Purchase a copy on IndieBound. Read the Wikipedia article on GEB.
Background
I first encountered this book by Douglas Hofstadter in high school and became enraptured. The concepts from Gödel, Escher, Bach that have most influenced my thinking and artistic practice are summarized in the notes to "Creativity and Constraint: Queering the Formal System."
Gödel, Escher, Bach (also abbreviated as GEB) usually proves too difficult for me to adequately summarize in conversation; I'll try to be succinct here.
A Brief (Incomplete) Summary
Hofstadter is a computer scientist and a polymath and he's interested in investigating concepts such as artificial intelligence, recursive thinking (what he calls "strange loops"), consciousness, and creativity among many other topics. To get to these abstract, philosophical concepts, he draws on the music of J.S. Bach, the visual art of M.C. Escher, and the work of mathematician Kurt Gödel. Bach's music, Escher's art, and Gödel's proofs all embody principles of formal systemsFormal System
What is a formal system?
Start with an [[Axiom]]
Continue with a rule or self-imposed constraintFormal Constraint
Quick Definition
Mathematics
A formal constraint (what I consider to be analogous with Hofstadter'sGödel, Escher, Bach- An Eternal Golden Braid
Purchase a copy on IndieBound.
Read the Wikipedia article on GEB.
Background
I first encountered this book by Douglas Hofstadter in high school and became enraptured. The concepts from Gödel, Escher, Bach that have most influenced my thinking and artistic practice are summarized in the notes to "Creativity and Constraint: Queering the Formal System."
Gödel, Escher, Bach (also abbreviated as GEB) usually proves too difficult for me to adequately summarize in conversation; I'll try to be... "rule of production" or "rule of inference") is a method for transforming one result of a formal system into another result. In mathematics, we can use mathematical induction to derive a theorem from an axiom or from another theorem. Because we trust that our rules of induction will preserve the truth of our mathematical statements, following t...
Apply the rules and constraints to the axiom and see what happens!
This definition is loosely based on Hofstadter’s [[Gödel, Escher, Bach- An Eternal Golden Braid]] (GEB) and on my own experience making and breaking formal systems.
Start with an axiom (in mathematics) or idea/material (creative disciplines like the arts).
A rule or constraint develops ... and properties of recursion (or self-reference), so they form complementary lenses through which to understand these concepts.
Hofstadter describes formal systemsFormal System
What is a formal system?
Start with an [[Axiom]]
Continue with a rule or self-imposed constraintFormal Constraint
Quick Definition
Mathematics
A formal constraint (what I consider to be analogous with Hofstadter'sGödel, Escher, Bach- An Eternal Golden Braid
Purchase a copy on IndieBound.
Read the Wikipedia article on GEB.
Background
I first encountered this book by Douglas Hofstadter in high school and became enraptured. The concepts from Gödel, Escher, Bach that have most influenced my thinking and artistic practice are summarized in the notes to "Creativity and Constraint: Queering the Formal System."
Gödel, Escher, Bach (also abbreviated as GEB) usually proves too difficult for me to adequately summarize in conversation; I'll try to be... "rule of production" or "rule of inference") is a method for transforming one result of a formal system into another result. In mathematics, we can use mathematical induction to derive a theorem from an axiom or from another theorem. Because we trust that our rules of induction will preserve the truth of our mathematical statements, following t...
Apply the rules and constraints to the axiom and see what happens!
This definition is loosely based on Hofstadter’s [[Gödel, Escher, Bach- An Eternal Golden Braid]] (GEB) and on my own experience making and breaking formal systems.
Start with an axiom (in mathematics) or idea/material (creative disciplines like the arts).
A rule or constraint develops ... before going into detail to explain how Kurt Gödel arrived at his celebrated Incompleteness TheoremsGödel's Incompleteness Theorems
The Foundational Crisis of Mathematics
Disclaimer: I'm not a mathematician or historian of mathematics. This description is that of a layperson; corrections and pushback are welcome.
