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Articles related to "mathematics"


Nicolas Bourbaki: The greatest mathematician who never was

  • For example, it is alleged that, while visiting Finland at the outset of World War II, French mathematician André Weil was investigated for spying.
  • The name Nicolas Bourbaki first appeared in a place rocked by turmoil at a volatile time in history: Paris in 1934.
  • The nine mathematicians in attendance agreed to write a textbook “to define for 25 years the syllabus for the certificate in differential and integral calculus by writing, collectively, a treatise on analysis,” which they hoped to complete in just six months.
  • Over time, the name Bourbaki became a collective pseudonym for dozens of influential mathematicians spanning generations, including Weil, Dieudonne, Schwartz, Borel, Grothendieck and many others.
  • Now in “his” 80th year of research, in 2016 “he” published the 11th volume of the “Elements of Mathematics.” The Bourbaki group, with its ever-changing cast of members, still holds regular seminars at the University of Paris.

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In Mathematics, It Often Takes a Good Map to Find Answers

  • Depending on who you ask, for example, present-day mathematicians have nearly as much chance of solving the Riemann hypothesis — the most famous unsolved problem in math — as da Vinci had of building a machine that could actually fly.
  • “It’s my belief that we might solve both at the same time, even if they seem to be quite far from any island I know how to reach with my math techniques,” Maynard said.
  • Other times, though, it’s not even clear what it would take to solve a problem — it’s only evident mathematicians can’t do it.
  • Given all this uncertainty, mathematicians work to develop a sense for what kinds of problems they have a chance of solving using the centuries of techniques available to them.

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How to use science to decide when you don’t know what’s going to happen

  • The mathematics of decision making under uncertainty (what we call imprecise probabilities) is concerned with making the best decisions given that you don’t know exactly what the worst thing that will happen is or how likely it is.
  • Mathematicians define the best decision as the one that induces the highest gain, a number that measures the overall benefit of making a given decision.
  • It turns out there is a precise mathematical answer, it comes from calculating the “expected value” which is simply the cost or benefit of an outcome times its probability.
  • If you want to win at any game of chance, all you have to do is figure out what the expected value is and make sure you can play many, many times in a row.
  • You subtract their gains and compute the worst case expected values of the decisions as a pair.

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How to use science to decide when you don’t know what’s going to happen

  • The mathematics of decision making under uncertainty (what we call imprecise probabilities) is concerned with making the best decisions given that you don’t know exactly what the worst thing that will happen is or how likely it is.
  • Mathematicians define the best decision as the one that induces the highest gain, a number that measures the overall benefit of making a given decision.
  • It turns out there is a precise mathematical answer, it comes from calculating the “expected value” which is simply the cost or benefit of an outcome times its probability.
  • If you want to win at any game of chance, all you have to do is figure out what the expected value is and make sure you can play many, many times in a row.
  • You subtract their gains and compute the worst case expected values of the decisions as a pair.

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Science without Validation in a World without Meaning

  • Modern scientific knowledge, while rejecting commonsense conceptual models, has always depended upon mathematically expressed theories that could be validated by prediction and observation.
  • Kline, writing in the twentieth century, was looking back from a post–quantum theory world and saw the full import of Newton’s basic assumption: “For I here design only to give a mathematical notion of these forces, without considering their physical causes and seats.” As of 1687, scientific knowledge was constituted in mathematics; “hypotheses” such as causality were no longer part of science.
  • If the meaning of the subject matter cannot be conceived in terms of ordinary categories of understanding, and cannot be expressed in ordinary human language, then how can leaders untrained in mathematics and scientific epistemology make informed decisions?

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Penrose: From mathematical notation to beautiful diagrams

  • Rather than rely on a fixed library of visualization tools, the visual representation is user-defined in a constraint-based specification language; diagrams are then generated automatically via constrained numerical optimization.
  • In contrast to tools that specify diagrams via direct manipulation or low-level graphics programming, Penrose enables rapid creation and exploration of diagrams that faithfully preserve the underlying mathematical meaning.
  • By specifying diagrams in terms of abstract relationships rather than explicit graphical directives, they are easily adapted to a wide variety of use cases.
  • A language-based design makes it easy to build tools on top of Penrose that provide additional power.
  • shaded disks), but allow one to use a completely different visual representation (e.g., a tree showing set inclusions).
  • A language-based specification makes it easy to visually inspect data structures or assemble progressive diagrams with only minor changes to program code.

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