Introduction

Consider a screwdriver. Can you enumerate all the possible uses of a screwdriver, alone or with other objects or processes? No. The possible uses of a screwdriver is both indefinite in number and can't be ordered. This means that no algorithmic procedure can list all the uses of such a screwdriver alone or with other objects or processes.

Next, consider one use of a screwdriver, say, opening a can of paint. Can you enumerate all the other objects and processes that can open a can of paint, i.e., carry out the "function" of opening a can of paint? Again, no. The objects or processes that might open a can of paint, carry out that function, are, again, indefinite in number and can't be ordered. Again, no algorithm can list all these objects and processes.

Information Theory Does Not Apply To The Evolution Of The Biosphere
Stuart Kauffman
National Public Radio

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 The most beautiful mathematics was that which had no practical applications in the outside world.
 
A view attributed to
Godfrey Harold "G. H." Hardy

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 Applied Mathematics - the branches of mathematics that are involved in the study of the physical or biological or  sociological world. The measure of the value of Applied Mathematics is its usefulness.
 
Various Sources

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 The topics covered in this book were first studied by the outstanding mathematicians of the eighteenth and nineteenth centuries. Among the many who devoted themselves to these studies are Gauss, Euler Fourier, Legendre, and Bessel. These men did not recognize the modern and somewhat artificial distinction between pure and applied mathematics.
 
The Functions of Mathematical Physics
Harry Hochstadt

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 'When I use a word,' Humpty Dumpty said, in rather a scornful tone, 'it means just what I choose it to mean — neither more nor less.'
'The question is,' said Alice, 'whether you can make words mean so many different things.'
'The question is,' said Humpty Dumpty, 'which is to be master — that's all.'

Through the Looking Glass
Lewis Carroll

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Batting Averages as Information:

• Choosing a pinch hitter.

• The movie Moneyball and the book, "the signal and the noise" by Nate Silver.

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Information Theory, The First Diagram:

Where:

• The Message is a string of symbols from some alphabet.
• Each symbol is encoded as, for example, a string of 1's and 0's.
• The result is transmitted.
• The string of 1's and 0's received is decoded into the alphabet, or perhaps a different alphabet

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The Questions:

1. Suppose we know that data can be transmitted without error.
    
   
   • Can we develop an "efficient" encoding technology?

    

2. Suppose we know that there might be errors in transmission, but we also know something about the error rates.
    
   
   • Can we develop an encoding technology that detects errors?
   • Can we develop an encoding technology that corrects errors?

    

3. Suppose we wish the message to be private.

4. What are appropriate encryption/encoding technologies?

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Information Theory, The Second Diagram:

• Can the square root of 2 be described in a string consisting of a finite number of 1s and 0s?

• Is there a "program" that   describes any outcome of a 1000 flips of a coin in less than 1000 1s and 0s? That is, A flips a coin 1000 times and encodes the result in a sequence of less than 1000 1s and 0s. A gives the encoded string to B and B decodes it and retrieves the original string.

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                                                                                            From David MacKay's Lectures (Lecture 1 the first 20+ minutes)

Watching the  David MacKay Lecture videos:

1. I will also focus on Binomial Distributions but I will  postpone discussing various topics until there is time to review Probability. I will  spend some class time reviewing the subject.

2. This course will diverge from the Lectures somewhere after 6+ Lectures.

3. There are differences in notation  and terminology .

4. I am going to present a more computationally robust version of Hamming Codes [7,4,3] codes.