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Simply Gödel
“Tieszen’s Simply Gödel is a remarkable achievement—a handy guide with the impact of a philosophical tome. It’s all elegantly lucid discussions of Kurt Gödel’s epochal discoveries, a sympathetic account of the eccentric genius’s life, focused discussions of his encounters with his astonished peers, and a visionary peek into the future of mathematics, philosophy, and the on-rushing specter of robots with minds. A compact masterpiece, brimming with fresh revelations.”—Rudy Rucker, author of Infinity and the Mind
Kurt Gödel (1906–1978) was born in Austria-Hungary (now the Czech Republic) and grew up in an ethnic German family. As a student, he excelled in languages and mathematics, mastering university-level math while still in high school. He received his doctorate from the University of Vienna at the age of 24 and, a year later, published the pioneering theorems on which his fame rests. In 1939, with the rise of Nazism, Gödel and his wife settled in the U.S., where he continued his groundbreaking work at the Institute for Advanced Study (IAS) in Princeton and became a close friend of Albert Einstein’s.
In Simply Gödel, Richard Tieszen traces Gödel’s life and career, from his early years in tumultuous, culturally rich Vienna to his many brilliant achievements as a member of IAS, as well as his repeated battles with mental illness. In discussing Gödel’s ideas, Tieszen not only provides an accessible explanation of the incompleteness theorems, but explores some of his lesser-known writings, including his thoughts on time travel and his proof of the existence of God.
With clarity and sympathy, Simply Gödel brings to life Gödel’s fascinating personal and intellectual journey and conveys the lasting impact of his work on our modern world.Simply Gödel
An accessible explanation of the incompleteness theorems An exploration of Gödel’s lesser-known writingsA sympathetic account of Gödel’s life and his repeated battles with mental illness Discover the remarkable legacy of Kurt Gödel in Simply Gödel, a compact masterpiece brimming with fresh revelations. With this book, readers will gain insight into the eccentric genius that changed mathematics, philosophy, and our understanding of robots and their minds. Simply Gödel will open readers’ eyes to Gödel’s pioneering theorems and his deep thoughts on time travel, and even the existence of God. Unlock the timeless brilliance of Kurt Gödel and be inspired to reach new heights of understanding. Buy now before the price changes!
Kurt Gödel (1906–1978) was born in Austria-Hungary (now the Czech Republic) and grew up in an ethnic German family. As a student, he excelled in languages and mathematics, mastering university-level math while still in high school. He received his doctorate from the University of Vienna at the age of 24 and, a year later, published the pioneering theorems on which his fame rests. In 1939, with the rise of Nazism, Gödel and his wife settled in the U.S., where he continued his groundbreaking work at the Institute for Advanced Study (IAS) in Princeton and became a close friend of Albert Einstein’s.
In Simply Gödel, Richard Tieszen traces Gödel’s life and career, from his early years in tumultuous, culturally rich Vienna to his many brilliant achievements as a member of IAS, as well as his repeated battles with mental illness. In discussing Gödel’s ideas, Tieszen not only provides an accessible explanation of the incompleteness theorems, but explores some of his lesser-known writings, including his thoughts on time travel and his proof of the existence of God.
With clarity and sympathy, Simply Gödel brings to life Gödel’s fascinating personal and intellectual journey and conveys the lasting impact of his work on our modern world.Simply Gödel
An accessible explanation of the incompleteness theorems An exploration of Gödel’s lesser-known writingsA sympathetic account of Gödel’s life and his repeated battles with mental illness Discover the remarkable legacy of Kurt Gödel in Simply Gödel, a compact masterpiece brimming with fresh revelations. With this book, readers will gain insight into the eccentric genius that changed mathematics, philosophy, and our understanding of robots and their minds. Simply Gödel will open readers’ eyes to Gödel’s pioneering theorems and his deep thoughts on time travel, and even the existence of God. Unlock the timeless brilliance of Kurt Gödel and be inspired to reach new heights of understanding. Buy now before the price changes!
