In your opinion. Or feel free to quote some people on their views of the answer.
"Number" doesn't have a definition. It's just a colloquialism.
And thats why number theory/set theory will be at a standstill until we choose to define it.
What makes you think number theory and set theory is at a standstill??
Are you just making things up?
Number theory studies the integers, which are perfectly well defined. It would be hard to imagine a more thriving and successful area of research than modern number theory.
I should have said "number theory will progress at an amazing rate, but far slower then it could, should it choose to continue ignoring to define certain terms."
All terms in number theory are perfectly defined.
And really, if you think one definition is going to progress a field, then you are being pretty naive.
Where are you getting your quotes??
I was told a few posts up, number doesn't have a definition. Sorry, care to define it for me?
Number doesn't have a definition. But number theory isn't about numbers, but about integers. And integers are well-defined.
Thank you for your patience, but If i may..
You are saying that number theory is about the study of integers...which are...numbers?
So then it is the study of something which is defined to be a type of "number."
Wouldn't it seem plausible then to define number?
And may I ask why number doesn't have a definition, then?
There is no such thing as a number. A number is just a loose term which signifies nothing.
You do have things like
All of these things are perfectly well-defined. The word number is just used to give it a common name. In the same way, complex numbers are not really complex and rational numbers are not really rational.
"The word number is just used to give it a common name."
Okay, by calling them all "numbers" you are giving them all something in common. You are saying you are only giving them a common name, but are you not also placing them under a common group, mainly being "numbers?" A complex number, though different then a transfinite number falls under the class of both being numbers, no?
So then they fall under the collection of both being....what? define that last part, is what I'm asking.
Its kind of like the word "mammal." People are mammals, dogs are mammals, whales are mammals. When you study humans, some of the stuff you study might give you information about mammals in general, but most of your focus is going to be on humans. Similarly, a number theorist studies the properties of integers - he might be able to generalise properties about integers to different "types" of numbers, but this doesn't mean every "type" of number is the same.
but yet again you are saying "every type of number", and so categorizing it under a certain conception within your mind, namely all being the idea of a number (regardless of being different types) , define that conception if possible.
It's a psychological thing. It's just something you say for psychological reasons. The word number doesn't have any mathematically deep meaning. It's like using "x" for variables and not "f".
A number should be something that can be added and multiplied to other numbers and it should in some way be an extension of the natural numbers. That's about all we can say.
Perhaps there is no good definition of this general concept. BUT, this does not mean anything about the field of number theory. Again, number theory is the study of integers, so we don't need to give a definition of "number" to study integers.
Exactly, its a psychological thing. Don't we all conceive of this "variable" in different ways, in a way in which each person conceptualizes it differently, but somehow conceives of it in a general way so that others can also know what one is referring to while talking about this conception?
We say "number" for psychological reasons, yes, and what are those reasons? So that whomever is listening can picture the idea within their intellect in a way, specific enough to themselves, that it makes complete sense uniquely to them, but in a way general enough, so that it can be talked of by all.
I was curious what your definitions of what this "picture/idea within your intellect" is and how you would describe it.
There is no definition.
Anyway, this thread has been sufficiently answered, so I'm going to lock it.
Separate names with a comma.