• Pussycat
    226
    Given Tarski's undefinability of truth, any system has no other choice but to receive its fundamental truths from a higher meta-system.alcontali

    But Tarski's and Godel's theorems work within a very strict - formal - mathematical framework. Do you think we can extrapolate them to the real world?
  • alcontali
    1.2k
    But Tarski's and Godel's theorems work within a very strict - formal - mathematical framework. Do you think we can extrapolate them to the real world?Pussycat

    Well, rather: extrapolate them to how we perceive the real world. Stephen Hawking lectured the following on the subject:

    What is the relation between Godel’s theorem and whether we can formulate the theory of the universe in terms of a finite number of principles? One connection is obvious. According to the positivist philosophy of science, a physical theory is a mathematical model. So if there are mathematical results that can not be proved, there are physical problems that can not be predicted.

    Some people will be very disappointed if there is not an ultimate theory that can be formulated as a finite number of principles. I used to belong to that camp, but I have changed my mind. I'm now glad that our search for understanding will never come to an end, and that we will always have the challenge of new discovery.
    Stephen Hawking on Gödel and the End of Physics
  • A Seagull
    255
    Given Tarski's undefinability of truth, any system has no other choice but to receive its fundamental truths from a higher meta-system.alcontali

    Or perhaps the more sensible thing to do would be to abandon any attempt to define 'truth' or even to use it in any formal system.
  • Pussycat
    226
    Well, rather: extrapolate them to how we perceive the real world. Stephen Hawking lectured the following on the subject:alcontali

    How we perceive scientifically the real world. I mean, if the only means of perception we have is science, but is it? This is scientism, which may be right of course.

    And after all, in both Tarksi and Godel, both concepts of proof and truth are extremely well defined. In the case of the real world however, even from a scientific outlook, they are completely vague: you can conjure them as you see fit. What is truth? What is proof? (well, not the TPF user)

    Stephen Hawking on Gödel and the End of Physicsalcontali

    Yeah, I remember reading it some years back. But the next sentence in his lecture, I think is important: "Without it, we would stagnate".

    And also: "Godel’s theorem ensured there would always be a job for mathematicians. I think M theory will do the same for physicists".

    So, in the above, Hawking draws the analogy between Godel's theorem and M theory, believing that M theory is to the real/physical world what Godel's theorem is to the mathematical equivalent. And therefore we will ad infinitum be looking for answers, which is a good thing, because otherwise we would stagnate. Stagnation, that comes from complete knowledge of how stuff works, is for Hawking the worst that can happen to us. And therefore he is relieved.

    This lecture was given in 2002. But then in 2010, after the publication of his book, "The Grand Design", he has a change of heart.

    http://edition.cnn.com/2010/WORLD/europe/09/11/stephen.hawking.interview/index.html

    "Science is increasingly answering questions that used to be the province of religion," Hawking replied. "The scientific account is complete. Theology is unnecessary."

    Wow! "The scientific account is complete"!

    Putting theological statements aside (or maybe not), he now believes that M-theory gives a complete description of reality! So I couldn't help it back then and send him an e-mail, well actually not to him because the probabilities of an answer would be next to zero, but to his co-author Leonard Mlodinow, and referring to the 2002 lecture, I asked him what made him change his mind, but I didn't get an answer, duh. :) So I am still curious.
  • Pussycat
    226
    The pursuit of wisdom. Wisdom, in turn, does not merely mean some set of correct statements, but rather is the ability to discern the true from the false, the good from the bad; or at least the more true from the less true, the better from the worse; the ability, in short, to discern superior answers from inferior answers to any given question.Pfhorrest

    Yeah, I think Aristotle was along the same lines, if I remember correctly. Socrates also. After all, if you don't praise your own house, it will fall down on you, like they say. But saying "to any given question", this opens philosophers up, it makes them vulnerable to ridicule. And there you have Aristophanes in his "Clouds", having Socrates wondering about a flea's long jump.

    To that end, philosophy must investigate questions about what our questions even mean, investigating questions about language; what criteria we use to judge the merits of a proposed answer, investigating questions about being and purpose, the objects of reality and morality respectively; what methods we use to apply those criteria, investigating questions about knowledge and justice; what faculties we need to enact those methods, investigating questions about the mind and the will; who is to exercise those faculties, investigating questions about academics and politics; and why any of it matters at all.Pfhorrest

    For sure, all these are part of our public and private investigations. But what of philosophy? What is its agenda? What does philosophy want?

    The tools of philosophy can be used against that end, but I prefer to call that "phobosophy" instead.Pfhorrest

    One could also use the term foolosophy.
  • 180 Proof
    715
    One could also use the term foolosophy.Pussycat
    :smirk:

    Thanks for the malaprop! It complements my own foolery.

    Perhaps the topic question of this thread should be (more prosaically - pardon the heideggerian echo) reformulated:

    Is thinking dead? and if so can we revive it? :chin:
  • Pfhorrest
    1.2k
    But saying "to any given question", this opens philosophers up, it makes them vulnerable to ridicule.Pussycat

    How so? Pursuing some question may be ridiculous, but philosophy is just about finding ways to pursue questions, not about picking which questions to pursue.

