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Roger comments on The mathematical universe: the map that is the territory - Less Wrong
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Comments (119)
Kharfa,
o You seem to be asserting anti-realism and trying to arrive at a correct ontology using an Aristotelian application of deductive logic. Would you like some help with that?
I'm not denying the reality of anything that you can show me. Please show me where "2+2=4" is or where it exists. Using that type of argument that things like this exist is like saying Santa Claus exists. That's possible, but we can't prove it or disprove it, and you can't show him to me. There's no point in discussing it. And, by the way, I don't need any help with that. Patronizing attitudes especially when not backed up by sound reasoning are of no interest to me.
o there's good evidence that minds are made out of math, instead of the contrary position.
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.
If I flip this switch on the left, two lights come on, and then if I flip this other switch the other half come on, and then there are four lights.
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?
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.)
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".
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).
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.
Those interested in a rigorous proof are advised to examine Principia Mathematica.
Not disagreeing, but fleshing out part of what it seems you're trying to say:
Numbers don't exist, that much ought to be clear. I think Eliezer says that numbers are in our minds, and our minds exist, but this is not the case: it's not numbers that are in our minds but representations of numbers.
Mathematical Platonism is, to me, religion for intellectuals. Mathematicians as esteemed at Kurt Goedel have even gone so far as to postulate that mathematics exists in an alternate universe. This is a basic error or at least wildly unparsimonious, akin to saying that modus ponens exists in an alternate universe.
To see how silly this is, it helps to realize that a sufficiently intelligent being would find all our mathematical theorems just alternative ways of stating the axioms, and all our mathematics just axioms and definitions with a bunch of obvious rephrases of the same. It would find our most advanced theorems as simple and obvious as modus ponens is to us - as just rewordings of the axioms and definitions.
From the perspective of a sufficiently intelligent being, mathematics is just a set of initial statements (axioms and definitions), along with humans' silly little demonstrations to help each other realize that a bunch rewordings of those statements (theorems) all mean the same thing.
From the perspective of a sufficiently intelligent being, physics is just a set of initial statements, along with a silly demonstration that history is what you get when you apply those statements over and over. How dull!
Amanojack,
One thing I've noticed that is probably covered somewhere in LW archive (I hope!) is that really smart and rational people can sometimes just be really good at hiding the truth from themselves. In other words, the smarter you are, the better you are at Dark Arts, and the easiest person to trick with Dark Arts is sometimes yourself.
[Heads up: your comments are displaying as one long line requiring side-scrolling instead of with natural line breaks.]