(DRAFT: Liable to change)
When working on a general way to mechanise computation in 1936, Alan Turing did not propose a test for whether a machine has the ability to do computations. That would have been a silly thing to do, since no one test, e.g. with a fixed set of problems could be general enough. Neither did he propose what might be called a "test-schema" or "meta-test", namely calling in average people to give the machine computational problems and letting them decide whether it had succeeded, as might be done in some sort of reviewing process for a commercial product to help people with computation.
Instead, he proposed a deep THEORY, starting with an analysis of the sorts of things a computational machine needs to be able do, based on the sorts of things human computers already did, e.g. when doing numerical calculations, algebra, or logic. He came up with a surprisingly simple generic specification for a class of machines, now known as Turing Machines, and demonstrated not by building examples and testing them, but mathematically, that any Turing machine would meet the requirements he had worked out. Since the specification allowed a TM to have any number of rules for manipulation, it followed mathematically that there are infinitely many types of Turing Machine, with different capabilities. Surprisingly, it also turned out that a special subset of TMs, the UNIVERSAL TMs (UTMs) could each model any other TM. Wikipedia provides a useful overview: http://en.wikipedia.org/wiki/Universal_Turing_machine
Doing something similar to what Turing did, but for machines that have human-like, or some other sort of intelligence, rather than for machines that merely (!) compute, would require producing a general theory about possible varieties of intelligence and their implications. This is totally different from the futile task of trying to define tests for machines with intelligence (or human-like intelligence) -- as Turing himself understood. In Sloman (mythical) evidence based on the contents of Turing's 1950 paper is presented against the hypothesis that Turing intended to propose a test for intelligence: he was too intelligent to do any such thing. He was doing something much deeper.
I'll summarise the arguments below and describe an alternative research agenda, inspired by Turing's publications, including his 1952 paper, an agenda that he might have worked on had he lived longer, namely The Meta-Morphogenesis project, using evidence from multiple evolutionary trails to develop a general theory of forms of biological information processing, including many forms of intelligence.
Understanding the varieties of forms of information processing in organisms (including not only humans, but also microbes, plants, squirrels, crows, elephants, and orangutans) is a much deeper and more worthwhile scientific and philosophical endeavour than merely trying to characterise some arbitrary subset of that variety, as any specific behavioural test will do.
This discussion note is
A slightly messy, automatically generated PDF version is:
(Or use your browser to generate one from the html file.)
Mary-Ann Russon interviewed me while it was being written and provided a summary in her International Business Times column.
For interest, and entertainment, here's a video of two instances of the same chatbot engaging in an unplanned theological discussion (at Cornell): http://www.youtube.com/watch?v=WnzlbyTZsQY
A partial index of discussion notes is in
NOTE (Extended 19 Jun 2014)
By the time you read this there will probably already be hundreds, or thousands, of web pages presenting comments on or criticisms of the Turing test in general, or the particular testing process at the Royal Society on 6-7 June 2014.
The announcement that one of the competing chatbots, Eugene Goostman, had been the first to pass the Turing Test produced an enormous furore. Here's a small sample of comments on the world wide web:
For some reason several reports in printed and online media referred to the winning contestant as a "supercomputer" rather than a program. Even if someone involved with the test produced that label in error, the inability of journalists to detect the error should be treated as evidence of poor journalistic education and resulting gullibility.
I believe my main criticism of the idea of any sort of expanded behavioural test for intelligence below, based on a comparison with trying to devise a test for computation, is new, but would be grateful for links to web sites or documents that make the same points about the Turing Test as I do, and any that point out errors in this analysis.
Adam Ford (in Melbourne, Australia) interviewed me about this event using skype,
on 12th June 2014 (2am UK time!) and has made the video available here (63 mins):
It attempts to explain some of the points made below, in a question/answer session, but suffers from lack of preparation for the interview. Our interview at the AGI conference in Oxford in December 2012 is far superior: https://www.youtube.com/watch?v=iuH8dC7Snno
Turing himself did not propose a test for intelligence, as a careful reading of his 1950 paper shows, and building machines to do well in such tests does not really advance AI.
