Roughly a year ago Edmund Harriss asked me to respond to his post regarding what he called “Mathematical Scales”, that is–
I am in love with this idea of training, taking someone who has proved incredibly able in an area and taking them back to the most basic ideas. I started to wonder what the equivalent might be for mathematics.
Here is a test of counting in the natural numbers. The difficulty increases as the test goes on.
1. If 3 were dropped from the counting sequence 0, 1, 2, 3, 4, 5, … what would 2 + 2 equal?
2. The Yukis were located in modern day California and used a base 4 and a base 8 system. This came about from a natural form of body counting — how?
3. Subitizing is counting without having to enumerate; for small quantities this means a “a feeling of immediately knowing how many items lie within the visual scene” and for larger quantities this becomes an estimate. For example, it is possible to note how many triangles are depicted
without having to stop and mentally think “one, two, three” but rather “three” straightaway.
Explain how subitizing and the Roman numeral system are related.
4. Describe how counting and the commutive property of addition () are related.
5. Early in a child’s development counting tends to be just “recite the numbers in order.” Later they understand that counting enumerates a set. For example, one principle is that the last number counted equals the number of items. Name three more such principles.
6. Count from 1 to 10 (where each number represents our usual base 10 understanding of 1 to 10) in base .
7. The number of sense of babies is tested by checking for essentially, level of surprise. In an experiment of Karen Wynn, one puppet was moved behind a curtain followed by a second puppet; the curtain opened to reveal either two or three puppets. The babies were more startled by three puppets than two, essentially thinking that two was the “correct” solution to 1 + 1.
In a different experiment by Xu and Carey, babies were startled by the situation depicted below
but not by this one:
Explain the ramifications for the number sense of babies.
[Diagram from The Number Sense by Stanislas Dehaene.]
8. Explain the following quote:
As Russell also helpfully pointed out, one of the reasons it took so long to discover a “definition” of the natural numbers, is that the names of the natural numbers — e.g. “3” — are used both as adjectives and nouns, and this distinction was not sooner realized.
— From Smullyan and Fitting’s Set Theory and the Continuum Problem
9. Here is a typical (although not the only) formulation of the Peano axioms:
I. 0 is a natural number.
II. If is a natural number, so is .
III. For any natural number , does not equal 0.
IV. For any and , if then .
V. Mathematical induction: For any set , these conditions are sufficient to show contains every natural number. a.) . b.) For every natural number n, implies .
What happens if the statement “0 is a natural number” is omitted?
10. Demonstrate why “every set of natural numbers has a smallest member” can substitute for the Principle of Induction in the usual Peano axioms.
11. The (admittedly radical) philosophy of ultrafinitism denies that the infinite set of natural numbers exists. Give one possible justification for this philosophy.
12. Number can be defined without sets or classes.
What is the major downside of this formulation?
13. The now-standard formulation of natural numbers in set theory (due to John von Neumann) is , , and so on. Give an alternate formulation.
14. Design a set of axioms (without referring to previously formulated counting axioms) so it produces the sequence 0, 0, 1, 0, 1, 2, 0, 1, 2, 3, 0, 1, 2, 3, 4, 0, 1, 2, 3, 4, 5, …
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