Recent content on Paradigm Sage
Chapter 26 - Engineering Portent To iterate. To change. To violate idempotency. If something is true, should it not stay true? One way to make sense of self-reference is to treat it as an iterator. Under this interpretation, the Liar’s paradox is simply a short cycle iterator: T, F, T, F, T… Iterators can be studied; mathematically, logically, symbolically, graphically. The scaling…
Chapter 25 - Laws of Form Portent A formal system of Nonlinear Logic (NLL) is daunting. Incorporating imaginary truthvalues as the resolution of logical paradox, demonstrating that they form a logical basis conjugate to the Boolean truthvalues, defining axioms, rules of inference, equality, transitivity, etc. - is a challenging goal. However, demonstrating that formal systems can model the…
Chapter 24 - Frame Portent The creative process; inspiration, progress, setback, breakthrough, catastrophe. It can be an emotional rollercoaster; it is not for the faint of heart. Science is a communal endeavour, we all stand on the shoulders of giants, we all need a support network, friends, colleagues, students, opponents - even nemeses! Fortunate are those investigators who experience such a…
Chapter 23 - Critique Portent “What are friends for?” Thesis: self-referential closed causal loops in spacetime built with spacelike causes, that self-collapse in the one and only basis in which the loop is indeterminate, deftly avoiding temporal paradox, while acting as nature’s censor mechanism, to prevent causality paradoxes with the side effect of allowing superluminal…
Chapter 20 - FTL by Indistinguishability Portent Measurements on distinguishable particles can’t leverage quantum entanglement and nonlocality to achieve superluminal signaling. What about indistinguishable particles? Or more preciesly, what if Alice and Bob can independently choose to make either a distinguishable or an indistinguishable measurement on a double pair of EPR entangled…
Chapter 21 - Now and Then Portent You can’t have FTL. You can’t even have superluminal signaling. If you could, you could travel into the past, or at the very least, send information to your past self. FTL will violate conservation of energy and conservation of mass. Even superluminal signaling denies isolation, preventing cutting off an experiment from all outside influence. Isolation…
Chapter 22 - Quantum Temporal Indeterminacy Portent The strongest argument aginst FTL, against any kind of superluminal signaling, against spacelike causality in general, is the threat of temporal paradox, a closed causal loop in spacetime that denies its own causality. Yet here we have a metaphor for spacelike causality which deftly avoids temporal paradox by collapsing the elements of such a…
Chapter 19 - Quantum Optics Portent Can’t use quantum entanglement and nonlocality, ala a single EPR pair, because no change in measurement base yields a signal. Two pairs looked promising, but even then, quantum randomness thwarts the effort. But what if, what if…you could change the odds. What if, distinguishable versus indistinguishable is locally choosable, but globally…
Chapter 18 - Indistinguishable Portent Flip two fair coins. Odds of each are independent, thus a 25% chance of each of the four outome permutations. HH HT TH TT Equal odds that the coins match to they do not match. Because…the coins are distinguishable, you can tell which is which. What happens when they are indistinguishable? Primary Docs Paradigm Discourse: Open the Bingo Bos discourse in…
Chapter 17 - Teleportation Portent It would seem that nature keeps superluminal signalling all to herself. If we can’t use quantum entanglement for FTL, what can we use it for? Primary Docs Paradigm Discourse: Open the Cloning discourse in the next tab Technical Chapter: Open the Teleportation chapter in the next tab Significance Quantum teleportion is the slick utilization of quantum…
Chapter 16 - Bowtie Portent Inspiration. “In spirit”, the odd phenomenon where insight just happens . Primary Docs Paradigm Discourse: Open the Art Show discourse in the next tab Technical Chapter: Open the Bowtie chapter in the next tab Significance A thought experiement, about closed causal loops in spacetime, using superluminal signaling down symmetric spacetime intervals. Nature…
Chapter 15 - Symmetric Spacetime Intervals Portent Objects are point like , waves are distributed . The arrival of an object can be abstracted as a spacetime event, two numbers; when and where. But when a wave arrives, the point-like abstraction is insufficient. Is there an abstraction that would do better? Primary Docs Paradigm Discourse: Open the Surf Beach discourse in the next tab Technical…
Chapter 14 - World Ribbons Portent The conventional metaphor for breaking out of a paradigm is thinking outside the box . The allusion is that the walls of the box block a view of what is outside the box. This is harder than it looks, it requires inspired guessing, and ruthless pruning. Most ideas are…junk, one can find oneself not just out of the box, but off the table and onto the floor.…
Chapter 13 - 12 Light Seconds Portent Physicists love their invariants. They hunt for them like buried treasure. They lovingly name them, “Conservation of Energy”, “Conservation of Mass”, etc. An invariant is that which doesn’t change while all around it does. They are reliable guideposts in the search for the laws of physics. Primary Docs Paradigm Discourse: Open the…
