On Platonism – Second Continuation (Column 385)
In the previous two columns I defined Platonism, and then distinguished between Socratic and Platonic emotions. I will now conclude this mini-series with a discussion of specific aspects of the Platonic outlook—in science, mathematics, halakhah, and ethics.
Scientific Platonism
It is commonly thought that modern science follows Aristotle, since scientists tend not to accept the existence of entities that have not been empirically observed (this is likely also why various surveys show that among scientists the percentage of those who believe in God is lower than in the general population). Yet on a closer look, there are Platonic dimensions at the foundation of scientific thinking. A salient manifestation of this is the debate in the philosophy of science over the status of theoretical entities. Physical theories deal with entities that are not directly observable (some of them, over time, reach a state where they can be observed more directly). Theoretical entities such as an electric field, an electron, a quantum wave function, a quark, and the like are the building blocks of central theories in physics, but one cannot truly observe them directly. Physicists posit their existence only because it provides a (relatively) simple, elegant, and convenient explanation for the phenomena we have observed.
For this very reason, some philosophers of science treat them as nothing but useful fictions—that is, theoretical notions whose referents do not exist in reality. Using them is helpful at the theoretical level because they allow us to formulate the theory in a way that fits and is convenient for our thinking and for various applications. Everyone agrees that there are always other possible explanations for the same set of facts, but they are less elegant, convenient, and efficient, so we do not choose them and instead prefer the simple and efficient explanation (Occam’s razor). From this you can understand that physicists have no indication that this explanation is true and that the entities it includes exist, apart from the fact that they constitute an efficient and simple explanation of the phenomena. (Remind you of the physico-theological proof for the existence of God? No—that’s just religious, primitive, and benighted.) This matter is examined more in terms of efficiency than in terms of truth. The approach that sees these entities as useful conventions (like Aristotle’s categories) is an anti-realist or instrumentalist view of science. In contrast to this view, realists regard theoretical entities as existing entities, just like a stone or a chair (see about this here. Three main arguments against scientific realism are presented there).
This dispute rages mainly among philosophers of science. In my impression, most scientists belong to the realist school, thereby holding a certain kind of Platonism. Theoretical entities are abstract objects not subject to (at least for now) direct observation, and yet through their fingerprints we conclude that they exist. The efficiency of the theory, according to adherents of this view, constitutes an indication of the existence of these entities—that is, that they are not mere fictions—just as for Plato the concrete entities provide an indication for the existence of Ideas. Indeed, as I already noted, there have been cases in which, over time, technology and science developed and theoretical entities became subject to direct empirical observation, and then it turned out that they are indeed real entities.[1] Such phenomena corroborate the realist-Platonic thesis.[2]
One must understand that the theoretical and meta-scientific debate between instrumentalists and realist-Platonists has implications for the conception of science in general—for example, for the question of what scientific research should focus on, and especially for its aims. Realists usually hold that the goal of scientific research is to uncover the hidden deep dimensions of the world (those not given to our direct observation). Laboratory observations are made on events and observable objects, but this is done in order to arrive at the formulation of general laws that concern theories and theoretical entities, which are the primary goal of scientific research. The facts do not serve the theory; rather, the facts serve the theory. According to this approach, even if we had all possible scientific facts, science would still not have begun its activity. Science is supposed to use those facts to build theories. It is difficult to accept such a view if you are not a Platonist—that is, if you think these entities and theories are no more than a useful human invention.
Instrumentalists, by contrast, maintain that theory is nothing but an efficient and convenient way of organizing the facts, and is entirely our own invention. Its value lies in the service it provides for retaining and using information about the totality of facts. On their view, it is plausible that the goal of science is the collection of all facts, and theory serves us for the simple, efficient, and useful organization of this informational corpus. Theory is not science’s goal but a tool in the service of gathering facts, which is its main aim.
Scholastic Platonism
I have written and said more than once in the past that a similar dispute is conducted regarding Torah study. Rabbi Ovadia and his students/sons hold that “lamdanic” reasonings (=theories) serve halakhah—that is, at most they are means to organize and streamline the use of halakhic data in order to rule correctly for any given case. Once we know the proper ruling for every situation, our scholarly work is done. In their view, scholastic analysis (lamdanut) has no value in and of itself; at most it serves the collection of halakhic information. This is lamdanic-halakhic instrumentalism. In contrast, yeshiva-style scholasticism assumes (sometimes unconsciously) that the goal of study is the lamdanic theory and reasonings. The “facts,” namely the halakhic rulings in specific cases, merely serve this goal and do not constitute the target of scholarly activity. This is scholastic realism.
In my article, “What is ‘Chalut’?,” I addressed this difference somewhat. You can see there that in the scholastic-realist approach, theoretical notions in the lamdanic world are the main goal of halakhic-talmudic study and analysis, and I further showed there the common scholastic assumption that chaluyot are existing entities—practically Platonic Ideas (the “chalut of eshet ish,” a married woman, is the lamdanic expression of the Idea of “married-woman-ness”). On this view, the dispute is between a scholastic Platonism, which strives to understand the Ideas—that is, the theories and general concepts (the reasonings)—and an Aristotelianism that sees theory merely as an organizing instrument and not as truth in itself.
