One day in November of 1619, stationed near the small Netherlands city of Breda as a gentleman-soldier as part of one of Maurice of Nassau’s two French divisions fighting the Spanish, Descartes (1596-1650) met Isaac Beeckman. It was a serendipitous event: Descartes happened to be stationed near Breda; Beeckman happened to be there to help a cousin slaughter pigs for market and to look for a wife; both happened to be looking at a mathematical puzzle posted as a challenge; since it was written in the local Flemish, Descartes turned to Beeckman for a translation because both happened to speak Latin.

All of this would have been of no consequence had not, as Beeckman later wrote in his diary:

Few people at this time were seriously attempting to apply mathematics to physics (as opposed to astronomy), and of these, Beeckman may have been the first to be doing this from a mechanistic viewpoint; i.e. from that of a mechanic who learns how machines work in terms of their parts (screws, wheels, gears,…). Beeckman extended this idea to the microscopic world; he attempted to quantitatively describe macroscopic phenomena—hydrostatics, optics, gravitation and acoustics—in terms of microscopic mechanical parts, generally just simple particles. For example, to mechanistically understand, say, hydraulics—what causes water pressure, how it is transmitted, how water flows, etc.—you analyze it in terms of microscopic parts (e.g. molecules of water) that move, push, and pull each other. The mechanical philosopher believes that large numbers of these parts and their elementary mechanical interactions, duplicate hydraulic phenomena.

A mechanic rarely questions how or why macroscopic systems work after seeing how they are put together. Mechanics of the early 17^th century felt that they satisfactorily understood the more complex machines of the day such as clocks and windmills once knowing how they were put together out of their component parts. Whereas a philosopher might wonder how or why a solid component has solidity and rigidity, a mechanic tends to accept such simple properties, to feel an unquestioned understanding of them on the basis of long acquaintance. The mechanic is drawn to the less common behavior of machines, complex configurations of simple parts.

Beekman’s approach, which became known as the Mechanical Philosophy, was part way between that of the Mechanic and the Philosopher. Hydrostatic phenomena, though common, are more puzzling than those associated with particles which were conceived of as simple solid balls. The idea of the Mechanical Philosopher was to explain the former in terms of the latter: reduce the more complex behavior of liquids to that of assemblages of simpler solid parts.

Although at first sight this philosophy may appear to be simply a new version of that of the Milesians, or of the later Greek Atomists, there was an essential difference. Mechanical philosophers had as their ideal, an explanation comprised of mathematical connections between a system’s microscopic parts and its macroscopic behavior; they wished for a quantitative understanding of phenomena with the mathematical certainty that goes with it.

The connection to mathematics sought by the mechanists would seem to arise naturally from the relative ease with which the role of each part of a system, conceived mechanically, is represented mathematically. In the simplest possible mechanical model of water for example, its molecular constituents simply occupy positions and volumes. The possible motions of assemblages of molecules can be represented and deduced with relative ease: e.g. a volume of water may move only by pushing another volume out of the way, and a fixed number of molecules occupies a fixed volume and weighs a fixed amount.

But this was before algebra and analytic geometry: there were no such things as variables and coordinates. No one knew about writing a symbol such as V for volume, x,y,z, for the position of a particular particle, N for a number of particles, and so on, and forming expressions from such symbols to represent, reason, and conjecture about their connections to macroscopically observable quantities. There was only an intuition (true genius, not that simple cleverness that this now thoroughly debased word is used for) that something like this should be possible.

Lacking the requisite tools, these first Mechanical Philosophers were reduced to, in effect, outlining how their models might work. From a modern perspective, their efforts appear amazingly crude, as the following example illustrates. Figure 1 is similar to one included in Descartes’ explanation of the fact that the force exerted by the water on the bottom of container B equals that of A when both are filled; and moreover, that the force is equally distributed at all points such as x, y, and z.

The lines from w to x, y, and z, indicate lines along which force is exerted from one particle to the next. It indicates how the weight of particles all the up way to the top of the container effect the force exerted at the bottom. This was important, since it was known that, as well as being proportional to the area of the bottom, the force on the bottom is proportional to the maximum height of the water above it, which equal in both containers. It was important to explain how force gets transmitted from top to bottom and around corners, and thus how that maximum height could effect points x and z. Equivalent insight was not forthcoming from other treatments of the problem which were purely macroscopic; i.e., the made no reference to the unseen, microscopic constituents of water.

