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I noticed that all the paradox-related articles were happily pointing to paradox, which however was less than a stub. So I tried to promote it at least to stub-status.
Slightly expanded.
Thanks. I have added an “on first sight”:
A paradox is an example of reasoning that seems perfectly valid on first sight, but which results in a contradiction or inconsistency.
On the other hand, there are other famous “pardoxes” which seemed paradoxical only on first sight, and were later understood to be not actually contradictory. For instance Zeno’s paradox ceases to be a paradox once one knows about differential calculus.
3: The way I learned about the concept of paradox in my education is in fact not needing quote for Zenon’s paradox as the paradoxes are exactly that kind of thing. Zenon’s paradox is a paradox and not a “paradox”. One makes a setup, way of thinking, model of something, which seems legitimately founded on some theory (or view of the world) and arrives at contradiction. However if one examines the foundations of the thinking than one learns that something has been assumed about the presumptions, way of concluding etc. which is like in some legitimate situations but is not in fact legitimate. I do not think that Zenon’s example has to do with specific of existence of differential calculus; it is rather just about infinite sums possibly being finite, as something not understandable from the point of view of grasp by an observer of finitely many finite partial sums. The consumer of that paradox has the wish to grasp nature from that perspective rather than resorting to possibility of consistency of thinking which is beyond that grasp.
The way I learn is that paradox is a line of thought leading to a contradiction but which can be removed by rethinking the setup of premises and allowed moves of reasoning. In short it is a seeming or removable contradiction, rather than a true one. So for me putting quotes “paradox” is against the very nature of the concept of paradox.
Let me highlight the difference again, then:
the contradictions of the logical paradoxes like Russell’s etc. follow by correct logical reasoning from the given axioms. The conclusion is that the axioms are inconsistent.
the contradiction in Zeno’s paradox follows from incorrect resoning from given axioms. The faulty reasoning being the incorrect discussion of a limiting process. The axioms are in themselves okay.
In Zenon’s paradox limiting process (in modern sense) is not a concept in the model. I do not think that it leads to a contradiction. It is not in agreement with everyday experience. And truly, the observer of finitely many finite sums can never reach the grasp of the sum. I think that only reinterpretation is needed. The reasoning is correct, just its applicability is about the grasp of that kind of observer. It is not about the nature. It is IMPLIED in the “axioms” of Zenon’s paradox that the world is graspable by that kind of intellect-observer, so it is in contradiction with experience. When you go into the exposition of the paradox you do follow the logic rules which are acceptable. You do not make an error. So it is correct. Incorrect is externally in the sense that one must learn that in truly applicable analysis one does not allow such a setup, but a setup free of a paradox exists. When you say that the reasoning is incorrect you suggest that within the accepted system of a paradox author he slips the stupid error. Allowing that the nature has infinite sums or that sets of all sets do not exist is of the same nature. You see a difference because at the time of Russell people had the habit of listing all assumptions, concepts and steps of reasoning into the formal list of letters, variables, axioms, rules of inference. I can easily make a formal system of description of the world of measuring line which agrees with Zenon’s and suffers Zenon’s paradox with in that case classical logic applied.
Zenon can divide line in pieces and compare finite sums and grasp intervals and finite sums of the intervals.
The reasoning is correct,
The reasoning leads from correct assumptions to clearly false conclusions. If the reasoning were correct, we would by now reject the assumptions. But we don’t.
Conclusion of Zenon is correct within his model. In the grasp of an observer of finitely many finite sums of the kind he described, the haze meeting turtoise never happens. The assumption that this description exhausts the events in nature is in contradiction with everyday experience. The critical event is outside of the medium. Like in Copenhagen interpretation of quantum mechanics things which we can think as a process of wave function reduction are out of the medium of observable. No measurement can ever reach it, and we define reality by this. Maybe there is another level of reality, another ontology which is wider and there is a model which allows something more than Copenhagen. A godly being seeing that would say in your spirit that anybody comparing their wider reality to conclusions of Copenhagen school makes a mistake.
Do you really believe this? Or are you just trying to have an argument with me? You believe Zenon found it a correct statement that nobody can ever overtake a turtle, that no arrow can ever move?
I will not argue further here. I don’t feel this is helping us make paradox a better article. But maybe somebody else can help us.
Yes, I believe this. He made a valid statement within the system of description of the world he knew. This does not describe the real world, because his implied axiomatics excludes some possibilities. His finite observer does not know your real line of time and its mathematics. It knows another, smaller medium. In my philosophy classes nobody ever stated that Zenon would make an error. It is just not applicable to real world. Within Zenon’s world no arrow will ever move, indeed. You know very well that your 20th century real line and observer who is a machine verifying the statements of modern arithmetic would never occur in Zenon’s implied axiomatics. One has to interpret the arguments within his system, without amendments from 20th century. Like some statements in nonstandard analysis about elements of real lines are not true in standard analysis as infinitesimals don’t exist so some possibilities of solutions of equations and so on, do not exist. Or some statements on properties of world with aether are not applied to the real world as there is no aether.
I do consider that agreement on these issues can make paradox a better article. All paradoxes assume similar mechanisms.
What the hell are the axiom of Zeno’s Paradox? When his detractors recorded his arguments, they did not list axioms.