It is worth mentioning here that in my article on okimtot (not for nothing titled “A Platonic Look at Okimtot”) I demonstrated the significance of this approach. There too I showed that the aim of scholastic analysis is to uncover deep structures, theories, and general reasonings of which the specific halakhot are only external expressions. I note that in both articles I did not only present the Platonic approach and its implications but also adduced proofs for it from the Talmud and its commentators. Also in volume XI of the Talmudic Logic series, devoted entirely to the Platonic nature of the Talmud, I discussed various sugyot that point to the Platonic character of talmudic discussions in general (through examples of discussions with no practical ramifications and situations that cannot actually occur, and more).[3]
Ethical Platonism
A similar question can be raised regarding the ethical domain. I have argued in the past (see, for example, in the third part of the fourth booklet) that there is no possibility of holding a consistent ethical position unless one is an ethical realist. One cannot speak of valid and binding values—moral or halakhic—without positing the existence of the Idea of the Good, observation of which yields the distinction between good and evil. Of course one can behave well without ideologies (simply because I feel like it, or because that’s how I’m built), and also without demanding of others or judging others for their behavior. But morality in its full sense entails judging others and demanding moral behavior from them. Such an approach requires ethical realism—that is, positing the existence of Ideas, comparison to which yields these distinctions.[4]
In the terms of the first column (383), one can say that here too we are dealing with a kind of taxonomy: a classification of actions and attitudes, and their arrangement on a hierarchical map of good and evil. Observation of the Idea of the Good is what gives validity to this classification and to the demands it places upon us. This is Platonism in the same sense as the scientific Platonism described above, and also in the same sense as the Platonism I described in that column regarding taxonomies of various kinds. Comparison or classification requires a standard against which and by which one compares, and that standard is a theoretical entity—an Idea. It is not observable by the senses, hence not exposed to the tools of science; yet it does exist in an objective sense outside each of us, and is exposed to Platonic observation by the eyes of the intellect (“ideatic seeing,” in Husserl’s terminology).
Mathematical Platonism
In the second part of my book, That Which Is and That Which Is Not, I dealt with the question of Platonism in mathematics. Mathematics differs from the sciences mainly in that it is not empirical. Mathematical theorems are not the result of observation but of thought, and therefore I showed there that they are also not subject to empirical refutation. This raises the question whether mathematical concepts and structures are existing entities, or whether they are only forms of our abstract thought. Are number, group, set, space, point, line, triangle, and the like entities—or merely mental concepts? One can formulate the question differently: does mathematics deal with something outside us, or only with ourselves? It seems easy to agree that mathematics is necessary, but its necessity could be tied to the structure of our thinking and not to properties of reality. On this view, there could be other beings who would not agree with our mathematics and would have a different mathematics, if any at all.
Mathematical Platonism holds that this is impossible. It sees mathematics as a field that investigates dimensions of objective reality, even if these are dimensions not observed by the senses but by the “eyes of the mind.” It is a philosophical science—that is, a field based on observation, but not observation by the senses; rather, intellectual observations. I note that this is precisely how I defined philosophy in the series devoted to the topic (see columns 155–160). In this conception, mathematics is a formal branch of philosophy.
As is well known, Kant defined mathematical theorems as synthetic-a priori claims—claims which, on the one hand, are not the results of observation (a priori), and on the other hand are not mere analyses of concepts but assert something about reality beyond what is contained in the concepts by their very definitions (that is, they are synthetic). This reflects a Platonic conception of mathematics, since he sees mathematics as an investigation of something outside us and our concepts. In a certain sense, mathematics makes claims about the world, even though it is not scientific since it is not subject to empirical refutation (see on this in column 50 and column 318).
It is important to understand that the Platonic claim is not that there are triangles or sets in the world. That point is clear and agreed upon even by those with a non-Platonic view. The heart of the Platonic view is that concepts like “triangle,” “set,” or “the number 5” are entities—that there are such Ideas in the world of Ideas.
A Look at Geometry
At first glance, geometry is a mathematical field that deals directly with the world, since its theorems describe various relations in our concrete world. But that is an incorrect description. In the book mentioned above I explained that physical space is at most a model for the abstract geometric theory. Unlike the abstract mathematical theory itself—which is the mathematician’s concern—the question of what the geometry of the world is, is an empirical question that occupies the physicist. Thus, for example, one can measure the sums of the angles in triangles we draw in our world, and if we indeed get 180° in all the measurements, we can make an inductive generalization about all triangles and establish an empirical law: the sum of the angles in every triangle in our real world is 180 degrees (i.e., that the geometry of our world is Euclidean).[5] But this is an empirical claim and not a mathematical one. Mathematics states that given certain assumptions this is the sum of the angles in a triangle, but it says nothing about the geometry of our world.
Thus the geometric theory belongs to mathematics, and the debate over Platonism does not concern the existence of triangles (which is a claim in physics) but the existence of the Idea of the triangle (which is the subject of the mathematician’s research and occupation). One can repeat these arguments with respect to other domains in mathematics, such as numbers in arithmetic, the concepts of logic, and so on.
Yet there is an aspect of geometry that may strengthen precisely here the Platonic intuition that geometry speaks about something in reality itself. Euclidean geometry does not necessarily deal with the space of our world (as noted, that is the physicist’s business), but it seems that it does deal with some theoretical space. Unlike other areas in mathematics, here we can even visualize it before our eyes. It has a visual presence and not just a conceptual-abstract one like the rest of mathematics. It seems obvious, regardless of the distinctions I offered above, that there is an imaginary, or theoretical, space—very similar to the concrete one—in which the sum of the angles in a triangle is certainly 180°. Even if our real space is somewhat curved and Euclidean geometry’s theorems are not valid for it, with respect to that imaginary space the theorems are necessarily true (that is, not subject to refutation). There is a strong sense that even if this theoretical-imaginary space does not exist in our world in a tangible sense, it is indeed a real entity. One can observe it in imagination, or with the eyes of the mind.[6] In this sense geometry is experienced by us as the result of “observation,” and not merely of abstract thought. Platonists claim this with respect to all fields of mathematics, but in geometry it is very easy to realize and be persuaded by this intuition.[7]
Therefore, at least with respect to geometry it is hard to escape the understanding that its theorems concern, in some sense, a reality that belongs to the Platonic world of Ideas. In geometry we actually encounter it in our cognition. The simple experience is that these theorems are not fully a priori. We do not create this imaginary world ex nihilo; rather, it is present in our thought—that is, impressed within the thought of every person—and we observe it. The Platonic feeling is very strong with respect to geometry. This, of course, raises the possibility that in other areas of mathematics as well we are observing Ideas that exist in some sense, and not merely a subjective mental structure of ours.
