Ontic and Epistemic Doubt A: A Scientific and Conceptual Introduction (Column 322)
With God’s help
In the coming columns I wish to address the relationship between epistemic (cognitive) doubt and ontic (factual, in-reality) doubt, in halakhic contexts and more generally. In this column I will first offer a scientific and conceptual preface to the topic.1 You don’t need prior knowledge to follow along, though I realize some readers shy away from scientific material and mathematical notation. Such readers can focus on my explanations; coefficients and equations aren’t truly necessary for understanding (even if they help).
Inability to Predict in Chaotic Systems
Duane Farmer, one of the pioneers of chaos theory and a member of the original Santa Cruz group that first conceptualized chaos, describes how he and his colleagues felt when they began to grasp that a physically meaningful and far-reaching theory was in their hands (see James Gleick’s Chaos, p. 253):
“Philosophically, it seemed to me a practical way to define free will, one that allows you to reconcile ‘permission is granted’ with ‘all is foreseen.’ The system is deterministic, but you can’t say what it will do the next moment… There’s a coin with two sides here. On one side there’s order from which randomness emerges, and one step away there’s randomness grounded in order.”
Despite the excitement, the linkage Farmer makes between randomness and free will is flawed for several reasons. First, free choice is not randomness; it is a different mechanism entirely—a philosophical misunderstanding regarding the meaning of free will. More seriously, there is a physical misunderstanding of what chaos means. A chaotic system is not truly random. In such a system we indeed can’t predict future behavior from the present state, but that’s only due to computational complexity and sensitivity to initial conditions. There is no genuine freedom here. Farmer conflates unpredictability with freedom. What leads him astray is the fact that if a person truly has free choice, we cannot predict their behavior. But the converse doesn’t hold: the mere fact we cannot predict behavior doesn’t necessarily imply freedom (i.e., non-determinism).
Consider tossing a coin or a die. Practically, we can’t predict the outcome, so many treat such phenomena as random and analyze them with probabilistic tools. But of course nothing “random” is actually going on. A coin toss or die roll is entirely deterministic, even conceptually simple—it’s just Newton’s laws. The difficulty of calculation stems both from enormous sensitivity to initial conditions (the direction and speed of the toss) and from the technical computation itself (shape of the objects, air currents, etc.). These prevent us from computing the result, but in principle such a computation exists. In Talmudic parlance one might say that “before Heaven it is known” what the outcome will be—God knows it. If you gave me a computer with infinite power and memory (plus exact data about the coin/die and initial conditions), I could tell you the outcome with certainty. True, I lack all that, so I can’t perform the computation—but that still isn’t freedom. A system that includes “freedom” it is not.
Epistemic Doubt and Ontic Doubt
What did Duane Farmer miss? Why the gap between unpredictability and freedom? The fact that I can’t predict the outcome of a coin toss stems from certain lacks, ambiguities, or unknowns on our side; it’s not a feature of reality itself. For any given state there is exactly one well-defined outcome determined by the present state. The lacuna is ours. In philosophical terms, this is epistemic doubt (a lack of knowledge). When it comes to chaotic systems, someone equipped with complete information and unlimited computational capacity could in principle know everything about them.2
To expose the error more sharply, recall that libertarian free will (the view that genuine free will exists) maintains that reality itself is not single-valued. Even given a particular set of circumstances, a person can still freely choose whether to do X or Y. Hence this is not a case of “we don’t know,” but of a reality that itself is not uniquely determined. In Talmudic terms, even granting an infinite-power knower does not help—there is no single right answer to know. We shall call such a situation ontic ambiguity (from ontology, the study of being): indeterminacy in reality itself, as opposed to epistemic doubt which concerns our knowledge of reality rather than reality itself. To sharpen the distinction, I’ll reserve “doubt” for a lack of human knowledge, and use “ambiguity” for a lack of determination in reality. As a rule of thumb: doubts are modeled by probability; ambiguities by fuzzy logic.
Quantum Physics: Ambiguity Within Physics
It is told of the British astronomer Arthur Eddington that when told there were three people in the world who understood relativity (him and Einstein, of course), he asked who the third one was. Richard Feynman—later, a younger and renowned physicist—quipped that relativity isn’t really that hard to understand, but quantum mechanics nobody understands; at best you get used to it.
One of quantum theory’s strange features is that, according to standard interpretations, it’s the only domain in physics where ambiguity appears in reality itself (not merely in limits to our computation). There is an ontic margin in physics, not only an epistemic one. I’ll explain the relevant point briefly (relying on the common interpretations of quantum theory).
The Double-Slit Experiment
As Feynman suggested, the best way to grasp quantum mechanics is via the double-slit experiment. The experiment has a fascinating, flip-flopping history. Already in Newton’s time, there was a debate among physicists about the nature of light: is it particles (Newton) or waves (Huygens/Fresnel)? In 1801—74 years after Newton’s death—Thomas Young performed the original double-slit experiment to settle this. The setup sharply distinguishes waves from particle beams. It proved equally useful in the next century for an analogous debate about the nature of particles like electrons (wave-like or particle-like?).
With a single slit, both waves and particles give a central maximum on the screen with diminishing intensity to the sides. But with two slits, a particle beam would simply sum the contributions from each slit, giving two equal peaks; a wave beam produces an interference pattern of alternating bright and dark fringes, including destructive interference (two strong waves summing to zero) and constructive interference (total intensity exceeding the simple sum).
