Some thoughts on generalization

The area of generalization, extrapolation, abstraction, analogies, etc., can be quite interesting—as can the question of how to best handle it. (For simplicity, I will mostly speak in terms of just “generalization”, and the examples might be tilted towards specifically generalization, but the word should be taken in a widened meaning of “generalization, extrapolation, […]”.)

For instance, with most* texts that I write, I find that the contents apply (in whole or in part; with or without relevant modifications; with or without abstraction; whatnot) to other areas. When should I mention this in the respective text resp. when would the mention bring value to the reader? This is a judgment call, which usually turns out as “don’t bother” with me. Factors to ideally consider include how obvious the generalization is (mentioning the too obvious can be a waste of the reader’s time and/or seem insulting), how far-going a generalization might give a benefit, how many possible generalizations there are (the more there are, the more work there is likely to be), the issue of how much text is needed to make the generalization worthwhile,** etc. Then there is the observation that spending more time thinking about a topic might bring forth a slew of new generalizations, which could lead to never-ending work.

*And chances are that the exceptions arise from my not having spotted the generalization(s) yet—not from an absence of possible generalizations.

**I might just mention that “X generalizes to Y”, but this need not be that interesting without a deeper discussion of the details and consequences. (This, especially, in areas like math. An advantage of usually writing about politics and similar topics, not math, is that the additional discussion needed can be much smaller.)

In most cases, however, I do not so much engage in a deep analysis as rely on the mood of the moment—if I at all remember that there was a generalization that could be mentioned. A recurring factor in my decisions is that I often am a little tired and/or tired-of-the-topic when I have a text done, and adding another paragraph to discuss generalizations is not very enticing at this stage. Here we also see an example of how trivial or obvious the generalizations can be. For instance, the same idea applies to virtually any tiring activity, or situation of being tired, that has an optional continuation—but that goes without saying for most readers. And where should the generalization be stopped? Replace “tiring” with “boring” and something similar applies. Ditto “painful”. Ditto even normally positive things, once we enter “too much of a good thing” territory. Abstracting and generalizing to find some single formulation might bring about the tautological “when I am disinclined too continue, I am disinclined too continue”, which truly is too trivial to bother with, might be too detached from the original situation (as being “tired” does not automatically imply “disinclined too continue”), and might still not be the end of the scale. (For instance, a similar idea might apply to a great many other contexts; for instance, a “when X, then X” is a further generalization, just of a different type.)

Similarly, that I might “rely on the mood of the moment” over a deep analysis is not unique to this situation. It can also affect e.g. what I buy in a grocery store, and the generalization starts again. But now we have two different ideas that both generalize, which allows us a generalization about generalizations…

An interesting complication, in a generalization (!) of an older text ([1]), is that adding a generalization to some ideas could conceivably raise an expectation in the reader that I add generalizations whenever I am aware of one. If there is a too obvious generalization of another idea, or a second of the first idea, that I do not mention, then I might look foolish in front of this reader. Of course, the fact that I occasionally have such concerns, while the typical reader is unlikely to even care or notice, generalizes another portion of [1]. Potential further generalizations of this generalization include that “many pay too much attention to the opinions of others”, “many overestimate how much others might care”, and “many fear non-existent threats”, with further generalizations of these possible. Then we have the conflict between my intellectually knowing that few readers will care/notice and my instinctually imagining that they will, usually followed by a quick suppression of my instinct by my brain—which, you guessed it, generalizes. (I will not mention further cases in the continuation, but they are plentiful.)

Of course, the amount and direction of generalization that is appropriate in a given context need not be the same in another context. For instance, if someone working on a specific physics problem makes a novel mathematical observation, this observation is likely to have an analogue in other problems and other areas, where the equivalent math appears, but this might simply not be of immediate interest. For someone working on such another problem, the situation might be different, but it is not a given that more than a one-off generalization to that single other problem is wanted. However, once a mathematician with the right interests gets his hand on the original observation, it might be generalized one or two steps fairly rapidly—and another one or two steps when some mathematicians with other, especially more abstract, interest gets involved. Etc.

It might even be argued that the ability to find the right level of generalization for the task at hand is more important than the ability to find generalizations. (And this level might in many contexts be “no generalization”.)

