Topics in Talmudic Logic, Lecture 2

Table of Contents

• Material logic and formal logic, and Aristotelian conceptualization
• Analogy, induction, and deduction as one inference broken down
• The philosophical uncertainty principle: certainty versus information
• Mathematics and logic as necessary inferences, and the status of assumptions
• Science as the accumulation of information, and the distinction from mathematics
• Mathematics as empty of empirical content, and Popper’s criterion
• Axioms: mathematical arbitrariness versus truth about the world, and the problem of observation
• Three-stage philosophical maturation: skepticism, fundamentalism, and synthetic maturity
• The history of ideas in the West: from positivism to postmodernism and fundamentalism
• The criterion distinguishing the first stage from the third: plausibility and criteria for soft thinking
• The priority of theory over facts: Carr, Hempel, and Semmelweis
• The goal: developing a logic of plausibility through Talmudic hermeneutic principles
• The convexity example: a mathematical proof and shifting the difficulty to the definition

Summary

General Overview

The lecture presents an introduction to logic as a lead-in to Talmudic logic, and distinguishes between material logic, which depends on content, and formal logic, which depends only on form. It argues that mathematics and deductive logic deal with necessary implication that adds no information beyond what is already contained in the assumptions, whereas science is based on analogy and induction, which do add information and therefore are not certain and are subject to falsification. From this, a conception is presented of a “philosophical uncertainty principle” between the amount of information added and the degree of certainty, and a model of “philosophical maturation” is described in three tracks: skepticism, fundamentalism, and synthetic maturity, which is willing to accept plausibility without certainty. At the end of the introduction, the goal is presented: to develop a “logic of plausibility” through the Talmudic hermeneutic principles, and to establish criteria for non-deductive thinking that do not collapse into postmodernism and do not depend on exalted sources of certainty.

Material Logic and Formal Logic, and Aristotelian Conceptualization

Logic deals mainly with inferences, and inferences that depend on content are called material logic, while inferences that depend only on form are formal logic. An example of formal validity is an argument of the form “Every X is Y; A is X; therefore A is Y,” regardless of what you substitute for the letters. Aristotle, in the Organon, did not invent logic, but his conceptualization created a systematic toolbox that makes it possible to identify fixed patterns and use them where needed. The direction sought later on is to do something similar in the Talmudic context: to identify general forms of thought and conceptualize them.

Analogy, Induction, and Deduction as One Inference Broken Down

Three modes of inference are presented: analogy, induction, and deduction. From a broader perspective, it is argued that analogy can really be broken down into two stages: induction and then deduction. The example of the table describes a move from the particular to the generalization, “Apparently all tables are brown,” and then back from the general to the particular in order to infer something about another table. Abduction is mentioned, but set aside at this stage.

The Philosophical Uncertainty Principle: Certainty Versus Information

An analogy is brought from the uncertainty principle in physics, where precise knowledge of one quantity comes at the expense of another, and a parallel principle is proposed between the degree of certainty and the amount of information. Deduction is a necessary inference because the conclusion adds no information beyond what is already included in the assumptions, and therefore anyone who accepts the assumptions cannot deny the conclusion. Analogy and induction add information, and therefore they are not secure. It is argued that, in principle, the amount of information an inference adds multiplied by its degree of certainty is a constant. Arguments differ according to how much additional information there is in the conclusion beyond the assumptions, and accordingly by the level of certainty that can be attributed to them.

Mathematics and Logic as Necessary Inferences, and the Status of Assumptions

Logic and mathematics are identified here as tools of necessary inference, where necessity exists only in the “if-then” relation, not in the assumptions themselves and not in the conclusion itself. The mathematician is “responsible only for the if-then,” and so if asked about the sum of angles in a triangle, he will answer, “It depends on your assumptions,” and will tie it to Euclidean or non-Euclidean geometry. A proof is defined as extracting the theorem from the assumptions and showing how it is hidden within them, so that someone who holds the axioms, in principle, already holds all the theorems. From this it is argued that begging the question is not a fallacy, but rather the criterion for a valid argument, because a valid argument is one in which the conclusion is included in the assumptions.

Science as the Accumulation of Information, and the Distinction from Mathematics

Science is defined as the accumulation of information, and therefore it relies on analogies and inductions rather than deductions, and scientific conclusions may turn out to be incorrect. Science is subject to the test of falsification, whereas mathematics is not subject to falsification; at most, errors in a proof may be discovered. A complementary role is presented, in which the physicist or scientist provides the basic assumptions and general laws מתוך observations and generalizations, and the mathematician/logician derives particular applications from them by deduction. The claim is that the truly scientific steps are not mathematical but always analogical or inductive.

Mathematics as Empty of Empirical Content, and Popper’s Criterion

It is argued that mathematics, in and of itself, does not make claims about the world and is “empty of empirical content,” and therefore does not meet Popper’s criterion of falsifiability for a scientific theory. The example of “two plus three equals five” shows that an experiment with oranges would not lead us to abandon the statement even if the counting came out strangely; rather, the experiment would be suspected. The example of adding forces shows that the failure is not in the calculation “ten plus ten equals twenty,” but in the physical assumption that adding forces is arithmetic addition rather than vector addition. Mathematics becomes relevant only after the scientific study of the world determines what mathematical model is appropriate to reality.

Axioms: Mathematical Arbitrariness Versus Truth About the World, and the Problem of Observation

A distinction is presented between axioms in mathematics, which are seen as arbitrary and structure-constituting, and assumptions about the world, for which there is such a thing as true and false. It is argued that many assumptions about the world are not the result of direct observation, such as the claim that exactly one straight line passes through two points, or that parallels do not meet, because observation cannot confirm universal claims out to infinity. General assumptions are formed as generalizations and inductions that add information, and therefore they are open to falsification but do not receive deductive certainty. Criticism of an argument can focus on the assumptions or on the inference, but the assumptions themselves originate in “soft” tools such as induction and analogy, not in mathematical proof.

Three-Stage Philosophical Maturation: Skepticism, Fundamentalism, and Synthetic Maturity

A model is described of a dogmatic child who takes things for granted, a positivist teenager who demands proof for everything, and a crisis in which it becomes clear that nothing can be proven without basic assumptions that are themselves unproven. From that crisis emerge three “paths of maturity”: skepticism, which holds that “only what is certain is acceptable” and “nothing is certain,” and therefore accepts nothing; fundamentalism, which also holds that “only what is certain is acceptable” but claims that there are certain transcendent sources, such as “it is written in the Torah” or “the rebbe said so”; and synthetic maturity, which accepts that there is no certainty but rejects the demand that only certainty is acceptable, and accepts plausibility as well. It is argued that science and common sense can exist only for the synthetically mature person, because they rely on soft inferences that are not certain. Skepticism is described as a deification of logic, because one who accepts only logic as a legitimate tool is left without information, since logic alone can be valid even with false assumptions.

The History of Ideas in the West: From Positivism to Postmodernism and Fundamentalism

An analogy is drawn between the stages of maturation and the history of civilization: an early dogmatic period, the Enlightenment revolt seeking proofs, and a twentieth-century crisis in which positivism broke down and postmodernity was born. Postmodernism is described as an offspring of modernism, because it concludes that if only logical consistency determines things and there is no proven foundation for assumptions, then everyone is trapped in his own narrative and there is no standard of right and wrong. At the same time, the appeal of fundamentalism and New Age thinking is also explained as the same search for alternative certainty after logic and science fail to provide certainty. It is argued that the way to confront fundamentalism is not postmodernism, but rather giving up the demand for certainty while still insisting on reasonable beliefs and on the ability to say, “You’re wrong,” even without a proof.

The Criterion Distinguishing the First Stage from the Third: Plausibility and Criteria for Soft Thinking

The difference between dogmatic acceptance and synthetic maturity is located in the concept of plausibility and in tests that are not rigorous yet are not arbitrary. It is argued that the central challenge of synthetic maturity is to justify criteria for plausibility that distinguish between different non-certain claims, as opposed to the position that every unproven claim has the same status. What is needed is a “logic of plausibility,” not a logic of certain validity, but rather a classification of claims along scales of more plausible and less plausible. This problem is described as unresolved to this day. Francis Bacon is presented as someone who tried to propose an inductive logic for conducting science, but the discussion moves on to criticize the picture of “first collect facts and then build a theory.”

The Priority of Theory over Facts: Carr, Hempel, and Semmelweis

In Carr’s book What Is History?, it is argued that the historian cannot begin by collecting facts without a prior theory of relevance, because there are infinitely many facts and only a theoretical framework determines what is worth collecting. A process is described of “back and forth,” in which one begins with a hypothesis, gathers facts in its light, refines the theory, and then returns again to the facts, as opposed to the Baconian picture of naive induction. Hempel’s example of Semmelweis and the two maternity wards describes attempts to gather facts in an almost completely random way, without understanding the cause of the mortality, until the discovery that requiring students who had been dissecting corpses to wash their hands changed the mortality rate and led to the understanding of microorganisms. The conclusion is that there is no sharp division between theory and facts, and that at the basis of science lie a priori assumptions that are not the result of observation.

The Goal: Developing a Logic of Plausibility through Talmudic Hermeneutic Principles

The stated goal is to develop, in a systematic way, a logic of plausibility, a logic of accumulating information rather than analyzing information already present. It is argued that a fortiori reasoning, paradigm construction, and verbal analogy, along with paradigm construction from two verses and refutations, lay out the full range of non-deductive thought at different levels of complexity. These tools are presented as general tools of thought that do not belong only to the Talmud, but their distinction and systematic treatment are found in the Talmud, and therefore the analysis will be based on it. Two outputs of the process are presented: importing logical tools from the broader world in order to decipher Talmudic passages, and exporting rules from the passages into general fields of thought.