161 pages, Kindle Edition
Published April 11, 2017
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Displaying 1 - 6 of 6 reviews
December 31, 2019
After having read Jim Holt's excellent - When Einstein Walked with Gödel: Excursions to the Edge of Thought I was on the lookout for a book for the general reader with a little bit of mathematical depth in regard to Gödel's work and this fit the niche perfectly. Tieszen does an excellent job of explaining the mathematics on a popular science level without bogging the reader down with needless nuance that would only interest a trained mathematician. The best thing, as the author mentions in the beginning of the book, is that the sections dealing with the technicalities can also be completely skipped for a purely non-technical explanation of why Gödel's work is among the greatest achievements of human thought.
An excellent book to read if you are already familiar with Gödel and want to get into a bit more of the nitty-gritty.
An excellent book to read if you are already familiar with Gödel and want to get into a bit more of the nitty-gritty.
July 22, 2018
Simply Gödel by Richard Tieszen covers Kurt Gödel’s life and work in an accessible manner. Considering the accomplishments of Mr. Gödel this is no mean feat. Around the Turn of the Twentieth Century, that is, around 1900 or so, the Mathematician David Hilbert established his famous 23 problems that he felt needed to be solved, or to prove that no solution would be forthcoming. Namely, his second problem was to “Prove that the Axioms of Arithmetic are Consistent.” Gödel managed to show that this was impossible using the tools of Arithmetic alone.
However, the book does not merely cover this element of Gödel’s life; it discusses in great detail his relationships with other thinkers of the time and how his opinions were formed. This leads to Gödel’s perfectionism and his holding back on publishing. As I mentioned in the opening sentence of this review, this book covers the life and work of Kurt Gödel; this also includes his personal relationships, especially the one with his wife. Gödel met Adele Porkert Nimbursky 10 years before they married, but Kurt’s parents had some issues with her. The book does not dance around from topic to topic but treats each event in a chronological manner.
The book contains thirteen chapters, with three of those chapters focusing on the Incompleteness Theorems. It also develops the idea of what a System of Logic is and the underlying principles to that idea. With a lucid style and careful prose, Tieszen shows the depth of his research into Gödel’s life. Furthermore, I enjoy the fact that Tieszen goes over the symbols of Formal Logic and explains their meanings. This is extremely helpful to me since I don’t know what a backward E is supposed to indicate, for example. I also like how the problem is laid out by Tieszen. He defines the issues of crafting something in formal logic and compares it to a Computer Programming Language. The syntax and grammar must be established beforehand and so on.
Speaking of computers, Turing is mentioned several times as well. The Turing machine is the classic mathematical construct that defines what any computer is capable of. Limitations on size and time are disregarded. Inconsistency is just another word for Contradiction in the case of logic. As the book covers Kurt Gödel’s life in detail, it also discusses his Completeness Theorem, the one he submitted for his Doctoral Thesis.
The book does its best to explain Gödel’s work in a format that is understandable, but I don’t know if many people would understand. If you have heard of Kurt Gödel, I imagine that you are somewhat familiar with Logical Systems, Zermelo-Fraenkel Set Theory, and other things. However, if you are someone that just wants a mindlessly enjoyable biography, this book might not be for you. I enjoyed the book quite a bit, especially since it does not stop at the Incompleteness Theorems, but covers the aftermath of them as well.
While I have read “On The Formally Undecidable Propositions of Principia Mathematica and Related Systems,” from what I recall of it, I only knew that it was important, I couldn’t make head nor tail of it. This book, Simply Gödel, helps to define what Gödel says and how he says it. That is the long and short of it.
However, the book does not merely cover this element of Gödel’s life; it discusses in great detail his relationships with other thinkers of the time and how his opinions were formed. This leads to Gödel’s perfectionism and his holding back on publishing. As I mentioned in the opening sentence of this review, this book covers the life and work of Kurt Gödel; this also includes his personal relationships, especially the one with his wife. Gödel met Adele Porkert Nimbursky 10 years before they married, but Kurt’s parents had some issues with her. The book does not dance around from topic to topic but treats each event in a chronological manner.