However, generalization is often something positive, for instance as a means too avoid reinventing the wheel, which can all too easily happen when workers in different fields encounter similar problems. Consider e.g. how often different physical phenomena are, at least too a decent approximation, governed by the same differential equations and how wasteful it would be to develop the same methods of solution in the case of each individual phenomenon—possibly, including the repeated development of the idea of differential equations… Mathematicians are particularly keen on such generalization, e.g. by showing that a certain set and associated operators match a known “algebraic structure”, after which they know that all results of that algebraic structure applies equally to the new case.

In other cases, a failure to abstract can be outright wasteful or harmful in other ways. Consider various arts, including painting and the theatre, where there has been a long history of new artists trying to outdo the previous generation in e.g. the breaking of norms, the “shock value”/provocation, and where to draw the border between art and non-art.* But why? If someone manages to find/create something truly thought-worthy, truly original, truly unforeseen, truly value-bringing as an extrapolation, whatnot—by all means. This has rarely been the case, however: most of what has been considered provocative has been well within what even the layman has been able to imagine on his own, has been an natural extrapolation of previous provocation,** has long been exceeded in less “artsy” contexts,*** or similar. I have, e.g., encountered fictional depictions of artists that have gone as far as to consider murder an art, incorporate murder in performance art, murder for artistic provocation, use body parts of a murder victim as art, and similar.**** What could a real-life artist do that would move me beyond the borders of what I have already seen in fiction or could myself conceive? Splash a bucket of pig’s blood on an empty canvas and call it art? Please! Why not just draw the natural conclusion that this type of provocation, escalation of provocation, whatnot is pointless and will often do more harm than good to the art at hand.*****

*As opposed to e.g. just experimenting further in some direction for more artistic purposes, say, to find out what the effect on a canvas is when a certain school of painting is pushed further and whether the result is worthwhile.

**In principle, if not in detail. For instance, if we start in a very straight-laced era of movie-making and have a “shocker” of showing a young woman in a still-covers-a-lot bikini, the next escalation of shock might consist of a showing a young woman in a modern bikini, showing a topless young woman from behind, or similar. To predict the exact escalation is hard; to predict the general nature of the escalation (and the risk of an escalation) is a different matter. Even an escalation to, say, a completely naked young woman having sex on the beach would be more a matter of quantity than quality, of taking several steps of escalation at once. Going in another direction, imagining a young man or an old woman in a bikini does not take a revelation either—but why would anyone wish to see them?

***Contrast e.g. a sexually explicit art movie with a porn movie.

****Note e.g. portions of “Dexter”, but the idea is somewhat common.

*****Unless the artist follows some disputable non-artistic agenda, e.g. to change societal norms, to ruin this or that art form for the “wrong” persons, or, even, to ruin art. While I do not think that such an attitude is common, it is certainly possible, compatible with the behavior of many political activists in other areas, and compatible with some other excesses. Consider e.g. how some seem to take the laudable attitude of “function should take priority over aesthetics” and amend it with a despicable, unreasonable, and irrational “ergo, we should deliberately strive to make buildings ugly”.

More generally, it is often the case that certain ultimate extrapolations and generalizations follow immediately to the reasonably intelligent, but that mid- or nitwits, who are themselves poor at generalization, try to take each individual step at a time. A good (fictional) example is found in Lewis Carroll’s “What the Tortoise Said to Achilles”, where just one or two iterations should have been enough to prove the point, but Achilles was too dense to understand this—and, maybe, the turtle too dense to understand that he had just caught himself in a trap, where his best bet would be to wait for Achilles to fall asleep and then make a crawl for it. An interesting parallel to this is the idea of plus-one, infinity, and infinity-plus-one:

Consider two kids arguing over the size of something, say, who has the greatest desire for a last piece of cake. A stereotypical dialogue might then include something along the lines of “I want it more than you!”, “Hah! I want it twice as much you!”, “Hah! I want it thrice as much you!”, etc., until someone drops the bomb of “I want it infinity times as much as you!”. Exactly how to consider the introduction of infinity is a tricky question, but something like “extrapolation of what events are likely, followed by an attempt to defeat the extrapolation” might be close to the mark, which could be seen as a case of successful generalization (in the extremely wide sense used in this text). Moreover, infinity as such could* be seen as an extremely interesting extrapolation of large numbers. However, we also see a failed generalization, as both kids, unless first-time participants, should have realized that infinity was coming and that whoever first dropped the bomb would “win”.