The Convexity Example: A Mathematical Proof and Shifting the Difficulty to the Definition

A mathematical theorem is presented according to which the intersection of two convex shapes is convex, and it is explained that the proof becomes trivial once convexity is defined so that for any two points in the shape, the straight line segment connecting them lies entirely within the shape. The proof shows that any two points in the intersection lie in each of the convex shapes, and therefore the segment lies in each of them and thus in the intersection. It is then argued that this move does not prove a claim about the world, but about the defined structure, because there is no proof that the mathematical definition captures exactly the intuitive concept of “convexity” in the world. A “law of conservation of difficulty” is presented, according to which the difficulty not resolved in the proof gets pushed into the definition, and whenever mathematics is applied to the world, a non-mathematical assumption is always required, namely that the definition fits reality. The conclusion is that every use of mathematics in empirical fields loses certainty because of the assumptions about the fit of the model, and therefore at most one can say that it is plausible, not that it is proven.

Okay, last time I started with a kind of introduction to logic, with the goal ultimately being to get to Talmudic logic, right? Topics in Talmudic logic—that’s our subject. But in order to do that we need to give some principled introduction to logic, and I hope to finish that today. Last time I started by talking about material logic and formal logic. Material logic—or logic in general—deals mainly with inferences, not only but mainly. When you have inferences that are connected to or dependent on content, that’s what is usually called material logic. There are inferences that depend only on form, formal inferences—yes, form means shape—so that’s formal logic. For example, if every X is Y and A is X, then the conclusion is that A is Y. And that doesn’t depend on what we substitute for A, X, and Y, and that is formal validity. All right, your name? I called on you before. Yosef Adoni Maroz? Okay. So I talked about the conceptualization Aristotle made in his Organon and what its significance is. Obviously Aristotle didn’t invent logic; people used it before him too. But the conceptualization matters, because it helps us notice that there is some fixed pattern here, some fixed tools, and it places them at our disposal so that we can use them where we need them. As opposed to the situation before the conceptualization, before Aristotle, when if someone had some inspiration or some idea he could use it, but it wasn’t systematic. He didn’t have some built toolbox from which he could choose the relevant tool. And as I told you looking ahead, that’s basically what we’re going to try to do in the Talmudic context as well. To try to identify forms of reasoning that of course people ordinarily use—I’m not going to invent them—but I’ll try to show that they exist and that they are general. In other words, to conceptualize them in some way. Fine, so that’s just the framework of the discussion.

Now within that framework I spoke about the meaning of logic and mathematics essentially as drawing conclusions from premises, where logic and mathematics deal with necessary inferences. Meaning, someone who accepts the premises cannot avoid accepting the conclusion. I spoke about three—well, four, but let’s say three—modes of inference: analogy, induction, and deduction. And I said that when you look at this more broadly… where’s the marker and the board? Okay. When you look at it more broadly, basically let’s write the formula on the board, yes, up top. A is X—no, not that plus sign there, yes. In a broader view, what we’re really doing is only analogies. Except that an analogy breaks down into two components: the first is induction and the second is deduction. When we make an analogy between this table and that table, we’re basically saying: this table is brown, so apparently all tables are brown. This is a table, so if all tables are brown then this table too is brown. I basically made an analogy between this table and that table, but I did it in two stages. First I took this table and generalized from it—that’s induction. Then I have a general statement about all tables, and I did deduction, I descended from that to the specific table before me. So in that sense there aren’t really three modes of inference, but one mode of inference that breaks down into two others. I also mentioned abduction, but let’s leave that aside for now.

I also said there—I spoke about the philosophical uncertainty principle, you could say, by analogy to the uncertainty principle in physics, or in quantum theory, where what is uncertainty? The uncertainty principle says that we have pairs of quantities in physics such that the more I know about one, the less I know about the other, and vice versa. Meaning, say position and momentum, or time and energy, there are such pairs that go together. So for example position and velocity—yes, momentum is velocity. So if I know exactly the position of a body, I know nothing about its velocity. If I know exactly its velocity, I know nothing about its position. That’s part of quantum theory; it doesn’t matter, if you don’t know it then no matter. But that’s what’s called the uncertainty principle. The uncertainty principle means there are two paired quantities between which there is this kind of game, where one always comes at the expense of the other. It’s a kind of zero-sum, or a constant product if you like: if you increase one, it comes at the expense of the other. Like… yes. Like health and taste, right? Foods—the degree of healthiness they have and how tasty they are, their product is constant. Meaning, the healthier it is, the less tasty it is, and vice versa. So again, that’s the gastronomic uncertainty principle. If you want, add price there too; really it’s the product of three things.

In any case, the philosophical uncertainty principle basically says there is the same kind of game between the degree of certainty and the amount of information. Meaning, an inference that adds a lot of information will have a very low degree of certainty. An inference with a high degree of certainty will add very little information. Tell me your name? Avichai Berger. Avichai Berger. How long have you been here already? Right. Okay. So the degree of certainty and the amount of information—yes, sorry, what’s your name? Rafael Khachatourian. So in what sense are we talking about this? I said that in deduction the inference is necessary. The conclusion follows necessarily from the premises. If all humans are mortal, Socrates is human, conclusion: Socrates is mortal. Going from the general to the particular is playing it safe. Why really is it so safe? Why can’t someone who accepts the premises deny the conclusion? Because the conclusion contains no information beyond what was already in the premises. This inference adds no information. Right? If I know all humans are mortal and I know Socrates is human, then I basically know that Socrates is mortal. The conclusion that Socrates is mortal is already embedded in the premises. It doesn’t add any information beyond what is in the premises.

Analogy and induction are not like that. This table is brown, that thing is a table—those are two premises. Conclusion: that thing too is brown. The fact that that thing is brown is not embedded in the premises. That is additional information beyond what I had in the premises. Right? That’s an inference that added information, and because of that it isn’t secure. An analogy may be correct, and it may not be correct. Deduction cannot be incorrect. Why? Because it is certain. It is certain because it adds no information. All right? Deduction adds no information; the conclusion was already latent in the premises. So basically what I want to claim is that… the amount of information the inference adds—the part of the conclusion that goes beyond what was in the premises—multiplied by the degree of certainty of the inference is constant. An inference that adds me a lot of information has low certainty. It’s speculative, right? It adds a lot of information beyond what I had at the beginning, so it’s very speculative. It’s not certain at all. An inference that adds little information is less speculative. I can place more trust in it. An inference that adds no information at all—deduction—is certain. Like in the stock market? What in the stock market? Risk and return. Right. So the claim is basically that different arguments differ from one another in the amount of information they add, and therefore also in the level of certainty I have in the argument. Deduction is absolute certainty, analogy and induction less so. Last time I spoke about whether it’s possible to create a hierarchy among them and what exactly that hierarchy is, but I won’t return to that here.

That’s the general introduction. After that I spoke about logic and life. Logic usually deals first of all with deduction. Logic, mathematics—for me right now that’s the same thing. Let the mathematicians forgive me; they don’t like statements like that. So logic and mathematics deal with necessary inferences. When I prove a theorem in geometry or number theory or whatever mathematical field, the conclusion follows necessarily from the premises. Anyone who accepts the premises has to accept the conclusion. Right? There’s no way not to accept it. It’s a certain inference. Why is it certain? Because it adds no information. Right? I spoke about that with the hot-air balloon: the two defining traits of the mathematician are that what he says is absolutely precise and totally useless to us. It adds no information, and therefore it can be completely certain.

Mathematics and logic deal with necessary inferences. The responsible mathematician, when asked what the sum of the angles in a triangle is, can’t answer me one hundred and eighty degrees. He has to answer: it depends what your premises are. If your premises are the premises of Euclidean geometry, then the sum is one hundred and eighty degrees. If the premises are different, then the sum is whatever you want. You can build a geometry for any number you want. So the mathematician is responsible only for the if-then. He is responsible neither for the “if” nor for the “then,” not for the premises and not for the conclusion, only for the relation between them. Because only that relation is necessary. The premises are not necessary and the conclusion is not necessary. Right? The relation between the conclusion and the premises is what is necessary. Okay? Here too in this argument, yes? Every X is Y, A is X, conclusion: A is Y. The top statement is not necessary, the second is not necessary, and the bottom is not necessary. The only thing that is necessary is the derivation of the third from the first two. And you cannot accept the first two without accepting the third. Whoever accepts the first two must accept the third. Okay? That’s mathematics and logic.

Science is built from claims of two other kinds: analogies and inductions. Because science deals with accumulating information. Mathematics deals with analyzing premises; it doesn’t add information beyond what is in the premises. It simply tries to unpack the premises and see all the information contained in them. To extract from the premises all the information that is there. Therefore it doesn’t add information; it just clarifies for me more fully the information I already have. If I have the premises, I have all the information that comes out of them. Whoever holds the axioms of geometry in fact holds all the theorems; you just have to show now how the theorems are hidden inside the premises. That is what’s called a proof. A proof means showing how the theorem is hidden inside the premises, how I can extract it from them. So once I have a proof, I have basically shown that the conclusion contains no information beyond what is in the premises. Okay?

That’s why I said that usually people think begging the question is a fallacy, but that’s not true. Begging the question is the criterion for a valid argument. Every valid argument begs the question. If it doesn’t beg the question, it isn’t a valid argument. To beg the question means that the conclusion is part of the premises, it is embedded within the premises. And a valid argument is always like that. Every valid argument begs the question.

So this is basically what comes out if I summarize everything I’ve said so far: mathematics deals with the derivation of a conclusion from premises, a necessary derivation of a conclusion from premises. A necessary derivation exists only if the conclusion contains no information beyond what was already in the premises. Or in other words, mathematics doesn’t help me accumulate information. Mathematics is not information-gathering. Mathematics just helps us understand the information we already have better. A logical argument simply cannot add information. There is information in the premises; the logical argument extracts the information from them. So who is responsible for information? How do we accumulate information? Science. All right? Science is basically, for our purposes, a general term for the accumulation of information. Not necessarily scientific information, of course—if I know that this wall is white, that’s non-scientific information. I accumulated it, it doesn’t matter. On the principled level: observation, yes? Scientific methods, observation, generalization—the scientific methods are methods of accumulating information. And therefore by their nature they are not deductive methods. That’s why science is not mathematics and not logic. Science is built mainly on analogies and inductions, not on deductions. Therefore the scientific conclusion can turn out to be incorrect. Science stands the test of falsification. Mathematics does not stand the test of falsification. You can’t falsify a mathematical theorem. You can find a mistake in the proof. But to find a counterexample doesn’t falsify a theorem; at most it turns out you made a mistake in the proof.