Experiments with light yielded the wave-pattern result and settled the old debate in favor of light’s wave nature. But early in the 20th century, evidence accumulated suggesting particles behave like waves. Louis de Broglie (1924) proposed that electrons have wave character and even supplied a mathematical description (wave function). Electron double-slit experiments were performed: the surprising result was an interference pattern—apparently electrons behave like waves.
To rule out the hypothesis that interference arose from electrons disturbing one another within the beam, the experiment was repeated with an extremely low emission rate so that only one electron was in flight at any time. Astonishingly, the pattern remained wave-like. A single electron interferes with itself. That is, an electron isn’t a tiny billiard ball at a definite place, but a wave spread out in space—just like light.
But if a single electron interferes with itself, it must “pass through” both slits. That seems intolerable. How can one particle go through two slits at once? To check, a detector was placed at one slit (call it slit A). When the detector registered the electron passing A, the pattern on the screen turned particle-like (two bright lobes) rather than wave-like. When a detector was placed at slit B and registered there, again the particle-like pattern appeared. In short: once we measure which slit the electron goes through, the interference pattern disappears; the electron stops behaving like a wave and behaves like a well-mannered particle. With tennis balls, by contrast, the pattern is always particle-like whether or not you “watch.”
Thus, as long as there’s no which-path measurement, the electron behaves like a wave; put a detector, and it behaves like a particle. Historically, after quantum theory was formulated, it also turned out that light isn’t purely a wave either. In some setups it behaves as discrete particles (photons). Newton returned from the dead, as it were.
The prevailing “Copenhagen interpretation” (associated with Niels Bohr and his school) concluded that the ontological distinction between “wave” and “particle” has lost its meaning. In micro-physics we don’t have two different kinds of beings, but one kind of being that sometimes behaves this way and sometimes that way—two different states of the same entity (photon or electron).
Quantum Theory: A Primary Interpretation
The picture physicists drew is this: as long as we don’t look (don’t measure), the electron is in a superposition—a sum of pure particle-states. A pure particle-state is one where the electron behaves like a tiny tennis ball: it has a definite position at every moment, and if we plot position versus time we get a well-defined trajectory. For example, the pure state “passes through slit A” we write as |A⟩, and the pure state “passes through slit B” as |B⟩. In both cases the electron behaves like a tennis ball.
But when we do not measure its position at the slits, the electron is in a superposition of these pure states: its “trajectory” is the sum of the trajectories, as if the particle goes through both slits. Hence it interferes with itself like a wave. Generally, an electron’s ordinary state is a combination of many such pure particle-states (not just two), each corresponding to a possible path. That’s why the overall pattern looks wave-like—the electron is “spread out” across space.
Once we measure its position, the state “collapses” into one pure particle-state making up the superposition (this is the “collapse of the wave function”). The wave function assigns probabilities to each pure state. Placing a detector at a slit is effectively conducting a lottery that selects one pure state from those composing the wave function, with probabilities given by the state’s coefficients. Without a detector, the particle remains “smeared” over all the pure states at once.
Symbolically we may write a two-state superposition as:
|ψ⟩ = α·|A⟩ + β·|B⟩
If we place a detector, the particle will be found going through |A⟩ with probability |α|² or through |B⟩ with probability |β|².
Ontic Ambiguity in Quantum Theory
Consider Schrödinger’s famous “cat in the box” thought experiment: a sealed vial of poison is triggered by a quantum process. The quantum state is a superposition of two simple (classical) states—vial open (poison released) and vial sealed. If open, the cat is dead; if sealed, the cat is alive. The cat’s state is thus a sum of “dead” and “alive,” not because it is both dead and alive in a simple mixture, but because its wave function is the sum of those classical alternatives.
Now note the difference from chaos. According to the usual interpretation, quantum mechanics here does not reflect epistemic doubt (our ignorance). It’s not that we don’t know which slit the particle passed or whether the cat is alive. Rather, the particle actually is in a sum of both alternatives, and the cat is in a sum of “alive” and “dead.” Probability in chaos reflected lack of knowledge; here, the distribution is a feature of reality itself. Even with a detector, the particle retains a non-zero chance for each outcome until measured; it has not been determined classically. Thus unlike chaos, quantum physics contains an ontic ambiguity: genuine indeterminacy in reality.
A helpful analogy: imagine a particle that can be yellow or blue. Measuring it yields either yellow or blue. Without measurement it is in a superposition of yellow and blue—not literally green (a mixture), but a sum of the pure color-states. If you pour a liquid containing many such particles into a beaker, you will of course see green (by the law of large numbers), but for a single particle the superposition is not a classical mixture.
Back to Determinism
In chapter 2 of my book The Science of Freedom I discussed whether quantum theory can “smuggle” free will into physics. Here we’ve found an ontic margin (ambiguity) within physics, not just an epistemic one as in chaos. Many claim the answer is yes. Briefly, I think not, for two main reasons. First, quantum effects manifest only on very small scales. A single neuron’s firing already lies far above the quantum scale. Second, quantum theory at most yields randomness (with a distribution over outcomes), but as noted, free choice is not randomness. In the quantum scheme, outcomes are governed by the coefficients (α, β in the equation above), not by human free will.
The upshot is: a libertarian must abandon strict physicalism—admit there is more in the world than matter and physical law—because within physics we don’t find room for genuine free will.
For Our Purposes
For our needs here it suffices to define an epistemic margin (doubt) and an ontic margin (ambiguity) in physics. I will use these in upcoming halakhic discussions, where this preface proves quite helpful not only in understanding the halakhic phenomena at issue, but also in making sense of the “mess” that reigns in quantum theory.
Notes (as referenced in the Hebrew original):
1) See chs. 9–10 of my book The Science of Freedom for more on free will.
