Algorithms and Truth

Fixing a broken lamp

In an earlier post, I introduced the concept of algorithms as a way of solving problems using a step-by-step process.  Most of us understand algorithmic thinking when it is applied to everyday tasks.  Fixing a broken lamp is a good example of an algorithm, because it involves taking different paths, depending on the outcome of several questions.  A recipe is a simple form of an algorithm:

Sausage and Peppers
1 package of Italian sausage (mild or hot, your choice)
2 bell peppers (red, orange, yellow, green, your choice), sliced
1 sweet onion, sliced
1 Tbsp minced garlic
fresh basil (or dried)
2 Tbsp olive oil
1 28 oz can whole tomatoes
1 package angel hair or penne pasta

1. Cook the sausage whole, on medium heat, until browned, using 1 Tbsp of olive oil. Alternatively, broil the sausage in the oven.  Slice into pieces and reserve
2. In the same pan, cook the bell peppers, onion over medium heat using 1 Tbsp of olive oil, until onion is lightly browned.
3. Add the garlic to the pan and simmer until slightly browned.
4. Add the sausage back to the pan.
5. Puree the tomatoes by pulsing, so that there are still small chunks.
6. Add the tomatoes to the pan, along with the basil.
7. Simmer the sauce on medium-low until reduced and thickened, about 10 or 15 minutes.
8. Serve over angel hair or penne pasta, cooked to package instructions for al dente.

Algorithms are used in computer science to devise a method for a computer to solve a problem.  One of the classic problems that has been solved in many different ways is the task of sorting a set of numbers.  One solution is the quicksort algorithm, which looks like this:

function quicksort(array)
     var list less, greater
     if length(array) ≤ 1
         return array
     select and remove a pivot value pivot from array
     for each x in array
         if x ≤ pivot then append x to less
         else append x to greater
     return concatenate(quicksort(less), pivot, quicksort(greater))

Quicksort Algorithm in action (Wikipedia)

The basic idea is this: (1) start with a list of numbers, (2) choose one number to be the pivot, (3) separate the remaining numbers into two lists, one with all the numbers smaller than the pivot, and one with all the numbers larger than the pivot, (4) repeat the quicksort algorithm on the list of smaller numbers, then on the list of larger numbers, (5) combine the results into a new list, with the sorted numbers less than the pivot, followed by the pivot, and then by the sorted numbers greater than the pivot.  This is illustrated at right.

One way of understanding Francis Bacon's Novum Organum is to view it as the development of a new algorithm for advancing scientific knowledge.  An illustration of Bacon's algorithm is found in the second part of Novum Organum, where he conducts an investigation into the form of heat.  He first lists the all the instances in which heat occurs, then lists all the instances in which heat does not occur, and finally lists the instances in which heat varies by degree.  Inductive reasoning can then be used to discover a law, based on direct observation of nature.

One of the advantages of algorithms is that they are easily compared.  Which recipe produces the best pizza?  Which sorting algorithm works fastest?  Which scientific method does the best job of discovering truth, and in which circumstances -- Aristotle's deductive reasoning, Bacon's inductive reasoning, or something else entirely?  What metrics do we use to even begin to answer this question?