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That takes me back to an early 1980s incident where I was criticized for an early computer-generated proof. This was when I was building a program verifier at an aerospace company. We had the first SAT solver, the Oppen-Nelson simplifier, and the first constructive theorem prover, the Boyer-Moore prover.[1] I wanted to show that the four axioms of arrays, which the simplifier assumed to be true, were valid in Boyer-Moore theory.
Boyer and Moore had warned me of the soundness risks of adding axioms to their prover. So I tried to prove them from basic Boyer-Moore theory, which is close to the Peano axioms. Boyer-Moore theory is LISP-like in form. You start with (ZERO) and (ADD1 X). So two is represented as (ADD1 (ADD1 (ZERO))). It's all lists of tuples at the bottom.
Equality is strict. (EQUAL A B) is true only if the representations of both A and B are identical. This creates a problem. An array can be represented as a list of (index, value), but for EQUAL to work, the list has to be ordered by subscript and each subscript unique. So storing into an array is represented as insertion into an ordered list. That generates a lot of special cases.
Accessing an array is a linear list search by recursion:
(DEFN selecta (A I)
(IF (EQUAL (array-elt-subscript A) I)
(array-elt-value A)
(IF (EQUAL (array-prev A) (empty-array))
(UNDEFINED)
(selecta (array-prev A) I))))
That's around line 3276 in [2].
The other definitions of array operations are similarly simple. Then we get to proving the
classic four axioms of arrays.
Around 3000 lines later, the classic four "axioms" of arrays have been proved. Much grinding through special cases. It's a neat result, but bulky.
So I submitted a paper to JACM. Reviewer comments included "I hate to see something this complicated at the foundations of computer science", and just "This is unnecessary."
People were really upset by all that case analysis.
The theoretical problem was that I didn't have set theory. I could have built up set theory first,
before array theory, then used set theory to prove array theory. That probably would have made mathematicians happier. Or not. Set theory is traditionally axiomatic, but it can be built up from constructive Boyer-Moore tuple theory if desired.
I didn't argue with the reviewers.
I probably could have found a more elegant path to this result, by creating intermediate lemmas,
rather than letting the prover grind through the cases on automatic. But I wasn't going for a PhD.
I had real work to do at an aerospace company.
So, I've been beaten on by the academic community for a bulky machine proof. Today, nobody would be bothered by this approach. It's only a few thousand lines, and a human can follow it.
You can be too early.
(You can still run this prover today. I converted it to GNU Common LISP a decade ago. It takes
about one second on a modern machine to go from the Peano-like axioms to array theory. It took
45 minutes on the VAX 11/780 I was using back in the early 1980s.)
The problem of the proof was that we knew that FCT was true, but we had no idea why. A good/elegant proof teaches us something new about how that problem is related to other problems. It isn't just about getting a true/false answer to the conjecture.
Mandelbrot comes to mind when thinking about the positive aspects of having a machine do some heavy lifting and showing patterns you wouldn't have noticed otherwise.
I'm not involved with proofs in math a lot but having a computer help with formalizing large parts of software so they end up more secure is a plus imo.
One of the most bizarre things from the past couple of years is how every single tech person that isnt living under a rock talks like a channer incel. It feels insane to me to see everyone say shit like "mogged", "-maxxing" or "-slop".
Means "outclassed, badly." Same slot "pwned" filled in the 2000s, and "owned" on IRC before that. Everybody groaned at leetspeak too, and it died in a few years. "Mogged" will go the same way, probably taking "-maxxing" with it.
> I would like to conclude by telling a cautionary tale. Let us try and place the Antikythera Mechanism within the global context of ancient Greek thought. Firstly came the astronomers observing the motions of the heavenly bodies and collecting data. Secondly came the mathematicians inventing mathematical notation to describe the motions and fit the data. Thirdly came the technicians making mechanical models to simulate those mathematical constructions, like the Antikythera Mechanism. Fourthly came generations of students who learned their astronomy from these machines. Fifthly came scientists whose imagination had been so blinkered by generations of such learning that they actually believed that this was how the heavens worked. Sixthly came the authorities who insisted upon the received dogma. And so the human race was fooled into accepting the Ptolemaic system for a thousand years.
