The maturation of purely mathematical mind

That period before Euclid Greek and Roman mathematicians

The Greeks divided mathematics into two areas, both of which had their origins in practical applications: arithmetic (the study of “multitude,” or discrete quantity) and geometry (the study of “magnitude,” or continuous number). According to Proclus’s Commentary on Euclid, ancient Egyptian surveying procedures gave rise to geometry (which literally translates to “measure of land”) because yearly flooding along the Nile prompted the Egyptians to redraw property borders. In a similar vein, it was Phoenician merchants that advanced the field of mathematics. Even though Proclus wrote his novel in the fifth century CE, his concepts may be traced back to the likes of Herodotus (mid-fifth century BCE) and Eudemus (a student of Aristotle) from even earlier periods (late 4th century BCE). If you are struggling with percentages or any other mathematical process, Visit their website to solve percentage queries.

Since there is so little evidence of applied mathematics from the early Greek era, the idea is plausible but difficult to test (roughly, the 8th through the 4th century BCE). Stone tablets, for example, attest to the widespread adoption of a numerical system conceptually comparable to the widely used Roman numerals. There are perhaps a dozen abacuses made of stone that date back to the fifth and fourth centuries BCE, indicating that Herodotus was likely aware of the abacus’s use as a calculation instrument in both Greece and Egypt. When surveying new cities in Greek colonies in the sixth and fifth centuries BC, standard lengths of 70 plethra (one plethron equals 100 feet) were frequently used as the diagonal of a square of side 50 plethra; in reality, the diagonal of the square is 50Square root of2 plethra, so using 7/5 (or 1.4) as an estimate for Square root of2 is equivalent to using 7/5 (or 1.4) as an estimate for Square root of Eupalinus of Megara, an engineer from the sixth century BCE, is credited for channelling an aqueduct through a mountain on the island of Samos, although the method he used is controversial among historians. In his Laws, Plato seems to argue that the Egyptians’ method of teaching math to youngsters via practical issues from everyday life was an example the Greeks should follow.

These hypotheses on the character of early Greek practical mathematics are supported by later sources, such as the arithmetic problems in papyrus writings from Ptolemaic Egypt (starting in the third century BCE) and the geometric manuals by Heron of Alexandria (1st century CE). Essentially, this Greek habit was quite close to that of ancient Egypt and Mesopotamia. There is little doubt that the Greeks borrowed ideas and concepts from much earlier civilizations.

The Greeks are commonly credited as the “creators of mathematics,” yet it was the theoretical foundations of their discipline that truly set them apart. This means that mathematical statements hold true everywhere and can be proven correct. For instance, the Mesopotamians created techniques for testing whether or not the equation a2 + b2 = c2 holds for a given set of whole integers a, b, and c. (e.g., 3, 4, 5; 5, 12, 13; or 119, 120, 169). The Greeks demonstrated a general way for producing such sequences, commonly known as Pythagorean triples: for every pair of whole integers p and q, even or odd, a = (p2 + q2)/2, b = pq, and c = (p2 + q2)/2. Such numbers satisfy the relation for Pythagorean triples, as established by Euclid in Book X of the Elements. This conclusion was proved by the Greeks as part of a more comprehensive explanation of the characteristics of flat geometric forms (Euclid proves it twice, once in Book I, Proposition 47, and once in Book VI, Proposition 31). It appears that the Mesopotamians knew that right triangles contain sides that consist of sets of the numbers a, b, and c.

While Euclid’s The Elements (about 300 BCE) is often seen as a seminal work in theoretical geometry, the transition from practical to theoretical mathematics may be traced back to the fifth century BCE at the earliest. Others, including Pythagoreans Archytas of Tarentum, Theaetetus of Athens, and Eudoxus of Cnidus, expanded the theoretical form of geometry by building on the foundations laid by Pythagoras of Samos (late 6th century) and Hippocrates of Chios (late 5th century) (4th century). There are no surviving copies of any of these men’s writings, therefore what we know about them comes from the opinions of other authors. While this sliver of evidence does demonstrate the extent to which Euclid relied on them, it does nothing to explain what motivated their research.