In science, since the conclusion is not necessary, the conclusion is what seems right to me. If I find a refutation, then I’ll throw out the conclusion, I’ll give it up. Science is not necessary. Science accumulates information, and the currency you pay for information with is the currency of certainty. There cannot be certain accumulation of information. No such creature exists. Okay?

So if I now return to the equation written above—deduction, analogy divided into induction and deduction—I can present it in the same way as the complementary roles of science and mathematics, or science and logic. Basically, if I want to know, when I draw a triangle on this sheet of paper, what the sum of the angles in the triangle is, what I need to do is first know what the axioms of the geometry of this sheet are, the sheet on which I drew the triangle. Then ask the mathematician: assuming those are the axioms, what is the sum of the angles in such a triangle in such a space? Okay? So those are complementary roles. The physicist gives me the basic assumptions—or the scientist—and the mathematician or logician derives the conclusions from them. The use of mathematics in scientific fields is exactly that. The scientific field gives me the rule, the general law, whether from observations, generalizations, whatever; and mathematics tells me how to derive applications to particular cases from that general law—deductions. Of course that adds no information; those are only applications. Okay, so mathematics is an instrument used by science, but the genuinely scientific steps are never mathematical steps, not ever. They are always steps of analogy or induction. Once I’ve arrived through induction at a general law, deduction—mathematics and logic—will tell me what that general law means in a specific case. Therefore there is really a process here like analogy, which is built from induction and deduction. When I want to infer some information about the world, I need to build an argument that has premises and an inference deriving from them a conclusion. I take the premises from science, and I make the inference using logical and mathematical tools. Okay? Therefore science and mathematics too are really processes that complete one large inference; there are two stages: the scientific stage and the logical-mathematical stage. Okay.

I said last time that mathematics by itself, if I isolate only this component of thought, says nothing about the world, contains no information, in the terms I used before. Right? For example, the sum of the angles in a triangle is one hundred and eighty degrees—that is not a statement about the world. It’s a statement about some abstract Platonic geometry. Whether our world obeys that geometry or not—that’s a question in physics. A physicist needs to know what the metric of our world is, whether our world is Euclidean or non-Euclidean or what, what the assumptions of the geometry are, what axioms of geometry are true in the geometry of our world. After we know the assumptions, we go to the mathematician and he tells us what the theorems relevant to that geometry are. Okay? So mathematics by itself says nothing about the world; it is empty of empirical content.

Sometimes they look at things in the world, and then measure them in meters. A meter is a definition that depends first of all on… no, no connection, I wasn’t speaking here about the scale. I wasn’t speaking about metric in that sense. I said “metric” before—metric is not the question of what scale you measure in; metric is a mathematical concept. But the one hundred and eighty degrees in a triangle couldn’t have been reached in advance without prior observations of what a triangle is. Right, that space is flat. If space is flat, then the sum of the angles in a triangle is one hundred and eighty degrees. Fine, but how do I know that our specific space is flat? That’s the physicist’s job to say, not the mathematician’s, because it’s a matter of observation. By the way, physicists really do say no. Our space isn’t flat; the sum of the angles in a triangle is not one hundred and eighty degrees. It’s almost—meaning, but not exactly one hundred and eighty degrees.

In any case, the claim I want to make is that mathematics or logic doesn’t make any claims about the world. They are responsible only for the if-then, not for the “if” and not for the “then.” All right. I gave examples of this, yes. I asked how you would put the claim two plus three equals five to the test of falsification. Popper’s criterion for a scientific theory is that it is a theory you can put to the test of falsification. So let’s ask: is two plus three equals five a scientific theory? If it is, then there ought to be some test that could refute or confirm it, right? There is no such test. Someone suggested last time: take two oranges, put them in a basket, add three more oranges, and let’s count. If it comes out five, it’s confirmed; if it comes out eighteen, it’s refuted. But that’s not right. Because if it comes out eighteen, we will never give up the statement that two plus three equals five. We’ll say there was a mistake in the experiment. The statement two plus three equals five really doesn’t depend on observations. It doesn’t stand the test of falsification.

As I said, yes, I gave the example of forces as vectors. You apply a force of ten newtons northward, and another force of ten newtons westward, okay—what is the resultant force on the body? What is the total force? Fourteen point something, not twenty. So have we refuted the proposition that ten plus ten equals twenty? No. Ten plus ten remains equal to twenty. What we refuted is the proposition that combining forces is described by arithmetic addition. You need vector addition, not arithmetic addition. Okay? That proposition is a proposition in physics, not in mathematics. The physicist needs to know which theory, which mathematical doctrine, is suitable to describe the addition of forces. The mathematical doctrine itself is the mathematician’s mandate: from premises, whatever follows from them is all fine. But what in reality corresponds to some mathematical doctrine and serves as a model for it—that is the scientific language. Mathematics only tells me: give me the results of observation, the rules, and I can show you all sorts of things that follow from them.

Okay, now I want to give an example. Up to this point this is more or less a summary of what we did last time. Before I give the example maybe I want to talk a bit about the meaning of axioms of inference. Usually mathematicians are used to thinking that axioms are arbitrary. Tell me the axioms, choose whatever axioms you like, and given those axioms I’ll tell you what theorems follow from them, okay? Which axioms you choose—as a mathematician I have nothing to say about that, okay? Therefore mathematicians often, and philosophers of mathematics too, tend to think that axioms are arbitrary; there is no true or false here. I’m not speaking about the world, I’m speaking about some structure, and I can define it however I want. The axioms define the structure I’m speaking about. Once it is defined, I can derive all kinds of conclusions about it, but I’m not speaking about anything in the world, therefore it isn’t a proposition in terms of true or false.

That is true of mathematics in principle, but it is not true of logical arguments of the kind we deal with in philosophy, arguments about the world—there it is not true. Suppose the axiom we’re speaking about is: one straight line passes through two points. If I am speaking about it as a constitutive axiom as in mathematics, where it defines the mathematical space, then there is no true or false here. I chose it as an axiom and I can derive from it the conclusions that follow. But if I am claiming this about the world, then there is true and false here. The question whether in our world one straight line passes through two points, or more, or fewer, I don’t know—that’s a question that does have true and false. Okay? So when I posit assumptions about the world, they are not arbitrary. That’s basically what follows from what I’ve said so far. Assumptions about the world are not arbitrary; there is true and false there. How do I know what is true and what is not true? I obviously won’t have a mathematical proof, right? Because mathematics says nothing about the world. So how do I know? By observation. And what is not accessible to direct observation—which is to say almost everything—is the result of some generalization based on observations.

For example, the claim that one straight line passes through two points is not the result of observation in the world—not in Platonic mathematics, in Platonic space—but in our world. It is not the result of observation. No observation can yield that result. How could I know by observation that one straight line passes through? Maybe there are another seventeen I haven’t encountered, I haven’t thought of? I can see that there is one—that I can see. How do I see that there aren’t eight? I don’t know. It seems very sensible to me, yes, it’s obvious, but I didn’t actually see it. It is not the result of observation. Or that parallel lines never meet—is that the result of observation? Of course not. Did someone walk all the way to infinity to check that it never happens at some point? No, it seems very sensible, right? It sounds very sensible to us. Therefore we assume it, but it’s not the result of observation. Maybe partially observation—we do see something—but it is not only observation. We infer conclusions on the basis of what we see. Those conclusions are generalizations, analogies, generalizations, hypotheses, whatever you want to call them; it isn’t deduction, it isn’t mathematics. It’s part of science, just as science generalizes, makes a general law out of particular examples that I observed, okay? That is how scientific laws are formed.

The law too that one straight line passes through two points in our world, not in mathematical space, is a scientific law. Therefore it is a result open to falsification, but on the other hand it does not arise from observation. It can be refuted by observation, but not confirmed by observation. That’s a very important point, because when we use logic—as far as I’m concerned this is using logic, we are not studying logic, we are using logic—that means I’m applying my logical tools to premises I believe are true. Just to play around with if-then, taking absurd premises and deriving another absurd conclusion from them—that’s an amusing mathematical-logical game, not interesting. Okay? When I want to make an argument, I have to use true premises and a correct argument. And someone who attacks an argument either attacks one or several of the premises, or attacks the inference—that is, how I move from the premises to the conclusion. Those are the two ways to attack the conclusion of a logical argument.

How do I really know my premises? How do I know them? From analogies, inductions, all sorts of very not—yes, but observation usually doesn’t give them. So where do I know it from? How do I know that every two masses attract one another? I saw a few cases like that, yes, that’s fine. But how can I know that it is always like that? That’s a generalization, right? Scientific induction. From the particular examples I make a generalization. How do I know the generalization is true? I have no proof, right? Induction can be mistaken. It’s not a certain tool. Deduction is a certain tool, induction is not. And analogy isn’t either. It’s enough that one thing be wrong. Right, but so far I haven’t found anything wrong. Who says it’s right? So far it hasn’t been falsified. I’ve also never found a fairy with three wings. So is the proposition that every fairy has two wings true? So far. Everything is only so far. The question is how I say that it is true, not “so far.” What? Because people hold by it, meaning they hold by it until it is falsified. On what basis? On the basis of the cases we saw. First of all, the cases we saw can be generalized in infinitely many ways, that doesn’t help you at all. But even suppose there’s only one way—so what? Why assume it’s true in general and not only for the cases we saw? The cases we saw, fine. You brought an entirely different theory. No, not entirely, but how do I know? Deduction constitutes a proof. But here I don’t have a proof in the mathematical sense. I have softer routes, analogy, induction, those are routes that add information. Remember, right? That’s how I accumulate information. Such routes are never certain.

At this point people can come—and do come, not just can come, many people do—and claim: indeed we have no trust at all in those routes. What is proven, I accept; what is not proven in the severe, rigorous, logical sense is arbitrary. And everyone with his own generalizations, and I have no way to determine a position about that. Unless you have some set of premises that are completely obvious, like “I know Torah” and things of that sort. And then… I know one and a half people who don’t accept that supposedly completely obvious premise. What is completely obvious there? Nothing is completely obvious. What does “completely obvious” mean? There are premises—clear or unclear, I don’t know what those words mean. We don’t have proofs; we have all sorts of softer routes, not logic and mathematics, through which we maneuver within the scientific world. Okay? That’s how we arrive at our basic assumptions.