A Look at Logic
A similar claim can be made regarding logic. Yuval Steinitz, in his book Invitation to Philosophy, notes that we tend to infer from the existence of a logical contradiction in a concept or in an event their impossibility—that is, their non-existence in the world. Quine brought the example of the round square dome of Berkeley College, which we know does not exist in reality simply because the “square-round” conjunction is contradictory. Steinitz argues that there is an apparently unjustified leap here from logic, which occurs in our mind, to conclusions about the objective world. These are negative synthetic-a priori claims. The assumption is that logical negation and contradiction concern reality itself, and thus by contemplating Ideas (for this is an a priori argument) one can infer conclusions about the tangible world (which is an expression or instantiation of the Ideas).
The claim is that these entities or worlds are not only considered by us abstractly; rather, this thinking also has certain visual dimensions (we discern them by a procedure that is quasi-observational). Hence it seems that these Ideas have existence in a Platonic sense.
It is worth noting that, as we saw regarding scientific Platonism, in mathematics too the debate over Platonism rages mainly among philosophers of mathematics. Most mathematicians themselves incline to Platonism (I once conducted a poll among mathematicians who attended a lecture I gave at Bar-Ilan’s Department of Mathematics).
Mathematics and Empirical Science
We now arrive at a somewhat surprising conclusion: mathematics is not so different from empirical science. The theoretical entities of empirical science are not “out there” in the concrete world, in a way very similar to mathematics’ entities. Both fields involve observations of the non-sensory kind and investigate the abstract concepts that are the objects of such observation. True, science also uses sensory observations, and therefore it is usually classified as empirical—unlike mathematics.
The conclusion is that, according to the synthetic position presented here, the claims of empirical science are synthetic-a priori, and mathematical theorems are also synthetic-a priori. Science makes claims about the world via claims about an “as-if world,” and mathematics makes claims about an “as-if world” too (a Platonic world of Ideas). The former are inferred by sensory observation of concrete objects (along with generalization and observation of theoretical, ideatic entities), and the latter are inferred by “as-if observation” alone. The resulting difference is that scientific claims are refutable, while mathematical theorems are not, and yet the procedures are quite similar.
Note that according to the account offered here, geometry also asserts the axiom “Through two points there passes exactly one straight line” itself, and is not concerned only with the entailment relations of the theorems derived from it (i.e., that if we accept this axiom it follows that the sum of the angles in a triangle is 180°). The “as-if observation” in the world of Ideas yields the axioms and the definitions of the concepts, and the mathematical proof only derives from those “observations” the conclusions that follow.
A Proof Against Platonism in Mathematics
Being mathematics, attempts were made even to examine logically the claim that mathematical concepts are existing entities. Russell, for example, raised the following claim against the Platonic existence of sets. Suppose there are N entities in the world (N may also be infinite). Every subset of N is also a set, and according to the Platonic assumption it is an existing entity. Yet there is a theorem in mathematics that says that the number of subsets of a set with N elements is: N2. Mathematicians have proved that always (even when N is infinite) it holds that: N2 > N. But if all sets are existing entities, then the meaning of this conclusion is that the number of entities in the world is greater than N, which contradicts our assumption. In other words, Platonism leads to a contradiction.
It seems to me that this argument is problematic irrespective of Platonism. The indication is that one can raise it with respect to the concept of set in its anti-realist (non-Platonic) sense as well. Even if we treat sets as concepts that exist only in our thought and not in any world of Ideas, we can still restate this argument and arrive at a contradiction (think of N as the quantity of concepts we have in the mind). In a sense, this argument undermines the very concept of set, not necessarily in its Platonic meaning. As is well known, Russell himself, in his famous paradox, undermined the intuitive definition of set and thereby led to the development of an axiomatic, formal set theory. I therefore doubt to what extent his earlier discussion, conducted within the intuitive framework, can yield conclusions against Platonism.
In any case, even if Russell’s argument were correct, at most it would prove that not every set is an entity; but one can still remain with a more minor Platonic view—that there are also sets that exist (even if not all), or that the concept of set (the Idea of set-hood) is an existing entity—and that is the heart of the Platonic position. Platonism speaks of the existence of the Idea of set-hood and its investigation by observing it, and not necessarily of the existence of the Idea of some particular set, such as the set of numbers {1,2,5}, or of a set of five cats in my yard.[8] Mathematical research concerns properties of sets as such, not of particular sets, and if such research is observation of Ideas with the eyes of the intellect, then the Platonist need only posit the existence of the Idea of set and its ramifications, and not necessarily the existence of Ideas that represent concrete sets.
From this comes an answer to Tolginus’s question here about the existence of the function x squared. I can now answer that there is no need to say that, according to Platonism, this function exists. The claim is that the concept of function has a Platonic Idea, but not this particular function. Indeed, there is no difference between a function and a group or a set (as he wondered further there). In all these contexts only the general concepts are expressions of existing Ideas, not a particular group, set, or function.[9]
An Implication Regarding Something Not Yet in Existence
In my article on intellectual property I distinguished between an abstract property that is incidental to a concrete thing (like the smell of an apple or the “eye” of honey; see Maimonides, Laws of Sales 22:14) and the absolutely abstract—like intellectual property. I argued there that Maimonides’ ruling, that there is no ownership of an abstract property, applies only when dealing with something abstract that is incidental to a concrete thing. The reason is that ownership of the apple entails ownership of its smell; therefore one cannot speak of ownership of the apple’s smell as such (perhaps ownership of an apple for its fragrance is possible, but not ownership of the smell itself). But, as I argued there, ownership of an idea—that is, intellectual property—is not negated by that claim. An idea is entirely abstract, not something incidental to a concrete thing. There is no impediment to ownership of such an abstract thing.
On the Platonic approach to halakhah, it seems that an idea is something that exists, and therefore ownership applies to it. This is in contrast to a property of a concrete thing, which is not an entity in itself. The speed of a body is not an entity, not even an abstract entity. Likewise for its redness. Concepts such as speed or redness are Ideas, and Platonists will say that they are existing entities—but not the redness or speed of a particular object. A particular property of a particular object does not necessarily have an Idea (as Russell’s argument above may show). This is quite similar to the distinction I made above between the Idea of set or function and particular sets or functions.
[1] One can dispute, regarding various kinds of observation, to what extent this is really observation in the full sense, since in most cases it is still indirect acquaintance through complex measuring systems.
