How far does a student need to go to school in order to be recognized as a scientist?
Recognized by whom?
I personally recognize anyone who is genuinely interested in science and actively investigates the universe to be a scientist no matter what their level of education.
Typically you can be recognized as a scientist by society in general (especialy the media) based more on your profession than by your education. If you are actually working on a science project the media will probably interview you as a "scientist" without even asking for your educational credentials. Although, typically to be hired to work on a scientific project most companies would require either a BS degree in science, or at least some history of involvement in scientific endeavors.
The more education the better of course. However, even people who have a Ph.D. in a science aren't always thought of as "scientists". This is especially true if they take a mundane job where they are doing more engineering work than scientific research. Although, many engineers might also consider themselves to be scientists especially if they are working on projects that are at the cutting edge of technology.
Typically, if you want to be considered for research grants the Ph.D. is the way to go. Or, you could self-study, save the college tuition, and then use that as a personal grant to yourself.
Finally, if you make a profound scientific discovery you are automatically consider to be a famous scientist regardless of your educational background.
In all fairness, I feel obliged to add the following. While it is entirely possible to self-study, it is also very unlikely to lead to the proper understanding required to advance the field. From here on, I will be talking about physics since I am not qualified to evaluate the ease of research in other fields; the situation is unlikely to be significantly better. Other fields, such as archeology and related, may depend on dumb luck as well; the recent discovery of hominid fossiles that may change the picture of human evolution is a case in point.
Making a "profound scientific discovery" is incredibly difficult. The breaking information in the field is not available in textbooks; it is found in journal articles and conferences that are not open to the public. A discovery without a profound knowledge of the field is a virtual impossibility, since physics has advanced far beyond F=ma. Who can possibly dream up the electroweak Lagrangian without any prior knowledge? The best way to keep on top is by actively pursuing research, either with a university or a research laboratory; both private and public labs are highly competent. To secure such a position, the only path is through undergraduate education and graduate school.
Another downside is that all scientific advances that can be done with everyday resources have already been done. In physics, the research is now being made in labs with very specialized equipment; the costs of doing particle physics are astronomical. Even theorists need large-scale computational power that cannot be provided by desktop PCs. Access to such resources is again the exclusive privilege of those employed in research labs or universities.
In light of this, many people who seek to make contributions in physics end up on a very different and unfortunate path. Without the education or resources to perform research, they study as best they can but inevitably do not achieve a profound grasp of the field, or they find some aspect of a theory too repulsive, counterintuitive or otherwise objectionable to accept. Then they try to come up with their own theories based on a limited view of what the original theory describes or does not describe. To quote someone's words from this site, imagination without knowledge is ignorance waiting to happen. A look through the Theory Development archives should clearly indicate what happens to home-baked theories.
In a nutshell: saving the college tuition is extremely dangerous, since odds are either you won't get the equivalent on your own, and even if you do, no one will be willing to take a chance on you. I am not saying this is impossible, but I am very skeptical of the odds of success.
I argue that "Scientist" is a state of mind, independent of whether you are professionally a scientist.
The Scientist is not afraid of doubt, or ignorance. He knows what he knows and what he does not know. He knows to doubt the veracity of any unsupported claim. He knows to keep asking, why and how, methodically. He knows to never accept something based on arbitrary assumptions as a final answer, only as temporary hypothesis. He knows that what he thinks is true is probably wrong, and his next revision will be better, but still, fundamentally wrong. But he isok with this,because he is getting closer to the right answer.
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