Philosophy, Film, Politics, Etc.

Tuesday, August 25, 2009

Logic and Reference

I want to better explain why I reject the idea that logic refers to something, such as abstractions or Platonic forms.

Words and sentences, of themselves, do not refer to anything. Rather, people can use words and sentences to refer to things. (This should be clear when we remember that the same words and sentences can refer to different things, depending on the context of utterance.) Furthermore, the meaning of a sentence is not always its referent; for we can understand sentences even when a referent is unspecified, and also in cases where the referent is non-existant. (E.g., "The King of France is bald.") From these points it follows, first, that the referent of a sentence depends on how it is used in a particular context; and, second, that sentences can be meaningful even if they have no known referent.

When we look at the meaning of a syllogism, we may easily find referents. For example,

  1. All men are mortal.
  2. Socrates is a man.
  3. Therefore, Socrates is mortal.

Taken by themselves, each of these sentences can be (and normally would be) used to refer to things, namely men, mortality, and Socrates. However, they can also be used to illustrate a certain logical form. The fact that there are referring terms is incidental. We could easily replace "Socrates" with "Alfred," and the logical validity would remain intact, even if nobody had any idea who (or what) Alfred might be.

The use of these sentences as an illustration of a valid syllogism is not their usual referring use. We can say that the meaning of each sentence, in this illustration, is based on their grammatical form, and not on what extra-linguistic entities they might be used to point to. We most clearly present rules of inference when we use symbols which have no referring use in our common language. We thus can say,

  1. All members of A are members of B.
  2. x is a member of A.
  3. Therefore, x is a member of B.

By taking out words like "men" and "Socrates," we help avoid the confusion of thinking that the meaning of our logical demonstration somehow depended on the referring use of our terms.

Of course, some students of logic might ask, "What does 'x' refer to?"

The correct answer is, nothing. The letter 'x' is a place-holder, and anything could be substituted for it. This does not mean that 'x' refers to anything, as though anything were something specific we could point to. And it doesn't mean that 'x' refers to some abstract category of 'logical thingness' or what have you.

Again, the meaning of these sentences, as they are used, is to demonstrate a form of logical deduction. That function does not depend on any references (except perhaps a reference to logic itself; but this reference is not found in any of the three sentences used in the syllogism). The very reason we use arbitrary symbols without referential meaning is to make this clear.

So, when people say that logic must refer to something, such as Platonic forms, not only is it not clear what they are getting at; it is not even clear why they feel the need to get anywhere.

Thursday, August 20, 2009

The Language of Consciousness

There is no good definition of "consciousness"--at least, not in any rigorous philosophical or scientific sense. There are just lots of ways we use the term in everyday life. For example, we use it to distinguish between sleep and wakefulness, or to indicate that we are focusing our attention on something, or that we remember something, or that we know something. These aren't all the same, or even necessarily similar, processes. So the idea that there is some unique thing called "consciousness" is perhaps an error. And so the idea that there are "conscious processes" in the brain is also perhaps an error.

The word "consciousness" does not pick out anything specific, but has meaning only in so far as it provides some structure to our discourse--specifically, our discourse about ourselves. It is a grammatical construction without extra-linguistic referent.* Once we've understood the language, we've understood consciousness. There is nothing left to understand. Thus, as Dennett says, there is nothing to understand about consciousness beyond verbal reports. (But this does not mean there is nothing else to understand about brains or behavior.)

If a person says "I am hungry," we know what they mean, because we've learned the language. And no investigation into their brain or stomach will explain the meaning any better to us. Of course, by looking at their brain and stomach we can get a better idea of why they've expressed that sentiment. But the meaning of the sentiment is no better understood by such an investigation.

Consider, if I say "John has malaria," you can understand me a little bit, even if you don't know who I am talking about. But if we are engaged in conversation, you will want to know who I'm talking about so that you can understand me fully. You assume that "John" refers to somebody specific.

