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Not Real, René – A giant historical fraud Introduction René Descartes is probably the most famous modern western philosopher in the world. His writings (done in Holland) are still studied closely to this day and some claim that much subsequent philosophy is a response to him (most of the rest is regarded as a “footnote to Plato”). In fact, his contributions to mathematics are viewed by many professional mathematicians as possibly even more significant than his philosophy. Indeed, his mathematical contributions form the deepest layers of modern theoretical physics. He was the first to link algebra to classical geometry, producing what he called a “Universal Mathematics”; today, its core is referred to as analytic (or Cartesian) geometry. The key to this activity was his invention of the concept of ‘real’ numbers. We shall show this was a flawed construction that puts the foundations of modern science at jeopardy – hence the challenging title of this essay, which is based on dramatic resumarizations of a series of ten essays written by a few of the many academics, who still investigate Descartes’ writings closely. These essays were published in a recent book “Descartes: Philosophy, Mathematics & Physics” edited by Stephen Gaukroger; published in 1980 by Harvester Press. Gaukroger is emeritus professor of history of philosophy and science at the University of Sydney (Australia); he was educated at London and Cambridge, where he was a research fellow when the book first appeared. As a scholar, he specializes in the rise of modern science in the 17th and 18th centuries with particular attention paid to the writings of Descartes. In this book, he also contributed a chapter titled: “Descartes’ project for a mathematical physics”.
PUBLIC LIFE Personal René Descartes (1596–1650) was born in a small village in Brittany; his mother died when he was one, (so like Newton) he was raised by his grandmother. His family was Catholic but the local area was controlled by the protestant Huguenots. He first went to the Jesuit College at La Fleche, where he was introduced to Greek mathematics and Galileo’s physics (which were under attack then by the Jesuit intellectuals). After graduating at 18, he studied law (like his father) for two years at the University of Poitiers. Initially, his ambition then was to become a military officer, so he joined the rebel Dutch States Army and undertook a formal study of military engineering (like Napoleon, later). Here he met the principal of the Dordrecht school, Isaac Beeckman. In 1620, he visited the laboratories of astronomer, Tycho Brahe and then Johannes Kepler in Regensburg. He claimed that he had three visions in November 1619 of a Divine Spirit, who revealed a New Philosophy to him. Awakening, he began sketching his new theory of Analytical Geometry along with the idea of applying mathematics to philosophy. He became convinced that for him, the study of science would prove to be the path to true wisdom, where all truths were linked to one another, so that logic would open the way to all science. Descartes left the army in 1620, sold his family home and invested in government bonds – this provided a comfortable income for the rest of his life. He returned to the Dutch Republic in 1628 becoming a full-time student. His affair with a servant girl resulted in the birth of a daughter in 1635; her death in 1640 was traumatic, confirming his views on the importance of the passions (unlike Newton). Descartes was invited in 1649 by Queen Christina of Sweden to her court to organize a new scientific academy and tutor her in his ideas about love. Christina wanted her lessons to begin at 5am three times a week in her badly heated castle; after only about five lessons, Descartes caught a cold that turned nasty and he died soon after; there are rumors that he might have been assassinated.
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Philosophy Descartes’ philosophy, (like his friend and fellow Catholic intellectual, Marin Mersenne) were motivated by a desire to defend Catholicism against the rising tides of Neo-Platonism and skepticism. Descartes’s views are readily accessible through his extensive writings, which include: Meditations, Passions of the Soul, The World and Principles of Philosophy. Descartes believed he had refuted skepticism using his famous statement: ‘Cogito, ergo sum’ or “I think, therefore I am”. Unfortunately, this ignores the fact that all humans (including Descartes) learn their language (and hence the ability to think symbolically) from their cultural context (especially, their family) while all forms of rationalisms are subject to the charge of Solipsism (the belief that only one’s own mind exists – external reality is just a mental fiction). Descartes was forced to admit two kinds of ‘stuff’ in his model of the world (or dualism): mind/soul stuff (like God) and ‘real’ stuff (like us). In humans, Descartes had these very different substances interact via the pineal gland deep in the center of the brain. Ultimately, Descartes had to emerge from behind his rational cloak and make his Catholic faith explicit, relying on his faith in God to guarantee the validity of His Truth and provide Descartes with the certainty he desperately sought. Over emphasis on thoughts (or words) has been the characteristic of verbal intellectuals for millennia; those who discount the physical world (of the senses) completely are known as philosophical idealists or ‘rationalists’. Those who adopt the exactly opposite view are called empiricists, materialists or physicalists (or even physicists!!). Ironically, the Catholic Church banned his books in 1663 and, in 1671, Louis XIV prohibited all lectures in France that involved any Cartesianism, perhaps because he shifted the ancient, objective philosophical debate from “what is true?” to the individualistic question of “of what can I be certain?” Natural Philosophy Descartes’s science was based on Beeckman’s idea of touch or mechanism (see Newton essay). Although Descartes’s mathematics and philosophy continue to influence modern thinking, his approach to Natural Philosophy (science) – using only logic or reason, has been overtaken by Newton’s ideas. Mathematics Descartes’s mathematical writings are much less well known than his philosophical ones, such as: Rules for the Direction of the Mind (or Regulae in Latin – 1626), Discourse on Method (1637), Geometry (1637) and Metaphysical Meditations (1641). Scholars now recognize that Descartes influenced Newton more than any other mathematician or natural philosopher. A deeper critique of the new mathematics of Descartes is reserved for a subsequent essay, entitled: “Empty Symbolism”.
