From Norvig and Russel definitions of rationality:
- Thinking Rationally - The Greek philosopher Aristotle was one of the first to attempt to codify “right thinking,” that
is, irrefutable reasoning processes. His syllogisms provided patterns for argument structures
that always yielded correct conclusions when given correct premises—for example, “Socrates
is a man; all men are mortal; therefore, Socrates is mortal.” These laws of thought were
supposed to govern the operation of the mind; their study initiated the field called logic.
- Acting Rationally - An agent is just something that acts. Of course,
all computer programs do something, but computer agents are expected to do more: operate
autonomously, perceive their environment, persist over a prolonged time period, adapt to
change, and create and pursue goals.
A rational agent is one that acts so as to achieve the
best outcome or, when there is uncertainty, the best expected outcome.
In the “laws of thought” approach to AI, the emphasis was on correct inferences. Making
correct inferences is sometimes part of being a rational agent, because one way to act
rationally is to reason logically to the conclusion that a given action will achieve one’s goals
and then to act on that conclusion. On the other hand, correct inference is not all of rationality;
in some situations, there is no provably correct thing to do, but something must still be
done. There are also ways of acting rationally that cannot be said to involve inference. For
example, recoiling from a hot stove is a reflex action that is usually more successful than a
slower action taken after careful deliberation.
Clearly and also intuitively, rationality is well defined.
Intelligence as seen form mathematical and computational approach:
Intelligence can be the ability for an agent to make rational or irrational decisions, on a varying time frame and also choose the level of rationality (strictly in a computational sense). For example, I have exams and I want to watch TV, on a time frame of a week/month the rational decision would be to study so that I can enjoy the fruits of my labor which will be much more than the instantaneous pleasure of TV (also I can watch reruns). But for a time frame of an hour watching TV is definitely the most rewarding thing. So intelligence can be defined as the capability in deciding the length of time frame to be rational (what we call visionaries those who can see rewards far in the future).
Also as Game Theory or economics suggest, we can have different definitions for rationality depending on our needs. Thus, watching TV to gain knowledge might be more important to someone than studying, so effectively he has a different rationality function (arbitrary made-up term) to satisfy. Thus, intelligence can be deciding our rationality function, based on our needs and external experiences (learning in a nutshell). Also we may decide to minimize our rationality cost function or leave it in an intermediate state (which can be thought of as the minima for a different rationality cost function, thus only rationality functions are the true variable and not the intelligent decision to minimize it or not).
Lets take an example of bees (I am not sure whether this is the correct interpretation though): Bees can hardly be called intelligent (no foresight), but they are rational. They perform the task assigned to them with efficiency and toil (even though this does not reward the bee itself, it rewards the genes carried by the bees and ensures it survival through the queen - can be thought of as evolutionary coded intelligence). Bees perform these jobs in an apparently rational way, which has been decided by 1000's of years of evolution. Though bees individually are of hardly any importance, together they create a truly intelligent community - taking smart decisions and actions albeit their farsightedness is only within a smaller time frame compared to humans. Thus in common terms, it can be thought that rationality always almost leads to intelligence, but the same cannot be said for vice versa (as for intelligence you now have a choice for rationality function and you can choose not to satisfy it or choose an irrational function).
An important consequence of bees not being intelligent is that they are always performing rational actions as hard coded in their genes, which is causing the entire colony to behave in an intelligent and a very optimized energy efficient way (but there maybe better strategies, we can never know whether they follow the best strategy unless we account for all the variables).
TL;DR : Intelligence can be thought of the ability of an agent to choose the amount of rationality the agent wishes to satisfy. Using mathematics we can always find one or more completely rational methods of solving a problem with variables being time and the environment. But intelligent beings can add more variables like their experience, needs and motivation. Intelligent beings to some extent can single-handedly manipulate the external environment to suit their needs.
From psychological viewpoint:
Here are a few definitions of different types of intelligence and learning - Quite good and concise.
IQ - In science, the term intelligence typically refers to what we could call academic or cognitive intelligence. In their book on intelligence, professors Resing and Drenth (2007)* answer the question 'What is intelligence?' using the following definition: "The whole of cognitive or intellectual abilities required to obtain knowledge, and to use that knowledge in a good way to solve problems that have a well described goal and structure."
Intelligence - Wikipedia - Actually has some good definitions.
Intelligence Quotient - Wikipedia
Emotional Intelligence - Emotional intelligence is the ability to identify and manage your own emotions and the emotions of others. It is generally said to include three skills: emotional awareness; the ability to harness emotions and apply them to tasks like thinking and problem solving; and the ability to manage emotions, which includes regulating your own emotions and cheering up or calming down other people.
Emotional Intelligence - PsychCentral
Emotional Intelligence - Wikipedia
Hope this is of some insight!