon has confirmed the existence of aircraft capable of maneuvers that are not known to be possible for man made aircraft, and which they are unable to identify.
The Pentagon’s confirmation of this is not new. What is new is that it’s showing up in mainstream platforms. Here are some of the many questions that this raises, and some preliminary observations.
Most basically, are the news sources telling the truth about what the Pentagon said? I say yes, since the news sources provide references that check out, and are not known to manufacture references. (To disagree at this level takes us into “fake news” territory, and I’d be curious to know why you’d suspect such a complete fabrication of information.)
Next, is the Pentagon telling the truth that these aircraft exist and maneuver as described? For this, we have the testimony of the Air Force and Navy personnel included in the reports about protocol for verifying a flying object’s existence and trajectory. Some reporters speculate on the footage based on what it looks like to them, but this is a moot point. Of course special effects are capable of producing virtually any video image (including grainy low resolution ones) but what matters here is that people who are trained in interpreting the video and radar data, and who’s ability to do so is relied upon by our military, are saying that the objects are real. Thus, I take the answer here to be another yes.
Then we ask, does the Pentagon know the observed maneuvers to be impossible for man made craft? That part is more difficult to answer because the military does manage confidential information, and the personnel interviewed may not have had clearance to know this. A statement from “the Pentagon” begs the question of who is responsible for the release, what level of clearance they have, and what level of classification the information is at. Looking into who is making the statements, with limited knowledge of these things, I find it hard to answer this question.
Finally, are the ships of extraterrestrial origin? As tempting as it may be to jump to this question, we have very little to go on based on the information given in the references currently being discussed. Once we classify phenomena as “unidentified”, all that does is put it among the vast majority of phenomena, on Earth and otherwise.
Fact: according to the news sources linked above, the Pentagon’s intelligence committee has ordered the director of national intelligence and the secretary of defense to deliver a report on the sightings by next month (2021 June).
It’s possible that this report won’t resolve the ambiguities we already have. It’s possible the report will raise more questions than it will answer. What will be important is that we maintain our ability to think critically about the information presented to us. That means looking out for internal logical consistency and looking out for logical fallacies.
Another thing one might do is go hunting for information about UFO’s on the internet. There you can find people saying a wide variety of things and the challenge of thinking about it critically is far greater. Once you’ve become good at spotting logical fallacies, you might find you can cut through a lot of the nonsense, but I wouldn’t recommend this as a first exercise.
One perspective I’ve been especially interested in is that of Dr. Steven Greer of The Disclosure Project. I’m very interested in other people’s impressions about all of this, and about that project in particular. In the meantime, I continue working on some playful access points to theories in extraterrestrial life for our video game Gardens of Danu. Please don’t hesitate to respond to share your thoughts if you so desire.
Have fun. Keep thinking. Stay curious.
2021 April 11 — The Shapes of the Numerals
In our last newsletter, I posed the following challenges:
- Why do you think the symbols 1, 2, 3, 4, 5, 6, 7, 8, 9 look the way they do?
- Invent your own symbols for ten and eleven that extend the pattern.
I received some great questions and answers about this. Tracy from Wingdale gave the following answer to challenge (B). Let X represent ten and let ∂ represent eleven (you’ll see why later). Her answer to (A) happens to be exactly my answer to (A), which makes me think we may be on to something! I see nothing like it anywhere online, so would love to see more investigation of this. I’ll come back to that answer at the end of this letter. Like last time, please feel free to respond to this email with any relevant questions, comments or ideas.
Recall, we were thinking about base twelve as an important innovation for early human civilizations. The idea is that base twelve makes it easier to count geological time (being close to the length of a lunar cycle), and that twelve makes it easier to do mental arithmetic (being divisible by a lot of smaller numbers). We noted that this was the system used by the Sumerians, who had a way of counting in base twelve on their hands, who built the earliest known human cities (c 4500 BCE) and who invented the earliest known written language (c 3400 BCE). I suggested that moving from base ten to twelve may have been the crucial leap forward that led to their impressive ability to communicate mathematics, and hence their technological advances.
