I am just back from the G4G conference in Atlanta, where I gave a six-minute talk on how to organize a waiting line. The video of the actual talk will appear on the web eventually, but in the meantime, here is video of my practice run from the night before:
Or, for higher quality video, click here.
Ethan Siegel, writing in Forbes, concludes that No, the Universe is Not Purely Mathematical in Nature.
His argument, unless I’ve badly misunderstood him, is that many purely mathematical models of the universe have turned out to be wrong, and one needs observations to guide the building of better models.
I think he has this exactly backward.
If our Universe is uniquely woven from some special fabric, then it at least might (or might not) be possible to discern some of its properties by pure reason.
But if our Universe is a purely mathematical structure, then it is surely one of a great many purely mathematical structures (we know this, because we are familiar with a great many purely mathematical structures). This means that only observations can help us determine which of those mathematical structures we inhabit.
Siegel’s article is well written and fun to read. But I think his arguments constitute evidence for exactly the opposite of the conclusion he wants to draw.
Nearly 9 years ago, Steven posted this brainteaser:
Here I have a well shuffled deck of 52 cards, half of them red and half of them black. I plan to slowly turn the cards face up, one at a time. You can raise your hand at any point — either just before I turn over the first card, or the second, or the third, et cetera. When you raise your hand, you win a prize if the next card I turn over is red.
What’s your strategy?
Read no further if you want to try and solve this brainteaser on your own first!
Continue reading ‘Using a Brainteaser to Discover a Theorem’
I was recently asked to speak at the awards ceremony for the winners of the Witwatersrand math competition. This presented a particular challenge, because there were winners in age groups ranging from nine-year-olds to college students. Here is the talk I ended up giving:
The poster to the left hangs on the wall of my office. Can you figure out the pattern to the sequence? Now can you estimate the size of the nth entry?
John Horton Conway died yesterday, a victim of Covid-19. His unique mathematical style combined brilliance and playfulness in equal measure. I came across his “soldier problem” at an impressionable age, and was astonished by the beauty of the solution. You’ve got an infinite sheet of graph paper, with one horizontal grid line marked as the boundary between friendly and enemy territory. You can place as many soldiers as you like in friendly territory, at most one to a square. Now you can start jumping your soldiers over each other — a soldier jumps horizontally or vertically, over an adjacent soldier (who is then removed from the board) into an empty square. The goal is to advance at least one soldier to the fifth row of enemy territory. Conway’s proof that it can’t be done struck me then as utterly beautiful, utterly unexpected, and a compelling reason to learn more about this mathematics business.
He invented the Game of Life. He invented the system of surreal numbers, a vast generalization of the usual real numbers, designed for the purpose of assigning values to positions in games but adopted by mathematicians for purposes well beyond its original design. (Don Knuth’s book on the subject is a classic, and easy reading). His Monstrous Moonshine conjecture is the reason I own a t-shirt that says:
which I am prepared to argue is the most remarkable equation in all of mathematics. He was the first person to prove that every natural number is the sum of 37 fifth-powers. More than a century after mathematicians first gave a complete classification of two-dimensional surfaces, Conway (together with George K. Francis) found a much better proof. He worked in geometry, analysis, algebra, number theory and physics. And reportedly, he could solve a Rubik’s Cube behind his back, after inspecting it for a few seconds.
Gone now, along with John Prine — another icon of my youth — and too many others. I got the word about Conway just as I was about to go to bed, and am typing this in a state of exhaustion. If I were more awake, it would be more coherent, but it will have to do.
Christmas week seems like a good time to share this video of my talk on “Truth, Provability and the Fabric of the Universe” delivered in March, 2018 in Madison, Wisconsin. The venue was the Free Thought Festival sponsored by a student group that goes by the umbrella title of “Atheists, Humanists and Agnostics” (AHA for short).
The video is below (or will be soon if you’re patient; it might take a minute or two to load). (Edited to add: I believe I’ve fixed things so it loads quickly now; please let me know if there are any problems.)