2) “Chaotic” unpredictability is epistemic: with complete information and unbounded computation, a perfect predictor could in principle know the outcomes.
Article Contents
With God’s help
Ontic and Epistemic Uncertainty I: A Scientific and Conceptual Introduction
In the coming columns I would like to address the relationship between epistemic uncertainty (uncertainty in our knowledge) and ontic uncertainty (uncertainty in reality itself, at the factual level), both in the context of Jewish law and more generally. In this column I will preface the discussion with a scientific and conceptual introduction to the subject.[1] The discussion does not require prior knowledge, but I assume there are readers who are put off by scientific material and mathematical notation. Such readers can read on and focus on my explanations. Coefficients and equations are not really necessary in order to understand the issue (though they are helpful).
Unpredictability in Chaotic Systems
Doyne Farmer, one of the pioneers of chaos theory and a member of the Santa Cruz group that first identified chaos, describes his feeling when he and his colleagues began to understand that they had in their hands a theory of far-reaching physical significance (see James Gleick’s book Chaos, p. 253):
On the philosophical level, it seemed to me like a practical way to define free will, in a manner that allows you to reconcile ‘freedom of choice is granted’ with ‘everything is foreseen.’ The system is deterministic, but you cannot say what it will do in the next moment… Here there is one coin with two sides. Here there is order out of which randomness emerges, and one step away there is randomness with order underlying it.
Farmer argues that this gap in physicalism, which allows us to find unpredictable phenomena within hard physics, is an enclave of randomness within the laws of physics, and he sees in it a possible opening for understanding the phenomenon of free will.
But despite his enthusiasm, the connection Farmer draws here between randomness and free will is baseless, for several reasons. First, freedom of choice is not randomness, but a different mechanism. This is a philosophical misunderstanding of the meaning of free will, which is certainly possible in a physicist. But more serious is a physical misunderstanding of the meaning of chaos. A chaotic system is not truly random. In such a system we indeed have no ability to predict its future behavior on the basis of its present state, but that is only because of the complexity of the calculation and the sensitivity to initial conditions. There is no real freedom here. Farmer is confusing unpredictability with freedom. What leads him to this error is the fact that if a person truly has free choice, then his behavior cannot be predicted. But the converse does not hold: the fact that his behavior cannot be predicted does not necessarily mean that it is free (that is, that it is not deterministic).
Think of tossing a coin or a die. We have no practical way to predict the result we will get, and therefore many people treat such phenomena as random and analyze them using probabilistic tools. But of course there is nothing random here. Tossing a coin or a die are completely deterministic processes, and in principle even simple ones. It is a matter of Newton’s laws and nothing more. The reason the outcome is difficult to calculate is both the great sensitivity to the initial conditions (in what direction and at what speed you threw the coin or the die) and the technical calculation itself (which depends on the shape of those objects, the influence of air currents, and so on). These aspects prevent us from carrying out the calculation, but it is clear to everyone that in principle such a calculation exists. In Talmudic language one could say that ‘it is revealed to Heaven’ what the result will be, that is, God knows it. To use Laplace’s phrase, one may say that if you give me a supercomputer with infinite computational power and memory (together with precise data about the coin, the die, and the initial conditions), I can tell you with certainty what result you will get. True, I do not have all that, and therefore I cannot perform the calculation, but it is still clear that such a calculation exists. This is not a system that contains any kind of freedom.
Epistemic Uncertainty and Ontic Uncertainty
What is the reason for the gap between unpredictability and freedom? What exactly did Doyne Farmer miss? The fact that in cases such as tossing a die or a coin I cannot predict the result stems from deficiencies on my part, not from indeterminacy in reality itself. Reality itself is uniquely determined; that is, for every given state there is one and only one outcome, well defined on the basis of the present state. The lacuna is in us, human beings.
From here on I will say that this is uncertainty (= lack of information) in our knowledge, and in philosophical terminology this is epistemic uncertainty (epistemology is the theory of knowledge). In this terminology, chaotic systems involve epistemic uncertainty. Someone equipped with all the information and all the computational ability can know everything about them.
To understand Farmer’s mistake, we must remember that a libertarian view that upholds the existence of free will (as opposed to determinism) maintains that reality itself is not uniquely determined. Even given a certain set of circumstances and full information about it, a person can still choose freely whether to do X or Y. If so, we are not dealing here with a lack of information on our part (an inability to predict), but with a reality that is itself not uniquely determined. In the Talmudic terminology mentioned above, even God does not truly know what a person will do, and certainly a supercomputer with infinite computational power will not help us here. In such a situation, the point is not that we do not have the right answer, but that there simply is no single answer.
Such a state I will call ontic uncertainty (ontology is the theory of being, a branch of metaphysics), that is, something that is not uniquely determined in reality itself, as opposed to epistemic uncertainty, which concerns our cognition of reality and not reality itself. To sharpen the distinction further, I will not refer to such a state as uncertainty but as indeterminacy. From here on, uncertainty will describe for us a state of lack of information on the part of the person, whereas indeterminacy is a description of the lack of definiteness of reality itself in such cases. As an aside, I note that states of uncertainty are described by probability, while states of indeterminacy are described by fuzzy logic.
Indeterminacy Within Physics: Quantum Theory
It is told of Arthur Eddington (a British astrophysicist and one of the early figures in relativity theory) that when he was told there were three people in the world who understood relativity, he asked who the third was (besides himself and Einstein, of course). Another famous physicist, younger than he was, Richard Feynman, said that relativity is actually not so hard to understand, but quantum theory no one understands. At most, one gets used to it.
