These are the fundamentals of cognitive power:
Analysis means being able to swap concepts and ideas (structures of concepts) in and out of working memory with absolute fidelity. Someone lacking analysis (over half of the population) suffers a dyslexia of sorts, the concepts in their minds swim and distort on their own. There exists a direct test for this phenomenon. The consequence of analysis is that a person can build up large conceptual models in their conscious mind, ones that far exceed the capacity of working memory, and reason about (comprehend) those models by following along their chains. This is what is necessary to master mathematics, formal logic, programming, and any other purely formal system. As a result, lack of analysis leads to thinking based on magic instead of logic.
Synthesis means the spontaneous generation of new original abstract concepts. This is done by multidimensional decomposition in the subconscious. Someone lacking synthesis (over 90% of the population) can stare at a dataset and distinctions will not spontaneously occur to them in a process commonly known as "intuition". Someone high in synthesis will have distinctions enter their mind entirely uninvited and so will possess concepts (understanding) of nearly everything they are familiar with. They will talk to people and suddenly realize, without any prompting or priming or even the vocabulary to articulate what it is they realize, that there are two distinct groups of people they are talking to. When talking to someone, they will spontaneously categorize that person as X and not Y and Z if they know those categories. They will jump meta-levels and question someone's motivations, goals and values. They will judge not just right and wrong (whether something accomplishes a goal) but good and evil (whether the goal is valuable). All spontaneously and without any need for conscious thought or even a vocabulary.
Intellectualism means that one believes ideas have value independent of any application. Someone lacking intellectualism will use 'philosophical' as a term of abuse, as synonymous with 'not worth thinking about'. Or more subtly they will believe that all ideas have an application even if this is unknown presently. This is blatantly false since the infinity of math certainly can't be squeezed into a finite universe. These people believe that all ideas have application because they don't want to bother thinking about the opposite because this opposite view has no practical value. When hostile, anti-intellectuals will demand to know what you have done, what you have accomplished, what status you have achieved, how much money you've been given, that gives you the authority to think. Because of course, thinking has no value except as an adjunct to doing. In extreme cases, anti-intellectuals do not believe that abstractions even exist on their own terms. They do not believe that mathematics is real. Only that it is useful as a "representation".
Intelligence means only memorization ability. It refers both to the size of working memory and the rate at which things are transferred from working memory to long-term memory (ie, are memorized). There is some overlap between intelligence and analysis.
Intelligence varies but its variation is consigned to the extremes and not the middle so it doesn't seem to vary that much in practice. IQ tests measure knowledge linearly but knowledge is acquired logarithmically so intelligence must be exponential. Yet, it doesn't really matter because paper notes for storage and now computer searches for retrieval have made memorization overwhelmingly obsolete. And further software can make it more obsolete still.
Synthesis and analysis vary by many orders of magnitude. I estimate that synthesis varies by 3 orders of magnitude from high functioning autistics to creative geniuses. And the average functioning person is very close to an autistic as far as synthesis is concerned.
Analysis may vary even more and all of the variation occurs smack in the middle, neatly separating the general population between the haves and the have nots, between rationals and magical thinkers.
Intellectualism? Very difficult to tell, or to detect for that matter, so I don't know.
Analysis and synthesis are somewhat independent of each other. Intelligence is entirely independent of synthesis. As for the rest, it's quite difficult to tell.
Which traits are most valuable? I would say analysis then synthesis then intellectualism. Of course, combinations of traits trump individual traits so that synthesis + intellectualism (philosophers judging good and evil) seems about as valuable as analysis alone (engineers creating bridges and landmines). And analysis + synthesis far surpasses analysis alone. Intelligence is least important for raw cognitive ability. High intelligence and nothing else means you'll get nowhere faster.
Doctors, lawyers and medical researchers are overwhelmingly high in intelligence and low in analysis and creativity. This is because their subjects (biology and the law) are ad hoc and artificial. Since it's impossible to reason about biology or to intuit the inner workings of biological systems, people with high analysis or creativity have an atavistic repulsion against it.
Philosophers are overwhelmingly high in intellectualism & creativity and low in analysis. With very few exceptions, philosophers are incapable of dealing with philosophical matters on a purely formal basis. In fact, symbolic logic is effectively beyond the mastery of most philosophers.
This is why despite philosophers having originated symbolic logic, the subject had to be adopted out by mathematicians. It is also why you can find philosopher professors teaching symbolic logic and even writing books on the subject in an entirely incoherent and disorganized manner. They do not comprehend logic and students of philosophy frequently contradict themselves in the most blatant and appalling manner.
This is also why philosophers obsess about nonsensical concepts and gibberish distinctions. And it is why the most ardent nonsense and gibberish (eg, Rawls, Dennet) confers upon philosophers high stature and respect instead of derision and scorn. While the very few philosophers capable of analysis, giants such as Quine, are accorded so little stature.
Philosophers don't use logic to analyze anything because they are incapable of analysis. Instead they substitute resonance and repetition, things that will lead the reader to synthesize concepts and become comfortable with them. John Rawls says as much in the opening chapters of his book, moral reasoning to him is not a matter of formal reasoning but a matter of altering ideas until they fit" against each other.
It is also why philosophers are so obsessed with historico-linguistic garbage. The words verbal diarrhea and intellectual sewer come to mind. It is complete nonsense (who cares what Kant or Descartes thought, who cares what Rousseau and Hobbes said, when they were wrong) but there sure is a lot of it.
Artists and designers have high synthesis, low intellectualism, and varying amounts of analysis.
The difference between an artist and a designer is that artists are self-centered -- they express their own thoughts and feelings, not someone else's. Method acting relies on the ability to feel what you wish to act out. And musicians that play mechanically without expressing their own feelings are simply bad artists.
Different kinds of designers have different secondary traits. A fashion designer doesn't have the analysis of a graphics designer or industrial designer. And only systems designers are intellectuals. Yes there are some, but they are few.
Engineers and exact scientists are overwhelming analytical and non-synthetic. Theoretical physicists are intellectual. Experimental physicists are anti-intellectual. A very small number of theoretical physicists are synthetic.
Anthropologists are overwhelmingly magical thinkers incapable of logic. The incoherence and blatant illogic of their field's central assumptions proves it. "Cultural beliefs make sense in the cultures that spawn them"? You might as well say that psychotic beliefs make sense to the psychotics that hold them. It is just as true and just as meaningless. And yet this is the central tenet of anthropology. Or at least, this is what is taught as the central tenet of anthropology. The real foundation of anthropology is this: love the little psychotic bastards.
Programmers range all over, from non-analytical coders to analytical programmers to intellectual developers to synthetic designers to intellectual and synthetic systems designers.
Autistics are lacking in the residual concrete synthesis that every normally functioning person possesses. Normal people generate concrete concepts of: angry, happy, sad, distant, intimate, so on and so forth. This is all done very early in life and is more or less hardwired in the most primitive parts of the brain which is why they are not abstractions per se. Austistics lack this. Whether concrete synthesis is entirely absent in their brain or simply inaccessible is immaterial -- it doesn't work, period. However, I have met at least one certified abstract synthetic person who claimed to be autistic. So assuming they weren't lying, it may be that abstract and concrete synthesis are somewhat independent.