Or you can either:
Or:
A big hat tip to Lisa Talpey for cleaning up the video and making it possible to see both my face and the slides at the same time.
The answer to Friday’s puzzle is YES. If I am a logic machine who only states what I can prove, and if I say “If I can prove there is no God, then there is no God”, it does follow that I can prove there is no God.
Once again, it was our commenter Leo who got this first (in Friday’s comment thread, graciously rot-13’d).
As with Thursday’s solution to Wednesday’s puzzle, there are two key relevant background facts:
A) An inconsistent system can prove anything.
B) A sufficiently complex consistent system cannot prove its own consistency. (This is Godel’s second incompleteness theorem.)
Here’s the logic:
1) I’ve asserted that “if I can prove there is no God, then there is no God”. We know that I assert only things that I can prove. Therefore I can prove this assertion.
2) That means I can also prove the equivalent assertion that “if there is a God, I cannot prove otherwise”.
3) Therefore, if I take my axioms and add the axiom “There is a God”, then I can prove that there is something I cannot prove.
4) Therefore, if I take my axioms and add the axiom “There is a God”, then I can prove that my axiom system is consistent (by Background Fact A.)
5) Therefore, if I take my axioms and add the axiom “There is a God”, my axiom system is inconsistent. (Because only an inconsistent system can prove its own consistency — that is, Background Fact B.)
6) Therefore the statement “There is a God” must contradict my axiom system.
7) This can happen only if my axiom system is able to prove that “There is no God”.
So yes, I can prove there is no God.
Continue reading ‘Monday Solution’
Here are the answers to yesterday’s puzzle. The first correct solution came from our commenter Leo (comment #18 on yesterday’s post).
The assumptions of the problem were: Everything I say out loud can be deduced from my axioms. My axioms include the ordinary axioms for arithmetic, among other things. And I recently said out loud that “I cannot prove that God does not exist”.
The questions were: Can I prove there is no God? Can I prove there is a God? And is there enough information her to determine whether there actually is a God?
The answers are yes, yes and no: Yes, I can prove there is no God. Yes, I can also prove there is a God. And no, you can’t use any of this to determine whether there is a God.
To explain, I’ll use the phrase “logical system” to refer to a system of axioms sufficiently strong to talk about basic arithmetic (and perhaps a whole lot of other things), together with the usual logical rules of inference. It’s given in the problem statement that I am a logical system.
Here are two background facts about logical systems:
A. An inconsistent logical system can prove anything at all. That’s because it’s tautological that if P is self-contradictory, then any statement of the form “P implies Q” is valid. If I’m inconsistent, that means I can prove at least one statement (call it P) that’s self-contradictory. Then if I want to prove, say, that the moon is made of green cheese, I note that:
B. No consistent logical system can prove its own consistency. This is Godel’s celebrated Second Incompleteness Theorem.
Now here’s the argument:
Continue reading ‘Thursday Solution’
I’ll be speaking this Saturday at the Freethought Festival in Madison, Wisconsin (follow the link to register!) on the topic “Truth, Provability and the Fabric of the Universe”. I’ll be glad to see you there.
Time is scarce, and lately I’ve been devoting it mostly to things other than blogging — but I feel the need to emerge from hiding just long enough to acknowledge the shocking death today of the extraordinary mathematician Vladimir Voevodsky at the age of 51.
Voevodsky is best known for finding the right definition of motivic cohomology sometime around the year 2000. This was a Holy Grail, the quest for which had been set in motion by the earth-shattering vision of Alexander Grothendieck. I happen to have been leafing through my well-worn copy of Voevodsky’s book on Cycles, Transfers and Motivic Homology Theories (co-written with Andrei Suslin and Eric Friedlander) when I heard of his death.
More recently, Voevodsky had turned his attention to logic and the foundations of mathematics. Here is video of his talk “What if the Current Foundations of Mathematics Are Inconsistent?”
A brief obituary is here. A brief mathematical autobiography, written by Voevodsky, is here. I might return and add a few personal reminscences.
The most exciting news of the past week had nothing to do with James Comey or Donald Trump.