One of the strange features of quantum theory is that, at least according to its accepted interpretations, this is the only area in physics in which one can find indeterminacy in reality itself (and not merely limits on our computational ability). There is an ontic gap within physics here, and not only an epistemic gap. I will try here to explain briefly this aspect of quantum theory (again, relying here on the accepted interpretations).
The Double-Slit Experiment
As Richard Feynman used to say, the best way to explain quantum theory is through the double-slit experiment. This experiment has a very interesting and highly changeable history. It begins already in Newton’s time, when a debate took place among physicists about the nature of light. Some claimed that it is composed of particles (like Newton), and others saw it as a wave (the Fresnel-Huygens theory). In 1801, seventy-four years after Newton’s death, Thomas Young conducted the first double-slit experiment in order to decide this issue. In the experimental setup Young used there is a clear difference between a wave and a beam of particles, and precisely for that reason it was useful in deciding the dispute about light. That is exactly also the reason this experimental setup would prove astonishingly useful in the following century as well, for deciding a similar dispute—this time about the nature of particles such as the electron (whether they are wave-like or particle-like).
To understand this, let us begin with the simpler case of a single slit. In the figure below a source (the cylinder on the right) is shown sending a beam of particles or a wave (the thick dashed line) toward a barrier with a slit in it. Behind the barrier there is a screen (depicted by a double line), which is in effect a kind of photographic film sensitive to the impact of the wave or the particles.
Figure 1: The experimental setup of a single slit
The graph on the left side of the figure describes the results of the experiment in the case of a single slit. Its height (on the y-axis) represents the number of particles or the intensity of the wave received at each point on the screen (each point on the x-axis of the graph represents the corresponding point on the screen).
One can see that the greatest amount is received exactly opposite the slit, and on both sides the amount gradually decreases (this represents some of the light or particles that deviated slightly to the side). This picture is correct whether we are dealing with a beam of particles or with a wave. Young’s idea was to distinguish a beam of particles from a wave by means of a similar experiment, except that in it one places a barrier with two slits. To understand this we must know something characteristic of wave phenomena: interference. When two beams of particles move through space, the total number of particles at each point is the numerical sum of the particles from the two beams at that point. By contrast, when two light waves move through space, the total light intensity at each point is not a simple sum of the light intensities of the two waves at the point in question. The intensity at each point in space rises and falls as a result of the interference effect between the waves.[2] Thus, for example, it is possible for two high-intensity waves to sum to zero, since one is positive and the other negative. In some places the intensity of the two waves cancels one another out, and in other places they reinforce one another (the overall intensity is greater than the sum of the intensities).
Now think about the experimental setup proposed by Young, which includes a barrier with two slits. If the source on the right emits a beam of particles (such as electrons, tennis balls, or elephants), some of them will pass through each of the two slits. Around each slit the number of particles will be distributed as in the graph of the single slit (see the graph in Figure 1). If so, in the two-slit experiment, at each given point on the screen the number of particles that arrives there will be the sum of the particles that arrived from the right slit and from the left slit. Therefore the picture on the screen will look like the sum of the pictures of the two single slits; that is, we will obtain a graph with two equal peaks, as described in Figure 2:
Figure 2: The double-slit experiment with a beam of particles
By contrast, when the source sends a wave beam, the phenomenon of interference will cause the picture to look completely different, as described in Figure 3:
Figure 3: The double-slit experiment with a wave beam
In the wave case described in Figure 3,[3] it is דווקא at the center between the two slits (where the particle picture almost vanishes; see Figure 2) that the maximum intensity appears. On both sides of the center there are side lobes whose peaks gradually decline. Thus, the double-slit experiment gives us a sharp distinction between a beam of particles and a wave beam.
In Young’s experiment, conducted as noted at the beginning of the nineteenth century, the double-slit experiment was carried out with a light wave, and the result was unequivocal: a wave picture was obtained there (Figure 3). Thus the dispute between Newton and Huygens (who had by then long been among the dead) was decided, and it was determined that light is a wave phenomenon. But do not worry about Newton; he will return. More than a hundred years later, at the beginning of the twentieth century, various pieces of evidence and arguments began to accumulate in favor of the picture that sees particles as waves. In 1924, a French prince named Louis de Broglie, in his doctoral dissertation, proposed seeing the electron as a wave and even offered a mathematical description of its wave function.
Now a debate begins that is the reverse of the one conducted between Newton and Huygens: are objects we had grown used to seeing as particles (for example, the electron) in fact wave-like by nature? To examine this question, double-slit experiments were carried out again, but this time with a beam of electrons. The expectation was that the picture would look like Figure 2 above, since apparently electrons are particles. The results were astonishing. It turned out that the picture obtained was similar to Figure 3. It followed from this that electrons too are in fact wave-like in nature. If so, electrons behave like light waves, and the distinction between particle and wave seemingly disappeared. It was not clear how a beam of particles could interfere with itself. As I noted, interference is a property of waves and not of particles.
At some stage the hypothesis was raised that the interference is a result of the fact that we are dealing with a beam of electrons, and not with a single electron. Different electrons interfere with one another, and from their meeting a kind of interference pattern is created. The advantage of physics is that it is an empirical science, and so, in order to test this proposal, the double-slit experiment was repeated while the beam of electrons was fired from the source at a very slow and sparse rate. In such a situation, each time one electron struck the screen, and at the same time there was no additional electron there. Therefore interference between two electrons could not have been created. To our astonishment, it turned out that a wave picture was still obtained (like Figure 3). That is, the single electron is also a wave, since it interferes with itself. The astonishing conclusion is that the electron is not a tiny particle located in a particular place (a kind of little ball), as people had thought until then, but rather a wave spread throughout space, exactly like light (and of course we still remember Fresnel-Huygens and Young).