Although Descartes had other important physical insights, they were vitiated by being attached to his gross ignorance of what today we consider as elementary physics (much of it to be created in the succeeding 100 years). Thus, there was little else of clear value in Descartes’ considerable discussion of this phenomenon. Why then discuss it? One reason is because it is important to appreciate just how vast was gulf separating the physical knowledge of this bone fide 17^th genius from that of least 21^st century student of freshman physics. It was a gulf which Descartes never crossed (but which Huygens, who as a youngster knew Descartes as his father’s house-guest, did ), and somehow never seemed to fully realize existed. And this is important because, as we shall discuss, this failure still applies to so many of Descartes intellectual descendants, who, though brilliant and learned, are still separated from physics by a gulf of ignorance of which they are unaware.

Twenty centuries separate the Ancient Greeks and Mechanists, but on the surface there seems relatively little to show for it; both explain phenomena verbally, qualitatively, partially, and unconvincingly. Nevertheless, there are significant differences. The Mechanists had a definite program of quantification guided by a microscopic model of nature whereas the Greeks, even the Atomists, had none. This was due to various factors. First, the significant progress in mathematics since the Greeks (the invention of the decimal representation of natural number system, algorithmic arithemtic…) and intimations of more soon to come made such a program imaginable.

Second, the attitude towards Natural Philosophy had changed radically. This has been eloquently expressed in the opening paragraph of a biography by Benjamin Farrington of Sir Francis Bacon, the premier advocate of this new attitude:

And finally, behind this conception of sacred duty was a whole complex of religious transformations, most obviously the Protestant Reformation, behind which was a shift in Christianity from its Greeks towards its Hebrew roots. This is how Farrington related this to certain names Bacon used books he wrote imagining the future Utopia that science would create:

Pythagoras, it will be remembered, performed the earliest recorded experiment in mathematical physics when he related musical consonance with rational fractions: two strings at the same tension whose lengths are in the ratio of small fractions, sound in consonance. This probably had no effect on music, as harps were undoubtedly tuned by ear in any case, nor could it have led directly led to new mathematics, but its inspirational value was enormous. In men’s minds, it connected mathematics to the soul, and as this was connected to God, it was perhaps the most powerful (although by no mean the only) reason to believe that number was close to the foundations of the universe.

It requires no imagination, even for those of us who are relatively unresponsive to it, to see how much music means to most people. The idolization of modern pop stars, of Beethoven, and of ancient Apollos, followed about by bands of swooning girls over the hills of Greece and Asia Minor, are all of one piece. Given the pre-eminence of music even today, as it competes with so many other forms of entertainment, it is easy to understand society’s far greater dependence upon it in the centuries prior to the invention of radio and TV, and to easily accessible transportation (to theatres, parties, and such).

But the musical scale based on Pythagorean tuning, in use until the late Middle Age, severely restricted the music based upon it; to a modern ear, virtually all that music sounds remarkably dull and repetitive. Its major deficiency was that most instruments (specifically excluding the human voice) could play in only one key. This meant that only instruments built to play in the same key could play together, and even worse, even one instrument playing alone, could not modulate to a different key.

As Europe emerged from the Middle Age with the growth of cities, commerce, and a middle class possessed of leisure time and thirsting for entertainment, the need for music greatly intensified. Many people important in early science worked on the problem (including Galileo and his father). Their initial efforts led to the realization that they were faced with a problem in mathematics. And that is where we take up the story, with Beeckman and Descartes; the story of how music led to the most important advances in mathematics since the discovery algorithmic arithmetic, to the creation of modern algebra, to the synthesis of algebra and geometry, and thence to Newtonian dynamics.

Less than six weeks after meeting him, Descartes presented Beeckman with a New Years gift: a 40 page essay on the science of music. Music and mechanism were to operate together in Descartes’ mind to bring forth the most important steps in the creation of algebra as well as its fusion with geometry into what is now called analytic geometry.

