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> You can derive most of mathematics from a few simple axioms. Yeah, nothing is stopping you from building towers of abstractions that are N degrees from any r

> You can derive most of mathematics from a few simple axioms. Yeah, nothing is stopping you from building towers of abstractions that are N degrees from any real experience or data. But those heavily-derived concepts are prohibitively difficult to teach and communicate, because each level in the abstraction hierarchy adds semantic noise. A major reason why classical, "pure" mathematics pedagogy is infamously ineffective. Ultimately, every great mathematician learned to count with rocks or apples -- not by internalizing Peano's axioms.

marginalia_nu 5y ago
That is how it is commonly taught, but can it really be argued it's the only way it can be taught? Regardless which method is easier, why is it possible to interact with negative, irrational, or complex numbers without anyone having ever seen them, but not integers? There of course needs to be a common set of ideas to communicate, but I'm not convinced you need one particular set. In many cases, having some of the ideas means you can synthesize the rest. You could for example reason about numbers by drawing upon size rather than quantity. Then you basically end up with the Euclidean method, a mathematics of proportion that goes a surprisingly long way.
- pphysch 5y ago
  Abstract quantities, even positive integers, don't exist in reality. Heck, even "objects" don't really exist, because "boundaries between things" don't concretely exist except in our perception and imagination. If there is any primal, axiomatic concept, it is "object"/"thing". From object you can derive quantity (many objects), from quantity you can derive quality (two objects are different), and so on and so forth.
  - marginalia_nu 5y ago
    You can reason about the same reality in various different terms, and the choice in terms shapes how you see the world. I don't think it's easy to argue that one concept is the true concept from which all concepts inevitably derive. You can just as easily think of things as parts of a whole as objects in an emptiness. In some languages the name for door is the same as the name for mouth, and a door is a bit like a mouth so it makes sense to relate them. All language is metaphorical like that, except not always as explicitly. When we say a thing is an object, we say that our concept of that thing shares similarities to the concept we have for objects.

marginalia_nu 5y ago

That is how it is commonly taught, but can it really be argued it's the only way it can be taught? Regardless which method is easier, why is it possible to interact with negative, irrational, or complex numbers without anyone having ever seen them, but not integers? There of course needs to be a common set of ideas to communicate, but I'm not convinced you need one particular set. In many cases, having some of the ideas means you can synthesize the rest. You could for example reason about numbers by drawing upon size rather than quantity. Then you basically end up with the Euclidean method, a mathematics of proportion that goes a surprisingly long way.

pphysch 5y ago
Abstract quantities, even positive integers, don't exist in reality. Heck, even "objects" don't really exist, because "boundaries between things" don't concretely exist except in our perception and imagination. If there is any primal, axiomatic concept, it is "object"/"thing". From object you can derive quantity (many objects), from quantity you can derive quality (two objects are different), and so on and so forth.
- marginalia_nu 5y ago
  You can reason about the same reality in various different terms, and the choice in terms shapes how you see the world. I don't think it's easy to argue that one concept is the true concept from which all concepts inevitably derive. You can just as easily think of things as parts of a whole as objects in an emptiness. In some languages the name for door is the same as the name for mouth, and a door is a bit like a mouth so it makes sense to relate them. All language is metaphorical like that, except not always as explicitly. When we say a thing is an object, we say that our concept of that thing shares similarities to the concept we have for objects.