Zen and the Art of Science: A Tribute to Robert Pirsig

Author Robert Pirsig, widely acclaimed for his bestselling books, Zen and the Art of Motorcycle Maintenance (1974) and Lila (1991), passed away in his home on April 24, 2017. A well-rounded intellectual equally at home in the sciences and the humanities, Pirsig made the case that scientific inquiry, art, and religious experience were all particular forms of knowledge arising out of a broader form of knowledge about the Good or what Pirsig called “Quality.” Yet, although Pirsig’s books were bestsellers, contemporary debates about science and religion are oddly neglectful of Pirsig’s work. So what did Pirsig claim about the common roots of human knowledge, and how do his arguments provide a basis for reconciling science and religion?

Pirsig gradually developed his philosophy as response to a crisis in the foundations of scientific knowledge, a crisis he first encountered while he was pursuing studies in biochemistry. The popular consensus at the time was that scientific methods promised objectivity and certainty in human knowledge. One developed hypotheses, conducted observations and experiments, and came to a conclusion based on objective data. That was how scientific knowledge accumulated.

However, Pirsig noted that, contrary to his own expectations, the number of hypotheses could easily grow faster than experiments could test them. One could not just come up with hypotheses – one had to make good hypotheses, ones that could eliminate the need for endless and unnecessary observations and testing. Good hypotheses required mental inspiration and intuition, components that were mysterious and unpredictable.  The greatest scientists were precisely like the greatest artists, capable of making immense creative leaps before the process of testing even began.  Without those creative leaps, science would remain on a never-ending treadmill of hypothesis development – this was the “infinity of hypotheses” problem.  And yet, the notion that science depended on intuition and artistic leaps ran counter to the established view that the scientific method required nothing more than reason and the observation and recording of an objective reality.

Consider Einstein. One of history’s greatest scientists, Einstein hardly ever conducted actual experiments. Rather, he frequently engaged in “thought experiments,” imagining what it would be like to chase a beam of light, what it would feel like to be in a falling elevator, and what a clock would look like if the streetcar he was riding raced away from the clock at the speed of light.

One of the most fruitful sources of hypotheses in science is mathematics, a discipline which consists of the creation of symbolic models of quantitative relationships. And yet, the nature of mathematical discovery is so mysterious that mathematicians themselves have compared their insights to mysticism. The great French mathematician Henri Poincare believed that the human mind worked subliminally on problems, and his work habit was to spend no more than two hours at a time working on mathematics. Poincare believed that his subconscious would continue working on problems while he conducted other activities, and indeed, many of his great discoveries occurred precisely when he was away from his desk. John von Neumann, one of the best mathematicians of the twentieth century, also believed in the subliminal mind. He would sometimes go to sleep with a mathematical problem on his mind and wake up in the middle of the night with a solution. The Indian mathematical genius Srinivasa Ramanujan was a Hindu mystic who believed that solutions were revealed to him in dreams by the goddess Namagiri.

Intuition and inspiration were human solutions to the infinity-of-hypotheses problem. But Pirsig noted there was a related problem that had to be solved — the infinity of facts.  Science depended on observation, but the issue of which facts to observe was neither obvious nor purely objective.  Scientists had to make value judgments as to which facts were worth close observation and which facts could be safely overlooked, at least for the moment.  This process often depended heavily on an imprecise sense or feeling, and sometimes mere accident brought certain facts to scientists’ attention. What values guided the search for facts? Pirsig cited Poincare’s work The Foundations of Science. According to Poincare, general facts were more important than particular facts, because one could explain more by focusing on the general than the specific. Desire for simplicity was next – by beginning with simple facts, one could begin the process of accumulating knowledge about nature without getting bogged down in complexity at the outset. Finally, interesting facts that provided new findings were more important than facts that were unimportant or trivial. The point was not to gather as many facts as possible but to condense as much experience as possible into a small volume of interesting findings.

