|Numbers, in other words, are a mathematical construction. Everything else that comes along with numbers is pure coincidence.|
and helios replied:
|This is unsatisfactory. It doesn't address at all why mathematics appears to occur naturally in the universe. If it's a purely human invention, it makes more sense that it'd be inapplicable to the natural world.|
I agree with helios here. The abstract mathematics of number systems (Peano's postulates, the rationals, the reals, etc.) has its genesis in the human mind of something that is perceived to be part of nature, not so much as an object that exists, but rather as a property or attribute of certain configurations. For example, if there is a box containing five apples, then "fiveness" is a property of the box and the apples taken collectively. In a primitive sense, how would we quantify the apples in the box? By "counting" them with our fingers. We match each apple with one of our fingers. If we run out of apples before we run out of fingers there are less apples then fingers and if we run out of fingers before apples then there are more apples than fingers. If we run out of apples at the same time that we run out of fingers then there are the same number of apples and fingers. This is exactly what occurs mathematically in comparing the cardinality of two sets. Two sets have the same number of elements if there exists a one-to-one onto function (a bijection) that maps one set on to the other. Eventually we give names to these quantities that we intuited (one, two, three, four, etc.). It seems to me that the practitioners of a language would need to have developed well above the level of a small primitive tribe before they would invent words for very large numbers such as million or billion. Even today we abandon the process of inventing new words for very large numbers and use scientific notation with powers of ten giving us the ability to represent arbitrarily large numbers, but when we do so we lose any intuitive understanding of the quantity that such numbers represent. For example, no one has any conception of how 1010000
differ in quantity since there is nothing in nature to compare them with. Though these two numbers can be represented, we are forced to revert to primitive notions of cardinality and the most that we can say is that the second number is bigger than the first. (You need more fingers to count the second than the first. A lot more.)
If quantity (and hence numbers) is something in nature that can be perceived by the human mind, is that capability unique to humans? I think not. Consider a wolf (not a domesticated animal to avoid influence by man) with a liter of five pups. She moves a short distance away and her pups naturally follow but one of them straggles. She suddenly recognizes that one is missing so she backtracks, finds the straggler, resumes her move with all five now in tow. Clearly to recognize that one is missing she must in some sense be counting. Even if one attempts to evade the argument by saying she's not really counting but perhaps had stored in her brain a collection of the scents of her five pups and simple matches those stored scents with the four pups that are present and thus discovers that one is missing, but she doesn't really have any conception of the idea of "fiveness" so she's not really counting. But if she can recognize that one of the five stored scents is not matched up she must in some manner be engaging in some type of cardinality comparison, primitive as it may be. I conclude therefore, that numbers or quantity are inherent and inseparable from nature and are not something created by the human mind but something that is perceived by mind.
helios: [of numbers]
|What is their ontological status?|
I believe that is a question that cannot be answered within the framework of naturalism and the metaphysical assumptions of modern science. For me, as a Christian, the answer to the question as to why the natural world and the Universe are the way they are, as to why the human mind can perceive such order and structure in the Universe as revealed by science, can only be found in the Creator and beyond that it is impossible to push.