There is also the notion that the language we use may actually structure the way in which we perceive the world (I guess it is arguable whether this equates to influencing "the kind of thoughts we can have", but on the face of it it seems even stronger.)
The classic example is the umpteen different Eskimo words for snow. Because they have all these different words, the theory goes, Eskimos are better able to distinguish correctly between bits of snow with different properties.
Another example that is cited concerns the visible spectrum. If you look at a spectrum, you're looking at something that varies continuously - there's a smooth increase in the frequency of the light from red to violet. But it often looks as though the spectrum is divided into bands of colours, with fuzzy transition areas between them. The full-blown Hypothesis applied to this would state that the locations of these bands that we see are strongly influenced by what colour words we have in our language. If we had a set of colour words that divided up the spectrum differently, we would actually see it differently.
Here's a table showing how colour words in English correspond to those in the Tiv language (spoken in Nigeria):
English | Tiv
-------------------
|\
green | \ pupu
________| \ (light)
| \
blue | \
________|ii \
| \
grey |(dark) \
| \
------------------
|
brown |
________|
|
red | nyian
________|
|
yellow |
________|__________
Obviously, the physiological basis of
colour perception is (usually) the same for speakers of both languages. But a case may still be made that the associations and groupings which feed into our perceptions are strongly influenced by the way our language divides things up, in this case, so that a Tiv speaker wouldn't see the same bands we seem to see when looking at a spectrum. Perhaps this is a case of "seeing as", similar to when we see a
duck-rabbit as a
duck. Perhaps someone fluent in both languages would be able to see both bandings, or groupings (though not at the same time) just as we can focus on the duck or the rabbit, when looking at the
duck-rabbit
Research has shown that cultures with more complex technologies and economies will generally tend to have a more variegated colour vocabulary.
The example of the snow words (also actually used by Sapir and Whorf) has been contested, but one can easily imagine other examples, for example words for different types of waves known by surfers. When the surfer looks at an upcoming wave, the effort she's put into learning how these behave, which is reflected in the more advanced vocabulary of the surfers, lets her draw more information out of the same perceptual 'input data'.
Or, when you deliver to your grandparent an intricate lecture on 18 forms of modern electronic music, distinguishing carefully between each, and are met with the blank statement that "it's all just noise, thud thud thud", this can be seen as an example of the Sapir-Whorf Hypothesis in action.
It seems that when a community of people make a concerted effort to investigate a feature of the environment, or are continually interacting with it, the extra knowledge they accumulate is filtered into their language, and in turn this becomes useful for preserving and communicating that knowledge, and hence the perceptual insights which accompany it.
Since thoughts that can't be expressed in any language are difficult to consider, in a scientific approach, probably the best way to evaluate the Sapir Whorf hypothesis is to consider thoughts which are easily expressed in some other language, but resist expression in our own.
Learning to think the thoughts of the speakers of this different language, or to make the equivalent perceptual distinctions, might be seen as equivalent in difficulty to learning that language (or the part of it concerning those particular thoughts), however. I think we can take it that the former is at least not more difficult than the latter.
So, the strong version of the Hypothesis, which implies we can't ever properly understand some meanings of a language originating in a different culture, takes an unduly negative view of the difficulties of learning fluency in such a language.
On this view, we could perhaps restate the weaker Hypothesis as the idea that at least for some terms from the different language, understanding thoughts expressed using these terms may be just as hard as actually learning to use the terms in the language in question, with whatever other learning of the language that that entails, and so expressing them may be equivalent in difficulty to teaching someone else how to use the terms (a task we can accomplish, however awkwardly, in our own language.) This is helpful, because it allows us to appreciate in greater depth that there is more to learning a language than just being able to remember the vocabulary and syntax. An even weaker version would be the uncontroversial idea that languages vary in their expressive strengths and weaknesses, and what is easily expressed in one language may take more effort in another one.
But the full-blown strong form will either need to provide some criterion for understanding another language (and it will then fail, because experience shows us that people may understand foreign languages arbitrarily well) or it puts itself beyond scientific evaluation, because it will be forced to claim that, despite meeting every practical test that's applied, a fluent speaker who's learned a different language somehow doesn't "really" understand it properly.
This choice between empirical failure and a mystical sense of "not understanding" is enough to show that the strong form (as originally proposed by Sapir and Whorf) is just wrong or vacuous.
Diagram and other information from:
http://anthro.palomar.edu/language/language_5.htm
"Snow-words" are thoroughly debunked at
http://www.stg.brown.edu/~sjd/mymusings/eskimo.html
and discussed here.
See also: colour terms in language.