[2] I do not mean to say that there is certainty about their existence, or that these claims are immune to error. My claim is that we are dealing with a reasonable belief, and as long as it has not been refuted, there is no bar to holding it. There may certainly be cases where we discover that some theoretical entities do not exist (and perhaps the theory built from them will also be refuted, though this is not necessary; they can still serve as useful fictions).
[3] See also in the Q&A here the discussion about ownership in something not yet in existence.
[4] Admittedly, if one believes in the existence of God, one can perhaps see the duty to do good as an obligation to His command, and then perhaps the assumption of ethical realism is not required. Ethical contemplation is a gaze inward at the soul (the conscience), in which the divine demands are imprinted.
[5] Today it is known that there are various measurements of physical space indicating that it is not exactly Euclidean—that is, that the sum of the angles in a triangle in physical space differs slightly from 180. Different interpretations may be proposed, but on the face of it there is here an apparent proof that the statement “the sum of the angles in a triangle is 180” is an empirical statement that can be refuted, and as such it does not belong to mathematics.
[6] Note that even physical space as such is not really observable. One can examine various properties of it, like the sum of the angles in a triangle drawn within it, and so on. Real space appears to us through various intermediaries. Even in physics we speak of it only through bodies that dwell within it. There are philosophical debates among physicists about the possibility of defining motion within empty space. Empty space is not observable, and some have argued that it does not exist in itself. It is a useful fiction for describing real phenomena. Kant tried to rescue even this notion by defining it as transcendental. Here too he “threw out the baby with the bathwater,” since in his doctrine this notion does not reflect objective reality, and therefore his arguments do not manage to rescue the view that posits the real existence of space and time.
[7] Of course this is not necessary, since anti-realists can claim that this is imagination and not observation of Ideas.
[8] This is somewhat similar to the distinction I made in the previous column between the Idea of horseness and the Idea of a particular horse, or even of an ideal horse. The former is not a horse and does not have the properties of a horse, while the latter does.
[9] Incidentally, I promised a column on this topic there, and here I am fulfilling it.
Discussion
It seems to me that these examples do Livio’s book a disservice. These are three foolish arguments.
1. Livio. The fact that there are different geometries does not mean that geometry is an invention. I explained this in the column (and that is my view on the matter, in answer to your question). The question of what the geometry of the world is is a question in physics. Lobachevsky and Riemann could not have invented another geometry that also describes the world, because there is only one geometry that can describe the world. That is an empirical question handled by physics, not mathematics. Geometry as a mathematical field deals with some abstract space (which Platonists think exists in the world of ideas), not with our world; and when there are different geometries, they simply describe different abstract worlds. That is all. It has nothing whatsoever to do with the question whether Platonism is correct or not.
2. Gardner. Gardner’s argument is really a glaring misunderstanding. That too follows quite simply from what I wrote in the column. I assume there is no need to explain it again.
3. Atiyah. The fact that a jellyfish cannot develop mathematics while a human can does not mean that mathematics is an invention. A jellyfish also does not know how to develop relativity, quantum theory, or Newtonian mechanics—so are those inventions too? This is just nonsense.
These are three excellent examples through which any ordinary person can see what I keep arguing all the time: even very smart people who deal with the philosophical significance of their field of expertise can talk nonsense and present it as conclusions supposedly demanded by their expertise. These really are common arguments even among mathematicians, which teaches you the same point.
A. Scientific Platonism.
A1. It is not clear to me how you connected theoretical entities to Platonism. No one disputes that one can infer a finger from fingerprints, and that is not a common denominator unique to those two positions. Ideas are intended only for the sake of explanation; theoretical entities also serve for prediction, meaning they are something entirely different, and one need not arrive at Platonism in order to accept theoretical entities. Nor is your definition of the dispute clear to me—in the case of an electron it is enough to think that there is ‘something’ there that behaves in certain situations like a particle-like body, and that ought to be agreed on by everyone, so what exactly is the dispute? As for a field: what does one need to know in order to hear the idea (which sounds very strange to me) that a force field is a thing? In a standard electricity and magnetism course I do not think anyone would ever have considered such a thought.
A2. I heard from you about the distinction between a phenomenological theory and an essential theory (like the distinction between Kepler’s laws and Newton’s laws). Here in the column you said that even if there were convenient and useful phenomenological theories, still ‘science had not yet begun its activity.’ You wrote that ‘semantics sometimes reflects essence,’ so I would not call that science. If one is looking for a theory that is intelligible and explained by reason and logic, then there is reason to continue beyond phenomenology, and that is what explains the learned (Platonist) process even if all the conclusions were already known. But if there is no hope of any further prediction, then even the principled distinction between phenomenology and essence is undermined.
B. Ethical Platonism.
B1. The three questions are: what defines the moral imperative, how human beings discover it, and what gives it validity. You tie all three to realism, and then the third one (validity) to God. As for discovery, I understand what Platonism adds. What the additions of definition and validity contribute is not clear to me. As for validity, you concluded that one does not need an ethical object and that a divine object is sufficient. On what basis is the speculation that God demands moral behavior, and does it rest on the assumption that God is good? You wrote this in several places and I also heard it from you, but I still do not grasp how facts (God’s command, the existence of a ‘laden fact’) bridge the naturalistic gap (you wrote that whoever does not understand this does not understand what morality is. That may be so. I don’t know). I definitely think morality obligates everyone, but it is not clear to me what a good object adds with respect to validity and with respect to definition. My thoughts are vague here, and this is something I need to reflect on. If there is no question, then no answer is needed.
B2. An incidental question. From your remarks in several places I got the impression (perhaps even explicitly, though I have no reference) that your moral outlook is casuistic—meaning that the stable moral judgment is about a single case, and that is like an empirical datum, and then one has to formulate a theory that fits all the data (the judgments themselves may also be influenced and change, but in the end one reaches an equilibrium where the theory and the judgments fit). And this accords with your view that the Idea of the Good is reflected somehow in concrete cases. But the opposite is possible: one may think or recognize general principles, and this power is not related at all to the power that judges concrete cases (so I believe Mill thought, and I think so too). Did I describe your position correctly, and if so, is this in your view reasoned or is it a matter of direct feeling—how morality works for you (and by analogy for others).