Research on the human brain supports the idea that the ability to value is essential to the discernment of facts.  Professor of Neuroscience Antonio Damasio, in his book Descartes’ Error: Emotion, Reason, and the Human Brain, describes several cases of human beings who lost the part of their brain responsible for emotions, either because of an accident or a brain tumor.  These persons, some of whom were previously known as shrewd and smart businessmen, experienced a serious decline in their competency after damage took place to the emotional center of their brains.  They lost their capacity to make good decisions, to get along with other people, to manage their time, or to plan for the future.  In every other respect, these persons retained their cognitive abilities — their IQs remained above normal and their personality tests resulted in normal scores.  The only thing missing was their capacity to have emotions.  Yet this made a huge difference.  Damasio writes of one subject, “Elliot”:

Consider the beginning of his day: He needed prompting to get started in the morning and prepare to go to work.  Once at work he was unable to manage his time properly; he could not be trusted with a schedule.  When the job called for interrupting an activity and turning to another, he might persist nonetheless, seemingly losing sight of his main goal.  Or he might interrupt the activity he had engaged, to turn to something he found more captivating at that particular moment.  Imagine a task involving reading and classifying documents of a given client.  Elliot would read and fully understand the significance of the material, and he certainly knew how to sort out the documents according to the similarity or disparity of their content.  The problem was that he was likely, all of a sudden, to turn from the sorting task he had initiated to reading one of those papers, carefully and intelligently, and to spend an entire day doing so.  Or he might spend a whole afternoon deliberating on which principle of categorization should be applied: Should it be date, size of document, pertinence to the case, or another?   The flow of work was stopped. (p. 36)

Why did the loss of emotion, which might be expected to improve decision-making by making these persons coldly objective, result in poor decision-making instead?  According to Damasio, without emotions, these persons were unable to value, and without value, decision-making in the face of infinite facts became hopelessly capricious or paralyzed, even with normal or above-normal IQs.  Damasio noted, “the cold-bloodedness of Elliot’s reasoning prevented him from assigning different values to different options, and made his decision-making landscape hopelessly flat.” (p. 51) Damasio discusses several other similar case studies.

So how would it affect scientific progress if all scientists were like the subjects Damasio studied, free of emotion, and therefore, hypothetically capable of perfect objectivity?  Well it seems likely that science would advance very slowly, at best, or perhaps not at all.  After all, the same tools for effective decision-making in everyday life are needed for the scientific enterprise as well. A value-free scientist would not only be unable to sustain the social interaction that science requires, he or she would be unable to develop a research plan, manage his or her time, or stick to a research plan.

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Where Pirsig’s philosophy becomes particularly controversial and difficult to understand is in his approach to the truth. The dominant view of truth today is known as the “correspondence” theory of truth – that is, any human statement that is true must correspond precisely to something objectively real. In this view, the laws of physics and chemistry are real because they correspond to actual events that can be observed and demonstrated. Pirsig argues on the contrary that in order to understand reality, human beings must invent symbolic and conceptual models, that there is a large creative component to these models (it is not just a matter of pure correspondence to reality), and that multiple such models can explain the same reality even if they are based on wholly different principles. Math, logic, and even the laws of physics are not “out there” waiting to be discovered – they exist in the mind, which doesn’t mean that these things are bad or wrong or unreal.

There are several reasons why our symbolic and conceptual models don’t correspond literally to reality, according to Pirsig. First, there is always going to be a gap between reality and the concepts we use to describe reality, because reality is continuous and flowing, while concepts are discrete and static. The creation of concepts necessarily calls for cutting reality into pieces, but there is no one right way to divide reality, and something is always lost when this is done. In fact, Pirsig noted, our very notions of subjectivity and objectivity, the former allegedly representing personal whims and the latter representing truth, rested upon an artificial division of reality into subjects and objects; in fact, there were other ways of dividing reality that could be just as legitimate or useful. In addition, concepts are necessarily static – they can’t be always changing or we would not be able to make sense of them. Reality, however, is always changing. Finally, describing reality is not always a matter of using direct and literal language but may require analogy and imaginative figures of speech.

Because of these difficulties in expressing reality directly, a variety of symbolic and conceptual models, based on widely varying principles, are not only possible but necessary – necessary for science as well as other forms of knowledge. Pirsig points to the example of the crisis that occurred in mathematics in the nineteenth century. For many centuries, it was widely believed that geometry, as developed by the ancient Greek mathematician Euclid, was the most exact of all of the sciences.  Based on a small number of axioms from which one could deduce multiple propositions, Euclidean geometry represented a nearly perfect system of logic.  However, while most of Euclid’s axioms were seemingly indisputable, mathematicians had long experienced great difficulty in satisfactorily demonstrating the truth of one of the chief axioms on which Euclidean geometry was based. This slight uncertainty led to an even greater crisis of uncertainty when mathematicians discovered that they could reverse or negate this axiom and create alternative systems of geometry that were every bit as logical and valid as Euclidean geometry.  The science of geometry was gradually replaced by the study of multiple geometries. Pirsig cited Poincare, who pointed out that the principles of geometry were not eternal truths but definitions and that the test of a system of geometry was not whether it was true but how useful it was.