So, why did I accept the invitation to be a judge this time? Because I naively thought it might be a good opportunity to sample advances in chatbot developments, on the assumption that only high quality contenders would be selected for the occasion. It turned out that the conditions of the test did not allow much probing, though I gained the impression that the five chatbots I interacted with had not advanced the state of the art much. If you know roughly how most chatbots are programmed, it is not too difficult to come up with questions or comments that the designer has not anticipated and for which a suitable response cannot be concocted by a quick search of the internet. Often a question about physical behaviours of some household material or object in a new context will suffice, e.g. "Will my sock stop these papers blowing about in the wind?" A chatbot with a very large database and flexible pattern matching capabilities may come up with some irrelevant answer referring to windsocks. A human in our culture may be able to produce a clever joke referring to windsocks in reply, but is more likely show common sense understanding of properties of papers, wind, and requirements for paperweights. But no one test is guaranteed to be decisive.
It is clear from Turing's 1950 paper that he did not propose his "imitation game" as a test for intelligence, though though he occasionally slipped into calling his non-test a test! What he actually did with the game was set up a prediction about particular capabilities of future computers so that he could demolish arguments that had previously been published against the possibility of machine intelligence. That's why, in previous publications I have referred to the mythical Turing Test.
Providing such a generic design, with potentially infinite generative power, because it can be instantiated in a huge class of possible individuals, each of which extends itself in a unique way in response to successive challenges during its lifetime, is comparable to, though much more difficult than, Turing's specification in 1936 of a generic design for computing machinery. Turing's specification for what we now call Turing machines, defining a class of computational systems, was in turn much deeper than any attempt to specify a set of tests for a machine to have computational capabilities. For example, it is impossible to specify a set of behavioural tests that will decide whether something is a universal turing machine.
(A proof of this is presented in a separate discussion of black-box tests. Martin Escardo informed me that this is actually a special case of Rice's theorem -- which has the consequence that for any 'interesting' computational property it is impossible to determine by behavioural tests whether a Turing machine has that property.)
The question whether some generic design (like the human genome, or elephant genome) has the ability to produce the kind of variety of developmental trajectories that humans (or elephants) are capable of is a much deeper and more interesting task than producing one system that can pass some extended sequence of tests, even over a lifetime.
Moreover it is mathematically impossible to produce a behavioural test that will determine whether any observed individual is an instance of something like the human genome, since over any finite number of tests two very different designs could produce the same behaviours. Although Rice's theorem was not proved in Turing's lifetime (as far as I know) it is clear from the Mind 1950 paper that he did not think that 'intelligence' or 'thinking' could be defined precisely and he did not think it sensible to try to devise a test for intelligence or thinking.
By 1936 he had already done something much deeper and more interesting, namely he produced a general theory about properties of a class of machines with computational powers, and a subset with universal computational powers. We still don't have a comparable theory for machines with the powers provided by any of the sophisticated genomes produced by evolution, including genomes for humans, elephants, orangutans, squirrels, or weaver birds. So we lack a theory of the kinds of intelligence that those organisms have.
What I've called The Meta-Morphogenesis project, partly inspired by Turing's ideas, may lead to a collection of such theories, though that will not happen soon.
Many more tests are proposed regularly, often ignoring previous proposals. A test proposed recently by Gary Marcus in his New Yorker blog, is an example, namely getting a machine to answer questions or make comments after watching a TV show or online video. Many other "improved" versions of the test have been proposed, usually based on the mistaken assumption that Turing's intention was to propose a test for intelligence.
For reasons given below, and elaborated in a separate discussion of black-box tests (purely behavioural tests) for intelligence, proliferating tests may be useful for entertainment or engineering purposes, but something totally different from a new intelligence test is required to provide deep answers to scientific or philosophical questions. (As Turing knew: he did not propose a test for intelligence, as his 1950 paper makes clear, though unfortunately most who discuss the test and propose improvements have never read his paper.) Instead of a test for intelligence, we need a theory, and tests for good theories.
On the basis of the sorts of deep theories covering varied phenomena that Turing himself had already produced (one of which was later presented in his 1952 paper on Morphogenesis mentioned below), I suspect Turing understood how much more important production of a theory, and tests for a theory was than production of tests for a special case of an instance of the theory. However, I suspect Turing had not thought that through when he wrote the 1950 paper.
That raises the question how the program was able to fool 10 of the 30 judges. No doubt the organisers of the event will be studying the transcripts, and perhaps questioning the judges about what they did and did not notice. But my main point is that all of this misses the point that there cannot be a good behavioural test for intelligence, just as there cannot be a good behavioural test for computation. I repeat: Turing's paper makes it very clear that he was not proposing such a test.
