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The Mathematical Experience: A National Book Award Winner – Exploring Mathematics: History, Philosophy, and Personalities for Lay Readers
This is the classic introduction for the educated lay reader to the richly diverse world of mathematics: its history, philosophy, principles, and personalities.
464 pages, Paperback
First published January 1, 1980
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May 17, 2022
Coming to Conclusions
The Mathematical Experience is just about as close to a phenomenology of mathematics as we’re ever likely to have. It is not about theories or principles, or arcane arguments and proofs but about what the authors do, and what others have recounted before them did. It is not even an explanation since many of the ‘greats’ and great events in mathematics are themselves unexplainable. The contents may constitute a kind of philosophy but if so it is a philosophy of educated tolerance and respect rather than normative or dogmatic prescription.
For me the real value of the book is its insights into what ‘mathematical method’ (and by extension all of scientific method) is about. There are many who insist that method is some attribute of rational thought which produces objective, robust, and incontrovertible facts about the world. That this is what any science does is at least questionable. But that these results are the consequence of following some fixed procedure is categorically wrong. There is no such thing as scientific method. Or more precisely, the criteria of what constitutes good science change, sometimes slowly sometimes almost overnight. Method is something learned right along with the results produced by method. Method cannot be distinguished from a more general culture:
Yes, for the edification of all the fundamentalists in the world (and there are many more of these than the religious sort) scientific knowledge is relative. Even in mathematics, the queen of the sciences, truth is a variable feast. You may think that the internal angles of a triangle have always summed to 180 degrees no matter where you are in the universe. But of course such a conclusion is warranted only is a universe with certain defined characteristics. It turns out that the universe that Euclid described isn’t definitive at all as demonstrated by the Bernard Riemann* and other 19th century mathematicians who simply dropped Euclid’s presumption that parallel lines never intersect. Using different presumptions triangles can have more than 180 degrees internally.
With this in mind, it’s important to note that Euclid’s geometry is not a less accurate or less general description of the world than Riemann’s. Neither geometry is a description of anything other than itself. And each is an entirely different and incompatible world. The fact that one may be transformed into the other does not mitigate their differences. Just that as in physics, Newton’s theory of gravitational forces is not an approximation of Einstein’s space-time, so the ‘principles’ of different geometries are not ‘generalisations’ but very different mathematical life forms with the equivalent of distinct atomic structures. They are relative because they start with different theoretical premises. They are relative by definition. Only when they are used practically does a criterion emerge to judge one against the other.
That criterion is really what is meant by scientific method. When one is conducting research, when results are reviewed with one’s colleagues, when a paper is being prepared for submission to a journal, and when the referees decide it is or is not worthy of publication, the criterion of what constitutes good science prevails. If anything it is this process which has the historic right to be called scientific method. It includes the acculturation of individuals into an instinctive mode of thought which varies by discipline. It also establishes a sort of intellectual hierarchy in which senior members have the final say about the work of more junior members. As in any other such hierarchy, it’s control over recognition, advancement and professional status is almost absolute.
But here’s the paradox: no one knows what that criterion is. Or rather more precisely, there is no agreement among practitioners about what the criterion should be. As the authors note about their own work: “We find that our judgment of what is valuable in mathematics is based on our notion of the nature and purpose of mathematics itself.” And there are widely different views about both the philosophy and the pragmatic usefulness of mathematical thought. So mathematicians, like all scientists, adopt a live and let live attitude. The consequence is that the criteria of what constitutes good mathematics is left purposely vague.
Even the criterion of what constitutes mathematics tout court is indeterminate except as what is produced by those who are accepted as part of the mathematics profession. “The definition of mathematics changes. Each generation and each thoughtful mathematician within a generation formulates a definition according to his lights,” the authors say. This too is part of scientific method, the continuous reconsideration of what the term ‘science,’ that is to say, ‘reason’ actually means. Thus, “In the final analysis, there can be no formalization of what is right and how we know it right, what is accepted, and what the mechanism for acceptance is.”
What must be called the essential ‘tolerance’ of science (despite its hierarchical structure) is captured well in the Preface: “Mathematics, like theology and all free creations of the Mind, obeys the inexorable laws of the imaginary.“ Reason, mathematical or otherwise, guarantees nothing. At best Reason is a continuing conversation among a changing group of people. The conversation is always flawed but there is hope in its continuation:
The American mathematician, C.S. Peirce is cited by the authors as defining "mathematics as the science of making necessary conclusions.“ this seems to me an apt summary of the book, keeping in mind that Peirce’s understanding of necessity was always relative to some changing end or purpose.