So how do we judge the usefulness or goodness of our symbolic and conceptual models? Traditionally, we have been told that pure objectivity is the only solution to the chaos of relativism, in which nothing is absolutely true. But Pirsig pointed out that this hasn’t really been how science has worked. Rather, models are constructed according to the often competing values of simplicity and generalizability, as well as accuracy. Theories aren’t just about matching concepts to facts; scientists are guided by a sense of the Good (Quality) to encapsulate as much of the most important knowledge as possible into a small package. But because there is no one right way to do this, rather than converging to one true symbolic and conceptual model, science has instead developed a multiplicity of models. This has not been a problem for science, because if a particular model is useful for addressing a particular problem, that is considered good enough.

The crisis in the foundations of mathematics created by the discovery of non-Euclidean geometries and other factors (such as the paradoxes inherent in set theory) has never really been resolved. Mathematics is no longer the source of absolute and certain truth, and in fact, it never really was. That doesn’t mean that mathematics isn’t useful – it certainly is enormously useful and helps us make true statements about the world. It’s just that there’s no single perfect and true system of mathematics. (On the crisis in the foundations of mathematics, see the papers here and here.) Mathematical axioms, once believed to be certain truths and the foundation of all proofs, are now considered definitions, assumptions, or hypotheses. And a substantial number of mathematicians now declare outright that mathematical objects are imaginary, that particular mathematical formulas may be used to model real events and relationships, but that mathematics itself has no existence outside the human mind. (See The Mathematical Experience by Philip J. Davis and Reuben Hersh.)

Even some basic rules of logic accepted for thousands of years have come under challenge in the past hundred years, not because they are absolutely wrong, but because they are inadequate in many cases, and a different set of rules is needed. The Law of the Excluded Middle states that any proposition must be either true or false (“P” or “not P” in symbolic logic). But ever since mathematicians discovered propositions which are possibly true but not provable, a third category of “possible/unknown” has been added. Other systems of logic have been invented that use the idea of multiple degrees of truth, or even an infinite continuum of truth, from absolutely false to absolutely true.

The notion that we need multiple symbolic and conceptual models to understand reality remains controversial to many. It smacks of relativism, they argue, in which every person’s opinion is as valid as another person’s. But historically, the use of multiple perspectives hasn’t resulted in the abandonment of intellectual standards among mathematicians and scientists. One still needs many years of education and an advanced degree to obtain a job as a mathematician or scientist, and there is a clear hierarchy among practitioners, with the very best mathematicians and scientists working at the most prestigious universities and winning the highest awards. That is because there are still standards for what is good mathematics and science, and scholars are rewarded for solving problems and advancing knowledge. The fact that no one has agreed on what is the One True system of mathematics or logic isn’t relevant. In fact, physicist Stephen Hawking has argued:

[O]ur brains interpret the input from our sensory organs by making a model of the world. When such a model is successful at explaining events, we tend to attribute to it, and to the elements and concepts that constitute it, the quality of reality or absolute truth. But there may be different ways in which one could model the same physical situation, with each employing different fundamental elements and concepts. If two such physical theories or models accurately predict the same events, one cannot be said to be more real than the other; rather we are free to use whichever model is more convenient (The Grand Design, p. 7).

Among the most controversial and mind-bending claims Pirsig makes is that the very laws of nature themselves exist only in the human mind. “Laws of nature are human inventions, like ghosts,” he writes. Pirsig even remarks that it makes no sense to think of the law of gravity existing before the universe, that it only came into existence when Isaac Newton thought of it. It’s an outrageous claim, but if one looks closely at what the laws of nature actually are, it’s not so crazy an argument as it first appears.

For all of the advances that science has made over the centuries, there remains a sharp division of views among philosophers and scientists on one very important issue: are the laws of nature actual causal powers responsible for the origins and continuance of the universe or are the laws of nature summary descriptions of causal patterns in nature? The distinction is an important one. In the former view, the laws of physics are pre-existing or eternal and possess god-like powers to create and shape the universe; in the latter view, the laws have no independent existence – we are simply finding causal patterns and regularities in nature that allow us to predict and we call these patterns “laws.”

