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.

 

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Those Who Walk Away from Omelas

In The Brothers Karamazov, after numerous examples of the torture of children and other horrors, Ivan Karamazov rejects theodicy with this argument:

“Besides, too high a price is asked for harmony; it’s beyond our means to pay so much to enter on it. And so I hasten to give back my entrance ticket, and if I am an honest man I am bound to give it back as soon as possible. And that I am doing. It’s not God that I don’t accept, Alyosha, only I most respectfully return him the ticket.”

“That’s rebellion,” murmured Alyosha, looking down.

“Rebellion? I am sorry you call it that,” said Ivan earnestly. “One can hardly live in rebellion, and I want to live. Tell me yourself, I challenge your answer. Imagine that you are creating a fabric of human destiny with the object of making men happy in the end, giving them peace and rest at last, but that it was essential and inevitable to torture to death only one tiny creature — that baby beating its breast with its fist, for instance — and to found that edifice on its unavenged tears, would you consent to be the architect on those conditions? Tell me, and tell the truth.”

“No, I wouldn’t consent,” said Alyosha softly.

Ivan’s argument is that a decent human being would not be willing to bring good out of evil in the particular way that happens in the universe, and therefore much less should a good God be willing to do that.

I will leave aside the theological argument for the moment, although it is certainly worthy of discussion.

Ursula Le Guin wrote a short story or thought experiment about this situation called The Ones Who Walk Away From Omelas. There is supposedly a perfectly happy society, but it all depends on the torture of a single child. Everybody knows about this, and at a certain age they are brought to see the child. Two very different responses to this are described:

The terms are strict and absolute; there may not even be a kind word spoken to the child.

Often the young people go home in tears, or in a tearless rage, when they have seen the child and faced this terrible paradox. They may brood over it for weeks or years. But as time goes on they begin to realize that even if the child could be released, it would not get much good of its freedom: a little vague pleasure of warmth and food, no doubt, but little more. It is too degraded and imbecile to know any real joy. It has been afraid too long ever to be free of fear. Its habits are too uncouth for it to respond to humane treatment. Indeed, after so long it would probably be wretched without walls about it to protect it, and darkness for its eyes, and its own excrement to sit in. Their tears at the bitter injustice dry when they begin to perceive the terrible justice of reality, and to accept it. Yet it is their tears and anger, the trying of their generosity and the acceptance of their helplessness, which are perhaps the true source of the splendor of their lives. Theirs is no vapid, irresponsible happiness. They know that they, like the child, are not free. They know compassion. It is the existence of the child, and their knowledge of its existence, that makes possible the nobility of their architecture, the poignancy of their music, the profundity of their science. It is because of the child that they are so gentle with children. They know that if the wretched one were not there snivelling in the dark, the other one, the flute-player, could make no joyful music as the young riders line up in their beauty for the race in the sunlight of the first morning of summer.

Now do you believe in them? Are they not more credible? But there is one more thing to tell, and this is quite incredible.

At times one of the adolescent girls or boys who go to see the child does not go home to weep or rage, does not, in fact, go home at all. Sometimes also a man or woman much older falls silent for a day or two, and then leaves home. These people go out into the street, and walk down the street alone. They keep walking, and walk straight out of the city of Omelas, through the beautiful gates. They keep walking across the farmlands of Omelas. Each one goes alone, youth or girl man or woman. Night falls; the traveler must pass down village streets, between the houses with yellow-lit windows, and on out into the darkness of the fields. Each alone, they go west or north, towards the mountains. They go on. They leave Omelas, they walk ahead into the darkness, and they do not come back. The place they go towards is a place even less imaginable to most of us than the city of happiness. I cannot describe it at all. It is possible that it does not exist. But they seem to know where they are going, the ones who walk away from Omelas.

Some would argue that the ones who walk away are simply confused. In the real world we are constantly permitting evils for the sake of other goods, and as a whole the evils included here are much greater than the torture of a single child. So Omelas should actually be much better and much more acceptable than the real world.

This response however is mistaken, because the real issue is one about the moral object. It is not enough to say that the good outweighs the evil, because a case of doing evil for the sake of good remains a case of doing evil. This is a little more confusing in the story, where one could interpret the actions of those who stay to be merely negative: they are not the ones who brought the situation about or maintain it. But in Ivan’s example, the question is whether you are willing to torture a child for the sake of the universal harmony, and Ivan’s implication is that if there is to be a universal harmony, God must be willing to torture people, and in general to cause all the evils of the world, to bring it about.

In any case, whether people are right or wrong about what they do, it is certainly true that we are much more willing to permit evils in a vague and general way to bring about good, than we are to produce evils in a very direct way to bring about good.

Not All Things are Water

The basic point of the post on Thales was that material things are composed of one or more material elements. The ancient materialists were accustomed not only to maintain this position, but also to assert that it also followed that there was nothing but those elements. Thus Democritus is said to have said,

By convention, sweet; by convention, bitter; by convention, hot; by convention,
cold; by convention, color; but in reality, atoms and void.

