The Home of Mathematical Knitting
Posts tagged mathematics
Mathematical notation provides perhaps the best-known and best-developed example of language used consciously as a tool of thought. Recognition of the important role of notation in mathematics is clear from the quotations from mathematicians given in Cajori’s A History of Mathematical Notations [2, pp.332,331]. Nevertheless, mathematical notation has serious deficiencies. In particular, it lacks universality, and must be interpreted differently according to the topic, according to the author, and even according to the immediate context. Programming languages, because they were designed for the purpose of directing computers, offer important advantages as tools of thought. Not only are they universal (general-purpose), but they are also executable and unambiguous. Executability makes it possible to use computers to perform extensive experiments on ideas expressed in a programming language, and the lack of ambiguity makes possible precise thought experiments. In other respects, however, most programming languages are decidedly inferior to mathematical notation and are little used as tools of thought in ways that would be considered significant by, say, an applied mathematician.
I’m just archiving this Asian Age summary of a lecture from 9th April 2015, because the newspaper webpage has vanished. [Photos]
Time Out listing: How would you design an object for a world that does not exist? What does such an object say about the world in which we actually live? This idea tugs at the core of ‘design fiction’ practice. For instance, the iPad first appeared in Kubrick’s 2001: A Space Odyssey. Its writer Arthur C. Clarke was also the first to imagine geostationary satellites. The impact of Minority Report on human-computer interfaces cannot be overstated. Even outside of fully formed fictional worlds, a standalone object can trigger many unexpected narratives, such as the famous 3D-printed gun or the US Army’s “indestructible sandwich”. We will discuss these and many other examples of speculative design in this talk.
Asian Age Article: (16 Apr) For Rohit Gupta, the essential question isn’t “why” but “why not”. He held forth on the concept of “design fiction” at a talk in the city recently. His previous projects include trying to figure out a way to fit astronomical contraptions on top of auto-rickshaws and coming up with a mechanism to type through walking (in which one could type out a whole text message in no less than seven hours!). While many around him may wonder “why”, for Rohit Gupta aka Compasswala aka fadesingh, the only question is “why not”. Giving a talk on design fiction at the Maker’s Asylum, the researcher who studies the history of science and mathematics explained why for him fiction was everywhere, not just in the depiction of future, but even the past. Speaking about what exactly design fiction is, Rohit says, “It’s about the objects. Design fiction deals with how to create objects that describe or imply a story or an aspect about a world that doesn’t exist.” Going on to give us an example in his own style, Rohit says, “Let us consider hypothetically that there was a catastrophic event in Mumbai in 1960 that entirely changed the city. Now let us take a map of Mumbai in 2015 that shows how it looks now in that scenario. We don’t have to describe everything that happened in the time frame between the disaster and now, but just the map, which is an object of design fiction can show or tell us a huge number of details about that world. ‘That’ is design fiction.” Rohit adds, “Design fiction has existed for a long time. Now we may have sci-fi movies and earlier there were books. But those were just the interfaces. It has existed for long before these interfaces came about.” While sci-fi and fiction is usually considered to depict the future or altogether different realities, Rohit contends, it is equally relevant and present in describing the past as well.
He explains, “Not many might have heard about the Ishango bone. Now the Ishango bone is considered to be the oldest mathematical instrument known to man. But basically it is just a simple bone with hand carved lines drawn on it in varying sequences. Now what these prehistoric humans were trying to do with those lines we don’t know, but researchers have interpreted various reasons ranging from calculating menstrual cycles to lunar calendars. But this is our modern interpretation of what this particular object tells us. It could well have been something else but these are the stories we are interpreting from it. So this is design fiction as well, only in the past.” Design fiction, says Rohit, varies from the miniscule to the astronomical. “You could create a simple toy in a workshop or you could even create an enter solar system like Asimov (Isaac) did in Nightfall.” But while the potential of design fiction could be limitless, it is upto us to ask the questions from whence we can derive the answers says Rohit. “This is increasingly becoming a trend. Researchers in top institutes are taking questions that may sound ridiculous and are coming up with the most scientific explanations for them. For example, 'How does a Muslim astronomer face Mecca while in space’ but believe it or not the Malaysians have actually come up with an entire manual for it.” And progress, says Rohit is all about not shying away from doing what may sound crazy. “One of my friends, a poet named Christian Book is now engaged in a project to create the world’s first indestructible book. How he’s doing it is the most interesting part. He actually took a strain of this microbe called Dienococcus Radiodurans, which is an extremophile (Something which can survive in extreme conditions such nuclear blasts, volcanoes or even in space) and imprinting a poem into its very DNA and is planning to launch it off into space. Now whom he is writing for or what the poem itself is irrelevant. But the only question is 'Why the hell not’,” concludes the Compasswala.
