I'm planning to get double honours in CS and Pure Math. I'm confused what's the biggest benefit of Ph.D. will be for me? People say you learn to do research, but I can do research(as I have the appropriate background, CS and Math) sitting at home or in industry. Then what's the biggest benefit of Ph.D.? Why should I do Ph.D.?
EDIT1: The only thing that is stopping me from getting a PhD is time.
Read a few research papers published in highly regarded venues in your field of interest. Are you capable of producing that kind of work at this point in your training?
Do you know how to identify important and original research questions?
Do you know how to select an appropriate methodology for answering a particular research question?
Do you have a "toolbox" of methods you can apply to different research questions as appropriate?
Are you able to critically evaluate your own and others' work as experts in the field would?
If not: a PhD is an apprenticeship in which you learn how to do the things mentioned above under the guidance of an expert advisor, in an environment that is designed for learning those skills, hopefully with a salary that allows you to focus on improving your research abilities.
a workgroup and peer students to collaborate with.
a salary/stipend (hopefully) while you do it.
shiny title on your business card.
being a student, you can obtain discounts in conferences. It is often assumed that all "early-career participants" are students, because it's the most likely path.
in the industry, having a phd might unlock higher salaries.
if you want a career in academia, it's a sort of certification that you had some research training. Not having one is unusual. Of course, once you have a few journal papers published it starts to matter a lot less.
Hypothetically, you can prove theorems at home and submit them to journals and review journal papers from home. The industry is unlikely to support pure maths during your work time. But there are not so many people worldwide who are able to do research without verbal communication. The majority of researchers need a social working environment to some extent, at least every while and then. And get financed as well. (Submitting a paper without an institution name on it looks strange, btw.)
PhD is sometimes related to science (always in pure maths). PhD is something you cannot do without being associated to a university. What you get from a PhD degree has been mentioned in other answers.
Within a startup as you suggest you may hit a problem that requires research.
Without research training you may not have enough breadth or depth in your field to estimate the scope of the problem i.e whether the solution is well known in the field, the solution is at the edge of current knowledge or whether the solution is currently intractable.
The issue that bedevils all computer science / software engineering problems both in academia and from an entrepreunurial perspective is estimating how long it will take to solve a given problem and how much it will cost.
Typically the issue here is the less knowledge one has about a given problem domain the lower the time estimate given and the higher the potential for getting the estimate wrong. Development time estimation is an almost intractable problem in it's own right for various well known reasons, that the agile methodology has grown up to address.
So what are the benefits of a PhD: wider and deeper domain knowledge obtained from the literature review, a set of research tools, ability to formally present solutions, a measure of mentoring and having been able to tackle a problem in a relative "soft environment" of academia, together with the enhanced credibility of the qualification.
If of course the problem you want to solve is not at at or beyond the current research wavefront for your field, has a tractable solution you may be able to achieve a quick implementation and get to market in short order. Clearly only you can asses that in first instance. However if you have no commercial experience and no higher academic qualification where will you get your venture funding from?