Causality: B. The Problem of Formalization (Column 461)
In the previous column I surveyed the question of causality in broad strokes. Among other things, I argued there that the principle of causality has no empirical source, and that the causal relation has no mathematical-logical representation at all. In the first comment to that column, Roi Yozebitch linked to an interview he conducted with Prof. Judea Pearl, a computer scientist, in which they discuss an article Pearl published that deals precisely with this point. His main claim there is that probability—indeed even conditional probability and a full Bayesian picture—cannot represent a causal relation. He then proposes a formalism that can represent causality. In this column I will use his model to sharpen the claims I made last time.
Probabilistic measures for causality
Probability offers a quantitative estimate of the weight of different possibilities under conditions that include uncertainty. For example, when you throw a die there are six possible outcomes, and therefore there is uncertainty about the expected result. Probability theory gives us a quantitative estimate for each of the possibilities and, in the case of a fair die, the weight is 1/6 for every outcome. Probability as such, of course, has nothing to do with causality. If we wish to get closer to notions of causality and relations between events, we can speak about correlation.
Correlation between two events A and B measures the connection between them. If there is no connection between the events, then the occurrence of one does not affect the other, which means the probabilities are independent. The probability of a compound event—i.e., the probability that both A and B occurred—is denoted P(A∗B)P(A*B). When there is no dependence between the events, the following equality holds:
P(A∗B)=P(A)P(B)P(A*B) = P(A)P(B).
When there is a connection between the events, the equality does not hold, and the strength of the connection is determined by the ratio between the two sides of this equation (the farther apart they are, the stronger the connection).
If we wish to approach a description of a causal relation more closely, it is customary to employ conditional probability and Bayes’ formula. In Column 402 I defined conditional probability and presented Bayes’ formula (see also Columns 144–145 and elsewhere). Briefly, conditional probability describes a change in probability when additional information is introduced. For example, the probability of obtaining a 5 on a fair die is P(A)=1/6P(A)=1/6. But if we also know that the outcome was greater than 1 (call this information BB), then the probability of having gotten a 5 increases, of course: P(A∣B)=1/5P(A\mid B)=1/5. Translation: the probability of obtaining a 5 given that the outcome is greater than 1 is 1/5. If we know that the result was odd (call this CC), then the conditional probability of getting a 5 in that case is P(A∣C)=1/3P(A\mid C)=1/3. And yet it is clear that conditional probability is not necessarily causality. It is hard to say that the fact that the result is odd is the cause of getting a 5 (and not only because in this case the conditional probability is not 1).
You will sometimes hear or read the claim that “correlation is not causation” (as noted last time), but that conditional probability is tied to causation. Thus, for example, there is a causal connection between rain (A) and clouds (B), which means that the overall probability of rain P(A)P(A) is smaller than the probability of rain given that there were clouds, P(A∣B)P(A\mid B). True, even if there are clouds it doesn’t necessarily rain, and therefore the conditional probability is not 1. What then will you say about the reverse conditional probability, P(B∣A)P(B\mid A)? If there is rain then clearly there are clouds. Therefore P(B∣A)=1P(B\mid A)=1. But no one would say that the rain causes the clouds (it’s merely an indication of them; cf. the fat-and-diet example in the previous column). Logically we would say that rain is a sufficient condition for clouds, but not their cause. This reflects another claim we encountered earlier: logic is indifferent to the time axis—and now we see that probability is as well. The probabilistic formula is indifferent to the question of what is cause and what is effect; it only tests their connection. In other words, even conditional probability is not a good measure of causality. In the previous example (with the die) we also saw that conditional probability does not necessarily express a causal connection.
To be sure, a causal connection between events can be described in terms of conditional probability. For example, suppose that striking a match (A) necessarily (i.e., it is a sufficient condition) ignites a fire (B). Then the conditional probability P(B∣A)=1P(B\mid A)=1. If it is not strictly necessary but still tends to ignite, then P(B∣A)P(B\mid A) will be greater than the unconditional P(B)P(B), but not equal to 1. Even in such a case there is a causal relation between the events, only it also depends on the circumstances.