Yes. But if you don’t accept anything, then you also won’t know, you won’t accept the premise that tomorrow the sun will rise. Correct. And there’s no… they won’t accept that either, right? No, I’m not sure there are people who really… there are people where I don’t know what goes on in their hearts, but there are many people who indeed claim they don’t accept it, but they act as if it were so because they are used to it; in any case they have nothing else to do. So that is what they’re used to. So at least philosophically—leave psychology aside—but philosophically, that is their position. I’m not judging what goes on in their hearts internally. Okay? I also have my suspicions.

But the point is that the softer modes of inference, the non-deductive ones, analogies and inductions, are methods that arouse suspicion in very many people, and not without reason, because they are not certain. You make an analogy between two things; maybe you’re right, maybe not, you can never be sure, right? And induction, generalization, something like that. And then what happens is basically this—I once described it as a three-stage maturation process. I’ll formulate it in a very schematic way, with no psychological pretensions. This is philosophical maturation, all right? Not psychological. Psychologists don’t know anything about psychology either, so I certainly know nothing. So I’m talking about a schema for philosophical maturation.

Okay? The small child when he is born, or a little after he is born, when he is small, generally takes things for granted. They tell him it’s so, so apparently it’s so. “Dad, why is the wall white?” “Because they painted it with white paint.” “Oh, I understand, now I know.” Who told you Dad told the truth? If Dad said it, it’s probably right. Okay? So he’s dogmatic. Meaning, he accepts what he is told, simple.

The first crisis is created, let’s say, when the child begins to mature a bit and becomes a teenager. Then you have the crisis of adolescence. In the crisis of adolescence the child starts to say: wait a second—Dad said it’s white because they painted it white. Who says? Maybe it was created white? They didn’t paint it white, it was always white. Prove it. How do you know? He asks his father, yes. How do you know? Prove it. He wants proofs. Right? Without proofs he’s a rational person; he’s not a fool like his father who accepts things without proof. He wants proof. Fine.

So he lives with that hope of being a rational human being, not a fool like his father who accepts things without proof, until he reaches the second crisis. The second crisis is the end of adolescence and the beginning of adulthood. And the second crisis is there when he comes to the conclusion that nothing can be proven. There is nothing in the world that can be proven. Why? Because every proof is built on basic assumptions, right? And the basic assumptions themselves you cannot prove. If you don’t accept anything that hasn’t been proven, then you cannot accept the basic assumptions. And if you don’t accept the basic assumptions, you also won’t accept the conclusion, because the fact that the conclusion follows logically from the basic assumptions doesn’t interest you. Because you don’t accept the basic assumptions. I don’t care that you derive conclusions from them if I don’t accept them in the first place. Meaning, the teenager who demands proof for everything and without proof accepts nothing reaches a crisis when he understands that with this approach he won’t be able to accept anything. Nothing. He won’t be able to know anything about the world.

And then there are three ways before him to mature. That’s the third stage. He was a child, a teenager, and now an adult. Three ways. The first is to remain with his adolescent assumptions. A: only something proven is admissible. He accepts things only with proofs. B: nothing can be proven. Nothing is proven. Conclusion: nothing is admissible. Right? Which is what was to be proven. Okay? That is the first maturation; let’s call it skeptical maturation. All right? He becomes a skeptic. “Who told you? Prove it. Nothing can be proven, so I accept nothing.” Okay?

The second maturation, the second route to maturity, is fundamentalist maturation. Fundamentalism means he accepts the assumption that only something certain is admissible, but he does not accept the assumption that nothing can be proven. Why not? Logically nothing can be proven because logic is built on basic assumptions. But he has some transcendent sources from above, as David said before, that are certain, beyond all doubt, and he accepts them. The rebbe said, the Holy One, blessed be He, said, Moses said, it is written in the Torah—whatever you want. Some source that is not open to doubt. Okay? An external source, not subject to critical examination, not subject to doubt, and it will give me the longed-for certainty. Notice: the fundamentalist does not give up the demand for certainty. He too accepts only certain things. What he claims is that he understands certainty cannot be attained by logical means, so you need some exalted source that gives me certainty. From here fundamentalism emerges. Because fundamentalism is not just the people who murder each other—that is only a result of fundamentalism. Fundamentalism, in my opinion, in its philosophical sense, is beliefs that do not stand up to critical scrutiny. Now if by chance you believe that you should kill everyone who is not like you, then in the news you look like a fundamentalist. But if, say, you believe in a fundamentalist way that one ought to benefit other people, then you are a fundamentalist but in the news you’ll look like Mother Teresa. Since you don’t do bad things, you’re not the horned fundamentalist. Okay? But philosophically you’re just as much a fundamentalist. He too does not subject his beliefs to critical examination. Only he is nicer, meaning his beliefs are more sympathetic, fortunately. Okay? But in principle, for me on the philosophical level, he is a fundamentalist. Meaning, he does not subject his beliefs to critical examination.

So that is the second maturation. In short, the first maturation—the skeptical one—accepts both assumptions of the adolescent. Assumption A, only something certain is admissible. Assumption B, nothing can be certain because proofs always rely on basic assumptions. The second maturation, that of the fundamentalist, accepts the first assumption—only something certain is admissible—but not the second assumption, that nothing is certain. There are certain things. It is written in the Torah, it is self-evident.

The third type of maturation is the maturation that does not accept the first assumption. It accepts the second assumption but not the first. The previous one accepted the first but not the second. Now I accept the second but not the first. What does that mean? I accept the second assumption: nothing is certain. I am not a fundamentalist; I do not recognize the existence of exalted sources that give me certainty, and I don’t think logic gives certainty because it is based on basic assumptions. I accept the fact that nothing in our world is certain. There is nothing I can know with certainty. On the other hand I do not accept the assumption that only something certain is admissible. I am willing to accept things even if they are not certain. If something seems reasonable to me, I will accept it too. Not only what is certain. So I reject the first assumption and accept the second.

This maturation I call synthetic maturation, for reasons that don’t matter right now. This maturation basically says that what I am doing, in the language I described earlier, is that I am willing to accept inferences that are analogy and induction, not only deductions. Right? Once I am willing to accept inferences that are analogies and inductions, I can support basic premises even though I have no proof for them. It seems reasonable to me, so I accept it, and consequently I can also accept the conclusions derived from those premises, because I accepted the premises too. And then the possibility opens before me of accumulating information about the world. Basically, the third option lets me accumulate information about the world without being a fundamentalist. Meaning, to accumulate it by scientific tools of analogy and induction, not from exalted sources, but as uncertain information—that’s the price. I know that nothing I think is secure. But I oppose views that say if it’s not secure, I don’t accept it, that only certainty is admissible. That is not true. I am willing to accept even something that is not certain. I lower the threshold of admissibility; the threshold of admissibility is not as high as it was for the adolescent. That is the change I undergo from adolescence to adulthood. All right? Each person undergoes a different change. One gives up, one stays with both assumptions and becomes a skeptic. One stays with the second assumption and becomes a fundamentalist. One stays with the first assumption—sorry, no—with the rejection of the first assumption and becomes a synthetic adult.

In principle, science can exist only for the third kind of adult. Because science is based on soft modes of inference. Not logic, not mathematics, but analogies, inductions, common sense if you like, plausibility, things like that. Not an exalted source, but observation—and of course also not certainty. So if you don’t accept anything that isn’t certain, you can’t accept any scientific claim. Skeptics, those who don’t accept things unless they are certain, cannot accept science either, not only belief in God or whatever. Nothing. Science too is not certain. The possibility of living with common sense exists only if you mature in the third way. Common sense, science—that is, soft inferences.

Now you need to understand that almost all the inferences that are relevant, interesting, non-trivial, that we know are soft inferences. Those are the inferences that bring me information. Suppose we have an argument, for example. Very rarely will I defeat you by knockout. I’ll just prove there is a contradiction in what you say, knock you down deductively. Meaning, I’ll prove that you are wrong. Many times I will raise softer considerations in order to persuade you that you are wrong. For skeptics, those considerations are uninteresting, because that isn’t certainty. They speak only in the language of logic. It looks a bit contradictory to the first glance, but that’s how it is. Skepticism rests on giving logic too high a status. Not contempt for logic, but the opposite, deification of logic. Why? Because if you think only logic is a tool whose conclusions you are willing to accept, then you are left with nothing. Remember? Logic contains no information; its certainty is bought precisely by that. Because if the premises aren’t true, then what do I care that the conclusion follows from the premises?

Yes, I spoke last time about a valid logical argument. I said: all tables have wings, this pen is a table, therefore this pen has wings. That is a valid argument. Its conclusion follows necessarily from the premises. Both premises are false and the conclusion is also false. But the argument is valid. The conclusion follows necessarily from the premises. Meaning, logic by itself is empty of information. And someone who accepts only logic as an admissible instrument is left without information. That’s basically what I’m saying.

Actually these three stages of maturation describe not only the maturation of an individual person, the philosophical maturation of an individual, but I think to some extent also the history of our civilization. If I divide it crudely into three parts, like everything in the army, then at first I’m speaking of a dogmatic period. People accept things because, I don’t know, because the sages said so, because the tribal sorcerers announced that you have to dance the rain dance and it will bring rain. So everybody dances the rain dance and it’s obvious. No one says wait a second, who told them that this dance brings rain? No, they say, they know. Fine? That is the dogmatic period, the childhood of our civilization.

Then comes the revolt of the Enlightenment. The revolt of the Enlightenment is when logic was born in Greece. There were seeds of it a little earlier, but it was conceptualized in Greece. And what does that mean? The concept of proof is created. Right—systematic logical thought and proof. Then people begin to ask: wait, who told you? Prove it. Now you need arguments in order to make claims. It’s no longer enough—who told you? Prove it. You need arguments in order to make claims; it’s not enough to say that some sage said it, he has a long white beard so he knows. Fine? Who says? Maybe not? Like adolescent rebellion. Yes, adolescent rebellion is the search for proof. “I don’t accept things without proof.” So here too, exactly the same. Always wanting to see your proof. Without that, I don’t accept what you say. Okay? That is the transition from childhood to adolescence.

