The French and American nationality, Mandelbrot, named and developed the fractal theory as a mathematical way to capture the infinite complexity of nature.
Fractals are used for measuring natural phenomena, that were regarded as non-measurable, such as clouds or coastlines. These discoveries have applications in many fields such as geology, medicine, astronomy, mechanical engineering, but also economics and anatomy.
According to his family, Benoît Mandelbrot died in Cambridge, Massachusetts from pancreatic cancer.
French President Nicolas Sarkozy in a statement paid tribute to the great mathematician, “a strong spirit, authentic, never hesitated to make innovations and to fight against established views.
Benoît Mandelbrot was born in Warsaw on November 20, 1924, in a Jewish family of Lithuanian origin. To escape the Nazi threat fled to France with his family, and then moved to the United States after the Second World War.
A very good video about fractals and Mandelbrot
Mandelbrot set Tools
What is random? If we could define it, it wouldn’t be random;
It is impossible to produce an arbitrarily long string of random digits and prove it is random. Strangely, it is also very difficult for humans to produce a string of random digits, and computer programs can be written which, on average, actually predict some of the digits humans will write down based on previous ones.
According to physcists now we are able to touch genuine randomness right at the source; quantum mechanics
René Descartes (1596–1650) was a French philosopher who is widely regarded as thefather of modern philosophy. He is also known as a great mathematician and the father of Analytic Geometry. Young Descartes in 1918 joined the International College of War of Maurice of Nassau in the Dutch Republic. In November 1620 was present at the Battle of the White Mountain outside Prague. On the night of 10–11 November 1619, Descartes experienced a series of visions which he later claimed as the reason to become a scientist. Three years later he decided to sell all of his property and invest in bonds which provided him with a comfortable income for the rest of his life. He worked on providing a philosophical framework for the natural sciences as these began to develop. In 1637 he published the Discourse on the Method attempting to arrive at a fundamental set of principles that one can know as true without any doubt.
Descartes defines knowledge in terms of doubt, which was an epistemological innovation. He conceives of knowledge as advancing truth, from unshakable first principles. Science is conceived as organized knowledge in the manner of a well-structured, architectural edifice which owes its structural integrity to two kinds of features: a firm foundation and a superstructure of support beams firmly anchored to the foundation. This knowledge organization is inspired by Euclid’s Geometry;
“Those long chains composed of very simple and easy reasoning, which geometers customarily use to arrive at their most difficult demonstrations, had given me occasion to suppose that all the things which can fall under human knowledge are interconnected in the same way.”
In order to start to build this structure we should examine those things which we think to be true and reject all those beliefs of which there might be some doubt¹ (which can be a very long -possibly impossible- process). In order to be more effective Descartes proposes to group beliefs by focusing on the faculty, such as the senses, the imagination or reason, from which beliefs are derived. Then he proposes to deploy a series of skeptical hypotheses which call into question the knowledge derived from these faculties.
In his work Meditationes de prima Philosophia² in 1641 he described this method (The method of doubt) which is an extended exercise in learning to doubt about everything, considered at three distinct levels:
1. Perceptual Illusion; the testimony of the senses with respect to any particular judgment about the external world may turn out to be mistaken.
2. The Dream Problem; it is possible that everything I now “perceive” to be part of the physical world outside me is in fact nothing more than a fanciful fabrication of my own imagination.
3. A Deceiving God; whenever I believe anything, even if it has always been true up until now, a truly omnipotent deceiver could at that very moment choose to change the world so as to render my belief false.
¹ Here Descartes tried to defeat skepticism on its own ground; He begin by doubting the truth of everything, not only the evidence of the senses. This is a very difficult work but he hopes that if any particular truth about the world can survive this extreme skeptical challenge, then it must be truly indubitable.
² Some believe that the purpose of the method of doubt was to offer to contemporary theologians his proofs of the existence of god and the immortality of the human soul. Others believe that even if Descartes did that, he did it because he hoped to preserve a distinct arena for the church while securing the freedom of scientists to develop mechanistic accounts of physical phenomena.
(22 April 1724 – 12 February 1804)
German philosopher, Professor of Logic and Metaphysics, he is considered the last influential philosopher of modern Europe during the Enlightenment. The purpose of his work was to establish philosophy as a pure science.
According to Kant there is no rational basis for the belief that the natural world is (or is not) arranged according to some purpose by a Designer. He believed though that the real incentive behind a scientist is the search for purpose in nature. Specifically for natural science he didn’t share Hume’s view that the root of all ideas is experience and he tried to show that there are a priori judgments in humans that provide the necessary foundations for human knowledge. His argument was that qualities like the truths of mathematics cannot be explained by experience. For example, our knowledge the interior angles of any triangle add up to a straight line is synthetic judgment , Kant held, since it contributes significantly to our knowledge of the world; the sum of the interior angles is not contained in the concept of a triangle.
But then there is another question rising; how do we come to have such a-priori knowledge? Kant’s answer is that this knowledge is the product of the process of the conformity to the truths of mathematics that we impose to every object of our experience. So in other worlds there is a “filter” that we filter our experiences with and this filter is actually what Kant called a-priori knowledge¹. He argued that humans in order to perceive an object, the object must be regarded as being uniquely located in space and time. Space and time, Kant argued, are the “pure forms of sensible intuition” under which we perceive what we do².
conditions for science
Kant believed that in order for scientific knowledge to be possible the world must be not only perceivable but thinkable as well, and in order to be thinkable he set two conditions that must be fulfilled;
1. In principle it should be possible to trace the connections between our sensory images in order to arrange and organize the experiences. (synthetic unity of the sensory manifold)
2. It must be possible for a single subject to perform this organization by discovering the connections among perceived images. (transcendental unity of apperception.)
¹ Of course the term ‘knowledge’ gives to the the “filter” a rather positive meaning, on the contrary, some can say, that it is barrier between humans and the physical word, which can mean also that mathematics is a also a barrier.
² Understanding mathematics in this way Kant answered to the old controversy between rationalists and empiricists regarding the nature of space and time; Leibniz argued that space and time are a product of our minds. Newton, on the other hand, insisted that space and time are absolute. Kant now declares that both of them were correct! As synthetic a priori judgments, the truths of mathematics are both informative and necessary.