Challenge B: new symbols for ten and eleven
We were interested in new symbols for ten and eleven so that we can write and do computations in base twelve. Some of you asked: why don’t we need a new symbol for twelve? Good question! The reason is basically the same as why we don’t have a symbol for ten in base ten. When we write 10, we think of it as “one ten and zero ones ”. If instead we were working in base twelve, we’d have a twelves place instead of a tens place. For example, the numbers nine, ten, eleven, twelve would be written: 9, X, ∂, 10. For another example, 23 would mean “two twelves plus three ones”.
On a related note, why do we have the words eleven and twelve, rather than saying “oneteen” and “twoteen”, like how we write them? I bet that this is from a linguistic root in base twelve that was already deeply entrenched before literacy was widespread, and that this is related to how the numerals got their shapes.
It turns out there’s a whole organization devoted to switching back to base twelve, which I didn’t know about when writing the last newsletter, called The Dozenal Society. An interesting result I read from them is that some of the unusually long life spans of people described in the bible could be explained by accounting for alternative bases (and not always bases ten or twelve) that were preferred by the original writers. The Dozenal Society opted for an inverted 2 and 3 to represent eleven and twelve, respectively, but I didn’t see any literature from them on the topic of why the numbers look the way they do. I’m actually more interested in this question about the shapes’ origins than I am in advocating a global reform to base twelve. Before putting that topic aside, I’ll just suggest learning about alternative bases to build your flexibility with math, and to just be aware of the arbitrariness of base ten beyond counting fingers.
Challenge A: Why do 1, 2, 3, 4, 5, 6, 7, 8, 9 look the way they do?
To start us off, we have a great suggestion from Amertah in San Diego: take a cross-cultural approach. How were the numerals written by other cultures beforehand? Let’s take this all the way back — and the dawn of written language isn’t even close to the beginning. There’ve been tally marks found on bones dating back to 35,000 BCE! And even there, you find different techniques for organizing the marks. Moving up to more formal written language, one sees a lot of variation between the Sumerian, Egyptian, Chinese and Mayan systems. The origin for our current symbols is attributed to Indian mathematicians in the first few hundred years CE, where there was a lot of comparing of notational systems that came in from different sources and looking for common ground. Another piece of the puzzle, shared by Victoria from Brooklyn, is the role of printing and typesetting in stabilizing the hand drawn variations. She shared this great article and video from Radio France, which also debunks a fake interpretation of the numeral shapes:
https://www.franceculture.fr/histoire/pourquoi-nos-chiffres-ont-la-forme-quils-ont
It’s an interesting activity to look up records of how different people drew the numerals over time in different places. One notices certain common shapes that have no apparent relationship to the quantities they represent. So... why those shapes?
Tracy and I think it comes from sign language, particularly from the Sumerian hand-counting system. They would count to twelve on the right hand in the following way (you can also find pictures of this online). For 1, touch your thumb to the bottom segment of your pinky. For 2, touch your thumb to the next segment up. For 3, touch your thumb to the tip of your pinky. For 4, 5, 6, do the same thing on your ring finger. For 7, 8, 9, the same thing on your middle finger and for X, ∂, twelve, the same thing on your pointer finger. Now, as you do this, look at the shapes your fingers make. In particular, think about how you might orient your hand if you were showing the number to someone else. The similarities for 4, 5, 6, 7, and 9 are compelling. (Extending this to ten is a little awkward, so we opted for the recognizable X, but ∂ follows the pattern nicely.)
I remain curious about the symbol 8. When you look at cross-cultural records, it seems to be one of the more consistent ones, yet bears no obvious relationship to the quantity or to the hand symbol. I have other ideas about this but they are stranger. I’d love to hear from you about what you think!
Have fun. Keep thinking. Stay curious.
