To me, as an engineer and inventor, the whole discussion about abstract ideas in relation to processors makes little sense. A computer is a machine, yet there is an ongoing trend to “anthropomorphize” computers. That is: functions that are performed by humans are said to be able to be performed by computers. Anyone who has done any serious programming knows that is not how it works. Let me explain.
Steps that humans can do almost mindlessly, for instance changing paragraph numbers in a text, may be excruciatingly difficult as programming steps. That is because computers are machines that process signals that follow very strict and inflexible routines that have no concept of what the signals mean.
Processors that are used to perform calculations are switching devices. Arithmetic is, of course, a human activity and outside humankind radix-n arithmetic does not exist. The calculating machine itself is a hard-core piece of equipment that will do whatever it is supposed to do, whatever it is told to do, no matter if a user sees, uses, describes or even understands its results or workings.
A calculator does not “calculate,” at least not in the way that humans mean when they use that term. Instead, a calculator merely switches. There is no concept or awareness of a number in a binary digital calculator. A digital calculator is merely a switching machine that processes discrete signals that place the machine in defined states. The meaning of signals is provided by humans assigning a number to a series of binary signals as representing a number, for instance via an icon or character on a key or as a character displayed on a screen. No matter if we paint a 3 or a 5 on a key, to a processor activating that key is merely entering a series of pre-defined signals.
The history of computing demonstrates how extraordinarily difficult it has been to develop a device that performs operations similar to arithmetic. Most of the early calculating devices even in electronic computers were counting or tabulating devices, not logic devices, and were usually realized with rotating wheels or strips of material or pulse counters or other forms of proportionally counting devices. Those machines cannot be characterized as “abstract ideas.” They overwhelm by their complex physical structure.
Modern calculating machines, including computers, use switching devices that are described by binary logic or switching tables also called truth tables that define Boolean functions. One of the first discrete calculating circuits described by Boolean algebra was presented by Claude Shannon of Information Theory fame in his MIT M.S. thesis in 1937. The calculating circuit of Shannon is a binary adder realized by electromechanical relays and is described by a Boolean algebra expression.  Shannon does not use truth tables, but provides “postulates” that correspond to a truth table.
The “logic” of Shannon is a description of the behavior of relays. The 0 and 1 are symbolic representations of what Shannon calls the “hinderance” of a switching circuit. If a relay circuit is closed it has zero “hinderance” or zero impedance and is assigned the symbol 0. If a relay circuit is open it is said to have full “hinderance” or infinite impedance and is assigned the symbol 1. The 0 and 1 are not real values of signals but are symbols assigned by a human to a state of a device. Shannon represents the “hinderances” as algebraic variables and describes the physical switching behavior of relays of the “hinderances” with “postulates” or algebraic expressions. The “postulates” of Shannon are thus a descriptive tool of a physical behavior of electrical relays. Shannon concludes that the calculus for switching devices has an equivalence with the calculus of propositions originated by George Boole.
The Boolean algebra is a (happy) coincidence to the physical structure and behavior of relays. The relay structure does not perform logic, but merely switches between states of “hinderance.” By using certain representative tools, a functional behavior of relays can be described by Boolean algebra. To say that the relays network “performs” logic is the tail wagging the dog and is wrong.
In a similar way a feedback op-amp circuit with a capacitor at an input can be described as a differentiator and with a capacitor in the feedback loop as an integrator. However, neither of these circuits actually performs calculus, which is strictly a human activity. In that same sense, a calculator does not actually calculate, nor does it perform logic.
The switching circuit realized from the logic expressions in Shannon is still a physical machine and not an abstract idea expressed in a computer. In a similar way an electrical filter is a physical device of which a performance is described by a complex transfer function, and is not a device that performs the transfer function.
All computing machines and processors are switching machines that process signals. They are not abstract ideas and they do not perform an abstract idea. The abstract idea is read into the machine by humans who find similarity to human activity by translating signals generated by a machine into a shape that humans understand, like symbols, text, images and sound.
 I apply a bit of a shortcut in technology from Shannon to Boolean logic described adders in modern computers. Most adders in early computers, such as the ENIAC, used pulse counters and/or flip-flop counters. The use of binary “level” devices such as XOR and AND gates in calculators was a technologically different and later form of switching.