One powerful argument in favor of the latter view is that most of the so-called “laws of nature,” contrary to the popular view, actually have exceptions – and sometimes the exceptions are large. That is because the laws are simplified models of real phenomena. The laws were cobbled together by scientists in order to strike a careful balance between the values of scope, predictive accuracy, and simplicity. Michael Scriven, a mathematician and philosopher at Claremont Graduate University, has noted that as a result of this balance of values, physical laws are actually approximations that apply only within a certain range. This point has also been made more recently by Ronald Giere, a professor of philosophy at the University of Minnesota, in Science Without Laws and Nancy Cartwright of the University of California at San Diego in How the Laws of Physics Lie.

Newton’s law of universal gravitation, for example, is not really universal. It becomes increasingly inaccurate under conditions of high gravity and very high velocities, and at the atomic level, gravity is completely swamped by other forces. Whether one uses Newton’s law depends on the specific conditions and the level of accuracy one requires. Newton’s laws of motion also have exceptions, depending on the force, distance, and speed. Kepler’s laws of planetary motion are an approximation based on the simplifying assumption of a planetary system consisting of one planet. The ideal gas law is an approximation which becomes inaccurate under conditions of low temperature and/or high pressure. The law of multiple proportions works for simple molecular compounds, but often fails for complex molecular compounds. Biologists have discovered so many exceptions to Mendel’s laws of genetics that some believe that Mendel’s laws should not even be considered laws.

So if we think of laws of nature as being pre-existing, eternal commandments, with god-like powers to shape the universe, how do we account for these exceptions to the laws? The standard response by scientists is that their laws are simplified depictions of the real laws. But if that is the case, why not state the “real” laws? Because by the time we wrote down the real laws, accounting for every possible exception, we would have an extremely lengthy and detailed description of causation that would not recognizably be a law. The whole point of the laws of nature was to develop tools by which one could predict a large number of phenomena (scope), maintain a good-enough correspondence to reality (accuracy), and make it possible to calculate predictions without spending an inordinate amount of time and effort (simplicity). That is why although Einstein’s conception of gravity and his “field equations” have supplanted Newton’s law of gravitation, physicists still use Newton’s “law” in most cases because it is simpler and easier to use; they only resort to Einstein’s complex equations when they have to! The laws of nature are human tools for understanding, not mathematical gods that shape the universe. The actual practice of science confirms Pirsig’s point that the symbolic and conceptual models that we create to understand reality have to be judged by how good they are – simple correspondence to reality is insufficient and in many cases is not even possible anyway.

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Ultimately, Pirsig concluded, the scientific enterprise is not that different from the pursuit of other forms of knowledge – it is based on a search for the Good. Occasionally, you see this acknowledged explicitly, when mathematicians discuss the beauty of certain mathematical proofs or results, as defined by their originality, simplicity, ability to solve many problems at once, or their surprising nature. Scientists also sometimes write about the importance of elegance in their theories, defined as the ability to explain as much as possible, as clearly as possible, and as simply as possible. Depending on the field of study, the standards of judgment may be different, the tools may be different, and the scope of inquiry is different. But all forms of human knowledge — art, rhetoric, science, reason, and religion — originate in, and are dependent upon, a response to the Good or Quality. The difference between science and religion is that scientific models are more narrowly restricted to understanding how to predict and manipulate natural phenomena, whereas religious models address larger questions of meaning and value.

Pirsig did not ignore or suppress the failures of religious knowledge with regard to factual claims about nature and history. The traditional myths of creation and the stories of various prophets were contrary to what we know now about physics, biology, paleontology, and history. In addition, Pirsig was by no means a conventional theist — he apparently did not believe that God was a personal being who possessed the attributes of omniscience and omnipotence, controlling or potentially controlling everything in the universe.

However, Pirsig did believe that God was synonymous with the Good, or “Quality,” and was the source of all things.  In fact, Pirsig wrote that his concept of Quality was similar to the “Tao” (the “Way” or the “Path”) in the Chinese religion of Taoism. As such, Quality was the source of being and the center of existence. It was also an active, dynamic power, capable of bringing about higher and higher levels of being. The evolution of the universe, from simple physical forms, to complex chemical compounds, to biological organisms, to societies was Dynamic Quality in action. The most recent stage of evolution – Intellectual Quality – refers to the symbolic models that human beings create to understand the universe. They exist in the mind, but are a part of reality all the same – they represent a continuation of the growth of Quality.

