3 ms·
If you follow the construction of numbers, integers are defined as how many elements are in a set. So integers kind of exist: if you have an empty bag, and add
by 101008 6y ago
If you follow the construction of numbers, integers are defined as how many elements are in a set. So integers kind of exist: if you have an empty bag, and add an apple, 1 is the cardinality of that bag. If you add another apple, then the cardinality is 2.
Then you define all the operations between integers as operations of cardinality between set. I really enjoyed that semester when I was getting my degree on Maths, by the way, although I use the knowledge I got there as a curiosity at parties, because it didn't prove useful (not that it is bad).
- random314 6y ago"God made the integers; all else is the work of man." Kronecker
- germanier 6y ago(I assume you meant natural numbers, as "-1" is not a useful cardinality for a set) The construction you refer to is due to Frege and Russell but it has its own kinds of subtleties. In particular it does not work in ZFC which we often base upon, so we usually use Peano's axioms instead to define them (depending on your context of course). While they are not all too surprising, I wouldn't call them absolutely obvious.
- 101008 6y agoYes, sorry, you are right. Natural numbers, not integeres. My bad.
- GoblinSlayer 6y agoIf you split one apple in two, you get two apples?
- viklove 6y agoTechnically you have 0 apples, and you have 2 halves of an apple.
- emergie 6y ago"Split" means that you have defined division, which implies having rational numbers. In the beginning there were only Natural numbers. -1 appeared the moment one decided to define complementary operation to addition. -1 in reality is only a shorthand for "0-1", 0 comes here as neutral element of addition. Addition is well defined on Natural numbers. Subtraction generates Zahl numbers. Similarly multiplication is well defined on Zahl numbers. It's the complementary operation - division - that generates Quotient/rational numbers.
- andrewla 6y ago> if you have an empty bag, and add an apple, 1 is the cardinality of that bag. If you add another apple, then the cardinality is 2. The artificiality is high here -- would you trade your bag of "2" apples for someone else's bag of two apples without asking about the size and quality of the apples? While counting numbers are wonderful things for examining things like "sets" it seem to me to be kind of silly to assign any "reality" to them. Discrete identical things are pretty much nonexistent at human scales. Continuous identical things, however, are numerous, where continuous here is not strict, but practical. This is not realistic as a foundation for mathematics. But I think the idea of using propositional logic and set theory was a mistake that we are only now starting to correct. Intuitionalists and category theorists (and finitists, frankly) are leading us in a better direction now.
- nullserver 6y agoHow do the apples get into the bag? Do they magically show up? Or do they transition from not in the bag to in the bag in some unit? What is the minimum possible unit of Apple change? Is it discrete or continuous? Maybe a X plank units of apple transitions every second.