Do you recall the exact moment you recovered from an illness? Or the exact moment you lose a memory? Or even the exact moment you feel tired? The likely answer is no. It seems impossible to name the exact time you recovered from a fever. However, it seems as if there should be, posing the question: Is there an exact moment where our body changes in state whether it be physically or mentally?
Though this question seems more like a philosophical question, it may still be possible to answer it using mathematical reasoning considering its fascinatingly close relationship with math. In introductory calculus, there is a theorem that branches off the intermediate value theorem known as Bolzano’s theorem. This theorem states that for a continuous function f(x) on the interval [a, b], where f(a) and f(b) are of opposite signs, there exists a value c where f(c) = 0. Though this theorem seems quite basic, it can be used to model this seemingly thought-provoking question.
If we were to make the assumption that our mind can be modeled as a continuous function, then Bolzano’s theorem can be applied. With this mathematical lemma, we can take the previous illness example. If there was a point in time where one was sick and a point in time where they were fully recovered, by Bolzano’s theorem, there must also be a point where they recovered (x = c) and the value of the model changes sign. Once again, this is only true with the assumption that the human body can be modeled as a continuous function.
Even with this evidence, a solution to this question seems difficult to prove. For example, it is possible that we tend to recover from illnesses while asleep. If this were true, psychologically, we will not remember the exact moment we recovered. The inherent mathematical nature does not account for the psychological or physical aspects of each unique individual.
It is also important to consider the scenario with the opposite assumption that the mind cannot be modeled as a continuous function, but rather one with jumps. If this were true, it still seems possible to prove a solution using piecewise functions. In this case, it is important to take the limit as x approaches c rather than finding f(c) since the function contains jump continuities. Mathematically, if the limit does not exist but the individual limits from the positive and negative sides contain opposite signs, the value c would still represent the exact moment where the body recovered despite the jump continuities since the values represent a change of state.
There could be a wide range of utilities for this kind of problem. In the future, artificial intelligence may be able to predict when one recover from sickness, loses a memory, experiences mood swings, and more. Additionally, this problem can lead to further questions such as at what point do we begin to recover from an illness? With the knowledge of derivatives and concavity, this question can easily be answered given the fundamental concept provided by the previously proposed question.
