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\title[ 4040: Reflections on Mathematics and its Role in Society ]{The Art of Equations}

\author{Dr. Sarah: Controversy in Equations}

\date{January 7, 2014}

\begin{document}


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\begin{frame}


 \frametitle{MAT 4040: Senior Capstone 
  \newline Today: Intro \& Controversy in Equations}
 

\begin{center} 

\includegraphics[width=3.8 in]{cg4fba76ef1f648.jpg}   
 \end{center}

 
\vspace*{-0.2cm}
Think of an equation that is important or interesting.

 {\color{purple} Write down:}
\begin{itemize}  
\item  The equation or its name
\item Why you choose this equation
\item  What it is trying to tell us about mathematics or the world
\end{itemize}
\pause
\vspace*{0cm}
In groups of 3, share and rank them in order of their \emph{truth}.




\end{frame}


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\begin{frame}

 \frametitle{$$\mbox{\emph{Controversy: Empirical versus Platonic Equations}  }$$ }

\vspace*{-1cm}


\begin{itemize}
\item Empirical: approximate relational fits to experimental data
\item Platonic: derived from  axioms
\pause \newline [G\"odel's incompleteness theorems]
\end{itemize}


\begin{center} \includegraphics[width=2.5 in]{godel.jpg}   
 \end{center}

% Austrian/American
%Play Godel's Incompleteness theorems
%Nintendo entertainment system.  you lose the ability to decide whether certain statements are true or false. This is rather shocking and you may wonder why G�del's result hasn't wiped out mathematics once and for all. 

\end{frame}
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\begin{frame}

 \frametitle{$$\mbox{\emph{How would you define an equation?  }}$$ }
 
\vspace*{-1cm}
 
 
 \begin{center} \includegraphics[height=1.5 in]{3equations.jpg}  \includegraphics[height=1.5 in]{tattoo2.jpg} \\
 \end{center}
 %Born Oppenheimer Approximation, 3-D Schr�dinger Equation and Schr�dinger Equations
 
 \pause




 



Aristotle's Thoughts on Equals: 
\emph{What is really peculiar to quantities is that we compare or contrast them in terms or on grounds of equality. We predicate `equal', `unequal' of all of the quantities mentioned�. Of nothing, moreover, save quantities can we affirm these two terms.  For we never say this disposition is `equal' to that or `unequal'�. Such things are termed `like' and `unlike'}






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 \vspace*{-1cm}

\begin{frame}

 \frametitle{$$\mbox{\emph{Controversy: Comparing  Infinite Quantities}  }$$ }


 \vspace*{-.85cm}


\begin{itemize}


\item Galileo Galilei (1564-1642):  \sout{sizes of infinite sets}

\item Georg Cantor (1845-1918) 
\begin{itemize}
\item revolutionary ideas on the comparison of  infinite sets 
\item basis of many ideas in modern mathematics, including analysis and calculus
\item mathematicians and theologians strongly objected that the work challenged the uniqueness and infinity of God 
\item no recognition during his lifetime
\end{itemize}
\end{itemize}
\begin{center} \includegraphics[width=2.3 in]{georgeanddiagonal.jpg}   \end{center}

%\begin{center} \includegraphics[width=1.5 in]{hilberts_hotel.jpg}   \end{center}
%Consider a hypothetical hotel with countably infinitely many rooms, all of which are occupied � that is to say every room contains a guest. One might be tempted to think that the hotel would not be able to accommodate any newly arriving guests, as would be the case with a finite number of rooms.  It is also possible to accommodate a countably infinite number of new guests: just move the person occupying room 1 to room 2, the guest occupying room 2 to room 4, and in general room n to room 2n, and all the odd-numbered rooms will be free for the new guests.
\end{frame}



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\begin{frame}

  \frametitle{$$\mbox{\emph{MAT 4040: Senior Capstone Goals}  }$$ }
\vspace*{-1cm}
 \begin{itemize}

\item \color{purple}Professional writing, speaking and reflections on mathematics and its role in society\color{black}
\item Investigate mathematical knowledge - different  areas 
\item Relate mathematics to other disciplines and society 
\item Understand something of the breadth of mathematics
\item Reflect on ideas from the past and from abroad
\item Reflect on personal development in the educational goals:
\begin{itemize}
\item	Thinking critically and creatively
\item	Communicating effectively
\item	Making local to global connections
\item	Understanding responsibilities of community membership. 
\end{itemize}

 \end{itemize}










\end{frame}

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\vspace*{-1cm}
\begin{frame}

