Self Reference Paradox Summarized

Hilary Lawson is right to connect the issue of the completeness and consistency of truth with paradoxes of self-reference.

As a kind of summary, consider this story:

It was a dark and stormy night,
and all the Cub Scouts where huddled around their campfire.
One scout looked up to the Scout Master and said:
“Tell us a story.”
And the story went like this:

It was a dark and stormy night,
and all the Cub Scouts where huddled around their campfire.
One scout looked up to the Scout Master and said:
“Tell us a story.”
And the story went like this:

It was a dark and stormy night,
and all the Cub Scouts where huddled around their campfire.
One scout looked up to the Scout Master and said:
“Tell us a story.”
And the story went like this:

It was a dark and stormy night,
and all the Cub Scouts where huddled around their campfire.
One scout looked up to the Scout Master and said:
“Tell us a story.”
And the story went like this:
etc.

In this form, the story obviously exists, but in its implied form, the story cannot be told, because for the story to be “told” is for it to be completed, and it is impossible for it be completed, since it will not be complete until it contains itself, and this cannot happen.

Consider a similar example. You sit in a room at a desk, and decide to draw a picture of the room. You draw the walls. Then you draw yourself and your desk. But then you realize, “there is also a picture in the room. I need to draw the picture.” You draw the picture itself as a tiny image within the image of your desktop, and add tiny details: the walls of the room, your desk and yourself.

Of course, you then realize that your artwork can never be complete, in exactly the same way that the story above cannot be complete.

There is essentially the same problem in these situations as in all the situations we have described which involve self-reference: the paradox of the liar, the liar game, the impossibility of detailed future prediction, the list of all true statementsGödel’s theorem, and so on.

In two of the above posts, namely on future prediction and Gödel’s theorem, there are discussions of James Chastek’s attempts to use the issue of self-reference to prove that the human mind is not a “mechanism.” I noted in those places that such supposed proofs fail, and at this point it is easy to see that they will fail in general, if they depend on such reasoning. What is possible or impossible here has nothing to do with such things, and everything to do with self-reference. You cannot have a mirror and a camera so perfect that you can get an actually infinite series of images by taking a picture of the mirror with the camera, but there is nothing about such a situation that could not be captured by an image outside the situation, just as a man outside the room could draw everything in the room, including the picture and its details. This does not show that a man outside the room has a superior drawing ability compared with the man in the room. The ability of someone else to say whether the third statement in the liar game is true or false does not prove that the other person does not have a “merely human” mind (analogous to a mere mechanism), despite the fact that you yourself cannot say whether it is true or false.

There is a grain of truth in Chastek’s argument, however. It does follow that if someone says that reality as a whole is a formal system, and adds that we can know what that system is, their position would be absurd, since if we knew such a system we could indeed derive a specific arithmetical truth, namely one that we could state in detail, which would be unprovable from the system, namely from reality, but nonetheless proved to be true by us. And this is logically impossible, since we are a part of reality.

At this point one might be tempted to say, “At this point we have fully understood the situation. So all of these paradoxes and so on don’t prevent us from understanding reality perfectly, even if that was the original appearance.”

But this is similar to one of two things.

First, a man can stand outside the room and draw a picture of everything in it, including the picture, and say, “Behold. A picture of the room and everything in it.” Yes, as long as you are not in the room. But if the room is all of reality, you cannot get outside it, and so you cannot draw such a picture.

Second, the man in the room can draw the room, the desk and himself, and draw a smudge on the center of the picture of the desk, and say, “Behold. A smudged drawing of the room and everything in it, including the drawing.” But one only imagines a picture of the drawing underneath the smudge: there is actually no such drawing in the picture of the room, nor can there be.

In the same way, we can fully understand some local situation, from outside that situation, or we can have a smudged understanding of the whole situation, but there cannot be any detailed understanding of the whole situation underneath the smudge.

I noted that I disagreed with Lawson’s attempt to resolve the question of truth. I did not go into detail, and I will not, as the book is very long and an adequate discussion would be much longer than I am willing to attempt, at least at this time, but I will give some general remarks. He sees, correctly, that there are problems both with saying that “truth exists” and that “truth does not exist,” taken according to the usual concept of truth, but in the end his position amounts to saying that the denial of truth is truer than the affirmation of truth. This seems absurd, and it is, but not quite so much as appears, because he does recognize the incoherence and makes an attempt to get around it. The way of thinking is something like this: we need to avoid the concept of truth. But this means we also need to avoid the concept of asserting something, because if you assert something, you are saying that it is true. So he needs to say, “assertion does not exist,” but without asserting it. Consequently he comes up with the concept of “closure,” which is meant to replace the concept of asserting, and “asserts” things in the new sense. This sense is not intended to assert anything at all in the usual sense. In fact, he concludes that language does not refer to the world at all.