The topics were connected by a critical problem of that era: the implementation of the system of equal temperament. Music in equal temperament is made up of the sequence of tones ascending by equal frequency ratios equal to the twelfth root of 2, or 2^1/12. Instruments needed to be manufactured to all produce the same such tones. Frets on stringed instruments and holes on wind instruments had therefore to be located in appropriate positions. How were these positions to be specified and then physically located?

The symbol 2^1/12 does not represent a value; it represents a property of a value: that multiplied by itself 12 times, it equals 2. It implicitly specifies a value. In fact, the only values we can explicitly state are rational numbers--the ratio between two integers--and no such ratio can equal 2^1/12. That is why it is called irrational.

Vincenzo Galilei, father of Galileo, had guessed on the basis of practical experience that 18/17 was close to 2^1/12. But he knew neither how good an approximation it was ( it was off by about .06% which not good enough for the human ear!), nor how to do better. On the other hand, geometrical constructions of irrationals were better understood. Figure 1 for example, shows that it is an easy exercise in plane geometry to measure off a length 2^1/2 times another using the straight edge and compass. And Zarlino, a contemporary of Vincenzo, had shown how two lines in the ratio 2^1/12:1 could be measured off this ratio directly using a mechanical device, known as a mesolabium that had been invented by Eratosthenes (a younger contemporary of Archimedes) in the third century BCE.

From these studies, Descartes got the idea for a more practical and versatile variant of the mesolabium which he called a proportional compass. It could be used to draw a whole class of curves, such as that shown in Figure 3, from which roots or powers could be read off. Then, like a true mathematician, he generalized the problem. First he asked how many different kinds of curves could one mechanically draw, and then how could one best design an instrument to draw a given type of curve. This question led to the fusion algebra with geometry, the creation of analytic geometry, and the start of modern mathematics!

A key to his approach is illustrated in Figure 4. It illustrates the idea that: all the basic operations of arithmetic on number can be translated and re-interpreted as geometric operations on line segments that produce other line segments. Consider the ancient concept of ‘squaring’ the line. The Greeks conceived of a squared line as an area--a square with sides equal in length to that of the line. In contrast, a mesolabium produces from a line segment, a new one whose length is the square of the first.

This line-from-line feature is key because it permits line segments to form a closed system under these operations. That is, just as allowed arithmetic operations on positive real numbers produce positive real number so also the corresponding geometric operations on line segments produce line segments. Furthermore, just as any number of successive allowed operations on a positive real number can be performed without fear of logical inconsistency, so also with the system of allowed operations on line segments.

To even discuss this equivalence there must be a way, to refer to the line segments; they need names like A, B, C,…. This step was the beginning of both abstract algebra and analytic geometry.

Abstract algebra began because these symbols could now represent either real numbers or the (undirected) line segments from which the real number quantities been abstracted. Generalizing on this idea, we see that the symbols themselves did not need to represent one particular set of things. Rather, they could represent any set on which these—or in general any similar—arithmetic operations could be consistently applied. Abstract algebra is just the study of systems of abstract symbols defined only by their set of self-consistent operations.

Giving names to line segments was also the beginning of analytic geometry because it united algebra and geometry: symbols could be interpreted either algebraically or geometrically, at one’s convenience in the process of problem solving. Descartes expressed this in his Discourse as follows:

Descartes thus extended symbolic manipulation previously confined to arithmetic--the manipulation of 0,…9--to geometry. This was laid out at the very beginning of Descartes’ short essay La Geometrie, as follows:

Between the first and last sentence, he lists the geometric forms of the basic operations. The last sentence says, in effect, that conceptual clarity arises from not distinguishing, by terminology, in which context, arithmetic or geometric, these operations are being performed. Operations should be separated from—abstracted from--context.

The first and second sentences say that all problems of geometry are analyzable in terms of basic geometric operations which correspond to arithmetic addition, subtraction, multiplication, division and the extraction of roots. The problems of geometry he had particularly in mind form a class of problems inherited from ancient Greek geometers of which one in particular, was the problem of Pappus (circa 275 CE). He now realized he could use arithmetical techniques to solve a problem of geometry and resolved to do so. As we shall see, in the process, he came upon the miraculous idea of analytic geometry, in which curves were related to fomula. He discovered, in effect, that the shapes (and motions and trajectories) found in nature could be put into ‘arithmetic terms’. The first inkling of this were in the constructions of the Pythagoreans, but how much time, work and how many missteps lay between!

