The History of Mathematics and Exact Sciences in Antiquity: Discrimination and Aposiopesis
3rd. enhanced
ed.
Athens 2026
© Δ. Ν. Κονιδάρης
Ιστορία των θετικών τεχνών και επιστημών κατά την Αρχαιότητα: αποσιώπηση και μεροληψία
γ’ εμπλουτισμένη & αναθεωρημένη έκδοση
Dimitrios Konidaris
Copyright © Copyright Year Δημήτριος Ν. Κονιδάρης
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ISBN 978-618-88756-1-6
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Imprint: Δημήτριος Ν. Κονιδάρης
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https://www.lulu.com/spotlight/dnkonidarisExtended Summary
10.1 Purpose and central thesis
The work proposes an alternative interpretation of the history of
mathematics and the exact sciences in antiquity. Its stated purpose is to
examine the historical record critically and to challenge what the author
considers selective or unbalanced narratives. The third edition incorporates
additions and revisions to the earlier work.
A
fundamental theme is the role of historical narratives in shaping perceptions
of cultural achievement. The author argues that histories of science can
reproduce prevailing intellectual, cultural, and ideological models, sometimes
resulting in the neglect or marginalization of particular civilizations,
regions, or traditions. The book therefore examines not only mathematical
achievements themselves but also how those achievements have been interpreted
by modern historians of science.
10.2 Historiography, bias, and the problem of cultural
priority
The opening section focuses on the historiography of ancient
mathematics. Particular attention is given to interpretations associated with
scholars such as B. L. van der Waerden and Otto Neugebauer.
Van
der Waerden's interpretation emphasizes the importance of Babylonian
mathematics as a starting point for developments that were subsequently
transformed within Greek mathematics. In this account, figures such as Thales
and Pythagoras are presented as having encountered Babylonian mathematical
traditions but having given them a distinctively Greek character. The
development subsequently proceeds through the Pythagoreans, Theaetetus,
Eudoxus, and ultimately Euclid.
Neugebauer's
work is presented as particularly influential in redirecting attention toward
Egyptian and Mesopotamian mathematics. His study of cuneiform mathematical
texts contributed substantially to the modern understanding of Babylonian
mathematics. At the same time, the author questions some of Neugebauer's
interpretations of Greek mathematics and criticizes what is regarded as an
excessive separation between Greek mathematics and its broader intellectual
context.
A
major historiographical controversy concerns the concept of “geometrical
algebra.” Van der Waerden and Neugebauer associated certain Greek geometrical
procedures with Babylonian algebraic techniques. The book examines this
hypothesis through parallels between Babylonian problems and propositions in
Euclid's Elements and Data.
The
discussion also presents the opposing position of scholars such as David Fowler
and Sabetai Unguru, who challenged the legitimacy of describing Greek
mathematics in terms of modern algebra. Unguru, in particular, argued that the
concept of algebra should not be projected anachronistically onto ancient Greek
mathematics. The controversy ultimately contributed to a significant
reconsideration of the historiography of Greek mathematics.
10.3 Egypt and Egyptian mathematics
The Egyptian section examines both the achievements and the limitations
attributed to ancient Egyptian mathematics.
The
document emphasizes that Egyptian mathematical practice was strongly connected
to practical requirements, including measurement, construction, administration,
and calculation. The principal surviving mathematical sources are identified as
the Rhind Papyrus and the Moscow Mathematical Papyrus. Egyptian mathematics
employed integers and unit fractions, with the notable exception of 2/3, and
included methods for arithmetic operations and the solution of linear
equations.
The
author distinguishes between mathematical technique and mathematical science,
arguing that Egyptian mathematics did not develop the same emphasis on
generalization and formal proof that later characterized Greek mathematics.
The
discussion nevertheless recognizes the considerable technical competence
required for Egyptian monumental architecture. It also highlights evidence
suggesting contacts between Egypt and the Aegean, including measuring
instruments associated with the Minoan system of measurement discovered in an
Egyptian architectural context.
Thus,
Egypt is presented neither simply as the origin of all subsequent mathematics
nor as an insignificant mathematical culture. Instead, the document stresses
the importance of distinguishing practical mathematical knowledge from the
later development of formal deductive mathematics.
