## Archive for the ‘**Math**’ Category

## Strange Numbers

I found this video fascinating. The problem Derek Muller mentions — that p-adic systems based on composite numbers — non-primes — can result in zero being the product of two non-zero numbers — is something one encounters in abstract algebra. If two non-zero numbers have zero as a product, they are called divisors of zero and they mess up a system — that is, they introduce issues that don’t arise if the system has no zero-divisors.

At any rate, here’s the video. It has some cute things.

## What Number Comes Next? The Encyclopedia of Integer Sequences Knows.

Siobhan Roberts has a very interesting article (gift link, no paywall) in the *NY Times*. It begins:

Some numbers are odd:

## 1, 3, 5, 7, 9, 11, 13, 15 …

Some are even:

## 2, 4, 6, 8, 10, 12, 14, 16 …

And then there are the puzzling “eban” numbers:

## 2, 4, 6, 30, 32, 34, 36, 40 …

What number comes next? And why?

These are questions that Neil Sloane, a mathematician of Highland Park, N.J., loves to ask. Dr. Sloane is the founder of the On-Line Encyclopedia of Integer Sequences, a database of 362,765 (and counting) number sequences defined by a precise rule or property. Such as the prime numbers:

## 2, 3, 5, 7, 11, 13, 17, 19 …

Or the Fibonacci numbers — every term (starting with the 3rd term) is the sum of the two preceding numbers:

## 0, 1, 1, 2, 3, 5, 8, 13 …

This year the OEIS, which has been praised as “the master index to mathematics” and “a mathematical equivalent to the FBI’s voluminous fingerprint files,” celebrates its 50th anniversary. The original collection, “A Handbook of Integer Sequences,” appeared in 1973 and contained 2,372 entries. In 1995, it became an “encyclopedia,” with 5,487 sequences and an additional author, Simon Plouffe, a mathematician in Quebec. A year later, the collection had doubled in size again, so Dr. Sloane put it on the internet.

“In a sense, every sequence is a puzzle,” Dr. Sloane said in a recent interview. He added that the puzzle aspect is incidental to the database’s main purpose: to organize all mathematical knowledge.

Sequences found in the wild — in mathematics, but also quantum physics, genetics, communications, astronomy and elsewhere — can be puzzling for numerous reasons. Looking up these entities in the OEIS, or adding them to the database, sometimes leads to enlightenment and discovery.

“It’s a source of unexpected results,” said Lara Pudwell, a mathematician at Valparaiso University in Indiana and a member of the OEIS Foundation’s board of trustees. Dr. Pudwell writes algorithms to solve counting problems. A few years ago, thus engaged, she entered into the OEIS search box a sequence that arose while studying numerical patterns:

## 2, 4, 12, 20, 38, 56, 88 …

The only result that popped up pertained to chemistry: specifically, to the periodic table and the atomic numbers of the alkaline earth metals. “I found this perplexing,” Dr. Pudwell said. She consulted with chemists and soon “realized there were interesting chemical structures to work with to explain the connection.”

Sequence serendipity provides what Russ Cox, a software engineer at Google, called “amazing cross-connective tissue for the sciences.” Dr. Cox, based in Cambridge, Mass., is the president of the OEIS board. He submitted his first sequence, which emerged from a programming contest puzzle, as a high-school student in 1996. He has twice rewritten the software for the database, which he thinks of as “the collective wisdom of math and science in this interesting numerical form.”

Donald Knuth, a computer scientist at Stanford, known for his analysis of algorithms, among other things, has also chanced upon breakthroughs. Working on a new problem, he always searches the OEIS. “It finds my bedfellows,” he said. “The beautiful thing is that you can compute your way into the literature.” . . .

## Justice Sam Alito’s strong argument that the attack on SCOTUS is fair

I think Sam Alito’s argument is interesting. It was in an article in Jezebel.

Two key statements are presented by Justice Alito:

A = SCOTUS is unfairly attacked.

B = The organized bar will defend SCOTUS

Alito then states: **If A, then B.**

And he also states: **Not B**

He leaves the conclusion to the reader, but this is a simple syllogism. The obvious, logical, and necessary conclusion is:

**∴ Not A**

Thus, Alito argues that SCOTUS is *not* unfairly attacked. By the Law of the Excluded Middle, Alito argues that SCOTUS is fairly attacked.

I’m a little surprised that he feels this way, but when he’s right, he’s right.

