When Hofsteder states that the implications of Godal’s thrm tore mathematics apart he brushes over the horrifing time mathematicisans had with Geometry, and it brings to mind one of my fav stories in math history.

Once upon a time (1800s) there was an old mathematician named Farkas Bolyai who had a son named Janos Bolyai.  Farkas’ main focus in his mathematical carreer was geometry, and in particular Euclid’s Fifth postulate.

Euclid, a Platonic, wrote his most famous text, Elements, in 13 volumes in 300 BC. “His aproach to geometry has dominated the teaching of the subject for over two thousand years.  Moverover, the axiomatic method used by Euclid is the prototype of all of what we now call ‘pure mathematics’.”*  When i growup and have a rare book collection you can be sure Elements will be in it.

Euclid’s Elements has five nice and neat postulates (or axiom) to build from, i won’t list them but i’m sure you can ask wikipedia.  Now the first 4 are dandy, but it is the last one that drove people to madness.  We can start with a definition “Two lines, l and m, are parallel if they do not intersect, i.e., if no point lies on both of them.” (notice the assumption that all lines are in the same plane, also note all parallel lines don’t need to be equidistant)

HERE IS THE DOOSEY: THE EUCLIDEAN PARALLEL POSTULATE: For every line l and for every point P that does not lie on l there exists a unique line m through P that is parallel to l.

I mean this seeems clearly true right? but even thinking about it as a pen and paper project we only ever draw lineSEGEMNTS.  Who knows if the actual lines will ever meet or not.

So back to the 1800s and the Bolyais.  Farkas spent all his life vainly trying to prove this puzzling postulate, and when he saw his son heading into the same path he wrote him a letter

You must not attempt this approach to parallels.  I know this way to its very end. I have traversed this bottomless night, which extinguished all light and joy of my life. I entreat you, leave the science of parallels alone…I thought I would sacrifice myself for the sake of the truth. I was ready to become a martyr who would remove the flaw from geomety and return it purified to mankind. I accomplished monstrous, enourmous labors; my creations are far better than those of others and yet I have not achieved complete satisfaction…I turned back when I saw that no man can reach the bottom of the night.  I turned back unconsoled, pitying myself and all mankind

I admit that I expect little from the deviation of your lines. It seems to me that I have been in these regions; that I ahve traveled past all reefs of this infernal Dead Sea and have always come back with broken mast and torn sail. The ruin of my disposition and my fall date back to this time. I thoughtlessly risked my life and happieness

Janos was not deterred by his father’s failures though. Instead he found himself in a completly new geometry, AWESOME. in 1823he writes to his father

It is now my definite plan to pubblish a work on parallels as soon as I can complete and arrage the material and an oppertunity presents itslf; at the moment I still do not clearly see my way through, but the path which I have folloed gives positive evidence that the goal will be reached, if it is at all possible; I have not quite reached it, but I have discovered such wonderful things that I was amazed, and it would be an everlasting piece of bad fortune if they were lost. When you, my dear Father, see them, you will understand; at present I can say nothing except this: that out of nothing I have created a strange new universe. All that I have sent you previously is like a house of cards in comparison with a tower. I am no less convinced that these discoveries will bring me honor than I would be if they were completed.

His father got excited about this thing his son had made (although he never really understood it). So Farkas had it published as a 26page appendix to his book, it was titled “The Science of Absolute Space with a Demonstration of the Independence of the Truth of Falsity of Euclid’s Parallel Postualte (Which Cannot Be Decided a Priori) and, in Addition, theQuadrature of the Circle in Cas of Its Falsity” and sent a copy to his friend from university Carl Friedrich Gauss, “the formost mathematician of his time”, hoping Gauss would help it to be published in its own right.

To end our majestic yet tragic tale, Gauss was a huge dick to the Bolyais.  He wrote them stating that the appendix was pretty sweet but he had already been working on the same conclusion and hadn’t intended it to be published in his life time but wanted to write it all down eventually. Janos was really hurt and never tried to publish his findings and Farkas still didn’t get it and published another flawed proof of Euclid’s Fifth.

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*All my quotes are from my favorite text book of all time, Euclidean and Non-Euclidean Geometries, Developemnt and History, by Marvin Jay Greenberg.  This book srsly made me love the maths.