There is an interesting concept humans used during almost the whole of their written history, at least starting from Sumer: numbers… But there is a quite large difference in how we perceive numbers today and how numbers were seen 4000 years ago or more. Let’s start with the beginnings. Numbers were used to count things. A sumerian farmer had to know how many cows and sheeps he has, how to sell them, what they require as food etc. All these concepts were encoded via symbols for whole (natural) numbers. This is why the concept of “natural” number was so … natural… and was found everywhere in nature. If a farmer had a cow and an ox they were not as surprised to say they have 2 animals and the number may even go up to 3, 4, etc. There was no need for a too deep philosophical understanding for why there are these jumps (quantum leaps) in the total number of animals the farmer possesed. Of course, after some progress in science and technology one found out that a new animal somehow evolves from small cells that organize themselves until they grow up as another animal so the evolution appeared to be somehow “continuous”. So, one had to invent fractionary numbers and then, with the discover of the circle one found out that some question regarding the circle have answers that cannot be represented using a finite number of digits. After all these extensions one has arrived today at several types of numbers: natural, integer, rational, irrational, real, complex, etc. What one has forgotten is that they do not encode “nature”. They encode possible answers to some questions related to nature. So, the sumerian farmer was not very surprized while seeing natural numbers in nature… well… don’t worry, the modern student in quantum mechanics, after learning complex analysis is very surprized to see that some questions have answers that are encoded solely as natural numbers. In order to understand this better the modern student has to unlearn modern analysis in one-year courses of general topology and algebraic topology. After this, our student finally should understand that there is no problem or contradiction between having some quantities giving a continuum of values and others a discrete set of values… How many times do you have to turn around a circle to come to the same place? Any integer number of times. The trajectory is continuous and if the question is “what is the trajectory one goes on when turning on a circle” the answer will obviously involve some real or complex numbers described in a continuous way. That doesn’t change the fact that when the question is “how many times do you have to turn around a circle to get to the same place” the answer will be an integer… see, integers and reals can live peacefully together…

# Numbers

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