B3. If morality is objective, then it would be valid even without people and animals. Even in such an empty world there would, for example, be an imperative not to cause suffering. If so, it is a bit surprising that all the objective moral imperatives known to us are such as pertain to our world. Are we led to assume that there are many more rules of objective morality of which we are simply unaware (and perhaps they are not relevant in our world).
C. Mathematical Platonism.
C1. What is the problem with there being infinitely many abstract objects? The problem with a concrete infinity is that if every object has some minimal size, then there will not be enough room. Likewise for energy and other things. But with abstract objects—what is the problem? As for the number of concepts we have in thought, that is something else entirely and merely a manner of speaking; in practice one thinks only of a finite number (the principle that there are infinitely many sets of various kinds is one principle). To speak directly about infinity in mathematics is simply not well defined, and therefore people work with limits instead. But if someone finds something clear to say about it, good for him—who is stopping him? (For example, proving that there are infinitely many different levels of infinity. Unfortunately I never got to fields that deal explicitly with infinity. Once I asked someone casually: I understand the distinction between countable and uncountable infinity, and several proofs and insights rest on it, etc., but what does one do with distinctions between two infinities that are both uncountable? He took a long sip of his coffee, fixed me with a hostile stare, and said that he didn’t think such a question deserved an answer, but if I insisted then here, take an example—and then he said something about some obscure property of Fourier series, which I neither understood nor remember. Since then my desire to study the topic went out, though presumably it is no less interesting than other fields.)
C2. Mathematical theorems apparently do not come from observation but from thought, even if there exists an object about which the theorems are concerned. One can observe the concept of a triangle until tomorrow and not know that the medians meet at one point, even though this is true of every Euclidean triangle as such. There are intuitive feelings about, for example, ‘how the solution of the differential equation behaves,’ and there is also the use of visual imagination, but to call this observation seems detached to me.
C3. Even if there is observation, it cannot come from the object at too high a level of generality. Observation of the concept of a function will yield very little information. Only after one assumes various properties of the function does it become interesting to investigate it. Seemingly one deals more with the properties of certain sets (for example, the set of solutions to a certain differential equation) than with the properties of sets ‘as such.’ Whoever discovered the concept ‘group’ (or linear map, or convex function, or whatever) did so precisely in order to capture an exact level of generality that would be both encompassing and rich (and this is a common matter in many problems: to look for the required level of generality. Too general will fail to deliver the goods; too specific will be noisy and inelegant and require duplicated work in different domains. In the Cauchy–Schwarz inequality, many versions were known until it was grasped that it is a general property of inner-product spaces, and then a proof was found—an ugly one—that indeed uses only those general properties and does not require in each domain that domain’s particular properties). But soaring higher and higher empties out all the properties, and it is hard to know where to stop on the line between a concrete set–a set–an object. Why in physics would you not say that the concept of being is an idea, but there is no concrete theoretical entity? The concept concept is an idea, but the concept good as a concrete idea does not exist. And so on.
C123. Summary. Why is the idea that only general objects exist not an emptying-out of the whole matter, since a general object does not carry enough information; and even when it does carry the information (as with medians in a triangle), I know of no ‘observational’ way to extract it from it; and nor do I understand why, if there is already a general set, there should not also be infinitely many concrete sets as well (it comes for free).
D. Association. The move you made regarding Russell (you claimed he only proved that not every set is an entity; that is basically what he himself said with the restrictions on which sets are legitimate sets) reminded me of an old argument of yours in an article on pluralistic halakhah. There you said that one cannot accept pluralism regarding the principled question whether pluralism or monism is correct, and therefore pluralism is false (though I do not question at all the conclusions of that article). First, regarding that argument: who, exactly, ever held a monistic position there? Second, pluralism need not accept every position in the world, but only that several positions can be correct on the same issue. Third, although this is probably obvious, I did not understand how this differs from another dispute, such as whether a creeping thing is pure or impure, where one can indeed accept pluralism. Fourth, like your move here regarding Russell, what is the problem with saying that in this dispute there is no pluralism (because there is an argument that proves pluralism here wrong, or simply because in this dispute there is no pluralism, period), but in some of the other disputes there is? With Russell you did not need an essential argument in order to suggest such a thing, whereas there you explicitly did not settle for such a ‘technical’ solution.
E. I read the article on intellectual property. As a halakhic article it is truly beautiful—as an ivory tower standing at a busy gate. Do you think this is also the moral-juridical anchor for intellectual property outside halakhah? Or in general law do you revert to the (normal..) view that ownership is only an agreement.
F. Thank you for the columns. It is stated in Midrash HaNe’elam: What is the meaning of what is said, “Is not My word like fire” [Jeremiah 23:29]? Could it mean actual fire? Rather, just as when fire is placed under a pot, the water immediately begins to roil and move from side to side and bubbles rise and new surfaces appear, so it is with these words. It is not easy to avoid being long-winded (without resorting to the diet of cutting off a leg), but “the bashful person cannot learn,” etc. By the way, it seems to me that the main substance of Steinitz’s remarks was not in Invitation to Philosophy (which I have not read) but in another book.
Such length is difficult, especially when several questions arise here and it will be hard afterward to conduct a discussion on them in parallel. So for now I will respond, but I ask that if you want to continue discussing, we break this up into a separate discussion for each topic.
A1. I associate this with Platonism because fingers are a physical body familiar to us, and therefore inferring conclusions about their existence is a simple and ordinary inference. By contrast, Platonism infers conclusions about the existence of abstract entities that are not familiar to us and are not observable (Ideas), and that is the character of the inference to theoretical entities.
A2. I did not write that phenomenological theories have no scientific value. They are not the end of the road, but it is not true that they are not even the beginning. I was speaking about the full collection of facts, which is not even the beginning of the scientific path. I also do not see experiments as the whole essence of scientific and learned achievement, but I certainly do see value in them. Nor do I think that if there are no predictions, an essential theory becomes phenomenological. Absolutely not.
