Mathematical Platonism—and Beyond (Column 435)

With God’s help

Disclaimer: This post was translated from Hebrew using AI (ChatGPT 5 Thinking), so there may be inaccuracies or nuances lost. If something seems unclear, please refer to the Hebrew original or contact us for clarification.

In the previous column I discussed whether mathematics is an invention or a discovery. There I distinguished among three different claims/questions:

1. Any mathematical theory is objectively true, in the sense that if there is a domain in the world in which its assumptions hold (it constitutes a model for it), then it will operate according to that theory’s laws. Hence it is clear that mathematics is not a human invention but a discovery. This does not mean that its entities exist or that there are angels producing the relations it stipulates, but these relations are necessary—even in our real world.
2. It follows that the mathematical theory itself, even if it has no realization in the world—that is, we have not (yet?) found a domain that would serve as its model—is true in an objective, Platonic sense. It is a truth about relations in an ideal realm. It is objective in that these are the necessary relations among the entities discussed in it, and there is no room to dispute this.
3. The question of whether such an abstract ideal realm actually exists is the question of Platonism. Personally, I tend to think it does, since in my view mathematical truths are the products of our observation of the Platonic world of ideas (and I think this holds for many of our concepts as well. I am an essentialist, not a conventionalist; see the end of the column). But here there is room for debate. Others will argue that these entities and relations reflect necessary structures and relations among entities, and not a reality that exists in and of itself.[1]

My conclusion there was that the first two questions have clear answers, and I don’t see much room for debate about them. Regarding the third question, there is definitely room for discussion, and as I noted, my stance is Platonist. The rationale lies on the philosophical plane (that is, neither scientific nor mathematical), and it can, of course, be disputed. In one of the talkbacks I was asked about this rationale, and I replied that in that column I mainly intended to bring order to the discussion and the concepts, not to ground my own position. I added that I might write a separate column on this—so here it is.

This morning my friend Shmuel Keren sent me another video on the topic, and I saw that it really focuses mainly on question 3 (though you can also find in it a bit of the conflations I described in the previous column). Already here I’ll say that I still haven’t found what my soul loves, but I’ll touch on this video at the end of the column. In any case, I decided it would be worthwhile to add another column that tries to address question 3 (the question of Platonism).

After writing, I remembered that I already had a series of columns on Platonism (383–385) that ended with a discussion of its significance in science and mathematics (385). The last column there dealt precisely with these subjects, but I decided to leave this one because of some novelties that arose here.

In the previous column I presented two positions on the question of Platonism, and the question is which of them is correct. To understand the question properly, we must understand the two positions well, and then we can try to discuss it.

The Platonist Position

In the first two claims I showed that mathematics is true in a Platonic sense, even without any model that realizes it in the concrete world. I argued that everyone agrees that any realization of the assumptions in any world whatsoever (including ours, of course) necessarily entails the theorems. Therefore, the mathematical theory is true even without being realized in practice. In yeshiva parlance, this is a sevara with no practical implication (nafka mina—or only a nafka mina for kiddushei ishah). Up to this point, this should be entirely uncontroversial. The debate concerns whether the mathematical theory also exists in some sense, or whether it is merely a logical structure.

Here, however, we must distinguish between mathematical concepts (number, a geometric object—such as a triangle or ellipse, and the like) and the principles they satisfy (the mathematical laws). At the level of the concepts themselves, a specific instance is merely a realization of the idea and has no existence in its own right. The angles of a right triangle satisfy the Pythagorean theorem. An angle is a mathematical entity, but the particular angles of this specific triangle are not a mathematical entity; they are a possible realization of the idea “angle.” That is for mathematical concepts. If we consider the relations among these angles, even if such relations have a root in some ideal realm, they certainly are not entities. The Pythagorean theorem is a relation among sides, and the sum of the angles is a relation among angles, but such relations are not objects. The Platonist position says that Platonic entities bear such relations among themselves, not that the relations themselves are existing entities. One can, of course, say that mathematical relations “exist” in the world of ideas in the same sense that a car’s speed exists in our world (speed is a relation between the car and the road). Speed is not an entity, but neither is it an artificial fiction. In that sense, it exists.