Pearl’s proposal
In §4.1 of his article, Pearl discusses whether the probabilistic language plus the time axis suffices to describe a causal relation. He describes the prevailing view, according to which these two components do indeed suffice to describe causality. But, he argues—contrary to what many think—the answer is negative (exactly as I argued in the previous column).
The example he brings to sharpen the difference is this: suppose a switch XX turns on two bulbs YY and ZZ. Assume the bulbs are at very different distances from the switch, and therefore bulb ZZ lights up a second before bulb YY. When we flip the switch XX, after some time ZZ will light, and a second later YY will light. Now compare this to the following situation: the switch XX turns on bulb ZZ, which in turn turns on bulb YY. From the logical and temporal points of view, the two situations are identical. But it is clear that from a causal point of view they are not. The first situation can be described as Y←X→ZY \leftarrow X \rightarrow Z, whereas the second is X→Z→YX \rightarrow Z \rightarrow Y. The difference between the two situations is easily demonstrated by examining interventions: if I intervene artificially and turn bulb ZZ off, in the first case this will not affect YY, but in the second case it will also extinguish YY. In the first case there is no causal relation between ZZ and YY (only correlation); in the second there is.
In my terminology, his conclusion can be put this way: to describe the causal relation fully, we must add to logic (and probability) and to time also the physical element (the production of the effect). The arrows in the diagrams express this third component. Pearl uses this representation to propose a formalism that better describes the causal relation.
Without entering the mathematical details, in §3.2 Pearl proposes using a description similar to the arrow diagrams in place of standard logical and probabilistic descriptions. He defines causal functions that take account of environmental conditions (circumstances), and thus allow the degree of influence to be included as well. For example, clouds (B) do indeed produce rain (A), but as we saw one cannot mark this causally as B⇒AB \Rightarrow A (i.e., “if B then necessarily A”), because it depends on additional circumstances (weather, the nature of the clouds, humidity, and so on). He therefore writes it as a function
A=f(B,w)A = f(B,w).
That is: there is a function ff that receives as inputs the circumstances ww and the clouds BB, and determines whether there will be rain AA. But do not be misled: this is not an ordinary mathematical function equality, because—as we saw last time—plain equality has no directionality. For Pearl this “equality” operates only one way, from right to left. This means that if we change AA artificially (say, we make rain from an airplane), it does not follow that anything in the input values of the function has changed. In other words, this is not an equality but a causal entailment, the one we indicated above with an arrow.
Two problems
I see two main problems with Pearl’s proposal. The first is that it makes no progress toward a full mathematical treatment of causality. The second is that it does not truly capture the concept of causality in its entirety. I will spell them out now.
The question that always arises for me when I encounter a logical symbolization—or any formalization—of some phenomenon is: what have we gained from this notation? Up to now we said there is a causal relation between AA and BB. Pearl suggests that, instead of saying “there is a causal relation,” we symbolize it with an arrow or a one-way function. What is the gain from this regular mathematical function notation? In other domains we have accumulated a lot of knowledge about the meanings of functions and how to use them. There is therefore obvious logic in describing natural phenomena using mathematical functions; the benefit of that notation is clear. But here we merely translated a Hebrew word into a semi-mathematical symbol—and I don’t see what we gained from the whole story. Pearl’s preliminary analysis does have value, of course, since it shows that standard mathematical and logical notation does not cope with causal phenomena. But his constructive proposal, in my view, brings little benefit to the discussion. We used to think that, instead of mathematics, we have to speak Hebrew; now we mark a Hebrew word with an arrow. I do not see what we gained by this. This is the first problem I find in Pearl’s proposal.¹
The second problem that arises here concerns whether this formalism truly captures the concept of causality in full. If you look at the switch-and-two-bulbs example above, you will see that Pearl addresses the causal relation by way of its phenomena. He distinguishes correlation from causation through the operation of intervention (a do-operation, in his terminology). But as I noted earlier, David Hume posed a more severe philosophical problem concerning the causal relation: it has no empirical source—i.e., it cannot be identified via observed phenomena and occurrences in the world. As is known, Hume argued that even if one sees a kick cause a ball to fly or clouds (under given circumstances ww) cause rain, we have no empirical way to ascertain that what we have here is a relation of production rather than mere correlation. This is a philosophical-scientific problem, not a problem of mathematical-logical symbolization. Even if Pearl’s notation solves the formal problem, it leaves the scientific problem in place. We have no way to express, in mathematical terms, the claim that there is between AA and BB a relation of production—that the one brings about the other. At most we can express the fact that a change in AA will change BB, i.e., the connection between the phenomena. But that still belongs to the phenomenological plane and not to the essential one. The fact of the connection is phenomenological (concerned with phenomena and their relations), but it does not touch causality in its essential philosophical sense.