B1. My claim is that if I feel validity in the moral imperative, this tacitly presupposes belief in God who commands and legislates it. Without that, there is no validity to moral rules. If there is still some question here, I did not understand it.
B2. I do not recall such a statement. I wrote in the past that ethical discussion usually proceeds through examples, but one extracts rules from them and back again. I do not see a clear and absolute order between the cases and the rules.
B3. The imperative can exist even without people, but it would have no addressees. It is like the law of gravitation in a world devoid of objects with mass.
C1. I did not understand the question. Who said there is a problem with infinitely many abstract objects? Russell’s paradox is not based on infinitude but on a contradiction regarding the number of entities.
C2. Sometimes mathematical theorems come from observation (as in axioms), and sometimes from inferences based on those observations. What is the question?
C3. The assumptions you posit can also be the product of observation. A mathematician’s intuition that these assumptions will turn out to be fruitful is not merely arbitrary, otherwise fields in mathematics would not develop (because people would propose various systems of assumptions and usually it would turn out to be neither here nor there and not interesting. The chance of arriving at fruitful assumptions randomly is negligible. Intuitive observation is involved here). I am not claiming that all concepts exist Platonically, only that some do. It is possible that there are invented concepts (though I am not sure there really are such, but it is possible).
D. Regarding pluralism, in my view the situation is exactly the opposite. I have written more than once that as a monist I do not rule out the possibility that there may be several correct answers to some questions, but I claim that there are incorrect answers, and that there are questions that have one answer. The substantive pluralist claims that there is no right and wrong (at least as long as no contradiction is involved).
My claim regarding the debate over pluralism itself is not only pure logic. On the logical plane, what I claim is that if you accept that with respect to this question there are mistaken positions, then by that very fact you have proved that in your own view there are mistaken positions, and this contradicts substantive pluralism (see the previous paragraph). But beyond that I argue against the motivation for halakhic pluralism. The motivation is the unwillingness to accept that there is halakhic error among the sages (or the sages of the Talmud, if you wish). If we have reached the conclusion that here there must necessarily be error among those who hold a pluralist position, then your motivation has been undermined. Hence there is no reason to assume pluralism.
E. I think so. Jurists usually do not write this, because they are unwilling to acknowledge such “mystical” reasons. But here, as in several other legal issues, one can see that the reasons do not really succeed in grounding the claims themselves. Therefore, in my view this is tacitly the assumption underlying conceptions of intellectual property, certainly those that speak of essential ownership (rather than beneficial social convention—and there are such approaches).
A1. Ideas are not merely entities unfamiliar to us but also entities of a different kind. Is your argument only meant to explain why one should not apply Ockham’s razor against Ideas, or more than that? Because I see that it does only that, and that does not seem to me at all like a central argument against Ideas.
A2. Is it true that Kepler’s laws (for example, that the line connecting a planet to the sun sweeps out equal areas in equal times) are phenomenology relative to Newtonian mechanics (the laws and the conservation laws)? If so, then the difference I see is only that Newton found something more general from which one can derive things about cannonballs as well and not only planets. In other words, he offered predictions. Otherwise what is the difference? Because in science every theory in the world is only phenomenological, and the basic laws cannot be understood. So the difference between a collection of facts and a phenomenological theory and an essential theory is only the number of principles. In halakhah, by contrast, one can distinguish clearly between a description of all the facts, a description of principles (= somewhat fewer facts), and an understanding of the principles. To illustrate: a halakhic theory with three intelligible principles is preferable to one with two unintelligible principles.
C1 I was mistaken
A1. I did not understand the distinction. Ideas are entities whose type is unfamiliar to us. Which argument are you referring to?
A2. Indeed. It seems to me that in That Which Is I pointed out that the distinction between phenomenology and essence is relative. There will be theories that are phenomenology relative to other theories. But I did not understand how you leapt from here to the claim that the difference is only in the number of predictions. That is simply not true. There can be a phenomenological theory that yields more predictions than another theory that is essential. True, usually the essential theory that underlies the phenomenology will yield at least the same experiments that the phenomenological theory yields.
Do you not understand the difference between saying that bodies fall toward the earth according to Newton’s law of gravitation and saying that there is a force that pulls them there? (That is so even if there were no additional predictions here, or if it were more general.) The best example of this difference is Einstein’s contribution to blackbody radiation (although there too the number of predictions, potentially at least, is greater. It is not exactly the number of predictions, because he added the existence of photons, but that in itself has no predictions; it only opened the way to broader theories that would yield more predictions).
A1. You present the picture as though (A1) one understands what an Idea is. (A2) It is clear that if we assume the existence of Ideas, the explanatory situation improves. (B) Rejection of Ideas stems from Ockham’s razor—not adding entities without evidence. Then you say the evidence for their existence is that the explanatory situation improves. This is without doubt a valid argument. But I do not think opposition to Ideas is connected to metaphysical stinginess; rather it concerns the fact that the meaning of the Idea itself is not understood, and it is also not clear how the existence of an Idea contributes to solving problems. Therefore this differs from opposition to theoretical entities, where one is not trying to understand anything at all (the understanding is: there is something with such-and-such properties), and where it is also agreed that if we assume a certain existence, the explanatory situation improves. I think there is really something to this point, although it is not clear to me (and therefore all the more so not clear in writing), and I need to find a recent book dealing with Platonism. So I will leave the matter for now. Just please confirm that I described your position (A1-A2-B) correctly, even if it is trivial.
A2. We studied blackbody radiation in statistical thermodynamics, but to my regret I remember almost nothing (and anyway for me it was one of those courses with an incomprehensible gap between the ability to get a good grade and the need actually to understand what is really going on there. Perhaps that too is a gap between phenomenology and essence). Therefore I am forced to try to make the distinctions using more basic knowledge about Kepler and Newton.
