Is Mathematics an Invention or a Discovery? (Column 434)

Dedicated to my friend Shmuel Keren

(and thanks for sending me Dekofsky’s video)

Many have dealt with this question, but time and again I feel it’s not clear whether it even makes sense. Beyond that, the arguments that usually come up in these discussions strike me as irrelevant and reflect a misunderstanding either of mathematics or of the question itself. Yesterday (Monday) I was sent a short video in which a man named Jeff Dekofsky addresses this question, and I decided it’s a good opportunity to touch on it a bit more systematically. Someone needs to bring some order here, no?

Presenting the Question(s)

Dekofsky opens by posing the question: Are mathematical statements true even before they are discovered? Are polygons or numbers existing entities, or representations of theories that are nothing but our inventions? Toward the end of the video he formulates a set of similar questions that ostensibly remain hanging (though his answer to them is quite clear; see below): Is mathematics an invention or a discovery? An artificial construct or a universal truth? A human product or a divine creation? All these questions are adjacent and similar, but they are not the same. For at least some of them, I doubt they have any clear meaning at all, and in any case their answers are obvious—so the debate is unnecessary.

Mathematical statements are of course true even before they are discovered, and even if they were never discovered at all. Here I simply don’t understand the question. This would be true even if I thought mathematics is a human invention. Does anyone imagine that the Pythagorean theorem wasn’t true until the sixth century BCE? Even if I accept for the sake of argument the claim that if there were no human beings in the world there would be no mathematics (or that there would be a different mathematics), it still seems clear that the Pythagorean theorem has been true ever since there were human beings in the world, even if none of them had yet heard of it. What happens in a world inhabited by different creatures (who don’t think as we do)? I’ll touch on that later.

The second question is that of Platonism (the existence of Ideas). It stands at the root of the discussion, since if Ideas exist then it’s quite clear they are not the work of our hands (unless you claim that we created the realm of Ideas, or that inventing a mathematical idea creates an Idea in the world). This question is therefore genuinely relevant to whether mathematics is an invention or a discovery.

But there is a fundamental point here that bears on most of these questions: if mathematical entities do not exist, yet mathematical relations accurately describe phenomena in the world, does that mean mathematics doesn’t exist? Take, for example, a moving car. The car’s speed is not an existing “thing,” but it is not our invention either. It is a property or feature of the car, and unless you are extreme skeptics it is clear that this is a real phenomenon in the world. Therefore, the question of whether the entities of mathematics exist and the question of whether mathematics’ claims are true (about the world) are independent—at least in one direction. If mathematical entities exist, then statements about them certainly describe their (own) reality. But even if they don’t exist, it is still possible that mathematical statements are objectively true. Even if the numbers 2, 3, and 5 do not exist as entities, there is no impediment to saying that the statement 2+3=5 is true and not an artificial fiction that we invented. When we put two apples into a bowl and then add three more, everyone will have to agree that there are altogether five apples in the bowl. The mathematical content is true even if the language in which it is described is our invention.

This brings me to the question I formulated above: what would happen in a world inhabited by people who think differently from us (not in our mathematical languages). If there is someone whose language lacks numbers or the operation of addition, he will still have to agree to the fact that there are five apples in the basket. He would only state it in his own language (see Column 381 and the series around it on linguistic representations in general, and on numerical languages that include only three numbers: 1, 2, and “many”).

In my book Two Wagons I critiqued Gadi Taub’s claim in his book The Slumped Revolt, who raised a similar contention regarding physics. Taub cited radical feminist claims that physics is a “male conspiracy,” meaning its formulation is masculine, and therefore it’s no wonder women succeed less in these fields. Taub noted that he wouldn’t want to fly in a plane built on a different physics, feminine or not. I argued that he is mistaken. There is room for feminist critique with respect to the language of physics. It is possible that a physics that had been created and developed by women would look entirely different. The language there would be different. This does not mean that the content would be different, or that the plane would be constructed in a way that would in practice make it crash. They might build a different plane that would also fly just fine, or they might even build the same planes as ours, but the planning and the theory underlying them would be formulated in a different language. It is very important to distinguish between the content and the language that represents and expresses it. Even if the language is subjective, that does not necessarily mean the content is an invention rather than a discovery. We shall return to this distinction below.

