How did biological evolution produce
abilities to discover and prove theorems
that are not derivable from any default
biological design feature.
A partial index of discussion notes in this directory is in
There are many different ways in which the notion of something infinite enters mathematics. Pre-school children may spontaneously discover some of them, though much will depend on the environment in which they live and play (e.g. their games, their older siblings, their parents, and nowadays also the things they find on internet sites or watching TV): there's no "standard" order of learning determined either by human psychology (or brain mechanisms), or the mathematical structures themselves, though there may only be a limited (non-infinite) variety of significantly different trajectories towards thinking about some form of infinity. Exploring that idea requires analysis and comparison of a large variety of examples, some listed below.
What's more important than the order in which most or all children take these steps, is what the information-processing mechanisms are that they require in order to be able to take them. I think it's fair to claim that very little is known about that, mainly because the people who come up with theories have had no experience of building working information-processing systems.
There are probably several major and very many minor stages in the development of young information-processing architectures that extend what can be learnt -- though there need not be a fixed order through which children pass through those stages.
This morning I heard a BBC Radio 4 recording of Tony Benn reading some of his diary records ("Free at Last - The Benn Diaries 1991 to 2001 Episode 3 of 5", unfortunately available to listeners for only a week). Benn reports on 5th June 1997, that a child relative who was 5, had said to his father: 'Daddy what is the biggest number in the world? Is it a hundred and sixty eight thousand?' When his father replied 'Well, there's a hundred and sixty eight thousand and one', the child responded 'I got very close to it, didn't I dad?'
Some structure had not yet been built in the child's mind recording the observation that whatever number has been reached in counting another one can be added. Or perhaps he lacked the information processing mechanisms required for seeing the implications, and merely asks 'So What?' as indicated below.
o If I've got to here . . . I can add one dot and get to here . . . . o If I've got to here . . . . I can add one dot and get to here . . . . . o If I've got to here . . . . . I can add one dot and get to here . . . . . . o and I can do "that sort of thing" for any initial pattern of dots. o So what???? Someone, somewhere in human history (or maybe many different people, or children), must have noticed the pattern and drawn the conclusion some of us now find so obvious. That requires their reasoning engines need to be what mathematicians call "omega-complete"! (However, that notion is usually applied to systems based on some axiomatic variant of predicate calculus, and I don't think there's a shred of evidence to suggest that toddlers or even five year olds use that, though it's not yet clear what alternatives are available.)(Some readers may be helped by the answer to the question posed here: http://math.stackexchange.com/questions/466465/omega-consistency-and-related-terms )
It's possible that the child had not yet grasped the general pattern, in all the examples of adding one more to a collection of items.
Or perhaps he had grasped the pattern but not the implications of applying it indefinitely.
Or perhaps he had grasped it for concrete operations, e.g. extending a row of dots, but not for the more sophisticated operations involved in repeatedly generating a new number name.
It was a great achievement when humans found a number notation that made that obvious, unlike Roman numerals and English number names, which require newly invented names (ten, twenty, hundred, thousand, etc.) instead of using a few primitives from which all possible names can be constructed, as in decimal or binary number notation.
Note that the most primitive Pre-Roman notation, which represents numbers merely by collections of strokes (or other discrete items) -- like these representations for 1, 2, 3, 4, 5, 6
/, //, ///, ////, /////, //////, .... etc,-- have adequate expressive power in principle, but become grossly unreadable (by humans) and consume vast amounts of space as numbers represented grow. For example, the number 52 would be
////////////////////////////////////////////////////instead of only two symbols '5' followed by '2'.
It is also possible that the child had already developed the mechanisms required for that observation without having deployed them in this case. No doubt the insight was not much delayed. (I don't think Benn saw this as relevant to anything deep about minds and mathematics. For him it was merely a delightful anecdote.)
As Immanuel Kant noticed, and Zeno long before him, there are several different kinds of infinity implicit in our perception of the world and in addition to the boundless ability to add one to any whole number reached by counting. A line with some thickness can in principle become thinner, e.g. by halving its width. That line can become thinner. A child who notices that may, like the child Benn reported, wonder if there's a minimum thickness beyond which no further reduction is possible. One answer is "Yes, a line that's perfectly thin". Why then isn't there a similar number at the end of the line of numbers?
Similar facts can be noticed about curvature and straightness: a line that is curved can become straighter, and as long as it retains some curvature further straightening is always possible. Is there a limit: a perfectly un-curved, i.e. straight, line?
An expanded entry on this topic would need to discuss several more types of what might be called "limiting case infinity", including areas, volumes, speeds of motion, speeds of rotation, points reached by unbounded shrinking of lines, areas, or volumes -- do they all end up at the same sort of point, or are there some linear, directed points, planar oriented points, and 3-D points occupying an infinitely small volume? [Need to look up research by Piaget on this. Did he know about omega-completeness??]
Do the same information-processing mechanisms allow children, eventually, to apply the same process of generating an infinite case to weights of objects, speeds of motion (steady state, or amount of deceleration, or amount of acceleration), to variations in smell, taste, pitch of sounds, differences of pitch, volume of sounds?
(Note that in some cases there are two infinities: infinitely small and infinitely large, but not all. Has anyone ever thought about infinitely large curvature and found that useful? It makes a difference whether you think of increasing curvature of a line of fixed curvature passing through a point, or increasing curvature along a line. The first notion generates a space of shrinking circles passing through the point, shrinking to that point in the limit. The case of a line that starts with some finite curvature -- e.g. zero curvature and then becomes increasingly curved as it extends produces a spiral, though what's going on at the 'end' of the spiral is probably quite hard for most people to think about?)
I suspect very few adults have a coherent set of beliefs about such matters. Some cognitive mechanisms are capable of generating questions without providing resources for reaching answers.
A Vi Hart Interlude on infinity
Happy Pi Day? NOPE
Why Pi is no more infinite than 3, 4, 3.5, 1/3, etc....
Note: 'grinch' appears to be a North American label for a kill-joy, spoil-sport, or -------- ??