Tiff

10 Jan 2009 311 views
 
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photoblog image YES IT'S NOT AUTUMN

YES IT'S NOT AUTUMN

Mr Evans & I were having a conversation over the fence this morning. He explained that in mathematical logic, Gödel's incompleteness theorems, proved by Kurt Gödel in 1931, are two theorems stating inherent limitations of all but the most trivial formal systems for arithmetic of mathematical interest. The theorems are of considerable importance to the philosophy of mathematics. They are widely regarded as showing that Hilbert's program to find a complete and consistent set of axioms for all of mathematics is impossible, thus giving a negative answer to Hilbert's second problem

Gödel's first incompleteness theorem, perhaps the single most celebrated result in mathematical logic, states that:

Any effectively generated theory capable of expressing elementary arithmetic cannot be both consistent and complete. In particular, for any consistent, effectively generated formal theory that proves certain basic arithmetic truths, there is an arithmetical statement that is true, but not provable in the theory

Gödel's second incompleteness theorem can be stated as follows:

For any formal recursively enumerable (i.e., effectively generated) theory T including basic arithmetical truths and also certain truths about formal provability, T includes a statement of its own consistency if and only if T is inconsistent.

Authors including J. R. Lucas have debated what, if anything, Gödel's incompleteness theorems imply about human intelligence. Much of the debate centers on whether the human mind is equivalent to a Turing machine, or by the Church-Turing thesis, any finite machine at all. If it is, and if the machine is consistent, then Gödel's incompleteness theorems would apply to it.

Hilary Putnam (1960) suggested that while Gödel's theorems cannot be applied to humans, since they make mistakes and are therefore inconsistent, it may be applied to the human faculty of science or mathematics in general. If we are to believe that it is consistent, then either we cannot prove its consistency, or it cannot be represented by a Turing machine.

I thought about this and thought yes: aren't human beans interesting.

YES IT'S NOT AUTUMN

Mr Evans & I were having a conversation over the fence this morning. He explained that in mathematical logic, Gödel's incompleteness theorems, proved by Kurt Gödel in 1931, are two theorems stating inherent limitations of all but the most trivial formal systems for arithmetic of mathematical interest. The theorems are of considerable importance to the philosophy of mathematics. They are widely regarded as showing that Hilbert's program to find a complete and consistent set of axioms for all of mathematics is impossible, thus giving a negative answer to Hilbert's second problem

Gödel's first incompleteness theorem, perhaps the single most celebrated result in mathematical logic, states that:

Any effectively generated theory capable of expressing elementary arithmetic cannot be both consistent and complete. In particular, for any consistent, effectively generated formal theory that proves certain basic arithmetic truths, there is an arithmetical statement that is true, but not provable in the theory

Gödel's second incompleteness theorem can be stated as follows:

For any formal recursively enumerable (i.e., effectively generated) theory T including basic arithmetical truths and also certain truths about formal provability, T includes a statement of its own consistency if and only if T is inconsistent.

Authors including J. R. Lucas have debated what, if anything, Gödel's incompleteness theorems imply about human intelligence. Much of the debate centers on whether the human mind is equivalent to a Turing machine, or by the Church-Turing thesis, any finite machine at all. If it is, and if the machine is consistent, then Gödel's incompleteness theorems would apply to it.

Hilary Putnam (1960) suggested that while Gödel's theorems cannot be applied to humans, since they make mistakes and are therefore inconsistent, it may be applied to the human faculty of science or mathematics in general. If we are to believe that it is consistent, then either we cannot prove its consistency, or it cannot be represented by a Turing machine.

I thought about this and thought yes: aren't human beans interesting.

comments (17)

  • zed
  • Australia
  • 10 Jan 2009, 00:14
Lovely colours, but the theorems gave me a headache... Human beans are interesting smile
Chris: Autumn colours Zed: down in Hampshire in the south of England.

I like to have a light hearted chat about philosophy with my neighbour
  • Larry Bliss
  • Raleigh, North Carolina, USA
  • 10 Jan 2009, 02:13
Ooh, my brain hurts... I'll just look at the pretty picture. smile
Chris: You're welcome Larry!
  • Philine
  • Germany
  • 10 Jan 2009, 07:11
Uhhhh, my morning brain could be a living example of "incompleteness"- but without any reading and reflecting I totally agree with your main thesis: "human beans are interesting"- without any exception! Nice that you have given us such a lovely view of a forest in autumn days (I am so free to ignore your title) in order to relax from some brain acrobatics and to walk along the green path snaking without any mathemathical logic through the wood. Is it also possible to talk with Mr. Evans about simple and banal things like weather marmalade cooking -or simply to be silent in order to hear the birds singing?
Chris: I'll ask Mr Evans about marmalade next time we meet Philine
  • Alan Rolfe
  • Great Britain (UK)
  • 10 Jan 2009, 07:13
Ooerr.. my 'ead 'urts! All too much for a Saturday morning, so I'll I'll simply say "Yeah.. whatever" wink
Chris: Sorry to have jolted you into the weekend Alan
De heer Evans en ik hadden een gesprek over het hek van vanmorgen. Hij legde uit dat in wiskundige logica, Gödel's onvolledigheid stellingen, bewezen door Kurt Gödel in 1931, zijn twee stellingen vermelding inherente beperkingen van alle behalve de meest triviale formele systemen voor rekenkundig van wiskundige belang. De stellingen zijn van groot belang voor de filosofie van de wiskunde. Zij worden algemeen beschouwd als blijkt dat Hilbert's programma op zoek naar een volledige en consistente set van axioma's voor het geheel van de wiskunde is onmogelijk, zodat een negatief antwoord op de tweede Hilbert probleem

