The Measure of Cancer

Part 2. Analysis

by Ian Magrath

Such harmony is in immortal souls; but whilst this bloody vesture of decay doth grossly close it in, we cannot hear it. Lorenzo, Merchant of Venice Act V, Scene I, William Shakespeare.

Fibonacci spirals (see Figure 2) are found everywhere in nature since they provide optimal spacing and packaging. In daisies, marigolds, chrysanthemums and sunflowers, the seed arrangements in the flowers form such spirals, in which the numbers of seeds in clockwise and anticlockwise spirals are adjacent in the Fibonacci series.

Plato taught that there are several levels of comprehension, ranging from those based purely on sensory impressions to eventual understanding, through dialectic, of the first principles, or Eidos (Ideas, or Forms), which he took to be the unchanging constants underlying the varying manifestations of the objects we perceive (aisthetón). He illustrated these mental states by the use of his now well-known similes - the Sun, the Divided Line and the Cave (Figure 1). Scientific understanding is perhaps the modern equivalent of Plato’s Eidos, but present ideas of the “underlying truths” are scarcely less tangible than the Forms, being enveloped in words which conjure up only vague images (such as quark, or anti-matter), or expressed as mathematical equations, sometimes complex and sometimes disarmingly simple (e.g., E = mc²). Effectively, the words and equations are the understanding. But in contrast to the imprecise nature of Plato’s Forms, the equations of modern physics can be tested to an exquisite degree of particularity, thanks to the emergence of scientific methodology some 2000 years after Plato’s death. Henceforth, the dialectic of natural philosophers, renamed “scientists” by William Whewell in the 19th century, would derive from far more substantial fare than that available in ancient Greece. Thought alone, however erudite, can never approach reality unless founded upon fact. Such facts we refer to as data. Data (singular, datum, from the Latin for something “given”), are derived from the observation of natural events (such as the occurrence of specific cancers at different incidence rates in different environments), or from experiments, whereby the results of carefully designed and executed interventions (such as the uniform administration of a particular therapy) are equally carefully measured. Plato recognized the importance of mathematics - that is, the rules of numerical calculation, whether arithmetical or geometrical - as the foundation of logic. But he lived in an era when numbers dealt with specific circumstances and were largely fixed to the tangible objects they enumerated, when the concept of zero did not exist, and when even movement (and hence change of any kind) was seen as mathematically paradoxical, at least by one school of thought (that of Parmedides and Zeno). Logic, inevitably, remained as tightly tethered to the world of the senses as were the numbers from which it derived, prohibiting anything more than the most superficial penetration beyond outward appearances. The very idea of “laws” of nature, in essence fundamental generalizations that apply to a broad, if not infinite, range of circumstances, was inconceivable. Modern science, in which elementary particles or unimaginably small effects on a gravitational field are sought entirely on the basis of predictions derived from mathematical equations, stands in dramatic contrast. As such, its emergence became possible only after a profound revolution in the very concept of number, and hence in calculation - a slow, stuttering revolution that occurred in the course of many centuries, eventually giving birth to a third major branch of mathematics - algebra.

This seminal step exceeded the capacity of a single human mind, however great. It came about, therefore, as series of smaller steps, each taken without a clear sense of where it might lead. The embryo of the central idea - that of abstract numbers - had lurked in the minds of the ancient philosophers for centuries. It underwent further development by mathematicians in the great library of Alexandria who continued in the tradition of the ancient Greeks. But the critical element was the development of much more flexible number systems in India in the course of the 4th and 5th centuries CE. The brilliant Muslim cultures, geographically interposed between the two main branches of the Indo-European races and nourished by the wisdom of their Near Eastern forebears, including Babylonians, Egyptians and Persians, as well as mathematicians from distant China, were ideally placed, physically and mentally, to bring these intellectual worlds together. After a lengthy gestation period in the courts of the Caliphs, algebra was born in its modern form in renaissance Europe and grew to maturity as successive mathematicians developed new analytic methods, bringing together the once separate domains of arithmetic and geometry and leading to new mathematical approaches to the analysis of data. The logical foundation of scientific understanding is truly a child of all of mankind.

