Linguist Geoffrey Pullum calls this to my attention. What say you logicians and philosophers?
My former colleagues at another university in Middle East have also been moved to online teaching indefinitely, with the students…
12 responses to “A proof without content”
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I think it’s worth noting the similarity to Tyron Goldschmidt’s “Demonstration of the Causal Power of Absences” (2016, Dialectica): https://philpapers.org/rec/GOLADO-4.
If I remember the discussion around that piece, one line of criticism (from David Oderberg?) was that the “demonstration” relies on there being a frustrated expectation generated by a title/metadata. Seems like a similar thing happens here: the first panel functions like a title, without which the blank panel would be meaningless. So maybe there is some content?
Having written that, I’m reminded of the quotation about explaining jokes and dissecting frogs…
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Mustn’t any proof contain the proposition allegedly proved (as its last line)?
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I think something like this must be the right response. But maybe such a formal statement applied to the empty box below the conjecture feels unsatisfactory. (After all, one might think this is just an arbitrary rule about proofs!) So here is a nearby point that is semantic rather than formal.
A proof contains (at least) the content of the theorem proved, or else the statement of the theorem expresses a tautology. That is, following Aristotle and Hilbert, there is some content intrinsic to any theorem, such that one standard of rigor is that of purity of proof—whether the proof only uses concepts/axioms, which fit that intrinsic content. Otherwise, if there is some theorem without intrinsic content, then that theorem must express a tautology (i.e., so that the empty set rigorously proves it).
Next, consider the statement from the comic: “it’s possible to construct a convincing proof without words, pictures, or content of any kind”. Suppose that this is a theorem. If it expresses a tautology, there is nothing more to say here. Supposing it does not express a tautology, then one can construct a proof, which contains (at least) the content of that statement. Yet this state of affairs contradicts the statement itself.
Therefore, the statement is tautologous. But this means that it simply doesn’t say anything at all—certainly not what we take it to mean on a natural language interpretation. (Hence, the joke.)
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Having started out with the prompt “Is this a case of zero reflecting back zero by other means?,” I toyed with this problem on Gemini and got this partial response:
‘The “zero reflecting back zero” is mathematically elegant but rhetorically silent. If you start with zero content in your premise (the idea that content isn’t needed), and you provide zero content in your conclusion, you have successfully maintained consistency, but you *haven’t moved the needle of human knowledge*. You’ve just drawn an equals sign between two voids.
‘3. The “Aesthetic” Proof
There is a concept in physics and math called Supersymmetry or Mirror Symmetry, where certain structures are reflected in others.‘In those cases, the “reflection” provides deep structural insight. In this comic, however, the reflection is destructive. By reflecting “zero” back onto “zero,” the comic strips away the medium of communication entirely. You aren’t left with a proof; you’re left with a tautology: “This proof is empty because it is a proof of emptiness.”
‘The “Silent” Counter-Argument
If I were to play devil’s advocate against your (and my) skepticism, one might argue this is an Inductive Proof:Base Case: Can we convey a thought with one word? (Yes).
Step: Can we convey it with fewer?
Limit: As content approaches zero (x→0), does the “convincingness” remain?
‘The comic argues that at the limit, the conviction remains because the reader completes the circuit. You are the “other means” by which the zero reflects; the proof happens in your brain, not on the paper.
‘Do you think a proof can ever truly exist independently of an observer, or is the “content” always something we project onto the symbols (or lack thereof)?’ [Here ends Gemini’s response].
However, perhaps more to the point, it seems the author of the comic is referencing a book titled ‘Proofs Without Words: Exercises in Visual Thinking,’ by Roger B. Nelsen, and boils down to a joke about non-thought in a pseudo-Zen (non)sense. In which case, one might argue that ‘truth’ is and is not proof-dependent, in a verbally ‘effable’ or even mentally non-spontaneous sense (i.e. it can manifest even in the case of a complex mathematical ‘truth’ in such a ‘non-thought’ way), but ultimately such ‘truth’ will almost certainly be subject to a formal logical or pragmatic ‘proof’ when/if it is communicated to another, who’s capable of appreciating its implications.
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In “The Possibility of Empty Fictions” (JAAC 2029), Nathan Wildman uses more or less the same strategy to generate an empty fiction.
I’m not entirely sure how I feel about the fiction case, though I tend to skepticism on such issues. As far as the proof goes, as a visual demonstration, it seems to have at least some content! (More generally, I wonder whether something without content can be _about_ anything, and, thus, whether it can be a proof in the first place.)
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Ugh. 2019.
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I think the comic doesn’t establish what it intends to establish. First, we can notice that the strategy in the comic can be used to show that a proof could be any arbitrary thing whatsoever. For example:
claim1: it’s possible to construct a convincing proof consisting solely of a picture of an orange
proof1: [a picture of an orange]
claim2: it’s possible to construct a convincing proof consisting solely of the sequence “atgjkhadjgahtkeatak”
proof2: atgjkhadjgahtkeatak
this is clearly fishy. What’s going on is that the proof only works assuming that the proof is indeed a proof. This is a self-referential scenario, structurally analogous to the truthteller sentence:
P: P is true
we can coherently think of P as true or as false, and it’s not clear that one truth value assignment is a priori preferable to the other. Similarly, in all the above proof cases, we can coherently think both that the proof is a proof, in which case the claim is true, or that the proof is not a proof, in which case the claim is false.
