Modern science began with the discovery of experiment, and it was Galileo (1564-1641) who first demonstrated this art^,,. In the process, he gave science three gifts. The first, the telescope extended the range of the eye, bringing the heavens thirty times closer to the earth–-just close enough to psychologically connect them. The other two gifts, the barometer and thermometer, extended senses of touch.

In mid July of 1609, Galileo, professor at the University of Padua since 1592, and visiting friends in Venice, heard rumors of a device accidentally discovered a year before by a Dutch spectacle maker. News spread rapidly; a letter reached Galileo’s friend Fra Sarpi in Venice by November; telescopes were for sale in Germany by December and in Paris by April of 1609. These were little more than expensive toys, but within three weeks during August, starting with a second hand description, Galileo developed a far superior instrument, a true telescope. On the 24^th he offered it to the Venetian senate. For this work, he received lifetime tenure, a huge raise and instant fame.

Galileo had converted a toy into an 8 power telescope; by January of the next year, 1610, he was observing the moon with a 30 power telescope and then the stars, and then the moons of Jupiter. All brought revelations. He rushed to claim priority for his discoveries and published his findings as early as March in a short, best selling book entitled, The Starry Messenger. In it, he said:

When turned to the stars, the telescope immediately multiplied their visible number and revealed

Most importantly, Galileo discovered a revolving system of four satellites around Jupiter, as well as revolving sunspots on the Sun. The satellites about Jupiter, seen at a distance from spaceship Earth, formed a miniature version of the solar system described by Copernicus. The sunspots showed that not only was the Sun ‘imperfect’, or not god-like, but that it revolved on its own axis as per Copernicus’ claim for Earth. Therefore both Copernican hypotheses–-that of the planets about the sun, and that of the Earth about its axis–-were supported by direct observation of other bodies; for the first time, Galileo, publicly and forcefully, proclaimed allegiance to Copernicanism.

Why did Galileo, and then the educated public, convert to Copernicanism after the discovery of the telescope? For one thing, the discovery of heavenly imperfections contradicted the metaphysical basis of previous beliefs It broke up a whole network of ideas, or template, into which ideas previously had to be fit in order to be taken to be true. Data, previously ignored, discarded, could then be re-conceived, data previously or unsought, sought, these to be converted into information and knowledge: a new template could be constructed.

We do not accept such a change easily…nor should we. When a strange phenomena appear, we do not immediately accept them as being real; we check: the more significant the phenomena appear to be, the more extensive the checks. Thus with the telescope; it had to be checked.

It required skill to use. On several occasions Galileo gave demonstrations at which few if any observers were able to see what he claimed they should. Many allowed that though the telescope worked "wonderfully" for terrestrial vision, it failed or "deceived" in the celestial realm.

Checking also called for some knowledge of its operation: a knowledge of optics which Galileo did not have. The little spots, and surface imperfection seen were possible in the instrument itself–in the glass of the lens, for example. Finally, there were possibly alternative explanations to the observations, and these needed checking. For example, the imperfections on the surface of the moon (seen by naked eye though less in far less detail long ago) and the sun could be due to ‘clouds’ of some sort floating within the line of sight.

Why then, with all these difficulties, was Galileo himself converted? A fundamental requisite for conversion is rational coherence amongst observations that, individually, might well be doubtful. If, for example, Galileo had just seen one or more dots near planets, he might well have thought they were due to irregularities in the lenses of the eye or telescope. But he observed them night after night, and plotted their regular positions and movements; such observations cohered amongst themselves in a way that an illusion would not. Coherence is the most powerful form of checking: cross-checking. We check all perception this way; for example, we cross-check a possible visual illusion with our sense of touch.

The wide coherence Galileo observed between all his observations, and especially its relation to the Copernican picture, also assured him that his observations were already part of a new template. Whether, the explanatory power of a new template is, in some sense, equal to that which is lost by discarding an older template is always, for some period of time, a matter of judgment: part of the judgment that distinguishes the more brilliant scientist (or more generally, thinker) from the less so.

Over and above this kind of coherence was the promise of a far greater one: the promise of a connection between astronomy (and hence mathematics) and physics, between phenomena in the heavens and on earth. If heavenly bodies were like earthly ones, it suggests that the laws governing both were the same. This is a very fundamental connection--between science, philosophy, and metaphysics: one that revolutionizes everyone’s total worldview.