Apart from the evident absurdity, exacerbated by my own realist description of his position, we can see from the general account of self-reference why this is the wrong answer. The man in the room might start out wanting to draw a picture of the room and everything in it, and then come to realize that this project is impossible, at least for someone in his situation. But suppose he concludes: “After all, there is no such thing as a picture. I thought pictures were possible, but they are not. There are just marks on paper.” The conclusion is obviously wrong. The fact that pictures are things themselves does prevent pictures from being exhaustive pictures of themselves, but it does not prevent them from being pictures in general. And in the same way, the fact that we are part of reality prevents us from having an exhaustive understanding of reality, but it does not prevent us from understanding in general.

There is one last temptation in addition to the two ways discussed above of saying that there can be an exhaustive drawing of the room and the picture. The room itself and everything in it, is itself an exhaustive representation of itself and everything in it, someone might say. Apart from being an abuse of the word “representation,” I think this is delusional, but this a story for another time.

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Being and Unity II

Content warning: very obscure.

This post follows up on an earlier post on this topic, as well on what was recently said about real distinction. In the latter post, we applied the distinction between the way a thing is and the way it is known in order to better understand distinction itself. We can obtain a better understanding of unity in a similar way.

As was said in the earlier post on unity, to say that something is “one” does not add anything real to the being of the thing, but it adds the denial of the division between distinct things. The single apple is not “an apple and an orange,” which are divided insofar as they are distinct from one another.

But being distinct from divided things is itself a certain way of being distinct, and consequently all that was said about distinction in general will apply to this way of being distinct as well. In particular, since being distinct means not being something, which is a way that things are understood rather than a way that they are (considered precisely as a way of being), the same thing applies to unity. To say that something is one does not add something to the way that it is, but it adds something to the way that it is understood. This way of being understood is founded, we argued, on existing relationships.

We should avoid two errors here, both of which would be expressions of the Kantian error:

First, the argument here does not mean that a thing is not truly one thing, just as the earlier discussion does not imply that it is false that a chair is not a desk. On the contrary, a chair is in fact not a desk, and a chair is in fact one chair. But when we say or think, “a chair is not a desk,” or “a chair is one chair,” we are saying these things in some way of saying, and thinking them in some way of thinking, and these ways of saying and thinking are not ways of being as such. This in no way implies that the statements themselves are false, just as “the apple seems to be red,” does not imply that the apple is not red. Arguing that the fact of a specific way of understanding implies that the thing is falsely understood would be the position described by Ayn Rand as asserting, “man is blind, because he has eyes—deaf, because he has ears—deluded, because he has a mind—and the things he perceives do not exist, because he perceives them.”

Second, the argument does not imply that the way things really are is unknown and inaccessible to us. One might suppose that this follows, since distinction cannot exist apart from someone’s way of understanding, and at the same time no one can understand without making distinctions. Consequently, someone might argue, there must be some “way things really are in themselves,” which does not include distinction or unity, but which cannot be understood. But this is just a different way of falling into the first error above. There is indeed a way things are, and it is generally not inaccessible to us. In fact, as I pointed out earlier, it would be a contradiction to assert the existence of anything entirely unknowable to us.

Our discussion, being in human language and human thought, naturally uses the proper modes of language and thought. And just as in Mary’s room, where her former knowledge of color is a way of knowing and not a way of sensing, so our discussion advances by ways of discussion, not by ways of being as such. This does not prevent the way things are from being an object of discussion, just as color can be an object of knowledge.

Having avoided these errors, someone might say that nothing of consequence follows from this account. But this would be a mistake. It follows from the present account that when we ask questions like, “How many things are here?”, we are not asking a question purely about how things are, but to some extent about how we should understand them. And even when there is a single way that things are, there is usually not only one way to understand them correctly, but many ways.

Consider some particular question of this kind: “How many things are in this room?” People might answer this question in various ways. John Nerst, in a previous discussion on this blog, seemed to suggest that the answer should be found by counting fundamental particles. Alexander Pruss would give a more complicated answer, since he suggests that large objects like humans and animals should be counted as wholes (while also wishing to deny the existence of parts, which would actually eliminate the notion of a whole), while in other cases he might agree to counting particles. Thus a human being and an armchair might be counted, more or less, as 1 + 10^28 things, namely counting the human being as one thing and the chair as a number of particles.