* Reimann also provided the mathematical foundation for the integral calculus developed by Leibniz and Newton, which had had no proof of its validity since the 17th century except its usefulness. So much for mathematical rigour. On a related topic, remember the very first thing you learned in Geometry, that the circumference of a circle is equal to its diameter multiplied by π? A cast iron Mathematical Truth, right? But only in Euclid’s world. No one in this world has ever, nor could ever, verify this truth empirically. Π is a transcendental number whose mathematical existence once again was only proven in the 19th century (by Joseph Liouville) although it had been in use for over two millennia. Nevertheless, π does not exist in our world at all. By definition, no measurement of π will ever be entirely accurate. No matter what level of precision we can achieve, there are an infinite number of additional levels that will foil our attempts to know what it really is. Something like God perhaps?.
The Mathematical Experience is just about as close to a phenomenology of mathematics as we’re ever likely to have. It is not about theories or principles, or arcane arguments and proofs but about what the authors do, and what others have recounted before them did. It is not even an explanation since many of the ‘greats’ and great events in mathematics are themselves unexplainable. The contents may constitute a kind of philosophy but if so it is a philosophy of educated tolerance and respect rather than normative or dogmatic prescription.
For me the real value of the book is its insights into what ‘mathematical method’ (and by extension all of scientific method) is about. There are many who insist that method is some attribute of rational thought which produces objective, robust, and incontrovertible facts about the world. That this is what any science does is at least questionable. But that these results are the consequence of following some fixed procedure is categorically wrong. There is no such thing as scientific method. Or more precisely, the criteria of what constitutes good science change, sometimes slowly sometimes almost overnight. Method is something learned right along with the results produced by method. Method cannot be distinguished from a more general culture:
“What was in Archimedes' head was different from what was in Newton's head and this, in turn, differed from what was in Gauss's head. It is not just a matter of ‘more,’ that Gauss knew more mathematics than Newton who, in turn, knew more than Archimedes. It is also a matter of ‘different.’ The current state of knowledge is woven into a network of different motivations and aspirations, different interpretations and potentialities.”
Yes, for the edification of all the fundamentalists in the world (and there are many more of these than the religious sort) scientific knowledge is relative. Even in mathematics, the queen of the sciences, truth is a variable feast. You may think that the internal angles of a triangle have always summed to 180 degrees no matter where you are in the universe. But of course such a conclusion is warranted only is a universe with certain defined characteristics. It turns out that the universe that Euclid described isn’t definitive at all as demonstrated by the Bernard Riemann* and other 19th century mathematicians who simply dropped Euclid’s presumption that parallel lines never intersect. Using different presumptions triangles can have more than 180 degrees internally.
With this in mind, it’s important to note that Euclid’s geometry is not a less accurate or less general description of the world than Riemann’s. Neither geometry is a description of anything other than itself. And each is an entirely different and incompatible world. The fact that one may be transformed into the other does not mitigate their differences. Just that as in physics, Newton’s theory of gravitational forces is not an approximation of Einstein’s space-time, so the ‘principles’ of different geometries are not ‘generalisations’ but very different mathematical life forms with the equivalent of distinct atomic structures. They are relative because they start with different theoretical premises. They are relative by definition. Only when they are used practically does a criterion emerge to judge one against the other.
That criterion is really what is meant by scientific method. When one is conducting research, when results are reviewed with one’s colleagues, when a paper is being prepared for submission to a journal, and when the referees decide it is or is not worthy of publication, the criterion of what constitutes good science prevails. If anything it is this process which has the historic right to be called scientific method. It includes the acculturation of individuals into an instinctive mode of thought which varies by discipline. It also establishes a sort of intellectual hierarchy in which senior members have the final say about the work of more junior members. As in any other such hierarchy, it’s control over recognition, advancement and professional status is almost absolute.
But here’s the paradox: no one knows what that criterion is. Or rather more precisely, there is no agreement among practitioners about what the criterion should be. As the authors note about their own work: “We find that our judgment of what is valuable in mathematics is based on our notion of the nature and purpose of mathematics itself.” And there are widely different views about both the philosophy and the pragmatic usefulness of mathematical thought. So mathematicians, like all scientists, adopt a live and let live attitude. The consequence is that the criteria of what constitutes good mathematics is left purposely vague.