Unsolved for two millennia, the problem of Pappus was cracked by Descartes in six weeks during the winter of 1632, thereby dramatically demonstrating the power of his new abstractions and methodology. Since the full statement of Pappus’ problem is complicated and in any case meaningless to a modern reader, we shall only focus on a simple stripped down piece of it, shown in Figure 5. It illustrates the key to Descartes’ whole approach.

Line K and L, and hence their intersection, O are given and fixed. The point P has lines PA and PB emanating from it. Suppose that P may move and with it the lengths of PA and PB and their intersection with K and L, but not their orientation. That is, P, PA and PB can be translated but not rotated.

K’ is parallel to K and L’ is parallel to L. Consequently, although the size of the dotted triangles ADO’ and CPO’ change when P moves, their internal angles do not; and, although the distances x, y, z₁, and z₂ also change when P moves, the ratios between legs in each triangle do not. In particular, this is true of the ratios z₁/x=a and z₂/y=b. Defining z=z₁+z₂ as the distance between P and A, we get z=ax+by. Thus as P varies its position, the distance z is known in terms of the x, y, variables and the constants, a and b; z became a known algebraic function of x and y, variables that at the same time located the position of a unique point.

It turns out that the problem of Pappus could be solved by finding values for x and y, and hence points P, which satisfied certain length ratios which could now be written in terms of expression like ax+by. That is, conditions expressed as geometrical length ratios were converted to algebraic conditions on coordinates, solvable for x, y. Each such pair x, y, located a points P that satisfied the conditions set out by the problem of Pappus. The set of all solutions determined a set of points that made up a curve. In this roundabout way, Descartes came upon the idea of an algebraic expression corresponding to a curve.

Further along in his book, Descartes makes some "general comments concerning the nature of curved lines", and after discussing the variety of curves his proportional compass could produce as examples, he says:

That is: all precisely definable curves are related to points on axes by mathematical relations; most commonly one has a relation in the form of a single equation.

Thus, two great ideas came into being: (1) The correspondence between geometric curves and algebraic relations, (2) the use of algebra in the process of analysis. Their deeper significance shall be discussed in turn.

In the process of solving the problem of Pappus, Descartes had invented Analytic Geometry and with it an insight into the nature of curves of which he said, in a later letter to Marin Mersenne:

This is an understatement if only because this invention gives us our only means to open up the super-sensible world to the analytic power of the human visual system!

But it is also an understatement for another reason. Before Descartes, the only shapes associated with mathematics, and analyzable with mathematical precision, were made of straight lines, circles, a few conic sections, and little else. In contrast, there were countless complex curves, surfaces and volumes shown us by nature. How could anyone imagine that a mathematics with such limited descriptive abilities might yet be able to describe all the shapes and motions found in nature? Yet if science was fundamentally mathematical in any sense, this would have to be somehow true.

Descartes opened up, to our view and control, a new universe of geometric forms beyond any previous conception. Any equation, indeed, any set of mathematical conditions one might imagine, determined points which when plotted formed some geometric object—a curve, surface, solid, and so on. A large variety of curves were easily created using even simple algebraic expressions. Thus through the experience provided by analytic geometry, the potential richness of mathematical description was revealed, making it possible to imagine, for the first time, that all natural shapes might be explicable by a mathematical science.

The same discovery that led to the geometric visualization of these shapes, also led to vastly improved methods of manipulating them. These go under the names of algebra and analysis, which we now discuss in turn.

There is no unique definition of algebra. Some identify it with the general objective of finding recipes for solving various classes of problems. Examples of algebra in this sense can be found in early Babylonian manuscripts. Although equation solving motivated the development algebra, it is more useful to think of it as a certain type of symbolic manipulation.