The book contains thirteen chapters, with three of those chapters focusing on the Incompleteness Theorems. It also develops the idea of what a System of Logic is and the underlying principles to that idea. With a lucid style and careful prose, Tieszen shows the depth of his research into Gödel’s life. Furthermore, I enjoy the fact that Tieszen goes over the symbols of Formal Logic and explains their meanings. This is extremely helpful to me since I don’t know what a backward E is supposed to indicate, for example. I also like how the problem is laid out by Tieszen. He defines the issues of crafting something in formal logic and compares it to a Computer Programming Language. The syntax and grammar must be established beforehand and so on.
Speaking of computers, Turing is mentioned several times as well. The Turing machine is the classic mathematical construct that defines what any computer is capable of. Limitations on size and time are disregarded. Inconsistency is just another word for Contradiction in the case of logic. As the book covers Kurt Gödel’s life in detail, it also discusses his Completeness Theorem, the one he submitted for his Doctoral Thesis.
The book does its best to explain Gödel’s work in a format that is understandable, but I don’t know if many people would understand. If you have heard of Kurt Gödel, I imagine that you are somewhat familiar with Logical Systems, Zermelo-Fraenkel Set Theory, and other things. However, if you are someone that just wants a mindlessly enjoyable biography, this book might not be for you. I enjoyed the book quite a bit, especially since it does not stop at the Incompleteness Theorems, but covers the aftermath of them as well.
While I have read “On The Formally Undecidable Propositions of Principia Mathematica and Related Systems,” from what I recall of it, I only knew that it was important, I couldn’t make head nor tail of it. This book, Simply Gödel, helps to define what Gödel says and how he says it. That is the long and short of it.
July 31, 2018
This book was written with me in mind, I am its target reader, so I was delighted when I was invited to read a complimentary copy in exchange for an honest review. As it turns out, the book is easier to read than to review. I decided to judge the book by its cover, and evaluate its claim to be simple, which is not at all the same as being brief or succinct.
The book covers Gödel’s entire life story in such a brief way as to be perfunctory. It gives enough to leave readers curious but dissatisfied, though at the end it recommends further reading which certainly does appeal to me, but it only achieves a limited amount of human interest for this reader. There has to be an implication that, like mathematics, this is not what the reader is looking for here. Simplifying a biography too much does not make it more engaging.
The book omits all mathematics, which makes it accessible to general readers. This can be justified because a general reader, while interested to know that Gödel was a mathematician of the highest stature, is never going to travel further into that maze. It does , though, provide a tourist’s guide to some of Gödel’s work, which covered an impressive range of topics, and it includes a lot of quite entertaining material, notably his speculations on the nature of time.
Having resolved to omit all formal mathematics, the author does decide to present a few simple examples of some standard formulas and arguments in what is called “predicate logic” or “quantificational logic.” This is the logic for which Gödel proved his completeness theorem. [p30] Here we see the term “simple” used in a relativistic sense. It is simple in the sense of being very basic but it is not simple material for the general reader, encountering a language of symbols for the first time. However, it is tolerable and it does end and I found I could proceed without any need to refer back to this section. I am not convinced that it was necessary at all.
Let’s get to the point then. The core of the book just accepts that its target reader really wants to read about Gödel in order to understand better his Incompleteness Theorems and their place in philosophy. This material is not simple, I don’t know why it ought to be or ever could be simple and it would be patronising as well as futile to pass it off as such. Instead of being simple, the explanation is patient, well structured and clearly expressed in plain language. However, the concepts are slippery and I found it necessary to review my highlights several times and to read the full text a number of times before I felt confident that I had a clear impression of what is being said.
