At a young age I thought that solving algebraic equations on paper was lame. I much prefer working things out mentally and simply put down the value for $x$ in the second step.

It is elegant and it made a socially awkward kid feel competent. So I gradually developed the inclination to do everything in my mind. There is something magical about the moment when you arrive at the final answer, the moment when you can let your mind be free as you no longer bear the responsibility to cache the data in your random-access memory. But we are humans after all so we don’t really have RAM and when things get really complicated, we would have to resort to writing the working down. For example, there is no way I can solve the equation above without pen and paper. It is too complex. (Interestingly, even Wolfram Alpha can’t quite solve it, likely a consequence of Abel–Ruffini theorem: the best it can do is to use some approximation algorithm. Perhaps the most efficient approach to obtain a solution for $x$ is through trial and error - checking integers that are some powers of 2 and for which $\sqrt{\frac{x}{4}}$ returns a whole number - but that requires one to have access to the information that $x$ has an integer solution.)

As humans we can retrieve stored information more efficiently when we receive certain sensory inputs (often in the form of EM radiation or air vibration) that we associated the information with previously. Suppose a person is given the task to compute the product of two random 4 digit numbers. Let’s say the person is not well trained in the art of multiplication, it would be a lot more difficult for her or him to do it mentally than doing it on a piece of scrap paper. Even having the two 4 digit numbers displayed in front of her or him as she or he works on the problem would ease things out slightly, comparing to merely hearing the numbers for once and relying on memories.

We have already been accustomed to interpreting those Arabic symbols as numbers, a concept in which morphisms can happen. So when we do the morphism (or transformation of abstract entities according to pre-defined rules preserving internal structure) in our mind, it is much easier to recall the information that would be necessary at the next phase by looking at the symbols we have written, than to assign some part of the brain to keep track of the information. It is not that the later cannot be done when the amount of information reaches a certain threshold: Kim Peek had clearly demonstrated that such threshold does not quite exist. The human brain is a powerful computational device. It’d be more logical to conjecture that in general it tends not to do so as a trade-off to achieve better performance in other areas (e.g. abstract thinking) due to its limited capacity. That is why I believe the ability to forget plays a much more important role than the ability to memorise.

There are clearly evolutionary advantages in relying on sensory stimuli to retrieve stored information rather than employing other mechanism to be able to retrieve them on a whim. Perhaps this is how the notion of semantics arises at a higher abstraction layer.