More on the Relation between Theory and Reality: The Meaning of Logic and Mathematics (Column 50)
With God’s help
In the previous two columns I discussed the relation between theory and reality. We saw there that the main importance of a theory lies in the principled insights it provides, sometimes non-trivial ones, but using it requires great caution. A theory is indeed simplistic by its very nature, but that is not an accusation. That is the nature of theories; that is their power and also their weakness. The accusation of oversimplification is relevant and justified when it is directed at someone who thinks that theory accurately describes reality and can be straightforwardly applied to it in practice. The oversimplification lies in the view or assumption that theory will solve every difficulty and problem. In this column I will broaden the discussion somewhat regarding the relation between logic and reality.
Arguments and statements: on logic and facts
A theory is a concept used in the sciences, and its point is to provide an integrated and coherent description of facts within a single conceptual and logical framework. Thus the theory of gravitation explains a whole collection of phenomena (the tides, the paths of the stars, the fall of objects to the earth) by placing them all within one conceptual and logical framework: every two bodies with mass attract one another with a force proportional to the product of their masses and inversely proportional to the square of the distance between them. This theory succeeds in explaining all the facts I mentioned above (and many others as well). But the fact that this theory has a logical structure and uses the language of mathematics does not mean that it is logic. Logic is, in itself, devoid of factual content, since it deals with various relations among facts and not with the facts themselves. Facts are the business of science, and there logic and mathematics are, at most, languages.
A logical argument can claim the following: if all tables have legs, and the object before me is a table, then the object before me has legs. This argument does not say that all tables have legs, nor that the object before me has legs. It claims only that there is a relation between the first two statements (the premises) and the third (the conclusion): the latter necessarily follows from them. That is, if they are true, then it too is necessarily true (it cannot be that they are true and it is not). By the same token one can offer the following logical argument: if all tables are kind-hearted, and the chair beside me is a table, then the chair beside me is kind-hearted. The three statements participating in this mad tea party are, of course, false, and yet the argument they compose is valid, because its conclusion follows necessarily from its premises (one cannot accept its premises and reject its conclusion).
A statement deals with facts and is evaluated in terms of truth and falsehood. By contrast, a logical argument deals with the relation between statements, and is judged in terms of validity (= its conclusion follows necessarily from the premises) and invalidity (= not valid). There is almost no connection between the truth or falsehood of statements and the validity or invalidity of the argument they compose.[1] Science (and observation in general) deals with the truth and falsehood of statements, whereas logic (and mathematics, as we shall see below) deals with the validity and invalidity of arguments. A logical argument has no factual content. It is empty of factual content, because it deals only with relations among facts and not with their truth or falsehood themselves. Such claims, empty of factual information and true by virtue of their logical structure, are called tautologies. The factual content lies in the statements that make up the argument, and, as noted, science is what deals with that.
Mathematics and life
A mathematical proof is a logical argument. It proceeds from certain premises (axioms) and shows that whoever adopts them cannot reject the conclusion proved on their basis. Therefore, at least in this sense, mathematics belongs to logic.[2] It too deals with arguments that are necessarily true, and they too are devoid of scientific content. Therefore the one who engages in it is not the scientist but the mathematician. From this perspective, the theorem that the sum of the angles in a triangle is 180 degrees is devoid of factual content. What it says is not that the sum really is 180, but that if you adopt the premises of Euclidean geometry, then you are necessarily committed to the conclusion that the sum of the angles in a triangle is 180. You may deny the premises, and thereby be freed from the yoke of the conclusion.
Even so, this mathematical argument has very great practical importance. If we are indeed convinced that the axioms of geometry correctly describe our world, then we will necessarily have to conclude that in every triangle we draw in this world the sum of the angles will be 180. Here we have moved from logic to life, for we are now speaking about statements with factual content. But it is important to understand that we did not learn this conclusion from logic alone. We first had to be convinced of the truth of the premises (the axioms), and only the combination of the axioms with the mathematical-logical inference brings us to that conclusion.
A person, then, can deny that the sum of the angles in a triangle in some world is 180, but in order to remain logically consistent he will have to deny at least one of the axioms of Euclidean geometry. And indeed, in the nineteenth century other geometries were developed (non-Euclidean ones) that assume different premises and still remain consistent. In the mathematical-logical sense they are no less correct than the Euclidean one, but the question of which of them describes our world has only one correct answer. The question of correctness, as well as the answer, belongs to physics and not to mathematics or logic. Observation will decide it, and mathematics can at most help us describe our observational findings. As an aside, it is interesting to note that following Einstein’s general theory of relativity it turns out that our world is actually not really Euclidean (but only approximately so. The deviation from its Euclidean structure is small, and usually entirely negligible).
An example of the role of logic in science: natural selection
If we now return to the example of gravitation discussed above, there too mathematics is the language in which the theory is written. The description of gravitational force is given in a mathematical formula, but the source from which we drew the information that this law correctly describes our reality is observation, not logic or mathematics. Mathematics is only the language in which we formulate these empirical findings.
The conclusion is that there are logical and mathematical dimensions in our thinking about reality, but this is only one component, necessary but not sufficient, of a scientific theory. It of course must be logically consistent, but consistency, like logic in general, does not mean that it is true. There are many consistent theories, even where only one theory is true (see the example of geometries in the previous section).
Natural selection is a very good example of this confusion. In my book God Plays Dice I argued that natural selection is a logical tautology, that is, a logical claim that is necessarily true and empty of factual information.[3] It can be derived a priori from reason alone, without any need for observation. In other words, it is a branch of mathematics and not of science. This was one of the claims that aroused the greatest opposition and anger among readers, and yet I still insist forcefully that it is correct (necessarily).
Natural selection essentially says the following: given several kinds of creatures (mutations), some of them will become extinct because of constraints and harsh circumstances. Those that survive are the ones more resilient to those circumstances. Hence, those that pass on to the next generation will be the more resilient (more developed, or more successful) ones. This is the principle of the survival of the fittest. If something survived, then obviously it was fit, and therefore the fit survived. If there were something that was not fit and survived, that would be a sign that it is in fact fit (our assumption was wrong). If there were something that was fit and did not survive, the conclusion would be that it was not really fit (our assumption was wrong). But the claim that the fit survives is a tautology, that is, it belongs to logic and not to physics. I will elaborate on this a bit more below.