At the start of the 20th century, mathematicians became increasingly worried about the foundations of mathematics – the subjects and concepts that formed the basis for more advanced topics in the field. Various logicians had uncovered paradoxes arising from principles of logic and set theory (two of the fun.... (You might want to read both of those notes before continuing here).
Disclaimer: I'm not a mathematician or philosopher, the following descriptions of Gödel's work are those of a layperson. Please feel free to offer your corrections.
Consistent or Complete?
The practical upshot of Gödel's work is that any sufficiently sophisticated mathematical formal system can either be complete or consistent but not both. This is due to the fact that such systems are able to operate upon themselves, using the tools and symbols of logic and mathematics to make statements about mathematics. Such metamathematical statements will uncover paradoxes that will destroy the system's consistency unless rules are put in place to limit the system's completeness
A complete formal system is one where we are able to express every true statement of the system. A consistent formal system is one without any contradicting statements. (For example, in arithmetic we don't assert that 1 + 1 = 2 AND 1 + 1 = 3 at the same time – only one of these is true).
Because we are interested in making mathematical systems that are consistent (without paradoxes or contradictions), we are left with systems that are incomplete – we are unable to adequately capture all the true statements that our system could conceivably express. Because of this shortcoming, we continually develop more and more advanced forms of mathematical reasoning to "reach" the truths that our earlier systems could not attain. In this way, mathematics builds upon itself in an infinite mannerAxiom
Quick Definition
An axiom is a logical presupposition that cannot be proven to be true – instead, we take it on faith that it is true.
Use in Mathematics
In mathematics, subjects like Geometry rely on a set of axioms that mathematicians "take on faith" in order to begin using mathematical induction to draw conclusions and develop more advanced theorems. Changing these axioms can lead to dramatically different results – for example, you can obtain spherical geometry and hyperbolic geometry ... through a process of abstraction.
Significance
In my own work in music composition, GEB has been a constant inspiration and touchstone. I have not written a piece of music without using the axiom/inference/proof model of the formal systemFormal System
What is a formal system?
Start with an [[Axiom]]
Continue with a rule or self-imposed [[Formal Constraint|constraint]]
Apply the rules and constraints to the axiom and see what happens!
This definition is loosely based on Hofstadter’s [[Gödel, Escher, Bach- An Eternal Golden Braid]] (GEB) and on my own experience making and breaking formal systems.
Start with an axiom (in mathematics) or idea/material (creative disciplines like the arts).
A rule or constraint develops ... (musical material/development technique/finished composition) as a guide.
For me, this way of thinking (and thinking about thinking) has helped with all sorts of problems – musical, ethical, or otherwise.
Related Note: formal systemsFormal System
What is a formal system?
Start with an [[Axiom]]
Continue with a rule or self-imposed constraintFormal Constraint
Quick Definition
Mathematics
A formal constraint (what I consider to be analogous with Hofstadter'sGödel, Escher, Bach- An Eternal Golden Braid
Purchase a copy on IndieBound.
Read the Wikipedia article on GEB.
Background
I first encountered this book by Douglas Hofstadter in high school and became enraptured. The concepts from Gödel, Escher, Bach that have most influenced my thinking and artistic practice are summarized in the notes to "Creativity and Constraint: Queering the Formal System."
Gödel, Escher, Bach (also abbreviated as GEB) usually proves too difficult for me to adequately summarize in conversation; I'll try to be... "rule of production" or "rule of inference") is a method for transforming one result of a formal system into another result. In mathematics, we can use mathematical induction to derive a theorem from an axiom or from another theorem. Because we trust that our rules of induction will preserve the truth of our mathematical statements, following t...
Apply the rules and constraints to the axiom and see what happens!
This definition is loosely based on Hofstadter’s [[Gödel, Escher, Bach- An Eternal Golden Braid]] (GEB) and on my own experience making and breaking formal systems.
Start with an axiom (in mathematics) or idea/material (creative disciplines like the arts).
A rule or constraint develops ...
Last modified on 01-28-2022.