The Approaches and the Arguments: the Wicked Greeks

Dekofsky explains that the Pythagoreans believed numbers are existing entities with consciousness that even act in the world (a kind of angels). Plato held that numbers and mathematical entities are entities with reality like the world itself (though of course they are non-material Ideas). Euclid held that the world and nature are realizations of abstract mathematical laws.

The first two approaches are essentially Platonist, that is, they construe numbers and mathematical entities as abstract objects that have real existence (though presumably not in space and time—like a soul, angels, or God: spiritual entities that do not exist in space and time). But the Euclidean approach, at least as he describes it, says nothing relevant to our issue. Before that, however, I should clarify it a bit more.

The notion that “nature is written in the language of mathematics” was not so clear in Euclid’s day, and I don’t think he meant the modern claim. It seems to me that he intended to say that the laws of physics are a branch of mathematics. As a conclusion from his words, in principle no observation is required in order to know them; logical thought suffices. The laws of physics arise from necessary relations between objects and phenomena, and those relations are the subject matter of mathematics. This is a completely different claim from the claim that nature is written in the language of mathematics. The prevailing scientific view today is that observation is required to assert anything about the world (this is empiricism as against rationalism). The laws of nature are the result of observation (which makes use of mathematical tools), not of pure mathematical analysis. And yet today it is far clearer than in the past that nature is written in the language of mathematics. Everything is formulated mathematically and it is hard to imagine physics without mathematics (unless you are women, apropos Taub). But mathematics merely serves us as a convenient language to describe the results of our observations and the generalizations drawn from them. Mathematics is only a language, while physics is based on observations of the world. The observations are the raw material described in mathematical language, and that language allows us to process them and draw conclusions. Even the drawing of conclusions from observations is not pure mathematics. It is scientific induction, generalization—only the language in which we often do this is mathematical. That is the contemporary scientific outlook. If we now return to Euclid, he claims there is no need for observation. Mathematical understanding, given correct definitions of the concepts, will yield all of science.

Yet even if the world is a realization of mathematical laws, as Euclid thought, that has nothing whatsoever to do with the claim that mathematical entities are existing entities. It only says that the relations among objects in our world are well described by mathematics, and that this is not accidental. Mathematics governs them and therefore describes them well. This certainly says that mathematics is a kind of truth about the world, but not an ontic truth. The entities of mathematics need not exist in the world, as Plato believed. It is a truth like the truth of the statements that there are five apples in the bowl or that the car’s speed is 100 km/h. These statements do not represent abstract or concrete objects, but relations and properties among (usually concrete) objects. Yet these relations are not an invention but a discovery of truths about the objective world.

The Approaches and the Arguments: the Modern Era – Kronecker

Dekofsky then brings views according to which even if numbers and mathematical objects are existing entities, mathematical statements certainly are not. The reason is that their truth rests on laws created by human beings. He explains that according to these views mathematics is a language that handles relations between abstract entities that are products of our brain.

Already here I would say that, in light of what we saw above, this claim is very problematic. If anything, it would be more accurate to say the exact opposite: the entities of mathematics perhaps do not exist, but the mathematical relations are certainly correct in the world itself (though they do not exist in the sense in which objects exist).

Dekofsky says that one representative of this view is Leopold Kronecker (a well-known nineteenth-century German mathematician). Kronecker said that God created the natural numbers, but everything else is man’s handiwork. I don’t know what he meant, but the claim that mathematics is the work of man can refer either to the content of mathematical statements or to the language in which they are expressed. Did he mean to claim that the Pythagorean theorem is the work of man? That it is an artificial fiction? That’s absurd. Draw a triangle on a sheet of paper, measure, and you will see that it is true. It is not an abstract object, but it is a true statement about the world. It is true first of all in the world of ideas, and of course true in any Euclidean space that exists in our world or in any other world (in mathematical parlance: in any world that is a model of that mathematical theory). The language of angles and relations, squaring or addition, may be a language we created. It is possible that mathematical relations can be described in other languages (feminine languages?). But the mathematical content, not the language in which it is represented, is certainly true. I don’t think any mathematician or philosopher disputes this (bizarre skeptics aside).

