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Gödel And The Limits of Rationality

From the conclusion of “Cantor’s Diagonalization Method” by Alexander Kharazishvili (and I’ll assume the reader has some familiarity with Gödel’s famous theorems):

Tarski’s theorem leads to Gödel’s first incompleteness theorem. Let Pr denote the set of natural numbers corresponding to the provable sentences in formal arithmetic. Pr is recursively enumerable. Since any recursively enumerable set of integers is arithmetical, Pr cannot coincide with Tr [“the set of all natural numbers that correspond to those sentences true in the standard model of formal arithmetic”]. It can only be a proper subset of Tr. Truth and provability are not the same. That this discovery was revolutionary hardly does justice to its significance. Thus, there is a sentence S that is true, but not provable, in the standard model of formal arithmetic. Neither S, nor its negation, ¬S, is provable and formal arithmetic is incomplete. A similar argument is applicable to any recursive mathematical theory that is stronger than formal arithmetic (e.g., ZFC set theory).

We’ve talked about this many times: the idea that reason and rationality are enough, or are the ultimate (as in best) form of thought, are obviously false ideas. They are useful and necessary, but alone they are insufficient.

Reason, in the way I am thinking about it, is almost mechanistic. It takes a pile of thoughts, processes them by known rules, and spits out ideas. The processing is like a machine, which can be souped up or in poor repair. However complex it is, it isn’t anything more than brute mental force.

The ore at the base of the pile of thoughts must be supplied from outside. Reason needs fuel it can’t provide itself. Reason operating only itself is like a perpetual motion machine, an impossibility.

It is not reason that concludes Reason is reasonable. It is not Reason that proves truths that cannot be proved by Reason. “Truth and provability are not the same.” Provability by Reason, that is. Proof by other mechanisms is still possible; indeed, necessary.

All mathematics proceeds from two things: (1) unproveable-by-Reason propositions (axioms) and (2) unproveable-by-Reason belief Reason will work to generate new true propositions. This sequence does not apply to mathematics alone, but all thought.

Therefore there must be something beyond Reason providing our deepest and most consequential truths.

22 thoughts on “Gödel And The Limits of Rationality Leave a comment

  1. Well, we have the Scientific Method which proceeds from unprovable (by reason or any other method) conjectures to testable deductions.

    In science, however, it is a physical fact that an infinite number of measurements is impossible, whereas in mathematics, the restriction that proofs must be finite seems an arbitrary restriction.

    There is also no evidence that the laws of physics require anything more than Presburger arithmetic, in which case it seems that all of physical Reality is amenable to reason.

  2. Tom – The electro-magnetic force incorporates opposing forces. Opposing forces require negative numbers to represent their repelling and attracting properties. Thus, it lies outside Presburger arithmetic. Therefore your statement is false. QED.

  3. This was the nugget behind my novella, “Places Where the Roads Don’t Go.” Roads is a metaphor for proofs. So the title is there are true statements that can’t be reached by the roads of proof. However, Mac, the narrator proves a theorem that the set of provable theorems is dense in the topological space of all true theorems. (That is the SFnal part of the story.) There is no such theorem, but there is at present no proof that there cannot be. That is, for any true theorem, there is a provable theorem arbitrarily close to it.

  4. McChuck – Every calculation that has ever been carried out – that can be carried out – by a finite state machine, is equivalent to calculations with addition and subtraction only.

    If you consult Presburger’s original thesis, you will find that his proof dealt with the INTEGERS, not just natural numbers.

  5. In systems simpler than ‘formal arithmetic’ is is still possible to prove all formal statements in that simpler system, apparently.

    So, for a given formal system you need to prove that it is at least as complicated as formal arithmetic to get the benefits of Gödel.

  6. THIS DOESN’T MAKE SENSE:

    “Reason, in the way I am thinking about it, … takes a pile of thoughts, processes them by known rules, and spits out ideas. However complex it is, it isn’t anything more than brute mental force.

    “The ore at the base of the pile of thoughts must be supplied from outside. Reason needs fuel it can’t provide itself.”

    ISN’T the “ore at the base” the the “known rules”?

    AND aren’t some of those “known rules” known to be facts? (rhetorical question)

    HERE’s WHERE IT BREAKS DOWN:

    All mathematics proceeds from two things:
    (1) unproveable-by-Reason propositions (axioms) and
    (2) unproveable-by-Reason belief Reason will work to generate new true propositions.

    The error there is that some of those allegedly “unproveable-by-Reason propositions (axioms)” ARE proven by other means, and used by mathematics, and ditto for #2 there. AND, some other facts proven by other means are, sometimes woven into a mathematical (or other) analysis.

