Unfortunately, not all proposed proofs of a statement in mathematics are actually correct, and so some effort needs to be put into verification of such a proposed proof. Broadly speaking, there are two ways that one can show that a proof can fail. Firstly, one can find a “local”, “low-level” or “direct” objection to the proof, by showing that one of the steps (or perhaps a cluster of steps, see below) in the proof is invalid. For instance, if the implication is false, then the above proposed proof “” of “” is invalid (though it is of course still conceivable that could be proven by some other route).
Sometimes, a low-level error cannot be localised to a single step, but rather to a cluster of steps. For instance, if one has a circular argument, in which a statement is claimed using as justification, and is then claimed using as justification, then it is possible for both implications and to be true, while the deduction that and are then both true remains invalid. (Note though that there are important and valid examples of near-circular arguments, such as proofs by induction, but this is not the topic of this current discussion.)
Another example of a low-level error that is not localisable to a single step arises from ambiguity. Suppose that one is claiming that and , and thus that . If all terms are unambiguously well-defined, this is a valid deduction. But suppose that the expression is ambiguous, and actually has at least two distinct interpretations, say and . Suppose further that the implication presumes the former interpretation , while the implication presumes the latter interpretation , thus we actually have and . In such a case we can no longer validly deduce that (unless of course we can show in addition that ). In such a case, one cannot localise the error to either “” or “” until is defined more unambiguously. This simple example illustrates the importance of getting key terms defined precisely in a mathematical argument.
The other way to find an error in a proof is to obtain a “high level” or “global” objection, showing that the proof, if valid, would necessarily imply a further consequence that is either known or strongly suspected to be false. The most well-known (and strongest) example of this is the counterexample. If one possesses a counterexample to the claim , then one instantly knows that the chain of deduction “” must be invalid, even if one cannot immediately pinpoint where the precise error is at the local level. Thus we see that global errors can be viewed as “non-constructive” guarantees that a local error must exist somewhere.
A bit more subtly, one can argue using the structure of the proof itself. If a claim such as could be proven by a chain , then this might mean that a parallel claim could then also be proven by a parallel chain of logical reasoning. But if one also possesses a counterexample to , then this implies that there is a flaw somewhere in this parallel chain, and hence (presumably) also in the original chain. Other examples of this type include proofs of some conclusion that mysteriously never use in any essential way a crucial hypothesis (e.g. proofs of the non-existence of non-trivial integer solutions to that mysteriously never use the hypothesis that is strictly greater than , or which could be trivially adapted to cover the case).
While global errors are less constructive than local errors, and thus less satisfying as a “smoking gun”, they tend to be significantly more robust. A local error can often be patched or worked around, especially if the proof is designed in a fault-tolerant fashion (e.g. if the proof proceeds by factoring a difficult problem into several strictly easier pieces, which are in turn factored into even simpler pieces, and so forth). But a global error tends to invalidate not only the proposed proof as it stands, but also all reasonable perturbations of that proof. For instance, a counterexample to will automatically defeat any attempts to patch the invalid argument , whereas the more local objection that does not imply could conceivably be worked around.
(There is a mathematical joke in which a mathematician is giving a lecture expounding on a recent difficult result that he has just claimed to prove. At the end of the lecture, another mathematician stands up and asserts that she has found a counterexample to the claimed result. The speaker then rebuts, “This does not matter; I have two proofs of this result!”. Here one sees quite clearly the distinction of impact between a global error and a local one.)
It is also a lot quicker to find a global error than a local error, at least if the paper adheres to established standards of mathematical writing.
To find a local error in an -page paper, one basically has to read a significant fraction of that paper line-by-line, whereas to find a global error it is often sufficient to skim the paper to extract the large-scale structure. This can sometimes lead to an awkward stage in the verification process when a global error has been found, but the local error predicted by the global error has not yet been located. Nevertheless, global errors are often the most serious errors of all.
It is generally good practice to try to structure a proof to be fault tolerant with respect to local errors, so that if, say, a key step in the proof of Lemma 17 fails, then the paper does not collapse completely, but contains at least some salvageable results of independent interest, or shows a reduction of the main problem to a simpler one. Global errors, by contrast, cannot really be defended against by a good choice of proof structure; instead, they require a good choice of proof strategy that anticipates global pitfalls and confronts them directly.