Discussion centres on the how and why of this theoretical change. Commonly cited is the finding of irrational numbers. The early Pythagoreans held the concept that “all things are number” as a fundamental principle. Even though the Greek word for number, arithmos, can only be used to describe whole numbers and, in some cases, ordinary fractions, it is possible to assign a number to any geometric measure (that is, some whole number or fraction; in modern terminology, rational number). In common speech, this is commonly taken for granted, as when the length of a line is expressed as a full number of feet plus a fraction of a foot. The lines that form the square’s sides and diagonal are an exception to this rule. (For instance, assuming that the ratio of two whole integers can be stated for the side and diagonal ratios, it can be demonstrated that they must be even. Since any fraction may be expressed as the ratio of two whole numbers with no common denominator, it is obvious that this cannot occur. This has the geometric implication that no length may be used as a unit of measure for both the side and the diagonal; that is, the side and the diagonal cannot both equal the same length multiplied by (different) whole integers. This is why the Greeks used the term “incommensurable” to characterise such comparisons of lengths. (Modern mathematicians use the word “number” to refer to irrational numbers like Square root of 2, which the Greeks did not.)

Although this was already widespread knowledge by the time of Plato, some late writers, such as Pappus of Alexandria (4th century CE), claim that it was discovered inside the school of Pythagoras in the 5th century BCE. By 400 BCE, it was generally accepted that lines corresponding to the square root of 3, the square root of 5, and other square roots are not directly equivalent to a standard unit of length. An much more complete discovery, that square root of p is irrational whenever p is not a rational square integer, is attributed to Plato’s companion Theaetetus. Book X, Section II, Proposition 115 of the Elements demonstrates that the effort of their pupils finally consolidated into a cohesive system, building on the foundation laid by Theaetetus and Eudoxus.

The discovery of irrationals unquestionably changed the trajectory of early mathematical investigation, regardless of any assumptions made in practical practise. As the irrationals demonstrated, mathematics on its own couldn’t accomplish what geometry needed to do. All mathematical assumptions were theoretically rendered suspect once seemingly obvious ones, such as the commensurability of all lines, were revealed to be incorrect. A minimum level of justification was required for all mathematical claims. The need to identify what makes a certain chain of reasoning worthy of the label “evidence” arose as a more basic issue. Evidently before his death in the fifth century BCE, Hippocrates of Chios and his contemporaries began gathering geometric findings into textbooks dubbed “elements” (meaning “fundamental outcomes” of geometry). A century later, when Euclid was writing his comprehensive textbook, they would be among his key sources.

There was fierce rivalry among the early mathematicians, who were part of a larger intellectual community that included pre-Socratic philosophers in Ionia and Italy and Sophists in Athens. Parmenides, a Greek philosopher from the fifth century BCE, challenged the basic basis of knowing when he claimed that only unchangeable objects could have actual existence. Heracleitus (c. 500 BCE) claimed, on the other hand, that the stability of our senses is an illusion created by a balance of opposing forces. Knowledge and proof both have their respective meanings questioned as a consequence.

In several of the disagreements, mathematical issues served as a focal point. The Pythagoreans (and Plato, who came after them) used the certainty of mathematics as a model for deducing truths in areas such as politics and ethics. Yet others thought that mathematics was riddled with contradictions. Paradoxes about motion and quantity have been attributed to Zeno of Elea, who lived in the fifth century BCE. The premise that a line may be bisected an endless number of times gives rise to a paradox since the outcome can be either a set of points of zero length (in which case the total of an infinite number of such points is zero) or a collection of minuscule line segments (in which case the sum is infinite). In actuality, the length of the provided line must be both and. In the fifth century BCE, Democritus and other atomist philosophers attempted to answer this question by positing that everything in the cosmos is made up of infinitesimally small particles called “atoms” (from the Greek atomon, meaning “indivisible”). However, the idea of incommensurable lines in geometry ran counter to this view since atoms would then be employed to quantify all lines. Tangents to circles can be confusing; not even Sophist Protagoras and Democritus could agree on whether they meet the circle at a point or a line. During the fifth century BCE, Socratic philosophers Antiphon and Bryson grappled with the issue of equating the circle and the polygons that may be made within it.

The pre-Socratics were the first to point out the problems with fundamental ideas like “existence” and “proof,” as well as more specific ones like “infinitely many” and “infinitely tiny.” These philosophical considerations may or may not have impacted mathematicians’ technical research, but they definitely made them more careful of making overly broad claims about their field’s breadth.

Any examination of the possible ramifications of such circumstances is, at best, hypothetical due to the incoherence of the sources and the lack of clarity with which the mathematicians responded to the issues posed. Greek mathematics is distinctive due to its meticulous examination of fundamental assumptions and emphasis on strong proofs. While it would be impossible to offer a comprehensive analysis of the causes of these changes, we may look to the technical advances and cultural climate of the early Greek tradition as two possible explanations.