Explanation

Fire, Water, and Numbers

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Fire vs. Water

“All things are water,” says Thales.

“All things are fire,” says Heraclitus.

“Wait,” says David Hume’s Philo. “You both agree that all things are made up of one substance. Thales, you prefer to call it water, and Heraclitus, you prefer to call it fire. But isn’t that merely a verbal dispute? According to both of you, whatever you point at is fundamentally the same fundamental stuff. So whether you point at water or fire, or anything else, for that matter, you are always pointing at the same fundamental stuff. Where is the real disagreement?”

Philo has a somewhat valid point here, and I mentioned the same thing in the linked post referring to Thales. Nonetheless, as I also said in the same post, as well as in the discussion of the disagreement about God, while there is some common ground, there are also likely remaining points of disagreement. It might depend on context, and perhaps the disagreement is more about the best way of thinking about things than about the things themselves, somewhat like discussing whether the earth or the universe is the thing spinning, but Heraclitus could respond, for example, by saying that thinking of the fundamental stuff as fire is more valid because fire is constantly changing, while water often appears to be completely still, and (Heraclitus claims) everything is in fact constantly changing. This could represent a real disagreement, but it is not a large one, and Thales could simply respond: “Ok, everything is flowing water. Problem fixed.”

Numbers

It is said that Pythagoras and his followers held that “all things are numbers.” To what degree and in what sense this attribution is accurate is unclear, but in any case, some people hold this very position today, even if they would not call themselves Pythagoreans. Thus for example in a recent episode of Sean Carroll’s podcast, Carroll speaks with Max Tegmark, who seems to adopt this position:

0:23:37 MT: It’s squishy a little bit blue and moose like. [laughter] Those properties, I just described don’t sound very mathematical at all. But when we look at it, Sean through our physics eyes, we see that it’s actually a blob of quarks and electrons. And what properties does an electron have? It has the property, minus one, one half, one, and so on. We, physicists have made up these nerdy names for these properties like electric charge, spin, lepton number. But it’s just we humans who invented that language of calling them that, they are really just numbers. And you know as well as I do that the only difference between an electron and a top quark is what numbers its properties are. We have not discovered any other properties that they actually have. So that’s the stuff in space, all the different particles, in the Standard Model, you’ve written so much nice stuff about in your books are all described by just by sets of numbers. What about the space that they’re in? What property does the space have? I think I actually have your old nerdy non-popular, right?

0:24:50 SC: My unpopular book, yes.

0:24:52 MT: Space has, for example, the property three, that’s a number and we have a nerdy name for that too. We call it the dimensionality of space. It’s the maximum number of fingers I can put in space that are all perpendicular to each other. The name dimensionality is just the human language thing, the property is three. We also discovered that it has some other properties, like curvature and topology that Einstein was interested in. But those are all mathematical properties too. And as far as we know today in physics, we have never discovered any properties of either space or the stuff in space yet that are actually non-mathematical. And then it starts to feel a little bit less insane that maybe we are living in a mathematical object. It’s not so different from if you were a character living in a video game. And you started to analyze how your world worked. You would secretly be discovering just the mathematical workings of the code, right?

Tegmark presumably would believe that by saying that things “are really just numbers,” he would disagree with Thales and Heraclitus about the nature of things. But does he? Philo might well be skeptical that there is any meaningful disagreement here, just as between Thales and Heraclitus. As soon as you begin to say, “all things are this particular kind of thing,” the same issues will arise to hinder your disagreement with others who characterize things in a different way.

The discussion might be clearer if I put my cards on the table in advance:

First, there is some validity to the objection, just as there is to the objection concerning the difference between Thales and Heraclitus.

Second, there is nonetheless some residual disagreement, and on that basis it turns out that Tegmark and Pythagoras are more correct than Thales and Heraclitus.

Third, Tegmark most likely does not understand the sense in which he might be correct, rather supposing himself correct the way Thales might suppose himself correct in insisting, “No, things are really not fire, they are really water.”

Mathematical and non-mathematical properties

As an approach to these issues, consider the statement by Tegmark, “We have never discovered any properties of either space or the stuff in space yet that are actually non-mathematical.”

What would it look like if we found a property that was “actually non-mathematical?” Well, what about the property of being blue? As Tegmark remarks, that does not sound very mathematical. But it turns out that color is a certain property of a surface regarding how it reflects flight, and this is much more of a “mathematical” property, at least in the sense that we can give it a mathematical description, which we would have a hard time doing if we simply took the word “blue.”

So presumably we would find a non-mathematical property by seeing some property of things, then investigating it, and then concluding, “We have fully investigated this property and there is no mathematical description of it.” This did not happen with the color blue, nor has it yet happened with any other property; either we can say that we have not yet fully investigated it, or we can give some sort of mathematical description.

Tegmark appears to take the above situation to be surprising. Wow, we might have found reality to be non-mathematical, but it actually turns out to be entirely mathematical! I suggest something different. As hinted by connection with the linked post, things could not have turned out differently. A sufficiently detailed analysis of anything will be a mathematical analysis or something very like it. But this is not because things “are actually just numbers,” as though this were some deep discovery about the essence of things, but because of what it is for people to engage in “a detailed analysis” of anything.

Suppose you want to investigate some thing or some property. The first thing you need to do is to distinguish it from other things or other properties. The color blue is not the color red, the color yellow, or the color green.

Numbers are involved right here at the very first step. There are at least three colors, namely red, yellow, and blue.

Of course we can find more colors, but what if it turns out there seems to be no definite number of them, but we can always find more? Even in this situation, in order to “analyze” them, we need some way of distinguishing and comparing them. We will put them in some sort of order: one color is brighter than another, or one length is greater than another, or one sound is higher pitched than another.