Chapter 12 - By Contradiction Portent There is no shortage of proofs that quantum nonlocality cannot be used to send superluminal signals. Eberhard’s paper of 1989 is probably the definitive treatment, comprehensive, succinct and devastating. Randomness defeats any possible combination of measurements, no matter how clever. But the problem with impossibility proofs is that they require a…
Chapter 11 - Quantum Tic-Tac-Toe Portent This, then that. This event caused that event: cause -> effect. In relativity, events are spacetime events, tightly local in both location (space) and duration (time). What are we to make of happenings that are not events, that are not localized? Can we use games to dig under the paradigm of events (point-like causes) as the only possible type of cause?…
Chapter 10 - Objections Portent Ambiguity. Most of us have a low tolerance for it. We like the cognitive clarity of black and white - it’s efficient. But on the eve of a paradigm shift, ambiguity is our friend. A deliberate strategy to let the web of ideas find a new local minimum. Primary Docs Paradigm Discourse: Open the Journal discourse in the next tab Technical Chapter: Open the…
Chapter 9 - R&D Portent Risk. How do new theories come into existence? How does someone, anyone, let go of a successful past to chart a new and uncertain future? The lure of the quest; challenge, purpose, reward…and risk. Perhaps evolution has built this algorithm into us as well. Primary Docs Paradigm Discourse: Open the Quest discourse in the next tab Technical Chapter: Open the R&D…
Chapter 8 - Quantum Temporal Paradox (QTP) Portent Entanglement cannot be explained by timelike causality nor by common cause. But those are the only two types of causes we know about. Well, except for spacelike causality (and its equivalent common cause variation), but that can’t be allowed - can it? If it is, the specter of causal paradox cannot be ignored. A choice must be made: either…
Chapter 7 - Entanglement Portent Entanglement is the unexpected phenomenon that individual particles can lose their individuality. They no longer have independent states, but only a joint state. They have become entangled . Mathematically, their states are no longer separable . This disturbs the statistics of their possibilities from classical expectations. More jarringly, it violates our…
Chapter 6 - Interference Portent Logical linear western thought. If a allows c , and b allows c , then a and b allows c . What could be more natural, more obvious, it’s simple logic - commonsense demands this conclusion. And it works - in classical systems. Not so much in quantum ones. The problem is not interference, that is a well understood wave phenomena. The problem is destructive…
Chapter 5 - Uncertainty Portent There is tension between observation and abstraction. Which one should dominant depends on where one is along the historical arc of a technical field. Sometimes more is less - and in perfect symmetry - sometimes less is more. Primary Docs Paradigm Discourse: Open the Ignoring discourse in the next tab Technical Chapter: Open the Uncertainty chapter in the next tab…
Chapter 4 - Quantum Mechanics Portent The uncertainty principle is not about measurements disturbing a system, although it is often taught that way. The real situation is much deeper, and more mysterious. The uncertainty principle is a consequence of conjugate bases . Conjugate bases do not have to have the same units. When they do, we find them tolerable, even understandable. When they…
Chapter 3 - Relativity Portent The speed of light is more fundamental than just a velocity, it is the thread that weaves space and time together. In relativity (Minkowski space), space and time are supposed to be treated on an equal footing. Yet, spacetime presents a unique symmetry/assymmetry signature: past and future break the isotropic nature of time, but left and right do not break the…
Chapter 2 - Paradox Portent Self-reference has been at the heart of the major limit theorems of the last century, from Russell’s paradox in set theory, to Gödel’s incompleteness theorems in math, to the halting problem in software engineering. In any sufficiently complex system, self-reference intrudes, unwanted, unwelcome, and unexpected, violating assumptions we were absolutely…
Chapter 1 - Paradigms Portent Paradigm is a technical term, introduced by Thomas Kuhn in his seminal work The Structure of Scientific Revolutions . It is derived from the Greek word for pattern. The modern idiom is thinking out of the box . Many puzzles are paradigm puzzles, they are quite opaque until you question the blocking assumption. This is a skill which can be developed. The great…
Rationale For those who want to know “why invest the time?” Problem: Chess in three dimensions doesn’t feel like chess. The pieces lose too much power, a one pawn advantange no longer yields the game. Every contest is a draw. Boring, frustrating; in equal measure. Conundrum: Over the last hundred years lots of rule sets have been proposed, a few are interesting, a couple even…
Out of the Box Premise Thinking out-of-the-box is hard. Questioning paradigms is not generally a part of our formal education. Indeed, in many cases, the whole enterprise is designed to innoculate us into very specific paradigms. Paradigms have their purposes, but they readily become thought traps, and steal the joy of discovery. As a culture we believe, intellectually, that thinking…
Clues Hints for those who like to figure things out for themselves. To achieve play that feels like chess requires: a formal line of development with geometrical and logical rigor that respects the game and requires the penetration of two paradigm barriers. It is a great exercise in thinking out-of-the-box, of breaking paradigms . Clues 4/08/26 - If the 2D board is a square of squares (8x8), then…