*The conditionality hinges on whether it is an extrapolation, not on whether it is interesting.

In a next step, we could have the other kid either conceding or trying some variation of “infinity plus one”—and their argument might then have turned to whether this “infinity plus one” was or was not larger than infinity. Here we land at a very interesting question, as mathematicians consider infinity-plus-one in the sense of addition equal to infinity, meaning that attempting to trump infinity with infinity-plus-one is as pointless as trying to trump three with three (resp., above, thrice with thrice). In this sense, the argument could finally degenerate into whether an “infinity vs. infinity” standoff should be considered a draw or a victory for the first invoker of infinity. But, while mathematicians consider infinity + 1, infinity + 2, and even infinity x 25 equal to infinity, they also have a generalized version of the “successor operator” implied by plus-one.* Here we have a generalization arguably bringing something more interesting than even infinity—the idea of a number larger than infinity.** In the unlikely event that infinity-plus-one was intended as this successor operator, not as a mere addition of one, the other kid would have kept the game alive.

*As in 1 + 1 = succ(1), 2 + 1 = succ(2) = succ(succ(1)), etc., for a suitable operator succ, which for integers is simply that same thing as adding one—but where succ(infinity) is something different. By such generalization of the successor operator, we then have a hierarchy of non-equivalent infinities. The “vanilla” infinity presumably intended by the triumphant kid would then carry the more specific name aleph0. (Assuming the most common approach of using “cardinal numbers” and with some oversimplification, as we potentially begin to compare apples and oranges.)

**But this idea might have originated from another line of reasoning, e.g. Cantor’s famous “diagonal proof” that there are more real than rational numbers, and its generalizations.

However, from here on, it is trivial to infer the possibility* of applying a successor operator to infinity infinitely often to form a super-infinity, a version of the successor operator that finds an even larger successor to the super-infinity, a super-duper-infinity and a successor to the super-duper-infinity, etc. Even trying to break out of this by e.g. constructing a super-duper-whatnot-infinity where the “whatnot” incorporates an infinity of terms stands the risk* of failing due to an even more generalized successor operator.

*As case has it, all these successors, super-duper-infinities, and successor operators exist, but their existence is not a given from the above. Without further investigations, we cannot infer more than the possibility. (This with some reservations for what qualifies as “existing” and what has what semantics.)

Disclaimer:
Apart from some minor editing, the above was written some weeks ago. At the point where the text ends, I was distracted by a text from my backlog, with more mathematical content, which fit well in context and would have clarified a few points above. Having written most of it, I found some issues that I wanted to mull over before finalization and failed to get around to it, which has led to these “some weeks” of delay. To avoid further delays of the current text, I have decided to put the other text back in the backlog. (Especially, as I could benefit from improving my markup language with regard to math before proceeding.) It is possible that some additional thoughts or sub-topics that I intended to include in the current text have been forgotten during this delay. Certainly, trying to go easy on the mathematically unknowledgeable, I run the risk of being sufficiently approximative with the truth as to annoy the mathematically knowledgeable, while not giving enough details for the unknowledgeable to be truly helped.

To, however, give two core ideas of the other text: (a) When we generalize a certain type of number, a certain algebraic structure, whatnot, there is rarely or never one single generalization, and statements made under the assumption of a single generalization can be faulty or simplistic. (E.g the claim that the square-root of -1 is i and/or -i, which truly amounts to something like “the field of complex numbers has the field of reals as a subfield and the number i from the field of complex numbers has the property that i^2 = -1 and (-i)^2 = -1”, which does not automatically preclude that some other field or other algebraic structure than the complex numbers has similar properties and provides another set of “roots”.) (b) The discussion of whether e.g. complex numbers and various infinities exist is in so far pointless as we can just abstractly define some set of elements and operations on these elements, use them when and where they happen to be useful, and forget questions like whether e.g. i is something real (non-mathematical sense) and/or something that “belongs” with the real (mathematical sense) numbers. For instance, the field of complex numbers can be quite useful in dealing with, say, calculations on electricity and magnetism, regardless of what nature we consider i to have—and there are fields equivalent to the complex numbers that do not even mention i.