In other words, the debate with Hume about the nature of the causal relation will not be settled by the formal tools Pearl proposes, and it is doubtful that it can be settled in such a way at all. Those tools express a difference between correlation and phenomenological causality, but they do not touch essential causality. There may be no formal way to treat essential causality. What science and mathematics can do is at most treat phenomena and occurrences in the world and the logical-temporal connections between them. The causality that produces those connections (in Kantian terms: the noumenon, as opposed to the phenomenon) belongs to metaphysics and is therefore not accessible to observation, science, or mathematical formalization.
To see this, consider the law of gravitation. This law describes the motion of a body of mass mm under the influence of another body of mass MM at a distance rr from it. The law of gravitation describes the acceleration of the body mm given the other mass MM (which is the factor BB) under the given circumstances (the distance rr, and the absence of other masses and influences). But one cannot derive from it the existence of a gravitational force that produces that motion. The motion of mm is correlated with the presence of mass MM—and that is a description on the phenomenological plane. The claim that a force is the causal producer of the motion, and that the equation linking the acceleration with the data (MM and rr) even hints at its existence—none of this follows.² The claim that there is a force producing this acceleration is our interpretation of the law of gravitation, just as the claim that force causes acceleration and not the other way round is our interpretation of Newton’s second law (see the previous column).
My second remark is not necessarily a critique of Pearl’s proposal. I meant only to sharpen the difference between the planes (phenomenological and essential) and to highlight the additional element in the causal relation beyond logic and the time axis—the element I called above the physical (productive) component. My claim is that a mathematical and logical formalism, including Pearl’s, does not succeed in describing that physical dimension of causality.
Summary
I have shown here that there is a fundamental problem in any logical or mathematical formalization of the causal relation. This is not an argument against one or another concept of causality. On the contrary: my claim is that even if my concept is not dismissible and can be formalized, it suffers from a lack of formalizability, whereas Pearl’s concept may be formalizable—but that does not refute mine. It merely shows that there is something about causality that does not belong to the scientific-mathematical stratum, but rather to the philosophical stratum. When I say that a force is the cause of acceleration, that is an interpretation and not a scientific statement. It does not arise from the equation but from our interpretation of it.
Science can manage without that interpretation. A scientist wants to know whether a force is acting or not, and according to the second law of Newton the existence of acceleration proves that a force is acting—even if the acceleration is not the cause of the force (which is not true, even by the prevailing interpretation that force is the cause of acceleration). A scientist wants to predict an earthquake, and for this it suffices to find correlations and Pearl-style causality; that is, to find what is expected to appear before the quake and can predict its occurrence. Whether that thing is the cause of the quake or not is a question of interpretation and therefore not pure science but rather philosophy. Science gets along without it. Think of Pearl’s bulbs example: a scientist will want to predict whether bulb YY will light, and for this it suffices to know the antecedent conditions—even if these are not causes in the full physical-philosophical sense. But I do not see this, by itself, as an explanation for the bulb’s lighting. An explanation for some event requires pointing to the cause of its occurrence. Therefore, to speak of an explanation for the bulb’s lighting we must add to the empirically found correlations that we have formalized the causal interpretation.
Links mentioned
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Pearl, “Causality and…” (reprint): https://ftp.cs.ucla.edu/pub/stat_ser/r284-reprint.pdf
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https://mikyab.net/translated-articles-rabbi-michael-abraham/post-72555
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https://mikyab.net/translated-articles-rabbi-michael-abraham/post-7616
¹ It should be noted that Pearl is a computer scientist, and his explicit goal is to enable computers to handle causal relations. He is not proposing philosophical or scientific solutions to the kind of issue I raised here. In that sense, his proposal should be judged by outcomes. Perhaps such a notation is more accessible for logical encoding and formalization for a computer. From this angle, the problem I described may not constitute a critique of his proposal at all; for our purposes, however, this analysis is important for sharpening my own claims about the causal relation and the role of the third component (production—what I called in the previous column “the physical component”).