Even the criterion of what constitutes mathematics tout court is indeterminate except as what is produced by those who are accepted as part of the mathematics profession. “The definition of mathematics changes. Each generation and each thoughtful mathematician within a generation formulates a definition according to his lights,” the authors say. This too is part of scientific method, the continuous reconsideration of what the term ‘science,’ that is to say, ‘reason’ actually means. Thus, “In the final analysis, there can be no formalization of what is right and how we know it right, what is accepted, and what the mechanism for acceptance is.”
What must be called the essential ‘tolerance’ of science (despite its hierarchical structure) is captured well in the Preface: “Mathematics, like theology and all free creations of the Mind, obeys the inexorable laws of the imaginary.“ Reason, mathematical or otherwise, guarantees nothing. At best Reason is a continuing conversation among a changing group of people. The conversation is always flawed but there is hope in its continuation:
“There is work, then, which is wrong, is acknowledged to be wrong and which, at some later date may be set to rights. There is work which is dismissed without examination. There is work which is so obscure that it is difficult to and is perforce ignored. Some of it may emerge later. There is work which may be of great importance such as Cantor's set theory-which is heterodox, and as a result, is ignored or boycotted. There is also work, perhaps the bulk of the mathematical output, which is admittedly correct, but which in the long run is ignored, for lack of or because the main streams of mathematics did not choose to pass that way.”
The American mathematician, C.S. Peirce is cited by the authors as defining "mathematics as the science of making necessary conclusions.“ this seems to me an apt summary of the book, keeping in mind that Peirce’s understanding of necessity was always relative to some changing end or purpose.
* Reimann also provided the mathematical foundation for the integral calculus developed by Leibniz and Newton, which had had no proof of its validity since the 17th century except its usefulness. So much for mathematical rigour. On a related topic, remember the very first thing you learned in Geometry, that the circumference of a circle is equal to its diameter multiplied by π? A cast iron Mathematical Truth, right? But only in Euclid’s world. No one in this world has ever, nor could ever, verify this truth empirically. Π is a transcendental number whose mathematical existence once again was only proven in the 19th century (by Joseph Liouville) although it had been in use for over two millennia. Nevertheless, π does not exist in our world at all. By definition, no measurement of π will ever be entirely accurate. No matter what level of precision we can achieve, there are an infinite number of additional levels that will foil our attempts to know what it really is. Something like God perhaps?.
July 2, 2013
One of my pet peeves is the belief that "creative" people are those who study the humanities and that "analytical" types (as if "analytical" must somehow stand in opposition to creativity) are those who study sciences and mathematics. Perhaps this belief stems from a mathematics education grounded in rote, memorization, and dull exercises. The Mathematical Experience is about mathematics, but takes a much more philosophical tone, exploring the concept of proof and truth, the history of math, and the process of discovery/invention of new ideas and theorems. I enjoyed it very much.
The book takes the form of several essays, averaging a few pages long each. I particularly enjoyed:
- "Unorthodoxies," about crank ideas but also the possibility of valuable ideas appearing to be cranks at first. "The doors of the mathematical past are often rusted. If an inner chamber is difficult of access, it does not necessarily mean there is treasure to be found therein."
- The chapter on the Chinese Remainder Theorem, a really fantastic survey of the same idea as seen through different mathematical lenses over history.
- "Nonstandard Analysis" was a thought-provoking essay about the history of the infinitesimal dx - like most present-day people I had learned it as expressed in terms of limits (the Weierstrassian "reformation," as it were) but had no idea of the infinitesimal's controversial use in other proofs. Challenges the concept of rigor, as practically useful results were "proved" in non-rigorous ways; of what use, and how much use, is rigor, and what is the role of intuition? (This was a far better exploration of the topic than the chapter devoted explicitly to intuition.)
- The essays towards the end hammer a little over-long on the distinctions between Platonism (mathematics is universal, immutable truth "out there" that we discover), constructivism (we invent mathematics, and objects must be constructed for their existence to be proved), and formalism (math is just a set of rule manipulations that does not encompass philosophical questions). But I did find the initial discussion of it to be very interesting.
- The two computer-oriented essays, "Mathematical Models, Computers, and Platonism," and "Why Should I Believe a Computer?" I think it would be wonderful if a mathematician with writing talents would update this book with more about how computers have influenced the development of mathematics.
The book takes the form of several essays, averaging a few pages long each. I particularly enjoyed:
- "Unorthodoxies," about crank ideas but also the possibility of valuable ideas appearing to be cranks at first. "The doors of the mathematical past are often rusted. If an inner chamber is difficult of access, it does not necessarily mean there is treasure to be found therein."
- The chapter on the Chinese Remainder Theorem, a really fantastic survey of the same idea as seen through different mathematical lenses over history.
