IV. Examples of Higher-order

1. I find that the things I say to the people I talk to normally follow some rules; for example the words I am writing have grammaticality, semantics, and maybe other properties given by respecting sets of rules. But I am not fully conscious of the order or the rules by which I speak about different topics. In other words, I am now talking about, let's say, self-reference, and I do it in a (I hope) meaningful and grammatical manner. I then change the subject and start to talk about what I have done yesterday in a meaningful and grammatical manner. But between the things I talk about there seems to be no set of rules to which they all should subscribe. This set of rules would be of a higher order. An act of speaking may have internal coherence, but all the acts taken as a set have no evident order; an act of speaking may have coherence internally, but no external coherence. I am not sure if it is possible for an act of speaking to have no internal coherence but have external coherence. Also, the rules governing the external coherence of an act of speaking cannot possibly be grammatical; but I am not sure whether they may or may not be semantical.

2. I've said in my first post that (almost all) concepts do not contain their own description. Language is used to describe the world and only exceptionally itself. This is what interests me the most, because I tend to take a description of something, or the sign for something else to be the thing it describes or signifies. There are some things which I am not sure whether I like or I like to like them.

3. Tim Maudlin, Truth and Paradox:

"[...] suppose that human reasoning capacity can be reduced to some sort of algorithmic procedure, such that sentences about, say, arithmetic that one takes to be provably true with certainty can be characterized as a recursively enumerable set of sentences. Godel's procedure then shows how to identify a sentence which, if the system is consistent, is certain to be both true and not identified by the system as a truth. The idea is now this: We can recognize the Godel sentence of the system as true even though the system itself cannot. Therefore our insight into what is certainly true outruns that of the system. But the system has not been characterized in any way except to say that it is consistent and, in some sense, algorithmic. Therefore our insight is, in principle, more extensive than that of any consistent algorithmic system. Therefore, our insight cannot be the result of any consistent algorithmic system. Therefore, the power of our minds cannot be captured by any algorithm. Therefore we are not like computers. Yet further, according to Penrose, it follows that the physics that governs our brains cannot even be computable, otherwise our insights would, in the relevant sense, be the output of an algorithm [...]."

4. M.C. Escher, Waterfall, 1961

This image plays with the dimensions. From a 3D perspective, the shape is impossible, also tridimensionality is only an ilusion here; but from the two-dimensional perspective you are lead to believe that water falls down from a higher spot. So the depiction of the shapes is based on the fact that it is a 2D depiction imitating a 3D perspective.