Convoluted reasoning

There’s a general trend to convoluted reasoning, which I think captures a wide range of common logical flaws. Convoluted reasoning captures a frighteningly common pattern even among brilliant thinkers: It’s all too easy with informal reasoning to follow a chain of intuition, where each step seems reasonable, only to end up at a totally incorrect conclusion. This is often used maliciously to convince people of nearly anything, as a defense, giving rise to “motte-and-bailey doctrine”/”strategic equivocation”, internally as rationalisation, and in the most sinister case, as a route for smart people to do something truly stupid. It is also the root of the map/territory conflation.

The mark of convoluted reasoning is implicit conversion between distinct objects which are related but follow very different logics. Let’s be more rigorous:

Consider a collection of categories and a chosen isomorphism between each. Take, for example, the Mind / Space categories, or many copies of the category of (your favorite) verbal logic, connected by the isomorphisms swapping between alternative definitions of words, or the categories induced by different metrics on the same type of object. The connections in an individual category compose, just as logical deductions should, but the isomorphisms on the underlying sets don’t induce functors – they don’t preserve the arrows! Of course, to our naive human brain, they’re all just the same sort of connection – I’d guess that most people don’t explicitly represent the categorical structure, preferring instead to represent everything more like simple set functions. In fact, I’d wager that at the lowest level, our brain saves space by collapsing the multiple categories into one space, so that (set) isomorphic objects (like multiple definitions of “privilege”) are literally identical, until there is a need to disambiguate them. Similarly, in the Mind / Space problem, we don’t often think of mental things being different from physical things, but rather that everything is either mental or physical (depending on which side of the wall you stand on).

So your’e probably already familiar with one example of how things can go wrong with convoluted reasoning, and that’s equivocation. We dance around in one category (collection of definitions for word), drawing all sorts of conclusions, like that “privileged” people should all shut their trap and maybe die in an excruciating fire for good measure, and then secretly jump into the other category (change definitions), so that you can include everyone you don’t like under the umbrella of “privilege”. Now substitute “privilege” for any demographic term you can think.

Don’t be fooled though, convolution isn’t just about switching definitions, it’s more about switching contexts. To see this, recognize that the same problem can arise where your isomorphism is literal identity, like so:

One very frequent case where we have many different categories on the same object are the categories induced by different metrics. In this case, the arrows represent “similarity” or “closeness”. You can often think of each possible metric as the “natural metric” on its own space, but they all happened to get mixed together in the space you’re working with. For example, the euclidian metric is the “natural metric” for \mathbb{R}^n, while the taxicab metric is the extension of the “natural metric” on a rational grid, embedded in $\mathbb{R}^n$. It is not always so obvious what the “natural metric” you’re looking for is (Hopefully I will address this in a future post).

A natural scenario for this to arise in is machine learning / pattern recognition. I’ll talk about a specific case that has interesting implications:

Consider the biological tree of life. Clearly, some species are “more related” than others. What’s not so clear is what we actually mean by “related”. Do we mean “has the most similar function”? This is clearly wrong, and has mislead biology for a long period before the study of genomics was invented. Then do we mean “has the fewest number of ancestors separating them” (so brothers would be closer than cousins, etc.)? This seems reasonable until you realize that not all mutations cause equal deviation. For example, some organisms, such as Ctenophore, have remained relatively basal throughout evolution, they haven’t deviated too far in “phenotype space” from their ancestors. So what’s the “right” metric? I won’t try to solve this here because I think it’s a hard problem (and it may be that there is no unqualified “right answer”, and the feeling that there should comes from built-in patter seeking heuristics). Rather, I’ll point out that computational biology has developed (read: apprehended from mathematicians some 30 years after the fact) a large collection of different metrics for this purpose, many of which resolve to metrics on genomic or protein sequences, such as edit-distance and its more biologically relevant varieties. Because of the complexity of data, we often need to use slightly different metrics for different types of data, for example, adding different sorts of normalizations to make everything match up. We often want to chain these comparisons up, so that if gene expression profile A is similar to gene expression profile B, and gene expression profile B is similar to drug response profile C, then expression profile A should be similar to drug response profile C, and we’ve found drugs to cure cancer, yay.

This tends to turn out badly for a few reasons. Barring the fact that some of our similarity measures don’t even obey the triangle inequality, the main reason this doesn’t work out is that they’re mutually incompatible measures. Sure, they kinda-sorta compose, we benchmark them zealously until something sticks, and you can still discover useful things with them (or at least be convincing enough to get published in a nice journal, and really friends, isn’t that the definition of useful discoveries?) but they lack the theoretical niceness to even put error bounds on how badly they can fail to compose. This is fundamentally because they are arrows in a different category.

 

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