Download Marlee Mines

Marlee Mines
Logic is more focused on deductive reasoning
and proof.
 Personally, I really thought that for math, logic
was kind of fun. I liked that there was no test,
and the questions were kind of difficult but not
necessarily as much number stuff, more common
sense stuff.
 I believe that logic is important, because people
need to see proof, and have a real reason for
believing things. In my life, I won’t believe most
things without some sort of proof





In 18th century Europe (among other countries) there
were some developments of logic, but few were
recorded
In the mid 19th century, George Boole and then
Augustus De Morgan began to present mathematic
treatments of logic
They built on work of those who worked mostly on
algebra to extend a traditional doctrine of math into
the frameworks for a foundation of mathematics
Charles Sanders Pierce worked on the work of George
Boole, and created a logical system for relations and
quantifiers which were published occasionally during
a 15 year period starting in 1870.




Gottlob Frege worked independently on logical
quantifiers for his Begriffsschrift, which was then
published in 1879. This was generally considered a
turning point in the history of logic.
Most of Frege’s work remained unclear until Bertrand
Russell began promoting it near the turn of the
century
The work of Frege is unused in contemporary texts
and was never widely used
From 1890-1905, Ernst Schröder published his work in
three volumes which summarized the work of
previous mathematicians and became a reference for
symbolic logic at the end of the 19th century
 The
importance of this is that it is mainly
about formal proofs
 Formal systems are another part of this
 Mathematic logic has contributed to the
study of foundation of mathematics
 In the early 20th century it was used to prove
the consistency of foundational theories
Sudoku
Pascal’s Triangle
If it’s not the day after Monday or the day
before Thursday and it isn’t Sunday tomorrow,
and it wasn’t Sunday yesterday, and the day
after tomorrow isn’t Saturday, and the day
before yesterday wasn’t Wednesday, then
what day is it?

The sum of n consecutive integers is divisible by n when n is odd. It is not divisible by
n when n is even.

Proof: Case 1 - n is odd: We can substitute 2m+1 (where m is an integer) for n. This lets us
produce absolutely any odd integer. What is the sum of any 2m+1 consecutive integers? It
is an arithmetic series (like 13+17+21+25 which has a common difference of 4). The sum of
an arithmetic series is:

a + a+d + a+2d + a+3d + ... + a+(n-1)d = n(first+last)/2
There are other equivalent formulas. In our problem, the common difference is 1:


a + (a+1) + (a+2) + ... + (a+2m) = (2m+1)(2a+2m)/2 = (2m+1)(a+m)
It is obvious that this is divisible by 2m+1, our original odd number. That proves case I.

Case 2 - n is even: We can substitute 2m for n. Again we have an arithmetic series:

a + (a+1) + (a+2) + ... + (a+2m-1) = (2m)(2a+2m-1)/2 = m(2a+2m-1)
At first glance, this would seem to not be divisible by 2m, as 2a+2m-1 is odd. But xy can be
divisible by z, even if neither x nor y is divisible by z. This sum is 2am+2m^2-m, which is m
less than a multiple of 2m. So this sum cannot be a multiple of 2m. You might want to
figure out why that is so. In other words, the sum is not divisible by 2m, our original even
number. And that proves case 2.


After Aristotle, logic was further worked on by
the Stoics and medieval scholastic philosophers
 The late 19th century began an explosive growth
in logic
 This growth continues today as we find further
information on logic
 Aristotle had gone through many difficulties in
order to establish the basics of logic in a neutral
way, away from the ideals of philosophers.
 Logic was developed to help understand our
reasoning, but can only go so far

About two thousand years (nearly exact) modern
mathematics were being developed based on
Aristotle’s theories
 Late 1600s Gottfried Leibniz (a contributor to
calculus) began to try to develop a systematic
language of reasoning to solve well defined
problems, logic being the answer
 Two more centuries passed before Augustus De
Morgan and George Boole began developing
Leibniz’s ideals

Early names for mathematic logic were
“symbolic logic” and “metamathematics”
 Logic is divided into four separate fields
according to Barwise's "Handbook of
Mathematical Logic“

Set theory
 Proof theory
 Model theory
 Recursion theory (also known as Computability
theory)







During this project, I’ve learnt quite a bit about how far
back this branch of math goes
I also discovered that there were many mathematicians
involved in creating the logic we know today
Logic is constantly expanding, at a rapid rate,
especially the past while to present
I found out that all of logic was based on the works of
Aristotle
There is more than just one type of logic, which is
something I never knew before now
Logic is not only a math thing, it can also be classified
as a philosophical branch as well, or at least when it
began to be viewed it was





Anonymous, . N.p.. Web. 4 Jun 2013.
http://en.wikipedia.org/wiki/Mathematical_logic
N.p.. Web. 5 Jun 2013.
http://www.math.psu.edu/simpson/papers/philmath
/philmath.html
. N.p.. Web. 5 Jun 2013.
http://www.csus.edu/indiv/d/dowdenb/160/s13/His
tory of Mathematical Logic.pdf
http://www.newworldencyclopedia.org/entry/Mathe
matical_logic . N.p.. Web. 5 Jun 2013.
Jim, Loy. N.p.. Web. 6 Jun 2013.
http://www.jimloy.com/number/consec0.htm