Where does the transition from adolescence to adulthood happen? In the West it happens somewhere around the middle of the twentieth century, it seems to me. Because during the twentieth century something very strange happened that historians of culture don’t really know how to explain, I think. Until around the middle of the twentieth century, the positivist view ruled the day. The view that says that basically—logical positivism, say the extreme part of positivism—says that we accept only things for which we have proof or which are the result of direct observation. Meaning only certain things. Let’s leave aside for the moment whether direct observation is really certain or not; they assumed it was, most of them did. Okay?

In the middle of the twentieth century some kind of crisis occurs and what was later called postmodernism is born. Postmodernism basically accepts nothing. A kind of skepticism: everyone is trapped within his own narratives and forms of discourse and basic assumptions, and there is really no way to set a standard—true and false, beautiful and ugly, good and bad—there is no standard. The loss of the standard. Now, this transition from modernism—the great faith in science, in logic, yes, in the first half of the twentieth century—to postmodernity in the second half of the twentieth century is a great puzzle. How does this happen? And it happens in the same circles. It’s not that there was one empire and then another empire came and defeated it in war. No, it’s some metamorphic transition from the first part to the second. Something like that.

I think the logical root of this transition is what I described before. It is basically the process of maturation, the transition from adolescence to adulthood. What happens in the adolescent? He says: I will be a rational person, I accept only things that are proven. A positivist, in other words. Right? What does he discover at some point? That nothing can be proven, because every proof depends on basic assumptions, and how do you know the basic assumptions? If you don’t accept things without proof, then you also won’t be able to accept things with proof, because the basic assumptions you won’t be able to accept. And this crisis gives birth to several forms of maturity. I said there were three. The first way of maturing is skepticism. Because if only a certain thing is admissible and nothing is certain, then nothing is admissible. That is postmodernity. Therefore postmodernity is born; it is indeed the offspring of modernity. It is not its antithesis. It is simply the continuation. It is the deduction of conclusions from the assumptions that existed in the adolescent, or in the previous period. And it breaks, because once all you want is logical consistency, beyond that nothing interests you, there is no true and false apart from logical consistency, then if I have one set of assumptions and you have another, and each of us is consistent with his assumptions, then there is no right or wrong here and no way to argue and no way to win or lose an argument. Right? And thus is born this narrativism of postmodernity. Everyone with his own narrative, everyone with his own basic assumptions. It is simply a product of an exaggerated attitude toward logic, toward the idea that logic is everything and only logic is a tool for admissible claims—and then suddenly you discover that this tool contains no claims. And then you become postmodern. Therefore modernism is the father that begot postmodernism. It is not an antithesis.

But after all, physics, following experiments, showed us that there are performances—results. Results do not mean certainty. Lorentz equations. Results do not mean certainty. Results mean that in my eyes it is reasonable. But someone who doesn’t accept something just because it is reasonable… no, but you prove it in reality. For example, this electricity here… it didn’t prove anything, it is confirmed in reality. It is realized. But confirmation is not enough in the eyes of the positivist. It has to be proven. Then you see… you see. So what if you see? Maybe there just happens to be a demon here making light for you? How do you know there is a general law of electromagnetism here? Scientific laws are never certain. Earlier I told you there is no certainty in what is written in the Torah or in all sorts of things, and now I am telling you that in science there isn’t either. Rest easy. There is certainty nowhere. They introduced the concept of probability. I would say plausibility. Probability is something else, because you need a sample space. But plausibility, not probability. Okay.

The alternative maturation, by the way, if we slide a bit into current events—the alternative maturation, by the way, if we slide a bit into current events, what is it? Fundamentalism. Why was fundamentalism born in the second half of the twentieth century? For the same reason. Once positivism breaks down, either you become a skeptic or you become a fundamentalist. You become a skeptic if you accept only certain things and you think only logic is the path to certainty. But logic gives certainty about nothing. So you become a skeptic. But if you think only certain things are admissible and you think there are ways to reach certainty—not by logical tools but by mystical, exalted tools of one kind or another—then you are a fundamentalist, okay?

And that is also, by the way, the source of the great appeal of fundamentalism in the West. Why do sane people from Europe go fight with ISIS? The phenomenon is unbelievable. What is the appeal of New Age, what is the appeal of all these crazy and senseless things in our world? It is the same rupture. This rupture that says logic and science are not leading us anywhere. We can’t reach certainty. So what do you do? You look for some other sources—either communicating with aliens or communicating with Muhammad. But you need to accept some external source that will give you the certainty you lack so badly in this crisis.

And therefore I think the West also doesn’t know how to deal with fundamentalism, since they share the same assumption. Both the West and the fundamentalists assume that only something certain is admissible. Their argument is over the question of how one reaches certainty. So the West doesn’t know how to reach certainty; look, these guys reached certainty, I can only admire them. Look, they reached self-sacrifice, I can’t get there. Okay? So it’s very hard to deal with fundamentalism using those tools.

Therefore I think the third maturation has not only philosophical importance—because without it there is no science, and without it you cannot really believe what you think, in common sense, in considerations of plausibility and the like, in science—it also has current, even political importance, I would say. The only real way to deal with fundamentalism is not to be postmodern. The postmodernists thought that if they created a world in which everyone has his own narrative, then the wolf would dwell with the lamb. Everyone would enjoy themselves, no one would fight with anyone else because everyone is right. The problem is that tango takes two. If you have someone else who thinks he is right, then you haven’t succeeded in convincing him that he isn’t right just as you aren’t right, and he will kill you even though you won’t kill him.

The only way to deal with this is to say: I too am right, even though maybe you are right—there is no certainty. But the fact that I have no certainty does not mean I don’t believe in my way. And if you threaten me, then I will destroy you. Or if you argue with me philosophically—leave destruction aside now, leave politics aside—then I will argue with you, because I think you are wrong. I don’t have a proof. I don’t have a proof—common sense—and still I say that you are wrong. Because my common sense tells me you are wrong. That is an illegitimate statement, neither among fundamentalists nor among postmodernists. For postmodernists, to say to someone that he is wrong is heresy against the faith. What do you mean wrong? Who appointed you? How can you know who is wrong and who is right? Who are you anyway? You are as right as he is or as wrong as he is; there is no right and no wrong and everything is fine. The circle of differences—we dance in Rabbi Shagar’s circle of differences, okay? Everyone. And that is nonsense, of course. This whole thing is a result of the shared assumption of those two seemingly opposites, the fundamentalists and the postmoderns. They are opposites, but opposites in the sense of two sides of the same coin; in that sense they are opposites.

And in order really to deal with these phenomena, the only way is to give up the first assumption. I accept things even though they are not certain. Common sense, plausibility, considerations of analogy, induction, softer tools. Not necessary tools. For me those too are admissible. I am willing to accept that too and even fight for it, because I believe it is true. Not certain—nothing is certain. After all, that too is the assumption that only logic can prove things. I said: I don’t accept that. Because if only logic can prove things, then nothing is proven, because there are assumptions in logic. And those assumptions you don’t know. That is exactly the point. So I say no: I don’t think only logic can prove, but it is not only what is proven that is admissible. Something unproven can also be admissible for me if I have common sense. Which is really the basis of science. Science began to advance from the moment people gave up the yearning for certainty. That didn’t stop science; that is what advanced science.

Science, the moment it gave up the level of certainty of logic—once people thought science and logic were the same thing. Aristotle, for example, concluded that a heavy body falls faster than a light one, which is of course factually incorrect. Okay? They fall at the same speed, acceleration g. Now, how did he… You don’t need a particle accelerator to see the difference. The technology existed in Aristotle’s day. Take a heavy stone and a light stone, go up to the first floor and throw them down, and see which arrives first. Right? Like the old comedy sketch: “You send your son to the Weizmann Institute, I’ll send mine to the Shimshon Institute, and we’ll see who gets there first.” Right, so that technology existed; you didn’t need particle accelerators for that.

So how did Aristotle not understand this? He never bothered to run the experiment. Why? Because it was obvious to him that it was true. He had a proof, some philosophical proof or other; obviously if it is heavier then it falls faster—what was to be proven. He didn’t bother to check it, because for him science was a branch of philosophy and logic. It was not yet a separate discipline. Science became a separate discipline in the sixteenth century. Why? Because then they suddenly understood that with philosophy and mathematics you get nowhere. Information is accumulated through other tools—softer tools, less certain ones—and therefore they can give me information. And once they understood that, science began to advance. But there is a catch. Science advances, but one must always remember: it is not certain. Because that is the price. In order to advance and accumulate information, I have to pay in the currency of certainty. Because certainty belongs only to logic, but with logic you don’t accumulate information. Okay? Therefore the third synthetic maturation is the only philosophical basis that can allow belief in science, and belief in law, and belief in all the softer, everyday modes of inference—not the mathematical-logical ones.

Okay? A synthetic skeptic—a person cannot really be a skeptic. Because the very fact that he lives and gets up in the morning and goes to work and comes back… I already told you, there are those who will explain to you: “No, I really am a skeptic, but I do it because I’m used to it; I have nothing better to do. In any case I don’t really think anything about anything.” Fine, so I do whatever comes to me. I have no justification, I don’t really believe in it. Don’t ask me again what goes on in his inner psychological heart. That’s what he claims, that’s the position he presents. For me, that is his position; I don’t care at the moment what he thinks in his heart. As I said, I suspect what you do that he thinks otherwise in his heart. Fine, I don’t care; I’m not judging him.

And where does the certainty of skepticism sit? Meaning, skepticism itself—is that… This is a famous argument against skepticism: how are you so sure of that? Why are you certain that only something certain is admissible? Where did you get that from? Is that certain? Why do you accept that? Fine, these are known arguments against skepticism, but I’m not entering into arguments against skepticism right now, I’m just describing at this stage.

Okay? So the great puzzle and the great crisis, or challenge, of synthetic maturation is the following challenge. Let’s return for a moment to the three-stage maturation model. The father standing opposite the child, yes? And the child asks him, “Tell me, Dad, why is this wall white? Why is the sky blue? Why does the sun rise?” and all sorts of questions like that. So in childhood the father tells him and he accepts it, everything is fine. When he becomes a teenager he says, “Wait, who told you? Prove it! Unless you prove it I don’t accept it. I’m a rational person, a positivist, I don’t accept.” Okay? And the father tears his hair out because he doesn’t know how to prove it. But he understands that it is true because it’s reasonable, right? It’s obvious, common sense. But the son—only proofs. Meaning, what he has no proof for, I don’t accept.

