A Brief History: Mathematical Development

INPG 353

Jacob Knight

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7 Periods of Development

Proto- Mathematics

Ancient Mathematics

Middle-Age Mathematics

Classical Mathematics

Pre-Modern Mathematics

01

02

04

05

03

06

07

Modern Mathematics

Post-Modern Mathematics

Proto-Mathematics

01

40,000 BCE - 2,000 BCE

40,000 BCE

2,000 BCE

Critical Details:

1. Defining Characteristics:
1. Empirical Math
2. Basic Math
3. Non-Abstract Math

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• Most math in this period pertained to day-to-day life or the natural world such as counting, keeping time, displaying shape and symmetry through arts and crafts, and measuring and building.

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• 40,000 BCE is the starting point since the most ancient artifacts that display evidence of mathematical thinking or reasoning date back to around 30,000-35,000 BCE. However, if more ancient evidence is displayed in the future, then the timeline can easily be shifted back to incorporate these findings.

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Mathematical and/or Scientific Significance:

By recognizing It is important to recognize where mathematics started to better appreciate the beauty of its growth over time.

Proto-Mathematics

40,000 BCE

2,000 BCE

Critical Details:

• The earliest evidence of mathematics to date. (35,000 BCE)

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• 29 distinct notches carved into a Baboon’s Fibula.

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• Resembles calendar sticks that are still used today by the San or Bushman people of Namibia.

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Mathematical and/or Scientific Significance:

This artifact implies that people were thinking mathematically as far back as 35,000 BCE and gives us a good idea of the type and scope of mathematics that these people were doing.

Lebombo Bone

35,000

40,000 BCE

2,000 BCE

Critical Details:

• The second earliest evidence of mathematics to date. (25,000 BCE)

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• Initial observations led historians to believe that row (a) correlates to a basic numeration system based on 10, row (b) correlates to the prime numbers between 10 and 20, and row (c) illustrates a method for multiplication by 2 which resembles a method used later in Egyptian multiplication.

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• After further microscopic scrutiny, scholars believe that this bone also served as a lunar phase counter of sorts.

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Mathematical and/or Scientific Significance:

This bone gives us further evidence of the type and scope of their mathematical practices. We can use this as a later reference to see the growth and development of the field.

Ishango Bone

25,000

40,000 BCE

2,000 BCE

Critical Details:

• The accounting tokens were small three-dimensional clay objects that represented quantities, units, or goods. (4,000 BCE)

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• The Mesopotamian tablet from Sumer is the oldest known clay tablet that displays mathematical computations. Shows a multiplication table in cuneiform. (2,700 BCE)

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• The Sumerian Multiplication table displays three columns with the first two representing distances from 6 meters to 3 kilometers and the third column representing the product of the first two. The Sumerians also developed the base-60 number system. (2,600 BCE)

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Mathematical and/or Scientific Significance:

Now, these artifacts are important since they display the depth of Mathematics during this period. Also, the Sumerian development of the base-60 number system is still used today in how we track time.

Mesopotamian Accounting Tokens, Sumerian Multiplication Tables, and Mesopotamian Tablets

2,700

3,200

2,600

4,000

Ancient Mathematics

02

2,000 BCE - 800 BCE

2,000 BCE

800 BCE

Critical Details:

1. Defining Characteristics of Ancient Mathematics:
1. Empirical Math
2. Transition to abstract numbers and figures
3. Not Axiomatic
4. Some advanced civilizations (Babylonians and Egyptians)

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• In concert with the empirical mathematics of the past, people used newfound ideas of abstraction to develop early arithmetic and geometry. The Babylonians and Egyptians then took these basic ideas and consistently sophisticated them to levels where they could build great architectural feats (pyramids) and solve quadratics and some cubics.

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• Development from the non-abstract to some abstraction with numbers and figures.

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Mathematical and/or Scientific Significance:

This period marks a significant advancement in mathematical understanding since abstracting numbers and figures is a key aspect of later mathematics such as calculus. In addition, the advanced civilizations of the time were able to apply their understanding of mathematics to develop other fields like astronomy, land surveying, and construction.