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It is a convincing proof of a different conjecture, namely the conjecture “It is possible to leave a rectangular region on a piece of paper blank.” So it is a convincing proof of something. Which means it is also a convincing proof of the stated conjecture.
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It doesn’t quite work the way it’s set up, but it could work if the proposition to be proved were something like, “It’s possible to construct a convincing proof that doesn’t employ words, pictures or content of any kind.”
The cartoon’s conjecture to be proved is a natural language proposition that makes an empirical claim, so not pure formal language. The claim is a claim about empirical possibility, so it can be proved by providing an instance of what is expressed in making that claim. We are presented with a square with no further visible shapes inside it, which appears to exemplify the idea, “without words, pictures or content of any kind”. But that same “empty square” could just as well exemplify the idea of “a cow eating grass”. So what differentiates these? If they can be differentiated, there is a difference in content, according to the usual definitions of the term ‘content’. What differentiates them is a difference in “how they are presented”, which Frege recognized as distinguishing the sentential properties of sense and reference (i.e., “referent”), which would count as a distinction in content, as opposed to form. “How they are presented” is given here by the expression “proof:” and the square frame, which usually is understood as a presentation of a picture. It doesn’t say, “picture of a cow eating grass”, but, via the colon, “an instance of the notion ‘proof’”. No intentional object (an object referred to) has any significance until it’s integrated to a categorical structure, and that particular “empty square” picture goes to the latter, and not the former. So the picture has content, beyond that of “being a picture”.
In the above the expressions “proof:” and “picture” could be taken as referentially identical, i.e., as both referring to the one object in the cartoon, the picture, understood as a proof. So if the proof consists of what is within the frame of the picture, then it could be said to not employ (the use of) words, pictures or content of any kind, and thus to be an instance of my revised conjecture in para one. But then it seems to lose its sense of paradox. The sentence labeled as the “conjecture” in the cartoon is ambiguous, but it leaves open the possible interpretation that the situation referred to includes the frame, i.e., is outside the proof, as opposed to the revised sentence I gave above, which refers to only what is inside the frame, the proof itself in its logical role of “instance”. (“construct … without words”, etc., vs “proof without words”, etc.) The sentence interpreted with this sense has a referent; the sentence interpreted in the other sense (“construct … without words”), so far, can not, as I suggested above, have an empirical instance.
BTW, I’m a linguist, not a philosopher.
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Note also that there is a “mouseover title” that says, “There is also a proof without content of a conjecture without content, but it’s left as an exercise for the reader.”
But the proposition expressed by the clause, “There is also a proof without content of a conjecture without content” is necessarily false, because it assumes the existence of “a conjecture without content”, and it is not possible for there to be such an object, nor even an act of successfully making a conjecture without content. Any conjecture, as an expression that is intended, eventually, to be judged true or false, would have to have the form of a proposition, and the logical form of any proposition already necessarily involves a distinction in content. That content, like all creation of content, is created by the reasons for making the (formal) distinction. So it’s not possible to have an object that is both a proposition and without content. (A tautology is always true, but any instance of a tautological expression will involve a lot of distinctions in content.) William James’s “blooming, buzzing confusion” could be an example of a situation without content, considered as prior to an attempt by language to understand it. “Judgement,”, said philosopher David Bell (in Frege’s Theory of Judgement (1979), p.1), “it was urged, is the primordial act in terms of which we make sense of the world”, so the question is, what is the principle according to which that initial division in the propositional form was made?
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The whole cartoon looks suspiciously like a picture to me. (The smaller blank rectangle wouldn’t seem like a proof at all if the viewer wasn’t primed by the words above it.) And, for that matter, the lower rectangle only functions because of the expectation that it will display (a representation of) a proof: so it, in the context, functions as a picture: a picture of a wordless and pictureless proof.
That said… It seems to me that there are, even in purely formal reasoning, proofs that work, not by inferring something, but by observing something. If, in a natural deduction system, one proves A->B by displaying a deduction of B from A, the reader is not convinced of the truth of A->B by inferring it from any proposition (or propositions) stated in the proof: one infers it from the observed existence of a certain derivation.
(But somehow it feels inappropriate to do nit-picking logic chopping of something as funny as XKCD.)-
In a logically valid argument the premises can’t be true and the conclusion false. The empty set of premises can’t be true. So the empty set of premises can’t be true and some random conclusion X false. Hence there is a logically valid argument from the empty set of premises to any proposition whatsoever. But it’s worse than that. In a logically valid argument the premises can’t be true and the conclusion false. A conclusion set consisting of the empty set can’t be false. So there are valid arguments from arbitrary sets of premises to the null-conclusion since the premises can’t be true and the conclusion false. Furthermore since the empty set of premises can’t be true and the empty conclusion set false there is a valid argument from the empty set of premises to the empty set of conclusions. This is perhaps unsurprising since they constitute the same set.
Other proofs make us gape ,with their symbols and shapes,
But we’ve got Doctor Nihil to thank,
So we all can attest, he has brought us the best,
A perfect and absolute blank.(With apologies to Lewis Carrol)
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