As soon as you find some ordering of that sort (brightness, or greatness of length, or pitch), it will become possible to give a mathematical analysis in terms of the real numbers, as we discussed in relation to “good” and “better.” Now someone defending Tegmark might respond: there was no guarantee we would find any such measure or any such method to compare them. Without such a measure, you could perhaps count your property along with other properties. But you could not give a mathematical analysis of the property itself. So it is surprising that it turned out this way.

But you distinguished your property from other properties, and that must have involved recognizing some things in common with other properties, at least that it was something rather than nothing and that it was a property, and some ways in which it was different from other properties. Thus for example blue, like red, can be seen, while a musical note can be heard but not seen (at least by most people.) Red and blue have in common that they are colors. But what is the difference between them? If we are to respond in any way to this question, except perhaps, “it looks different,” we must find some comparison. And if we find a comparison, we are well on the way to a mathematical account. If we don’t find a comparison, people might rightly complain that we have not yet done any detailed investigation.

But to make the point stronger, let’s assume the best we can do is “it looks different.” Even if this is the case, this very thing will allow us to construct a comparison that will ultimately allow us to construct a mathematical measure. For “it looks different” is itself something that comes in degrees. Blue looks different from red, but orange does so as well, just less different. Insofar as this judgment is somewhat subjective, it might be hard to get a great deal of accuracy with this method. But it would indeed begin to supply us with a kind of sliding scale of colors, and we would be able to number this scale with the real numbers.

From a historical point of view, it took a while for people to realize that this would always be possible. Thus for example Isidore of Seville said that “unless sounds are held by the memory of man, they perish, because they cannot be written down.” It was not, however, so much ignorance of sound that caused this, as ignorance of “detailed analysis.”

This is closely connected to what we said about names. A mathematical analysis is a detailed system of naming, where we name not only individual items, but also various groups, using names like “two,” “three,” and “four.” If we find that we cannot simply count the thing, but we can always find more examples, we look for comparative ways to name them. And when we find a comparison, we note that some things are more distant from one end of the scale and other things are less distant. This allows us to analyze the property using real numbers or some similar mathematical concept. This is also related to our discussion of technical terminology; in an advanced stage any science will begin to use somewhat mathematical methods. Unfortunately, this can also result in people adopting mathematical language in order to look like their understanding has reached an advanced stage, when it has not.

It should be sufficiently clear from this why I suggested that things could not have turned out otherwise. A “non-mathematical” property, in Tegmark’s sense, can only be a property you haven’t analyzed, or one that you haven’t succeeded in analyzing if you did attempt it.

The three consequences

Above, I made three claims about Tegmark’s position. The reasons for them may already be somewhat clarified by the above, but nonetheless I will look at this in a bit more detail.

First, I said there was some truth in the objection that “everything is numbers” is not much different from “everything is water,” or “everything is fire.” One notices some “hand-waving,” so to speak, in Tegmark’s claim that “We, physicists have made up these nerdy names for these properties like electric charge, spin, lepton number. But it’s just we humans who invented that language of calling them that, they are really just numbers.” A measure of charge or spin or whatever may be a number. But who is to say the thing being measured is a number? Nonetheless, there is a reasonable point there. If you are to give an account at all, it will in some way express the form of the thing, which implies explaining relationships, which depends on the distinction of various related things, which entails the possibility of counting the things that are related. In other words, someone could say, “You have a mathematical account of a thing. But the thing itself is non-mathematical.” But if you then ask them to explain that non-mathematical thing, the new explanation will be just as mathematical as the original explanation.

Given this fact, namely that the “mathematical” aspect is a question of how detailed explanations work, what is the difference between saying “we can give a mathematical explanation, but apart from explanations, the things are numbers,” and “we can give a mathematical explanation, but apart from explanations, the things are fires?”

Exactly. There isn’t much difference. Nonetheless, I made the second claim that there is some residual disagreement and that by this measure, the mathematical claim is better than the one about fire or water. Of course we don’t really know what Thales or Heraclitus thought in detail. But Aristotle, at any rate, claimed that Thales intended to assert that material causes alone exist. And this would be at least a reasonable understanding of the claim that all things are water, or fire. Just as Heraclitus could say that fire is a better term than water because fire is always changing, Thales, if he really wanted to exclude other causes, could say that water is a better term than “numbers” because water seems to be material and numbers do not. But since other causes do exist, the opposite is the case: the mathematical claim is better than the materialistic ones.

Many people say that Tegmark’s account is flawed in a similar way, but with respect to another cause; that is, that mathematical accounts exclude final causes. But this is a lot like Ed Feser’s claim that a mathematical account of color implies that colors don’t really exist; namely they are like in just being wrong. A mathematical account of color does not imply that things are not colored, and a mathematical account of the world does not imply that final causes do not exist. As I said early on, a final causes explains why an efficient cause does what it does, and there is nothing about a mathematical explanation that prevents you from saying why the efficient cause does what it does.

My third point, that Tegmark does not understand the sense in which he is right, should be plain enough. As I stated above, he takes it to be a somewhat surprising discovery that we consistently find it possible to give mathematical accounts of the world, and this only makes sense if we assume it would in theory have been possible to discover something else. But that could not have happened, not because the world couldn’t have been a certain way, but because of the nature of explanation.

Structure of Explanation

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When we explain a thing, we give a cause; we assign the thing an origin that explains it.

We can go into a little more detail here. When we ask “why” something is the case, there is always an implication of possible alternatives. At the very least, the question implies, “Why is this the case rather than not being the case?” Thus “being the case” and “not being the case” are two possible alternatives.

The alternatives can be seen as possibilities in the sense explained in an earlier post. There may or may not be any actual matter involved, but again, the idea is that reality (or more specifically some part of reality) seems like something that would be open to being formed in one way or another, and we are asking why it is formed in one particular way rather than the other way. “Why is it raining?” In principle, the sky is open to being clear, or being filled with clouds and a thunderstorm, and to many other possibilities.