But there is no absolute difference between incorrect reasoning from correct axioms and correct reasoning from incorrect axioms. Axioms are just a special case of rules of inference (as constants are a special case of functions), and there is nothing in reasoning but rules of inference. In a strong enough ambient logic, any additional rule of inference can be turned into an axiom. So we may reconstruct Zeno’s axioms by observing his reasoning. Conversely, we may rephrase Russell’s axioms rules of inference and criticise him for incorrect reasoning.
Specifically: Zeno (implicitly) assumes that a sum of infinitely many positive numbers is infinite; this is wrong (at least, it is false in the models that we use today), but his reasoning from this axiom is correct. Conversely, if you don’t take unrestricted comprehension or the existence of the set of all sets as an axiom, then Russell’s reasoning in constructing his paradox is invalid; clearly, he needs to go back to school and learn how to use the axiom of separation properly!
What the hell are the axiom of Zeno’s Paradox?
The statement clearly proceeds by first listing the statements taken to be universally accepted, and then drawing conclusions.
The universally accepted statement is: A path from a to c has to go via b.
Given this, the argument starts to draw conclusions. But they turn out to be wrong.
Russell’s paradox is an actual contradiction: if you assume you can quantify over all sets, then you derive a contradiction.
Zeno’s paradox is not an actual contradiction: if you assume that motion is continuous, it does not actually follow that there is no motion, unless you make a mistake in the reasoning.
But you guys please decide how to handle the entry paradox. I’ll bow out of this discussion here. It seems to me we get bogged down too much in arguments about the obvious. I’d rather reserve energy for more contentful arguments. So let’s assume I am all wrong here and move on.
The statement clearly proceeds by first listing the statements taken to be universally accepted, and then drawing conclusions.
It does? Whose version are you reading? There is no definitive source for Zeno’s paradoxes.
if you assume you can quantify over all sets, then you derive a contradiction
Assuming some further axioms, yes.
if you assume that motion is continuous, it does not actually follow that there is no motion
But assuming some further axioms, it does.
Zeno’s paradoxes are not resolved by calculus alone because they rely on physical assumptions, and they have physical implications; here is a good brief discussion: http://mathpages.com/rr/s3-07/3-07.htm.
how to handle the entry paradox
I am perfectly happy with ’at first sight’.
I separated the examples into foundations (or possibly logic more broadly) and physics (or possibly natural science more broadly). (And if anybody cares about my opinion, Zeno’s paradoxes definitely belong under physics.)
I’ll see if I can add something here later, but for now there’s a relevant MO discussion.
Yeah, Banach–Tarski doesn’t really fall under either of my categories (although as I note in a comment at M.O., it could have fallen under foundations, in an alternate history).
I guess Skolem’s paradox is in the same category as Banach-Tarski. These don’t seem to fit the general description “reasoning that seems perfectly valid on first sight, but which results in a contradiction or inconsistency” either — they are not formally contradictory, only unintuitive.
Re #14, do you think a case can be made that without considering any of the evidence which will lead to relativity theory and quantum mechanics, there’s something conceptually odd about classical mechanics. But perhaps all one finds through Zeno-like considerations is that we were wrong to consider a body merely as inhabiting a 3D space or as being a point in 6D configuration space, but that, in a given moment, it is a point in 12D phase space.
Hegel was troubled by the mechanics of his day. Perhaps with the benefit of hindsight, can you think yourself out classical mechanics?
Re #18, I like Toby’s thought of an ’alternate history’ reducing the difference between Banach-Tarski and Russell. Ultimately, all there is is inconsistency, how to deal with it if you find it and how to avoid it in the first place.
So what is the story of alternative-universe Skolem?
Hmm, well if the problem is that
while this conclusion is true internally, it is not true externally
couldn’t another universe have had us axiomatise that internal truths be externally true?
couldn’t another universe have had us axiomatise that internal truths be externally true?
It's always possible, but I have trouble imagining how anybody could understand the concept of internal vs external truth and then make that conjecture. The Skolem paradox really has the flavour of a student's mistake to me, like the proofs that $1 = 2$ that subtly (or not so subtly) involve division by zero. (But technically, those can also be made into true paradoxes, by adding the axiom that one can after all divide by zero.)
In contrast, the axiom that every subset of the real line is Lebesgue measurable is a quite reasonable axiom, one of those adopted in dream mathematics. It just contradicts the axiom of choice.
added a stubby pointer to ultraviolet catastrophe to the entry paradox in the section “Examples - From natural science”.
(Am pointing to this from quantization where I am currently adding a motivation-section.)
Coming across this entry again for no particular reason, I felt that its Idea-section ended somewhat abruptly. I have expanded as follows:
Mathematical paradoxes of this form (such as Russell’s paradox) are resolved by requiring stronger well-formedness rules to admissible sentences, for instance by requiring strict typing and/or guarded recursion which prevents infinite self-reference.
In contrast, paradoxes in physics may not necessarily be logical inconsistencies and tend to be resolved by rejecting unjustified assumptions and adopting more accurate laws of nature. For instance Zeno’s paradox of motion is resolved by recognizing the notion of limit of a sequence; and the hole paradox is (or was) effectively a misunderstanding of the rules of differential geometry applied to spacetime.
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