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he authors jonathan bartlett and asatur zh khurshudyan describe an interesting proposal for representing higher order derivatives the argument is basically this as is well known the chain rule for first derivatives seems to follow algebraically if you use leibniz notation for the derivatives however there s a written by mkoconnor november 9 2019 hoping 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 of the two open lanes close i think you should hope to see written by mkoconnor october 27 2019 october 27 2019 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 evaluation of it s actually possible to compute these derivatives this without really remembering any specific facts about trigonometry written by mkoconnor october 16 2019 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 but i don t want to jump all the way to the full definitions without a rigorous basis for reasoning about infinitesimals it can be written by mkoconnor may 29 2017 may 29 2017 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 forth but this feels like a leap beyond my understanding i wanted an account of how it is even possible in written by mkoconnor may 7 2017 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 to make more videos and get better at their production as time goes on i d been interested in written by mkoconnor july 24 2016 july 24 2016 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 discusses a variety of subtleties in the notion of continent and concludes that it is not well defined enough to provide an answer let s written by mkoconnor may 19 2016 may 19 2016 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 a couple of simple rules about entropy the setup is as follows imagine that there is some set of states of the world called the written by mkoconnor may 18 2016 may 18 2016 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 undetectable or maybe only detectable with modern laboratory equipment but i only thought that because i never bothered to actually plug in any numbers using written by mkoconnor may 14 2016 may 15 2016 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 definition is as follows and are both equal to the set of sentences such that a computer can determine the truth or falsity of in finite written by mkoconnor may 9 2016 may 9 2016 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 rule to convert angular momenta into vectors then adding the vectors this is bad not just because the right hand rule is confusing but because it written by mkoconnor may 9 2016 may 9 2016 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 compelling patterns that turn out to be false when you look at larger numbers using complexity theory we written by mkoconnor may 5 2016 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 is approximately for almost every this means that if you were to pick for convenience say it s between 0 and 1 written by mkoconnor may 3 2016 may 4 2016 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 a sense in which an individual string like 10101001110000100101 is more random than 0000000000000000000 even though both strings are equally written by mkoconnor august 21 2015 august 21 2015 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 multiplication and differentiation correspond to meaningful operations on your sequence of coefficients i ve known about the gist of generating functions for a written by mkoconnor february 13 2012 february 13 2012 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 make it simpler or more complicated written by mkoconnor december 24 2011 december 24 2011 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 ordinary generating functions can be thought of entirely analogously with set maps replacing sets there is a written by mkoconnor december 15 2011 december 15 2011 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 resolve function the resolve function is called like resolve formula domain where domain gives the domain for the quantifiers in formula since we ll always be working over in written by mkoconnor july 9 2011 july 10 2011 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 structures you can interpret topological spaces as being a particular type of logic and give a purely logical non spatial written by mkoconnor september 18 2010 september 19 2010 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 preorders on if logically entails if sue considers at least as likely to be true as is let be the equivalence written by mkoconnor march 13 2010 march 13 2010 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 how can we represent the situation of this branching function topologically written by mkoconnor february 22 2010 march 14 2010 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 input all orderings and produces an output ranking of all candidates arrow s impossibility theorem states that there is really no satisfactory voting procedure written by mkoconnor february 17 2010 february 17 2010 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 different arenas ethics decision theory and physics in his chapter on quantum mechanics he defends the many worlds interpretation written by mkoconnor july 19 2009 july 20 2009 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 functions is easy as any injective function provides a trivial example we can written by mkoconnor may 14 2009 may 15 2009 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 that the first order theory of the integers without multiplication is decidable in the course of john harrison s logic textbook handbook of practical logic written by mkoconnor april 9 2009 april 9 2009 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 local university the first professor you come across knows what means the second professor you come across written by mkoconnor march 23 2009 april 2 2009 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 this later in particular is stronger than but certainly given that we believe that everything proves is true we believe that does not prove a written by mkoconnor december 20 2008 december 21 2008 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 it has one it was the consideration of this question that led cantor to the invention of set theory written by mkoconnor december 11 2008 december 13 2008 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 sometimes called a quantum group is although it is not strictly related to logic i ll write up what i learned here written by mkoconnor november 14 2008 may 5 2016 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 contains the reals and also some infinitesimals prove that some statement holds of and then use a general transfer principle to conclude that written by mkoconnor november 9 2008 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 board the problem was then to analyze the computational complexity of the function which takes and tells you which written by mkoconnor november 3 2008 november 6 2008 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 initial set up is an np complete problem you may have also heard of computational complexity results for minesweeper see here for example there are a written by mkoconnor october 26 2008 october 26 2008 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 point on i e the numbers which are eventually put in the bag by the program suppose written by mkoconnor october 16 2008 november 6 2008 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 computably enumerable but not decidable puzzle 1 describe a winning strategy for the following game you are written by mkoconnor october 13 2008 october 14 2008 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 computationally and philosophically interesting to determine for what purposes you need the real numbers and for what purposes you need only the written by mkoconnor october 13 2008 april 2 2009 what would the world look like if everything was computable an introduction to hyland s effective topos 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 numbers are the most basic computational objects there are notation i ll use to refer to when we re considering it as part of written by mkoconnor october 7 2008 october 9 2008 a 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 l...