“My first impression on seeing him (Grothendieck) lecture was that he had been transported from an advanced alien civilization in some distant solar system to visit ours in order to speed up our intellectual evolution.”
Marvin Greenberg
I was very hesitant about writing this article, for the fear that my ignorance of logic and philosophy might prompt me to say something stupid and embarrass myself (which I probably will anyway). Nevertheless, I couldn’t resist the temptation. These ideas have circulated in my mind for too long. Therefore, I ask the reader to indulge me in spilling some nonsense, which, in my defense, even if worthless mathematically, may still serve as a fun sci-fi project or thought experiment. The question I’m interested in is the following. Imagine an alien civilization on a distant planet in a solar system millions of lightyears away which is advanced enough to have developed their own “mathematics” (whatever it means). What would their mathematics be like? To break this down into several smaller questions: firstly, what would their formal logical system (to make things interesting, we assume they do use one) be like; secondly, what would the mathematical objects they study be like; and thirdly, what would their theorems be like?
Before we start discussing these questions, allow me to explain why I believe these questions are worth thinking about. In human history, different cultures all developed the notion of natural numbers, albeit obviously having different representations of them. For the Arabs, it’s 1, 2, 3, 4, 5, 6 and so on, for the Romans, it’s I, II, III, IV, V, VI, and so on. While the Arabic numerals use the position of a symbol to denote its value (place-value system), the Roman numerals have fixed values for each symbol (I = 1, V = 5, X = 10, L = 50, C = 100, and so on). They are both fully faithful representations of natural numbers, but some theorems are easier to discover and prove if you are working in the “right” representation. For example, I’d imagine that it would be easier for the Arabs (or any other culture that use place-valued based 10 representation of natural numbers) to discover and prove the theorem that a natural number is divisible by 3 if and only if the sum of its digits is divisible by 3, since this theorem is due to the fact that 10 equivalent to 1 modulo 3, which is easier to observe in arabic numbers. Of course, the theorem is still true for roman numbers, as we can easily translate between the two representations. Another example from elementary mathematics is different coordinate systems of the plane, such as Cartesian coordinates vs. polar coordinate. There are many examples from elementary calculus on the plane, which are easier to calculate when you swap to a different coordinate system. Similarly, I think working in the “right” foundations makes some theorems easier to discover and prove.
Therefore, you can tell that I actually lied a little in my motivating question: it’s not the aliens I care about. I only phrased the question in terms of aliens as an excuse to provoke thoughts on a body of hypothetical mathematics, along with a foundation thereof, independent of human intuition, experience, and practical needs. You can think of “aliens” as a metaphor for an abstract agent, not necessarily literal aliens living in our universe, capable of advanced reasoning but possibly in a drastically different fasion. Perhaps the foundations we use today, such as Zermelo-Fraenkel set theory, are too successful, which has maybe made us lazy to think about other possibilities. Granted, we have played with many other set theories and type theories, but those are more or less motivated by the mathematics of our species. I firmly believe that there’s a lot more room for exploration, and that if we wish to, we could be much more imaginative in our foundations. Perhaps, unbeknownst to us, there is a foundation of mathematics out there radically different from all foundations we have which could shed light on areas of math in ways we couldn’t fathom yet. On an unrelated note, Michel Demazure once said about Grothendieck, that “I did not view him as I did other great mathematicians, who were made of the same fabric — better fabric, to be sure, as they were brighter, faster, harder workers. Grothendieck always seemed essentially different; he was an ‘alien.’” I think it is worthwhile to adopt a similar attitude: to think like an alien.
The Mathematics of Aliens
I would like to begin with some naïve thoughts, both to reinforce the points I made in the previous paragraphs, and to pave the way for the next section. A fundamental observation is: human experience, intuition, and societal influences, has played a major role in the development of human mathematics, and consequently in human foundations of mathematics. Naturally, one wonders what if these variables are different for the aliens?
Firstly, what if the aliens experience the world differently? I’d imagine if the alien’s body is made of gas, plasma, or some spooky quantum-y matter, they would probably have a much more esoteric perception of the notion of distance. Perhaps even non-archimedean distance is more natural to them? Maybe they discover \(p\)-adic numbers before they discover real numbers? In human history, real numbers and the rigorization of calculus (real analysis) is central to the development of foundations. Perhaps \(p\)-adic numbers would play a similar role in the alien’s development of mathematical foundations, leading to different definitions of these structures? On the other hand, what if the aliens live in a universe where the laws of physics is different? This may affect their concept of many fundamental notions, such as space or shape. Their perceptions of causality may also affect their concept of probability, and so on.
