How do mathematicians know when to stop?

You have pointed out that in a single time (not even of relevance to our society), a single development in mathematics would not have improved things. This in no way serves to back up your claim that almost all maths is unbeneficial. I suspect that the reason you are struggling to back your claim up is that actually it is false.

How about "Number theory of Roman numerals". Do you think it is worth being developed?

But any branch of mathematics is just a human invention. Since we could (in theory) invent infinitely many branches of mathematics, how do we decide which branch is more worthy of study than the other? It has to be based on the usefulness.

Suppose we were living in primitive times and had just invented the decimal system for counting. Inventing the binary system then would not benefit society.

Number theory does not depend upon notation - the theory in Roman Numerals is identical to the theory in decimal. It has already been developed.

Perhaps the reason you are asking these questions is that you don't actually know anything about Maths?

At risk of repeating myself, I would also like to point out that you have simply offered another example. No individual example or set of examples will suffice to demonstrate that almost all Maths is useless to us. In order to back up the point you have made, you need to say something about the nature of Mathematics rather than the nature of narrow parts of Mathematics.

Inventions are invented. Science (eg: Maths) is discovered. Mathematicians aren't just pulling numbers out of their arse and acting all smart, they discover abstract mathematical concepts that describe the world around us. So yes, learning about maths is learning about life.

What I meant was being able to do all sorts of calculations using Roman numerals.

Please don't resort to personal insults.

Just like in mathematics, we see if something is true for a few special cases and then try to prove it in general.

Regardless, such an oddly specific example tells us nothing about the nature of Mathematics. Furthermore, we can perform calculations in numerals by converting the numbers to decimal, performing our calculation, then converting them back. I have just created a complete theory of your problem, you are focussing on something of a triviality.

I am not resorting to personal insults. It is increasingly clear that you know nothing about mathematics and if this is the case then it is a waste of both our times to discuss Mathematics with each other. You should not try to comment on issues that you don't know anything about. At the very least you should try to take a brief amount of time to understand them. Whether you understand how Maths works is very significant to discussion with you in this thread. If we find that you know nothing about Maths then your questions can be answered simply by you learning more about Maths.

Finally, that is not what we do in Maths. It is simply not. That is not how Maths is done. Specific cases tell us nothing. Furthermore, you have incorrectly used the word 'special'. It is true that sometimes in Maths we will take special cases and generalise, but the special cases are almost always themselves general in some way, and the examples you have chosen are actually not in any way special. They are oddly random cases that are nothing to do with the nature of Maths and everything to do with fitting the specific criteria you have selected to try to prove your point. Again, and for the last time, I will tell you that if you cannot say something about the nature of Maths and if all you are capable of doing is giving (stupid) examples then you have already defeated your own argument.

Let me turn the question around: Can you prove that most (if not all) branches of mathematics that can be developed will benefit society?

Easy, by induction - the scientific kind and not the mathematical kind. Most if not all branches of mathematics have consistently proved useful in the past, therefore it is reasonable to expect that future developments will also be useful.

That's a circular argument because most branches of mathematics have been developed in response to scientific needs. For example, calculus was invented by Newton to solve mechanics problems.

Do you have a similar problem with, say, music or other art forms?

What's the difference? Music is much less of a benefit to society than maths is.

Unlike complicated maths, music has therapeutic value.

Doing maths is very therapeutic in itself, but even disregarding that, are you implying that those aspects of music outweigh the myriad benefits mathematics has had on society?

Obviously not. But I think any kind of music is more "beneficial" than useless mathematics.

The world needs to be shot of people such as yourself.

Read this and re-evaluate your existence:
http://thebestpageintheuniverse.net/c.cgi?u=math

Regards,
The somewhat more intelligent half of the human race.

FYI, I'm a student of mathematics (see some of my posts way back in '08 and '09) and am very passionate about the subject. The reason I do it is for the intellectual challenge. The question in the OP is just something I was musing upon.

FYI, I'm a student of mathematics (see some of my posts way back in '08 and '09) and am very passionate about the subject. The reason I do it is for the intellectual challenge. The question in the OP is just something I was musing upon.

Seems like you have answered your own question there. Unless of course you think that the intellectual challenge is an acceptable motivation only for you, but not for other people?