Comments on: Real Numbers

By: Steve Landsburg

Steve Landsburg — Sun, 17 Jan 2010 19:16:19 +0000

CIP: I understand your discomfort with the lack of a clear and fully articulated criterion. But I also think there is a clear difference between Newton’s law of gravity, which is subject to revision, and the various truths about the natural numbers, which are decidedly non-revisable.

By: CapitalistImperialistPig

CapitalistImperialistPig — Sun, 17 Jan 2010 18:56:13 +0000

The problem with questions like “Were the natural numbers discovered or were they invented?” is that there is no obvious criteria for resolving the question. What do you mean by each? The distinction is hardly simple.

Consider an easier case. Was Newton’s law of gravity discovered or invented? The only sensible answer is “both.” Newton invented a mathematical model which captured much (but not all) of the behavior of gravity in the Universe. The inexactitude of the law proves that he didn’t just discover a pre-existing reality, he invented something a lot – but not exactly – like that reality. Is math – the natural numbers, say – like that? Come back when you can think of a criterion that might answer the question.

By: Daniel Morgan

Daniel Morgan — Fri, 15 Jan 2010 18:57:14 +0000

@Steve Landsburg

I definitely agree with what you just said, but I still don’t want to use the word “exists.”

From where I am in the book, it seems like doing so has empirical consequences, but I’m not sure how this jives with the requirement of nonfalsifiability usually given to empirical claims.

Also, if math exists because it has properties, does it follow that nonsense math exists because it also has properties? “2 + 2 = 5″ has the property of being incorrect. Does this mean that, if we accept your origin of the universe arguments from the book, that illogical mathematical systems are manifested as inconsistent parallel universes?

And, if so, how do we know that we’re not living in one now?

By: Al V.

Al V. — Fri, 15 Jan 2010 15:11:01 +0000

@Steve Landsburg, re. your statement “Fermat’s Last Theorem … would be true in any Universe, whatever the laws of physics in that Universe might be.” Tegmark hypothesizes different levels of universes. Level I are universes that obey our laws of physics. Level II are those that have the same laws of physics, but diffent physical constants. Level III are those that have different laws of physics, but the same mathematical foundation. And Level IV are those have have different mathematics. Thus, Fermat’s Last Theorem would be valid in Levels I, II, and III, but not at Level IV.

Of course, this is completely hypothetical.

By: autogen

autogen — Fri, 15 Jan 2010 07:04:38 +0000

You are quite right.. This is a religious question. :)

By: Steve Landsburg

Steve Landsburg — Fri, 15 Jan 2010 03:40:30 +0000

SJA:

Do mean to say that we know numbers were discovered rather than invented because we have reason to believe certain things about numbers that we cannot prove deductively?

SJA: Yes, this is precisely what I mean to say.

By: SJA

SJA — Fri, 15 Jan 2010 03:37:02 +0000

What is an example of something that has properties that *are* entailed by reasoning from a set of axioms? Tautologies? Do mean to say that we know numbers were discovered rather than invented because we have reason to believe certain things about numbers that we cannot prove deductively?

By: Steve Landsburg

Steve Landsburg — Fri, 15 Jan 2010 03:23:56 +0000

Ben: Fermat’s Last Theorem is an example.

By: Ben

Ben — Thu, 14 Jan 2010 18:49:01 +0000

@Steve Landsburg:
“the notion that the natural numbers exist only in a sense entailed by axiomatic reasoning runs up against the fact that they have recognizable properties that are *not* entailed by reasoning from any set of axioms. This is really the crux of the matter”

I’m not really sure what you have in mind. Surely it is either assumed axiomatically, entailed by reasoning from axioms, or is an analogical correspondence with an object in the real world, and so not actually a property of the number.

Is there anything else? Perhaps an example would help?

By: CapitalistImperialistPig

CapitalistImperialistPig — Thu, 14 Jan 2010 16:58:17 +0000

I myself am sure of the Platonic existence of the Natural Numbers and even the Integers, Rationals, Reals and Complex (no matter what Kronecker said). Quaternions too – but Octonions – not so much.