Tag Archives: Interpretation

Narrative and Characterisation in Proof – The Case against Sontag

Posted in Logic, Mathematics, Things that might be true by

We’ll argue that like good narratives, the better proofs tend towards the surprising or the inevitable. The latter proofs have a pleasing finality. The former make you doubt your own capacities; so shocking are the methods or the result that you cannot quite believe another human mind formulated such a conclusion. Nevermind, proved it!

In both cases the proof must make you confident of the connection between the premises  and conclusion. We must be left with a concrete impression that the narrative has been played out, and the players retired. The details of each step can be made more or less explicit so long as they are suitably suggestive of the narrative intended by the author, and its culmination is recognised as an ending.

In short, we want to “show not tell” but the details we do show have to be “telling”.

We owe an example here. Given that we have spent the last few posts proving properties of \lim, we shall now show that the underlying concept can be characeristed in terms of converging sequences.

Converging Sequences

To do this we need to introduce a new character. A sequence \{x_{n}\} is a list of n numbers. We say that \{x_{n}\} converges to x_{0} if and only if the distance between x_{n}, x_{0} tends to 0 as n gets increasingly larger.

This can be put as follows:

x_{0} is the accumulation point of a sequence \{x_{n}\} if and only if for every positive measure of closeness  \epsilon to x_{0} (no matter how small) there exists a natural number N, such that if n > N, we have |x_{n} - x_{0}| < \epsilon

since this definition ensures that the distance between x_{n}, x_{0} becomes vanishingly small as the sequence increases.

First Steps: A Suggestive Fact

Now as sequences enter our narrative their features become relevant. So we might come to suggest that the sum of two sequences is the same as the sum of their accumulation points. Formally:

\{ x_{n} + y_{n} \} \rightarrow x+y

We calcify this suggestion by a quick proof using the triangle inequality of absolute values. Assume the accumulation points x, y exist for our sequences. See first that

|(x_{n} + y_{n}) - (x + y)| = |(x_{n} - x) + (y_{n} - y)| \leq |x_{n} - x|+ |y_{n} - y|,

then appealing to the definition of a sequence we need to pick an \epsilon > 0. Now by assumption there are \epsilon_{1}, \epsilon_{2} measures of closeness. We specify these so that there sum is equal \epsilon, i.e. \epsilon_{1} = \epsilon_{2} = \epsilon/2. Then we have

|x_{n} - x| < \epsilon/2 (\forall n > N_{x}) \\ |y_{n} - y| < \epsilon/2 (\forall n > N_{y})

Now we pick N = max\{N_{x}, N_{y}\} We need to show that that the the distance between the sum of our nth terms  and the sum our accumulation points is strictly less than \epsilon. But by our choice of \epsilon_{1}, \epsilon_{2} we know this to be true, since

 |(x_{n} - x) + (y_{n} - y)| \leq |x_{n} - x|+ |y_{n} - y| < \epsilon/2 + \epsilon/2 = \epsilon

as desired.

If this looks familiar, we’re on the right course. The lemma above should be suggestive of the summation property of limits since both proofs rely on similar manipulations of the \epsilon distance measure. Only now, with this familiarity established, can we make a compelling suggestion about the relationships between limit points and accumulation points. This hint, like the revealed motive of a shadowy villain in a spy thriller sheds light on the character.

The Characterisation Theorem

The most memorable characters tend to lodge themselves in our mind as personalities, we can ask what would Jesus do, and have concrete notions as to how he would act in any fanciful circumstance. This is to say, that somewhere along the way we internalise the idea of Jesus the person. No matter what you think of that character, you think of the individual as a character equipped with various capacities and propensities. Whatever is true for gods is true for mathematical objects too.

A characterisation theorem supplies a way to describe the mathematical behavior of a poorly understood notion in terms of the limitations of a well understood notion. In this way we make new questions tractable. What would Jesus do? How do sequences behave under composition? Both questions presuppose characterisation.