The University of Montpelier has released high-quality scans of about 18,000 pages of notes and scribbles by Alexandre Grothendieck. If you’re competent in both French and the art of deciphering handwriting that was never meant to be readable except to the author, you can while away some hours sifting through them here.
It would be an understatement to say that Grothendieck was never shy about revealing and publicly analyzing his thought processes, but these notes are presumably less filtered than the tens of thousands of pages he chose to share in his lifetime. FOr the many who are already sifting through them, and for the many more who are waiting to hear the reports of the sifters, they will yield new insights into one of the most extraordinary minds in human history.
It’s said that Pythagoras had a man put to death for blabbing in a public bar that the square root of two is irrational. Today I hope I can post this without fear of reprisals. I don’t know who first drew this beautiful proof, which works equally well in modern English and in ancient Greek:
I never met Alexander Grothendieck. I was never in the same room with him. I never even saw him from a distance. But whenever I think about math — which is to say, pretty much every day — I feel him hovering over my shoulder. I’ve strived to read the mind of Grothendieck as others strive to read the mind of God.
Those who did know him tend to describe him as a man of indescribable charisma, with a Christ-like ability to inspire followers. I’ve heard it said that when Grothendieck walked into a room, you might have had no idea who he was or what he did, but you definitely knew you wanted to devote your life to him.
And people did. In 1958, when Grothendieck (aged 30) announced a massive program to rewrite the foundations of geometry, he assembled a coterie of brilliant followers and conducted a seminar that met 10 hours a day, 5 days a week, for over a decade. Grothendieck talked; others took notes, went home, filled in details, expanded on his ideas, wrote final drafts, and returned the next day for more. Jean Dieudonne, a mathematician of quite considerable prominence in his own right, subjugated himself entirely to the project and was at his desk every morning at 5AM so that he could do three hours of editing before Grothendieck arrived and started talking again at 8:00. (Here and elsewhere I am reporting history as I’ve heard it from the participants and others who followed developments closely as they were happening. If I’ve got some details wrong, I’m happy to be corrected.) The resulting volumes filled almost 10,000 pages and rocked the mathematical world. (You can see some of those pages here).
I want to try to give something of the flavor of the revolution that unfolded in that room, and I want to do it for an audience with little mathematical background. This might require stretching some analogies almost to the breaking point. I’ll try to be as honest as I can. In the first part, I’ll talk about Grothendieck’s radical approach to mathematics generally; after that, I’ll talk (in a necessarily vague way) about some of his most radical and important ideas.
Continue reading ‘The Generalist’
Around 1970, Alexander Grothendieck, the greatest of all modern mathematicians and arguably the greatest mathematician of all time, announced — at the age of 42 — the official end of his research career. Another great mathematician once told me that he thought he knew why. Following two decades of discoveries and insights that, one after the other, stunned the mathematical world, Grothendieck had, for the first time, achieved an insight so unexpected and so consequential that he himself was stunned. Grothendieck had discovered his own mortality.
I am told that just a few hours ago, his vision proved accurate. But the notion of Grothendieck as a mortal seems hard to swallow. He dominated pure mathematics not just through the force of his ideas — ideas that seemed eons ahead of everyone else’s — but through the force of his personality. When, around 1960, he announced his audacious plan to solve the notoriously difficult Weil conjectures by first rewriting the foundations of geometry, dozens of superb mathematicians put the rest of their careers on hold to do their parts. The project’s final page count, including the twelve volumes known as SGA (Seminaire de Geometrie Algebrique) and the eight known as EGA (Elements de Geometrie Algebrique) approached 10,000 pages. The force and clarity of Grothendieck’s unique vision scream forth from nearly every one of those pages, demanding that the reader see the mathematical world in a new and completely original way — a perspective that has proved not just compelling, but unspeakably powerful.
