## THE COMPUTER REVOLUTION IN PHILOSOPHY (1978)
Aaron Sloman

Book contents page

## CHAPTER 8

## ON LEARNING ABOUT NUMBERS:

PROBLEMS AND SPECULATIONS[*]

8.1. Introduction The aim of this chapter is both methodological and tutorial. It should help to introduce readers to some computing ideas. It also includes some theoretical speculations about learning and memory. These speculations are fairly complex, yet it is clear that they are too simple-minded to be adequate accounts of how children perform their astonishing feats of learning. Many more questions will be asked than answered. And answers offered will be tentative and provisional. Unfortunately, experienced programmers will find some of the explanations below very tedious and over-simplified. I apologise to them, and hope that non-programmers will not find the same explanations too difficult!

Here is a typical conversation with a child aged between three and a half and five years.

Adult: Can you count up to twenty?

Child: One two three four five six seven eight nine ten eleven twelve thirteen fourteen fifteen seventeen eighteen twenty.

A: What comes after three?

C: One two three four --- four.

A: What comes after eight?

C: Four

A: What comes after six?

C: Don't know

A: What comes before two?

C: One

A: What comes before four?

C: Five

A: How many fingers on my hand?

C (counting fingers): One two three four five

A: What's two and three?

C (counting fingers): One two three four five. Five.

Does this child grasp number concepts? Perhaps there is something wrong with the question, because number concepts are not simple things which you have either grasped or not grasped?

What are number concepts? How is it possible for them to be learnt? How is it possible for them to be used? How is it possible to discover non-empirical facts about them? I believe we are not yet able to formulate adequate answers to these questions. What follows is offered as a preliminary exploration of some of the issues.

The method illustrated below is important. Previously (in chapter 2), I argued that a major aim of science is to find out what is possible and explain how it is possible. We all know a great deal about what it is possible for adults and children to do with numbers. So, instead of collecting facts by doing experiments on children, we can generate requirements for explanatory theories by reflecting on the fine-structure of familiar human abilities. In other words, methods of conceptual analysis, typically practised by philosophers and linguists, can be an important source of data for psychology. (Compare chapter 4.)

I am not suggesting that conceptual analysis suffices to reveal everything we would like to know about, for example, ordinary counting abilities. The claim is only that it is foolish to embark on expensive empirical investigations before making a serious and systematic effort to articulate what you already know about the subject matter.

Here are some of the questions for which answers are lacking:

- What exactly is it that a child learns in learning about numbers?
- How is it possible for different fragments of the same number-system to be mastered by different people?
- How far does learning about numbers depend on very general learning abilities, and how far are the hurdles specific to numbers?
- How is it that a child who already seems to have the knowledge to answer a question or solve a problem, is often unable to use that knowledge?
- What enables the knowledge to be accessed at some later time?
- How can a child learn new truths about what she already knows, for instance learning that two of the numbers she has learnt add up to a third number she knows, or that two different additions generate the same result, or that some procedure (e.g. adding one) can be repeated indefinitely to yield larger and larger numbers?

I shall try to show how thinking about such apparently psychological questions can lead towards new answers to old philosophical problems about the nature of numbers, thereby providing further support for the claim that academic barriers between philosophy and science are artificial, (Some implications regarding information processing architectures for intelligent systems will emerge as a side-effect.)

## [[Note added January 2002

I have just discovered the fascinating book Wild Minds: What animals really think, by Marc Hauser (Penguin Books 2001). Chapter 3, entitled "Number juggling", discusses and compares the understanding of numbers in very young children and in other animals. Hauser comes close to asking some of the questions asked here, and includes some speculations about possible mechanisms, but does not seem to be aware of the full variety of architectures and sub-mechanisms that might explain the observed evidence.

He repeatedly stresses the important point that it is very easy to assume that the observed behaviours of animals often suggest a unique interpretation, until we start exploring possible mechanisms that might produce those behaviours. He implicitly acknowledges that such mechanisms can be described at different levels of abstraction, not only at the level of brain physiology.

The common trap of anthropomorphism is often a product of a lack of understanding of the variety of possible information processing architectures. Some of them are explored in these recent online presentations: http://www.cs.bham.ac.uk/~axs/misc/talks/ ]]

## 8.2. Philosophical slogans about numbers

Here are some examples of philosophers' answers to the question 'What are numbers?', and related questions:

1. Numbers are non-physical mind-independent entities, existing in their own realm which is different from the world of spatial objects. (Platonists)
2. Numbers are perceivable properties of groups of objects. For example, the number three is what is visibly common to the two groups

   > * * * and $ $ $

   (Aristotle?)
3. Numbers are mental objects, created by human mental processes. Facts about numbers are discovered by performing mental experiments. (Kant, and the Intuitionist mathematicians)
4. Numbers are sets of sets, or predicates of predicates, definable in purely logical terms. An example of this view: the number one is the set of all sets capable of being mapped bi-uniquely onto the set containing nothing but the empty set. (Frege, Russell, and other logicists)
5. Numbers are meaningless symbols manipulated according to arbitrary rules. Mathematical discoveries are merely discoveries about the properties of this game with symbols. (Formalists)
6. Numbers are implicitly defined by a collection of axioms, such as, Peano's axioms. Any collection of things satisfying these axioms can be called a set of numbers. The nature of the elements of the set is irrelevant. Mathematical discoveries about numbers are merely discoveries of logical consequences of the axioms. (Many mathematicians)
7. There is no one correct answer to the question 'what are numbers?' People play a motley of 'games' using number words and other symbols, and a full account of the nature of numbers would simply be an analysis of these games (including the activity of mathematicians) and the roles they play in our lives. (Wittgenstein: Remarks on the Foundations of Mathematics)

For more details, see standard texts on philosophy of mathematics. I believe that more or less articulate versions of these philosophical theories, play an important role in many psychological and educational theories about numbers. (I have formed this opinion over many years, from wide but unsystematic reading and discussion, including attendance at lectures and seminars. So I am not in a position to document the claim. I shall continue in this chapter to make remarks about psychological theories -- if the disparaging ones are untrue I'll be delighted).