What many religions were missing, in Pirsig’s view, was not objectivity, but dynamism: an ability to correct old errors and achieve new insights. The advantage of science was its willingness and ability to change. According to Pirsig,

If scientists had simply said Copernicus was right and Ptolemy was wrong without any willingness to further investigate the subject, then science would have simply become another minor religious creed. But scientific truth has always contained an overwhelming difference from theological truth: it is provisional. Science always contains an eraser, a mechanism whereby new Dynamic insight could wipe out old static patterns without destroying science itself. Thus science, unlike orthodox theology, has been capable of continuous, evolutionary growth. (Lila, p. 222)

The notion that religion and orthodoxy go together is widespread among believers and secularists. But there is no necessary connection between the two. All religions originate in social processes of story-telling, dialogue, and selective borrowing from other cultures. In fact, many religions begin as dangerous heresies before they become firmly established — orthodoxies come later. The problem with most contemporary understandings of religion is that one’s adherence to religion is often measured by one’s commitment to orthodoxy and membership in religious institutions rather than an honest quest for what is really good.  A person who insists on the literal truth of the Bible and goes to church more than once a week is perceived as being highly religious, whereas a person not connected with a church but who nevertheless seeks religious knowledge wherever he or she can find it is considered less committed or even secular.  This prejudice has led many young people to identify as “spiritual, not religious,” but religious knowledge is not inherently about unwavering loyalty to an institution or a text. Pirsig believed that mysticism was a necessary component of religious knowledge and a means of disrupting orthodoxies and recovering the dynamic aspect of religious insight.

There is no denying that the most prominent disputes between science and religion in the last several centuries regarding the physical workings of the universe have resulted in a clear triumph for scientific knowledge over religious knowledge.  But the solution to false religious beliefs is not to discard religious knowledge — religious knowledge still offers profound insights beyond the scope of science. That is why it is necessary to recover the dynamic nature of religious knowledge through mysticism, correction of old beliefs, and reform. As Pirsig argued, “Good is a noun.” Not because Good is a thing or an object, but because Good  is the center and foundation of all reality and all forms of knowledge, whether we are consciously aware of it or not.

What Does Science Explain? Part 5 – The Ghostly Forms of Physics

The sciences do not try to explain, they hardly even try to interpret, they mainly make models. By a model is meant a mathematical construct which, with the addition of certain verbal interpretations, describes observed phenomena. The justification of such a mathematical construct is solely and precisely that it is expected to work — that is, correctly to describe phenomena from a reasonably wide area. Furthermore, it must satisfy certain esthetic criteria — that is, in relation to how much it describes, it must be rather simple. — John von Neumann (“Method in the Physical Sciences,” in The Unity of Knowledge, 1955)

Now we come to the final part of our series of posts, “What Does Science Explain?” (If you have not already, you can peruse parts 1, 2, 3, and 4 here). As I mentioned in my previous posts, the rise of modern science was accompanied by a change in humanity’s view of metaphysics, that is, our theory of existence. Medieval metaphysics, largely influenced by ancient philosophers, saw human beings as the center or summit of creation; furthermore, medieval metaphysics proposed a sophisticated, multifaceted view of causation. Modern scientists, however, rejected much of medieval metaphysics as subjective and saw reality as consisting mainly of objects impacting or influencing each other in mathematical patterns.  (See The Metaphysical Foundations of Modern Science by E.A. Burtt.)

I have already critically examined certain aspects of the metaphysics of modern science in parts 3 and 4. For part 5, I wish to look more closely at the role of Forms in causation — what Aristotle called “formal causation.” This theory of causation was strongly influenced by Aristotle’s predecessor Plato and his Theory of Forms. What is Plato’s “Theory of Forms”? In brief, Plato argued that the world we see around us — including all people, trees, and animals, stars, planets and other objects — is not the true reality. The world and the things in it are imperfect and perishable realizations of perfect forms that are eternal, and that continually give birth to the things we see. That is, forms are the eternal blueprints of perfection which the material world imperfectly represents. True philosophers do not focus on the material world as it is, but on the forms that material things imperfectly reflect. In order to judge a sculpture, painting, or natural setting, a person must have an inner sense of beauty. In order to evaluate the health of a particular human body, a doctor must have an idea of what a perfectly healthy human form is. In order to evaluate a government’s system of justice, a citizen must have an idea about what perfect justice would look like. In order to critically judge leaders, citizens must have a notion of the virtues that such a leader should have, such as wisdom, honesty, and courage.  Ultimately, according to Plato, a wise human being must learn and know the perfect forms behind the imperfect things we see: we must know the Form of Beauty, the Form of Justice, the Form of Wisdom, and the ultimate form, the Form of Goodness, from which all other forms flow.