- "Nonstandard Analysis" was a thought-provoking essay about the history of the infinitesimal dx - like most present-day people I had learned it as expressed in terms of limits (the Weierstrassian "reformation," as it were) but had no idea of the infinitesimal's controversial use in other proofs. Challenges the concept of rigor, as practically useful results were "proved" in non-rigorous ways; of what use, and how much use, is rigor, and what is the role of intuition? (This was a far better exploration of the topic than the chapter devoted explicitly to intuition.)
- The essays towards the end hammer a little over-long on the distinctions between Platonism (mathematics is universal, immutable truth "out there" that we discover), constructivism (we invent mathematics, and objects must be constructed for their existence to be proved), and formalism (math is just a set of rule manipulations that does not encompass philosophical questions). But I did find the initial discussion of it to be very interesting.
- The two computer-oriented essays, "Mathematical Models, Computers, and Platonism," and "Why Should I Believe a Computer?" I think it would be wonderful if a mathematician with writing talents would update this book with more about how computers have influenced the development of mathematics.
July 28, 2015
All in all, I'd say this is worth reading, although with some reservations. I can't complain too much, since I found it in the trash. It does a pretty great job introducing historical figures. Also, there are plenty of neat puzzles and proofs, if you can follow them.
Here are my nitpicks:
It definitely feels a bit dated, for one thing.
The authors suffer terribly from the curse of knowledge; they obviously have no idea what is meant by "general reader".
It is far too polemical. The authors get less and less objective as it goes on, vis a vis Platonism vs Formalism, as well as religion.
Ultimately, I recommend flipping through and only reading what interests you. It is quite fortunately written so as to make that possible.
Here are my nitpicks:
It definitely feels a bit dated, for one thing.
The authors suffer terribly from the curse of knowledge; they obviously have no idea what is meant by "general reader".
It is far too polemical. The authors get less and less objective as it goes on, vis a vis Platonism vs Formalism, as well as religion.
Ultimately, I recommend flipping through and only reading what interests you. It is quite fortunately written so as to make that possible.
Want to read
August 17, 2009My second dip into this book was routed by an encounter with Godel, Escher, Bach. I'm sorry Mathematical Experience, but your womanly wiles are no match for the allures of recursion and paradoxes that is GEB. Perhaps one day we will meet again...
Despite our early parting of ways, I can highly recommend this book to (a) young students getting interesting in long-term studies in mathematics and (b) civilian math fanboys who want to know more about the culture of mathematics.
The book reads like an alien biologist's account of the Earth species, "Mathematician" - his typical behavior, how he relates to other organisms (such as the "Physicist" or "Engineer"), and how he thinks. It's an honest and seemingly accurate account of the world of the mathematician. The book also grapples with deep mathematical questions like "What is math?", "Why does it work so well in describing reality?", and "What does it mean to 'prove' something?" The authors take a democratic approach and introduce many mathematicians' attempts to answer these questions, which makes for an interesting mix of viewpoints and avoids giving the reader a false sense of security with "the" answer to these (very) open questions.
Despite our early parting of ways, I can highly recommend this book to (a) young students getting interesting in long-term studies in mathematics and (b) civilian math fanboys who want to know more about the culture of mathematics.
The book reads like an alien biologist's account of the Earth species, "Mathematician" - his typical behavior, how he relates to other organisms (such as the "Physicist" or "Engineer"), and how he thinks. It's an honest and seemingly accurate account of the world of the mathematician. The book also grapples with deep mathematical questions like "What is math?", "Why does it work so well in describing reality?", and "What does it mean to 'prove' something?" The authors take a democratic approach and introduce many mathematicians' attempts to answer these questions, which makes for an interesting mix of viewpoints and avoids giving the reader a false sense of security with "the" answer to these (very) open questions.
Read
February 22, 2017Interesting book. I pick it up and put it down whenever. There's some cool info and history in there.
June 18, 2016
Completely accessible. Nicely written and not at all dated (despite being published in 1981). Compelling in its satire of the foundations of mathematics.
February 4, 2018
Doing intermediate level Mathematics alone in cold nights of winters, i felt strange sensation of entering into the magical world of abstract math. The Math which has nothing to do with the mundane reality of daily life. later in life i left abstract math and philosophy for its application in different unrelated field. Although application of math was very interesting but my mind remained stuck in abstract world and that strange sensation.
Years later reading and watching biographies of great mathematicians like Hardy, ramanujan, Riemen, Kurt Godel, Dr Ziaddun, Paul Erdos, Wittgenstein, John Nash, and Laktos i somehow began to understand that strange sensation. Perhaps i began to understand myself better. Now i understand why mathematicians are so eccentric and devout their life to something which very few people understand. let me describe this strange sensation by an analogy of reading books.
