Alice’s adventures in inverse tan land – mathematical argument, language and proof

Andrew Palfreyman’s article [1] reminds us of the result (1) $${\rm{ta}}{{\rm{n}}^{{\rm{ - 1}}}}{\rm{ + ta}}{{\rm{n}}^{{\rm{ - 1}}}}\,2{\rm{ + ta}}{{\rm{n}}^{{\rm{ - 1}}}}{\rm{ 3 = }}\,\pi {\rm{, }}$$ having been set the challenge of finding the value of the left-hand side by his head of department at the start of a departmental meeting.

The prescient power of indifference

Many a secret that cannot be pried out by curiosity can be drawn out by indifference.Sidney J. HarrisLack of information is generally considered a hindrance to inquiry. Surprisingly, a simple mathematical argument, relying on the Principle of Indifference, shows there are situations where the opposite holds. Even more surprisingly, this indifference allows one to guess, with a success rate greater than 50%, the outcome of a coin toss or any other experiment having two equiprobable outcomes. The scheme is based on work by American statistician David Blackwell (1919–2010) and a principle of mathematical probability attributed to Swiss mathematician Jakob Bernoulli (1655–1705).

THE MATHEMATICAL ARGUMENTATION ABILITY AND ADVERSITY QUOTIENT (AQ) OF PRE-SERVICE MATHEMATICS TEACHER

The Mathematical argumentation has been studied before, but no research has a focus on mathematical argumentation and adversity quotient of the pre-service mathematics teacher. This study is experimental research that aims to know and examine in depth about the influence of AQ of pre-service mathematics teacher toward the achievement of mathematical argument ability. The population of this study is the pre-service mathematics teacher in Cimahi City, West Java, Indonesia; while the sample is 60 pre-service mathematics teachers selected purposively. The instruments of this study are tests and non-tests. They are based on the assessment of good characteristics towards students' mathematical argumentation abilities, while the non-test instrument is based on the assessment of good characteristics towards AQ. The results of this research show that: (1) AQ gives positive influence to the development of mathematical argumentation ability of pre-service mathematics teacher with the influence of 60.2%, while the rest of it (39.8%) is influenced by other factors outside AQ; (2) The ability of mathematical argumentation of pre-service mathematics teacher is more developed on AQ of Climber type; (3) Students with the Quitter AQ type still tend to have less ability of mathematical argumentation.

Formally Proving the Boolean Pythagorean Triples Conjecture

In 2016, Heule, Kullmann and Marek solved the Boolean Pythagorean Triples problem: is there a binary coloring of the natural numbers such that every Pythagorean triple contains an element of each color? By encoding a finite portion of this problem as a propositional formula and showing its unsatisfiability, they established that such a coloring does not exist. Subsequently, this answer was verified by a correct-by-construction checker extracted from a Coq formalization, which was able to reproduce the original proof. However, none of these works address the question of formally addressing the relationship between the propositional formula that was constructed and the mathematical problem being considered. In this work, we formalize the Boolean Pythagorean Triples problem in Coq. We recursively define a family of propositional formulas, parameterized on a natural number n, and show that unsatisfiability of this formula for any particular n implies that there does not exist a solution to the problem. We then formalize the mathematical argument behind the simplification step in the original proof of unsatisfiability and the logical argument underlying cube-and-conquer, obtaining a verified proof of Heule et al.’s solution.

Sources of Evidence

This chapter examines the relations between patterns of causes and outcomes. The popular way to phrase Niels Bohr's principle that the validity of every conclusion depends on its source of evidence is to write that scientists assign a probability to the validity of each statement in accord with the evidence cited to support it. Statements that refer to exactly the same observation can have more than one validity if they are based on different evidence. It is useful to distinguish among the validity of a statement about nature that is based on certain observations, the truth of a conclusion that is based on the coherence of a logical or mathematical argument, and the rightness of a moral proposition based on a feeling.