In my quest to read all the pop neuroscience available online, I read this fascinating article on Gerald Edelman. It was full of profound quotes like
We don’t have goals. We just have values.
More importantly, it talked about the concept of polymorphous sets as proposed by Wittgenstein.
“Typical Wittgenstein,” Edelman mused. “There is a kind of ostentation in his modesty. I don’t know what that is. He provokes you and it’s very powerful. It’s ambiguous, sometimes, and it’s not cute. It’s riddle, it’s posturing around the riddle.”
A little girl playing hopscotch, chess players, Swedish sailors doing naval exercises, rugby players are all playing games, Edelman continued. To most observers, these phenomena seem to have little or nothing to do with each other, and yet they are all members of the set of possible games.
“This defines what is known in the business as a polymorphous set. It’s a very hard thing. It means a set defined by neither necessary nor sufficient conditions.”
This is something I’ve read about before, and it didn’t strike me that hard before, as it did on reading this article. Most “things” that we talk about in life are polymorphous sets. There is no necessary and sufficient condition for a fruit to be an apple. Does it have to be red? No it can be green. Does it have to be sweet? No it can be sour. Does it have to be of a particular size? No, if you saw a freakishly small apple, you’d still agree that it is an apple.
Why teaching machines how to identify objects is hard
A neural network can be taught to recognize a cat if it is fed millions and millions of images of cats, and asked to choose suitable parameters so that it is able to recognize whether a particular object is a cat or not. I, on the other hand, learned to recognize cats by looking at one cat. Maybe a couple more. And this is true for almost every other human I know. We haven’t seen all cats in the world. However, if you show me an animal, I can almost always tell you right away if it is a cat or not. Does that mean we might have as much computational power as an actual neural network? Not even close.
The fact that we are able to recognize cats after being shown just a handful of cats follows rom the fact that when we see a cat, we automatically form a polymorphous set of cats in our minds. There is no necessary or sufficient condition to be a cat. A cat doesn’t have to be large or small, black or white, agile or lazy. We will just know whether something is a cat when we see one.
This sounds like a spectacularly bad strategy. What if my polymorphous set of cats differs substantially from yours? It is completely possible. Society would stop functioning if our polymorphous sets of various notions were different enough. However, they’re not. Our brains are similar enough that our polymorphous sets are very very similar for most things, if not the same. You and I may have only seen a handful of tigers in our lives, and probably different individual tigers. However, if we’re walking down a road together and see a tiger, we will instantly recognize it as one.
But hang on. Is the animal that we saw really a tiger? What if it is a slightly mutated version of a tiger (it probably is)? Can we still call that a tiger? For instance, humans are mutations of chimps. If humans (mutated chimps) are not the same as other chimps, why are mutated tigers the same as other tigers? Well….you may say that humans have had millions of mutations by now, but that mutated tiger only probably has a small number of mutations. How many mutations does it take for a tiger to stop being a tiger, and become a different species? Well….we don’t know. All we know is that when we see an animal, we will know when it is a tiger and when it is not.
As one can see above, these polymorphous sets that we have formed in our minds are just horrible by design. They’re not precise, and leave way too much ambiguity. But they work. This is because humans are very similar to one another, and form pretty much the same polymorphous sets. Hence, as long as you and I both agree that that particular animal is a tiger, we are fine.
However, machines obviously have very different “minds” as compared to ours. It is not capable of forming polymorphous sets, leave alone polymorphous sets that are the same as ours. Hence, it has to be fed millions of data points for it to have the same sets as our polymorphous sets. This is not really the machine’s fault. We are terribly imprecise in how we name and define things. The machine wasn’t born as a human, and hence does not possess our particular brains that would help it form the same polymorphous sets.
Is there any hope then? Can we ever have machines that would be able to form the same polymorphous sets as humans? One solution is that if we are able to determine all the parameters inside the human brain and pass them on to a machine, we will never have to teach a machine anything ever again. We will possess perfect communication with it. However, this seems like something that can only be realized in the distant future. Maybe we could also somehow communicate our polymorphous sets’ forming apparatus to machines? Again, I am not aware of any research happening in this direction.
Mathematics aims to make things precise. It does so by doing away with the concept of polymorphous sets.
For example, what is a measurable set? It is a set with the property that for any other set , we have the property that . Of course one would have to define the concept of outer measure before describing this. The point of this definition is that it helps us capture an important property of measurable sets, and explicitly tells us whether a set under consideration is measurable or not. Hence, there is a necessary and sufficient criterion for a set to be a member of the class of measurable sets.
Obviously, no one really started with this definition. Here is a possible development of the concept: do you see this nice set? The unit interval? This has some reasonable properties. What’s crazy is that very broken sets like also have these reasonable properties! Alright, so our intuition tells us that very very broken sets can also have all of these properties that we’re talking about. Then someone comes up with the Vitali set. This very broken set clearly does not have the reasonable sounding property you were talking about!
Do we now see that prevents us from forming a polymorphous set of measurable sets? What forces us to come up with precise and bothersome definitions instead? When we form polymorphous sets, we merely attach a nebulous description. We don’t specify any property that the elements of a polymorphous set have to satisfy. For example, the polymorphous set of cats consists of animals that are “feline” in some vaguely defined sense. We don’t say that the polymorphous set of cats has the property that cats are able to jump 1 ft in the air vertically. If we do so, then we are prescribing a necessary condition to the polymorphous set.
In other words, definitions are necessary (and sufficient) conditions. If is a measurable set, then it should satisfy these nice properties. If it does not, then it cannot be a member of the class of measurable sets.
As Mathematics is built on definitions, which are necessary and sufficient conditions of membership to a class, it cannot have polymorphous sets. But why not? Why does this concept work in the real world, and not in Mathematics? This is because in the real world we want to just describe something, while in Mathematics we want to derive useful properties of things. For instance, in the real world, when we try to classify animals as cats or not, we don’t want to understand the properties of cats (that they should be able to jump a certain height, or weigh this much, etc). However, in mathematics, when we talk about measurable sets, we want to be able to derive useful properties that they possess (that it can be approximated by an open set, that its measure is countably additive, etc). Prescribing any useful properties leads one to form a necessary and sufficient condition for membership in a set, that does away with polymorphous sets and leads to precise definitions.
Hence, although there is perhaps some value in having polymorphous sets of concepts in Mathematics, when it comes down to deriving interesting properties, the vicious cycle of properties-> necessary and sufficient conditions for membership-> non-intuitive definitions seems inevitable. What is slightly tragic about the way that Mathematics is generally taught at the college level and beyond is that this sequence of events is not made explicit. We’re not told that we want our objects to have these nice properties, and hence we need this long winding definitions. We start with the definition, and are then told that these very nice and non-intuitive definitions miraculously lead to these nice properties.
Perhaps in a future post, I can explain how polymorphous sets can still be useful in developing mathematical intuition. However, it seems irrefutable that precise definitions are absolutely integral to mathematical consistency.
Thanks for reading!