I think quite a lot about conceptual engineering in the context of math cognition. What can we get out from a careful study of conceptual content of a mathematical expression at different stages of its development? Is there any relation between the conceptual content of expressions at the very beginning and the content of mature concepts?
My first motivation to look carefully into math cognition came from observing that in many books on philosophy of math the authors – while presenting the epistemological aspect of their proposal – recall some fact from folk psychology. It struck me as very arbitrary, and hence methodologically fallacious. In the same time I had a feeling that exploring the amazing world of research in number cognition could lead towards interesting observation relating conceptual structure of mathematical concepts.
What I propose is a move similar to that proposed by Shapiro in his text on open-texture: instead of looking at the development of the mathematical concept from a historical perspective, we will be looking at an ontogenic path of individuals. Shapiro claims that certain mathematical concepts, such as computability or natural number, reached today a stage of closed-texture. This stage could have been reached thanks to careful scrutiny of conceptual engineering. Together, these two lines of concept development enable improvement of our understanding of what natural numbers really are.
Finally, before I start my analysis of the conceptual content, let me say that reaching a close-texture stage reminds me Aristotelian idea of a form that eventually reaches the stage of maximum realization in the given material.
In the talk I gave recently at a Festival of Philosophy at Ischia, I started engineering into concepts present already in the earliest stages of number concept development relating not individual objects, but the relational structure of natural numbers.
You all know which numbers are natural, they are sometimes called positive entire numbers and they are usually called: “one”, “two”, “three”, “four”, etc. Some other people call them: “uno”, “due”, “tre”, “quattro”, etc., when I was at school I counted: “jeden”, “dwa”, “trzy”, “cztery”, etc., my kids when they are at school they count: “ett”, “två”, “tre”, “fyra”, etc.
Some people are very keen on claiming that zero is a necessary element of natural numbers, as in this famous joke about a famous polish mathematician Wacław Sierpiński. Sierpiński was known for being distracted when it came to everyday life. It was his wife who was helping him organizing. Once she prepared his baggage for the travel and she advised him to count the suitcases when he leaves the train. “You have six suitcases”, she told him. “Count them before you leave the train to make sure you have all of them”. Sierpiński once arrived to his destination called his wife in a complete panic. “I lost one of the suitcases”, he told his wife. “There is just five of them”. “Are you sure”, asked the wife, “have you counted them?” “Yes, I did! Let me count again: Zero, one, two, three, four, five!”
Another, even more striking example, about names for natural names comes from a novel by Jorge Luis Borges entitled “Funes the Memorious”. Funes was a guy who was known for being weird and having strange philosophical ideas thinking about things other people do not usually think of. He was a self-made philosopher, we might say. One day Funes got a brain injury in an accident and his perception of the world got even more unusual. For instance, when Funes saw a dog in the sun light standing in front of him, and then the same dog in the afternoon running on the street, he would not consider this is the same dog. Funes also thought that leafs on the tree are different leafs at every time they change position. He could remember every single detail of what he saw and heard and describe the difference long before it happened. However, Funes was not able to synthesise or generalise his knowledge.
What is interesting for us in this context, is that Funes also invented his own numeric system.
He told me that in 1886 he had invented a numbering system original with himself, and that within a very few days he had passed the twenty-four thousand mark. (…) Instead of seven thousand thirteen (7013), he would say, for instance, “Máximo Pérez”; instead of seven thousand fourteen (7014), “the railroad”; other numbers were “Luis Melián Lafinur,” “Olimar,” “sulfur,” “clubs,” “the whale,” “gas,” “a stewpot,” “Napoleon,””Agustín de Vedia.” Instead of five hundred (500), he said “nine.” Every word had a particular figure attached to it, a sort of marker; the later ones were extremely complicated…. I tried to explain to Funes that his rhapsody of unconnected words was exactly the opposite of a number system. I told him that when one said “365” one said “three hundreds, six tens, and five ones,” a breakdown impossible with the “numbers” Nigger Timoteo or a ponchoful of meat. Funes either could not or would not understand me.
Now, some could object me that I promised to speak about natural numbers, but I have not said a world about them yet wasting all my time on speaking of names of numbers, or numerals. This person wouldn’t be entirely wrong. It might be the case that all these systems of names are equally good to name natural numbers. It might be the case that even the system of names of Funes could be used to refer to natural numbers.
