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Chapter 2 What is Math 13
4. Comment on the following experiments:
sort=:/:~
w=:'SPOT'
sort w
table=:(i.!#w)A. w
# table
sort table
5. A table with more rows than columns may be displayed more compactly by
transposing it. Try the following:
transpose=:|:
transpose table
The function A. applies to lists of numbers as well as to lists of letters (words),
and when applied to lists such as i.3 and i.4 produces tables that show its
behaviour more clearly. The following experiment uses the link function (;) to
box tables and link them together for more convenient comparison:
i=:i.24
(i A. 'SPOT');(i A. 'ABCD');(i A. 0 1 2 3)
+----+----+-------+
|SPOT|ABCD|0 1 2 3|
|SPTO|ABDC|0 1 3 2|
|SOPT|ACBD|0 2 1 3|
|SOTP|ACDB|0 2 3 1|
|STPO|ADBC|0 3 1 2|
|STOP|ADCB|0 3 2 1|
|PSOT|BACD|1 0 2 3|
|PSTO|BADC|1 0 3 2|
|POST|BCAD|1 2 0 3|
|POTS|BCDA|1 2 3 0|
|PTSO|BDAC|1 3 0 2|
|PTOS|BDCA|1 3 2 0|
|OSPT|CABD|2 0 1 3|
|OSTP|CADB|2 0 3 1|
|OPST|CBAD|2 1 0 3|
|OPTS|CBDA|2 1 3 0|
|OTSP|CDAB|2 3 0 1|
|OTPS|CDBA|2 3 1 0|
|TSPO|DABC|3 0 1 2|
|TSOP|DACB|3 0 2 1|
|TPSO|DBAC|3 1 0 2|
|TPOS|DBCA|3 1 2 0|
|TOSP|DCAB|3 2 0 1|
|TOPS|DCBA|3 2 1 0|
+----+----+-------+
B. Proofs
Although proofs are an important (and many would say the essential) part of
mathematics, we will spend little time on them in this book.
In introducing his book Proofs and Refutations: The Logic of Mathematical
Discovery [4], Imre Lakatos makes the following point:
Its modest aim is to elaborate the point that informal, quasiempirical, mathematics does not grow through a monotonous
increase of the number of indubitably established theorems but