The Purpose Of Doubt In Knowledge Philosophy Essay
words (5 pages) Essay
Published: 1st Jan 2015 in
The more I know about knowledge, then the more doubt will
develop into my mind. That is what I felt about doubt through
experiences in my life. We usually have doubt when we find it
hard to belief our accept something as the truth. Doubts exist
when our mind is in process of accessing something whether to
belief or not to believe. When we are 100% confident about
something that we think is true, then we will never
misunderstood the idea that had been brought in the first place.
Maybe without doubt, we would never come across the word
“evolution” because evolution is about some facts that are due
to the feeling of doubt. Key is something that can bring us
nearer to the truth and as generator to the knowledge that we
seek for. Doubt will lead us to find two form of knowledge.
First is the doubt in knowledge which can help us to get clearer
knowledge when we try to prove the claim made is true or not,
second is the doubt in knowledge will lead us to discover and
explore a new knowledge. Persian proverb says that “doubt is the
key to knowledge” because from doubt, we have the feeling to
know about something and perhaps will lead us to get clearer
knowledge or new knowledge. We need to fulfill certain criteria
before we can acknowledge something as a knowledge, such as it
must be justified by empirical evidence, logic, memory or public
authority. Without this criteria, a claimed knowledge made by
scientist or historian can be myth. History and Mathematics are
two different area of knowledge that can help us to find at what
extent doubt is used as an element to discover new knowledge.
History is a subjective parts of area of
knowledge and its talks about people and events of the past. It
is not just the compilation of the past that is being recorded
solely without any research and discussion, but after the facts
are being interpreted and analysed into something significant.
Lets take an example of a history about Piltdown man. For
instance, in 1912, a paleoantropologist, Charles dawson claimed
he have found human skull in England belong human ancestor aged
about 500 thousand years ago or known as Dawson’s down-man.
According to him, it is a pioneer evidence of human evolution.
Most experts agreed that these specimens age about 500 thousand
years old and being displayed in museums for more than 40 years.
But out of sudden a few scientist doubt about the specimens was
from human ancestor because the skull fragment and the jawbone
obviously from two difference species. A few attempts were made
by scientists to prove this claim was wrong but they did not
succeed. Later, because of the advance in technology, a
paleonthologist, Kenneth Oakley has tried the technology of
flourine testing on Piltdown man in order to determine the
actual age of the fossils. He applied the technology on the
fossil and he found that the fossil is just a few thousand years
not 500 thousand years as claimed by Dawson. Further
investigation was made and it is found that the fossil is
combination of human skull and ape. A conclusion is made by
paleontologists that Piltdown man is a join together of a piece
of jigsaw puzzle with the right colour but wrong shape and was
immediately removed from the museum. Piltdown man is an example
that shows us that doubt can lead to get a clearer knowledge and
can bring us and nearer to the truth.
In order to agree the knowledge in history
can be gained because of doubt, we have to realize that it has
it own limitation and weakness. The weakness is doubts solely
cannot solve any puzzles that we are wonder about. It must be
accompanied by scientific method. The scientific method itself
still can be doubted because human who created method can still
make mistake. Maybe in the next few years someone will discover
that fluorine technique used to investigate the date of the
fossil is not accurate and valid enough to test for age of the
specimens. We have to be remembered that science always evolve
as time passes by and it is not impossible that more advance
techniques will be discovered until the actual date and time the
fossil belong to can be revealed. Henry Fairfield Osborn said
that “We have to be reminded over and over again that Nature is
full of paradoxes”. His statement is made to remind people that
those things that we see and expect in this world does not
necessary true as a lot more thing we don’t know about nature
and the most important thing is we did not exist at the time of
the history real event mentioned. Sometimes, historian only made
a speculation or lucky guess on the specimens they found as long
as it met their theoretical expectations. So it can be
understood that even an evolutionist will get the closest
interpretation on the fossil findings when they doubt and
willing to discuss among themselves.