What about the claim that there is no greatest prime (more precisely: for every prime number there is another prime greater than it; it follows that there are infinitely many primes)? This is a mathematical theorem concerning relations among prime numbers. It asserts a claim of non-existence about the Platonic world of ideas (you will not find there an entity X that has the property of primality such that there is no other entity Y, also prime, that is larger than it). Suppose there were a theorem that there exists a greatest prime; then we would have a positive existential claim about the world of ideas: there exists a number X that is prime such that no other prime number is greater than it. This is a claim about Platonic entities, but the claim itself is not a Platonic entity.

There is another important point. The Platonist position says that mathematical concepts are a kind of entities that exist in the world of ideas—not, of course, in our space and time. Beyond that, they also do not interact with entities that exist and act in our space-time. I add this because there are physical entities or phenomena that most non-Platonists will agree exist even though they are not located in space-time, because they interact with ordinary physical entities (those that do exist in space-time).

Take, for example, a photon. A photon, in the standard definition of quantum theory, is an object that has a definite energy and therefore, by the uncertainty principle, does not exist at any particular instant. In addition, it has a definite momentum, and thus, by the same principle, it does not exist at any particular point in space. A photon also has no mass (what is called “rest mass 0”), and so one might have thought to claim that it is a Platonic entity. What makes it a physical entity is the fact that it interacts with concrete entities. An atom can absorb or emit a photon, and its state will change as a result. One can also say that the photon does exist in space-time, but not at any particular point in it: it is spread over the entire time axis and over all of space.[2] One can also speak of concepts such as energy or momentum, but it seems these are more phenomena than concepts (recall the example of a car’s speed from the previous column). The concept of speed (as opposed to the speed of a particular car) and the concepts of momentum are Platonic ideas. A quantum wave function of an electron, a field of some sort, or the potential of that field, likewise exist in that sense; their abstract conceptualization can be a Platonic idea.

The Non-Platonist Position

What does the opposing view say? As noted, it agrees with the first two claims. It disputes only question 3, contending that mathematical entities are our invention and do not exist anywhere. If we had not invented them, they would not exist. And what about the relations among them? I think non-Platonists will also agree that these are necessary relations among those (fictitious) entities; however, since the entities themselves do not exist anywhere or in any sense, the relations among them likewise do not exist. They are relations among mental constructs; hence one cannot say the constructs are artificial inventions while the relations among them do exist (not even in the attenuated sense in which relations “exist,” as a car’s speed does). On this view, inventing the entities includes inventing their relations: those relations are part of the very definition of the concepts and were invented together with them—yet they remain necessary relations about which one cannot disagree.

Methodology

How can we even approach clarifying question 3? It would seem to be a debate about the existence of realms not accessible to us and that do not interact with anything around us. By its very nature, such a question cannot be settled on the basis of scientific facts or mathematical facts. As noted above, a stance on such a question can be formed, if at all, only on the basis of philosophical arguments and intuitions. It’s no wonder it is hard to reach an agreed-upon conclusion here (unlike with the first two questions). Still, as I wrote, I tend toward Platonism, and my aim in this column is to explain why (in column 383 I have already discussed several of these points).

Definitions by Extension and by Intension/Essence

To move forward, I wish to set the question in a broader context. The question of Platonism touches many concepts, not only from mathematics. Even regarding ideas like horseness, democracy, or Judaism, one can argue whether these are mental constructs (Aristotle) or existing ideas (Plato). Aristotle saw all these as abstractions: we look at concrete objects and, from the common features of different subsets among them, we create different categories. According to Aristotle, contemplating horses yields the idea of horseness by abstracting away irrelevant features and retaining the shared set of features. Plato, by contrast, thought we proceed in the opposite direction: from knowing the Idea of horseness we identify that there is a horse before us, and from all horses there arises the set of concrete objects called “horses.”

Triangles, circles, or numbers are also ideas to which the same disagreement applies. For Aristotle, these are abstractions created from contemplating reality. We see several different objects with triangular form, recognize something common to them, and call it “triangularity.” Likewise with the collection of sets with five elements: what they have in common forms the category “five,” which we denote as: 5. Thus an abstract category is formed, but it is itself a mental fiction. For Plato, by contrast, we grasp that there is a category “5,” and from it—and by it—we identify the sets that contain five elements and see that there is something in common among them. Likewise with triangles.