## A Number System Invented by Inuit Schoolchildren Will Make Its Silicon Valley Debut

Inuit schoolchildren invented an interesting number system, with cleverly designed numerals. It’s base 20 (vigesimal) rather than base 10 (decimal). Instead of the basic digits being 0 through 9, as in Arabic numerals (base 10), or 0 through F, as in hexadecimal (base 16) numerals, the Inuit system has numeral digits representing the numbers 0 through 19. Since the system is base 20, each position represents a power of 20 rather than a power of 10 (as in decimal numerals) or 16 (as in hexadecimal numerals):

Decimal place values: 10⁰, 10¹, 10², 10³, …

Hexidecimal place values: 16⁰, 16¹, 16², 16³, …

Vigesimal place values: 20⁰, 20¹, 20², 20³, …

Amory Tillinghast-Raby writes in *Scientific American*:

In the remote Arctic almost 30 years ago, a group of Inuit middle school students and their teacher invented the Western Hemisphere’s first new number system in more than a century. The “Kaktovik numerals,” named after the Alaskan village where they were created, looked utterly different from decimal system numerals and functioned differently, too. But they were uniquely suited for quick, visual arithmetic using the traditional Inuit oral counting system, and they swiftly spread throughout the region. Now, with support from Silicon Valley, they will soon be available on smartphones and computers—creating a bridge for the Kaktovik numerals to cross into the digital realm.

Today’s numerical world is dominated by the Hindu-Arabic decimal system. This system, adopted by almost every society, is what many people think of as “numbers”—values expressed in a written form using the digits 0 through 9. But meaningful alternatives exist, and they are as varied as the cultures they belong to.

The Alaskan Inuit language, known as Iñupiaq, uses an oral counting system built around the human body. Quantities are first described in groups of five, 10, and 15 and then in sets of 20. The system “is really the count of your hands and the count of your toes,” says Nuluqutaaq Maggie Pollock, who taught with the Kaktovik numerals in Utqiagvik, a city 300 miles northwest of where the numerals were invented. For example, she says,

tallimat—the Iñupiaq word for 5—comes from the word for arm:taliq. “In your one arm, you havetallimatfingers,” Pollock explains.Iñuiññaq, the word for 20, represents a whole person. In traditional practices, the body also serves as a mathematical multitool. “When my mother made me a parka, she used her thumb and her middle finger to measure how many times she would be able to cut the material,” Pollock says. “Before yardsticks or rulers, [Iñupiat people] used their hands and fingers to calculate or measure.”During the 19th and 20th centuries, American schools suppressed the Iñupiaq language—first violently and then quietly. “We had a tutor from the village who would help us blend into the white man’s world,” Pollock says of her own education. “But when my father went to school, if he spoke the language, they would slap his hands. It was torture for them.” By the 1990s the Iñupiaq counting system was dangerously close to being forgotten.

The Kaktovik numerals started as a class project to adapt the counting system to a written form. The numerals, based on tally marks, “look like” the Iñupiaq words they represent. For example, the Iñupiaq word for 18, “

akimiaq piŋasut,”meaning “15-3,” is depicted with three horizontal strokes, representing three groups of 5 (15) above three vertical strokes representing 3. . .

## Aperiodic tiling with a single shape

Periodic tilings of the plane — a pattern repeated indefinitely that covers the plane — are common: the checkerboard pattern using squares, the public-restroom-floor tiling of black and white hexagons, various tilings of equilateral triangles, and so on. And you can force aperiodicity — nonrepeating — on those by using colors at varying intervals. But what about a tiling that is *necessarily* aperiodic (nonrepeating)?

Craig S. Kaplan in a Mastodon thread writes:

How small can a set of aperiodic tiles be? The first aperiodic set had over 20000 tiles. Subsequent research lowered that number, to sets of size 92, then 6, and then 2 in the form of the famous Penrose tiles. https://youtu.be/48sCx-wBs34

I have blogged that video before, and it’s fascinating — I just watched it again.

Kaplan continues:

Penrose’s work dates back to 1974. Since then, others have constructed sets of size 2, but nobody could find an “einstein”: a single shape that tiles the plane aperiodically. Could such a shape even exist? https://en.wikipedia.org/wiki/Einstein_problem

The whole thread is worth reviewing, and he includes some interesting links.

