About Mathematics

A mathematics novice rushes in where most folk fear to tread.

Under this heading you will find a mathematical novice’s impressions of what mathematics is all about. It is a long article in several parts, accessed through sub-menu headings.

There will be no actual mathematics done here. This is more to remind me of why I might be tempted to think mathematically, and about the environment I will be entering when I am so moved.

Below are definitions of some concepts that will be mentioned in this article.

modus ponens inference rule (syllogism)
major and minor premisses entail conclusion
generalization inference rule
proposition with variable x: p(x) entails, for all x, p(x)
proposition
a truth-functional statement in mathematical logic
axiom
a non-provable proposition whose truth value can be assumed in an appropriate context (model)
axiomatic theory
a set of propositions, either axioms or derived propositions, together with valid rules of inference. For the axiomatic first-order theories forming the foundation of classical mathematics, the two valid inference rules are modus ponens and generalization
interpretation
a transformation of the variables, functions, and relations in the statements of a logical theory, into corresponding entities from a domain of context
model
an interpretation of a logical theory for which all axioms are true
tautology
a proposition that is always true under any interpretation. In a 2-value logic, p ∨ ¬p (read p or not p) is a tautology.
valid argument
a sequence of propositions that follows the rules of inference for the encompassing theory
sound argument
a valid argument where all propositions (hypotheses) are true
argument model
the set of true premises that make an argument sound
proof (deduction)
a sound argument establishing a proposition as true
theorem
a proposition of importance to a theory
lemma
a minor helping proposition useful for the proof of a theorem
corollary
a proposition that is an immediate consequence of a theorem
conjecture
a proposition for which no proof has yet been assigned; a plausible inference (hypothesis aka educated guess). A conjecture generally is based either on evidence or analogy, or both for the strongest conjectures. Great conjectures are usually tackled piecemeal, bottom-up through computation, mechanically searching for confirmations or counterexamples; sideways, looking at analogous situations; top-down, reaching for leaps of inspiration that may relate disparate facts into an integrated proof strategy. Two great conjectures related to ANT from the past are the Riemann Hypothesis and Fermat’s Last Theorem. The latter conjecture defied proof for some 350 years, but was proved by Wiles in 1995 and is the inspiration for the book Fearless Symmetry. The former is still a standing conjecture, but parts (under varying assumptions and analogies) are known to be true. The biggest conjectures generate interest that energizes generations of mathematicians
pattern
a discernible regularity of an arrangement of entities, governed by some logical rule that facilitates prediction and contemplation
symmetry
a function (transformation) that preserves what is important about some object. A starfish has 5-fold symmetry because 5 rotations of {72^\circ} brings the object back to its original position. Humans merely have 2-fold or {180^\circ} symmetry (our mirror image, and then the mirror image of our mirror image). The symmetry of the book title Fearless Symmetry refers to preservation of the mathematical properties of objects under transformations
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