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Set Theory and Weather Prediction
Here’s a puzzle: You and Bob are going to play a game which has the following steps. Bob thinks of some function (it’s arbitrary: it doesn’t have to be continuous or anything). You pick an . Bob...
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Almost a Number-Theoretic Miracle
An arithmetic statement is one made up of quantifiers “,” “,” the logical connectives “and,” “or,” “not”, function symbols , , constants , , and variables which are bound by the aforementioned...
View ArticleA Curious Application of Ambiguity with Respect to the Possessive Form
Why did the chicken cross the island on Lost? To get to the Others’ side. (Composed by Tim Goldberg.)
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Non-Rigorous Arguments 1: Two Formulas For e
I’m a big fan of non-rigorous arguments, especially in calculus and analysis. I think there should be a book cataloging all the beautiful, morally-true-but-not-actually-true proofs that mathematicians...
View ArticleLots of Fun Math Papers
In the course of looking up a link for my last blog entry, I discovered the MAA Writing Awards site, which collects many pdfs of articles that have won MAA writing awards. From browsing it a bit, it...
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Integrability Conditions (Guest Post!)
Please enjoy the following guest post on differential geometry by Tim Goldberg. A symplectic structure on a manifold is a differential -form satisfying two conditions: is non-degenerate, i.e. for each...
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A Geometrically Natural Uncomputable Function
There are many functions from to that cannot be computed by any algorithm or computer program. For example, a famous one is the halting problem, defined by if the th Turing machine halts and if the th...
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The Undecidability of Identities Involving Sine, Exponentiation, and Absolute...
In the book A=B, the authors point out that while the identity is provable (by a very simple proof!), it’s not possible to prove the truth or falsity of all such identities. This is because Daniel...
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Avoiding Set-Theoretic Paradoxes using Symmetry
Intuitively, for any property of sets, there should be a set which has as its members all and only those sets such that holds. But this can’t actually work, due to Russell’s Paradox: Let , and then you...
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Playing Games in the Transfinite: An Introduction to “Ordinal Chomp”
Chomp is a two-player game which is played as follows: The two players, A and B, start with a “board” which is a chocolate bar divided into small squares. With Player A starting, they take turns...
View ArticleA language which does term inference
Many strongly typed languages like OCaml do type inference. That is, even though they’re strongly typed, you don’t have to explicitly say what the type of everything is since a lot of the time the...
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What Would the World Look Like if Everything was Computable?: An Introduction...
Suppose that we wanted to construct a mathematical universe where all objects were computable in some sense. How would we do it? Well, we could certainly allow the set into our universe: natural...
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When are the Real Numbers Necessary?
The natural numbers can all be finitely represented, as can the rational numbers. The real numbers, however, cannot be so represented and require some notion of “infinity” to define. This makes it both...
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Two Puzzles in Recursion Theory: Verbose Sets and Terse Sets
Let be the set of all such that the th Turing machine halts. (For these puzzles, we will assume that Turing machines are always run on a blank initial state, i.e., they take no input.) Recall that is...
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Another Puzzle in Recursion Theory: n-Enumerable Sets
We can think of a computably enumerable (or c.e.) set as a bag which some computer program puts more and more numbers into over time. The set then consists of all numbers which are in the bag from some...
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How to Show that Games are Hard
Peg Solitaire is a pretty popular game, often found in restaurants (including Cracker Barrel, if I remember correctly). It’s also NP-complete (by which I mean determining a winning strategy given the...
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Games Which are Impossible to Analyze
In the last post, I mentioned the computational complexity of various games. To be explicit, we consider each “game” to actually be a sequence of games for . For example, would be checkers played on an...
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Doing Calculus on the Rationals (with the help of Nonstandard Analysis)
Nonstandard Analysis is usually used to introduce infinitesimals into the real numbers in an attempt to make arguments in analysis more intuitive. The idea is that you construct a superset which...
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A Simple Introduction to Quantum Groups
In the course of reading some background material for an article by James Worthington on using bialgebraic structures in automata theory, I was led to finally reading up on what a Hopf algebra...
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Trigonometric Series and the Beginnings of Set Theory
Let be a -periodic function. It may or may not have a representation as a trigonometric series A natural question to ask is whether or not the representation of as a trigonometric series is unique, if...
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What Happens When You Iterate Gödel’s Theorem?
Let be Peano Arithmetic. Gödel’s Second Incompleteness Theorem says that no consistent theory extending can prove its own consistency. (I’ll write for the statement asserting ‘s consistency; more on...
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An Interesting Puzzle in Propositional Logic
Suppose that you’re translating an ancient text, and in this text you come across three words whose meaning you are unsure of: , , and . So, you head down to the ancient language department of your...
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A Suite of Cool Logic Programs
You may have heard about the Tarski-Seidenberg theorem, which says that the first-order theory of the reals is decidable, that the first-order theory of the complex numbers is similarly decidable, or...
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Functions with Very Low Symmetry and the Continuum Hypothesis
A function from to is called even if for all , . We might call it even about the point if, for all , . Conversely, we can call a function strongly non-even if for all , , . Finding strongly non-even...
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Quantish Physics: A Discrete Model of Quantum Physics
In the book Good and Real, author Gary Drescher, who received his PhD from MIT’s AI lab, defends the view that determinism is a consistent and coherent view of the world. In doing so, he enters many...
