Here is my version of the history of mathematics.

Invention in early civilisation of summerian of early base counting system or number representation, early arithmetic operation, and land measuring systems with ropes and rods for standard unit length. Early version of pythagore theorem and early algebriaic problems expressed in prose. This knowledges is an engineering type of knowledge. A set of effective procedures but these procedures are not systematically organized into whole.

The first pre-socratic philosopher such Thales and pythagore have learned the early counting techniques and measurement techniques from their commercial patners , the Egyptians, and gradually invented axiomatic geometry whose primary axioms are the simplification of the complex land measuring technique and use simple logical combinations in such a way that the measurement procedures could be logically built. It is the unification of the implicit logic that exist in human language and cognition and to merge these with spatial concepts. So this provided a language where spatial measurement procedure could be mechanically built with a mechanic called logic. Geometrical language was invented and a new notion of truth, logical truth was created into an ideal mechanical mechanical world. The development of this language climax in the ancient world with Euclid and Archimedes around 300BC. Mathematical modeling was invented and also science in the form of the mental mechanic of Euclid and Archimedes.

Here I will go very fast on the next 2000 years that mostly took place in India and the muslin world. The decimal numerical system was invented simplifying enormously the arithmetic operations. The algebriaic notation using constants and variable allowed to create an algebriaic language of equations. Formal algebra was invented and procedures for equation solutions.

The next major break through was Galileo. He spatialized time. The biggest scientific invention of all time.

Descartes then invented analytical geometry: the merging of geometry with algebra. Geometry and the axiomatic method became subsumed under an algebraic Euclidean geometry.

This is the creation of the algebrian space time. This is the creation of the equivalence between a form and an equation. Modern science started there as the project of geometrizing the whole world.

What was missing from the Cartesian scientific method was the mechanics of Archimedes for calculating surface, volume and length in this new framework and Leibniz and Newton invented it in the forms of Calculus. Dynamic in the form of differential equations was invented and here is completed the modern mental framework of modern science. The consequence of the Newtonian paradigm will be work out in the next two hundred years.

Mechanics will be reformulated in terms of Hamiltonian mechanics in terms of trajectories of minimum action in configuration space along the initial conceptions of Leibniz, Fermat, Cauchy, Laplace.

Most of the mathematics of Engineering have been invented in this fertile period by all the names which we are familiar: Fourier, Dalembert, etc etc

The greatest innovation of this period was done by Galois. His idea will give rise to the concept of group which will become a privilege frameword structuring the whole of algebraic geometry. Physics and mathatics will be restructured based on group. The group theoretical way to express algebriaic geometry allowed it to distance itself from Euclidean geometry, in fact to realized that there exist all kind of geometry that can be invented in this language. Non-Euclidean geometries were invented. The Erlander program of Felix Klein allowed to classified all the new geometry into a systematic framework. Then the revolutionary epistemology of Kant could take new forms where the apriori of the physical science could be one of these new geometry. This has lead Gauss and Rieman , Poincarre and Lorentz and Einstein towards the new physics of GR. This has lead the creation of the new quantum physics on Hilbert Space.

Math's origins have commonly been linked to the study of our immediate planetary cycles. For example Chinese an\d Mayan astronomers used modular arithmetic that modeled movements of our moon, earth, Mars, Venus, Mercury and other planets. Floyd Lounsbury, Yale, wrote up aspects of Mayan math and astronomy as included in this 1997 master's thesis http://www.scribd.com/doc/34812625/A-New-View-on-Maya-Astronomy-by-Christopher-Powell .

To answer the question, you may find it helpful to consider a topic (e.g., set theory or

geometry) and check the beginnings of views of the topic in different countries. Two obvious choices are Greece (geometry) and India (set theory). In Greece, you might start by considering how Euclid approached geometry. In India, you might want to consider how India mathematicians viewed sets. Here is a quote from the Wiki article on infinity that you might find interesting:

In the Indian work on the theory of sets, two basic types of infinite numbers are distinguished. On both physical and ontological grounds, a distinction was made between asaṃkhyāta ("countless, innumerable") and ananta ("endless, unlimited"), between rigidly bounded and loosely bounded infinities.

According to the article below their is more than one answer to the origin to mathematics:

What are The Origins of Math?

http://www.kavlifoundation.org/science-spotlights/kavli-origins-of-math#Origins_of_Math

I hold the view that Nature is fundamentally mathematical and is giving us hints that the universe is mathematical. See the question :

What is the relationship between Mathematics,Science and Nature?

https://www.researchgate.net/post/What_is_the_relationship_between_Mathematics_Science_and_Nature

Max Tegmark takes this to the extreme by proposing that our entire physical reality isn’t just described by math, but that it is a mathematical structure, having no properties besides mathematical properties.

James' points are valid for linear math, the 'arrow of time' math that Greeks used for arithmetic, algebra and geometry. However the oldest math seems to have been modular in clock arithmetic as Chinese and Mayan astronomers formalized modeling our immediate universe. A Mayan math and astronomy study attempts to parse pages 65to 69 of the Dresden Codex. The season table http://planetmath.org/mayanseasonalalmanac offers three levels of data that expose Mayan thinking in subtle ways.

Maths originated when human stated counting

Milo Gardner: James' points are valid for linear math...

Apart from the cardinality of sets, perhaps you will agree set theory is not linear. Consider, for example, set theoretic topology, where a subset A in X is dense, provided the closure of A equals X. If we agree that set theory has its origin in ancient India, then what we have is a non-linear view of sets.

@Issam Sinjab: from the observation by Simon Hellerman in your first link, this still leaves open the question "What is mathematics" posed in

https://www.researchgate.net/post/What_is_mathematics#shares

Correction: see quote from SIMEON HELLERMAN at

http://www.kavlifoundation.org/science-spotlights/kavli-origins-of-math#Origins_of_Math

the history of science exposes several origins of math: http://en.wikipedia.org/wiki/History_of_science ... calendars and astronomy may be the oldest sciences.

This question and responses piqued my curiosity to google prehistoric mathematics, so confident I am in maths' priority, and find at http://www.storyofmathematics.com/prehistoric.html: "The Ishango bone, a tally stick from central Africa, dates from about 20,000 years ago"

James

Though I don't expect you to read all my answers, I have actually asked the question "what is mathematics?" much earlier back in Feb 2013.

In the thread:

https://www.researchgate.net/post/What_is_the_relationship_between_Mathematics_Science_and_Nature

I asked and answered the question "what is mathematics?" on Feb12 2013. This is what I said:

What is mathematics? It was only within the last thirty years or so that a definition of mathematics emerged on which most mathematicians now agree: mathematics is the science of patterns. What the mathematician does is examine abstract patterns-numerical patterns, patterns of shape, patterns of motion, patterns of behavior, voting patterns in a population, patterns of repeating chance events, and so on. These patterns can arise from the world around us or from the depths of space and time.

Feb 12, 2013

And the next day, on Feb 13, I then gave a more detailed answer:

Earlier, I have asked the question "What is mathematics?" I have answered that mathematics is the science of patterns. Different kinds of patterns give rise to different branches of mathematics. For example:

-Arithmetic and number theory study patterns of numbers and counting.

-Geometry studies patterns of shape.

-Calculus allows us to handle patterns of motion.

-Logic studies patterns of reasoning.

Probability theory deals with paterns of chance.

Topology studies patterns of closeness and position.

In his 1940 book 'A Mathematician's Apology', the accomplished English mathematician G. H. Hardy wrote:

"The mathematician's patterns, like the painter's or the poet's, must be beautiful, the ideas, like the colors or the words, must fit together in a harmonious way. Beauty is the first test; there is no permanent place in the world for ugly mathematics...It may be very hard to define mathematical beauty, but that is just as true of beauty of any kind-we may not know quite what we mean by a beautiful poem, but that does not prevent us from recognizing one when we read it."

The beauty to which Hardy was referring to is a beauty that can be observed , and appreciated, only by those sufficiently well trained in the discipline. It is a beauty "cold and austre," according to Bertrand Russel, the famous English mathematician and philosopher, who wrote, in his 1918 book Mysticism and Logic:

"Mathematics, rightly viewed, possesses not only truth, but supreme beauty-a beauty cold and austre, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of stern perfection such as only the greatest art can show."

Mathematics' greatest success has undoubtedly been in the physical domain, where the subject is rightly referred to as both the queen of the (natural) sciences by Carl Friedrich Gauss (see my introduction to the main question of this thread) and as the queen and servant of the (natural) sciences( As noted by Eric Bell, a Scottish American mathematician).

In an age when the study of the heavens dominated scientific thought, Galileo said:

"The great book of nature can be read only by those who know the language it was written. And this language is mathematics."

In a much later era and striking a similar note, when the study of the inner workings of the atom had occupied the minds of many scientists for a generation, the Nobel Prizewinner Richard Feynman had this to say about mathematics:

"To those who do not know mathematics it is difficult to get across a real feeling as to the beauty, the deepest beauty, of nature ... If you want to learn about nature, to appreciate nature, it is necessary to understand the language that she speaks in".

Modified Feb 13, 2013 by the author

Ethnomathematics is the study of the relationship between mathematics and culture.

http://en.wikipedia.org/wiki/Ethnomathematics

@Issam Sinjab: ...I have actually asked the question "what is mathematics?" much earlier back in Feb 2013.

Many thanks for pointing this out. Mea culpa.