We can summarize this as follows: in logic there are two kinds of definitions for concepts—definition by intension/essence and definition by extension. An intensional (essential) definition describes a concept’s content, and from it we can diagnose which objects deserve to be called by that name and which do not. A definition by extension defines a concept by listing all the objects that realize it. For example, the concept “democracy” can be defined by its content as: a government formed by citizens’ voting every few years, having separation of powers and civil rights, etc. From this we can understand that England and Israel are democracies, while North Korea is not. But one can also define democracy by extension—by presenting a list of all democratic countries. In these terms, it seems Aristotle uses definitions by extension, whereas Plato uses definitions by intension. Indeed, for Plato the concept’s essential content mirrors an Idea that also exists; but it is very reasonable to see this as a natural corollary of essentialism. If a concept has an essence, it seems straightforward to assume that the concept exists. Otherwise, in what sense are certain features essential to it while others are not?! Distinguishing the essential features of a concept is done by eidetic intuition (Husserl’s term) of the concept—or of the Idea—and by distinguishing features that appear in it from features that appear in the various realizations (the horses) but not in the Idea (horseness).

Essentialism and Conventionalism

These two ways of defining allow us to determine, for any country that comes before us, whether it is a democracy. In the first way we examine its features and compare them to the essential (intensional) definition; in the second, we go over the list and see whether that country appears on it.

The question now presses: how does Aristotle generate his formative list? How does he know that these particular objects are horses? Is this merely an arbitrary definition? If so, why doesn’t he lump Horse Yankel together with the tree in my yard and the cloud above my head into a single group, give it a name, and make it a category? It seems that, for some reason, among horses he perceives something shared that is relevant to be called by its own name, whereas the other trio does not. Countries with certain features deserve their own category (democratic countries), but countries longer than 200 km do not form a category. Why do some features ground a category while others do not? On what basis is this determined?

Importantly, for Plato this question does not arise: eidetic intuition yields the fundamental categories (horseness, number, ellipse, 5, the color red, etc.), and the classification of objects proceeds by comparing them to those Ideas. But for Aristotle we are the ones who create the ideas/categories—so how does the whole process begin? Borges’s descriptions cited in column 383 vividly illustrate this problem.

Aristotelians will likely say it is arbitrary. More precisely, it is determined by our brain structures, which for some reason perceive a commonality among all horses or among all five-element sets, but not among the three objects I described above (that cloud, that horse, and that tree). Perhaps one could add considerations of efficiency: a more fruitful classification (one that yields more, and more significant, conclusions) is the one we will use. Other classifications are not less “correct,” but there is no point in using them. The question is whether this seems plausible to you—that the three objects I described constitute a category to the same degree that democratic countries do. Or that the category “human beings” has no more standing than some arbitrary collection of objects, with or without any common denominator. To me, not really.

I believe I have already brought here the comparison between the ancient division of the elements into four—fire, air, water, earth (abbreviated in Hebrew ARM“A)—and the modern periodic table. Is there a right and wrong here? Not necessarily. The modern classification uses one coordinate system and the ancient one another. Both could be equally “right.” The ancient classification need not have intended anything practical (that to produce a plant one needs a particular mixture of the four elements). Perhaps some did think so, but one can view it as a way of principled description: a plant has some striving upward (like fire or air); it is less heavy (so it has less earth), and so on. On this, I too tend to agree—though it is quite clear to me that chemical elements do exist (that is a real category), whereas ARM“A does not.

Clearly, a person with a different cast of mind or brain structure would use different categories and classify objects differently. If someone is more comfortable thinking of objects in terms of the four ancient elements, he will describe them that way; and if someone resonates more (or is more fruitful) with the modern classification, he will use it. Therefore, Aristotelians claim there are no existing Ideas here but fictions that are our artificial creations. The Platonic view, however, is based on the sense that there is something more here than convenience or arbitrary brain structure. It is an essentialist view, whereby the definition describes an essence that exists in the objects and constitutes their Idea. According to Aristotle, our concepts were built by conventions. They have no basis in the objective world as such, for it can be described in many ways; everything depends on the observer (us).

Reframing the Question

In the end, the question of Platonism expands here beyond mathematics. The question “Why am I a Platonist about mathematics?” broadens into “Why am I a Platonist regarding concepts in general?”—that is, why do I think concepts have an objective essence and are not merely conventions.[3] Mathematical concepts are just a special case of this broader question.

