- #1
Paulibus
- 203
- 11
Let me start by writing about the natural or counting numbers. The story of how, where and when we invented them is lost in the misty dawn of history. But perhaps our African ancestors, like our living simian cousins (and some other animals) evolved the ability to distinguish between few and many; in our case with articulated labels that gradually developed in parallel with language. Or perhaps numbers were invented in “... the age of stone when men stood up on their hind legs and began to count time by the sun”, as Lew Archer put it in Ross MacDonald’s story "The Moving Target".
We really don’t know how all this happened, but the world-wide language of mathematics that is based on abstract numbers (algebra?) and perceived shapes (geometry?) is still evolving. Viva new inventions still to be made!
Only a few millennia ago, in the Middle East, the counting numbers: 1,2,3... etc., were up and running, as it were, serving the practical purpose of quantifying resources, such as territory, livestock and crops. But the counting numbers proved inadequate to satisfy our hard-wired-by-evolution need to talk about and describe everything we see or sense. Once resources have been quantified, they can be exchanged, lent and borrowed; practices that are lubricated by that abstract medium of exchange; money. Money has many representations; some quite real (hard or spot cash) and some not quite so real (binary 0's and 1's, easily exchanged but dangerously ephemeral).
By classical times (I mean times that span the hinge between B.C. and A.D.) there already existed a venerable dichotomy between negative and positive numbers, perhaps dating also from the “age of stone”. I suspect that the invention of money and its social world of debits and credits, together with the ancient practice of stock theft, may have enhanced the need for the concept of negative numbers. Numbers that quantify debt are conveniently labelled negative. They can then be added to and algebraically cancel positive credit, so quantifying “flat broke” ith another invented number, zero. Margaret Atwood in her book "Payback", has argued persuasively that the ancient dichotomy of debtors and creditors reflects our innate sense of justice and fairness, hard-wired into us by Evolution.
Also in Classical times, the gaps between the stepped sequences of counting numbers, both positive and negative, separated by the number zero, had been quantified and partly filled by fractions. Perhaps fractions were devised to quantify fair sharing of resources, like apple pie. The filling of gaps was completed when it was reluctantly recognised that there were numbers (like ∏) that weren’t fractions (a.k.a. rational numbers; the ratio of counting numbers). Such numbers, like ∏, are written with an endless, never-repeating sequence of digits, and are known as irrational numbers. Finally the continuous set of all numbers, negative, zero and positive, rational and irrational, including those used for counting (a.k.a. whole numbers) are the abstract concept on which much of mathematics (and hence physics, our quantified and predictive description of reality) is based. Mathematicians call this set of numbers simply “the Reals”; in the 17th century the mathematician John Wallis introduced the idea of an infinitely long number line as a useful geometrical analogue of the Reals. This concept still seems to be in use.
Now consider how numbers can be used to quantify space or how we decorate space with imagined numerical position labels. As a start think first of a distant object in front of you, say a tree near your horizon. The natural but primitive way to quantify its distance from you might be to estimate how many steps it would take you to walk to it. For this purpose counting numbers suffice. You could construct a table of locations for such objects by associating with each its estimated count of steps and its bearing from some convenient direction, for instance North. Such a table would be a primitive set of polar coordinates R (say with the number of steps, ranging from 0 to about 10,000) and Φ (bearing, ranging from 0 to 2 ∏). I suggest that
such a simple way of quantifying the two dimensions of space on an (assumed flat) Earth may have persisted from “the age of stone” until 1657, when the philosopher Réne Descartes put two and two together and made 3 (dimensions); replaced by 4 in the Annus Mirabilis of 1905.
Descartes counted distances along oppositely directed lines with oppositely signed numbers, negative and positive, from an origin denoted zero. He did this along three mutually perpendicular lines that he called axes. Points in three dimensions thus acquired three numeric labels, now called the Cartesian coordinates of that point. Ever since this Cartesian system of coordinates was invented some 350 years ago, its adoption has greatly simplified calculations in physics, mainly because its axes are mutually orthogonal.
But there is no such thing as a free lunch, and Cartesian coordinates that use negative numbers required a new algebraic invention to fully conform with long-established algebraic practice. In particular the algebraic operation of taking the square root of a number creates a problem with negative numbers. The problem was solved by the Gordian knot method: the label “imaginary” (denoted by i) is substituted for the square root of minus one, and the set of real numbers is replaced by a set of “complex” numbers for purposes of calculation. Complex numbers are the sum of two entities; a real number and a real number multiplied by i. This arcane construct, serving as “payment for lunch” --- lunch being algebraic consistency --- has
enriched mathematics and physics in a way that Roger Penrose, in his tome Road to Reality (A complete Guide to the Laws of the Universe) describes as "magical".
The introduction of complex numbers seems to me to have deep connections with one of the two symmetries of space, namely that of rotation symmetry. (The other is translation symmetry, whether it be in space or in time). The workings of the universe can be described in the same way by all observers who perceive it from perspectives distinguished only by their orientation with respect to any inertial (freely falling) reference frame. For instance i may be regarded as an operator active in a plane perpendicular to any potential rotation axis, in which distance along an axis (the real axis, say x) is proportional to a positive real number. (The number line) Suppose this plane is regarded as the complex plane x + iy (often called an Argand diagram) --- originally by Casper Wessel who first invented this representation of complex numbers in 1797 (Jean Argand also did in 1806). Then multiplication of z = x + iy twice by i rotates the real axis by ∏ and transforms the positive real axis into its collinear negative self, completing the Cartesian form of this axis.
