Log or Linear? Distinct Intuitions of the Number Scale …

Log or Linear? Distinct Intuitions of the Number Scale in Western and Amazonian Indigene Cultures
Science 30 May 2008: 320(5880), pp. 1217-1220
Stanislas Dehaene, et al.
http://www.sciencemag.org/content/320/5880/1217

The mapping of numbers onto space is fundamental to measurement and to mathematics.
Is this mapping a cultural invention or a universal intuition shared by all humans regardless of culture and education?

At all ages, the Mundurucu mapped symbolic and nonsymbolic numbers onto a logarithmic scale, whereas Western adults used linear mapping with small or symbolic numbers and logarithmic mapping when numbers were presented nonsymbolically under conditions that discouraged counting.

This indicates that the mapping of numbers onto space is a universal intuition and that this initial intuition of number is logarithmic.
The concept of a linear number line appears to be a cultural invention that fails to develop in the absence of formal education.

Gödel’s incompleteness theorems

Gödel’s incompleteness theorems
http://en.wikipedia.org/wiki/G%C3%B6del’s_incompleteness_theorems

The theorems, proven by Kurt Gödel in 1931, are important both in mathematical logic and in the philosophy of mathematics.

For any such system, there will always be statements about the natural numbers that are true, but that are unprovable within the system.

The second incompleteness theorem, an extension of the first, shows that such a system cannot demonstrate its own consistency.

Relation to the liar paradox
The liar paradox is the sentence “This sentence is false.” An analysis of the liar sentence shows that it cannot be true (for then, as it asserts, it is false), nor can it be false (for then, it is true). A Gödel sentence G for a theory T makes a similar assertion to the liar sentence, but with truth replaced by provability: G says “G is not provable in the theory T.”

Extensions of Gödel’s original result
… it is common to state the effectiveness and expressiveness conditions as hypotheses for the incompleteness theorem, so that it is not limited to any particular formal theory. The terminology used to state these conditions was not yet developed in 1931 when Gödel published his results.

the epistemological relevance of the second incompleteness theorem

Examples of undecidable statements
There are two distinct senses of the word “undecidable” in mathematics and computer science. The first of these is …

Limitations of Gödel’s theorems
The conclusions of Gödel’s theorems are only proven for the formal theories that satisfy the necessary hypotheses.
Not all axiom systems satisfy these hypotheses, …

Gödel’s theorems only apply to effectively generated (that is, recursively enumerable) theories.
If all true statements about natural numbers are taken as axioms for a theory, then this theory is a consistent, complete extension of Peano arithmetic (called true arithmetic) for which none of Gödel’s theorems apply in a meaningful way, because this theory is not recursively enumerable.

Minds and machines
Authors including J. R. Lucas have debated what, if anything, Gödel’s incompleteness theorems imply about human intelligence. Much of the debate centers on whether the human mind is equivalent to a Turing machine, or by the Church–Turing thesis, any finite machine at all. If it is, and if the machine is consistent, then Gödel’s incompleteness theorems would apply to it.

Adolescents’ Functional Numeracy Is Predicted by …

Adolescents’ Functional Numeracy Is Predicted by Their School Entry Number System Knowledge
David C. Geary mail, Mary K. Hoard, Lara Nugent, Drew H. Bailey
January 30, 2013
http://www.plosone.org/article/info:doi/10.1371/journal.pone.0054651

One in five adults in the United States is functionally innumerate; they do not possess the mathematical competencies needed for many modern jobs. We administered functional numeracy measures used in studies of young adults’ employability and wages to 180 thirteen-year-olds. The adolescents began the study in kindergarten and participated in multiple assessments of intelligence, working memory, mathematical cognition, achievement, and in-class attentive behavior. Their number system knowledge at the beginning of first grade was defined by measures that assessed knowledge of the systematic relations among Arabic numerals and skill at using this knowledge to solve arithmetic problems. Early number system knowledge predicted functional numeracy more than six years later (ß = 0.195, p = .0014) controlling for intelligence, working memory, in-class attentive behavior, mathematical achievement, demographic and other factors, but skill at using counting procedures to solve arithmetic problems did not. In all, we identified specific beginning of schooling numerical knowledge that contributes to individual differences in adolescents’ functional numeracy and demonstrated that performance on mathematical achievement tests underestimates the importance of this early knowledge.

cited in:
Who Says Math Has to Be Boring?
December 7, 2013
http://www.nytimes.com/2013/12/08/opinion/sunday/who-says-math-has-to-be-boring.html