What does “causality” mean, and how can you represent it mathematically? How can you encode causal assumptions, and what bearing do they have on data analysis? These types of questions are at the core of the practice of data science, but deep knowledge about them is surprisingly uncommon.
a paper by Marko Rodriguez called A Methodology For Studying Various Interpretations of the N,N-dimethyltryptamine-Induced Alternate Reality, [suggesting] among other things that you could prove DMT entities were real by taking the drug and then asking the entities you meet to factor large numbers which you were sure you couldn’t factor yourself. So to that end, could you do me a big favor and tell me the factors of 1,522,605,027, 922,533,360, 535,618,378, 132,637,429, 718,068,114, 961,380,688, 657,908,494, 580,122,963, 258,952,897, 654,000,350, 692,006,139? “Universal love,” said the cactus person. “Transcendent joy,” said the big green bat. The sea turned hot and geysers shot up from the floor below. First one of wine, then one of brine, then one more yet of turpentine, and we three stared at the show.
a paper by Marko Rodriguez called A Methodology For Studying Various Interpretations of the N,N-dimethyltryptamine-Induced Alternate Reality, [suggesting] among other things that you could prove DMT entities were real by taking the drug and then asking the entities you meet to factor large numbers which you were sure you couldn’t factor yourself. So to that end, could you do me a big favor and tell me the factors of 1,522,605,027, 922,533,360, 535,618,378, 132,637,429, 718,068,114, 961,380,688, 657,908,494, 580,122,963, 258,952,897, 654,000,350, 692,006,139?
“Universal love,” said the cactus person.
“Transcendent joy,” said the big green bat.
The sea turned hot and geysers shot up from the floor below. First one of wine, then one of brine, then one more yet of turpentine, and we three stared at the show.”
Here, we study the characteristics of functional brain networks at the mesoscopic level from a novel perspective that highlights the role of inhomogeneities in the fabric of functional connections. […] The results show that the homological structure of the brain’s functional patterns undergoes a dramatic change post-psilocybin, characterized by the appearance of many transient structures of low stability and of a small number of persistent ones that are not observed in the case of placebo.
Popping peyote buttons with his assistant in the laboratory, Klüver noticed the repeating geometric shapes in mescaline-induced hallucinations and classified them into four types, which he called form constants: tunnels and funnels, spirals, lattices including honeycombs and triangles, and cobwebs. In the 1970s the mathematicians Jack D. Cowan and G. Bard Ermentrout used Klüver’s classification to build a theory describing what is going on in our brain when it tricks us into believing that we are seeing geometric patterns. Their theory has since been elaborated by other scientists, including Paul Bressloff, Professor of Mathematical and Computational Neuroscience at the newly established Oxford Centre for Collaborative Applied Mathematics.
I very much enjoy reading the “What Is…?” column in the Notices of the AMS. Unfortunately, there seemed to be no index to this column. I have therefore created this one in the hope that it’ll be helpful to others as well.
Excellence comes from qualitative changes in behavior, not just quantitative ones. More time practicing is not good enough. Nor is simply moving your arms faster! A low-level breaststroke swimmer does very different things than a top-ranked one. The low-level swimmer tends to pull her arms far back beneath her, kick the legs out very wide without bringing them together at the finish, lift herself high out of the water on the turn, and fail to go underwater for a long ways after the turn. The top-ranked one sculls her arms out to the side and sweeps back in, kicks narrowly with the feet finishing together, stays low on the turns, and goes underwater for a long distance after the turn. They’re completely different!
This post is a crash course on the notation used in programming language theory (“PL theory” for short). For a much more thorough introduction, I recommend Types and Programming Languages by Benjamin C. Pierce and Semantic Engineering with PLT Redex by Felleisen, Findler, and Flatt. I’ll assume the reader is an experienced programmer but not an experienced mathematician or PL theorist. I’ll start with the most basic definitions and try to build up quickly.
Attention conservation notice: Over 7800 words about optimal planning for a socialist economy and its intersection with computational complexity theory. This is about as relevant to the world around us as debating whether a devotee of the Olympian gods should approve of transgenic organisms. (Or: centaurs, yes or no?) Contains mathematical symbols (uglified and rendered slightly inexact by HTML) but no actual math, and uses Red Plenty mostly as a launching point for a tangent.
Description of the method, with corresponding parts of the DFT highlighted (via http://www.altdevblogaday.com/2011/05/17/understanding-the-fourier-transform/)
I’ve always been intrigued by the sensation of movement in music. And it is fair to say that it was my first calculus class that led me to graduate study in mathematics because, for the first time, I saw movement in mathematics. My fascination with each of these was nudged again by an interview with jazz pianist Vijay Iyer that I heard on NPR’s All Things Considered.