Book Report of Visions of infinity by Ian Stewart

Visions of Infinity

by:  Ian Stewart

            Visions of Infinity by Ian Stewart is a book about how the field of mathematics is advanced not just about the field of mathematics.  This book was much more about how mathematicians go about discovering their ideas about mathematics then about the proofs themselves.  The book gives the reader insight into how mathematicians think and how they approach problems.  It also teaches us about the struggle between the field of mathematics as a whole wanting more recognition, and the fact that most mathematicians would rather live without publicity. Mathematicians went a long time without any sort of large accolades being given to the study of mathematics. Unlike other fields which had the Nobel Prize for their scientists to strive for.  The book also covers many different theorems and conjectures in mathematics, though it focuses on how the mathematicians focused on the problem.  The book also goes over the seven Millennial Problems which are seven problems in mathematics that were so difficult that a $1,000,000 reward was offered for a correct solution.  To date only one of these problems the Poincare Conjecture has been solved.

            The book starts off with explaining how surprised the BBC was when they presented a TV show about mathematics and people actually watched it with interest.  This is because most individuals were, and still are to the most part, that mathematics had stopped growing a long time before.  The problem with the program was that it proposed a solution to Fermat’s Theorem which had been proposed 300 years before, so many individuals wrongly assumed that mathematics hadn’t been furthered in 300 years.  This could be because while other fields that relied on mathematics received prizes for their works received prestigious prizes, such as the Nobel Prize, and mathematics most prestigious reward was the Fields Medal which is largely unknown about outside of the mathematical community.

            Mathematicians are also very cautious about giving out accolades.  When Grigori Perelman first proposed his solution to the Poincare conjecture it took eight years for him to be offered the $1,000,000 reward for his solution to the Millennial Problem. He then declined the prize since the recognition and the reward was not important to him after waiting for so long.  No mathematician wanted to throw their weight behind Perelman for fear about him being incorrect.  This shows how cautious mathematicians can be and how many mathematicians will try and refuse the spotlight if given the chance.  Many mathematicians also want their opinions to be accepted on merit and not based upon public opinion, which is problematic because the solution needs to be publicized for other mathematicians to know to look at it.  The private nature of most mathematicians prevents the field of mathematics.

            The book Visions of Infinity did a very good job about explaining how mathematicians approach problems and why it takes so long for them to receive recognition for their work.  It also shows accolades are necessary for public recognition of the advances of mathematics.  The book also shows how the nature of mathematicians private lives can contradict what needs to be done in order to achieve greater recognition for the field of mathematics.  Overall this book was worth the read for anyone curious about how mathematicians think or about the field of mathematics as a whole. 

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Why a Proper Number System is Important in Mathematics

A proper numbering system in mathematics is a very important part of being able to do math.  We might be able to do algebra using letters, but without the numbers these letters are meaningless.  The numbers themselves might start off as arbitrary but they do gain value once it is figured out how they work.  A simple number system can make predicting how the world will react to different things possible.  Trying to do complex math with roman numerals sounds to cumbersome to even attempt.  A simple number system can even make learning math easier, while a complex numbering system would make it difficult to grasp the more complex parts of mathematics.

In mathematics numbers act as placeholders in an arbitrary way.  They are arbitrary in the sense that at one point someone said this squiggle line is worth more than that squiggle line.  This holds true for the numbers 0-9 and then the numbers have a pattern that they follow.  Without these placeholders doing math would be impossible it would lead to statements like finding the derivative of that big number multiplied by that slightly smaller number.

Having placeholders in math is important, but it also matters how easy the placeholders are to work with.  Using a cumbersome number system like roman numerals would make doing math significantly more difficult.  Trying to do even simple division with roman numerals would take an incredibly long time.

It has been shown that places with simpler number systems tend to have better test scores than countries with a complicated numbering system.  Students in China have an easier time learning math because instead of having eleven their number translates closer to one-ten which would make learning how to count in the teens easier.

So while the shape of the numbers might not be important, how easy they are to use and the fact they exist is very important.  The easier the numbering system itself is the easier it will be to learn and the more complex the math can be that uses the numbering system.

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Tesselation

Tesselation made using Geogebra

Tesselation

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