Question

What is a number?

Answer 1

A number is a mathematical object used to count and measure.

Answer 2

In my older questions, we've proved a lot of obvious things about numbers such as 1 is not equal to 0, (and maybe some less obvious things such as |a+b| leq |a| + |b|), but I still don't understand what a number actually is.

Answer 3

I used numbers in my previous questions in a very abstract way; I haven't counted or measured anything. So numbers are more than just mere counting/measuring objects.

Answer 4

what was the proof 1 is not equal to zero ?

Answer 5

a number can be defined as the equivalence class of sets that are in 1-1 correspondance with it. so for instance, 1 = { {a}, { elephant} , { calculator} , {tree} , ... }

Answer 6

Does that really answer the question? I still don't know what a number truly is :(

Answer 7

we can have numbers without set theory, right?

Answer 8

well, see how i just shifted the question to , what is a 1-1 correspondance. well its a bijection between sets. what is a set... and on and on we go .

Answer 9

but the nice part about math is that we define the RELATIONS between objects, even if we do not know what the objects actually are. we can only intuit it. and we model numbers , so for instance what is '2 trees', or '3 hydrogen atoms' . thats a model of what a number is , the property we have in mind

Answer 10

2 trees , we are not interested in the texture, color , of the trees. we are interested in the 'twoness'. what all twoness things have in common. even if we cannot define it without going in circles, at least we are consistent in our relations and about these twoness and threeness things

Answer 11

numbers have become so abstract as math evolved.

Answer 12

well, thats because we found more fundamental ways to describe numbers. i dont know if they are more abstract, since they were always abstract to begin with. can you show me a number 3? where is it? 3 is not the symbol we refer to 3 as , '3'. thats just the label for it . so where is it?

Answer 13

I can show you 3 camels, but what about 0 camels?

Answer 14

math is a construct of the mind, evolved by millions of years of environmental pressure. math does not 'exist' outside the mind, but the relationships do exist in nature which have molded our brains over evolutionary pressure and selection. ie, people who could count sheep got laid more ;) (without oversimplifying)

Answer 15

can you show me 2/3 of a camel, or -1 camel, or 0.20982397327810039135871322225343... camels, or 3 + 2i camels?

Answer 16

yes you can show me 3 camels, but you cant point to '3' in nature

Answer 17

you can say, i hope you agree that all these triplets of things have something in common, that which is 'threeness'

Answer 18

but I've never encountered the other kinds of numbers (besides natural numbers) in nature

Answer 19

3 camels, 3 donkeys, 3 beetles, 3 horses, ... what do all these things have in common?

Answer 20

there are 3 of them?

Answer 21

lets say i am an alien and i dont know what 3 is

Answer 22

nope, 3 is chinese to me. so what do they have in common . i see they have fur, they smell funny ..

Answer 23

so there is a way to explain to an alien what 3 horses, 3 trees, 3 alligators, .. have in common . besides for being animals. we could mix it up and throw in 3 rocks

Answer 24

first the alien has to agree that these are 'collections', groups or assemblages

Answer 25

we can show him that pattern

Answer 26

what is the pattern of 3 ?

Answer 27

remember, i am alien, dont know what 3 is

Answer 28

groups of 3 stuffs

Answer 29

but i do know what sets are , they are like bags with stuff in side. i know how to seperate things into groups

Answer 30

what is 3 stuffs?
you are going in circles

Answer 31

ok, here is a way to explain to an alien what 3 is. its a roundabout way, but it does the job

Answer 32

will the alien figure out the remaining numbers?

Answer 33

dunno. but you can use functions or 'mappings'. so you have a bag of little pebbles that you use to 'correspond' to objects.

Answer 34

first we have to define what it means =, < , >

Answer 35

= means that given a barn of animals , and you have a pouch of marbles say. suppose that as each animal exits the barn, you take one marble out of the pouch. if there are no more animals in the barn, and there are no more marbles in the pouch, then the 'size' or cardinality of the marbles in the pouch is the cardinality of the animals in the barn

Answer 36

true

Answer 37

the barn thing reminds me of a joke.... I can't remember it :(

Answer 38

now, we can also define more easily if we use the marbles in the pouch example again

Answer 39

suppose that i run out of marbles to take out of my pouch and there are still animals exiting the barn. so the cardinality of the pouch of marbles is < than the cardinality of animals

Answer 40

suppose that all the animals have exited the barn, and there is still marbles in my bag. then i can say the cardinality of marble pouch is greater than cardinality of barn animals. notice that in my 3 cases, i never use the word 'more', equal, or less. then my definition would be circular

Answer 41

so there are three cases with my pouch of marbles. i run out of marbles , i have marbles left, or there are no marbles in the pouch. these are the three conditions , < , > , and = , with respect to the set of barn animals and the set of marbles

Answer 42

mathematicians call them, injection, surjection, and bijection

Answer 43

so injection = <, surjection = >, and bijection = = ?

Answer 44

not quite , injection is actually <=, surjection is >=

Answer 45

the function (or mapping)
M1 -> horse
M2-> sheep
M3-> rabbit
M4-> mouse
donkey
is an injection

Answer 46

actually this answers a different question, why is 3 > 2, for instance. to define 3 , we use the set of all sets that are bijective with { {}, {{}}, {{{}}}} }

Answer 47

A standard construction in set theory, a special case of the von Neumann ordinal construction, is to define the natural numbers as follows:

We set 0 := { }, the empty set,
and define S(a) = a ∪ {a} for every set a. S(a) is the successor of a, and S is called the successor function.
By the axiom of infinity, the set of all natural numbers exists and is the intersection of all sets containing 0 which are closed under this successor function. This then satisfies the Peano axioms.
Each natural number is then equal to the set of all natural numbers less than it, so that
0 = { }
1 = {0} = {{ }}
2 = {0, 1} = {0, {0}} = { { }, {{ }} }
3 = {0, 1, 2} = {0, {0}, {0, {0}}} = { { }, {{ }}, {{ }, {{ }}} }
n = {0, 1, 2, ..., n-2, n-1} = {0, 1, 2, ..., n-2,} ∪ {n-1} = {n-1} ∪ (n-1) = S(n-1)
and so on

Answer 48

so you can construct all of math from nothing, literally ;)

Answer 49

yay!

Answer 50

this may just be a rumor, but I hear that 1 is the loneliest number.

Answer 51

A number is a thing that is used to count.