A successful explanation will be a complete explanation when it says “once you take the origin into account, the apparent alternatives were only apparent, and not really possible.” It will be a partial explanation when it says, “once you take the origin into account, the other alternatives were less sensible (i.e. made less sense as possibilities) than the actual thing.”

Let’s consider some examples in the form of “why” questions and answers.

Q1. Why do rocks fall? (e.g. instead of the alternatives of hovering in the air, going upwards, or anything else.)

A1. Gravity pulls things downwards, and rocks are heavier than air.

The answer gives an efficient cause, and once this cause is taken into account, it can be seen that hovering in the air or going upwards were not possibilities relative to that cause.

Obviously there is not meant to be a deep explanation here; the point here is to discuss the structure of explanation. The given answer is in fact basically Newton’s answer (although he provided more mathematical detail), while with general relativity Einstein provided a better explanation.

The explanation is incomplete in several ways. It is not a first cause; someone can now ask, “Why does gravity pull things downwards, instead of upwards or to the side?” Similarly, while it is in fact the cause of falling rocks, someone can still ask, “Why didn’t anything else prevent gravity from making the rocks fall?” This is a different question, and would require a different answer, but it seems to reopen the possibility of the rocks hovering or moving upwards, from a more general point of view. David Hume was in part appealing to the possibility of such additional questions when he said that we can see no necessary connection between cause and effect.

Q2. Why is 7 prime? (i.e. instead of the alternative of not being prime.)

A2. 7/2 = 3.5, so 7 is not divisible by 2. 7/3 = 2.333…, so 7 is not divisible by 3. In a similar way, it is not divisible by 4, 5, or 6. Thus in general it is not divisible by any number except 1 and itself, which is what it means to be prime.

If we assumed that the questioner did not know what being prime means, we could have given a purely formal response simply by noting that it is not divisible by numbers between 1 and itself, and explaining that this is what it is to be prime. As it is, the response gives a sufficient material disposition. Relative to this explanation, “not being prime,” was never a real possibility for 7 in the first place. The explanation is complete in that it completely excludes the apparent alternative.

Q3. Why did Peter go to the store? (e.g. instead of going to the park or the museum, or instead of staying home.)

A3. He went to the store in order to buy groceries.

The answer gives a final cause. In view of this cause the alternatives were merely apparent. Going to the park or the museum, or even staying home, were not possible since there were no groceries there.

As in the case of the rock, the explanation is partial in several ways. Someone can still ask, “Why did he want groceries?” And again someone can ask why he didn’t go to some other store, or why something didn’t hinder him, and so on. Such questions seem to reopen various possibilities, and thus the explanation is not an ultimately complete one.

Suppose, however, that someone brings up the possibility that instead of going to the store, he could have gone to his neighbor and offered money for groceries in his neighbor’s refrigerator. This possibility is not excluded simply by the purpose of buying groceries. Nonetheless, the possibility seems less sensible than getting them from the store, for multiple reasons. Again, the implication is that our explanation is only partial: it does not completely exclude alternatives, but it makes them less sensible.

Let’s consider a weirder question: Why is there something rather than nothing?

Now the alternatives are explicit, namely there being something, and there being nothing.

It can be seen that in one sense, as I said in the linked post, the question cannot have an answer, since there cannot be a cause or origin for “there is something” which would itself not be something. Nonetheless, if we consider the idea of possible alternatives, it is possible to see that the question does not need an answer; one of the alternatives was only an apparent alternative all along.

In other words, the sky can be open to being clear or cloudy. But there cannot be something which is open both to “there is something” and “there is nothing”, since any possibility of that kind would be “something which is open…”, which would already be something rather than nothing. The “nothing” alternative was merely apparent. Nothing was ever open to there being nothing.

Let’s consider another weird question. Suppose we throw a ball, and in the middle of the path we ask, Why is the ball in the middle of the path instead of at the end of the path?

We could respond in terms of a sufficient material disposition: it is in the middle of the path because you are asking your question at the middle, instead of waiting until the end.

Suppose the questioner responds: Look, I asked my question at the middle of the path. But that was just chance. I could have asked at any moment, including at the end. So I want to know why it was in the middle without considering when I am asking the question.

If we look at the question in this way, it can be seen in one way that no cause or origin can be given. Asked in this way, being at the end cannot be excluded, since they could have asked their question at the end. But like the question about something rather than nothing, the question does not need an answer. In this case, this is not because the alternatives were merely apparent in the sense that one was possible and the other not. But they were merely apparent in the sense that they were not alternatives. The ball goes both goes through the middle, and reaches the end. With the stipulation that we not consider the time of the question, the two possibilities are not mutually exclusive.

Additional Considerations

The above considerations about the nature of “explanation” lead to various conclusions, but also to various new questions. For example, one commenter suggested that “explanation” is merely subjective. Now as I said there, all experience is subjective experience (what would “objective experience” even mean, except that someone truly had a subjective experience?), including the experience of having an explanation. Nonetheless, the thing experienced is not subjective: the origins that we call explanations objectively exclude the apparent possibilities, or objectively make them less intelligible. The explanation of explanation here, however, provides an answer to what was perhaps the implicit question. Namely, why are we so interested in explanations in the first place, so that the experience of understanding something becomes a particularly special type of experience? Why, as Aristotle puts it, do “all men desire to know,” and why is that desire particularly satisfied by explanations?

In one sense it is sufficient simply to say that understanding is good in itself. Nonetheless, there is something particular about the structure of a human being that makes knowledge good for us, and which makes explanation a particularly desirable form of knowledge. In my employer and employee model of human psychology, I said that “the whole company is functioning well overall when the CEO’s goal of accurate prediction is regularly being achieved.” This very obviously requires knowledge, and explanation is especially beneficial because it excludes alternatives, which reduces uncertainty and therefore tends to make prediction more accurate.