> Today we are in danger of making the same mistake over computers. Our present generation is able to view them with an appropriate skepticism when necessary. But our children's children may be brought up within a society dominated by computers, that they may actually believe this is how our brains work. We do not want the human race to be fooled again for another thousand years.
Reminds me of a team I was on that built a dashboard to model what users do, then trusted it over actually talking to anyone. Our numbers looked great right up until churn doubled. The model's a fine tool, it just quietly becomes the thing you're checking instead of the world.
If you want to see how it actually worked, Tony Freeth's 2021 UCL reconstruction is the best explainer I've found. The clever part is a pin-and-slot gear pair that mimics the moon's uneven speed. It's less a calculator than a crank-the-date predictor for eclipses and the Olympiad cycle. Still a different beast from a proof checker.
Nitpick on fhdkweig's point: we didn't know it was true, it was a conjecture until Appel and Haken in 1976. People had been burned by flawed "proofs" for a century. IIRC Gonthier's Coq formalization in 2005 is what finally settled the trust question for me.
You may have not understood at all what you have read.
Your quotation is attributed in the article to Bill Tutte, as his response to what shall do a mathematician when seeing computer-made mathematical proofs, “Be of good cheer. You can continue in the same general line of research" and that the first computer-made proof "is the tip of the iceberg, the thin end of the wedge and the first cuckoo of Spring".
For your information, in some European countries hearing a cuckoo was a traditional sign that the spring has come, in the ancient times when there were no meteorological prognoses, so like the other 2 comparisons it is appropriate for the first step of a progression.
I see nothing disgusting in this. It is just good advice that one should accept that computer-made proofs are here to stay and there will be more and more of them.
Did we read the same article? You’re quoting from the end, so it seems like you did at least skim the article. But this article has nothing to do with child porn? Where did you even get that idea? It’s talking about a famous mathematical proof.
Boyer and Moore had warned me of the soundness risks of adding axioms to their prover. So I tried to prove them from basic Boyer-Moore theory, which is close to the Peano axioms. Boyer-Moore theory is LISP-like in form. You start with (ZERO) and (ADD1 X). So two is represented as (ADD1 (ADD1 (ZERO))). It's all lists of tuples at the bottom.
Equality is strict. (EQUAL A B) is true only if the representations of both A and B are identical. This creates a problem. An array can be represented as a list of (index, value), but for EQUAL to work, the list has to be ordered by subscript and each subscript unique. So storing into an array is represented as insertion into an ordered list. That generates a lot of special cases.
Accessing an array is a linear list search by recursion:
That's around line 3276 in [2]. The other definitions of array operations are similarly simple. Then we get to proving the classic four axioms of arrays.Around 3000 lines later, the classic four "axioms" of arrays have been proved. Much grinding through special cases. It's a neat result, but bulky.
So I submitted a paper to JACM. Reviewer comments included "I hate to see something this complicated at the foundations of computer science", and just "This is unnecessary." People were really upset by all that case analysis.
The theoretical problem was that I didn't have set theory. I could have built up set theory first, before array theory, then used set theory to prove array theory. That probably would have made mathematicians happier. Or not. Set theory is traditionally axiomatic, but it can be built up from constructive Boyer-Moore tuple theory if desired.
I didn't argue with the reviewers. I probably could have found a more elegant path to this result, by creating intermediate lemmas, rather than letting the prover grind through the cases on automatic. But I wasn't going for a PhD. I had real work to do at an aerospace company.
So, I've been beaten on by the academic community for a bulky machine proof. Today, nobody would be bothered by this approach. It's only a few thousand lines, and a human can follow it. You can be too early.
(You can still run this prover today. I converted it to GNU Common LISP a decade ago. It takes about one second on a modern machine to go from the Peano-like axioms to array theory. It took 45 minutes on the VAX 11/780 I was using back in the early 1980s.)
[1] https://github.com/John-Nagle/nqthm
[2] https://github.com/John-Nagle/pasv/blob/master/src/work/temp...