My husband described to me a version of the cosmological argument based on explanations. Everything in the world requires an explanation. Since the chain of explanations can’t go on forever, there must be an ultimate Explanation for it all. And that Explanation wants us to stop being gay. At least, that’s how I assume the rest of the argument goes. I don’t actually care.
We were joking about this, because the argument imagines that there is a cosmic pointer from each object to its explanation. But the very idea of an explanation is really quite squishy. What even is an explanation? It depends on social context.
So I dug up an old abandoned blog draft on the subject of explanations.
When you get a degree in physics, there’s a funny change you can see between introductory physics and advanced physics. Specifically, in the way you’re supposed to solve homework problems.
In introductory physics homework, you’re provided with equations of motion (e.g. velocity, acceleration), and the solution is to describe how objects move over time. In advanced physics homework, you’re given a system (e.g. a pendulum on a spring), and the solution is merely to write down the equations of motion. Often, you do not actually solve the equations of motion, and you may not know how to solve them. Or the only way to solve them is to take a separate class in computational simulations. So you just write down the equations and call it a day.
On the one hand, equations of motion are the solution; on the other hand, they’re a statement of the problem. So there’s a certain amount of social construction that goes into the notion of a “solution”. Likewise for the notions of “explanation” or “proof”.
An explanation is something that conveys an understanding of “why”. But to whom is it conveying understanding, and in what context? Some people may have great difficulty understanding, and require more of an explanation. In other cases, the person already understands, but you still need to explain it to them because that’s the homework assignment.
Also, what even is “why”? Why does it happen? Why do we believe it? What chain of events led to this?
The notion of “explanation” is so squishy, we’re left grasping for any hard truths, any at all. Here’s a proposal: in order for something to count as an explanation, the thing it explains must be true. If you explain something, but that thing is not true, then we can say that the explanation has objectively failed.
Now suppose that an attempted explanation explains something that is true. But you could also take that same explanation word for word, and apply it elsewhere to explain something that is false. Is it really an explanation?
For example, suppose I make the observation that people who lack a skill tend to overestimate their own skill. We ask why that happens, and then we propose that there is a psychological effect that we dub the “Dunning-Kruger” effect. As an explanation, this is already dubious. But supposing this explanation is correct, then we would conclude that it applies to other people with psychology, such as East Asians. It does not. If our explanation did not reference race or culture, then the same explanation also explains something that is false. Objectively, it is not an explanation.
Follow this line of reasoning, and we might end up thinking that an “objective” explanation would look something like a proof. Most fields do not deal in proofs. But at least mathematicians deal in proofs, so let’s follow that thread and hope we get somewhere.
In fact, mathematicians will tell you that even proofs are based on social agreement.
Consider the Four Color Theorem. This mathematical theorem states that if you divide a map into regions, it is possible to assign one of four colors to each region, such that no adjacent regions share a color. The Four Color theorem was proven in 1976 by Appel and Haken, using computer assistance. While there are an infinite number of distinct maps to be checked, the authors managed to reduce it to a mere 1,834 configurations. Each configuration was checked by a computer over a thousand hours of computation. The proof contained 400 pages of microfiche.
A mathematical proof is often considered undeniable, and perhaps only more so when it’s checked by computation. In reality, the original proof contained multiple errors, later corrected. But mathematicians still give credit for the proof to the original authors, rather than the last person to have spotted an error in the proof because c’mon, they basically had it.
Setting aside whether the original 400 pages of microfiche constituted a proof, we can say that this is well and far away from any conventional notion of explanation. I just don’t think the typical person would gain any understanding of the proof by sitting down to read it. In fact, I don’t think anything could convey an understanding of the proof. I can imagine writing a popular explanation of the general outline, but if the explanation does not contain the full proof of the theorem, then does it really qualify as an explanation? There is a sense in which no explanation exists.
Bit of language games, there.
Depends. One needs to define ‘proof’ and to define ‘explanation’.
The former, I think, is a formally valid derivation from axioms, the latter is a way to understand something.
