Post
by **Science Fan** » Tue Jun 13, 2017 4:23 pm

The Arabs did not invent Arabic numerals, the Hindus did, and the Arabs got the idea from the Hindus. Europeans got the "Arabic numerals" from the Arabs, and assumed that the Arabs invented the system, not knowing that the Arabs borrowed from the Hindus.

You can count without any number system by matching. A quick example is when a person sees that every seat in an arena is filled by a person sitting in each seat, regardless of what the actual number of seats or people in the arena is, we know that the number is the same. So, one can "count" in this manner without using natural numbers, but without being able to match two different sets of objects together, counting would have been done by the use of natural numbers. However, my point was, and remains, that natural numbers have limitations, which is why other numbers were developed, the last being complex numbers. In fact, all numbers can be thought of as complex numbers in disguise, and we even have a theorem that tells us there will be no need to develop any additional numbers beyond complex numbers. Natural numbers are useful for addition and multiplication, where we have closure --- add or multiply any two natural numbers together and you get a natural number, so one remains in the set of natural numbers when using the operations of addition and multiplication. However, this closure property fails when doing simple subtraction. Then, we need to develop integers, adding zero and negative numbers, in order to have closure. But, integers fail when we add in the operation of division. Then we have to add in rational numbers. But, there are more irrational numbers than rational, so we have to add in real numbers. Then, we have difficulty finding roots, so we add in complex numbers.

I think Phil X's general point is an interesting one, however, concerning the nature of numbers. I'll expand on his point from natural numbers to all numbers, including complex numbers. Complex numbers give us some unexpected surprises. For example, an exponential function grows and grows at an accelerating pace, anyone can see this when graphing e raised to the x power, and just plug in natural numbers for x, one will see that curve's graph take off towards infinity. We can also graph sin(x) and cos(x) and see that we get a graph that bounces back and forth between 1 and minus 1. So, whoever would have thought that an exponential function would equal a function consisting of a sin and cos function? I sure as hell wouldn't; however, they do equal each other when we plug in complex numbers. I think this is an illustration of the point Phil X is making ---- that we cannot just manipulate the numbers we come up with in any old manner, the logic surrounding them contains a life of their own.

As far as the metaphysical question, I'm not convinced that the only two options are a realism based on some form of Platonism versus purely made--up fiction from a human mind. I think rather it's a combination. Until the numbers are invented, by the human mind, I fail to see how they could exist, because they do not exist outside of us in the universe outside our minds. However, once invented, then they do have a logical structure, and do have an existence that is at least somewhat independent from us, as we cannot merely manipulate the numbers any way we please.

The thing is the issue being raised here cannot be answered within mathematics, because it is a philosophical question, which is why even top mathematicians disagree with each other on this issue.