Change the objects under study. Remove a condition, or add new conditions. Remove or add context. Each of these potential changes is discussed in more detail below.
However, this ebook will answer that question in a more complicated way. I am interested in how mathematics "works" -- in some sense I am interested in how Mathematics is implemented as a language, whereas most just "use" it and are not interested on examining it in detail or systematically.
On the other hand, I am very well aware of the many efforts within and outside of mathematics looking at notions of complexity or theoretical characterizations of logic that have been applied to mathematics and logic before. For example, Kolmogorov and Chaitin complexity or Topos theorymodel theorydomain theoryand paraconsistent logic are interesting and useful knowledge domains that can contribute to an examination of mathematics.
Description. BackupWordPress was created by our friends at HumanMade but is now under new ownership. We’re committed to opensource and WordPress and will provide free support for the many BackupWordPress fans. "Ah, that makes sense." You say. Indeed, but what's cool is that we then have a pedantic way of specifying the Sierpinski triangle. Write a recursive function to generate a pattern of stars such as the following: Also, write a program that prompts the user to enter the number of lines in the pattern and uses the %(2).
However, what is the point of mathematics? Mathematics is a language.
This issue will become more important as time goes on. The crisis in Physics, with string theory having no experimental basis other than what has been discovered before e. Mathematicians would say string theory is not mathematics rather it's physics, but where's their proof? All they can do is throw words at the problem -- those fuzzy meaning things -- those slippery things -- in the form of natural language, no different than lawyers and politicians or similar criminals.
Of course, they can ignore or be oblivious to the problem, like politicians. Albert Einstein I quickly came to recognize that my instincts had been correct; that the mathematical universe had much of value to offer me, which could not be acquired in any other way. I saw that mathematical thought, though nominally garbed in syllogistic dress, was really about patterns; you had to learn to see the patterns through the garb.
I learned that it was from such patterns that the insights and theorems really sprang, and I learned to focus on the former rather than the latter.
Richard Feynman In theory, there is no difference between theory and practice. In practice there is. Yogi Berra Much of mathematics was developed by "non" mathematicians -- Archimedes, Newton, and Gauss are considered the giants of mathematics, significantly used the natural world to create their ideas in mathematics.
It has been primarily "physics" and "engineering" that propelled mathematics in the last three centuries, but the methods that worked in the past usually don't work as effectively for harder and new problems of the future, such as -- what is the future "evolution" of humankind and the Web?
The problem of addressing "function" as opposed to structure has not been done well in mathematics. The field of economics has tried to use conventional mathematics, and has generated many baroque "theories" of little use except creating academic empires or meteoretic financial groups e.
Biology and evolution involve more dynamic and functional questions, and mathematics will need to go beyond structural ideas to progress significantly. No doubt the vast majority of mathematicians will not be interested, hence it might be better to not characterize the development as mathematics or metamathematics.
Maybe a neo-logism is more appropriate: There are many measures and characterizations of complexity and power of expressiveness that have been applied to both to computation, logic, and mathematics, but most of the time those characterizations or measures have concentrated on the thing that mathematics and logic tries to represent, rather than trying to look very closely at the thing mathematics and logic that it can represent -- the thing -- reality.
In other words, I am more interested in Mathematics Itself, not its canonical forms or its various instances of representations, whether that be Number theoryTopos theory in the form of Closed cartesian categoriesLie AlgebrasIntuitionist's logicTuring MachinesModel TheoryInconsistent Mathematics or Domain Theory.
I will address these various incarnations as to their role, later. In fact, one could argue that mathematics is a poor man's version of information science a more proper name for computer sciencefor it is missing a good sense of "process.
It should be noted that the more complex the numbers, as in quaternions or octonions, the more information that is embedded implicitly. However, how much of that information is random information?
And what does one MEAN by "random" -- again "random" is a "word. As a start, there seems a balance of information between order and disorder when generating mathematical ideas and looking at their consequences.
Gregory Chaitin has shown that within number theory some facts or "theorems" are essentially "random. Abandon the Shannon I am interested mathematics from a global informational and semantic point of view.
Of course, some would immediately ask what about Chaitin complexity, or Crutchfield's approachor even Shannon information.Write a C++ program that does the following: Write a recursive function that takes as a parameter a nonnegative integer and generates the following pattern of stars.
If the nonnegative integer is 4, then the pattern generated is: **** *** ** * * ** *** **** Also, write a program that prompts the user to enter the number of lines in the pattern and uses the recursive function to generate the.
It prototype should be like void Recursive pattern(int n, int c) where n number represent Asterisk, and c represent column. Following is a text-based fractal pattern of asterisks and spaces.
my task is to write a function that can generate patterns like this. I am trying to do this exercise. Write a recursive function to generate a pattern of stars such as the following: * ** *** **** **** *** ** * It seems simple but is giving me a lot of problems.
I sit down to write when there is an inspiration within to do so and then the words just flow through. These writings are not meant to create belief systems, dogmas, agendas or techniques.
The point is, if you the science fiction writer postulate lots of technological advances in your novels, you must at least pay lip service to the sad fact that it will make a sizable segment of your society very angry.
Write a C++ program that does the following: Write a recursive function that takes as a parameter a nonnegative integer and generates the following pattern of stars.
If the nonnegative integer is 4, t .