If the electron really interferes with itself, that means that the single electron is in fact a wave that passes through both slits at once. Therefore the two parts of the electron wave (the wave of the single electron) reach the screen from different places and create an interference pattern (Figure 3). This already seems unbearable. A tennis ball passes each time through only one of the slits. How can a particle pass through both slits simultaneously?
To examine that question, we must determine each time through which of the two slits our electron passes, or whether it passes through both. So we naturally return to the laboratory. The experiment was now repeated, but this time a detector was placed near one of the slits (let us call it slit A). When the electron passes through the slit, the detector detects it and reports to us that the electron passed through slit A. If the detector says nothing, the conclusion is that the electron passed through slit B. The double-slit experiment was repeated in the presence of the detector, and here the astonishment was complete. The detectors indeed showed us through which slit the electron had passed, but this time the picture obtained on the screen was a particle picture, namely the graph in Figure 2. The phenomenon of the electron’s interference with itself (Figure 3) disappeared. The electron stopped being a wave and went back to behaving like a well-behaved particle (like a tiny tennis ball).
Clearly, with tennis balls the situation is different. If we perform a double-slit experiment with tennis balls, with or without a detector, the result will always be particle-like (Figure 2). The conclusion is that a tennis ball is not a quantum particle. But the electron, at least so long as there is no detector near one of the slits, behaves like a wave. Therefore the electron is a quantum particle. When there is a detector checking it, it goes back to behaving like a respectable tennis ball, but when nobody is looking at it, it really runs wild. But note that even when there is a detector near the slit, the electron is still not entirely a tennis ball. Even if we fire the electron from the source toward slit A, unlike a tennis ball there is still some chance that the electron will nevertheless pass through slit B. When there is a detector, the single electron does not pass through both slits, but you still cannot know with certainty through which of them it will pass. Only after the detector has measured its location does it begin to behave like an ordinary classical particle (that is, if it passed through slit A, it will of course strike the screen opposite that slit and not on the other side). But as long as it has not actually been measured, there is some chance that the detector will detect it at A and also some chance that it will be detected at B.
Before continuing, I will just remark on the wonders of the historical swing of the pendulum, and say that after quantum theory was formulated, it became clear that light too is not entirely a wave. There are situations in which it behaves as a collection of particles (photons), and there too it depends on whether one places photon detectors near one of the slits or not. They embalmed the embalmer for nothing—Newton returned from the dead.
In the accepted interpretation that later developed for quantum theory (called the ‘Copenhagen interpretation,’ following the Danish physicist Niels Bohr and his school), the conclusion reached was that the meaning of these results is that the distinction between particle and wave lost its ontological significance. In reality itself there are not truly two such kinds of entities. Microscopic physical entities sometimes behave this way and sometimes that way. In other words, wave and particle are different states of the same entity (photon or electron), and not two different kinds of entities.
Quantum Theory: An Initial Interpretation
The conclusion physicists drew from this confusing picture was that so long as one does not look at the electron (does not place a detector), the electron really does pass through both slits at once. This means that the electron is in a state called superposition, which is a sum of pure particle states.
A particle state is a state in which the electron behaves like a little tennis ball, and this is the conception of particles in classical physics. A particle has a defined place in space at every given moment, and if we draw a graph of its position as a function of time we will obtain a definite trajectory in space, exactly as in the case of a tennis ball. In one pure particle state (= a classical state), the trajectory of the electron passes through slit A, that is, it moves along a trajectory like a tennis ball from the source to the screen. Let us denote this state as follows: |A>. In another pure particle state, which we denote |B>, the trajectory of the electron passes through slit B, in exactly the same way. In both these cases the electron behaves like a little tennis ball. But such a pure particle state is obtained only when we measure the electron’s position by means of a detector: if the detector shows us that the particle passed through slit A, that means that the electron is in the first pure particle state. And if the detector shows that the electron passed through slit B, that means that the electron is in the second pure particle state.
What happens if we do not measure the electron’s position at all (by means of a detector)? In that case the electron is in a superposition of such pure particle states: its trajectory is the sum of trajectories A and B, and its state is the sum of the two particle states. In everyday language people usually describe this as though the particle passes through both A and B, and therefore also interferes with itself like a wave. The ordinary state of an electron is composed of a combination of very many particle states (trajectories), and not only two. This combination of pure particle states creates a picture that resembles a wave. The electron, as it were, splits, and each part of it follows a different trajectory. Thus our electron follows both the trajectory that passes through A and the trajectory that passes through B, and many other trajectories as well, and therefore it is in effect spread throughout space. Just like a wave (or, as Natan Yonatan put it, ‘just like a shore’).
That is the case as long as we have not measured the particle’s position. But when we do measure its position, it suddenly ‘collapses,’ that is, it takes on the form of one pure particle state from among those that made up its superposition (its wave function). Which of the particle states will it be? The wave function defines the probability of obtaining each particle state. Placing the detector near the slit is in effect carrying out a lottery that selects one particle state from among those that make up the wave function, with the distribution for obtaining each state defined by the values of the function. If we did not place a detector, the particle is spread across all the particle states together.
