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Computers, mathematics, and proof

Each day brings greater involvement of computers in every facet of our day to day life. Mathematics is no different.

Use of computers for computation in mathematics is now very well-established. For example, suppose we want to figure out if $2^{15485862}$ is divisible by $15485863$. Computing the power and then long dividing by hand would take a person awhile. (But there is a secret shortcut if you know some abstract algebra). But a well-written computer program can do it in milli-seconds.

Another use, relevant to us directly, is the encoding mathematical ideas and proof in computer programs. Proofs become data which the computer can then check for correctness.

Mathematicians and computer scientists have worked on this idea for decades but it is only relatively recently that computer checked proofs have reached a threshold of usability to see widespread adoption by mathematicians.

We will use a tool called Lean to aid us in our investigations of mathematics and proof in particular. To begin with, we will see how to represent all of proposition logic in Lean.

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