The role of falsification tests
Logic and mathematics are never subject to empirical falsification tests. No one can refute the claim that 2+3=5, or the theorem that in Euclidean space the sum of the angles is 180 degrees. By contrast, a scientific theory is supposed to stand up to falsification tests. Moreover, it is scientific only if in principle it can be subjected to such tests. If it withstands them, then it is a corroborated scientific theory (or, in the view of the minimizers: a theory not yet refuted), and if not, then it is an incorrect scientific theory (a theory that has been refuted).
Now think about whether and how one could refute the theory that the sum of the angles in a triangle is 180 degrees. Ostensibly the answer is simple: we must draw a triangle, measure the angles, and add them up. If we get 180, we have corroborated the theory; if not, we have refuted it. What would happen if we drew such a triangle and indeed found that the sum is 317 degrees? The theory would have been refuted. But this is mathematics (geometry) and not science, so how can it be empirically refuted? Mistake. We are not dealing with geometry but with physics. What we refuted was not the mathematical claim that in Euclidean space the sum of the angles in a triangle is 180, but the physical claim that the space in our world is Euclidean. That is a claim in physics, and as such it is no wonder that it must stand up to falsification tests.
Once, when I taught mechanics, I asked the students whether they could propose an experiment that would put the theory 2+3=5 to an empirical falsification test. Some answered: take 2 oranges and place them in a bowl. Take 3 more and place them in the bowl as well. Now count the total number of oranges in the bowl. If you got 5, the theory has been corroborated; if not (for example, if you counted and got 8), it has been refuted. The conclusion, apparently, is that the theory 2+3=5 is not mathematical but scientific, since it is open to refutation.
But here too this is a mistake. Try to imagine a situation (utterly impossible, of course) in which you performed the experiment, and indeed, though you are rubbing your eyes in amazement, you repeatedly count 8 oranges in the bowl. Would you now tell yourselves that a new arithmetic must be developed (because the old one has been refuted)? Of course not. You would certainly look for what happened there; perhaps someone else put oranges into the bowl. Perhaps there were already oranges in it and you did not notice, and so on (an error in the experiment). Suppose you find no explanation. There is a camera following the bowl, and one can see that it was empty and that no one else approached it. What then? Well, one can also say that the camera malfunctioned or was switched, and so on. Suppose we rule that out as well; what would we say then? Would we give up the theory 2+3=5? Certainly not. At most we would give up the physical assumption that adding oranges to a bowl is well described/represented by algebraic addition. That assumption is a physical claim, not a mathematical one, and as such it stands up to empirical falsification tests (like the Euclidean nature of the world in the example above).[4]
The discussion with the students took place as an introduction to the study of vector calculus, which is the calculus that describes the addition of forces, velocities, or accelerations in mechanics. I said to the students there: take a point mass. Apply to it a force of 10 newtons (the accepted unit of force in mechanics is the newton) northward and another force of 10 newtons eastward. What is the total force acting on the body? Not everyone knew the answer, but everyone understood that it is not 20 newtons. The answer is 14 and something (2√10). Have we refuted the arithmetic law 10+10=20? Certainly not. We have refuted the physical assumption that the addition of forces in physics is described by arithmetic addition. The correct description is vector addition.
The conclusion is that physics is what decides which logic or which mathematics correctly describes the world. There is no claim about the world that has mathematical validity. There are only mathematical or logical arguments that are applicable to the world: that if it is Euclidean (that is, if it satisfies the premises of Euclidean geometry), then the sum of the angles in a triangle drawn in it is 180. Only the "if-then" has certain logical validity, but never the conclusion about the world itself. The use of mathematical language does not mean that we are dealing with mathematics. Physics too uses the language of mathematics.[5]
Back to natural selection
We can now examine whether natural selection can be refuted. If it is a scientific theory, it is supposed to stand up to empirical falsification tests. Many indeed claim that it is scientific and empirically falsifiable, but I have not been persuaded. Suppose we performed the following experiment: we took two entities, A and B, where A is fitter than B. We placed them in an identical given situation, and found that specifically B survived and A became extinct. Ostensibly we have refuted the theory of natural selection (that the fitter survives better). Again, mistake. What we have refuted is the assumption that A is fitter than B (or, more generally, the definition of fitness that we used). If A is fitter than B, then necessarily it is supposed to survive better than B (under those given circumstances, of course). And if B survived better, then it is clear that it is also the fitter one.
The conclusion is that as long as we have not poured content into the term "fit," we have said nothing beyond a logical tautology. What is fit survives, and what is not fit does not. This theory becomes scientific only when it pours concrete content (content open to empirical falsification) into the term "fit." If someone claims that the one with stronger muscles is fitter, then natural selection can now be put to an empirical test, and we can see whether this thesis will be refuted or corroborated. If the one with stronger muscles becomes extinct, that is a sign that the claim that muscles give greater fitness is incorrect. But even now the fit always survives; it is only the empirical content that we poured into the concept "fit" that must be updated.
So what is the importance of such a trivial theory if it is not open to refutation? Why are such tautological statements useful at all? Because they give us a theoretical framework within which we can think, investigate, and advance. This is done through falsification tests. After we understood the evolutionary principle that the fit survives, we can now ask which traits are helpful for survival and empirically test the different answers (subject them to falsification tests). The insight that this tautology gives us is enormously important. But one must be careful not to see it as a scientific claim. By the same token, it is equally important not to identify scientific claims with the logic that underlies them.
The Vandervelde Law
One of the principles of rabbinic hermeneutics is a fortiori reasoning. An a fortiori inference appears several times in Scripture itself. For example: Behold, the children of Israel have not listened to me; so how will Pharaoh listen to me?, meaning: if the children of Israel did not listen to me, how much less will Pharaoh listen to me. An a fortiori inference is based on the assumption of a relation of lesser and greater stringency between the inferred case and the source case. Israel are supposed to be more obedient than Pharaoh, and if they do not listen to Moses then Pharaoh certainly will not listen to him. But an a fortiori inference can be rebutted by raising a counterconsideration. For example, an argument that explains why Pharaoh might actually be found more obedient than Israel for some reason. If so, an a fortiori inference is not a logical argument, since we do not have complete certainty of its validity. One can adopt its premises and reject the conclusion. Of course, if we adopt the premises that the people of Israel did not obey God’s voice, and also the premise that Pharaoh is less obedient than Israel, then the conclusion that Pharaoh too will not obey follows necessarily. But clearly the assumption that he is less obedient is open to interpretation, for it is possible that under certain circumstances he will actually obey more.
