#turing-machine — Public Fediverse posts

“Things that are so far removed from our daily experience… are inherently hard to understand”*…

That’s certainly true of numbers. And as the numbers grow, the cognitive challenges grow with them. (Indeed, by way of example: 1 million seconds, is roughly 11.5 days; 1 billion seconds is almost 32 years.)

We’ve looked before at the mysterious extremes of math: zero and infinity [and here]. But as Dan Falk reminds us, the numbers in between can seem pretty strange as well– especially the extremely large ones. In a review of Richard Elwes‘ Huge Numbers: A Story of Counting Ambitiously, From 4½ to Fish 7, Falk spotlights some of the largest numbers humans have ever contemplated…

… Aficionados of huge numbers are called “googologists,” a reference to the number 10100, known as a googol. Such numbers have a peculiar sort of existence. For the vast majority of us, they’re of limited everyday value. Calculations at the supermarket checkout, or at tax time in April, typically involve far more modest figures. Perhaps we’ve read that the U.S. national debt is in excess of $38 trillion — a mind-numbing figure, to be sure, but it’s not as though any one individual needs to count it up in stacks of $20 bills.

And yet, much larger numbers await those who seek them out. Consider the kinds of numbers that crop up in problems involving combinations and permutations. For example, in how many distinct ways can one shuffle a deck of cards? Elwes takes us through the calculation, and we end up with a figure of about 8×1067. Compared to that number, the odds of getting a royal flush when dealt a five-card poker hand seem pretty decent, sitting at a mere 1 in 649,740 (still rare enough that many poker players have never held such a hand). Or consider that famous 1980s cultural touchstone, the Rubik’s cube. In how many ways can one scramble the cube? It turns out that the figure is about 43 quintillion, or 4.3×1019 — but in spite of that ridiculously large figure, people do routinely solve the puzzle, and champions can do it in mere seconds. In fact, as Elwes explains, no Rubik’s cube arrangement is more than 20 moves away from any other arrangement.

Or consider the age of the universe, estimated to be about 13.8 billion years. This may seem like a lengthy span of time, but our cosmic future is where the really big numbers come up. Elwes examines the so-called heat death of the universe, in which all matter has broken down into subatomic particles. We may reach this point in [10 raised to the 10th power, raised again to the 120th power] years — this dizzying figure is 10 raised to the power of 10120 — at which point, Elwes says, the universe will have ballooned up to a diameter of 10 to the power of 10 to the power of 10120 light years. (Yes, that’s [10 raised to the 10th power, again to the 10th power, then to the 120th power] light years.) Elwes adds a footnote: “At this point, the choice of units hardly matters; the distance is so immense that whether we choose to measure it in Planck lengths or giga-light years makes little difference.” Let that sink in!

As mind numbing as such figures are, the highest numbers contemplated by humans come not from physics but from pure mathematics and computer science. Like “Graham’s number” — an immense figure put forward as the upper-bound for solutions to a problem in a branch of mathematics known as Ramsey theory. Some readers may find the ensuing discussion of multi-dimensional hypercubes a bit challenging, but one can enjoy the payoff regardless: We end up with a number that can’t even be expressed in conventional notation, and which earned a mention in the 1980 edition of the “Guinness Book of World Records” as “the highest number ever used in a mathematical proof.”

Reading this book is a little bit like sitting in the back row of an auction house where a rare Picasso (let’s say) is up for grabs: How high is this thing going to go? And indeed, Elwes keeps going. We eventually meet the so-called busy beaver numbers, a set of numbers that crop up in theoretical computer science, when one tries to deduce whether a particular computer program will eventually stop, or keep going forever — a conundrum known as the “halting problem.” As Elwes explains, it’s not at all straightforward to distinguish the two types of programs (and if it was, it would help mathematicians tackle some of the most vexing problems in their field).

The fifth busy beaver number, known as BB(5) — associated with a computer program that can access five internal states — works out to 47,176,870. And that’s as far as we’ve gotten, Elwes explains. No one has worked out the value of BB(6), but he assures us that it’s beyond the range of any physical computer; and BB(16) leaves even Graham’s number in the dust.

But wait, there’s more! “Rayo’s number,” concocted by Agustín Rayo — a dean and professor at MIT — using set theory, is bigger still (here’s a fun video about it); and “Fish 7,” mentioned in the book’s subtitle, named for a Japanese googologist who goes by the pseudonym “Fish,” builds on Rayo’s number, and … well, the details are not easily digested, but the mind-melting nature of these numbers comes across as a feature, not a bug, of Elwes’s story… the narrative is enlivened by explorations of the peculiarities of math history…

… Archimedes tried to estimate how many grains of sand would be needed to fill up the known universe, back in the third century B.C. Did he simply have too much time on his hands? Not at all, insists Elwes: The Greek thinker was articulating an important idea — that no matter how unfathomably large a quantity may be, we can describe it with precision, thanks to mathematics. “Archimedes,” he writes, “was penning a manifesto for the expressive power of large numbers.”…

… [Elwes focuses] on numbers that are ridiculously large and yet finite. In the end, perhaps this is the most mind-boggling fact of all: that these enormous numbers, from Graham’s number to Fish 7 and beyond, fall as far short of infinity as does the humble number 1…

The mysteries of the massive: “The Mind-Boggling Science of Enormous Numbers,” @danfalk.bsky.social on @richardelwes.bsky.social in @undark.org.