You ask great questions. The question about the nature of mathematics surfaced when I saw that we have been talking about the relation between mathematics, science and nature, but with the understanding that we know the answer to the question What is mathematics? I think you will agree that there other perspectives (besides your very good perspective) that can be considered in answering the question. That is, there are aspects of mathematics itself not address in your answer.

Totally agree James. This is the type of question where there is no one single answer.

According to Aristotle when we come to know something, the mind (nous) becomes one with the object of thought, mind and object are informed by the same eidos. Being informed by the same eidos, the mind participates in the being of the known object, rather than simply depicting it. So the forms in the objects become active in our mind.

The forms comes alive in a certain manner in Aristotle philosophy.

Louis,

Aristotle, a student of Plato, was mainly a biologist with many other skills. Aristotle's view of forms provides a basis for a Platonist's view of mathematics. In which case, a mathematician discover's mathematics via forms. Is that what you intened?

James,

In my theory of vision and imagination, the structures (forms) of the image world are actualized within a schemata tree withing the visual system; this actualization is similar to the process of actualization of the object surface during ontogeny as in Aristotle's model of perception. If you see a frog, there is an schemata tree actualization of the frog that is similar to the surface of the frog formation taking place in the visual system; So aristotle who took the ontogenic development in all life forms as similar to the perceptual understanding taking place in our mind.

Louis,

Have you formulated axioms that provide conditions (bases) for this view of perceptual understanding?

James, your question goes to the heart of numeration systems, the context in which arithmetic, algebra and geometric conditions were first stated. Pre-Columbian Californians used individual and combinations of base 2, 4, 5, 8, 10, 12, 16 and 20. There were three groups that included base 20 because trading of obsidian and turquoise took place with Mesoamerica (the California numeration systems were decoded by .A L. Kroeber in 1923)

Milo,

Very interesting. Please suggest an article where numeration systems in Pre-Columbian California are discussed.

James, a free google ebook version of "Handbook of California Indians" by Kroeber

http://books.google.com/books/about/Handbook_of_the_Indians_of_California.html?id=xW8cn5mdu6wC

covers cosmology, where base 4 was spiritual to all northern American and central American native cultures , and other related topics.

Milo,

Many thanks for the reference.

This just out on geometric origins from today's Harvard Gazette: http://news.harvard.edu/gazette/story/2013/09/a-higher-plane/?utm_source=SilverpopMailing&utm_medium=email&utm_campaign=09.24.daily%2520%281%29

So intrinsically mathematical origins are geometric, the physical, abstracted for communication to the symbolic, always essentially answering the fundamental question of quantity, or how much (including how far, or distance, etc.) -- knowledge needed for life to survive, thrive, and evolve.

@Cj Nev: here is a paragram from the Harvard Gazette article:

Researchers in earlier studies were able to show that animals possess two basic geometric abilities: the ability to use distance and directional information to navigate their world, and the ability to use angle and length information to recognize shapes.

Both distance and directional information as well as angle and length information have underlying sets of points that are part of a perceived pattern recognized at an early age.

And not only is there the perception of how far but also how close. From that, one can conjecture topology (nearness of points and sets) and proximity space theory (nearness and remoteness of sets) have their origin in such perceived patterns in nature.

Hi James Peters, yes I just had a chance to read the article, expecting more from it such as you have added, extending lengths to topology, nearness of points, essentially in my mind even to CURVED or warped (angular) spacetime perhaps. I think the next abstraction (anchored in reality), my research area, is to ask HOW (and why) this spacetime topology is warped (gravitational patterns) everywhere continuously from points (or Planck length) to the cosmos.

Among other formulations, I suspect or intuit the shape or form of space involves an underlying logarithmic pattern. I also perceive the natural log, pi, Fibonacci factorials, continued fractions, and pi, etc., may ultimately converge self-similarly (fractals-aflack!) perhaps and multidimensionally, at the rate of phi, to phi, the golden ratio, everywhere, since like pi and other constants, no matter the length of anything, the ratio of the whole segment to a larger part of that segment ALWAYS equals the ratio of the larger segment to the smaller part of that same segment, lending a certain reliable predictability to our observations or measurements.

Interestingly and in a self-referencing (mirroring, reflecting, symmetric, fractal-aflack!) manner, the eye itself is also, it shouldn't be surprising (awesome but not surprising), also structured, i.e., 'evolved,' according to phi, e.g., the cones, etc., or in the sense of hearing, the cochlea, both spiraling anatomies of and from various focal 'point' lengths through to the cosmos, again in my current research I suspect all at the rates of phi, including self-referentially (fractal-aflack!) phi (or phi^2) itself, multidimensionally everywhere.

Some scientists consider that the mathematics occurred from hens because hens are able to consider to three, and some scientists consider that the mathematics occurred from ants because ants are able to consider to sixty. And I consider that the mathematics occurred from a binomial. Here my monograph on this subject: A.Yurkin. New binomial and new view on light theory. About one new universal descriptive geometric model//Lambert Academic Publishing, 2013. ISBN 978-3-659-38404-2

The earliest records of mathematics show it arising in response to practical needs in agriculture, business, and industry. In Egypt and Mesopotamia, where evidence dates from the 2d and 3d millennia B.C., it was used for surveying and mensuration; estimates of the value of π (pi) are found in both locations. There is some evidence of similar developments in India and China during this same period, but few records have survived. This early mathematics is generally empirical, arrived at by trial and error as the best available means for obtaining results, with no proofs given. However, it is now known that the Babylonians were aware of the necessity of proofs prior to the Greeks, who had been presumed the originators of this important step.

Greek Contributions

A profound change occurred in the nature and approach to mathematics with the contributions of the Greeks. The earlier (Hellenic) period is represented by Thales (6th cent. B.C.), Pythagoras, Plato, and Aristotle, and by the schools associated with them. The Pythagorean theorem, known earlier in Mesopotamia, was discovered by the Greeks during this period.

During the Golden Age (5th cent. B.C.), Hippocrates of Chios made the beginnings of an axiomatic approach to geometry and Zeno of Elea proposed his famous paradoxes concerning the infinite and the infinitesimal, raising questions about the nature of and relationships among points, lines, and numbers. The discovery through geometry of irrational numbers, such as 2, also dates from this period. Eudoxus of Cnidus (4th cent. B.C.) resolved certain of the problems by proposing alternative methods to those involving infinitesimals; he is known for his work on geometric proportions and for his exhaustion theory for determining areas and volumes.

The later (Hellenistic) period of Greek science is associated with the school of Alexandria. The greatest work of Greek mathematics, Euclid's Elements (c.300 B.C.), appeared at the beginning of this period. Elementary geometry as taught in high school is still largely based on Euclid's presentation, which has served as a model for deductive systems in other parts of mathematics and in other sciences. In this method primitive terms, such as point and line, are first defined, then certain axioms and postulates relating to them and seeming to follow directly from them are stated without proof; a number of statements are then derived by deduction from the definitions, axioms, and postulates. Euclid also contributed to the development of arithmetic and presented a geometric theory of quadratic equations.

In the 3d cent. B.C., Archimedes, in addition to his work in mechanics, made an estimate of π and used the exhaustion theory of Eudoxus to obtain results that foreshadowed those much later of the integral calculus, and Apollonius of Perga named the conic sections and gave the first theory for them. A second Alexandrian school of the Roman period included contributions by Menelaus (c.A.D. 100, spherical triangles), Heron of Alexandria (geometry), Ptolemy (A.D. 150, astronomy, geometry, cartography), Pappus (3d cent., geometry), and Diophantus (3d cent., arithmetic).

Chinese and Middle Eastern Advances

Following the decline of learning in the West after the 3d cent., the development of mathematics continued in the East. In China, Tsu Ch'ung-Chih estimated π by inscribed and circumscribed polygons, as Archimedes had done, and in India the numerals now used throughout the civilized world were invented and contributions to geometry were made by Aryabhata and Brahmagupta (5th and 6th cent. A.D.). The Arabs were responsible for preserving the work of the Greeks, which they translated, commented upon, and augmented. In Baghdad, Al-Khowarizmi (9th cent.) wrote an important work on algebra and introduced the Hindu numerals for the first time to the West, and Al-Battaniworked on trigonometry. In Egypt, Ibn al-Haytham was concerned with the solids of revolution and geometrical optics. The Persian poet Omar Khayyam wrote on algebra.

Western Developments from the Twelfth to Eighteenth Centuries

Word of the Chinese and Middle Eastern works began to reach the West in the 12th and 13th cent. One of the first important European mathematicians was Leonardo da Pisa (Leonardo Fibonacci), who wrote on arithmetic and algebra ( Liber abaci, 1202) and on geometry ( Practica geometriae, 1220). With the Renaissance came a great revival of interest in learning, and the invention of printing made many of the earlier books widely available. By the end of the 16th cent. advances had been made in algebra by Niccolò Tartaglia and Geronimo Cardano, in trigonometry by François Viète, and in such areas of applied mathematics as mapmaking by Mercator and others.

The 17th cent., however, saw the greatest revolution in mathematics, as the scientific revolution spread to all fields. Decimal fractions were invented by Simon Stevin and logarithms by John Napier and Henry Briggs; the beginnings of projective geometry were made by Gérard Desargues and Blaise Pascal; number theory was greatly extended by Pierre de Fermat; and the theory of probability was founded by Pascal, Fermat, and others. In the application of mathematics to mechanics and astronomy, Galileo and Johannes Kepler made fundamental contributions.