All the foregoing reasons–qualities of coherence and the ability to form new knowledge templates--are valued only because they lead to data reduction. Is there another reason they might be valued?

The telescope triggered a widespread conversion from a geocentric to a heliocentric description of the universe. It was the first of a long series of subsequent steps in which the domain of the natural world of systematized and predictable phenomena has been increased at the expense of that of the supernatural: the part of the world that is incomprehensible, irreducible to law, by chaos, and by the free-will of all to whom such power is ascribed.

Superstition is associated with a belief in the supernatural, but what is considered to be supernatural varies in time. Should educated people who believed in supernatural components to the motion of the heavens before Galileo be called superstitious? And consider the many pray and fervently believe that the soul’s movement in the creation of and response to a great work of art will never be understood in a scientific sense, never be reduced to the workings of natural law. Do they not believe, by definition, in the supernatural? Is this then a modern superstition?

The design of every instrument necessarily builds upon that of its predecessors and all are manufactured by skilled artisans. But the pace and direction of development of scientific instruments also depend upon the guidance of a mind trained and bent towards science. It required a Galileo to change a toy into a telescope–and he did so in a matter of days. And it required a Galileo to turn it towards the heavens and invest the time and effort to develop the necessary skill to use it for great discovery. The oft stated view that science somehow grows naturally out of the labor of unlearned men, out of their desire for immediate economic gain, out of pure technology, does not fit the story of the telescope, or of the other instruments we shall be discussing.

Nothing illustrates more clearly the hopelessness of natural philosophy--of the attempt to understand nature without the aid of mathematics and experiment--than the history of the controversy over the existence of the vacuum.

Parminedes deduced that existence is a plenum, that the world,--everything that exists,--is everywhere filled, that a vacuum cannot exist. The reasoning assumed that a vacuum was the absence of every-thing, thus the presence of no-thing.

But we cannot perceive nothing because we perceive properties and only things have properties. We may have the concept of nothing, but cannot actually perceive it. (Just as we can have concepts like God, or The Infinite, but cannot actually perceive either.)

But we do perceive the space around us.

Thus, that space must be, everywhere, some-thing: it must always and everywhere be filled with some thing. Hence, no vacuum.

After Parmenides, to whom the use of strict logic in natural philosophy is attributed, this was also the view of Aristotle, the greatest logician of antiquity, and of his followers, the Aristotelian philosophers who eventually became infamous as opponents of the scientific revolution.

Does the vacuum not exist? If not, how is it that most people are taught nowadays that it does. Is pure space a vacuum? Does a vacuum mean the presence of no-thing? Is space no-thing? Is no-thing in space? Answers obviously depend on the meanings of words like 'vacuum' and 'thing'. The point is that these words have various and changing connotations.

It would be interesting to disentangle such questions, and see just how ambiguities in words, and intuitive but unwarranted assumptions, interacted to mislead philosophers, but that takes a book in itself. It will be sufficient to simply note our present understanding of the vacuum to see how far off track they were led by words and intuitions. To wit: today’s wisdom--the meaning given these words today--is that a vacuum exists but it is also a plenum; it is not nothing, but no thing is in it; and although it is full of something, things pass through it unimpeded! Words have changed their meaning to fit the expanded universe open to us by the scientific expansion of our senses.

The first experimental steps necessary to move beyond natural philosophy produced the barometer. It happened as follows:

In 1630 a Genoese engineer had written Galileo asking his opinion about why a siphon failed to conduct water over a 20 meter high hill from a lake to a lower point on the other side of the hill. A pipe stopped at both ends had been filled with water and laid over the hill. When the ends were then unstopped, water had not flowed, as expected from normal siphoning experience, from the higher lake to the lower point. The engineer also noted that each end of the pipe retained a height of about 10 meters (32ft) of water between which the pipe contained neither water nor air. Presumably it therefore contained a vacuum. How was this possible?

Galileo had seen a similar thing happen with water pumps, which failed to lift water beyond that same height of 10 meters. This was a serious problem for the burgeoning mining industry that had to clear ever more water seepage from ever deepening mines. Galileo thought that the water columns simply broke like cords but then did not fall below 10 meters because the vacuum (nothing?) somehow pulled on them.

