Picture of a squirrel on a fence

Ineffable facts, deep ignorance, and the sub-algebra hypothesis: Part 2

This post originally appeared on OUPblog and appears here as part of our partnership with them.

by Thomas Hofweber

If there are any ineffable facts, then it is striking that they essentially are nowhere to be found. It is natural to think of ineffable facts as rare, radical exceptions, something unusual, maybe something tied to consciousness, or art, or paradoxes. But why would that be justified? There might be lots of ineffable fact, and they might be all around us. The natural motivation for why we should think that there are any ineffable facts would seem to support this more ubiquitous conception of them. The squirrel is surrounded by facts ineffable for it. It even might live on the window sill of the Athens unemployment office, but the fact that there is an economic crisis in Greece is beyond what it can represent. It would be deeply ignorant of facts pertaining to where it lives and what is going on right around it. And similarly, we might be in the middle of something that we are deeply ignorant of and which is beyond what we can represent. But if so, this somehow doesn’t seem to come up much, or matter.

We don’t normally have to face that we are limited in what we can represent, and how this limitation is holding us back. We don’t normally encounter such a limitation in a way we would recognize. For all practical and theoretical purposes it seems that we can represent all there is to represent, and that our ignorance is never deep. But this should be surprising, if we accept that some facts likely are ineffable for us. That a fact is ineffable only means that we can’t represent it in thought or language, it doesn’t mean that we can’t encounter it in other ways. We might even see one, and at the same time realize that what we are seeing is so weird and different that our concepts and language give out when trying to represent it. We might run into such facts in ordinary life or in scientific endeavors, not just once, but all the time. But neither one of these seems to happen, at least not with the frequency and significance that one might expect. The ineffable, if it is there, seems to be more systematically hidden than should be expected. Maybe we got lucky after all then, and we can represent all there is to represent. But we shouldn’t count on such luck. There is also another hypothesis, one that accepts that there are ineffable facts, but explains why they are systematically hidden from us. I would like to suggest that we should seriously entertain this hypothesis, and that it is to be favored over having gotten lucky. It is the “sub-algebra hypothesis.”

To illustrate, consider the world from the point of view of an integer, that is, one of the (positive or negative) whole numbers:


Let’s assume that the integers can talk about each other, and they can think about addition, subtraction and multiplication. The integers so animated could ask various questions. They can ask what 7 minus 4 is, or what 8 times 3 is. And in each case they can state the answer. They can talk about all the integers, and for each question they can ask, the answer is always an integer. The sum or product of any two integers is always another integer. It would seem natural to the integers that the integers is all there is to the world. And this would be so even if the world they live in is much richer than that. The world might consist of not just the integers, but the rational numbers, that is, all the fractions of integers:


The integers are spread out among the rational numbers, with infinitely many rational numbers between any two integers. But these other rational numbers would be completely hidden from the integers. With their language and thoughts they can never reach them. If they could only talk about one more number, for example 1/2, then they could reach more, but still not everything. If they only had one more concept — division — then they could reach everything. But the integers don’t have division or a name for 1/2, and so they will think that all there is to the world is what they can talk about: the integers under addition, subtraction and multiplication.

The reason why the rest of the world is so systematically hidden from the integers is that the integers with addition, subtraction and multiplication form a sub-structure, or sub-algebra, of the rational numbers with those operation as well as division. Adding, subtracting or multiplying two integers always leads back to an integer, and so the integers are closed under these operations. Even though the integers are surrounded by other rational number, which get arbitrarily close to them, the integers are completely unaware of how rich the world is that they live in. All this extra structure and further glory of the world is completely and systematically hidden from them.

I would like to suggest that this might in essence be our situation. We might be like the integers, surrounded by facts that go beyond what we can think or say, while at the same time it seems to us that we can represent all there is to represent, and that all of our ignorance is just ordinary ignorance, but never deep ignorance. The facts we can represent might form a sub-algebra of all the facts that obtain, and it might just be a small one.

The sub-algebra hypothesis would, in outline, make sense of that, on the one hand, we are limited in what we can represent about the world, while, on the other hand, ineffable facts seem to be completely hidden and irrelevant for inquiry and other activities. Our sub-algebra might be closed under causal and explanatory relationships, which is to say that if we can represent an event then we can also represent its cause, and if we can represent a fact then we can also represent its explanation. In ordinary inquiry, where we ask for causes and explanations, we will thus never have to go beyond our sub-algebra. But when we try to understand the world as a whole, then our representational limitation might well mislead us into thinking that the part of reality we can represent is all there is to reality. Just as the integers might think that all there is to reality are the integers, so we might think that all of reality is just the part we can represent. Reality might be much richer than what we can represent about it, but the rest of it would be completely hidden from us, and irrelevant for our ordinary and even regular scientific concerns. Except, of course, for the question what reality as a whole is like. For that question and its consequences our deep ignorance will matter.

Thomas Hofweber is professor of philosophy at the University of North Carolina at Chapel Hill. His research specializes in metaphysics and the philosophy of language. He studied for his undergraduate degree at the University of Munich, before completing his PhD at Stanford University. Before moving to North Carolina, he taught at the University of Michigan, Ann Arbor. He is the author most recently of Ontology and the Ambitions of Metaphysics.


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