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February 11 2013

The Mathematical Journey | Life Skills | National Numeracy

Numeracy is a key skill necessary to allow each of us to make informed choices and decisions in all aspects of everyday life.

 

See it on Scoop.it, via oAnth's day by day interests - via its scoop.it contacts
Reposted from02mysoup-aa 02mysoup-aa

January 18 2012

The Man of Numbers: How Fibonacci Changed the World | brainpickings.org

What Medieval mathematics have to do with remix culture, publishing entrepreneurship, and gamification.

 

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 // oAnth

 "The change in society brought about by the teaching of modern arithmetic was so pervasive and all-powerful that within a few generations people simply took it for granted. There was no longer any recognition of the magnitude of the revolution that took the subject from an obscure object of scholarly interest to an everyday mental tool. Compared with Copernicus’s conclusions about the position of Earth in the solar system and Galileo’s discovery of the pendulum as a basis for telling time, Leonardo’s showing people how to multiply 193 by 27 simply lacks drama.” ~ Keith Devlin

Original URL -


November 16 2011

Talk Nondually & Mathematics Without Appearing Ludicrous


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oAnth:

recommendation - talk ~35 min - math as a factor of cultures and socialisation.
Reposted fromsigaloninspired sigaloninspired

October 26 2011

02mydafsoup-01

September 06 2011

Philosophia Scientiæ - Travaux d'histoire et de philosophie des sciences

Philosophia Scientiæ est une revue scientifique à comité de lecture qui publie des travaux en épistémologie, en histoire et en philosophie des sciences. Elle accueille notamment des études traitant des mathématiques, de la physique et de la logique, mais elle est ouverte aux travaux portant sur les autres disciplines scientifiques.

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// oAnth

préface est librement disponible

http://philosophiascientiae.revues.org/354



Reposted from02mysoup-aa 02mysoup-aa

June 29 2011

02mydafsoup-01

April 13 2011

CAPTCHA chaos by Marianne Freiberger | plus.maths.org

[...]

The new method, proposed by T.V. Laptyeva and S. Flach from the Max-Planck-Institute for the Physics of Complex Systems in Germany and K. Kladko from Axioma Research in the US, also uses standard encryption algorithms, like AES, but it works some magic with the password. It uses a long password consisting of two parts. One part is an easily memorable short password (SP) and the other is longer and known as the strong key. The confidential text is encrypted using a standard algorithm like AES and the combination of short password and strong key as the password.

Image embedding

The strong key — in this example the word "chaos" — is embedded in an image.

The strong key doesn't have to be memorised by the user. Instead the strong key is then embedded in an image in a similar way as for the familiar CAPTCHA test, which shows the user a distorted image of a word which computers used to find difficult to identify (though good image recognition software can these days do the trick).

But here's the trick: the image can also be interpreted as a snapshot in time of a dynamical system — a system that evolves over time according to a set of mathematical equations. The state of the system at each individual time step can be described using numbers and represented by an image.

The particular dynamical system used in this case comes from physics, where it is used to model the behaviour of ferroelectric materials, such as barium titanate. These are akin to magnetic materials, but rather than responding to magnetic fields, they respond to electric fields, becoming electrically polarised when an electric field is applied.

Ferroelectric materials lose their ferrorelectric property at a certain temperature. Such a sudden change in a material's porperties is called a phase transition. Typically a system undergoing a phase transition moves from an ordered state — electric polarisation in the case of ferroelectric materials — to a disordered state. The dynamical system the researchers chose for their encryption method is a simple model of the behaviour of ferroelectric materials close to the phase transition. One of the properties the system exhibits near this order-disorder transition point is chaos: parameters describing the system fluctuate wildly and the system is so sensitive to small changes that it is impossible to predict exactly what state it will be in after a number of time steps.

Image evolution

The image containing the strong key (called the ISK) is evolved in time (downward direction). Even the smallest change in the resulting image will cause you to miss the original when you go back in time.