    The key thing to consider that warps the entire line of thinking (or “reasoning”) is this tip:

    “…in the way I am thinking about it…”

    If ‘the way you’re thinking about it’ essentially postulates that certain sorts of facts and other proven data are ignored/excluded, then of course you will facilitate your ability to conclude that some of the unprovable beliefs one wishes to be true are true, or may well be true (even when deep down inside you know they ain’t).

    This is an excellent example of how seemingly sophisticated analysis can be used to help retain a belief despite the onslaught of evidence to the contrary.

  7. except there are no “deepest and most consequential truths”… none… there are only beliefs and tautologies… and, wishful thinking.

  8. “The ore at the base of the pile of thoughts must be supplied from outside. Reason needs fuel it can’t provide itself.”

    You mean, facts?

    “It is not reason that concludes Reason is reasonable.”

    We conclude reason is reasonable based on the fact that it works.

    “It is not Reason that proves truths that cannot be proved by Reason.”

    Are you suggesting that Gödel didn’t use reason to create his incompleteness theorem? If not, what truths cannot be proven by reason?

    “All mathematics proceeds from two things: (1) unproveable-by-Reason propositions (axioms) and (2) unproveable-by-Reason belief Reason will work to generate new true propositions. This sequence does not apply to mathematics alone, but all thought.”

    It doesn’t apply to all thought, as all thought isn’t about starting with axioms and working from there without reference to the outside world.

  9. Shecky R,

    Like your namesake, you always crack me up. So. Is it a belief, tautology, or wishful thinking, that there are only beliefs, tautologies, and wishful thinking? Whichever it is, it can’t be a deep or consequential truth. Thus, you may as well stop talking.

  10. Give Dr. Briggs a break, OK? He’s obviously been studying up and has made significant progress from his howlers about how Gödel’s theorems mean that you can’t prove axioms. His attempt to extract philosophical significance from his newfound rough understanding is perfectly reasonable, even if it doesn’t improve on countless dorm-lounge sessions and similar beginner’s essays. People are naturally eager to share their excitement about a new-found understanding, and as long as they aren’t mocking others (and our host today shows a restraint unusual for him), we shouldn’t mock him.

  11. “Reason, in the way I am thinking about it, is almost mechanistic. It takes a pile of thoughts, processes them by known rules, and spits out ideas”

    Surprising statement from a non-naturalistic writer. It is reason as viewed by Hobbes, not by Aquinas. Hobbes sought to reduce all reason to addition and subtraction. But it can’t be done.
    A mathematical fruitful concept like “infinity” can not be so reduced. Never mind non-mathematical concepts.

    I doubt Briggs’s view of “mechanistic” processes. I don’t know what is “mechanistic” about taking “a pile of thoughts” and process them by “known rules” (which rules?) and spit out “ideas”?
    I fail to see any meaning in this sentence and fail to see any connection with Goedel’s theorem.

  12. YOS,
    How do you distinguish a true but not provable theorem from a merely provable theorem?
    How is truth of a proposition different from its provability?

  13. The Fesian galahs (a galah is a type of parrot, fairly ubiquitous in Australia, that can be taught to repeat sounds that seem like words) are still trying to confuse philosophy with fantastic conjectures detached from the science of logic.

    Logic is the rules for consistent and coherent reasoning that is based on “the law of non-contradiction”: A “thing” (say an idea) and its contradiction cannot both be true.

    Reason should rest on “unproveable” but “self evident” (one should say “commonsense”) premises, which are propositions for which the only alternative is their contradiction and which contradiction is self-contradictory and thus absurd. A couple of simple examples: “I exist”; “the whole is greater than the part”; and my favourite, “a thing that does not exist cannot cause itself to exist”… there are more.

    Anything “provable” relies on such premises because without them nothing is constant and, therefore, knowable in any way at all. “Modern philosophy” is all about rationalising fantastic conjectures (like “Evolution” where Nothing is assumed to turn itself into Everything without any first or final cause).

    The nutcases proposing this nonsense from their academic ivory towers (a fortification designed to prevent the ingress of commonsense) have devised another con-job they call “para-consistent logic” whereby they ditch inconvenient commonsense premises that do not allow their desired ideology in favour of conveniently assumed premises that will facilitate the desired “conclusion” or ideology.

  14. Now Lee, if you only weren’t so determined to find fault, you’d realize this post had nothing whatever new to say about Godel, or about any kind of mathematics, or even about the unprovability of axioms. Instead, it is a proof against empiricism, and a pointer to the argument that since our deepest truths are given to us, they must be given by Someone.

  15. @Briggs, October 17, 2018 at 5:27 am
    With logic (and maths) we attempt exemplary exercises on unperilous simulations of confrontation with reality.

  16. So, for a given formal system you need to prove that it is at least as complicated as formal arithmetic to get the benefits of Gödel.

    Since theoretical physics is largely higher mathematics, it bodes ill for a Theory of Everything.