Was it the introduction of Cartesian coordinates that provided the dominant motive behind the introduction of complex numbers? or was it the fascination mathematicians once had with the use of formulae to solve equations (e.g. quadratic equations) that popularised i? This seems to be the view taken in Mario Livio's book "The equation that couldn't be solved"?
Can somone please unravel my muddled thinking?
We really don’t know how all this happened, but the world-wide language of mathematics that is based on abstract numbers (algebra?) and perceived shapes (geometry?) is still evolving. Viva new inventions still to be made!
Only a few millennia ago, in the Middle East, the counting numbers: 1,2,3... etc., were up and running, as it were, serving the practical purpose of quantifying resources, such as territory, livestock and crops. But the counting numbers proved inadequate to satisfy our hard-wired-by-evolution need to talk about and describe everything we see or sense. Once resources have been quantified, they can be exchanged, lent and borrowed; practices that are lubricated by that abstract medium of exchange; money. Money has many representations; some quite real (hard or spot cash) and some not quite so real (binary 0's and 1's, easily exchanged but dangerously ephemeral).
By classical times (I mean times that span the hinge between B.C. and A.D.) there already existed a venerable dichotomy between negative and positive numbers, perhaps dating also from the “age of stone”. I suspect that the invention of money and its social world of debits and credits, together with the ancient practice of stock theft, may have enhanced the need for the concept of negative numbers. Numbers that quantify debt are conveniently labelled negative. They can then be added to and algebraically cancel positive credit, so quantifying “flat broke” ith another invented number, zero. Margaret Atwood in her book "Payback", has argued persuasively that the ancient dichotomy of debtors and creditors reflects our innate sense of justice and fairness, hard-wired into us by Evolution.
Also in Classical times, the gaps between the stepped sequences of counting numbers, both positive and negative, separated by the number zero, had been quantified and partly filled by fractions. Perhaps fractions were devised to quantify fair sharing of resources, like apple pie. The filling of gaps was completed when it was reluctantly recognised that there were numbers (like ∏) that weren’t fractions (a.k.a. rational numbers; the ratio of counting numbers). Such numbers, like ∏, are written with an endless, never-repeating sequence of digits, and are known as irrational numbers. Finally the continuous set of all numbers, negative, zero and positive, rational and irrational, including those used for counting (a.k.a. whole numbers) are the abstract concept on which much of mathematics (and hence physics, our quantified and predictive description of reality) is based. Mathematicians call this set of numbers simply “the Reals”; in the 17th century the mathematician John Wallis introduced the idea of an infinitely long number line as a useful geometrical analogue of the Reals. This concept still seems to be in use.
Now consider how numbers can be used to quantify space or how we decorate space with imagined numerical position labels. As a start think first of a distant object in front of you, say a tree near your horizon. The natural but primitive way to quantify its distance from you might be to estimate how many steps it would take you to walk to it. For this purpose counting numbers suffice. You could construct a table of locations for such objects by associating with each its estimated count of steps and its bearing from some convenient direction, for instance North. Such a table would be a primitive set of polar coordinates R (say with the number of steps, ranging from 0 to about 10,000) and Φ (bearing, ranging from 0 to 2 ∏). I suggest that
such a simple way of quantifying the two dimensions of space on an (assumed flat) Earth may have persisted from “the age of stone” until 1657, when the philosopher Réne Descartes put two and two together and made 3 (dimensions); replaced by 4 in the Annus Mirabilis of 1905.
Descartes counted distances along oppositely directed lines with oppositely signed numbers, negative and positive, from an origin denoted zero. He did this along three mutually perpendicular lines that he called axes. Points in three dimensions thus acquired three numeric labels, now called the Cartesian coordinates of that point. Ever since this Cartesian system of coordinates was invented some 350 years ago, its adoption has greatly simplified calculations in physics, mainly because its axes are mutually orthogonal.
But there is no such thing as a free lunch, and Cartesian coordinates that use negative numbers required a new algebraic invention to fully conform with long-established algebraic practice. In particular the algebraic operation of taking the square root of a number creates a problem with negative numbers. The problem was solved by the Gordian knot method: the label “imaginary” (denoted by i) is substituted for the square root of minus one, and the set of real numbers is replaced by a set of “complex” numbers for purposes of calculation. Complex numbers are the sum of two entities; a real number and a real number multiplied by i. This arcane construct, serving as “payment for lunch” --- lunch being algebraic consistency --- has
enriched mathematics and physics in a way that Roger Penrose, in his tome Road to Reality (A complete Guide to the Laws of the Universe) describes as "magical".
The introduction of complex numbers seems to me to have deep connections with one of the two symmetries of space, namely that of rotation symmetry. (The other is translation symmetry, whether it be in space or in time). The workings of the universe can be described in the same way by all observers who perceive it from perspectives distinguished only by their orientation with respect to any inertial (freely falling) reference frame. For instance i may be regarded as an operator active in a plane perpendicular to any potential rotation axis, in which distance along an axis (the real axis, say x) is proportional to a positive real number. (The number line) Suppose this plane is regarded as the complex plane x + iy (often called an Argand diagram) --- originally by Casper Wessel who first invented this representation of complex numbers in 1797 (Jean Argand also did in 1806). Then multiplication of z = x + iy twice by i rotates the real axis by ∏ and transforms the positive real axis into its collinear negative self, completing the Cartesian form of this axis.
Was it the introduction of Cartesian coordinates that provided the dominant motive behind the introduction of complex numbers? or was it the fascination mathematicians once had with the use of formulae to solve equations (e.g. quadratic equations) that popularised i? This seems to be the view taken in Mario Livio's book "The equation that couldn't be solved"?
Can somone please unravel my muddled thinking?
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