In other area of knowledge such as
Mathematics doubt is still exist but in different view and
perspective. This is because Mathematics differs from history in
a way that it is more to objective, less ambiguous, less open to
interpretation and more clearly defined. Math is more to numbers
and can be differentiate very clearly which one is wrong and
which one is right or in other word the answer is not
subjective. Mathematics knowledge arises from problems that we
face everyday and the solution in Mathematics is free from both
observation and induction. In mathematics, people not always
doubt whether the knowledge claim by mathematician is true or
not because mathematics have the ability of self-proving and its
objectivity, but people do doubt whether the claim made by
mathematician about certain theorem can be modified in order to
form a more perfect theorem. The simplest example is prime
number formula. The definition of prime number is the number
which can divide by 1 and itself. As a student, I myself also
have doubt why number 1 did not count as prime number while
number 1 can be divided by 1 and itself. But for mathematicians,
they doubt whether prime number have a formula so that it can
cover all prime numbers until infinity. After fail to solve this
problem, Mathematicians claim that this problem is unsolvable.
This unsolvable problem remains for 2300 years. Although this
problem remains unsolved for so long, that doest mean
mathematician had give up. The doubt grows deeper until in year
2003, Professor.S.M.R Hashemi Moosavi found the formula for
prime number. This shows that from doubt that prime number
formula is unsolved, we now can enjoy the fruitful of the doubt
although it takes decades or centuries.
However, no ones would know that claim
made by Professor S.M.R Hashemi Moosavi is the most perfect one
to suit the formula for prime number because at this time no one
can prove his theorem is wrong. Unlike the previous
Mathematician, they also once had claimed that they have found
this formula, but it only last for a certain period because
someone had proven it only applicable up to certain numbers, not
included until the infinity. We also can doubt what the last
number of prime number is since the ending of infinity itself is
Having discussed doubt is the key to knowledge from the
perspective of two areas of knowledge namely history and
mathematics makes me realized that doubt do exist in the process
of searching for knowledge. Although the process obtaining
knowledge though the feeling of doubt is differ in History and
Mathematics due to its subjectivity, objectivity and its nature,
the final ending will bring us to only one destination; gaining
knowledge. Therefore, doubt is the key to knowledge is agreed in
these two areas of knowledge.
Reference : UKEssays.
(November 2018). The Purpose Of Doubt In Knowledge Philosophy
Essay. Retrieved from
What is a prime number?
A prime number is
a positive integer that has exactly two positive integer
factors, 1 and itself. For example, if we list the
factors of 28, we have 1, 2, 4, 7, 14, and 28. That's
six factors. If we list the factors of 29, we only have
1 and 29. That's 2. So we say that 29 is a prime number,
but 28 isn't.
Another way of
saying this is that a prime number is a whole number
that is not the product of two smaller numbers.
Note that the
definition of a prime number doesn't allow 1 to be a
prime number: 1 only has one factor, namely 1. Prime
numbers have exactly two factors, not "at most
two" or anything like that. When a number has more than
two factors it is called a composite number.
Here are the first
few prime numbers:
2, 3, 5, 7, 11,
13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61,
67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113,
127, 131, 137, 139, 149, 151, 157, 163, 167, 173,
179, 181, 191, 193, 197, 199,
The Largest Known Primes
on the Web
What are Mersenne
primes and why do we search for them?
A Mersenne prime is a prime
of the form 2P-1. The first Mersenne primes are 3, 7,
31, 127, etc. There are only 41 known Mersenne primes.
GIMPS, the Great Internet
Mersenne Prime Search, was formed in January 1996 to discover
new world-record-size Mersenne primes. GIMPS harnesses the power
of thousands of small computers like yours to search for these
"needles in a haystack".
Most GIMPS members join the
search for the thrill of possibly discovering a record-setting,
rare, and historic new Mersenne prime. Of course, there are many
more information about Mersenne primes , History ,Theorems and
click here .
Why do people find these primes?
"Why?" we are often asked,
"why would anyone want to find a prime that big?"" I often now
answer with "did you ever collect anything?"" or "did you ever
try to win a competition?"" Much of the answer for why we
collect large primes is the same as why we might collect other
rare items. Below I will present a more complete answer divided
into several parts.
