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Comment by tambourine_man | Hacker News Reader

I was very relifed to discover later in life that I wasn't alone in disagreeing with Cantor. That, along with a few other proofs and axioms, were a major source of angst for me as a kid in school.

I find Wittgenstein's take on it very interesting.

http://plato.stanford.edu/entries/wittgenstein-mathematics/

I wish some teacher had told me that other maths are possible if we agree with different rules, even though schools are always going to teach and require the status quo.

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It's true that mathematics is based on rules, and different rule sets can lead to different results, one of which could be a system with no irrational numbers. However, I think it's pretty difficult to deny the sheer usefulness of the "status quo" and its acceptance of irrational numbers.

Yeah, me too.

Witggenstein has a good take. René Guénon also,

http://www.sophiaperennis.com/books/rene-guenon-series/the-m...

It seems that mathematics is presented in this sort of way that makes it seem free of the doubts and struggles and vagaries of other disciplines.

How do you define real numbers without uncountable sets though? I'm not sure most people are willing to give up the real numbers just because they are a bit unintuitive.

You have to make a distinction between the intensional and extensional domain. Real numbers are merely infinite decimal expansions, where "infinite" is a convenience term (semantically no different from "really big").

I believe the mantra is "π is a rule, not a thing".

real numbers are a closure of the set of rational numbers under the limit operation. It just happens that the closure is uncountable.

It is pretty standard device in mathematics - having a set and an operation, to produce and explore another set built as a closure of the first set under the operation. Seems pretty intuitive to me :)

What is really unintuitive, puzzling is the starting point of it all - natural numbers... 1, 2, 3... why there is a one stone, a one star, ... why this discretization?

You can start lower, if you want to.

{} == 0

{{}} == 1

...

Computable (real) numbers are a countable subset of the reals and represent the largest set we can analytically reason about; however, equality of computable numbers is not itself computable-- you can have computable c1 and c2 where cmp(c1, c2) is nonterminating (which means that c1 == c2 but it cannot be shown to be so in finite time).

I'd argue that, for most people's tastes, computable analysis is wonkier than real analysis, even if the latter has uncountable infinities (which is rarely considered a problem).

You might find Norman J Wildebergers Youtube videos on the subject interesting: http://www.youtube.com/watch?v=REeaT2mWj6Y He argues quite convincingly that real numbers like pi shouldn't be seen as numbers and that infinity does not exist. It's very fascinating reasoning to listen to even if you don't agree with all of it.

"Language is the universal whore that I must make into a virgin."

> schools are always going to teach and require the status quo.

Rephrase this: "Schools are always going to teach and require the mathematics that most people use, especially given that most people are going into fields where math is a tool, not an end in itself."

A different example of this is how Nonstandard Analysis remains 'nonstandard' even though it was essentially how both Newton and Leibniz originally imagined calculus: Until Robinson showed how to put infinitesimals on a rigorous footing, the only axiomitization of calculus involved epsilon-delta proofs and so that became, and still remains, the standard form of analysis.

Now, most people learning calculus will never need to know how it came to be, but axioms give a structure to guide thinking, and that is why they're taught to non-mathematicians. The precise axioms which are chosen is of somewhat secondary importance compared to the habit of mind formed by internalizing some axiom system and learning to think on that basis at least some of the time.

Note that nonstandard analysis is formalized by a constructive equivalence to 'standard' analysis; it's not a good example of the idea "other maths are possible if we agree with different rules", since there is no difference anywhere (except potentially in the mental model, where NSA is superior -- but all the math is the same).

For "other maths are possible if we agree with different rules" I always think of euclidean vs noneuclidean geometry, where you actually get different results. But that's not a good fit with saying the reals are uncountable; as far as I know people who are unhappy with different infinite cardinalities don't have an alternate system (and in fact cannot have an alternate system, since the diagonalization theorem is a theorem where the parallel postulate is an axiom); they just deal with their unhappiness by ignoring the idea.

And often it's quite useful to understand the "standard" to see why deviations were thought up. An example would be non-classical logic.

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