² In mechanics we do speak of gravitational force, but that is at most a definition. Instead of saying that a mass MM causes a mass mm to accelerate at aa, we say that MM exerts a force of magnitude mama on mm. We have no way to know that a force is acting except via the fact that an acceleration develops. That is not accessible to observation. I have often noted that discovery of gravitons might change the situation, as photons express the electromagnetic force (field). But the existence of gravitons would also be an interpretation of the equation, and even if it were confirmed empirically, it would not follow from Newton’s law of gravitation.
Article Contents
With God’s help
Causality: B. The Problem of Formalization
In the previous column I surveyed the issue of causality from a general perspective. Among other things, I pointed out there that the principle of causality has no empirical source, and that the causal relation has no mathematical-logical representation at all. In the first comment, Roi Yuzvitz posted a link to an interview he conducted with Prof. Judea Pearl, a computer scientist, in which they discuss an article Pearl published that deals with exactly this point. His main claim there is that probability, and even conditional probability and a Bayesian picture, cannot represent a causal relation. He then proposes a formalism that can represent causality. In this column I will use his model to sharpen the claims I made in the previous column.
Probabilistic Measures of Causality
Probability offers a quantitative estimate of the weight of different possibilities under circumstances that involve uncertainty. For example, in the roll of a die there are six possible outcomes, and therefore there is uncertainty about the expected result. Probability theory offers us a quantitative estimate for each of the possibilities, and in the case of a fair die the weight is 1/6 for each outcome. Probability in itself, of course, has absolutely nothing to do with causality. If one wishes to come closer to the concepts of causality and connection between events, one can speak about correlation.
A correlation between two events A and B measures the connection between them. If there is no connection between the events, then the occurrence of one does not affect the other, and that means that the probabilities are independent. The probability of a compound event, that is, the probability that both A and B occurred, is denoted: P(A*B). In the case where there is no dependence between the events, the following equality holds: P(A*B) = P(A)P(B). When there is a connection between these events, the equality does not hold, and the degree of connection between the events is determined by the ratio between the two sides of this equation (the farther they are from one another, the stronger the connection between the events).
If one wishes to come even closer to a description of a causal relation, it is customary to think that we should use conditional probability and Bayes’s formula. In column 402 I defined conditional probability and presented Bayes’s formula (see also columns 145–144 and others). In brief, conditional probability describes a change in probability when additional information becomes available to us. For example, the probability of getting a 5 on a fair die is: P(A)=1/6. But if we know additional information, for example that the result of the roll was greater than 1 (let us denote this information by B), then the probability that we got a 5 naturally increases: P(A/B)=1/5. Translation: the probability of getting a 5 given that the result is greater than 1 is 1/5. If we know that the result was odd (C), then the conditional probability of getting a 5 in that case is: P(A/C)=1/3. And yet it is clear that conditional probability is not necessarily causality. It is difficult to say that the fact that the result is odd is the causal factor behind the outcome being 5 (and not only because the conditional probability in this case is not 1).
Sometimes you may hear or read the claim that correlation is indeed not causality (see the previous column), but conditional probability is connected to causality. Thus, for example, there is a causal connection between rain (A) and clouds (B), and that means that the overall probability of rain P(A) is smaller than the probability of rain if it is known that there were clouds: P(A/B). Of course, even if there are clouds it does not necessarily rain, and therefore the conditional probability is not 1.
What would you say about the reverse conditional probability: P(B/A)? If there is rain, then clearly there are clouds. Therefore this is indeed equal to 1. But no one would say that the rain is the cause of the clouds (it is an indication of them. See the example of fat and dieting in the previous column). Logically speaking, we would say that rain is a sufficient condition for clouds, but not their cause. This reflects another claim we encountered in the previous column, namely that logic is indifferent to the time axis, and now we find that probability is as well. The probabilistic formula is indifferent to the question of which is the cause and which is the effect, and examines only the connection between them. That is, conditional probability too is not a good measure of causality. In the previous example as well (that of the die outcomes), we saw that conditional probability does not necessarily express a causal connection.