Ancient Mathematics

2,000 BCE

800 BCE

Critical Details:

• Many Babylonian tablets such as Plimpton 322 and YBC 7289 reveal that they understood and

could use Pythagorean Triples as well as construct right triangles. (1750 BCE & 1,800 BCE)

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• In addition, some tablets such as YBC 11120 display their ability to calculate the area of a circle using 3 as an approximation for 𝝅. (1,600 BCE)

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• Further, some tablets such as YBC 7290 reveal their ability to calculate the area of trapeziums by multiplying the averaging side length by the average base length. (1,600 BCE)

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Mathematical and/or Scientific Significance:

The Babylonian tablets reveal much about their understanding of mathematics, and in particular, they give further context to what we now consider the Pythagorean Theorem. It is important to recognize that the Babylonians were using Pythagorean triples 1000 years before Pythagoras became known for further developing this concept. Granted the formality at which the Babylonians were using these concepts is less sophisticated than those in the classical era, but it can be argued that without their early development of these ideas, then Greeks like Pythagoras would not have been able to further them hundreds of years later.

Babylonians

1,800

1,750

1,600

2,000 BCE

800 BCE

Critical Details:

• Based on the angle used during the construction of the Pyramids, it is very likely that the Egyptians knew of the special Pythagorean relationship and were able to construct right angles with rope.

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• The Tomb of Menna, a scribe, reveals various methods of measuring and calculating which mainly connect the daily life. One such example is a measuring tactic where long distances are being measured using ropes that have knots tied with consistent intervals, similar to our modern-day rulers. (1420 BCE)

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• The Rhind Papyrus contains 84 multiplication, division, fraction, and geometry problems. One significant part of this document is the (2/n) table which displays rational numbers being converted to exact fractions. (1,550 BCE)

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Mathematical and/or Scientific Significance:

The evidence of Egyptian Knowledge furthers our conception that many of the Theorems and Processes that we use today were not created at one time but rather build upon and sophisticated from more basic forms. In particular, the Egyptians give us further evidence of the development of the Pythagorean theorem.

Egyptians

1,550

1,420

Classical Mathematics

03

800 BCE - 300 CE

800 BCE

300 CE

Critical Details:

1. Defining Characteristics of Classical Mathematics:
1. More Sophisticated Geometry
2. Some advancements in abstraction
3. Axiomatic Geometry (Greeks)

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• This period focuses mainly on the Greek contribution to mathematics; however, one should note that although the Greeks were the main contributor during this time, they were not the only contributors.

​

• During this time, the Greeks began to regard numbers as useful but not always reliable; therefore they wanted an abstract representation of a “number” to replace the concrete numbers for measurements.

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Mathematical and/or Scientific Significance:

This is the period where a majority of the ideas that had been floating around being used by all different groups became more formalized and information was spread between many disjoint groups. However, at this time there was no way for the world’s countries and nations to communicate quickly and effectively with one another. This is why it required additional time for ideas to correlate and coalesce into the more unified mathematics that we know today.

Classical Mathematics

800 BCE

300 CE

Critical Details:

1. Pythagoras of Samos is a Greek philosopher and mathematician who lived from 570 BCE to 495 BCE

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• He is credited with proving the Pythagorean theorem

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• He also tried to show that music is mathematical and discovered that two tones sound pleasing if the ratio of their frequencies is a simple fraction.

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Mathematical and/or Scientific Significance:

The Pythagorean theorem continues to be fundamental for our geometry today, so we are lucky that Pythagoras compiled the ideas that drove the relationship used by Babylonians, Egyptians, and other peoples so that we can teach it and understand it nowadays.

Pythagoras

570

495

800 BCE

300 CE

Critical Details:

1. Established by Plato in 387 BCE because he believed that knowledge could be sought through observation and therefore taught to others.

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• It is widely considered the world’s first university and served as a hub for great mathematicians, philosophers, and scientists to gather together and share ideas.

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• The academy was free to be a part of which allowed anyone who wanted to become invested could.

