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Any physical system can be described by the case of Gödel, e.g., a model of a car is formal axiomatic system.
Not circular, but seemingly obvious - although when they're used as axioms what seems obvious needs to be stated explicitly. They were popularly known as the Laws of Thought in philosophical circles, but as the article suggests, Russell wasn't a fan of the soundbite title 'Laws of Thought'. Russell was exploring them and others for his formal logical foundations and formally derives ii and iii in Principia Mathematica.Yes .. 'twas a very important phase of history in that they were probing the limits of math's applicability to 'giving access to' what they accepted as being 'reality' (aka Realism).
I'm not a real fan of Russell's philosophy. Wasn't he the one who distilled his so-called Law of Thought?:
i) The law of identity: 'Whatever is, is.'
ii) The law of non-contradiction (alternately the 'law of contradiction'): 'Nothing can both be and not be.'
iii) The law of excluded middle: 'Everything must either be or not be.'
What a pile of truism word-salad! All depends on what is meant by 'is' and 'be'.
Circular reasoning!
What is 'the case of Gödel' ?Any physical system can be described by the case of Gödel, e.g., a model of a car is formal axiomatic system.
Yes they are ... and they're untestable.Not circular,
I'd suggest that Godel might treat these axioms as unprovables outside of the system .. (and thus they may, or may not, 'be true' .. which isn't saying much) ... which makes them useless (low value) outside of that particular system.FrumiousBandersnatch said:I think the meaning of 'is' and 'be' depends on unstated specifiers, e.g. 'true', or 'red'; and as long as they're used consistently, it doesn't matter.
You don't have to believe axioms, they're just fundamentals on which to build logical systems; geometry, for example, has multiple variants, depending on the axioms chosen.Yes they are ... and they're untestable.
Very low usefulness on their own ... although I guess one could believe in them if one wants to ... then put 'em on the pile over there in the corner .. along with every other belief humans have ever come up with.
The point is that they're simply taken to be true (i.e. brute fact) within the system in question, and then their implications can be explored.I'd suggest that Godel might treat these axioms as unprovables outside of the system .. (and thus they may, or may not, 'be true' .. which isn't saying much) ... which makes them useless (low value) outside of that particular system.
I have no disagreements with what you're saying there.You don't have to believe axioms, they're just fundamentals on which to build logical systems; geometry, for example, has multiple variants, depending on the axioms chosen.
The point is that they're simply taken to be true (i.e. brute fact) within the system in question, and then their implications can be explored.
Any physical system can be described by the case of Gödel, e.g., a model of a car is formal axiomatic system.
Ha! .. In the 'Gödel's Incompleteness (extra footage 2)' follow-up, he talks about possible implications for the limitations of science and where theologists usually jump in (which takes us right back in giving some sensible answers to the OP question).For the record and for anyone interested in exploring what Gödel's Incompleteness Theorem is all about in a relatively easily digestable YouTube, try Marcus du Sautoy's 14 minute explanation here.
(Du Sautoy is an excellent presenter on Math topics .. highly recommended):
Gödel's Incompleteness Theorem - Numberphile
Wiki said:Du Sautoy is an atheist but has stated that as holder of the Simonyi Chair for the Public Understanding of Science his focus is going to be "very much on the science and less on religion," perhaps suggesting a difference of emphasis compared to his predecessor in the post, Professor Richard Dawkins. He has described his own religion as being "Arsenal – football", as he sees religion as wanting to belong to a community. Du Sautoy is a supporter of Common Hope, an organisation that helps people in Guatemala.
OK.I have no disagreements with what you're saying there.
The point to notice here however, is that axioms imply a pretense which includes the assumption that we play no role in giving meaning to their terms such as those involving existence (denoted by the word 'is'), and thence that truth exists (ie: 'is true'). This then implies that there is some kind of logical necessity that is independent of our process of generating the 'fuzzy' meanings we hold for those axiomatic statements.
An alternative mind dependent reality (MDR) hypothesis viewpoint, (previously presented), however, diametrically opposes that notion, because it begins by noticing the process where we, (aka humans), control/assign our meanings, rather than pretending it away.
Descartes said "I think therefore I am" (which begs the question: "Don't you mean you think therefore you think you are?"). This is not Solipsism either because it would say: "I know I am because I think, but I don't know anything else" (which the begs the question: "Do you count what you mean by 'I' and by 'existing', as among the things you do know, or among the things you don't know?"
The MDR hypothesis says: "What I mean by 'thinking', and what I mean by 'being', involve a process of meaning generation that depends sensitively on how my mind works, including what I mean by 'how my mind works'. It may be less impressive, but that's the price of actually being 'true' (in the scientifically (objectively) demonstrable sense).
You won't find this scientifically valid viewpoint anywhere in any of these long past philosopher's writings. Russell, Descartes, Popper, Hume (etc) and all of history's philosophers I've investigated all skipped over this fundamental point. History in this instance, serves only past history .. and I think the MDR viewpoint is becoming a preferred more scientific approach (eg: along the similar lead set by Hawking/Mludinow).
Its coming from the MDR perspective that the ridiculousness of the circular arguments in Russell's axioms become apparent. Whilst axioms obviously work in adding tremendous value in the systematic field of mathematics (and formal logic), when used in a general philosophies, they produce nothing different from tightly held, entirely ambiguous and subjective religious beliefs (aka: Religions!)
Ha! .. In the 'Gödel's Incompleteness (extra footage 2)' follow-up, he talks about possible implications for the limitations of science and where theologists usually jump in (which takes us right back in giving some sensible answers to the OP question).
(Du Sautoy took over from Dawkins in his role as the Simonyi Chair for the Public Understanding of Science. Du Sautoy also openly declares his atheisim):
Not quite sure of who/what your complaint is directed towards(?)Atheism has nothing to do with Godel or his Incompleteness Theorems so I see no reason to bring it up. Furthermore, neither Dawkins or Du Sautoy or the Simonyi Chair for the Public Understanding of Science impress me at all. None are relevant to Godel at all.
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