Still, a causal connection between events can be described in terms of conditional probability. For example, suppose that striking a match (B) necessarily causes (that is, is a sufficient condition for) the ignition of fire (A), then one may say that the conditional probability P(A/B) = 1. If it does not necessarily cause it, one can still say that the conditional probability P(A/B) is greater than the unconditional probability P(A), though it is not equal to 1. In such a case too there is a causal connection between the events, except that it also depends on the circumstances.
Pearl’s Proposal
In section 4.1 of his article, Pearl discusses the question of whether probabilistic language together with the time axis is sufficient to describe causality. He presents there the accepted approach, according to which those two components are indeed sufficient to describe a causal relation. But, he argues, contrary to what many think, the answer is negative (precisely as I argued in the previous column).
The example he brings there in order to sharpen the difference is the following. Suppose a switch X turns on two bulbs, Y and Z. Suppose the distance of the bulbs from the switch is very different, and therefore bulb Z lights up one second before bulb Y. When we turn on switch X, after some time bulb Z will light up, and one second later bulb Y will light up. Now compare this to the following situation: switch X turns on bulb Z, and that in turn turns on bulb Y. From the standpoint of the logical and temporal relations, the two situations are identical, but clearly from a causal point of view there is no identity between the two states. The first state can be described as follows: Y 🡨 X 🡪 Z, whereas the second answers to a different description: X 🡪 Z 🡪 Y. The difference between the two situations is very easy to demonstrate through checking acts of intervention. For example, if I intervene artificially and turn off bulb Z, in the first case this will not affect bulb Y, but in the second case it will turn off Y as well. This means that in the first case there is no causal relation between Z and Y (only correlation), whereas in the second case there is.
In my terminology, his conclusion can be described as follows: in order to describe the causal relation fully, one must add to logic (and probability) and time the physical element as well (causation). The arrows in these descriptions express this third component. Pearl uses this representation to propose a formalism that will describe the causal relation in a better way.
Without entering into mathematical details, in section 3.2 of his article Pearl proposes using a description similar to that of the arrows, instead of the accepted logical and probabilistic descriptions. He defines there causal functions that also take account of the environmental conditions (the circumstances), and thus it is also possible to introduce the degree of influence. For example, clouds (B) do create rain (A), but as we have seen it is impossible to denote B🡪A in the causal sense (that if there is B then there must also be A), since this depends on additional circumstances (weather conditions, the character of the clouds, humidity, and the like). Therefore he denotes this as a function fA(B,w) = A. The meaning is that there is a function fA that receives as input the circumstances (w) and the clouds (B), and determines whether there will be rain (A). But do not make a mistake. This is not ordinary mathematical notation for a function, since, as we saw in the previous column, mathematical equality cannot do the job because it has no directionality. For Pearl this “equality” works only in one direction, from right to left. This means that if we change A artificially (make it rain from a plane), this does not mean that anything in the values of the function has changed. In other words, what we have here is not an equality but causal entailment, the one denoted above by an arrow.
Two Problems
I see two main problems in Pearl’s proposal. One is that there is no progress in it toward a mathematical treatment of causality. The second is that it does not truly capture the concept of causality in full. I will now spell them out in greater detail.
The question that always arises for me when I encounter a logical-mathematical notation for some phenomenon is whether we have gained anything from that notation. Until now we said that there is a causal relation between (B,w) and A, and we had no way to symbolize it. Pearl proposes that instead of saying there is a causal relation, we symbolize it by an arrow or by a one-way function. In what way does this notation help? With ordinary functions, we have accumulated a great deal of knowledge about their meaning and the way to use them. Therefore it makes sense to describe natural phenomena באמצעות mathematical functions, and the gain from that notation is obvious. But here we have merely translated a word in ordinary language into a logical-mathematical symbol, and so I do not see what we have gained from the whole story.