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Mathematical and/or Scientific Significance:

Since this academy served as a hub for knowledge, the members were able to develop concepts through collaboration and spread math and science topics. In addition, they began to formalize the different methods and concepts that had been developed in both math and science.

387

Platonic Academy of Athens is Established

(i.e. Plato’s School of Athens)

800 BCE

300 CE

Critical Details:

1. Euclid of Alexandra is a Greek philosopher and mathematician who lived from approximately 325 BCE to 265 BCE.

​

• He is considered the Father of Geometry

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• His book The Elements introduced the world to Euclidean Geometry which was based on axioms and outlined several important proofs in geometry and number theory.

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Mathematical and/or Scientific Significance:

Euclid is perhaps the most important person from the Classical period of Mathematics. He is the reason why we can consider this period axiomatics, and his Euclidean geometry is our standard geometry to this day.

Euclid

325

265

800 BCE

300 CE

Critical Details:

• We can track the depth of mathematics being done by other advanced countries by looking at China.

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• Historians have uncovered a Bamboo multiplication table that is in base 10. This table is the oldest known decimal multiplication table that has been uncovered. (300 BCE)

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• The Suàn shù shū consists of 69 problems on topics such as arithmetic, fractions, integer factorization, geometric sequences, inverse proportions, unit conversion, and error handling. In addition, there are volume problems where they use 3 as an approximation for 𝝅, similar to the Babylonians in the past. (200 BCE)

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Mathematical and/or Scientific Significance:

Although it isn’t discussed as often as European mathematics, Chinese mathematics was on the same level as and perhaps more advanced in some areas. We must recognize that mathematics today was not developed from one source or only from one area and rather is an amalgamation of various cultures and civilizations all putting their work together.

China

300

200

800 BCE

300 CE

Critical Details:

• Hipparchus of Nicaea is a Greek astronomer and mathematician who lived from 190 BCE to 120 BCE.

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• He is considered the Father of Trigonometry that is connected to astronomy

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• He created the first comprehensive star catalog in the western world, invented the astrolabe, and solved problems in spherical trigonometry.

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Mathematical and/or Scientific Significance:

Hipparchus may be considered the father of trigonometry, but it is important to recognize that his trigonometry was connected to astronomy. This is not to take away from his contribution but rather puts his work into further context.

Hipparchus

190

120

Middle-Age Mathematics

04

300 CE - 1,300 CE

300 CE

1,300 CE

Critical Details:

• Defining Characteristics of Middle-Age Mathematics:
• Sophisticated abstraction in algebra
• Algorithmization of arithmetic

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• Although the main contributors of this period were Indian, Arabic, and Central Asian, there also existed significant European contributors such as Fibonacci from Italy.

• This period also suffered from a lack of wide-scale communication, similar to the classical period.

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Mathematical and/or Scientific Significance:

This period’s advancement in algebra and arithmetic resembles the advancement made in Geometry during the Classical Period. Mainly, the modern notation for algebraic expression solutions was developed and formed into an algorithmic process. This algorithm was then applied to astronomy, engineering, and commerce.

Middle-Age Mathematics

300 CE

1,300 CE

Critical Details:

1. Brahmagupta is an Indian mathematician who lived from 598 to 668 CE

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• He is credited with inventing the rules for addition, subtraction, and multiplication with zero as well as with negative numbers

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• Most of his writings do not contain proof, so it is difficult to determine exactly how he reached his results and conclusions

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Mathematical and/or Scientific Significance:

Operations with zero and negative numbers are vital for our modern algebraic system to function without issue. Thus, his work in this field allowed later mathematicians to compile formalized operations into what we now know as algebra.

Brahmagupta

598

668

300 CE

1,300 CE

Critical Details:

• Muhammed Al-Khwarizmi is a Persian mathematician who lived from 780 to 850 CE

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• Many consider him to be the Father of Algebra

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• His book The Compendious Book on Calculation by Completion and Balancing shows how to solve linear and quadratic equations and the word Algebra can be traced back to the Arabic title of this work fī ḥisāb al-jabr waʾl-muqābala.