However, my account also raises new questions. If explanation eliminates alternatives, what would happen if everything was explained? We could respond that “explaining everything” is not possible in the first place, but this is probably an inadequate response, because (from the linked argument) we only know that we cannot explain everything all at once, the way the person in the room cannot draw everything at once; we do not know that there is any particular thing that cannot be explained, just as there is no particular aspect of the room that cannot be drawn. So there can still be a question about what would happen if every particular thing in fact has an explanation, even if we cannot know all the explanations at once. In particular, since explanation eliminates alternatives, does the existence of explanations imply that there are not really any alternatives? This would suggest something like Leibniz’s argument that the actual world is the best possible world. It is easy to see that such an idea implies that there was only one “possibility” in the first place: Leibniz’s “best possible world” would be rather “the only possible world,” since the apparent alternatives, given that they would have been worse, were not real alternatives in the first place.

On the other hand, if we suppose that this is not the case, and there are ultimately many possibilities, does this imply the existence of “brute facts,” things that could have been otherwise, but which simply have no explanation? Or at least things that have no complete explanation?

Let the reader understand. I have already implicitly answered these questions. However, I will not link here to the implicit answers because if one finds it unclear when and where this was done, one would probably also find those answers unclear and inconclusive. Of course it is also possible that the reader does see when this was done, but still believes those responses inadequate. In any case, it is possible to provide the answers in a form which is much clearer and more conclusive, but this will likely not be a short or simple project.

Explaining Causality

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A reader asks about a previous post:

a) Per Hume and his defenders, we can’t really observe causation. All we can see is event A in spacetime, then event B in spacetime. We have no reason to posit that event A and event B are, say, chairs or dogs; we can stick with a sea of observed events, and claim that the world is “nothing more” but a huge set of random 4D events. While I can see that giving such an account restores formal causation, it doesn’t salvage efficient causation, and doesn’t even help final causation. How could you move there from our “normal” view?

b) You mention that the opinion “laws are observed patterns” is not a dominant view; though, even though I’d like to sit with the majority, I can’t go further than a). I can’t build an argument for this, and fail to see how Aristotle put his four causes correctly. I always end up gnawing on an objection, like “causation is only in the mind” or similar. Help?

It is not my view that the world is a huge set of random 4D events. This is perhaps the view of Atheism and the City, but it is a mistaken one. The blogger is not mistaken in thinking that there are problems with presentism, but they cannot be solved by adopting an eternalist view. Rather, these two positions constitute a Kantian dichotomy, and as usual, both positions are false. For now, however, I will leave this to the consideration of the reader. It is not necessary to establish this to respond to the questions above.

Consider the idea that “we can’t really observe causation.” As I noted here, it does not make sense to say that we cannot observe causation unless we already understand what causation is. If the word were meaningless to us, we would have no argument that we don’t observe it; it is only because we do understand the idea of causation that we can even suggest that it might be difficult to observe. And if we do have the idea, we got the idea from somewhere, and that could only have been… from observation, of course, since we don’t have anything else to get ideas from.

Let us untie the knot. I explained causality in general in this way:

“Cause” and “effect” simply signify that the cause is the origin of the effect, and that the effect is from the cause, together with the idea that when we understand the cause, we understand the explanation for the effect. Thus “cause” adds to “origin” a certain relationship with the understanding; this is why Aristotle says that we do not think we understand a thing until we know its cause, or “why” it is. We do not understand a thing until we know its explanation.

Note that there is something “in the mind” about causality. Saying to oneself, “Aha! So that’s why that happened!” is a mental event. And we can also see how it is possible to observe causality: we can observe that one thing is from another, i.e. that a ball breaks a window, and we can also observe that knowing this provides us a somewhat satisfactory answer to the question, “Why is the window broken?”, namely, “Because it was hit by a ball.”

Someone (e.g. Atheism and the City) might object that we also cannot observe one thing coming from another. We just observe the two things, and they are, as Hume says, “loose and separate.” Once again, however, we would have no idea of “from” unless we got it from observing things. In the same early post quoted above, I explained the idea of origin, i.e. that one thing is from another:

Something first is said to be the beginning, principle, or origin of the second, and the second is said to be from the first. This simply signifies the relationship already described in the last post, together with an emphasis on the fact that the first comes before the second by “consequence of being”, in the way described.

“The relationship already described in the last post” is that of before and after. In other words, wherever we have any kind of order at all, we have one thing from another. And we observe order, even when we simply see one thing after another, and thus we also observe things coming from other things.

What about efficient causality? If we adopt the explanation above, asserting the existence of efficient causality is nothing more or less than asserting that things sometimes make other things happen, like balls breaking windows, and that knowing about this is a way for us to understand the effects (e.g. broken windows.)

Similarly, denying the existence of efficient causality means either denying that anything ever makes anything else happen, or denying that knowing about this makes us understand anything, even in a minor way. Atheism and the City seems to want to deny that anything ever makes anything else happen:

Most importantly, my view technically is not that causality doesn’t exist, it’s that causality doesn’t exist in the way we typically think it does. That is, my view of causality is completely different from the general every day notion of causality most people have. The naive assumption one often gets when hearing my view is that I’m saying cause and effect relationships don’t exist at all, such that if you threw a brick at glass window it wouldn’t shatter, or if you jumped in front of a speeding train you wouldn’t get smashed to death by it. That’s not what my view says at all.

On my view of causality, if you threw a brick at a glass window it would shatter, if you jumped in front of a speeding train you’d be smashed to death by it. The difference between my view of causality vs the typical view is that on my view causes do not bring their effects into existence in the sense of true ontological becoming.