The greatest mathematical advances of the 17th cent., however, were the invention of analytic geometry by René Descartes and that of the calculus by Isaac Newton and, independently, by G. W. Leibniz. Descartes's invention (anticipated by Fermat, whose work was not published until later) made possible the expression of geometric problems in algebraic form and vice versa. It was indispensable in creating the calculus, which built upon and superseded earlier special methods for finding areas, volumes, and tangents to curves, developed by F. B. Cavalieri, Fermat, and others. The calculus is probably the greatest tool ever invented for the mathematical formulation and solution of physical problems.

The history of mathematics in the 18th cent. is dominated by the development of the methods of the calculus and their application to such problems, both terrestrial and celestial, with leading roles being played by the Bernoulli family (especially Jakob, Johann, and Daniel), Leonhard Euler, Guillaume de L'Hôpital, and J. L. Lagrange. Important advances in geometry began toward the end of the century with the work of Gaspard Monge in descriptive geometry and in differential geometry and continued through his influence on others, e.g., his pupil J. V. Poncelet, who founded projective geometry (1822).

In the Nineteenth Century

The modern period of mathematics dates from the beginning of the 19th cent., and its dominant figure is C. F. Gauss. In the area of geometry Gauss made fundamental contributions to differential geometry, did much to found what was first called analysis situs but is now called topology, and anticipated (although he did not publish his results) the great breakthrough of non-Euclidean geometry. This breakthrough was made by N. I. Lobachevsky (1826) and independently by János Bolyai (1832), the son of a close friend of Gauss, whom each proceeded by establishing the independence of Euclid's fifth (parallel) postulate and showing that a different, self-consistent geometry could be derived by substituting another postulate in its place. Still another non-Euclidean geometry was invented by Bernhard Riemann (1854), whose work also laid the foundations for the modern tensor calculus description of space, so important in the general theory of relativity.

In the area of arithmetic, number theory, and algebra, Gauss again led the way. He established the modern theory of numbers, gave the first clear exposition of complex numbers, and investigated the functions of complex variables. The concept of number was further extended by W. R. Hamilton, whose theory of quaternions (1843) provided the first example of a noncommutative algebra (i.e., one in which ab ≠ ba). This work was generalized the following year by H. G. Grassmann, who showed that several different consistent algebras may be derived by choosing different sets of axioms governing the operations on the elements of the algebra.

These developments continued with the group theory of M. S. Lie in the late 19th cent. and reached full expression in the wide scope of modern abstract algebra. Number theory received significant contributions in the latter half of the 19th cent. through the work of Georg Cantor, J. W. R. Dedekind, and K. W. Weierstrass. Still another influence of Gauss was his insistence on rigorous proof in all areas of mathematics. In analysis this close examination of the foundations of the calculus resulted in A. L. Cauchy's theory of limits (1821), which in turn yielded new and clearer definitions of continuity, the derivative, and the definite integral. A further important step toward rigor was taken by Weierstrass, who raised new questions about these concepts and showed that ultimately the foundations of analysis rest on the properties of the real number system.

In the Twentieth Century

In the 20th cent. the trend has been toward increasing generalization and abstraction, with the elements and operations of systems being defined so broadly that their interpretations connect such areas as algebra, geometry, and topology. The key to this approach has been the use of formal axiomatics, in which the notion of axioms as "self-evident truths" has been discarded. Instead the emphasis is on such logical concepts as consistency and completeness. The roots of formal axiomatics lie in the discoveries of alternative systems of geometry and algebra in the 19th cent.; the approach was first systematically undertaken by David Hilbert in his work on the foundations of geometry (1899).

The emphasis on deductive logic inherent in this view of mathematics and the discovery of the interconnections between the various branches of mathematics and their ultimate basis in number theory led to intense activity in the field of mathematical logic after the turn of the century. Rival schools of thought grew up under the leadership of Hilbert, Bertrand Russell and A. N. Whitehead, and L. E. J. Brouwer. Important contributions in the investigation of the logical foundations of mathematics were made by Kurt Gödel and A. Church.

http://www.infoplease.com/encyclopedia/science/mathematics-development-mathematics.html#ixzz2geEcveyY

From 2,050 BCE Egyptian mIddle kingdom scribes like Ahmes extended base 10 arithmetic to finite arithmetic. Scribes measured money, lands and agricultural products grown on land ( grain), the central unit of payments to labor .in hekats.. http://ahmespapyrus.blogspot.com/2009/01/ahmes-papyrus-new-and-old.html

Greeks used the Egyptian unit, fraction system in arithmetic, algebra and geometry adding improved limits of pi per Archimedes, as Plato also described in the "Republic"

Plato did not discuss pi per Archimedes. Plato 's review of Greek math provided strong links to Egyptian unit fraction origins per http://planetmath.org/platosmathematics

Milo,

Although all the basis of the mathematical measurements technique came from Babylonean civilisation and more directly from Egypt, Thales (635-543 BC) of Miletus was the first to whom deduction in mathematics is attributed. There are five geometric propositions for which he wrote deductive proofs, though his proofs have not survived. Pythagoras (582-496 BC) and his school greatly expanded axiomatic geometry and they discovered of incommensurable lengths (irrational numbers).. The full axiomization of geometry took a long time and was completed by Euclid (300BC) creating the base prototype for the axiomatic mathematical modeling of the world, a transformation of the measurement procedures into a logical geometrical modeling language.

Louis, I am glad that you have placed deduction and Euclid as ideal forms of math. My ideal math exactly converted rational numbers to concise unit fraction series in a numeration system used by Egyptians, Greeks,, Arabs, medieval,s as late as Galileo. Galileo, a continuous period over 3,600 years.

For example Archimedes used the numeration systrem in a square root method that generally estimated irrational numbers and pi to 6-decimal places in three steps per: http://planetmath.org/squarerootof3567and29.

From my point of view Babylonian square root and measurements were not as exacting compared to Egyptian methods, points that may be surprising. .

Milo and Louis,

You may want to consider carrying forward by 20 centuries to find the origins of current-day mathematics. I am thinking of

Poincare, theory of the physical continuum, 1895: tolerance space theory

Frechet, thesis on metric spaces, 1906: metrics

Young, The Theory of Sets of Points, 1907: sets independent of Cantor numbers

Efremovic, infinitessimal geometry, 1910

Hausdorff, Set Theory, 1914: topological spaces

Smirnov, Proximity spaces, 1952 (1964): advances in Efremovic's theory of proximity

I mention these works because point to origins of mathematics familiar to us nowadays.

Louis, let me add exacting specifics to Egyptian geometry and weights and measures per http://planetmath.org/egyptiangeometryareascalculatedincubitskhetsandsetats

Babylonian rational number representations rounded off 1/p and 1/pq to the closest 1/60n factorization. For example 1/91 was recorded as 1/90, a small round off error.

Many cuneiform tablets report this class of rounding off.

Egyptians easily factored 2/91 = 2/7(1/13) = (1/4 + 1/28)(1/13) = 1/42 + 1/364

and more concise fraction series. as Ahmes reported by subtraction

2/91 - 1/70 = 1/130

and a scaling factor that reached 2/91 = 1/70 + 1/130

all with zero error., methods that would have been used to exactly converted 1/91,

as outlined above. Want to try using the exact Egyptian methods?

James, I am keenly aware of the math building blocks created from1900 to 1910 period ... http://www-groups.dcs.st-and.ac.uk/~history/Chronology/1900_1910.html ... without set theory the deeper joys of math may not have become life long interests.

All that said , did you and others know that most colleges only required college arithmetic, a form of ' 1801 "Discussions on Arithmetic" for most liberal arts degrees. in 1910? Only after WW I did French, BrITISH and US politicians and education remove German math topics like college arithmetic and bring back Euclid as Classical Greek math metaphors ...math education continues to follow fads every generation or so> .My view is that long term math topics are most worthy of being label as 'original'.

Milo,

Very good as well as interesting and important observations. I remember tutoring concerning Euclid's elements for high school students when I was a graduate student in California. I was amazed at the lucidity and incredible beauty of Euclid's work. So simple and yet so elegant.

The theme of the thread is the history of mathematic. It is a vast subject. In order to focus on what is important in this history I would sudgest to focus on the math that are used at the level of undergraduated course in engineering which include everything we were taught at school from pre-school. I invite everyone to summarize in a short post the history of the most important inventions that has led to this body of knowledge. I will put my post to this effect tomorrow.

Louis,

Your suggestion is excellent! The story probably starts with Plato, who had intuition about geometry, and then Euclid (his Elements) and Archimedes (his circles and his approach to approximation). Archimedes seldom gets the credit he deserves in his approach to approximation.

James,

I sudgest that you go backward from what you find the most important in what you use today and find out when it was invented. Then you tell the story in a chronogical order. This backward method is the method we use unconsciously when we simplify theoretical construct that are complicated and it is the method by which mathematics and all the science are constantly deconstructed and reconstructed in a simpler fashion. In fact my theory of vision and imagination is the application of this principle to images. An illustration of the general principle that: Structural Hiearchy is History. There is another way to come up with the history of mathematics; it is to find the sequence in which all the mathematical concepts were teached to you as clue to when it was invented in history. The principle of recapitulation or the principle of gradual structural graduation.

Louis and James, electrical engineering math offers the highest form of theoretical arithmetic, algebra. geometry and physics. The remaining engineering fields water down modular arithmetic and other math used to design computers by EE's . Hence the math taught to MEs , CEs and other engineering fields are not as shining examples, compared to EE math

Milo,

Perhaps your observation about EE math can be enriched by including functional analysis and mathematical analysis.