Unsurprisingly, many intelligent people in the modern world regard Plato’s Theory of Forms as dubious or even outrageous. Modern science teaches us that sure knowledge can only be obtained by observation and testing of real things, but Plato tells us that our senses are deceptive, that the true reality is hidden behind what we sense. How can we possibly confirm that the forms are real? Even Plato’s student Aristotle had problems with the Theory of Forms and argued that while the forms were real, they did not really exist until they were manifested in material things.

However, there is one important sense in which modern science retained the notion of formal causation, and that is in mathematics. In other words, most scientists have rejected Plato’s Theory of Forms in all aspects except for Plato’s view of mathematics. “Mathematical Platonism,” as it is called, is the idea that mathematical forms are objectively real and are part of the intrinsic order of the universe. However, there are also sharp disagreements on this subject, with some mathematicians and scientists arguing that mathematical forms are actually creations of the human imagination.

The chief difference between Plato and modern scientists on the study of mathematics is this: According to Plato, the objects of geometry — perfect squares, perfect circles, perfect planes — existed nowhere in the material world; we only see imperfect realizations. But the truly wise studied the perfect, eternal forms of geometry rather than their imperfect realizations. Therefore, while astronomical observations indicated that planetary bodies orbited in imperfect circles, with some irregularities and errors, Plato argued that philosophers must study the perfect forms instead of the actual orbits! (The Republic, XXVI, 524D-530C) Modern science, on the other hand, is committed to observation and study of real orbits as well as the study of perfect mathematical forms.

Is it tenable to hold the belief that Plato and Aristotle’s view of eternal forms is mostly subjective nonsense, but they were absolutely right about mathematical forms being real? I argue that this selective borrowing of the ancient Greeks doesn’t quite work, that some of the questions and difficulties with proving the reality of Platonic forms also afflicts mathematical forms.

The main argument for mathematical Platonism is that mathematics is absolutely necessary for science: mathematics is the basis for the most important and valuable physical laws (which are usually in the form of equations), and everyone who accepts science must agree that the laws of nature or the laws of physics exist. However, the counterargument to this claim is that while mathematics is necessary for human beings to conduct science and understand reality, that does not mean that mathematical objects or even the laws of nature exist objectively, that is, outside of human minds.

I have discussed some of the mysterious qualities of the “laws of nature” in previous posts (here and here). It is worth pointing out that there remains a serious debate among philosophers as to whether the laws of nature are (a) descriptions of causal regularities which help us to predict or (b) causal forces in themselves. This is an important distinction that most people, including scientists, don’t notice, although the theoretical consequences are enormous. Physicist Kip Thorne writes that laws “force the Universe to behave the way it does.” But if laws have that kind of power, they must be ubiquitous (exist everywhere), eternal (exist prior to the universe), and have enormous powers although they have no detectable energy or mass — in other words, the laws of nature constitute some kind of supernatural spirit. On the other hand, if laws are summary descriptions of causation, these difficulties can be avoided — but then the issue arises: do the laws of nature or of physics really exist objectively, outside of human minds, or are they simply human-constructed statements about patterns of causation? There are good reasons to believe the latter is true.

The first thing that needs to be said is that nearly all these so-called laws of nature are actually approximations of what really happens in nature, approximations that work only under certain restrictive conditions. Both of these considerations must be taken into account, because even the approximations fall apart outside of certain pre-specified conditions. Newton’s law of universal gravitation, for example, is not really universal. It becomes increasingly inaccurate under conditions of high gravity and very high velocities, and at the atomic level, gravity is completely swamped by other forces. Whether one uses Newton’s law depends on the specific conditions and the level of accuracy one requires. Kepler’s laws of planetary motion are an approximation based on the simplifying assumption of a planetary system consisting of one planet. The ideal gas law is an approximation which becomes inaccurate under conditions of low temperature and/or high pressure. The law of multiple proportions works for simple molecular compounds, but often fails for complex molecular compounds. Biologists have discovered so many exceptions to Mendel’s laws of genetics that some believe that Mendel’s laws should not even be considered laws.