One last closing remark: as error-testing is the complementary exercise to proof-building, it is not surprising that the standards of rigour for the two activities are dual to each other. When one is building a proof, one is expected to adhere to the highest standards of rigour that are practical, since a single error could well collapse the entire effort. But when one is testing an argument for errors or other objections, then it is perfectly acceptable to use heuristics, hand-waving, intuition, or other non-rigorous means to locate and describe errors. This may mean that some objections to proofs are not watertight, but instead indicate that either the proof is invalid, or some accepted piece of mathematical intuition is in fact inaccurate. In some cases, it is the latter possibility that is the truth, in which case the result is deemed “paradoxical”, yet true. Such objections, even if they do not invalidate the paper, are often very important for improving one’s intuition about the subject.
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8 February, 2022 at 10:39 pm
Dwight Walsh
Dear Prof. Tao,
I have been a participant on your forum titled “Why global regularity for Navier-Stokes is hard” but seem to be wearing out my welcome there. In February 2020, I came up with what I believe is a valid proof of the Navier-Stokes Millennium Problem. Since I am totally unknown in the mathematics community, however, I was unable to find a reputable mathematician able and willing to sponsor me as an author for a moderated archive. [I have a Ph.D. in physics along with some background in the Navier-Stokes equation through a few research grants]. Anyway, I first learned about the CMI sponsored Millennium problems in late 2001, and thought my background might offer me a good start in solving the Navier-Stokes Millennium Problem. So I considered the problem intermittently over the next 20 years, and made some breakthroughs in early 2020. Since I couldn’t post my paper to a moderated archive, I build my own website and posted all versions of my paper to this website as evidence of my participation in this millennium problem. This website is located at
http://www.navierstokessmoothsolutions.com
I first announced my website and possible solution to the NS Millennium Problem on the forum “Why global regularity for Navier-Stokes is hard” in September 2021. If you visit this forum, you will see that my reception there has been quite hostile over objections to my proof. While the dialog in this forum is rather confusing due to poor rendering, misplaced replies, and sometimes my misunderstanding of terms in the article, I believe the problem with the objections to my proof becomes much clearer starting with my entry on 7 Feb 2022, 2:36 PM. In fact, it is actually quite simple to understand.
The problem starts with a certain function q(x,t) that I use as a tool in the proof. I won’t get into its exact meaning, but it is precisely defined in the paper. I then make a series of “local” arguments (ie. straight-forward mathematical steps) that proves there is no blowup, and therefore the conjectured blowup time T_b does not exist. At this point, Antoine Deleforge comes up with his version of q(x,t) which he got from who-knows-where and uses it to claim that I failed to show the solution was (uniformly) bounded on the semi-open time interval [0,T_b). Now, I already proved that there is no blowup time if we use the version of q as defined in the paper, If we use Deleforge’s version, however, we most certainly do have a blowup time T_b and a blowup! Based on this, I can see why you might call NS global regularity an “impossible problem”
So my question is does Deleforge (and several others) have a valid objection to my proof.
9 February, 2022 at 11:05 am
Dwight Walsh
Update on objection to proof
Just recently, Antoine Deleforge updated his objection to my proof of existence and smoothness of solutions to the NS equation. He now acknowledges that I did define q(x,t) in equations (106) and (108) but claims that I did not use these equations in arguing equation (118), and therefore are irrelevant to the discussion. This, however, simply isn’t true since I did use equations (106)-(108) to prove inequality (109), and inequality (109) was used to established that the integrals in equations (115) and (118) are finite. This, in turn, is used to prove that the time integral of |grad p| is bounded over all time. [One thing I might note is that I forgot to specifically mention inequality (109) in my arguments for (115) and (118). This may have caused some confusion but I believe it should have readily been resolved by context.]
Please understand that in the interest of resolving the NS global regularity issue, I believe it is CRUCIAL that this objection to my proof be addressed ASAP.