Louis,

Your idea is great! Your suggestion about exploring the history of mathematics is in keeping with an approach to history by Miguel de Unamuno in what he called intra-history. Since we understand the present and near-present best (we experienced them), history makes more sense when we start with the present and work backwards to remote eras in the past.

I think, the origin of mathematics go far back as waaaaaaay-pre-Euclid.

It originates from people's need to explain natural phenomena and PREDICT what else could happen by using the human intelligence, so, it can gain an advantage over other species !!! It is a TOOLSET that humans build. Although, there is an origin, there will never be an END, since the natural phenomena we need to explain will never come to an end. As we build the tools, we will keep discovering more natural phenomena. The complex numbers were invented 200 years ago. Can you believe what can be built on top of complex numbers ? Endless. We are ions away from discovering a good portion of the natural phenomena.

Euclid is the first concrete example we know, because of the book ELEMENTS. I wouldn't be surprised if there were "mathematicians" 5000 years before that. May be 10000. When we, humans, were hunter gatherers, I am sure somebody (a mathematician ?) tried to formulate how many fish he can catch if he stood by the water on SUNSET vs. SUNRISE. This is Mathematics, in fact, the starting point of probability theory !!! Why aren't we giving that poor cave-man the credit he deserves ? Possibly, not as much as Claude Shannon :) but, something ! Because, there was no written proof, like EUCLID's Elements.

So, when you ask the question : "What is the origin of mathematics ? " there are two ways to interpret this: "what we think is the origin" ... or, may be, "when did we start caring that, there should be a formal discipline called mathematics" ? since the only thing the poor caveman cared about was staying alive by catching the fish, not the THEORY behind it ! Although, he used the THEORY he built to gain an advantage in his survival process. And, since what he knows in his mind was never disseminated to the others, the progress of mathematics (or any science, for that matter) was exponentially slower ... if not ZERO.

In the beginning , How did a man count his fingers ? Number of Sheep .....

Simon, North and Central Americans spiritually counted in base four. The four directions were linked to practical systems that added bases 5 ,13, 18 and 20 in Central American calendars,; and combinations of bases 2, 5, 6, 8, 10, 12, 16 and 20 in California trading systems.. Numeration systems preceded higher math forms that built astronomical counting systems in China, Mesoamertica and elsewhere in modular arithmetic. My view is the first abstract math was modular arithmetic. Abstract and deductive algebraic and geometric systems came later.

I am just 3 years old in research. I am happy to listen mathematics.

I think mathematics originated when human beings started to look for a quantitative assessment of the world around them; i.e., the emergence of mathematics is closely related to the emergence of the measurements. First humans started making the simplest quantitative assessment - to compare two objects. Then they began counting "countable" (discrete) objects; thus the simple non-positional notations are incurred. Later they started to count uncountable (analog) objects by introducing the concept of reference...

Here is my version of the history of mathematics.

Invention in early civilisation of summerian of early base counting system or number representation, early arithmetic operation, and land measuring systems with ropes and rods for standard unit length. Early version of pythagore theorem and early algebriaic problems expressed in prose. This knowledges is an engineering type of knowledge. A set of effective procedures but these procedures are not systematically organized into whole.

The first pre-socratic philosopher such Thales and pythagore have learned the early counting techniques and measurement techniques from their commercial patners , the Egyptians, and gradually invented axiomatic geometry whose primary axioms are the simplification of the complex land measuring technique and use simple logical combinations in such a way that the measurement procedures could be logically built. It is the unification of the implicit logic that exist in human language and cognition and to merge these with spatial concepts. So this provided a language where spatial measurement procedure could be mechanically built with a mechanic called logic. Geometrical language was invented and a new notion of truth, logical truth was created into an ideal mechanical mechanical world. The development of this language climax in the ancient world with Euclid and Archimedes around 300BC. Mathematical modeling was invented and also science in the form of the mental mechanic of Euclid and Archimedes.

Here I will go very fast on the next 2000 years that mostly took place in India and the muslin world. The decimal numerical system was invented simplifying enormously the arithmetic operations. The algebriaic notation using constants and variable allowed to create an algebriaic language of equations. Formal algebra was invented and procedures for equation solutions.

The next major break through was Galileo. He spatialized time. The biggest scientific invention of all time.

Descartes then invented analytical geometry: the merging of geometry with algebra. Geometry and the axiomatic method became subsumed under an algebraic Euclidean geometry.

This is the creation of the algebrian space time. This is the creation of the equivalence between a form and an equation. Modern science started there as the project of geometrizing the whole world.

What was missing from the Cartesian scientific method was the mechanics of Archimedes for calculating surface, volume and length in this new framework and Leibniz and Newton invented it in the forms of Calculus. Dynamic in the form of differential equations was invented and here is completed the modern mental framework of modern science. The consequence of the Newtonian paradigm will be work out in the next two hundred years.

Mechanics will be reformulated in terms of Hamiltonian mechanics in terms of trajectories of minimum action in configuration space along the initial conceptions of Leibniz, Fermat, Cauchy, Laplace.

Most of the mathematics of Engineering have been invented in this fertile period by all the names which we are familiar: Fourier, Dalembert, etc etc

The greatest innovation of this period was done by Galois. His idea will give rise to the concept of group which will become a privilege frameword structuring the whole of algebraic geometry. Physics and mathatics will be restructured based on group. The group theoretical way to express algebriaic geometry allowed it to distance itself from Euclidean geometry, in fact to realized that there exist all kind of geometry that can be invented in this language. Non-Euclidean geometries were invented. The Erlander program of Felix Klein allowed to classified all the new geometry into a systematic framework. Then the revolutionary epistemology of Kant could take new forms where the apriori of the physical science could be one of these new geometry. This has lead Gauss and Rieman , Poincarre and Lorentz and Einstein towards the new physics of GR. This has lead the creation of the new quantum physics on Hilbert Space.

Louis,

Your summary of the history of mathematics, with a few gaps, is very good. Going back to what you suggested earlier, we can start from what we know best (present topics in mathematics) and work our way backwards. Here are a couple of paths to consider:

nearness of points and sets:

topology viewed as set theory (Naimpally and others, 2012) ->

Bourbacki (1980+) -> ... ->

Hausdorff (1914) ->

Young (1907) ->

Cantor (1890) -> ...

nearness of sets:

proximity space theory viewed as Infinitessimal geometry (Naimpally et al., 2013)-> ... -> Leader (1959+)-> ...->

Smirnov (1952-1959) ->

Cech (1930+) ->

Efremovic (1910+) ->

Reisz (1908) -> ...

Peter,

Could you provide brief description of the math inventions associted with these names.

I should have added Gauss'differential geometry which is the context for the invention of the non-Euclidean geometries.

Milo, this was an awesome summary of the number systems. I was, though, desperately looking for when anyone would talk about ZERO ! It sounds fascinating that, although cultures toyed around with the idea of ZERO, it wasn't really formally introduced into mathematics centuries after these number systems were invented. I don't know who is credited for the ZERO ? I think the Arabic and the Indian world in somehow two parallel efforts ... Back in those days, there was no INTERNET :) so, the diffusion of knowledge wasn't that fast !

In Arabic, zero is sifr, which almost sounds identical to what it is in my mother language, Turkish, SIFIR. So, I assume that, Turks learned it from the Arabs during the heavy interaction between 900-1500 ...

I was going through Wikipedia, and it sounded like, the base 10 and ZERO were invented right around the same time. It is looking like the 900's and 1000's as the clear time ...

Tolga,

When adding in base ten with an abacus and translating the result in writing, you need a place holder for the position even if that row of the abacus has not beed to the right. The zero was the place holder in the writing system. Because a base ten positional system need to have a place holder, thus a zero.

So, you are saying, it was invented way before 900 ? But, was never really an integral part of the mathematics. It was an after-thought kind of thing. It wasn't really truly understood to be an important mathematical object up until the 900's ...

Louis,

Here is the first part of what you asked for:

nearness of points and sets:

topology viewed as set theory (Naimpally and others) : graph topology ->

Bourbacki (1980+): general topology -> ... ->

Kuratowski (1958-...): closure operator -> ... ->

A. Taimanov (1952): extension of continuous mappings ->...->

Alexandroff and Hopf (1935-): separation spaces ->...->

Hausdorff et al. (1914-...): compactification ->...->

Hausdorff, Alexandroff, Wallman, Smirnov et al. (1914): topological spaces -> ... ->

Young (1907): theory of sets of points, derived sets w/o Cantor numbers -> ...

Cantor (1879-1886) : well-defined sets, transfinite numbers-> ...

Thank James

Tolga, in a Eurocentric sense zero did not become a place-holder until Simon Stevin's two books, one for science and one for business in 1585AD. However, in a world-wide sense Mayans used a positional zero 2,000 years earlier, and likely earlier Olmecs that built the long count.

Tolga, and others, Arabs obtained numerals from India in 800 AD http://planetmath.org/arabicnumerals without accepting zero as a place-holder, an abstract step that took place almost 800 years later.

In ancient time, people did begin calendar at number 1. We are in year 2013 and so we live 2012 years after Jesus's birth in year 1. In the modern practice, coordinate axis start at 0.

Dear researchers , I proud to be an Indian as Arabs obtained numerals from India in 800 AD. And also Ramanujan was from our State. (Kumbakonam., Tamil Nadu, India)

Milo, that's a cool chronological ordering you brought up. I understand that, zero was a "placeholder" way before it was officially a NUMBER. When I said "invention of zero," there are two interpretations :

1) Calculating its value in computations, i.e., missing bead in the calculator (placeholder), Example --> "if you have 2 apples, and I have NO apple, we have two total between the two of us. This is implying zero, but, not explicitly giving zero a meaning ,,, NOTHING is not ZERO. ... and ...