2021 February 22 — The Moon & Our Hands
There is a discrepancy between Earthling astronomy and primitive counting methods. That was the first of several prompts to enter into an ancient aliens theory, which I will not be doing here, but if you wish to do so on your own, you have my blessing. I’ll save mine for another time. (Full disclosure, this doesn’t have nothing to do with my story ideas for Gardens of Danu.) Here, I’ve stuck to verifiable historical facts, and tried to be clear about where I’m getting creative, but please check me on them anyway — if nothing else, they’re fun things to look up.
Earthling astronomy at its most basic has to do with counting the number of days in a lunar cycle (or “lunation”), and the number of lunations in a year. That’s the most fundamental method of counting time in human history. Now, the internet tells us that the average lunation is a little over 29 and a half days, and that there are a little under 12 and a half lunations per year. If you round those off to whole numbers (which we do), and compensate with some error correction (e.g., leap year), this is still a tad inconvenient for an animal that counts on its 10 fingers.
Did you know that 5 fingers per hand has been the evolutionary standard number since before people were homo sapiens? Did you know that if you trace back our evolutionary origins, every species in our ancestry had 5 fingerlike things per hand-like thing, all the way back to the Devonian Period, over 300,000,000 years ago?! I assume that base 10 thinking is pretty deeply ingrained in our neurology. Yet, Earth time has a different preference.
Perhaps the ability to adapt our numbers to the sky is directly related to the forming of civilization. If you want to master your environment, you need to be able to predict it (When will the nights be lighter? When will it be warm or cold? When will the tide be high? etc). Then there’s another, perhaps more coincidental bonus for counting by 12’s. It avoids everyone’s least favorite topic from elementary school: fractions. You can divide 12 by 1, 2, 3, and 4 without ever having to learn a separate notation for those abominable non-whole numbers. Maybe that’s enough for everyday life, and maybe we save our more robust division methods for more complicated work. In the meantime, 12 is friendly, even to young children as they build their number sense. With 10 — the jerk! — you can’t even divide it by 3 before running into an infinite decimal expansion. Maybe adapting our biological preference for base 10 to the astronomical bases 12 and 30 were the key to getting it together as homo sapiens.
Archeological findings support this theory. The first civilization was in the Sumer region of Mesopotamia, about 6,000 years ago. The Sumerians had base 12 and 60 systems of counting. Using 60 gives even more divisibility than 30 does, and works for lunar cycles if we count days and nights as separate events. Not only that, but evidence suggests they had an efficient method for counting this way on their hands. (Look this up!) Something I find especially striking is that their method uses the opposable thumb as a tool for counting on the hand, perhaps a precursor for using a writing instrument on paper. Upgrading the thumb in this way has profound implications. It happens that the Sumerians also developed the earliest written language.
Okay, so what about today? Nearly everything has gone back to base 10. We let computers handle most (perhaps too many) computations, and most of us throw our hands in the air when it comes to fractions. It seems like sharing has become problematic, on a metaphysical level! But check this out. What time is it? How big is an angle in a regular triangle? Those base 12 and 60 systems trace all the way back to the origins of the written word! I think it’s worth thinking about why.
Subscribers were Invited to Participate in the Following Challenge
I say we bring back base 12. Who’s down? We’ll need two things: a new symbol for ten, and a new symbol for eleven.
To make this more interesting, let’s try to make our new symbols consistent with the symbols we already have for the digits 1 through 9. The hardest part, perhaps, is figuring out why the heck the symbols for 1 through 9 look the way they do in the first place! Actually, nobody knows the answer to this for sure, so use your imagination. I have my own idea, which I’ve never even told anyone publicly. Maybe I’ll tell you next time.
So, to participate, send me your own invented symbols for the numbers ten and eleven. Bonus challenge: give me your reason why you think the symbols 1, 2, 3, 4, 5, 6, 7, 8, 9 look the way they do, and explain why your new symbols continue the pattern. (If sending an image attachment, please keep the file size small.)
Have fun. Keep thinking. Stay curious.