We need to think about intelligence in something like the way Turing had previously thought about computation: namely by analysing requirements for various kinds of intelligence, including a wide variety of types of animal intelligence, various types of possible machine intelligence, and discussing which kinds of machinery are capable of explaining which sort of intelligence. That requires a deep theory about products of biological evolution, acknowledging that the concept of "intelligence" required has the feature known to computer scientists as "parametric polymorphism" discussed in more detail here.
On the day, I felt the time available did not permit me to evaluate progress in chatbot design, though deciding which was the human seemed to me to be very easy in each case. I'll find out later whether I was fooled by any of the chatbots, but I was pretty sure that I managed to identify all five of them by the first, second or third response. However, one of my tests was 'failed' by all the humans as well as all the machines. In response to "My new hearing aids should help us communicate" not one of them pointed out the irrelevance of hearing aids to textual interaction. One gave a one-word response: 'brilliant' which might have expressed pleasure at improved communication or an obscure compliment to the tester. Later responses convinced me that was a human.
Despite the shortage of time, the differences between human and machine responses usually seemed clear. Of course I may be wrong. I'll update this when I have been told how many decisions I got right: However, I don't think that I meet Turing's requirement that the judges should be "average interrogators" (see below). That would rule out someone who had built a (toy) chatbot, namely the Birmingham Pop11 Eliza (based on a toy chatbot developed about 35 years ago at Sussex university as a teaching demonstration for undergraduates, who played with it then learnt to build simple chatbots of their own, as precursors to deeper work on language understanding). If not being average rules me out as a participant, that increases the proportion of participants who were fooled by any machine I identified!
The short time did not really allow me to probe any of the chatbots in depth, so I was not able to learn much about their strengths and weaknesses. The short time limit was required in order to fit enough separate judging sessions into the time available, though I suspect Turing's reference to five minutes for his "Imitation game" was intended to allow five minutes for each player, especially as most humans (in particular those whom he referred to as "average") are not high-speed typists. His actual words were:
"I believe that in about fifty years' time it will be possible to programme computers, with a storage capacity of about 109, to make them play the imitation game so well that an average interrogator will not have more than 70 per cent. chance of making the right identification after five minutes of questioning. The original question, `Can machines think?' I believe to be too meaningless to deserve discussion."
(His guess regarding memory capacity by the end of the century was remarkably accurate. I suspect that in the year 2000 a higher proportion of the 'average' human population would have been fooled by some of the more sophisticated chatbots than now because since then a lot more humans have learnt about what computers can and cannot do and the sorts of stupidity they often display. But it is too late to test that hypothesis.)
That was one of the key teaching ideas when we introduced courses in Artificial Intelligence, including programming, for students in Arts and Social Science subjects at Sussex University, around 1976, led by the late Max Clowes http://www.cs.bham.ac.uk/research/projects/cogaff/sloman-clowestribute.html His ideas about teaching had a deep impact on our teaching of programming and AI: http://www.cs.bham.ac.uk/research/projects/cogaff/sloman.beginners.html
See also the tutorials on chatbots, pattern matching, linking a chatbot to a changeable database -- as in the SHRDLU program in Winograd (1972) -- and related educational topics here: http://www.cs.bham.ac.uk/research/projects/poplog/cas-ai/video-tutorials.html
More importantly, we already know how to create things that are able to behave like human beings over many years, in very many different environments, including all the environments in which humans have survived. That's what we do when we produce babies, as many others have pointed out. But we don't know how they work, and we don't know all the design requirements that a typical human (or other intelligent animal) needs to satisfy in order to live a human life.
This illustrates my claim that merely being able to produce a machine that performs over any (bounded) length of time as a human would does not indicate that we have increased our understanding of human intelligence, or any other kind of intelligence. It depends on the intellectual (scientific, philosophical and engineering design) knowledge used in the process. Making babies uses less of such knowledge than making chatbots, but neither requires deep understanding of how human or animal minds work.
For example, we don't really know much about what the functions of human and animal vision are, although many people who think they do know have begun trying to build robots that can satisfy their supposed functional requirements (e.g. segmenting the environment and recognising/labelling the fragments segmented), or taking in binocular 2-D images, or moving images, and creating 3-D models that can be displayed on a screen in 'fly through' mode. My brain cannot do that. I can't even draw static versions of most of the things I can see very clearly. (My wife is much better at this.) So many of the supposed capabilities involved in human or animal vision are inventions of researchers, not based in deep understanding of the functions of biological vision systems.