For the sake of what follows, let us denote the superposition state between two particle states as follows:
When the state |A> is a trajectory in which the particle behaves like a point particle that passes through slit A, and the chance of obtaining it (if one places a detector near the slit) is defined by a coefficient, and the state |B> is a state in which the particle behaves like a point particle that passes through slit B, and the chance of obtaining it (if one places a detector near the slit) is defined by its coefficient. The general state is the superposition of these two states. If we place a detector, the particle will pass either only through slit A (and this will happen with the corresponding probability) or only through slit B (and this will happen with the corresponding probability).
*It is worth noting that the superposition state does not mean that the particle passed through both slits at once, but rather that its wave function is composed of two functions, each of which describes a state of a particle that passes through one of the slits. In everyday language one may say that part of the particle passes through each slit.*
Quantum Theory: Ontic Indeterminacy
In the well-known thought experiment called ‘Schrödinger’s cat,’ one places a cat in a closed box with a sealed vial of poison. The stopper of the vial is controlled by a quantum process, and its state is described by a sum of two simple (classical) states, as in the equation above. Except that here the state |A> is that the vial is open, and the state |B> is that it is closed. But if the vial is open, the poison spreads through the box and kills the cat, that is, the cat is dead. And if the vial is sealed, then the cat is alive. The meaning of a superposition state is that the cat is in a state that is a sum of a live state and a dead state. Again, it is not accurate to say that the cat is both alive and dead, but rather that its quantum state (its wave function) is composed of two classical states: |alive> + |dead>.
Now we must pay attention to the difference between quantum theory and what we saw in chaos. At least according to the accepted interpretation, in the case of quantum theory this is not epistemic uncertainty, that is, a lack in our information. It is not only that we do not know through which slit the particle passed, or whether the cat is alive or dead. This is indeterminacy in reality itself, that is, the particle really does pass through both slits and the cat is in the sum of the two states. This does not describe a probability that stems from our lack of information about reality. This is the state of reality itself. As noted, even if there is a detector, the particle still has some probability of ending up in either of the two states, which means that it is still not a simple classical particle. Thus, unlike what we saw regarding chaos, quantum theory is a physical context in which one can encounter indeterminacy in reality itself.
To understand this better, think of a particle that can be yellow or blue (a superposition of |yellow> + |blue>). When we measure it, we may obtain either of two results: yellow or blue. If we do not measure it, it is in a superposition of yellow and blue, but that is not exactly the color green, because it is not correct to say that the particle is a mixture of blue and yellow. The more accurate description is that its state is a sum of a blue-particle state and a yellow-particle state. By contrast, when you pour a liquid made up of many such little balls into a large container, the color of the liquid as a whole will of course always be green, by the law of large numbers.[4]
Back to the Question of Determinism
In my book The Science of Freedom (in chapter ten) I dealt with the question whether free will can be introduced into physics through quantum theory. Here we have already found an ontic gap (indeterminacy) within physics and not only an epistemic gap, as in chaos, and therefore many argue that it can. Briefly, I would say that in my opinion quantum theory cannot be used to do this, for two main reasons: quantum phenomena appear only on very small scales. Our choice does not occur on those scales (even a single neuron is very large relative to the quantum scale). Beyond that, quantum theory gives us at most a space for randomness (with a distribution function for the results) within physics, but as we saw, free choice is not randomness. In other words, according to quantum theory the result is determined by the distribution (the coefficients in the equation above), and is not given over to the free choice of human beings.
The conclusion is that if someone advocates libertarianism, he must give up physicalism; that is, he must assume that there is something in the world beyond matter and the laws of physics. It seems that free will cannot be inserted into the laws of physics.
For our purposes here, it is enough for me to define the concepts of an epistemic gap (uncertainty) and an ontic gap (indeterminacy) in physics. I will use them in subsequent columns in Jewish-legal contexts, and there we will see that this introduction is very helpful for understanding the legal phenomena under discussion, and perhaps no less so for understanding the mess that prevails in quantum theory.
1.
Footnotes
- See chapters 9–10 of my book The Science of Freedom.
- The waves are added by summing the values of the function that describes the wave, but the intensities are the squares of the values of the function. Hence it is clear that the sum of the intensities of the two waves (the sum of the squares of the values of the two functions) is not equal to the intensity of the wave composed of both of them (the square of the sum of the values of the function).
- For the intensity distribution to be as described here in Figure 3, a certain distance between the two slits is required, depending on the wavelength of the wave beam. I do not need those details here, so as not to complicate the description.
- That is one of the reasons that in large systems, that is, systems composed of many particles, one does not see quantum phenomena. They are smeared out by the law of large numbers, and the result obtained is the classical result. Therefore a tennis ball, which is a body composed of masses of tiny particles, behaves the way we know and does not go wild as we saw with electrons.
Discussion
Indeed. We’ll get to that in the next column.
Thanks for the fascinating column.
If I understood Frumer correctly, he argues that chaos is not computable (as opposed to your claim that it is simply complicated to compute), and nevertheless things can be calculated.
And then he projects the case onto the religious language of knowledge and choice.
And that is how he tries to escape the contradiction, by way of proving that this is possible and even happens in reality.
(That is, that there exists in reality a situation in which you do not know what the tiny causal factor will do, and nevertheless,{perhaps via the law of large numbers} you know what the outcome will be.)
If that is what he claims, he is mistaken. But he did understand chaos, so that is not what he claimed.
If that is what he claims, he is mistaken. He understood chaos, so that is not what he claimed.