At the end of 1638, Galileo’s new book, Dialogs Concerning Two New Sciences, arrived in Rome where it was eagerly read by a group of his disciples. It contained his thoughts on water columns and two members of this group, Rafael Maggiotti and Gasparo Berti, decided to see for themselves. They performed an experiment some time in 1641.

A tube, plugged at the bottom, was filled with water and then plugged at the top, taking care that no air was trapped inside, and that no air could enter through or around the plug, as shown on the left of Figure 1. Then it was unplugged below and the water drained out until, as had been often reported, a 32 feet (10 meter) column remained. The question was, what was in the top region of space the water had previously occupied: was it a vacuum, dilute (continuous) air, or some type of different more ‘subtle’ matter, an ‘aether’? After some careful discussion, they concluded that the question could not be decided without further work: a common enough conclusion to a scientific paper now, but not then. The apparatus they used was the first barometer.

The new spirit of the times shines through when reading about this venture. Maggioti and Berti did not merely take the word of either the Genoese investigator or even their master, Galileo, but looked for themselves. They did not merely re-observe the effect in its natural and original setting; they attempted to eliminate uncontrollable factors, which could influence their observations. This required an 'unnatural' setting, and hence a special apparatus. As Figure 2 reminds us, anything influencing the conclusion drawn from an experiment acts as a logical predicate. A valid conclusion drawn on the basis of an experiment is certain only in the absence of uncontrolled factors; otherwise, it is contingent upon unknown or uncontrolled factors and hence uncertain,.

Some of their considerations were as follows:

1) Leakage of small residues of air into the bulb from water. This was difficult to control. The height to which the water fell was contingent on this residual air

2) The possible existence of a subtle aether. Some fluid-like substance might exist that was ‘subtle’ enough to flow through glass. The bulb on top would not be empty when the water dropped but be filled with aether flowing in from its surroundings. This aether might not effect the height to which the water fell but it would influence the conclusion that a vacuum existed within the bulb. That conclusion was therefore contingent on the predicate of the non-existence of such aether.

3) One of the observations that they made was to ring a bell inside the bulb and listen for its sound. They heard nothing. They concluded that even if such aether did exist, it could not carry sound.

The young Genovese investigators discussed whether their observations, taken together, meant that a vacuum (as the word was then understood) existed in the bulb. They concluded that this was not so, that they needed further observations and precautions in order to reach certain conclusions as to the existence of a vacuum.

But the judgement required to determine how to eliminate contingencies is not simple. Small residual quantities of air in the bulb can never be eliminated; and subtle aethers have never been eliminated, they have only been successively redefined. Furthermore, what about factors unknown to them? How can an experimenter eliminate an unknown factor, one she is not even aware of?

Such questions can help us understand why the very idea of scientific experiment came so late in history. Common sense drawn from everyday observations was that all conclusions about phenomena are necessarily contingent. The idea that certainty was attainable, defied, and still defies common sense.

Initially, we can only have faith in order to defy common sense and attempt an apparently irrational goal. Only after achievement, do such goals tend to appear perfectly rational.

This group of disciples of Galileo had a new faith: a vision of certainty that inspired them. Just as a vision of certainty had inspired the care in which axioms were formulated, and logic was applied, to the creation of Mathematics in the time of Thales and Pythagoras, so also it inspired these disciples of Galileo, the modern Thales, to take extraordinary care in the control of physical contingencies in order that mathematical logic be applicable to observation. This control is the essence of experiment. Galileo was the first to show that this kind of control was possible and that logic based on experimental results could lead to certainty about the physical world.

Evangelista Torricelli, a member of this group of young Romans, who seems to have left to become Galileo’s assistant shortly before the experiment began, was kept informed by his friend Maggiotti. Galileo had commented in unpublished notes that he believed that columns of other liquids would also break to a vertical height inversely proportional to their density. In particular he mentioned mercury, which has density 14 times greater than water and so would be supported only to a height of (32 ft.)/14 = 27 inches. Torricelli decided to try this out.

