Isn't it strange that the animal we used to be developed into the creature that we now are? How – and why – did human intelligence and culture evolve? How did we evolve minds, philosophies and technologies? And now that we have them, where are they taking us?

The orthodox answer to these questions looks inside our brains to see what they are made of and how the various components operate. This leads to a story based upon DNA biochemistry, the evolution of nerve cells as pathways for sensory information, and their organisation into complex networks – brains – that can manipulate neural models of natural objects and processes. Mind is seen as a property of an unusual brain – complex enough to develop culture – but here the ‘reductionist’ story starts to lose its thread. Many people see mind as something that transcends ordinary matter altogether. Philosophers worry that the universe around us may be a figment of our own imagination.

In Figments of Reality we explore a very different, but complementary, theory: that minds and culture co-evolved within a wider context. Every step of our development is affected by our surroundings. Our minds are rooted in ordinary matter; they are complex processes – or complexes of processes – that happen in material brains. Our brains are linked to reality by their molecules; but they are also linked to reality on another level, their ability to model reality within themselves.

We end our journey through human mind and culture by trying to answer some of the questions that we raised in the opening chapter. How did such a peculiar animal as the human gain such a grip upon the planet? What is it that makes us the way we are? And where are we going next?

Let us first take stock.

We are genuinely remarkable members of the animal kingdom.

In The Philosophical Review for October 1974 Thomas Nagel wrote a celebrated essay: ‘What is it like to be a bat?’ In it he examined the difference between an external observer's understanding of the physical processes that occur in a bat's brain, and the bat's own mental perceptions. He argued that no amount of external observation can tell us what being a bat feels like to the bat.

Human minds do more than just recognising various bits and pieces of the universe. They look for patterns in what they recognise, and do their best to understand how the universe works. The universe, however, is very complex: in order to understand it we must also simplify it. Indeed the whole point of understanding something is that you can grasp it as a whole, and that necessitates some kind of simplification or data-compression. An explanation of the universe that was just as complex as the universe itself would merely replace one puzzle by another. In this chapter we shall argue that the brain organises its perceptions of the world into significant chunks, which we shall call ‘features’. As usual we shall take an evolutionary and contextual view of this ability, as well as asking about the internal structure of the brain. Not just ‘how does it work?’ but ‘how did it arise?’ And to get started, we shall take a look at two simpler creatures: the mantis shrimp and the octopus.

Both the octopus and the mantis shrimp are effective organisms, even if they never meet another of their own kind to learn from.

The background theory and philosophy is now out of the way, and we are ready to begin the journey from molecules to minds. It is a journey which, at every stage, involves the concept of evolution. Evolution is a general mechanism whereby systems can ‘spontaneously’ become more complex, more organised, more startling in their abilities.

Although in the second third of the seventeenth century Oughtred's Key and Harriot's Praxis converted key English thinkers to the analytic way, neither work seemed to satisfy the needs of less talented students wanting to pursue algebra. The Key was too brief, and both it and the Praxis were too dependent on symbols for the general student. The Praxis, designed as a supplement to Viète's Analytic Art, was not a general algebra textbook. The Key – with its focus on quadratic equations and positive real roots – had been, to a certain extent, obsolete as an introductory algebra textbook even at the time of its original publication. Furthermore, subsequent editions of The Key, which continued to be reissued through 1702, took no explicit account of Renè; Descartes's Gèomètrie of 1637 or other later algebraic developments.

Of all the analytic mathematicians in England during the second half of the seventeenth century, John Collins (1625–1683) most persistently pushed for the preparation of an English algebra textbook that both elaborated the principles of the subject in a fashion appropriate for university men and sophisticated practitioners, and covered the main algebraic developments of the century. Collins coaxed Isaac Newton and, at times, John Wallis to produce the desired book. He encouraged John Pell to complete his English edition of a German algebra as well as John Kersey to finish his textbook on the subject.

Viète's symbolical style so captured the imaginations of the Englishmen William Oughtred and Thomas Harriot that at least the former seems to have taken as the mission of his advanced years the “inciting, assisting, and instructing others” in the analytic art. Both produced textbooks on the subject, which brought early modern algebra, under the guise of the analytic art, to England. Published in 1631, these symbol-laden books struck a responsive chord in an English scientific community that was beginning to favor, among other things, plain prose. By the third quarter of the century, then, there was a growing school of English analytic mathematicians; there were also the beginnings of a related drive toward the acceptance of mathematics, including algebra, as a scholarly pursuit.

