This is Math's Fatal FlawSo this discovery was one of the greatest accomplishments of the XX century, and yet it is a "fatal flaw"?Not everything that is true can be proven. This discovery transformed infinity, changed the course of a world war and led to the modern computer.
The video is actually pretty good, but I object to all the explanations that say that Mathematics is somehow deficient because the consistency of the axioms cannot be proved from the axioms.
Nobody would ever want the consistency to be provable from the axioms anyway. Such a proof would mean nothing. Inconsistent systems allow such proofs, and nobody wants that.
It would be nice to have an algorithm to determine whether a given math statement is true or false. The above discovery shows that it is not possible. But again, this is not a fatal flaw. It is what makes Math interesting.
Dear Roger,
ReplyDeleteI haven't watched the video fully. (I rapidly went some half way through, and postponed watching it.) However, I can say this much:
Actually, I don't have a problem with this statement:
"Not everything that is true can be proven",
*provided* that you also add the following implicit premise:
"within the context of the currently accepted set of axioms".
They always drop the second part, and only highlight the first part. And, taking this artificially curtailed part as an absolute truth, they then attempt to make celebrities out of people who otherwise produced work which wasn't so extra-ordinary in the first place. (I mean folks like Godel, pronounced like "girdle". He was smart, but not super-smart, as they habitually make him out to be.)
There *is* some deficiency in there. But the deficiency to be pointed out should pertain to the *error* of insisting upon the idea that the currently developed set of mathematical axioms ought to remain fully intact regardless of any future expansion in mathematical knowledge. Sticking to a concrete stage of development is the error involved here.
Mathematics is not deficient. It *is* valid. But it is valid only within a certain conceptual context. It might be possible to expand the mathematical knowledge beyond its current boundaries, which might require a revision in the set of statements currently accepted as axioms.
So, what these paradoxes reveal is the need to acknowledge that currently accepted axioms don't fully capture the currently available conceptual context. ...That, man is able to *induce* such new truths from fresh observations of reality which cannot be *deduced* from the axioms of the currently formulated theory.
Instead of saying that, they take some new insight which shows some limitations of a *specific* set of axioms, and then, instead of advocating a revision in the axioms, they use the observed limitations in order to prove that the very idea of proofs itself is invalid. Then they go further and use this "example" to undermine certainty, even knowledge.
That's exactly like saying: Because the Maxwell-Lorentz ontology cannot explain, say, the observed phenomenon of the quantization of energy, therefore, at the most fundamental level, the science of physics cannot say anything with certainty. Philosophers, and some babblers in philosophy like Bohr and Heisenberg, did say that---and highlighted it as *the* feature of QM.
Practical physicists, taken as a whole, were smarter. They simply *revised* the postulates of physics as carefully as they could. Of course, they too didn't acknowledge the role of philosophy in physics, and hence, didn't take ontology seriously, and in any case, weren't fully consistent. That's why, the postulates they formulated do fall short. But being less abstract than maths, the operative scene in physics isn't so bad---comparatively speaking.
Mathematicians... Oh well...
Best,
--Ajit
Math's fatal flaw is primarily the limitations of the humans who practice it, particularly those who claim to teach it.
ReplyDeleteWith each passing year, the mathematical field grows larger, more complex, and increasingly esoteric to the point of rendering it mostly useless except for pointless academic bragging rights and outright obfuscation at the higher levels. For the most part, my math classes were quite frankly, poorly taught. I found most mathematical classes a uselessly complicated joke crammed into a semester as tightly as possible, with no time to absorb, examine, question, or test anything whatsoever, with a shallow 'just trust me it works, let's move on' as being the most common proof or explanation given so we could rush mindlessly onward to the next increasingly rushed concept which just HAD to be covered, even though absolutely no one was retaining much of the material just a week after an exam. The mad rush to the finish line mentality of mathematical instruction renders most of it functionally useless and without purpose other than to drive people away from trying to understand it, much less 'find it interesting'.
"Science is the belief in the ignorance of the experts." — Richard Feynman