Bird Brain Part 2

Bird Brain Part 2

In my last blog, I described how some birds' brains outperform human brains. I concluded by describing some of the outstanding abilities of Alex, an African Grey Parrot. One of Alex's feats caused some confusion among my readers: I wrote that I was astounded Alex understood the concept of zero, an abstract notion that was not fully integrated into mathematics until a few hundred years ago.

The first known symbol for the number 0 

Zero plays two distinct roles in mathematics: 1) it is a placeholder, e.g., it is used to distinguish the number 1 from the number 10; and 2) it symbolizes the concept of nothingness. Indeed much confusion between these two uses of zero could have been avoided throughout history if the two distinct notions had been given different names and different symbols. Alex displayed a rudimentary understanding of zero as "nothingness" -- the absence of anything that exists.

Before proceeding, let's look more closely at how Alex processed information and how he displayed what he knew. Alex, of course, was frequently asked to perform for researchers and for the media. When Alex tired of performing, he either would say "wanna go back" [to his perch] or deliberately give incorrect answers to questions to end the session. In the case I cited in the earlier blog, Alex had become bored with performing and tried to end the session by avoiding correct answers.

"Once, Alex was given several different colored blocks (two red, three blue, and four green ...). Pepperberg asked him, "What color three?" expecting him to say blue. However, as Alex had been asked this question before, he seemed to have become bored. He answered "five!" This kept occurring until Pepperberg said "Fine, what color five?" Alex replied 'none'." (Wikipedia)
The word "none" previously had been used by Alex to say there was no difference between two identical objects.

"If asked what the difference was between two identical blue keys, Alex learned to reply, “None.” (He pronounced it “nuh.”)" In both cases, the word nuh was used by Alex to mean "nonexistence" rather than for purposes of counting. There is a long history of philosophical and mathematical controversy about zero. Indeed there are more than a dozen books devoted to the subject (my favorite is Zero, The Biography of a Dangerous Idea by Charles Seife). Unless you want to dig a little deeper into the academic side of the literature about zero, stop reading now. What follows is a simplified glimpse into the strange nature of the concept of zero and its relation to set theory from which all numbers are logically constructed. I include the discussion only because it underscores how sophisticated Alex's concept of "nuh" was.

0, 1, 2, 3... looks like a natural sequence of numbers. Zero didn't always occupy its place in this list. We saw an unintentional sleight of hand when we crossed the "Y2K" threshold on January 1, 2001, and not January 1, 2000. The reason was simply that the calendar most used by international standards were conceived when the sequence of years was numbered -2, -1, 1, 2, 3... (the negative numbers represent BC and the positive numbers represent AD). Zero was not recognized as holding its current place in the list of integers: ...-2, -1, 0, 1, 2, 3... and with good reason. Early civilizations naturally used natural numbers that could be match one for one with things (e.g., human fingers).

To be precise for current purposes, natural numbers are used to count things and begin with 1 continuing towards infinity (1, 2, 3,...,). Whole numbers are natural numbers with 0 added (0, 1, 2, 3...), Integers are whole numbers with their negative counterparts (-2, -1, 0, 1, 2, 3...).

The reason mathematics developed without zero until a few centuries ago zero was a misfit. Any natural number can be divided by another natural number but not by zero. Thus zero is a misfit, it misbehaves in other ways as well. Add a number to itself and you always get a different number (1+1=2); but not so with zero (0+0=0). Zero cannot make any number bigger; add zero to any number as many times as you like and the original number stubbornly remains the same, unlike any other number. Zero reduces any other number to zero by multiplication as if to reproduce itself indiscriminately. And there are many other anomalies that prevent zero from being a part of a simple set of rules that apply to all numbers.

Numbers can be constructed from set theory. The contents of a grocery cart is a set and its contents can be counted using whole numbers: an empty cart has zero items, I can use the express line if the set contains 15 or fewer items. But to construct numbers from set theory requires another misfit. An empty cart cannot be assigned the number zero to build numbers from sets. I will skip the technical details and ask your intuition to take a leap of faith. We must begin building numbers starting with the empty set, a set that consists of nothing. The cart contained zero items; there is a difference.

Examples of an empty set: the set containing all triangles with four corners, the set of whole numbers that are larger than 3 and less than 2. The empty set -- also called the null set -- is different than zero. A set containing zero is written as {0}. This set has one member, viz. the whole number, 0. The null set contains no members and is written {}. Alex did not declare "what color five" to be zero blocks on the tray; he said there is no group of five blocks; what color five is an empty set. That was a pretty sophisticated response for a bird brain.