Fortunately, in the case of sequences, we can provide one:

For f:X \rightarrow Y with x_{0} an accumulation point in X,  then \lim_{x \to x_{0}}f(x) = y_{0}  if and only if \text{ for all } \{ x_{n} \} \rightarrow x_{0} \text{ then } \{f(x_{n}) \} \rightarrow y_{0}

The idea is to define the limit of a function in terms of the relationship between two convergent sequences. We prove each direction separately.

\Rightarrow

First suppose that (i) \lim_{x \to x_{0}}f(x) = y_{0}  and (ii) \text{ for all } \{ x_{n} \} \rightarrow x_{0}

We want to show that \{f(x_{n}) \} \rightarrow y_{0}. In other words,  for any distance \epsilon to x there exists a natural number N_{\delta}, such that if n > N_{\delta}, we have |f(x_{n}) - y_{0}| < \epsilon.

Let \epsilon > 0.

Then by (i) we know that when 0 < |x - x_{0}| < \delta_{1} there is an \epsilon_{1}-distance, arbitrarily small, such that |f(x) - y_{0}| < \epsilon_{1}.

and by (ii) it follows, in particular that we can find an n > \delta_{1} , which ensures |x_{n} - x_{0}| < \epsilon

Now by these two facts we can specify N_{\delta}  = \delta_{1}, and the claim follows by the fact that the absolute value of |f(x_{n}) - y_{0}| ~ |x_{n} - x_{0}| < \epsilon. This proves the right to left direction.

\Leftarrow

For this direction we prove the contrapositive. That is to say we need to show that:

If \lim_{x \to x_{0}}f(x) \neq y_{0} then \exists \{ x_{n} \} \rightarrow x_{0} such that \{ f(x_{n}) \} \nrightarrow y_{0}

So assuming the antecedent we know that:

\neg\forall\epsilon(\epsilon > 0)\exists\delta(\delta > 0) such that if 0 < |x - x_{0}| < \delta, then |f(x) - y_{0}| < \epsilon

Rolling negation through the quantifiers we get:

\exists\epsilon(\epsilon > 0)\forall\delta(\delta > 0) such that \exists (x) where 0 < |x - x_{0}| < \delta, then |f(x) - y_{0}| \geq \epsilon

Pick some \epsilon > 0. To see that the sequence \{ f(x_{n}) \} does not converge to y_{0} we need to see that

There is a positive measure of closeness  \epsilon to x for all natural numbers N, such that there exists n > N, we have |f(x_{n}) - y_{0}| \geq \epsilon

Now consider a sequence of \delta-distance measures  \{ \delta_{ 1/n} \} around x_{0} By our construction we have a sequence where \{ x_{n} \} \rightarrow x_{0} such that we have for every positive n > 1/n and by our initial assumption |f(x_{n}) - y_{0}| \geq \epsilon. This proves our claim and finishes the proof.

The Ramifications of the Reveal

This is not a difficult proof, but its primary benefit is that it confirms an implicit suspicion that most readers will have when encountering the \epsilon , \delta definition of limits, that there must be a more intuitive way! In the same way that you might find it difficult to relate to Darth Vader before the “Daddy” revelation, \epsilon , \delta proofs are harder to comprehend before their characterisation in terms of convergent sequences.

But once the reveal occurs, the behavior of sequences is more predictable. They are seen to be well behaved under the standard arithmetic operations. This is a direct corollary of the fact that \lim respects the standard algebraic operations. Similarly Vader’s temptation of his son is more excusable than a faceless bad guy’s  attempted seduction of Skywalker. The insight of our proof is not surprising, but given our suggestive fact about the sum of sequences it has a pleasing inevitability and provides some conceptual clarification of the nature of \epsilon , \delta statements. Because characterization determines expectation, we expect the Scorpion to sting the frog in Aesop’s fable and we are somehow sadistically pleased to see the result. Similarly once a mathematical entity has been characterised they become familiar and reassuringly predictable.

Mathematics as Art

I have spent far too much ink on this result, but the point I want to make relates not to this particular proof, but proof in general. While here I explained the relevance of each connection, it is neater and more engaging to leave a proof somewhat inexplicit. The sequence of statements should be suggestive and coaxing; they should inspire intellectual curiosity rather than simply satiate the readers desire for knowledge. Seen as a work of art a proof demands engagement first, rather than mute appreciation.