Pearl’s preliminary analysis certainly has value, since it shows that mathematical and logical notation fails to cope with phenomena of causality, but his constructive proposal, in my opinion, does not really bring any benefit to the discussion. Until now we thought that instead of mathematics one had to speak ordinary language, and now we denote a word from ordinary language by an arrow. I do not see what we have gained from that. This is the first problem I see in Pearl’s proposal.[1]
The second problem that arises here concerns the question whether this formalism really captures the concept of causality in full. If you look at the example of the switch and the two bulbs presented above, you will see that Pearl approaches the causal relation through its phenomena. He distinguishes between correlation and causality through the act of intervention (a do-operation, in his terminology). But as I mentioned in the previous column, David Hume posed a more severe philosophical problem regarding the causal relation: that it has no empirical source, that is, that it cannot be identified through phenomena and occurrences in the world. As remembered, Hume argued that even if one sees that a kick causes a ball to fly, or that clouds (under given circumstances w) cause rain, we have no empirical way of verifying that what is present here is indeed a relation of causation and not mere correlation. This is a philosophical-scientific problem and not a problem of mathematical-logical notation. Even if the notation proposed by Pearl solves the formal problem, it leaves the scientific problem intact. We have no way to express mathematically the claim that a relation of causation exists between A and B, that is, that one produces the other. At most one can express the fact that changing A will change B, but this is still a mode of reference that belongs to the phenomenological plane and not to the essential one. The connection between the phenomena is a phenomenological fact (it deals with phenomena and the relation between them), but it does not touch causality in its essential philosophical sense.
In other words, the dispute with Hume regarding the nature of the causal relation will not be decided by the formal tools Pearl proposes, and it is doubtful whether it can be decided in such a way at all. These tools express a difference between correlation and phenomenological causality, but they do not touch essential causality. It may be that there is no formal way to deal with essential causality. What science and mathematics can do, at most, is to deal with phenomena and occurrences in the world and with the logical and temporal connections between them. The causality that produces these connections (in Kantian terminology: the noumena, as opposed to the phenomena) belongs to metaphysics and is therefore inaccessible to observation, to science, and to mathematical notation.
To understand this, let us look at the law of gravitation. This law describes the motion of a body of mass m under the influence of another body of mass M that is at a distance r from it. The law of gravitation describes the acceleration of the body m given the other mass M (which is the factor B) under the given circumstances (the distance r, and the absence of other masses and additional influences), but one cannot derive from it the existence of a gravitational force that produces this motion. The motion of m is correlated with the existence of a mass M, and this is a description on the phenomenological plane. But the force is the causal factor of the motion, and the equation that links the acceleration with the data (M and r) does not even hint at its existence.[2] The claim that there is a force that causes this acceleration is our interpretation of the law of gravitation, just as the claim that the force causes the acceleration and not the reverse is our interpretation of Newton’s second law (see the previous column).
The second remark is not necessarily a criticism of Pearl’s proposal. My intention was only to sharpen the difference between the planes (the phenomenological and the essential) and the additional element that exists in the causal relation beyond time and logic, which touch only the phenomenological plane. My claim is that mathematical and logical formalism, including the one Pearl proposes, does not succeed in describing the physical dimension of causality.
Summary
I have shown here that there is an essential problem in the logical or mathematical formalization of the causal relation. This is not a claim against one causal conception or another. On the contrary, my claim is that although Hume’s conception of the causal relation may perhaps be susceptible to formalization, whereas my conception suffers from an inability to be formalized, this does not invalidate my conception and does not prove or confirm his. It only means that there is something in causality that does not belong to the scientific-mathematical level but to the philosophical level. When I say that force is the cause of acceleration, this is an interpretation and not a scientific determination. It does not arise from the equation but from our interpretation of it.
Science can manage even without this interpretation. A scientist wants to know whether a force is acting or not, and according to Newton’s second law the existence of an acceleration proves that a force is acting here, even if the acceleration is not the cause of the force (which is not true even according to the accepted interpretation, according to which the force is the cause of the acceleration). A scientist wants to predict an earthquake, and for that it is enough for him to find correlations and causality in the sense of Judea Pearl, that is, he must find what is supposed to appear before the quake and can predict its occurrence. Whether that thing is the cause of the quake or not is a question of interpretation, and therefore it is not pure science but more philosophy. Science manages without it as well. Think about Pearl’s example of the bulbs. A scientist will want to predict whether bulb Y will light or not, and for that it is enough for him to know the antecedent conditions, even if these are not causes in the full physical-philosophical sense. But I do not see this, in itself, as an explanation of the bulb’s lighting. An explanation of any event requires pointing to the cause of its occurrence. Therefore, in order to speak about an explanation of the bulb’s lighting, we must add to the correlations that we found empirically and represented formally, the causal interpretation.
