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When reading about some open problems, a lot of them have quotes by renowned mathematicians that “[the conjecture] cannot be solved using the current technology” or something along these lines. What do they mean by that? Are they talking about the axioms? Or are they generally speaking in terms of intelligence and mathematical abilities?

These ones are just at the top of my mind:

https://terrytao.wordpress.com/2011/08/25/the-collatz-conjecture-littlewood-offord-theory-and-powers-of-2-and-3/

https://m.youtube.com/watch?v=MXJ-zpJeY3E (skip to the end where the talk about the Riemann Hypothesis)

But I’ve seen many, many more examples

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    $\begingroup$ Paul Erdős said about the Collatz conjecture: "Mathematics may not be ready for such problems" $\endgroup$ – J. W. Tanner Jul 14 '19 at 23:05
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    $\begingroup$ @J. W. Tanner That’s one of the examples of what I mean $\endgroup$ – user668217 Jul 14 '19 at 23:09
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    $\begingroup$ Yes, I gave that, because someone asked for examples, in a comment that has now been deleted $\endgroup$ – J. W. Tanner Jul 14 '19 at 23:11
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    $\begingroup$ Are you sure you did not misheard "current techniques" as "current technologies"? $\endgroup$ – Giacomo Alzetta Jul 15 '19 at 10:40
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    $\begingroup$ @GiacomoAlzetta People really do say "current technology" just as the poster describes - no reason to think it's a mishearing. You can see it in paragraph 2 of the link to Terry Tao's blog that the poster provides, for example. $\endgroup$ – James Martin Jul 15 '19 at 11:53
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This doesn't have any formal meaning. It just means that they believe the problem can't be solved with the techniques that mathematicians have already developed and instead some big new idea will be necessary to solve it. That is, "the current technology" refers to the collection of proof methods that mathematicians have discovered already. It's meant as a sort of metaphor between mathematics and engineering: some engineering problems can be solved by just finding a clever way to put together already existing technologies, while others require major new inventions. In the same way, some mathematics problems can be solved by just finding clever new uses of ideas that are already known while other mathematics problems require something more novel.

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    $\begingroup$ +1 Ultimately, though, every "novel proof technique" can be expressed in terms of the "old proof techniques". Indeed, how else would one prove the validity of it? (Unless, of course, you vary the foundation by adding axioms for instance.) So formally, what can be proven using those novel techniques can already be proven now. However, that proof may be significantly longer and out of reach for human minds. The same way you can put a nail into some piece of wood with your feet vs. with a hammer. Tools (proof techniques) bridge the gap. $\endgroup$ – ComFreek Jul 15 '19 at 7:36
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    $\begingroup$ I would be a bit more careful with "doesn't have any formal meaning". For very famous conjectures such as P ?= NP there are large classes of techniques that have been proven to be fruitless or proven to need to show something so much stronger that it's unlikely to work. $\endgroup$ – orlp Jul 15 '19 at 14:04
  • $\begingroup$ And of course the only way we know that something can be proven using current techniques is if there is a proof. $\endgroup$ – Acccumulation Jul 15 '19 at 22:36
  • $\begingroup$ @ComFreek: Sure, if you go down to the level of "proof techniques" like modus ponens and other basics, you can say that anything provable is provable with current techniques, but that's a lot lower-level than people mean when they say this stuff. It's like saying that the Roman Empire could have built a helicopter with the already-established "technology" of interacting with the physical world using their bodies, because some sequence of physical actions they could do with their bodies would have created a working helicopter. $\endgroup$ – user2357112 supports Monica Jul 16 '19 at 2:50
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It means that the methods they have don't suffice to solve the problem and the technology they researched wont give back a mathematic answer that satisfy their conjecture.

The problem in general (which is the amount of times “[the conjecture] cannot be solved using the current technology” is cited) its not a factual (some would say 'engineering') problem because the question is generally not a finite and descriptive one (refer to 'How do I ask a good question?' in stack overflow https://stackoverflow.com/help/how-to-ask ).

Many mathematic problems of the sort don't seek a concrete answer but rather a general solution. Meanwhile, technology works great to get concrete solutions to specific problems and therefore technology would have to be directed/tailored to solve that specific problem or to give broader answers that can answer such question, which is something not done yet for the problem in question.

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  • $\begingroup$ If you guys send a downvote, please tell the user how to improve his answer. $\endgroup$ – Luis Felipe Aug 7 '19 at 14:07
  • $\begingroup$ Or at least write down why you downvoted. $\endgroup$ – deags Aug 7 '19 at 15:24
  • $\begingroup$ I have not downvoted, but I suspect that this is downvoted I suspect for multiple reasons. First, it doesn't answer part of the question, what mathematicians mean by "technology" in the sense. Second it doesn't make sense to talk about whether technology "satisfy their conjecture" - satisfying a conjecture has a very different meaning than proving a conjecture. The second paragraph demonstrates a lack of understanding of what the people quoted in the OP mean when they are talking about "current technology" and the final paragraph shows similar confusion. $\endgroup$ – JoshuaZ Sep 25 '19 at 13:15
  • $\begingroup$ Technology by definition is the sum of techniques, skills, methods, and processes used in the production of goods or services, or in the accomplishment of objectives, such as scientific investigation. The tech exists to solve and answer the conjectures, the error is in normally in the conjectures and assumption about technology available. What seems clear is the lack of understanding about technology. $\endgroup$ – deags Sep 26 '19 at 15:37
  • $\begingroup$ @deags You are using the term "technology" in a way that is not what mathematicians mean when they use the term in this context. The meaning in this context is much closer to how Eric Wolsey answers the question. That a profession is using a term in a way other than how you would use the term doesn't mean there is a "lack of understanding" - it means that the term is being used in a specific meaning. This is no different than how math the term "ring" doesn't mean a small circular object which fits around the finger, or the term "field" doesn't mean a flat outdoor area. $\endgroup$ – JoshuaZ Sep 27 '19 at 1:53

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