For example the following is my solution (obviously I don't claim the method itself is my own! It's too incredibly basic to be new - just that I also used it to answer a test question) for establishing the integer part of any addition of square roots of integers. In this case it's for the integer part of the sum sqrt5+sqrt6+sqrt7, which using ruler and compass (or alternatively graph paper) is easy to establish at being at 7. With full accuracy of ruler and compass, the decimal part can also be established in detail. (scroll down for an example drawn with far better detail/accuracy; the method is the same)

https://preview.redd.it/8m6b787smaaf1.png?width=792&format=png&auto=webp&s=8d18e94b82929214931cd30f1e1650af838c5e14

It rests on the known method of constructing square roots - likely of ancient Greek origin (?) and the opening example in Descartes' book on geometry (https://kids.britannica.com/students/assembly/view/67611, https://americanhistory.si.edu/ru/collections/object/nmah_694637 etc). What I am asking is if you know any prominent historical examples where this way was used to calculate (or approximate) sums of many roots.

important note: the circles were approximately drawn - with compass they'd be accurate and so the full (integer+decimal) part of the sum would have optimal (visually checked) accuracy (in the case of this sum, be at around 7.33)

Here is an example with better accuracy of decimal:

sqrt5+sqrt6 is approximately 4.68 - so here approximated by around 98% (loss due to drawing inaccuracies)

Does anyone know any good books that go over history of mathematics? As in random stories of famous mathematicians/how certain theorems were created/any interesting math based stories that one could bring up in a classroom when introducing topics.

Did Pythagoras, Plato, and early Christians tap into Egyptian wisdom? This question explores how ancient Egypt shaped Pythagoreanism, Platonism, Neoplatonism, and the rise of Christianity. From Thoth’s esoteric knowledge to Alexandria’s syncretic hub, Egypt’s fingerprints are on these philosophies and faith, tied to Hermeticism and Gnosticism.

Pythagoreanism: Proto-Monotheistic Roots

Around 530 BCE, Pythagoras founded a secretive brotherhood in Croton, structured like a mystery cult with mathematikoi (inner initiates) and akousmatikoi (outer followers). Its theology leaned toward unity:

• Monotheistic Tendencies: The Monad, the number one, was a divine source, and the tetractys symbolized cosmic harmony. The “music of the spheres” implied a unified cosmic intelligence.
• Egyptian Influence: Iamblichus (Life of Pythagoras) claims Pythagoras studied in Memphis with priests. The brotherhood’s secretive structure mirrors Egyptian priesthoods, like those of Amun, who guarded esoteric texts. Egyptian metempsychosis parallels the Book of the Dead’s soul journey. Geometry for pyramids influenced Pythagorean number mysticism (e.g., Pythagorean theorem). Akhenaten’s monotheism (c. 1350 BCE) and Amun’s henotheism may have shaped the Monad, per Porphyry.
• Polytheistic Context: Pythagoreanism honored Apollo, blending proto-monotheistic unity with Greek polytheism.

Extent: Moderate Egyptian influence, shaping structure, reincarnation, and monotheistic leanings, but Orphic and Ionian roots also key.

Platonism: Henotheistic Framework

Diogenes Laertius claims Plato studied in Egypt (c. 390 BCE) in Heliopolis or Sais, encountering priests’ mathematics and theology. Platonism (c. 427–347 BCE) centers on the Theory of Forms:

• Monotheistic Tendencies: The Form of the Good (Republic) is a transcendent source, akin to a supreme deity. The Demiurge (Timaeus) crafts the cosmos, suggesting a single design principle.
• Egyptian Influence: Ptah, Memphis’ creator god, parallels the Demiurge, crafting through thought. Ma’at’s cosmic order aligns with Plato’s harmonious cosmos. Akhenaten’s monotheism prefigures the Good. Egyptian geometry, via priests, shaped Plato’s view of mathematics as truth (Meno)). Pythagoreanism, possibly Egypt-inspired via Archytas, mediated the Monad’s influence. Plato’s nod to Egyptian records in Timaeus suggests cosmological familiarity.
• Polytheistic Context: Plato references Greek gods, making Platonism henotheistic, prioritizing one principle within polytheism.

Extent: Subtle Egyptian influence, enhancing cosmology, soul theory, and monotheistic-like principles, but Greek roots (Socrates) dominate.

Neoplatonism: Monotheistic Core

Neoplatonism, founded by Plotinus (204–270 CE) in Alexandria, blends Platonism with mysticism:

• Monotheistic Tendencies: The One (Enneads)), an ineffable source, emanates Nous and Soul. The soul’s ascent to The One mirrors monotheistic salvation.
• Egyptian Influence: Amun’s henotheism and Hermeticism’s divine Mind (Corpus Hermeticum), tied to Hermes Trismegistus, parallel The One. Isis and Osiris cults emphasized universal divinity. Plotinus studied under Ammonius Saccas in Alexandria, where the Library and Catechetical School amplified Egyptian ideas. Iamblichus’ theurgy reflects Memphis’ rituals. The Nag Hammadi Codices link Neoplatonic and Gnostic soul journeys.
• Polytheistic Context: Later Neoplatonists incorporated gods, making it inclusive monotheism.