So the real benefit of the book’s brevity is that it leaves more time for second and third readings. It would be a shame to enjoy a quick and unruffled reading without taking the time to reflect on its content. That said, the aim of simplicity might actually have been better served by being a bit (even a lot) less succinct at times. Let’s face it: Gödel’s Incompleteness Theorems stopped philosophy in its tracks – it had an immense impact. This introduction for general readers may be set out in simple terms but it is worth taking the time out to work through all the arguments and appreciate their significance. The good thing, based on my own notes and highlights, is that there is more than enough material here to get a decent grounding and the explanations are clear and helpful.
Some quotes to get a flavour:
The axiomatic method has been with us for a long time. More than 2,000 years ago, Euclid applied it to the geometry of his day. Since then, axiomatization has been viewed as an ideal way to systematically organize and unify mathematics and logic.[p28] ... From the small kernel of truths about the domain that the axioms express, the idea is to derive, using valid principles of reasoning, the domain’s other truths...[p29] ... The notion of an axiomatic formal system can thus be made so exact that generating theorems from axioms, based on the rules of inference, is purely mechanical or algorithmic. [p41]
...It is often said that the first incompleteness theorem tells us that mathematical truth is not equal to formal proof. Formal proof is arithmetic in nature, but truth is not. It is also worth keeping in mind that axiomatic formal systems are Turing machines, so we are speaking of the limitations on computers that can do some arithmetic. Such computers cannot be both consistent and complete... As a Platonic rationalist might put it, we uncover more and more arithmetic truths. These truths transcend the axiomatic formal system at each stage. The Platonist would hold that truth is independent of the finite, concrete axiomatic systems or machines and that more truths are, as it were, waiting to be discovered. We can keep ascending on the basis of reason to more truths, but we will always fall short of grasping them all. [p58]
The realm of mathematical truth cannot be regimented into systematic order, as originally intended, by setting out a single fixed set of axioms and rules of inference. We can formally deduce an endless number of truths from any given set of mathematical axioms and rules of inference that are independent of those axioms and rules if the system is consistent. Hence, the theorems come as a blow to anyone who thought the essence of mathematics was, in the end, axiomatic formal reasoning.[p71]
...formal or computational exactness does not always yield certainty. To think otherwise is an illusion. The alleged clarity associated with formal systems is always clarity relative to some background against which the formal system is interpreted. It cannot be strict formalism—blind computation—all the way down. What could clarity amount to without meaning? [p79]
The incompleteness theorems are rigorously proved mathematical theorems about the scope and limits of precisely defined axiomatic formal systems, or Turing machines, in which one can do some arithmetic... Thus, they do not apply to many kinds of texts, such as religious texts—the Bible, Koran, Vedas, Buddhist sutras—and typically not even to theories in the natural sciences. The incompleteness theorems do not show that there are non-material souls, or that some kind of mystical intuition must replace cogent proof in mathematics. They do not provide grounds for despair or mystery-mongering. They do not show that the world in general, or the world of mathematics and logic in particular, is chaotic. They do not imply that truth is relative or accidental. They do not imply that there are only many different interpretations of things and no truth or objectivity. It is correct to say that formal proof is always relative to the axiomatic formal system in which it is defined but the incompleteness theorems drive a wedge between formal provability and mathematical truth. The theorems do not imply that there are absolutely unsolvable mathematical problems. [p82]
Proofs in mathematics have been around for millennia, while strict formalism about proof has existed for only about a hundred years. The theorems do not imply any kind of anti-rationalism or irrationalism. [p83]
The book covers Gödel’s entire life story in such a brief way as to be perfunctory. It gives enough to leave readers curious but dissatisfied, though at the end it recommends further reading which certainly does appeal to me, but it only achieves a limited amount of human interest for this reader. There has to be an implication that, like mathematics, this is not what the reader is looking for here. Simplifying a biography too much does not make it more engaging.
The book omits all mathematics, which makes it accessible to general readers. This can be justified because a general reader, while interested to know that Gödel was a mathematician of the highest stature, is never going to travel further into that maze. It does , though, provide a tourist’s guide to some of Gödel’s work, which covered an impressive range of topics, and it includes a lot of quite entertaining material, notably his speculations on the nature of time.