Once again you can see that a discussion about mathematics is really a discussion about philosophy. The question is whether philosophy—which deals with abstract concepts (justice, collective, morality, state, concepts, propositions, definitions, etc.)—deals with reality or with our artificial constructs. This becomes sharper in light of my conclusion from the previous column: mathematics is a branch of philosophy. Both deal with contemplating concepts—that is, analyzing and drawing conclusions from such contemplation. The question is whether we are contemplating ourselves—as Aristotle claimed—or an ideal reality outside us—as per Plato.

Having broadened the question, we can now discuss it more generally.

Two Kinds of Definition

In column 108 (see also the entire series there) I distinguished between constitutive definitions and regulative definitions. Many mathematicians will tell you that a definition is arbitrary. We can define concepts as we please; there is no right and wrong. That is, of course, a possible situation—but the question is whether all definitions are like that. It is very plausible that not all our concepts are grounded in existing Ideas, and even a Platonist will agree that we can define concepts artificially for our own purposes. The question is whether it is always so, or whether there are also other definitions. In particular, this concerns the definitions of mathematical concepts. Are those definitions our creations, or do they attempt to capture something that exists outside us? It is now easy to see the connection to our discussion of Platonism.

If all definitions are arbitrary, then there is no sense in speaking of a correct or incorrect definition. At most, we could speak of a definition that accords with some dictionary or with standard usage—or does not. But if there are correct and incorrect definitions, i.e., definitions for which the question is not lexicographic (what appears in the dictionary), then a dictionary entry will not persuade me I’m wrong merely because it records it thus. From my perspective, the definition in question attempts to capture something objective; if I have not captured it, then I am wrong—even if my definition matches common convention. In effect, the “convention” reflects the intuition of the Idea; otherwise it would not have been adopted in usage—perhaps it would not even have been equally comprehensible to all of us.

It is important to understand that I am not speaking here about terminology alone. Suppose that, in my view, the definition of a triangle is: an object with three angles summing to 180°. Someone else says it is an object with four angles. In that case, there is no disagreement between us: he uses the word “triangle” to describe what I call a “quadrilateral.” But if he says that a triangle is an object with three angles summing to 250°, then he is wrong. The error is not only that one can prove the sum of the angles is otherwise, but first and foremost because that is not what lies under the term “triangle.” This is essentially the parable of the elephant in Maimonides. When someone employs a definition that differs in an essential feature (e.g., defining an elephant as a winged creature), the conventionalist will say he is not disagreeing with me—he is simply speaking about a different object. The essentialist will say he is indeed disagreeing with me and that he is wrong.

Arguments from Intuition

I think most of us (perhaps all of us) sometimes have a very clear sense that a certain definition is mistaken—not lexically or semantically, but in reality. Two indications can be adduced: disputes about definitions (see column 251 and the references there) and changes in definitions.

When we argue about the definition of a concept—say, “Who is a Jew?”—one might think the dispute could be resolved by inventing an additional term and repartitioning the terms. I will call my concept “Israeli,” and you will call yours “Jew,” and we will part as friends. Why fight over the use of the word “Jew”? Note that, without addressing who is right, the very fact that we argue indicates that each of us has a clear sense that he is right and the other wrong. That is, we both agree there is a right and a wrong in defining the concept “Jew.” The secular Jew will say that the true heir of Abraham is the one who speaks Hebrew, reads Amos Oz, and serves in the army; the religious Jew will say it is the one who keeps the commandments. Is there a dispute here? I, as a Platonist, claim yes. We cannot part ways with me calling my concept “Israeli” and you calling yours “Jew.” There is a dispute about a concept, and we are both speaking about that very concept; the linguistic term that expresses it is not the issue.