In Grothendieck, modesty would have been ridiculous, and he was never ridiculous. Here, in his own words — words that ring utterly true — is Grothendieck’s own assessment of how he stood apart (translated from French by Roy Lisker):
Most mathematicians take refuge within a specific conceptual framework, in a “Universe” which seemingly has been fixed for all time – basically the one they encountered “ready-made” at the time when they did their studies. They may be compared to the heirs of a beautiful and capacious mansion in which all the installations and interior decorating have already been done, with its living-rooms , its kitchens, its studios, its cookery and cutlery, with everything in short, one needs to make or cook whatever one wishes. How this mansion has been constructed, laboriously over generations, and how and why this or that tool has been invented (as opposed to others which were not), why the rooms are disposed in just this fashion and not another – these are the kinds of questions which the heirs don’t dream of asking . It’s their “Universe”, it’s been given once and for all! It impresses one by virtue of its greatness, (even though one rarely makes the tour of all the rooms) yet at the same time by its familiarity, and, above all, with its immutability.
If you want to compute the circumference of the observable universe to within, say, the width of a human hair, you’ll need to know about 35 digits of π, though this never seems to deter a certain sort of person from memorizing the first 100, 200 or 500 digits. But it turns out there’s no need to memorize anything at all! You can recover any number of digits you like from a simple little physics experiment that I just learned about, though it was invented over ten years ago by Professor Gregory Galperin of Eastern Illinois University. His lovely little paper is here.
To see how it works, start with two identical billiards lined up in front of a wall like so:
Now push Ball 2 toward Ball 1 and count the collisions: First Ball 2 collides with Ball 1 and pushes it toward the wall. (At this point Ball 2 has transferred all its momentum to Ball 1 and stops moving). Then Ball 1 collides with the wall and bounces back toward Ball 2. Then Ball 1 collides with Ball 2 and pushes it off to a far-away place. Three collisions. That tells you that π starts with a 3.
If you want more accuracy, make Ball 2 exactly 100 times as heavy as Ball 1. This time the sequence of events is a little more complicated, but it turns out there are exactly 31 collisons. That tells you that π starts with 3.1.
Or if you prefer, make Ball 2 exactly 10,000 times as heavy as Ball 1. You’ll get exactly 314 collisions. π starts with 3.14.
Continue reading ‘Follow the Bouncing Ball’
Last week, I posted video of my talk to the undergraduate math students on truth, provability and the fabric of the universe — and heard from several readers who requested that I post it in a non-flash format.
My readers’ wish is my command. Here is the file for download in an m4v format.
This is relatively low resolution. The best viewing is still the flash version here.
Here is my talk to the University of Rochester’s Society of Undergraduate Math Students on “Truth, Provability and the Fabric of the Universe”. The audience was great, and except for a couple of slips of the tongue (like “Sir William of Ockham” for “William of Ockham”), I thought it went very well.
Faithful readers will recognize multiple themes from the book The Big Questions, and from numerous past blog posts, including:
Continue reading ‘Truth, Provability and the Fabric of the Universe’
Max Tegmark is a professor of physics at MIT, a major force in the development of modern cosmology, a lively expositor, and the force behind what he calls the Mathematical Universe Hypothesis — a vision of the Universe as a purely mathematical object. Readers of The Big Questions will be aware that this is a vision I wholeheartedly embrace.
Tegmark’s new book Our Mathematical Universe is really several books intertwined, including:
In 1706, the British astronomer John Machin calculated π to 100 digits (by hand of course). His trick was to notice that π = 16A – 4B where A and B are given by
If you’re computing by hand, this is an excellent discovery, because the series for A involves a lot of divisions by 5, which are a lot easier to calculate than, say, divisions by 7, and the series for B converges very fast, so just a few terms buys you a whole lot of accuracy. (Try using, say, just the first four terms of A and just the first term of B to see what I mean.)
Machin’s 100 digits were a substantial improvement over the 72 digits obtained just a little earlier by Abraham Sharp, using the far less efficient series
In 1729, a Frenchman named de Lagny got all the way to 127 digits, but, in the words of the scientist/engineer/philosopher/historian Petr Beckmann (of whom more later), de Lagny “sweated these digits out by Sharp’s series, and so exhibited more computational stamina than mathematical wits.”
