<p>The times have passed when people were doing science lying on sofas and philosophizing about the structure of the Universe. Nowadays, people judge you by the things you have actually done, in addition to the words you have spoken. As lizzardfire pointed out, if you do not have anything tangible on your resume to support your interest in math/science, the admissions committee will have a hard time choosing your application over someone’s with clearly demonstrated and supported passion for math/science. </p>
<p>I am not trying to prevent you from applying to Caltech: if you think you can convince the admission people that you are capable of handling hard coursework, despite not actually being exposed to any of that (if I understand correctly from your posts elsewhere, your school does not offer any AP classes? Correct me if I am wrong), then apply by all means. However, another perfectly acceptable course of action would be to go to a bigger school with many non-science oriented departments. That way, if you find out you are not into academia, you would be able to switch and not suffer 3-4 years. And if you are still passionate about science when you graduate, you would then actually be able to support your passion when applying to grad schools.</p>
<p>Remember, if you want to go into science/math, your undergraduate institution serves merely as means of getting into grad school.</p>
<p>I might have gone completely off-topic, but I really wanted to say it. So there :)</p>