About one year later, Torricelli and Viviani, who had also been working for Galileo, performed the experiment that included a further test of Galileo’s vacuum pull hypothesis. In June of 1644, he described what happened in a letter. It starts,

Torricelli had concluded that the weight of the air transmitted as pressure to the mercury column, was sufficient to push the column up; further, that any extra pull from the vacuum was (on the basis of his experiment) unlikely and, in any case, was a superfluous (vain) hypothesis. Nature, it seems, did not abhor, but was indifferent to the vacuum.

Torricelli could then start the letter with an important change in emphasis, proposing that the height of a mercury column be used to measure atmospheric pressure. This height was seen as an image of that of the atmosphere pressing from above on the surface of the mercury. It is because of this, and the consequent sentence: We live submerged at the bottom of an ocean of air,-- his vision of the atmosphere (supported with physically reasonable numerical estimates of its size and weight), a vision which ultimately led to this idea’s verification--that Torricelli is honored as the inventor of the barometer.

On September 19, 1648, Florin Perier, a lawyer living in Clermont-Ferrant, France near the base of the Puy-de-Dome, at the instigation of his brother-in-law, the great physicist and philosopher Blaise Pascal, arose at 5 in the morning and sent messengers to 5 distinguished local citizens. They assembled at 8 o’clock in a monastery garden situated in the lowest part of town. With 16 pounds of carefully distilled mercury and two similarly constructed glass tubes, each sealed at one end, he recreated Toricelli’s experiment–filled the tubes with mercury, inverted them in the same bowl, and compared the heights of the columns side by side. They were identical; both approximately 26 inches high. He emptied and refilled and rechecked three times.

Leaving one barometer behind in the monastery garden to be periodically observed by a monk, the group ascended through Torricelli’s ocean of air, about 3,000 feet, to the top of the Puy-de-Dome where, having less air above them, according to Torricelli, the mercury column should be pushed to a lower height. They re-assembled the second barometer and it was about three inches lower. Perier relates their reaction:

Thus the discovery of the greatest ocean on earth–the atmosphere. They rechecked part way down the mountain and found to their great pleasure that the barometer proportionally recovered its height; they checked back at the monastery and were told that the barometer left there had not changed all day; and finally they emptied and refilled and rechecked the barometer twice more. These checks helped confirm that it was a reliable measuring device.

Comparing the sketch of Galileo’s thermoscope in Figure 3 to that of the barometer; the difference between them is (i) air trapped in the closed end of the Thermoscope instead of a vacuum, and (ii) he use of water instead of mercury. The water’s height varies directly with its temperature, but also with the pressure of the air, both that of the atmosphere and that trapped inside the tube.

Air pressure had not yet been discovered when Galileo made his demonstration. Until it was, and its contingent effects eliminated, the height of the water had an ill-defined meaning and was therefore not a useful measure. The air pressure acted as an uncertain premise in a logical deduction. It effected conclusions to be drawn from the behavior of the system in uncertain ways. The definition of the instrument had to be sharpened (just as were definitions in a Socratic dialog) before its reading could univocally define the quantity we call temperature.

By 1664, by the Grand Duke Ferdinand II of Tuscany (another disciple of Galileo) and his Academy of Science in Florence, had taken the first important steps. His instrument, shown in Figure 4, had no air trapped within and was completely sealed by the glass from atmospheric pressure –an otherwise contingent environmental factor. Now, only the liquid's temperature could change its height.

The liquid's volume changed with temperature causing the surface within the stem to move. Glass beads were attached to the stem to measure this. Their presence converted a mere ‘scope’ into a ‘meter’--something which measured rather than merely displayed an effect.

To be scientifically useful, however, an instrument must also be objectively defined–the society of educated people had to be able to agree upon its utility; they had to agree that temperature ‘existed’. That means, temperature’s existence it had to be socially created. This happened when the temperature measured by different thermometers could agree with one another. That did not occur until the work of Fahrenheit 1714, 50 years after Ferdinand.

