A Brief History of Mathematics

11th July 2011

Illustration of a fractal.

The Mandelbrot set. Image credit: Wolfgang Beyer/CC-SA.

1. Ancient arithmetic

There's evidence that prehistoric people understood simple mathematics, as well as astronomy. The oldest evidence comes from the Lebombo bone, which is about 37,000 years old and was found in Swaziland[1]. It has 29 notches carved into it, which could have been used to record numbers, making it a tally stick. Stronger evidence comes from the Ishango bone, which was found on what is now the border between Uganda and the Democratic Republic of the Congo[2].

The Ishango bone is about 20,000 years old and has a series of notches carved into it in three columns. Patterns in these numbers may show that they were made by someone who understood addition, subtraction, multiplication, division, and prime numbers.

Prime numbers are numbers greater than one that are only divisible by themselves and one. They are the 'building blocks' of mathematics in a similar way to how atoms are the building blocks of chemistry. The first ten prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23, and 29.

Photograph of the Ishango bone, and a diagram showing the three rows of notches.

The Ishango bone. Image credit: Ben2/CC-SA, Albert1ls/CC-SA, Albert1ls/CC-SA, and Albert1ls/CC-SA.

Row (a) of the Ishango bone begins with 3, which is doubled to make 6, 4 is then doubled to make 8. 5 is doubled to make 10. The last two numbers in the row, 5 and 7, are the third and fourth prime numbers.
Row (b) continues with the next four prime numbers, showing all of the prime numbers between 10 and 20.
Row (c) consists of the numbers 10+1, 20+1, 20-1, and 10-1.

The Sumerians in southern Mesopotamia, which is part of modern-day Iraq, developed a written language in about 3000 BCE[3]. This was around the same time that they developed the first school mathematics. People understood geometry and algebra by about 2000 BCE, by which time Sumer had become part of Babylon[4].

Around this time, both the Babylonians and Ancient Egyptians were aware of the number π (pi) the ratio of a circle's circumference to its diameter[5].

Circle

Image credit: modified by Helen Klus, original image by Kjoonlee/Papeschr/Public domain.

Diagram of a circle with the area labelled. Circumference = pi × radius<sup>2</sup>.

Image credit: Papeschr/Public domain.

By about 1500 BCE, the Babylonians were also aware of Pythagoras' theorem, which shows how the lengths of the sides of right-angled triangles are related[6].

Diagram of a right-angled triangle with the three sides extended to make squares. The area of the largest square is equal to the sum of the other two.

Pythagoras' theorem: in a right-angled triangle, the square of the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides. Image credit: Michael Hardy/Wapcaplet/Utilisateur/Leo2004/CC-SA.

2. The only true knowledge

Ancient Greek mathematics began with Thales, who was born in about 624 BCE and contributed to geometry[7], and Pythagoras, who was born in about 570 BCE[8]. Both were inspired by the Babylonians and Ancient Egyptians.

Pythagoras was fascinated both by the connections between mathematics and nature, and by the certainty that mathematical knowledge provides, especially when compared to sensory knowledge, which is sometimes unreliable.

Pythagoras is thought to have been the first to discover that music can be expressed mathematically. He believed that objects in space obey the same physical laws as the Earth, and so suggested that planets create music as they follow trajectories determined by similar mathematical equations[9].

Pythagoras coined the term 'mathematics', which meant 'learning', and founded a religious movement called Pythagoreanism. Pythagoreans believed that the whole universe is composed of mathematics, and that numbers are real entities that do not exist in space and time[10].

The Pythagoreans were reportedly shocked to discover irrational numbers. These are numbers that cannot be expressed as a fraction or written down in full because they contain an infinite amount of numbers with no known repeating pattern. They are said to have considered irrational numbers to be a flaw in nature, and had their discoverer killed[11].

A century after the founding of Pythagoreanism, Ancient Greek philosopher Socrates repeated Pythagoras' argument, that numbers are abstract entities that represent real things. Socrates' student, Plato, claimed that we can only be certain of knowledge that originates from the mind, not the senses, which can be tricked[12].