In contrast Susan Sontag has argued that works of art should not be interpreted, the choice and motivation of the artist are not particularly relevant to the experience of artistic appreciation. This is not true in the case of mathematical proof.

She writes:

…interpretation is the revenge of the intellect upon art. Even more. It is the revenge of the intellect upon the world. To interpret is to impoverish, to deplete the world – in order to set up a shadow world of “meanings”…It is always the case that interpretation of this type indicates a dissatisfaction (conscious or unconscious) with the work, a wish to replace it by something else. (Against Interpretation)

Considered in the case of mathematical beauty this view is strikingly silly. Firstly the quote assumes an availability of a plethora of “meanings”, but in mathematical proof the semantics of each statement are much more rigidly defined than, for instance, in a novel. The hunt for meaning in mathematics is simply the search for understanding. It is certainly not some dramatic pursuit of vengeance upon an artist presumptuous enough to try and imbue their own creation with a purely personal significance. It is a conversation with an artist who happens to be capable of cogently discussing their own work.

While I am sympathetic to view that artistic criticism is often just windy rhetoric coupled with inane interpretive efforts, there is no call for the wholesale renunciation of interpretation as a method of artistic engagement.

Free Indirect Style

It’s perhaps unfair to Sontag, who wrote in a very different context, but her call for an “erotics of art” and the dismissal of interpretive endeavor risks ignorance of the arts which are premised upon the audience’s interpretive engagement.

Where artists rely on the audience’s engagement, there can be a beautiful dynamic by which the artist coaxingly leads their audience to the intended revelation, and inspires the onset of understanding. This is true as much in good fiction as it is in mathematics. Although the techniques are markedly different, and much more subtle in fiction.

The best argument to this effect is described by James Wood in the book How Fiction Works. He writes of the narrative technique called indirect free style which allows the voice of the narrator to inhabit various roles (e.g.  reliable, unreliable narrator or foil of a particular characters) as the narrative demands. Each role is freely adopted to better communicate an aspect of character or situation so as to make both more concrete and relatable.

Free indirect style is at its most powerful when hardly visible or audible: “Ted watched the orchestra through stupid tears.” In [this] example, the word “stupid” marks the sentence as written in free indirect style. Remove it and we have standard reported thought: “Ted watched the orchestra through tears.” The addition of the word “stupid” raise the question: Whose word is this? It’s unlikely [he] would want to call [his] character stupid for merely listening to some music in a concert hall. No, in a marvelous alchemical transfer the word now belongs partly to Ted. (How Fiction Works, pg10)

It is, in part ,Ted’s judgement of his own tears that allows for this evaluation. So in this description the author hints at a greater emotional and intellectual depth to the character of Ted. It becomes true of him that he would be embarrassed by such needless emotional displays. We know more now about him than we did before. Ted is more predictable, more relatable because we have discovered a truth about him. This kind of revelation is only inspired by a profound intellectual attention to technique, usage of language and engagement with an imagined intent of the writer.

What is true of fiction is true too of proof. Sontag’s call for an “erotics of art” fails here, we suggest, simply because neither are particularly erotic. The primary aesthetic value comes from understanding a proof, not merely appreciating that it holds. This is not to say that understanding is the only aesthetic value, or even that understanding is the primary such value in all domains. As there are many kinds of art, there are many ways to enjoy Art, but a proof is nothing but ugly scribblings without interpretation and understanding.

Conclusion

To write a stylistically good proof, we must write to be understood. We must write to inspire curiosity and engagement; we should aim for nothing less. As a consequence we are appealing to the desire of the reader to test their own suspicions. Whether they want to be confirmed or overthrown, a good proof will have to be surprising or tinged with a sense of inevitability. Proof should be written like noir fiction, there should be a hinted mystery that allows the reader to engage, and each subsequent step should prompt further speculation while simultaneously leading us to a reveal, inevitable in retrospect.