I see a lot of confusion between intelligence and knowledge. There's this cultural perception that a well-educated person is intelligent - and wise to boot. However, I've met plenty of people who are incredibly intelligent, yet know very little; or people who have memorised lots of information, but don't really understand it because they're not very intelligent.
So there's definitely a distinction between intelligence and knowledge, and I think there's also a third intellectual strength: wisdom. Note that I'm only analysing core intellectual strengths, too - skills like empathy are also important, but beyond the scope of this article; and thinks like mathematics are a matter of training the mind to do certain things quickly that can only be done if you're intelligent enough to master it in the first place, and have the knowledge - many skills, such as mathematics, being able to paint or to accurately outguess defenders and get a football into a goal, are also mental skills, but ones that are learnt by building upon the basic attributes of knowledge, intelligence, and wisdom.
(Players of role playing games will now recognise what got me thinking about all this in the first place)
Knowledge is stuff you can memorise directly from some external source. It might just be facts - the name of the first president of the United States, the melting point of lithium, who won the last World Cup.
Intelligence is a bit harder to define. It's about being able to perceive patterns, I think. An IQ test only really tests certain aspects of intelligence, but I think it's mainly pattern recognition and being able to hold complex structures in your short-term memory that you can then recognise the patterns in.
This can be likened to being able to gain a high-altitude view of things in your mind. The gifted artist looks at people's perceptions of the world, and sees them as all parts of a bigger picture; by spotting the bits of the picture that few people are seeing, and then managing to portray that to people (which is a skill), they manage to surprise and excite us, and to broaden our horizons by drawing our attention to things we had overlooked. The gifted mathematician looks at the properties of the Lambda calculus and of Hilbert systems and notices a shared pattern, and thus comes up with the Curry–Howard correspondence, and thereby realises profound facts about the processes of computation and reasoning that are driving the development of programming languages into the future.
And, yet, I have met people with great knowledge, and great intelligence who, nonetheless, are definitely fools.
There is a certain stereotype of the short-sighted baffled boffin; the kind who invents the nuclear weapon, then after it is used in anger, splutters "But... but... I thought it would be an end to warfare! I didn't think anybody would be so stupid as to use it!". The kind who sits there cranking out great work in their narrow field, yet without even being able to comprehend a reason why, yet alone wanting to.
And, also, I've met plenty of people who aren't intelligent, know very little, yet somehow manage to find their way peacefully and happily through life, bringing something undeniably good to everything they are involved with. They clearly have some positive core attribute, but what is it?
This is why I introduce the concept of wisdom. My hypothesis is that wise people have a grasp of certain fundamental patterns that underlie everything. Not the specific patterns that intelligence focuses on; more things like 'when two powerful forces are in opposition, things can slip and suddenly come out sideways'. This basic principle applies to physical forces, as well as to things like ideologies in opposition. They're patterns, and it's easy to mistake them for the rules that intelligence seeks to understand within complex systems; but there's some important differences. The patterns wisdom finds are broad. To find them, you need to look at a lot of things, rather than to look deeply at one. You can look at these things shallowly, and indeed, doing so can help you to spot the patterns without access to intelligence, by letting you "see the wood for the trees" rather than being tangled in details.
Also, they are weak correlations. They point to vague tendencies of systems, rather than to definite rules. They do not apply in all cases, even. They are more gut feelings, or intuitive hunches.
A truly great scientist combines all three attributes. They have knowledge of their field, the intelligence to understand it well enough to spot the rules, and wisdom that provides them with hunches; certain properties of a physical system may, based on past experience of such properties, lead the scientist to wonder if those properties will be conserved under rotation? Then they can use their knowledge and intelligence to do the maths and work it out, which may lead them to an interesting conclusion.
Similarly, the great artist has knowledge of the world, intelligence to spot interesting concepts - and wisdom that lets them guess how the viewers will react to their work. Nobody can know how the world will respond to something new, as we cannot know what's going on in other people's heads. No matter how intelligent you are, you'll never be able to reason the behaviour of millions of people. The best we can do is to draw on wisdom, to form hunches.
There is a stereotype of wisdom, but it's often confused with intelligence or knowledge. Sherlock Holmes is, perhaps, the stereotypical intellectual genius; his knowledge and intelligence are focussed on, but there's clearly wisdom as well. The purest expression of wisdom we find in popular media is the "wise old sage" stereotype. They're typically portrayed as old-fashioned and technophobic; they rarely exhibit vast stores of knowledge, or even great intelligence. They're usually an old-looking wispy-haired white male in a robe, spouting seemingly meaningless phrases that nonetheless turn out to be strangely insightful and useful.
This is a bit of a caricature, but with some vague connection to the reality, I think. A purely wise person would need to have been somewhat isolated from modern life in order to avoid ending up gaining knowledge, and the absence of knowledge would give them precious little complex mental structures to practise intelligence upon. But a long life would give an active mind time to figure out the deeper patterns, and build up wisdom. However, unless that wisdom didn't involve people much, I think a truly wise person would tend to have better communication skills than the traditional portrayal!