Plato believed that there are two realms, the realm of the 'forms', which contains perfect concepts, and the physical realm we perceive with our senses. The physical world contains imperfect copies of the true forms. Since mathematics could be understood without sensory experience - i.e. we do not need to see 100 objects in order to know what 50+50 is - it represents certain knowledge, and so Plato believed numbers exist as real entities in the realm of the forms[13].

In about 387 BCE, Plato set up an academy in Athens, which became the mathematical centre of the world. Its students included Ancient Greek philosopher Aristotle[14], Ancient Greek astronomer Eudoxus of Cnidus, and possibly Ancient Greek mathematician Euclid[15].

Euclid made a number of discoveries in geometry shortly after 300 BCE, and was the first to suggest that there are an infinite amount of prime numbers. Euclid's proof is contained in Elements, which became one of the most popular geometry textbooks for the next 2000 years[16].

The concept of infinity had previously been discussed by Ancient Greek philosopher Zeno of Elea. Zeno was about twenty years older than Socrates, and is said to have visited him in about 450 BCE, when Socrates was about twenty years old. Zeno considered how one infinity could be larger than another - there are twice as many numbers than there are even numbers, for example, but both are infinite. He stated that if you walked across a room, halving the distance you travel with each step, then you would never reach the other side[17].

Ancient Greek astronomer Hipparchus may have discovered trigonometry in about 150 BCE[18]. Trigonometry shows how the angle inside a triangle changes as the lengths of its sides change.

The sine of an angle is the ratio of two lengths: the length opposite the angle, and the longest length, the hypotenuse.

Sine(angle) =
Opposite length/Hypotenuse
.
Diagram of a right-angled triangle. The largest side is labelled ‘c’. The shortest ‘b’, and the other side is labelled ‘a’. The angle opposite side ‘a’ is labelled ‘A’, the angle opposite side ‘b’ is labelled ‘B’, and the angle opposite side ‘c’ is labelled ‘C’. From the perspective of angle ‘A’, length ’a’ is the opposite length, length ‘c’ is the hypotenuse, and length ‘b’ is the adjacent length.
In the triangle above, Sine(A) =
Length a/Length c
.

The sine of an angle will be a number between 1 and -1. The sine of 90° is 1 because the opposite length will also be the longest. In the triangle shown, sine(C) would equal length a/length a, which equals 1.

In the same way,

Cosine(A) =
Adjacent length/Hypotenuse
, or  
Length b/Length c
, and Tangent(A) =
Opposite length/Adjacent length
, or  
Length a/Length b
.

Indian mathematician Brahmagupta was the first to use zero as a number in 628 CE[19]. This was a controversial idea since it suggested that 'nothing' represented something that is real.

An even stranger new number, i, was devised almost one thousand years later. i is equal to the square root of minus 1.

i = √-1      i × i = -1

The number i seems impossible, after all the square root of a number (e.g. 4) is equal to another number that, multiplied by itself, becomes the first (e.g. 2 × 2 = 4 and so the square root of 4 is 2), and any number that is multiplied by itself should be positive. i multiplied by i is taken to equal -1 because this assumption helped solve mathematical problems, like cubic equations.

Italian mathematician Rafael Bombelli was the first to introduce the laws for multiplying i and -i in 1572[20]. Although the symbol was not introduced until the 1700s[21], French natural philosopher Rene Descartes first referred to i as an imaginary number in 1637[22a].

That same year, Descartes[22b] and French mathematician Pierre de Fermat[23] independently devised the Cartesian coordinate system, which is used to plot points on a graph. This helped English natural philosopher Isaac Newton and German mathematician Gottfried Leibniz develop calculus in the latter half of the century[24].

Cartesian coordinate system.

Cartesian coordinates. Image credit: K. Bolino/Public domain.

A sine wave in Cartesian coordinates, where the x-axis is the angle, and the y-axis is the sine of the angle, a value between 1 and -1.

The sine wave in Cartesian coordinates. Image credit: Helen Klus/Public domain.

Descartes did not accept Pythagoras' argument, that mathematical knowledge is certain. He devised the 'dream argument', which states that we can never be certain of anything because there's no way to prove that what we perceive with our senses is real. It could be the case that only our mind exists, and the external world is like a dream or hallucination[25a].