Alo definitely read this post by Jason Kottke, which goes further into it — and includes this fascinating video:

## Exponential growth is messing with our minds

The gif above shows the peculiarity of exponential growth, as does the chart at the right. Both are from a very interesting post by Kevin Drum, well worth reading, on why we are feeling disoriented by the rate of technological change in general and the increasing capabilities of AI in particular. (We who are science-fiction fans have been aware of this phenomenon for some time by reading novels about the Singularity, when AI becomes subject to managing its own improvement. Even now AI is designing better circuitry to implement better AI.)

## The Hypercube: Projections and Slicing

The original was Betamax, and the audio is flakey in a few places.

by Thomas Banchoff and Charles Strauss, Brown University, was awarded the Prix de la Recherche Fondamentale at the International Congress of Scientific Films in Brussels in 1978, the year it was produced. It was featured in a plenary lecture at the International Congress of Mathematicians in Helsinki in that same year.

## Useful password idea from xkcd

Full disclosure: I don’t like to remember dozens of different passwords — well, more like I can’t — so I use a password manager. I used to use LastPass, but they have had too many data breaches and have not been forthcoming about what all they lost, so I switched to 1Password, which I like a lot. LastPass was free but bad; 1Password is less than 75¢ a week (paid annually) and very good — plus they maintain it.

But with a password manager, you still need a master password — something long, so that it’s hard to break, but also easy to remember. For that, the idea above is excellent. A master password of (say) correct+horse+battery+staple or correct.horse.battery.staple or the like would work well and also be easy to remember.

## Emmy Noether and the conservation of momentum

Emmy Noether is a big name in mathematics (e.g., Noetherian rings), so that title of a post by Kevin Drum caught my eye. Drum writes:

Yesterday I asked why there’s no name for a unit of momentum. Today I have answers. Plus, if you read all the way to the end, I have a genuinely constructive suggestion.

First things first, in case you have no idea what I’m talking about. In the metric system—officially known as SI—there are three basic quantities: the meter, the kilogram, and the second.¹ Everything else is derived from those three. For example,

force = mass * acceleration, so:F = ma

a = distance / seconds²

Therefore, F = mass * distance / seconds²

One unit of force = 1 kg * 1 meter / 1 second²

This quantity is called a

newton, named after Isaac Newton. Lots of other things have names too: ohm, watt, lumen, joule, and so forth. Click here for a list.Momentum is a critically important quantity, equal to

mass * velocity. So why wasn’t it ever given a name? I did several minutes of research on this question, and the most authoritative sounding answer came from a commenter at Stack Exchange called Conifold. He or she explains that there were two waves of standardization and naming:The second wave, started in the 1860s and formalized by the 1880s in both SI and its competitor CGS, was meant to catch up with developments in thermodynamics and electromagnetism,

and gave us ohms, volts, farads, watts, etc.Kilograve was renamed into kilogram and became the unit of mass. The unit of force was named dyne in CGS (from Greek dynamis — force) and newton in SI.….The unit for power, watt, was suggested even before joule, by Siemens in 1882, to replace Watt’s own horsepower used to measure the output of steam engines. Siemens was an electric engineer. Joule himself was honored by a unit name for determining the mechanical equivalent of heat.

Momentum was out of luck.In other words, momentum has no name because no one ever bothered to give it one. However, another commenter, jkien, tells us that it

wasgiven a name in the CGS systemIn 1887 . . .

## Animating the Mercator projection to correct size and shape

Fascinating animation that takes countries enlarged by the Mercator projection and shrinks them down to their actual relative size. The lighter a country’s color, the more its size has been enlarged by the Mercator projection (so the more it shrinks in the animation to get down to its correct relative size).

## List of common misconceptions

Wikipedia has a listicle of common misconceptions.

- 1Arts and culture
- 2History
- 3Science, technology, and mathematics
- 4See also

## Can God Be Proved Mathematically?

Spoiler alert: You already know…

Manon Bischoff writes in *Scientific American*:

Who would have thought about God as an apt topic for an essay about mathematics? Don’t worry, the following discussion is still solidly grounded within an intelligible scientific framework. But the question of whether God can be proved mathematically is intriguing. In fact, over the centuries, several mathematicians have repeatedly tried to prove the existence of a divine being. They range from Blaise Pascal and René Descartes (in the 17th century) to Gottfried Wilhelm Leibniz (in the 18th century) to Kurt Gödel (in the 20th century), whose writings on the subject were published as recently as 1987. And probably the most amazing thing: in a preprint study first posted in 2013 an algorithmic proof wizard checked Gödel’s logical chain of reasoning—and found it to be undoubtedly correct. Has mathematics now finally disproved the claims of all atheists?