This question has been investigated in the paper “What Natural Numbers Could Not Be”, by Paul Benacerraf (1965) that is most often referred to as the founding paper of a position in philosophy of mathematics called mathematical structuralism.
Benacerraf written his paper to oppose Frege’s position according to which every natural number has its particular essence and can be considered separately. For instance the number five can be understood independently of the number three, or the number seventeen.
Benacerraf starts his paper by imagining two boys: Ernie and Johnny. Those boys’ parents are mathematicians living in a close intimacy with abstract objects (boy’s names come actually from Ernst Zermelo and John von Neumann), and before they explain their children which abstract objects are natural numbers, they explain them various things about the mathematical universe in general.
Only after few years of study they tell their kids which elements from the mathematical universe are natural numbers. They tell them which properties these objects have. And… the boys discover they think that numbers are different mathematical objects! There is however one thing that their numbers have in common: they share the same structure. Both structures have the starting element and then elements succeed in a regular way into the infinity. Benacerraf concluded that it is not important which mathematical objects stands for natural numbers, it can be anything which can be put into this kind of progression.
Structuralism in mathematics says that mathematics is the science of structures. That all the properties of mathematical objects are first of all structural or relational properties. That you can think about mathematical structures as if it were systems of placeholders. Imagine a soccer team, you can give two types of descriptions, you can say that the squad consists of Millik, Mertens, Allan, Soderlund, and Reina etc. But you can also say that it consists of a goal-keeper, left-forward, left-midfielder, center-back etc. In this second description this is the position in a structure that comes first, and who is occupying this position is only the secondary feature.
Structuralism is a very attractive proposal, solving problems in philosophy of mathematics, in particular related to mathematical practice allowing to smoothly change one structure to another. For instance we can explain how does it work that numbers have so many different representations. In set theory these are particular sets, in computer science these are sequences of 0s and 1s, natural numbers are also instantiated by an Arabic system starting with zero.
Problem with saving the intuition that there is something intrinsically wrong with numbers of Funes can be solved by introducing a clear distinction between syntactical and semantical level. In his another paper published in 1995 entitled “Recantation or any old omega-sequence will do after all” Benacerraf address this issue. He draws a clear distinction between syntactical and semantical level. Benacerraf claims that natural numbers, understood as abstract objects, just form a progression, and can be represented can be any folks, whereas the system of names in our culture needs to be recursive.
Structuralims is a pretty attractive standpoint, and what I am interested in is to see which conceptual content is crucial for a structuralist viewpoint. What I am interested in is to look for conceptual content in cognitive sciences. Structuralists themselves are basing the epistemological aspect of their research on ideas from psychology saying that their position is grounded on the ability of humans to recognise patterns. I was always strongly disappointed in this very arbitrary proposal and I started looking into what happens in labs studying number cognition.
What I am going to present now is one of the leading paradigms from the contemporary cognitive science. It has been developed by such people as Elisabeth Spelke, Stanislas Dehaene, or Susan Carey.
The way in which I like to think about it, and which is not the standard way, consists in formulating a two-level model of math cognition. The first level is a purely syntactical level of names of natural numbers, the second is the level of numbers considered to be abstract objects. Interactions between the two levels exist and are integrated part of the whole picture, but unlike in the standard approach, it is possible and even advisable to study the two levels separately. The situation looks more or less like the one described by Carnap in the Introduction to Semantics:
If in an investigation explicit reference is made to the speaker, or, to put it in more general terms, to the user of a language, then we assign it to the field of pragmatics. (Whether in this case reference to designata is made or not makes no difference for this classification.) If we abstract from the user of the language and analyze only the expressions and their designate, we are in the field of semantics. And if, finally, we abstract from the designate also and analyze only the relations between the expressions, we are in (logical) syntax. [Carnap 1948, 9]
Both syntax and semantics can be studied from two perspectives [Carnap 1948, 12]:
- a descriptive approach, strongly entwined with pragmatics, consisting of empirical investigations of the semantical or syntactical features of historically given languages,
- a pure approach, fully independent from all pragmatic considerations, consisting of lying down ”definitions for certain concepts, usually in form of rules”, and studying ”the analytic consequences of these definitions”. These rules can be formulated with intention of providing a model of existing ”pragmatical facts”, but they can also be chosen in an arbitrary manner as a consistent set of sentences. [Carnap 1948, 13, compare also page 155].
