In this article we try to identify those ancient mental attitudes that had to change before experimental science could become the primary means for learning about our world. We begin by distinguishing experiment from measurement and from experimental problem-solving.
1. What is Experiment?
Measurement is a process by which we assign numbers to quantities . Measurement is usually part of an experiment, but only a part. In contrast, experimental problem-solving is a physical trial-and-error strategy for solving a problem . For example: I have six keys on a ring; do any of them open this particular lock? I may answer this by trying each key in the lock. This is a legitimate problem-solving procedure, and it might be called an experiment of a certain kind, but it is not the kind of experiment that advances understanding.
Rather than solve problems or simply quantify variables, we consider experiments to be activities that are planned and executed so as to lead to understanding of observed phenomena . Understanding typically means establishing relations, usually in quantitative forms, among variables. In particular, an experiment involves these five steps:
(a) Establishing a theoretical context that motivates and provides structure for the experiment. A theoretical context identifies relevant quantities to be controlled and measured, establishes the size and scope of equipment, and suggests relationships among quantities.
(b) Designing and constructing a physical situation-usually an apparatus-that is consistent with the theoretical context and that is expected to be capable of clarifying relations among quantities.
(c) Observing, which usually means measuring, although qualitative observations might also be used. Data are collected.
(d) Interpreting the data, which involves assigning meaning to the data, drawing inferences, and making judgments. Meaning provides physical explanations for the relations that are found among quantities.
(e) Communicating results.
Note that measurement and experimental problem-solving may be aspects of an experiment, but an experiment also requires a theoretical context for motivation and interpretation.
2. The Ancients (before 1000 AD)
The development of experimental science is traditionally attributed to Galileo (1564-1642) and Newton (1642-1727); certainly, the activities of Galileo and Newton illustrate a break with earlier approaches toward learning about the world. So, what attitudes had to change in the 17th century to promote this break? Recall the ancient Babylonians, Egyptians, and Greeks, who mastered certain kinds of knowledge  but failed to develop an experimental component to their studies. Their accomplishments seem to divide into three kinds:
(a) Ancient engineering-experimental problem solving that produced the Egyptian pyramids (c.2500 BC), the war machines of Archimedes (287-212 BC), Roman bridges, roads, and aqueducts (c.1st century BC), and the automata of Heron at Alexandria (1st century AD) . These accomplishments were largely motivated by political and religious concerns, and they failed to produce a systematic strategy for learning about the world.
(b) Pure empiricism, that is, systematic measurements that produced quantities of data sufficiently vast that they could be correlated-reduced to models-and used for predictions. The best examples are the astronomical observations of the Babylonians and Ptolemy . We view these empirical models as unscientific because they were not formulated nor tested within a theoretical context: the Babylonians could predict a lunar eclipse but they could not explain the cause of one.
(c) Speculative science, which maintained that we can learn about the world only by pure thought-experiment and calculation are deceptions to be avoided. These attitudes are exemplified by the writings of Plato and Aristotle and influenced thinkers down into the Middle Ages .
Why were measurement and calculation avoided? For several reasons, but here are two important ones : First, some things obviously come to us in discrete entities, so we can naturally count them. We have fingers on a hand, cattle in a herd, and pebbles on the ground: the Latin word “calculus” meant pebbles for counting. But many other things are continuous, not discrete-things like time, the heat of day, the speed of a runner. The ancient mind did not see how such things could be made discrete, particularly since these continuous quantities were often classified with things that we still do not recognize as measurable-things like beauty, goodness, and justice. To measure continuous quantities, we must (a) divide them into discrete chunks and (b) identify an arbitrary unit chunk as a standard for the measurement. The ancient attitude tended to view such arbitrary divisions as unnatural (true) and therefore unable to lead us to deeper understandings of the natural world (false) .