I will give two examples to illustrate the point. As is well known, there are two equivalent ways to describe the two-dimensional plane: the better-known one is the Cartesian coordinate system (a system of axes in which the X and Y axes are perpendicular to each other, and each point is described by its two projections on the two axes). There is also another kind: polar coordinates, in which each point in the plane is described by its distance from the origin and the angle that distance makes with the X-axis. Thus, for example, the point (3,4) in Cartesian coordinates would be described in polar coordinates as a point whose distance from the origin is 5 and whose angle with the X-axis is 53⁰. We have several modes of description for the set of points in the plane; does that mean the two-dimensional plane is our invention? Not necessarily. The points in the plane exist in the world (on the sheet of paper, for instance), but the language we use to describe them is our creation.

Take another example (common in analytic philosophy). Think of “the center of mass of the Milky Way galaxy on 14.12.2021 exactly at 12:00 noon.” To understand this description, a person must understand a great many concepts: center, center of mass, galaxy, the Milky Way galaxy, hour, noon, numbers and numerals like 12:00, and so on and so forth. Each of these in turn requires understanding many other concepts. The description presupposes a great deal of knowledge that we barely notice. But if you consider the referent of this description (to use Russell’s terminology, in his famous essay “On Denoting”), we are dealing with one particular point in space that one can point to with a finger (incidentally, one can point to it at any moment and from any place, except that at other moments it will no longer be the center of mass of the Milky Way). Such ostension requires no prior knowledge, and a person who sees it can identify that point in space without any prior knowledge of any of the concepts I used and certainly without the heavy freight contained in them. Incidentally, it is of course possible to describe that very point in space in countless other ways (we have returned here to the coordinate systems discussed earlier). Does that mean that this point doesn’t really exist? That it is our invention? Certainly not. The language in which we describe it may be our creation.

Likewise, if you consider the concept of “center of mass” itself, it may be our creation. But it has an objective meaning in the world itself that can be explained to anyone. Therefore even this concept is not an invention but a discovery—though of course this does not mean it refers to some abstract object existing in the world (the Idea of “center of mass”). That already depends on whether you are a Platonist or not.

If we return to Kronecker, it seems to me that, by the principle of charity, his words should be interpreted in a way that makes sense. He likely did not mean that the mathematical content is an invention. The mathematical language is an invention, but the relations we have discovered regarding those invented concepts are real and objective. Their truth is not necessarily in our world but in some abstract world (the world of ideas). But the meaning of saying that this is truth rather than invention is that if there is, in our concrete world, a model that realizes that mathematical theory, then its statements will be true of it.

A Very Important Interim Summary

Note that everything from here on operates at this level of discussion. That is, we are not dealing with Platonist claims as if mathematical entities have existence in some world of Ideas. The claims of modern mathematicians, physicists, and philosophers concern whether mathematics is an invention or a discovery—in other words, whether it is an objective truth about the world itself or an artificial human invention. But not necessarily in a Platonist sense. The indication of such objectivity is that if we find in the world a domain that constitutes a model of a given mathematical theory, that domain will behave according to the laws that the mathematical theory dictates. This is in fact the definition of the relation between a theory and a model (in mathematical model theory), and therefore you can see that the question has essentially emptied itself of content. No one disputes this, bizarre skeptics aside. At most, one may treat the mathematical language and the entities it discusses as inventions, but the mathematical content is an objective truth (in this sense) by everyone’s account.