Abacists and Algorists

The remarkable tool of mathematical analysis is named after a book imposingly entitled Hisab Al Jabr wa’l Muqabala (Transposition and Reduction) written by the Persian/ Arab astronomer and mathematician, Muhammad Al-Khwarizmi, born around 783 CE. Al-Khwarizmi lived at the court of Caliph Mu’Am and worked in the famed “House of Wisdom”- the equivalent, in medieval Baghdad, of a scientific academy - translating the works of the ancient Greeks, Romans and Byzantines. Al-Khwarizmi’s book described two important steps in the solution of equations, and although his descriptions were in words rather than symbols, his book made a sufficient impression for an abbreviation of its title to become the name of the new branch of mathematics. Al-Khwarizmi also studied Indian texts and it is from the title of the extant Latin version of his book Algoritmi de Numero Indorum (Al-Khwarizmi on the Hindu Art of Reckoning) that the use of the word algorithm to describe a series of steps in a mathematical or logical operation is derived. Al-Khwarizmi and his followers, who became known as algorists, eschewed the abacus (used widely in those days as a calculating machine by mathematicians known as abacists) in favor of calculations performed with Indian numerals, including the zero, written in dust or sand. Today, the sand has transformed into silicon chips, for algoriths are the building blocks of computer programs.

Our modern numerical notational system, consisting of nine symbols and zero, derives from the numerals invented by Indian astronomers and mathematicians at the time of the 4th century CE Gupta dynasty. The zero, as well as indicating no-thing at all, was used for the first time as a place holder in a place-value system. Such systems use multiplication of a base number to generate different sized sets of numbers which are then enumerated, the value of each set in any given number being represented by a different position. They differ markedly from additive systems in which each number simply follows the next and has its own name and symbol. This works well for small series of numbers, but would require prodigious feats of memory for large series. In contrast, place-value systems require only one less numeral than the base. Such systems are intrinsically more abstract than additive systems, since numbers tend to lose their individual characteristics and an infinite series of numbers can be generated by a simple rule. Moreover, zero itself is an abstract concept. Most place-value systems are decimal, that is, based on ten (the original “digital” system!) and its multiples. Many other bases have been used in the past, and their remnants persist today - for example, the sixty-based Babylonian system has given us our circle of 360° and triangle of 180°. In the decimal system, the nine numbers and zero are used repeatedly to indicate, in any given number, the value (number) of the sets of hundreds, tens, units, etc., or the absence of a particular set. In the late Middle Ages, this system of counting was transmitted to Europeans by Arab scholars, along with the works of Greek mathematicians such as Archimedes, Euclid and Diophantus and their own important contributions. For this reason our modern numerals are still referred to as Arabic.

Figure 1. Plato’s Divided Line simile showing the various mental states, from lowest to highest. Modern science, as shown on the right, has no room for unsupported opinion.

One of the most important, and earliest links between the Hindu-Arabic number system and European mathematics was the Italian mathematician Fibonacci, otherwise known as Leonardo of Pisa (1170-1250). He began and spent much of his life in Pisa, Italy, but was brought up and taught mathematics in North Africa, in a port town called Bugia (now Bejaia) in modern Algeria, where his father represented Pisan merchants to the customs authorities. In his book, Liber Abaci (1202), which had little or nothing to do with the abacus (the title was, in its day, religiously correct, for the church frowned upon any form of islamic influence), he described the nine Indian numerals and zero, which he had learned about in Bugia. He also provided a large collection of problems aimed at merchants and described the famous Fibonacci series thus: A certain man put a pair of rabbits in a place surrounded on all sides by a wall. How many pairs of rabbits can be produced from that pair in a year if it is supposed that every month each pair begets a new pair which from the second month on becomes productive? This series, in which each number is the sum of the two preceding numbers, occurs frequently in nature, being the mathematical basis of a type of spiral curve which occurs in the Nautilus shell (Figure 2), as well as in various seed, leaf and petal arrangements. The numbers in a Fibonacci series also lead to the Golden Proportion or Ratio (1.618034), said to have been derived by the Greek sculptor, Phidias, (hence its symbol, the Greek letter, phi) from the proportion of limb length to height of women “pleasing to his eye.” The golden ratio is converged upon as the Fibonacci series tends to infinity, being the ratio between one number and the preceding number. Because it allows a limit to be reached in the smallest number of steps, a modified Fibonacci series has often been used as the basis for dosage increments in “phase I” clinical studies in which the maximally tolerated dose of a new drug is sought.