For the by-products of the
People collect rare and beautiful items
For the glory!
To test the hardware
To learn more about their distribution
This does not exhaust the
list of reasons, for example some might be motivated by primary
research or a need for publication. Many others just hate to see
a good machine wasting cycles (sitting idle or running an inane
Perhaps these arguments
will not convince you. If not, just recall that the eye may not
see what the ear hears, but that does not reduce the value of
sound. There are always melodies beyond our grasp.
Euclid may have been the
first to define primality in his Elements approximately 300 BC.
His goal was to characterize the even perfect numbers (numbers
like 6 and 28 who are equal to the sum of their aliquot
divisors: 6 = 1+2+3, 28=1+2+4+7+14). He realized that the even
perfect numbers (no odd perfect numbers are known) are all
closely related to the primes of the form 2p-1
for some prime p (now called Mersennes). So the quest for
these jewels began near 300 BC.
Large primes (especially of
this form) were then studied (in chronological order) by Cataldi,
Descartes, Fermat, Mersenne, Frenicle, Leibniz, Euler, Landry,
Lucas, Catalan, Sylvester, Cunningham, Pepin, Putnam and Lehmer
(to name a few). How can we resist joining such an illustrious
Much of elementary number
theory was developed while deciding how to handle large numbers,
how to characterize their factors and discover those which are
prime. In short, the tradition of seeking large primes
(Especially the Mersennes) has been long and fruitful It is a
tradition well worth continuing.
2. For the
by-products of the quest
Being the first to put a
man on the moon had great political value for the United States
of America, but what was perhaps of the most lasting value to
the society was the by-products of the race. By-products such
as the new technologies and materials that were developed for
the race that are now common everyday items, and the
improvements to education's infrastructure that led many man and
women into productive lives as scientists and engineers.
The same is true for the
quest for record primes. In the tradition section above I
listed some of the giants who were in the search (such as
Euclid, Euler and Fermat). They left in their wake some of the
greatest theorems of elementary number theory (such as Fermat's
little theorem and quadratic reciprocity).
More recently, the search
has demanded new and faster ways of multiplying large integers.
In 1968 Strassen discovered how to multiply quickly using
Fast Fourier Transforms.
He and Schönhage refined and published the method in 1971. GIMPS
now uses an improved version of their algorithm developed by the
long time Mersenne searcher Richard Crandall .
The Mersenne search is also
used by school teachers to involve their students in
mathematical research, and perhaps to excite them into careers
in science or engineering. And these are just a few of the
by-products of the search.
collect rare and beautiful items
Mersenne primes, which are
usually the largest known primes, are both rare and beautiful.
Since Euclid initiated the search for and study of Mersennes
approximately 300 BC, very few have been found. Just 37 in all
of human history--that is rare!
But they are also
beautiful. Mathematics, like all fields of study, has a
definite notion of beauty. What qualities are perceived as
beautiful in mathematics? We look for proofs that are short,
concise, clear, and if possible that combine previous disparate
concepts or teach you something new. Mersennes have one of the
simplest possible forms for primes, 2n-1. The
proof of their primality has an elegant simplicity. Mersennes
are beautiful and have some surprising applications.
4. For the
Why do athletes try to run
faster than anyone else, jump higher, throw a javelin further?
Is it because they use the skills of javelin throwing in their
jobs? Not likely. More probably it is the desire to compete
(and to win!)
This desire to compete is
not always directed against other humans. Rock climbers may see
a cliff as a challenge. Mountain climbers can not resist certain
Look at the incredible size
of these giant primes! Those who found them are like the
athletes in that they outran their competition. They are like
the mountain climbers in that they have scaled to new heights.
Their greatest contribution to mankind is not merely pragmatic,
it is to the curiosity and spirit of man. If we lose the desire
to do better, will we still be complete?
5. To test
Since the dawn of
electronic computing, programs for finding primes have been used
as a test of the hardware. For example, software routines from
the GIMPS project were used by Intel to test Pentium II and
Pentium Pro chips before they were shipped. So a great many of
the readers of this page have directly benefited from the
search for Mersennes.