I am in fact making three claims, which are of course related:

1. Every mathematical theory is an objective truth, in the sense that if there is in the world a domain in which its assumptions hold (it constitutes a model for it), then it will behave according to its laws. Therefore 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 effecting the relations it lays down, but those relations are necessary also in our real world.
2. Hence the mathematical theory itself, even if it has no realization in the world—that is, we have not (yet?) found a domain that would be a model for it—is true in an objective Platonist sense. It is a truth concerning relations in an ideal realm. It is objective in the sense that these are the necessary relations among the entities it discusses, and there is no way to argue about that.
3. The question whether such an abstract ideal realm actually exists is the question of Platonism. Personally I tend to think it does, because in my view mathematical truths are products of our observation of the Platonic world of ideas (as, in my opinion, are many of our concepts. I am an essentialist, not a conventionalist. See the end of the column). But here there may be room for debate. Others will argue that these relations reflect the structure of our brain rather than something in the world itself. Then the question will arise how it happens that these brain structures prove effective in science, which deals with the world—and here we enter the realms of evolution and claims about human conceptualizations (this is how we conceptualize what happens in the world; it is not the world itself). See more on this below, in the discussion of Wigner.

The dispute regarding the third question concerns whether the abstract ideal realm addressed in claim 2 exists in some sense (without entering into Plato’s definition of existence. In any case it seems clear to me that existence does not require occupying space and/or having mass)—and that is Platonism—or whether it is merely a subjective structure existing in our intellect, through which alone we can apprehend the world, which is the conceptualist approach. The debate is whether mathematical structures are ontic or epistemic. But according to both approaches we are dealing with an objective truth in the sense of the first two claims.

In any event, I claim that the third question is the only one that has meaning in the context of our discussion. As for the previous two, they have a literal meaning, but their answers are obvious. Therefore the two questions—and certainly the debate about them—are rather meaningless.

We can now move on to more modern positions and see that everyone is just churning water. Much ado about nothing. They all essentially agree to the first two claims I set out here, and at most they argue about the third (though, if you read attentively, you’ll see it doesn’t really arise in their discussions). It is important to understand that the term “mathematical Platonism,” which originally pertains only to claim 3, is used today by extension mainly to express the first two claims. This is very confusing, and one might say that this is the principal confusion that gives rise to this barren debate. As I said, regarding the first two claims there is broad agreement (I think any reasonable person agrees to both), and therefore there may be a debate about Platonism (question 3), but there is really no debate about them. Platonism in the confusing sense I described is entirely agreed upon.

The Approaches and the Arguments: the Modern Era – Hilbert and Axiomatization

Next Dekofsky tells us that the renowned German mathematician David Hilbert (late 19th to early 20th century) and his colleagues created axiomatic structures for all branches of mathematics (as we all know from geometry), and from this he infers that they viewed them as a deep logical game—and still a game.

I am not familiar with the details of Hilbert’s views, but again I will invoke the principle of charity. The fact that one creates an axiomatic structure for a given mathematical field does not mean it is a game. It is well known that one can arrange the statements in a given mathematical field in different orders. For example, in geometry one can choose certain statements as axioms and derive all the others from them (including statements that usually serve as axioms). So too in logical rule systems (see, for example, here). The choice among the different arrangements of the statements (which will be axioms and which will be derived theorems) is indeed the result of our considerations of convenience, but the truth and character of the set of statements itself can be entirely objective. Again, you can see the distinction I noted above between the content itself and the modes of presenting it. Note that the question of Platonism (question 3) does not arise here at all, and in principle one could be a thoroughgoing Platonist and still play Hilbert’s “games.” Even if mathematical statements and entities exist in some Platonic world of Ideas, there is no impediment to arranging them in different orders among themselves.