Now suppose you have arrested this dynamical system at a particular moment in time, say at time $t_0$. The encryption method works by embedding the strong key in the corresponding image using something akin to the CAPTCHA method. The system is then allowed to evolve for a certain number of time steps, until it reaches time $t_ n$. The state of the system at time $t_ n$ will have similar overall features for whatever initial state $t_0$ we start with, but due to the chaotic dynamics the precise nature of these features is impossible to predict and they appear random. The numerical information which encodes the state of the system at time $t_ n$ can therefore be split into two parts. One part describes the regular features of the system — this can be stored in plain text. The other describes the random-looking detail and is encoded using the short password.

Now imagine that you're the legitimate user in possesion of the correct SP. You enter the SP, which decrypts the random-looking part of the information. This is then combined with the information that encodes the regularities of the dynamical system, to give you all the information about the system at time $t_ n$. The system is now evolved backwards in time until it gets back to time $t_0$. In theory, such a backward evolution does not always get you back to the exact state you started from: because the system is chaotic, the smallest inaccuracy in your information (coming for example from rounding off numbers) can amplify as you move backwards in time and cause you to miss the original image. But by making sure you don't use too many time steps (i.e. that the number $n$ isn't too large) you can guarantee that you end up with something nearly identical to the original image. Then, instead of having to remember it, you can just read the strong key off that image. It is now combined with the short password to give you the overall password which decrypts the original data.

But what if you're a hostile attacker systematically trying out a large number of short passwords? Suppose that you have access to the plain-text information describing the regular features of the dynamical system at time $t_ n$ and to the encrypted data representing the random-looking part. First you use a particular short password guess to get a candidate decryption of the random-looking part. Usually the candidate decryption would be searched for patterns and correlations which indicate if it's the true text. But in this case the true text itself looks random, so any such search is futile. You just have to plough on: you evolve the system backwards until time $t_0,$ hoping that the resulting image contains the correct strong key.

[...]

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April 11 2011

02mydafsoup-01
Play fullscreen
ran-dom – Shell sort algorithm performed by a hungarian folk...
Reposted fromyetzt yetzt viasofias sofias

March 30 2011

02mydafsoup-01
For Deleuze, the distinction between problematics and axiomatics is reflected in the distinction between two different conceptions of the multiple: differential of continuous multiplicities (problematics) and extensional or discrete sets (axiomatics). As we have seen, royal or ‘major’ mathematics is defined by the perpetual translation or conversion of the latter into the former. But it would be erroneous to characterize differential multiplicities as ‘merely’ intuitive and operative, and extensional sets as conceptual and formalizable. ‘The fact is’ writes Deleuze, ’ that the two kinds of science have different modes of formalization […]. What we have are two formally different conceptions of science, and ontologically, a single field of interaction in which royal science [axiomatics] continually appropriates the contents of vague or nomad science [problematics], while nomad science continually cuts the contents of royal science loose.’ One of Badiou’s most insistent claims is that Deleuze’s theory of multiplicities is drawn from a ‘vitalist’ paradigm, and not a mathematical one. But in fact, Deleuze’s theory of multiplicity is drawn exclusively from mathematics - but from its problematic pole. Badiou implicitly admits this when he complains that Deleuze’s ‘experimental construction of multiplicities is anachronistic because it is pre-Cantorian. More accurately, however, one should say that Deleuze’s theory of multiplicities is non-Cantorian.’ Cantor’s set theory represents the crowning moment of the tendency toward ‘discretization’ in mathematics; Deleuze’s project, by contrast, is to formalize the conception of ‘continuous’ multiplicities that corresponds to the problematic pole of mathematics. Problematics, no less than axiomatics, is the object of pure mathematics; just as Weierstrass, Dedekind and Cantor are the great names in the discretization programme, and Hilbert, Zermelo, Frankel, Godel and Cohen the great names in the movement toward formalization and axiomatization, it is Abel, Galois, Riemann and Poincaré who appear among the great names in the history of problematics. ”
— Blog: concrete rules and abstract machines 2011-03-29 | For Deleuze, the distinction between problematics... from -  Peter Hallward: Think Again. Alain Badiou and the Future of Philosophy.

April 21 2010

02mydafsoup-01
Computer, sagt der amerikanische Mathematiker Steve Strogatz, berechnen mittlerweile Dinge, die auch die brillantesten Mathematiker nicht mehr überprüfen können.
Macht der Simulation: Plötzlich sind wir alle Zuschauer - Digitales Denken - Feuilleton - FAZ.NET

January 21 2010

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