Machin’s methods were ingenious, but no more ingenious — and certainly no more striking — than John Wallis’s 1655 discovery that
Continue reading ‘A Pi-Day Treat’
My mother, who reads this blog, reports that she’s lost a few nights’ sleep lately, tormented by thoughts of Knights, Knaves and Crazies. Serves her right. Once when she and I were very young, she tormented me with a geometry puzzler that I now know she must have gotten (either directly or indirectly) from Lewis Carroll; you can find it here. If she remembers the solution, she should be able to sleep tonight.
Herewith, a proof that a right angle can equal an obtuse angle. The puzzle, of course, is to figure out where I cheated.
But wait! Let’s do this as a video, since I’m starting to fool around with this technology and could use the practice. Consider this more or less a first effort. If you prefer the old ways, you can skip the video and read the (identical) step-by-step proof below the fold.
Or, if you prefer to skip the video, start here:
Big news from the math world:
One of the oldest problems in number theory is the twin primes problem: Are there or are there not infinitely many ways to write the number 2 as a difference of two primes? You can, for example, write 2 = 5 -3, or 2 = 7 – 5, or 2 = 13 – 11. Does or does not this list go on forever? There are very strong reasons to believe the answer is yes, but many a great mathematician has tried and failed to find a proof.
Here’s a related problem: Are there or are there not infinitely many ways to write the number 4 as a difference of two primes? What about the number 6? Or 8? Or any even number you care to think about? It seems likely that the answer is yes in every case, though no proof is known in any case. But….
Continue reading ‘News From The Math World’
When you met the late Armen Alchian on the street, he used to greet you not with “Hello” or “How ya doin’?”, but with “What did you learn today?” Today I learned that there are contexts in which the most ludicrous reasoning is guaranteed to lead you to a correct conclusion. This is too cool not to share.
But first a little context. The first part is a little less cool, but it’s still fun and it will only take a minute.
First, I have to tell you what a tree is. A tree is something that has a root, and then either zero or two branches growing out of that root, and then either zero or two branches (a “left branch” and a “right branch”) growing out of each branch end, and then either zero or two branches growing out of each of those branch ends, and so on. Here are some trees. (The little red dots are the branch ends and the big black dot is the root; these trees grow upside down.)
A pair of trees is, as you might guess, two trees — a first and a second.
There are infinitely many trees, and infinitely many pairs of trees, and purely abstract considerations tell us that there’s got to be a one-one correspondence between these two infinities. But what if we ask for a simple, easily describable one-one correspondence? Well, here’s an attempt: Starting with a pair of trees, you can create a single tree by creating a root with two branches, and then sticking the two trees from the pair onto the ends of the two branches. Like so:
Continue reading ‘Seven Trees in One’
 |
 |
 |
 |
 |
 |
You and a stranger have been instructed to meet up sometime tomorrow, somewhere in New York City. You (and the stranger) can decide for yourselves when and where to look for each other. But there can be no advance communication. Where do you go?
Me, I’d be at the front entrance to the Empire State Building at noon, possibly missing my counterpart, who might be under the clock at Grand Central Station. But, because there are only a small number of points in New York City that stand out as “extra-special”, we’ve at least got a chance to find each other.
A Schelling point is something that stands out from the background so sharply that we can expect people to coordinate around it. Schelling points are on my mind this week, because I’ve just heard David Friedman give a fascinating talk about the evolution of property rights, and Schelling points play a big role in his story. But that story is not the topic of this post.
Instead, I’m curious about the Schelling points that say, two mathematicians, or two economists, or two philosophers, or two poets, or two street hustlers might converge on. Suppose, for example, that you asked two mathematicians each to separately pick a number between 200 and 300, with a prize if their answers coincide. I’m guessing they both go for 256, the only power of two within range.