I am going to leave aside the discussion of “true ontological becoming,” because it is a distraction from the real issue. Does Atheism and the City deny that things ever make other things happen? It appears so, but consider that “things sometimes make other things happen” is just a more general description of the very same situations as descriptions like, “Balls sometimes break windows.” So if you want to deny that things make other things happen, you should also deny that balls break windows. Now our blogger perhaps wants to say, “I don’t deny that balls break windows in the everyday sense, but they don’t break them in a true ontological sense.” Again, I will simply point in the right direction here. Asserting the existence of efficient causes does not describe a supposedly “truly true” ontology; it is simply a more general description of a situation where balls sometimes break windows.

We can make a useful comparison here between understanding causality, and understanding desire and the good. The knowledge of desire begins with a fairly direct experience, that of feeling the desire, often even as physical sensation. In the same way, we have a direct experience of “understanding something,” namely the feeling of going, “Ah, got it! That’s why this is, this is how it is.” And just as we explain the fact of our desire by saying that the good is responsible for it, we explain the fact of our understanding by saying that the apprehension of causes is responsible. And just as being and good are convertible, so that goodness is not some extra “ontological” thing, so also cause and origin are convertible. But something has to have a certain relationship with us to be good for us; eating food is good for us while eating rocks is not. In a similar way, origins need to have a specific relationship with us in order to provide an understanding of causality, as I said in the post where these questions came up.

Does this mean that “causation is only in the mind”? Not really, any more than the analogous account implies that goodness is only in the mind. An aspect of goodness is in the mind, namely insofar as we distinguish it from being in general, but the thing itself is real, namely the very being of things. And likewise an aspect of causality is in the mind, namely the fact that it explains something to us, but the thing itself is real, namely the relationships of origin in things.

Necessary Connection

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In Chapter 7 of his Enquiry Concerning Human Understanding, David Hume says about the idea of “necessary connection”:

We have looked at every possible source for an idea of power or necessary connection, and have found nothing. However hard we look at an isolated physical episode, it seems, we can never discover discover anything but one event following another; we never find any force or power by which the cause operates, or any connection between it and its supposed effect. The same holds for the influence of mind on body: the mind wills, and then the body moves, and we observe both events; but we don’t observe– and can’t even conceive– the tie that binds the volition to the motion, i.e. the energy by which the mind causes the body to move. And the power of the will over its own faculties and ideas– i.e. over the mind, as distinct from the body– is no more comprehensible. Summing up, then: throughout the whole of nature there seems not to be a single instance of connection that is conceivable by us. All events seem to be entirely loose and separate. One event follows another, but we never can observe any tie between them. They seem associated, but never connected. And as we can have no idea of anything that never appeared as an impression to our outward sense or inward feeling, we are forced to conclude that we have no idea of ‘connection’ or ‘power’ at all, and that those words– as used in philosophical reasonings or in common life– have absolutely no meaning.

This is not Hume’s final word on the matter, as we will see below, so this has to be taken with a grain of salt, even as a representation of his opinion. Nonetheless, consider this caricature of what he just said:

We have looked at every possible source for an idea of mduvvqi or pdnfhvdkdddd, and have found nothing. However hard we look at an isolated physical episode, it seems, we can never discover anything but events that can be described by perfectly ordinary words; we never find any mduvvqi involved, nor any pdnfhvkdddd.

We could take this to be making the point that “mduvvqi” and “pdnfhvdkdddd” are not words. Other than that, however, the paragraph itself is meaningless, precisely because those “words” are meaningless. It certainly does not make any deep (or shallow for that matter) metaphysical or physical point, nor any special point about the human mind. But Hume’s text is different, and the difference in question is a warning sign of Kantian confusion. If those words had “absolutely no meaning,” in fact, there would be no difference between Hume’s passage and our caricature. Those words are not meaningless, but meaningful, and Hume is even analyzing their meaning in order to supposedly determine that the words are meaningless.

Hume’s analysis in fact proceeds more or less in the following way. We know what it means to say that something is necessary, and it is not the same as saying that the thing always happens. Every human being we have ever seen was less than 20 feet tall. But is it necessary that human beings be less than 20 feet tall? This is a different question, and while we can easily experience someone’s being less than 20 feet tall, it is very difficult to see how we could possibly experience the necessity of this fact, if it is necessary. Hume concludes: we cannot possibly experience the necessity of it. Therefore we can have no idea of such necessity.

But Hume has just contradicted himself: it was precisely by understanding the concept of necessity that he was able to see the difficulty in the idea of experiencing necessity.

Nonetheless, as I said, this is not his final conclusion. A little later he gives a more nuanced account:

The source of this idea of a necessary connection among events seems to be a number of similar instances of the regular pairing of events of these two kinds; and the idea cannot be prompted by any one of these instances on its own, however comprehensively we examine it. But what can a number of instances contain that is different from any single instance that is supposed to be exactly like them? Only that when the mind experiences many similar instances, it acquires a habit of expectation: the repetition of the pattern affects it in such a way that when it observes an event of one of the two kinds it expects an event of the other kind to follow. So the feeling or impression from which we derive our idea of power or necessary connection is a feeling of connection in the mind– a feeling that accompanies the imagination’s habitual move from observing one event to expecting another of the kind that usually follows it. That’s all there is to it. Study the topic from all angles; you will never find any other origin for that idea.

Before we say more, we should concede that this is far more sensible than the claim that the idea of necessity “has absolutely no meaning.” Hume is now conceding that it does have meaning, but claiming that the meaning is about us, not about the thing. When we see someone knock a glass off a table, we perhaps feel a certainty that it will fall and hit the floor. Experiencing that feeling of certainty, he says, is the source of the idea of “necessity.” This is not an unreasonable hypothesis.

However, Hume is also implicitly making a metaphysical argument here which is somewhat less sensible. Our feelings of certainty and uncertainty are subjective qualities of our minds, he suggests, not objective features of the things. Therefore necessity as an objective feature does not and cannot exist. This is not unrelated to his mistaken claim that we cannot know that the future will be similar to the past, even with probability.