The Approaches and the Arguments: the Modern Era – Poincaré and Geometry

Dekofsky then moves on to Henri Poincaré (a noted French mathematician, physicist, and philosopher, also from the 19th–20th centuries), one of the creators (discoverers?) of the non-Euclidean geometries. These are geometric theories that proceed from assumptions different from those of the familiar Euclidean one; while Euclidean geometry describes spatial relations in a straight (Euclidean) space (for example, a flat sheet of paper), those geometries describe spatial relations in curved spaces (like the surface of a sphere). According to Dekofsky, Poincaré thought that the existence of such geometries proves that Euclidean geometry is a kind of logical game of ours and not an objective truth.

This is a very common argument in this discussion. Many raise it to bolster philosophical pluralism (a plurality of truths), and likewise to portray mathematics as an artificial human invention. But it is very easy to see that this argument is groundless. As I noted, each such geometry describes a different kind of space. In a Euclidean space, only Euclidean geometry is true. In any other space (like the surface of a sphere), a different geometry is true. Does this mean that none of the geometries is objectively true? Nonsense. On the contrary, the existence of the different geometries follows from the fact that there can be spaces of different types. But for any given space there is exactly one correct geometry, and therefore, if anything, this is a paradigm example of the objectivity of mathematical theories.

To be sure, this is a claim about abstract (Platonic, or simply abstract conceptual) worlds, which is what mathematics deals with. And of course the question of application to the world arises again. It is clear to any sensible person that if there is in the world a space with a Euclidean character, its geometry will be Euclidean and nothing else. Therefore the first two claims (1 and 2 above) are certainly true even after the discovery (!) of the non-Euclidean geometries—and, in fact, even more so after it.

Einstein’s theory of relativity does indicate that our physical world is not Euclidean (even this is not entirely accurate, but I won’t get into that here), but that says nothing at all for our purposes. The geometry of the world is not Euclidean, but it is certainly a very specific mathematical geometric theory—and not any other. What has that to do with the relativity of mathematics, or with its being an artificial human creation?! Beyond that, whatever the geometry of our world may be, any consistent geometry is objectively true in the sense I described in the first two claims above (1 and 2), and perhaps also in the third (depending on whether you are a Platonist). None of them is a subjective, artificial human invention, and none can be rejected—i.e., you cannot draw different conclusions.

The Approaches and the Arguments: the Modern Era – Wigner

It is time for physicists to enter this discussion, since it actually belongs more to physics than to mathematics (or, more precisely: more to the philosophy of physics than to the philosophy of mathematics). Eugene Wigner was an important physicist in the 1960s. Dekofsky notes that Wigner claimed mathematics is real, and his justification is, to the best of my knowledge, the most common argument for mathematical realism. Wigner argued that in many cases mathematical theories that were created in an entirely abstract way at some period later turn out to be useful in describing various physical and scientific phenomena. Needless to say, the creators of these mathematical theories did not dream of the scientific fields in which they would find use. According to him this proves that mathematics is discovered by people rather than invented by them. If it were a human invention, what are the odds that it would turn out to be useful in the scientific description of the objective world?

Since I have already shown that the question under discussion is banal and cannot be argued about, it is not clear what Wigner’s argument adds. Against what is he arguing?! A mathematical theory is true in the sense of the first two claims (1 and 2) even if no model were ever found to realize it in practice. So why should actual realization change anything? Regarding the third question, those who hold the conceptualist position can argue that since these structures are forms of human thought it is no wonder that we grasp the world within them. This is precisely Kant’s synthetic-a priori thesis.

I should note that I do not agree with it, since it could have been the case that nothing in the world would fit our forms of conceptualization and thought, and therefore we would not be able to advance in understanding science. The very progress shows that these are not merely subjective conceptualizations—but this is already a dispute with Kant, which is not (or: Kant)[1] our concern here. In any case, if Wigner’s argument has any significance, it is only at this level, except that here it is unnecessary. In my books Two Wagons and Truth and Not Stable I have already shown that Kant’s claims collapse on their own terms, irrespective of the history of science and its complex relations with mathematics.

What Has All This to Do with Zen Koans?