Having resolved to omit all formal mathematics, the author does decide to present a few simple examples of some standard formulas and arguments in what is called “predicate logic” or “quantificational logic.” This is the logic for which Gödel proved his completeness theorem. [p30] Here we see the term “simple” used in a relativistic sense. It is simple in the sense of being very basic but it is not simple material for the general reader, encountering a language of symbols for the first time. However, it is tolerable and it does end and I found I could proceed without any need to refer back to this section. I am not convinced that it was necessary at all.
Let’s get to the point then. The core of the book just accepts that its target reader really wants to read about Gödel in order to understand better his Incompleteness Theorems and their place in philosophy. This material is not simple, I don’t know why it ought to be or ever could be simple and it would be patronising as well as futile to pass it off as such. Instead of being simple, the explanation is patient, well structured and clearly expressed in plain language. However, the concepts are slippery and I found it necessary to review my highlights several times and to read the full text a number of times before I felt confident that I had a clear impression of what is being said.
So the real benefit of the book’s brevity is that it leaves more time for second and third readings. It would be a shame to enjoy a quick and unruffled reading without taking the time to reflect on its content. That said, the aim of simplicity might actually have been better served by being a bit (even a lot) less succinct at times. Let’s face it: Gödel’s Incompleteness Theorems stopped philosophy in its tracks – it had an immense impact. This introduction for general readers may be set out in simple terms but it is worth taking the time out to work through all the arguments and appreciate their significance. The good thing, based on my own notes and highlights, is that there is more than enough material here to get a decent grounding and the explanations are clear and helpful.
Some quotes to get a flavour:
The axiomatic method has been with us for a long time. More than 2,000 years ago, Euclid applied it to the geometry of his day. Since then, axiomatization has been viewed as an ideal way to systematically organize and unify mathematics and logic.[p28] ... From the small kernel of truths about the domain that the axioms express, the idea is to derive, using valid principles of reasoning, the domain’s other truths...[p29] ... The notion of an axiomatic formal system can thus be made so exact that generating theorems from axioms, based on the rules of inference, is purely mechanical or algorithmic. [p41]
...It is often said that the first incompleteness theorem tells us that mathematical truth is not equal to formal proof. Formal proof is arithmetic in nature, but truth is not. It is also worth keeping in mind that axiomatic formal systems are Turing machines, so we are speaking of the limitations on computers that can do some arithmetic. Such computers cannot be both consistent and complete... As a Platonic rationalist might put it, we uncover more and more arithmetic truths. These truths transcend the axiomatic formal system at each stage. The Platonist would hold that truth is independent of the finite, concrete axiomatic systems or machines and that more truths are, as it were, waiting to be discovered. We can keep ascending on the basis of reason to more truths, but we will always fall short of grasping them all. [p58]
The realm of mathematical truth cannot be regimented into systematic order, as originally intended, by setting out a single fixed set of axioms and rules of inference. We can formally deduce an endless number of truths from any given set of mathematical axioms and rules of inference that are independent of those axioms and rules if the system is consistent. Hence, the theorems come as a blow to anyone who thought the essence of mathematics was, in the end, axiomatic formal reasoning.[p71]
...formal or computational exactness does not always yield certainty. To think otherwise is an illusion. The alleged clarity associated with formal systems is always clarity relative to some background against which the formal system is interpreted. It cannot be strict formalism—blind computation—all the way down. What could clarity amount to without meaning? [p79]
The incompleteness theorems are rigorously proved mathematical theorems about the scope and limits of precisely defined axiomatic formal systems, or Turing machines, in which one can do some arithmetic... Thus, they do not apply to many kinds of texts, such as religious texts—the Bible, Koran, Vedas, Buddhist sutras—and typically not even to theories in the natural sciences. The incompleteness theorems do not show that there are non-material souls, or that some kind of mystical intuition must replace cogent proof in mathematics. They do not provide grounds for despair or mystery-mongering. They do not show that the world in general, or the world of mathematics and logic in particular, is chaotic. They do not imply that truth is relative or accidental. They do not imply that there are only many different interpretations of things and no truth or objectivity. It is correct to say that formal proof is always relative to the axiomatic formal system in which it is defined but the incompleteness theorems drive a wedge between formal provability and mathematical truth. The theorems do not imply that there are absolutely unsolvable mathematical problems. [p82]
Proofs in mathematics have been around for millennia, while strict formalism about proof has existed for only about a hundred years. The theorems do not imply any kind of anti-rationalism or irrationalism. [p83]
July 29, 2018
Suffering from insomnia at 4 a.m, reasoning about the consistency of axiomatic systems and tautologies, about Gödel, Russell, Wittgenstein. Then you ask a fundamental question : is my insomnia creating my keen reasoning or vice versa? A perpetual question without answer.