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Mathematical and/or Scientific Significance:

We notice that Al-Khwarizmi did not come up with all the rules for algebra on his own and rather compiled the work of previous and current mathematicians along with some of his knowledge of the subject to define and form the first standardized form of a field of mathematics that is fundamental to our modern mathematics.

Al-Khwarizmi

780

850

300 CE

1,300 CE

Critical Details:

• Bhaskara II is an Indian mathematician and astronomer who lived from 1,114 to 1,185 CE

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• He is credited with using his understanding of zero and negative numbers to develop some of the basic concepts of calculus 500 years before Newton or Leibniz

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• His ideas about dividing by zero align with modern concepts of a non-zero “infinitesimal”

Mathematical and/or Scientific Significance:

He was able to build off the works of Brahmagupta to develop a deeper understanding of zero and negative numbers when at the time not many people were comfortable operating with either of them. This allowed him to contribute to the beginning of the development of Calculus.

Bhaskara II

1,114

1,185

300 CE

1,300 CE

Critical Details:

• Leonardo Pisano (i.e. Fibonacci) is an Italian mathematician who lived from 1,175 to 1,250 CE

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• He is credited with publishing Liber Abaci which explains the decimal system, defines a new method for divisions, and serves as a practical textbook for merchants

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• He is also known for popularizing the Arabic numerals over the Roman numerals in Europe.

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Mathematical and/or Scientific Significance:

Fibonacci was able to merge both Arabic and European culture by popularizing Arabic numerals in Europe. In addition, his book helped spread ideas involving number sequencing which is important now for sequences, series, and computer summations.

Fibonacci

1,175

1,250

300 CE

1,300 CE

Nasir al-Din Tusi

1201

1274

Critical Details:

• Nasir al-Din Tusi is an architect, philosopher, physician, scientist, writer, and theologian who lived from 1,201 to 1,274 CE

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• He is considered to be the Father of Trigonometry disjoint to astronomy, notice the difference from Hipparchus.

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• He also conceptualized and studied the Tusi couple which is a device where a circle rolls around the inside of a larger circle with twice the diameter.

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Mathematical and/or Scientific Significance:

Tusi’s work on Trigonometry differed from those in the past which allowed the field to develop in its own right. Tusi’s Trigonometry more closely resembles our modern-day trig than Hipparchus’ does which implies that Tusi’s contributions allowed trig to develop to a point that is more useful for us in the field of mathematics.

Pre-Modern Mathematics

05

1,300 CE - 1,685 CE

1,300 CE

1,685 CE

Critical Details:

• Defining Characteristics of Pre-Modern Mathematics:
• Functions
• Analytical geometry
• Calculus

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• During this period, different mathematical fields such as classical geometry and algebra which had always been separate began to merge thanks to new developments

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• Thanks to trade and mercantile mathematics during the previous two periods, people were much more comfortable with negative numbers and arithmetic algorithms. Numbers during this time began to have less real meaning that modeled natural phenomena.

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Mathematical and/or Scientific Significance:

This was the first point in time where the most advanced field of classical geometry came into contact with other fields such as algebra to begin to create new fields that would last into the modern and Post-modern era. These new fields would lead to further development and study in the discipline of mathematics and ultimately result in a new birth for math.

Pre-Modern Mathematics

(Renaissance Mathematics)

1,300 CE

1,685 CE

Critical Details:

1. Nicole Oresme is a French mathematician, philosopher, and bishop who lived from 1,323 to 1,382 CE

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• He is credited with inventing a form of coordinate geometry even before Descartes

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• He is also considered to be the first to use fractional exponents and during his life wrote about economics, physics, and astronomy.

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Mathematical and/or Scientific Significance:

He set the groundwork for Descartes’s future work in establishing the field of coordinate geometry. This means that he was a vital part of the merging of algebra and geometry.

Oresme

1,323

1,382

1,300 CE

1,685 CE

Critical Details:

• Niccolò Fontana Tartaglia is an Italian mathematician, engineer, and bookkeeper who lived from 1,499 to 1,557 CE.