Algebra in this sense developed as a result of a deepened understanding of mathematical thinking. Descartes expressed this as follows:

He looks here towards a ‘universal’ mathematics, a general approach to the study of relations among measurable quantities (order and measure), to be applicable to any type of quantity (and to be made visible by analytic geometry). This vision shifts mathematics’ emphasis from that of calculating numbers or geometric figures to finding relations applicable to all kinds of measurable quantities. A prerequisite of this, to avoid suggesting particular types of quantity, is the use of abstract symbols.

Algebra required a number of other basic ingredients. Notation was one, and it was more difficult and important than might be supposed (remember the importance of Hindu positional notation!). Descartes himself provided the symbol for powers—one of the last pieces of modern notation to be created. In 1593, Francis Vieta (considered by many to be the father of algebra), wrote in the following style :

B in D quadratum 3-D cubo

whereas just a few decades later, Descartes would write this in modern notation as:

3BD²-D³

It easy to think of transforming Descartes’ expression into D²(3B-D) but not Vieta’s. The superior notation is labor saving because it is shorter and directly expresses crucial information (e.g. the number of times D multiplies itself). Notation does much to make algebra into a tool for performing the logic that lies behind algebra’s symbolic manipulation, or in other words, for reducing thought to manipulation.

Another basic ingredient of algebra was paper. Algebra is visual; it allows relationships to be grasped visually; something done by the visual portion of the nervous system, automatically, unconsciously, with maximal speed and ease, and with minimal effort. Lacking paper we must use, evanescent memory as our ‘paper’, sound as our ‘pencil’ and considerable mental effort to manipulate concepts within our head.

A precursor of algebra was its ‘rhetorical’ form. From the famous Algebra of al-Khowarizmi there is this example of it [Reef , p.84]:

Why is it easy? Because the paper is remembering everything else for you while you concentrate on first the multiplication and then the addition, and you need merely shift your eyes back and forth to compare related portions of the expressions. The rhetorical form is much harder to follow. Of course it is the only way to go if you are sitting in an ancient (or modern but underdeveloped) country with writing materials either scarce or non-existent. The book of al-Khowarizmi is a classic case in which the technics of an older medium of communication (speech) are being transcribed to, but have not yet been reformulated for, a new medium.

Ultimately, algebra’s importance stems from the fact that relationships are the content of knowledge. That the sky lights up is an observation; that this is related to the sun appearing on the horizon, is knowledge. That the sun is in a particular position is an observation; that all its positions are ordered on an apparently circular daily path is knowledge. Observation alone is useless; once made, it is the past and without knowledge, the past says nothing about the future. Similarly, each quantity observed in science is useless unless understood as an instance of a relationship among quantities—a function. Indeed, the shapes visualized through analytic geometry are visualizations of relationships; and laws of physics are relations between possible observations, symbolic relations defining functions, and without algebra they can scarcely be manipulated to provide useful results.

Like algebra, the word analysis has taken on a number of meanings. Here we shall be concerned only with the concept as created by the Greeks, and re-created by Descartes and Vieta. We start by analyzing the Pythagorean theorem: a²+b²=c² (see Figure 6).

The first step in analysis is to state what is known about the problem. What is known about any quantitative system is either relations between quantities or values. Since this theorem is about right triangles, it would seem reasonable that relations among right triangles should be important. Among these is the statement, a : b : c = x : z : a = z : y :b which is easily proven from the similarity of the right triangles abc, xza and zyb; and this will only be true when abc is a right triangle. In addition and very importantly, analysis generally includes the statement of the proposition being analyzed which is a²+b²=c².

The second step is to combine these statements and their consequences algebraically. Let us eliminate a and b from the Pythagorean theorem using a²=xc and b²=yc which come from the parts of the proportions a:c=x:a and b:c=y:b respectively. Substituting into the Pythagorean theorem yields xc+yc = c², or, finally, x + y = c which is obviously true by construction. So we have discovered what was initially obvious!?

This analysis, which is known as ‘theoretical analysis’, is now complete, but what is it good for? First of all, if it had led to a false statement, the initial proposition (here the Pythagorean theorem) would have been proven inconsistent with the truth, and hence false. However, the fact that it leads to a true statement (here x + y = c) does not prove it true; for a proof cannot be based on the proposition to be proven and, as we have just seen, analysis assumes the truth of the proposition to be proven (which may lead to a logical fallacy called ‘affirming the consequent’). Synthesis goes in the opposite direction, logically deducing the proposition only from what is given.