    But what of philosophy? What is its agenda? What does philosophy want?Pussycat
    What does running want? An activity isn't the kind of thing that has wants. It's a means. Why run? To get somewhere fast, or for exercise maybe. Why do philosophy? I already answered that.
  • alcontali
    1.2k
    How we perceive scientifically the real world. I mean, if the only means of perception we have is science, but is it? This is scientism, which may be right of course.Pussycat

    Mathematics has no direct empirical take on the world. Its models are always abstract Platonic worlds. It is through its influence on empirical disciplines (such as science) that it affects our real-world view. There are obviously other empirical disciplines such as history with its historical method. However, in my impression, history does not use the language nor the invariants of mathematics.

    And after all, in both Tarksi and Godel, both concepts of proof and truth are extremely well defined. In the case of the real world however, even from a scientific outlook, they are completely vague: you can conjure them as you see fit. What is truth? What is proof? (well, not the TPF user)Pussycat

    There is no proof in empirical disciplines, simply because proof about the physical universe is impossible. The regulatory framework in use in science with which they attempt to maintain correspondence between their logic sentences and the physical universe is obviously far from perfect. Falsificationism is merely a best-effort endeavour.

    "Science is increasingly answering questions that used to be the province of religion," Hawking replied. "The scientific account is complete. Theology is unnecessary." Wow! "The scientific account is complete"!Pussycat

    Any link to that?

    I would be surprised if Hawking has ever repeated the "God of Gaps" ideological conjecture.
  • alcontali
    1.2k
    Or perhaps the more sensible thing to do would be to abandon any attempt to define 'truth' or even to use it in any formal system.A Seagull

    I personally think that Tarski's convention T is an elegant and adequate workaround for the undefinability of truth. The video below explains convention T in approximately 10 minutes and in a surprisingly simple way:

  • A Seagull
    255
    Or perhaps the more sensible thing to do would be to abandon any attempt to define 'truth' or even to use it in any formal system. — A Seagull
    I personally think that Tarski's convention T is an elegant and adequate workaround for the undefinability of truth. The video below explains convention T in approximately 10 minutes and in a surprisingly simple way:
    alcontali

    Well I watched your video. It seems that the main aim of the T convention was to avoid the so called 'liar paradox'.

    But as mentioned/discussed in another thread (Statements are true?), there is no paradox if the assumption that statements are 'true' or 'false' is not made.

    Without the requirement for statements to be 'true' or 'false' but that instead 'true' or 'false' are merely labels that can be appended to a statement, there is no paradox nor problem. It would not even be paradoxical for a statement to be labelled as both 'true' and 'false'.
  • alcontali
    1.2k
    Well I watched your video. It seems that the main aim of the T convention was to avoid the so called 'liar paradox'.A Seagull

    The liar paradox is not used in the proof strategy for the undefinability the truth. The main consideration is Carnap's diagonal lemma:

    • There will be true sentences for which the predicate is false.
    • There will be false sentences for which the predicate is true.

    So, let's try to define a truth predicate. We will now face the following situation:

    • There will be true sentences for which the truth predicate is false.
    • There will be false sentences for which the truth predicate is true.

    So, we will know of a particular sentence that it is true but the truth predicate will say that it is false, and the other way around. That is clearly inconsistent.

    Without the requirement for statements to be 'true' or 'false' but that instead 'true' or 'false' are merely labels that can be appended to a statement, there is no paradox nor problem.A Seagull

    In Carnap's diagonal lemma, the truth of a sentence is an externally supplied label. The whole question is whether this externally supplied label can be replaced by a predicate. It cannot, because that would lead to contradictions.

    It would not even be paradoxical for a statement to be labelled as both 'true' and 'false'.A Seagull

    Actually, it isn't.

    There are logic systems that are many-valued (additional values other than just true and false) or where the truth status of a sentence is many-valued.

    I am not sure what the impact of many-valued logic would be on Carnap's diagonal lemma, which the underlying reason for the undefinability of truth. At first glance, it may mean that all combinations of truth value for the sentence and the truth value for a truth predicate will be populated. In that case, a truth predicate will still be undefinable.

    Concerning incompleteness, making fixes to the logic will also not fix the problem. At the core you have the consideration that a theory with infinite model size will have a model for each infinite cardinality and therefore have an infinite number of models (Löwenheim–Skolem theorem). That allows for facts to be true in one model but false in another. That kind of true facts will never be provable from the theory, because provability requires this fact to be true in all models.
  • Pussycat
    226
    Mathematics has no direct empirical take on the world. Its models are always abstract Platonic worlds. It is through its influence on empirical disciplines (such as science) that it affects our real-world view. There are obviously other empirical disciplines such as history with its historical method. However, in my impression, history does not use the language nor the invariants of mathematics.alcontali

    There is no proof in empirical disciplines, simply because proof about the physical universe is impossible. The regulatory framework in use in science with which they attempt to maintain correspondence between their logic sentences and the physical universe is obviously far from perfect. Falsificationism is merely a best-effort endeavour.alcontali

    And because of this distinction between the formal/mathematical/non-empirical/logical world and the real world which is nothing like the other, we should be really suspicious of attempts made to reconcile the two.

