Could a Man Fully Understand Every Single Part of His Mind? – a quick mathematical argument

It has been an endless journey in humanity throughout the whole history to find out about who actually we are and why on earth we are here. Many scholars argue for the best theory regarding what human is and what it means to be a human. One of its key point to understand the issue is another circular question: can someone fully understand about himself? To make it fancier, we could remake the wording: is the mind able to understand how the mind works?

It seems that we never heard somebody claiming that he or she literally knows everything. Even the wisest and the most intelligent person we’ve ever known sometimes looks unsure about something, in particular about him/herself. At this point, we will just agree that “Noone knows everything”. In most of the time, people will just agree with the statement since we have never learned that there is someone knowing everything in every knowledge. In some sense, it feels counterintuitive to fight against the statement since we face a wildy growing size of knowledge today (some people like to call it the age of exponential rate of information). However, as our scientific understand is also growing, could it be possible that the speed of human mind understanding may someday outperform the growth rate of information (and thus yielding a super human who could know everything)? Although we may be quick to agree with the first statement, the latter question is definitely not a trivial one. Never experiencing a fire in your house does not prove that your house is fireproof. Although we might have no history of human’s understanding of all things, it is still legitimate to question the possibility of a person who will know everything.

The purpose of this article is to give some mathematical arguments which assure that we will never ever meet any omniscient human anywhere anytime. Basing on two basic assumptions, we will prove that no one is ever able to understand all things. In fact, we prove the stronger one: no human could fully understand how his mind work. The first assumption is that human’s lifetime is finite. The second will be introduced after we define several terminologies. We will prove this by contradiction. Let we pick any human, say human X, altogether with his mind and lifetime. Now let X be the first human who knows everything. Let us take any subject of human’s understanding, say the knowledge k_0: and let T_0 be the time needed for X to understand k_0. Since X knows everything, he also need to understand how his mind is able to understand k_0. Let us denote this knowledge as knowledge k_1 and we iterate this by the following recursive definition: for i\ge 1, knowledge k_i is the knowledge in which X understand k_{i-1}. Next, we define also for i\ge 1T_i as the time needed for X to understand knowledge k_i. Here we assert our second assumption: 0< T_0 < T_1 < \dots < T_i < \dots . The assumption may need to be verified more clearly, but I think it should appear in a very natural way. We see for j>i, knowledge k_j must be more complicated than k_i and therefore needs more time to learn (since k_j includes k_i in some sense, it follows that the assumption feels very natural). Now we investigate the time needed for X to understand this chain of knowledges:

\sum\limits_{i=0}^\infty T_i

However, we can observe quickly that the real sequence (T_i)_{i\ge 0} is monotonically increasing and therefore, is bounded away from zero. On the other hand, a basic theorem in calculus tells us that for arbitrary real sequence (a_i)_{i\ge 0}, if \sum\limits_{i=0}^\infty a_i converges to a bounded real value then (a_i)_{i\ge 0} must converge to zero. By taking the contraposition of the theorem, we get \sum\limits_{i=0}^\infty T_i diverges, in particular goes to infinity. This contradicts the first assumption and we are done. QED.

Now that we have proved our assertion, we can observe that the legitimation of this theorem depends heavily on the two assumptions. In fact, we may accept these assumptions by our natural intuition. However, questioning the truth value of the statements may provide another interesting issue. The first one may be understood not as an a posterior fact, since no one who has been experiencing either death or eternal life can be able to proclaim the statement. The reason is that a dead man cannot make a statement and an everlasting man cannot talk to us unless we are an everlasting being as well (and to prove its lasting property empirically, you must observe it in all time forever). Therefore, it ought to be an apriori fact.

The second assumption is more difficult to be proved since the knowledge k_i with i>2 seems to be extremely far away from our daily discussions. A branch of philosophy called epistemology is a very complicated science of knowledge which mainly discussed k_2. Therefore, k_3 may become the science on how we could understand epistemology. It is ridiculous to say that k_3 is easier to understand than k_2, the epistemology, and that is why we define our second assumption. To refute our second assumption is just the same as to create a supernatural human phenomenon in which the learning time of one’s knowledge is reciprocal to its subject difficulty.

To sum up, although it is valid to make a question on human’s chance to absorb all knowledge, the answer is a big no due to some mathematical arguments. Moreover, the arguments are strongly influenced by two basic assumptions which sound very natural to accept. In fact, refuting the truth value of the assumptions may bring an extremely great difference to humanity we know these days, which means that the probability of their fallacies is extremely small. Thus, we may safely arrive at a strong conclusion that nobody can be able to fully understand his/her mind.

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