a structuralist definition, in that numbers are whatever it takes to get certain number-like operations - like those that preserve certain global symmetries, such as commutativity or associativity. — apokrisis
There is an advantage to this approach. Mathematicians are not constrained by a definition of number, which allows them to discover new types of numbers all the time. — fishfry
If you don't like the fact that numbers are defined in terms of set theory — tom
I guess you won't like the fact that numbers are also defined in terms of field axioms. — tom
And no, new types of number are not "discovered all the time". — tom
Good idea and a very natural attempt; but arithmetic properties aren't sufficient. Weirder still, there are numbers that lose associativity as well, such as the octonions. — fishfry
In short though, mathematical structuralism is more subtle than just listing arithmetic properties like associativity.The kinds of properties that they use in category theory are ... well, they're kind of weird and nonintuitive when you first see them. The structural relations they have in mind are various types of universal mapping properties. It's hard to do justice to what this means in a simplified format but I might take a run at it once I get into responding in detail to your earlier post on structuralism. — fishfry
I did not write the quote you attributed to me. What is your attitude problem? — fishfry
Nobody has provided a counterexample and you now seem to agree. — fishfry
If you choose category theory as your foundation, there's still no general definition of number. — fishfry
Your inability to discuss the foundations of maths is noted. — apokrisis
Your complete misunderstanding and lack of comprehension of category theory and mathematical structuralism was evident to several other posters the last time we discussed this. — fishfry
I'm done responding to your posts on this site. — fishfry
The question then arises, 'what is the nature of number?' Conjecturally, one might say, number is a series of equal values (quantity). Hence, Pythagoras' and other ancient mathematicians' inclination to render number as equal, whole values. If this is an accurate description of number, then it follows, the concept of number is tied to the idea of a 'unity' value (unit measure). — cruffyd
Temporal construction is such that inequality defines its nature. Equality, on the other hand, can only be outside of temporality. One might say: 'equality can only exist eternally.' — cruffyd
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