Note: a concept, as opposed to a term, is not a linguistic matter but a matter of content. Even if someone decides to call democracy by another name, it is still the same concept—only expressed by another linguistic term (as happens when translating to another language). But if we argue about whether democracy can permit a state’s religious identity, we are not speaking about two different concepts; we are disputing. The conventionalist should regard all such disputes as mere confusion: you call your model “democracy,” and I will call mine “theocracy-democracy.” Why argue? The very existence of the dispute indicates both sides agree there is right and wrong here. There is a correct and an incorrect definition of this concept. That implies it is not merely a convention. We are both speaking about the same concept, and the question in dispute is who of us hits upon the true meaning of the concept “democracy.” That concept resides somewhere in the objective world, not only within us (otherwise there is no dispute).[4]

So much for disputes about concepts and their definitions. There is another phenomenon that indicates a Platonist sense about concepts: processes in which a concept’s definition changes over time. We discussed the dispute about “Who is a Jew?”; now I will look at it from another angle. The secular proponent himself agrees that in the past the concept “Jew” was not defined as he proposes. He must concede that in the past the definition was religious: there was no state then, no army, and no Hebrew literature, etc. Therefore, it is irrelevant to attack his proposal using examples from the past (to say that in the past it was religious). He agrees to that; he simply claims the definition changed over time.

How can a definition change over time? From a conventionalist perspective, there is no reason to change a concept’s definition. Coin a new concept and give it that meaning. In what sense is the concept described by the new definition the same concept as before? If the definition changed, it is not the same concept, for the definition is what determines the concept. Moreover, practically speaking, if the earlier concept was defined in a certain way, there is no reason not to leave it in place and to define a new concept—just as in the earlier discussion about disagreements: there is no point in changing and updating such definitions. The Platonist will, of course, say that even after the change it is the same concept: the same Idea appears in the contemporary world in a different manner. What the earlier concept and its new definition have in common is that both address the same Idea (they are directed toward the same object).

Conventionalists offer convoluted explanations to justify the existence of disputes and of changes in definitions. I assume some such explanations will surface in the talkbacks (if anyone reads these columns at all). But none seems truly convincing to me. In my view, they are merely attempts at post-hoc rationalization, whereas their intuitive sense is exactly like mine: there is right and wrong in defining concepts; hence we argue about them. When one hears a dispute about democracy, the clear impression is that this is not a semantic dispute but a value-laden one. That implies both parties are speaking about the same concept—hence both use the same term—yet they dispute the character of that concept and therefore its definition.

In short, when we hear someone propose a definition that we do not accept, at least in some cases our objection is not based merely on the dictionary and common usage, but on a clear sense that he is wrong. In our view, his “observation” of the concept failed to capture its meaning correctly. The same happens in the mathematical context. When someone proposes a definition there for some concept—such as set, group, convexity, etc.—one who hears the definition understands that a concept has been captured that has an essence, one worth engaging with. It is not merely an arbitrary definition that just happens, afterward, to be fruitful. From the outset it seems substantial—that we have captured something. Moreover, sometimes there is a dispute about the definition, and this is not only a question of fruitfulness but of truth.

Furthermore, in the mathematical context one can show this in another way. We could, in principle, have defined innumerable systems of concepts or collections of properties and attached a linguistic term to each. In most cases, such systems would be insignificant, and no interesting theorems could be proved about them. If our choice of systems of features and concepts (group, set, space, ring, etc.) were arbitrary, then only in very few cases would we hit upon a fruitful system. The number of fruitful systems among those we might try would be negligible. Moreover, I am not aware of many attempts to create mathematical systems that failed. I don’t think there are many. The conclusion is that mathematics’ success is probably grounded in an initial sense that a certain system of properties or concepts will likely be fruitful—even before we have analyzed it and succeeded in showing this. The Platonist will say that this sense rests on an eidetic observation of that system. Therefore, it is reasonable that this is discovery and not invention, and that the basis of discovery is observation of some Platonic reality in the world of Ideas. This is an argument akin to what I raised above regarding Borges, and also to the argument from the graph discussed in column 426.

It is important to understand that this argument differs from Wigner’s argument, discussed in the previous column. Wigner (and many others) claimed that the success of mathematics in explaining various physical phenomena corroborates Platonism. I rejected this, arguing that it only shows mathematics is objective (claims 1–2) but does not necessarily speak to question 3. Seemingly, here I am arguing something similar: the success of our choices of systems of features or concepts indicates that the choice was made by observation and not by arbitrary invention. But there is a big difference between the arguments. Regarding Wigner, one could say that among the mathematical systems already developed, it is reasonably probable that some will prove apt to describe reality—especially since those systems were developed in connection with reality (for we live in reality). But with my argument, the situation differs. An arbitrary choice of a system of concepts/properties should have yielded nothing, because the fruit in question here is intellectual fruit (interesting theorems in mathematics), not empirical-scientific fruit as with Wigner. I see no reason why an arbitrary choice or invention would, with such high probability, hit upon fruitful systems.