Descartes stated: "whether I am awake or asleep, two plus three makes five, and a square has only four sides", but we can never be certain this is true as a God, or a "malicious, powerful, cunning demon" could trick us into believing that incorrect mathematics is correct[25b].

3. The language of the universe

During the 1600s, astronomers such as Newton, German astronomer Johannes Kepler, and Italian natural philosopher Galileo Galilei, used mathematics to accurately describe the motion of the planets. Like Pythagoras, Galileo believed that the universe is composed in the language of mathematics.

In 1623, Galileo stated:

"philosophy is written in this grand book, the universe, which stands continually open to our gaze. But the book cannot be understood unless one first learns to comprehend the language and read the letters in which it is composed. It is written in the language of mathematics, and its characters are triangles, circles, and other geometric figures without which it is humanly impossible to understand a single word of it; without these, one wanders about in a dark labyrinth"[26].

Kepler was also inspired by Pythagoras, and believed that the motion of the planets produces music. He used mathematics to show that the planets orbit the Sun in ellipses and, by 1619, he was able to determine the time it takes each planet to orbit and their relative distances from the Sun[27].

In 1687, Newton published his law of universal gravitation[28]. This was groundbreaking because it showed, not just that abstract mathematical principles, such as the newly invented calculus, could be applied to what we observe in nature, but that the laws responsible for the movement of the planets are also responsible for the movement of objects on Earth. Newton also believed that the universe could be understood as a mathematical object, and described God as "skilled in mechanics and geometry"[29].

Newton's contemporary, Leibniz, discovered another link between mathematics and nature when he first considered the idea of fractals[30]. These are shapes that exhibit the same type of structures on all scales, such as frost, some plants, and coastlines. Twentieth century mathematicians, such as French mathematician Gaston Julia[31] and Polish-French-American mathematician Benoit Mandelbrot[32], were inspired by Leibniz to create complicated fractals of their own.

By this time, quantum mechanics, and German-Swiss-American physicist Albert Einstein's theories of special and general relativity, had shown that nature obeys the laws of mathematics, even when this contradicts our common sense understanding of the world.

Illustration of part of the Mandelbrot set, a fractal.

Part of the Mandelbrot set. Image credit: Wolfgang Beyer/CC-SA.

As well as independently developing calculus and inspiring the discovery of fractals, Leibniz was also one of the first to consider that a new number was special - the number e[33]. e is related to the laws of logarithms, which were devised by Swiss mathematician Jost Bürgi[34] and Scottish mathematician John Napier in 1614[35].

Logarithmic scales are used to show quantities that get rapidly larger.

Bürgi and Napier showed that if x = by then logb(x) = y.

A quantity that increases from 10 to 100 to 1000, for example, uses a base of 10:

If x = by then logb(x)= y, and so:

If 10 = 101 then log10(10) = 1.
If 100 = 102 then log10(100) = 2.
If 1000 = 103 then log10(1000) = 3...

After the invention of Cartesian coordinates, a graph could be drawn that allows quantities from one to one billion, for example, to be plotted on the same axis.

Table
Plot of numbers in the table in Cartesian coordinates, the x-axis ranges from 0 to 100 million, and so it is hard to distinguish between numbers below 100,000.
Plot of numbers in the table in Cartesian coordinates on a logarithmic scale, the x-axis ranges from 0 to 100 million but because the scale is logarithmic, all the numbers are visible.

Plots showing the numbers in the table. The second plot utilises a logarithmic scale. Image credit: Helen Klus/CC-NC-SA.

Logarithms to the base of ten are common, but any number can be used, and Bürgi and Napier made tables of logarithms in different bases. One base that is of particular interest is the base of about 2.718. Swiss mathematician Leonhard Euler first referred to this number as e in 1731[36] and, in 1748, he showed that e is an irrational number that is fundamentally connected to many laws of mathematics[37].

Euler showed that e = 1 +
1/1
+
1/1 × 2
+
1/1 × 2 × 3
+
1/1 × 2 × 3 × 4
... with the sequence going on forever.