Those who formulate these rules point out that there is a special kind of a fortiori inference that they call A maneh is included within two hundred. ("if two hundred is included, one hundred is certainly included"), that is, a quantitative a fortiori inference in which the a fortiori reasoning cannot be rebutted. These are cases in which the inferred case is contained within the source case, and therefore the conclusion there is necessary. Thus, for example, there is a verse dealing with a person’s liability for damage caused by his pit in the public domain: When a man opens a pit, or when a man digs a pit.. The Mekhilta expounds from here: If one is liable for opening it, then for digging it all the more so. ("if one is liable for opening it, then certainly for digging it"). That is, it would have been enough for the verse to write liability for opening (removing the cover from a covered pit), and from this we would infer liability for digging (the digging that creates the pit). Why? Not only because opening is less severe than digging (as in an ordinary a fortiori inference), but because opening is included in digging under the principle of A maneh is included within two hundred. (two hundred certainly includes one hundred). If there is liability for opening, then there will necessarily be liability for digging, not because digging is more severe than opening, but because every digging includes, in particular, an opening as well (when we dig a pit, as part of that process we also remove the upper layer that covers it). Therefore one can impose liability for digging by virtue of the opening that it contains. This is a necessary inference, since what is imposed on one who opens is certainly imposed on one who digs, because every digger is, in that respect, also an opener.
Such an a fortiori inference is considered by some writers on these rules to be a logical deduction, that is, an argument that cannot be refuted. Therefore some of them wrote that if there is a law learned by such an a fortiori inference, one may punish someone who violates it, even though in laws learned from an ordinary a fortiori inference one does not punish (Punishments are not derived by inference.). The reason is that punishment is not imposed because of the concern that perhaps there is a rebuttal,[6] but in an a fortiori inference of the A maneh is included within two hundred. type there is no concern for rebuttal, and therefore one can punish.
But it turns out that this is not the case. Although this logical structure cannot be rebutted, when we speak about its applications in the world they suddenly can be refuted. Chaim Perelman, a Belgian Jewish philosopher of law, brings the following example. There was in Belgium a law called the Vandervelde Law (the name of a place). The law forbade selling in taverns a quantity of two liters of wine. The reason was that the legislator wanted workers to bring their weekly wages home and not spend the money in the tavern. And then a crafty worker came to the bartender and asked him for no less than ten liters of wine. The seller refused on the grounds that this was illegal. The matter reached court, and ostensibly we would expect the seller to have the upper hand. Note that we are dealing here with an a fortiori inference of the A maneh is included within two hundred. type. As we saw, ostensibly this is pure logic (deduction): after all, a sale of ten liters is, in particular, a sale of two liters, except that eight more liters are being sold as well. Could the addition of another eight liters permit the sale of the first two liters?
You can already guess that, surprisingly, the judge ruled in favor of the buyer. His reasoning was that the law indeed forbids selling two liters so that the wage reaches home, but the law does not forbid a person to invest his money in wine for trade. Clearly a worker may decide that in addition to his regular work he also wants to be a wine merchant (a basic right of occupational freedom). Therefore buying quantities beyond a worker’s weekly wage is not prohibited.
Conclusions
This argument is nothing but a rebuttal of an a fortiori inference of the A maneh is included within two hundred. type. How does such a miracle happen? How can a valid logical inference be refuted? The answer is that what was refuted was not the logic but the legal content poured into it. Although on the physical plane two liters are certainly contained in ten liters, legally they are not. On the legal plane there is something in two liters that is not present in ten. This is yet another example showing that even inferences and arguments that seem to us to be pure logic, when we come to apply them in the world, lose their certainty and their logical necessity. The logical structure is of course immune to mistakes and rebuttals, but its application in the world involves assumptions beyond logic, and these assumptions (whether legal or physical) are always open to refutation.
The significance of this is that the use of logic in relation to life and reality always contains additional components beyond logic. Logic is a necessary but not sufficient part of scientific theory and of our insights regarding reality. The logical structure of the theory casts a spell over us and seems airtight, certain, and unassailable. It is no accident that we speak of "exact sciences." But that is not so. Certainty and absoluteness do indeed characterize logic in abstract Platonic worlds, but in life the situation is always more complex, and less certain and decisive.
In the previous two columns I spoke about the fact that reality displays recalcitrance toward every theory, scientific or otherwise. Here we saw a much more far-reaching aspect: reality brazenly defies even logic and mathematics, although their arguments are rightly perceived as certain. The very application of logic and mathematics to reality is enough to take them out of the category of pure logic and to impair their certainty and validity.
That is just what we said: logic could have been perfect if only the facts did not interfere…
[1] The only connection between the validity of an argument and the truth of the statements that compose it is the following: if the argument is valid, then if the first two statements (the premises) are true, the third is necessarily true as well.
[2] This is without entering into the controversial question whether mathematics can be grounded in logic, a question that has been the subject of fierce disputes over the past hundred years.
[3] My book Two Wagons is devoted almost entirely to the connection between certainty and the informational vacuum.
[4] In my book God Plays Dice I referred to this claim as a practical tautology. Presumably nothing will cause us to think that adding oranges to a bowl is not described by simple arithmetic. It seems to me that even if no explanation is found, we will assume that such an explanation exists but has eluded us. We will never agree to give up the assumption that adding oranges to a bowl is described by arithmetic addition, even though this is a physical assumption (that is, an assumption that asserts something about the world, and not merely a relation among claims).
[5] This is considered one of the great philosophical riddles since the founding of modern science. How is it that mathematics, which does not describe the world but rather the form of our thinking, turns out to be an excellent tool for describing the world itself? How did the miracle happen that "the universe is written in the language of mathematics"?
[6] This explanation is not agreed upon, and in my opinion it is incorrect. But here it is only an example.
Discussion
An invalid argument is also a kind of incorrect claim. It is not true that certain claims entail some other specific claim. That is because an argument is a kind of claim about claims. We have an intuitive sense that an invalid argument is incorrect. That is, the world against which this claim about those specific claims is tested is found in our consciousness, not outside it. But that too is part of the larger world (the external world plus the internal one [consciousness]).