In the beginning, You start reading books for time pass or gaining knowledge or to impress people or countless other reasons. But after reading few hundred books , every other purpose disappears and you read books only for the sake of books. It seems you are alive only while reading books. Infact world of books becomes more real than the real world. In a sense you become abnormal or a book addict.
After above monologue, but their is a reason for it. This book is also about mathematical experience of mathematicians. There are interesting definitions and concepts for both laymen and experienced mathematicians. My interest in this book was to know the foundation concepts like formalism, constructivism and platonism. Based on my feelings i am biased towards platonism.
Years later reading and watching biographies of great mathematicians like Hardy, ramanujan, Riemen, Kurt Godel, Dr Ziaddun, Paul Erdos, Wittgenstein, John Nash, and Laktos i somehow began to understand that strange sensation. Perhaps i began to understand myself better. Now i understand why mathematicians are so eccentric and devout their life to something which very few people understand. let me describe this strange sensation by an analogy of reading books.
In the beginning, You start reading books for time pass or gaining knowledge or to impress people or countless other reasons. But after reading few hundred books , every other purpose disappears and you read books only for the sake of books. It seems you are alive only while reading books. Infact world of books becomes more real than the real world. In a sense you become abnormal or a book addict.
After above monologue, but their is a reason for it. This book is also about mathematical experience of mathematicians. There are interesting definitions and concepts for both laymen and experienced mathematicians. My interest in this book was to know the foundation concepts like formalism, constructivism and platonism. Based on my feelings i am biased towards platonism.
August 15, 2017
Wow, from Plato to Polya, this venerable work looks very very much like it is worth reading, taking notes, and reading again just for the pleasure of it, once I actually have time to enjoy all of the citations (like the Myth of Euclid?! and Chinese mathematics!! -cool!!!).
I so enjoy the study of teaching mathematics, pity I don't enjoy the students nearly as much, the vast majority of the time (ok, nearly all of the time, but I do enjoy planning my lessons!).
And let us not forget Pacioli, of double-entry book keeping fame, no?
I so enjoy the study of teaching mathematics, pity I don't enjoy the students nearly as much, the vast majority of the time (ok, nearly all of the time, but I do enjoy planning my lessons!).
And let us not forget Pacioli, of double-entry book keeping fame, no?
July 13, 2015
This book does a good job coming to terms with the fundamental dilemma of mathematics: what in the world is it? While I don't agree with all their conclusions, the book really does convey a sense of the difficulties lying behind the mathematical experience. Highly recommended for anyone interested in the philosophy of mathematics or in epistemology in general.
January 14, 2008
A new perception of math
Currently reading
April 24, 2008by Philip J. Davis and Reuben Hersh
-
Am reading this book slowly
and in no particular order.
Am heavily into "The Prime Number Theorem" chapter.
Am getting some new insights.
-
Am reading this book slowly
and in no particular order.
Am heavily into "The Prime Number Theorem" chapter.
Am getting some new insights.
May 16, 2008
Somewhat technical but still accessible for the educated layman. Probably the most useful and interesting math book I have ever read.
May 25, 2008
Good overview of the history of math and how it has shaped what mathematicians are researching today.
February 15, 2013
Wonderful book for anyone interested in mathematicians and their place in society. The imaginary dialogs are hilarious.
March 8, 2017
I can't say enough about how much I enjoyed this book while a math major in college. Highly recommended for math fans.
April 2, 2020
Авторы книги очень начитанные ребята. Только за список ссылок на достойные источники нужно иметь ее на примете. В общем она о математике в очень широком смысле слова. Много нестандартных точек зрения. Понравилось.
July 1, 2022
Content was good, albeit it got boring at multiple points in the book.
December 25, 2021
Took me back this years ago, to all my favourites in group theory, prime numbers and others, while giving an almost philosophical insight to the field and it's wizards.
January 13, 2020
Mathematics is certainly useful to science, engineering and even our daily lives. We never stop and think about accepting the legitimacy of the basis for it. Can we really prove that 1+1=2? Sure, math is useful, but is it a man-made creation that helps us understand the universe, or is it something we discover that is abstract that exists independent of us and our language, thought, and practices just as stars exist independently of us? If you enjoy math and the history of math or if you enjoy philosophy and logic you will like this book. Some parts will be difficult if you haven't had at very least algebra and you really cannot skip those parts if you want to understand the concepts. 440 pages. Good glossary.
