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
:
- http://en.wikipedia.org/wiki/The_Story_of_Maths re. Jan
25, 2014
- http://en.wikipedia.org/wiki/Paul_Cohen_(mathematician)
re. Jan 25, 2014
- 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
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