1.
Footnotes
- It should be noted that Pearl is a computer scientist, and his declared aim is to enable a computer to handle causal relations. He is not proposing solutions to philosophical or scientific problems. In that sense, his proposal should be judged by its results. It is possible that such notation is more accessible for logical encoding and formalization for a computer. In that sense, the problem I described here may not constitute a criticism of his formalism, but for our purposes this analysis is important in order to sharpen my own claims regarding the causal relation, the need for all three components, and to clarify the role of the third component (causation, what I called in the previous column “the physical component”).
- True, in mechanics people speak about the force of gravity, but that is at most a definition. Instead of saying that mass M causes mass m to accelerate with acceleration a, one says that mass M exerts on it a force of magnitude ma. We have no way of knowing that a force is acting here except through the fact that an acceleration develops. That is not accessible to observation. I have already remarked more than once that finding gravitons may perhaps change the situation, just as photons are an expression of the electromagnetic force (the field). But the existence of gravitons is an interpretation of the equation, and even if this is confirmed, it is not a result of the equality in Newton’s law of gravitation.
Discussion
The intervention test is phenomenological. David Hume does not accept it as an indication of causation. Therefore I argue that it is a good test on the phenomenological plane, but philosophically it does not contain the dimension of causation. It seems to me that this concept is well understood by all of us; otherwise this discussion would not be taking place. If you are looking for a phenomenological definition of it, then you have thrown the baby out with the bathwater, since my whole claim is that it has none and yet it exists.
I accept the central claim emerging from the post, namely that formalization has a price (it is “blind” to physical and metaphysical meanings or essences, for example the causal connection).
At the same time, I assume you would agree with me that it also has a gain. The gain is of course the ability to create a science that is communicative for human creatures like us. If we imagine a possible world in which there are creatures that do not share with us the ability to create formal systems (mathematical symbols, etc.), but do have an intuitive ability to grasp directly abstract essences such as the causal connection.
Now the question is: what kind of world and what kind of “history,” if any, would such creatures have?
After all, it is hard to imagine that they could even begin to create any kind of joint enterprise (material, spiritual, or intellectual). It may be that from the outset, for them, “everything” is shared (for they directly grasp the same objects). In fact, it seems that even in terms of values and motivation they would have nothing pushing them toward joint activity (and perhaps even toward activity “just because”).
I understand that this question takes the topic in a new direction that perhaps does not interest you in this framework. As you see fit.
I completely agree.
The discussion about the world those creatures would have does not seem useful to me. The question is what other means they have to communicate with one another instead of our formalization. Think about people who lack formal ability nowadays, or people from four thousand years ago. Did they not have a history or an understanding of causality? In short, this is a discussion we have no tools to conduct.
“And the causal relation has no mathematical-logical representation at all”
The derivative with respect to time represents development over time, and from the expression one can derive what we want to call cause and effect.
All the rest is mere illusion.
Thank you for the interesting article, and also for its predecessor. From what the rabbi noted about logic’s lack of dependence on time, the symbolization of the expression “only if” into logical notation became sharper for me.
And when the rabbi wrote that nothing is gained here from the formalization, I was immediately reminded of the Kuzari –
73. The Rabbi said: But their knowledge of Him is like our knowledge. The philosopher defined Him as the beginning and the cause by which the thing that is in it essentially and not accidentally comes to rest and moves.
74. The Kuzari said: As if he were saying that a thing which moves by itself and comes to rest by itself has some cause by which it comes to rest and moves, and that cause is nature.
75. The Rabbi said: This is indeed what he meant to say, with great precision and exactness, and with a distinction between what acts accidentally and what acts by nature; and these formulations startle the hearers, but this is what emerges from their knowledge of nature.
Why do you choose to use the causal interpretation as condition 3 and not the “intervention test” you mentioned above? Isn’t it more intuitive? In addition, doesn’t the “causal interpretation” have the problem of “telya tanya belo tanya”? The claim is that for something to be called a cause there must be a relation of “causing,” but that concept too is not defined any more clearly (certainly not empirically, or even by way of a thought experiment) than the concept of cause. תודה