The earliest philosophy of science (advocated by Francis Bacon, for
example) was simple or naive inductivism. The idea was that the
scientist would collect data about some phenomenon and then “induce” a
theory or hypothesis explaining the data. It turned out that this
only worked in simple cases. More commonly scientists already had
a hypothesis in mind, otherwise they would not know what data to
collect in the first place. Also, as scientific theories became
more abstract and further removed from direct observation, the
connections between hypothesis and observation became more indirect,
and so it was difficult to go from the data to the theory.
(Examples: effects of wavelength on observed refraction of light,
gravitational law and observed orbits of planets.)
Therefore the next dominant methodology of science was “hypothesis and confirmation.” The idea was that you would come up with some hypothesis about how nature works (e.g. a theory), deduce observable consequences of that hypothesis (so it is called the hypothetico-deductive method), and then test (via an experiment) if the observations matched the predictions. Each time you did this and the experiment succeeded, it was considered a confirmation of the hypothesis or theory, and a well-confirmed theory was accepted as true.
Unfortunately, it was eventually noticed that it was not so hard to get confirmations of a theory, and some competing theories and hypotheses could be equally well confirmed. Another weakness of this approach is that proponents of a theory are more likely to report successful experiments than unsuccessful ones. Therefore in the mid-20th century Karl Popper and some other philosophers of science proposed an alternative methodology, “hypothesis and refutation.” The idea is that theories are subjected to more rigorous tests by opponents trying to refute them than by their proponents. That is, an opponent of a theory or hypothesis deduces observable consequences of the theory and sets up an experiment, hoping that the predictions will not match the observations. The fundamental idea is that while confirmations incrementally add to the credibility of a theory, a single refutation can demolish it. (Of course, in practice, proponents of a theory can generally find excuses for the failure or tinker with the theory to accommodate the refutation. Eventually, however, such excuse-making collapses under its own weight.) A theory that has withstood many attempted refutations is considered well tested and likely to be true.
Notice that hypotheses-and-refutations is more psychologically realistic. A believer in a theory will see confirmations everywhere they look, so they cannot be relied upon to test it rigorously. A disbeliever, however, will seek out the theory’s weak points and attempt to expose them in as obvious a way as possible. This approach to science is essentially a social process, and it depends on there being a diversity of (informed!) opinion in the scientific community, so that theories are adequately tested.
Common to both of these hypothetico-dedective philosophies is that (in contrast to inductivism) it doesn’t matter where the hypotheses or theories come from. They may come from the scientist's ideas about how nature works, from philosophical or religious convictions, from fever dreams, or from reading tea leaves! Indeed, Popper and others have stressed that the bolder the conjectures are, the better, for this is how science advances quickly. What makes a theory or hypothesis potentially scientific is that it is capable of being tested and perhaps refuted by empirical data, and what makes it a good scientific theory or hypothesis in actuality is that it has withstood many attempted refutations. In very broad strokes, the foregoing is a picture of “textbook” philosophy of science.
This is all quite different from Goethe’s approach. To reiterate, according to the conventional philosophy the explanatory theory can come from anywhere, and so long as it stands up to testing it is provisionally accepted. It could be replaced at any time by a bolder theory that explains more phenomena and stands up to testing at least as well. For Goethe, however, it is very important where the theory comes from. In conventional science, the hypothesis is a free invention, and the scientist poses questions, deduced from the hypothesis, to nature and extracts an answer. The answer either supports the theory or not. So the experimenter takes the lead and nature is obliged to answer the questions posed. (A police interrogation is an apt analogy.)
Goethe’s “delicate empiricism” is a more cooperative approach. The scientist begins by observing nature and noticing patterns and regularities. This may seem like naive induction, but Goethe recognizes that the perceived pattern is co-created by nature and the observer (also a part of nature). Using a developing intuitive grasp of the phenomena, the scientist makes series of observations (experiments in a Goethean sense) until he or she begins to grasp the underlying pattern, the “inner nature” behind the phenomena. As I remarked last week, this inner nature is not completely revealed in the phenomena, any more than a person’s character is completely revealed in their words and deeds. However, because nature and the observer (who is part of nature) co-create the phenomena, eventually, when enough experiments are done, the observer will internalize the inner nature of the phenomena in their own mind. In some ways it is as though they have acquired an image of the inner nature behind the observed phenomena, much as they might acquire an understanding of a person’s character. (There need be no abstract theory remote from the phenomena, any more than a deep understanding of another person presupposes an explicit, abstract psychological model of their personality.)