Now how does the son look at the father? Not how the father looks at the son—he tears his hair out. But how does the son look at the father? Notice: the son, the teenager, right? has gone through two stages of his biography so far. Right? Childhood and adolescence. He does not yet know the third stage. He has not yet gone through it. His father is in the third stage, assuming the father matured in the third way, the synthetic way. So his father is in the third stage. Because the postmodern father really shouldn’t answer his son anything; he’s in the same condition as the son. So the point is that the father—the child asks him, “Tell me why the sun rises,” and he says, “Because the earth rotates and therefore the sun rises.” “Who told you? Prove it!” I don’t have a proof, but it sounds reasonable to me. There are all sorts of indications. Okay? This is a scientific result, so there is no proof, there is confirmation.

So how does the son view the father? The son assigns the father to stage one. Right? Because the father accepts things even though they are not certain, there is no proof for them. And he says them with full confidence as though he knows them for certain. Because for the teenager, only what is certain is admissible. If his father accepts these things, then apparently in his eyes they are certain. How are they certain? The man is an idiot. Meaning, he is apparently still stuck in childhood, where whatever he is told he accepts absolutely, without asking like I do. I am a rational person; I seek proof. Without that I accept nothing. You know Mark Twain’s saying? A saying I’m very fond of. He has many beautiful sayings; a very witty man. He said: when I was fourteen, my father spoke such nonsense that your ears would ring. Five, six, seven years passed—how much that fellow learned in those five, six, seven years, unbelievable. Same thing. The teenager, when he looks at his father, since he does not yet know the third stage, is in the second stage and has passed the first. So he assigns his father to the first stage, because he accepts things even though he has no proof for them. My father tells me things without having proof and is convinced they are true, he thinks they are true. He is simply stuck where I was in the previous stage; I’ve already gotten out of that, and that fellow is apparently still stuck there. Okay? That’s how children look at their parents. It’s not a metaphor; that’s how it is. I think in many cases it really is like that.

And the father tries to explain to him: listen, there is a third stage, I’m not stuck in the first stage. But the child hasn’t gone through it. You know, a person eventually shaves on his own beard. What you haven’t gone through yourself, you don’t… “A person does not fully grasp words of Torah unless he has stumbled in them.” Meaning maybe the child genuinely cannot understand this until he gets there himself. When he goes through the crisis of maturation and understands that he has to choose one of three paths—synthetic, skeptical, or fundamentalist—he won’t understand what his father is talking about. After he gets there, there is some chance he too will become that kind of person. Fine? And then of course he too will get stuck with his own son, everything is fine; everyone gets back all that they fed others. In that we can all take comfort: our children will get it from their children. So our children and grandchildren, exactly.

Now what is the point? What really is the difference between the third stage and the first? Where is the teenager mistaken in his perception of the adult? Why is he not right that the adult is really stuck in stage one? What is the difference? After all, in both places you accept something that has not been proven, just like that, without proof. So what’s the difference? Why is stage three different from stage one? Why is it not just regression to the womb, a return to stage one? Why is it stage three? The answer: plausibility. Exactly. The concept of plausibility is crucial here. Meaning, I don’t accept things just because someone said so. That is the dogmatist of the first period, the child. I accept things that have passed certain tests. Not rigorous logical arguments, but tests of common sense, analogies, inductions, scientific falsification tests, scientific generalizations, different methods we have developed to manage our softer kind of thinking, not our mathematical-logical thinking. That is the difference between stage one and stage three—understand, that is the essence of the difference. Synthetic maturation means developing non-deductive ways of navigating between uncertain truths, or between uncertain claims. In order to make decisions in a world where there is no certainty. Because logic is a world of certainty. But in a world where I don’t have certainty, how do I know I’m accepting something not just because I’m dogmatic, but because it is reasonable to accept it? I need to develop criteria for what counts as reasonable. How do I define reasonable and unreasonable? Logic defines what is certain. But I’m not looking for that; I’m looking for criteria for what is reasonable. That is not a simple question at all. To this day it has not been solved. That is really the great challenge of synthetic thought, of synthetic maturation.

Now in this context, what is the difference between synthetic maturation and postmodernism? You could define postmodernism in exactly the same way. No—postmodernists don’t recognize reasonable and unreasonable. Whatever feels right. No, the concept of plausibility doesn’t work there; you cannot reach a decision. No, that is the difference. You’re making it sound as though it’s the same thing, and it isn’t. That is exactly the difference. The postmodernist is not willing to accept something as more or less plausible; if it isn’t proven, its status is equal to every other claim. By contrast, the synthetic adult says: this is not proven and that is not proven, but this seems more plausible to me, so I accept it.

And in the end the postmodernist also lives according to something, he gets up in the morning. No, no, that’s what I said before—you’re mistaken. No? He gets up in the morning but he doesn’t really believe in it. A Jewish postmodernist. Leave Jewish aside; there is no such thing as a Jewish postmodernist. None. Why? If he’s Jewish then he’s not a postmodernist. There isn’t one. There isn’t. If he prays just because he feels like it, not because he really thinks so, then he isn’t Jewish, and he also isn’t praying. He doesn’t think, he believes in it. Ah, does that make it more plausible? Then he isn’t a postmodernist, he’s synthetic. No, that’s exactly synthetic. Right. No, but that’s not postmodern, that’s synthetic. The postmodern one is the one who doesn’t do that. The postmodern one says: I accept nothing as admissible. So what if he does? He doesn’t accept it as admissible, like they asked me before. Suppose someone, a skeptic, comes and says to you: look, I don’t believe anything that isn’t proven. Wait, listen, listen, listen—I’ll explain why. I say to you: I don’t believe in anything that isn’t proven, okay? You ask me: tell me, will the sun rise tomorrow? I have no idea. So why do I set my alarm clock for sunrise tomorrow morning? Okay? Wait, listen. You know why? The answer I’ll give you is: because that’s what I’m used to. I don’t believe that the sun will rise tomorrow, but that’s how I’m used to operating; what do I care, I have no better time to set it for, you understand? That is a true, pure postmodernist. If someone… define him however you like, that’s how I define him. I’m defining two concepts now. You can define Moshe as Yankel if you want; there’s no point. Let’s synchronize our language. When I say postmodernist, I mean this one. When I say synthetic thinker, I mean that one. Okay? There’s no point in giving them both the same name. And no, it isn’t the same thing; these are two different things. You can call them by the same name, but they’re two different things. You can define a circle as a square—does that make them the same thing?

Even if he sets it for eight in the morning, from his perspective he could set it for ten or twelve. It doesn’t matter, but he does behave. There is no person in the world who doesn’t behave as though he believes things. He asks what time it is; you tell him eight; then he understands he has a meeting in half an hour because it’s at eight-thirty. So ask him: wait, who are you? Maybe the clock is lying, maybe the time is actually eleven? Fine, so he says: all right, I don’t know, I believe in nothing, but this is how I’m used to conducting myself. I have no better way, so I conduct myself this way. That is a true, pure postmodernist. In reality there are also postmoderns who reject assumptions most people accept. No, no, that’s not postmodern. Leave it—I don’t want to get into the discussion; it’s just concepts, semantics, not worth talking about. I defined my types and gave them names. If you want to call them Moshe and Yankel, call them that. Fine? It doesn’t matter. I think my definition is also correct, but that’s another discussion.

So the challenge facing the synthetic adult is to explain how he differs from the postmodernist, in my terms. That is, the question whether you can point to criteria of plausibility that you are willing to stand behind and say: this I believe—not just act by it. Anyone can act however he likes; that’s psychology. But this I believe, in my eyes this is true, and someone who doesn’t say so is mistaken. Statements a postmodernist cannot say. Okay? But I say: if you think otherwise, you are wrong—not because that’s what I’m used to, but because I think this is true. Do I have a proof? No. But I’m not just a dogmatist. That was stage one—just because I feel like it, because someone said so. No, I have criteria. Like scientific criteria, say. I have criteria of plausibility. That is their importance.

You need to understand—I’ll put it as sharply as possible—in order to fight ISIS, you need to find criteria of plausibility. Do you understand? You cannot, you have no philosophical infrastructure with which to deal with these powerful views—postmodernism or fundamentalism—unless you set up another philosophical infrastructure that says: I have a structure, I have a logic that tells me what is more plausible and what is less plausible. Not what is certain—that is the logic everybody knows—but what is plausible. I have criteria. I can know. Contrary to what the teenager thinks about me, I do not accept everything because someone said so. He thinks that because he thinks that anything not proven, I just accept arbitrarily. No, I don’t accept everything. I accept things that are plausible. What is the criterion? What is a criterion of plausibility? That is our starting point.

All right, I have more or less finished the introduction. What? You can never fight ISIS from the standpoint of certainty. By “fight ISIS” I mean from the standpoint of my own inner backbone, not in the sense of persuading them, because otherwise I become null before them, you understand? I can’t stand against them.

The attempts to propose criteria of plausibility begin, not by chance, at the beginning of the age of modern science, at the beginning of the modern era. Francis Bacon was essentially the first to try to propose an inductive logic. A logic for the conduct of science, okay? A logic of elimination and all sorts of things like that. How do you make inductions intelligently? Meaning: how can I know that this lectern is brown and the sheep outside is also brown, therefore all objects in the world are brown? That sounds implausible, right? Why? How is that different from saying this lectern is brown, therefore all lecterns are brown? Both are uncertain. But the second is more plausible than the first—also not true, but at least more plausible. Why? What is the criterion that makes the second more plausible than the first? That’s what I’m talking about, and I’m looking for a logic that enables me to sort things on scales of plausibility, not on scales of certainty, valid and invalid like classical logic. All right? A logic of plausibility.