One can dispute this as well and claim that the sense rests on self-observation: what we succeed in developing depends on our abilities and thought-patterns, and thus no wonder we have an initial intuition about a system’s fruitfulness. I think the clear sense of those who work in these fields is otherwise; but, as noted, there is certainly room for debate on the question of Platonism.

The Role of Intuition

I will go further. I think that intuitively—if we could manage to ignore our rationalist culture—we all have a very clear Platonist sense. People’s tendency to recoil from Platonism (as a kind of mysticism) stems from its sounding groundless. If it is not sensed by the senses, there is no place to treat it as observation. My claim is that eidetic observation is a reasonable explanation, and it fits our intuitive feelings; hence there is no reason to abandon them. We are able to grasp things about the world not by means of the senses (see again column 426 and the sources I cited there).

An example is the problem called “the philosophers’ chestnut” (discussed several times on this site; see, e.g., columns 251, 99, and others). When you and I report seeing an object in red, do we mean the same color? It may be that when you speak of red you mean what I call green, except that from birth you have become accustomed to everyone calling that color “red,” so you too call it thus. There is no way to test synchronization between the internal images in different people’s consciousness. The belief that we do, in fact, mean the same color rests on intuition, for we have no observational way to confirm or refute it. (This holds, of course, for any term we use, not only color. One could go much further and say that when you speak of red you are actually experiencing what I call listening to Mozart’s flute concerto—or you are experiencing some experience unknown to me altogether.) Our synchronization nonetheless leads us to think it is real—that when we speak of the same thing we also experience the same thing. Any synchronization of concepts and insights is like that. If there were not something external that underlies our concepts, it would be hard to see what our assumption of synchronization is based on. How could we assume such synchronization at all? It is natural to say that such synchronization rests on observation of something objective outside both of us, not on invention.

What reinforces this claim is Wittgenstein’s argument about following a rule (see here, ch. B). The crux is the difficulty of understanding how we manage to act in accordance with rules, or how Reuven can explain to Shimon the meaning of some rule. I addressed this in column 83 and in column 385 already mentioned. There I showed that Wittgenstein’s explanation, as if this were a game we all understand because we are built the same way, does not really answer satisfactorily. Thus, for example, on his view there can be no such thing as a clever or a foolish person (see there the example about solving a sequence on a standardized test). The more plausible alternative is to see rule-following as the product of observing the rules with the “mind’s eye” (that is, observing a Platonic realm of entities and the relations among them). This observation synchronizes us all and enables us to speak about rules and assume that we understand one another, to pass them from one to another (to teach them), and to follow them. The problem Wittgenstein raises (which led him to the thesis of language games—which is, in my view, highly implausible) indicates that mathematics should be conceived Platonically.

Now I can briefly address the claims raised in the video mentioned at the start of the column.

Critique of the Video’s Claims

The video is part of the series closer to truth, hosted by the investment banker Robert Lawrence Kuhn, who deals quite a bit with intellectual questions (this is video 409 in the series; his site has over 4,000 videos). The videos feature interviews with researchers and thinkers from various fields; here, naturally, we have mathematicians, physicists, and a philosopher.

I preface by saying that most of the answers mainly expressed feelings, more than presenting arguments. This is not necessarily a criticism, since, as I have shown in this column, the source from which one can form a view on this issue is primarily intuition. But beyond a few distinctions that contribute to the discussion, most of the arguments were unpersuasive and sometimes outright mistaken (some repeat the confusions described in the previous column). I will briefly address the main arguments.

The video begins—how could it not?!—with Roger Penrose. Penrose presents a Platonist position and grounds it on the claim that the mathematical description of the world is remarkably accurate (he offers numbers whose provenance and reference I do not fully understand). I am not sure I grasped his argument. If we used the Hebrew language to describe reality, would that say something about our language? Reality dictates the result, and the language is merely the medium of expression we use. The form of description is a function of our ways of thinking, and so it is reasonable that thought is the tool we use to describe the world. The import of “accuracy” is unclear to me. First, if the description is not entirely accurate, then it would seem to mean the description is not true but merely a reasonable approximation. That, in fact, reduces my confidence in Platonism, and it suggests that science is the result of good use of our modes of thought (perhaps evolution explains why we have such good tools for understanding reality). Perhaps he means that it is perfectly accurate but that we have only tested it to, say, 10 to the minus 14. Even then, I don’t see how this attests to Platonism. It only says that we have good tools for handling reality. How does one get from here to the claim that mathematical entities exist in some Platonic sense?!