* Steven Strogatz

###

As we enumerate enormity, we might spare a thought for a seminal mathematician, Alan Turing; he died on this date in 1954. He was a foundational computer science pioneer (inventor of the Turing Machine (an influential model for the general-purpose computer), creator of the “Turing Test” (only too relevant in these AI-infected times), inspiration for “The Turing Award” (the “Nobel Prize of computing“), and cryptographer (leading member of the team that cracked the Enigma code during WWII).  

source

“Things that are so far removed from our daily experience… are inherently hard to understand”*…

That’s certainly true of numbers. And as the numbers grow, the cognitive challenges grow with them. (Indeed, by way of example: 1 million seconds, is roughly 11.5 days; 1 billion seconds is almost 32 years.)

We’ve looked before at the mysterious extremes of math: zero and infinity [and here]. But as Dan Falk reminds us, the numbers in between can seem pretty strange as well– especially the extremely large ones. In a review of Richard Elwes‘ Huge Numbers: A Story of Counting Ambitiously, From 4½ to Fish 7, Falk spotlights some of the largest numbers humans have ever contemplated…

… Aficionados of huge numbers are called “googologists,” a reference to the number 10100, known as a googol. Such numbers have a peculiar sort of existence. For the vast majority of us, they’re of limited everyday value. Calculations at the supermarket checkout, or at tax time in April, typically involve far more modest figures. Perhaps we’ve read that the U.S. national debt is in excess of $38 trillion — a mind-numbing figure, to be sure, but it’s not as though any one individual needs to count it up in stacks of $20 bills.

And yet, much larger numbers await those who seek them out. Consider the kinds of numbers that crop up in problems involving combinations and permutations. For example, in how many distinct ways can one shuffle a deck of cards? Elwes takes us through the calculation, and we end up with a figure of about 8×1067. Compared to that number, the odds of getting a royal flush when dealt a five-card poker hand seem pretty decent, sitting at a mere 1 in 649,740 (still rare enough that many poker players have never held such a hand). Or consider that famous 1980s cultural touchstone, the Rubik’s cube. In how many ways can one scramble the cube? It turns out that the figure is about 43 quintillion, or 4.3×1019 — but in spite of that ridiculously large figure, people do routinely solve the puzzle, and champions can do it in mere seconds. In fact, as Elwes explains, no Rubik’s cube arrangement is more than 20 moves away from any other arrangement.

Or consider the age of the universe, estimated to be about 13.8 billion years. This may seem like a lengthy span of time, but our cosmic future is where the really big numbers come up. Elwes examines the so-called heat death of the universe, in which all matter has broken down into subatomic particles. We may reach this point in [10 raised to the 10th power, raised again to the 120th power] years — this dizzying figure is 10 raised to the power of 10120 — at which point, Elwes says, the universe will have ballooned up to a diameter of 10 to the power of 10 to the power of 10120 light years. (Yes, that’s [10 raised to the 10th power, again to the 10th power, then to the 120th power] light years.) Elwes adds a footnote: “At this point, the choice of units hardly matters; the distance is so immense that whether we choose to measure it in Planck lengths or giga-light years makes little difference.” Let that sink in!

As mind numbing as such figures are, the highest numbers contemplated by humans come not from physics but from pure mathematics and computer science. Like “Graham’s number” — an immense figure put forward as the upper-bound for solutions to a problem in a branch of mathematics known as Ramsey theory. Some readers may find the ensuing discussion of multi-dimensional hypercubes a bit challenging, but one can enjoy the payoff regardless: We end up with a number that can’t even be expressed in conventional notation, and which earned a mention in the 1980 edition of the “Guinness Book of World Records” as “the highest number ever used in a mathematical proof.”

Reading this book is a little bit like sitting in the back row of an auction house where a rare Picasso (let’s say) is up for grabs: How high is this thing going to go? And indeed, Elwes keeps going. We eventually meet the so-called busy beaver numbers, a set of numbers that crop up in theoretical computer science, when one tries to deduce whether a particular computer program will eventually stop, or keep going forever — a conundrum known as the “halting problem.” As Elwes explains, it’s not at all straightforward to distinguish the two types of programs (and if it was, it would help mathematicians tackle some of the most vexing problems in their field).

The fifth busy beaver number, known as BB(5) — associated with a computer program that can access five internal states — works out to 47,176,870. And that’s as far as we’ve gotten, Elwes explains. No one has worked out the value of BB(6), but he assures us that it’s beyond the range of any physical computer; and BB(16) leaves even Graham’s number in the dust.

But wait, there’s more! “Rayo’s number,” concocted by Agustín Rayo — a dean and professor at MIT — using set theory, is bigger still (here’s a fun video about it); and “Fish 7,” mentioned in the book’s subtitle, named for a Japanese googologist who goes by the pseudonym “Fish,” builds on Rayo’s number, and … well, the details are not easily digested, but the mind-melting nature of these numbers comes across as a feature, not a bug, of Elwes’s story… the narrative is enlivened by explorations of the peculiarities of math history…

… Archimedes tried to estimate how many grains of sand would be needed to fill up the known universe, back in the third century B.C. Did he simply have too much time on his hands? Not at all, insists Elwes: The Greek thinker was articulating an important idea — that no matter how unfathomably large a quantity may be, we can describe it with precision, thanks to mathematics. “Archimedes,” he writes, “was penning a manifesto for the expressive power of large numbers.”…

… [Elwes focuses] on numbers that are ridiculously large and yet finite. In the end, perhaps this is the most mind-boggling fact of all: that these enormous numbers, from Graham’s number to Fish 7 and beyond, fall as far short of infinity as does the humble number 1…

The mysteries of the massive: “The Mind-Boggling Science of Enormous Numbers,” @danfalk.bsky.social on @richardelwes.bsky.social in @undark.org.

