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Aesthetic Measure 1933
1933. An attempt to bring the basic formal side of art within the purview of simple mathematical formula defining aesthetic measure. the basic formula; polygonal forms; ornaments and tilings; vases; diatonic chords; diatonic harmony; melody; musical quality in poetry; earlier aesthetic theories; art and aesthetics. Over 20 plates and illustrations.
- GenresMathematics
292 pages, Paperback
First published January 1, 1933
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December 3, 2018
Rapped Up & Hopping Out
Recently someone mentioned to me that he found rap and hip-hop music to be less than to his taste. I asked what he meant. “Well,” he said, “I just don’t like it.” This is about as seriously as most people, I suspect, take the issue of aesthetics, as if it were a trivial consideration that has to do with unchallengeable preferences about art or what handbag goes with which shoes.
An aesthetic is a criterion, a standard by which we assign something (or someone) a significance, importance or value. “I don’t like rap music.” isn’t an aesthetic, it’s a judgement which is the consequence of an aesthetic, a judgement which is typically completely unaware of the criterion on which it is based. This sort of unconscious and unconsidered judgement is usually a matter of habit.
There are several problems with aesthetic habit. First since the criterion in play is never stated, it’s not possible to learn, to improve one’s aesthetic choices. The different aesthetic of the rapper, and there is one, remains as invisible as one’s own. The judgement about rap, or anything else, is then merely a prejudice. Second, this lack of awareness makes all aesthetics appear as absolutes, unfounded preferences, which we each have a right to maintain simply because there is no way to compare them.
It is this second problem, really a fundamental political problem, which creates an urgency about aesthetics. If our aesthetic criteria are hidden, they cannot be compared. This creates unresolvable conflict. More importantly it prevents the discovery of criteria which are more encompassing than the ones we already have. Such ‘synthetic’ criteria are real and discoverable and in a very practical sense incorporate apparently conflicting criteria into one that is more general, while simultaneously maintaining the integrity of the less general.
There are examples of this sort of aesthetic synthesis everywhere if one chooses to look for them. In physics, for example, the (sensory) aesthetic of relativity - time/space conflation - doesn’t make the (mystical) aesthetic of Newtonian Physics -instantaneous action at a distance - wrong, it simply shows where it is appropriate, namely for things that are not very small or very fast.*
The mathematical expression known as EuIer’s Identity is a spectacularly elegant example of the synthesis of at least four apparently incommensurable numerical universes into a coherent whole driven by a very specific mathematical aesthetic (pattern). In my own field of Christian theology, there are countless examples of this sort of aesthetic synthesis - for example in the Catholic doctrine of the Immaculate Conception which reconciles the apparently contradictory doctrines of Original Sin and the Incarnation (the aesthetic is clearly one of hidden consistency).
I have also experienced the power of aesthetic synthesis throughout my career in business. Typically, for example, in any large company, the senior executives don’t share a criterion of value. The marketing manager knows that everything depends on market share (a competitive aesthetic): If we increase share, we can spread our fixed costs and thereby compete more effectively. And the way to increase share is to invest in advertising and an expanded sales force. Meanwhile, the operations manager knows that the way to be cost effective is to invest in the latest technology (a reputational aesthetic): This will lower unit cost and allow us to ‘buy’ market share with lower prices. These are fundamentally incompatible and cause endless fruitless argument.
It might appear that this sort of practical issue is merely a technical economic problem. After all, isn’t the overall criterion of a business ‘profit’? Shouldn’t the chief executive make his decision based on this aesthetic? Perhaps... if the chief executive could get a consensus on what constitutes profit. Economists (an aesthetic of efficiency) have a very different aesthetic of profit than Accountants (an aesthetic of orderliness), who have little time for the aesthetic of Financial Theorists (a mathematical aesthetic of beauty believe it or not), who don’t necessarily agree with the range of criteria of shareholders from pension funds to pensioners (quite possibly an aesthetic of predictable stability). In fact I have never encountered a CEO who treated profit, however it was defined in the company, as anything but a constraint within which he or she had to establish their own aesthetic. In any case the chief executive still has to cope with the radical conflict between marketing and operations.