Quick Start A brief guide for those who like to dive right in. Launch a Board Use the Setup Panel to make a board and place the pieces in their starting positions (two button clicks). Create an Advancement Square (advsq) Use the Advsq Panel to grow or shrink an advsq. Use Next Piece to see how it differs across the base pieces. Click on a tile and use Place to select its origin. Explore the panel…
Guide A complete guide for those who prefer to look before they leap. Theory A playable implemetation of 3D Chess with planar moves and advancement squares . The 3D board is a cube of cubes (8x8x8). Each tile is the bottom of a cube. An 8 color board includes the 2 bishop colors (tile faces) and the 4 duke colors (tile edges). Bishop: white | black Duke: gold | silver | ruby | jade Conventions…
MetaPhysics Speculations for those who want a sense of the implications. 1. Purpose Explore metaphysical implications of the universe as game metaphor. 2. Description This section will grow, but slowly. The basic idea is to ask what advantages a game metaphor of reality has over the machine metaphor of classical physics. The machine metaphor fails badly in quantum physics, essentially all our…
Development 7/13/26 - Release of v0.6, a minimal viable product (MVP). All the tools to understand the rules are done. However, lots of polish remains to be done. UI improvements Feature additions Bugs (yes there are a few annoying edge cases - worst case, refresh the page, sorry.) Code quality improvements Graphical improvements Enforce the rules Target date for version 1.0 - 1/1/27. Vision A…
Comments A place to share thoughts. 3D Chess Comments This page collects questions, suggestions, bug reports, and observations from people exploring the game. Email allangoff [at] gmail [dot] com (Please include a sentence or two about what you were studying and why you decided to reach out.) Questions Suggestions Bugs Interesting Positions General Discussion
Of all the paradigm shifts that science has muddled through, quantum mechanics has to rank as the most traumatic. It violates our common sense again and again, and then yet again in some completely new and unexpected way, constantly challenging rational, logical, fundamental assumptions about the nature of reality. It challenges taxonomies — is it particle or is it wave? It challenges observation…
Peer Reviewed Paper This paper introduced Quantum Tic-Tac-Toe as a pedagogical game for illustrating superposition and collapse. It is included here as prior work, not as a statement of the current QT3 program. The present site revisits the game’s deeper conceptual implications, which could not be explored in the original venue. Published Paper: Download the original 2006 AJP paper (PDF)
Expositions A place for short articles on the principles, premises, promises, and perils of QT3. Intrepid Reader Ask a question. What is confusing, unclear, intriguing, or questionable. Or even “why didn’t you think of ’that’?” Send an email to contact [at] paradigmsage [dot] com Who knows, a member of Understanding’s team just might post a response here. SamIam…
Instructions Rules There are two players: X moves first, his spooky marks are subscripted with odd integers. O moves second, her spooky marks are subscripted with even integers. There are two types of moves: Placement moves: click in two squares to place a pair of spooky marks. Collapse moves; click on a spooky mark on the loop (purple) to collapse that move to that square. The other player makes…
Limitations Critical discussions of where QT3 is limited as a toy universe. TOC Bases Oscillation Relativity Weights Randomness Phase Only One Basis Quantum tic-tac-toe asks the players to place each move into a superposition of two squares. The nine squares form the basis states of the quantum system. It is the only basis, there are no conjugate bases. One could envision an alternative basis, say…
Play An implementation of QT3. This is where the Intrepid Reader can interact with a playable version of QT3. Ask your own questions of the game. Demo Copy this state string X1+(1,2); O2+(2,3); X3+(3,1)[132]; O3@X1(1)!X1(1)!O2(2)!X3(3); O4+(4,5); X5+(4,6); O6+(4,6)[56|4]; X6@X5(4)!O4(5)!X5(4)!O6(6); X7+(7,8); O8+(8,7)[78]; X8@X7(7)!X7(7)!O8(8); {X=1,O=0.5} and paste it into the state box (black…
Research History At the time of the original publication of qt3, it was known that the maximum number of simultaneous classical games was limited to 27. Thus the SW only allowed space for 27 games in the classical ensemble. In building this version, it became clear that there where a total of 512 possible configuration locations for those classical games. That invites a question - are they all…
Superposition Quantum Tic-Tac-Toe was invented with three strategic objectives: Demonstrate superposition in its simplest possible form. Demonstrate that collapse can be forced by self-reference . Demonstrate that an objective measurement process is conceivable. Spooky Marks The key to superposition is multiple possibilities. The smallest multiple possibilities is two . Therefore, just two…
Variations Present variations on QT3, such as N-way QT3 Unitary evolution Probabilistic collapse Hybrid Indistinguishable QT3 N-Way QT3 There are two possibilities here: fixed N (like 3) and variable N (N>1). The simple cyclic entanglements of classical quantum tic-tac-toe (gotta love language) are no longer the only kind of entanglement that requires collapse. Unitary Evolution In this variation,…