Developing Projects
Gardens of Danu
This video game takes place on the moons of our solar system, where you meet a population of creatures called The Liralunes. On each moon, you interact with a Liralune by conversing with them and playing a puzzle game against them. You are trying to find out who the Liralunes are, and what is the nature of non-human intelligent life, but you must work to attain information and you must think critically to decide for yourself what is real. This is a collaboration with software developer Michael Adams, featuring dolls by Tracy Pecarina of 2Ts in a Pod, animations by Allen Cordell, and music by Dylan Sparrow. Play testing is being conducted with the help of students from Bridge to Enter Advanced Mathematics.
Nothing
This picture book follows two characters who test the boundaries of logic in an attempt to “Think about nothing”, intended for imaginative readers of ages 8 to ∞. This is a collaboration with Dennis Ryan.
Banisher
This is an apocalyptic magic themed tabletop board game featuring an original game mechanic that uses concepts from geometric group theory. It’s a strategy game of complete information for one to four players that can be played competitively or collaboratively. Currently, the game exists as a collection of meticulously handcrafted sets that are of personal significance to the creater. These are being experimented with to determine the appropriate presentation for an affordable mass-producible fascimile.
Voyages
A Hypothesis Voyage is a virtual play testing session, conducted live over video chat. These involve interactive digital content, and sometimes common household materials. They give you a glimpse of our games in development, and your feedback helps us continue making them awesome. Sessions are held in both English and Spanish.
Forms to sign up for Voyages will appear on this page. To get notified when we’re booking a new Voyage, sign our mailing list. The intended ages, prerequisites, etc, vary by session and are specified in the invitations.
Past Voyages
Note: in the entries below, the language is fixed to agree with the language of the session.
2020.11.6 — Viernes Sin Cargo: Maker vs Breaker (esp)
Con participantes de México, exploramos la historia del teorema de cuatro colores. Entonces, enfocamos en un juego desarrollado para hacer más accesible los principios involucrados, y que tiene sus propios problemas no resueltos.
2020.10.10, at Ohio State University BAMM — Pipes and Pentagrams (eng)
For middle and high school students. Division is probably the least popular topic in grade school. Topology can be pretty hard too. But put them together and you get an awesome all-ages game that gets us thinking about one of the most important theorems in graph theory. This also resulted in some awesome post-session worksheets created in collaboration with BAMM.
2020.10.2 — Free Fridays: Maker vs Breaker (eng)
In a fun evening with fellow educators from Italy and the US, we explored the story of The Four Color Theorem, and how we use that as a source for games that teach math as a developing process. We shared methods for online engagement and how we are all rising to the challenge of online teaching, including methods that work better online than they do in the classroom.
2020.9.11, at Ohio State University BAMM — When the Maker Wins (eng)
For faculty and graduate students of the math department. We discussed the roll of artificial intelligence in mathematics, and Thurston’s notion of what constitutes a proof, focusing on the example of the Four Color Theorem. We explored a game invented in the late 80s in order to study that theorem, and played it against each other on our interactive boards via Google Docs. Finally we talked about some open problems related to the game.
2020.9.4 — Viernes Sin Cargo: El Problema de Futurama (esp)
Con una audencia de edades mezcladas de los EEUU, Francia y México. Conductor Doctor Roldan compartió unas diapositivas nuevas y chidos para introducir el problema. Doctor Quinn mangó tableros interactivos por Google Docs. Había varias soluciones e ideas interesantes sobre el problema y pensamos en visitarla de nuevo en el futuro.
2020.8.7 — Free Fridays: The Futurama Problem (eng)
Conductor Doctor Quinn came into this Voyage ready with brand new interactive virtual game boards and more (soon to be regular staples of our Voyages). But the weather was not on our side! Power outages across the northeastern US prevented many from joining, and thus we embarked with just one brave pilot, Luis, who solved The Futurama Problem all on his own!
2020.6.5 — Viernes Sin Cargo: Paradromia (esp)
Viaje de Hypothesis número cuatro, y el primero en español. Gracias a nuestros amigos de Matemorfosis y de Music Math, había mucho interés de México — ¡tuvimos 46 solicitantes para solo 16 asientos! Los pilotos exploraron patrones aritméticos en anules paradrómicas, y también algo de topología, y nos divertimos mucho.