The pure approach to syntax opens up a possibility to make an abstraction not only from the users of the language, but also from meaning, and in consequence enable studying syntax without any appeal to semantic.
Carnapian approach adapted to our situation is the following: imagine a child learning first mathematical concepts. Such a child in our culture is very quickly exposed to various counting-out games like: “Eeny, meeny, miny, moe, Catch a tiger by the toe. If he hollers, let him go, Eeny, meeny, miny, moe”. Children eventually learn to repeat names of numbers that are used in their culture (Fuson 1988).
However, as highlight it developmental psychologists, knowing to enumerate names for numbers is not enough to understand what these names mean.
There is a famous experiment called Give-N Task (Wynn K. Children’s acquisition of number words and the counting system. Cognitive Psychology. 1992;24:220–251). A child is asked to feed a stuffed animal with, let’s say, four oranges. The experimenter instructs a child participating in the experiment: “give four oranges to the anteater” (anteaters are pet-toys of UC-Irvine where Sarnecka’s Cog Lab made a lot of work involving Give-N Task). The chid takes a bunch of oranges and gives to the anteater. Let’s say, seven oranges. “Oh,” says the experimenter, “are you sure that it is four? Shall we count them? One, two, three, four, five, six, seven. Have you given four oranges to the anteater?” The child looks surprising and… add another three to the bunch of oranges.
The Give-N Task and other experiments We realised that children learn meanings names of numbers one at the time and in order.
We also realised that in addition to cultural interactions humans are equipped with two cognitive systems that allows us to process quantitative information.
The hypothesis of the existence of core cognition has been formulated by Mehler and Bever who in 1967 conducted an experiment to explain a strange thing observed by Piaget in 1952. Piaget claimed that children get the concept of number from interaction with the external world. He presented to kids two lines of marbles. In the first part of the experiment children were shown two equal lines of marbles and were asked if there is one line where there is more marbles or whether these lines are equally long. Then the experimenter would not change the quantity of marbles but would extend one of the lines in a way that spaces between marbles are bigger and as the question again. Young children would fail saying that there is more in a longer line.
Many explications have been proposed, for instance that a child felt that if the experimenter repeats the same question twice, then she expects a different answer.
But the experiment of Mehler and Bever shown a very interesting result. Mehler and Bever had two versions of experiment. In the first version they used marbles, in the second candies. They shown children two lines of marbles/candies. In their version they had four elements in the upper line and six in the lower. Four formed a longer line. The children replied in a way coherent with Piaget’s predictions in the first case and in the second… they took the line with more candies!
This was the beginning of studying the cognitive system called Approximate Number System. This system allows humans to assess quantities in an approximate manner. When I get in this room my brain provided me information relating the approximate number of people in this room. I didn’t have to count my public, but I could moderate my voice in such a way that everybody could here me.
We also have another system of processing quantities, which is used for small numbers. It enables humans to process information about small quantities up to four.
Before I reach my conclusions let me say few words about Munduruku people. The tribe Munduruku has not developed stable number names for numbers bigger than 5.
Now, let me get back to the point that I was making at the beginning of this paper. As you remember I was telling you about a structure of names and a structure of abstract objects.
It is not my objective to claim that structuralists are right. I have a much modest objective. I just want to say that when you look at the research in number cognition you realise that structural properties of the sequence of natural numbers are an extremely important factor of learning what numbers are for kids from our culture.
Let me now look at how this picture is related to the picture that want to draw mathematical structuralists.
A child learns by heart a system of names for natural numbers, which forms a progression that she learned from her parents. It is a place-holder structure, a child at least at the beginning has no clue which elements occupy which place.
A child learns meanings of names one on time and in order.
A child from our culture needs to know how to generate a successor in order to make sense of bigger numbers. This is not true for Munduruku children, nor for those people that many years ago had systems of names of natural numbers based on body parts.
However, in our contemporary culture where we know a lot about optimal notations, like decimal of binary, structural properties are the most crucial and hence it supports the structuralist view on mathematics (at least to mathematics as we know it today).
The hypothesis that I would like to formulate at the end of this preliminary investigation is the following: it seems to me that structuralism couldn’t have developed without the discovery and consciousness of recursivity of mathematical systems.