A wide variety of topics, modeling, prime number theory, myths, religion, philosophy, pure vs. applied mathematics...and much more. I read this a long time ago and just reread several chapters.
A wide variety of topics, modeling, prime number theory, myths, religion, philosophy, pure vs. applied mathematics...and much more. I read this a long time ago and just reread several chapters.
June 14, 2020
This is the Portuguese translation of "The Mathematical Experience". An interesting attempt to convey the nature and importance of Mathematics to the lay reader, the text digresses through a variety of topics in a clear and, at times, inspired prose. It is not a mathematical text though, and apart from Chapter 5 and some examples spread through Chapters 4 and 6, not much mathematical culture is required from the reader, although someone lacking a mathematical education at the level of the first two years at the University will probably miss the better parts of the arguments and is likely not to make much sense of the rest. In spite of some odd choices (such as the emphasis in the example of Non-standard Analysis, a clearly marginal subject in present day mathematics) this is a book worth reading that tries to portrait the mathematical activity as part of the large human effort to understand and make sense of ourselves and the world.
Read
January 24, 2019packed with interesting stuff, first half mostly of historical interest, middle parts have well-written expositions about a variety of topics across many subdomains of math, and towards the end talks about pedagogy and philosophy.
Has discussions about formalism vs. Platonism (as well as Brouwer's constructionism/intuitionism) and how "philosophy of mathematics" is typically about foundations, i.e., Russell, Frege, and Hilbert, whose formalist-foundational projects were blown up by Goedel and associated paradoxes. The authors argue that "philosophy of math" should be more about teasing out the nature of mathematics as being something invented yet objective: "true facts about imaginary objects".
only took me 2 years and 3 months to finish it :)
Has discussions about formalism vs. Platonism (as well as Brouwer's constructionism/intuitionism) and how "philosophy of mathematics" is typically about foundations, i.e., Russell, Frege, and Hilbert, whose formalist-foundational projects were blown up by Goedel and associated paradoxes. The authors argue that "philosophy of math" should be more about teasing out the nature of mathematics as being something invented yet objective: "true facts about imaginary objects".
only took me 2 years and 3 months to finish it :)
February 18, 2015
This is a great book. If you are at all interested in mathematics you should read this. It's written in a pleasant stile. Its not overly popularized and not pedantic. It covers a wide range of subjects some things you will undoubtedly know, some things you know but never really thought about and some things completely new ( to me at least ). They don't force there opinion on you I wish more science writing was done in this style. As you see I don't want to give away to much because reading the book is ( to me at least ) like a a voyage of discovery. It took me some time reading the book but I enjoyed every minute of it. Have fun.
July 17, 2007
this is an excellent book that discusses the history of mathematics, as well as the evolutions of mathematical philosophy, citing clear understandable examples.
Reading this book helped me understand my discipline better and helped me understand where the field of mathematics has been and where it is heading.
I think having a math background helped me understand this book. I don't know if someone without a math background would like this book as much.
Reading this book helped me understand my discipline better and helped me understand where the field of mathematics has been and where it is heading.
I think having a math background helped me understand this book. I don't know if someone without a math background would like this book as much.
January 11, 2022
Intellectual dissonance! The history lesson is so important! Loved it as a 19 year old. And, now reading it as a 50-year old I feel even more inadequate in my knowledge of mathematics.
I will be reading this a lot more in the coming days and years.
I will be reading this a lot more in the coming days and years.
April 3, 2021
It Covers major developments in the history of mathematics. I was quite taken by it on first reading a long time ago and it is good but this is not my first time at the mathematics rodeo. It still is solid and if you are more of a newbie with math you may enjoy it.
September 3, 2012
What do mathematicians DO all day? This book provides some answers. Here is my review on Yahoo Voices: http://voices.yahoo.com/book-review-m...
September 21, 2021
If you like mathematics and you like to read, then this book is for you.
January 9, 2018
"Our knowledge of what exists may go far beyond what we are able to calculate or even approximate. Here is a simple instance of this. We are given a triangle with three unequal sides. We ask, is there a vertical line which bisects the area of the triangle? Within algorithmic mathematics one might pose the problem of finding such a line, by ruler and compass, or by more generous means. Within dialectic mathematics, one can answer, yes, such a line exists without doing any work at all. One need only notice that if one moves a knife across the figure from left to right, the fraction of the triangle to the left of the knife varies continuously from 0% to 100%, so there must be an intermediate position where the fraction is precisely 50%.