Continue reading ‘A Sip of Monstrous Moonshine’
Over at Less Wrong, the estimable Eliezer Yudkowsky attempts to account for the meaning of statements in arithmetic and the ontological status of numbers. I started to post a comment, but it got long enough that I’ve turned my comment into a blog post. I’ve tried to summarize my understanding of Yudkowsky’s position along the way, but of course it’s possible I’ve gotten something wrong.
It’s worth noting that every single point below is something I’ve blogged about before. At the moment I’m too lazy to insert links to all those earlier blog posts, but I might come back and put the links in later. In any event, I think this post stands alone. Because it got long, I’ve inserted section numbers for the convenience of commenters who might want to refer to particular passages.
1. Yudkowsky poses, in essence, the following question:
Yudkowsky phrases the question a little differently. What he actually asks is:
This, I think, threatens to confuse the issue. It’s important to distinguish between the numeral “2”, which is a formal symbol designed to be manipulated according to formal rules, and the noun “two”, which appears to name something, namely a particular number. Because Yudkowsky is asking about meaning and truth, I presume it is the noun, and not the symbol, that he intends to mention. So I’ll stick with my version, and translate his remarks accordingly.
2. Yudkowsky provisionally offers the following answer:
Continue reading ‘Accounting for Numbers’
The (really really) big news in the math world today is that Shin Mochizuki has (plausibly) claimed to have solved the ABC problem, which in turn suffices to settle many of the most vexing outstanding problems in arithmetic. Mochizuki’s work rests on so many radically new ideas that it will take the experts a long time to digest. I, who am not an expert, will surely die with only a vague sense of the argument. But based on my extremely limited (and possibly mistaken) understanding, it appears that Mochizuki’s breakthrough depends at least partly on his willingness to abandon the usual axioms for the foundations of mathematics and replace them with new axioms. (See, for example, the first page of these notes from one of Mochizuki’s lectures. You can find other related notes here.)
That’s interesting for a lot of reasons, but the one that’s most topical for The Big Questions is this: No mathematician would consider rejecting Mochizuki’s proof just because it relies on new axiomatic foundations. That’s because mathematicians (or at least the sort of mathematicians who study arithmetic) don’t particularly care about axioms; they care about truth.
There’s a widespread misconception that arithmetic is about “what can be derived from the axioms”, which is a lot like saying that astronomy is about “what can be discovered through telescopes”. Axiomatic systems, like telescopes, are investigative tools, which we are free to jettison when better tools come along. The blather of thoughtless imbeciles notwithstanding, what really matters is the fundamental object of study, whether it’s the system of natural numbers or the planet Jupiter.
Mathematicians care about what’s true, not about what’s provable; if a truth isn’t provable, we’re fine with changing the rules of the game to make it provable.
Continue reading ‘Simple as ABC’
I mentioned earlier this week that I’d been crafting a long post on the fabric of the Universe when I was sidetracked by relatively mundane political events. Now I’ve been sidetracked again by the entirely unexpected (to me) news of the death from melanoma, at age 65, of the Fields Medalist Bill Thurston, who devoted his life to understanding the shape of space.
One-dimensional topology is the study of curves and two-dimensional topology is the study of surfaces. Both subjects are quite well understood. Thurston was the king of three-dimensional topology, which gains additional interest from the fact that we perceive ourselves as living in a three-dimensional Universe. Three-dimensional topology attempts to classify all the possible shapes for that Universe.
One of course is also interested in four, five, six and many-dimensional topology, four dimensions being of particular interest because they can be used to model space together with time. But although three dimensions are more complicated than two and two are more complicated than one, it turns out that when you go much higher, a lot of things get simpler. Consider knots, for example. There are no knots inside a one or two dimensional space; a knot needs three dimensions in which to pass over and under itself. But in more than three dimensions, you can untie any knot just by pulling on its ends — roughly because the additional dimensions give it so much space in which to untangle itself. For those and related reasons, topology is often hardest in three and four dimensions — coincidentally (or maybe not) the very dimensions most relevant to the way we experience the world.
