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I know about the Plimpton 322 tablet and Pythagorean triples. But can we be sure that this is the first instance of mathematical reasoning? I am talking about notions, propositions, or questions about mathematics.

I ask you if you can say what the milestone of mathematical reasoning or mathematical knowledge is, as accepted by the scientific community.

I have two ideas as to what ancient notions of mathematics could be like. Let's say that humans were a band of Paleolithic hunters or gatherers. If the group gets $N$ items of food, then, leaving out social synergies, the appropriate distribution is the Euclidean division between the participants. If there are paintings of two animals in a cave, does this mean that Paleolithic men/women had the notion of the integer $2$? Are there scientific thoughts from the viewpoint of history of mathematics?

What is the first occurrence of mathematical reasoning or mathematical knowledge?

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  • $\begingroup$ Too, I am interesting in the following question, add a comment if you want: in all historical meetings between different cultures (I imagine for example the meeting between chinese civilization and Marco Polo, or more current/modern meetings with isolated civilizations in jungles or islands), did also knew/had, I say these isolated cultures, notions about the euclidean division? $\endgroup$ – user3537 Jan 16 '16 at 17:17
  • $\begingroup$ If for this second question, this is the question that I've written in previous comment, is better that I edit a new post/question in this site, please say me. $\endgroup$ – user3537 Jan 16 '16 at 17:33
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    $\begingroup$ Hi, JuanLG. I've heavily edited the grammar on this post to make it clearer. It would be great if you could check my edits to make sure that I didn't accidentally change your meaning. Regarding your second question: Yes, it might be better to ask that separately. $\endgroup$ – HDE 226868 Jan 16 '16 at 17:50
  • $\begingroup$ Very thanks much for your edits that improve my grammar. Thanks @HDE226868 $\endgroup$ – user3537 Jan 16 '16 at 21:19
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To address this question, one has to define more precisely what "mathematical reasoning" is. When you start to count things, this already can be called "mathematical reasoning". And there is no doubt that this kind of mathematical reasoning is much older than the invention of writing. So it does not belong to history in the proper sense of this word, and there are no written records by definition. Therefore, it is impossible to say when "mathematical reasoning" really started.

There are old Egyptian texts, like Rhind's Papyrus, and even older "Moscow Papyrus", dated roughly as 1850 BC. They certainly contain some sort of mathematical reasoning. Manipulation with integers and fractions, measuring of areas and volumes, as well as the fact that some numerical relations between side lengths of triangles imply a right angle, all this belongs to this kind of mathematical reasoning. The observation that some numbers are even and others are odd, and that the sum of even numbers is even, as well as the sum of two odd numbers, is also a mathematical reasoning. If one sheep exchanges for two widgets, then five sheep exchange for 10 widgets, this is also a mathematical reasoning, and it is certainly very ancient.

So mathematical reasoning is done practically by all humans and is probably as old as humankind itself.

This has to be distinguished from Mathematics in the narrow sense. Mathematics as mathematicians understand it is a special kind of reasoning which involves proofs. As far as we know this was invented by ancient Greeks. There is no slightest evidence that this was invented anywhere else independently. No primary written record survives, but according to tradition which goes back to Hellenistic times, Thales and Pythagoras (or his disciples) in early 6 century BC are credited with this invention.

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  • $\begingroup$ I am mathematician but I have some heretical thoughts , I say this to excuse my strange question . Thanks very much for your answer @AlexandreEremenko $\endgroup$ – user3537 Jan 16 '16 at 21:25
  • $\begingroup$ As far as I understand other answers here, Pythagoras himself did little or no mathematics, His followers seem to have done some. $\endgroup$ – vonbrand Jan 16 '16 at 22:25
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    $\begingroup$ @vonbrand: Little is known about Pythagoras and his early followers. As I said: "according to Hellenistic tradition...". There is no other primary source. $\endgroup$ – Alexandre Eremenko Jan 16 '16 at 23:35
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    $\begingroup$ @vonbrand: same applies to Thales. People who wrote about them in Hellenistic times, also had some distorted knowledge. Nothing survives from earlier times. $\endgroup$ – Alexandre Eremenko Jan 16 '16 at 23:37
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I agree that there are types of reasoning that have been in use for many millenia before writing and which could be described as "mathematical".

There are notions, such as transitivity, that seem fundamental to human reasoning. For example, suppose I owe Juan a share of the next mammoth hunt. If I give Alexandre a mammoth steak and ask him to give it to Juan, then it is reasonable for me to expect my debt to Juan to have been fulfilled. Certain human transactions are commutative, others are not.

I'll hazard a guess that certain kinship structures could be viewed as a type of mathematical reasoning. These relationships (e.g. incest taboos, marriage eligibility, inheritance, reciprocity etc.) have been in use for a very long time, and can involve some very intricate, sophisticated reasoning and analysis.

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Cave paintings in Lascaux, a late Upper Paleolithic cave complex in southwestern France, dated back to c.15–18,000 B.C. and associated with the Magdalenian culture are often seen as some of the earliest artifacts displaying the beginnings of scientific and mathematical thought. "Mathematical aspects in late Upper Paleolithic mythopoetics derived from observational astronomy. Some cave paintings recorded an understanding of the path the moon takes around the sun – the ecliptic. An understanding of the ecliptic leads to the discovery of the zodiac, the annual path of the sun through the celestial sphere... An important cave painting at Lascaux, France depicts six large dots above a magnificent portrait of an Auroch that is known as Bull #18... At Lascaux in 15,300 B.C. the Pleiades were very near the point of the autumn equinox... The six stars in the Salle des Taureaux therefore represent a striking and excellent heavenly marker for the beginning of autumn and of spring". There is some speculation involved here, but the helical rise and fall of the Pleiades has been used for calendaric purposes in many later cultures.

Ishango bone dating back to about 9000-6500 BC (Upper Paleolith) is another candidate. It has an intricate arrangement of notches that suggests mathematical facility that goes beyond simple counting. Gerdes writes in On the History of Mathematics in Africa South of the Sahara:"The bone's discoverer, De Heinzelin, interpreted the patterns of notches as an "arithmetical game of some sort, devised by a people who had a number system based on 10 as well as a knowledge of duplication and of prime numbers". Marshack, on the contrary, explains the bone as early lunar phase count". Pletser recently argued that it is a primitive slide rule.

Much later, but still before Egyptian papyruses, Sumerian and Babylonian sources show a relatively advanced practical mastery of mathematics. Muroi's Oldest Example of Compound Interest in Sumer traces the origins of compound interest back to c. 2400 BC, when an inscription on the foundation cone of a temple reads that the city of Lagash lent barley at interest to its neighbor Umma, which Umma did not repay. The interest calculation involves computing the seventh power of four-thirds. A cuneiform tablet YBC 3879 from the third Sumerian Ur period c. 2000 BC was studied by Freiberg, and shows proficiency with elementary geometry, arithmetic, measurement relation between the two, and even ability to solve quadratic equations, see The origin of quadratic equation in actual practice.

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    $\begingroup$ Very thanks much @Conifold, your words are very nice and incredible, I say your references about the ecliptic in paleolithic ages, notions about more complex counting and arithmetic that I believe, and after your notes about computations for egypcian's interest when they lent barley between cities. Sorry by my english, and very thanks much for this remark, I will read in more detail. $\endgroup$ – user3537 Jan 19 '16 at 7:59

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