In particular, most, and perhaps nearly all, vision researchers assume that visual systems acquire metrical information and attempt to create information structures in which metrical values are used, or, when there's not enough information for precise metrical inferences, probability distributions over metrical values are constructed instead.
So if your visual system cannot tell whether a gap is exactly 45 cm wide, many researchers assume that instead it produces a sort of internal graph of the probabilities of all the possible values around 45 cm that are consistent with the retinal stimulation. That's a very complex process and if it is being done for all estimates of distance, direction, curvature, gap sizes, speeds, etc. then the requirements for the brain to handle all that probabilistic information sensibly become computationally and mathematically very demanding -- or perhaps totally intractable. This may explain why the competences of current robots seem to be so much more restricted than those of very young children and many other animals.
(I find it particularly mathematically implausible that brains representing spatial structures, relations, and processes of change in structures and relations, in terms of collections of probability distributions could have discovered the beautiful theorems and proofs in Euclidean geometry that were first discovered without the help of mathematics teachers thousands of years ago. No current machine that I know of comes close to that. For examples of the kinds of reasoning required see: http://www.cs.bham.ac.uk/research/projects/cogaff/misc/triangle-theorem.html Hidden Depths of Triangle Qualia http://www.cs.bham.ac.uk/research/projects/cogaff/misc/triangle-sum.html Old and new proofs concerning the sum of interior angles of a triangle.)
Probability based vision researchers tend to ignore the biologically much more plausible alternative of making use of large amounts of definite (i.e. not merely probable) information about partial orderings (nearer, bigger, more curved, sloping more steeply, heavier, faster, etc.) that provide an adequate basis for a large amount of animal action, especially in connection with servo-control mechanisms (using visual feedback). It is often possible to be absolutely certain that A is taller than B when you can see both, even if you can't estimate the estimate the height of either with much precision. Likewise you can often tell with certainty whether your hand is moving away from, or towards an object or neither, even if you cannot estimate the speed of movement or the distance accurately. For more on this see this web page.
The ability to make and use such comparative perceptual judgements and to reason about their consequences is an important aspect of natural vision, and seems to be part of the basis of some human mathematical competences, for example in reasoning that if A is further away than B, and B is further away than C, then A is further away than C, and noticing that this is not an empirical generalisation but can be understood as an example of an inviolable constraint on partial orderings. (It is not at all clear how brains do this or how to give robots the ability to make such mathematical discoveries.) So this is an example of the kind of challenging requirement that needs to be met by a new generic design for a class of spatially competent machines. And explaining how that mathematical reasoning ability might be implemented in either animal brains or future machines is part of the requirement for the kind of deep research that should replace either attempts to improve the (mythical) turing test or make machines that appear to pass the test, e.g. if asked or if given a practical task that requires such reasoning. More examples of such requirements are concerned with proto-mathematical discoveries made and used by very young children, before they know what they are doing, i.e. "Toddler Theorems": http://www.cs.bham.ac.uk/research/projects/cogaff/misc/toddler-theorems.html
Vision researchers in AI/Robotics, neuroscience and psychology mostly ignore the deep connections between human spatial perception and the ability to do mathematics, especially abilities required for making the sorts of geometric mathematical discoveries reported in Euclid's Elements over two thousand years ago, which includes many theorems and proofs of the theorems that must have been discovered originally when there were no mathematics teachers. How? Partial answers are suggested in Sloman MKM08 and AISB-10.
Turing was far too intelligent to claim that the sort of ability displayed by a competent performer in his imitation game was adequate for anything like human intelligence, apart from a very narrow subset. His purpose was very different, namely to refute a specific set of arguments other thinkers had produced about the impossibility of machine intelligence.
I have sat through many research seminar presentations that allude to a test for intelligence allegedly proposed by Turing, after which, when asked, the speaker confesses to not having read what Turing actually wrote.
It should be clear to anyone who has read the Mind 1950 paper, that Turing did not propose any test for intelligence. The 1950 paper has been reprinted in many places, most recently in the prize-winning 2013 collection of papers and commentaries, with contents listed here.
The 2013 collection also includes my paper 'The mythical turing test', available also as a 'preprint' that will be revised from time to time Sloman (mythical).