Having arrived at the solution, we may notice with a shock that the specific properties of the triangle weren't used at all; the same argument would work for any kind of an area! And so we assert the existence of a vertical bisector of any given figure, without knowing how to find it, without knowing how to compute the area cut off by the knife, and without even needing to know how to do it.
The mathematics of Egypt, of Babylon, and of the ancient Orient was all of the algorithmic type. Dialectical mathematics - strictly logical, deductive mathematics - originated with the Greeks. But it did not displace the algorithmic. In Euclid, the role of dialectic is to justify a construction - i.e., an algorithm.
It is only in modern times that we find mathematics with little or no algorithmic content, which we could call purely dialectical or existential.
One of the first investigations to exhibit a predominantly dialectic spirit was the search of the roots of a polynomial of degree n. It was long surmised that a polynomial of degree n, pn(z) = a0zn + a1zn-1 + ...
an, must have n roots, counting multiplicities. But a closed formula like the quadratic formula or the cubic formula was not found. (It was later shown that it is not possible to find a similar formula for n greater than 4.) The question then became, what other sources can we bring to bear on the problem of finding approximate roots? The theorems which guarantee this, originally provided by Gauss, are dialectic. The algorithmic aspect is still under discussion.
In most of the twentieth century, mathematics has been existence-oriented rather than algorithm-oriented. Recent years seems to show a shift back to a constructive or algorithmic view point.
Henrici points our that
'Dialectic mathematics is a rigorously logical science, where statements are either true or false, and where objects with specified properties either do or do not exist. Algorithmic mathematics is a tool for solving problems. Here we are concerned not only with the existence of a mathematical object, but also with the credentials of its existence. Dialectic mathematics is an intellectual game played according to rules about which there is a high degree of consensus. The rules of the game of algorithmic mathematics may vary according to the urgency of the problem on hand. We never could have put a man on the moon if we had insisted that the trajectories should be computed with dialectic rigor. The rules may also vary according to the computing equipment available. Dialectic mathematics invites contemplation. Algorithmic mathematics invites action. Dialectic mathematics generates insights. Algorithmic mathematics generates results.'
There is a distinct paradigm shift that distinguishes the algorithmic from the dialectic, and people who have worked in one mode may very well feel that solutions within the second mode are not 'fair' or not 'allowed.' They experience paradigm shock. P. Gordan who worked algorithmically in invariant theory reputedly felt this shock when confronted with the brilliant work of Hilbert who worked dialectically. 'This is not mathematics,' said Gordan, 'it is theology.'"
Having arrived at the solution, we may notice with a shock that the specific properties of the triangle weren't used at all; the same argument would work for any kind of an area! And so we assert the existence of a vertical bisector of any given figure, without knowing how to find it, without knowing how to compute the area cut off by the knife, and without even needing to know how to do it.
The mathematics of Egypt, of Babylon, and of the ancient Orient was all of the algorithmic type. Dialectical mathematics - strictly logical, deductive mathematics - originated with the Greeks. But it did not displace the algorithmic. In Euclid, the role of dialectic is to justify a construction - i.e., an algorithm.
It is only in modern times that we find mathematics with little or no algorithmic content, which we could call purely dialectical or existential.
One of the first investigations to exhibit a predominantly dialectic spirit was the search of the roots of a polynomial of degree n. It was long surmised that a polynomial of degree n, pn(z) = a0zn + a1zn-1 + ...
an, must have n roots, counting multiplicities. But a closed formula like the quadratic formula or the cubic formula was not found. (It was later shown that it is not possible to find a similar formula for n greater than 4.) The question then became, what other sources can we bring to bear on the problem of finding approximate roots? The theorems which guarantee this, originally provided by Gauss, are dialectic. The algorithmic aspect is still under discussion.
In most of the twentieth century, mathematics has been existence-oriented rather than algorithm-oriented. Recent years seems to show a shift back to a constructive or algorithmic view point.
Henrici points our that
'Dialectic mathematics is a rigorously logical science, where statements are either true or false, and where objects with specified properties either do or do not exist. Algorithmic mathematics is a tool for solving problems. Here we are concerned not only with the existence of a mathematical object, but also with the credentials of its existence. Dialectic mathematics is an intellectual game played according to rules about which there is a high degree of consensus. The rules of the game of algorithmic mathematics may vary according to the urgency of the problem on hand. We never could have put a man on the moon if we had insisted that the trajectories should be computed with dialectic rigor. The rules may also vary according to the computing equipment available. Dialectic mathematics invites contemplation. Algorithmic mathematics invites action. Dialectic mathematics generates insights. Algorithmic mathematics generates results.'