Oughtred, Harriot, and their disciples shared first and foremost a deep and irrevocable commitment to symbolical reasoning. As the two English algebraic pioneers fostered Viète's symbolical style, they also perpetuated his hesitancy about the expanding algebraic universe. They and most of their immediate successors reasoned symbolically, wrote algebraic equations for geometric problems, but largely ignored negative and imaginary roots or – in the most daring of cases, including Harriot's speculations in unpublished manuscripts – struggled to make some sense of them. In short, the algebra that was imported into England through Oughtred's and Harriot's textbooks was Viète's, not Cardano's. In England, a deep appreciation of the symbolical style came first; openness to the expanding universe of algebra followed slowly.

Thus the very advertisement to Girolamo Cardano's Great Art (Ars magna) suggested the dawning of a new epoch in the algebra of the Western world. Although less original than implied here, The Great Art – published in 1545, just two years after Copernicus's De revolutionibus and Vesalius's De fabrica – was an exciting scientific classic. Exciting in the mathematical way, the work announced the solution of two hitherto unsolved problems, finding the roots of cubic and quartic (or biquadratic) equations. As many other classics, it helped to redefine its field. In particular, it fostered an expanding universe of algebraic objects through its consistent acknowledgment of negative roots as well as its brush with imaginary roots.

Still, The Great Art was not solely responsible for the major reconstruction that Western algebra underwent in the sixteenth and early seventeenth centuries. Algebra's reconstruction involved new results, objects, language, methodological justification, and a changing relationship with geometry. It was as much due to Francois Viète's Analytic Art as to Cardano's Great Art.

The British intellectual tradition produced three major thinkers, in addition to Newton, who published on algebra in the first half of the eighteenth century: George Berkeley, Colin MacLaurin, and Nicholas Saunderson. Berkeley's writings on algebra were the most philosophical of the works produced by the three men and, in their time, the most neglected. Berkeley was also the only of the three to draw significant inspiration from Wallis, for MacLaurin and Saunderson began their algebras as commentaries on Universal Arithmetick. Berkeley wove Wallis's algebraic reflections with the arithmetic insights of Barrow and possibly Hobbes into a coherent philosophy of arithmetic and algebra as sciences of signs, in contrast to his philosophy of geometry as the science of perceptible finite extension.

As Berkeley's general philosophical concerns affected his understanding of early modern mathematics, so mathematics presented him with problems and insights that helped to shape his philosophy. His earliest notebooks referred to Barrow's view of number as a “note” or “sign,” and number, so understood, became one of his stock examples against abstraction. Berkeley, moreover, was the first major British (and perhaps European) philosopher to come to terms with the symbolical style of early modern algebra. His early acceptance of symbolical reasoning and Barrow's view of number helped to raise in him “semiotic consciousness” – or “the explicit awareness of the role of the sign as that role is played in a given respect” – and thus contributed toward his innovative theory of language.

In late-seventeenth-century England there developed a geometric and synthetic backlash to the algebraic tradition established by Oughtred, Harriot, Pell, Kersey, and Wallis. Wallis, in particular, became entangled in an extended debate over the relative ranks of geometry and arithmetic, the appropriateness of using algebra to solve geometric problems, and the legitimacy of reasoning on symbols. The restive, pensive mood of the Scientific Revolution, combined with the lack of clear disciplinary lines, made such a debate possible and permitted its participants to roam freely over the mathematical, philosophical, and educational aspects of the new algebra.

The debate peaked prior to the publication of Wallis's Treatise, with critiques of his early writings by Thomas Hobbes (1588–1679) and Isaac Barrow (1630–1677). It was perhaps significant that Hobbes and Barrow were royalists, opponents of the Puritan movement with which Wallis had been so intimately connected. Moreover, both men craved certainty, in real life as well as in mathematics, and both men found that certainty in traditional geometry. Both saw themselves simultaneously as defenders of an enduring mathematical tradition with roots back to ancient Greece and as enlightened thinkers of the new age of science. Favoring geometry over arithmetic and algebra, and synthesis over analysis, they helped to lay the foundations for the preoccupation with geometry that was to mark British mathematical education through the early nineteenth century.

Still, neither thinker was a naive defender of the old mathematics. Hobbes's mathematical views were shaped by seventeenth-century respect for experience as a source of knowledge and his nominalist inclinations.