10.4 Chinese mathematics
The Chinese section examines the mathematical tradition represented
particularly by the Jiuzhang suanshu, or Nine Chapters on the Mathematical
Art.
The
text compares the importance of this work within Chinese mathematics to that of
Euclid's Elements in the Western tradition. The work consists largely of
mathematical problems organized according to subject and accompanied by
procedures for obtaining solutions.
The
author discusses calculations involving the constant π and a
geometrical treatment of the Pythagorean rule. At the same time, the chronology
and extent of the mathematical material are treated cautiously, with the author
questioning claims that some of its contents should be dated considerably
earlier.
The
overall interpretation in the document is that Chinese mathematics represents
an important historical tradition, although the author considers its
contribution during the pre-Christian period relatively limited compared with
other ancient mathematical traditions.
10.5 Babylonian mathematics
Babylonian mathematics constitutes one of the most important components
of the book.
The
evidence demonstrates a highly developed computational tradition extending from
the Old Babylonian period to the Seleucid period. Babylonian mathematicians
employed a sexagesimal positional system and could solve equations of the
first, second, and higher degrees, sometimes involving more than one unknown.
They also worked with geometrical problems involving triangles, trapezia,
circles, and related figures.
The
famous Plimpton 322 tablet is discussed in connection with Pythagorean triples,
although the author explicitly acknowledges that interpretations of the tablet
are disputed.
A
central historiographical question is therefore not whether Babylonian
mathematics was sophisticated—it clearly was—but how its relationship to Greek
mathematics should be understood.
Neugebauer
is presented as emphasizing the dependence of aspects of Greek mathematics upon
earlier Mesopotamian traditions. However, the work also introduces the
alternative interpretation associated with Reviel Netz: rather than imagining a
simple transfer of theoretical mathematical knowledge from Babylon to Greece,
mathematical transmission may have involved the movement and transformation of
individual problems, techniques, and intellectual practices across a long
cultural continuum.
The
author consequently argues for a more complex model of transmission than a
simple “Babylon → Greece” sequence.
10.6 Neolithic Aegean numeracy and accounting
One
of the book's distinctive contributions is its attention to Neolithic Aegean
numerical practices, which the author argues have frequently received
insufficient attention in conventional histories of mathematics.
Evidence
includes:
- clay
tokens or tokens;
- marked
rods or tallies;
- numerical
marks;
- possible
proto-writing;
- impressed
or incised signs;
- accounting
devices.
Such
objects occur in Thessaly, Macedonia, Central Greece, Thrace, the Aegean
islands, and the wider Balkan region, with some dating as early as the seventh
millennium BCE.
The
work emphasizes the close historical relationship between accounting and
writing. Numerical notation could develop alongside systems for recording
information, as happened elsewhere in the ancient world.
An
important historiographical issue is whether these developments resulted
primarily from Near Eastern influence (Ex Oriente Lux) or represented
indigenous developments in southeastern Europe (Ex Balcanis Lux). The
book presents archaeological evidence that has been interpreted in support of
local development, while recognizing that the question remains part of a
broader scholarly debate.
10. 7 Minoan and Mycenaean mathematics and exact sciences
The
sixth major section examines mathematical and scientific knowledge in the
Minoan and Mycenaean Aegean.
The Minoan-Mycenaean numerical
system was decimal and employed unit fractions, including 2/3. The surviving
Linear A and Linear B tablets, however, provide only limited direct information
about the broader mathematical knowledge of these societies.
The
architecture of Mycenaean and Minoan societies is used as indirect evidence for
mathematical knowledge. The study of Mycenaean tholos tombs has led researchers
to propose practical knowledge of mathematical curves such as the parabola and
possible knowledge of Pythagorean triples.
The
palace architecture of Phaistos is also discussed in connection with numerical
proportions and, according to the cited research, relationships resembling the
Fibonacci sequence. The author presents these interpretations as evidence of
sophisticated proportional and geometrical practices, while the references also
acknowledge that some specific claims—such as Egyptian knowledge of the
Fibonacci sequence—remain disputed.
Minoan
buildings are further described as having been planned using intersecting axes,
grids, ropes, stakes, and orientation techniques connected with solar
observation.