* Steven Strogatz

###

As we enumerate enormity, we might spare a thought for a seminal mathematician, Alan Turing; he died on this date in 1954. He was a foundational computer science pioneer (inventor of the Turing Machine (an influential model for the general-purpose computer), creator of the “Turing Test” (only too relevant in these AI-infected times), inspiration for “The Turing Award” (the “Nobel Prize of computing“), and cryptographer (leading member of the team that cracked the Enigma code during WWII).  

source

“Things that are so far removed from our daily experience… are inherently hard to understand”*…

That’s certainly true of numbers. And as the numbers grow, the cognitive challenges grow with them. (Indeed, by way of example: 1 million seconds, is roughly 11.5 days; 1 billion seconds is almost 32 years.)

We’ve looked before at the mysterious extremes of math: zero and infinity [and here]. But as Dan Falk reminds us, the numbers in between can seem pretty strange as well– especially the extremely large ones. In a review of Richard Elwes‘ Huge Numbers: A Story of Counting Ambitiously, From 4½ to Fish 7, Falk spotlights some of the largest numbers humans have ever contemplated…

… Aficionados of huge numbers are called “googologists,” a reference to the number 10100, known as a googol. Such numbers have a peculiar sort of existence. For the vast majority of us, they’re of limited everyday value. Calculations at the supermarket checkout, or at tax time in April, typically involve far more modest figures. Perhaps we’ve read that the U.S. national debt is in excess of $38 trillion — a mind-numbing figure, to be sure, but it’s not as though any one individual needs to count it up in stacks of $20 bills.

And yet, much larger numbers await those who seek them out. Consider the kinds of numbers that crop up in problems involving combinations and permutations. For example, in how many distinct ways can one shuffle a deck of cards? Elwes takes us through the calculation, and we end up with a figure of about 8×1067. Compared to that number, the odds of getting a royal flush when dealt a five-card poker hand seem pretty decent, sitting at a mere 1 in 649,740 (still rare enough that many poker players have never held such a hand). Or consider that famous 1980s cultural touchstone, the Rubik’s cube. In how many ways can one scramble the cube? It turns out that the figure is about 43 quintillion, or 4.3×1019 — but in spite of that ridiculously large figure, people do routinely solve the puzzle, and champions can do it in mere seconds. In fact, as Elwes explains, no Rubik’s cube arrangement is more than 20 moves away from any other arrangement.

Or consider the age of the universe, estimated to be about 13.8 billion years. This may seem like a lengthy span of time, but our cosmic future is where the really big numbers come up. Elwes examines the so-called heat death of the universe, in which all matter has broken down into subatomic particles. We may reach this point in years — this dizzying figure is 10 raised to the power of 10120 — at which point, Elwes says, the universe will have ballooned up to a diameter of 10 to the power of 10 to the power of 10120 light years. (Yes, that’s light years.) Elwes adds a footnote: “At this point, the choice of units hardly matters; the distance is so immense that whether we choose to measure it in Planck lengths or giga-light years makes little difference.” Let that sink in!

As mind numbing as such figures are, the highest numbers contemplated by humans come not from physics but from pure mathematics and computer science. Like “Graham’s number” — an immense figure put forward as the upper-bound for solutions to a problem in a branch of mathematics known as Ramsey theory. Some readers may find the ensuing discussion of multi-dimensional hypercubes a bit challenging, but one can enjoy the payoff regardless: We end up with a number that can’t even be expressed in conventional notation, and which earned a mention in the 1980 edition of the “Guinness Book of World Records” as “the highest number ever used in a mathematical proof.”

Reading this book is a little bit like sitting in the back row of an auction house where a rare Picasso (let’s say) is up for grabs: How high is this thing going to go? And indeed, Elwes keeps going. We eventually meet the so-called busy beaver numbers, a set of numbers that crop up in theoretical computer science, when one tries to deduce whether a particular computer program will eventually stop, or keep going forever — a conundrum known as the “halting problem.” As Elwes explains, it’s not at all straightforward to distinguish the two types of programs (and if it was, it would help mathematicians tackle some of the most vexing problems in their field).

The fifth busy beaver number, known as BB(5) — associated with a computer program that can access five internal states — works out to 47,176,870. And that’s as far as we’ve gotten, Elwes explains. No one has worked out the value of BB(6), but he assures us that it’s beyond the range of any physical computer; and BB(16) leaves even Graham’s number in the dust.

But wait, there’s more! “Rayo’s number,” concocted by Agustín Rayo — a dean and professor at MIT — using set theory, is bigger still (here’s a fun video about it); and “Fish 7,” mentioned in the book’s subtitle, named for a Japanese googologist who goes by the pseudonym “Fish,” builds on Rayo’s number, and … well, the details are not easily digested, but the mind-melting nature of these numbers comes across as a feature, not a bug, of Elwes’s story… the narrative is enlivened by explorations of the peculiarities of math history…

… Archimedes tried to estimate how many grains of sand would be needed to fill up the known universe, back in the third century B.C. Did he simply have too much time on his hands? Not at all, insists Elwes: The Greek thinker was articulating an important idea — that no matter how unfathomably large a quantity may be, we can describe it with precision, thanks to mathematics. “Archimedes,” he writes, “was penning a manifesto for the expressive power of large numbers.”…

… [Elwes focuses] on numbers that are ridiculously large and yet finite. In the end, perhaps this is the most mind-boggling fact of all: that these enormous numbers, from Graham’s number to Fish 7 and beyond, fall as far short of infinity as does the humble number 1…

The mysteries of the massive: “The Mind-Boggling Science of Enormous Numbers,” @danfalk.bsky.social on @richardelwes.bsky.social in @undark.org.