My experience is that it is possible to find a synthetic criterion that brings together these sorts of seemingly contrary criteria of value. But the essential ingredients for such a synthesis is the articulation of the hidden aesthetics which have become habitual and therefore inhibiting to consensus, or even to reasonable discussion. I have witnessed transformations in groups of senior executives through the courageous expression of aesthetics like: ‘we’ll know how well we’re doing by monitoring revenues from new products’ (an aesthetic of innovation); or ‘our real corporate duty right now is to ensure people don’t start killing each other in the streets’ (an aesthetic of urgent social care); and ‘our real measure of success is not return on assets but the amount of assets we share with the government.’ (a cooperative aesthetic). Each of these is situation-specific, somewhat emotional, not the least obvious, in need of much further articulation, impermanent and yet very effective in generating a new shared aesthetic consensus.
This is why Aesthetic Measure is important. It is a rather Quixotic but exciting attempt to formulate a universal criterion for aesthetic choice (certainly less Quixotic but more exciting than say economic utility) . It is also risky, because like any such formulation it opens up the possibility of criticism, even mockery. While fundamentally flawed, however, Aesthetic Measure is not a book, or an effort, to be mocked.
Like Bertrand Russell’s Principia Mathematica or Fred Hoyle’s Evolution From Space, George Birkhoff’s Aesthetic Measure is a heroic failure. By trying to do what can’t be done - create a universal measure of value - it points a way toward progress by pinpointing something of vital importance: The more articulate we become about what we like, the more what we like can be appreciated by others; and the more that what we like is appreciated, the greater the chance that we can enlarge our appreciation.
The goal of Aesthetic Measure is the formulation of a universal criterion of aesthetic value, a standard that can be used to compare the importance, worth, and intrinsic beauty among diverse objects. That it fails is its most important contribution to understanding what aesthetics is about and how the way in which we make judgements of value can be improved.
Birkhoff proposes a very specific and concise formulation for his universal standard: M=O/C, where M is the aesthetic measure, O is composed of the various ‘formal’ elements of order (like repetition, rhyme, sequence, balance, etc., in poetry or music), and C is the complexity of the object being evaluated, for example the number of notes in a melodic score or syllables in a poem. What the equation implies is that ‘better’ is that which employs less to do more. Birkhoff then applies it in a wide range of objects from mathematical polygons, to Chinese vases, with an appendix that touches even on architecture.
This odd looking equation may be misguided and wrong, but it is not nonsensical. It is a very personal statement that summarises Birkhoff’s own experience as it stood at the time he wrote the book. It is the product of a life, one of whose main concerns was the aesthetic, trying to understand itself. In isolation, Birkhoff's aesthetic formula is indefensible. Anyone could formulate an alternative. Many have.
But that is exactly the strength of an expression like Birkhoff's aesthetic measure: it provokes a response which must be at least as articulate in order to be effective. It brings the aesthetic into the realm of conversation and discussion. It has power. The more articulate, the more powerful. Sometimes because it captures what others are trying to express as well. But always because it demands a response. I feel a degree of sympathy for Birkhoff’s measure but its real significance for me is that I feel like I want to modify it: Precisely its point.
The bulk of Aesthetic Measure is taken up with an explanation of why Birkhoff believes his particular formulation is both appropriate as well as better than others that he has considered. The construction of the measure, therefore, has not been in any way arbitrary.** It is not whimsical. It's very existence is a challenge to anyone who disagrees to come up with something better. But that something better must include Birkhoff's as a special case. There can be no 'best' except temporarily until a more inclusive measure is proposed.
I don't know what the aesthetic of modern rappers is, but (I extemporise!) it might be something like a repetitive iambic hexameter that uses the form to explore the variations on contemporary romantic engagement. If so, it's not all that distant from a Shakespearean sonnet. Birkhoff's aesthetic measure could be applied both to rap as well as sonnets. While I'm not willing to risk an evaluation of each genre on that measure, I submit that it is at least possible to do so if one had a serious interest in the matter. ***
Thanks to Birkhoff, therefore, we progress somewhat beyond the dead end of "I don't know much about art, but I know what I like."
* I don’t think an explanation of Newton’s mystical tendency is needed given his religious, alchemical, and other occult interests. However my use of ‘sensory’ for Einstein’s aesthetic might seem strange given the apparently counter-intuitive intellectual demands of Relativity. Nevertheless Einstein himself was keen to explain both his special and general theories in terms of sensory reality - acceleration as indistinguishable from gravity, the perception of bursts of light at different points in a moving train, etc.
** It should be clear at this point that measurement of any kind - social, scientific, commercial - is itself an essentially aesthetic activity. The choice of a metric is the operationalisation of an aesthetic. See: https://www.goodreads.com/review/show...
*** An interesting synthesis of two aesthetics - from maths and music - is discussed at length here: https://www.goodreads.com/review/show...
