math-historyreference-request
Many years ago a lecturer of mine had a photocopy of a page from a book containing a math problem ( I think it was a simple quadradic equation ) that was stated/solved in Cuneiform, Arabic, Latin scripts and Finally in modern math notation.
I have contacted my lecturer but he has no idea where it was from, nor I have been able to find it using google books searches etc.
Does anyone know where to find it?
Thank you in advance
Here I'd like to address three books with a dedicated historical development perspective. Since OP's question is highly opinion based I think it might be helpful to look at some statements from the authors themselves, in order to get an impression how this subject should be treated.
A Radical Approach to Real Analysis by David Bressoud
(From the Preface): The task of the educator is to make the child's spirit pass again where its forefathers have gone, moving rapidly through certain stages but suppressing none of them. In this regard, the history of science must be our guide. (Henry Poincaré).
This course of analysis is radical; it returns to the roots of the subject. It is not a history of analysis. It is rather an attempt to follow the injunction of Henry Poincaré to let history inform pedagogy. It is designed to be a first encounter with real analysis, laying out its context and motivation in terms of the transition from power series to those that are less predictable, especially Fourier Series, and marking some of the traps into which even great mathematicions have fallen. ...
... Careful definitions mean nothing until the drawbacks of the geometric and intuitive understandings of continuity, limits, and series are fully exposed. For this reason, the first part of this book follows the historical progression and moves backwards. It starts with infinite series, illustrating the successes that led the early pioneers onward, as well as the obstacles that stymied even such luniaries as Euler and Lagrange.
There is an intentional emphasis on the mistakes that have been made. These highlight difficult conceptual points. That Cauchy had so much trouble proving the mean value theorem or coming to terms with the notion of uniform convergence should alert us to the fact that these ideas are not easily assimilated. The student needs time with them.
The highly refined proofs that we know today leave the mistaken impression that the road of discovery is straight and sure. It is not. Experimentation and misunderstanding have been essential components in the growth of mathematics. Exploration is an essential component of this course.
In the same spirit the author has also written A Radical Approach to Lebesgue's Theory of Integration.
The next one is a classic from a famous mathematician which is already an essential part of mathematical historical development.
Number Theory - An approach through history from Hammurapi to Legendre by André Weil
(From the Preface): ... Until rather recently, number theory, or arithmetic as some prefer to call it, has been conspicuous for the quality rather than for the number of its devotees; at the same time it is perhaps unique in the enthusiasm eloquently expressed in many utterances of such men as Euler, Gauss, Eisenstein, Hilbert.
Thus, while this book covers some thirty-six centuries of arithmetical work, its bulk consists in nothing more than a detailed study and exposition of the achievements of four mathematicians: Fermat, Euler Lagrange, Legendre. These are the founders of modern number theory The greatness of Gauss lies in his having brought to completion what his predecessors had initiated, no less than in his inaugurating a new era in the history of the subject.
Our main task will be to take the reader, so far as practicable, into the workshops of our authors, watch them at work, share their successes and perceive their failures.
... The method to be followed here is historical throughout; no specific knowledge is expected of the reader, and it is the author's fond hope that some readers at least will find it possible to get their initiation into number theory by following the itinerary retraced in this volume.
The last one is from an author which was (and is) important for me when I was a student.
Theory of Complex Functions by Reinhold Remmert
(From the Preface): ... The present book contains many historical explanations and original quotations from the classics. These may entice the reader to at least page through some of the original works.
"Notes about personalities" are sprinkled in "in order to lend some human and personal dimension to the science" (in the words of F. KLEIN on p. 274 of his Vorlesungen uber die Entwicklung der Mathematik im 19. Jahrhundert - see [Hs]).
But the book is not a history of function theory; the historical remarks almost always reflect the contemporary viewpoint. Mathematics remains the primary concern. What is treated is the material of a 4 hour/week, one-semester course of lectures, centering around Cauchy's integral theorem. Besides the usual themes which no text on function theory can omit, the reader will find here ...
In the same spirit the author has also written Classical Topics in Complex Function Theory, which can be seen as volume II of the former book.
[Add-on]: Some thoughts
OP's quote: "... read the author, reproduce the results in your own way, and think about it to internalize, and repeat for the entire history ..." is not a challenge to reinvent some theorems, mathematical structures or concepts.
It is an invitation to
read the author's text
try to understand it in some depth
try to extract the essential ideas and internalize them
... and then check if you really got some understanding by
trying to reproduce it, but now without some additional means.
You could imagine a friend which is curious about the stuff and you try to write down and verbally explain the single steps. This way you can test if you really got a proper understanding.
You could also reflect about the historical approach and think about how the current approach differs (in terms of definitions, methods, ...).
The books referred in this answer provide many examples in appropriate historical context which can be formidably used to study and train this way the own abilities by grasping the essential ideas and trying to reproduce them independently.
The development of mathematical ideas is by far not only a single thread connecting a point $A$ in the past and another point $B$ in later times.
A nice characterization of mathematical development is given by Will Dunham in The Calculus Gallery: Masterpieces from Newton to Lebesgue:
- In a continuing ebb and flow, mathematicians develop grand theories and then find pertinent counterexamples to reveal the boundaries of their ideas. This juxtaposition of theory and counterexample is the logical engine by which mathematics progresses, for it is only by knowing how properties fail that we can understand how they work. And it is only by seeing how intuition misleads that we can truly appreciate the power of reason.
... and even this is a simplified description, since there are many superpositions of ebb and flow with respect to different ideas in mathematics.
There are many sources that claim no inconsistencies have been found in ZF (or ZFC, which is known to be consistent if ZF is consistent.)
Ershov and Palyutin on p. 92 of Mathematical Logic write: "What can be said about the consistency of ZFC? No inconsistencies have been discovered in ZFC itself this far."
Ebbinghaus, Flum and Thomas on p. 12 of Mathematical Logic write "Nevertheless, the fact that ZFC has been investigated and used in mathematics for decades and no inconsistency has been discovered, attests to the consistency of ZFC."
Bloch on p. 120 of Proofs and Fundamentals: A First Course in Abstract Mathematics 2nd ed writes "However, even if no one has definitively proved that the ZF axioms are consistent, we observe that these axioms have been designed to remove the known problems of naive set theory such as Russell’s Paradox. ... Ultimately, the ZF axioms seem reasonable intuitively; they work well in providing a framework for set theory as we would want it; the known problems with naive set theory have been eliminated by the ZF axioms; and experts in the field have not found any new problems that arise out of these axioms. Hence, we can feel confident that the ZF axioms are a very reasonable choice as the basis for mathematics. We simply cannot do any better than that."
There are many more texts which say essentially the same thing.
As many authors note, including all I mentioned above, the incompleteness theorem prevents us from having a proof of the consistency of ZF that is formalizable in ZF - which many authors feel precludes any sort of finitistic consistency proof.
So we are left with the situation that, although we can only prove the consistency of ZF by assuming sufficiently strong axioms, few mathematicians treat ZF as if they expect it to be inconsistent.
Best Answer
It looks like it could be this book : A History Of Mathematical Notations Vol I (1928) Florian Cajori http://www.archive.org/details/historyofmathema031756mbp
It would probably interest you even if it's not the exact same one.