A Review on the Story of Maths 4: To Infinity and Beyond

We must know, we will know.  That mentality urged late nineteenth to the twentieth century mathematicians to uncover mysteries in this field that has remained unanswered throughout the ages. The last episode of The Story of Maths featured less than ten outstanding geniuses who shared their thoughts throughout the entire world in an effort to give birth to the concept of infinity and reshape the mathematics far and better from what it was before.

David Hilbert, like any other ordinary man, has many problems that haunted him for the rest of his life. But unlike any of us, these problems were beyond financial or work, school and family; in fact these became the answers to the bigger problem of considering infinity’s existence  and trying to contain such information in a box and represent it by means and symbols already known to us.  This French guy opened up 23 problems faced by some mathematicians that attracted the attention of billions across the globe. Among his accomplishment are the Hilbert Inequality, Hilbert Space, Hilbert Classification and proving that infinity can be expressed in finite set.

One day, Georg Cantor spent his lazy afternoons at the University and outstripped himself from the mundane things that have always enveloped his being. While contemplating on his life, math called him and he had to respond. Is there infinity of fractions? Is it bigger than those of whole numbers? If the larger infinity of decimals were on the right and fractions on the left, is there infinity in the middle of this two? As said in the documentary, these questions were the bases for his Continuum Hypothesis which states that there exists no set whose power is greater than that of the naturals and less than that of the reals (or equivalently, the cardinality of the reals is exactly aleph-one, rather than just at least aleph-one).¹ This was Hilbert’s first problem.

Mr. Sautoy told us no bedtime stories on Henri Poincare whose versatility caused him to be on top of math and in the field of engineering. He was famous for stating that two shapes have similar rubber-sheet geometry if they can be molded to each other’s shape. Interesting. He then formulated his Poincare conjecture which had not been solved until the early 2000’s. He did not live a happily-ever-after after dying from a prostate problem in 1912.

Some people do not normally leave their tasks undone. But in math, there are things which can never be finished if the beginning itself is not that well-defined within the measures of the existing approved truths or if it is too vague because of its absolute broadness. Well that viewpoint corresponds to what Kurt Gödel believed. In his Incompleteness theory, he said that some parts in math cannot be explained within its system. He showed that some statements which were logically true cannot be proved. With that, mathematics takes both ugly and beautiful faces. Ugly because it can be not so clear at times (the juxtaposition of abstract concepts and concrete representations somehow confuse the scene for mathematicians and ordinary people). Beautiful because math enables everyone to always think wider than the sky.

In the name of mathematics, Paul Cohen used his God-given reasoning and mathematical abilities to prove certain statements. Cohen is noted for developing a mathematical technique called forcing, which he used to prove that neither the continuum hypothesis (CH), nor the axiom of choice, can be proved from the standard Zermelo–Fraenkel axioms (ZF) of set theory. In conjunction with the earlier work of Gödel, this showed that both of these statements are logically independent of the ZF axioms: these statements can be neither proved nor disproved from these axioms. ² Other important mathematicians like Julia Robinson and Yuri Matiyasevich were also mentioned on the show.

Algebra and geometry got married when Andrew Weil added Galois’ work with his work on topology, algebra, and geometry and etc. The latter believed mathematics should be the study of structure as opposed to number and shape. Galois had discovered new techniques to tell whether certain equations could have solutions or not. The symmetry of certain geometric objects was the key. ³

Without the above-mentioned mathematicians, there would be no ordinals, set and number theory, trigonometric series, concept of one-to-one correspondence and significant hypotheses; most of all, “Nicolas Bourbaki” would not be born.

References

:

1. http://en.wikipedia.org/wiki/The_Story_of_Maths re. Jan 25, 2014
2. http://en.wikipedia.org/wiki/Paul_Cohen_(mathematician) re. Jan 25, 2014
3. Ibid

Glory to God

Mathematics of Life, a review

Encarta defined science as the study of anything that can be tested, examined and verified. Using that definition, we can say that math is a science – science of the omnipresent patterns surrounding us. Since it is a science, its scope is very wide and perhaps it has something to do with the other existing sciences as well namely physics (mother of all sciences), chemistry (study of matter), biology (study of life) and other related fields.

  The Mathematics of Life’s nineteen chapters showed the undeniable relationship between math and biology. The first chapter highlighted the invention of microscope by a Dutch scientist Leuwenhoek, which paved the way to the discovery and the greater understanding of the unknown and unseen microscopic creatures ( protists for example) and the further developments ( bacteria culture) that  were created later on. Chapter Two is about the naming, classification and groupings of the organisms into ( from  broadest to specific) kingdom, phylum, class, order, family, genus and species  also known as the Linnaean System of Classification , named after Carlos Linnaeus a botanist.