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• He is credited with finding a formula for any depressed cubic equation by incorporating complex solutions

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• He also published the first Italian translations of Archimedes and Euclid

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Mathematical and/or Scientific Significance:

Tartaglia aided in the spread of mathematical knowledge by translating the classical works of Archimedes and Euclid into Italian. Further, by diving into complex solutions, he set the stage for further development and study in the imaginary number system.

Tartaglia

1,499

1,557

1,300 CE

1,685 CE

Critical Details:

• Gerolamo Cardano is an Italian mathematician and scientist who lived from 1,501 to 1,576 CE

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• He is known for expanding upon Tartaglia by discovering a way to turn any cubic equation into a depressed cubic then solve it by applying Tartaglia’s algorithm

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• He published the Artis Magnæ which was an updated compendium of mathematics that contained the full solution for solving any cubic equation.

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Mathematical and/or Scientific Significance:

Cardano not only improves upon the mathematical methods that were being developed at the time (solving cubics) but also then tries to immediately share these ideas and spread them through his publication. This spread of information is critical for the continual development of the field.

Cardano

1,501

1,576

1,300 CE

1,685 CE

Critical Details:

• Johannes Kepler is a German astronomer and mathematician who lived from 1571 to 1630 CE.

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• He is most famous for his developing the Laws of Planetary Motion

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• His most notable mathematic contribution was developing ideas that would be essential for integral calculus in the future. For example, he treated integration in terms of the area bounded by a curve and developed procedures that are similar to our modern numerical integration methods.

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Mathematical and/or Scientific Significance:

Kepler contributed greatly to our new understanding of a heliocentric model of the universe as well as applied his new mathematical methods in Nova Stereometria Doliorum Vinariorum to determine the volume of various complicated solids of revolution such as wine barrels. Future scholars would incorporate these ideas into the creation of infinitesimal integral calculus.

Kepler

1,571

1,630

1,300 CE

1,685 CE

Critical Details:

1. René Descartes is a French mathematician and philosopher who lived from 1,596 to 1650 CE.

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• Considered to be the father of analytical geometry which required the development of a coordinate geometry system which is why the cartesian coordinate system is named after him

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• He is also credited with first using superscripts for exponents

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Mathematical and/or Scientific Significance:

His advancements in the field of analytical geometry paved the way for Newton and Leibniz to develop calculus in the following decades. In addition, his work essentially merged the fields of geometry and algebra which had previously been quite disjoint.

Descartes

1,596

1,650

1,300 CE

1,685 CE

Critical Details:

• a. Sir Isaac Newton is an English physicist, mathematician, and astronomer who lived from 1642 to 1726 CE.

b. Gottfried Wilhelm Leibniz is a German mathematician and philosopher who lived from 1646 to 1716 CE

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• The two men together are considered the fathers of Calculus

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• In addition, Newton defined the laws of motion and gravity and Leibniz created some of the first mechanical calculators.

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Mathematical and/or Scientific Significance:

The two men’s contributions to the discipline of mathematics are almost indescribable; however, their significance doesn’t stop there. Newton’s laws went on to be the foundations for physics itself.

Newton/Leibniz

1,642

1,646

1,300 CE

1,685 CE

Critical Details:

• Jacob Bernoulli is a Swiss mathematician and scientist who lived from 1,655 to 1705 CE

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• He developed Newton and Leibniz’s calculus to create the field of calculus of variations

• Also, he is credited with developing techniques for solving differential equations to the extent that we learn the Bernoulli method in Differential Equations 1.

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Mathematical and/or Scientific Significance:

First, we must recognize that his solving techniques in differential equations continue to be used to this day. It is also important to recognize that although calculus was a relatively new field, he was able to expand on it and give room for further growth in Calculus and in mathematics itself.

Bernoulli

1,655

Modern Mathematics

06

1,685 CE - 1,950 CE

1,685 CE

1,950 CE

Critical Details:

• Defining Characteristics of Modern Mathematics:
• Abstract analysis, algebra, and geometry
• Modern Logic
• Detachment from paradoxes and issues uncovered during the Classical Period

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• This period could be considered to be a comprehensive synthesis of all previous periods’ mathematical knowledge

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• Scholars at this time also had to deal with overcoming challenges from the past such as Paradoxes from the Classical Period

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Mathematical and/or Scientific Significance:

By facing the challenges of the past along with additional issues that popped up as scholars worked in this period, mathematics was enriched and reworked so that new fields were discovered, old fields were formalized, and modern mathematics became more unified and abstract than before.