After Dekofsky describes several examples of mathematical theories that became useful long after their creation (continuing Wigner’s line of argument), he says that at this stage these questions sound to him like a Zen koan (see Columns 17 and 211): If in some forest there is a certain number of trees, does that number exist even if no one is looking at that forest and counting them? I assume his intention was to say that, in light of Wigner’s arguments and the like, it is clear that this is nonsense. Mathematics is objective, and it exists even without us.

At this stage there is no need to say that I agree with the conclusion but not with the arguments. The conclusion follows of itself, irrespective of the arguments he presented. But even regarding his koan I have a few comments. First, if he is talking about the number of trees and not about the trees of the forest themselves, then we are dealing with a Platonic entity (=the number), and indeed it is not clear that it truly exists. The forest certainly exists without us, but the debate about the number can remain unresolved. One need not be a Zen adept for that.

Beyond that, I have often noted the error in the original koan. The original koan asks: if a tree falls in a forest and no one is there to hear, does it make a sound? The answer everyone expects is: of course it does. But, ironically, the correct answer is: of course it does not. Sound is a cognitive phenomenon that exists only within us. In the world there is an acoustic wave, and only when it strikes our eardrum does it create within us an audio experience of sound. In the world itself there are no sounds and no sights. There are objects and phenomena there that create in us, in consciousness, sights and sounds.

If we return to the number of trees in the forest, it is clear that the forest exists even without us. But any description we give of it is a product of our language and our conceptualization. If that is the intention, then even after his koan one can easily say that the mathematical language is an invention. If we mean the mathematical contents that it describes, they are certainly objectively true (claims 1 and 2 above). I can tell you with certainty that any forest we find in the future that has ten rows of fifteen trees will contain 150 trees. Moreover, even a Platonic or conceptual forest constructed that way “contains” 150 trees. Unsurprisingly, the first two claims (1 and 2) hold here as well, and as for the third one—of course we can debate it.

Summary: What Is Mathematics

The upshot of all the above is this: the foundations of mathematics (the axioms) and its basic entities (the definitions of the concepts) are formed by our observation of an abstract ideal realm (Platonic or conceptual). The derivation of theorems from those axioms and concepts proceeds by valid formal logic and is not open to dispute. On this view, mathematics is a kind of empirical science—except that we do not apprehend it through the senses but through the “eyes of the mind,” or our intuition. Here I intend to claim that this is a different kind of cognition, and not mere thinking. Thinking takes place entirely within us and has no connection to what happens outside (that is probably the situation according to the conceptualist view). Here we are engaged in cognition of something that is outside us—albeit a non-sensory cognition (“the eyes of the mind”).

No scientific finding can confirm or refute mathematics, and therefore, in the discussion about its objectivity, there is no point in adducing arguments from the world of science—and certainly not from mathematics. This is a meta-domain discussion, and the considerations here can only be philosophical. I presented some of them in the course of this column. Therefore the entire methodology of Dekofsky’s video is flawed. His arguments are irrelevant to the discussion and, as I have shown, unnecessary. One can reach clear conclusions without them. And what remains unclear (question 3) remains unclear even after them.

I suppose many of you can guess my next sentence. In light of the definition I proposed here, mathematics is a branch of philosophy (see the series of columns 155 and on, and in more detail the series of lectures in the two courses I taught on philosophy—accessible only to those who registered). Mathematics deals with deriving conclusions from premises that are not the product of scientific observation, just like philosophy. The difference between them is that mathematics is a formal branch and its inferences are necessary, unlike other areas of philosophy that are not necessarily formal and whose inferences can admit probability and not only certainty. Thus, mathematics is a particular kind of philosophy. As I argued in that series of columns and lectures, philosophy too, like mathematics, is not an artificial human invention but a set of claims about the world. Moreover, in my estimation there is very little genuine disagreement about its principles. I mean, of course, only those parts of it that have some substantive content, and not all the nonsense and fluff that accompany them in droves (see, for example, Column 223 on executing judgment on the French and their helpers, and on the meaning of disputes in philosophy).

[1] My study partner in Bnei Brak used to call me a “Physi-Kant.”