How would calculus be possible in a finitist axiom system? I am interested in learning a little more about finitism, currently about which I only know a few encyclopedic paragraphs.  
I know that during some time, some mathematicians like Kronecker thought that finitism is the right choice, so I guess that an important theory such as calculus would somehow be obtained in such an axiom system. 
So I have two questions along these lines:
1) Is there a construction of calculus within a finitist axiom system? If so, does it include  the important theorems that are taught to a first year student, (like the extreme value theorem, and fundamental theorem of calculus, with an appropriate definition of function)?  Are the proofs much more complicated than the standard calculus?
2) Could you give some fundemantal axioms, and define what a function means in such a system?  I am especially curious about constructing some real numbers with a definition like this Wikipedia example: http://en.wikipedia.org/wiki/Constructivism_%28mathematics%29#Example_from_real_analysis , but I don't know what a function would mean.  
 A: It is not completely clear for me what is the intended meaning of "a finitist axiom system". AFAIK, Kronecker was not a finitist, but rather a semi-intuitionist. Do you mean something similar to Primitive Recursive Arithmetic (PRA) (which is considered by some experts to correspond to Hilbert's finitism?). Do you consider first-order Peano Arithmetic (PA) as a finitist axiom system?                                         
If you mean a system that does not accept existence of infinite objects but only finite numbers/strings/..., then there are various approaches toward mathematical analysis, which would satisfy this condition. For example there is Markov/Russian School of constructivism, there are computability schools, ... . One important school which is completely compatible with classical mathematics is Bishop school, see books by Errett Bishop and Douglas Bridges.
A: The book 
Simpson, Stephen G.
Subsystems of second order arithmetic.
Perspectives in Logic. Cambridge University Press, Cambridge; Association for Symbolic Logic,  ISBN: 978-0-521-88439-6  MR2517689
will tell you far more than you want to know about this topic. It explains exactly what assumptions have to be added to a basic finitisitic system  to prove various common theorems of calculus. The idea is to start with a basic form of second order arithmetic equivalent in strength to primitive recursive arithmetic (which is what is sometimes meant by finitisitic mathematics) and show that theorems of calculus are equivalent over this weak system to various axioms (such as weak Konig's lemma). You can also check http://en.wikipedia.org/wiki/Reverse_mathematics for some details.
A: Digital computers "do" calculus.
Digital computers are always used to find numerical solutions to problems in analytic mathematics. Unless the solution is by rare chance a rational number (0 or 1 perhaps) that is the only way. “The demand for continuous description was encouraged by the fact that the mathematician claims to be able to indicate simple continuous descriptions of some of his simple mental constructions… Physical dependences can always be approximated by this simple kind of functions (the mathematician calls them ‘analytical’, which means something like ‘they can be analysed’). But to assume that physical dependence is of this simple type, is a bold epistemological step, and ptobably sn inadmissible step.” (Erwin Schrödinger, 1951, “Science and Humanism.)
