That mathematics has been so useful in describing nature is a puzzle that many thinkers have struggled to answer. Galileo showed his deep respect for this surprising relationship between mathematics and nature when he said: “The laws of nature are written by the hand of God in the language of mathematics.” The physicist, Eugene Wegner in 1960, wrote that the “unreasonable efficiency of mathematics in science was a wonderful gift which we neither understand nor deserve.” (1960 article The Unreasonable Effectiveness of Mathematics in the Natural Sciences). This is the reason that some fluency in mathematics is essential to understand physics beyond a superficial level. It is interesting to consider how this language or mathematics emerged.
Even though humans seem to have an innate sense of quantity, surprisingly it is not much better than that of birds and reptiles. What do we mean by this? Before a counting system was developed, we relied on our sense of quantity. A measure of this sense of quantity would be determined by what it would take to notice a change in the quantity of something. Clearly, if one bean were removed from a pile of beans, we would not detect the difference. Just how small a pile would it take before we would be aware of a change in quantity? It turns out it’s not until the pile of beans is less than ten that we would notice a difference. Birds and lizards do just as well. Hence, it is clear that we need something other than an innate sense of quantity to determine changes of quantity. We need a counting system.
But humans are clever and long before inventing numbers and counting, they devised operational techniques for dealing with quantity. In the case of the gatherers, any dispute about fairness could have been resolved very simply by lining up objects in adjacent rows, providing a one-to-one correspondence between quantities. This correspondence method of comparing the quantity of one group with that of another lays the foundation for counting. When it became clear that this method is valid, independent of the objects in the collections, so that one set of convenient objects could become the standard set for comparison, humans moved closer to inventing a number system.
Even with this very simple counting system, some basic arithmetic operations can be accomplished. Adding and removing objects from collections are the operational equivalent of addition and subtraction. The operational idea that one could toss each family’s share back into a pile that would contain the original quantity reflected, operationally, that there is a conserved quantity, a quantity that remains constant despite all the operations. That is to say that there is some quantity that is conserved, or invariant, as long as nothing is removed or added to the whole system. This idea of a conserved quantity in a system underscores the so-called laws of conservation, which have come to permeate the fundamental laws of physics, as we know them today.
Separating the objects into equal piles is the operational equivalent of division while counting by piles is the operational equivalent to multiplication. It is not very likely that the division will come out even. So what happens when there is a left over that is too small to make another complete go-around? If an equal distribution is required, then this could be operationally accomplished by cutting each of the remaining beans into a number of parts corresponding to the number of families, and then deal out the parts. Underscoring this operation is the mathematical concept of fractions. Thus this scenario shows how humans could have dealt with problems involving quantity without a numerical system and at the same time lay the foundations for basic mathematical operations. But something else was required before mathematics could emerge.
Today it is difficult to imagine dealing with quantities without a counting system based on ordinal numbers. Measurement requires counting and science depends on measurement. So before science could develop, a usable number system was needed. What is surprising, considering how fast linguistic skills developed, how long it took to develop a numbering system. How it happened is a very complicated story that is not completely understood. If mathematics is indeed the “queen of science,” then we should consider its origins.
Despite these advances, these systems were still comparison systems in which the names of the numbers simply replaced the pebbles. Such correspondence systems that make no distinction between the individual items in the comparison group are known as cardinal number systems. Not until the order in which the symbols were used mattered could a method of computation with the symbols emerge. The idea that one number precedes another in a definite order seems to be so trivial to us today that it is easy to forget that there could be no mathematics without an ordinal number system. Once the names of numbers changed from being modifiers of objects, i.e., two apples, to a more abstract idea of “twoness,” then patterns and relationships between numbers began to emerge. A form of abstract thinking was now possible. In the words of Tobias Danzig: “Correspondence and successions are the two principles which permeate all mathematics – nay, all realms of exact thought are woven into the very fabric of our number system.”