The contest for the Lucasian professorship in 1760 symbolized the state of English mathematics at the beginning of the second half of the eighteenth century, in more ways than one. Unlike Saunderson and Colson, Maseres and Waring were products of Cambridge University. Their mathematics was shaped, albeit with different results, by the Cambridge mathematical traditions in place by the midcentury. Like many Cambridge mathematical scholars of their period, both men were influenced by Saunderson's Elements of Algebra, which “was long the standard treatise on the subject.” In his major mathematical work, A Dissertation on the Use of the Negative Sign in Algebra of 1758, Maseres praised certain aspects of Saunderson's Elements even as he severely criticized the book's liberal use of negative quantities. In 1760 Waring reported that everyone at Cambridge was reading Saunderson and cited the latter as the “Authority” for some of his own mathematical manipulations.

Maseres and Waring were not ordinary Cambridge graduates. They were high wranglers, fourth and senior (or first) wranglers, respectively. That is, the election of 1760 was the first Lucasian election to be contested by Cambridge graduates who had distinguished themselves in the mathematical honors course that solidified at the university around the mid-eighteenth century. This course was a natural outgrowth of the university's emphasis on mathematics as a logic. If mathematics was really the best instrument to exercise and train the human mind, and Cambridge was charged by the Elizabethan statutes to teach logic to all second- and third-year undergraduates, the argument ran, then the undergraduate curriculum should center on mathematics.

Whereas MacLaurin worked comfortably in the geometric and algebraic traditions, some of his mathematical contemporaries at Cambridge University developed the algebraic tradition more so than the geometric. A bias toward algebra characterized the lectures and writings of Nicholas Saunderson, who held the Lucasian professorship from 1711 to 1739, and those of his Lucasian successors, John Colson and Edward Waring.

Published posthumously in 1740, Saunderson's Elements of Algebra in Ten Books was the English counterpart of MacLaurin's Treatise of Algebra. Both textbooks evolved from classroom lectures that began as commentaries on Newton's Universal Arithmetick. Both tried to show the reasonableness of, if not demonstrate, some of the algebraic rules that Newton had stated without proof. Both singled out the negative numbers for lengthy explanation while largely accepting Newton's view of imaginary numbers as impossible. Still, there were important differences between the textbooks that spoke to Saunderson's relative detachment from the British geometric tradition. Saunderson not only fostered algebra's independence from geometry, as did MacLaurin, but also boldly declared the superiority of analysis over synthesis. No earlier British algebraist had defended algebraic analysis as strongly and explicitly as he now did in his Elements. He argued that analyis was a method of demonstration; as such, analysis was the equal of synthesis; and, furthermore, the analytic method was actually to be preferred to the synthetic because “a synthetical demonstration only shews that a proposition is true; whereas an analytical one shews not only that a proposition is true, but why it is so.”

Even if Berkeley's philosophy of the abstract sciences of arithmetic and algebra had little effect on British mathematics through the mideighteenth century, Newton's bifocal mathematical legacy helped to assure that algebra as well as geometry continued to be cultivated in Great Britain. Probably most early-eighteenth-century British mathematical thinkers were strongly attracted toward the geometric focus of Newtonian mathematics; some, however, felt at least an equal pull toward the arithmetico-algebraic focus.

In Scotland there were two schools of thought on algebra: the somewhat anti-Newtonian school, led by Robert Simson (1687–1768), and the largely Newtonian school, led by Colin MacLaurin (1698–1746). Simson – the “father-figure” of Scottish mathematicians of the first half of the eighteenth century – was primarily attracted to the geometric focus of Newtonian mathematics. By his later years, if not in his early career as well, he came near abandoning early modern algebra. He preferred geometric analysis to analytic geometry; perhaps questioned the symbolical style; and rejected the negative and imaginary numbers. MacLaurin, on the other hand, published first on geometry, then prepared a manuscript on algebra, and next devoted eight years of his life to answering Berkeley's Analyst with his Treatise of Fluxions, a largely geometric work with, however, a significant algebraic section.

By all accounts, MacLaurin was a brilliant mathematician; in many respects, he was the greatest of Newton's mathematical disciples. Early meetings with MacLaurin seem to have convinced Newton that he and the young Scotsman were kindred mathematical souls, who appreciated both the new and the old mathematics.