2020.5.2 — Free Fridays: Heads, Tails, Tricks (eng)
The third Hypothesis Voyage was piloted by Juniper and Toby, and new pilots Mason and Kartik. Our pilots solved a coin tossing paradox, then pushed into some tougher territory where the problem got a little too hard for us. Nonetheless, we learned a lot about using trees to analyze games and probabilities, which apply to a broad range of scenarios.
2020.4.24 — Free Fridays: Paradromia (eng)
The second Hypothesis Voyage featured two pilots: Juniper and Toby, and was extremely successful. They conducted paper cutting experiments to study arithmetic patterns in paradromic rings. This lead us unexpectedly into some knot theory and some really fun visual thinking. According to their dad, our pilots continued their experiments for the next week, decorating the house with Möbius strips and other paradromic rings.
2020.4.3 — Free Fridays: Square Deal (eng)
First Hypothesis Voyage ever! Thanks to pilots Ben, Johnny, Juniper and Toby for getting this space ship off the ground. They solved a card game (meaning, figured out how to win the game no matter what the opponent does), then learned how to use the same principle to perform an impressive magic trick. Thanks to parents Jason and Curtis for the great feedback and for helping to make this happen!
Past Projects
2024 November 11
Online lecture Mathematical Modeling of the Paranormal for the Parapsychological Research Society of the Experimental Middle School Affiliated to Beijing Normal University, hosted by Yiyang Li
2023 May 29
Banisher play test session hosted by Eden Morris in Brooklyn, New York
2023 April 21
Workshop Infinity and the Fourth Dimension at the New York Branch of the Anthroposophical Society in America
2023 February-May
Gardens of Danu puzzle playtesting workshop with students from BEAM Next
2022 August 18
Banisher play test session hosted by Jeff Pachman in Brooklyn, New York
2022 June 11
Banisher play test session hosted by Brian Hargraves in Danbury, Connecticut
2021 September - 2022 June
Writing activities and games for the Entry Points program of Bridge to Enter Advanced Mathematics
2021 September-December
Wrote and taught the online Saturday enrichment courses Infinity and Imagination for BEAM Next
2021 February-April
Taught online courses and developed new activities to foster growth mindsets in the Entry Points program of Bridge to Enter Advanced Mathematics
2021 February-May
Conceived, wrote, and taught the online Saturday enrichment course Infinity for BEAM Next
2020 October 9
Led online Graph Games workshop for university instructors and students for Buckeye Aha! Math Moments at Ohio State University
2020 September 11
Led online Pipes and Pentagrams workshop for middle school students for Buckeye Aha! Math Moments at Ohio State University
2020 September-December
Conceived, wrote, and taught the online Saturday enrichment course The Shape of the Universe for BEAM Next
2020 April
Developed the Math for Pirates course on logical reasoning for Bridge to Enter Advanced Mathematics
2020 April 1
Hypothesis filed its articles of organization with the State of New York.
People
Dennis Ryan
Visual Artist
Dennis has a remarkable ability to communicate emotion via control of a wide variety of media such as watercolor, photoshop, faux finishes, color design, Tromp l’ceil, gilding and more. Dennis holds a BA in painting from SUNY New Paltz and works independently with architects, designers, decorators and (lucky for us!) Hypothesis. Samples of his work can be found at dennis-ryan.com.
Dylan Sparrow
Artist
Dylan is a NYC based visual artist, musician and writer whose work is at once nostalgic and futuristic. His hobbies are street hiking, making mixtapes, and playing Quarth. Dylan holds a BA in Illustration from the School of Visual Arts and an MA in Teaching of Writing from LIU Brooklyn. For more about his work, see www.dylansparrow.com.