Modern Mathematics

(Enlightenment Mathematics)

1,685 CE

1,950 CE

Critical Details:

• Leonhard Euler is a Swiss mathematician who lived from 1707 to 1783 CE

​

• He is known for inventing much of our modern mathematical terminology and notation

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• He also synthesized mathematical knowledge from the past to make discoveries in various fields such as calculus, analysis, graph theory, physics, astronomy, etc.

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Mathematical and/or Scientific Significance:

By utilizing much mathematical knowledge, Euler was able to take mathematics at this time to a new level. He combined analytical methods to solve number theory problems which resulted in a new field of analytic number theory. Further, his work on notation helped unify mathematics by developing a form of expression that a majority of the world latched onto.

Euler

1,783

1,707

1,685 CE

1,950 CE

Critical Details:

• Augustin-Louis Cauchy is a French mathematician and physicist who lived from 1,789 to 1,857 CE

​

• He is considered the father of Complex analysis

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• Also, he took it upon himself to reformulate and reprove results in areas where mathematicians may have been careless and imprecise with their findings

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Mathematical and/or Scientific Significance:

His reworking of previous mathematicians’ work helped him formalize the fields of calculus and analysis. This meant that there was now a version that could be both shared widely and taught in a standard way to students.

Cauchy

1,857

1,789

1,685 CE

1,950 CE

Critical Details:

• Bernhard Riemann is a German mathematician who lived from 1,826 to 1,866 CE

​

• He is who we first credit with rigorously defining integration.

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• He also worked in the field of differential geometry.

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Mathematical and/or Scientific Significance:

Riemann sums are now the basis today for teaching students about integration. In addition, through his work in the field of differential geometry, he was able to help lay the groundwork for general relativity.

Riemann

1,866

1,826

1,685 CE

1,950 CE

Critical Details:

• Georg Cantor is a German mathematician who lived from 1,845 to 1918 CE

​

• He is credited as the inventor of set theory

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• During most of his life, many of his colleagues opposed his discoveries which may have contributed to his mental health issues, but in the end, they recognized the importance of his work

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Mathematical and/or Scientific Significance:

Cantor’s set theory became the foundation for all mathematics moving forward. From it came the axioms that were necessary and sufficient for further advancement. In addition, he helped us differentiate between different sizes of infinity which opposed our view of infinity at the time.

Cantor

1,918

1,845

Post-Modern Mathematics

07

1,950 CE - Present

1,950 CE

Present

Critical Details:

• Defining Characteristics of Post-Modern Mathematics:
• A reflection of the complex and in-depth structures that envelop math
• Topology, Modern geometry, Axiomatized abstractions
• New applications with technology

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• Full of analytical and set theory language

​

• As of now, this is the last period which means it currently contains the most recent development as well as future advancements; however, it may not be long before there is an entirely new period that emerges from discoveries in this post-modern era

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Mathematical and/or Scientific Significance:

Although mathematics has seemingly discarded its connection with the real world do not be fooled. At the heart of mathematics is this longing and yearning to understand the real world by first being able to represent it abstractly. This has contributed to further and further abstraction over time. This period reveals the power of our abstraction and how it has allowed us to develop a much more complex understanding than we had back in the age of proto-mathematics.

Post-Modern Mathematics

(Current Mathematics)

1,950 CE

Present

Critical Details:

• Benoit Mandelbrot is a mathematician born in Poland, raised in France, and lived in the United States and who lived from 1924 to 2010 CE.

​

• He studied fractal geometry where he was interested in how roughness and chaos appear in the real world

​

• He is famous for discovering the Mandelbrot set in 1980

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Mathematical and/or Scientific Significance:

Mandelbrot achieved one of his lifelong goals when working for IBM. He was able to create a graphical representation of fractals which is this visual representation of roughness or chaos. This not only displays how technology has allowed mathematics to advance but also displays how even though math is studying the abstract, at its core, we are trying to learn more about the real world. Also, his work helped lead to the development of the Julia set which I used last semester to graph fractals of my own.

