Multiple Kinds of Reason in Mathematics - Essay I
There are three kinds of rule-based intelligence in mathematics, logic
and most pattern-based subjects.
The first kind met in primary school arithmetic consists of skills with
repeatable, reproducible and therefore verifiable results - results that
are then considered right or wrong.
The second kind also met in primary school consists of pattern or rule
recognition. The development or exploitation of the ability to recognize
or suggest simply patterns in order to predict the next element in a
sequence. If the prediction fails, another pattern is required.
The third kind, assumption-based, deductive reason, appears after
inductive mastery of logic, that is mastery of implication rules If A
then B and their use. The third kind follows the use of implication rules
and definitions and assumptions, one at a time and one after another, to
arrive at logical conclusions. Here chains of reason how to be posed in a
readable, repeatable, reproducible and therefore verifiable manner.
For third kind of thinking in mathematics, there was a search for secure
assumptions, so that deductive reason could proceed in a consistent and
reliable manner. Unfortunately, uncertainty results in mathematical
logic imply more can suggested than proven in mathematical theories which
are not finite. So the assumptions made for the third kind of reason stem
from experience or trial and error over time. That identifies modern pure
mathematics as another empirical art. But mathematics by providing a
format for measurement and calculations remains the queen of science, a
queen in the hierarchy of empirical arts.
Pre-coordinate Euclidean geometry, the original model for pure reason in
mathematics, with its assumptions and deductive chains of reason is still
worth presenting in part if not in full in high school mathematics in a
selective manner to build algebraic-deductive skills and geometric skills
and sense. However, the empirical nature of pre-coordinate and hence
coordinate-free Euclidean Geometry is implied by diagrams with subtle
faults that imply incorrect conclusions - subtleties detected with the
use of coordinates in advance mathematics courses.
Further Reading: Logic
chapters 1 to 5 (Français)
in Volume 2, Three Skills for
Algebra introduce the Euclidean logic methods and questions in
mathematics free manner. . The use of logic in the form of direct or
indirect use of implication rules B if A or equivalently, If
A then B, informally or within axiomatic (assumed rules and
patterns) frameworks leads to further rules and patterns to accept and
use. See to the last chapters and postscripts of Volume 1A, Pattern Based Reason,
for a further discussion of consistency questions and indirect chains
of reason in general, and not just in mathematics.
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