James Petz
Philosopher
James is a wealth of inspiration for anyone having the bandwidth to keep up with him. He’s usually reading several books at a time and grokking information on everything from the history of counterculture to esoteric meanings of contemporary mythology. He acts as a sounding board for Hypothesis concepts on a weekly basis and more broadly volunteers as an advocate for neurodivergent perspectives.
Joe Quinn
Director & Co-Founder of Hypothesis *
Joe has been inventing fun and interesting games for his friends since he was a teenager. He has also been an underground video and performance artist, a mathematician, a curriculum designer and a teacher. In 2004 he won a can of gandules from The Arlene’s Grocery Picture Show, in 2016 he earned a PhD in Topology from The City University of New York, and in 2018 he made a dent in the locally famous MoMath Problem. He makes his money working at the College of Staten Island, Bridge to Enter Advanced Mathematics and the Pre-College Programs of Columbia University.
Kat Byrne
Artist / Poet
Kat has been writing poetry, drawing, and sculpting since she was a kid. Her poetry has been published by Slope. Kat’s creations are like warm spoonfuls of honey that can cure even the illest of conditions. She holds a BA in Creative Writing from SUNY Purchase, and works as a Barista.
Katrina Colletti
Data Analyst
Katrina is a brilliant and detail oriented creator whose interests include teaching, scientific research, animal care and rescue, making math fun for the general public, data science for social good, and advocating that everyone looks like a scientist. Most of her science research pursuits have centered around theoretical and experimental high energy physics. Katrina holds an MS in Physics from Texas A&M University, where she was part of the SuperCDMS Collaboration.
Marcel Alexander
Marketing Coordinator
Marcel has been studying innovative marketing techniques since he was a high school student, working in everything from door-to-door sales to online marketing strategies. In his free time, he enjoys sci fi, hanging out with his friends, and composing music. Marcel holds a BA in Classical Composition from Purchase College, and studied orchestration under the tutelage of Richard Milan Simons in Westchester County, NY.
Michael Adams
Software Engineer
Michael is a software engineer currently based in Paris, who also works in math outreach. He served for three years in the Peace Corps teaching math in a rural primary school in South Africa. While there, he helped set up a science centre in the village before spending a year as the Mathematics Program Coordinator at the Unizulu Science Centre in Richards Bay. Michael holds a BA in Mathematics from Bucknell University.
Sylvia Miles
Accountant
Sylvia is a long time lover of and successful advocate for math and education. As a teacher, she taught math to all ages, from first grade through college calculus. She currently works as a CPA in sunny Southern California, where you can find her hiking or playing math games with her nephew. Sylvia holds a BS in Mathematical and Computational Sciences from Stanford University, and an MEd in Secondary Education and Teaching from UCLA.
Tracy Pecarina
Visual Artist
Tracy is known by many for the beautiful original paintings, drawings, and dolls. She forms one half of the mother-daughter art and woven wares team 2 Ts in a Pod (www.2tsinapod.com), based in Pawling, New York. She works with repurposed and up-cycled yarns and fabrics to create one-of-a-kind dolls, each with their own unique names, stories, and personalities, all looking to find their rightful adoptive homes. Tracy holds a BA in Communications from SUNY New Paltz.
Victoria Manning
Visual Artist
Victoria is a Brooklyn based artist who received her BFA in Photography from the School of Visual Arts. Her projects explore connections between history, science, art, and literature through the use of photography, sculpture, needlework, and other mediums (www.victoriamanning.com). She is also a museum registrar and currently works at the Museum of Modern Art.
* Hypothesis is a single-member LLC owned and directed by Joe Quinn, founded by Érika Roldán Roa and Joe Quinn, and which works alongside various like-minded artists, contractors, and volunteers.
Donations
We welcome and greatly appreciate any donations. With your help, we strive to provide affordable access to our content for as many people as possible.
Past Donations
2024 January 23 — $100 USD
2023 February 21 — $150 USD
2021 February 11 — $100 USD
2020 December 8 — $700 USD
2020 October 28 — $100 USD — “Looking forward to seeing your initiative grow. It’s needed.”