INNOVATIONS
Descartes’ mathematical breakthrough was the invention of the two-dimensional reference frame, now called the Cartesian co-ordinate system, allowing reference to a point in space as a set of numbers indicating the horizontal and vertical separations from the origin (the intersection of the two axes). This ‘trick’ allowed Descartes to relate new modern symbolic mathematics (algebra) to the ancient science of geometry.
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REAL NUMBERS The concept of number is one of the oldest ideas in the western intellectual tradition. It is believed to have originated with merchants in Babylonia, who needed to track their trading “deals”, using one ‘mark’ for each object. Like so many skills, the intellectuals of Classical Greece abstracted out the idea of Quantity (see related essay) that they recognized as spanning discrete objects (such as pebbles) and continuous stuff (such as water). It was not long before the mystic Pythagoras discovered the numerical basis of harmonious harp strings mapped to simple numerical ratios. Immediately, he over generalized this discovery to make the claim that: “All was Number”. Before long, he had a cult of followers, who were quite fanatical about these numerical insights. Unfortunately, one of his followers (Hippasus) proved that the hypotenuse on a triangle of unit length adjacent sides (i.e. of length the square root of 2 or √2) could not be represented by a fraction; for this radical discovery of “irrationals”, he was put to death and Pythagoras’s cult soon dissolved (except for influence on Plato).
Notation Today, everyone is familiar with numbers involving a single decimal point, such as: 3.1414. This is called the decimal representation, which is convenient for creatures with ten fingers and suits its widespread use for representing money. What is not realized (by non-mathematicians) is that this requires an infinite set of digits to adequately map common fractions, such as 1/3 etc. Mathematically, a ‘small’ real number R (with magnitude less than one) is shorthand for the summation series: R = 0. d1 d2 d3 … = d1/10 + d2/100 + d3/1000 + … =
∑
n=N n=1
dn / (10)n
with N → ∞ .
where … (ellipsis) means unending (or infinite) and dn is a single digit in the range (0,1,2 thru 8,9).
Definition Ever since Descartes, mathematicians have defined these infinite, decimal numbers as “real” numbers and, following Descartes, regard them as any length of a line i.e. the continuous quantities representing the distance from the origin along one of the Cartesian axes. This reflects one of the ancient definitions of a line as “an infinite number of points”, so that ‘real’ numbers can be thought as the points on an infinitely long line called the ‘number line’ or ‘real’ line, with the special points corresponding to integers (the counting numbers) equally spaced, with the unit length at point “1”. The real axis is usually regarded as the horizontal axis, while the vertical axis is called the “imaginary” axis, together defining the complex plane. Complex (or imaginary) numbers are all the points (off the real axis) in this ‘complex’ plane. Extending these ideas to three dimensions needed the invention of quaternions (using 4 numbers) by the Irish genius, W. R. Hamilton in 1843. [See my technical paper UET1.] Descartes learned about the decimal representation from one of his teachers in Holland, Simon Stevin, who insisted that there was no difference between rational and irrational numbers. Descartes pulled off a major rhetorical coup by calling these “real” numbers, implying this was Nature’s natural number scheme. Indeed, all of these ideas have been absorbed into the foundations of mathematical physics, where they are used to represent and measure all continuous quantities, even though none of the numerous points drawn on a physical line can be distinguished, one from another. Descartes also redefined words to make his case stronger, so that he defined ‘dimension’ as any object that could divided into several identical parts (like space or time) so that he could apply this definition to the ideas of weight (heaviness) and speed (motion). His training as a lawyer had not all been wasted. 3
Critique The key mathematical mistake made by Descartes was to forget that geometry is scale-less; it’s theorems work, no matter how large are the figures considered: within one page or a sports field. This is because geometry is investigating the definitions of a few simple figures or, effectively, relationships between a small finite number of points, such as the three corners of a triangle or the center of a circle and a few of the points around the circumference. As such, there is no real ‘object’ as a unit length; these are purely arbitrary (such as one inch, one centimeter or one mile). Descartes vitally needed in his mapping from geometry to numbers that there be a critical unit length, as the following example of multiplying two line lengths together in Geometrie [p.370], where he tries to multiply AE by AC claiming AB=AE*AC B E Since AED & ABC are similar triangles, then ratios same. ∴ AB : AC = AE : AD ∴ AB = AE * AC / AD. But, desired result is only true if AD = 1. A C D This is similar to the error in the Pythagorean ‘proof’ that √2 cannot be a rational fraction. The mistake is thinking that lengths can be mapped to countable numbers, so that 1.0 has no reality, while 1 does. As the Ancient Greeks knew, arithmetic is objective: everyone can count the same number of pebbles as we all share similar senses, so that if the objects retain unchanging properties we can all agree on their individual existence (subjectivity is confirmed