2) Treating ZERO as a NUMBER. --> 2+0=2 ...

So, NOTHING (i.e., placeholder) came way before ZERO, but, ZERO wasn't given its own spot, its own name, and its own personality up until 800 ...

Louis, do you agree with me that, NOTHING is different than ZERO? The placeholder was invented a while ago, but, it doesn't have the same meaning as zero ? do you agree ?

Simon, that is real cool what the Indian folks did ... Due to the diffusion of this knowledge to the Arabs, and to the Turks eventually, Indians might have implicitly help the Ottoman science in a big way ! This eventually propelled them to be one of the biggest empires 700 years later :) Back then, scientific advances were taking so long, since the INTERNET speeds were real low (camels and horses did not have high data throughput rates :)

I am curious about one thing. Just like German's put Gauss on their 10 Deutschemark, and, Russinans put Chebyshev on their stamp, I even remember back in Turkey 30 years ago, the Turkish philosopher Mevlana being on a Turkish stamp , did Indians put any of their scientists on a stamp ? or, money ? I would vote for Brahmagupta and Ramanujan ...

Our language which includes our number system set certain limit to what we can express and imagine. A people with a very primitive numbering system will not be able to imagine hiearchies of organisation involving relation among big numbers. A people with a cyclic positional exponential numbering system can imagine a complex cyclic universe. Just look at Hindu mythology, they could not imagine it without their

sophisticated numeric system:

'According to Hindu system, the cosmos passes through cycles within cycles for all eternity. The basic cycle is the kalpa, a “day of Brahma”, or 4,320 million earthly years. His night is of equal length. 360 such days and nights constitute a “year of Brahma” and his life is 100 such years long. The largest cycle is therefore 311, 040,000 million years long, after which the whole universe returns to the ineffable world-spirit, until another creator god is evolved.'

http://en.wikipedia.org/wiki/Hindu_mythology

Tolga,

I have not verified the following speculation but here what I think. First positional numbering system is invented and this implies a place holder that is not a number itself, just an artefact of the notation. The numbering system was invented for counting and for doing accouting (arithmetic) . Remember that the invention of coordinate axes (analystical geometry) is recent where numbers are also lengts on the axis and that we need an origin. This origin needs to be a number and lenghts have to be calculated by substraction of numbers. You need a number for origin that substract nothing, here you have zero. Basically I think that back in India some geometer transformed the place holder into a point of origin that need to be 0.

Tolga, zero in the Mayan world followed two definitions. In (260)365) = 18980 day calendars there were no zero days. Lunar and solar calendars bases 13, 18 and 20 counting were 1-13, 1-18 and 1-20, respectively. However in modular situation bases 4, 5, 13 , 18, and 20 Mayans counted 0-3, 0-4, 0-12, 0-17 and 0-19, respectively, discussed in the broader Mayan math context http://planetmath.org/mayanmath. Note that the positional number system was defined by the Olmec long count 18(20)^n, with n = 0, 1, 2, 3, ...

Dear Tolga, Indian government had introduced Ramanjan stamp in 1962 on Dr.Ramanujan 's 75th birthday (22 december 1962). . In December 2011, in recognition of his contribution to mathematics, the Government of India declared that Ramanujan's birthday (22 December) should be celebrated every year as National Mathematics Day, and also declared 2012 the National Mathematics Year. In Hindustan university , we celebrate Mathematics day on every 22nd of December through organising Mathematics guest lectures .

The following link give some images of Ramanujan stamp

https://www.google.co.in/search?q=ramanujan+stamp&es_sm=93&tbm=isch&tbo=u&source=univ&sa=X&ei=OI1RUtu4I8qErgfKh4GgCQ&ved=0CCwQsAQ&biw=1366&bih=667&dpr=1

Ramanujan Hardy Number 1729 can be found in the following link. http://en.wikipedia.org/wiki/1729_(number)

Zero was invented by Aryabhatta in (5th century) India.

A practical zero was used in 2,000 BC Babylon and Egypt. Egyptians used the word sfr to denote zero accounts in double entry bookkeeping. Olmecs used a theoretical zero in the long count over 1,000 years before Aryabhatta. R.C. Gupta suggests Mayan math diffused to India and Brahmguta, a huge story validated.

Milo, thank you for that note, who would have known that, NOTHING (ZERO) would have such a history behind it :) Possibly, NOTHING-ness fascinated people for centuries as much as INFINITY.

Simon, it is good to know that, Ramanujan gets such a credit.

One of the computer nodes in our university is named "ramanujan" by my Indian collaborator ! but, I was a big fan of Ramanujan way before I met my collaborator :)

Ramanujan and Erdos are possibly the most interesting mathematical figures due to being so unusual (and, so capable). I am wondering who gave the honorary Dr. title to Ramanujan, I assume,. Oxford ? or, University of Madras ?

University of Madras gave Dr degree to mathematician Ramanujan.. ...now the madras city is called Chennai city.

Comments would be appreciated on http://planetmath.org/mayanmath . The Planetmath entry discusses zero and astronomy based math systems as mentioned above.

Bulls and mathematics.

My mother-in-law breeds bulls, pigs of goats, hens, cats and dogs. I made experiments on mathematics with a bull. As all people were occupied with agriculture, the dog of breed of "Like" or "Huskies" was the only witness of my experiments with a bull.

Categorically I ask not to repeat my experiments because the bull is very strong and uncontrollable animal!

My supervision and scientific results.

1) The bull distinguishes such mathematical concepts as "Small "and "Big".

The bull of the person at all isn't afraid but when I unexpectedly got up before a bull on a chair, he strongly was frightened and escaped, on - seen he decided that I am an elephant.

2) The bull is able to consider to two.

The bull at all isn't afraid of the person, the bull isn't afraid of the person with a stick. Became clear that the bull is afraid of the doubled stick that is a pitchfork. The bull escaped from a pitchfork. Concerning this effect I had disagreements to my father-in-law. My father-in-law G.N.Bobrov

I considered that the horned animal is afraid of other horned animal, that is a pitchfork. However, I have other, mathematical justification of this effect. The bull counted to two and understood that with two horns to him won't cope therefore escaped.

3) Safety measures and labor protection for animals.

We three together: the bull, dog and I made mathematical experiments on a meadow under light of the sun, but casually bull, the hoof stepped on a paw to a dog. The dog cried. The dog bit a muzzle of a bull. The bull in horror escaped. On it mathematical experiments ended.

Sincerely yours,

Alexander

Ishango Bone :

Archaeological dating places the bone between c. 25,000 BP and 16,000 BP.

This interpretation of the Ishango bone provides evidence that people during earlier prehistory could have developed significant proto-mathematical knowledge beyond simple counting. This includes the selection of certain numeral bases (10, 12), specific kinds of numbers (odd, even, prime numbers), and rules of multiplication and division by two. An object such as the Ishango bone could have been used for time reckoning, for special games or other purposes.

Ishango Bone on UNESCO site

http://www2.astronomicalheritage.net/index.php/show-entity?identity=4&idsubentity=1

UNESCO Astronomy and World Heritage Webportal - Show entity

Louis,

A good observation. I have seen evidence of proto-math that you mention in pictographs painted on isolated rocks in the middle of lakes in remote regions of Minnesota.

Decoding the numbers and implied math taken from walls is important ... for example Guatemala: The long count cited four distance numbers on a 419 AD wall (near Tikal. Guatemala) with four super-numbers. The first is divisible by 117, 260, 360, 365, 584, 585 and 780: Mercury, lunar, earth, Venus, Mars and 18 periods of the Mayan 52 year calendar round period of 18980 days; the second divisible by 117, 260, 360, 364, 365, 780 and 63 (18980); the third by 117, 260, 365, 780 and 91(18980), and the fourth by 117, 260, 365, 780 and 129(18980).

The planets Mercury, Venus, Mars, Saturn, and Jupiter, plus the moon lined up in ”string of pearls” combinations that aligned Chinese calendars to Feb. 2, 1951 BCE (facts seen on Stellarum and other astronomical programs), related evnets that Mayans placed at the center of their mythic and scientific worlds that double checked Mayan calendars.

Mayan rational numbers scaled super-number distance numbers in exacting ways. Planetary almanacs divided calendar round periods that defined rational number quotients and day remainders.

Mayan calendars were recorded in day quotient and day remainders. Solar calendars used in China, India, and the Hellene world only used remainder arithmetic. Ancient Near East lunar calendars and weights and measures were also written in quotient and exact remainder arithmetic.

I want to offer some comments about the Ishango bone earlier mentioned by Louis Brassad. Indeed, the Ishango bone, about 20,000 years old , and currently at the Museum d'IHistotoire Naturelle , Brussels is one of the earliest numerical recording devices. What has not yet been mentioned relative to this discourse on the origin of mathematics is the fact that de Heinzelin. the archaeologist who excavated the bone , suggested in 1992 the possibility of the Ishango numeration system travelling as far as Egypt to influence its numeration system---the earliest written decimal system in the world..

