Thomas Hobbes has suggested that the life of people living without any form of organized government was ‘nasty, brutish and short,’ but human beings seem to have organized themselves in a wide variety of ways since the dawn of humanity. As civilizations grew and became more complex, the resulting economic systems allowed the development of groups of people who had time to think about things in the world around them.

Among the early civilizations, that of ancient Greece seems to have attained a situation where a reasonably large group of people had the leisure to think about and discuss the world around them. While this situation also existed in most of the other great civilizations of the same period, the ancient Greeks seem to have been able to advance further than the others.

If we consider the period up to about 500 BC, there had been many developments in mathematics that attempted to solve the problems of building structures. That these methods succeeded can be seen in the longevity of the Pyramids and the magnificence of other structures of the period. But the people of the time recognized that their connection to the world was imprecise: measurements never closed perfectly, for example. So the mathematics of the time had this imprecision built into it. It was very much developed from experience and practice, not derived from theory. Indeed, the idea of ‘theory’ would have made little sense at the time.

This lack of basic numeracy is reflected in languages, even today. Many languages lacked words that related to numbers beyond two or three. The idea of number beyond physical objects didn’t exist, so the description of the number of objects often varied with the type of objects. English has a number of words that mean ‘two’ but are applied to a limited range of objects or today in limited circumstances: couple, pair, brace, twin, twain; and borrowed from other languages: dyad, duo, duet, duplet. But there are very few words that deal with three, four or five objects that have not been borrowed from other languages, and mostly from the counting numbers of those languages. Similarly, while most modern languages have singular and plural forms of words, some languages have three number forms for one, two, or more than two objects, e.g., Ancient Greek.

So the ancient Greeks were able to have groups of people with time to think, as did other civilizations of the day. However, the ancient Greeks introduced several important thinking developments or tools, which allowed them to develop their thinking in new and highly productive directions. Among these tools were:

- Hypothesis;
- Deductive Logic; and
- Abstraction.

It has been speculated that the primary difference between Ancient Greek and Chinese thinking of this period was that the Greeks had the notion of the hypothesis, while the Chinese did not; and the lack of the ability to hypothesize limited the advances that the technologically and organizationally more advanced Chinese could accomplish.

The purpose of hypothesis is to advance an idea in a speculative manner, to say ‘What if…’ and continue thinking about the question without having to worry about the reality of the situation. Even the ancient Greeks had their own limitations to using hypotheses, of course, in that they understood mathematical powers of two and three very well, as they were direct representations of 2-D and 3-D geometry, but the lack of apparent higher spatial dimensions meant they did not contemplate, even hypothetically, powers higher than three.

Deductive logic allowed an idea or concept to be tested and then used as the basis for a subsequent idea. The carefully constructed edifice of Euclidean Geometry, a masterpiece of deductive logic, is a testament to the power of this tool.

But it is perhaps in abstraction that the ancient Greeks have given the world one of the most powerful and pervasive of thinking tools. In a stroke they established the foundation of modern mathematics, the need for statistics, the basis for error theory, and one of the reasons why mathematics today is considered so difficult by so many people. We can’t really blame them for the last, but abstraction is a difficulty for many people.

**Abstraction**

How does abstraction work? With abstraction, we can separate the numbers from the objects being counted, and then consider the numbers by themselves. That this was once ‘unnatural’ was reflected in the deep connection between numbers and objects in older languages. What do four stones and four people and four cows and four mountains have in common, apart from the abstract concept of number?

A more important abstraction was the idea that behind the imperfections of the real world there might be (hypothetically) a perfect world, where everything fell into its correct place and relationship. By overlooking the irregularities and unpredictable details of the real world, perhaps there was something that could be learnt in this abstract world. Plato’s discussion of the prisoners in the cave who had only shadows cast on the wall to help them deduce the nature of reality is a picture of such abstraction.

So the ancient Greeks were able to deduce that the Earth was approximately spherical, even as they realized that there were mountains that clearly made it non-spherical. The presence of this spherical abstraction meant that it was sensible to deduce that north-south location could be determined from the position of the sun and stars in the sky, and that it was possible to measure the size of the Earth, as Eratosthenes did.

By creating a field of thought that allowed for mathematical ‘perfection,’ unsullied by the uncertainties, imperfections and imprecisions of the world of real objects and measurements, the ancient Greeks allowed the development of deterministic mathematics. By ‘deterministic’ we mean that if we work through a process or calculation, we get the same answer every time, and that there is, in general, only one answer; or, as with inverse trigonometric calculations, that the multiple answers are tightly defined.

Abstraction allowed the mathematical side to flourish, allowing incomparable advances over the last two millennia. But abstraction brought with it additional problems. Why doesn’t the world work exactly, like its mathematical representation? How do we deal with the differences between the abstraction and the reality? Are there any underlying patterns in these differences?

It is in the gap between the abstraction and the reality that most of the interesting things happen in the geospatial sciences (which include surveying, geomatics and all those geo-things). It is where this series of blog posts will focus their attention, as this is perhaps the least well understood area of the geosciences, and perhaps also the most important.