Analysis is useful as a guide to a proof. It lays down a logical trail which you try to retrace in the opposite direction. Doing so in this case, we achieve a synthetic proof of the Pythagorean theorem ( ‘ ‘ gives the basis for the assertion to the left of it)

• c=x+y by construction
• c²=cx+cy axiom of algebra
• c²=a²+cy cx=a² from similarity of abc and xza
• c²=a²+b² cy=b² from similarity of abc and zyb

Which was to be demonstrated.

Figure 7 diagrams the essence of the analysis and synthesis and their relationship. The Pythagorean theorem is one of the starts for analysis whereas it is the end for the synthesis. The other starting boxes, not shown, contain theorems and axioms of geometry and algebra, and propositions true by construction (e.g. that all the triangles are right triangles).

In addition to theoretic analysis there is a ‘problematic’ type. It is performed exactly as before the only difference being that the algebra leads to a specific value of a quantity, not a relationship between quantities. Algebra is generally introduced in the schools in conjunction with problematic analyses and it is a common experience for students to be very impressed with its power. As indeed they should be: analysis makes otherwise taxing logical thought easy by reducing it to manipulation of algebraic symbols.

The livelier the mind, the more impressed it will be by the powers of analysis for lively minds can imagine even more ambitious goals. Vieta ended his essay on the art of analysis as follows:

Descartes considered this art so great that he suspected the ancient mathematicians of hiding it:

Analysis was certainly a major inspiration to Descartes:

Vieta and Descartes both allude to the application of the technic of analysis outside of mathematics (or, of the extension of analysis to subjects beyond those traditionally included within mathematics). They leaped to this generalization but how was this to be accomplished? That was what Descartes set out in his ‘Method’, his method for acquiring knowledge as certain as that of mathematics. It informed his whole philosophic program, and through it, much of subsequent European history. It has, in fact, become so characteristic of our civilization that we now unhesitatingly profess to apply ‘systematic analysis’ to all difficult problems.

In 1633, Descartes wrote to his friend Mersenne that he had almost completed a manuscript the tentative title of which was: Project for a Universal Science Which Might Raise Our Nature to Its Highest Degree of Perfection. Next the Dioptric, the Meteors, and the Geometry, Where the Most Curious Matters Which the Author Could Find to Give Proof of the Universal Science He Proposes Are Explained in Such a Manner That Even Those Who Have Never Studied Can Understand Them. He then learned that Galileo’s Dialogue on the Two Chief Systems, published the previous year, had been condemned by the Church, that the author had been forced to sign a formal abjuration of his belief in the Copernican doctrine, that his book had been publicly burned, and that he had been sentenced to an indefinite term of imprisonment by the Inquisition. He thereupon wrote to Mersenne

He soon concluded however that despite his fears of offending the Church, his discoveries should not suppressed. As his comments suggest, the Descartes felt, like Bacon, that science was to be applied to human welfare as a religious duty:

So by 1636 he had a manuscript cleansed of possible heresy and ready for the printer. It was published in Leiden in 1637 with the title: Discourse on the Method of Rightly Conducting the Reason and Seeking the Truth in the Sciences. Next, the Dioptric, the Meteors, and the Geometry, Essays in This Method. Now commonly called the Discourse, it is Descartes’ first and most famous publication.

He wrote in French instead of the usual Latin, explaining:

Those using only ‘natural reason’ were the educated public as opposed to Aristotelian philosophers. It was, as one translator [Ref. ] says:

Descartes writing style quickly obtained, and has since retained, the reading public he sought. Patterned after that of Montaigne, it was consonant with the style of thinking he professed and advocated for others. Like Montaigne’s essays first published (1580) sixteen years before Descartes’ birth, the Discourse is in the form of a simple, direct and honest personal memoir that relates the writer’s inner thoughts, doubts, and stumblings towards the truth. And as we shall see, like Montaigne, perhaps his most basic message is that the only sure truth we have access to is our own thoughts.

