A Quote by Robert L. Devaney
It can be argued that the mathematics behind these images [of the orbit diagram for quadratic functions and the Mandelbrot set] is even prettier than the pictures themselves.
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The most complex object in mathematics, the Mandelbrot Set ... is so complex as to be uncontrollable by mankind and describable as 'chaos'.
. . . the membership relation for sets can often be replaced by the composition operation for functions. This leads to an alternative foundation for Mathematics upon categories -- specifically, on the category of all functions. Now much of Mathematics is dynamic, in that it deals with morphisms of an object into another object of the same kind. Such morphisms (like functions) form categories, and so the approach via categories fits well with the objective of organizing and understanding Mathematics. That, in truth, should be the goal of a proper philosophy of Mathematics.
Mathematics is much more than computation with pencil and a paper and getting answers to routine exercises. In fact, it can easily be argued that computation, such as doing long division, is not mathematics at all. Calculators can do the same thing and calculators can only calculate they cannot do mathematics.
When we talk about the impact inside mathematics, and applications in the sciences, [Mandelbrot] is one of the most important figures of the last 50 years.
The Huygens images were everything our images from orbit were not. Instead of hazy, sinuous features that we could only guess were streams and drainage channels, here was incontrovertible evidence that at some point in Titan's history - and perhaps even now - there were flowing liquid hydrocarbons on the surface.
Not a lot of people know this, but I'm very good at mathematics. When I was an angry teenager, I used to sit in my room and do quadratic equations to calm myself down.
The Mandelbrot set is the most complex mathematical object known to mankind.
All over China, parents tell their children to stop complaining and to finish their quadratic equations and trigonometric functions because there are sixty-five million American kids going to bed with no math at all.
Others of them employ outward marks ... They style themselves Gnostics. They also possess images, some of them painted and others formed from different kinds of material. They maintain that a likeness of Christ was made by Pilate at that time when Jesus lived among them. They crown these images, and set them up along with the images of the philosophers of the world, such as Pythagoras, Plato, and Aristotle, and the rest. They have also other modes of honoring these images just like the Gentiles.
Aspiring to these wide generalizations, the analysis of quadratic functions soars to a pitch from whence it may look proudly down on the feeble and vain attempts of geometry proper to rise to its level or to emulate it in its flights.
Images exist; things themselves are images... Images constantly act on and react to one another, produce and consume. There is no difference between images, things and movement.
Distinction between species and specimen is very much like the distinction between images and actual pictures, or, you know, objects that have a definite material identity. The classifications, the categories, the stereotypes, and the images are on one side, and the material pictures, statues, texts, and so forth are on the other.
Mystery is an inescapable ingredient of mathematics. Mathematics is full of unanswered questions, which far outnumber known theorems and results. It's the nature of mathematics to pose more problems than it can solve. Indeed, mathematics itself may be built on small islands of truth comprising the pieces of mathematics that can be validated by relatively short proofs. All else is speculation.
We know that mathematicians care no more for logic than logicians for mathematics. The two eyes of science are mathematics and logic; the mathematical set puts out the logical eye, the logical set puts out the mathematical eye; each believing that it sees better with one eye than with two. Note that De Morgan, himself, only had sight with only one eye.
When I was at drama school, people weren't taking pictures of themselves every five minutes. So I didn't realise how I looked. It was only when people started taking pictures of themselves that I looked at myself and thought: 'Oh my God, I look really miserable.' Even when I'm happy I look sad.
I remember researching a really complicated article and having trouble keeping track of all the different perspectives. I ended up drawing a diagram to help myself follow how the ideas were interrelated. I looked at the diagram when I had finished and thought, "Oh, maybe I don't need to write the article now - maybe I've done my job as a journalist. I can convey my understanding through the diagram."
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