When he explains what Platonic existence means, he says that the claim “there is no greatest prime” is true regardless of us and was true before us. His point is that we do not impose these structures on reality; they are there. That is correct—but it only shows that the relations among ideas are objectively true, which speaks to claims 1–2, not necessarily to question 3.

In reaction to Penrose the Platonist, he interviews Mark Balaguer, a philosopher of mathematics. To the question of how it can be that abstract mathematical structures are effective in the scientific description of the world, Balaguer answers that there are many mathematical systems and only a small subset is useful to us in science. That indeed addresses claims 1–2, but the implications for question 3 are clear. Even if all mathematics is invention, success is a reasonable accident of one system among several. I have already addressed this above, arguing that the number of systems is too large for this to be merely a successful accident. Moreover, when we are interested in question 3, the important issue is the fruitfulness of the systems, not their scientific effectiveness—and, as I explained, that holds for almost all mathematical systems, not only a subset.

He then divides the positions on Platonism into four: (1) mathematics exists in the physical world—I did not understand that (perhaps he means applications, but that is physics, not mathematics; see columns 50 and 318); (2) it exists in the mental sphere—this is the Aristotelian view; (3) it has Platonic existence—neither in space-time nor interacting with physical reality; (4) it is fictitious—it has no existence at all. I did not understand the difference between (4) and (2). It seems, as philosophers are wont to do, that he maps positions and enjoys presenting a rich landscape—even if the richness is fictitious. This is like a Talmudic “iyun” shiur that lays out different commentators’ views and draws distinctions among them to enrich the map, even when often these are merely different formulations of the same position.

In the end he juxtaposes the Platonist and the fictionalist views and claims there is no way to decide between them. When Kuhn asks whether he agrees it is either this or that, he answers no, because he thinks there is no correct answer—there is no clear definition of the concept “abstract object.” So long as it is unclear what we are talking about, it is hard to say there is an answer. This is an interesting claim, but I incline to think it is incorrect.

Suppose we have no way to decide (in my view, intuition is a legitimate way to decide) which of the two answers is correct; still, the inability to define does not prevent the claim that one answer is correct. I also do not know how to define in what sense God exists, yet I believe He exists. Moreover, one could even say that, by his reasoning, the photon does not exist, since we have no way to define its existence. That it interacts with the world does not tell us in what sense it exists or what it is exactly. If so, on his view we should regard the phenomena associated with it as phenomena with no source in the real world. If you think further, you’ll see that even the existence of a material object, like a table, cannot be defined (philosophers define existence via what can be quantified over—circular and content-less). Will he then say that the table does not exist? The existence of an abstract entity is defined exactly as that of a table or a photon. The difference among them is not in their mode of existence but in their different properties. A table has a location in space and time; a photon does not. And a Platonic Idea does not even interact with material reality. So what? Why is its existence less well-defined than that of a table? The concept “existence” has no definition, but it is clear to us all; that suffices. As I understand it, whatever stands before us and we apprehend—exists. This does not depend on our ability to define it, nor even on our ability to define what “existence” itself means, with respect to it or in general.

The next interviewee is the mathematician Gregory Chaitin. He describes the Platonist feelings many mathematicians have. In his remarks he distinguishes between discoveries regarding which a mathematician feels that if he had not discovered them, someone else would have—that they have Platonic existence—and discoveries that seem like his personal game (with no indication anyone else would have “played” it). He says that without Platonism the mathematician is essentially wasting his life on fictions. But that is not so, since even if one adopts claims 1–2, there is discovery of something real—even if there is no Platonic world of mathematical entities. He admits that his feeling is strongly Platonist, despite the discomfort this arouses in him, as it seems to him like religious belief in abstract entities (which he apparently does not accept).

