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Hodeg’s conjecture

There’s a particular book in THE LIBRARY that contains a proof of Hodge’s conjecture, but I don’t know if the proof is correct. I’d need to first find the book and then turn it over to experts that could verify it.

There’s another particular book in THE LIBRARY, and this one contains a correct proof of Hodeg’s conjecture. It’s beautiful in its simplicity of arguments and marvelous in its sagacity. While the proof is by no means trivial, it’s not beyond what mathematicians today in the beginning of the 21st century can understand. This result is able to drastically change the way we see mathematics and, in less than a century, it can lead to exciting new theories in physics. With these tools in hand, the general solutions to the flow differential equations will be found and from then a much better theory of chaotic systems, although by then it will have a vastly different name.

By this point you think I’m talking fiction, a marvelous book whose contents will revolutionize humankind, and that it belongs in the same category as The Book as discussed by Pál Erdős

Paul Erdos liked to talk about The Book, in which God maintains the perfect proofs for mathematical theorems, following the dictum of G. H. Hardy that there is no permanent place for ugly mathematics. Erdos also said that you need not believe in God but, as a mathematician, you should believe in The Book (Aigner & Ziegler, 2010).

…but I’m not. The proof of Hodeg’s conjecture is real and exists in THE LIBRARY, our inability to find it has no bearing in its existence (after all, Bielefeld exists despite you knowing no one who has actually been there (Wikipedia contributors, 2021))

The problem—a LIBRARIAN would argue—is that we haven’t even conceived yet Hodeg’s conjecture.

Some would argue that by this measure, we’re facing the same problem posed by Douglas Adams when his Deep Thought spat out the number forty-two; namely that we don’t know the question. This is a mistake.

You see, as it has been pointed out before, for every “original” book in THE LIBRARY, there are 1.4096... × 101, 810, 837 copies of it that have only one typographical error. Unless that one typo is made in a critical place and never corrected1 all of these copies are trustworthy ways of accessing the proof of Hodeg’s conjecture. There are many copies of it floating around.

So, where’s the problem? By not knowing the question, we’re faced with the problem of having too many answers and no good way to discern between the true and useful ones, and the false and useless ones. For instance, Hodeg’s conjecture could hinge on the exact value of the 20th Busy Beaver number. Before me I have two books—A and B—one which states that such value is:


…and the other book stating that the value in question is:


…two strings of the exact same length.2 There’s 101, 810, 837 almost perfect copies of both A and B, each with elegant arguments that lead us to wildly different consequences. Or take these other two books—C and D—that have the following sentences in the middle:

is, in fact, irrational

and the other one states:

is, in fact a rational

It’s possible that the context of these proofs would be enough to determine which of these books is incorrect, but there’s no way to be sure unless we know the context, and to know the context we need the question.

Without questions, there’s no good way to establish an answer. Which of A, B, C or D above could be the “right” answer to our problem? Arguably, there’s many more books with countless alterations similar to the ones I explained above. We’re in excess of answers, and in want of good questions.

Is three the answer? Or is it thirty?

The answer is already there, and it’s practically at our fingertips–considering the vast size of THE LIBRARY and the sheer number of near-perfect copies of the book containing Hodeg’s Conjecture. How would you know it’s the one you have in your hands?

* * *

An out-of-character note

Andy, the writer of the above paragraphs says:

Before the internet, almost no human had an easy way of communicating their ideas to a massive audience. Of course, humans have always had those closest to them to express their ideas to, starting with immediate family and friends. From there, some positions of authority (like a teacher, a mayor or a pastor) had a way to influence their ideas to a larger audience, but these were confined mostly by geography, so the impact was related to physical proximity. Larger influences came with larger presences. Papal Bulls and Imperial Edicts are two examples of influence that are still hard to grasp by human standards.

I have no direct proof, but I’m human and I know some humans. I’m willing to bet this influence has been misused more than once since humans have been humans.

Fast forward to today. Lots of people in the so-called «western civilization» have access to the internet, which differs from its predecessing media channels by virtue and vice of being very symmetrical. Having a television does not mean one can transmit thoughts through television, but the same is not always true on the internet.

We—those with enough privilege to have easy access to the internet—adopted massive channels of communication without first adopting basic rules of communication; but that’s hardly our fault since those rules did not exist and could not exist.3 We started broadcasting our thoughts in whatever way we found to work, at varying degrees of thought, with deep gaps in information processing, with only a modicum of actual research skills.4 Look at any scientific congress and you’ll find that sharing information is not a trivial matter, and that’s among a group of people with very similar backgrounds, understanding on a particular topic and critical skills. If such a group finds difficulties in their communications, one can imagine that the problem only exacerbates as the group grows and the differences in kind and kinds gets larger and larger. Enter economic interests that weren’t part of the early internet and you get the Mighty Algorithm lumping and categorizing communications in the name of corporations after realizing they could exert influence on large audiences the same way it happened before the internet (newspapers, TV, et al.)

We’re still learning how to use the internet, both as incoming and outgoing data. The existence of large corporations has only made matters worse, but is not the one and only source of our misuses of information flow.

If you’re reading this you’re part of the problem, in the sense that we’re all inside this thing called internet5 This means, you’re also part of the solution: we have lots and lots of answers out there, not just as a metaphorical library. But not all of them are true and we need good questions (and good questionings) to filter them out. My musings around Borges’ Library are a pet obsession of mine, but this topic in particular I hold as a parable. This is a way for me to think about things and sometimes even get to a palpable conclusion.

THE LIBRARY holds oceans of garbage, and only a small fraction of readable books. Among these, only a small fraction holds Truth. Knowing the Questions is key in knowing the difference.

Something similar happens on THE INTERNET.

SciFiQuest 3021: The Quest From the Black Lagoon

References and Bibliography

  1. Unlikely, since every book in THE LIBRARY contains a lot of information and it seems unlikely that a tome with that much text would be so fragile, information-wise.

  2. In THE LIBRARY, as described by Borges, there’s only 22 letters and three other characters: comma, period, and space. One could conceive of another LIBRARY containing more characters and digits and in this second bookspace, the two strings of equal length could well be two numbers, like 1024 and 2048. My usage of letters in the above example is just to align it to the existing image in our minds, and the inclusion or exclusion of numerals doesn’t change the argument.

  3. To this day they do not exist fully, and some could argue that a good list of rules for communicating on the internet is a Quixotic Quest much like, say, a list of rules for communicating in Spanish. There’s grammar and syntax, but beyond linguistics there’s no way to establish «How to responsibly inform your audience of a topic in which you have some degree of knowledge marginally above the average person without sounding like an insufferable asshole».

  4. And no: googling something up barely counts as a check mark in the large list of things that make up true Research skills. This is why lots of heated arguments that use the phrase «do your own research» aren’t fruitful: «doing research» is used in this context as a lazy proxy for actual research, merely indicating that «I have read things that back up what I’m saying» without further consideration for the solidness of such sources, the extent of its claims, its place on larger discussions and even the other person’s accesibility to such sources.

  5. And, by extension, human communication in general.

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