Natural selection is not a tautology in the way you present it, because it contains a factual claim about the world: resources are limited. Without this claim (which, being factual, is falsifiable), the whole question of survival has no meaning. Let everyone survive and enjoy themselves.
Darwin’s innovation is (among other things) precisely such a view of the world, and not, for example, the natural worldview of “Bless the Lord, O my soul,” in which there is enough for everyone.
I read your book (God Plays with Dice), and as best I remember you did not address this there either.
Eilon, that’s semantics. Of course any argument can be turned into a claim. For example, the argument:
Premise A: A
Premise B: B
Conclusion: C
can be written as a compound claim: (A and B) entails C.
But if there is a valid logical connection from A and B to C, then the compound claim is a tautology.
Uziya,
Indeed, the assumption is that there are constraints (challenges), but if that is the empirical content of the theory then it is a very banal empirical content. By the same token, you could say that this theory contains the assumption that something exists in the world. That too is a factual assumption that removes it from the category of tautology, but it is banal. That is not what we are discussing.
Beyond that, in a more precise formulation natural selection is really saying: if only one individual (or only some individuals) survives, it will be the strong one (the survivor). Or more accurately: if someone goes extinct, it is the weak one.
There are lots of examples like that, and in my book I brought them in order to show the unfalsifiability of natural selection. Whenever something seems to contradict it, they explain that precisely the weaker trait is the one with survival value. By the way, I have no complaint, since those rejoinders are completely correct. A tautology has to work. Its being unfalsifiable does not mean it is not true, of course. In this case the opposite is true: it is unfalsifiable because it is necessarily true (and therefore will never be refuted. Like a mathematical theorem).
It is not banal at all, and that is why I brought the example of “Bless the Lord, O my soul.” Humanity’s worldview throughout the generations was precisely that. Even today, our intuitive view of nature is that every animal gets, all in all, what it needs, and that there is wonderful harmony.
To claim that this is not so is a revolutionary flash of insight, one that turns the point of view upside down and now makes tautologies like natural selection possible.
If there is banality in the claim, it is the kind of banality characteristic of genius—a claim that, once it is stated, becomes trivial, but no one thought of it beforehand.
Dear Uziya, I can’t understand what is unclear here, but I will explain again.
Everyone understands that creatures become extinct, both because of constraints and because of lack of resources. This has been understood forever, and no psalm of “Bless the Lord, O my soul” says or ever dreamed of saying otherwise. Was there ever anyone who thought that no creatures go extinct because of problems? This is nonsense, begging your pardon. That is not Darwin’s innovation. For that tiny innovation, which every child understands, you don’t need scientific revolutions and brilliant minds.
On the other hand, you are definitely right—and I have written this more than once (also in my book)—that sometimes a tautology is brilliant. Every mathematical theorem is a tautology, and it is brilliant. In addition, you are also right that there are tautologies which, once stated, seem self-evident (which is not true of most mathematical theorems), but are hard to come up with. Absolutely true. So what? Did I say anything different?
I did not say Darwin was not a genius, and I did not say that he innovated nothing. I say the opposite of both claims. My claim is something entirely different: this is a claim that cannot be falsified, that is, a tautology. That is all. Natural selection is a tautology just like Fermat’s theorem, and just like it, once proved it is a tautology, and yet one still has to be a genius to formulate it and certainly to prove it.
I understood your remarks about the genius in tautology from the book (and from the thread that preceded it in 5788).
I claim that what you think “everyone understands” was in fact not understood at all, and that the psalm “Bless the Lord, O my soul” and the like did indeed imagine and say otherwise. An example off the cuff: there are medieval sages who claimed that the Holy One, blessed be He, watches over animals in order to preserve the species. It was obvious to them that even if a spider or a bird dies or is preyed upon, overall someone up there is taking care of the species as a whole, and therefore in the end all is well for everyone.
Hello, and thank you for the interesting columns.
Could you elaborate on what you wrote: [5] “This is considered one of the great philosophical riddles since the founding of modern science. How is it that mathematics, which does not describe the world but rather the form of our thought, turns out to be an excellent tool for describing the world itself? How did the miracle occur that ‘the universe is written in the language of mathematics’?”
At first glance, I do not see what miracle there is in our being able to describe the world in words and relations.
If there is a miracle here, perhaps it lies in the human intellect’s ability to grasp material things in an abstract way.
What difference does it make whether it is preyed upon for the sake of the species or for some other reason? If one has to be preyed upon for the sake of the species, then that means the whole species cannot survive. No one anywhere, ever, denied that creatures do not survive for various reasons, whether “Bless the Lord, O my soul” or not. I truly do not understand what is unclear or what you are claiming. To me this is literally Chinese.
Galileo already marveled at the fact that the book of nature/the universe is written in the language of mathematics.
First, it is indeed remarkable that we can describe the world in words and relations. It certainly could have been the case that we did not have the tools to do so (just as ants apparently cannot do this). Second, we succeed in using mathematics with astonishing effectiveness. This requires a much longer discussion, and perhaps I will write about it someday. In the meantime, see for example here: http://www.sciam.co.il/archive/archives/4647
If I may, I will slightly alter one of the key paragraphs in your remarks (the one that begins with the words “Natural selection is really saying the following”):
Relativity theory is really saying the following: given several bodies with mass, they will curve spacetime in proportion to their mass. The curvature of spacetime will restrict the motion of bodies in spacetime to certain directions only. Hence: bodies moving in spacetime will be able to move only in certain directions. This is the principle of restricted restricted motion. If something is restricted in the directions of its motion, then obviously the directions of its motion are restricted, and therefore that which is restricted in its directions of motion has restricted motion. If there were something not restricted in its motion whose motion was restricted—that would be a sign that it really is restricted in its motion (our assumption was mistaken). If there were something restricted in its motion whose motion was not restricted—the conclusion would be that it is not really restricted in its motion (our assumption was mistaken). But the claim that what is restricted in its motion has restricted motion is a tautology, that is, it belongs to logic and not physics. Therefore, relativity theory is a tautology (logical, practical, or whatever other sense you feel like thinking about).
I should note that I am fairly sure I could make analogous changes to this paragraph for every scientific theory I know. Is relativity theory a tautology (in some sense)? Alternatively, perhaps the absurdity above strongly suggests that you made a mistake somewhere? Or perhaps I made a mistake somewhere in the example above? What do you think?
