2014

April

It is impossible to symbolize the actual mechanics of thought. Consider the argument, “it is asserted ‘p’, it is asserted ‘p implies q’, therefore ‘q’ is also asserted,”. It may be rendered symbolically using the assertion sign \(\vdash\) and the conclusion sign \(\therefore\),

\[ \begin{align}\begin{aligned}\vdash p\\\vdash p \implies q\\\therefore q\end{aligned}\end{align} \]

The law of modus ponens is a primitive notion in logic [1] and it must be accepted at face value in order to go any further. If one asks why this law is so, there can be no explanation. It is simply the case and one must be satisfied at that. There is no greater ballast upon which to rest the truth of this proposition. The reason cannot be captured in symbols; the symbols which express modus ponens can only stand and present the meaning for an attentive mind to decipher. If it is not accepted, the possibility of communication has ceased.

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The human mind is not a truth machine. It is a thought machine.

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Here is a problem no computer will ever solve: Select a random number between 0 and infinity [2]. The computation intractability of this problem is twofold:

1. No computer can represent a continuum. It must always represent the space of numbers discretely.

2. No computer can generate truly random numbers, except through its malfunction.

The second point is subtle, because much of our everyday understanding of probability is interwined with the intuition acquired through the digital representation of randomness. It is through pseudo-random numbers that abstractions like the “Law of Large Numbers” or “Sample Spaces” become conceptually realized in modern day classrooms. However, this intuition tacitly assumes the number returned by a random number generator is actually a representative of the class “random things”. In reality, pseudo-random numbers are the result of intricate recursive formulae of modular arithmetic. Every random number generator implemented through pure logic is an approximation of randomness.

“Approximate randomness”. The phrase itself is almost nonsense. Randomness is not something that is approximated, but is what causes the phenomenon of approximation. Randomness is exactly the reason that precise measurement cannot be ascertained. Approximation is the physical act of engaging with randomness.

The idea that randomness has a formula is absurd. The idea that randomness can be generated through formal representation is antithetical to its very conception.

[1]

There are other decompositions of logic that take as primitive other modes of inference besides modus ponens, but the essential idea here is that logic requires a “starting point”. Logic must start with a premise. Either one accepts that premise as “true” or they do not, and in the latter case there is no further recourse except to point and say, “see?”. – April 2025

[2]

The exact same problem can be restated in simpler terms: Select a random number between 0 and 1. In fact, the endpoints make no difference. Select a random number between a and b for any \(a,b \in \mathbb{R}\) such that \(a \neq b\). All such problems form a class of problems no computer built on binary logic will ever solve. – April 2025