Indeed, archaelogical and early written records confirm African origins of Egyptian civilization. The ancient Egyptians, according to knowledgeable Greeks such as Herodotus, Aristotle, and Appolonius, were descendants of people from Ancient Ethiopia, which at that time referred to the hinterland of Africa inhabited by black

people. Moreover, Egyptians were essentially Africoid (black) until around 1300 B.C

and the Pharaohs who were Africod include Narmer, the Phraoh who unified upper and lower Egypt; Zoser, the Pharaoh of the third Dynasty who inaugurated large architectural projects in hewn stone, and Cheops, the builder of the great pyramid. In fact, some Scholars (see the references below) have further established genetic relationships between African languages (especially Wolof ) and the Egyptian Language. It is well known that by 3000 B.C., there were already well developed written number systems in Egypt (decimal based) and Mesopotamia (sexagesimal-base 60 )

For further information see 1) G. G. Joseph. The Crest of the Peacock: Non-European Roots of Mathematics. Penguin Books 1991: 2) C. A. Diop.. African Origin of Civilizations: Myth or Reality. Laurence Hill and Co, New York 1974 3) G. Moktar. General History of Africa, Ancient civilization of Africa UNESCO.Heineman, paris London 1981: 4) J. Van Sertima . Blacks in science: Ancient and Modern. Transaction Books 1983.

Aderemi, I have been reviewing your story line for 20 years. Much of what you have suggested is true. The second edition of Crest of the Peacock includes a few early words on decoding Egyptian base 2 and base 5 only recorded in cursive base ten after the merger of Upper and Lower Egypt in 3,000 BCE.

George Joseph was kind to include a few words on my efforts to decode hieratic rational number math, a topic that is also not fully decoded.

As you may know , returning to your thread the core African contribution was base 2, a deep spiritual love for parental pairs as Ron Eglash began to decode over 25 years ago recorded in physical African (fractal) villages per

http://www.ted.com/talks/ron_eglash_on_african_fractals.html

Thank you for your thoughtful contribution. Much needs to be learned about the earliest African base 2 merging with base 5 that created base 10 arithmetic that morphed into hieratic finite arithmetic by 2,950 BCE, two threads that created classical Greek and medieval finite arithmetic as late as Galileo's square root method.

oops, typo: 2,950 BCE should be 2050 BCE

I feel I should elaborate a bit more on the contributions of Egyptian and Babylonian mathematicians long before the classical period of Greek civilization and the golden age of European Mathematics--just to show that the early contributions from Egypt and Mesopotamia were much closer to the origin of mathematics--the topic of this discourse. Indeed, by 2000 B.C. Egyptian mathematicians had discovered multiplication, fraction, geometric and arithmetic series, the area of a circle, the value of the volume of a pyramid ,the area of a curved surface--which were recorded in Hames Papyri (written around 1650 B.C.) and now housed in the Museum of fine Arts in Moscow.

The contributions of Babylonian mathematicians --the construction of tables for multiplication and reciprocals, the solution of linear and non-linear equations in one, two or three variables, the solution of Pythagorean triples, the areas of rectangles, triangles, trapeziums e.t. c ---are contained in numerous clay tablets scattered in museums all over Europe, and at Yale University, Columbia University, and University of Pennsylvania in the USA as well as in the Iraq Museum in Baghdad.

The sophistication of the technology and mathematics available in ancient Egypt can be imagined from the great pyramid of Giza, built around 2600 B.C. from individual stones, each weighing about 70 tons. (Actually I have been to Giza myself several times and marveled at these wonders of Ancient Egyptian technology). With a volume of at least twice that of the Empire State Building in New York, and thirty times its mass, it is the most massive building of all time. (See G. G. Joseph: The crest of the Peacock: Non European Roots of Mathematics. Penguin Books, 1991.)

Aderemi,

Just an observation of the aesthetic of the Egyptian architecture, the obelisk, pyramid, the temples, even the elegant minimal forms used in their statues and their arts conveys a highly mathematical sense of aesthetic. Geometrical figures are everywhere and curves are simple like in the modern art and in curves produced by modern CAD system.

Louis, thank your for posts. Egyptian geometry was well developed within hieroglyphic and hieratic numeration systems. Hieratic numeration http://www.academia.edu/617613/Egyptian_Fractions_Unit_Fractions_Hekats_and_Wages_-_an_Update was the numeration and math that diffused to classical Greece, and modified by Arabs in 800 AD to Vedic numerals in a subtraction context.

@Milo Gardner:

Here is a capsule view of Babylonian mathematics:

In mathematics, the Babylonians were somewhat more advanced than the Egyptians.

Their mathematical notation was positional but sexigesimal.

They used no zero.

More general fractions, though not all fractions, were admitted.

They could extract square roots.

They could solve linear systems.

They worked with Pythagorean triples.

They solved cubic equations with the help of tables.

They studied circular measurement.

Their geometry was sometimes incorrect.

For enumeration the Babylonians used two symbols.

displaymath272

All numbers were forms from these symbols.

Example:

displaymath274

Note the notation was positional and sexigesimal:

displaymath276

displaymath278

tex2html_wrap_inline280 There is no clear reason why the Babylonians selected the sexigesimal system. It was possibly selected in the interest of metrology, this according to Theon of Alexandria, a commentator of the fourth century A.D.: i.e. the values 2,3,5,10,12,15,20, and 30 all divide 60.

tex2html_wrap_inline280 Remnants still exist today with time and angular measurement. In fact the Babylonians used a 24 hour clock, with 60 minute hours, and 60 second minutes.

http://www.math.tamu.edu/~dallen/history/babylon/babylon.html

Corrections to my previous post:

For enumeration, the Babylonians used two symbols:

James, thank you for suggesting that Babylonians were more advanced than the Egyptians. I will not argue the Old Kingdom period. Base 50 was more accurate than Horus-Eye bi9nary round off that threw away 1/64 units.

However, as my paper details in many respects hieratic rational numbers were exactly written ... zero round-off expect for pi and irrational numbers. The date that Egyptians switched from 256/81 = 3.16 to 22/7 is unclear in Ahmes' 1650 BCE text ... RMP 38 shows 3200 ro multiplied by 7/22 ...with the answer multiplied by 22/7 ... returned 3200 ro ... as a wonderful; proof.

many such proofs fill middle kingdom texts ... in ways that Babylonian scribes did not exactly prove.

more can be reported ... like inverse proportion pesu units that were used in the Berlin Papyrus (decoded in1900) used identical imagery to the solutions to two second degree equations. , by a square root method ... likely later used by Archimedes.

enough for today

.thank you fort the interesting discussion.

Milo,

We agree.

Egyptian math was more advanced, more accurate, than Babylonian math, in square root of irrational numbers weights and measures , math that continued in the Greek era as the 300 BCE Hibeh Papyrus, a time keeping text, discloses http://planetmath.org/hibehpapyrus . Chaldean astronomy may have been more advanced than Greek and Egyptian astronomy. Egyptian astronomy is officially unreadable, hence no direct comparisons can be made.

@Tolga,

I would like to comment your assertion

 "And, since what he (the caveman) knows in his mind was never disseminated to the others, the progress of mathematics (or any science, for that matter) was exponentially slower ... if not ZERO. "

As much as I know the first numbers were found as petrogrphs in the cave Chauvet-Point d'Arc in the Ardeche region of south France, which was inhabited since the paleolithic. The petrographs found here are belonging to the Aurignac culture (part of the superior paleolithic, which came after the musterian culture of Homo Neanderthaliensis, in the middle paleolithic).

The first number depicted looks like

. . .

.

. .

which reads as 6.

The second number looks like

...

III ...

which reads probably 6 + three time the base: 36 in 10-base, 27 in seven-base.

The 5-base is not probable, because in this case the 6 would have been noted as "I ."

The radiocarbon dating of the charcoal used to make the drawungs yielded the result of 31000 years before present (+/- 1300 y)

So, the petrograph of Point d'Arc demonstrates the use of a number system (with unknown base), publicly posted at a gathering place in order to disseminate this knowledge. Well, it was not as efficient as a Facebook post, but probably the members of the clan which inhabited that cave were aware of the origins of mathematics.

The second nr is depicted as three vertical lines followed by six points drawn in two rows.

@Milo, James, Aderemi,

I enjoyed reading your posts.

I would like however to emphasise that is not always beneficial if someone tries hard to demonstrate a linear and continuous evolution of mathematics based on artifacts organized upon temporal (successional) and geographical criteria. The majority of people in the upper paleolithic had two hands equipped with five digits (not counting the survivors of sabertooth tiger's hunting), so they have all the premises to use digital calculus on base 2, 5 and 10. Even if the first petrograph depicted was found in what is called now South France, that should not add fuel at the French's pride: probably the mathematics has a very wide origin throughout the whole world inhabited by cro-magnonians. Interestingly it seems that there is no evidence of mathematical activities in Neanderthalians and Denisovanians (or have we?).

In order to give an exemple of sentence (2) I would like to recommend the problem of using the romb (I will use for it) as a numeral.

The was used as a numeral meaning 1000 in two locations: in a tally-writing system of hungarian sheperds in medieval Transylvania – and in Crete until 3900 BP. Since 3900 BP the cretans started tu use the symbol of Sun for thousand (an O with four emergent rays).

The rest of the numerals (unit and tens) remained the same.

If we try to make a link based on the fact that a symbol is used to replace the same number in two widely separated locations (in space and time), we will end up wondering if the ancient symbol of 1000 was born as in Middle Europe, where it survived until XIX c, meantime it was navigated to Crete around 5000 BP where it was replaced with the symbol of the Sun – or the ancient symbol of 1000 = was born in Crete around 5000BP, the symbol was exported to the mainland Europe before 3900 BP and later the cretans replaced with the Sun, but the persisted in Europe.

But what if the use of meaning 1000 is just a coincidence?