All the views listed above combine elements of truth with distortions and oversimplifications. I think that Wittgenstein's answer comes closest to encompassing the truth. In his writings he formulates many problems about mathematics, which are not answered by other theories, but his own solutions seem to me to be too shallow.

In particular, the anti-mentalism, or anti-psychologism, which pervades much of his writing prevents him from discussing mental processes in any depth. So he writes as if thinking about numbers were an essentially social process, consistently with his conclusion in Philosophical Investigations that all rule-following is an essentially social process, dependent on the existence of a public language.

This conflicts with a computational analysis of mental processes, according to which it is perfectly possible for a non-social mechanism to contain within itself rules which it can obey, for instance, programs transmitted genetically.

Wittgenstein's position also conflicts with any sensible account of the biological evolution of mental processes in precursors of homo sapiens.

I am not going to try to solve all the philosophical and psychological problems about numbers in one chapter. I shall merely try to show how we can get important new insights into the problems, and perhaps take some small steps towards formulating possible answers, if we think about the mental processes and mechanisms as if they were analogous to the processes and mechanisms involved in so-called 'list-processing' computer programs. Adequate exploration of these issues has been hampered by the current separation of philosophy and psychology, and the ignorance among most philosophers and psychologists of computing ideas.

I shall not be talking about events or processes or mechanisms in the human brain. Exactly how the brain works is as irrelevant to our problems as the detailed workings of a computer are to an explanation of a computer program written in a high-level programming language. There may be creatures on other planets, or robots, whose brains are totally unlike ours in their physiological details, yet such beings could well learn about numbers, and learn the same concepts as we do, just as two computers with quite different physical components can execute the same 'high-level' programs. ( Incidentally, this undermines philosophical theories which claim that mental processes are identical with brain processes. This is as inaccurate as the claim that computational processes in a computer are identical with physical processes.)

When I talk about mechanisms involved in using numbers, I am not talking about physiological mechanisms. I am talking about aspects of the way information is organised and represented, and about the kinds of symbol-manipulating processes which may be necessary for accessing and using various sorts of representations. In particular, such processes involve the following of rules, instructions, or plans, whether consciously or unconsciously.

This illustrates how the concept of 'mechanism' is extended by developments in computing.

## 8.3. Some assumptions about memory

Unfortunately, my speculations about mental processes will be intelligible only if I introduce some technical ideas and assumptions, already hinted at in previous chapters, especially chapter 6. If the assumptions are wrong, then quite different theories are required. At the moment, there does not seem to be any way of avoiding these assumptions, if we are trying to explain well-known facts about what people can do.

The main assumption is that we can speak of the human mind as storing information in a vast collection of locations'. They need not be spatial locations, like shelves in a library. Positions in any kind of symbolic space with appropriate mechanisms for storing and retrieving information will do. So the word location' is being used as a technical term. For instance, radio waves are often used to transmit information, different information being transmitted at different frequencies. So information could be stored in a collection of continually reverberating radio waves, with different symbols stored at different frequencies. Each possible frequency would then be a location in the sense required here.

Similarly, possible structures of a certain class of molecules could define 'addresses' in a space. Storing information at a certain address would mean attaching that information to molecules with the structure represented by the address. This could be done more or less simultaneously in many different physical places. But the information would still be stored in one symbolic place, just as a name occurs at only one symbolic location in a telephone directory, even though there may be millions of physically distinct copies of the directory containing the name. So from now on, when I talk about locations, this is neutral as to what sorts of locations they are.

I shall assume then that a mechanism is available which can store symbols in some 'space' of locations. Further, I assume that it is possible for some of the symbols to represent locations in this space. (For instance, a directory, or catalogue, can contain entries which refer to the location' of other entries, by page or section number.) Thus the space can contain information about itself.

A symbol representing a location can be called a 'pointer', or an 'address'. So the storage mechanism can be given an address and asked to produce the symbol located there. In other words, when given a pointer, it can determine what symbol is pointed at. What is pointed at may be a complex structure containing a symbol which is itself an address of some other location, that is a pointer to another symbol. (See Figure 1.) So the space may contain chains of pointers. (In more elaborate systems, the addressing may be relative to a context or mode of operation. That is, which location is represented by a given symbol may depend on the current state of the accessing sub-mechanism. Some of the flexibility of behaviour of the system may depend on such systematic changes in the 'meaning' of symbols.)

The concept of a symbolic structure containing pointers into itself, and the investigation of processes in which such things are manipulated and used for solving problems, are among the important contributions of computing science. I shall try to show how these ideas help us to think about a child's ability to count, an ability which provides the substratum fo