As you probably already suspect, it has not. Gödel was indeed able to prove that the existence of

something, which he defined as divine, necessarily follows from certain assumptions. But whether these assumptions are justified can be called into doubt. For example, if I assume that all cats are tricolored and know that tricolored cats are almost always female, then I can conclude: almost all cats are female. Even if the logical reasoning is correct, this of course does not hold. For the very assumption that all cats are tricolored is false. If one makes statements about observable things in our environment, such as cats, one can verify them by scientific investigations. But if it is about the proof of a divine existence, the matter becomes a little more complicated.While Leibniz, Descartes and Gödel relied on an ontological proof of God in which they deduced the existence of a divine being from the mere possibility of it by logical inference, Pascal (1623–1662) chose a slightly different approach: he analyzed the problem from the point of view of what might be considered today as game theory and developed the so-called Pascal’s wager.

To do this, he considered two possibilities. First, . . .

## The Biggest Project in Modern Mathematics

The video description:

In a 1967 letter to the number theorist André Weil, a 30-year-old mathematician named Robert Langlands outlined striking conjectures that predicted a correspondence between two objects from completely different fields of math. The Langlands program was born. Today, it’s one of the most ambitious mathematical feats ever attempted. Its symmetries imply deep, powerful and beautiful connections between the most important branches of mathematics. Many mathematicians agree that it has the potential to solve some of math’s most intractable problems, in time, becoming a kind of “grand unified theory of mathematics,” as the mathematician Edward Frenkel has described it. In a new video explainer, Rutgers University mathematician Alex Kontorovich takes us on a journey through the continents of mathematics to learn about the awe-inspiring symmetries at the heart of the Langlands program, including how Andrew Wiles solved Fermat’s Last Theorem.

## So where is the number designated by “1”?

The comic strip at the left ponders a well-worn question: Is the subject of mathematics real? or not? Or, to put it another way, are mathematical entities discovered? or invented?

My view is that mathematics has a kind of intermediate reality. The reality of mathematics, unlike, say, the reality of the Moon, is restricted to human culture. Within human culture, the number one is real, but if there were no humans, there would be no number one.

In other words, math is as real as a human language. Sounds exist within nature, and humans can make complex sounds, but *language* — those sounds together with their meaning — exists only insofar as there are people who understand the meaning of sounds. The meaning is not “out there” — where the sounds are, in the vibration of air — but “in here,” where the brain extracts the meaning conveyed.

The meaning clearly exists in a sense, and indeed has consequences “out there” in the “real” world — the Industrial Revolution and the consequences (such as the climate change we now are experiencing) wold not have occurred without language. But once all those who understand some language are gone, the language is no more. There may be carvings in rocks or marks on vellum, but the meaning of those is absent, so the incisions and marks no long longer are language but just physical things, bereft of the meaning they once conveyed.

(For that matter, sound is not “out there.” What’s “out there” are vibrations in the air. Sound is the way our brain interprets air vibrations that have been fed to it as electrical impulses from the motion of tiny hairs in the liquid contained in the cochleae of our ears. Until that transition is made, there is no sound, only air vibrations. Thus a tree falling in a remote forest with no animals nearby will produce air vibrations but not sounds, because there’s no one to translate air vibrations to brain signals.)

So a sheep on a hillside is not “one” sheep unless it is observed by a person who has learned the human idea (the meme) of counting, and only such a person might observe that there are “zero” horses and “zero” cows on the hillside.

Math, like language, like music, like fashion, and like religion, is a cultural construct, a set of memes. Math has the reality of memes (as does, say Don Quixote or unicorns) but it is “in here,” not “out there.” And even “in here” there are problems, as Kurt Gödel pointed out.

And yet, consider this poem by Clarence R. Wylie Jr.:

Paradox

Not truth, nor certainty. These I forswore

In my novitiate, as young men called

To holy orders must abjure the world.

‘If…,then…,’ this only I assert;

And my successes are but pretty chains

Linking twin doubts, for it is vain to ask

If what I postulate be justified,

Or what I prove possess the stamp of fact.

Yet bridges stand, and men no longer crawl

In two dimension. And such triumphs stem

In no small measure from the power this game,

Played with the thrice-attentuated shades

Of things, has over their originals.

How frail the wand, but how profound the spell!