Not all explanations are proofs (creationism), and not all proofs are explanatory (e.g. tautologies).
I take it that the way you define “explanation”, an explanation does not need to be true? So what happens when we require that an explanation be true?
I think my discussion here is obviously pretty non-rigorous. As always I think it’s worth checking the Stanford Encyclopedia of Philosophy, and yes there is an article about explanations, with separate entries for scientific and mathematical explanations. I don’t have the time or interest to read that, but there it is.
While there may be an explanation for everything, I don’t think we have proof of that. 😉
But to the point about explanations, I think in some (probably many or most) cases the proof of an explanation is not contained in the explanation itself. Let’s look at the Four Color theorem again.
The theorem states that for any division of a plane into sections, only four colors are needed to clearly define the boundaries of each section, i.e. adjacent sections do not use the same color. This is the theory, but how can it be proved? In essence there are two ways to prove this theory, find an elegant formulation of this theory which conclusively shows (likely invoking previously established theories) that under all conditions this theory cannot be violated. Conversely, if it can be shown that ALL possible maps can be colored in such a way that only four colors are needed, then the theory has also been demonstrated to be true.
I think that’s where people trip up with the understanding of how the Four Color theorem was proved. They are incredulous that the mathematicians can prove that ALL possible maps have been tested. I am myself. However, a lot of mathematicians have reviewed the math which says that ALL possible maps can be reduced to a set of 1,834 configurations. That is, frankly, the proof of the Four Color Theorem, the math which reduces an infinity of maps to only 1,834. The rest of the proof is the brute-force check of every configuration.
Which means we have evidence that the Four Color Theorem is true, because we have tested all possible combinations where it might be false. We know it is true, that has been proven. If someone asks, “How do we know it is true?”, we can answer the “How” question. What we don’t have is an explanation as to why it is true.
This is different than what we were taught about mathematical proofs. A mathematical proof starts with axioms and says, “if those are true, this follows….” That form of proof allows us to say “Why”, at least until we descend to the level of the axioms. Then, all of a sudden, we are confronted with something we must accept as true either by definition, on trust, or by showing that in the infinity (and beyond) of possibilities there are no contradictions to this specific axiom. The entire field of non-Euclidian geometry was founded on people deciding that maybe there are alternative definitions, and we can’t just take Euclid’s theorems entirely on trust.
Any theorem can be proved by answering “how”. The “why” question can sometimes be answered, but only if all the evidence supporting the theorem is understood to be true. That is, you can prove the Pythagorean Theorem by several “how” answers. The most elegant proof I know is those rotating exhibits in science museums where the sides of a right triangle are extended into boxes and you can watch water fill the boxes as they rotate around, and directly see that the volume of water in the box adjacent to the hypotenuse exactly fills the boxes adjacent to the other two sides. That is a form of proof. You can also prove it mathematically with simple algebra. That is another form of proof. It can also be proved using geometry, and that proof is what we often use because we can say “why” as well. The “why” rests on simpler math, which has already been proved. So we can say, Z is proved because X is proved and Y and proved and Z is shown to be an unavoidable result of X and Y. But note that “unavoidable” in the last sentence. That is saying the same thing that Four Color theorem is saying about having tested all possibilities.
Which leads us to the final point. All explanations are tentative. Some can be established as more accurate representations of reality than others, explanations which have been tested are more accurate representations of reality than untestable, revealed, knowledge. But it may happen that sometime in the future an exception to the Four Color theorem is found, which would not fully invalidate the theory entirely, but it would put limits on what the theory explains. Explanations are models of reality, not reality itself.
Is there truth, or not, in one of my favorite lines in literature, from a story by Ring Lardner, Jr:
?
That is obscure, Pierce. I for sure don’t get you.
I took a look:
The Young Immigrunts (1920), Chapter 10, “N.Y. to Grenitch 500.0”
(https://en.wikiquote.org/wiki/Ring_Lardner)
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Siggy @2, for me, an explanation can be false, but a proof cannot be false.
Clearly two different categories.
cf. https://en.wikipedia.org/wiki/Alethic_modality