What is the correct account here? In fact we already know, from the beginning of the conversation, that “necessary” and “possible” are meaningful words. We also know that in fact we use them to describe objective features of the world. But which features? Attempting to answer this question is where Hume’s approach is pretty sensible. Hume is not mistaken that all of our knowledge is from experience, and ultimately from the senses. He seems to identify experience with sense experience too simplistically, but he is not mistaken that all experience is at least somewhat similar to sense experience; feeling sure that two and two make four is not utterly unlike seeing something red. We want to say that there is something in common there, “something it is like,” to experience one or the other. But if this is the case, it would be reasonable to extend what we said about the senses to intellectual experiences. “The way red looks” cannot, as such, be an objective feature of a thing, but a thing can be objectively red, in such a way that “being red,” together with the nature of the senses, explains why a thing looks red. In a similar way, certainty and uncertainty, insofar as they are ways we experience the world, cannot be objective features of the world as such. Nonetheless, something can be objectively necessary or uncertain, in such a way that “being necessary” or otherwise, together with the nature of our minds, explains why it seems certain or uncertain to us.

There will be a similarity, however. The true nature of red might be quite strange in comparison to the experience of seeing red, as for example it might consist of surface reflectance properties. In a similar way, the true nature of necessity, once it is explained, might be quite strange to us compared to the experience of being certain or uncertain. But that it can be explained is quite certain itself, since the opposite claim would fall into Hume’s original absurdity. There are no hidden essences.

Revisiting Russell on Cause

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We discussed Bertrand Russell’s criticism of the first cause argument here. As I said there, he actually suggests, although without specifically making the claim, that there is no such thing as a cause, when he says:

That argument, I suppose, does not carry very much weight nowadays, because, in the first place, cause is not quite what it used to be. The philosophers and the men of science have got going on cause, and it has not anything like the vitality it used to have.

This is absurd, and it is especially objectionable that he employs this method of insinuation instead of attempting to make an argument. Nonetheless, let me attempt to argue on Russell’s behalf for a moment. It is perhaps not necessary for him to say that there is no such thing as a cause. Suppose he accepts my account of cause as an explanatory origin. Note that this is not purely an objective relationship existing in the world. It includes a specific relationship with our mind: we call something a cause when it is not only an origin, but it also explains something to us. The relatively “objective” relationship is simply that of origin.

A series of causes, since it is also a series of explanations, absolutely must have a first, since otherwise all explanatory force will be removed. But suppose Russell responds: it does not matter. Sure, this is how explanations work. But there is nothing to prevent the world from working differently. It may be that origins, namely the relationship on the objective side, do consist of infinite series. This might make it impossible to explain the world, but that would just be too bad, wouldn’t it? We already know that people have all sorts of desires for knowledge that cannot be satisfied. A complete account of the world is impossible in principle, and even in practice we can only obtain relatively local knowledge, leaving us ignorant of remote things. So you might feel a need of a first cause to make the world intelligible, Russell might say, but that is no proof at all that there is any series of origins with a first. For example, consider material causes. Large bodies are made of atoms, and atoms of smaller particles, namely electrons, protons, and neutrons. These smaller particles are made of yet smaller particles called quarks. There is no proof that this process does not go on forever. Indeed, the series would cease to explain anything if it did, but so what? Reality does not have to explain itself to you.

In response, consider the two following theories of water:

First theory: water is made of hydrogen and oxygen.

Second theory: every body of water has two parts, which we can call the first part and the second part. Each of the parts themselves has two parts, which we can call the first part of the first part, the second part of the first part, the first part of the second part, and the second part of the second part. This goes on ad infinitum.

Are these theories true? I presume the reader accepts the first theory. What about the second? We are probably inclined to say something like, “What does this mean, exactly?” But the very fact that the second theory is extremely vague means that we can probably come up with some interpretation that will make it true, depending in its details on the details of reality. Nonetheless, it is a clearly useless theory. And it is useless precisely because it cannot explain anything. There is no “causality” in the second theory, not even material causality. There is an infinite series of origins, but no explanation, and so no causes.

The first theory, on the other hand, is thought to be explanatory, and to provide material causes, because we implicitly suppose that we cannot go on forever in a similar way. It may be that hydrogen and oxygen are made up of other things: but we assume that this will not go on forever, at least with similar sorts of division.

But what if it does? It is true, in fact, that if it turns out that one can continue to break down particles into additional particles in a relatively similar manner ad infinitum, then “water is made of hydrogen and oxygen” will lose all explanatory force, and will not truly be a causal account, even in terms of material causes, even if the statement itself remains true. It would not follow, however, that causal accounts are impossible. It would simply follow that we chose the wrong account, just as one would be choosing wrongly if one attempted to explain water with the second theory above. The truth of the second theory is irrelevant; it is wrong as an explanation even if it is true.

As I have argued in a number of places, nature is not in the business of counting things. But it necessarily follows from this that it also does not call things finite or infinite; we are the ones who do that. So if you break down the world in such a way that origins are infinite, you will not be able to understand the world. That is not the world’s problem, but your problem. You can fix that by breaking down the world in such a way that origins are finite.

Perhaps Russell will continue to object. How do you know that there is any possible breakdown of the world which makes origins finite? But this objection implies the fully skeptical claim that nothing can be understood, or at least that it may turn out that nothing can be understood. As I have said elsewhere, this particular kind of skeptical claim implies a contradiction, since it implies that the same thing is known and unknown. This is the case even if you say “it might be that way,” since you must understand what you are saying when you say it might be that way.

Consistency and Reality

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Consistency and inconsistency, in their logical sense, are relationships between statements or between the parts of a statement. They are not properties of reality as such.

“Wait,” you will say. “If consistency is not a property of reality, then you are implying that reality is not consistent. So reality is inconsistent?”

Not at all. Consistency and inconsistency are contraries, not contradictories, and they are properties of statements. So reality as such is neither consistent nor inconsistent, in the same way that sounds are neither white nor black.