Reading is easy, but reading for your brain attic's unsolvable questions is complex. This book is simple; is simply about Gödel ! - Gödel is not simple ! Gödel's simplicity is complex [thanks god].
An an intellectual viage in the mind of a polymath !
A biography about an idiosyncratic genius cannot be a conventional book.
The greatest logician after Aristotle cannot be presented through a normal biographical optic ; not because of his asociality and psychological turbulences, but because of his rationalisation of the world, of the world of truth, the world of material absence as a cardinal element, the world of objective truth.
The pages of this book create a precise viage from Vienna Circle, their logical positivism , Gödel's early signs of Platonic rationalism, through David Hilbert's Formalism, Principia Mathematica, Wittgenstein, Alan Turing's Machine ,P → ( Q → P), Gödel's completeness theorem , Gödel's Numbers, Gödel's incompleteness theorems, their consequences in Logic, Mathematics, Computer Science, Continuum problem in Set Theory, till Princeton, Einstein, Physics and Time.
Summa Summarum, gentlemen : Simply Gödel !
Reading is easy, but reading for your brain attic's unsolvable questions is complex. This book is simple; is simply about Gödel ! - Gödel is not simple ! Gödel's simplicity is complex [thanks god].
An an intellectual viage in the mind of a polymath !
A biography about an idiosyncratic genius cannot be a conventional book.
The greatest logician after Aristotle cannot be presented through a normal biographical optic ; not because of his asociality and psychological turbulences, but because of his rationalisation of the world, of the world of truth, the world of material absence as a cardinal element, the world of objective truth.
The pages of this book create a precise viage from Vienna Circle, their logical positivism , Gödel's early signs of Platonic rationalism, through David Hilbert's Formalism, Principia Mathematica, Wittgenstein, Alan Turing's Machine ,P → ( Q → P), Gödel's completeness theorem , Gödel's Numbers, Gödel's incompleteness theorems, their consequences in Logic, Mathematics, Computer Science, Continuum problem in Set Theory, till Princeton, Einstein, Physics and Time.
Summa Summarum, gentlemen : Simply Gödel !
February 11, 2021
Well, I tried it, but this area of thought is way outside my bailiwick. It's interesting, though, and I think that a person with some training in philosophy might really enjoy it.
July 22, 2018
Gödel was a complex person so any effort to draw him simply is welcomed. He stands tall in that fuzzy area between genius and madness. For example, he proved that any consistent system (no contradictions) is incomplete (e.g. math relies on axioms that cannot be verified with math) but died of self-starvation struggling with paranoia. What a great divergence as a function of time. And Richard did a good job putting together the bits of his life influencing his intellectual development (say personality, Vienna, Plato, Leibniz and Princeton) and his achievements as a scientist. Of course most of the book is about the incompleteness theorems including the background from where the idea germinated (say number theory, logic, Hilbert and Russell-Whitehead). But it goes further including his friendship with Einstein and the very sad end of his life in Princeton.
Displaying 1 - 6 of 6 reviews