definitions are specifications partitioned from the total body of all possible objects. we specify definitions, and discover the relationships between definitions.
Mathematics is discovered. When Cantor established his theory of cardinalities, he was inventing the idea of how to compare sets. But as a consequence, he discovered that some infinite sets have greater cardinalities than others. There are many other cases of this kind of thing. The concepts behind Group Theory were invented slowly and gradually refined. Eventually there was a large investigation into the classification of finite simple groups. If mathematics is invented, then I'd like to know who invented the Monster group!
fun fact cantor was specificaly trying to establish when a functions' fourier series is unique, and I think the idea was that it was unique when it had a countable amount of discutinuities or something like that
Quantities are discovered, numbers are invented. You dont invent a number system, you just build one with predefined attributes, like base 10 requires 123456789 and 0, or an equivalent representation. You discover things about that number system, and invent sigils to correspond. The question is really asking if mathematics exist before we know them, and yes they do. We invent the representation and notation, but the logic and information was always there to build from.
It sounds like you're saying that numerals are invented but numbers are discovered, and I completely agree! No one can doubt that any intelligent extraterrestrial civilization we encounter will use different symbols for their numbers and most likely a different counting system as well, but they'll still be working with the same numbers! Just like a rose by any other name smells just as sweet, a number by any other name is still the same number.
Discovered; maths are there before us and are applicable to the entire universe. According to Plato the only thing that is required is to access the idea and download it
Our methodologies and symbols are invented but the innate qualities of martens are relational and therfore preexistent to our debtors of math Our math is our method of inquiry. The tools we use to explore. What if we only represented all numbers as a binomial. The primes it is composed from?
logic is discovered, then all based on it is also discovered... :) ... and beyond math, i would say ::: everything is discovered... among logical possibilities of realization.
Imo, we invent definitions and discover theorems (or at least conjectures)
This is a very good insight!
definitions are specifications partitioned from the total body of all possible objects. we specify definitions, and discover the relationships between definitions.
Mathematics is discovered.
When Cantor established his theory of cardinalities, he was inventing the idea of how to compare sets. But as a consequence, he discovered that some infinite sets have greater cardinalities than others.
There are many other cases of this kind of thing. The concepts behind Group Theory were invented slowly and gradually refined. Eventually there was a large investigation into the classification of finite simple groups. If mathematics is invented, then I'd like to know who invented the Monster group!
fun fact cantor was specificaly trying to establish when a functions' fourier series is unique, and I think the idea was that it was unique when it had a countable amount of discutinuities or something like that
3:25 counting fruits: there's clearly 6 bananas and 3 kiwi fruits? Not sure how much watermelon there is, clearly less than 1 though ;)
It looks like they should have introduced fractions first before showing the fruit slices!
Quantities are discovered, numbers are invented. You dont invent a number system, you just build one with predefined attributes, like base 10 requires 123456789 and 0, or an equivalent representation. You discover things about that number system, and invent sigils to correspond. The question is really asking if mathematics exist before we know them, and yes they do. We invent the representation and notation, but the logic and information was always there to build from.
It sounds like you're saying that numerals are invented but numbers are discovered, and I completely agree! No one can doubt that any intelligent extraterrestrial civilization we encounter will use different symbols for their numbers and most likely a different counting system as well, but they'll still be working with the same numbers! Just like a rose by any other name smells just as sweet, a number by any other name is still the same number.
Is like the abstract idea of number 2 and all the different representations of 2, II, ..,
Discovered; maths are there before us and are applicable to the entire universe. According to Plato the only thing that is required is to access the idea and download it
So, it sounds like you're a mathematical Platonist like myself! Are you a Platonist about other things as well?
Discovered and there are many things still left to be discovered in mathematics
maths are a Definition of a concept of reality, like physics on an abstract plane, i say neither discovered or invented, but defined
This is an interesting point of view, which I need to look into some more. Thanks for sharing your thoughts.
What's the difference between inventing an axiom and defining an axiom? Once we have axioms we make discoveries about their interactions
Our methodologies and symbols are invented but the innate qualities of martens are relational and therfore preexistent to our debtors of math
Our math is our method of inquiry.
The tools we use to explore.
What if we only represented all numbers as a binomial. The primes it is composed from?
I don't know how you can represent all numbers as binomials other than in the trivial way, i.e., n = nC1.
good video....
Thanks! I'll make some new ones as soon as I can. Meanwhile, you might want to check out some of my other videos.
both
Would you mind elaborating on this a bit?
Inventing, discovery, design, they're all just search with arbitrary fitness functions.
Is this an AI point of view?
Lol no. I'm just a designer with a hobby in philosophy.
I think maths is discovered, but we invent the representation of it, the symbols and language used to embody them
logic is discovered, then all based on it is also discovered... :) ... and beyond math, i would say ::: everything is discovered... among logical possibilities of realization.