Why do physicists believe that particles are pointlike? String theory gives physicists reason to believe that particles are 1-dimensional strings because the theory has a purpose - unifying gravity with the gauge theories. 
So why is it that it's popular belief that particles are 0-dimensional points? Was there ever a proposed theory of them being like this? And why? 
What reason do physicists have to believe that particles are 0-dimensional points as opposed to 1-dimensional strings?
 A: Occam's razor suggests that in the simplest explanation is the most probable. Physicists will assume that elementary particles are point-like, until they have evidence to suggest otherwise.
A: Pointlike and point are entirely different concepts. The planet Jupiter is pointlike to likely 6 or more decimals of precision when studying the dynamical evolution of the Solar system. Does not mean that Jupiter is a point!  Just because something behaves pointlike has always meant that we just don't know enough yet. String theory is one theory about a deeper level, there are others. 
So I don't think that many physicists actually think that the electron is a point. Its just that you don't need to worry about any structure when working at piddling energies of a 100GeV or less...
A: Your question is based on an assumption that the vacuum is empty, and matters 
(including particles) are things we placed in the empty vacuum.
But the Casimir effect shows that the vacuum is not empty but a dynamical medium.
This led to an emergence point of view of elementary particles: they are quantized collective motions of the vacuum medium.
In one approach, we regard the vacuum as a collection of qubits.
(ie the space is a qubit ocean.) If those qubit form a string-net liquid,
then the  quantized collective motions of qubits can give rise to photons, electrons, etc.
So
elementary particles, like photons and electrons, are not elementary in the sense that there are underlying theories, such as the quantum qubit model on lattice, from which they can be derived as an effective approximation (see for example our paper arXiv:hep-th/0302201). 
Under such an emergence picture, if we examine 
elementary particles closely, we see the qubits that form the whole space.
The question weather  elementary particles are point-like or not do not make sense within the emergence approach.
The string-net condensation provides a unified origin for gauge interactions and Fermi statistics:
Both elementary gauge bosons (such as photons, gluons) and elementary fermions (such as electrons, quarks) can emerge as quasi-particles in a quantum spin model on lattice
if the  quantum spin model has a "string-net condensed state" 
as its ground state. An comparison between the string-net approach and the
superstring approach can be found here.
There is a falsifiable prediction  from the  string-net theory:
all fermions (elementary or composite) must carry gauge charges (see our paper cond-mat/0302460).
The standard model contain composite fermions that are neutral for $U(1)\times SU(2)\times SU(3)$ gauge theory. So according to the string-net theory, the
standard model is incomplete. The correct model should contain extra gauge theory, such as a $Z_2$ gauge theory. So the string-net theory predicts extra
discrete gauge theory and new cosmic strings associated with the new discrete gauge theory.
The emergence approach may also produce (linear) quantum gravity from quantum spin models (see our paper arXiv:0907.1203).
However, the emergence approach (such as the string-net theory), so far, fail to produce the chiral coupling between the $SU(2)$ weak interaction and the fermions.
A: Elementary particles don't really have a shape or a size, these are emergent qualities that stem from interactions between particles. In quantum physics a particle is represented by its quantum state, and if you want to describe that in space you get a wave function which tells us how much of the particle is present at any given point in space. Because there is no theoretical limit for the size of the spatial region where the wavefunction is nonzero you cannot assign a finite size to the particle. You can imagine the particle either as infinitely small (i.e. point like), or just say that the concept of size is not very meaningful.
A: In the standard model (as in all traditional relativistic quantum field theory), particles are pointlike. All experimentally available facts about microphysics seem to be consistent with the standard model. This is the (fully sufficient) reason for believing that particles in Nature are pointlike.
Pointlike is a technical term that refers to the fact that in the standard model, the Lagrangian is a function of fields at the same point (rather than of integrals over fields in some small neighborhood of this point, described by form factors specifying the ''form'' of the particle). 
The main reason why, nevertheless, many physicists speculate that (at much higher resolution) particles might not be pointlike is that there is no known way how to harmonize quantum field theory with gravitational forces, whereas string theory (where particles are stringlike) seems to offer a potential way of doing so. Nobody knows to which extent these speculations will turn out to be correct.
A: I had been preparing an answer to the question made duplicate .
there were more questions than in the question above, so I am answering here:

Elementary particles are like mathematical points?

In the standard model of physics they are assumed so.

Does make sense in quantum mechanics and standard model think this way?

This is the table of  elementary particles of the standard model of particle physics.

All matter is a composite of these particles, and yes, they are modeled as point particles. Yes, the mathematical model of the standard model has been validated over and over again, and its quantum mechanics based predictions are fulfilled, as recently as the discovery of the Higgs.

Is is true that two elementary particles are indistinguible?

No, this is wrong  as a general statement. Different types of particles (electrons, quarks ...) are characterized by different quantum numbers and are distinguishable. 
The same type of  elementary particles are experimentally indistinguishable, two electrons are interchangeable,  except  by their quantum numbers in specific boundary conditions. In general one cannot attach an identity card on an elementary particle.
A: Canonical particles possess an actual radius of hardness, which is determined by the Compton's expression $\lambda_\text{Compton} = \frac{h}{mc}$. One can read more about it here 
http://inerton.wikidot.com/canonical-particle
Why do particle physicists speculate about point-like particles? It seems to me this is associated with their education; namely, their teachers told them wrong things and implanted an abstract tunnel vision approach to the reality. It is a pity but this is the truth. 