An important element of Fahrenheit’s technique was the use of two fixed points for calibration. A fixed point is a temperature singled out by nature. Fahrenheit’s first fixed point was the freezing point of brine, which he defined as the zero of his scale; his second was body temperature, which he set at 96 degrees (since changed). At a later date, Celsius assigned his first and second fixed points at the freezing and the boiling points of pure water to which he assigned the temperature values of zero and 100. The position reached by the liquid at each fixed point on the stem of the thermometer was marked. Equal intervals marked off between fixed points defined equal temperature difference. Thus the 96 equally spaced marks on Fahrenheit's scale defined degrees Fahrenheit, and similarly, 100 for Celsius.

How could people be certain that supposedly fixed points were actually fixed? Was the temperature of every healthy human body exactly the same, and was one healthy person’s temperature the same all the time? Were freezing and boiling points independent of where water was drawn from? If not, then different thermometers could still disagree with one another; the measure would still not be objective. In fact none of the ‘fixed’ points mentioned so far are exactly fixed: boiling and freezing points depend on altitude and chemical purity, and healthy body temperatures vary. Such questions were carefully investigated, and solutions took considerable time and effort.

The development of the thermometer can be compared to a Socratic dialog with nature. A perception was vaguely defined by a Thermoscope. The definition was then clarified, as in dialectic, by separating out extraneous factors. Separating out pressure effects clarified the measure of temperature just as separating religious piety from morality clarifies the meaning of the latter. Both separations are accomplished by questioning, one of nature, the other of persons. After clarification, it became necessary to establish objectivity. In one case everyone uses a word in the same way, in the other, we all use equivalent thermometers. Only when all this is accomplished can one use either a word such as morality or a measure such as temperature, as part of a logical argument.

and as containing

published in Leyden to avoid a ban imposed on his work by the Inquisition. The Preface to this book written by the publisher indicates not only Galileo’s status but also that of infant Science:

The book takes the form of a dialog between three men: Salviati, who represents Galileo; Sagredo who represents the audience of educated and influential people Galileo wished to reach; and Simplicio, who represents the Aristotelian establishment Galileo wished to overcome.

Amongst other things, the three discuss the properties of bodies falling through media such as water and air. Simplicio recounts Aristotle’s teachings as follows:

According to Simplicio, Aristotle said that Speedµ Weight when comparing the speed of two objects of different weights falling through the same medium; he also says subsequently that Speedµ 1/Density when comparing the speed of an object in media of differing densities. Interpreting these statements as peraining to typical speeds attained during falls from typical heights, Salviati points out that common experience indicates this is false and that he therefore doubts that Aristotle ever tested his assertion. Simplicio refers Aristotle’s wording to claim that he did.

This exchange presents both Aristotle and Simplicio as foolish. It makes it seem that Aristotle, whose observations of the structure of plants, animals and of human institutions have stood as models to be studied and emulated for two millennia, somehow missed a common observation. But it is in fact true that, comparing the final speed ( the speed eventually attained after being dropped and being accelerated by Earth's gravity) of bodies of different weights but of the same shape, that Speedµ Weight; and similarly, comparing the final speed of the same body in media of different densities, that final Speedµ 1/Density. It is very possible that Aristotle himself simply assumed these qualifications (final speed, same shape, and same body) to be understood.

Whatever the case, Aristotelians such as Simplicio, not Aristotle, were Galileo’s important targets. Aristotle’s followers--two thousand years of them--had typically based their views solely on their master's writings. Most of them, like most people at all times, had not themselves thought through questions to any depth but had just read and then repeated what might have been meant as only abbreviated descriptions of more careful observations. Galileo emphasizes their focus on knowledge gained only through reading by the fact that Simplicio’s response is based only on his interpretation of Aristotle’s language and not on any appeal to experience.

He demolishes the misconceptions of the Aristotelians concerning speeds of descent with a quantitative argument as follows:

Salviati: If then we take two bodies whose natural [presumably this means either average or final] speeds are different, it is clear that on uniting [i.e. attaching] the two, the more rapid one will be partly retarded by the slower, and the slower will be somewhat hastened by the swifter. Do you not agree with me in this opinion?

Simplicio: You are unquestionably right. Salviati: But if this is true, and if a large stone moves with a speed of, say, eight while a smaller moves with a speed of four, then when they are united, the system will move with a speed less than eight; but the two stones when tied together make a stone larger than that which before moved with a speed of eight. Hence the heavier body moves with less speed than the lighter; an effect which is contrary to your supposition. Thus you see how, from your assumption that the heavier body moves more rapidly than the lighter one, I infer that the heavier body moves more slowly.