We can however speak of consistency with respect to reality in an extended sense, just as we can speak of truth with respect to reality in an extended sense, even though truth refers first to things that are said or thought. In this way we can say that a thing is true insofar as it is capable of being known, and similarly we might say that reality is consistent, insofar as it is capable of being known by consistent claims, and incapable of being known by inconsistent claims. And reality indeed seems consistent in this way: I might know the weather if I say “it is raining,” or if I say, “it is not raining,” depending on conditions, but to say “it is both raining and not raining in the same way” is not a way of knowing the weather.

Consider the last point more precisely. Why can’t we use such statements to understand the world? The statement about the weather is rather different from statements like, “The normal color of the sky is not blue but rather green.” We know what it would be like for this to be the case. For example, we know what we would expect if it were the case. It cannot be used to understand the world in fact, because these expectations fail. But if they did not, we could use it to understand the world. Now consider instead the statement, “The sky is both blue and not blue in exactly the same way.” There is now no way to describe the expectations we would have if this were the case. It is not that we understand the situation and know that it does not apply, as with the claim about the color of the sky: rather, the situation described cannot be understood. It is literally unintelligible.

This also explains why we should not think of consistency as a property of reality in a primary sense. If it were, it would be like the color blue as a property of the sky. The sky is in fact blue, but we know what it would be like for it to be otherwise. We cannot equally say, “reality is in fact consistent, but we know what it would be like for it to be inconsistent.” Instead, the supposedly inconsistent situation is a situation that cannot be understood in the first place. Reality is thus consistent not in the primary sense but in a secondary sense, namely that it is rightly understood by consistent things.

But this also implies that we cannot push the secondary consistency of reality too far, in several ways and for several reasons.

First, while inconsistency as such does not contribute to our understanding of the world, a concrete inconsistent set of claims can help us understand the world, and in many situations better than any particular consistent set of claims that we might currently come up with. This was discussed in a previous post on consistency.

Second, we might respond to the above by pointing out that it is always possible in principle to formulate a consistent explanation of things which would be better than the inconsistent one. We might not currently be able to arrive at the consistent explanation, but it must exist.

But even this needs to be understood in a somewhat limited way. Any consistent explanation of things will necessarily be incomplete, which means that more complete explanations, whether consistent or inconsistent, will be possible. Consider for example these recent remarks of James Chastek on Gödel’s theorem:

1.) Given any formal system, let proposition (P) be this formula is unprovable in the system

2.) If P is provable, a contradiction occurs.

3.) Therefore, P is known to be unprovable.

4.) If P is known to be unprovable it is known to be true.

5.) Therefore, P is (a) unprovable in a system and (b) known to be true.

In the article linked by Chastek, John Lucas argues that this is a proof that the human mind is not a “mechanism,” since we can know to be true something that the mechanism will not able to prove.

But consider what happens if we simply take the “formal system” to be you, and “this formula is unprovable in the system” to mean “you cannot prove this statement to be true.” Is it true, or not? And can you prove it?

If you say that it is true but that you cannot prove it, the question is how you know that it is true. If you know by the above reasoning, then you have a syllogistic proof that it is true, and so it is false that you cannot prove it, and so it is false.

If you say that it is false, then you cannot prove it, because false things cannot be proven, and so it is true.

It is evident here that you can give no consistent response that you can know to be true; “it is true but I cannot know it to be true,” may be consistent, but obviously if it is true, you cannot know it to be true, and if it is false, you cannot know it to be true. What is really proven by Gödel’s theorem is not that the mind is not a “mechanism,” whatever that might be, but that any consistent account of arithmetic must be incomplete. And if any consistent account of arithmetic alone is incomplete, much  more must any consistent explanation of reality as a whole be incomplete. And among more complete explanations, there will be some inconsistent ones as well as consistent ones. Thus you might well improve any particular inconsistent position by adopting a consistent one, but you might again improve any particular consistent position by adopting an inconsistent one which is more complete.

The above has some relation to our discussion of the Liar Paradox. Someone might be tempted to give the same response to “tonk” and to “true”:

The problem with “tonk” is that it is defined in such a way as to have inconsistent implications. So the right answer is to abolish it. Just do not use that word. In the same way, “true” is defined in such a way that it has inconsistent implications. So the right answer is to abolish it. Just do not use that word.

We can in fact avoid drawing inconsistent conclusions using this method. The problem with the method is obvious, however. The word “tonk” does not actually exist, so there is no problem with abolishing it. It never contributed to our understanding of the world in the first place. But the word “true” does exist, and it contributes to our understanding of the world. To abolish it, then, would remove some inconsistency, but it would also remove part of our understanding of the world. We would be adopting a less complete but more consistent understanding of things.

Hilary Lawson discusses this response in Closure: A Story of Everything:

Russell and Tarski’s solution to self-referential paradox succeeds only by arbitrarily outlawing the paradox and thus provides no solution at all.

Some have claimed to have a formal, logical, solution to the paradoxes of self-reference. Since if these were successful the problems associated with the contemporary predicament and the Great Project could be solved forthwith, it is important to briefly examine them before proceeding further. The argument I shall put forward aims to demonstrate that these theories offer no satisfactory solution to the problem, and that they only appear to do so by obscuring the fact that they have defined their terms in such a way that the paradox is not so much avoided as outlawed.

The problems of self-reference that we have identified are analogous to the ancient liar paradox. The ancient liar paradox stated that ‘All Cretans are liars’ but was itself uttered by a Cretan thus making its meaning undecidable. A modern equivalent of this ancient paradox would be ‘This sentence is not true’, and the more general claim that we have already encountered: ‘there is no truth’. In each case the application of the claim to itself results in paradox.