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Figure 2. Fibonacci numbers, showing how the use of this sequence to create a series of squares leads to the Fibonacci spiral, which is seen in the Nautilus shell (a cephalopod related to octopuses and squids that has existed for 500 million years). See also the photograph on the front page. The Divided Line simile is superimposed.

Ars Analytica

Muslim and early European mathematicians lacked a means of symbolically representing generalized numbers, and it was Francisco Vieta (François Viète, 1540-1603), a French jurist who studied cosmology and astronomy in his spare time, who laid out the foundations of symbolic algebra in his work In Artem Analyticem Isogoge (Introduction to the Analytic Art). Vieta drew upon the writings of two Alexandrian mathematicians, Diophantus, whose Arithmetica dates from the 3rd century CE, and Pappus, whose somewhat later Synagogue (Mathematical Collection) dealt mainly with problems in geometry. He recognized that algebra was a general theory of proportions, expressed in equations, and that it could be applied to both geometry and arithmetic. He was well aware that “things that are new are wont to be set forward rudely and formlessly, and then must be polished and perfected in succeeding centuries,” but nevertheless, appreciated that algebra, the “Ars Magna,” had a fundamental place in the system of knowledge, enabling the world to be perceived in terms of its underlying harmonious structure or symmetry, and as a “lawfully” ordered course of events, rather than simply a collection of countable entities.

Scientific analysis is the necessary foundation of all attempts to improve socioeconomic conditions and to control disease, including cancer.

Mathematical analysis gradually came to be perceived as a process whereby solutions to specific questions can be obtained by inserting numbers (data points), into previously established equations that express general relationships between one thing and another. Improvements in symbolic representation and methods of calculation created ever more powerful means of expressing such relationships, and hence of predicting the outcome of a broad variety of interactions. In 1582, Simon Stevin of Flanders greatly simplified calculations by using decimal fractions in place of ordinary fractions. Pierre de Fermat, another French magistrate, and his contemporary, René Descartes, invented analytic algebra, whereby the sets of points that define lines, curves and surfaces could be represented by algebraic equations. Descartes’ treatise on this topic, Géométrie, appeared in 1637, but Fermat, who claimed to be content simply to discover the truth, failed to publish his work, thereby involuntarily ceding primacy to Descartes. Descartes used the later letters of the alphabet, x, y and z, as symbols for variables and the early letters, a, b and c, for constants in alge.braic equations. This convention superceded Vieta’s earlier system, and has persisted to the present day. Descartes also introduced the superscript notation to indicate positive powers, such as 10³, and a system of coordinates to represent points in space - a critical element in the graphical depiction of change. In the latter part of the 17th century Newton and Leibnitz invented differential and integral calculus - methods for analyzing continuous change (fluxion was Newton’s term), including shape, quantity, position, movement, speed and time. Ironically, Zeno’s paradoxes, which had appeared to show that motion, if considered as the successive halving of distance ad infinitum, is mathematically impossible, were finally resolved by a method based on the mathematical manipulation of infinitesimally small changes. As Plato had recognized, the natural world is about change, and the powerful mathematical techniques developed to express interrelationships quantitatively led to major advances in scientific understanding. But knowledge cannot be derived from the mere manipulation of numbers, or algebraic symbols. It relies heavily upon an additional ingredient - imagination. Science is driven by hypothesis, i.e., a proposed explanation for a set of observations (data). The ability to use reason to create hypotheses goes beyond mathematics, and corresponds, perhaps, to Plato’s dialectic. It is the testing of such hypotheses by seeking additional data (re-search), experimentally where necessary, that distinguishes scientific knowledge from beliefs that are not based on the rational interpretation of factual information.