* Steven Strogatz

###

As we enumerate enormity, we might spare a thought for a seminal mathematician, Alan Turing; he died on this date in 1954. He was a foundational computer science pioneer (inventor of the Turing Machine (an influential model for the general-purpose computer), creator of the “Turing Test” (only too relevant in these AI-infected times), inspiration for “The Turing Award” (the “Nobel Prize of computing“), and cryptographer (leading member of the team that cracked the Enigma code during WWII).  

source

“Things that are so far removed from our daily experience… are inherently hard to understand”*…

That’s certainly true of numbers. And as the numbers grow, the cognitive challenges grow with them. (Indeed, by way of example: 1 million seconds, is roughly 11.5 days; 1 billion seconds is almost 32 years.)

We’ve looked before at the mysterious extremes of math: zero and infinity [and here]. But as Dan Falk reminds us, the numbers in between can seem pretty strange as well– especially the extremely large ones. In a review of Richard Elwes‘ Huge Numbers: A Story of Counting Ambitiously, From 4½ to Fish 7, Falk spotlights some of the largest numbers humans have ever contemplated…

… Aficionados of huge numbers are called “googologists,” a reference to the number 10100, known as a googol. Such numbers have a peculiar sort of existence. For the vast majority of us, they’re of limited everyday value. Calculations at the supermarket checkout, or at tax time in April, typically involve far more modest figures. Perhaps we’ve read that the U.S. national debt is in excess of $38 trillion — a mind-numbing figure, to be sure, but it’s not as though any one individual needs to count it up in stacks of $20 bills.

And yet, much larger numbers await those who seek them out. Consider the kinds of numbers that crop up in problems involving combinations and permutations. For example, in how many distinct ways can one shuffle a deck of cards? Elwes takes us through the calculation, and we end up with a figure of about 8×1067. Compared to that number, the odds of getting a royal flush when dealt a five-card poker hand seem pretty decent, sitting at a mere 1 in 649,740 (still rare enough that many poker players have never held such a hand). Or consider that famous 1980s cultural touchstone, the Rubik’s cube. In how many ways can one scramble the cube? It turns out that the figure is about 43 quintillion, or 4.3×1019 — but in spite of that ridiculously large figure, people do routinely solve the puzzle, and champions can do it in mere seconds. In fact, as Elwes explains, no Rubik’s cube arrangement is more than 20 moves away from any other arrangement.

Or consider the age of the universe, estimated to be about 13.8 billion years. This may seem like a lengthy span of time, but our cosmic future is where the really big numbers come up. Elwes examines the so-called heat death of the universe, in which all matter has broken down into subatomic particles. We may reach this point in years — this dizzying figure is 10 raised to the power of 10120 — at which point, Elwes says, the universe will have ballooned up to a diameter of 10 to the power of 10 to the power of 10120 light years. (Yes, that’s light years.) Elwes adds a footnote: “At this point, the choice of units hardly matters; the distance is so immense that whether we choose to measure it in Planck lengths or giga-light years makes little difference.” Let that sink in!

As mind numbing as such figures are, the highest numbers contemplated by humans come not from physics but from pure mathematics and computer science. Like “Graham’s number” — an immense figure put forward as the upper-bound for solutions to a problem in a branch of mathematics known as Ramsey theory. Some readers may find the ensuing discussion of multi-dimensional hypercubes a bit challenging, but one can enjoy the payoff regardless: We end up with a number that can’t even be expressed in conventional notation, and which earned a mention in the 1980 edition of the “Guinness Book of World Records” as “the highest number ever used in a mathematical proof.”

Reading this book is a little bit like sitting in the back row of an auction house where a rare Picasso (let’s say) is up for grabs: How high is this thing going to go? And indeed, Elwes keeps going. We eventually meet the so-called busy beaver numbers, a set of numbers that crop up in theoretical computer science, when one tries to deduce whether a particular computer program will eventually stop, or keep going forever — a conundrum known as the “halting problem.” As Elwes explains, it’s not at all straightforward to distinguish the two types of programs (and if it was, it would help mathematicians tackle some of the most vexing problems in their field).

The fifth busy beaver number, known as BB(5) — associated with a computer program that can access five internal states — works out to 47,176,870. And that’s as far as we’ve gotten, Elwes explains. No one has worked out the value of BB(6), but he assures us that it’s beyond the range of any physical computer; and BB(16) leaves even Graham’s number in the dust.

But wait, there’s more! “Rayo’s number,” concocted by Agustín Rayo — a dean and professor at MIT — using set theory, is bigger still (here’s a fun video about it); and “Fish 7,” mentioned in the book’s subtitle, named for a Japanese googologist who goes by the pseudonym “Fish,” builds on Rayo’s number, and … well, the details are not easily digested, but the mind-melting nature of these numbers comes across as a feature, not a bug, of Elwes’s story… the narrative is enlivened by explorations of the peculiarities of math history…

… Archimedes tried to estimate how many grains of sand would be needed to fill up the known universe, back in the third century B.C. Did he simply have too much time on his hands? Not at all, insists Elwes: The Greek thinker was articulating an important idea — that no matter how unfathomably large a quantity may be, we can describe it with precision, thanks to mathematics. “Archimedes,” he writes, “was penning a manifesto for the expressive power of large numbers.”…

… [Elwes focuses] on numbers that are ridiculously large and yet finite. In the end, perhaps this is the most mind-boggling fact of all: that these enormous numbers, from Graham’s number to Fish 7 and beyond, fall as far short of infinity as does the humble number 1…

The mysteries of the massive: “The Mind-Boggling Science of Enormous Numbers,” @danfalk.bsky.social on @richardelwes.bsky.social in @undark.org.