Secondly, what if the alien’s intuition, that is, the way their brain works, is different? In Liu Cixin’s Remembrance of Earth’s Past scifi trilogy (spoilors!), he described an alien civilization, the trisolarans, who spontaneously and instantaneously communicate their thoughts through electromagnetic waves. To the trisolarans, thinking is the same as speaking, and therefore they have no concept of deception, which gave them a big disadvantage against humans when they invaded earth. Similar things might be true for math. When I was learning algebraic topology for the first time, I found the calculations of fundamental groups and homology groups using Seifert—van Kampen theorem and Mayer—Vietoris sequences particularly interesting. Compared to other proofs I did, they seem to depend highly on pictorial intuition. This kind of intuition is especially common in the study of low-dimensional topology, where mathematicians often work with pictorial intuition before writing down rigorous proofs. A general observation seems to be that the more intuition we have of something, the less reliant we are on its formalisms. One could imagine that there might be aliens who are able to perform this kind of informal intuitive argument on higher dimensional or non-archimedean spaces, which we mere humans can only approach with rigorous formalism.
Thirdly, what if the aliens have different societal influences on mathematics? For example, in human history, the development of various number systems, from \(\mathbb N\), to \(\mathbb Z\), to \(\mathbb Q\), and to \(\mathbb R\), is rooted in our demands for increasingly sophisticated mathematical structures in socioeconomic production such as accounting and engineering. From counting discrete finite things with natural numbers, to including negative and rational numbers for the purpose of representing money, and to including numbers arising in geometry such as \(\pi\) and square roots (arising as length of hypotenuse of right triangles), and so on. An alien society with possibly different socioeconomic demands may lead to a different line of development of number systems, or perhaps they didn’t start with \(\mathbb N\) to begin with! The study of number systems begat the study of abstract rings, hence one could imagine aliens having different axioms for rings. And this could be said about many other mathematical structures and their abstractions. On the other hand, the influence of natural languages in math can’t be ignored. What is fascinating to me is that first-order language bears striking similarities with our natural languages. It’s hard to believe that there isn’t a connection. One expects aliens to have different natural languages, which could influence the syntaxes of their formal languages.
The more I ponder about these possibilities, the more I believe that human mathematics is by no means natural or unique epistemologically — there are a lot of things in our math that are rather arbitrary. Aliens could disagree with us on what mathematical objects are more basic, which elaborations of the basic objects are more natural, which body of mathematics is more intuitive, which mathematical structures better capture fundamental notions, and so on.
The Universal Feature
In the previous section, we talked about what could be different for alien math, but a much more fundamental and much more challenging question is, what would be universal among all alien’s (and our) math? Here, I propose a candidate answer to this question: abstractions.
For example, natural numbers are already an abstraction. They are an abstraction of the notion of finite size (as cardinals) or finite well order (as ordinals). Here lets just focus on the former. From an early age we learn to count things with our fingers, but what we are really doing is establishing a bijection between our fingers and the things we are counting. We do this because we realize that counting is the same mechanism regardless of the thing we are counting. Counting our fingers and counting anything else are isomorphic! The natural numbers is the abstraction of this mechanism. There can also be abstraction of abstractions, e.g. various number systems are abstracted as “rings”. I would like to think of abstraction as philosophically different from generalization; it is a special kind of generalization that unveils the deeper nature of the object in study. This idea of abstraction is so fundamental that, in some sense, one could argue that abstraction is precisely what characterizes mathematic ontologically. I do not believe that there are aliens who do math by mechanically generating theorems from axioms using inference rules (and if so, it is debatable what they are doing is actually math). There must be a point where they introduce concepts and definitions of mathematical objects.
Why is this question of universal feature across alien maths important? I think it is because it is an attempt to peak into what is truely natural in math. Usually, we say something is natural if it doesn’t depend on any arbitrary elements. For example, any isomorphism of vector spaces \(V\rightarrow V^*\) is not natural because it depends on a choice of basis, but map that takes an element to its evaluation map \(V\rightarrow (V^*)^* \) is a natural isomorphism because it doesn’t depend on a choice of basis. Similarly, humans (and likely alien civilizations) have their own arbitrary “choices” in their math, as discussed in a previous section. Therefore, the elements that are universal, which do not depend on the arbitrary elements, would be more natural.
