Let’s admit it, most of us are intellectually lazy, and thinking is a strange and hard activity, because it requires effort. Even philosophy, the prime discipline devoted to thinking, mostly circulates the ideas of other people, and has some similarity to journalism. Real thinking is hard, and I will always respect Kant for the simple advice he gave to […]

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If you add up all the natural numbers from 1 to infinity, you should get a really large version of infinity, but this is not the case. What you get is  minus one/twelfth. Watch the following well-made video clip in order to understand how this happens. Follow @JBraungardt Related posts: Mathematical Infinity Newsfeed Prime Number […]

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Here are two computer-animated videos from Youtube that demonstrate some properties of knots, the relations between knots and space, and how we arrive at hyperbolic space. Easy to watch, and very informative.   Part Two:   Follow @JBraungardt Related posts: Klein Bottle Knot theory Topology Turning a Sphere inside-out 0 to 6 dimensions and back […]

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Source: Kurt Gödel, Collected Works, Volume III (1961) publ. Oxford University Press, 1981. The Complete lecture reproduced here. I would like to attempt here to describe, in terms of philosophical concepts, the development of foundational research in mathematics since around the turn of the century, and to fit it into a general schema of possible […]

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Below there is  an interesting video inspired by numbers, geometry and nature, created by Cristóbal Vila. Nature looks complex, but the underlying principles are simple, for instance the Fibonacci Series of numbers.  The natural beauty and complexity we see all around us is deeply astonishing, but what’s even more astonishing is all the math behind […]

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How do you turn a sphere inside out, without punching a hole into it? It is possible,as you can see in the transformations in this fascinating video: Follow @JBraungardt Related posts: Hyperbolic Space 0 to 6 dimensions and back – simple rotation. Klein Bottle Tesseracts: from 3 to 4 Dimensions. Kojève – hole and ring […]

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How does mathematics and reality relate to each other? Some mathematicians believe that mathematical entities are “real”, and this view can be characterized as a version of Platonic philosophy. Quine and Putnam developed an argument in support of mathematical realism, which starts with the observation that mathematics is indispensable for almost every other science. Here […]

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