I've read several papers in philosophy -- mostly philosophy of mind and philosophy of religion -- which claim that mathematical truth is an objective Truth (capital T). I disagree with this view, and I present a counter-example demonstrating that well accepted mathematical truths are not objective Truths. Instead, mathematical truths are: 1) the consensus of the mathematical community; 2) the consensus of the mathematical community is sometimes wrong, and can go uncorrected for a long time.

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Malfatti's Problem

In 1803, Gian Francesco Malfatti published an essay to answer:

How to arrange three non-overlapping circles of greatest total area inside of a triangle?

The mathematical details of this problem are not relevant for making the philosophical point, but if you want the messy details, read the background information as well as the "Mathematical Notes" at the bottom.

In essence, here is the timeline.

1803 Malfatti poses the problem and publishes his solution. The mathematical community is well aware of his publication and deems his solution optimal. ...126 years pass... 1929 Lob and Richmond show that Malfatti's solution is not always the optimal solution (published in 1930). 1967 Goldberg shows that Malfatti's solution is actually *never* correct. 1990 Zalgaller and Los solve the Malfatti problem (today called "the marble problem"). Published in Russian in 1992; published in English in 1994.

The important thing here is that between 1803 and 1929 for all intents and purposes it was a mathematical fact that Malfatti's solution was correct. His proof was not part of some obscure branch of mathematics that few mathematicians understood, this is a Euclidean geometry problem! Yet it took 126 years before anyone noticed the problem. This wasn't an small oversight mistake either. Malfatti's solution was not simply wrong some of the time, but it turned out that Malfatti's solution was wrong *every* time -- there is absolutely no triangle in which Malfatti's accepted solution is correct. Yet to merely notice the holes in this proof of Euclidean geometry took over a century.

The moral of the story

• Mathematical truth is not given to us from above. It is determined by the consensus of the mathematical community.
• Even when the mathematical community is paying attention, even in a domain as simple as Euclidean geometry, the community can make really, really bad mistakes which can go unnoticed for quite sometime.

Mathematical Notes

• Malfatti was not wrong in his calculations. He simply made the incorrect assumption that the circles with the maximum total area would each touch one another circle and touch two side of the triangle. Malfatti's calculations correctly found the circles with the largest area where each circle touches one other circle and touches two sides of the triangle.
• Malfatti originally restricted his interest only to circles inscribed within a right triangle, however, the correct solution for right triangles was discovered in the same paper (Zalgaller and Los, 1990) that found the solution for any triangle.
• The solutions by Chokuyen Naonobu Ajima (1732-1798) and Jakob Steiner (published in 1826) do not solve Malfatti's problem. Although they are elegant proofs, they both make Malfatti's original (wrong) assumption, that the optimal circles will each touch one other circle and two sides of the triangle.

References

1. Malfatti, G. "Memoria sopra un problema stereotomico." Memorie di matematica e fisica della Societé Italiana delle Scienze 10-1, 235-244, 1803.
2. Lob, H. and Richmond, H. W. "On the Solution of Malfatti's Problem for a Triangle." Proc. London Math. Soc. 2, 287-304, 1930.
3. Goldberg, M. "On the Original Malfatti Problem." Math. Mag. 40, 241-247, 1967.
4. Zalgaller, V. A. and Los', G. A. "Solution of the Malfatti Problem." Ukrain. Geom. Sb., No. 35, 14-33 and 161, 1992.
5. Zalgaller, V.A. and Los', G.A. (1994). "The solution of Malfatti's problem". Journal of Mathematical Sciences 72 (4): 3163–3177. Springer.

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Responses

I received this wonderful response from Michael Fox. He could very well be right. It seems odd that people simply would've forgotten about Malfatti's original problem, but it's certainly possible. He writes:

Hi, I saw your comments on Malfatti and his 1803 solution to his circle problem. I thinnk the history of this problem is rather more complicated. Let's call the problem of putting three circles in a given triangle so that each touches the other two and two sides of the triangle 'the circle problem', and the problem of fitting in three non-overlapping circles so that they have the greatest possible area 'the marble problem'. (Malfatti wanted to determine the three circular cylinders of greatest volume that could be cut from a triangular prism of marble. This is equivalent to finding the circles of greatest area ... .) Malfatti solved the circle problem correctly, but his assumption, as you say, was wrong. But independently, a French mathematician, Gergonne, had hit on the circle problem in around 1800, and took about 10 years finding a solution. Then, in 1810/11, he published the problem in the Annales de Gergonne, and a little later published his solution. He had also had a letter from a Professor Bidone of Turin telling him about Malfatti's solution, but with no mention of the marble problem. Other mathematicians, such as Steiner, heard of the problem through Gergonne's journal, and almost certainly were not aware of Malfatti's original marble problem. In the various mentions of Malfatti's problem that I have seen in 19th and early 20th century books, none link it to any other problem. My belief is that the marble problem was simply forgotten until more research was done early last century. Michael Fox