Simplicio: I am all at sea…

Figre 5 illustrates the point. Simplicio is being tripped up here because he has not taken into account contingent factors. In the present case, they are the shape and volume of the falling object. These influence the speed of descent through a medium. Salviati's argument showed just how contingent Simplicio’s conclusions were.

The real lesson of this dialog is, again, the necessity of careful quantitative experiment in order to eliminate hidden factors (in this case, the hidden factor was the dependence of speed of fall on the shape of the falling bodies) preventing the logical and mathematical analysis of nature.

Salviati’s ‘thought experiment’ took Simplicio, the representative of the typical Aristotelian, by surprise. Despite the fact that this would have been true for most Aristotelians, most does not mean all. Critical thinkers arise in all ages, and since antiquity, many people had noted that Aristotle was wrong in this as well as other matters. It has never been hard to see that, for example, a two pound rock normally falls not twice as but about as fast as a one pound rock. And simple observation aside, Galileo’s was not even the first thought experiment illustrating a logical inconsistency in Aristotle’s statement. A few decades before him, the Venetian mathematician Benedetti had published a similar discussion of the matter. Yet none of this seemed to have a big effect. Why?--and what had changed by the time of Galileo to make such things begin to matter?

There are always many answers to such questions and usually most have some validity; most point to some contributing factor. One underlying factor was a changing perception of the meaning and purpose of natural philosophy that was already suggested in part by the previously quoted remarks about the Venetian Arsenal.

In conjunction with Christianity--a conjunction effected in the 13th Century, most notably by St. Thomas Aquinas--Aristotelianism constituted a complete worldview for most Europeans of that era; it supplemented Christianity’s sparse views concerning natural phenomena. Like all worldviews, its purpose was to frame the world, to provide general guidelines and points of reference for thoughts, to provide the big picture. Differing worldviews today, for example, classify murderers as sinful or sick; morality as given or defined; social conflict as natural and inevitable, or unnatural and rectifiable; and so on. Choices determine subsequent thoughts on these topics.

Aristotelianism classified motions as natural and forced with falling and rising as natural. And it said that all things have intrinsic natures. Falling bodies have it in their nature to seek the center of the universe. It is their natural place. Light bodies seek the lunar sphere that is their natural place. In a well run society in which all citizens are happily in their natural place, and in the same way, in the cosmos all things seek their natural place. From the Aristotelian viewpoint, such views, the proper fruit of contemplating natural philosophy, stated the essence of the matter and further details were inessential.

One or more difficulties, such as Aristotelianism had concerning falling bodies, cannot disturb a whole worldview. In the eyes of its viewers, worldviews appear too set firmly rooted in the nature of things to be dislodged. Dijksterhuis has noted that the discovery of America earlier in that era had consequences destructive to its Aristotelian geography, geophysics, ethnology, and even its theology which were also blithely ignored. And our modern era provides its own examples of religions and ideologies maintaining themselves despite the existence of multiple internal contradictions and failures.

It was only when more was demanded of natural philosophy, when the purpose of natural philosophy was expanded to include predicting and controlling nature, that Aristotelianism’s failure to describe motion became fatal. Every society desires to predict and control nature; even the most primitive learn how to control fire and predict the seasons. What made Europe different was that somehow the idea emerged that control and prediction could be achieved via the Pythagorean/Platonic tradition within the natural philosophy inherited from ancient Greece–a tradition which, unlike all others, stressed logic, critical analysis, and especially mathematics.

Near the start of the Dialogs, the Aristotelian character Simplicio, says in various places:

Galileo, elsewhere, formulates the meaning of the Aristotelian’s attitude along the following lines: they believe it is impossible to achieve mathematical certitude because the nature of physical being is qualitative and vague. It does not conform to the rigidity and the precision of mathematical concepts. It is always "more or less", describable only by categories such as ‘natural’ or ‘violent’.