* Steven Strogatz

###

As we enumerate enormity, we might spare a thought for a seminal mathematician, Alan Turing; he died on this date in 1954. He was a foundational computer science pioneer (inventor of the Turing Machine (an influential model for the general-purpose computer), creator of the “Turing Test” (only too relevant in these AI-infected times), inspiration for “The Turing Award” (the “Nobel Prize of computing“), and cryptographer (leading member of the team that cracked the Enigma code during WWII).  

source

“Things that are so far removed from our daily experience… are inherently hard to understand”*…

That’s certainly true of numbers. And as the numbers grow, the cognitive challenges grow with them. (Indeed, by way of example: 1 million seconds, is roughly 11.5 days; 1 billion seconds is almost 32 years.)

We’ve looked before at the mysterious extremes of math: zero and infinity [and here]. But as Dan Falk reminds us, the numbers in between can seem pretty strange as well– especially the extremely large ones. In a review of Richard Elwes‘ Huge Numbers: A Story of Counting Ambitiously, From 4½ to Fish 7, Falk spotlights some of the largest numbers humans have ever contemplated…

… Aficionados of huge numbers are called “googologists,” a reference to the number 10100, known as a googol. Such numbers have a peculiar sort of existence. For the vast majority of us, they’re of limited everyday value. Calculations at the supermarket checkout, or at tax time in April, typically involve far more modest figures. Perhaps we’ve read that the U.S. national debt is in excess of $38 trillion — a mind-numbing figure, to be sure, but it’s not as though any one individual needs to count it up in stacks of $20 bills.

And yet, much larger numbers await those who seek them out. Consider the kinds of numbers that crop up in problems involving combinations and permutations. For example, in how many distinct ways can one shuffle a deck of cards? Elwes takes us through the calculation, and we end up with a figure of about 8×1067. Compared to that number, the odds of getting a royal flush when dealt a five-card poker hand seem pretty decent, sitting at a mere 1 in 649,740 (still rare enough that many poker players have never held such a hand). Or consider that famous 1980s cultural touchstone, the Rubik’s cube. In how many ways can one scramble the cube? It turns out that the figure is about 43 quintillion, or 4.3×1019 — but in spite of that ridiculously large figure, people do routinely solve the puzzle, and champions can do it in mere seconds. In fact, as Elwes explains, no Rubik’s cube arrangement is more than 20 moves away from any other arrangement.

Or consider the age of the universe, estimated to be about 13.8 billion years. This may seem like a lengthy span of time, but our cosmic future is where the really big numbers come up. Elwes examines the so-called heat death of the universe, in which all matter has broken down into subatomic particles. We may reach this point in [10 raised to the 10th power, raised again to the 120th power] years — this dizzying figure is 10 raised to the power of 10120 — at which point, Elwes says, the universe will have ballooned up to a diameter of 10 to the power of 10 to the power of 10120 light years. (Yes, that’s [10 raised to the 10th power, again to the 10th power, then to the 120th power] light years.) Elwes adds a footnote: “At this point, the choice of units hardly matters; the distance is so immense that whether we choose to measure it in Planck lengths or giga-light years makes little difference.” Let that sink in!

As mind numbing as such figures are, the highest numbers contemplated by humans come not from physics but from pure mathematics and computer science. Like “Graham’s number” — an immense figure put forward as the upper-bound for solutions to a problem in a branch of mathematics known as Ramsey theory. Some readers may find the ensuing discussion of multi-dimensional hypercubes a bit challenging, but one can enjoy the payoff regardless: We end up with a number that can’t even be expressed in conventional notation, and which earned a mention in the 1980 edition of the “Guinness Book of World Records” as “the highest number ever used in a mathematical proof.”

Reading this book is a little bit like sitting in the back row of an auction house where a rare Picasso (let’s say) is up for grabs: How high is this thing going to go? And indeed, Elwes keeps going. We eventually meet the so-called busy beaver numbers, a set of numbers that crop up in theoretical computer science, when one tries to deduce whether a particular computer program will eventually stop, or keep going forever — a conundrum known as the “halting problem.” As Elwes explains, it’s not at all straightforward to distinguish the two types of programs (and if it was, it would help mathematicians tackle some of the most vexing problems in their field).

The fifth busy beaver number, known as BB(5) — associated with a computer program that can access five internal states — works out to 47,176,870. And that’s as far as we’ve gotten, Elwes explains. No one has worked out the value of BB(6), but he assures us that it’s beyond the range of any physical computer; and BB(16) leaves even Graham’s number in the dust.

But wait, there’s more! “Rayo’s number,” concocted by Agustín Rayo — a dean and professor at MIT — using set theory, is bigger still (here’s a fun video about it); and “Fish 7,” mentioned in the book’s subtitle, named for a Japanese googologist who goes by the pseudonym “Fish,” builds on Rayo’s number, and … well, the details are not easily digested, but the mind-melting nature of these numbers comes across as a feature, not a bug, of Elwes’s story… the narrative is enlivened by explorations of the peculiarities of math history…

… Archimedes tried to estimate how many grains of sand would be needed to fill up the known universe, back in the third century B.C. Did he simply have too much time on his hands? Not at all, insists Elwes: The Greek thinker was articulating an important idea — that no matter how unfathomably large a quantity may be, we can describe it with precision, thanks to mathematics. “Archimedes,” he writes, “was penning a manifesto for the expressive power of large numbers.”…

… [Elwes focuses] on numbers that are ridiculously large and yet finite. In the end, perhaps this is the most mind-boggling fact of all: that these enormous numbers, from Graham’s number to Fish 7 and beyond, fall as far short of infinity as does the humble number 1…

The mysteries of the massive: “The Mind-Boggling Science of Enormous Numbers,” @danfalk.bsky.social on @richardelwes.bsky.social in @undark.org.