There is a distinct paradigm shift that distinguishes the algorithmic from the dialectic, and people who have worked in one mode may very well feel that solutions within the second mode are not 'fair' or not 'allowed.' They experience paradigm shock. P. Gordan who worked algorithmically in invariant theory reputedly felt this shock when confronted with the brilliant work of Hilbert who worked dialectically. 'This is not mathematics,' said Gordan, 'it is theology.'"
April 14, 2021
Lo leí y marqué bastante en su momento. Releyéndolo ahora veo que está muy bien hecho y que es de lo mejor que leí de divulgación: nivel de profundidad adecuado (ni muy complejo ni trivial o anecdótico), recorre temas interesantes y deja al pasar nombre de autores, artículos o libros para que podamos adentrarnos en los que nos más nos llamó la atención. No tiene un tono condescendiente sino que más bien neutro de buena revista de divulgación (de hecho varios artículos habían sido publicados antes en Scientific American). Tiene un glosario y una bibliografía muy completos y dos apéndices que me resultaron útiles sobre hitos de la matemática y la clasificación de sus distintas ramas (actualizado a 1979, imagino en estos cuarenta años se habrán multiplicado).
El contenido es una introducción a las matemáticas, una exploración de sus variedades, los aspectos externos (modelos, utilidad, mercado) e internos (símbolos, abstracciones, generalizaciones), cómo se pasó de la certeza a la falibilidad y sobre cómo enseñar y aprender esta disciplina. Tal vez por un compilado de artículos escritos en forma individual puede haber un salto entre tema y tema pero eso no impide que se pueda leer como un todo y sirva de primer abordaje a varios temas fascinantes como el infinito de Cantor o la intuición tetradimensional. Aprecio que miran otras tradiciones matemáticas como la china y dan bastante contexto como la insospechada cercanía en sus orígenes con la astrología y la religión (en ese sentido me pareció divertido que varios matemáticos se ganaran la vida haciendo horóscopos o escribiendo amuletos para rico y poderosos).
Más sobre esta reseña y otras en:
https://pablomariafernandez.substack....
El contenido es una introducción a las matemáticas, una exploración de sus variedades, los aspectos externos (modelos, utilidad, mercado) e internos (símbolos, abstracciones, generalizaciones), cómo se pasó de la certeza a la falibilidad y sobre cómo enseñar y aprender esta disciplina. Tal vez por un compilado de artículos escritos en forma individual puede haber un salto entre tema y tema pero eso no impide que se pueda leer como un todo y sirva de primer abordaje a varios temas fascinantes como el infinito de Cantor o la intuición tetradimensional. Aprecio que miran otras tradiciones matemáticas como la china y dan bastante contexto como la insospechada cercanía en sus orígenes con la astrología y la religión (en ese sentido me pareció divertido que varios matemáticos se ganaran la vida haciendo horóscopos o escribiendo amuletos para rico y poderosos).
Más sobre esta reseña y otras en:
https://pablomariafernandez.substack....
January 24, 2022
Seleção de ensaios nos quais dois professores de mtemática percorrem a história da matemática atentos às questões históricas e epistemológicas. Há uma preocupação com o ensino da matemática: como trabalhar de modo que o aluno pense por si mesmo, chegue ele mesmo à conclusão de um raciocínio, sem que o argumento de autoridade seja necessário. Para mostrar os atuais desafios da matemática, os autores selecionam alguns problemas atuais, isto é, de 1981. Naquela época, já se vão mais de quatro décadas, os programas de computador agilizavam o cálculo, mas geravam polêmicas: como considerar demonstrado o “teorema das quatro cores” com os cálculos exaustivos de uma máquina e não com uma fórmula? Segundo esse teorema, bastam quatro cores para colorir qualquer mapa de modo que as fronteiras entre países se diferenciem.
Para mim, o capítulo mais interessante é o “Realidade matemática” onde a noção de intuição passa por uma crítica que envolve também a psicologia e a filosofia. Essa discussão será importante para situar a matemática em três perspectivas: a platônica ou realista, a formalista e a construtivista. Ao final os autores propõem uma mediação: a matemática é uma invenção (perspectiva formalista), mas isso que criamos guarda mistérios que tentamos descobrir. Ou seja, a matemática é uma realidade objetiva independente da consciência, isto é, que podemos ou não descobrir. A matemática trabalha com raciocínios e argumentos sobre ideias cujos significados podem ser compartilhados. Assim, o social complexifica o que se convencionava reduzir a uma oposição entre matéria e mente.
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