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.