## Can computer simulations help fix democracy by curtailing gerrymandering?

Harry Stevens has a good article (gift link, no paywall) in the *Washington Post* on a good approach to help even mathematically illiterate judges — that is, the great majority of judges — to understand when gerrymandering has been done. The article begins:

After the release of the 2020 Census, legislatures across the country redrew their states’ congressional district maps, just like they do every decade. And, just like every decade, aggrieved citizens sued them for gerrymandering — the process whereby politicians craft district boundaries to ensure their own parties’ victory.

But this time around, something has changed. A technological revolution, decades in the making, has added a sharp new arrow to those citizens’ quiver of legal arguments. Known as algorithmic redistricting, the technology has persuaded judges to throw out gerrymandered maps in several states, including New York and Ohio.

And it will be part of a case before the Supreme Court in October that could play a role in the 2024 election and the future of voting rights.Here is how it works. . .

Continue reading. (gift link, no paywall)

It’s a very good article, with interactive graphics to explain the approach clearly.

## Prediction of China’s 9/11: “China’s ENTIRE economy will crash by September 11, 2022”

That’s the prediction made in the video below: that China’s economy will collapse 34 days after August 7, 2022. Perhaps by coincidence, that date is 9/11/2022. I have marked my calendar. In the meantime, it’s an interesting report and worth watching.

## The mathematical power of 3 random words

Mary Lynn Reed, Professor of Mathematics, Rochester Institute of Technology, writes in *The Conversation*:

It’s hard to imagine that three random words have the power to both map the globe and keep your private data secure. The secret behind this power is just a little bit of math.

What3words is an app and web-based service that provides a geographic reference for every 3-meter-by-3-meter square on Earth using three random words. If your brain operates more naturally in the English measurement system, 3 meters is about 9.8 feet. So, you could think of them as roughly 10-foot-by-10-foot squares, which is about the size of a small home office or bedroom. For example, there’s a square in the middle of the Rochester Institute of Technology Tigers Turf Field coded to brilliance.bronze.inputs.

This new approach to geocoding is useful for several reasons. First, it’s more precise than regular street addresses. Also, three words are easier for humans to remember and communicate to one another than, say, detailed latitude and longitude measurements. This makes the system well suited for emergency services. Seeing these advantages, some car manufacturers are starting to integrate what3words into their navigation systems.

## Ordered triples

Here’s how three random words in English or any other language can identify such precise locations across the whole planet. The key concept is ordered triples.

Start with the basic assumption that the Earth is a sphere, recognizing that this is an approximate truth, and that its radius is approximately 3,959 miles (6,371 kilometers). To compute the surface area of the Earth, use the formula 4πr

^{2.}With r = 3,959 (6,371), this works out to approximately 197 million square miles (510 million square kilometers). Remember: What3words is using 3-meter-by-3-meter squares, each of which contains 9 square meters of surface area. So, working in the metric system, Earth’s surface area is equivalent to 510 trillion square meters. Dividing 9 into 510 trillion reveals that uniquely identifying each square requires around 57 trillion ordered triples of three random words.An ordered triple is just a list of three things in which the order matters. So “brilliance.bronze.inputs” would be considered a different ordered triple than “bronze.brilliance.inputs”. In fact, in the what3words system, bronze.brilliance.inputs is on a mountain in Alaska, not in the middle of the RIT Tigers Turf Field, like brilliance.bronze.inputs.

The next step is figuring out how many words there are in a language, and whether there are enough ordered triples to map the globe. Some scholars estimate there are more a million English words; however, many of them are very uncommon. But even using only common English words, there are still plenty to go around. You can find many word lists online.

The developers at what3words came up with a list of 40,000 English words. (The what3words system works in 50 different languages with independently assigned words.) The next question is determining how many ordered triples of three random words can be made from a list of 40,000 words. If you allow repeats, as what3words does, there would be 40,000 possibilities for the first word, 40,000 possibilities for the second word, and 40,000 possibilities for the third word. The number of possible ordered triples would then be 40,000 times 40,000 times 40,000, which is 64 trillion. That provides plenty of “three random word” triples to cover the globe. The excess combinations also allow what3words to eliminate offensive words and words that would be easily confused for one another.

## Passwords you can actually remember

While the power of three random words is being used to map the Earth, the U.K. National Cyber Security Centre (NCSC) is also advocating their use as passwords. Password selection and related security analysis are more complicated than attaching three words to small squares of the globe. But a similar calculation is illuminating. If you string together an ordered triple of words – such as brilliancebronzeinputs – you get a nice long password that a human should be able to remember far more easily than a random string of letters, numbers and special characters designed to meet a set of complexity rules.