Thurston revolutionized three-dimensional topology in the 1980s with his geometrization conjecture, which says that any three-manifold (the three-dimensional analogue of a smooth curve or surface) can be cut up into pieces, each of which exhibits one of eight permissible geometries. The simplest of those geometries is the flat three-dimensional space you think you see around you, where you can draw three straight lines in mutually perpendicular directions and extend them forever. Another is the geometry of the three-dimensional sphere, which is an analogue of the two-dimensional surface of the earth, where any “straight” line eventually circles back to meet itself.
The geometrization conjecture was important, but what really mattered was the vast array of new techniques Thurston introduced for visualizing and understanding the structure of three-manifolds. When those techniques came on line in the early 1980s, he was widely acclaimed as the mathematician of the decade.
One thing that set Thurston apart was his insistence that mathematics is a human study, and that it’s the mathematician’s job to communicate not just theorems and proofs, but a unique way of thinking. Stories are often told of mid-twentieth century mathematicians (usually French) who, when asked a question about their work, would scribble a picture on the blackboard, deliberately stand in front of that picture to shield it from everyone else’s view, and then, having studied it a few minutes, erased the picture, turned around, and gave a purely formal explanation designed to obscure all of the motivation and insight. Nobody ever told a story like that about Bill Thurston. Here he is, talking about the mystery of three-manifolds; dip in at a random moment and chances are excellent you’ll hear him talking not about how he proved a theorem but about how he sees the world:
Continue reading ‘The Visionary’
 |
 |
If you study economics, or statistics, or chemistry, or mathematical biology, or thermodynamics, you’re sure to encounter the notion of a Markov chain — a random process whose future depends probabilistically on the present, but not on the past. If you travel through New York City, randomly turning left or right at each corner, then you’re following a Markov process, because the probability that you’ll end up at Carnegie Hall depends on where you are now, not on how you got there.
But even if you work with Markov processes every day, you’re probably unaware of their origins in a dispute about free will, Christianity, and the Law of Large Numbers.
Continue reading ‘Chain Reaction’
There’s an old puzzle popular among a certain type of schoolchildren that challenges the solver to write as many positive integers as possible using exactly four 4’s, together with some set of mathematical operations. (As is often the case with school children, the exact rules tend to get negotiated in real time as the puzzle is being solved.) Some examples are:
But when I became a man, I put away childish things. So here’s the grown-up version of the problem, which I got from Mel Hochster over 20 years ago, and still don’t know how to solve:
This being a grown-up problem, the rules are carefully specified:
Continue reading ‘Tuesday Puzzle’
In the comments section of Bob Murphy’s blog, I was asked (in effect) why I insist on the objective reality of the natural numbers (that is, the counting numbers 0,1,2,3…) but not of, say, the real numbers (that is, the numbers we use to represent lengths — and that are themselves represented by possibly infinite decimal expansions).
There seem to be two kinds of people in the world: Those with enough techncal backgroud that they already know the answer, and those with less technical background, who have no hope — at least without a lot of work — of grasping the answer. I’m going to attempt to bridge that gap here. That means I’m going to throw a certain amount of precision to the winds, in hopes of being comprehensible to a wider audience.
Continue reading ‘The Number Devil’
Stanley Tennenbaum was an itinerant mathematician with, for much of his adult life, no fixed address and no permanent source of income. Sometimes he slept on park benches. He didn’t have a lot of teeth.
But if you were involved with mathematics in the second half of the twentieth century, sooner or later you were going to cross paths with Stanley, probably near the coffee machine in a math department. He’d proudly show you the little book he carried in his breast pocket, with the list of people to whom he owed money. Then he’d teach you something, or he’d tell you a good story.
Stanley had little tolerance for convention. His one permanent job, at the University of Rochester, came to an abrupt end during a faculty meeting where he spit on the shoes of the University president and walked out. Surely the same personality trait had something to do with his departure from the University of Chicago without a Ph.D., though the paper he wrote there (at age 22) has acquired fame and influence far beyond many of the doctoral theses of his more conventionally successful classmates. I’d like to tell you a little about that paper and what I think it means for the foundations of mathematics.
Continue reading ‘That Does Not Compute’
