Introduction - will explain 18th century concept of proof, focusing on (probably), examples as proof of general by outlining algorithm/procedure for example (exactness/concreteness being valued over abstractness).
In
Arithmatick books in the 18th century, mathematical operations, such
as addition, subtraction, multiplication, division, etc. operate by means of an
explicitly defined procedure for proving, or solving. By following this algorithm, one is able to
solve any problem of that type (i.e. addition, division, etc.). Further, this algorithm, in order to arrive
at the correct solution, must be done in a certain order, and arrive at only
one solution. That is, up to equalities
(i.e. 1 gallon is 4 quarts), the solution is unique. For subtraction, say 5842 – 2751, as
outlined in Spence’s Arithmatick
compendiz’d..., “The Operation is thus. Begin at the right Hand, and say, 1
from 2, there remains 1, which set down. Next 5 from 4, I cannot but from 14 (borrowing
10 as was directed to make it 14) there remains 9. Then proceed, saying, 1 I borrowed and 7 is 8,
from 8 remains nothing, for which set down a Cypher [0]. Lastly, 2 from 5 , remains 3, which set down, and
the Work is finished” (12). This algorithm
has two parts. Let’s say we’re
subtracting
If the first number is bigger than the
second, “say,
from
, there remains
”. If the second is bigger
than the first, “say
from
, I cannot but
from 1
(borrowing 10 as was directed) there remains
, and proceed, saying 1 I
borrowed and
is
, from
is
, which set down.” For each column of numbers, this procedure
holds. What is not explained here, or in any other arithmatick treatise in the
18th century, is why this works.
However, because mathmaticians know that the algorithm works, following
it is the ‘reasonable’ approach to arithmatick.