With words for consciousness and feelings, we may similarly be tempted to look for hidden referents. Yet, this is a mistake. Not all nouns are names for things. In my understanding, the language of consciousness (notions of mind, thought, and feeling) is used to indirectly refer to unknown causes of behavior. The meaning of these terms is rooted in behavior, and yet it does not directly refer to the behavior itself, nor does it refer to any identifiable causes. They are floating signifiers, meaningful but without discernible referents.

With neuroscience, we can greatly improve our understanding of the brain and human behavior. But we won't understand consciousness any better, because there is nothing about consciousness hidden in the brain (or anywhere else). The word "consciousness" doesn't point to the brian. It doesn't point anywhere. It is not an extended finger, but more like a waving hand.

Consider an example. I am focusing on writing this post. That is a fact about my mind, right? Now, by studying the brain we can better understand how a human being goes about writing and thinking about philosophy. We can analyze the behavior. But we won't gain a better understanding of what it means to focus on writing something. We won't improve our understanding of that, because that is something we already understand by verbal report. The meaning of the expression "focusing on writing" is found in the act itself, in the behavior, which anybody can observe.

We can use verbal reports to guide our study of the brain, just as we can use any other behavioral cues. But in so doing, we are using the behavior to understand the brain, and not vice versa. It is only because we understand verbal reports that we can use them to analyze the brain. So how could analyzing the brain help us understand the reports any better?

Again, it can help us understand what caused them, but that does not help us understand what they mean.

* Edit: I should clarify this. Verbal reports, such as "I am hungry," are usually not references to observable behavior, but nor are they references to anything hidden behind observable behavior. They are not usually references at all. When I said that "consciousness" is a grammatical construction without extra-linguistic referent, I was ignoring the way we can use that term to analyze human behavior. The language of consciousness can be used to analyze behavior, and so "consciousness" can refer to things like wakefulness, readiness, and so on; however, the term "consciousness" is usually taken to mean something hidden behind those observable behaviors. This is what I reject. While the language of consciousness directly involves observable behavior, it is thought to indirectly point to something else. In my view, we couldn't possibly be pointing to anything else. So, words like "consciousness" can be taken to refer to observable behavior, but when we take them to refer to something else, such as a sense of self, or an "immediacy of experience," we are referring to linguistic constructions, and nothing outside of our discourse.

Friday, August 14, 2009

Mathematical Procedures and Incommensurability

I. Procedure and Representation

We can use numbers to perform calculations without having to stipulate that each number refers to anything outside of our mathematical operations. Number systems are tools for counting and performing other arithmetical functions. We can define arithmetic procedurally, and avoid wondering what sort of existence numbers might have on their own, perhaps in some Platonic realm. Numbers are symbols used to represent mathematical procedures.

I used to think that the existence of irrational numbers posed a problem for this view. To define rational numbers, we say they can be represented as a fraction between m and n (where m and n are not both divisible by two). Irrational numbers are defined as numbers which cannot be represented as fractions in this way. They seem to point to something beyond comprehension, beyond the possibility of finite containment.

Indeed, the fact is, we have symbols for irrational numbers; the numbers themselves are the referent. So in what sense can we say that irrational numbers are representations of procedures, if the numbers in question cannot be fully represented in any finite space?

This is a fundamental logical problem, or so it has been claimed.

II. Infinity and Impossible Objects

A long time ago I thought it was a good idea to define "infinity" as "the relationship between a discrete point and a continuous line." I don't know why that definition occurred to me, or why I liked it. Probably because of something to do with Zeno's paradoxes.

The sense is thus: A discrete point has no extension. A line has extension. Therefore, no matter how many discrete points you attempt to connect, you will never get a line. And no matter how many discrete points you attempt to extract from a line, you will never decrease its length. So, there is an infinite number of discrete points on any line, and a line is infinitely longer than any series of discrete points.

As a definition, this may not be very useful. But as a way of thinking about infinity, I find it very interesting. And implicit in this approach is an incommensurability; namely, that between extended and unextended objects.

