1. Number or label all displayed equations (Fisher’s Rule):
   

   The most common violation of Fisher’s rule is the misguided practice of numbering only those displayed equations to which the text subsequently refers back. … it is necessary to state emphatically that Fisher’s rule is for the benefit not of the author, but the reader.
   For although you, dear author, may have no need to refer in your text to the equations you therefore left unnumbered, it is presumptuous to assume the same disposition in your readers. And although you may well have acquired the solipsistic habit of writing under the assumption that you will have no readers at all, you are wrong.

2. When referring to an equation within the text, identify it by a phrase as well as a number (aka the Good Samaritan Rule):
   

   A Good Samaritan is compassionate and helpful to one in distress, and there is nothing more distressing than having to hunt your way back in a manuscript in search of Eq. (2.47), not because your subsequent progress requires you to inspect it in detail, but merely to find out what it is about so you may know the principles that go int othe construction of Eq. (7.38).

3. Punctuate the equation (aka the Math is Prose Rule):
   

   The equations you display are embedded in your prose and constitute an inseparable part of it.

   Regardless … of how to parse the equation internally, certain things are clear to anyone who understands the equation and the prose in which it is embedded.

   We punctuate equations because they are a form of prose (they can, after all, be read aloud as a sequence of words) and are therefore subject to the same rules as any other prose. … punctuation makes them easier to read and often clarifies the discussion in which they occur. … viewing an equation not as a grammatically irrelevant blob, but as a part of the text … can only improve the fluency and grace of one’s expository mathematical prose.

Owen talks in detail about how the Copernican model came to supplant the Ptolemaic model. In particular, he describes how Kepler went from Ptolemaic epicycles to elliptical orbits. Contrary to general impression, Kepler did not fit ellipses to Tycho Brahe’s observations of Mars. The ellipticity is far too small for it to be fittable! But rather, he used logical reasoning to first offset Earth’s epicyle away from the center in order to avoid the so-called Martian Catastrophe, and then used the phenomenological constraint of the law of equal areas to infer that the path must be an ellipse.

This process, along with Galileo’s advocacy for the heliocentric system, demonstrates a telling fact about how Astrophysics is done in practice. Hyunsook once lamented that astronomers seem to be rather trigger happy with correlations and regressions, and everyone knows they don’t constitute proof of anything, so why do they do it? Owen says about 39 1/2 minutes into the lecture:

Here we have the fourth of the myths, that Galileo’s telescopic observations finally proved the motion of the earth and thereby, at last, established the truth of the Copernican system.

What I want to assure you is that, in general, science does not operate by proofs. You hear that an awful lot, about science looking for propositions that can be falsified, that proof plays this big role.. uh-uh. It is coherence of explanation, understanding things that are well-knit together; the broader the framework of knitting the things together, the more we are able to believe it.

Exactly! We build models, often with little justification in terms of experimental proof, and muddle along trying to make it fit into a coherent narrative. This is why statistics is looked upon with suspicion among astronomers, and why for centuries our mantra has been “if it takes statistics to prove it, it isn’t real!”

(Marco Sirianni, from a conference on starbursts).