scientiae: Kurt Gödel's Incompleteness theorem (1931) proved there are statements in such any language of axioms and algorithms that cannot be proved, and still others that cannot be disproved, using it.
dtgreene: That's not what the theorem stats; in fact, your statement doesn't really say much.
What it says is that
there are statements that can't either be proven or disproven. …
scientiae: You must like the Zen Buddhist koan. :) (I note that you are contravening the specific mathematical principal which prohibits alternate transformations to each side of an equation.)
dtgreene: If I remember correctly the post this is a reply to, all I did was divide by zero. (The transformation on each side of the equation, mainly cancelling the zero (equivalent to dividing by it), is the same; it's just that this particular transformation isn't something that can be consistently defined without violating the axioms of arithmetic.)
Actually, you divided the left-hand side of the equation by one and the right-hand side by two. :)
The
koan reference is because the Zen Buddhists (who are not theists, they practice philosophy) are interested in the gaps in language, which is a symbolic representation of reality. You seem to be missing the point I am making here: a thought you have must be expressed in your mind in a process that outputs language for you and me to read it. This is a meme, if you like, and it is not a perfect representation of your mental process. (There are things that one can think that cannot be expressed, for instance, the perception of the colour red: what it "feels" like, or if you prefer "looks" like.)
For example, a famous koan is
the sound of one hand clapping.
Once the verbal representation of the electrochemical thought is comprehended (let's say successfully) by the recipient, it is then imperfectly represented in their brain's electrochemical wetware, using frangible memeory and neurotransmitters to store and process it.
Now, being symbolic, which is such a powerful system because it is not anchored to reality (like a token, or an index), the meaning of any linguistic fact is, as you say, dependent on the context. Hence, Derrida deconstructed traditional statements in novel ways (including their opposites). This is not a bug, it's a feature. But it can be manipulated to cause harm rather than create wisdom, too.
scientiae: It's true that in the future we might uncover knowledge of what we now regard as impossible. (Gödel's theorem doesn't preclude the determination of truth, it just means there is algorithmic way — that we know of — to do it; one has to iterate through the calculations manually.)
dtgreene: [1]Doing it manually is no help here; if a statement has no proof, there is no way, manual or otherwise, to derive the proofless statement from the axioms and rules of inference.
[2] In fact, one could write an algorithm to enumerate every theorem of a formal system, and since the statements and theorems of the system are countable, one could write an algorithm that will find a proof or disproof of any statement if there is one; that algorithm is guaranteed to terminate if (and I believe only if) the statement is decidable.
[3] Besides, truth is relative to the specific system of axioms in use. Case in point:
True or False: There exist non-degenerate triangles with two right angles. (A degenerate triangle is one where (at least) one of the angles is 0.)
scientiae: [Trick] Question: Is mathematics invented or discovered?
dtgreene: Sort of. The axioms are invented, in some way, and every other property is then discovered; those properties are only "true" relative to the system of axioms (and rules of inference) in use.
[1] When I said it was still possible to "manually" determine the proof of a statement (in a consistent system, etc.) I should have used the term "brute force";
i.e., it is still possible to determine the truth of something by calculating it, but this doesn't help us create an algorithmic "short-cut" using only the defined axioms.
[2] We are in violent agreement; you have just restated what I said, with more precise mathematical terminology (which I was trying to avoid to make a more general point about truth).
[3] The statement is true, if one takes Euclid's fifth postulate as an axiom. The statement is false if we use hyperbolic geometry, which changes this assumption to false. Allow me to elucidate.
Analytic propositions are true by virtue of their meaning: By definition, their predicates are contained within their subjects, a form of tautology. Synthetic propositions are true because their meanings correlate empirically,* thereby adding something to a concept. The distinction is important because, as Kant noted, mathematics produces metaphysical truths, or “sentences both informative and known, without recourse to experience”.**
Consider mathematical knowledge derived from
analytic axioms (like Euclid’s postulates of geometry). Nikolai Lobachevsky (simultaneously, but separately, with János (Johann) Bolyai, in 1830) inferred hyperbolic geometry by restating Euclid’s fifth postulate.†
This work was utilized Hermann Minkowski, in 1908, to restate James Clerk Maxwell’s equations (
A Dynamical Theory of the Electromagnetic Field; 1865) in four dimensions. In this way, Euclid has given to humanity (two-and-a-half millennia after he wrote it) the mathematics for atomic science, simply by inverting an assumption (that he couldn't "prove", anyway). So mathematics is synthetic
a priori. (It's a brain-frying concept.)
________
* Georges Rey, “The Analytic/Synthetic Distinction”,
The Stanford Encyclopedia of Philosophy (2017).
** James Garvey (London, UK: Ivy Press, 2009), “Grand Moments: Kant’s Synthetic A Priori”,
30-Second Philosophies, ed. Barry Loewer, p.130.
† For any given line
R and point
P not on
R, in the plane containing both line
R and point
P there are at least two distinct lines through
P that do not intersect
R.
Interestingly, Hofstadter noted that both Girolamo Saccheri (1667–1733), who published
Euclid Freed of Every Flaw, 1733, which was a treatise based on the assumption of the inverse of Euclid’s fifth postulate, to disprove it, and JH Lambert, a half-century later, also nearly discovered hyperbolic geometry; still forty years before the Hungarian Bolyai & the Russian Lobachevsky (see Hofstadter,
Gödel, Escher, Bach, p.91
f.).
scientiae: Politics and science don't mix.
Mafwek: Oh, but they do, merely do the fact that scientists are still people, and as such still political beings. That doesn't mean that scientist's personal political beliefs and interests can't help in development of science; but that often those political interests makes them attack theories which are against those interests, or simply discredit those those theories to further their own careers.
P. S. I'll have to decline invitation merely do the fact that I categorically decline every request if I haven't met someone IRL
As I have already noted, human (read: symbolic) reality, which is a virtual layer atop the reality of the cosmos, is a shared concept. Bearing this in mind, of course your statement is self-evidently true. (I suppose I should have put a smiley at the end of the statement.)
PS That is quite sane and acceptable. I merely provided an avenue to take our philosophizing off-topic. :)