* Steven Strogatz

###

As we enumerate enormity, we might spare a thought for a seminal mathematician, Alan Turing; he died on this date in 1954. He was a foundational computer science pioneer (inventor of the Turing Machine (an influential model for the general-purpose computer), creator of the “Turing Test” (only too relevant in these AI-infected times), inspiration for “The Turing Award” (the “Nobel Prize of computing“), and cryptographer (leading member of the team that cracked the Enigma code during WWII).  

source

oh dang Turing Machine IS fun as I hoped. Fast though.

#BoardGames #TuringMachine #BoardGameArena

oh dang Turing Machine IS fun as I hoped. Fast though.

#BoardGames #TuringMachine #BoardGameArena

oh dang Turing Machine IS fun as I hoped. Fast though.

#BoardGames #TuringMachine #BoardGameArena

oh dang Turing Machine IS fun as I hoped. Fast though.

#BoardGames #TuringMachine #BoardGameArena

#wiskunde #turingmachine

Soms ontdek je zomaar onverwachts echt van die diamantjes op Youtube.

Comprimeren van programma's in multi tape Turing machines:
https://youtu.be/8JuWdXrCmWg

Wat gebeurt er in een LLM? Zelden zo'n compacte goed geïllustreerde uitleg gezien.

https://youtu.be/VkHfRKewkWw

#wiskunde #turingmachine

Soms ontdek je zomaar onverwachts echt van die diamantjes op Youtube.

Comprimeren van programma's in multi tape Turing machines:
https://youtu.be/8JuWdXrCmWg

Wat gebeurt er in een LLM? Zelden zo'n compacte goed geïllustreerde uitleg gezien.

https://youtu.be/VkHfRKewkWw

#wiskunde #turingmachine

Soms ontdek je zomaar onverwachts echt van die diamantjes op Youtube.

Comprimeren van programma's in multi tape Turing machines:
https://youtu.be/8JuWdXrCmWg

Wat gebeurt er in een LLM? Zelden zo'n compacte goed geïllustreerde uitleg gezien.

https://youtu.be/VkHfRKewkWw

#wiskunde #turingmachine

Soms ontdek je zomaar onverwachts echt van die diamantjes op Youtube.

Comprimeren van programma's in multi tape Turing machines:
https://youtu.be/8JuWdXrCmWg

Wat gebeurt er in een LLM? Zelden zo'n compacte goed geïllustreerde uitleg gezien.

https://youtu.be/VkHfRKewkWw

#wiskunde #turingmachine

Soms ontdek je zomaar onverwachts echt van die diamantjes op Youtube.

Comprimeren van programma's in multi tape Turing machines:
https://youtu.be/8JuWdXrCmWg

Wat gebeurt er in een LLM? Zelden zo'n compacte goed geïllustreerde uitleg gezien.

https://youtu.be/VkHfRKewkWw

@beka_valentine I picture:

#TheBaroqueCycle

https://en.wikipedia.org/wiki/The_Baroque_Cycle

Especially the #TechnologickalCollege in

#TheSystemOfTheWorld

https://en.wikipedia.org/wiki/The_System_of_the_World_(novel)

By #NealStephenson in addition to his handling of #Turing in his

#Cryptonomicon

https://en.wikipedia.org/wiki/Cryptonomicon

We still use the purely theoretical #TuringMachine as a base measure of a computional system.

So often it takes a concentration of resources, enabling an inventor to try and fail, over and over again, often at great expense.

@beka_valentine I picture:

#TheBaroqueCycle

https://en.wikipedia.org/wiki/The_Baroque_Cycle

Especially the #TechnologickalCollege in

#TheSystemOfTheWorld

https://en.wikipedia.org/wiki/The_System_of_the_World_(novel)

By #NealStephenson in addition to his handling of #Turing in his

#Cryptonomicon

https://en.wikipedia.org/wiki/Cryptonomicon

We still use the purely theoretical #TuringMachine as a base measure of a computional system.

So often it takes a concentration of resources, enabling an inventor to try and fail, over and over again, often at great expense.

@beka_valentine I picture:

#TheBaroqueCycle

https://en.wikipedia.org/wiki/The_Baroque_Cycle

Especially the #TechnologickalCollege in

#TheSystemOfTheWorld

https://en.wikipedia.org/wiki/The_System_of_the_World_(novel)

By #NealStephenson in addition to his handling of #Turing in his

#Cryptonomicon

https://en.wikipedia.org/wiki/Cryptonomicon

We still use the purely theoretical #TuringMachine as a base measure of a computional system.

So often it takes a concentration of resources, enabling an inventor to try and fail, over and over again, often at great expense.

@beka_valentine I picture:

#TheBaroqueCycle

https://en.wikipedia.org/wiki/The_Baroque_Cycle

Especially the #TechnologickalCollege in

#TheSystemOfTheWorld

https://en.wikipedia.org/wiki/The_System_of_the_World_(novel)

By #NealStephenson in addition to his handling of #Turing in his

#Cryptonomicon

https://en.wikipedia.org/wiki/Cryptonomicon

We still use the purely theoretical #TuringMachine as a base measure of a computional system.