This reflects the pagan tradition of a Chaos co-equal with Mind; a world with a fundamentally chaotic component only on top of which can cosmic order exist. Equivalently, it reflects the reality of the limitations of logic as applied to the sublunary world if premises are contingent on unknown, and hence effectively chaotic, factors. The pagan was the commonsense tradition. Galileo’s work on projectile motion showed that, with the exercise of sufficient care, contingent factors could be controlled and that when this was done, logic revealed natural law both simple and mathematical. This is discussed in more detail in the next section.

The idea that nature was mathematical came from the Greeks, but they also elevated mind/soul to godlike status, and looked to it as the source of the basic truths of natural philosophy. Phenomena could not be that source because, at their most fundamental level, they were suffused with chaos, and subject to the uncertainty of multiple competing wills. Thus Archimedes, one of the few great minds of antiquity known to have attempted to learn from nature by experimentation, was criticized for doing so at the time.

The idea that the world was not a cosmos built upon chaos, but was created coherently down to the most microscopic level came as an implication of monotheism, as did the far lower status it afforded mankind relative to God. The human mind, made of and for earth, was not given insight as an inbuilt cosmic inheritance; it might therefore need controlled observation of phenomena as a guide to the understanding of nature. With these attitudes of mind, quantitative experiment could become a guide to the mathematical nature of nature once the required mathematics was in place; and that occurred just about in time for Galileo.

Here it is worth noting that in contrast to the Platonic academy, the Church, which held a somewhat equivalent position as guardian of the dominant worldview of its time, never discouraged experimentation. How could it? Is not the careful study of the world the study and appreciation of God's creation, and as such, an act of piety. Perhaps it is an expression of the greatest piety. In Jewish tradition, a rabbi deep in study of the Law is excused even from his daily prayers. Perhaps a scientist can be excused for the same reason?

Galileo was the inheritor of these two great traditions, Hebrew and Greek, which he wove together into the fabric of quantitative experiment. To do this, Galileo assumed the reality of ideals. He did this explicitly when he stated that God wrote the laws of nature in number: for what was more ideal than number and more real than God’s laws?

He also assumed, as did the Judaeo-Christian tradition, the ‘goodness’ of God's creation, in the sense, as Einstein put it, that God was subtle but not malicious when creating this world. God did not give us phenomena and brains to mislead us, but rather to lead us.

God’s subtlety, however, requires a matching human subtlety. In order to deduce ideal forms (or formula) from experimental observations of phenomena--deduce the world of Being from that of Becoming--all the predicates contributing to the deduction must be known. These predicates are created by environmental factors. So all these factors must be understood and under control, and this requires considerable subtlety.

In the case of projectile motion, for example, it is first of all necessary to understand that the medium surrounding the projectile (air, water, ...) is a complicating and not an essential factor. Then, it is necessary to know how to eliminate the effect of the medium on the projectile. Then it is necessary to know how to handle all sorts of errors inevitably made in the course of measurement. Two techniques based on these ideas and found in Galileo’s Two New Sciences will now be discussed.

The first idea is illustrated by Galileo’s treatment of the effect of air friction on falling bodies.

Salviati:… Having observed this I came to the conclusion that in a medium totally devoid of resistance all bodies would fall with the same speed.

Simplicio: This is a remarkable statement, Salviati. But I shall never believe that even in a vacuum, if motion in such a place were possible, a lock of wool and a bit of lead can fall with the same velocity.

Salviati: A little more slowly, Simplicio. Your difficulty is not so recondite nor am I so imprudent as to warrant you in believing that I have not already considered this matter and found the proper solution.… Our problem is to find out what happens to bodies of different weight moving in a medium devoid of resistance, so that the only difference in speed is that which arises from inequality of weight. Since no medium except one entirely free from air and other bodies, be it ever so tenuous and yielding, can furnish our senses with the evidence we are looking for, and since such a medium is not available, we shall observe what happens in the rarest and least resistant media as compared with what happens in denser and more resistant media. Because if we find as a fact that the variation of speed among bodies of different specific gravities is less and less according as the medium becomes more and more yielding, and if finally in a medium of extreme tenuity, though not a perfect vacuum, we find that, in spite of great diversity of specific gravity, the difference in speed is very small and almost inappreciable, then we are justified in believing it highly probable that in a vacuum all bodies would fall with the same speed.