In connection with the article (especially its beginning), I was always told that the difference between a case that is a psik reisha and a case that is not depends on whether there is uncertainty about what will happen in reality as a result of the act, or whether it is clear in advance what will happen. And I always found this difficult – after all, from the standpoint of reality itself there is no such thing as doubt, and if we knew all the data (for example – the weight of the bench, the moisture of the ground, and the angle of dragging) we would know how to calculate in advance whether there will be a furrow or not. So why does the point that we are not wise enough to calculate it prevent the case from being a psik reisha?
In the next columns
If we are discussing physics, it is worth noting that in the last ten or twenty years, the de Broglie–Bohm interpretation (the one that supports hidden variables, without indeterminacy, and is opposed to the Copenhagen interpretation) has been gaining significant momentum, mainly because of experiments that succeeded in reconstructing the trajectories of single photons fired through two slits—that is, obtaining the diffraction and interference pattern on the screen while at the same time knowing through which slit the photons passed (of course, the experiments “succeeded” subject to a few asterisks, footnotes, and certain reservations, but this is still significant progress). For details, you can google:
Observing the Average Trajectories of Single Photons in a Two-Slit Interferometer
Since I am not a physicist, I am taking my life in my hands to ask a question that may be completely foolish and stem purely from ignorance.
You claim that even a neuron is too large in quantum terms, and therefore quantum theory cannot serve as a basis for free choice. My question is whether this is necessarily so. That is, might it be that even though the neuron’s response to some stimulus expresses a process of chaos that is seemingly completely deterministic, perhaps at the beginning of the process there is some single particle that set the process in motion from the start, and that particle is fully quantum. The analogy I can think of is a machine that has several codes that activate it. Each code causes the machine to perform a different action (or to choose from a defined set of actions for that code through a chaotic process), but the choice of codes is made through a quantum process.
I now recall that many years ago I asked a physics professor (who has since retired) exactly this question, and he nodded his head and said “maybe,” but was not willing to say anything definite one way or the other. Can you guess why he was not willing to say anything more decisive? (From my perspective, as a layman, either the answer is yes or it is no. Can the answer to a question also be in superposition?…). Or do you have a definite answer and disagree with him?
There is such a proposal by Rabbi Professor Yehuda Levi. In my book The Science of Freedom I explained why this cannot be. In large systems, the small effects get smeared out. For that not to happen, all the degrees of freedom would have to be coordinated, and that does not happen in noisy systems, only in liquids or conductors, and even there it happens only at extremely low temperatures. That is not the situation in neurons.
I will only mention my second argument: even if quantum effects were relevant to the neuron, that would yield randomness, not choice.
I have two questions, which I hope can be answered without teaching me the entire theory of quantum mechanics on one foot –
A. Can it be explained that the collapse of the superposition into one of the states is actually caused by the detector? I do not know how the detector works, but presumably it affects in some way the things it detects, and perhaps on the scale of an electron that effect is already critical.
B. Are such superposition realities indeed an ontic doubt, or do they simply reveal to us a limitation in the categories of thought? That is, could it be that Schrödinger’s experiment (if it existed literally) does not show that the cat is alive-dead, but rather that alive and dead are insufficient categories, and creatures can belong to neither of them?
A. It is commonly thought that the detector is what causes the collapse.
B. It is commonly thought that this is indeed ontic. It depends on the interpretations, which is why I wrote that I am following the accepted interpretation. That does not contradict what you wrote at the end. I tend to agree that what really exists is neither a wave nor a particle but some entity that has two kinds of states. You can call that a limitation of the categories of thought. But that entity can still turn into a wave or a particle from that very same state. There is an ontic doubt here.
Wonderful article. Waiting for part 2 🙂
It seems to me (though perhaps I am mistaken) that the following story should be brought in here:
R. Baruch Ber Leibowitz was discussing, in learning, with his students the sugya of “one who betroths one out of five.” In the heat of the argument one of the students said to R. Baruch Ber, “Here even the Holy One, blessed be He, does not know who the one betrothed is…” R. Baruch Ber turned completely pale and responded immediately and sharply: “Slow down. We only say, ‘It is not revealed before Heaven.’ What is written is written; do not go too far with words.”
Nice 🙂
Not-a-physicist, keep in mind that the problem does not end with this explanation, and laying the blame on the detector creates a new problem, around which there is also debate as to where exactly the measurement process occurs. In the detector or in the person? And where in the detector? Or where in the person? Perhaps the entire detector is also part of the observed system, and the retina in the human eye is the detector? In short, where exactly does the collapse of the wave function occur? This is called the ‘Heisenberg cut,’ and here too there are interpretations as to where it lies. For example, the extreme von Neumann approach claims that even the information-processing system in the human brain is part of the observed system, and measurement occurs only when the person actually becomes conscious of the result, and not before that.
There are also other approaches—for example, Renninger’s thought experiment from 1953, which shows how a wave function can collapse even without a measurement (this is an experiment in which the measurement was not actually performed, and yet one can infer the state of the system and collapse the wave function). The Einstein–Rosen paradox also relates to this; there too a particle collapses into a certain state even though no direct interaction was performed with it.
What is nice?
Mordechai, I join your view that on the face of it this is possible.
I raised this claim in the past in a comment on another post on the site, and a bit of discussion developed around it.
If you want, you can read it here:
https://mikyab.net/posts/66535#comment-34284
Do not put words in my mouth. I did not express an opinion; I asked a question. I am not a physicist, and I do not tend to express opinions in fields in which I am not an authority.
I am also not playing at humility here. In fields in which I feel firmer ground under my feet, I have not hesitated to go against Rabbi Michi, and at times sharply so (and he too, astonishingly enough, was not intimidated by me and did not flee to a monastery of the silent…). In this field my knowledge is limited, so at most I ask questions but do not express opinions.