So often it takes a concentration of resources, enabling an inventor to try and fail, over and over again, often at great expense.

@beka_valentine I picture:

#TheBaroqueCycle

https://en.wikipedia.org/wiki/The_Baroque_Cycle

Especially the #TechnologickalCollege in

#TheSystemOfTheWorld

https://en.wikipedia.org/wiki/The_System_of_the_World_(novel)

By #NealStephenson in addition to his handling of #Turing in his

#Cryptonomicon

https://en.wikipedia.org/wiki/Cryptonomicon

We still use the purely theoretical #TuringMachine as a base measure of a computional system.

So often it takes a concentration of resources, enabling an inventor to try and fail, over and over again, often at great expense.

Every single talk that I saw from @rosa is just fantastic and highly recommended. 🤩

https://rosa.codes/machines2025/

#brightonruby #rubyconference #RubyProgramming #godelianriddle #goedelianriddle #hilbertsprogram #decidability #entscheidungsproblem #brightonruby2025 #turingmachine #smullyan #raymondsmullyan

Every single talk that I saw from @rosa is just fantastic and highly recommended. 🤩

https://rosa.codes/machines2025/

#brightonruby #rubyconference #RubyProgramming #godelianriddle #goedelianriddle #hilbertsprogram #decidability #entscheidungsproblem #brightonruby2025 #turingmachine #smullyan #raymondsmullyan

Every single talk that I saw from @rosa is just fantastic and highly recommended. 🤩

https://rosa.codes/machines2025/

#brightonruby #rubyconference #RubyProgramming #godelianriddle #goedelianriddle #hilbertsprogram #decidability #entscheidungsproblem #brightonruby2025 #turingmachine #smullyan #raymondsmullyan

Every single talk that I saw from @rosa is just fantastic and highly recommended. 🤩

https://rosa.codes/machines2025/

#brightonruby #rubyconference #RubyProgramming #godelianriddle #goedelianriddle #hilbertsprogram #decidability #entscheidungsproblem #brightonruby2025 #turingmachine #smullyan #raymondsmullyan

Every single talk that I saw from @rosa is just fantastic and highly recommended. 🤩

https://rosa.codes/machines2025/

#brightonruby #rubyconference #RubyProgramming #godelianriddle #goedelianriddle #hilbertsprogram #decidability #entscheidungsproblem #brightonruby2025 #turingmachine #smullyan #raymondsmullyan

I have no idea how this works, but I love it!

Turing Machine, a punch card computer puzzle game. Your three chosen numbers produce a punch card that can be used to test certain conditions (e.g Blue is even.)

Great fun, challenging, and unique.

#BoardGames #TuringMachine #Reviews

I have no idea how this works, but I love it!

Turing Machine, a punch card computer puzzle game. Your three chosen numbers produce a punch card that can be used to test certain conditions (e.g Blue is even.)

Great fun, challenging, and unique.

#BoardGames #TuringMachine #Reviews

I have no idea how this works, but I love it!

Turing Machine, a punch card computer puzzle game. Your three chosen numbers produce a punch card that can be used to test certain conditions (e.g Blue is even.)

Great fun, challenging, and unique.

#BoardGames #TuringMachine #Reviews

I have no idea how this works, but I love it!

Turing Machine, a punch card computer puzzle game. Your three chosen numbers produce a punch card that can be used to test certain conditions (e.g Blue is even.)

Great fun, challenging, and unique.

#BoardGames #TuringMachine #Reviews

I have no idea how this works, but I love it!

Turing Machine, a punch card computer puzzle game. Your three chosen numbers produce a punch card that can be used to test certain conditions (e.g Blue is even.)

Great fun, challenging, and unique.

#BoardGames #TuringMachine #Reviews

Turing-Drawings

https://github.com/maximecb/Turing-Drawings

#HackerNews #TuringDrawings #TuringMachine #GitHub #ArtTech #CreativeCoding #TechInnovation

Turing-Drawings

https://github.com/maximecb/Turing-Drawings

#HackerNews #TuringDrawings #TuringMachine #GitHub #ArtTech #CreativeCoding #TechInnovation

Turing-Drawings

https://github.com/maximecb/Turing-Drawings

#HackerNews #TuringDrawings #TuringMachine #GitHub #ArtTech #CreativeCoding #TechInnovation

Turing-Drawings

https://github.com/maximecb/Turing-Drawings

#HackerNews #TuringDrawings #TuringMachine #GitHub #ArtTech #CreativeCoding #TechInnovation

How to Optimize Rust for Slowness: Inspired by New Turing Machine Results

https://medium.com/@carlmkadie/how-to-optimize-your-rust-program-for-slowness-eb2c1a64d184

#HackerNews #Rust #Optimization #TuringMachine #Programming #SlowPerformance

How to Optimize Rust for Slowness: Inspired by New Turing Machine Results

https://medium.com/@carlmkadie/how-to-optimize-your-rust-program-for-slowness-eb2c1a64d184

#HackerNews #Rust #Optimization #TuringMachine #Programming #SlowPerformance

How to Optimize Rust for Slowness: Inspired by New Turing Machine Results

https://medium.com/@carlmkadie/how-to-optimize-your-rust-program-for-slowness-eb2c1a64d184

#HackerNews #Rust #Optimization #TuringMachine #Programming #SlowPerformance

How to Optimize Rust for Slowness: Inspired by New Turing Machine Results

https://medium.com/@carlmkadie/how-to-optimize-your-rust-program-for-slowness-eb2c1a64d184

#HackerNews #Rust #Optimization #TuringMachine #Programming #SlowPerformance

@ramin_hal9001 ooh. If you look at all possible computer programs only very simple ones can be written with branching (if else) and loops.