(1) Tycho Brahe collected data on the stars, (2) Kepler formulated a phenomenological theory that summarizes the data on the stars in several laws, (3) Newton formulated things differently—far more generally—with several laws as well, (4) and then one can go on to think that there is something that is the force of gravity, and perhaps something that is angular momentum, and so on. The move from data to theory is the transition 1-2 between Brahe and Kepler. I thought that the transition from phenomenology to essence was the transition 2-3 from Kepler to Newton, and therefore I asked how you define the difference (and I thought it concerned only the number of predictions). I now understand that the transition is within Newtonian mechanics itself, 3-4—that is, the transition from description to the addition of entities.
But if there is no additional implication at all, then this is rather pointless. There is an entity, there is no entity, there are five entities—what difference does it make, and why is that more intelligible? You write that Einstein opened the way to broader theories; do you think that even without that they would have been interested in his contribution? You explicitly wrote yes, but I am asking once more in order to hear the emphatic yes.
By contrast, in lomdus the last stage is not only adding entities but adding reasons/conceptual explanations. For example, (1) several laws are given. (2) Someone comes and summarizes them all in the rule that we follow the presumption of prior status. (3) Now you come and offer a legal explanation of why the presumption of prior status matters. That is a very intelligible move from data to phenomenology to essence.
E. Thank you. (I struggle with the connection between essential ownership and moral norms, but perhaps that belongs to the previous column.)
A1. Here we disagree. I do not see what is problematic about understanding what an Idea is. In my view, the main opposition is because of the “mysticism” and Ockham’s razor.
A2. Correct description. And indeed, in my opinion theory is important not only because of the number of predictions but because of the very addition of understanding itself (as distinct from explanatory power, which is usually terminology that describes explaining more phenomena, that is, increasing the number of predictions). That itself is my claim in the column.
What you described in lomdus fits exactly what happens in science.
Now that I have successfully understood regarding A (many thanks), I move on to B, with your permission. If subscription fees were charged here, I would of course pay them as a matter of law.
B1. Even a tacit assumption requires some sort of justification. I am asking: from where comes the tacit assumption that God commands morality (does it come from observation of something?), and is it preceded by an even more tacit assumption that God is good.
B2. If ‘one extracts rules from them,’ that means that the examples are logically prior to the rules—meaning that the basic determining power is the power that judges a concrete case, just as empirical data are prior to scientific generalization.
With your permission I return to my favorite topic, consequentialism, and to the example of pushing the person off the bridge. I think many people would accept the consequentialist principle (from which one can derive an obligation to push him off the bridge) as correct, but when the example arrives they recoil because it is clear to them that one must not push the person off the bridge. [For some reason, in my case the moral feeling actually aligns very well with the principle. Though perhaps I would have an emotional difficulty.] But what does one do when such a conflict arises? Which has the birthright? If there is a conflict between a sensory datum and a scientific theory, and the datum is checked and found correct again and again, then the theory must be rejected. Is this also the relation in morality between judgments about a case and principled rules?
By the way, in matters of moral luck too, I think many people accept the principle that there is no difference in “guilt” according to outcomes not under the person’s control, but in practice when the case arises they suddenly look for various devices such as ‘responsibility’ and ‘agent-regret.’ For example: two negligent engineers built a grandstand with a 30% chance of collapse, and in one case the collapse actually occurred, so he should feel more pained than his colleague. Two people donated a kidney, and in one case the transplanting doctor was more expert and therefore the donation succeeded, so the donor whose kidney succeeded naturally ought to rejoice and be more pleased than the other donor. [When I finish sorting out my own thoughts on moral luck I will raise a question on the subject, but here I request a short spoiler. Again, I have no reference, but this is how I think I understand your general position from your remarks: morality does not depend on luck, and the judgment and praise in the above cases are identical, but there is still responsibility in the sense that if the negligent engineer in whose case the damage occurred does not ‘feel bad about himself,’ then you judge him more harshly. As for ‘punishment for its own sake,’ I understand you would say he does not deserve it, but compensation he will indeed have to pay, because it is better to place the burden on him than on the injured parties.]
B3. If I am translating you correctly, then you think it is reasonable to assume that nowadays there exist (many) moral laws that we do not know, and perhaps they are not relevant to us. That is even stronger than the reasonable assumption that there are countless additional forms of life somewhere in the universe.
B1. It works the other way around (in the “theological” direction): I think that moral rules have binding validity. There is no way to grant them validity unless there is a commanding source. Conclusion: there is a source that commanded morality. This is essentially the argument from morality.
B2. No. As I wrote, the relation between rules and particulars is not one-way but back-and-forth. Without some understanding within you of the rules, you will not succeed in extracting them from the particulars (Wittgenstein, following a rule).
There are feelings that I have, but on second thought I reach the conclusion that this is emotion and not intuition. When the results are terrible, the emotion rebels, but reason overcomes and says that this is only emotion, and moral guilt is according to the action and not according to the result (as distinct from responsibility, which is indeed for the result. The engineer’s feeling of guilt reflects responsibility and not only guilt, and that is straightforward). I have no criterion that instructs me when this is emotion and when it is not, but I have an intuitive feeling (!) that distinguishes between them.
B3. I do not know how you inferred that from my words. I do not assume it, though it is possible (that is usually what is revealed when there is moral progress. The abolition of slavery or equality for women are the appearance of a moral principle that had been hidden until that time).
B1. I understand. Until today I thought that theology helps expose assumptions, but in the end a person has to check whether he does in fact recognize them by direct intuition (and if not, then to throw out the whole package, because if God does indeed command morality, why does a person not feel this directly? The prior presumption against the present presumption).
B2. I have not yet exhausted the matter, but I will think first.
B3. Because what is the logic of assuming that all morality that exists somewhere out there is only of the kind that pertains to living creatures? If additional morality is possible, then the chance that there is none is seemingly 0. But there is no difference between assuming this and assuming that this is possible: morality that does not pertain to human beings and animals. To me this seems on its face like nonsense.
B3. You’re riding on the back of your feelings. If it is logical—then all well and good. If not—then do not assume it.
Thank you.
Let me return for a moment to B1 to make sure of something regarding the mechanism in the theological direction.
Theological means: datum b, and the datum that a is necessary for b, so b entails a because it is sufficient. Now the question arises: if a is true, why did we not attain it as an assumption through direct recognition, just as we recognized b?
