Comment author:SilasBarta
14 April 2010 03:13:36PM
*
1 point
[-]

Please show me where "2+2=4" is or where it exists.

If I take two rocks, and put them in a cup where you can't see, and then put two more rocks in the cup, and then rattle it around and dump out the contents of the cup on the table, just look at it: two plus two equals four inside the cup.

That doesn't prove that 2+2=4. That proves that rocks obey a regularity that is concisely describable by reference to an axiom set under which 2+2=4. That's not the same thing. Math still "exists" only as a (human) representation of other real things.

(I will note in passing that Steven Landsburg, who promotes the "Math exists independently" belief -- indeed, pretty much defines his worldview by it -- argues for the position in his book The Big Questions essentially by cheating and slipping in the definition that "Math exists iff math is consistent." Go fig.)

I could list more examples, and you should have no trouble verifying them experimentally. Were you maybe expecting a stone tablet somewhere, that, if modified, would cause plusOf(2,2) to output five?

No, but I can think of experiences under which I would keep the belief

a) "rocks obey a regularity that is concisely describable by reference to the standard axiom set for math"

but discard the belief

b) "under that axiom set, 2+2=4"

However, to get better insight into why this would happen, you should replace "2+2=4" with "5896 x 5273 = 31089508".

Comment author:byrnema
14 April 2010 03:44:58PM
*
3 points
[-]

These discussions about whether 2+2=4 are confusing because 2+2 is 4 by definition of this operation of addition, and then we see in what cases real world phenomena are described by this operation. If you define 2+2 as anything but 4, then you're just describing a different operation. There are many operations where 2 and 2 give 5.

The problem, always, is that there's no causal connection between mathematics and reality. Suppose you try to force one and say that however many rocks you have in a cup when you combine two cups (each with two rocks), that is going to be THE operation of addition. Then asking what 2+2 has to be is asking how many rocks you can have in the final cup. Well, it's not logically impossible for rocks to follow a rule that every four rocks in a certain small area will make a new rock and increase their number to 5. It just happens, in our reality, that rocks satisfy Peano arithmetic (and don't resonate daughter rocks).

Comment author:SilasBarta
14 April 2010 04:14:33PM
*
2 points
[-]

It just happens, in our reality, that rocks satisfy Peano arithmetic (and don't resonate daughter rocks).

Well, it's complicated by the fact that rocks can break, which means that you need to go into a lot more detail to say the extent to which the standard axioms of math map to rocks. This is why simplistic proofs of 2+2=4 by reference to rock behavior are so misleading and unhelpful.

It's important to unpack exactly what is meant by "2+2=4". The most charitable unpacking I can give is that it means both:

a) There exists an axiom set under which (by implication, not definition), 2+2=4.
b) That axiom set has extremely frequent isomorphisms to (our observations of) physical phenomena.

But most people's brains, for reasons of simplicity, truncate this to "2+2=4". The problem arises when you try to take this representation and locate it somewhere in the territory, in which case ... well, you get royalties from The Big Questions, but you're still committing the mind-projection fallacy :-P

Under a sufficiently high temperature, they will coalesce into one rock.

I don't quite have the argument framed, but it's something like arithmetic applies in our world, but only under circumstances which have to be specified separately from arithmetic.

## Comments (119)

Best*1 point [-]That doesn't prove that 2+2=4. That proves that rocks obey a regularity that is concisely describable by reference to an axiom set under which 2+2=4. That's not the same thing. Math still "exists" only as a (human) representation of other real things.

(I will note in passing that Steven Landsburg, who promotes the "Math exists independently" belief -- indeed, pretty much defines his worldview by it -- argues for the position in his book

The Big Questionsessentially by cheating and slipping in the definition that "Math exists iff math is consistent." Go fig.)No, but I can think of experiences under which I would keep the belief

a) "rocks obey a regularity that is concisely describable by reference to the standard axiom set for math"

but discard the belief

b) "under that axiom set, 2+2=4"

However, to get better insight into why this would happen, you should replace "2+2=4" with "5896 x 5273 = 31089508".

*3 points [-]These discussions about whether 2+2=4 are confusing because 2+2 is 4 by definition of this operation of addition, and then we see in what cases real world phenomena are described by this operation. If you define 2+2 as anything but 4, then you're just describing a different operation. There are many operations where 2 and 2 give 5.

The problem, always, is that there's no causal connection between mathematics and reality. Suppose you try to force one and say that however many rocks you have in a cup when you combine two cups (each with two rocks), that is going to be THE operation of addition. Then asking what 2+2

has to beis asking how many rocks you can have in the final cup. Well, it's notlogically impossiblefor rocks to follow a rule that every four rocks in a certain small area will make a new rock and increase their number to 5. It just happens, in our reality, that rocks satisfy Peano arithmetic (and don't resonate daughter rocks).*2 points [-]Well, it's complicated by the fact that rocks can

break, which means that you need to go into a lot more detail to say the extent to which the standard axioms of math map to rocks. This is why simplistic proofs of 2+2=4 by reference to rock behavior are so misleading and unhelpful.It's important to unpack exactly what is meant by "2+2=4". The most charitable unpacking I can give is that it means both:

a) There exists an axiom set under which (by implication, not definition), 2+2=4.

b) That axiom set has extremely frequent isomorphisms to (our observations of) physical phenomena.

But most people's brains, for reasons of simplicity, truncate this to "2+2=4". The problem arises when you try to take this representation and locate it somewhere in the territory, in which case ... well, you get royalties from

The Big Questions, but you're still committing the mind-projection fallacy :-PUnder a sufficiently high temperature, they will coalesce into one rock.

I don't quite have the argument framed, but it's something like arithmetic applies in our world, but only under circumstances which have to be specified separately from arithmetic.