But all computer programs (no matter how hard or genius the algorithm is) can be written with branching and recursion. So I would just assume they’re doing something very smart by using a recursive algorithm 😆

#functionalprogramming #computerscience #turingmachine #algorithms #startrek

@ramin_hal9001 ooh. If you look at all possible computer programs only very simple ones can be written with branching (if else) and loops.

But all computer programs (no matter how hard or genius the algorithm is) can be written with branching and recursion. So I would just assume they’re doing something very smart by using a recursive algorithm 😆

#functionalprogramming #computerscience #turingmachine #algorithms #startrek

@ramin_hal9001 ooh. If you look at all possible computer programs only very simple ones can be written with branching (if else) and loops.

But all computer programs (no matter how hard or genius the algorithm is) can be written with branching and recursion. So I would just assume they’re doing something very smart by using a recursive algorithm 😆

#functionalprogramming #computerscience #turingmachine #algorithms #startrek

It is capable of simulating computations using 7-bit instructions over a total of 224 bits (about 14 bytes), allowing it to theoretically run 2^224 different programs.

#lego #ideas #legoideas #turing #turingmachine #machine #functional #model #computation #computer #mechanical #programming

It is capable of simulating computations using 7-bit instructions over a total of 224 bits (about 14 bytes), allowing it to theoretically run 2^224 different programs.

#lego #ideas #legoideas #turing #turingmachine #machine #functional #model #computation #computer #mechanical #programming

It is capable of simulating computations using 7-bit instructions over a total of 224 bits (about 14 bytes), allowing it to theoretically run 2^224 different programs.

#lego #ideas #legoideas #turing #turingmachine #machine #functional #model #computation #computer #mechanical #programming

It is fully mechanical, constructed from over 2,900 LEGO pieces, runs without electricity and is powered by a hand crank.

The machine has a moving head that reads and writes symbols on a physical tape that supports four symbols and eight states, allowing for 32 possible symbol-state combinations.

#lego #ideas #legoideas #turing #turingmachine #machine #functional #model #computation #computer #mechanical #programming

It is fully mechanical, constructed from over 2,900 LEGO pieces, runs without electricity and is powered by a hand crank.

The machine has a moving head that reads and writes symbols on a physical tape that supports four symbols and eight states, allowing for 32 possible symbol-state combinations.

#lego #ideas #legoideas #turing #turingmachine #machine #functional #model #computation #computer #mechanical #programming

It is fully mechanical, constructed from over 2,900 LEGO pieces, runs without electricity and is powered by a hand crank.

The machine has a moving head that reads and writes symbols on a physical tape that supports four symbols and eight states, allowing for 32 possible symbol-state combinations.

#lego #ideas #legoideas #turing #turingmachine #machine #functional #model #computation #computer #mechanical #programming

So, someone has built a working Turing machine entirely out of Lego bricks, and the project has advanced to the next stage of review by Lego Ideas after gaining 10,000 supporters.

#lego #ideas #legoideas #turing #turingmachine #machine #functional #model #computation #computer #mechanical

So, someone has built a working Turing machine entirely out of Lego bricks, and the project has advanced to the next stage of review by Lego Ideas after gaining 10,000 supporters.

https://ideas.lego.com/projects/10a3239f-4562-4d23-ba8e-f4fc94eef5c7

#lego #ideas #legoideas #turing #turingmachine #machine #functional #model #computation #computer #mechanical

So, someone has built a working Turing machine entirely out of Lego bricks, and the project has advanced to the next stage of review by Lego Ideas after gaining 10,000 supporters.

https://ideas.lego.com/projects/10a3239f-4562-4d23-ba8e-f4fc94eef5c7

#lego #ideas #legoideas #turing #turingmachine #machine #functional #model #computation #computer #mechanical

So, someone has built a working Turing machine entirely out of Lego bricks, and the project has advanced to the next stage of review by Lego Ideas after gaining 10,000 supporters.

https://ideas.lego.com/projects/10a3239f-4562-4d23-ba8e-f4fc94eef5c7

#lego #ideas #legoideas #turing #turingmachine #machine #functional #model #computation #computer #mechanical

This is probably the coolest #Lego build I've ever seen. They built a working #TuringMachine entirely out of Lego Technics!

https://ideas.lego.com/projects/10a3239f-4562-4d23-ba8e-f4fc94eef5c7/updates#content_nav_tabs

This is probably the coolest #Lego build I've ever seen. They built a working #TuringMachine entirely out of Lego Technics!

https://ideas.lego.com/projects/10a3239f-4562-4d23-ba8e-f4fc94eef5c7/updates#content_nav_tabs

This is probably the coolest #Lego build I've ever seen. They built a working #TuringMachine entirely out of Lego Technics!

https://ideas.lego.com/projects/10a3239f-4562-4d23-ba8e-f4fc94eef5c7/updates#content_nav_tabs

This is probably the coolest #Lego build I've ever seen. They built a working #TuringMachine entirely out of Lego Technics!

https://ideas.lego.com/projects/10a3239f-4562-4d23-ba8e-f4fc94eef5c7/updates#content_nav_tabs

This is probably the coolest #Lego build I've ever seen. They built a working #TuringMachine entirely out of Lego Technics!

https://ideas.lego.com/projects/10a3239f-4562-4d23-ba8e-f4fc94eef5c7/updates#content_nav_tabs