Consider pi, which defines the relationship between a circle's radius and its circumference, as well as between its radius and its area. Pi is an irrational number. Why?

Well, one way to look at it is to consider how we define a circle as such. We define a circle as the set of all points on a linear plane equidistant to a single point. Yet, in this case, a circle is defined in terms of discrete points, which have no extension. A circle is a continuous line. So we have an incommensurability, and this indicates a geometrical impossibility. Pi is irrational because it is impossible to produce a perfect circle.

This is not a human limitation. It is a geometrical fact.

Yet, in some sense, mathematicians say that pi exists. The word "pi" represents something real, even if it cannot be calculated completely. We can calculate it to any arbitrary degree of accuracy. The question is, what does "accuracy" mean here? What do our calculations signify?

From the procedural point of view, the calculations signify steps in our attempt to generate a circle.

Consider how we might go about constructing a circle in the real world. We might try any number of ways. The irrationality of pi indicates that, no matter what method we use, there will always be a better way. It's not that we are getting closer and closer to the true value of pi. Rather, it is that we are getting closer and closer to a perfect circle, even though such an object cannot exist. We can approach impossible objects with infinite precision; we just cannot make them. So, pi is irrational. This is a geometrical fact about circles, and not about whatever might be trying to generate them.

III. Geometry and Arithmetic

Irrational numbers are found when we attempt to produce arithmetical models of impossible geometrical operations. Irrational numbers indicate a tension between geometry and arithmetic.

I recently explored this idea by trying to find a geometrical procedure for creating a single perfect cube out of the parts of three identical cubes. If it is possible to create such a cube, then the cube root of 3 must be calculatable. Since the cube root of 3 is irrational, I suppose that one cannot, even in theory, combine three cubes into a single one.

Today I thought of another example: the square root of five. Consider a square with sides length 2 meters. We would thus say the area of this square is 4 square meters. I suggest that it is theoretically impossible, by any means imaginable, to increase the size of such a cube by exactly twenty-five percent, arriving at a square with an area of 5 square meters.

Then I did a couple Web searches on irrational numbers, and I was happy to find this page, which supports my understanding. It says irrational numbers surface at the intersection of arithmetic and geometry, and that they indicate incommensurability. The author of that site (Laurence Spector, a math instructor at a community college in New York) claims here that there is a "fundamental logical problem" concerning the existence of irrational numbers. He suggests that the tension between arithmetic and geometry exists because it is impossible to name or measure every length.

I think there is something wrong with that explanation. Are we to take it that irrational numbers represent unmeasurable lengths?

Now, Spector notes that in order for irrational numbers to be considered numbers at all, we must have a procedure to name, or measure, them to any arbitrary degree. That is, to say that pi is a number, we must have some procedure which allows us to place it on the continuum of Real numbers. This doesn't mean we designate a specific spot for it; rather, it means that we can place it between two rational numbers--two numbers which we know how to measure.

But why does Spector conclude that the existence of irrational numbers means that not every length is measurable?

His assumption, apparently, is that irrational numbers represent lengths. This is the problem.

Numbers do not represent lengths. People may represent lengths by using numbers. (Similarly, words do not refer to things. People refer to things using words.)

Am I referring to a specific length when I say "the square root of two meters?" No, I don't think so. Not unless somebody told me that some particular length was the square root of two meters. But calling any length "the square root of two meters" can only be arbitrarily justified, and not according to any definitive standard of measurement. There is no way to determine that any length is equal to the square root of two meters, but we can always say it's close enough. This doesn't mean that any particular length is unmeasurable. It means "the square root of two meters" doesn't pick out a particular length at all.

When we produce a longer calculation of pi, we are not getting a closer approximation to the true relationship between a circle's radius and its circumference. Rather, we getting a more detailed standard of measurement.

More detailed. Not more precise or more accurate.