Since my message appears to be nonsense, I will explain. To phrase it as “even God does not know” is a precise formulation and hits exactly the point. If one separates “Heaven” from God, one loses the whole idea (because if God knows, then it is an ordinary doubt). Admittedly, at first glance it sounds provocative, but it presents precisely the idea that the concrete knowledge (who is betrothed) simply does not exist (is not defined), and once you understand that, it completely stops being provocative. Did the Gemara not use similar formulations in order to clarify ideas? “My children have defeated Me,” “What does My son so-and-so say,” the dispute of the Holy One, blessed be He, and the Heavenly Academy, and Maimonides’ ruling there in halakhah 9 like the Heavenly Academy—is that lacking? Caution in correct formulations is relevant only if one fears mockery and demagoguery, but within the study hall that makes no sense, because there is no benefit in error. Therefore within the study hall they said without any problem, “Woe to me if I speak, for they will learn deceit and forgery; and woe to me if I do not speak,” and in the end they said, “for the ways of the Lord are upright,” etc. In short, not a nice story at all.
There is an article by Professor Yakir Aharonov in issue no. 4 of the scientific journal Odyssey that explains quantum theory in a way that, to the best of my understanding, matches the explanation here. He describes quantum theory mainly by a renewed explanation of the concept of time: unlike the usual understanding that the future develops from the present, the present and the past too are created as a result of the future. Consequently, although there is an ontic doubt regarding the present, this does not mean that one can actually choose in the present, but only that the future reveals retroactively what was in the past and in the present.
See there: https://www.teva.co.il/about-teva/social-responsibility2/odyssey/
The revered author here wrote as follows: “Nice,” end quote; and in the next column he wrote as follows: “Even the Holy One, blessed be He Himself, if asked, would not know how to tell us which of them is betrothed,” end quote. Understand this.
In the future, one should quote precisely, and then the question does not arise in the first place.
What the revered author wrote exactly was: Nice :).
Study this carefully.
If I understood correctly, the electron is an entity that has two states, and it cannot “turn into a wave or a particle from that very same state” – because in a situation without a detector it will turn into a wave, and in a situation with a detector it will turn into a particle.
If so, the only ontic doubt is in the situation with a detector – whether the entity will turn into a particle that passes through A or through B. Did I understand correctly?
This is a matter of interpretation. I tend to think as you do. In my opinion, in a situation where there is no detector, what we have before us is not a particle but a wave described by the wave function, and it indeed passes through both slits (more accurately: it is composed of the sum of the two particle functions). An ontic doubt exists only regarding which slit it will pass through after there is a collapse into a particle state. In that state we have a particle, and therefore there is room for doubt whether it passes through slit A or B, and that is an ontic doubt.
As for the story itself, I think it is right to direct our attention in another direction. R. Baruch Ber did not mean to emphasize that the information exists with the Holy One, blessed be He, in some personal secret vault and only “Heaven” and the Heavenly Academy are not updated. If that was indeed his intention, then the claim of the writer above is justified: we have done nothing except add a higher level to the epistemic lack of information. In my humble opinion, R. Baruch Ber meant to argue that our failure to grasp the way the Master of the Universe works means that it could still be that He has a position on the question of who is really betrothed, and yet this would still be a doubt about which there is no information anywhere. For the Master of the Universe is above place, time, and logic. He mainly intended to silence the student from saying things about which we have no tradition. (And see Rubin’s book, What God Can’t Do.)
That fits the distinction between “God does not know” and “the Holy One, blessed be He, would not know how to tell.” In my view it is still nonsense. Even the compromise Rubin proposed at the end of the book is nothing. Everything we say is mediated through our logic and our time. Nor is it clear how uprooting the difficulty with the little smiley helps. An ugly story. Only if it serves as a critique of etc. will it become nice. And one cannot say that the rummaging proves beauty in the manner of a peshar doubt.
The little smiley helps convey the criticism; sometimes it is preferable to a short, concise but opaque sentence.
How could the story be nice in your eyes?
The topic has been beaten to death and I have nothing to add. Everything written in the article here https://tinyurl.com/y2r8btkf is correct in my opinion. Just a few small Hinduizations on the margins. In Rubin’s entire book there is no additional theoretical contribution (that is correct) on this. The book is part of the above article with the history and genealogy of ideas.
A story that illustrates or argues something true in a pointed way is nice. A story that illustrates a failure is nice only as such, not in its content.
But I have troubled everyone quite enough on this marginal issue, and I hereby respectfully withdraw my hands.
Hello Rabbi,
There is something I did not understand – is a doubt that depends on a person’s choice, assuming there is free choice, an ontic doubt or an epistemological one? That is, is the doubt whether Reuven will decide to choose the tree or the pli (not what will come out in the end) an ontological one?
Epistemic. Consider the question regarding a choice that was made in the past; there it is certainly epistemic. There is no fundamental difference between the past and the future. If you ask what that person will choose, meaning a question about his current state before he chose, that is a somewhat different question, but here too there is no ontic indeterminacy. It is not a state in the present but nothing. When he chooses, there will be a choice.
Hello, can the page be fixed? The text does not appear, also in the follow-up column (323)
Strange. I passed it on to Oren the editor for him to take care of.
A beautiful column, thank you!
It seems to me that these points are difficult in relation to the laws of uncertainty, and to R. Shimon Shkop’s statement that when he betrothed several women, they are all betrothed; this is an ontic doubt and not merely an epistemic one. As opposed to a case where one betrothed one woman and it is not known who she is, which is an epistemic doubt, etc.