  \frametitle{$$\mbox{\emph{Grading and MAT 4040 Content}  }$$ }

\vspace*{-1cm}
 \begin{itemize}


\item {\small Participation 30\%
 \item Homework 30\%  {\tiny Late work will obtain at most half credit.  Must be turned in by assigned final.}

  \item Course Project 40\%  {\tiny Must participate to pass the class.}
  
 \vspace*{.5cm}
\item LaTeX typesetting and standard mathematical publishing formats 
\item Bibliographies and recognition of support and prior results 
\item Mathematics and society and stereotypes in mathematics 
\item Mathematical thought processes 
\item Discussion on success and career opportunities in mathematics 
\item Professional societies and their role in mathematics 
\item Mathematics and the general education 
\item Oral presentation styles, tools, and strategies 
\item Mathematics in other cultures and the culture of mathematics 
\item Topics in different areas, including ``pure" and ``applied" 
\item Topics in the history of mathematics and mathematicians

}


 \end{itemize}
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\vspace*{-1cm}
\begin{frame}
 \frametitle{$$\mbox{My own equation from \emph{Bender's Big Score}}$$ }

\vspace*{-.9cm}


 {\small {\bf Farnsworth}: \emph{So paradox free  time travel is possible after all.}
\newline {\bf Bubblegum}: \emph{Right on. But dig this multiplicand here.}
\newline {\bf Farnsworth}: \emph{The doom field? That must be what corrects the paradoxes.}
\newline {\bf Curly Joe}: \emph{When that momma rises exponentially, it could rupture the very 
                   fabric of causality.}}
\vspace{-.2cm}

\begin{center} \includegraphics[height=1.7 in]{Greenwaldianacopy.jpg}    \includegraphics[height=1.7 in]{DSCN0049_2.jpg} 


\vspace{-.2cm}

 {\scriptsize  \vspace{0.2cm}{\it Futurama}\texttrademark and \copyright{ } Twentieth Century Fox Film Corporation.\\ \vspace{-.2cm}
Educational and related content is not specifically authorized by Fox.}

 \end{center}

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\begin{frame}
 \frametitle{$$\mbox{\emph{Controversy with Fractions and Representing Equations}}$$ }

\begin{center} \includegraphics[width=3in]{cartoon4.jpg}  \end{center}

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\begin{frame}
 \frametitle{$$\mbox{\emph{Controversy with Fractions and Representing Equations}}$$ }
 
 \vspace*{-0.7cm}
 
 \includegraphics[width=0.11in]{lightbulb.jpg}  \hspace*{.3cm}  {\small Watch the video:  http://www.youtube.com/watch?v=AKYZhdbnOWM
%\newline Transcript: http://tech1.dccs.upenn.edu/home/news/trans\_deturck.html}
}\vspace*{0.2cm}

\pause {\bf Dennis Deturck}: \emph{Despite the fact that great historical and theoretical significance has been imported to fractions and rational numbers, its study should be deferred until it's really needed and can be appreciated, which may not be until after somebody learns calculus.}

\vspace*{0.2cm}
{\small
 \includegraphics[width=0.11in]{lightbulb.jpg}  \hspace*{.3cm}  
 Reflect on the speech. Share aspects that surprised you, aspects that you agreed or disagreed with, and aspects related to your own life


\pause

 \includegraphics[width=0.11in]{lightbulb.jpg}  \hspace*{.3cm}  What are topics that were once taught and are no longer in the curriculum?
 
 

 \includegraphics[width=0.11in]{lightbulb.jpg}  \hspace*{.3cm}  What other areas and equations do fractions impact?
 
 %Roman Numerals, Abacus, Log tables, ether - dark matter
 %baking, construction, expressing and solving equations that require algebra 1/3=.3333333 repeating, area in geometry or probability
}

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\begin{frame}

 \frametitle{$$\mbox{\emph{Beauty in Equations}  }:  e^{i \pi} = -1$$}


 \vspace*{-.8cm}

\begin{itemize}

{\small
\item \emph{It is absolutely paradoxical; we cannot understand it, and we don't know what it means, but we have proved it, and therefore we know it must be the truth.}
[Benjamin Peirce, 19th century] 

\pause
 \item \emph{Like a Shakespearean sonnet that captures the very essence of love, or a painting that brings out the beauty of the human form that is far more than just skin deep, Euler's Equation reaches down into the very depths of existence.} [Keith Devlin]}


\end{itemize}

 \vspace*{-.3cm}

\begin{center} \includegraphics[width=2 in]{Homerepiicopy.jpg} \end{center}

% A bird in the hand is worth 2 in the bush


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