Hello. My view is that you made a mistake somewhere, and I’ll briefly explain where.
The “small” difference between your example (relativity theory) and my claim about natural selection is that the basic claims of relativity theory are not tautologies. You can of course assume them, and then they will come out as tautologies. By contrast, natural selection itself is a tautology. It seems to me that this difference is not very complicated, but I will spell it out anyway.
Relativity theory states the proposition: (A) the presence of mass curves space.
Natural selection states the proposition: (B) the survivor survives.
The first proposition is full of empirical content, and is of course light-years away from being tautological. By contrast, the second is a simple tautology that follows from the meaning of its terms.
Of course, once you assume that the presence of mass does indeed curve space (as you did in your example), the conclusion that space is curved or that motion within it is restricted becomes a tautology. So what? If you assume that all masses fall to the ground, then the conclusion that some mass will fall to the ground is a tautology. And similarly, if you assume that all frogs are kind-hearted, you can derive that tautologically as well. If you assume X, then X is tautological (for X implies X is a tautology).
But what does this silliness have to do with what I wrote about natural selection? The tautological character of natural selection does not arise from assuming it and deriving it from itself. What I claimed is that its own basic assertion (proposition B) is a tautology.
And from another angle: in order to establish relativity theory we needed observations. Even after someone learns it, it is not at all obvious. By contrast, I can teach natural selection to any alien in a single instant a priori, without showing him anything on the empirical plane. And of course I mean only the natural-selection component within the neo-Darwinian picture (not genetics or the formation of mutations, which are also required components).
Again, let me clarify that I do not mean to claim that the thesis of natural selection is not useful, or that it is not clever. My only claim is that it is a tautology, albeit certainly an illuminating and very clever one.
With God’s help, 28 Tevet 5777
Not everyone survives in “Bless the Lord, O my soul” (Psalms 104). There is loss: “You take away their breath, they perish and return to their dust,” and there is coming-into-being: “You send forth Your spirit, they are created, and You renew the face of the ground.”
It is not necessarily the strongest that survives. Even the young lions roaring for prey need “to seek their food from God,” and from human beings more is demanded than that: not only to ask and pray, but also to act and work: “Man goes out to his work and to his labor until evening,” and to keep the ways of the Lord so that, God forbid, they not fall under the decree, “Let sinners cease from the earth, and let the wicked be no more.” And this is his purpose in the world: “to work it and to guard it.”
And the best choice is Beruriah’s choice, which enables a person to create for himself an improving “mutation,” taking the world out of the cycle of predators and prey into a harmonious world in which “let sins cease, not sinners”!
With blessings, S. Z. Levinger
(I am responding here because there is no way to add a comment below)
I do not understand what is not understood. People always saw a zebra being preyed upon or a spider being trampled, but it was basically clear that “there’s no need to shove; there is enough for everyone.” There would be enough carrots both for rabbits and for elephants, and enough zebras both for lions and for crocodiles, with some left over besides. True, the lame zebra would be preyed upon first, but it would never occur to anyone that the very existence of all zebras is not self-evident.
Isn’t that all of our intuitive worldview?
The view was that species are fixed from the six days of creation, and that it simply makes no sense to speak of the possibility that they compete with one another over limited resources. Whoever created them would see to feeding them, from the horns of wild oxen to lice eggs.
The utterly non-trivial innovation is that it is not like that. Rather, essentially there is not enough for everyone, and whoever pushes gets, and whoever does not push disappears. Zebras are not only competing among themselves (which one runs faster and is saved, and which one lags behind and is preyed upon), but all animals are in fact competing with one another over limited resources.
Is it clearer now?
An a fortiori argument of the type “included in two hundred is one hundred” does not always overlap with legal reasoning, as you say, but in Torah reasoning it is valid. The prohibited act is absolute and context-independent; its main function is not behavioral regulation but essential ordering.
Hello Michi.
If so, all I have to do is refute the claim that natural selection is a claim of the type “the survivor survives,” and then you will agree that natural selection is not a tautology (logically)? In other words, I need to show that what you describe as “natural selection” is not the thing evolutionary biologists are talking about when they talk about “natural selection,” and that what they describe is not a tautology (logically)?
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Certainly. This is indeed semantics. But it jars on my ears a bit to say that tautologies are not truths. They are trivial truths. Mathematicians too call them that (true in every world). I know the rabbi does not place much weight on their opinion (and precision) on this matter, but I do not think the language here is accidental. By the way, note that one can see the similarity between the tautologies
(the “u” in shuruk) in the modus ponens argument and that of the survival of the survivor, if we formulate it as a statement within the language, that is:
((A implies B) and A) implies B.
Then if we notice why this is a tautology, we will see that it follows from the nature of the implication relation. What this argument is really saying is that if A implies B, then A will imply B. Which is really an instance of A implies A. Which says (in the metalanguage) that if A is true, then A is true.
The theory of evolution did not grow in a vacuum. It came as a result of humanity’s experience in the centuries from the Renaissance until the nineteenth century, in which it became clear that the world is not static but developing and improving, and that new ideas and new discoveries enrich human knowledge and well-being.
Later came the renewed insight that the enrichment of the individual and the nation does not come at the expense of limited resources. On the contrary, the aspiration to become wealthy leads the energetic person to develop the world. He brings new raw materials from afar, develops efficient and economical methods of work, and provides employment for more workers. Free competition also compels others to improve, innovate, and develop, and the whole world develops.
What Darwin did was project this phenomenon onto the world of animals and plants as well, claiming that constant competition obligates all the “competitors” to develop and improve in order to continue existing, because “if you’re not going up—you’re going down.”
And just as in the human world, so too in nature—success does not depend on strength alone, but on the extent to which increased strength does not upset the balance of nature. For if the lion devours all the zebras and antelopes, and if they consume all the grass—then all will starve to death. The victory of the strong depends on preserving an adequate living space even for the weaker than he, and thus everyone can continue to exist and improve.
From here one must return to humanity and argue that indeed free competition benefits everyone; but just as competitiveness increases, “at the end of the day,” wealth and material well-being—so too there should be competitiveness in the pursuit and refinement of moral values, and the more we aspire to be better, the more we will foster in the world a “jealousy of scribes,” in which each person will seek to prove himself in better feelings and better deeds, and “at the end of the day” we will all be better!
