What ARE numbers?
The Blind Eternities forum
Posted on June 13, 2015, 3:42 p.m. by InfiniteParadoX
For you ChiefBell <3 So what are numbers? How do numbers function? Theoretical ways to demonstrate and explain things that without them we couldn't comprehend? I'd say they're pretty cool. How does everyone else perceive this kind of stuff?
The counting numbers, which I shall refer to as natural numbers, are derived from the empty set (the set with nothing in it). For example, we say that 0 is the empty set ( 0= ( ) ). Then 1 is the set that contains the empty set ( 1=( () ) ), and we proceed inductively making each number n defined as the set that contains all numbers less than it. So 3 is the set (0,1,2)= ( (), ( () ), ( (), ( () ) ) ). Yes I know it looks confusing, but it works.
From there you can define the Integers by putting the operations addition and subtraction onto the counting numbers. Then we can define Rational Numbers to be ratios between Integers p/q where q can't be zero. To then define the Real numbers we need a bit more work, which is usually done in Real Analysis, a 300-400 level Math courses at Undergradutate universities. Then the Complex numbers come from us defining what the square root of -1 is, and that's all the numbers you should care about.
And that, TappedOut, is what numbers are.
June 13, 2015 3:55 p.m.
canterlotguardian - Imaginary Numbers are a subset of Complex Numbers, so I did not forget about them :)
June 13, 2015 4:03 p.m.
There are two fundamental views of numbers.
1) They a real. Ie. they 'exist' in nature.
2) They are not real. They are nothing more than figments of our imagination.
There are multiple philosophical theories that deal with this. The platonic view is that numbers are 'divine'. We don't know where they come from, or why, but they exist in nature and everything we use them for is an extension of that. Evidence for this comes from the immutability and precise nature of mathematics. Problems with the system are problems with our understand of data, not of maths itself.
There is another view called formalism that essentially says that maths is nothing more than our made up realisation of certain relationships between points and integers. Without understanding and meaning maths cannot exist. This one does not need much explanation because its fairly simple. It has interesting views for computers though because it follows therefore that machines technically can't do maths because they don't understand relationships; only values.
Those, broadly are the two ideas behind numbers and whether they 'ARE' (ie. whether they exist) or whether they 'ARE NOT' (ie. they are made up).
Sorry I just read up on this.
For reference no-one can agree whether they are real or not.
June 13, 2015 4:24 p.m.
ChiefBell - If I may, I'd like to posit an argument for why numbers are real empirical objects. My expertise in this subject is based on the fact I majored in Mathematics in college, spent two years in graduate school for mathematics, and am now a high school math teacher.
People mainly use numbers for the sake of communication, so we can quantify how many objects we have. Now, the definition where I constructed our number system I used above does require some belief that numbers exist. Now why should that be? Really all the definition I used requires is the agreement that "nothing" is a valid concept. So I shall attempt to prove that "nothing" or "none" exists.
Let's say that there is a field, and the only non-plant life in the field is yourself, your parents, and a horse for each of you. So how many cows are there in the field? Based on the information I gave you, the only logical answer is "There are no cows in the field." Then if I asked you to list all the cows in the field, you should respond with a blank list. That blank list is the empty set which I linked to and quickly defined above. Now, since we can agree on the existence of the empty set the rest of my construction of numbers follows as I defined it.
Let me know if you have any disagreements with my logic.
June 13, 2015 7:41 p.m.
Ah but it if follows that maths is real because it is observed phenomena how do you explain complex relationships that you can't observe. You would argue that you can find 0 in nature. You can find 1. Can you find 1,000,000? You can extrapolate the data you already have but from that moment forward you are moving away from the tenet that it is real because it is a feature of the world. Can you find the binomial theorem? Throughout history it has long been argued whether maths is truly observable as you have described or whether it is not observable; only abstract. We know that high level mathematical concepts are fairly abstract and intangible - so where does that knowledge come from if they are real? This is a tricky problem for 'real' philosophers.
Besides, even if it IS observable that does not necessitate that the system you adopt to record those observations necessarily exists. Your conception of numbers and the relationships between them, whilst based on natural phenomena, do not necessarily have to be natural in themselves. Do you see what I mean?
In fact increasingly mathematicians and philosophers have argued that maths is totally unecessary. I believe in 1980 a mathematician was able to prove certain basic concepts such as pythagorus' law without using basic numeracy at all. He used only written laws and comparisons between objects. The idea that you can do things like science without maths suggest that it's not something that's an inherent feature of nature and we could have easily come up with an alternate system.
For reference I am not a mathematician, but a philosopher. My knowledge of maths doesn't really go beyond the ability to cqrry out statistical tests by hand and a good knowledge of mechanics (as in movement related) maths. I like to explore maths as a feature of the human experience.
canterlotguardian says... #2
Numbers are entirely subjective. I mean, their very nature demands the statement that someone had to come up with what they mean. If they had wanted something different, we could now have a monetary system where a $5 bill is worth more than a $100 bill but a $20 bill is still worth more than a $10 bill.
June 13, 2015 3:46 p.m.