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本帖最後由 compsognathus 於 15-2-23 22:57 編輯
I myself believe that the mathematical concept of ratios had developed in humans long before natural or real numbers.
Many tribal cultures do not have many words for the natural numbers. For example, the Munduruku people only have words for one up until five. Studies done with Munduruku people show that they place numbers on the number line in a compressed, logarithmic form, so that the graph of their perceived distance between consecutive numbers resembles the graph of a logarithmic function. To a Munduruku, five dots is five times as great as one dot, but they notice that ten dots is twice as big as five dots, leading to this logarithmic graph.
A similar study performed using young children show that those without formal mathematics education mapped numbers identically to the manner in which Munduruku do so. As the child is introduced to the natural numbers and arithmetic, the curve slowly progresses into a straight line, the answer an educated adult would be most likely to give, provided the numbers were small. Even as adults, humans tend towards a logarithmic distribution of numbers, especially with larger numbers. For example, the terms “millionaire” and “billionaire” are thrown around almost as synonyms. When a similar study was carried out with adults, the adults showed logarithmic mapping when provided with amounts of dots too large to count quickly.
This concluded that humans have a predisposition to ratios rather than enumeration of individual numbers.
This is a rather logical deduction, as in the wild, circumstances would call for the application of ratios and proportions, not counting. When a tribal human sees two trees with fruit on them, he has to quickly deduce which of the trees bears more fruit. He does not need to individually enumerate the amount of fruit. When an enemy tribe is attacking the tribe of said human, he needs to be quickly able to deduce which of the tribes has more people, and therefore whether it is worth it to stand ground and fight or simply flee and avoid catastrophic losses. The ability to calculate ratios (though not necessarily express them in written form) is a survival skill in nature.
Another study involved a number of lionesses and a recording. One lioness was passing bushes, when a recording of one lioness roaring was played. The lioness stopped, looked around and moved on. When five lionesses passed by, three roars were played. The five lionesses roared in response and made an attack. This indicates that lions also have a concept of ratios, as they seem to recognize that a one-on- one fight was not worth it, but in greater numbers, they would readily fight.
To quote Stanislas Dehaene, "It appears that we, as humans, can access two different methods of numerical mapping. The logarithmic, ratio-based method is the most intuitive; we inherit it from our primate evolution and we still access it in the absence of precise mathematical tools. Through education, we also acquire a linear mapping. However, this does appear to be a cultural construct."
Sources:
1: Alex’s Adventures in Numberland (Alex Bellos)
2: http://www.sciencedaily.com/releases/2008/05/080529141344.htm (Source: Harvard University)
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發表於 15-2-23 16:19