So Francis Bacon proposed something like that. It doesn’t really stand up to scrutiny; I’m not going into his logic now. I’ll perhaps just show one example through which I can demonstrate this issue. Look, in two different places you can see this—you know, in the world of ideas this is a very common phenomenon. As someone who deals with many fields, I encounter it a lot; anyone who deals with many fields can see it. Different fields discover the same thing, the same phenomenon, each in a different context, and no one knows about the other. But they are talking about exactly the same thing. Sometimes one is a bit earlier, one a bit later, but they move on parallel tracks; they don’t know that over there they already know this, because they don’t always understand it’s the same thing.

I’ll give you an example. This is not an example of the principle; the principle was just an introduction, I want the example. There is a book by a historian named Carr, a well-known British historian from the beginning of the twentieth century, and the book is called What Is History? He asks how the process of historical research works. According to Baconian logic, it works like this—the naive view of the historian’s work—you gather facts, analyze them, and arrive at your historical conclusion, your historical theory. For example, you want to know why Napoleon was defeated at Waterloo. Okay? So you need to gather the facts on the ground: what the morale of the soldiers was, what equipment they had, what strategy they used, how many soldiers there were, whatever, all sorts of things like that. Then you analyze the facts and arrive at the conclusion as to why Napoleon lost.

But, Carr says, that is a naive view, it cannot be correct. Why? Because as long as you are not equipped with a theory of what can affect victory in battle, you don’t know which facts to collect. After all there are infinitely many facts; how do you know which of the facts is relevant? Maybe the name of the mother of the adjutant of the seventeenth battalion in the two armies? This one’s mother was called Hannah and that one’s Yentel, so H comes before Y, so they won. Apparently that’s why they won. What’s wrong with that? You know it’s not true? Who told you? Why are you chuckling? Think of someone who knows absolutely nothing right now about military theory. He doesn’t understand what affects victory in battle and what does not. So he approaches the research tabula rasa. He knows nothing. Fine? Now he gathers—what does Francis Bacon say? How do you start research? You gather facts, information, analyze, and reach theory. Carr says that cannot be right. As long as you don’t know the theory, you don’t know which facts to collect. Obviously you assume things, even if you don’t know the theory fully; still, you roughly assume what may be relevant and what may not. The name of the mother of the adjutant of the seventeenth battalion probably won’t be relevant; the morale of the soldiers probably will, or their equipment, something like that. How do you know? You haven’t checked yet. You’re just beginning your military research, you’re now at year zero, military theory is just beginning to be built. Okay? So ostensibly I know nothing yet; I’m starting to gather facts. Not true that I know nothing. Obviously there are things I know, because otherwise I couldn’t even begin, I couldn’t collect the facts and start the process. There is some general hypothesis that I know. I collect facts, I test the hypothesis. It doesn’t work; I collect more facts, patch the hypothesis, test the revised hypothesis against the facts, go back, test whether it works on all battles, to see where they won and where they didn’t, whether it fits or not, build an even more refined hypothesis—and so back and forth between theory and facts, that is how one advances.

There is a kind of miracle here, you have to understand, because there is some starting point that can’t really be explained. Where do we get the initial framework from, even before the facts? How do we know which facts to collect? Okay? But from that point onward at least it goes back and forth, as opposed to what Francis Bacon said in his logic, in his inductive logic, that you gather facts, generalize, analyze, and reach theory. Not true. You begin with theory—a hypothetical theory, a theory you propose—collect facts, check them in light of the theory, revise the theory, go back and collect more facts, check, revise, and so on. That is how theories advance and become refined and improved and stand against reality. Falsification tests and so on. Okay? That’s in the context of history.

The same exact thing happens in philosophy of science. In philosophy of science there is a book by Hempel, Carl Hempel, a well-known philosopher of science. He brings there a canonical example—everyone has used it ever since he brought it. It appears also in the Open University’s introduction to philosophy of science; Hempel is the book that accompanies the course. He gives there an example from a hospital with two maternity wards. A doctor named Semmelweis, an Austrian Jew, about childbed fever. Yes, exactly, childbed fever: there are two wards. One ward was Semmelweis’s, he was the head of the ward, and there was a high mortality among women giving birth there, and in the second ward the mortality was much, much lower. Now they tried to understand why. Why in this ward was there higher mortality than in the second ward?

Now exactly the same move as there—it’s amazing how this happens in parallel, more or less even in the same period. Hempel is a little later than Karl Popper, no one mentions the other—and now Hempel, or rather Semmelweis himself, asks: okay, so now I need to check why it happens here and not there. Let’s gather facts. He has no idea which facts are relevant because he doesn’t know what causes the deaths. So the facts he gathered there were bizarre. He checked at what hours the priest entered each ward, whether he walked east to west or west to east, whether the windows were open or closed. He had no idea. It was shooting in the dark. After all, as long as you don’t know what causes the deaths, how do you know which facts to gather? The relevance of the facts is determined by the theory, but you won’t manage to reach the theory if you haven’t gathered the facts. So how does this business even start? Exactly like Napoleon’s defeat. How does it happen?

Miraculously, at some stage he discovered absolutely bizarre facts, and this is a wonderful example that without some idea of the theory, you won’t be able to gather facts. It’s not true that facts precede theory. Theory precedes facts. And true, afterward it undergoes articulation and refinement back and forth. You begin with theory, move to facts, refine the theory, go back to facts—but you begin from theory. You don’t begin from facts. There is something a priori. At the foundation of every science sits something a priori. That’s pretty amazing because science is usually considered empirical, the result of observation. It’s almost the example of something scientific that we build on observation, we don’t assume anything—nonsense. Science rests on very strong assumptions. Without these assumptions there is no science and nothing at all. Assumptions that are not the result of observation—assumptions that are a priori.

Now in the end there they discovered that there were students who did dissections of corpses, entered the ward, and worked on the women giving birth there without washing their hands. Then, among all the crazy experiments they tried, someone said let’s try having them wash their hands first. They had no idea that this had anything to do with mortality, you have to understand. Someone who doesn’t know the role of microorganisms in disease—for him, washing hands is about like saying Psalm 119, more or less, forgive the comparison; in my eyes washing hands helps a lot more, but I mean it’s the same kind of thing, like snakes and amulets. After all, you have no idea what helps and what doesn’t, right? So understand, this is entirely shooting in the dark, just like the direction the priest walked. No difference. And suddenly, whoosh, he discovered it worked. By chance. He could have searched to this day among all sorts of things; there are infinitely many facts, and for each fact start checking whether it affects things, and by chance he happened to hit the relevant fact after a year rather than after two thousand years. It could have taken two thousand years too, depending on the order in which he checks all the options. There are many options. Okay? Then he moved forward and suddenly understood that something about the hands and the washing had an effect. Then he checked what was on the hands, and little by little discovered the influence of microorganisms. Exactly as in Carr.

What does this mean, basically? It means there is a kind of symbiosis. This division between science and mathematics, or between theory and facts, is not sharp. Meaning, it always comes together, always. You must always use both. You need axioms, say, and then mathematics derives conclusions from them. You need analogy, you need induction, and afterward you descend by deduction. It always comes together. Therefore in the real world, when you deal with facts in the world, not with hypothetical logical or mathematical structures, you always use both things: theory, a priori elements, and observations. Meaning, it is some combination of those two things.

Now of course the question returns: all right, so what is the criterion? How do I decide what is plausible and what is not? It’s all shooting in the dark. How do I know which shot is nonetheless in a plausible direction even though I still don’t have information? I have to know how to analyze facts and think which are relevant and which are not, what to do with the facts. In short, in order to advance in the scientific world—not the logical-mathematical world, but precisely the world that deals with facts, scientific, historical, research of one kind or another—you need exactly what I called a soft logic. A logic of plausibility, not a logic of certainty and validity, but a logic of plausibility. Okay.

So what I want to do from this point on is to try to develop, in a systematic way, a logic of plausibility. This whole introduction was really meant to get us now to this starting point. I want to develop a logic of plausibility. And I will do—wait a second—I will do it through three of the hermeneutic principles: kal va-chomer, binyan av, and gezerah shavah, and binyan av from two verses, and the refutations of them. And my claim is that these three principles and the refutations, and all combinations of them at whatever level of complexity—and you can reach very high levels of complexity—basically lay out all non-deductive thought. That is my claim. They are the basis of non-deductive logic. Non-deductive? Yes. The soft logic, what I called earlier a logic of accumulating information. As opposed to the rigid logic, which is a logic of analyzing existing information. A logic by means of which I accumulate information, going from premises to a conclusion that contains additional information. Yes, scientific logic—the logic of science—is a logic of accumulating information. I’ll try to show this through those rules.

Now since I’m not going to start that now, there isn’t much time left, I’ll finish with an example that may illustrate the… Accumulating information in technology, for example, started from theory, say. Information isn’t exactly—technology isn’t exactly accumulating information. Technology is deductive. A radiator, for example, that came from designs. Yes, but technology in essence is deductive. Because once you have the scientific law, technology is just an application of the scientific law. Science is inductive. Science goes from particulars, from examples, to general laws. Once you have the general law, technology harnesses that law to a particular case and uses it. So technology in essence is deductive. The essential advances, the substantial progress, happen in science, not in technology. The progress of information. Technology is the application. But it follows from it; technology is a derivative. That is the deduction. The induction is the science, the deduction is the technology. Okay.

I want to demonstrate something that gives some sort of… it’s a nice example that always connects for me with this issue. Or maybe one more sentence before I go into the example. What I really want to do with the analysis of these Talmudic modes of inference—they are not Talmudic, of course. Kal va-chomer and binyan av from one verse and from two verses are modes of inference used in every field of thought, not only in the Talmud. This distinction between these three and the systematic treatment of them appears in the Talmud. Therefore I use the Talmud to analyze them, but in practice it will give us a logic whose scope is not only the context of Talmud or the Oral Torah, but in fact our thinking in general. Therefore there are really two kinds of outputs from the process I want to undertake. One output is import, the second is export. Basically, one can use logical tools developed in the broader world in order to decipher Talmudic passages—let’s call that import, if we see ourselves as located inside the study hall. And there is export. That is, there are things I analyze in the Talmudic discussions, I extract from them some set of rules, and it may be that they can be useful in other fields because I understand them to be general tools; they are not tools specific only to here. All right? Fine, so with that I’ve finished the introduction.