If you increase your word list beyond 40,000, you’ll get . . .

## A Mirror of Nature

In *Introspection* Mike Edmunds has an interesting essay on implications of the Antikythera mechanism. The essay begins:

THE ANTIKYTHERA MECHANISM, an astronomical calculator found in a first-century BCE shipwreck, has proven to be mechanically more sophisticated than anything known from the subsequent millennium. While many are amazed at such a discovery, a more appropriate response would be admiration, for the mechanism fits well into its historic context. Indeed, the ancient scholar Cicero offered contemporary accounts of similar devices, which he saw as embodying the peak of human ingenuity.

^{1}But the significance of the Antikythera mechanism extends beyond the elegance and complexity of its design. It may also represent a major development in our understanding of the universe.## Astronomical Mechanisms

IN THE SPRING of 1900, sponge divers working near the Greek island of Antikythera came across the wreck of a Roman cargo ship. Among the remains was a corroded shoebox-sized case with more than thirty bronze gear wheels in interlaced trains.

^{2}Those who attempted to reconstruct it over the next century would learn that it included annular dials on its front and large spiral dials on the back that represented the day in the year, the lunar month in the 235-month Metonic cycle, the lunar phase, the position of the sun and moon in the zodiac, and whether the month might contain a lunar or solar eclipse. Irregularities in lunar motion were incorporated by means of an ingenious pin-and-slot variable-speed device. Predicted eclipses and the lunar calendar were based on observed cycles passed down to the Greeks from the Babylonians. The device itself was probably constructed in Rhodes sometime between 150 and 160 BCE, though both the date—it might be as early as 205 BCE—and the source are subject to debate. Inscriptions on the device strongly suggest that its front face also displayed the positions in the zodiac of the known planets: Mercury, Venus, Mars, Jupiter, and Saturn.The significance of the Antikythera mechanism, as it came to be called, only began to be more broadly realized after the publication of Derek de Solla Price’s paper “Gears from the Greeks” in 1974. Price found the mechanism so sophisticated that it might “involve a completely new appraisal of the scientific technology of the Hellenistic period.”

^{3}Just four years earlier, Germaine Aujac had written a perceptive, although largely forgotten, review of several kinds of mechanical devices that could have influenced Greek views of the universe.^{4}Price may well have been unaware of Aujac’s article. He does not reference it and Aujac does not mention the Antikythera mechanism. It would take another thirty years before the publications of the international Antikythera Mechanism Research Project (AMRP) and of Michael Wright prompted more general awareness of the artifact as confirming the reality of such complex machines in the ancient world.^{5}The devices Aujac wrote about were

sphaerae—orsphéropéein the original French—mechanisms depicting the sky moving around the earth, with or without the planets.Sphaeraecould be three-dimensional terrestrial or celestial globes and armillary spheres, but also two-dimensional circular constructions like the Antikythera mechanism. According to Aujac, by combining observation with the theory and construction ofsphaerae, the Greeks were bringing models of the earth and heavens closer to their real equivalents. In his writings, Ptolemy acknowledged the existence ofsphaerae, although he seems to have thought that they were admired more for their craftsmanship than for their value as physical models.^{6}James Evans and Christián Carlos Carman have argued that geared technology may slightly predate, and might actually have inspired, the mathematical developments around 200 BCE in Greek planetary theory such as eccentrics and epicycles.^{7}## A Mechanical Universe

FROM MY OWN perspective, the deeper question concerning

sphaeraeis to what extent the development of this technology prompted the Greeks and Romans into a new worldview.^{8}The technology may have affected not only mathematics, but also the idea that the universe itself is in some sense mechanical—and long before the so-called scientific revolution of the Renaissance. For Samuel Sambursky, the question iswhether these models are only convenient means of illustration, devices adapted to our needs for an ordered description, or whether they represent to a greater or lesser degree some faithful image of a physical reality corresponding to them.

^{9}If meant as a faithful image of reality, there are several themes present in such an image. The first would be the . . .

## The Riddle That Seems Impossible Even If You Know The Answer

The key is that if you start with the box labeled with your number, then either you arrive at a box that contains that number or you open boxes endlessly. Given that there are a finite number of boxes, that’s impossible. So by starting with the box labeled with your number, you *know* that you are in the loop with a box that contains your number. The only issue is whether that loop is 50 boxes or fewer.