Plants in all the simplicity or grandeur of the design or the number of its leaves or petals showed the mathematical pattern that exists in its form in the third chapter. Chapter 12 is about the DNA, a short term for deoxyribonucleic acid, a polymer consisting of repeating nucleotides ( Nitrogen bases, pentose sugar and phosphate group) that carries instructions on how an organism would look like and other relevant information. Chapter 14 talked of an important principle in game theory showing the zero-sum games in the reality of the lizards’ mating schemes and the fact that lizards with orange throats beats blue, blue beats yellow and yellow beats orange. Chapter 15 stated that networks surround us—information transmission conducted by our brain to the different parts of the body and the organisms interdependence on each other shown in the food web.

Chapter 16 presented the paradox of the plankton: few niches, enormous diversity. Chapter 17 spoke of the Punett Square (which is used to determine the possible allele combinations in a zygote) and retackled the definition of life as biology defined it. Chapter 18 opened up the possibility of life but earth.  One of the lines found in the last chapter is this:  they (advances in the physical sciences and mathematics) don’t have to be an exact representation of reality to be useful. I buy that idea. Reality is beyond our five (or 6, they say) senses. All sciences are about that—thinking outside of the box.

Question. Where is mathematics there? Although the bond between math and biology was not obviously stated, one can see the applications of math in the image of microscopes in the logic of its creation, the concepts of Fibonacci numbers in chapter four, the concept of probability shown especially in chapter 17 and the experiments that were conducted  to prove theorems and make sense of the information acquired through observation and data gathering that made use of mathematical quantities, scientific notations, given properties of math and other axioms.

The book was written in a language that most non-enthusiasts will not find it that intimidating or hard to understand. Words were mostly simple.  The titles per chapter are catchy, creative and instantly give readers hints as to what that part specifically contains. Also, there were concrete examples ranging from vivid descriptions on microscopic creatures, the mention of Noah’s Ark in chapter two , typical number of petals Marigolds have up to the last when the author ended his book with the line: by the time we get to the twenty second century, mathematics and biology will have changed each other out of all recognition, just as mathematics and physics did in the nineteenth and twentieth centuries. The author’s ideas were organized. There may be moments of information overloads but it did not affect the good quality of the book.

Reference:

/www.dropbox.com/sh/qwobqaw4bcy58v7/8GfjiPbw5C/2013-2014-2/book%20review/Ian%20Stewart%20The%20Mathematics%20of%20Life%20%202011.pdf

Glory to God

What is Mathematics really, a review

Math is the language of the universe. The universe has always uttered syllables of the mysteries written in the rocks, moons, planets and stars and the non-living matter, organisms and humans that inhabit it—the seasons and the number of days in months of the year, the number of offspring a pair of rats, for instance, can produce given a specific period, our measure of time ( which is in 60’s: seconds, minutes  and hours), distances between heavenly bodies and all types of forces acting on it. Math and its principles became the tool to interpret and explain these fragments of syllables and turn it to a vocabulary of symbols and grammar of systematic mathematical operations.

  But can one contain the enormous information and complexities of math in a box and concretized everything that has to do with it? Do we see math close or exactly the same from the version of truth of the author who strongly define and see it, based on his arguments, in its superficial content? Hersh, in the book What is Mathematics Really, said that math must be understood as a human activity, a social phenomenon, part of human culture, historically evolved and intelligible only in social context¹.  He further expressed his opposition in Platonism stating that it should not have anything to do with hold-able, see-able, hear-able and smell-able mathematicians as it proposes abstract ideas on physical and mathematical things such as the concept of infinity. He also objected formalism and intuitionism and insisted on his “humanist” viewpoint.

Hersh’s work is very pleasing to read-he shows rather than tells and his choice of words and the way he put those in sentences showed how organized his thoughts are and it makes one, for me, feel  that he was directly talking in front of readers, sitting on old decent chair and taking a sip of coffee. He provided examples to support his beliefs such as the 4-cube incident in the beginning and a series of formulas in the latter parts. His usage of dialogues in the first part was effective because he was able to make himself clear on his stand on the concept of infinity without being forceful in feeding information and making use of heavy-looking paragraphs to make a point. It was a good way to arouse the human interest in readers and seduce them to read and maybe help them discern or consider the gist of what he’s saying.

However, I don’t agree to his main point of writing this book. For me, math is beyond concreteness and we are not to treat it in such a way explainable by our senses.

Reference

1. www.dropbox.com/sh/qwobqaw4bcy58v7/epqUsLqjUv/2013-2014-2/book%20review/Reuben%20Hersh%20What%20Is%20Mathematics%2C%20Really%20%201997.pdf

Glory to God