decoration decoration
Stories

GROKLAW
When you want to know more...
decoration
For layout only
Home
Archives
Site Map
Search
About Groklaw
Awards
Legal Research
Timelines
ApplevSamsung
ApplevSamsung p.2
ArchiveExplorer
Autozone
Bilski
Cases
Cast: Lawyers
Comes v. MS
Contracts/Documents
Courts
DRM
Gordon v MS
GPL
Grokdoc
HTML How To
IPI v RH
IV v. Google
Legal Docs
Lodsys
MS Litigations
MSvB&N
News Picks
Novell v. MS
Novell-MS Deal
ODF/OOXML
OOXML Appeals
OraclevGoogle
Patents
ProjectMonterey
Psystar
Quote Database
Red Hat v SCO
Salus Book
SCEA v Hotz
SCO Appeals
SCO Bankruptcy
SCO Financials
SCO Overview
SCO v IBM
SCO v Novell
SCO:Soup2Nuts
SCOsource
Sean Daly
Software Patents
Switch to Linux
Transcripts
Unix Books
Your contributions keep Groklaw going.
To donate to Groklaw 2.0:

Groklaw Gear

Click here to send an email to the editor of this weblog.


Contact PJ

Click here to email PJ. You won't find me on Facebook Donate Paypal


User Functions

Username:

Password:

Don't have an account yet? Sign up as a New User

No Legal Advice

The information on Groklaw is not intended to constitute legal advice. While Mark is a lawyer and he has asked other lawyers and law students to contribute articles, all of these articles are offered to help educate, not to provide specific legal advice. They are not your lawyers.

Here's Groklaw's comments policy.


What's New

STORIES
No new stories

COMMENTS last 48 hrs
No new comments


Sponsors

Hosting:
hosted by ibiblio

On servers donated to ibiblio by AMD.

Webmaster
Problem is | 380 comments | Create New Account
Comments belong to whoever posts them. Please notify us of inappropriate comments.
Problem is
Authored by: PolR on Monday, May 28 2012 @ 10:59 PM EDT
You forget the hypotheses of the theorems.

First these are not a theorems about math. These are theorems about formal systems with the capability of reproducing Peano arithmetic. This is a large portion of math but this is not all of math. These theorems don't apply to Boolean algebra for example.

Second, if you don't have consistency, you don't have math. The incompleteness theorems work because consistency is assumed in the hypotheses. Without an assumption of consistency the proofs mean nothing.

You wrote literally:

maths in its very foundations is built on incompleteness and inconsistency.
Maybe you wrote something that means more than you intended, but it is not possible to build math on inconsistency. You can't be confident your proofs mean anything when the conclusions are inconsistent. What Gödel's second incompleteness theorem said is you can't prove the consistency of Peano arithmetic from within the same system. This doesn't mean it isn't consistent, just that the proof is unavailable from within the same system. This is an incompleteness theorem because it is about a proof that can't be done. This is not about (in)consistency itself.

FWIW Gentzen has managed to prove the consistency of first order arithmetic using transfinite induction.

What Gödel proved is not that you cannot have a solid foundation underpinning mathematics. He proved that you cannot build this foundation and prove this it is solid (ie consistent) using the methods envisioned by David Hilbert. The possibility of having a solid foundation you can't prove to be solid is left open. Then empirical evidence of solidity may be available. If after a few decades of searching for the inconsistencies none are being found, mathematicians may have grounds to believe their foundations are consistent.

Mathematicians can't imagine a system of logic for writing some foundations of mathematics which won't be subject to the methods of Gödel. But how can we know whether this is a failure of human imagination or a fundamental limitation of mathematics? Most mathematicians think it is a fundamental limitation on the grounds that all attempts to find a work around have failed. This is drawing conclusions from strong empirical evidence. This is different from proven mathematical knowledge.

As Gentzen has illustrated, it may still be possible to discover some principles of logic or mathematics which will surprise us. Gentzen proof has its own assumptions and I am not familiar with it. I don't know how big of a work around Gödel it offers. But I think it hints that we should be careful not to overstate what Gödel theorems actually mean. They are very strong theorems but they are not without limits. And I also think you tend to overstate them because you are not familiar with their limits.

[ Reply to This | Parent | # ]

Groklaw © Copyright 2003-2013 Pamela Jones.
All trademarks and copyrights on this page are owned by their respective owners.
Comments are owned by the individual posters.

PJ's articles are licensed under a Creative Commons License. ( Details )