It argues, as this paper does, that Turing was far too intelligent to propose the sort of test that is attributed to him, and that he was merely making a fairly limited prediction about what he thought computers might be able to do by the end of the century. His main purpose was to analyse and refute previously published arguments that seemed to imply that his prediction could not succeed. For now, it's not important whether his arguments worked. The point is that he did not propose a behavioural test for intelligence and that attempting to do so would be misguided because it does not address the deep research problems.
All the proposed variants of the test fail to address the need to identify a design that is based on an explanatory theory rather than a design whose performance merely matches some observed behaviours.
A discussion of limitations of what can be learnt from "black box" tests of turing machines can be found here, including a brief mention of Rice's Theorem (Roughly: no "interesting property" -- in a technical sense of that phrase -- of a computational system C can be proved by a turing machine observing the behaviour or inspecting the rules of C).
One clue about what he might have thought about possible future developments is his aside regarding digital (discrete) computers (in the 1950 paper):
"Strictly speaking there are no such machines. Everything really moves continuously. But there are many kinds of machine which can profitably be thought of as being discrete-state machines."And this statement made in passing, but not elaborated:
"In the nervous system chemical phenomena are at least as important as electrical."I suspect those two comments and the examples in the 1952 paper suggest that Turing had started thinking about chemical information processing mechanisms which existed in a variety of organisms long before brains evolved, and continue to play important roles in animal bodies, e.g. fighting infection, repairing damage, and of course growing brains in embryos. One feature of chemical information processes in biological organisms is that they combine continuous change with molecules (or parts of molecules) moving together or apart, folding, twisting, unwinding, etc., with discrete processes such as formation or release of chemical bonds, and many catalytic and autocatalytic processes. It is possible that that combination can do things discrete computers (including Turing machines) cannot do. If so, AI and Robotics in future may have to extend the repertoire of available implementation mechanisms for their designs.
These ideas suggest a long term project of trying to identify major transitions in information processing in organisms, including changes in both what is done and how it is done, e.g. using chemical forms, neural forms, and forms of computation based on use of virtual machinery in more complex evolved species.
I call the project to investigate those evolutionary developments and their consequent developmental (epigenetic) processes the Meta-Morphogenesis (M-M) project, partly because it was inspired by Turing's 1952 paper.
The project includes attempting to specify a variety of forms of biological (human and animal) information processing, e.g. in visual perception, mathematical discovery (especially in geometry and continuous topology), nest building by many animals, including weaver birds, without assuming that they can all be implemented in digital computers. The project is outlined here: http://www.cs.bham.ac.uk/research/projects/cogaff/misc/meta-morphogenesis.html
If important aspects of human intelligence rest on such mechanisms and (a) we still have no deep and broad specification of the requirements to be met, (b) we don't yet know what mechanisms can meet those requirements and (c) we don't fully understand the evolved mechanisms, or the biological functions for which they are essential components, then we may be unable to replicate human-like intelligence in machines, in the foreseeable future.
There are certainly many machines performing impressively with fragments of human intelligence, and in some cases superhuman fragments, because of the speed and complexity of what they do. But there are also many aspects of human and animal intelligence that we are nowhere near emulating in machines. Examples include the mathematical abilities that must have led to the discoveries eventually collated in Euclid's Elements over two millennia ago, and many animal abilities including the weaving of long thin leaves to make hanging nests done by weaver birds, demonstrated in this video: https://www.youtube.com/watch?v=6svAIgEnFvw
What Turing had done previously provides clues as to the task he was addressing. In particular, in his ground-breaking work on computation in 1936 he did not provide a 'test for computation' by specifying how each computer should be tested by comparing it with a standard computer, or an 'average' sample of standard computers. (Before then, most computers were human, though there were some mechanical and electrical calculating and sorting devices, and before that there were Jacquard looms. See the Jacquard Loom Walkthrough in Stacey Harvey Brown's video).
Instead of proposing tests for whether computations are being performed, Turing did something much deeper in 1936. He produced an analysis of a class of competences exhibited by humans when doing arithmetical calculations or logical derivations -- by making successive sequences of marks on a surface, such as pencil marks on paper.
He then proposed an abstract schema, a generic specification, for a type of machine, now known as a Turing machine, that could be instantiated in infinitely many ways (so he was talking about properties of a class of machines, not of any one machine); and he demonstrated that the instantiations covered a very wide variety of sequences of manipulations of numerals and other symbols. By allowing the lengths of the sequences to be arbitrarily long (requiring a potentially infinite tape in the Turing machine) he showed that any such machine had the potential to perform any one of an infinite variety of such computations. In particular any known arithmetical calculation could be translated into a sequence of such operations that a turing machine could perform.