A Natural Foundation?
In my first undergraduate class, I learned about Zermelo-Fraenkel set theory — the lingua franca of modern mathematics. However, I never felt truely convinced by it. Don’t get me wrong, I have no doubt of its rigor or success as a foundation of mathematics. What I mean is that I get a sense that many things about it are incredibly “awkward” or “unnatural”, for lack of a better word. I’ll only discuss one example: the definition of cardinal numbers. Naïvely, one would define a cardinal number as the following
(Naïve) Definition. A cardinal number is an equivalence class of sets under bijection.
Of course, this is ludicrous, we can’t have the natural number 1 be a proper class. The actual definition of cardinal numbers is much more sophisticated: it is an ordinal number not equinumerous as any smaller ordinals, and ordinals are in turn defined as a set strictly well ordered by membership and every subset of which is an element. This definition, due to von Neumann, circumvents the set-theoretic issue by “restricting” the class of sets to its “skeleton”, ordinal numbers, in order to prevent cardinal numbers themselves from being too large. However, I argue that this naïve definition is what captures the idea of cardinals more naturally, because it emphasizes the conceptual meaning of cardinal numbers that they are avatars of the sizes of sets. This is not seen in von Neumann’s definition, in which the well-order structure comes first and is substantial to the definition. I was quite surprised to learn that in some type theories, the naïve definition actually works. This means that the crux of this problem really is of a set-theoretic nature. Other difficulties we encounter in set theory are usually of a similar fashion — the limitation comes from something being too large to be a set. For example, in category theory, and especially when higher structures are introduced, there are often meaningful things that could be too large to be a set, such as the functor category \(\mathrm{Fun}(\mathcal C,\mathcal D)\) between two categories. Of course, there are ways to circumvent this, such as restricting things to a Grothendieck universe, but, again, it feels contrived and not as natural as the naïve definition. Why are many of these “awkwardnesses” of this fasion? My guess is, this is because the notion of sets, i.e. a collection, is treat as a priori, and thus encoding other notions, when they are not usually thought of as a collection, as sets would lead to ad hoc modifications i.e. the awkwardnesses, especially when it “ranges over” a proper class.
In all human math foundations, one common theme seems to be to build mathematical objects from certain “a priori” or “atomic” or “basic” notions. In set theory, the basic notion is sets (and the basic relation between them, namely membership). Everything else is built from sets, even if the mathematical object isn’t usally understood as a collection, e.g. a tuple \((x,y)=\{x,\{x,y\}\}\), a natural number \(2=\{\emptyset,\{\emptyset\}\}\), or a function \(f\subseteq X\times Y\). I presume the justification for this philosophy is that it is an easy way to make sure everything is consistent with everything else (at least we hope so since we can’t prove this due to Gödel). And also because, well, we have to start somewhere, don’t we? I concur with this point, but still I’m not fully convinced by this philosophy, and I wager that there are alien civilizations who agree with me on this. The problem I have with it is, none of these “atomic objects” seem “canonical”. The notions of function, relation, and many others seem equally fundamental as the notion of sets, and one could imagine some aliens using “function theory” or “relation theory” as their foundation. It is difficult to assert that any of the notions of “function”, “relation”, “sets”, etc, are more basic than all others. I guess you could also take the utilitarian point of view: it doesn’t matter what the basic object is, as long as it works. However, I find this to be a lazy answer. I’m not satisfied by it.
This prompted the question: what is the canonical basic notion? First, what makes a notion “basic”? Let’s take a step back and first think about why do people usually consider the notion of collection, i.e. sets, a basic notion? The raison d’être for set theory is that it allows us to treat a collection of objects as one, a notion indispensible for mathematical practice. From the Bourbaki point of view, a “mathematical structure” is a set with possible extra structures, such as algebaric structures, topological structures, or order structures, and various combinations of them. A morphism between these structures would be a function of sets preserving said structures, and then we could talk about these objects up to isomorphisms. In this sense, sets are basic because all “mathematical structures” by definition are built from them. However, this point of view is restrictive. I don’t believe that the ubiquity of a notion would imply primitivity, eventhough assuming so would be good for practical reasons. From a maybe naïve point of view, perhaps a way to approximate primitivity is to see if it could encode all of hitherto human math. In this sense, a lot of other notions, not just sets, could be considered basic. This again is a very difficult question. What then would the canonical basic notion be? Let me take a radical stance at this question: there is no canonical basic notion at all!
