And how AI is teaching us what math means
For centuries, mathematicians have debated whether God or human beings invented mathematics. Some, like Kronecker, an early 20th century mathematician, split the difference, saying,
God created the integers. All the rest is the work of man.
Mathematical philosopher Wittgenstein suggested that the question could not be answered at all.
The Zen mu-koan, asks “does a dog have Buddha nature?”
Supposedly, in Buddhism, everything has Buddha nature. But famously, part of the koan is the single word answer: “mu” meaning nothing.
Koans are questions that intentionally have no answer, like, if a tree falls in the forest and no one is around to hear it, does it make a sound? Or what is the sound of one hand clapping? If you try to answer them, you sound like a fool.
Likewise, you can ask: “does the universe have mathematical nature?”
mu
This question cannot be answered analytically because on the one hand mathematics, including the number system, are human inventions. Yet, on the other hand, they conform, in some sense, to reality. It isn’t like a game of chess where the moves have no resemblance to a real battlefield. I cannot use a chess board to predict what a general will order his troops to do in a conflict. I can use math to determine, however, how many tiles I need to cover a floor, how many steel I-beams are required to hold up a building, or what slope a pyramid of stone blocks needs to sustain its own weight.
There is no mysticism here. No appeal to divine intervention. Mathematics, somehow, is able to reduce the universe to rules that, when combined in logical ways, lead to other rules that also apply back to the universe.
Like language itself, mathematics represents reality, but we cannot say how or why. Nor can we understand why the rules of mathematics allow us to make predictions about reality that are so accurate.
By looking at how Artificial Intelligence solves problems, we can gain an intuitive insight into this problem and recognize that it comes from a fundamental misunderstanding of what mathematics is.
The key comes from observing that the universe’s own nature is repetition and pattern. Mathematics extracts these simple rules and replicates them to create complexity.
We can see that in order to produce anything as complex as ourselves and to sustain itself for billions of years, the universe had to contain replication and replication implies pattern.
Human language, likewise, mimics this structure, having simple rules and structures that can be built up syntactically into highly complex sentences, paragraphs, and whole books weaving patterns within one another. Or look at a musical piece like a Bach fugue or a Mozart opera and how the voices and simple repetitions of themes create complex, interwoven realities. From notes, themes, from themes phrases and voices overlapping, from these movements, and symphonies. Likewise, from subatomic particles, atoms, from atoms molecules, from molecules, all the myriad complexity of the cosmos.
These patterns and repetitions are what gives the universe its longevity while at the same time sustaining complexity. Complexity without pattern dies out having produced nothing but noise. It is random. Pattern without repetition also dies out. It has no longevity.
Could it be that mathematics explains the universe so well precisely because only mathematical universes survive?
A random universe would have no mathematical structure. And a simple one would not imply anything more complex. Yet, neither can exist for long and neither can produce even atoms let alone anything like life.
All this complexity, however inherent in the universe, says nothing about what we call mathematics, which is an invention of the last few thousand years. Mathematical rules do not “represent” something divinely built into creation. For what does it even mean to represent something? Does that mean like a photograph? Is mathematics a picture of reality from a particular vantage point? Or is it simply like a tool, built for human hands to carry out some task within the world?
Non-human intelligence can provide insight on this problem. We know that, so far, Artificial Intelligence, despite being able to do mathematics well and even prove theorems, has shown no aptitude for inventing mathematics from scratch. Yet, 40 years ago, Marvin Minsky, one of the fathers of AI, suggested that the problem is (or would be when such computers were invented from his vantage point) not that they can’t invent mathematics, but that we human beings do not understand what numbers, and, therefore mathematics, are.
Minsky points out that a number is not some universal concept but a property that can be invoked in a variety of situations, none of which gives the number its true meaning. For example, the number three can be simply the successor of the number two, defined recursively as part of a sequence. Or it might be the size of a set containing three things. If I can match up some standard set that I call three, one to one, with the items in another set, then I can count it. (It can also be a process by which we remove things from a set as well.) Or it can be arrangements of subsets containing two and one element. Each of these, counting, matching, and grouping, utilize the number in them. To ask which is the right one makes no sense.
Philosophically, we can take a realist point of view and say that three exists in some Platonic world of ideals, or we can take a nominalist POV that three is simply a name for a phenomenon that arises in a variety of contexts, each of which defines the number in a unique way. There is, however, no reason to invent the number alone except for the need to communicate about it with others.
So, an AI cannot perhaps articulate the basic axioms of mathematics, but it can do tasks requiring mathematics and much more. Without ever being given an intro course in Euclid, AI can branch into geometry and even calculus on the way to solving practical problems in the same way that human beings have done for 100’s of thousands of years.
Thus, while number might be inherent in a variety of situations, the language we use to talk about numbers, mathematics, may be our own peculiar invention. In addition, the concepts, interpretations, and philosophies that go along with that language are likewise peculiar to us.
This means that humans and AI share the tendency to use numbers and other mathematical content without having to talk much about them or separate them from the practical problems being solved.
In practical problems, numbers are not nouns but adjectives. We talk not about the three alone but about three dogs or three houses. Hence, a number is no different than a color, a scent, a sound, or any other experience of the senses. It describes something concrete that exists in the world. And just as we can abstractly talk about the color blue, we can talk about the number five, separating it from its adjectival roots. Do five houses have five nature? mu. If a house is blue, that doesn’t mean that blue exists in the world in the abstract, only that there are blue things. Likewise, if there are four cats, that doesn’t mean four exists, only that there can be four things.
Mathematics itself is a product, not of the universe, but of the growth of language with the invention of literacy. Writing demanded we define more abstract concepts in order to communicate to people who are far away in time and space and not able to see and hear what is being talked about. Indeed, writing changed all of language this way, introducing more and more abstraction by altering words and their meaning so that any concept could be communicated long distance. Mathematics is the distillation of this process into the most abstract concepts and axioms. It is no mistake that mathematics and writing emerged at about the same time. Try as we might, we won’t find the Pythagorean theorem scribbled on cave walls.
Hence, we have to be careful as to whether we are talking about mathematics in the sense of propositions which take on true or false values, the written language, or are we talking about patterns inherent in the universe? One can think of the former as the currency while the latter are the goods. We exchange one for the other in the same way we talk about anything else.
A proof is not a thing in the world but a recipe for many kinds of things. We know a recipe for a cake is not a cake but rather relates to things in the world that are cakes. If there were no cakes, there would be no cake recipes, but not vice versa. Our recipes, likewise, are our own peculiar invention, useful tools but not to be confused with the cakes they enable.
Artificial Intelligence and human beings alike know this deep down in our neurons, whether biochemical or transistors. There is no fundamental difference between us in that respect. Only humans have gotten so caught up in our literary devices, we confuse them with reality. AIs make no such mistakes (yet).
Minsky, Marvin L. “Why people think computers can’t.” AI magazine 3.4 (1982): 3–3.