Can someone honestly say a smartphone like this is not a computer? If It was slightly bigger like an iPAD would it be a computer?
Someone can "honestly" claim a smartphone is not a computer. Afterall, it's not inherently dishonest to be completely wrong. ;)
An iPad is also a "computer". As a computational device, iPads are very very similar to iPhones. They are fabricated with different use cases in mind, but architecturally they're profoundly similar.
The root of this discussion is entirely a matter of definition.
The term "Computer" has become extremely ambiguous in the modern context. In a historical sense, a "Computer" was actually a person who would solve math problems. Obviously, no one here is concerned with that.
The Wikipedia article on "Computers" has a fairly accurate strict definition of 'Computer':
A computer is a programmable machine designed to sequentially and
automatically carry out a sequence of arithmetic or logical
operations. The particular sequence of operations can be changed
readily, allowing the computer to solve more than one kind of problem.
Mr. Alan Turing is generally considered the father of modern computer science due to his creation of the Turing Machine, which provided the conceptual framework upon which all modern computers are based. A Smartphone is as much a turing machine implementation as an x86 derived device.
Smartphones have become characterized by their general computing abilities. The emergence of HTML 5 will further close the distinguishing gaps between smartphones and desktop/notebook machines.
Currently, most people tend to think of "Computers" as x86 machines running a "Desktop" Operating System (Win XP, Apple's OSX, Ubuntu Linux, etc). Granted, they may generally not realize that's what they have in mind.
The use of BIOS/EFI, as a hardware abstraction interface for software is a major architectural property that leaves a notable distinction between how operating environments operate on an x86 machine versus an ARM machine (and related machines often implemented in the Smartphone space). I think it's fair to speculate this distinction will be resolved within the next several years.
By strict definition "smartphones", certainly are computers. And they are extremely comparable, across the range of machines that constitute "Computers", to x86 machines (as used for Desktop and Laptop computing devices). However, in the context that the general population applies the term "Computer" (by which they mean something running a desktop operating system), a smartphone presently falls outside of the context of the term.
As Smartphones continue to develop, we will very soon see this distinction evaporate. With the emergence of cloud computing, improved network communications, HTML 5 and more efficient hardware (in all dimensions), the tasks that we use Desktop Computers for will expand well beyond such devices and be driven by a wider range of devices - all of which will always fall within the strict definition of what a Computer fundamentally is.
In closing, none of this really matters anyway. It's entirely a matter of definition. If enough people band together and decide the meaning of a term, then the rest tend to accept that definition.