But if we understand that the question is not, and cannot be, purely about how things are, but is also a question about how things should be understood, then both of the above responses seem unreasonable: they are both relatively bad ways of understanding the things in the room, even if they both have some truth as well. And on the other hand, it is easy to see that “it depends on how you count,” is part of the answer. There is not one true answer to the question, but many true answers that touch on different aspects of the reality in the room.

From the discussion with John Nerst, consider this comment:

My central contention is that the rules that define the universe runs by themselves, and must therefore be self-contained, i.e not need any interpretation or operationalization from outside the system. As I think I said in one of the parts of “Erisology of Self and Will” that the universe must be an automaton, or controlled by an automaton, etc. Formal rules at the bottom.

This is isn’t convincing to you I guess but I suppose I rule out fundamental vagueness because vagueness implies complexity and fundamental complexity is a contradiction in terms. If you keep zooming in on a fuzzy picture you must, at some point, come down to sharply delineated pixels.

Among other things, the argument of the present post shows why this cannot be right. “Sharply delineated pixels” includes the distinction of one pixel from another, and therefore includes something which is a way of understanding as such, not a way of being as such. In other words, while intending to find what is really there, apart from any interpretation, Nerst is directly including a human interpretation in his account. And in fact it is perfectly obvious that anything else is impossible, since any account of reality given by us will be a human account and will thus include a human way of understanding. Things are a certain way: but that way cannot be said or thought except by using ways of speaking or thinking.

Idealized Idealization

On another occasion, I discussed the Aristotelian idea that the act of the mind does not use an organ. In an essay entitled Immaterial Aspects of Thought, James Ross claims that he can establish the truth of this position definitively. He summarizes the argument:

Some thinking (judgment) is determinate in a way no physical process can be. Consequently, such thinking cannot be (wholly) a physical process. If all thinking, all judgment, is determinate in that way, no physical process can be (the whole of) any judgment at all. Furthermore, “functions” among physical states cannot be determinate enough to be such judgments, either. Hence some judgments can be neither wholly physical processes nor wholly functions among physical processes.

Certain thinking, in a single case, is of a definite abstract form (e.g. N x N = N²), and not indeterminate among incompossible forms (see I below). No physical process can be that definite in its form in a single case. Adding cases even to infinity, unless they are all the possible cases, will not exclude incompossible forms. But supplying all possible cases of any pure function is impossible. So, no physical process can exclude incompossible functions from being equally well (or badly) satisfied (see II below). Thus, no physical process can be a case of such thinking. The same holds for functions among physical states (see IV below).

In essence, the argument is that squaring a number and similar things are infinitely precise processes, and no physical process is infinitely precise. Therefore squaring a number and similar things are not physical processes.

The problem is unfortunately with the major premise here. Squaring a number, and similar things, in the way that we in fact do them, are not infinitely precise processes.

Ross argues that they must be:

Can judgments really be of such definite “pure” forms? They have to be; otherwise, they will fail to have the features we attribute to them and upon which the truth of certain judgments about validity, inconsistency, and truth depend; for instance, they have to exclude incompossible forms or they would lack the very features we take to be definitive of their sorts: e.g., conjunction, disjunction, syllogistic, modus ponens, etc. The single case of thinking has to be of an abstract “form” (a “pure” function) that is not indeterminate among incompossible ones. For instance, if I square a number–not just happen in the course of adding to write down a sum that is a square, but if I actually square the number–I think in the form “N x N = N².”

The same point again. I can reason in the form, modus ponens (“If p then q“; “p“; “therefore, q”). Reasoning by modus ponens requires that no incompossible forms also be “realized” (in the same sense) by what I have done. Reasoning in that form is thinking in a way that is truth-preserving for all cases that realize the form. What is done cannot, therefore, be indeterminate among structures, some of which are not truth preserving. That is why valid reasoning cannot be only an approximation of the form, but must be of the form. Otherwise, it will as much fail to be truth-preserving for all relevant cases as it succeeds; and thus the whole point of validity will be lost. Thus, we already know that the evasion, “We do not really conjoin, add, or do modus ponens but only simulate them,” cannot be correct. Still, I shall consider it fully below.

“It will as much fail to be truth-preserving for all relevant cases as it succeeds” is an exaggeration here. If you perform an operation which approximates modus ponens, then that operation will be approximately truth preserving. It will not be equally truth preserving and not truth preserving.

