Tag Archives: Mathematics

How to play mathematics

BOOK REVIEW

The Pearly Gates of Cyberspace, Author: Margaret Wertheim.
Physics on the Fringe, Author: Margaret Wertheim.
African Fractals: Modern Computing and Indigenous Design, Author: Ron Eglash.

(glasbergen.com)By Margaret Wertheim – The world is full of mundane, meek, unconscious things materially embodying fiendishly complex pieces of mathematics. How can we make sense of this? I’d like to propose that sea slugs and electrons, and many other modest natural systems, are engaged in what we might call the performance of mathematics.

Rather than thinking about maths, they are doing it.

In the fibers of their beings and the ongoing continuity of their growth and existence they enact mathematical relationships and become mathematicians-by-practice. By looking at nature this way, we are led into a consideration of mathematics itself not through the lens of its representational power but instead as a kind of transaction.

Rather than being a remote abstraction, mathematics can be conceived of as something more like music or dancing; an activity that takes place not so much in the writing down as in the playing out.

Since at least the time of Pythagoras and Plato, there’s been a great deal of discussion in Western philosophy about how we can understand the fact that many physical systems have mathematical representations: the segmented arrangements in sunflowers, pine cones and pineapples (Fibonacci numbers); the curve of nautilus shells, elephant tusks and rams horns (logarithmic spiral); music (harmonic ratios and Fourier transforms); atoms, stars and galaxies, which all now have powerful mathematical descriptors; even the cosmos as a whole, now represented by the equations of general relativity.

The physicist Eugene Wigner has termed this startling fact ‘the unreasonable effectiveness of mathematics’.

Why does the real world actualize maths at all? And so much of it?

Even arcane parts of mathematics, such as abstract algebras and obscure bits of topology often turn out to be manifest somewhere in nature. more> https://goo.gl/ifKV2Z

How thinking about infinity changes kids’ brains on math

BOOK REVIEW

Remembering: A Study in Experimental and Social Psychology, Author: Frederic Bartlett.

By Sarah Scoles – No mathematical concept is more intense than infinity. Which makes infinity uniquely relevant to addressing some key concerns about modern education.

According to the constructivist philosophy of education, built on the ideas of the late philosopher Ernst von Glasersfeld [2, 3, 4, 5], this experience altered my brain’s perception of mathematics, even though it didn’t involve doing math in the traditional sense.

Students today often feel like they are drowning in a sea of standardized tests. A seemingly rigorous approach has left many of them rather good at math tests while leaving them bad at math as a concept, and at the crucial forms of logical thinking that comes with it.

Infinity provides an antidote. It has the power to create conceptual wows – and to do so even in minds that have not yet been exposed to algebra or any kind of number theory.

Infinity raises its fist to rote memorization and multiple-choice testing, because encounters with infinity are fundamentally conceptual in nature.

They correspondingly create conceptual (as opposed to procedural) knowledge – a foundational comprehension of, for example, what multiplication is, and the ability to understand its utility in a variety of situations. more> https://goo.gl/gbkU67

The Man Who Tried to Redeem the World with Logic

BOOK REVIEW

Principia Mathematica, Authors: Alfred North Whitehead and Bertrand Russell.
The Ego and the Id, Author: Sigmund Freud.

By Amanda Gefter – Their building block was the proposition—the simplest possible statement, either true or false. From there, they employed the fundamental operations of logic, like the conjunction (“and”), disjunction (“or”), and negation (“not”), to link propositions into increasingly complicated networks. From these simple propositions, they derived the full complexity of modern mathematics.

Which got Warren McCulloch thinking about neurons. He knew that each of the brain’s nerve cells only fires after a minimum threshold has been reached: Enough of its neighboring nerve cells must send signals across the neuron’s synapses before it will fire off its own electrical spike. It occurred to McCulloch that this set-up was binary—either the neuron fires or it doesn’t.

A neuron’s signal, he realized, is a proposition, and neurons seemed to work like logic gates, taking in multiple inputs and producing a single output. By varying a neuron’s firing threshold, it could be made to perform “and,” “or,” and “not” functions. more> http://goo.gl/ywv7iQ

Mathematical Imagery

Sphere packing 1 Sphere packing 2
Sphere packing 3 Sphere packing 4
Sphere packing 5 Sphere packing 6
NSF – How does one fill a sphere with smaller spheres of various sizes so that every possible void is filled? There are only five known configurations, all obtained by a sphere inversion transformation, the 3-D equivalent of a circle inversion. Initial spheres are positioned at the vertices of a platonic solid. The sphere containing and touching all these spheres is part of the constellation also, but is obviously never drawn in the images. Inversion spheres are placed so that they are orthogonal to the spheres at the faces of the platonic solid. One inversion sphere is at the center, orthogonal to all initial spheres. For the case of the tetrahedron, cube and dodecahedron, there is only one possible configuration, consisting of touching initial spheres. In the case of the octahedron, there is a constellation with touching spheres and one other with non-touching spheres. No packing can be obtained starting from the icosahedron. The colors of the individual spheres in the images correspond to the iteration count.

This mathematical imagery was produced by Jos Leys.

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NIST Releases Successor to Venerable Handbook of Math Functions

NIST – [VIDEO 2:57] The National Institute of Standards and Technology (NIST) has released the Digital Library of Mathematical Functions (DLMF) and its printed companion, the NIST Handbook of Mathematical Functions, the much-anticipated successors to the agency’s most widely cited publication of all time. These reference works contain a comprehensive set of tools useful for specialists who work with mathematical modeling and computation.

The two works comprise a complete update and expansion of the 1964 Handbook of Mathematical Functions, which upon its publication quickly became an indispensable reference for scientists and engineers who use the tools of applied mathematics. NIST embarked on the new work in response to the Internet revolution in information exchange as well as advances in mathematics itself.

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