Euler also showed that the number e is connected to the numbers i, π, 1 and 0:

e + 1 = 0,

and that e and i are connected to trigonometry using the formula:

eiθ = cosθ + i sinθ.

Within twenty years, Swiss mathematician Johann Heinrich Lambert had proven that π, like e, is an irrational number[38]. Irrational numbers cannot be written as fractions, yet π is the definition of the circumference of a circle, C, divided by its diameter, d (π = C/d). The irrationality of π means that either the circumference or the diameter of a circle, or both, must also be irrational. This shows that perfect circles can only exist if space is infinitely divisible.

4. A million dollar mystery

Since Euclid proved there are an infinite amount of prime numbers in around 300 BCE, the idea that we could list them all in a 'periodic table' of mathematics has been abandoned. Mathematicians instead searched for an equation that could be used to determine if a number is prime or not, without going through the long process of seeing if it's divisible by any number but itself and 1. This seemed impossible; primes appear to be completely random.

German mathematician Carl Friedrich Gauss was still a teenager in 1796 when he discovered that the distribution of prime numbers is related to logarithms[39]. Gauss found that prime numbers are more sparsely distributed the higher you count. Between 1 and 100, there are only about four numbers between any two primes, if you count up to 10,000, however, then this average doubles to 8. This becomes 12 if you count to almost a million, 16 at over ten million, and 20 if you count to over a billion.

Table: the first row has numbers ranging from 10 to 10 billion (N), the second has the average number between two primes from 1 to N, and the third has the difference between the two averages. For n=10-100, this is 1.5, for 100-1000 this is 2.0, by 10,000-100,000 this value is 2.3.

Image credit: Helen Klus/Public domain.

Plot of real numbers on the x-axis, and imaginary numbers on the y-axis.

Plot of real and imaginary numbers. Image credit: Allisonandvalerie/CC-SA.

Gauss saw that the average amount of numbers between any two primes increases by about 2.3 every time N increases by a factor of 10. This means that they increase on a logarithmic scale to the base of e.

If x = by then logb(x) = y, and so,

if 10 = e~2.3 then loge(10) = ~2.3.

Although Gauss' method is incorrect at first, he argued that it would become more and more accurate the higher you count. Belgian mathematician Charles de la Vallée-Poussin[40] and French mathematician Jacques Hadamard[41] proved Gauss correct in 1896. A logarithm to the base of e became known as a 'natural' logarithm.

Gauss was also responsible for creating a new coordinate system in order to map imaginary numbers.

Gauss' student, German mathematician Bernhard Riemann, liked to plot functions on this plot of imaginary numbers. He found that one function, known as the zeta function, creates a three-dimensional 'imaginary landscape' that's related to prime numbers[42].

Euler defined the zeta function as ζ(s) =
1/1s
+
1/2s
+
1/3s
+
1/4s
...+
1/ns
... where n is the next number in the sequence.
This is equal to ζ(s) =
1/1-2-s
+
1/1-3-s
+
1/1-5-s
+
1/1-7-s
+...
1/1-p-s
... where p is the next prime number in the sequence.

Riemann found that a plot of this function, with s as any complex number (a number with a real and imaginary component) other than 1, produces zeroes for many numbers. The zeroes associated with all of the negative even numbers are called trivial zeroes.

All of the other zeroes are called non-trivial zeroes. The real part of every non-trivial zero of the Riemann zeta function is 1/2, and the position of these zeroes is thought to be related to the 'error' between Gauss's formula for calculating the distribution of prime numbers and the real distribution. This is known as the Riemann Hypothesis, and no one has been able to prove it yet. Riemann may have had a proof but he died shortly after proposing the Riemann Hypothesis and much of his unpublished work was destroyed.

The Clay Mathematics Institute is offering a prize of US$1,000,000 to anyone who can prove the Riemann Hypothesis.

5. The interconnectedness of all things

In the 19th century, people begun to challenge Plato's argument, that mathematics contains true knowledge. In particular, it was not understood how Plato's abstract forms could interact with physical objects, since both obey different laws. Philosophers of science wanted to find another way to justify the certainty we have in mathematics, and many other schools of thought emerged, including intuitionism, formalism, and logicism[43a].