collectively in ‘objective’ knowledge). In fact, even geometry relied on intuitive physical features when each of us agreed on approximately real figures (such as triangles) being “close enough” to idealized (perfect) ‘objects’. Only Platonists insisted there had to exist idealized ‘forms’ for us to imagine similar ones in our minds. Interestingly, even our computers can only work with countable numbers (integers), anything else is always an approximation. Unfortunately, too many mathematicians are Crypto-Platonists, who believe that the mathematical objects they study have more reality than themselves; they confuse themselves by using the standard words of normal life, such as ‘exist’, ‘prove’ and (since Descartes:) ‘real’, to confuse ordinary folk. This is why each time mathematicians tried to extend their basic concepts, such as number, into new areas that did not match common reality, like negative or imaginary numbers, they were strongly resisted. With their control of the academic game across western society, their inventions have steamrollered along to the point where most high-school graduates are convinced that all mathematics is real. Scientists too have been reluctant to give up their continuous (fluid) models of reality, with its implicit inclusion of the powerful, religious concept of infinity, so they have been delighted to use Descartes’ concept of real numbers to link their own mathematical theories to their experiments involving measurements that also need this concept. Now, when their theories match their experimental measurements in the fourteenth decimal digit they are more firmly convinced than ever that their whole scientific venture is on the right road and not just some intellectual game being played in their heads. There are only 3 dimensions in physics: mass, length and time. We need real examples, like electrons, to be the unit exemplars; so that all mass is an arithmetic (integer) number of electron masses, unit time is the ‘ticking’ (chronon) of the rate of electron interactions and unit length is the distance between the closest interaction of two electrons (luxon). This implies that conventional units are related to the REAL units as follows: 1 gram = 1027 el.mass, 1 second = 1016 chronons, 1 cm = 328140 luxons. This scheme would ground physics in countable number ratios of real objects; thus: a REAL physics.
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CONCLUSIONS It is not a coincidence that Descartes restricted his new co-ordinate scheme to just two dimensions (a flat plane like a sheet of paper). This too had been the preferred approach to classical geometry as its proofs made its implicit appeal to the powerful human sense of vision that seems to ‘see’ everything at a glance – this has the benefit of not needing time (or memory) in the process. Descartes deliberately also emphasized the visual sense, in his research and in his theory of human senses (Dioptrics); this deliberate focus linked optics, psychology and brain physiology, giving him a theory of knowledge and a seemingly unchallengeable (until the late 20th century) theory of perception. This returns to Euclid’s trick of appealing to humans’ intuition for primitive shapes (point, line, circle) to ‘justify’ his axioms when his whole theory (“Elements”) derived from human graphic tools (ruler, compass), not logic. The heart of Descartes’ metaphysics (the ultimate foundations of reality) was his proposal that the very essence of a real body was its extension in space i.e. its observable size. In order to preserve this critical idea, he had to transform line lengths into “universal numbers”, so that he could justify his inventive linkage between timeless spatial analysis (geometry) and symbolic arithmetic (algebra); here he needed to smuggle in his hidden unit length and introduce his invention of infinite (hence mystical) decimal representation that he rhetorically called “real numbers”. This was the important step needed by many mathematicians and scientists to magnify their efforts exponentially (joke!). Accordingly, they will both defend this concept to the death (of any challenger), as theologians will defend the idea of “soul”. It is truly ironic that the six major concepts of this most famous philosopher have failed to materialize: a) “Size”: Observable objects are incapable of explaining the natural world (classical mechanics). b) “Corpuscles”: The ultimate material might well be electrons that may just be point particles. c) “Geometry”: Continuous models of the world (‘Strings’) are still needing real explanations. d) “Real Numbers”: We now know the world to be finite with discrete values (quantum theory). e) “Units”: Nature’s units (e, h, c or ‘alpha’) are not arbitrary and not visible. f) “God”: Theology has been abolished from Natural Science, (scientists become the High-Priests). At the end of the day, Descartes had to rely on his doubtful dualism to justify all his innovations so that he was reduced to falling back on the types of arguments and concepts of his scholastic and religious predecessors (like St. Thomas Aquinas and St. Augustine) that he had tried to supersede. Once again (see Newton), it is not surprising that a genius like Descartes failed to get beyond the verbal games of his generation. Religious and scholastic intellectuals had been locked in these word games for hundreds of years and human cultures are as much subject to inertial trajectories (difficult to change) as the heaviest of planets. Our civilization has exhausted simple ideas and we are now facing complexity everywhere with few guidelines to help make progress.
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