A second stumbling block seems to have been the confusing of mensuration with pure mathematics. Mensuration is the counting and measurement used in commerce, surveying, the military, and taxation. It led to the deductive geometry of Euclid (c.300 BC). In contrast, pure mathematics (number theory is a modern example) involves the manipulation of abstract symbols for no practical purpose. The ancients recognized pure numbers as abstract symbols, but they confused the levels of abstraction, and so attributed religious or mystical meanings to numbers. An extreme example is provided by the followers of Pythagoras (580-500 BC) who, for example, identified “justice” with the numeral 4. Plato and Aristotle were less extreme, but they still doubted that a one-to-one correspondence could be established between abstract numbers and physical objects. This ancient attitude was not merely for poetic effect; rather, it betrays a different-indeed alien-mind-set . As a result, we must be careful not to project our modern concepts of number onto the numerals appearing in ancient texts. Residuals of these mystic attitudes still appear today, such as in meanings attached to the numeral 13, the numeral 666, and the 40 days of the Old and New Testaments.
In many ancient societies, the distinction between mensuration and pure mathematics was often blurred to serve personal or political ends. For example, in ancient Egypt the ability to count on the fingers was considered to be magical , and hence the procedure was kept secret and revealed only to the properly initiated.
3. The Middle Ages (1000-1600 AD)
During the Middle Ages, ideas concerning the quantification of continuous variables began to change. As usual, the changes were slow, nonuniform, sporadic, and often implicit, though they were driven by definite concrete needs. For example, as late as the 15th century, cooking recipes in England rarely contained definite quantities or any numbers .
To illustrate how attitudes toward a continuous variable evolved into a discrete quantification, let’s consider time. To the ancients, time embodied two important characteristics : (a) It is continuous, so early time keepers utilized flows of water or sand. (b) It is cyclic: day follows day, season follows season, and the stars rotate about a pole. Time as a repeating cycle was a psychologically comforting interpretation to the ancients; time as the unfolding of a linear one-dimensional quantity is decidedly modern.
The shift from time continuous to time discrete can be dated to Benedictine monks of the 10th and 11th centuries. The monks had the obligation to observe the canonical hours of each day. Without clocks, how did the monks identify 3:00 am for Matins? By the simple expedient of counting: one monk stayed up and counted from bedtime to 3:00 am; a second monk stayed up to keep the counter awake. In this way was time made explicitly discrete.
The Benedictines also made a second contribution, driven by their invention of the Gregorian chant and the need to remember the chants and teach them to novices. Consequently, in the 11th century a Benedictine choirmaster invented the first x-y plot -one in which pitch (frequency) was plotted on the ordinate and time was plotted on the abscissa. On such a plot (a musical staff), the symbols (notes) used for pitch had finite-discrete-durations, making time discrete on a written record 100 years before the invention of a mechanical clock . Furthermore, the notation for pitch included symbols for discrete rests: a finite duration during which no sound is made. This important invention-a symbol for nothing but holding a place-is fully equivalent to the invention of the zero in the Hindu-Arabic number system.
4. The 17th Century
And so we come to the 17th century with these important, but unconnected, mental attitudes: (a) A speculative science for creating hypotheses (Plato, Aristotle). (b) An empiricism that attributes value to measurement and models (Ptolemy). (c) Deductive reasoning for learning new things from accepted things (Euclid). (d) Experimental problem-solving for applying knowledge (Archimedes, Heron). (e) Quantification for making continuous variables countable (Benedictine monks)
By the 17th century these attitudes were all in place, it merely remained for someone, like Galileo and Newton, to put them together in a productive way. The combination of these five activities forged a new approach to learning about the world: experimental science.
 P. Caws, “Definition and Measurement in Physics,” in “Measurement: Definitions and Theories,” C. W. Churchman and P. Ratoosh, eds., Wiley, New York, 1959.
 J. B. Conant, “Foreword to The Early Development of the Concepts of Temperature and Heat,” D. Roller, ed., “Harvard Case Studies in Experimental Science”, Case 3, Harvard University Press, Cambridge, MA, 1950.
 O. Neugebauer, “The Exact Sciences in Antiquity”, Harper & Brothers, New York, 1962.
 P. James and N. Thorpe, “Ancient Inventions”, Ballantine Books, New York, 1994.
 A. W. Crosby, “The Measure of Reality”, Cambridge University Press, New York, 1997.
 T. Austin, ed., “Two Fifteenth-Century Cookery Books”, Oxford University Press, 1964; reprinted in M. Black, The Medieval Cookbook, Thames and Hudson, New York, 1992, p. 104.