Extent: Strong Egyptian influence, via Hermeticism and Alexandria’s syncretism, shaped its monotheistic core.

Early Christianity: Egypt’s Crucible

Alexandria, under the Ptolemies, was a cultural hub where Egyptians, Greeks, and Southern Levantine monotheists shaped early Christianity, blending Egyptian spirituality with Hellenistic and Southern Levantine ideas.

• Alexandria’s Role: The Septuagint (c. 3rd century BCE), a Greek Hebrew Bible, shaped Christian scriptures. The Catechetical School synthesized Egyptian wisdom with Christianity.
• Egyptian Expatriates: Levantine monotheists, possibly Hyksos-descended (c. 1650–1550 BCE), carried Egyptian spiritual DNA, like Akhenaten’s monotheism, into the Exodus narrative. Philo (20 BCE–50 CE) fused Platonism with monotheism, influencing Christian thought.
• Saint Mark: Mark, born in Cyrene with Egyptian ties, founded the Church of Alexandria (c. 42 CE). His Gospel of Mark (c. 60–70 CE), the earliest Gospel, echoes Hermetic gnosis. The Coptic Church blended Egyptian motifs (e.g., ankh-like crosses).
• Luke and Paul: Luke, likely visiting Alexandria, wrote Gospel of Luke and Acts (c. 80–90 CE), preserved in Coptic manuscripts. Paul engaged Alexandrian Christians like Apollos (Acts 18:24–28), with epistles (Romans, c. 50–60 CE) resonating with Platonism and Gnosticism.
• Hermeticism and Gnosticism: The Corpus Hermeticum’s gnosis and Valentinus’ Egyptian dualism shaped Christian theology. The Nag Hammadi Codices preserve this esoteric legacy.
• Monotheistic Tendencies: Christianity’s strict monotheism, centered on one God, draws from Levantine traditions but was enriched by Egyptian henotheism (e.g., Amun) and Neoplatonic unity, via Alexandria’s syncretism.
• Egyptian Influence: Copts and Hellenized Egyptians spread Christianity via Pharos Lighthouse networks. Clement and Origen, trained in Alexandria, globalized Egyptian-infused theology.

Extent: Profound Egyptian influence, as Alexandria’s syncretism and expatriates like Mark and Origen made Egypt a Christian powerhouse, rooted in Thoth’s wisdom and Ma’at’s order.

Cultural Exchange and Expatriates

Egyptian expatriates, including Copts and Levantine monotheists, spread ideas via Alexandria’s trade networks. Philo, Origen, and Coptic scribes bridged Egyptian spirituality to Platonism, Neoplatonism, and Christianity. Southern Levantine ties, possibly Hyksos-related, carried Akhenaten’s monotheistic legacy, influencing all four systems.

Extent of Egyptian Influence and Monotheism

• Pythagoreanism: Proto-monotheistic (Monad); moderate Egyptian influence via Pythagoras’ studies, shaping structure and cosmology.
• Platonism: Henotheistic (Good, Demiurge); subtle Egyptian influence through Plato’s travels and Pythagoreanism.
• Neoplatonism: Strongly monotheistic (The One); significant Egyptian influence via Hermeticism and Alexandria.
• Christianity: Strictly monotheistic; profound Egyptian influence via Alexandria’s syncretism and expatriates, globalized by Augustine and Coptic traditions.

Conclusion

Egypt’s wisdom, from Thoth’s esoteric teachings to Alexandria’s melting pot, shaped Pythagoreanism’s proto-monotheism, Platonism’s henotheistic principles, Neoplatonism’s monotheistic core, and Christianity’s global faith. The Pythagorean brotherhood’s Egyptian-inspired structure fed Plato’s ideas, which Plotinus and early Christians like Mark wove into mystical and theological tapestries. Egypt’s legacy, carried by expatriates and syncretism, bridges ancient wisdom to modern thought.

Sources: Diogenes Laertius, Iamblichus’ Life of Pythagoras, Plato’s Timaeus, Plotinus’ Enneads, Corpus Hermeticum, Nag Hammadi Codices, Gospel of Mark.

I remember hearing somewhere that at a certain point in the history of mathematics, a certain mathematician came very close to discovering a branch of mathematics that would be very useful in advancing society to the point it was now. I might be misremembering, but I also recall hearing that if they didn't interrupt that process, our society would have been able to advance at a much quicker pace and starting at a much earlier point in history. Anyone know what branch of mathematics or what researchers were involved in this, or if I'm just not remembering properly and all of this isn't real?