Mandelbrot

1,980

1,950 CE

Present

Critical Details:

• William Paul Thurston is an American mathematician who lived from 1946 to 2012 CE

​

• He studied topology, manifolds, and geometric group theory extensively

​

• His Geometrization Conjecture describes the structure and geometry of different three-dimensional spaces which eventually helped him win a Fields medal

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Mathematical and/or Scientific Significance:

Thurston represents just one of the several different types of mathematicians who exist to learn more about the world around them in different ways. This is a testament to how mathematics has expanded to such a point that no single person can know it all or be an expert on it all. Rather it takes many individuals doing specialized work in all mathematical fields to continue to advance in this discipline

Thurston

2,012

40,000 BCE

21,011 BCE

Present

Final Overlay View

21,011 BCE

2,000 BCE

800 BCE

300 BCE

1,300 CE

1,685 CE

1,950 CE

Thank You!

References

Allen, D. (1997). The origins of Greek mathematics. Retrieved May 2, 2022, from https://www.math.tamu.edu/~don.allen/history/greekorg/greekorg.html

Boyer, C. B. (2012). The history of the calculus and its conceptual development: (the concepts of the calculus). Dover Publications.

Burton, D. M. (2011). The history of Mathematics: An introduction. McGraw-Hill.

Ebrahim, A. (2010). The development of mathematics. Mathematical Science Technologies. Retrieved May 2, 2022, from https://mathscitech.org/articles/development-of-mathematics#CITEBoyer/Calculus

Friberg, J. (2006). A remarkable collection of Babylonian Mathematical Texts. SpringerLink. Retrieved May 2, 2022, from https://link.springer.com/book/10.1007/978-0-387-48977-3

Gardner, M. (n.d.). Rhind Papyrus. Wolfram MathWorld. Retrieved May 2, 2022, from https://mathworld.wolfram.com/RhindPapyrus.html

Gullberg, J., & Gullberg Pär. (1997). Mathematics: From the birth of numbers. W.W. Norton & Company.

References

How Imaginary Numbers Were Invented. (2021). Youtube. Retrieved May 2, 2022, from https://www.youtube.com/watch?v=cUzklzVXJwo.

Katscher, F. (2011). How tartaglia solved the cubic equation - tartaglia's solution in modern notation. How Tartaglia Solved the Cubic Equation - Tartaglia's Solution in Modern Notation | Mathematical Association of America. Retrieved May 2, 2022, from https://www.maa.org/press/periodicals/convergence/how-tartaglia-solved-the-cubic-equation-tartaglias-solution-in-modern-notation

Kline, M. (1990). Mathematical thought from ancient to modern times. Oxford University Press.

Merzbach, U. C., & Boyer, C. B. (2011). A history of mathematics. Wiley.

NASA. (2021). Pythagorean theorem. NASA. Retrieved May 2, 2022, from https://www.grc.nasa.gov/WWW/K-12/rocket/pythag.html

Norman, J. (n.d.). The Lebombo bone, oldest known mathematical artifact. The Lebombo Bone, Oldest Known Mathematical Artifact : History of Information. Retrieved May 2, 2022, from https://www.historyofinformation.com/detail.php?entryid=2338

References

O'Connor, J., & Robertson, E. (1998). Fibonacci - Biography. Maths History. Retrieved May 2, 2022, from https://mathshistory.st-andrews.ac.uk/Biographies/Fibonacci/

O'Connor, J., & Robertson, E. (1999). Euclid - Biography. Maths History. Retrieved May 2, 2022, from https://mathshistory.st-andrews.ac.uk/Biographies/Euclid/

O'Connor, J., & Robertson, E. (2000). Bhaskara II - Biography. Maths History. Retrieved May 2, 2022, from https://mathshistory.st-andrews.ac.uk/Biographies/Bhaskara_II/

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