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We can imagine physical constants to be different in a different universe or even not be constant in our own universe.
We can imagine and simulate different physical and information-theoretical laws, e.g. regarding entropy or correlation of force and distance.

However, $1 + 1 = 2$ as we usually define addition over natural, integer, fractional and complex numbers will always and provably hold, regardless of context and numeral system (of course, the representation may change, but not the semantics).

Why is this?
Why can't $1 + 1 = 3$ given the same axioms?

And no, Peano's axioms do not define logic. This is a common misunderstanding. In fact, they are based on logic concepts.

Why are mathematical laws completely invariant such that logic is even possible and sort of inevitable?

From what I understand is that logic seems to be the only solution that works (and everything else breaks down at some point), because it is possible to conclude everything from a false statement (see also In classical logic, why is (p⇒q) True if both p and q are False?).

But then again, this leaves us right at where we left ("Why is it the only solution?").

Also, I don't see how Peano explains structures like this: http://www.nature.com/news/two-hundred-terabyte-maths-proof-is-largest-ever-1.19990

Arc
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  • Look up peano's axioms or peano arithmetic. – Eoin Feb 18 '15 at 23:08
  • What is “$2$” ? – Lubin Feb 18 '15 at 23:10
  • 2 is the result of 1 + 1 according to those axioms. But why is it that we can even work with such axioms and given a set of axioms, there will ever only be a deterministic outcome? Axioms can be changed (laws of physics might define some axioms), but mathematical and logic laws are invariant. – Arc Feb 18 '15 at 23:11
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    Maybe this is philosophy. Synthetic a priori as in Kant ... http://en.wikipedia.org/wiki/Analytic–synthetic_distinction . If so, maybe http://philosophy.stackexchange.com is a better place to ask. – GEdgar Feb 18 '15 at 23:12
  • I have doubts that there are more mathematically-apt philosophers on http://philosophy.stackexchange.com/ than philosophical mathematicians here. – Arc Feb 18 '15 at 23:14
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    You're assuming that the universe "obeys" some mathematical law; and that mathematical law is somehow dictated by the physical reality. That's your first mistake. – Asaf Karagila Feb 18 '15 at 23:18
  • @AsafKaragila so basically, our universe only works and mathematics emerge just because our perception and understanding is based on logic, so we are only ever able to live, work, understand and conclude within logic itself. – Arc Feb 18 '15 at 23:23
  • No, mathematics is one thing; our understanding of reality is another thing. They are two different things which have nothing to do with one another nowadays. – Asaf Karagila Feb 18 '15 at 23:24
  • @AsafKaragila What I meant was not what an individual understands but rather what any one could be capable of understanding or even just perceiving, because of causality and that the universe wouldn't work the way it does if you put two coins next to one another and suddenly you had three coins there, out of nothing and always, as a mathematical, not physical law (physical as in like, vacuum energy or virtual particles). – Arc Feb 18 '15 at 23:28
  • You're very incoherent. It seems that the motivation for your question lies in philosophical (or perhaps pseudo-philosophical) thoughts that lie far *far* beyond the scope of this website. – Asaf Karagila Feb 18 '15 at 23:34
  • What exactly is incoherent? Maybe I shouldn't have asked about why 1+1=2 as it automagically triggers a "Peano reply reflex" but rather why logic itself only has one outcome. Either you follow logic reasoning and arrive at what you can explain and proof, or you don't, and then can't conclude anything that makes sense. Without logic, you just cannot apply any axioms in any meaningful way, so yes, I would definitely say that our subjective reality, though maybe not our universe must obey logic. I did not say, though, that mathematics were rooted in physics, but quite the opposite. – Arc Feb 19 '15 at 08:08
  • Peano may have trouble explaining the structure of numbers and primes, yet there is some. It's this emergence of pattern and meaning that is so puzzling, and it applies to logic as well. https://www.quantamagazine.org/20160313-mathematicians-discover-prime-conspiracy/ – Arc Mar 16 '16 at 00:10

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With axioms you can prove that $1+1\neq 0$ and $1+1\neq 1$. Now, $2$ is only a shorthand for $1+1$. That is, $2$ is defined to be $1+1$.

ajotatxe
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  • Well, yes, it is defined as such. Nonetheless, there is also an "understanding", a concept, behind that definition, in a more concrete sense, intuitively and perceptionally. While you could just say 3 to what is the result of $1 + 1$, we will still understand that glyph 3 to mean our more common 2. – Arc Feb 18 '15 at 23:18
  • And given those very axioms, though you can swap glyphs for 2 and 3, you will never get (the concept of) 3 to be 1's successor and (the concept of) 2 to be 3's successor. Peano's axioms define a distinct, unique and strict order. – Arc Feb 18 '15 at 23:34