He also showed, surprisingly, that a subset of the instances, the Universal Turing machines, could each model all the (infinitely many) other TMs.
The capabilities emulated included a variety of different forms of symbolic reasoning that mathematicians and logicians had studied, which previously only humans could do.
That work required him to start with a precise specification of the requirements for what humans were able to do, so that he could prove mathematically that all the requirements could be satisfied in Universal Turing machines.
Later 'Universal' proved to be a misnomer because there are wider classes of types of computation (information processing) not covered, and Turing began to explore some of them. For example, when he died he had been working on Chemical mechanisms, reported in the 1952 paper on the chemical basis of morphogenesis and since chemical mechanisms use a mixture of discrete and continuous operations, they may be able to perform important tasks that a purely discrete machine cannot perform. Discrete operations can approximate continuous ones up to a point, but there are notorious unavoidable consequences of 'rounding errors' which in some cases can add up to huge errors. Moreover, Turing machines, and most of the systems about which theoretical computer science deals, are systems whose internal behaviour consists of a succession of discrete states. In contrast biological information processing systems may include continuous processes and generally include many different interacting processes that are not synchronised. Such possibilities have important philosophical implications that have not generally been understood by philosophers or cognitive scientists. See this discussion of "Virtual Machine Functionalism" http://www.cs.bham.ac.uk/research/projects/cogaff/misc/vm-functionalism.html
Similarly, if we want to understand human intelligence or something more general that includes human intelligence, we need a generic specification of a type of design that can be instantiated in many ways, with appropriate consequences, something like the human genome being instantiated in many new born babies who grow up in an enormously wide variety of environments and develop many different sorts of intelligence and competence and interests and achievements, etc. This idea was presented in Sloman (2007)and (2010) but in a way that made it hard for readers to understand. (That sort of general design is a special case of what can be produced by the far more general processes of biological evolution by natural selection operating on a sufficiently powerful medium of change, as discussed in the Meta-Morphogenesis project proposal.)
Simply trying to design one machine and then testing it may be fun, but it's really just "hacking", with little or no scientific or philosophical value, though it may have useful consequences, including educating future philosophers, scientists and and engineers about the technology -- and especially what does not work: a most important form of learning whose significance is under- appreciated by many teachers. (Good teachers, especially good mathematics teachers understand this.)
I find it very surprising that so many intelligent people take the (mythical) Turing Test project seriously as a way to specify what intelligence is, instead of attempting to specify a class of machines that can develop a wide variety of instantiations of the concept of intelligence. In his 1950 paper Turing, mistakenly in my view, hinted at a way of doing that by building a robot with a large empty memory except for some powerful general learning mechanisms, and then showing how such a robot could learn and develop with help from teachers. Many AI researchers have been seduced by similar ideas, but I think most of them fail to grasp the point made by John McCarthy in 1996, namely
Evolution solved a different problem than that of starting a baby with no a priori assumptions. ....... Instead of building babies as Cartesian philosophers taking nothing but their sensations for granted, evolution produced babies with innate prejudices that correspond to facts about the world and babies' positions in it. Learning starts from these prejudices. What is the world like, and what are these instinctive prejudices?I suspect that if Turing had continued the research begun in his 1952 paper on the Chemical basis of Morphogenesis he would have appreciated this point.
It should be obvious that the idea of ANY test for intelligence is silly, because there are so many varieties of intelligence, including e.g. weaver birds, though not all are equally intelligent: https://www.youtube.com/watch?v=6svAIgEnFvw
Human infants and 3 year old toddlers can grow up to be professors of quantum physics, composers, bricklayers, hurdlers, doctors, reporters, farmers, plumbers, parents, etc.
Yet all of them would perform poorly on most tests for intelligence in the first few years of life.
However some toddlers who would fail most intelligence tests seem to make mathematical discoveries unwittingly and most adults never notice: http://www.cs.bham.ac.uk/research/projects/cogaff/misc/toddler-theorems.html
The concept of intelligence, like many other philosophically puzzling concepts, exhibits something like the feature known to computer scientists as "parametric polymorphism" (probably discovered much earlier by mathematicians and given other labels). There's a brief tutorial on that here: http://www.cs.bham.ac.uk/research/projects/cogaff/misc/family-resemblance-vs-polymorphism.html