One could say that this is an example of the attitude of “this or nothing.” If everything is made of atoms, then either a thing is atoms, or it is nothing.

The correct answer here is not to say “this or that,” but “this and that,” that is, that things made of atoms are in some way atoms, but they are also things made of them, and the things made of them are not merely atoms.

 

More on Knowing and Being

I promised some examples of the point made in the previous post. I will give just a few here, although the point could easily be extended to many more.

Parmenides argues that nothing can come to be, since “what is not” cannot be or become. He also claims that “it is the same thing that can be thought and that can be,” and apparently this is intended to cover not only what is, but also the way that it is. Consequently, his position seems to imply a perfect identity between thought and being, even if it is ultimately inconsistent, since he says that human beings are wrong about change and the like, and this implies a discrepancy between thought and being.

Alexander Pruss argues that all words are sharply defined, at least in the mind of God.  He makes the argument, “Words are part of the world, so if there is vagueness in words, there is vagueness in the world.” This is no different, of course, from arguing that since words are part of reality, and some words are universal, there are universal things. There are universal things, if we mean by that universal terms or concepts, and there are vague things, if we mean by that vague words or concepts. But there are no universal cats or dogs, nor are there vague cats or dogs, despite the words “cat” and “dog” being vague.

C.S. Lewis argues, “Either we can know nothing or thought has reasons only, and no causes.” As I argued in the linked post, reasons in fact are a kind of final cause relative to their consequences, and they do not exclude efficient causes. This case might be somewhat less evident than the two previous cases, but I would argue that the cause of Lewis’s error here is the fact that, as St. Thomas says, the human mind can understand many things at once only by understanding them as one. Consequently, we can understand that an efficient cause can be for the sake of an end, but if the efficient cause and the final cause are presented as simply two causes, without the order that they actually have, they are not intelligible in this way.

These are examples of speculative errors resulting from confusing the mind’s way of knowing with the way that things are. I asserted in the last post, however, that practical errors can also result from this confusion. There is a very fundamental way this can happen: by nature we know things only if they have some relation to ourselves. The corresponding practical error would be to suppose that those things are real and important only in relation to ourselves. Look around you, and it appears that the world is centered on you. If you take this appearance and attribute an absolute truth to it, you will conclude that everything else has its being and importance in relation to you. Consider that you exist, and that all of the past has past out of existence. It might seem that the past only existed to bring you about.

St. Therese says about humility, “To me it seems that humility is truth. I do not know whether I am humble, but I do know that I see the truth in all things.” This is related to the examples I gave above. Since we know things in relation to ourselves, there is the temptation to suppose that things exist in the very same way. This leads to a false idea about our place in reality. Humility consists, on the contrary, in the truth about our place in reality, as I noted here.

The Cave

Book VII of Plato’s Republic begins with this conversation between Socrates and Glaucon:

And now, I said, let me show in a figure how far our nature is enlightened or unenlightened: –Behold! human beings living in a underground den, which has a mouth open towards the light and reaching all along the den; here they have been from their childhood, and have their legs and necks chained so that they cannot move, and can only see before them, being prevented by the chains from turning round their heads. Above and behind them a fire is blazing at a distance, and between the fire and the prisoners there is a raised way; and you will see, if you look, a low wall built along the way, like the screen which marionette players have in front of them, over which they show the puppets.

I see.
And do you see, I said, men passing along the wall carrying all sorts of vessels, and statues and figures of animals made of wood and stone and various materials, which appear over the wall? Some of them are talking, others silent.

You have shown me a strange image, and they are strange prisoners.
Like ourselves, I replied; and they see only their own shadows, or the shadows of one another, which the fire throws on the opposite wall of the cave?

True, he said; how could they see anything but the shadows if they were never allowed to move their heads?

And of the objects which are being carried in like manner they would only see the shadows?

Yes, he said.
And if they were able to converse with one another, would they not suppose that they were naming what was actually before them?

Very true.
And suppose further that the prison had an echo which came from the other side, would they not be sure to fancy when one of the passers-by spoke that the voice which they heard came from the passing shadow?

No question, he replied.
To them, I said, the truth would be literally nothing but the shadows of the images.

That is certain.

The human situation with respect to truth is somewhat like Plato’s cave dwellers. I argued in the previous post that all human knowledge is imperfect and vague. A more precise argument to the same effect would be the following.

There are two basic ways that we can learn the meaning of words. In one way, by having the meaning of the word explained to us. This requires the knowledge of other words. In another way, by learning the contexts in which the word is used. This does not necessarily require the previous knowledge of other words. Thus for example we learn the meaning of the word “red” by seeing it used in reference to red things, without a necessity of explaining the idea of red using other words. Third, we might learn the meaning of some words by a combination of the above methods.