  17. ISN’T the “ore at the base” the the “known rules”?

    No. Dr. Briggs clearly distinguished them. Consider the reasoning:
    M. All men are mortal.
    m. Socrates is a man.
    /.: Socrates is mortal.

    The rule is modus ponens.
    The ore is the mortality of human beings and the humanity of Socrates. These are processed through the rule to a conclusion.
    ###

    AND aren’t some of those “known rules” known to be facts?

    Facts are not rules any more than variables are an equation. But “R is a rule” can be a fact, even when R is not.
    “Fact” comes from factum est, “something which has the property of having been made.” As such it is cognate with “deed” and the word “indeed” is still used as “in fact.”
    OTOH, “truth” comes from triewð (W.Saxon), or treowð (Mercian), meaning “faithfulness.” Which is why a groom pledges to be true to his bride. And they are “be-trothed”
    Thus, natural science theories are supposed to be “true to the facts” while literature is supposed to be “true to life.” In the fable “Beauty adn the Beast” there are no facts; but there is a truth: viz., “Sometimes a person must be loved before he is lovable.”

    We conclude reason is reasonable based on the fact that it works.

    Whatever “works” means. Something is reasonable iff a reason can be given for it.

    Are you suggesting that Gödel didn’t use reason to create his incompleteness theorem?

    No, I think he is suggesting that Reason cannot prove truths that cannot be proved by Reason. For example, the Axiom of Choice in ZF set theory.

    If not, what truths cannot be proven by reason?

    “A sycamore once grew in my front yard.”
    “My father loved me.”
    “Wellington won the battle of Waterloo.”
    “This formula is unprovable-in-the-system”
    “Science works.”

    However, I would propose a distinction between Logid and Reason. Something could be reasonable [i.e., likely true] without being provable. “Proven by reason” is going a lot further than simply “reasonable.”

    his howlers about how Gödel’s theorems mean that you can’t prove axioms.

    More precisely, Gödel’s theorems mean that you must have axioms. Provability is always “within-the-system”. Hence, given the first four axioms of Euclidean geometry, we cannot prove the fifth; so it (or another like it) must be an axiom. Similarly, the Axiom of Choice cannot be proven within the remaining axioms of ZF set theory, which is why it is so often included as an axiom in itself. Oddly, the Continuum Hypothesis cannot be proven within ZF+AC, but AC can be proven within ZF+CH, which is why CH is so often taken as true-but-unprovable.

    How is truth of a proposition different from its provability?

    The proposition “This formula is unprovable-in-the-system” is true, but unprovable.

    Gödel’s theorem states that in any consistent system which is strong enough to produce simple arithmetic there are formulae which cannot be proved-in-the-system, but which we can see to be true. Essentially, we consider the formula which says, in effect, “This formula is unprovable-in-the-system”. If this formula were provable-in-the-system, we should have a contradiction: for if it were provablein-the-system, then it would not be unprovable-in-the-system, so that “This formula is unprovable-in-the-system” would be false: equally, if it were provable-in-the-system, then it would not be false, but would be true, since in any consistent system nothing false can be provedin-the-system, but only truths. So the formula “This formula is unprovable-in-the-system” is not provable-in-the-system, but unprovablein-the-system. Further, if the formula “This formula is unprovable-in- the-system” is unprovable-in-the-system, then it is true that that formula is unprovable-in-the-system, that is, “This formula is unprovable-in-the-system” is true.
    — J.R. Lucas, Philosophy, XXXVI, 1961, pp.(112)-(127)

  18. Apologies, Tom, I didn’t read the summary closely enough. Stupid summaries specifically state it uses only the natural numbers.

    Works for finite state machines. The universe is not a finite state machine.

  19. McChuck – The universe is a finite state machine. It is subject to the Bekenstein Bound.

    And yes, much on the internet regarding Presberger’s thesis is misleading for some reason. “Integer” is even in the title.

  20. ““Reason, in the way I am thinking about it, … takes a pile of thoughts, processes them by known rules, and spits out ideas. However complex it is, it isn’t anything more than brute mental force.
    “The ore at the base of the pile of thoughts must be supplied from outside. Reason needs fuel it can’t provide itself.””
    The ore, like the thoughts are information.
    That comes from somewhere. From empiricism and from the knowledge of the presence of God. For atheists it can only come from empirical evidence. All science that is at the furthest reaches from common understanding, is still based on information handed down, or rediscovered first hand. It all started with sensing, knowing, in it’s various forms; for that, consciousness is required.
    Knowing truth is a separate thing from empiricism. Knowing truth is innate given some information ‘fed in’.
    Reason is just the conduit for working out, manipulating and discovering truth. In science, verifiable with the outside world.
    There is no need to speak of non reality. I agree with Michael 2’s assessment. Obsessing about reality is a sign that reality has become somewhat distant and requires proof. Like natural law, it doesn’t need a cheer leader.

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