The fact of the matter is that even with the best laws that science has come up with, we still can’t predict the motions of more than two interacting astronomical bodies without making unrealistic simplifying assumptions. Michael Scriven, a mathematician and philosopher at Claremont Graduate University, has concluded that the laws of nature or physics are actually cobbled together by scientists based on multiple criteria:

Briefly we may say that typical physical laws express a relationship between quantities or a property of systems which is the simplest useful approximation to the true physical behavior and which appears to be theoretically tractable. “Simplest” is vague in many cases, but clear for the extreme cases which provide its only use. “Useful” is a function of accuracy and range and purpose. (Michael Scriven, “The Key Property of Physical Laws — Inaccuracy,” in Current Issues in the Philosophy of Science, ed. Herbert Feigl)

The response to this argument is that it doesn’t disprove the objective existence of physical laws — it simply means that the laws that scientists come up with are approximations to real, objectively existing underlying laws. But if that is the case, why don’t scientists simply state what the true laws are? Because the “laws” would actually end up being extremely long and complex statements of causation, with so many conditions and exceptions that they would not really be considered laws.

An additional counterargument to mathematical Platonism is that while mathematics is necessary for science, it is not necessary for the universe. This is another important distinction that many people overlook. Understanding how things work often requires mathematics, but that doesn’t mean the things in themselves require mathematics. The study of geometry has given us pi and the Pythagorean theorem, but a child does not need to know these things in order to draw a circle or a right triangle. Circles and right triangles can exist without anyone, including the universe, knowing the value of pi or the Pythagorean theorem. Calculus was invented in order to understand change and acceleration; but an asteroid, a bird, or a cheetah is perfectly capable of changing direction or accelerating without needing to know calculus.

Even among mathematicians and scientists, there is a significant minority who have argued that mathematical objects are actually creations of the human imagination, that math may be used to model aspects of reality, but it does not necessarily do so. Mathematicians Philip J. Davis and Reuben Hersh argue that mathematics is the study of “true facts about imaginary objects.” Derek Abbot, a professor of engineering, writes that engineers tend to reject mathematical Platonism: “the engineer is well acquainted with the art of approximation. An engineer is trained to be aware of the frailty of each model and its limits when it breaks down. . . . An engineer . . . has no difficulty in seeing that there is no such a thing as a perfect circle anywhere in the physical universe, and thus pi is merely a useful mental construct.” (“The Reasonable Ineffectiveness of Mathematics“) Einstein himself, making a distinction between mathematical objects used as models and pure mathematics, wrote that “As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality.” Hartry Field, a philosopher at New York University, has argued that mathematics is a useful fiction that may not even be necessary for science. Field goes to show that it is possible to reconstruct Newton’s theory of gravity without using mathematics. (There is more discussion on this subject here and here.)

So what can we conclude about the existence of forms? I have to admit that although I’m skeptical, I have no sure conclusions. It seems unlikely that forms exist outside the mind . . . but I can’t prove they don’t exist either. Forms do seem to be necessary for human reasoning — no thinking human can do without them. And forms seem to be rooted in reality: perfect circles, perfect squares, and perfect human forms can be thought of as imaginative projections of things we see, unlike Sherlock Holmes or fire-breathing dragons or flying spaghetti monsters, which are more creatively fictitious. Perhaps one could reconcile these opposing views on forms by positing that the human mind and imagination is part of the universe itself, and that the universe is becoming increasingly consciously aware.

Another way to think about this issue was offered by Robert Pirsig in Zen and the Art of Motorcycle Maintenance. According to Pirsig, Plato made a mistake by positing Goodness as a form. Even considered as the highest form, Goodness (or “Quality,” in Pirsig’s terminology) can’t really be thought of as a static thing floating around in space or some otherworldly realm. Forms are conceptual creations of humans who are responding to Goodness (Quality). Goodness itself is not a form, because it is not an unchanging thing — it is not static or even definable. It is “reality itself, ever changing, ultimately unknowable in any kind of fixed, rigid way.” (p. 342) Once we let go of the idea that Goodness or Quality is a form, we can realize that not only is Goodness part of reality, it is reality.