Defining a word by other words cannot lead to an absolutely precise meaning unless the other words that are used have an absolutely precise meaning, so such a definition cannot be a source of absolute precision. And unless one points to all possible cases in which a word might be used, which is a physical impossibility, experiencing the contexts in which a word is used will also never lead to an absolutely precise meaning. It is consequently unlikely that you can arrive at an absolutely precise meaning by any combination of these methods, and consequently unlikely that you can arrive at such absolute precision in any way.

Language and thought are very closely connected. There are perhaps not many real cases of feral children, but the experience we do have of such situations suggests that someone who does not learn a language is basically unable to think. This is perhaps not true absolutely, but it is surely true that such a child cannot think clearly. Clear thinking requires language. This at least suggests that perfectly clear thinking would require perfectly clear language. But this is impossible, by the above argument. Consequently perfectly clear thinking is impossible for human beings.

This argument does not refute itself. It it of course impossible to have a proof that there are no proofs. Likewise, if absolute subjective certainty is impossible, as I have suggested, then it is impossible to have such certainty about this very fact. There is no inconsistency here. Likewise, if all thoughts and all arguments are vague, then these very thoughts and this very argument is vague. There is no inconsistency here.

Vagueness

Vagueness comes in various kinds. In the first place, everything that human beings think or say is vague in principle, because of the weakness of human understanding. Our understanding is to the fullness of reality “as the eyes of bats are to the blaze of day,” and this weakness of our understanding is found within everything we think or say. Discussing why there is something rather than nothing, we saw that there must be a being which is necessary in itself. But even after making this argument, this does not become evident to us in itself, and this is because we do not know the nature of being.

Considering whether names said of God belong primarily to him or to us, St. Thomas says,

I answer that, In names predicated of many in an analogical sense, all are predicated because they have reference to some one thing; and this one thing must be placed in the definition of them all. And since that expressed by the name is the definition, as the Philosopher says (Metaph. iv), such a name must be applied primarily to that which is put in the definition of such other things, and secondarily to these others according as they approach more or less to that first. Thus, for instance, “healthy” applied to animals comes into the definition of “healthy” applied to medicine, which is called healthy as being the cause of health in the animal; and also into the definition of “healthy” which is applied to urine, which is called healthy in so far as it is the sign of the animal’s health. Thus all names applied metaphorically to God, are applied to creatures primarily rather than to God, because when said of God they mean only similitudes to such creatures. For as “smiling” applied to a field means only that the field in the beauty of its flowering is like the beauty of the human smile by proportionate likeness, so the name of “lion” applied to God means only that God manifests strength in His works, as a lion in his. Thus it is clear that applied to God the signification of names can be defined only from what is said of creatures. But to other names not applied to God in a metaphorical sense, the same rule would apply if they were spoken of God as the cause only, as some have supposed. For when it is said, “God is good,” it would then only mean “God is the cause of the creature’s goodness”; thus the term good applied to God would included in its meaning the creature’s goodness. Hence “good” would apply primarily to creatures rather than to God. But as was shown above, these names are applied to God not as the cause only, but also essentially. For the words, “God is good,” or “wise,” signify not only that He is the cause of wisdom or goodness, but that these exist in Him in a more excellent way. Hence as regards what the name signifies, these names are applied primarily to God rather than to creatures, because these perfections flow from God to creatures; but as regards the imposition of the names, they are primarily applied by us to creatures which we know first. Hence they have a mode of signification which belongs to creatures, as said above.

St. Thomas does not say it explicitly, but the principle he presents here, “this one thing must be placed in the definition of them all,” implies that according to his argument, God is contained in the definitions of creatures. And in this way the being which is necessary in itself must be in the definition of every being. Consequently our failure to understand the nature of being in itself implies a failure to fully understand the nature of any being.

Second, someone can say something vague for the sake of accuracy, namely because he knows that his knowledge is vague, and he wishes to express his knowledge as it is, rather than claiming to know more than he does. Freeman Dyson and others say that “it is better to be wrong than vague,” but they are, strictly speaking, wrong about this. It is better to be vague and right, rather than clear and distinct, but wrong.

Third, someone can say something vague because he does not primarily care about whether or not it is true, but about something else. In this case it may be left vague because there is no need for the effort that it would take to make it clear, since the person’s purposes can be achieved without that effort, or it may be left vague because those purposes are achieved even better when it remains vague. Thus there is a story about Hegel, which I was unable to track down while writing this post, which says that he explained to someone that the Phenomenology of Spirit had to be extremely difficult to understand, in order to ensure that Hegel would become famous. I do not find the story particularly credible, but I do find the motive credible.

I said “strictly speaking” above in discussing people who say that it is better to be wrong than vague, because in many cases such people are actually opposing the third kind of vagueness, and are simply intending to say that it is better to care about the truth but to make a mistake, than not to care about the truth at all, and therefore not to bother to say something which could turn out to be wrong.