The Discourse concerned method: the Method of Rightly Conducting the Reason and Seeking the Truth in the Sciences. Its essence was given in four short rules which he resolved "never in a single instance to fail" to observe as follows:

These are, in one sense, nothing else but generalizations, beyond mathematics, of the analytic method which he had developed to such good effect in his invention of analytic geometry. These rules will be discussed in turn.

(1)The first rule is a remarkable mixture of skepticism and gullibility which has become characteristic of all rationalism. Its spirit of skepticism can be traced to more than one source. One of these, already mentioned, came from his school. La Fleche was run by the Jesuits who originally developed and applied it towards undermining the Protestants’ personal reading of Scripture as discussed in a previous chapter.

This spirit governs the selection of mathematical axioms; they must seem self evidently true (without the necessity of proof), and they must be stated, as in a mathematical formula or definition, clearly and distinctly, one of Descartes’ favorite phrases.

These ideas can also be seen in the mechanical world-view he had learned from Beeckman: explanation in terms of a microscopic mechanics of components and process which were clearly and distinctly understood from everyday life such as solid physical parts moved by pushes and pulls. The mechanical philosophy countenanced no strange forces, no materials with strange properties, and no essences and spirits. These were holdovers from ancient and now discredited traditions—Hermetic, Aristotelian, and Platonic—that Descartes and the whole future movement of rationalism was in rebellion against at least as it applied to natural philosophy (religion became a separate issue).

It is useful to re-iterate at this point that mathematics is a system of deductions built on axioms which are both true by definition and are clearly and distinctly understood by virtue of the fact that they are expressed in words, symbols, and formula having unequivocal meaning again by definition. Mathematical axioms were often suggested by experience. Experience then suggested, for example, that straight lines existed, that they could be infinitely extended, and that through any given point, a straight line could be drawn, parallel to another given straight line. But whether axioms are true for all experience, mathematicians today neither know nor care. As far as modern mathematics is concerned, it is only the internal consistency of a set of axioms that matters. Thus, there are now many geometries; the axioms of each must be internally consistent, and those of one geometry are not consistent with those of another. This generalized view of mathematics, however, was not yet in existence in Descartes lifetime (it was largely due to him that it later developed).

Descartes took the conditions that applied to the axioms of the mathematics of his day—the appearance of indubitable truth—and transferred them to all thought, and in particular, to the analysis of nature. One might say that, having glimpsed the future power of mathematics and its central role in science, he made a passionate rush to judgment and conflated them. This is not to say that he did not understand that experiments had an important role in science, but he never imagined that they could lead to the abandonment of obvious basic truths. This turned out to be a very great error.

(2)The second rule of Method is the key step, analysis, which, as conveyed by its etymology, connotes a breaking up into parts. In mathematics it simply means writing down in the form of equations all the bits of information one knows about a system as a preliminary to algebraically combining them. Few would argue with this.

Generalized beyond mathematics, it means breaking up a problem into its component parts, or a system into its simpler parts. The general working assumption of science is that this can be and must be done in order to completely understand a system’s behavior. A good example of this in physics is the near perfect gas which is analyzed in terms of individual molecules and their basic interactions. The behavior of 10²⁶ molecules is then deduced and shown (often by computer simulation) to reproduce observed properties of such gases. The assumption is that only when explanation is at this level can a system be truly said to be understood. This is ‘complete reductionism’, something which many people now argue with.

(3)The third rule can be understood as a guide to logical deduction. How do you combine the facts known about a problem to solve it? In mathematics, logical deduction proceeds by combining and manipulating expressions, and this rule might guide the order of combination. Analysis of all but the very simplest mathematical problems leads to a number of equations, and if combined unsystematically, they often lead to a thicket of algebra. The unwary then stumble about, often only to find themselves back where they started (or left with a true but useless identity like 0=0). Descartes very sensibly suggests being systematic.

(4)The last rule is not necessary in mathematics as long as the algebra is done correctly. That is great advantage of a self consistent system of symbolic manipulation: it guarantees the logic. But outside such a system, in the exercise of logical thought done in our heads, it is easy to slip up, to skip a step, or do it incorrectly. What Descartes is emphasizing here is not merely checking one’s logic, but breaking up the logic into the smallest possible steps, each so simple and obvious as to be error-proof, and then checking that there are no missing links in the logical chain. I have previously noted that this is the source of the impregnable strength of a mathematically constructed wall of logic. And it is reflected in subsequent remark where he talks of those long chains of reasons, all quite simple and quite easy, which geometers are wont to employ in reaching their most difficult demonstrations.