With blessings, S. Z. Levinger
It really isn’t any clearer. But I am beginning to despair. I will try one last time, and if this doesn’t work I suggest we part as friends.
Some creatures go extinct. That is obvious to everyone since the six days of creation. Extinction can happen for all sorts of reasons: lack of resources, predation, natural disasters, and the like. Evolution does not really care why they go extinct; what matters is only that they go extinct for some reason. The cause of extinction has certain characteristics, and whoever is resistant to it will not go extinct. Whoever is not—will go extinct. That is natural selection, and all the rest is really unimportant. And this has been known forever. What Darwin noticed is that this extinction can lead to monotonic and directional changes, and thus life can develop. That, and only that, is his innovation. No connection whatsoever to depletable resources, “Bless the Lord, O my soul,” and all the rest of that stuff. There is not the slightest connection.
It seems to me that the matter of “the survivor survives” needs sharpening. Evolution says that a certain trait will cause its bearer to produce more offspring who reach maturity and produce offspring of their own, and whoever reproduces more and dies less will survive more than whoever reproduces less or dies more. This is a tautology because one cannot think of a world in which this would not be the case. But there is another important claim which is certainly not a tautology, and it is called the tree of life, but that is more archaeology than biology.
That is not correct. One who partially admits a claim is liable, while one who denies everything is exempt, even though that is a case of “included in two hundred is one hundred,” because of the presumption that a person does not brazenly deny his creditor to his face.
Ethologica,
Indeed, that is what you would have to do (but in my estimation you will not succeed, since that is precisely the definition of natural selection, in one wording or another, everywhere—from Darwin himself to our own day).
But this is not the place to get into the details of that discussion. Natural selection was only a side example here of a tautology (besides, it seems to me we have already had this argument once before).
Elchanan,
There are many other claims that are not tautological. But here I was dealing only with natural selection.
Yoav, I did not understand. An a fortiori argument of any kind in Torah logic is no different from its function anywhere else. In Talmudic logic too, the a fortiori of “included in two hundred is one hundred” is invalid, exactly as everywhere else. I brought examples of this in several places.
One of them is the rule that one may not derive punishments a fortiori. One can argue that even if we derive B from A by an a fortiori of “included in two hundred is one hundred” (that is, A is included in B), and even if one who violates A is liable to punishment X, there is still room to exempt from punishment one who violates B. The reason is that perhaps the severity of B requires a different punishment from X, a more severe one. And as is known, some later authorities explained the rule that one may not derive punishments a fortiori in this way. This explanation itself is a refutation of the a fortiori of “included in two hundred is one hundred” (by whose force we wanted to derive the liability of one who violates B to punishment X). So there you have it: despite the fact that A is included in B, one who violates B is not liable to the punishment imposed on one who violates A. QED.
Elchanan, you brought an interesting example. But my sense is that it is not such a successful one. True, partial admission is included in total denial, but his liability to an oath is precisely because he partially admitted, not despite it. Therefore the analogy to an a fortiori of “included in two hundred is one hundred” is out of place. See the example I brought above here from deriving punishments. It really shows this (because the punishment is not because of the lesser severity of the offense, but despite its lesser severity).
Uziya wrote (it was mistakenly posted above):
“I too have despaired. But so that we part not only as friends but with a matter of Torah, I would direct your honor to look a bit at the third chapter of On the Origin of Species, where one can see how our master of blessed memory himself presents his innovation, and what exactly the connection is to ‘Bless the Lord, O my soul’ and the other vegetables mentioned in the passage.”
And I, the insignificant one, replied:
“And so as not to leave the page blank, and because of my great affection for the Master, I will only add that my words do not depend on what his exalted honor, the illustrious saint of blessed memory, wrote in chapter 3 of his book, since what matters is not what Darwin thought he was innovating, but what he really innovated. That resources are limited and that animals become extinct for various reasons—this he did not innovate; go and read it in the elementary school. I am sure that already in Rabbi Akiva’s cheder all the children around him knew this.”
Michi, in the article you relied on the reasoning behind “one may not derive punishments a fortiori” out of concern that there may be a refutation, and it is understandable why in a case of “included in two hundred is one hundred” one may derive punishment a fortiori.
In the comment you relied on a different reasoning, one that truly refutes the basic logic of kal va-homer, at least regarding punishments, and indeed also refutes the logic of “included in two hundred is one hundred.” This approach really compels your conclusion that kal va-homer is not a logically necessary inference. Rather, it would be claimed that the Torah was written in a way suited to this kind of exegetical derivation.
Hello Yoav.
I did not understand what you are getting at. Are you claiming there is a contradiction in my words (regarding the explanation of the rule that one may not derive punishments a fortiori)?
As I explicitly wrote, the reasoning that perhaps there is a refutation is, in my opinion, incorrect (because with interpretations even more speculative than a kal va-homer midrash, we do not worry that perhaps there is a refutation). But I nevertheless used it in order to demonstrate the possibility of a refutation of a kal va-homer of “included in two hundred is one hundred.” The second reasoning (that perhaps the punishment is not strong enough) is that very refutation. So there is no contradiction here.
Hello again, Michi.
In my opinion, not only can I show that natural selection is not a tautology of the form “the survivor survives,” but it is actually quite easy.
Below is a basic description (from here on, “the description”) of natural selection from a fairly standard introductory textbook in evolutionary biology (Ridley’s Evolution, third edition, section 4.2). Please read it carefully.
Natural selection is easiest to understand, in the abstract, as a logical argument, leading from premises to conclusion. The argument, in its most general form, requires four conditions:
1. Reproduction. Entities must reproduce to form a new generation.
2. Heredity. The offspring must tend to resemble their parents: roughly speaking, “like must produce like.”
3. Variation in individual characters among the members of the population. […]
4. Variation in the fitness of organisms according to the state they have for a heritable character. In evolutionary theory, fitness is a technical term, meaning the average number of offspring left by an individual relative to the number of offspring left by an average member of the population. This condition therefore means that individuals in the population with some characters must be more likely to reproduce (i.e., have higher fitness) than others. […]
If these conditions are met for any property of a species, natural selection automatically results. And if any are not, it does not. […] But when the four conditions apply, the entities with the property conferring higher fitness will leave more offspring, and the frequency of that type of entity will increase in the population.