I have noted many times in the past, as for example here, here, here, and especially here, that following the rules of syllogism does not in practice infallibly guarantee that your conclusions are true, even if your premises are in some way true, because of the vagueness of human thought and language. In essence, Ross is making a contrary argument: we know, he is claiming, that our arguments infallibly succeed; therefore our thoughts cannot be vague. But it is empirically false that our arguments infallibly succeed, so the argument is mistaken right from its starting point.

There is also a strawmanning of the opposing position here insofar as Ross describes those who disagree with him as saying that “we do not really conjoin, add, or do modus ponens but only simulate them.” This assumes that unless you are doing these things perfectly, rather than approximating them, then you are not doing them at all. But this does not follow. Consider a triangle drawn on a blackboard. Consider which of the following statements is true:

  1. There is a triangle drawn on the blackboard.
  2. There is no triangle drawn on the blackboard.

Obviously, the first statement is true, and the second false. But in Ross’s way of thinking, we would have to say, “What is on the blackboard is only approximately triangular, not exactly triangular. Therefore there is no triangle on the blackboard.” This of course is wrong, and his description of the opposing position is wrong in the same way.

Naturally, if we take “triangle” as shorthand for “exact rather than approximate triangle” then (2) will be true. And in a similar way, if take “really conjoin” and so on as shorthand for “really conjoin exactly and not approximately,” then those who disagree will indeed say that we do not do those things. But this is not a problem unless you are assuming from the beginning that our thoughts are infinitely precise, and Ross is attempting to establish that this must be the case, rather than claiming to take it as given. (That is, the summary takes it as given, but Ross attempts throughout the article to establish it.)

One could attempt to defend Ross’s position as follows: we must have infinitely precise thoughts, because we can understand the words “infinitely precise thoughts.” Or in the case of modus ponens, we must have an infinitely precise understanding of it, because we can distinguish between “modus ponens, precisely,” and “approximations of modus ponens“. But the error here is similar to the error of saying that one must have infinite certainty about some things, because otherwise one will not have infinite certainty about the fact that one does not have infinite certainty, as though this were a contradiction. It is no contradiction for all of your thoughts to be fallible, including this one, and it is no contradiction for all of your thoughts to be vague, including your thoughts about precision and approximation.

The title of this post in fact refers to this error, which is probably the fundamental problem in Ross’s argument. Triangles in the real world are not perfectly triangular, but we have an idealized concept of a triangle. In precisely the same way, the process of idealization in the real world is not an infinitely precise process, but we have an idealized concept of idealization. Concluding that our acts of idealization must actually be ideal in themselves, simply because we have an idealized concept of idealization, would be a case of confusing the way of knowing with the way of being. It is a particularly confusing case simply because the way of knowing in this case is also materially the being which is known. But this material identity does not make the mode of knowing into the mode of being.

We should consider also Ross’s minor premise, that a physical process cannot be determinate in the way required:

Whatever the discriminable features of a physical process may be, there will always be a pair of incompatible predicates, each as empirically adequate as the other, to name a function the exhibited data or process “satisfies.” That condition holds for any finite actual “outputs,” no matter how many. That is a feature of physical process itself, of change. There is nothing about a physical process, or any repetitions of it, to block it from being a case of incompossible forms (“functions”), if it could be a case of any pure form at all. That is because the differentiating point, the point where the behavioral outputs diverge to manifest different functions, can lie beyond the actual, even if the actual should be infinite; e.g., it could lie in what the thing would have done, had things been otherwise in certain ways. For instance, if the function is x(*)y = (x + y, if y < 10^40 years, = x + y +1, otherwise), the differentiating output would lie beyond the conjectured life of the universe.

Just as rectangular doors can approximate Euclidean rectangularity, so physical change can simulate pure functions but cannot realize them. For instance, there are no physical features by which an adding machine, whether it is an old mechanical “gear” machine or a hand calculator or a full computer, can exclude its satisfying a function incompatible with addition, say quaddition (cf. Kripke’s definition of the function to show the indeterminacy of the single case: quus, symbolized by the plus sign in a circle, “is defined by: x quus y = x + y, if x, y < 57, =5 otherwise”) modified so that the differentiating outputs (not what constitutes the difference, but what manifests it) lie beyond the lifetime of the machine. The consequence is that a physical process is really indeterminate among incompatible abstract functions.