If non-recognition counts as recognition of nonexistence, then from the absence of recognition of a one can infer not-b. It follows that a theological proof rests on the idea that direct recognition (b) is more indicative than non-recognition (a). Correct?
If I saw someone at 2 o’clock at the end of Highway 6, I infer that at 1 o’clock he was at the beginning of Highway 6. But if everything is filmed for me and I do not see him at 1 o’clock at the beginning, while I do see him at 2 o’clock at the end, then I must assume a malfunction.
Sometimes there is an assumption within us such that, in order to notice it, we need other indications through which it appears. Such are the wonders of our psychology. Almost every persuasion in an argument is like this. A person does not notice that he believes in God, but it is clear to him that morality is valid. He does not hold the position that there is no God, only that there is no indication that there is. Now he has an indication that there is.
C. Mathematical realism.
There is, after all, a distinction between the question whether mathematical propositions concern real objects, and the question whether the process of mathematical thought involves observation or thinking. I am dealing with the process of thought. You wrote that axioms are attained by observation, so I am dealing with all the other propositions. For example, a theorem such as that the medians in a triangle meet. Was observation involved there? Did some mathematician sit, draw triangles in the sand, and see that the medians meet, and then say oops, maybe this is general, tried it and came up with a proof? By virtue of the rule-following argument, are you claiming that observation was involved here? And what did he observe? Did he observe this theorem directly? After all, observation of the Idea of the triangle cannot yield such a theorem. If even in a theorem like this observation was not involved, that means that almost the entire process of mathematical thought has nothing to do with observations. I do not know whether this stands against what you wrote in the column; I am asking whether this is correct or not.
I understand that you are saying that in order for a mathematician to manage to think of a concept like a group or a linear map or the derivative of a function (which captures diverse properties that appear in many contexts), it is reasonable to assume that he needed observation, because from thought alone there is no reason he would hit precisely on a useful definition. If that is the argument (the argument from fruitfulness), then it is more sophisticated than the argument from taxonomy, and it returns to the problem of rule-following (which, as usual, you solve by recognition and not by thinking). I am a little hesitant to understand this as the argument, because apparently this is an argument that did not appear in the three columns (and I also do not remember it from That Which Is, although my memory is not a reliable reflection. I read the book when it came out and it left its mark on me, but I remember its details less than those of the other parts. And in spirit I moved away from it over the years. But the recent columns opened something up for me, and now it is my duty to return to it.)
Quite possible. Sometimes it is observation and sometimes it is logical analysis. I assume that the inspiration for where to go with the logical analysis comes from observation of the ideal objects. But all this is unrelated to what I wrote in the column.
I do not see a necessary connection to the problem of rule-following, though there too I would say similar things. But I do not see a direct dependence.
D. On pluralism. In your reply above you wrote: "As a monist I do not rule out the possibility that there may be several correct answers to some questions, but I claim that there are incorrect answers, and that there are questions that have one answer. The substantive pluralist claims that there is no right and wrong so long as no contradiction is involved." This surprised me. I went back to read the article ‘Is Halakhah Pluralistic?’ and it turns out that I had not understood it correctly before.
Could you give an example of a question that has two correct answers? The examples I think of are, for instance, a person sees in the street two identical lost items belonging to his father, and he can return only one of them—which should he return? There are two correct answers: he can choose to return the one on the right or the one on the left. If that illustrates the claim, then the claim seems very weak. Clearly, “monism” does not mean that in every question there is one practical answer; it means that there is one theoretical answer. And the one and only theoretical answer is: “You may choose one of the two practical courses.” Even a ‘set of answers’ is an ‘answer’ in every respect. (Just as in automata theory, the standard trick is to turn a ‘set of states’ into a ‘state.’)
By that logic there is no pluralism in the world either. There is monism that holds that the answer is either A or B or C or D or E or F.
A somewhat more substantive example is a clash between two values of equal weight when there is no technical solution such as passive non-action being preferable, or leniency, or stringency. For example, what should an infant found in a city that is half Jewish and half gentile do with respect to Sabbath observance or Torah study? Here there is no leniency or stringency, since if he is a gentile then he is forbidden to study and to observe the Sabbath, and if he is a Jew then he is obligated to study and to observe the Sabbath. There is no side that is merely permissive.
I highly recommend Mario Livio’s book Is God a Mathematician?..
By the way, there he tended quite a bit toward the view that geometry, unlike mathematics, is a product of the human mind, since Lobachevsky and Riemann invented different geometries that also describe the world effectively.
— The main point of the book deals with the question whether mathematics is a discovery or an invention. After the author raises several questions about the two views, he concludes that mathematics is both invention and discovery. By the way, what is the rabbi’s opinion on the matter?
—
I’ll copy a very short passage here:
Martin Gardner, the famous author of many books in the field of recreational mathematics, also advocates the approach that sees mathematics as a discovery. In his view, there is no doubt at all that numbers and mathematics are endowed with an existence of their own, whether or not humans know them. As he put it wittily, "If two dinosaurs joined two other dinosaurs in a forest clearing, there were four of them there—even if there were no human beings around to observe them, and even if the animals themselves were too stupid to know it."….
Others disagree with this view…. The British mathematician Sir Michael Atiyah (who won the Fields Medal in 1966 and the Abel Prize in 2004) remarked:
……If the universe were one-dimensional, or even discrete, it is hard to see how geometry could have developed.
It may seem as though we are standing on firmer ground when we deal with integers, and that counting is truly a primitive concept. But let us imagine that intelligence had taken up residence not in the human brain but in some giant jellyfish, solitary and isolated in the depths of the Pacific Ocean. It would have no experience at all of separate objects as such, only of the water surrounding it. Motion, temperature, and pressure would be its basic sensory data. In such a pure continuum, the discrete would not appear at all, and there would be nothing to count.
Therefore Atiyah believed that "human beings created mathematics by idealizing and abstracting factors from the physical world." The linguist George Lakoff and the psychologist Rafael Núñez share his view. In their book Where Mathematics Comes From they state: "Mathematics is a natural part of human existence. It arises from our bodies, our brains, and our everyday experience in the world."