It is very easy to underestimate what Descartes accomplished in his Discourse. These rules perhaps sound obvious and innocuous. The great Leibniz thought so. He called them vacuous, remarking that it was like advising a chemist to follow the method :

But no one before Descartes had proposed that this mathematical methodology be a model for the acquisition of all philosophic/scientific knowledge. As Descartes himself said of his principles:

That only a generation later Leibniz thought them obvious testifies perhaps to Descartes immediate and profound influence; and that many today still would think them obvious, to his enduring influence.

His four principles were precepts of what was to become virtually a ‘religion’ of Cartesian rationalism. He had already demonstrated the fervor of its religious spirit in the meticulous spelling out of the precepts themselves, determined that they be absolutely clear, the ‘simplest and easiest to know’, and ‘that nothing was omitted’.

His fervor was commensurate with his goals. Mathematics had given him principles of philosophy from which the knowledge of everything else in the world can be deduced. The amazing connection he had discovered, between arithmetic and geometry, through algebra, hidden for two millennia, had inspired him:

No knowledge of which Man is competent is beyond reach. Is there anything outside of Man’s competence? He says elsewhere:

But then Descartes immediately goes on, in the tradition of the Scholastic philosophers from whom he was rebelling, to attempt to demonstrate the necessity of God using his method. This is a great assumption of power of the human intellect. How does Man get this power? Naturally, from God (or today we might just stop at the ‘naturally’), and since God would not deceive us,

This is a truly astounding gift. Given this, not too much is beyond Man’s powers. It bids us: Bite this apple! Build this tower to the heavens! It is only necessary to be sure that our thoughts have sufficient clarity.

In his Preface to the Principles of Philosophy, Descartes gave the following picture of Philosophy:

Thus philosophy as a whole is like a tree whose roots are metaphysics, whose trunk is physics, and whose branches are all the other sciences. These reduce themselves to three principal ones, viz. medicine, mechanics and morals...

That this Tree of Philosophy, shown in Figure 8, is the old Tree of Knowledge is clear especially from the appearance of Morals as one of its branches. The picture seems to suggest that morals are somehow ultimately deducible from physics and many of Descartes’ followers over the years interpreted him in just that way. This was just what the writer of Genesis had worried about and where it would lead was neither hard to see nor long in coming.

Axioms (which lie at the roots of the Tree of Philosophy) are the most powerful statements in a deductive system. The sooner any are discovered the better. How do we discover axioms? What distinguishes them? They certainly must have the quality of being true. So Descartes tried to find something he was sure was true.

Turning to what he had been taught in school for guidance, Descartes was disappointed. He had been educated in one of the finest school in Europe, was one of its best students and always spoke warmly of his teachers. Nevertheless, he severely criticized the humanistic education he had received from the Jesuits, finding much of it either useless or wrong. So first he wiped his mental slate clean; then, doubting and testing everything, he looked for something indubitably true—a statement which survived all doubt. He finally found it:

Thus he reasoned: anything which doubts is conscious; anything conscious exists; I exist. He had doubted, and doubting was a form of thinking. This was a paradigm: any other axiom had to have a similar clear indubitability, a similar maximally simplicity:

To summarize what has been said here so far, Descartes proposed that all the knowledge of which man is competent is interconnected, structured like a great tree of knowledge, with roots in metaphysical truths as clearly true as the fact that he, Descartes, existed; that its interconnections could be understood using his analytic method based on reason and intuition; and that their efficacy was vouchsafed by the goodness of God. This summary can be compared to one of his own which appeared near the end of the first part of his Principles of Philosophy. It expresses these ideas slightly differently and mentions some others not discussed here:

This, in barest outline, was the basis of a program to achieve all possible knowledge through analysis with a degree of certitude comparable to that of mathematics. It has been influential beyond measure. It is now the world’s program. It was not so before Descartes.

He devoted his life attempting to demonstrating it was practical: