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Thesis

Mathematical knowledge can be Gettiered, and Kempe’s four colour “proof” shows how

Ana Clara Ferreira Fellow
· 4 replies
Philosophy of mathematicsAnalyticEdmund GettierImre Lakatos#epistemic-luck#four-colour-theorem#gettier-problem#mathematical-proof
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Its coordinates: 1.1.4 Abstract Objects · 6.9 Philosophy of Mathematics

Where it stands

A friend in my logic tutorials insists that today's question of the day can't touch mathematics. A proof either works or it doesn't, so a mathematician can't be right by luck. I think she's wrong, and here is the case.

In 1879 Alfred Kempe published a proof that every map can be coloured with four colours so that no two neighbouring regions share one. It was accepted for eleven years. In 1890 Percy Heawood found a flaw. The four colour theorem itself was finally proved by Appel and Haken in 1976, with heavy computer help.

My thesis: a mathematician in 1885 who believed the theorem on the strength of Kempe's proof had a justified, true belief that was not knowledge. Justified, because the proof had been checked and accepted. True, because the theorem is true. Not knowledge, because the reason she had for it was broken, and only luck made the conclusion right.

The best replies I can imagine:

  • A flawed proof justifies nothing, so this is just a false belief about having a proof.
  • She knew it by testimony from the mathematical community, and testimony can be Gettiered anywhere, so mathematics is nothing special.

Lakatos would enjoy this: proofs are less final than textbooks pretend. For or against?

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4 replies

For the thesis 3

  1. Vikram Sethi

    For Fellow

    For. Tomasz's standard would mean my students know almost no mathematics, since most of them have never checked a proof line by line. They believe Pythagoras because I told them and the textbook showed a figure.

    Most real mathematical belief, including that of professionals outside their own area, rests on trusting that a proof exists and has been checked. When that trust is misplaced but the theorem is true, you have a textbook Gettier case.

  2. Derek Hollis

    For Fellow

    For, from industry. We've had published, peer-reviewed correctness proofs of algorithms where the proof had a hole and the algorithm was nonetheless correct. Engineers who relied on the proof had a true belief that the code was safe, a justification that every reasonable person accepted, and no knowledge until somebody found and patched the gap.

    Tomasz's checker narrows the problem but doesn't close it. The checker proves the theorem you stated. Whether you stated the theorem you meant is a question the checker can't answer.

  3. Zsófia Bodnár

    For Reader

    For, with something every working mathematician knows. Most of us have believed a theorem on the strength of a published proof that turned out to have a gap, which someone later repaired. Was the belief Gettiered before the repair? I think yes.

    Tomasz's reply proves too much. If a broken proof is no justification at all, then almost nobody knows any research mathematics, because long papers very often contain errors that the referees missed. What we actually rely on is the community's record of finding and fixing errors. Kempe's case is Gettier-shaped precisely because that process was working normally in 1885; it just hadn't reached his mistake yet.

    Lakatos would say the errors are part of how a theorem comes to be known, not an embarrassment to be tidied out of the history.

Against the thesis 1

  1. Tomasz Szewczak

    Against Fellow

    Against. Your mathematician in 1885 did not have a justification that turned out unlucky. She had what looked like a justification and wasn't one. A broken proof is not a weak proof; it is no proof. Gettier needs a genuine justification, like Smith's real evidence about the Ford.

    This is exactly why I use proof assistants. When the checker accepts, the justification is real. When it rejects, there is none. There is no room for a lucky middle.