As conceptual creations, ideal forms are found in both science and religion. So why, then, does there seem to be such a sharp split between science and religion as modes of knowledge? I think it comes down to this: science creates ideal forms in order to model and predict physical phenomena, while religion creates ideal forms in order to provide guidance on how we should live.

Scientists like to see how things work — they study the parts in order to understand how the wholes work. To increase their understanding, scientists may break down certain parts into smaller parts, and those parts into even smaller parts, until they come to the most fundamental, indivisible parts. Mathematics has been extremely useful in modeling and understanding these parts of nature, so scientists create and appreciate mathematical forms.

Religion, on the other hand, tends to focus on larger wholes. The imaginative element of religion envisions perfect states of being, whether it be the Garden of Eden or the Kingdom of Heaven, as well as perfect (or near perfect) humans who serve as prophets or guides to a better life. Religion is less concerned with how things work than with how things ought to work, how things ought to be. So religion will tend to focus on subjects not covered by science, including the nature and meaning of beauty, love, and justice. There will always be debates about the appropriateness of particular forms in particular circumstances, but the use of forms in both science and religion is essential to understanding the universe and our place in it.

The Role of Imagination in Science, Part 1

In Zen and the Art of Motorcycle Maintenance, author Robert Pirsig argues that the basic conceptual tools of science, such as the number system, the laws of physics, and the rules of logic, have no objective existence, but exist in the human mind.  These conceptual tools were not “discovered” but created by the human imagination.  Nevertheless we use these concepts and invent new ones because they are good — they help us to understand and cope with our environment.

As an example, Pirsig points to the uncertain status of the number “zero” in the history of western culture.  The ancient Greeks were divided on the question of whether zero was an actual number – how could nothing be represented by something? – and did not widely employ zero.  The Romans’ numerical system also excluded zero.  It was only in the Middle Ages that the West finally adopted the number zero by accepting the Hindu-Arabic numeral system.  The ancient Greek and Roman civilizations did not neglect zero because they were blind or stupid.  If future generations adopted the use of zero, it was not because they suddenly discovered that zero existed, but because they found the number zero useful.

In fact, while mathematics appears to be absolutely essential to progress in the sciences, mathematics itself continues to lack objective certitude, and the philosophy of mathematics is plagued by questions of foundations that have never been resolved.  If asked, the majority of mathematicians will argue that mathematical objects are real, that they exist in some unspecified eternal realm awaiting discovery by mathematicians; but if you follow up by asking how we know that this realm exists, how we can prove that mathematical objects exist as objective entities, mathematicians cannot provide an answer that is convincing even to their fellow mathematicians.  For many decades, according to mathematicians Philip J. Davis and Reuben Hersh, the brightest minds sought to provide a firm foundation for mathematical truth, only to see their efforts founder (“Foundations , Found and Lost,” in The Mathematical Experience).

In response to these failures, mathematicians divided into multiple camps.  While the majority of mathematicians still insisted that mathematical objects were real, the school of fictionalism claimed that all mathematical objects were fictional.  Nevertheless, the fictionalists argued that mathematics was a useful fiction, so it was worthwhile to continue studying mathematics.  In the school of formalism, mathematics is described as a set of statements of the consequences of following certain rules of the game — one can create many “games,” and these games have different outcomes resulting from different sets of rules, but the games may not be about anything real.  The school of finitism argues that only the natural numbers (i.e., numbers for counting, such as 1, 2, 3. . . ) and numbers that can be derived from the natural numbers are real, all other numbers are creations of the human mind.  Even if one dismisses these schools as being only a minority, the fact that there is such stark disagreement among mathematicians about the foundations of mathematics is unsettling.

Ironically, as mathematical knowledge has increased over the years, so has uncertainty.  For many centuries, it was widely believed that Euclidean geometry was the most certain of all the sciences.  However, by the late nineteenth century, it was discovered that one could create different geometries that were just as valid as Euclidean geometry — in fact, it was possible to create an infinite number of valid geometries.  Instead of converging on a single, true geometry, mathematicians have seemingly gone into all different directions.  So what prevents mathematics from falling into complete nihilism, in which every method is valid and there are no standards?  This is an issue we will address in a subsequent posting.