Intuitionism argues that mathematical statements are mental constructions. Formalism states that numbers do not exist, except as symbols that are not about anything, and that mathematics is built from the manipulation of these symbols by following rules. Logicism states that the whole of mathematics can be derived from the philosophical laws of logic.

The most famous set of logical mathematical axioms are the Peano axioms, named after the 19th century Italian mathematician Giuseppe Peano[44].

The Peano axioms state that:

1. Zero is a number.
2. If N is a number, the successor of N is a number.
3. Zero is not the successor of a number.
4. Two numbers, of which the successors are equal, are themselves equal.
5. If a set of numbers, S, contains zero, and also the successor of every number in S, then every number is in S.

Austrian mathematician Kurt Gödel proved that the whole of mathematics could not be derived from the Peano axioms in 1931. This is known as Gödel's first incompleteness theorem. Gödel's second incompleteness theorem states that the Peano axioms cannot be used to prove their own consistency[45].

Philosophers of mathematics have since developed mixtures of structuralism - which states that numbers are defined by their place within a mathematical structure - and formalism - which states that numbers, and other abstract entities, do not exist - in order to justify our belief in mathematics[43b].

In the 1980s, French neuroscientist Jean-Pierre Changeux argued that mathematics is a product of the human mind; it tells us more about how our brains function than the external world[46]. If mathematics is merely a human invention, then it would not provide a 'universal language' as many people hope.

While there is still no consensus on what numbers represent, most philosophers of mathematics agree that mathematics can be described as beautiful[47]. In the 5th century CE, Ancient Greek philosopher Proclus Lycaeus stated that:

"wherever there is number, there is beauty"[48].

British physicist Paul Dirac said that mathematical beauty was "like a religion" to him[49], and in 1939, Dirac suggested that:

"the research worker, in [their] efforts to express the fundamental laws of Nature in mathematical form, should strive mainly for mathematical beauty…It often happens that the requirements of simplicity and beauty are the same, but where they clash, the latter must take precedence"[50].

The following year British mathematician Godfrey Harold Hardy stated that he was:

"interested in mathematics only as a creative art…The mathematician's patterns, like the painter's or the poet's must be beautiful…Beauty is the first test: there is no permanent place in the world for ugly mathematics"[51].

Mathematicians describe some methods of proof as 'elegant', and describe results that connect two apparently unrelated concepts as 'deep'[52].

Euler's formulas, e + 1 = 0 and eiθ = cosθ + i sinθ, are considered beautiful as they show how the most important numbers are related, and their proof could be described as deep. American physicist Richard Feynman described the latter equation as "the most remarkable formula in mathematics"[53]. Proof of the Riemann Hypothesis could also be considered deep, as it will relate prime numbers to the numbers e and i.

Photograph of pollen taken using an electron microscope.

Pollen under an electron microscope. Image credit: Dartmouth Electron Microscope Facility/Public domain.

Photograph of stars in a globular cluster.

100,000 or so stars within the globular cluster Omega Centauri. Image credit: NASA, ESA, and the Hubble SM4 ERO Team/CC-A.

The pleasure that we find in mathematical structures is evident when we consider why we find beauty in things that are either too small, or too far away, to see without the aid of technology. We may find the shapes of pollen and stars to be beautiful, for example, because we recognise the mathematical structures they are composed of.

UPDATE: As of 2016, the Riemann Hypothesis has still not been proven. Some references have been updated.

6. References

  1. Bangura, A. K., 2012, 'African Mathematics: From Bones to Computers', University Press of America.

  2. Brooks, A. S. and Smith, C. C., 1987, 'Ishango revisited: new age determinations and cultural interpretations', The African Archaeological Review, 5, pp.65-78.

  3. Schmandt-Besserat, D., 1992, 'From Counting to Cuneiform', University of Texas Press.

  4. Robson, E., 2007, 'Mesopotamian mathematics' in 'The mathematics of Egypt, Mesopotamia, China, India, and Islam: A sourcebook', Katz, V. J. and Imhausen, A., 2007, Princeton University Press.