While it may sound rather droll or uninspiring to look at spiritual concepts in numerical terms, remember that Pythagoras was also a master initiate of the esoteric schools of his time. The numerical principles he taught symbolize divine principles or archetypes that can be tapped into during the mystical experience.

Thus, “the whole field of arithmetical speculation had to do with the internal growth of the person. It made possible the exploration of qualitative factors, not merely the assessment of quantities. Therefore, it was the foundation upon which must be built the most advanced type of” meditational practice.

The numerical principles depicted in the Tetractys represent mental or psychological archetypes. The world itself is conceived to exist as a projection of these archetypes: its various kingdoms being merely these archetypes clothing themselves in various garments of material form.

Archetypes pre-structure the design of all forms that come into existence within Space. Space is really Consciousness, and the world of form is really a “thought form” that exists within the Mind of this Consciousness. Archetypes represents the patterns of Mind by means of which Consciousness projects and builds its cosmic thought form.

In esoteric philosophy, Creation is understood to exist as the end-product of a divine process of meditation in which God “projects the world by visualization, or by the power of will and Yoga.” This projection takes place via the seven “builder gods” or “Elohim”, which in Mahayana Buddhism are termed as the Celestial Buddhas or Dhyani Buddhas (meditating Buddhas). The Tetractys is intended to depict this archetypal creation process in mandalic form. In its ten dots are captured the threefold nature of the Divine Self and the seven Rays or principles of Mind which are projected out from it at the onset of creation.

The idea is that the seven creator gods within the macrocosm move in the design of one great archetype. This archetype describes “the rhythm of consciousness moving through the world. This rhythm is the motion of Tao, symbolized in China by the undulations of the body of a cosmic dragon.”

By attuning themself to the cosmic rhythm of the Tao, the meditating mystic experiences an alignment in consciousness between their own inner psychology and the divine spiritual archetypes existing in Space. Through this process, the mystic “recapitulates the divine creation process” and "experiences the very mystery of creation.”

The disciple of the Pythagorean school, meditating upon the design of the Tetractys, stimulates its cosmic principles within himself and thus recapitulates within his own microcosmic consciousness the same process of creation as that which originally took place within the macrocosm or universe as a whole.

Through this method, “Pythagoras revealed a kind of yoga, a means of disciplining human consciousness by reference to the orderly procedures of number, which represent the systematic unfoldment of creative processes moving out from their own causes.”

Hall elaborates on the mystical disciplines that the Pythagoreans incorporated within their school: “the Pythagorean philosophy of number was held to be both scientific and intuitive. Once the disciple had been instructed in the principles of the teachings, he advanced by a process of personal discovery. New mathematical patterns and equations continuously presented themselves to his attention. These he interpreted according to his own unfolding insight.”

As we’ve discussed, the performance of mediation works through the stimulation of archetypes. Once awakened, these archetypes “come alive”. The challenge of meditation is simply to awaken them and bring them to life within ourselves.

Manly Hall describes this act of awakening divine powers within as one of “reminiscence”: “To Pythagoras, mathematics was a science built upon reminiscence. By this he meant that mathematical principles were not discovered, created, or invented by the mind, but had an eternal subsistence in the intelligible nature of all things. Reminiscence, therefore, is a recalling of that which is already known, usually by a process of association. One thing reminds us of another, leading to a refreshment of recollection. That which is recollected is drawn into the fore part of attention, where it becomes available again after being seemingly forgotten.”

By catalyzing this process of “reminscence” within its students, the Pythagoreans sought to “draw certain deep and hidden knowledge from within the individual and apply it to the deep and hidden mystery of the world.” Therefore, by practicing the philosophical disciples emphasized by the Esoteric School, by degrees, the wise man could perfect the grand concept of existence and establish his own proper place in the universal program.”

Full article: https://thewisdomtradition.substack.com/p/the-mandala-an-image-of-the-invisible-7b5

Hi, maths lovers!

I'm curious, were there any facts about the development of mathematics through the ages or tidbits from certain mathematician's biographies that piqued your interest or that you found funny or even weird?

I personally found fascinating that renaissance era mathematicians held maths duels where they exchanged problems and the one who managed to solve the most of them was considered a winner and often got some prestigious post as a result (while the other was publicly humiliated, naturally).

I've always heard remarks about Egyptian mathematicians knowing a lot about trigonometry before the Greeks learned about them, or about how Indian mathematicians knew about certain results in Calculus that would later be attributed to Liebniz.

With this idea in mind, does anyone have recommendations for books or places (websites) where one can learn about mathematical developments of non-European regions. I don't want to say something like "the true history" as that becomes a little aggressive.

I'd like to explore more in an effort to show my students that math is not a field that only sees old European people as mystical math gods when it comes to the things they might be learning in school.

I imagine some of these might just be local to their countries and only be in there language, but I'd still like to know.

Thank you!