Natural selection is described here as a mechanism that causes evolutionary change—four conditions such that, if they are met, a change will occur in the frequency of traits in a population over generations of offspring.
Let us compare this description to your description of natural selection. Your description began like this:
Natural selection is really saying the following: given several kinds of creatures (mutations), some of them will perish because of constraints and difficult circumstances. Those that survive are the more resistant to those circumstances. Hence: those that pass on to the next generation will be the more resistant (more advanced, or more successful). This is the principle of the survival of the survivors.
You describe the result of natural selection in terms of “survival,” “extinction,” and “resistance.” Your formulation really is “the survivor survives,” but it is your own handiwork. It does not appear in the description above. There, the result of the process is a change in the frequency of traits in the population. The result is caused, among other things, by variation in fitness. Fitness is not defined by “a change in the frequency of a trait in the population.” The result is clearly distinguished from the cause, even though it follows from it logically. This is not a claim of the type “the survivor survives,” neither semantically nor logically.
From all this it is clear that if we continue talking about the tautologies of natural selection, we are obligated to try to translate your claims into language that an evolutionary biologist would recognize as speaking about natural selection. You claimed:
(1′) If something survived, then obviously it is a survivor, and therefore the survivor survived.
(2′) If there were something that was not a survivor and survived—that would be a sign that it is in fact a survivor (our assumption was mistaken).
(3′) If there were something that was a survivor and did not survive—the conclusion is that it is not really a survivor (our assumption was mistaken).
And translated into the appropriate terms we get:
(1) If the frequency of some trait in a population increased—then clearly that trait confers higher fitness on the individuals who possess it than on those who do not possess it.
(2) If the frequency increased in a population of a trait that gives the individuals possessing it lower fitness than that of individuals who do not possess it—that is a sign that this trait raises the fitness of individuals in the population.
(3) If the frequency decreased in a population of a trait that gives the individuals possessing it higher fitness than that of individuals who do not possess it—the conclusion is that this trait lowers the fitness of individuals in the population.
To the ears of evolutionary biologists, claims (1), (2), and (3) sound completely strange. They do not follow from the description above, nor are they hinted at in it. More than that, anyone knowledgeable in evolutionary biology knows that claims (1), (2), and (3) are factually incorrect—they are known to be false. The reasons for this are surveyed in the book from which I brought the description. You are invited to look at it. There are whole chapters devoted to this. In any case, many things can be said about this description. To say that it is an argument of the type “the survivor survives”—that is not one of them.
All this leads to the question that interested you so much. What exactly is the innovation of natural selection? The description above is a logical formulation, in words, of natural selection—a mechanism capable of creating evolutionary change, a change in the frequency of traits over the generations. This mechanism can also be formulated mathematically. In each of these formulations, the innovation is that the formulations are intended to describe a certain aspect of nature. Specifically, the innovation is that the mechanism described above, natural selection, is capable of producing adaptations (matches between organism and environment), and indeed was involved in their evolution over the history of life on Earth. The logical validity of the logical formulation in the description will not determine whether this innovation is in fact accurate, even if you can teach aliens that this description is indeed logically valid. Evolutionary biologists know that because of the evidence they have gathered over the years.
If you have objections—I would be glad to hear them. If you want us to go back to talking a bit about the analogy to relativity theory—with pleasure. Want me to go into why I think this is a conversation that is critical דווקא for this article—with pleasure. Want to continue even without my explaining that—even more gladly. It is an interesting conversation. In any case, I’ve dug enough for one day. The time is short and the tasks are many.
Go from strength to strength. I’ve begun reading a bit on the site and I’m quite enjoying the material. Kudos. If only I had the free time to do things like this.
Hello.
As I said, I do not think this is the place for this discussion. I am also rather skeptical, in light of past experience (quite exhausting, I must say), about trying to convince someone about it. I think you are mistaken. And in my experience, long and convoluted discussions of this sort usually only obscure the issue rather than clarify it.
As I anticipated מראש (in light of past experience), the definitions you brought simply insert additional components of evolutionary theory into natural selection (genetics, mutations, etc.), and I already wrote that in my remarks about tautologies I am referring only to this component in the theory. That is what I imagined would happen, because that is what has always happened in such discussions in the past. Already the first characteristic you cited speaks about heredity, and thereby it is obviously going beyond tautology. What is the innovation in that?!
Instead, you can look at any basic text (even Wikipedia, forgive the banality) that defines the concept of natural selection, and you will see that it can easily be translated into what I wrote. I do not think you will find even one source on Google that will not translate exactly into my definition.
To conclude, I will note that this discussion has no practical importance. The fact that this is a tautology is said in praise of natural selection, not in condemnation of it. Because of its tautological character it is necessarily true. So why hairsplit about it?! And the question whether the component of natural selection includes dimensions of heredity or not is an unimportant methodological question. Clearly there is such a distinct component, and it is tautological. So if you divide the field of discussion differently and blur the fact that this is a tautology, you have accomplished nothing.
I think this is enough to show the misunderstanding in your argument, and so I hope you will forgive me if I do not go into it here in greater detail.
With God’s help, 2 Shevat 5777
Psalm 104 is an explanation and elaboration of the creation story in the Torah, according to the order of the six days of creation.
Beginning with the first day: “You are clothed with light as with a garment”; continuing on the second day: “stretching out the heavens like a curtain, laying the beams of His upper chambers in the waters…” A central place is given to the third day, on which He “founded the earth upon its bases,” and the waters flee before the Lord’s rebuke, “to the place You founded for them; You set a boundary they may not pass, that they may not return to cover the earth.” Yet the waters are also allowed to come into the realm of the land for blessing, to water the earth and its inhabitants—wild donkeys and birds of the heavens, beasts and human beings.
The great things on earth are not an end in themselves, but are there to give shelter to those smaller and weaker than they: cedars and cypresses—for birds and storks; the high mountains for the ibex, and the rocks a refuge for the hyraxes. Even the great luminaries created on the fourth day have no intrinsic value; the purpose of the moon is “for appointed times”; the sun, which comes after the moon, is meant to mark the changing of the shifts between the young lions seeking their food at night and man going out to his labor by day.
