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Natural Science
NATS 1700
Zbigniew Stachniak

Lecture 2. Where there is life, there are numbers: pre- history of computing (c) Z. Stachniak, 2011 informal and unedited notes, not for distribution Our present day reality is often referred to as the digital age; we often speak about ourselves as consumers of information. However, we dont need to look very hard to see that our civilization is based on more elementary concepts than computers and information, con- cepts such as numbers and counting. Numbers are everywhere, they surround us. We see numbers, hear num- bers, read them and write them many times a day. We qualify objects using numbers: we measure them, we weight them. We speak about progress and the luck of it with numbers. We build, buy and sale with numbers. Each individual carries an assigned set of unique numbers: social security number, student number, bank account number, passport number, drivers licence number, your age and weight, your height and size of shoes. Without numbers there would be no civilization as we know it, there would be no mathematics, no computers. It was the invention of numbers and counting that kick-started our civilization. If the history tape of our civilization was re-wound a few million years back to the point when humans began to build social ties, would we develop our civilization in the same way? Would we invent numbers, give them symbols, start to count and perform arithmetic operations? Would we invent calcu- lating machines and, eventually, computers? In this lecture I argue that this is exactly what would happen. 1 Man is a rational animal because he can count [Aristo- tle]. To count, one needs the concept of a number and a purpose. There are many important questions about our journey towards numbers and counting, and how they became inseparable from our every-day activi- ties (purpose), questions such as: When did we abstract the concept of number from quantity? When did we achieve the developmental state in which we began to comprehend that a pair of children, a pair of cows, and a pair of trees represent the same quantity: as many children as cows and as many cows as trees? When did we start to express these numbers using words and signs, when did we learn how to record them? Numbers are abstract concepts representing quantities. They are abstract since they do not exist in nature, only in our minds. We can write and read their symbols but we can neither see them nor touch them. Most likely, we invented numbers when there was a need to mentally op- erate with quantities without instantiating these quantities, when there was a need to separate the discrete quantity two from two eggs and from two trees. These two aggregates of objects: two eggs and two trees share a quantity: a pair. The ability to abstract quantities probably happened very early in human de- velopment but how earlyone million years ago? 10 million?nobody knows. 2 Let us imagine early human hunters, say 50,000 years ago, making prepara- tions for a hunt. Assuming that they didnt care too much about what kind of animals they wanted to trap, antelopes or wild pigs, how did they agree on the number of killed animals that would make their hunt a success? They had to agree on whether a single large animal would do, or a smaller animal and some other animala pairwould be enough. Or perhaps, they had to agree on capturing many animals, where many was a distinct quan- tity from single (or one) and from a pair (or two). Since the kind of animals they were planning to hunt was not an issue, what the hunters had to agree upon was the quantity: one, two, or many. To be able to do that, the hunters had to be able to abstract (or to separate) the concept of quantity from particular instances of quantities, that is they had to be able to envision numbers. That process of abstracting numbers took place long ago as ethnographers failed to discover a single language (ancient or modern) in which the sugges- tion of numbers does not appear, which did not provide some way to describe or dierentiate quantities. Some of primitive languages provide a very lim- ited set of numbers: just 1 and 2, or 1, 2, and many. Even those that have no words to describe numbers provide phrases to dierentiate between one and two, or one and many. 3 Does counting require higher intelligence? Since we dont see advanced technology-based civilizations of insects or horses, does this mean that we are the only species on Earth that understands num- bers and can count? Finding an answer to this question may solve other puzzles such as the level of intelligence required to perform some primitive forms of counting. Until a century ago or so, it was thought that man alone could count. But observations and research initiated in the rst half of the last century sug- gests another conclusion, a possibility that species other than ourselves may use, or may be trained to use, some forms of counting. Example 1: Birds in general It can be observed that if a birds nest contained say, four eggs, then one could safely be taken. But if two were removed, the bird generally aban- doned the nest. This experiment does not imply that birds can count. Birds could simply distinguish between full and half-empty nest. Or, perhaps, a bird simply observes that something is wrong with the nest, perhaps that it isnt its own, and ies away. However, another documented event may suggest that birds not only can distinguish dierent quantities but that they can also act upon their quantity- based observations. 4
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