  5. Bailey, D. H., Plouffe, S. M., Borwein, P. B., and Borwein, J. M., 1997, 'The quest for pi', The Mathematical Intelligencer, 19, pp.50-56.

  6. Robson, E., 2002, 'Words and pictures: New light on Plimpton 322', The American mathematical monthly, 109, pp.105-120.

  7. School of Mathematics and Statistics, University of St Andrews, 'Thales of Miletus', last accessed 15-02-16.

  8. School of Mathematics and Statistics, University of St Andrews, 'Pythagoras of Samos', last accessed 15-02-16.

  9. Caleon, I. and Ramanathan, S., 2008, 'From music to physics: The undervalued legacy of Pythagoras', Science & Education, 17, pp.449-456.

  10. Marcus, R., 2016, 'An Historical Introduction to the Philosophy of Mathematics: A Reader', Bloomsbury Publishing.

  11. Huffman, C., 'Pythagoreanism', Stanford Encyclopedia of Philosophy, last accessed 15-02-16.

  12. Plato and Jowett, B. (trans), 2012 (c. 380 BCE), 'The Republic', Project Gutenberg.

  13. Linnebo, Ø., 'Platonism in the Philosophy of Mathematics', Stanford Encyclopedia of Philosophy, last accessed 15-02-16.

  14. Chroust, A. H., 1967, 'Plato's academy: The first organized school of political science in antiquity', The Review of Politics, 29, pp.25-40.

  15. Hayhurst, C., 2006, 'Euclid: The Great Geometer', The Rosen Publishing Group.

  16. Apostol, T. M., 2013, 'Introduction to Analytic Number Theory', Springer Science & Business Media.

  17. Huggett, N., 'Zeno's Paradoxes', Stanford Encyclopedia of Philosophy, last accessed 15-02-16.

  18. Hosch, W. L., 2010, 'The Britannica Guide to Algebra and Trigonometry', The Rosen Publishing Group.

  19. Roy, R., 2003, 'Babylonian Pythagoras’ theorem, the early history of zero and a polemic on the study of the history of science', Resonance, 8, pp.30-40.

  20. Bombelli, R., 1572, 'L'algebra' ('Algebra'), Feltrinelli Edittore.

  21. Euler, L., 1794 (1777), 'De formulis differentialibus angularibus maxime irrationalibus, quas tamen per logarithmos et arcus circulares integrare licet' ('On forms of the differential equations of the angular, especially irrational, which, however, is permitted to integrate in terms of logarithms and circular arcs'), Institutiones calculi integralis, 4, pp.183-194.

  22. (a, b) Descartes, R., Smith, D. E. (trans), and Latham, M. L. (trans), 1925 (1637), 'Geometry', The Open Court Pub. Co.

  23. Stahl, S., 2012, 'Real Analysis: A Historical Approach', John Wiley & Sons.

  24. Sastry, S. S., 2006, 'The Newton-Leibniz controversy over the invention of the calculus', University of Wisconsin.

  25. (a, b) Descartes, R., Cottingham, J. (trans), Bennett, J. (trans), 2006 (1641), 'Meditations on First Philosophy', Cambridge University Press.

  26. Galilei, G., Drake, S. (trans), and Drake, S. (ed), 1957 (1623), 'The Assayer' in 'Discoveries and Opinions of Galileo', New York: Doubleday & Co.

  27. Kepler, J. and Aiton, E. J. (trans) and Duncan, A. M. (trans) and Field, J. V. (trans), 1997 (1619), 'The Harmony of the World', American Philosophical Society.

  28. Newton, I. and Motte, A. (trans), 1846 (1687), 'The Mathematical Principles of Natural Philosophy', Daniel Adee.

  29. Newton, I. and Motte, A. (trans), 1846 (1726), 'General Scholium' in 'The Mathematical Principles of Natural Philosophy', Daniel Adee.