Since there is no end to the possible digits you can place to the right of the decimal point, there is no sense in claiming that you're ever getting closer to the end. The longer the calculation, the more information we have; but this doesn't make the information better in any purely mathematical sense. It is only better if you have some purpose, some use, for all of those numbers.

IV. Conclusion

I see no sense in talking about unknowable, unmeasurable, or infinitely improbable lengths. Lengths are defined in terms of operations, procedures. We get numbers like "the square root of two" because we are devising procedures which have no geometrical correlates. There is no procedure for producing a perfect circle in geometry. There is no procedure for increasing a square by twenty-five percent, or of combining three identical cubes into one. These are theoretical limitations, not practical obstacles.

When we say irrational numbers are real, we mean that our procedures for calculating them are valid. I have no issue with that point. The question is not whether or not the calculations are valid; the question is what they mean.

I don't know if Spector's interpretation is common among philosophers or mathematicians. However, it seems to me that the supposed "logical problem" is only a problem of interpretation, of thinking that numbers indicate anything other than the procedures we have for calculating them.

Wednesday, August 12, 2009

Summarizing Dennett on Consciousness

A few days ago I posted the following in a discussion at PhilPapers:

Not far into Consciousness Explained (paperback, p. 23), Dennett writes: "Today we talk about our conscious decisions and unconscious habits, about the conscious experience we enjoy (in contrast to, say, automatic cash machines, which have no such experiences) -- but we are no longer quite sure we know what we mean when we say these things. While there are still thinkers who gamely hold out for consciousness being some one genuine precious thing (like love, like gold), a thing that is just 'obvious' and very, very special, the suspicion is growing that this is an illusion. Perhaps the various phenomena that conspire to create the sense of a single mysterious phenomenon have no more ultimate or essential unity than the various phenomena that contribute to the sense that love is a simple thing."

I think understanding this passage is critical to understanding Dennett's approach. Our talk of consciousness is not necessarily always about the same thing. The word is sometimes used to refer to a sort of "inner monalogue." At other times, a focus of attention or an act of the imagination. It may yet refer to a strong feeling or a vague sensation. I am not suggesting all of these concepts are clearly defined, mind you. Yet, they are defined enough to enjoy currency in everyday life. They make sense, even if they lack philosophical rigour. The point is that there is a great deal of phenomena that produces this talk of consciousness; that there is much sense and value in such talk; and that, if we want to understand what people are talking about when they talk about consciousness, we must understand what is motivating, underlying, and otherwise producing the language.

The point is not to first define what single phenomenon or entity is behind all of these processes, as though we even had a clear idea of what all of these various processes or phenomena entailed. If we did, there would be nothing left to discover. Rather, the point is to try to understand what such talk is about; what is going on to produce and ultimately justify such notions as feeling and thought. In the end, we may find that the term "consciousness" is unnecessary to explain humanity. This does not mean consciousness will have been "explained away." It only means that the term "consciousness" has come to serve a variety of functions in the absence of a robust model of humanity, and that once our understanding of humanity improves, the term may not seduce us into thinking it is so important. (And please remember I am talking about the term here, and not anything which it might signify in any particular situation.)


Soon after that, I posted this:

Dennett does not present Consciousness Explained as an explanation of consciousness. He presents it as an attempt to clear away some of the confusions in various disciplines, including philosophy, cognitive science, and neuroscience, which he believes hinder our progress towards explaining humanity. Perhaps the title is pretentious. No doubt it was chosen as an attention-getter. I don't read it as a statement of victory, but as a statement of focus. Dennett wants to overcome various philosophical arguments (such as Chalmers' zombie argument) which attempt to make consciousness out to be something inexplicable, something which cannot be scientifically explained. Dennett wants to change the way we approach discussions of "consciousness" (whatever that word is taken to mean) by deflating the presuppositions which stand in the way of a full, scientific explanation of humanity.