    Tarski's theorem is good for maths, brilliant even, but when it tries to apply itself to the real world, then it is an abomination.

    Any link to that?alcontali

    I've given the link in my post.
  • alcontali
    1.2k
    And because of this distinction between the formal/mathematical/non-empirical/logical world and the real world which is nothing like the other, we should be really suspicious of attempts made to reconcile the two.Pussycat

    Well, they are not being "reconciled". Science uses the language of mathematics to maintain consistency in what it says. Mathematics does not tell science what to say. It only tells science how to say it while eliminating quite a bit of the risk of contradicting itself.

    It is a bureaucracy of formalisms (mathematics) that helps maintaining consistency in another bureaucracy of formalisms (science).

    Tarski's theorem is good for maths, brilliant even, but when it tries to apply itself to the real world, then it is an abomination.Pussycat

    Mathematics never tries to apply itself to the real world.

    Mathematics is not an empirical discipline. It is deductive from first principles only.

    I don't know what scientists can do with Tarski's theorem within their own work. Scientists are otherwise really good at using mathematics to their benefit. Their use of mathematics is certainly not considered to be an abomination.
  • Pussycat
    226


    When Hawking says:

    What is the relation between Godel’s theorem and whether we can formulate the theory of the universe in terms of a finite number of principles? One connection is obvious. According to the positivist philosophy of science, a physical theory is a mathematical model. So if there are mathematical results that can not be proved, there are physical problems that can not be predicted. — Hawking

    ... he makes the error of applying Godel's theorem to physics and the real world. There is no connection, let alone an obvious one. If one thing is obvious, this is Hawking's misinterpretation of the theorem.

    EDIT: Ah yes, almost forgot. If in his 2002 lecture it was obvious for Hawking that Godel's theorem proved that scientific knowledge will never be complete, why then in 2010 he said that "the scientific account is complete"? Most probably another "obvious"! :D I am fed up hearing about obvious connections and conclusions. :worry:
  • alcontali
    1.2k
    he makes the error of applying Godel's theorem to physics and the real world. There is no connection, let alone an obvious one. If one thing is obvious, this is Hawking's misinterpretation of the theorem.Pussycat

    Well, it is still clearly his field that would need to make such connection, because mathematics itself will certainly not make any.

    He may indeed have incorrectly made the connection.

    Still, that can only be assessed by subjecting his connection to the empirical regulatory framework of his field. In my opinion, he may have wanted to provide the paperwork required by the regulations in his own field along with a mechanical procedure to verify the paperwork.

    Therefore, I would agree with a decision of the bureaucracy to reject his hypothesis about that connection for failing to submit the paperwork required for that purpose.
  • Pussycat
    226
    Yeah, well, too much bureaucracy and paperwork involved in the process I guess, thus making philosophy a ... bureaucratic enterprise! Ah those beerocrats! :beer: One way to see where philosophy took a wrong turn, I think.
  • Pussycat
    226
    But anyway, these are matters for the bureaucrats to sort them out themselves, whether there are any connections or not, I mean.

    But what of, what about philosophy?

    At the end of Book VII of Plato's Republic, Socrates discusses with Glaucon the current state of philosophy, and what needs to be done in order to have people trained in it:

    -And as to truth, I said, is not a soul equally to be deemed halt and lame which hates voluntary falsehood and is extremely indignant at herself and others when they tell lies, but is patient of involuntary falsehood, and does not mind wallowing like a swinish beast in the mire of ignorance, and has no shame at being detected?
    -To be sure.
    -And, again, in respect of temperance, courage, magnificence, and every other virtue, should we not carefully distinguish between the true son and the bastard? For where there is no discernment of such qualities, states and individuals unconsciously err; and the state makes a ruler, and the individual a friend, of one who, being defective in some part of virtue, is in a figure lame or a bastard.
    -That is very true, he said.
    -All these things, then, will have to be carefully considered by us; and if only those whom we introduce to this vast system of education and training are sound in body and mind, justice herself will have nothing to say against us, and we shall be the saviours of the constitution and of the State; but, if our pupils are men of another stamp, the reverse will happen, and we shall pour a still greater flood of ridicule on philosophy than she has to endure at present.
    -That would not be creditable.
    -Certainly not, I said; and yet perhaps, in thus turning jest into earnest I am equally ridiculous.
    -In what respect?
    -I had forgotten, I said, that we were not serious, and spoke with too much excitement. For when I saw philosophy so undeservedly trampled under foot of men, I could not help feeling a sort of indignation at the authors of her disgrace: and my anger made me too vehement.
    -Indeed! I was listening, and did not think so.
    -But I, who am the speaker, felt that I was. And now let me remind you that, although in our former selection we chose old men, we must not do so in this. Solon was under a delusion when he said that a man when he grows old may learn many things–for he can no more learn much than he can run much; youth is the time for any extraordinary toil.
    — Plato

    Oh, Socrates, you were a jokester, among many other things. The old will learn to run, and the young will toil. Cause it's true that you can't teach an old dog new tricks.

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