The
Akrotiri wall paintings provide another body of evidence. Their construction
involved geometrical organization of large painted surfaces, including
preliminary grids and mathematical curves. The document compares these
practices with Egyptian techniques while acknowledging that the direction of
influence is not always demonstrable.
10.8 Astronomy and cosmology
The discussion of Aegean science extends beyond mathematics into
astronomy and cosmology.
The
book considers evidence for astronomical observation and mathematical
organization in Minoan and Mycenaean culture and discusses possible
relationships between Aegean, Egyptian, Mesopotamian, and later Greek
astronomical traditions.
The
broader argument is that the development of ancient astronomy should not be
represented as an isolated process occurring in one civilization. Rather, it
emerged through long-term interaction, observation, transmission,
reinterpretation, and independent development.
The
work also connects mathematical concepts with ancient cosmological ideas,
including the concept of the cosmic axis or axis mundi, linking earth
and heaven.
10.9 Aegean–Mesopotamian interaction
A particularly important section
examines the extensive contacts between the Aegean and Mesopotamia during the
Bronze Age.
The
evidence includes:
- trade
networks;
- metals
and raw materials;
- weights
and measures;
- seals
and cylinder seals;
- artistic
motifs;
- ceramics;
- architectural
and decorative techniques;
- Aegean objects found in Mesopotamia;
- Mesopotamian objects found in the Aegean.
The
Aegean participated in a broad interconnected network extending from the
Mediterranean to Anatolia, Mesopotamia and, more widely, toward South Asia. The
author stresses that this network provides a context in which mathematical and
technological knowledge could circulate.
Particularly
important is the evidence from Mari, which indicates connections involving
Minoan products, metals, ships, craftsmen and trade. The document notes that
Minoan products acquired prestige in parts of the eastern Mediterranean and
that Aegean artistic forms were adopted in several Near Eastern palatial
centres.
Aegean-style
frescoes and artistic practices are discussed in relation to sites including
Avaris, Tell Kabri, Alalakh, Ebla, Qatna, Mari and Nuzi. The adoption of Aegean
artistic forms is interpreted as evidence of substantial cultural interaction
and of the prestige of Minoan-Mycenaean culture in parts of the eastern
Mediterranean.
10.10 Weights, measures and commercial mathematics
An
important component of the argument is the history of measurement and weighing.
The
document notes that systems of standardized weights appeared in Egypt and
Mesopotamia during the fourth millennium BCE and subsequently became attested
in the Aegean and Anatolia. Different traditions of weights could coexist
within the same commercial environment.
This
evidence illustrates that ancient merchants were capable of conducting
international trade using their own systems of measurement while establishing
equivalences between different standards.
The
conclusion is significant: the history of mathematics cannot be separated from
the history of commerce, administration, architecture, metallurgy and
technology.
10.11 Greek–Mesopotamian interaction in later periods
The relationship between Greece and Mesopotamia continued well beyond
the Bronze Age.
The
text discusses Greeks operating in Mesopotamia during Assyrian, Persian,
Babylonian and Seleucid periods. In the Seleucid period, interaction
intensified considerably. Greek names, Greek cultural practices and elements of
Greek architecture and artistic decoration appear in Mesopotamian contexts.
Uruk
is particularly important in this discussion. Members of local elite families
sometimes adopted Greek names and participated in the new Hellenistic cultural
environment. Archaeological evidence also demonstrates the incorporation of
Greek or Greek-inspired architectural and decorative elements into Babylonian
buildings.
The
author emphasizes that this process was bidirectional: Greeks were not merely
transmitters of culture to Mesopotamia. Greeks themselves encountered
Mesopotamian mathematical and astronomical traditions, and these encounters
contributed to the development of later Greek science.
10. 12 The question of Egyptian influence on Greek
mathematics
A
later section specifically addresses the widespread claim that Greek
mathematicians learned mathematics from the Egyptians.
The
document examines ancient testimony concerning Egyptian mathematical prestige
but questions whether such testimony should automatically be interpreted as
proof of direct intellectual dependence.
The
author differentiates between:
- knowledge that could have been transmitted
between cultures;
- practical
techniques;
- mathematical
problems;
- theoretical
concepts;
- formal
deductive systems.
The
important distinction is that contact or transmission does not necessarily
establish intellectual dependence in its strongest sense.