More precisely, the same inner nature is operating both behind the phenomena and in the scientist’s mind, so that when the scientist imagines variations in the phenomena or associated natural processes, they are not free or arbitrary imaginings (phantasia, “fantasy”), but rather they obey the same inner nature as the external phenomena (i.e., imaginatio vera, “true imagination”). They understand the phenomena because the inner nature has become a part of them. Thus Goethe’s science is participatory rather than observational. The observer and the observed come to dwell in each other. (So also, if you have acquired a deep, intuitive understanding of another person, you can in effect think their thoughts; you know what they are feeling, what they will do in a given situation, etc.)
This is why it is so important for Goethe that the scientist “listen” attentively and sensitively to nature without forcing nature to answer a predefined set of questions dictated by a theory. For Goethe, experimental science is more like a conversation between friends than an interrogation. The goal is mutual understanding, not coerced confession.
The following remarks are intended to help you understand Goethe’s
views on mathematics (as I understand them). The first thing to
notice is that we primarily perceive qualities (e.g., colors, shapes),
forms, objects, and arrangements, all of which we can consider
qualities in the broad sense. These qualities are given to us
immediately in experience.
We have a very limited ability to perceive quantity, by which I mean the ability to immediately recognize the number of things in the same way we can immediately perceive color, shape, etc. In general, we can perceive the number of at most perhaps five or six things immediately (that is, without counting). If they are specially arranged (for example, like the dots on dominoes), then we can perceive numbers up to about sixteen. (But unless the objects come to us arranged systematically, we will have to arrange them, which is already a counting procedure, that is, a kind of measurement, and not immediate perception.) We can of course perceive the size of things, and see, for example, that one thing is about twice as long as another, but this perception of quantity is also very limited. So, to a first approximation, it is not inaccurate to say that we perceive quality but not quantity.
In general, quantity is determined by a measurement process, which
converts a quantity, which is not directly perceivable, into a
qualitative difference that is. For example, if we measure
something with a ruler, we perceive how the edge of the object lines up
with a mark on the ruler (so we are perceiving an arrangement of
forms). Even when we count something, we move objects from the
uncounted to the counted pile, or move our finger from one to another,
while we say or think the numbers in order; when we reach the end (a
qualitative perception), we know the number. More sophisticated
measurements involving instruments also reduce to qualitative
perception, for example the alignment of a needle with the numbers on a
dial, or the forms of the digits on a digital readout. So (with
the exception of the few directly perceptible quantities already
mentioned), quantities cannot be directly perceived, but must be
The advantage of dealing with quantities is that they are more easy to determine objectively. For example, with complex and subtle qualities, it is difficult to eliminate the subjective element (although that was precisely Goethe’s goal). Different people see colors differently, and we may have differing perceptions of the warmth of an object. Quantities are measured by numbers, which are represented by qualities about which (presumably) it is easier to reach agreement, such as the number to which a needle is pointing. (Even here some perceptual training is necessary: you may remember learning in high school chemistry that you have to read a thermometer at the top of the mercury meniscus. Unless observers read the thermometer the same way, they will get different results.) In sum, measurement converts quantities into easily perceivable qualitative differences.
Many quantities measure directly perceivable qualities; examples are
length, weight, volume, temperature, brightness, loudness, pitch.
However, as you know, as science has progressed, the tendency has been
to explain these quantities in terms of more fundamental
quantities. So, matter and energy are explained in terms of
elementary particles’ rest mass, charge, spin, etc., and the four
fundamental forces (gravitational, electromagnetic, strong nuclear,
weak nuclear). (String theory takes the reduction even
deeper.) As a consequence, reality is explained in terms of
quantities, which we cannot perceive, and ultimately in terms of
quantities that are only very indirectly related to the qualities we do
perceive. So “ultimate reality” is remote from our experience,
and our everyday experience is explained as a sort of complex,
inescapable illusion with little similarity to true reality. The
result is that we are alienated from the natural world. Nature is
seen as something abstract, mechanical, and non-living to be
manipulated from afar. (It’s rather like trying to understand
other people and to relate to them in terms of their neurons and
physiological processes. If we were really capable to relating to
people in this way, the effect would be to alienate us from them, for
we would be viewing them as a kind of biochemical robot. To the
extent you could view yourself this way, it would be
self-alienating, for your intellectual understanding of yourself would
be disconnected from your experience of yourself.)
Goethe’s approach is different, for he tries to keep the explanation at the same level as the phenomena to be described, that is, everything is in terms of qualities (which is all that we can experience). In this way, the explanations have the same reality as the phenomena, so that neither undermines the reality of the other. Nature is understood in human terms and experienced as a part of us, as we are experienced as a part of nature.
You might ask how we could have modern technology without our quantitative scientific theories. Most likely, we could not, but Goethe is not suggesting that we abandon quantitative science. Rather he is advocating an alternative qualitative approach to science, which serves different goals. Its result is participatory understanding and appreciation of nature rather than domination of nature for the sake of technology (although, as a complement to quantitative science, Goethean science might lead to a more environmentally friendly technology).