Before I attempt to justify this claim, I would like to make a digression from the world of math to the world of the philosophy of science. There is an idea in science relevant to our discussion: reductionism. Reductionism, roughly speaking, is the belief that things can be explained by breaking them down into smaller more fundamental things. A quintessential reductionist belief is, chemistry is just applied physics, biology is just applied chemistry, etc.
Reductionism has been criticized by many, but most famously by theoretical physicist P. W. Anderson in his More is Different. In a nutshell, Anderson argues that “the main fallacy of this idea is that the reductionist hypothesis does not by any means imply a constructionist one: the ability to reduce everything to simple fundamental laws does not imply the ability to start from those laws and reconstruct the universe.” Another more sophisticated criticism comes from W. V. O. Quine, in his Two Dogmas of Empiracism. I argue that, similar to empirical sciences, the fact that we can reduce mathematical structures to more basic constructions, in the sense of analytic methods, doesn’t imply that we could start from the “most basic notion” and build everything on top, i.e. to implement all mathematics in some synthetic theory of a basic notion. Here, the meanings of “everything” and “all mathematics” are quite subtle. Let’s not get into the philosophical debate of whether mathematics exists outside foundations, and take the word to mean the collection of all math devised by all possible abstract alien civilizations. This hypothesis explains why we find it difficult to find a canonical basic notion — it may not exist. I believe it should be questioned whether we should build the foundations of math in this ethos.
It seems to me that, among the most working mathematicians (by this I mean those whose works lie in traditional fields such as number theory and geometry), perhaps excluding those more philosophically sophisticated, the foundations of mathematics are viewed with pure utilitarianism, if not sheer nonchalance. For them, the foundations are merely instruments to clarify and rigorize mathematics, and it is almost never the case that they consider a formal foundation of math an ontological characterization of mathematics. For example, in cases where category theory is involved, some mathematicians would assume axiomatically the existence of a strongly inaccessible cardinal in addition to the standard axioms. No mathematician in their right mind would claim that, because of this, category theory is not legitimate mathematics. It seems as if that many mathematicians assume the axiom of choice, or some other axioms of ZFC, primarily because the theories they already informally have, whose substance often has little or nothing to do with the set theoretic foundations on which they are built, demand them. In this sense, axioms of our foundations of math are ad hoc!
Mathematicians are justified in thinking this way. When doing mathematics, they do not typically think in pure logic. Instead, they operate on a level much beyond pure logic: they often come up, from intuition and experience, with a rough outline of (often ambiguous) reasoning prima facie, and then write them down formally and fill in the details to check if they are correct — much analogous to how physicists check their theories empirically. It is therefore easy to see why so many pure mathematicians are attracted to Platonism — it feels like they are discovering mathematics, not inventing it. I claim that this is the source of their utilitarianism in foundations — if what we are doing is discovery, then it makes no sense to demarcate what we could discover. Philosopher Gottlob Frege famously compares the work of a mathematician to that of a cartographer. The cartographer picks the parameters of the map (the scale of the map, the type of projection to use, the color of each region, and so on), but the cartographer does not invent the actual geography. I would like to add to this analogy that: the parameters of the map are to the geography what the axioms of our foundations are to mathematics. Altering these parameters will change subtlely how the geography presents itself to us, but it does not change the geography. Moreover, the cartographer chooses the parameters in a way to better present the geography to us, so it is the geography which motivates their choices of parameters. In the same vein, it is the content of the actual math which motivates our choices of axioms, and thus our axioms does not ontologize mathematics, i.e. we reject that math is the consequence of some chosen set of axioms.
Given that it is hopeless to find a canonical basic notion on which to build the entirety of mathematics, and the fact that mathematicians reject to ontologize math based on any particular axiomatic foundation, I wonder whether there is an “axiom-free” approach to the foundation of math, in contrast with the orthodox view that the edifice of math stands upon a solid agreed-upon base? Maybe we could forgo the view that a basic notion is needed to build others, and allow a multitude of “basic notions”? How do we ensure that one works well with another? Ay, there’s the rub! These questions have become too difficult for my current level of mathematical maturity, and I think this is a good point for me to stop rambling my nonsense.
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