The central idea here is that of the ideal limit: seeking a limiting situation in which the simplicity of the underlying world of perfect forms reveals itself when shorn of the complications of environmental contingencies. Here the limiting situation, never perfectly attainable, is that of a medium devoid of resistance. Galileo understood that media such as air or water are inessential complications that need to be eliminated to reveal the simplicity underlying free fall.

Neither Aristotle nor Simplicio understood this, and in fact, Simplicio wonders if motion in such a place were possible perhaps because of an old idea that an object in motion needed to be pushed along by the surrounding medium. Galileo changed the medium from a necessity to a complication that could be eliminated by finding the limiting behavior shown in Figure 6.

There are also uncontrollable contingent factors whose effects are illustrated by experimental results tabulated in Figure 7 and graphed in Figure 8. The first column (t) shows a sequence of measured times. The third column (d) shows measured distances an object drops in near vacuum. The unit of distance equals the distance the object drops in the first unit of time. The fifth column (s) is calculated as

Speed = (change in distance)/(change in time)

where the changes are between consecutive rows of measured values.

The experimental points do not fall on a straight line; many different curves can be drawn either passing or threading closely through them. How then can anyone draw one particular straight line and to say that it represents the points? This is what Galileo did, as shown in the last column of Figure 7, and this can serve as a paradigm for what experimentalists do with their data to this day.

The straight line chosen to thread the experimental data points describes a distance of fall from rest that increases as time²; this is as in contrast to, for example, time to some non-integer power such as time^2.001 that would fit the data equally well but would produce a non-integer series. Constant acceleration is the simplest kind of acceleration and it gives us integers. The straight line is therefore the simplest curve reasonably consistent with the data assuming the distribution of points about it is due to random (unknown and insufficiently controlled) sources of experimental error. Randomness can be checked by taking many more data, at other times and under different circumstances.

By taking an ideal limit that eliminated effects of surrounding media, by threading the simplest line through the data points, and by calling the conclusion resulting from these operations true—the truth about reality,—Galileo was teaching the creation of scientific reality. He was showing how science should create its Being from Becoming, and its Knowledge from Belief. Until he taught this, efforts at science had merely been natural philosophy.

But is this scientific concept of reality not strange? For, what do we really see? Are not the experimental points, rather than the line through them, the actual reality? Does not all motion take place within a real medium, and not in an imagined ideal vacuum? Science calls real a perfect straight line representing motion never seen and never measured, in a situation never encountered—more real than motion in real situations really seen. By doing so it defies common sense and seems to justify its critics.

The usual retort to this, that of most scientists as well as non-scientists, is that, nevertheless, these methods work. And indeed, their clear utility is generally sufficient to keep science funded, and to keep most people betting their lives on it (e.g. by flying on airplanes and visiting doctors). But yet, as a response to such critics, utility and success are clearly insufficient, for the critics of science are as strong as ever. Therefore, assuming one does not wish to merely brush them off with labels of intellectual and psychological inferiority and resentment, the usual retort is manifestly inadequate.

If the procedures initiated by Galileo and resulting in the true birth of science, were and remain neither intuitive nor obviously necessarily justified (except, after the fact, for the creation of science!), what moved him to create them in the first place? What permitted and suggested the idea that a manufactured perfectly straight line threaded through the data is more real than the real data? Clearly, the answer lies in the worldview of Platonism. From within it, such a strange (perverse?) idea is permitted and suggested.

From a viewpoint outside of Platonism + Monotheism = The Scientific Worldview (I cannot use the word Scientism since it has been already appropriated), that straight line seems to be merely a simplified description that we impose upon an intrinsically chaotic, complex and uncertain reality. And is this not the idea that most people have? From within the Scientific Worldview, however, that straight line is part of a simple reality obscured from our direct gaze by the effectively limitless complexity of actual instances of natural law as well as the limited capacity of the human mind to deals with them. From within, natural law exists (Platonism) and its instances are everywhere and always made manifest to us (Monotheism). Moreover, that natural law is written in number (Platonism) and we have no reason to suspect it to appear obviously reasonable (Monotheism). Instead, we can hope that we can be led to it, and should not be surprised if it reveals a continually more surprising universe. That is what we find: that natural law leads away from the naïve and comforting ideas of infancy, it leads ever outward from the sense of ourselves as the center of the universe.