@Andras, @Martin, are you saying that, Mathematics knowledge WAS disseminated even 10000's of thousands years ago, but, because of a STANDARDIZED symbolism, the knowledge dissemination was so inefficient that, it was almost non-existent ... The mathematical symbolism as we know started with Diophantus ... Before Diophantus, all we had was WORDS to describe mathematical equations (if three cows start walking towards four people, eight roads intersect, five streets ...) ...

Diophantus introduced 4x+5x^2-.....

If you have to INTERPRET what the cave carvings mean, this means that, every cave has its own STANDARD, which makes it pretty difficult to disseminate knowledge ...

Am I hearing what you said correctly ?

@Andras,

Your mention of the petrographs found in southern France reminds me of the petrographs that are found on rock out-croppings on some hillsides and in the middle of remoted lakes in Minnesota. One of the petrographs that I studied included rows of dots and paintings of animals that look like deer or elk. It is reasonable to conjecture that whoever painted the rows of dots was, in effect, recording a crude tally of the number animals found during a hunt.

Using a row of 5 dots to represent a tally or number is hugely different from writing the number 5 instead of the row dots.

@Tolga,

It is great what you have pointed out about Diophantus. There is an epitaph on the tomb of Diophantus:

This tomb hold Diophantus. Ah, what a marvel! And the tomb tells scientifically the measure of his life. God vouchsafed that he should be a boy for the sixth part of his life; when a twelfth was added, his cheeks acquired a beard; He kindled for him the light of marriage after a seventh, and in the fifth year after his marriage He granted him a son. Alas! late-begotten and miserable child, when he had reached the measure of half his father's life, the chill grave took him. After consoling his grief by this science of numbers for four years, he reached the end of his life.

And this epitaph can be written as a linear equation:

x = x/6 + x/12 + x/7 + 5 + x/2 + 4

For more about this, see

http://www.mlahanas.de/Greeks/Diophantus.htm

I like to comment on the contributions of the Chinese and the Mayas mentioned by Martin Kovar. Relative to the current discourse on the origin of mathematics, the Chinese contributions came much earlier because there was already some measure of numeracy around 3000 B,C during the River Valley civilization which lasted till 1,000 B, C. During this period, numeracy began in China with practical mathematics , e. g. the construction of magic squares. Indeed ,the world's earliest known magic squares were found in China and dates back to around 3,000 B.C. The earliest known Chinese numeral was decimal-based and appeared between 1500 and 1200 B,C, in form of Shang oracle bones--so named because the inscriptions on them show that they were used for divination or fortune telling. Around A. D. 100, during the Han dynasties, (B.C. 200 to A. D. 200 ) there appeared one of the oldest and most famous Chines mathematics texts. The book covered such topics as land surveying (involving the calculation of areas of fields of various shapes); business mathematics,( concerning the distribution of property and money according to prescribed rules which sometimes led to arithmetic and geometric progression); engineering mathematics, (involving the computations of volumes of three dimensional shapes familiar to builders of castles, houses and canals). However, by A.D. 500, the Mayas of Central America had also recorded great achievements in various fields including mathematics, art, sculpture and astronomy. For mathematics, the Mayas shared (with India) the credit for two independent fundamental developments--the principle of place value system and the use of zero.

Now, the early Indian contributions to mathematics paralleled those of China because there was evidence of numeracy in India found among the ruins of the River Valley civilizations dating back to 3,000 B.C. The early achievements of India between 3,000 B.C and 1,500 B.C. include a centralized system of weights and measures, brick technology, and the development of instruments for measuring length, Between 1500 and 200 B.C. the Indians developed numerals, arithmetic operations, some geometry, number theory, permutations and combinations. the binomial theorem, and astronomy--essentially in order to determine the auspicious day and hour for performing sacrifices.

Between 200 B.C. and 400 A.D. the Indians also discovered the decimal place notation, the use of zero, solutions of simultaneous and quadratic equations and how to represent unknown quantities and negative signs..

It is not what I meant, Tolga.

Between the quotation marks lies the final sentence of your post unleashed on Oct 4 – a very popular answer on RG. There you say that caveman's mathematics was not disseminated and this slowed the scientific progress.

My intention was to contest this approach and demonstrate that the 30.000 years old petrographs could serve as a model of dissemination of ideas. The numerals drawed on the walls are next to some artistic pictures of wild horses, bisons etc. So probably that cave was used for ceremonies, where the ideas corresponding to the drawings were hammered into the heads of the invited tribe-members (it's the paleolithic teaching method!): and the ideas at Chauvet – Point d'Arc were related to fine arts and mathematics.

I will also argue that symbols were not used before Diophantus.

Taking in account that two pictograms were used for the second number, where "dot" meant the unit and the pictogram of "line" symbolized the base of the number system (greater than 6 as presumed from the first number), probably here we have the first mathematical symbol used. We don't know if the base was 10 or 8 or 7 – but maybe that not really matters.

On the other hand, few millennia before Diophantos(!), the sumerians used four symbols to write numbers. They used a stylus with a narrow and a thick end. Narrow lines obtained by pushing the narrow side of the stylus in the clay tablet meant unit and imprinting the tip of the narrow end perpendicularly in the clay meant ten. Line imprinted with the thick end was 60 and dot made with the thick end was 3600 (60x60). The two ends of their stylus drove them to use a combination of two number-systems: on base ten and base 60.

Do you know, how they wrote 600 ?

Not in a diophantian way!

It was a wide line with a narrow dot in the center: wide line made with thick end of the stylus held oblique and a little hole made in the center of this wide line, made with the narrow end of the stylus held perpendicularly. That implies multiplication: 60 x 10 = 600. What you would write with nine diophantian symbols, the sumerians wrote with only two symbols. In Uruk, 5500 years before present. I find quite remarkable the power of these sumerian symbols.

You end your post asserting that every cave had its own standard which makes difficult to disseminate knowledge.

Tolga,

The University of Rochester is a modern "cave" where you use your standards to make science and teach your students – who should be proud to be taught there. They are taught not only by mural (blackboard) drawings, but by courses, lectures, online and interactive courses also. The standards used in Rochester are quite different from those used at Point d'Arc – but the essence is the same: knowledge has to be disseminated. Point d'Arc can be seen as a school where arts and the use of a number system were taught (hammered into paleolithical heads) and is hard to compare with your institution where You use now the standards worked out by Diophantos, Euclid, Descartes, Galilei and other postpaleolithical people. Of course, you are so much more efficient in disseminating knowledge that it can be difficult to realise the parallell between P d'Arc and Rochester. That's because some 32.000 years timelike difference.

James,

a raw of dots is completely different of a drawing with three lines and two rows of three dots, because on the Minnesota petrograph described by you there are only dots with one-to-one correspondence (one dot = one bison): unar number system, base-1.

At P d'Arc the line means (probably) 10 units, so the line symbolizes ten dots – and that's a different situation because it represents a more complex idea, a thought expressed in a base-10 number system.

In an other post you write-up the diophantin way of expressing fractions.

That is familiar to us, people educated in diophantine caves.

But before that, besides the sumerian way of noting multiplication, there were other writing systems also.

In China numbers wrote in same color had to be added.

If one of the numbers were in yellow (saffron), meant to multiply them.

In red: divide.

Under a hanging-post: extract.

Andras, I love every word of your post !!!

At University of Rochester, my lab actually does look like a cave with carvings of circuits everywhere :) :) :)

================

The idea of the humans inventing EVERYTHING we know now 10000's years ago is startling ! When I was a kid, I remember reading about how MARSIANs might have come to earth and brought technology, and left ... According to these STORIES, this is why we have certain advanced structures on this planet now, which were made 10000's years ago.

================

The alternative story to this one is as follows: YES. An advanced civilization WAS HERE 10000's of years ago, and had very advanced MATH, PHYSICS, and everything else to build such amazing structures on this planet. The only difference is that, they weren't MARSIANs , they were HUMANs !!! Due to the unfortunate events such as big wars among civilizations, they got wiped out before they could disseminate this advanced knowledge ! Sure, their symbolism might have been totally different, their numbers systems, etc ... might have been different, but,

*** a rectangle with side lengths a and b has an area a*b. It is a*b today, and it WAS a*b 300000 years ago !

*** PI is also PI, and was PI 1000000 years ago !

==================

There could have very well been an Euclid (like mathematician) 50,000 years ago, and an Euler (like mathematician) 47,000 years ago to build on that knowledge. I don't know if they invented Riemann hypothesis (equivalent) :) Who knows, may be some day, we will, find carvings that has the proof of Riemann hypothesis :) We can, then, give the Clay Institute Millennium Prize to that caveman :)

@Tolga,

So, you have found and alternate solution to Fermi's Paradox. Fermi's original story goes like this (from Wikipedia):

"The universe is vast, containing myriads of stars, many of them not unlike our Sun. Many of these stars are likely to have planets circling around them. A fair fraction of these planets will have liquid water on their surface and a gaseous atmosphere. The energy pouring down from a star will cause the synthesis of organic compounds, turning the ocean into a thin, warm soup. These chemicals will join each other to produce a self-reproducing system. The simplest living things will multiply, evolve by natural selection and become more complicated till eventually active, thinking creatures will emerge. Civilization, science, and technology will follow. Then, yearning for fresh worlds, they will travel to neighboring planets, and later to planets of nearby stars. Eventually they should spread out all over the Galaxy. These highly exceptional and talented people could hardly overlook such a beautiful place as our Earth. - "And so, " - Fermi came to his overwhelming question, - "if all this has been happening, they should have arrived here by now, so where are they ? " - It was Leo Szilard, a man with an impish sense of humor, who supplied the perfect reply to the Fermi Paradox: - "They are among us," - he said, - "but they call themselves Hungarians. "

I have read that story with Erdős Pál being the mathematician who gave this answer to Fermi – but that can be irrelevant.