Much of the book is critical. He discusses various ways philosophers and scientists get into trouble by assuming there is an underlying unity of consciousness. He also attempts to make a more constructive contribution to the study of humanity, producing a very rough, initial sketch of a Multiple Drafts model; but this is offered as little more than speculation, not as an answer but rather as a step towards changing the way we approach questions about human experience. He is asking us to stop assuming that there is some unitary and intuitively obvious thing called "consciousness." He is asking us to instead ask what various, complex processes might produce the illusion that there is some unified, intuitively obvious thing so many people are tempted to call "consciousness."


I was then asked to further explain my understanding of Dennett's approach. I just submitted the following, which will hopefully appear on the site in a few days:

I should note that I haven't read Dennett for a few years, and I only just glanced at Consciousness Explained to extract the quote I offered earlier. So I may not do him justice here, and I may exaggerate the places where his and my understanding meet. That said, here is an elaboration of my understanding of Dennett's approach to consciousness, since you asked.

As I understand it, his approach is to try to understand why people use the language as they do without presupposing anything about what would make such language true or false. He wants to understand why the notion of consciousness has a role in our language at all. I thus think he is very Wittgensteinian.

The first step is to refrain from assuming that there is anything to be explained beyond observable behavior. Second, Dennett takes claims about conscious states at face value. He calls this heterophenomenology, which I think is a version of "ordinary language" philosophy. He argues that, whatever consciousness is, there cannot be any facts about consciousness beyond what is expressed in verbal reports.

He argues that the language of consciousness is part of a general discursive strategy which he calls the intentional stance. That is, our conceptual framework for talking about consciousness--notions like want, love, expect, and so on--is not a representational model, but a predictive stragey for regulating our behavior. The language of intentionality is seen as a set of tools for dealing with the enormous complexity of human behavior, and not as a set of terms which correspond directly to any particular facts of existence.

Intentions, mental states, consciousness . . . According to Dennett, these concepts do not refer to specific processes or entities. For example, when I tell somebody, "I feel hungry," because I want to make plans to go have lunch, I may be indirectly talking about my digestive system or some processes in my brain, but there is not a particular fact about myself which corresponds to the words "I," "feel," or "hungry." Nor is there any fact about myself which corresponds to the sentence as a whole. Rather, the meaning of my utterance is defined by the situation. Saying "I am hungry" here is akin to playing a particular card in a game of bridge. It serves a function; it has meaning in the context of that game; but it does not refer to anything. (Wittgenstein made this same point when he said that verbal reports of pain do not refer to inner sensations, but simply replace crying.) When I say, "I am hungry," I am trying to move a social situation in a particular direction. I am not representing a fact about myself, as if such a fact could exist anywhere outside of the language-game.

Dennett postpones coming to any conclusions about what the term "consciousness" means. For example, he notes that when people talk about consciousness, they usually mean something which has a "point of view." He does not define "consciousness" as "having a point of view," but he notes that this is one of most popular ways the term is used. So he approaches an explanation of why people talk as if they had a point of view, ultimately regarding the notion of a point of view as a "theorist's fiction." He does not think that there is such a thing as a "point of view" which exists outside of our discourse; nor does he think there are some beings which just have consciousness or just have a discernible point of view, as if these were facts about bodies or minds which could be borne out of any investigation whatsoever. For Dennett, there is no fact of the matter here; there is no sense in questioning whether or not somebody really has a point of view, or really is conscious. Thus, for Dennett, the very notion of philosophical zombies is absurd. When we treat some things as having consciousness (or as having a point of view), we are employing an explanatory or predictive framework. We are not postulating facts which could be corroborated or falsified according to any scientific theory.

Dennett's concern is therefore not, "what beings have consciousness, and what beings don't?" Nor is it, "what constitutes consciousness? What is consciousness made of?" He does not say it is necessarily meaningless to ask these questions, however. He just doesn't want to presuppose that the term "consciousness" refers to anything. If somebody wants to define "consciousness" so that it has a particular, identifiable referent, then they can talk about whether or not and how it exists in any particular cases. But, according to Dennett, that is not how the language of consciousness has evolved.