Slowinski, who has help
find more Mersennes than any other, works for Cray Research and
they use his program as a hardware test. The infamous Pentium
bug was found in a related effort as
was calculating the twin prime constant.
Why are prime programs used
this way? They are intensely CPU and bus bound. They are
relatively short, give an easily checked answer (when run on a
known prime they should output true after their billions of
calculations). They can easily be run in the background while
other "more important" tasks run, and they are usually easy to
stop and restart.
learn more about their distribution
Though mathematics is not
an experimental science, we often look for examples to test
conjectures (which we hope to then prove). As the number of
examples increase, so does (in a sense) our understanding of the
Seyyed Mohammad Reza Hashemi Moosavi
Elliptic curves are a special kind of algebraic curves which
have a very rich arithmetical structure.
There are several fancy ways of defining them. but for our
purposes we can just define them as the set of points satisfying
a polynomial equation of a certain form. To be specific,
consider an equation of the form
integers (There is a reason for the strange choice of indices on
but we won't go into it here). we want to consider the set of
satisfy this equation.
To make things easier, let us focus on the special case in which
the equation is of the form
cubic polynomial (in other words, we're assuming
In this case (*), it's very easy to determine when there can be
singular points, and even what sort of singular points they will
be. If we put
Then we have
we know, the curve will be smooth if there are no common
solutions of the equations
we know, from elementary analysis, that an equation
a smooth curve exactly when there are no points on the curve at
which both partial derivatives of
in other words, the curve will be smooth if there are no common
solutions of the equations (**).
And the condition for a point to be "bad" be comes
which boils down to
other words, a point will be "bad" exactly when its
coordinate is Zero and its
coordinate is a double root of the polynomial
of degree 3, this gives us only three possibilities:
no multiple roots, and the equation defines an elliptic curve
(Three distinct roots),
example, elliptic curve
three distinct roots).
a double root (curve has a node), (For example, curve
a triple root (curve has a cusp), (For example, curve
the roots of the polynomial
the discriminant for the equation
out to be
This does just what we want:
If two of the roots are equal, it is Zero, and if not, not.
Further more, it is not too hard to see that
actually a polynomial in the coefficents of
which is what we claimed. In other words, all that the
discriminant is doing for us is giving a direct algebraic
procedure for determining whether there are singular points.
while this analysis applies specifically to curves of the form
actually extends to all equations of the sort we are considering
there is at most one singular point, and it is either a node or
with some examples in hand, we can proceed to deeper waters. In
order to understand the connection we are going to establish
between elliptic curves and Fermat's Last Theorem, we need to
review quite a large portion of what is known about the rich
arithmetic structure of these curves.
Elliptic curve of the form
is a elliptic curve of Non – Singular if
not a double root or
a triple root. In fact below equation
has three distinct roots, if
no multiple roots, and
Non –Singular cubic elliptic curve.
3 New Proof of Fermat's last Theorem by
HM Final Main Theorem
Seyyed Mohammad Reza Hashemi Moosavi
HM Final – Main Theorem
For every odd
Last Theorem ():
Special case is of an elliptic curve (Non – Singular Cubic
It is enough in the below general elliptic curve:
Or elliptic curve HM (*) we assume:
then after replacing in (*) or (**):
Elliptic curve (*) is Non – Singular.
First Fermat's equation is multiplied
We assume .
We know that Proofed
an elliptic curve.
New Proof of Fermat's Last Theorem by HM Final – Main
Elliptic curve HM (*) is non – Singular, because:
(Three different roots)
 S. M. R. Hashemi Moosavi, The Discovery of Prime Numbers
Formula & It's Results (2003).
 S. M. R. Hashemi Moosavi, Generalization of Fermat's Last
Theorem and Solution of Beal's Equation by HM Theorems (2016).
 S. M. R. Hashemi Moosavi, 31 Methods for Solving Cubic
Equations and Applications (2016). (ISBN: 978-600-94467-7-3).
By Seyyed Alireza Hashemi Moosavi
Click the link to download or open the PDF file of
New Proof of Fermat's
by Final - Main HM Theorem