10.13 Greek or Arabic algebra?
Another major section addresses the history of algebra, particularly the
conventional attribution of the decisive origins of algebra to the Islamic
mathematician al-Khwarizmi.
The
document begins from the familiar interpretation that Arabic mathematics of the
eighth and ninth centuries represented a major transition toward algebraic
thinking. It then investigates the extent to which this development was
genuinely new and the degree to which it incorporated earlier Greek
mathematical traditions.
The
discussion emphasizes the importance of Greek authors such as:
- Euclid;
- Diophantus;
- Apollonius;
- Pappus.
It
also examines the later development of symbolic mathematics and the emergence
of generalized algebraic notation. In this context, the history of algebra is
presented as a long process of transformation rather than the invention of a
single civilization or individual.
The
document therefore challenges simplified narratives that sharply separate
“Greek geometry” from “Arabic algebra.”
10.14 The broader conclusion of the work
The concluding section presents the history of mathematical knowledge as
a cumulative, non-linear process. Knowledge develops through successive stages,
but these stages can include interruptions, regressions, transformations and
intellectual leaps.
The
author emphasizes the importance of the material conditions that made
intellectual development possible. Surplus production, administration,
urbanization and early state organization created environments in which systems
of recording, accounting and calculation could emerge.
The
overall historical model proposed by the work is therefore not one of a single
cultural “origin” followed by passive transmission. Instead, mathematical
knowledge is portrayed as the product of:
observation
→ practical needs → recording → calculation → exchange between cultures →
abstraction → generalization → proof → further transformation.
10.15 Key Points
- Histories of mathematics are not neutral by
definition. The selection and interpretation of evidence can influence
perceptions of cultural achievement.
- The book challenges simplistic “origin”
narratives. Ancient mathematics developed through multiple traditions and
long-term interaction.
- Babylonian mathematics was highly sophisticated,
particularly in numerical calculation and the solution of complex
problems.
- Egyptian mathematics was highly practical, with
substantial competence in arithmetic, measurement and construction, but
the author distinguishes it from later deductive mathematical science.
- Greek mathematics introduced or developed a
distinctive emphasis on abstraction, generalization and proof, especially
in the tradition represented by Euclid and Archimedes.
- The concept of “geometrical algebra” is
controversial. The book discusses the disagreement between scholars who
emphasize Babylonian antecedents and those who reject the anachronistic
application of modern algebraic terminology to Greek mathematics.
- Neolithic southeastern Europe and the Aegean
deserve greater attention in histories of numeracy, accounting and
proto-writing.
- Minoan and Mycenaean societies possessed
significant practical mathematical knowledge, visible particularly in
architecture, measurement, artistic construction and possibly astronomy.
- Aegean–Mesopotamian interaction was extensive.
Trade, metallurgy, weights, seals, artistic motifs, architecture and
possibly scientific knowledge circulated across regions.
- Cultural transmission was multidirectional. Greek
culture influenced Mesopotamia in later periods, while Greek scholars also
encountered and incorporated older Egyptian and Mesopotamian traditions.
- The development of algebra was gradual and
interconnected. The transition from Greek mathematics to medieval Islamic
mathematics should not be reduced to a simple replacement of “Greek
geometry” by “Arabic algebra.”
- The central methodological message is
historiographical: the history of mathematics should be reconstructed from
the totality of available evidence, while avoiding both cultural exclusion
and simplistic claims of unilateral influence.
10.16 Overall synthesis
The attached work ultimately presents the history of ancient mathematics
as a complex network of interacting traditions rather than a linear succession
of civilizations. Egypt, Mesopotamia, the Aegean and Greece, China and later
Islamic mathematical cultures each contributed different forms of numerical,
geometrical, astronomical and practical knowledge. The author places particular
emphasis on the need to distinguish practical mathematical techniques from
formal mathematical science, and cultural contact from demonstrable
intellectual dependence.
At
the same time, the work argues that the distinctive Greek
achievement—especially the development of systematic proof, abstraction and
deductive organization—should remain visible within a broader history that also
recognizes the substantial achievements of earlier and contemporary
civilizations. The final perspective is therefore one of continuity,
interaction, transformation and cumulative development, rather than exclusive
cultural ownership of mathematics.