Recently someone mentioned to me that he found rap and hip-hop music to be less than to his taste. I asked what he meant. “Well,” he said, “I just don’t like it.” This is about as seriously as most people, I suspect, take the issue of aesthetics, as if it were a trivial consideration that has to do with unchallengeable preferences about art or what handbag goes with which shoes.
An aesthetic is a criterion, a standard by which we assign something (or someone) a significance, importance or value. “I don’t like rap music.” isn’t an aesthetic, it’s a judgement which is the consequence of an aesthetic, a judgement which is typically completely unaware of the criterion on which it is based. This sort of unconscious and unconsidered judgement is usually a matter of habit.
There are several problems with aesthetic habit. First since the criterion in play is never stated, it’s not possible to learn, to improve one’s aesthetic choices. The different aesthetic of the rapper, and there is one, remains as invisible as one’s own. The judgement about rap, or anything else, is then merely a prejudice. Second, this lack of awareness makes all aesthetics appear as absolutes, unfounded preferences, which we each have a right to maintain simply because there is no way to compare them.
It is this second problem, really a fundamental political problem, which creates an urgency about aesthetics. If our aesthetic criteria are hidden, they cannot be compared. This creates unresolvable conflict. More importantly it prevents the discovery of criteria which are more encompassing than the ones we already have. Such ‘synthetic’ criteria are real and discoverable and in a very practical sense incorporate apparently conflicting criteria into one that is more general, while simultaneously maintaining the integrity of the less general.
There are examples of this sort of aesthetic synthesis everywhere if one chooses to look for them. In physics, for example, the (sensory) aesthetic of relativity - time/space conflation - doesn’t make the (mystical) aesthetic of Newtonian Physics -instantaneous action at a distance - wrong, it simply shows where it is appropriate, namely for things that are not very small or very fast.*
The mathematical expression known as EuIer’s Identity is a spectacularly elegant example of the synthesis of at least four apparently incommensurable numerical universes into a coherent whole driven by a very specific mathematical aesthetic (pattern). In my own field of Christian theology, there are countless examples of this sort of aesthetic synthesis - for example in the Catholic doctrine of the Immaculate Conception which reconciles the apparently contradictory doctrines of Original Sin and the Incarnation (the aesthetic is clearly one of hidden consistency).
I have also experienced the power of aesthetic synthesis throughout my career in business. Typically, for example, in any large company, the senior executives don’t share a criterion of value. The marketing manager knows that everything depends on market share (a competitive aesthetic): If we increase share, we can spread our fixed costs and thereby compete more effectively. And the way to increase share is to invest in advertising and an expanded sales force. Meanwhile, the operations manager knows that the way to be cost effective is to invest in the latest technology (a reputational aesthetic): This will lower unit cost and allow us to ‘buy’ market share with lower prices. These are fundamentally incompatible and cause endless fruitless argument.
It might appear that this sort of practical issue is merely a technical economic problem. After all, isn’t the overall criterion of a business ‘profit’? Shouldn’t the chief executive make his decision based on this aesthetic? Perhaps... if the chief executive could get a consensus on what constitutes profit. Economists (an aesthetic of efficiency) have a very different aesthetic of profit than Accountants (an aesthetic of orderliness), who have little time for the aesthetic of Financial Theorists (a mathematical aesthetic of beauty believe it or not), who don’t necessarily agree with the range of criteria of shareholders from pension funds to pensioners (quite possibly an aesthetic of predictable stability). In fact I have never encountered a CEO who treated profit, however it was defined in the company, as anything but a constraint within which he or she had to establish their own aesthetic. In any case the chief executive still has to cope with the radical conflict between marketing and operations.
My experience is that it is possible to find a synthetic criterion that brings together these sorts of seemingly contrary criteria of value. But the essential ingredients for such a synthesis is the articulation of the hidden aesthetics which have become habitual and therefore inhibiting to consensus, or even to reasonable discussion. I have witnessed transformations in groups of senior executives through the courageous expression of aesthetics like: ‘we’ll know how well we’re doing by monitoring revenues from new products’ (an aesthetic of innovation); or ‘our real corporate duty right now is to ensure people don’t start killing each other in the streets’ (an aesthetic of urgent social care); and ‘our real measure of success is not return on assets but the amount of assets we share with the government.’ (a cooperative aesthetic). Each of these is situation-specific, somewhat emotional, not the least obvious, in need of much further articulation, impermanent and yet very effective in generating a new shared aesthetic consensus.