The supposed solutions, Lawson says, are like the one suggested above: “Just do not use that word.” Thus he remarks on Tarski’s proposal:

Adopting Tarski’s hierarchy of languages one can formulate sentences that have the appearance of being self-referential. For example, a Tarskian version of ‘This sentence is not true’ would be:

(I) The sentence (I) is not true-in-L.

So Tarski’s argument runs, this sentence is both a true sentence of the language meta-L, and false in the language L, because it refers to itself and is therefore, according to the rules of Tarski’s logic and the hierarchy of languages, not properly formed. The hierarchy of languages apparently therefore enables self-referential sentences but avoids paradox.

More careful inspection however shows the manoeuvre to be engaged in a sleight of hand for the sentence as constructed only appears to be self-referential. It is a true sentence of the meta-language that makes an assertion of a sentence in L, but these are two different sentences – although they have superficially the same form. What makes them different is that the meaning of the predicate ‘is not true’ is different in each case. In the meta-language it applies the meta-language predicate ‘true’ to the object language, while in the object language it is not a predicate at all. As a consequence the sentence is not self-referential. Another way of expressing this point would be to consider the sentence in the meta-language. The sentence purports to be a true sentence in the meta-language, and applies the predicate ‘is not true’ to a sentence in L, not to a sentence in meta-L. Yet what is this sentence in L? It cannot be the same sentence for this is expressed in meta-L. The evasion becomes more apparent if we revise the example so that the sentence is more explicitly self-referential:

(I) The sentence (I) is not true-in-this-language.

Tarski’s proposal that no language is allowed to contain its own truth-predicate is precisely designed to make this example impossible. The hierarchy of languages succeeds therefore only by providing an account of truth which makes genuine self-reference impossible. It can hardly be regarded therefore as a solution to the paradox of self-reference, since if all that was required to solve the paradox was to ban it, this could have been done at the outset.

Someone might be tempted to conclude that we should say that reality is inconsistent after all. Since any consistent account of reality is incomplete, it must be that the complete account of reality is inconsistent: and so someone who understood reality completely, would do so by means of an inconsistent theory. And just as we said that reality is consistent, in a secondary sense, insofar as it is understood by consistent things, so in that situation, one would say that reality is inconsistent, in a secondary sense, because it is understood by inconsistent things.

The problem with this is that it falsely assumes that a complete and intelligible account of reality is possible. This is not possible largely for the same reasons that there cannot be a list of all true statements. And although we might understand things through an account which is in fact inconsistent, the inconsistency itself contributes nothing to our understanding, because the inconsistency is in itself unintelligible, just as we said about the statement that the sky is both blue and not blue in the same way.

We might ask whether we can at least give a consistent account superior to an account which includes the inconsistencies resulting from the use of “truth.” This might very well be possible, but it appears to me that no one has actually done so. This is actually one of Lawson’s intentions with his book, but I would assert that his project fails overall, despite potentially making some real contributions. The reader is nonetheless welcome to investigate for themselves.

Real Distinction II

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I noted recently that one reason why people might be uncomfortable with distinguishing between the way things seem, as such, namely as a way of seeming, and the way things are, as such, namely as a way of being, is that it seems to introduce an explanatory gap. In the last post, why did Mary have a “bluish” experience? “Because the banana was blue,” is true, but insufficient, since animals with different sense organs might well have a different experience when they see blue things. And this gap seems very hard to overcome, possibly even insurmountable.

However, the discussion in the last post suggests that the difficulty in overcoming this gap is mainly the result of the fact that no one actually knows the full explanation, and that the full explanation would be extremely complicated. It might even be so complicated that no human being could understand it, not necessarily because it is a kind of explanation that people cannot understand, but in a sense similar to the one in which no human being can memorize the first trillion prime numbers.

Even if this is the case, however, there would be a residual “gap” in the sense that a sensitive experience will never be the same experience as an intellectual one, even when the intellect is trying to explain the sensitive experience itself.

We can apply these ideas to think a bit more carefully about the idea of real distinction. I pointed out in the linked post that in a certain sense no distinction is real, because “not being something” is not a thing, but a way we understand something.

But notice that there now seems to be an explanatory gap, much like the one about blue. If “not being something” is not a thing, then why is it a reasonable way to understand anything? Or as Parmenides might put it, how could one thing possibly not be another, if there is no not?

Now color is complicated in part because it is related to animal brains, which are themselves complicated. But “being in general” should not be complicated, because the whole idea is that we are talking about everything in general, not with the kind of detail that is needed to make things complicated. So there is a lot more hope of overcoming the “gap” in the case of being and distinction, than in the case of color and the appearance of color.

A potential explanation might be found in what I called the “existential theory of relativity.” As I said in that post, the existence of many things necessarily implies the existence of relationships. But this implication is a “before in understanding“. That is, we understand that one thing is not another before we consider the relationship of the two. If we consider what is before in causality, we will get a different result. On one hand, we might want to deny that there can be causality either way, because the two are simultaneous by nature: if there are many things, they are related, and if things are related, they are many. On the other hand, if we consider “not being something” as a way things are understood, and ask the cause of them being understood in this way, relation will turn out to be the cause. In other words, we have a direct response to the question posed above: why is it reasonable to think that one thing is not another, if not being is not a thing? The answer is that relation is a thing, and the existence of relation makes it reasonable to think of things as distinct from one another.

Someone will insist that this account is absurd, since things need to be distinct in order to be related. But this objection confuses the mode of being and the mode of understanding. Just as there will be a residual “gap” in the case of color, because a sense experience is not an intellectual experience, there is a residual gap here. Explaining color will not suddenly result in actually seeing color if you are blind. Likewise, explaining why we need the idea of distinction will not suddenly result in being able to understand the world without the idea of distinction. But the existence of the sense experience does not thereby falsify one’s explanation of color, and likewise here, the fact that we first need to understand things as distinct in order to understand them as related, does not prevent their relationship from being the specific reality that makes it reasonable to understand them as distinct.