I have a History and Philosophy of math course which covers everything from the beginning of math history, could anybody give me some IDEAS as to what topic I should write about?

Why don’t you do some research on historical mathematicians, like Archimedes.

Just make sure you don’t use Wiki as a research reference.

Fermat’s Last Theorem
Nicola Tesla
Fibonacci Sequences
The history of the calculus
Breaking the Enigma code and Alan Turing

This idea may be totally wrong for your paper, but what if you hypothisized a world in which it was discovered the entire system of math was somehow gotten wrong, and 5 really equals 2+2 or some other odd mathematical system?

What would have happened if humans had 8 fingers instead of 10.

how about a paper on the early life of einstein?

I’ve always wondered how the Romans got by without the concept of “0” in their numbering system. And how the hell did they teach addition, subtraction, multiplication and division using Roman numerals. “Let’s see, XLVII minus XL… I minus a nullity is I, I minus another nullity is another I, V minus another nullity is another V, and XL minus XL is nothing at all…”

I’ve also wondered just how Algebra was developed by Islamic scholars, and how that came to be accepted in Europe, when so little else of Moorish culture was much admired or replicated, and the people were driven out (of Europe).

Mathematics of ancient India – the ones who invented so called Arabic numbers
Impact of discovery of non-Euclidean geometry
Cult of Pythagoras

I really like @CyanoticWasp‘s suggestion of researching the formation of the Roman Numeral system.

the division by zero conundrum, of course

If it were up to me, I would do something about the concept of zero and the impact it had on computation. Alternately, I might look into the idea of infinities, the notion of pi, or the meaning of a two-dimensional plane. I don’t think I would take on the biggest mysteries of mathematics. Instead I’d look at the ones we take for granted.

One more suggestion:

Cantor’s work with infinity (nice introductory treatment of this in One,Two, Three…Infinity by George Gamow, also check link)

You can never go wrong with ancient Mexico. Aztecs, Mayans, those kind of people.
Other than that, I’d do the history on one of the two most unique shapes: the triangle, or the circle.

(But remember that “unique.” like “pregnant,” is not quantifiable and stands alone.)

@Jeruba‘s idea is good. Infinity is fascinating as all hell.

You can start with cantor and go from there.

why is dividing 0 by 0 undefined and not 1
when does 1 not equal 1