Even in the great and wide sea there is room for all. “There move innumerable creatures, living things both small and great.” Human beings too have representation in the sea: “There go the ships.” And even in the kingdom of the day, the fearsome Leviathan has no advantage, for it too is merely a plaything in the hands of its Creator.
Small and great are equal before the Lord. “They all look to You to give them their food in due season.” All of them grow weak and terrified when their Creator hides His face, and all of them are created anew by His will. In that divine harmony, made in wisdom, there is no place for those who violate the divine order: “Let sinners cease from the earth, and let the wicked be no more,” as wisdom instructs: “The soul that sins…” This is the order of the world.
But “repentance preceded the world,” and before the “Bless the Lord, O my soul” of Psalm 104 comes the “Bless the Lord, O my soul” of Psalm 103, which stands entirely under the sign of the storm of falling and rising again: man sins and falls, oppressed by his sin, but the Lord made known to Moses His ways of grace, that He is “compassionate and gracious, slow to anger and abundant in mercy.” Precisely man’s recognition of his weakness and insignificance—“for He knows our frame; He remembers that we are dust”—brings the Lord to show kindness to man, to forgive his sins, heal all his diseases, redeem his life from the pit, and crown him with kindness and mercy.
The penitent merits to bless his Creator, together with “His angels, mighty in strength, who do His word,” together with “His hosts, ministers of His who do His will,” and together with “all His works in all places of His dominion.” For man has revealed his strength in overcoming his fall and raising himself from the depths to the heights.
With blessings, S. Z. Levinger
In short: it is not the strong who survives—but the one who strengthens himself, who knows how to rise higher and improve!
With blessings, S. Z. Levinger
I was not convinced by the Belgian example that a quantitative a fortiori can be refuted.
What they did in that case was a limiting interpretation of the entire law: the law does not forbid buying 2 liters; rather, it forbids wasting a great deal of money in the tavern.
Even if the fellow had bought exactly 2 liters, he could have made his argument and been acquitted, so as best I understand there is no connection to the a fortiori.
What you called a “limiting interpretation” is a refutation. That is exactly what a refutation of an a fortiori of “included in two hundred is one hundred” looks like. You do not expect such a refutation to prove that one hundred is greater than two hundred (or not included in it). A refutation will always show that the relation between them is not on the relevant axis, that is, it will do what you called a “limiting interpretation.”
If the limiting interpretation had shown that 2 liters remains forbidden but 10 liters is permitted, I would have kept quiet.
After the limiting interpretation, both 2 and 10 liters are permitted for investment but forbidden for drinking, and therefore in my opinion there is no refutation here of the a fortiori.
To clarify my opinion, I will give an example of what I consider a good refutation:
The law forbids drinking more than one glass of whiskey (in order to prevent people from getting drunk).
A person drank twenty glasses, and not only did he not get drunk, but he also proved (physiologically) that the drunkenness curve does not rise monotonically with drinking, but begins to go down after the fifth glass and reaches zero after 20 glasses.
I do not see any difference between the examples. In your example too, if there is someone who drinks one shot and does not get drunk, that too will be permitted. But this discussion seems unnecessary to me. We both agree that there are refutations of an a fortiori of “included in two hundred is one hundred.” Take this example or that one, as you wish.
With God’s help, 4 Shevat 5777
I do not think it is correct to say that reality refutes logic, but rather that from the standpoint of reason there is a system of valid considerations that requires taking them into account, while giving each its place to be expressed. Reason requires restricting the sale of wine to laborers who drink in order to get drunk, and in this respect ten liters are clearly more serious than two liters; and reason requires allowing a wine merchant to buy quantities. Both logical considerations have room to apply. The wholesaler may sell a retail merchant wine in the quantity he needs, but one who buys in a tavern or a grocery store—we presume that he is buying for personal use and not for commercial purposes!
With blessings, S. Z. Levinger
Of course reality does not refute logic, since logic is not subject to refutation. It is not science but mathematics (an a priori domain). Rather, the correct logic does not operate in reality. What has been refuted is the logical model for reality (as in the example I brought from vector calculus. Clearly arithmetic was not refuted, but it is not suitable for describing forces in physics).
Hello Michi.
This issue is very practical. First, correcting common misunderstandings of this science is very practical in general. Second, the truth matters for this article too: in my view, natural selection is a wonderful example of a “logical” mechanism that is known not necessarily to “hold” in reality.
I will repeat the basis of your mistake. Boiled down to a short sentence, the essence of the description of natural selection in the above-mentioned book is “the frequency in a population of a trait that increases an individual’s fitness will increase.” The result is clearly distinguished from its cause. That is light-years away from the tautological “the survivor survives.”
In response, you claimed that the description adds heredity as a non-tautological component. Indeed, heredity is required in order to understand how a change in the frequency of the trait in the population comes about. But, as is clear from my remarks above, that is not the reason why at the heart of natural selection there is no claim of “the survivor survives,” and that reason remains unanswered. Despite this, you now claimed, similarly to before, that any “basic text” can be translated into your “the survivor survives” form. Please, try your hand. Show me where in the description above “the survivor survives” is expressed. Perform the translation. I’m all ears.
Here is an interesting argument: suppose we find a robot with the characteristics of a living creature—that is, it replicates itself and contains organic material. In that case, would we claim that it developed by a natural process? According to evolutionary logic the answer is yes, since it contains the characteristics of a living thing. But as far as we know this is impossible. In addition, there is no functional gradation on the way to the formation of a robot. Even a minimal locomotion mechanism would require a number of components simultaneously.
I have lost the context. But that is not correct. There is no such thing as evolutionary logic, and I know of no logic that would infer from something whether it developed naturally or artificially. Darwin did not infer evolution from the fact that there are living creatures that replicate. Evolution is the result of a logical consideration (natural selection: the survivor survives) joined by genetics and the formation of mutations, and finally applied to empirical findings.
This topic reminded me of the book The Survival of the Sickest by Dr. Sharon Moalem. It turns out there are phenomena in nature in which diseases and bacteria use their host (human beings) in order to pass on to future generations by giving him a certain advantage in transmitting his genes onward. So it comes out that people who are more susceptible to certain diseases and bacteria (that is, “weaker”) may דווקא receive a survival advantage over those who are not. This is also one of the reasons certain diseases survive for a very long time, even though one might ostensibly have expected the theory of “survival of the fittest” to lead to all diseases becoming extinct.