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Two Interesting Observations about Voting I Hadn’t Seen Until Recently
By “voting”, I mean the following general problem: Suppose there are candidates and voters. Each voter produces a total ordering of all candidates. A voting procedure is a function which takes as...
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Topology and First-Order Modal Logic
The normal square root function can be considered to be multi-valued. Let’s momentarily accept the heresy of saying that the square root of a negative number is , so that our function will be total....
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The Spectrum From Logic to Probability
Let be the set of propositions considered by some rational logician (call her Sue). Further, suppose that is closed under the propositional connectives , , . Here are two related but different...
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A Logical Interpretation of Some Bits of Topology
Edit: These ideas are also discussed here and here (thanks to Qiaochu Yuan: I found out about those links by him linking back to this post). Although topology is usually motivated as a study of spatial...
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Mathematica and Quantifier Elimination
In 1931, Alfred Tarski proved that the real ordered field allows quantifier elimination: i.e., every first-order formula is equivalent to one with no quantifiers. This is implemented in Mathematica’s...
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Generating Functions as Cardinality of Set Maps
There is a class of all cardinalities , and it has elements , and operations , , and so forth defined on it. Furthermore, there is a map which takes sets to cardinalities such that (and so on)....
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Complexity to Simplicity and Back Again
Generalizing a problem can make the solution simpler or more complicated, and it’s often hard to predict which beforehand. Here’s a mini-example of a puzzle and four generalizations which alternately...
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Why is the derivative of a generating function meaningful?
A generating function is a formal power series where the sequence of coefficients is the object of interest. Usually the point of using them is that operations on the power series (like addition,...
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A Good Definition of Randomness
Most mathy people have a pretty good mental model of what a random process is (for example, generating a sequence of 20 independent bits). I think most mathy people also have the intuition that there’s...
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Two Constants: Khinchin and Chaitin
Take a real number, . Write out its continued fraction: It’s an intriguing fact that if you look at the sequence of geometric means this approaches a single constant, called Khinchin’s constant, which...
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A Complexity-Theoretic Account of The Strong Law of Small Numbers
The Strong Law of Small Numbers (see also Wikipedia) says that “There aren’t enough small numbers to meet the many demands made of them.” It means that when you look at small numbers, it’s easy to see...
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YouTube Physics Explanations Shouldn’t Use the Right-Hand Rule
Popular explanations of physical phenomena like gyroscopes or magnetic fields often end up having to explain the right-hand rule to explain how rotational quantities add (say, by using the right-hand...
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The Arithmetic Hierarchy Meets the Real World
Mathematical logic has a categorization of sentences in terms of increasing complexity called the Arithmetic Hierarchy. This hierarchy defines sets of sentences and for all nonnegative integers . The...
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Gravity is Stronger Than I Thought
I’m not a physicist, and I’d always supposed that, while the Earth has a significant gravitational pull because it’s so massive, the gravitational pull between everyday objects must be completely...
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Thermodynamics is Easier Than I Thought
Actually, thermodynamics is hard and I don’t understand it. But even without totally understanding thermodynamics, it turns out its possible to do a surprising number of useful calculations with just...
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The CGP Grey Sheaf of Continents
CGP Grey is a youtuber with a variety of interesting videos, often about the quirks of geography and political boundaries. In this video, he asks the question “How many continents are there?”,...
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Making Money Disappear Through Infinite Iteration, Now In YouTube Form!
A while ago, I wrote a blog post called Making Money Disappear Through Infinite Iteration, and I just put out a video version of this post on youtube. Note: it’s very rough and unpolished, but I hope...
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How is it even possible for a sailboat to sail into the wind?
Until this morning, I didn’t really understand how it was possible for a sailboat to sail into the wind: popular descriptions like Wikipedia’s talk about keels and lift and the Bernoulli effect and so...
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How does the Infinitesimal Intuition About Lie Brackets Actually Work?
You can often get the gist of a mathematical subject via an informal explanation involving infinitesimals. But I often find that questions arise from that informal explanation that I’d like resolved,...
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Differentiating Sine Without Doing Any Work or Knowing Anything
If you do a google search for how to derive the facts that and , most of the derivations you’ll find rely on knowing something like the double angle formula and go through a direct, non-trivial...
View ArticleHoping for Lane Closures
Two lanes of a four lane highway are closed and you’re stuck in a traffic jam. If there are no on- or off-ramps, what should you hope to see on the road ahead of you: the other two lanes re-open or one...
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What does it mean to extend the manipulability of differentials?
In an interesting paper called Extending the Manipulability of Differentials, the authors Jonathan Bartlett and Asatur Zh. Khurshudyan describe an interesting proposal for representing higher-order...
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But Why Is Proof by Contradiction Non-Constructive?
We think of a proof as being non-constructive if it proves “There exists an such that without ever actually exhibiting such an . If you want to form a system of mathematics where all proofs are...
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A Nice Definition of “Field Theory”
Like most people, I don’t really know anything about quantum field theory. But the other day I stumbled across this paper by Stefano Gogioso, Maria E. Stasinou, and Bob Coecke that provides a very...
View ArticleThe Axiom of Choice Isn’t Always Non-Constructive
The Axiom of Choice is usually introduced as a non-constructive axiom that mathematicians used to care about but don’t really pay much attention to anymore. It’s true that mainstream mathematicians...
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