Extending the list of outputs will not select among incompatible functions whose differentiating “point” lies beyond the lifetime (or performance time) of the machine. That, of course, is not the basis for the indeterminacy; it is just a grue-like illustration. Adding is not a sequence of outputs; it is summing; whereas if the process were quadding, all its outputs would be quadditions, whether or not they differed in quantity from additions (before a differentiating point shows up to make the outputs diverge from sums).

For any outputs to be sums, the machine has to add. But the indeterminacy among incompossible functions is to be found in each single case, and therefore in every case. Thus, the machine never adds.

There is some truth here, and some error here. If we think about a physical process in the particular way that Ross is considering it, it will be true that it will always be able to be interpreted in more than one way. This is why, for example, in my recent discussion with John Nerst, John needed to say that the fundamental cause of things had to be “rules” rather than e.g. fundamental particles. The movement of particles, in itself, could be interpreted in various ways. “Rules,” on the other hand, are presumed to be something which already has a particular interpretation, e.g. adding as opposed to quadding.

On the other hand, there is also an error here. The prima facie sign of this error is the statement that an adding machine “never adds.” Just as according to common sense we can draw triangles on blackboards, so according to common sense the calculator on my desk can certainly add. This is connected with the problem with the entire argument. Since “the calculator can add” is true in some way, there is no particular reason that “we can add” cannot be true in precisely the same way. Ross wishes to argue that we can add in a way that the calculator cannot because, in essence, we do it infallibly; but this is flatly false. We do not do it infallibly.

Considered metaphysically, the problem here is ignorance of the formal cause. If physical processes were entirely formless, they indeed would have no interpretation, just as a formless human (were that possible) would be a philosophical zombie. But in reality there are forms in both cases. In this sense, Ross’s argument comes close to saying “human thought is a form or formed, but physical processes are formless.” Since in fact neither is formless, there is no reason (at least established by this argument) why thought could not be the form of a physical process.

 

Some Complaints about Parts and Wholes

In the comment here, John Nerst effectively rejects the existence of parts and wholes:

In my view, there must be a set of fundamental rules that the universe is running on and fundamental entities that doesn’t reduce to something else, and everything else is simply descriptions of the consequences of those rules. There is a difference between them, what we call it isn’t important. I don’t see how one could disagree with that without going into mystical-idealist territory.

The word “simply” in “simply descriptions of the consequences of those rules” has no plausible meaning except that wholes made out of fundamental particles, as distinct from the fundamental particles, do not exist: what really exists are the fundamental particles, and nothing more.

John denies that he means to reject the common sense idea that wholes exist by his statement:

I do mean different things by “humans exist” and “humans exist in the territory”, and you can’t really tell me what I mean against my saying so. I haven’t asserted that humans don’t exist (it depends on the meaning of “exist”).

But it is not my responsibility to give a plausible true meaning to his statements where I have already considered the matter as carefully as I could, and have found none; I do not see what his claim could mean which does not imply that humans do not exist, and I have explained why his claim would have this implication.

In a similar way, others reject the existence of parts. Thus Alexander Pruss remarks:

Parthood is a mysterious relation. It would really simplify our picture of the world if we could get rid of it.

There are two standard ways of doing this. The microscopic mereological nihilist says that only the fundamental “small” bits—particles, fields, etc.—exist, and that there are no complex objects like tables, trees and people that are made of such bits. (Though one could be a microscopic mereological nihilist dualist, and hold that people are simple souls.)

The macroscopic mereological nihilist says that big things like organisms do exist, but their commonly supposed constituents, such as particles, do not exist, except in a manner of speaking. We can talk as if there were electrons in us, but there are no electrons in us. The typical macroscopic mereological nihilist is a Thomist who talks of “virtual existence” of electrons in us.

Pruss basically agrees with the second position, which he expressed by saying at the end of the post, “But I still like macroscopic nihilism more than reductionism.” In other words, it is given that we have to get rid of parts and wholes; the best way to do that, according to Pruss, is to assert the existence of the things that we call wholes, and to deny the existence of the parts.

In effect, John Nerst says that there are no wholes, but there are fundamental things (such as particles) that have the power to act as if they were wholes (such as humans), even though such wholes do not actually exist, and Alexander Pruss says that there are no parts (such as particles), but there are simple unified things (such as humans) which have the power to act as if they had parts (such as particles), even though they do not actually have such parts.

To which we must respond: a pox on both your houses. In accord with common sense, both wholes and parts exist, and the difficulty of understanding the matter is a weakness of human reason, not a deficiency in reality.