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Comment by icen | Hacker News Reader

1+1=2

And no, apples are not the same things as numbers.

Some people would say that the relational patterns in numbers are a kind of definition, and that such statements aren't so much truths but consequences of those patterns - and that it's only when you're using the two '1's to represent real world details that the '2' is true or false.

(and BTW, there are real-world situations where '1' and '1' do not make '2'... e.g. putting a snake and a mouse in a box. We only apply maths statements like that to the kinds of entities/situations where they will apply).

That's an axiom. It doesn't say anything about the real world, it just introduces a framework of thought.

In other words, if you can imagine a reality where 1+1=2 for some meaning of '1', '2', '+' and '=', then a series of conclusions will hold true.

But all that says is that 1+1=2 is too obvious. How about some other, less straightforward mathematical theorem, then, like Fermat's Last Theorem.

You can hardly accuse that of being an axiom, or just introducing frameworks of thought.

Mathematics is based on logical deduction. You start with a set of rules and you prove additional rules.

Science is based on (lots of things but primarily) empirical induction. You observe some phenomena, decide that they are of the same type, generate a model that predicts how all phenomena of that type behave. By necessity you observe an infinitesimal fraction of the phenomena in your type.

In math proofs are universal and timeless and complete. There is no additional observation to be made. A proof can be wrong, but if we conclude that it is logically correct then that's it. (edit: well, there's the incompleteness theorem: it could be that a set of axioms is inconsistent, but then EVERYTHING based on those axioms becomes invalid. It's still quite different than scientific progress which changes models a little bit at a time.)

Scientific models are approximations of reality(very good approximations often!) but they get tweaked as new observations come in. Or just wholesale thrown out(or at least become obsolete), see Newtonian orbital mechanics vs relativistic orbital mechanics. Newton was right-ish, but relativistic solutions are right-er(possibly fully correct?)[1].

With deductive reasoning you are expanding a definition. With inductive reasoning you are guessing at connections between observed events.

[1] http://physics.stackexchange.com/questions/26408/what-did-ge...

Or just wholesale thrown out(or at least become obsolete), see Newtonian orbital mechanics vs relativistic orbital mechanics.

Newtonian orbital mechanics is not obsolete: Within the limits of its applicability, it's a perfectly fine framework and in use to this day.

It's not an axiom, it's a theorem (or maybe a definition of "2"). The more usual definition of "2" is "S(S(0))" (the successor of the successor of zero). "1" is "S(0)". "S(0) + S(0) = S(S(0))" is a theorem, a consequence of the axioms "for all x and y, x + S(y) = S(x + y)" and "for all x, x + 0 = x".

I called it an axiom for the sake of brevity. I felt that reciting the Peano axioms would only cloud my point.

Was I 100% precise? No. Did you add something meaningful to the discussion? That's for you to answer.

"axiom" is two letters shorter than "theorem", so no, brevity doesn't cut it as an excuse. The difference between an axiom and a theorem is, undoubtedly, germane to a thread about the different ways we can know things.

I would go further: is "Axioms exist" true? You don't find them lying around in nature. They exist only in people's minds. I would argue that this is an inherent metaphysical question.

The scientismist's (?) standpoint could be that it exists because it is manifested as a concept in human brains. But by the same argument, wouldn't they then need to acknowledge that God exists?

The crux of the matter is to do science, you need formalism to express your theories, to interpret measurement results etc. Those formalisms are what we usually call "math". But any mathematical system has axioms, i.e. something we need to just "believe" in in the first place, and those believes are not themselves provable by science.

Axioms can be shown to be useful or not. This is why some mathematicians reason with or without axiom of choice. Likewise, systems of axioms can be shown to be consistent or not.

But maybe one day we will know how our brains work, and we will understand what is the neural correlate of "1" , "+" and "2" and we will be able to clearly see that the only answer to 1+1 is 2 in an empirically falsifiable way.

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