  30. Rutherford, D. and Cover, J. A., 2005, 'Leibniz: Nature and Freedom', Oxford University Press.

  31. Julia, G., 1918, 'Mémoire sur l'iteration des fonctions rationnelles' ('Memoir on the iteration of rational functions'), Journal de Mathématiques Pures et Appliquées, 8, pp.47–245.

  32. Mandelbrot, B., 1982, 'The Fractal Geometry of Nature', W. H. Freeman and Co.

  33. Cajori, F., 1913, 'History of the exponential and logarithmic concepts', The American Mathematical Monthly, 20, pp.35-47.

  34. Waldvogel, J., 2012, 'Jost Bürgi and the discovery of the logarithms' in 'Seminar für Angewandte Mathematik', ETH-Zürich.

  35. Napier, J. and MacDonald, W. R. (trans), 1889 (1614), 'The construction of the wonderful canon of logarithms', William Blackwood and Sons.

  36. Euler, L., 1731, 'Letter to Goldbach, 25 November 1731', Correspondance mathématique et physique de quelques célèbres géomètres du XVIIIème siècle, P. H. Fuss.

  37. Euler, L. and Blanton, J. D., 1988, (1748), 'Introduction to infinitesimal analysis', Springer New York.

  38. Lambert, J. H., 2004 (1768), 'Mémoire sur quelques propriétés remarquables des quantités transcendantes circulaires et logarithmiques' ('Memory on some remarkable properties of transcendental quantities circular and logarithmic'), M ́em. de l’Acad. R. des Sci. de Berlin, 17, pp.265-332.

  39. Gauss, C. F. 1874 (1863), 'Werke' ('Works'), Gedruckt in der Dieterichschen Universitäts-Buchdruckerei.

  40. Poussin, C. J. D. L. V., 1896, 'Recherches analytiques de la théorie des nombres premiers' ('Analytical research of the theory of prime numbers'), Annales de la Societe Scientifique de Bruxelles, 20, pp.183–256.

  41. Hadamard, J., 1896, 'Sur la distribution des zéros de la fonction zeta(s) et ses conséquences arithmétiques' ('On the distribution of zeros of the zeta(s) function and its arithmetic consequences'), Bulletin de la Societé mathematique de France, 24, pp.199-220.

  42. Riemann, B. and Wilkins, D. R. (trans), 1998 (1859), 'On the number of primes less than a given magnitude', Monatsberichte der Berliner Akademie.

  43. (a, b) Horsten, L., 'Philosophy of Mathematics', Stanford Encyclopedia of Philosophy, last accessed 15-02-16.

  44. Peano, G., 1889, 'The principles of arithmetic presented by a new method', in 'From Frege to Gödel: A Source Book in Mathematical Logic, 1879-1931', 1967, Harvard University Press.

  45. Gödel, K., 1931, 'Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I' ('On formally undecidable propositions of Principia Mathematica and Related Systems I'), Monatshefte für mathematik und physik, 38, pp.173-198.

  46. Changeux, J. P., Alain Connes, A., and DeBevoise, M. B. (trans), 1999 (1989), 'Conversations on Mind, Matter, and Mathematics', Princeton University Press.

  47. McAllister, J. W., 2005, 'Mathematical beauty and the evolution of the standards of mathematical proof' in 'The Visual Mind II', MIT Press.

  48. Kline, M., 1990, 'Mathematical Thought From Ancient to Modern Times, Volume 3', Oxford University Press.

  49. Dirac, P. and Weiner, C. (ed), 1977, 'History of Twentieth Century Physics', Academic Press.

  50. Dirac, P., 1939, 'The Relation between Mathematics and Physics', Proceedings of the Royal Society, 59, pp.1938-1939.

  51. Hardy, G. H., 1967, 'A Mathematicians Apology', Cambridge University Press.

  52. Brockman, J., 2013, 'This Explains Everything: Deep, Beautiful, and Elegant Theories of How the World Works', HarperCollins.

  53. Feynman, R. P., Leighton, R. B., and Sands, M., 1965, 'The Feynman Lectures on Physics, Volume I', Basic Books.

Blog | Space & Time | Light & Matter | Mind & Multiverse | Timeline

RSS Feed | Images | About | Copyright | Privacy | Comments