Death, taxes and proof

 Today's "Thought of the Day" is missing attribution: 
THOUGHT OF THE DAY:     
"Our new Constitution is now established, and has an appearance that promises permanency; but in this world noth- ing can be said to be certain, except death and taxes." 

That is Benjamin Franklin, from a letter to French scientist Jean-Baptiste Le Roy, dated November 13, 1789.

The quotation is slightly ironic because Franklin was discussing the new U.S. Constitution and its apparent permanence, then added the famous observation about death and taxes. The wording in your Thought of the Day has the old-fashioned line break/hyphenation from the source:

    “Our new Constitution is now established, and has an appearance that promises permanency; but in this world nothing can be said to be certain, except death and taxes.”

So the attribution should be:

THOUGHT OF THE DAY:
“...except death and taxes.”
— Benjamin Franklin, 1789

And, appropriately, Franklin was himself rather fond of aphorisms. This one has certainly outlived quite a few constitutions.

Putting the two things together gets across the unpleasantness of taxes.  There are of course a lot of other things in this world that seem certain, mainly that nothing is forever and you can't prove a negative.

Exactly. The joke works because Franklin pairs two radically different kinds of certainty: death is an unavoidable fact of nature; taxes are an unavoidable fact of government. Putting them together makes taxation sound like a law of physics.

And your broader point is rather good. “Nothing is forever” is almost the inverse of Franklin's formulation: not what is certain, but the certainty that certainty eventually expires.

“You can't prove a negative” is a little more complicated, though. In strict logic you certainly can prove some negative propositions—you can prove, for example, that there is no integer satisfying a particular equation. What you generally can't do by simply failing to find something is prove that it doesn't exist. That's the practical epistemological version: absence of evidence isn't evidence of absence, particularly when your search is necessarily finite.

There is also a wonderfully nasty corollary: you can never prove that you have found the last exception. The moment you say, “There are no more,” the universe seems to regard that as an invitation.

Franklin's sentence, incidentally, has acquired a meaning he probably didn't intend. We now tend to read “death and taxes” as saying that taxes are as inevitable as death. But in 1789 the United States was still experimenting with what sort of taxation a constitutional government could impose. His “appearance that promises permanency” followed immediately by “but...” is almost a little joke about political permanence itself.

The real Franklin thought of the day might therefore be: everything is provisional, including the things we are most confident about.

Yea that's an interesting point.  "Anything is possible" is the physical obverse of you can't prove a negative, but with numbers you can.  The point here is they are abstractions and "proof" has to be carefully defined.

Yes. And that distinction is important because “possible” changes meaning depending on what universe you're talking about.

With physical reality, “anything is possible” is usually shorthand for we cannot rule it out from our present knowledge. That is a statement about epistemic possibility, not necessarily about what the laws of nature permit. A perpetual-motion machine isn't merely something we haven't happened to build; under our best physical theories, it is impossible in a much stronger sense.

Mathematics gives us a wonderfully clean contrast because we can construct an artificial universe in which the rules are explicit. Within that universe, negatives can be proved quite routinely. For instance, there is no largest integer: given any proposed largest integer n, n+1 is larger. That's a proof of a negative, and it doesn't require searching through all the integers.

But even there, your qualification about abstraction is crucial. “Proof” doesn't mean “absolute demonstration of truth floating free of assumptions.” A mathematical proof establishes a proposition relative to its axioms and rules of inference. Change the axioms and sometimes the answer changes.

Euclidean geometry is the classic example. In Euclidean geometry, through a point outside a line there is exactly one parallel. In other geometries, there may be none or infinitely many. The proof didn't fail; rather, the proposition belonged to a particular formal system.

And that gives us a rather delicious hierarchy:

Physical statement: “We haven't found an exception.”
→ evidence, perhaps very strong evidence.

Physical statement: “An exception cannot exist.”
→ requires a theory of nature that entails that conclusion.

Mathematical statement: “No such object exists.”
→ potentially provable within a specified formal system.

Logical statement: “This follows from these premises.”
→ depends on precisely defining both the premises and the inference rules.

So yes, “you can't prove a negative” is really too crude. The more interesting statement is:

You can't prove a negative merely by looking for the thing and failing to find it.

Mathematics escapes that problem because it can sometimes prove nonexistence without looking at the objects at all. It reasons about the structure of the entire abstract domain.

And then, naturally, Gödel comes along and spoils the party by showing that even sufficiently powerful formal systems contain propositions that cannot be proved or disproved within the system itself.

Which is perhaps the mathematical version of Franklin's joke: nothing is quite as certain as you thought it was.

I can imagine a world without taxes, and even one without death.

Certainly. And they're interestingly different cases.

A world without taxes is easy to imagine. You need only imagine a society that finances whatever collective functions it has by some other mechanism: voluntary contributions, user fees, ownership of productive resources, monetary creation under appropriate constraints, or simply a society with very little that requires collective financing. Whether such a world could remain stable is another question, but there's no logical contradiction in it.

A world without death is also perfectly imaginable, but here the physical problem becomes much more interesting. You don't even need to imagine immortality in the religious sense. Imagine organisms that continually repair molecular and cellular damage, never accumulate irreversible defects, and can replace worn-out components indefinitely. That's biologically conceivable as a concept, even if our universe may make it extraordinarily difficult or perhaps impossible in practice.

And there is a curious distinction between “without death” and “immortal.” A creature might be biologically immortal—having no intrinsic aging mechanism—while still being capable of being killed by an accident. To have literally no death, you'd need something considerably stronger: no accident, predation, catastrophic environmental change, etc. Or creatures sufficiently indestructible that those things don't matter.

In fact, if we're playing with Franklin's sentence, we can turn it around:

    Death and taxes are not logically necessary; they're contingently universal.

That's rather different from saying they're inevitable.

We happen to inhabit a world in which biological organisms die and governments have discovered taxation. Neither fact is a theorem of logic.

And this is where your earlier observation about abstractions becomes particularly apt. “Certain” is doing an enormous amount of work in Franklin's sentence. He meant something like certain given the human world we inhabit, not logically necessary in every conceivable world.

You can imagine a world without taxes. You can imagine one without death. You can even imagine one without Benjamin Franklin complaining about either.

The last one may be the least plausible.