At the end of his remarks he raises another interesting claim: a mathematician should relate to his work like an empirical scientist. If you run a computer experiment and see something holds in all cases you checked—even if you cannot prove it mathematically—then it is likely true, just like generalizing results of an empirical experiment in science. There is, admittedly, a possibility of later refutation, but in his view such scientific inference is relevant in mathematics as well. This seems to me an interesting expression of the reality of mathematics. If it were fiction, then so long as something is unproven it has no standing; in what sense is it “true” before it is proven? But if there is a reality that is true regardless of us, then there can be various indications that give us clues to its truth even before we have proven it. Sometimes we will err, as in scientific generalization; yet, in my view, this is a very interesting expression of a Platonist outlook.

The next interviewee is the well-known Wolfram, a physicist who is also a mathematician. He claims there are many “mathematicses,” and the mathematics we have is the result of a fortuitous development in the human chain of attempts to develop it. It could have been entirely different if history had unfolded otherwise. From the sequel it appears he means that one could have begun developing mathematics from different sets of axioms; but I have already written above: in my view, all those sets are mathematics. There are not several mathematicses, nor a competition among alternatives. All the “mathematicses” are merely parts of one whole. The fact that we have discovered only part of it says nothing. Exactly as our physics and the physics of the future—or quantum theory and thermodynamics—are not several different “physicses.” I do not really understand his talk of many mathematicses.

The last interviewee is Frank Wilczek, a Nobel laureate physicist. He maintains that both answers are correct: mathematics is both discovered and invented—but mainly discovered. His claim is that axioms are inventions, but deriving theorems from them is discovery. Besides the fact that this does not pertain to Platonism but to claims 1–2, it seems he means that the axioms are arbitrary and the theorems are true. But this distinction seems to me absurd. First, axioms are not necessarily inventions (though he is entitled to assume that), and in my view it is even plausible that they are discoveries (see the argument from fruitfulness above). Second, deriving theorems cannot be discovery if it concerns the meanings of axioms and concepts that are inventions. That would be mere wordplay. If you think you are discovering relations among the axioms or among the concepts they concern, then the axioms too are something that exists (question 3) or at least something that is true (claims 1–2).

At the end he says that non-Euclidean geometries are inventions, but inventions must come from somewhere, and the universe could be the source of that invention. This too seems problematic to me, since that invention occurred long before anyone considered that the universe is non-Euclidean (and as I already noted, even that is not entirely accurate). So it is hard to claim the universe is the source of that invention (unless he assumes mathematicians had some unconscious intuitive sense of the universe’s character). Beyond that, in this very distinction of his I see something significant. His claim is that even if geometry describes the universe, the universe’s structure is not geometry. Geometry does not exist in the universe itself; it is an abstract structure applied to the universe. The universe is its model. He is right that the question of Platonism is not about the appearance of that structure in the universe but about where, if anywhere, that structure itself is (see column 50 and elsewhere).

[1] I corrected the last sentence of question 3. There was an imprecision in the original formulation.

[2] A question for reflection: what about God with respect to space-time? Is He like a photon (existing throughout space-time), or is He not in space-time at all (the “literal” construal of tzimtzum)? For illustration, recall that in column 340 I argued that it is incorrect to say that a point’s length is 0; it is more correct to say that it has no length—i.e., the concept of length is not applicable to it (see some of the talkbacks there, where the possibility of defining its length as 0 was raised). Similarly, are the concepts of space and time inapplicable with respect to God, or is He in all of space-time (in which case He is simply like a body of immense size, with nothing special about Him)? Along similar lines I once distinguished (I think in a recorded lesson) between a being that is a necessary being and a being that exists at all times. Many confuse them, but they are not the same. A necessary being exists at all times, but if something exists at all times, it need not be a necessary being. Of a being that exists necessarily, the concept of non-existence does not apply. A being that exists at all times simply does not, in fact, fail to exist at any time. Incidentally, I think that in the same place I also discussed the oxymoronic term “self-cause,” which many confuse with necessary being (in medieval terminology this was very common). But that is merely a manner of speaking (which only confuses, as it has no meaning). Nothing is its own cause. It is more correct to say it has no cause.

[3] Note that I assume here that if a concept has an essence, then it exists in a Platonic sense (even though the two claims are not logically or conceptually identical). Otherwise, it is hard to understand what a concept has beyond convention. I remarked on this above.

[4] See column 371 for the discussion of C. S. Lewis’s arguments; it is very similar.