Gödel's eerste onvolledigheid theorema, misschien wel de meest gevierde resultaat in wiskundige logica, bepaalt:

Elk effectief gegenereerde theorie staat elementaire rekenkundige expressie kan niet worden zowel consistent en volledig zijn. In het bijzonder voor elk consistent, effectief gegenereerd formele theorie dat bewijst bepaalde elementaire rekenkundige waarheden, is er een rekenkundige verklaring dat waar is, maar niet te bewijzen in de theorie

Gödel's tweede onvolledigheid stelling kan worden gesteld als volgt:

Voor elke formele recursief enumerable (dwz daadwerkelijk gegenereerd) theorie T inclusief elementaire rekenkundige waarheden en ook bepaalde waarheden over formele bewijsbaarheid, T bevat een verklaring van zijn eigen consistentie als en slechts als T is inconsequent.

Auteurs waaronder JR Lucas hebben besproken wat, als er iets, Gödel's onvolledigheid stellingen impliceren over de menselijke intelligentie. Veel van de centra debat over de vraag of de menselijke geest is het equivalent van een Turing machine, of door de Church-Turing thesis, een eindige machine op alles. Als dat zo is, en indien de machine is in overeenstemming, dan Gödel's onvolledigheid stellingen zou van toepassing zijn.

Hilary Putnam (1960) stelde dat, terwijl Gödel's stellingen niet kan worden toegepast op de mens, omdat ze fouten maken en zijn derhalve strijdig is, kan worden toegepast op de menselijke faculteit wetenschap of wiskunde in het algemeen. Als we moeten geloven dat zij in overeenstemming is, dan kunnen we niet bewijzen de samenhang ervan, of het niet kan worden vertegenwoordigd door een Turing machine.

Ik dacht dat over deze en dacht ja: zijn niet menselijk jansbrood interessant.


Well it makes just as much sense to me in Dutchtongue

Nice picture though
Chris: Thank you O Great One
Lovely picture but the text is incomprehensible to a non-mathematician like me. However, I do make a lot of mistakes...getting up today was one of them
Chris: Well Oggers old thing: we are singing from the same hymnsheet on that one!
Wow ! Very very nice result ! Bravo and a likey !
Chris: Well thank you kindly Zeb
  • Chad Doveton
  • location location location.
  • 10 Jan 2009, 09:43
The gypsy rover came over the hill etc. as the song goes.
Chris: Before my time Mate
  • anniedog
  • United Kingdom
  • 10 Jan 2009, 10:18
I think you've got too much time on your hands Tiff - why don't you take up a useful hobby like macrame or basket-weaving?
Ingrid
Chris: Oh dear...
This really is a very nice picture Sir Tiffo.
Chris: It is actually on Natural Trust property somewhere near Romsey in Hants - but without looking at a map I cannot remember its name. I was with Ellie at the time but she seems to have temporarily deserted us on S/C
Philine has no right to call marmalade banal ... the 'Womens Institute' markets deprived of sales of the above product would collapse and and be out of existance quicker than Woolworths was. .... and then we would be left with just that atrocious Robinson's silver shred version ... Now that is a banal.

Nice picture of the way through the woods Chris ... but the blurb is pretty turgid.

richard
Chris: I like Frank Cooper's Oxford marmalade myself.

You mean you didn't teach the little sods about Godel's Incompleteness Theorems when you were teaching Richard!
  • Philine
  • Germany
  • 10 Jan 2009, 14:34
Oh, sorry, of course I know the difference between jam and marmalade and I myself prefer only thickly sliced and in England made marmalade (not: Marmelade) and I understand that marmalade must be one essential of English identity! - Oh yes, the decline of Woolworth and Wedgewood and our Rosenthal is really! Marble breaks and iron bends. But our marmalade will never end. I suppose that for Mr. Evans and Mr. Phillips II. also maramlade might be a philosophical question! I read a German wiki-article about Kurt Gödel and understood quite less than in the English text, consequently: nothing! We woman are said to prefer the "logic of the heart" (Pascal), but I'm not sure if Mr. E and Mr. Ph. would agree with that hypothesis! (Another possible talk-theme)
Chris: We shall return to these themes in the fullness of time Philine: thank you
  • Philine
  • Germany
  • 10 Jan 2009, 14:37
Sadly I forgot the word "sad" (really sad)!
Chris: Oh very!
Jolly nice picture, Chris,hazy and springlike.
Jolly incomprehensible text that I gave up on, since I only do simple plus, minus, by and divide. And if all human complied to a formula then SC would not exist.
Chris: How ghastly it would be if were were all the same Sheila. Meeting others would have limited appeal
This is absolutly superb Chris. Even better than a painting!
Chris: That is very kind of you to say so Richard
Another nice place to walk through Chris, you certainly know how to find them, as for the blurb, my brain hurts trying to understand itsmile
Chris: It was a very nice autumn day Brian. As for the blurb: don't try thinking too hard!
  • Ellie
  • England
  • 21 Jan 2009, 22:43
For the other Mr Phillips - it's "Great Copse" on the Mottisfont Estate - I took a picture of the sign wink
Chris: Thank you Ellie: it was a very nice day

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