This is why Aesthetic Measure is important. It is a rather Quixotic but exciting attempt to formulate a universal criterion for aesthetic choice (certainly less Quixotic but more exciting than say economic utility) . It is also risky, because like any such formulation it opens up the possibility of criticism, even mockery. While fundamentally flawed, however, Aesthetic Measure is not a book, or an effort, to be mocked.
Like Bertrand Russell’s Principia Mathematica or Fred Hoyle’s Evolution From Space, George Birkhoff’s Aesthetic Measure is a heroic failure. By trying to do what can’t be done - create a universal measure of value - it points a way toward progress by pinpointing something of vital importance: The more articulate we become about what we like, the more what we like can be appreciated by others; and the more that what we like is appreciated, the greater the chance that we can enlarge our appreciation.
The goal of Aesthetic Measure is the formulation of a universal criterion of aesthetic value, a standard that can be used to compare the importance, worth, and intrinsic beauty among diverse objects. That it fails is its most important contribution to understanding what aesthetics is about and how the way in which we make judgements of value can be improved.
Birkhoff proposes a very specific and concise formulation for his universal standard: M=O/C, where M is the aesthetic measure, O is composed of the various ‘formal’ elements of order (like repetition, rhyme, sequence, balance, etc., in poetry or music), and C is the complexity of the object being evaluated, for example the number of notes in a melodic score or syllables in a poem. What the equation implies is that ‘better’ is that which employs less to do more. Birkhoff then applies it in a wide range of objects from mathematical polygons, to Chinese vases, with an appendix that touches even on architecture.
This odd looking equation may be misguided and wrong, but it is not nonsensical. It is a very personal statement that summarises Birkhoff’s own experience as it stood at the time he wrote the book. It is the product of a life, one of whose main concerns was the aesthetic, trying to understand itself. In isolation, Birkhoff's aesthetic formula is indefensible. Anyone could formulate an alternative. Many have.
But that is exactly the strength of an expression like Birkhoff's aesthetic measure: it provokes a response which must be at least as articulate in order to be effective. It brings the aesthetic into the realm of conversation and discussion. It has power. The more articulate, the more powerful. Sometimes because it captures what others are trying to express as well. But always because it demands a response. I feel a degree of sympathy for Birkhoff’s measure but its real significance for me is that I feel like I want to modify it: Precisely its point.
The bulk of Aesthetic Measure is taken up with an explanation of why Birkhoff believes his particular formulation is both appropriate as well as better than others that he has considered. The construction of the measure, therefore, has not been in any way arbitrary.** It is not whimsical. It's very existence is a challenge to anyone who disagrees to come up with something better. But that something better must include Birkhoff's as a special case. There can be no 'best' except temporarily until a more inclusive measure is proposed.
I don't know what the aesthetic of modern rappers is, but (I extemporise!) it might be something like a repetitive iambic hexameter that uses the form to explore the variations on contemporary romantic engagement. If so, it's not all that distant from a Shakespearean sonnet. Birkhoff's aesthetic measure could be applied both to rap as well as sonnets. While I'm not willing to risk an evaluation of each genre on that measure, I submit that it is at least possible to do so if one had a serious interest in the matter. ***
Thanks to Birkhoff, therefore, we progress somewhat beyond the dead end of "I don't know much about art, but I know what I like."
* I don’t think an explanation of Newton’s mystical tendency is needed given his religious, alchemical, and other occult interests. However my use of ‘sensory’ for Einstein’s aesthetic might seem strange given the apparently counter-intuitive intellectual demands of Relativity. Nevertheless Einstein himself was keen to explain both his special and general theories in terms of sensory reality - acceleration as indistinguishable from gravity, the perception of bursts of light at different points in a moving train, etc.
** It should be clear at this point that measurement of any kind - social, scientific, commercial - is itself an essentially aesthetic activity. The choice of a metric is the operationalisation of an aesthetic. See: https://www.goodreads.com/review/show...
*** An interesting synthesis of two aesthetics - from maths and music - is discussed at length here: https://www.goodreads.com/review/show...
April 20, 2020
As far as I know, this is the stupidest book ever published.
Other bad books may be lazy, ill-considered, hackish, uninsightful, or boring. It takes a Harvard professor to bring this much effort and certainty to an idea that is totally and obviously wrong.
How is it that no editor pulled Birkhoff aside and save him a century of embarrassment?
Other bad books may be lazy, ill-considered, hackish, uninsightful, or boring. It takes a Harvard professor to bring this much effort and certainty to an idea that is totally and obviously wrong.
How is it that no editor pulled Birkhoff aside and save him a century of embarrassment?
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