I’ll give one more example, and with that I’ll end the class now, and next time we’ll begin the actual work. Look, there is a theorem in mathematics that I once saw in some nice little booklet, a theorem dealing with the concepts of convexity and concavity. Yes, so intuitively a convex shape is, say, a shape that bulges outward, all right? Say, a convex shape. A concave shape is, say, a shape like this, that has some kind of inward depression. All right? I’m saying everything with hand-waving, of course, but on purpose. Fine, this is called a convex shape, meaning convex in all directions. A concave shape also can’t really be concave in all directions; rather, it is not convex everywhere, not all its points are convex—that is called a concave shape. Is there a concave shape that is concave at all points? I seem to recall once seeing that there is, but that’s a mathematical pathology and only mathematicians know. For our purposes, a concave shape is a shape that is not convex at all its points, all right? Convexity must hold at all points. For concavity it’s enough that it be only at one point or in part of them.

Now there is a theorem in mathematics that says that the intersection of two—what are convex shapes? For example, this too is a convex shape, a triangle. Yes, it is not concave, so it is convex. Anything that is not concave is convex, all right? Even though there are no bulges in this part, it still counts. But here all these points do not bulge; it’s straight. Okay? But for us this is called convex. And only that is concave. Okay?

Now there is a theorem that says every intersection of two convex shapes is also convex. All right? For example, take the example of the intersection of these two, a triangle and a circle. Their intersection is this area. Okay? That area too is convex. And if you now intersect another one, a hundred, it doesn’t matter how many convex shapes of whatever kind you want—if they are all convex, the intersection comes out convex. That’s a theorem. Okay?

Now the question is how you prove that theorem. So I kind of wracked my brain over this. Wait, I didn’t need to—it’s a known theorem, but as a puzzle. I wracked my brain and didn’t succeed. I tried dividing convex shapes into various types because basically it’s clear the boundaries are convex, right? Because the original shapes are convex. But with the joins, you always have to see why no pathology can ever emerge at the joins. Okay? So I had no systematic way to do it.

So what does the little book do? I’m not going to give you the puzzle now—it would have been nice to give it, let you try and fail, and then I’d tell you the proof. But let’s spare that, because this is only a side discussion. So the claim is the following: what I do is first define what a convex shape is. Before, I was waving my hands. Now I have to define it. Mathematicians don’t work without defining things. So how do you define a convex shape? The accepted definition is: a shape such that for every two points in it, if you connect them with a straight line, the entire line lies inside the shape. Right? For example, here that doesn’t hold, because these two points, if you connect them with a straight line, not all the line lies inside the shape. You understand the logic, right? A convex shape—it’s very sensible that this defines our intuitive concept of convexity. We basically know what a convex shape is; we don’t need the definition for that. The definition tries to conceptualize what we know, meaning to define sharply what we know intuitively. So this conceptualization is not bad.

Okay. Once we’ve defined it, now the proof is really trivial. Here I was able to prove it, once he had given me the definition of a convex shape. It’s very simple. Let’s take this, say two convex shapes—the extension is by induction, of course, but let’s take two. All right? I want to prove that this shape, the one obtained by the intersection, is convex. Right? In order to prove that, I have to prove that for every two points inside it, if I connect them with a line, the line lies inside the shape. Right? Now since these two points lie in this shape, which is the intersection of these two shapes, then these two points lie both in this shape and in that shape, because they belong to the intersection. Right? So that means these two points lie in the circle, and the circle is a convex shape, so the line also lies in the circle, because the circle is convex. Now these two points also lie in the triangle, and the triangle too is convex. So the line lies in the triangle too, because the triangle is convex. But if the line lies both in the triangle and in the circle, then it also lies in the intersection. Therefore every two points that lie in the intersection, the straight line connecting them lies entirely within the intersection. Which is what was to be proven. Now of course add another shape, intersect this shape with another shape—it’s the same thing, it doesn’t matter, that’s already by induction. Okay? So it’s a very simple proof.

What prevented me from succeeding before? The definition, right? Since I hadn’t defined the concepts sharply, I had no way to proceed systematically. Now I want to ask you a different question related to what we discussed before. But that means that even if you take this shape and make a triangle inside it and then in their intersection do the same thing… it’s all the same, it’ll work forever, however many shapes you add. This shape is not such that every two points… Every two points in the intersection belong to all the intersecting factors because that’s what an intersection is; that’s the definition of intersection. So the line—this shape isn’t one where every two points are in it. He’s talking about the second shape, the other one, this shape. And this one is not; that’s why it is concave and not convex. We are talking about intersections of convex shapes. But if you put another triangle inside, what difference does it make? I’m talking about intersections of convex shapes. Take two convex shapes, place one on top of the other, and you get some intersection. Is that intersection convex? Or could it come out concave? I’m saying no, it cannot come out concave; it is convex. All right?

Now look, the point is this: I now want to ask you a question related to what we discussed before. I’m asking you now not a question in mathematics, but a question about the world. Is it true in every two convex shapes, intuitively, what we call convex, that their intersection is convex? Proven. Their intersection is convex—that’s what you proved here. Here I didn’t prove it about the world; here I proved it in mathematics. Why? Why? In representation, that’s not… it depends how you look at it. You can take two bubbles and intersect them. Or in two dimensions—leave bubbles aside, that’s three-dimensional, complicated—let’s talk two-dimensional. Klein bottle—I don’t know whether it is convex or concave, I don’t know.

Look, the answer is that you cannot. Why? Because notice the dirty trick I did here—or rather the book did. I actually asked a question from life, not a question in mathematics. I want to know: the shapes that we intuitively call convex, that we understand intuitively, if we intersect them, will the intersection be convex or not? Right? What did I do? I translated it into mathematics. I said: what we intuitively call a convex shape can be conceptualized in this way—that for every two points, if you connect them by a straight line, the entire line belongs to the shape. Right? And I assume this definition properly represents the intuitive notion of a convex shape. Do I have a proof of that? No. Can anyone show me, prove, that every shape I would intuitively call convex satisfies that definition? It sounds very reasonable to me, like “one straight line passes through two points,” but obviously there is no proof, right? There is no proof. I believe that this is a good conceptualization of the intuitive notion of convexity.

Okay? So basically what I did here was this: I couldn’t manage with the intuitive concept of convexity, so what did I do? I replaced it with another concept, and I proved it about that other concept. But the original question still remains. I did not prove that. Who says that the other concept correctly represents the intuitive notion about which I asked the question? I have no proof of that. Therefore this is not a proof of the proposition. It is a proof of a mathematical theorem, but it is not a proof of my claim about the world. Or in other words, there is what I discovered over and over again in my own flesh when I wrote my doctorate: there is a law of conservation of difficulty. You never get rid of the difficulty. If you found a way around it, you will meet the difficulty again at the end of the road. You never get rid of it. In other words, if you managed to prove something that before you couldn’t prove, then know that you didn’t really prove it. You simply hid the difficulty under the rug, swept the dust under the rug.

Where was the difficulty in proving the claim before the definition? The difficulty was how to define convex shapes. If I knew how to define them, I would know how to prove it. Now someone comes and offers me a definition. Who says that this definition captures precisely the intuitive notion “convex shape”? It seems right to me, but who says? I didn’t check all convex shapes in existence, right? I don’t know. As long as I don’t prove that, this theorem can’t tell me the claim about the world, that in the world every two convex shapes whose intersection is taken also yield a convex shape. To prove that about the world, I need to continue and prove that the conceptualization I made—that for every two points the line also belongs—fits what I intuitively call a convex shape. I don’t know how to prove that. And that is exactly what I also didn’t know when I tried to prove the theorem at the beginning. Mathematics did not solve the problem. It hid it inside the definition. Meaning, all the problems I don’t know how to deal with I push into the definition. Once I have a definition, from there on I proceed mathematically, everything is precise, absolute, certain, everything is wonderful. We haven’t really solved anything. All the difficulties are still there; it’s just hidden in the definition.

Meaning, overall you have some chain, you need to get through this path. Here there is some obstacle, I don’t know how to get over it. So let’s assume we got over it and got here, and now let’s continue walking from here. Let’s assume. That is basically what I did—I swept it, put a rug over it. I swept this dust under the rug, and now I can walk, no problem, I got over everything. But the dust is still under the rug; the house is not clean. Why? Because if I am interested in claims about the world, mathematics cannot help me with that—pure theoretical mathematics by itself. In order for mathematics to help, I need to assume another assumption, and that is an assumption in physics: that this mathematical concept, this definition of convexity, correctly describes the intuitive notion of convexity—just like vectors with forces, exactly the same thing. And that is already an assumption in physics, and someone can bring a counterexample and refute it. Someone may suddenly find a shape that everyone would say intuitively is convex, and yet it does not satisfy these conditions. Yes, I mentioned Klein bottles and such, and I suspect that there that kind of thing could happen. Klein bottles? Never mind, pathological shapes that only mathematicians know what to do with. There I have some feeling that all kinds of things like that may happen.

Okay, so what this means basically is that when I ask a question in value-laden fields, in fields dealing with information in the world—law, science, and so on—and not in logic, then using logical tools is basically just focusing on this part. But there is still this part. This part is really the transition from the intuitive concept to the definition. That is a generalization, basically, right? All convex shapes satisfy this definition. A generalization. Once I make that generalization, then if something satisfies it I can, in a logical deductive process, prove my theorem. So we have the illusion that mathematics solves problems; it solves nothing. Mathematics is absolutely precise and therefore doesn’t help us, like with the hot-air balloon. Except that if I assume a premise in physics, that these mathematical assumptions correctly describe the reality I am dealing with, then no problem. But that assumption will never be certain. I will never have a proof of it, because it is an assumption about the world; it is not mathematics.

Okay? Therefore whenever I deal in mathematics, I deal with something pure, pristine, certain, precise, wonderful—the conclusion follows necessarily, everything is excellent. Every time you bring it into the world—the world of law, the world of history, the world of science, whatever you want—it becomes uncertain. It stops being certain, because always in the background—and we’ll encounter this again and again later—there is some premise in the background that is not a premise in mathematics; it is a premise in law or a premise in physics or a premise in history or whatever else. Fine. Okay, you can say it’s plausible. Right. You can say it’s proven. Exactly, exactly. And that connects precisely.