However your post raises some interesting questions. If we discard the jokes related to paleoastronauts,we are left with some intriguing problems related to the orientation of those prehistoric monuments which could be used for astronomical (and astrological) purposes.

E.g. the Great Pyramid's descending passage is oriented toward a patch of sky where 4500 BP (when the pyramid was constructed) the actual North Star could be seen: the Tuban (Alpha Draconis) Or the exact orientation of the pyramids to the north-south line presumes some exceptional astronomical and mathematical calculations. It seems that the stars named Mizar (Zeta Ursa Majoris) and Kochab (Beta Ursa Minoris) would have appeared to revolve around the pole on almost (but not exactly) opposite sides, so that a line joining them would always pass very nearly through the pole. When these two were aligned vertically, the pyramid builders might have hoisted a long plumb line and fixed it at the moment when the two stars both lay on the line. The point where the vertical line touched the ground would indicate north.

To obtain these orientations, serious trigonometrical knowledge was necessary.

I tried to look after the Hames papyrus mentioned by Aderemi, but I've found only Ahmes-papyrus which is held at the British Museum. Ahmes was a scribe, he lived between 1680-1620 BC and recorded the mathematical knowledge of his time on a papyrus. This document was purchased by a Scottish health-turist in 1858 (his name was Mr. Rhind) and took it with him to Britain.

The Moscow Mathematical Papyrus was bought in 1892 by Golenischev, so now is housed by the Pushkin State Museum. This document is older than the Ahmes-p – it was created about 1850 BC and contains beside many others the formula to calculate the frustum of a square pyramid.

The problem arising from Tolga's post is: the Great Pyramid is much older, than these mathemathical papyruses. And the Khufu's is not the first pyramid built in Kemet.

Not to mention an ancient monument which is closer to the other Rochester, in South-east England: Stonehenge, built approximately at the same time as the Great Pyramid.

The origin of mathematics must antedate the emergence of these buildings.

Can we demonstrate this allegement by archeological findings?

Ahmes Papyrus' 2/n table and 87 problems were introduced by EMLR 1/p and 1/pq conversion methods, and the Akhmim wooden tablet that defined Egyptian multiplication and division of a hekat unity (64/64) by 1/3, 1/7, 1/10, 1/11 and 1/13 to a two-part quotient and remainder answers per five proofs that returned (64/64).. The two 200 years older documents are discussed in http://historyofegyptianfractions.blogspot.com/ as well as the 1202 AD "Liber Abaci", written in a closely related unit fraction numeration system.

The 2050 BCE Egyptian fraction math was continuously in use written in two numeration system until 1600 AD. The latest numeration system was created by Arabs in 800 AD and documented by Fibonaci's " Liber Abac":, Latin speaking and writing Europe 's primary arithmetic book for 250 years. .

Ok, Andras, here is my THEORY :) or, whatever you want to call it.

==================

*** I am not discarding Fermi's theory about the warm soup, water, sun, etc ... It is very possible that, there are other planets like earth, where there IS life ... However, we traveled all the way to MARS, and found NOTHING. So, if there is anything, it is in a GALAXY FAR FAR AWAY, like GEORGE LUCAS theorizes :) in his Star Wars movie :)

*** So, according to my "theory" :) this GALAXY FAR FAR AWAY is so far away that, there wasn't any travel possible to our galaxy.

*** So, where did these fantastic monuments come from ? on earth ?

*** What if there was a human civilization, 1,000,000 years ago (this is 1,000,000BC), and went all the way up to inventing a good portion of mathematics we know right now, and physics, and chemistry, and philosophy ... But, got destroyed, either by a meteorite, or, wars among themselves, or, simply HUNGER ? Before they could disseminate their knowledge ?

*** 50,000 years after that, the sun warmed up the oceans, simple life started again, evolved, evolved, and, next thing you know, there is another civilization 10,000 years later (this is 940,000 BC). They build advanced math, physics, everything again. However, since they start at a completely different point, their numbering system, mathematical symbolism, everything is different. But, it is the same math, just presented in a different way ... Unfortunately, the faith strikes them again. They vanish before they can disseminate the knowledge, and/or build enough tools to protect themselves, from either THEMSELVES, or, natural disasters ... This time may be it is a tsunami that destroys them.

*** This cycle repeats, until the human civilization can build a stable enough structure, where they can keep advancing ... It takes 942,000 more years, until 2013A.D.

*** I am having trouble believing that, in a planet with about a 4 billion year history, our current civilization with 6,7 billion people is the only civilization that lived here .. and, everything started about 5,000 or 10,000 years ago ...

*** What I am proposing is identical to Fermi's idea. The only difference is that, he described everything in "d" (distance), I am describing everything in "t" (time).

*** A civilization that is FAR FAR AWAY IN DISTANCE is as difficult for us to comprehend as a civilization that is FAR FAR AWAY IN TIME.

What do you think ?

OK, Tolga. Few minutes ago I have posted my oppinion about the role of space in this Universe. Probably a sequel for StarWars will follow observing these three roles of space: what seems to us "somewhere in a distant galaxy" could be closer than you can imagine. Exactly as Hamlet told to Horatio in an other old story...

So, human civilization a million year ago?

According to our evidences a million yBP the Homo habilis (cranial capacity 600 cm^3) was extinct, the world was ruled by H. erectus (CC1000 cm^3). Actually it was ruled by the top predators like sabertooth tiger and cave-bear - however the hominides, our brave ancestors were in the upper half of the trophic chain.

600.000 yBP H. heidelbergiensis appeared and CC rose to 1400 cm^3, so some tools were used to kill little animals.

350.000 yBP an other species appeared: H. neanderthaliensis, with a CC 1200-1900 cm^3. That is much more than your or mine - but that's where quantity is defeated by quality. H. neand used combined tools, symbolic logic and thinking - but probably could not talk as we can. Nor write or calculate more than three. That was a major obstacle in disseminating knowledge - so they were driven to extinction by a species which could talk, write and calculate: that was the H. sapiens. They (we) have appeared only some 200.000 yBP - and they are the only surviors of their species.

So I don't think that the origins of mathematics on this planet may go back to more hundreds of thousand years.

But there is a possibility at 20.000-10.000 years BP: the so-called Atlantis-civilization.

I have to recognize that I used heavily this theory in my novels - in order to fuel the story which explains special and general relativity, entanglement, time-travel, string-theory and so on. What's more, the action is directed by Hermos Trismegistos, an atlantidian king who became immortal and time-traveller. But that is part of the fiction: we don't have archeological evidences (yet) to document the existence of an advanced human culture before ours. The evidences from Point dÁrc cave can not explain the appearance of such a highly evoluated civilization.

We have to admit that for the moment there is a gap in our understanding related to the megalithic civilizations. It's better to admit that we dont'understand something than to come with some undemonstrable theory.

I have done some reading to answer the possibility of "advanced Mathematics before 20,000 yBP (years before present) or even 200,000+ yBP. Here are the results :

*** I am convinced that, if any civilization invented something advanced, it will be the HOMO SAPIENS trail ... Clearly, pre-Modern-humans, Homo erectus (1,800,000 - 140,000 yBP) , Neanderthal (600,000 - 35,000 yBP), or Cro-Magnon (43,000) weren't advanced enough ...

*** HOMO SAPIENS trail begins ~ 200,000 yBP in Africa. The chronology below uses the country names that are there NOW, rather than what WAS there. Sole purpose of this is to make it easy to visualize..

*** Up to 90,000 yBP, spreading across Africa.

*** 90,000 - reaching the Arabic Peninsula

*** 90k - 55k - one trail continues towards Iran -> Pakistan -> India

*** 55k - 30k - one separate trail moves North to Saudi Arabia -> Iraq -> Turkey -> Europe

*** 55k - 30k - A trail from Iran -> Afganistan -> China

*** 30k - 10k - Trails from China --> Russia, going through Kazakhistan and Mongolia

*** 55k - 30k - a trail from India continues -> Malaysia -> Australia.

*** 55k - 30k - a trail from China continues to reach Japan

*** 33k - 10k - a trail from Russia continues to go over the ice with the sled dogs, to reach Alaska, and continue down to Americas. The beginning of this period is approx. where the wolf might have significantly genetically diverged to form a domestic version, to be the human's best tool: the DOG. DOGs were instrumental in this journey to Americas

*** 55k-30k - European trail faces Neanderthals and Neanderthals become extinct at around 30kyBP.

*** 33k-10k - European trail moves to Italy -> France -> Spain. A separate trail to England

*** 10k - 1kyBP - This is where everything we discussed in this thread happens. Pretty much, every civilization mentioned above is stable at this point. It looks like, the Homo Sapiens needs a nice, stable 4K or 5K years without any natural disasters, or competitions from its own human rivals to prosper and get the mathematics from zero to where we are.

There doesn't seem to be too much uncertainty in this chronology ... right ? So, the origin of mathematics must go back to about 10,000 yBP. We have ...

Ancient Egypt (5kyBP), numeration using hieroglyphics

Middle Kingdon of Egypt (4kYBP), word problems, story problems,

Pythagoras (3kyBP), proof without using mathematical symbolism

Diophantus (2kYBP), using mathematical symbolism, only a single variable

Hindu-Arabic Numerals (1kYBP)

After that, there is an exponential improvement curve ...