The Maths History feed on Twitter noted that today, the 20th of September, is the anniversary of the death of Moritz Pasch, 1843 – 1930, which I admit isn’t a name that leapt immediately to my mind. But he was part of the big movement in the late 19th and early 20th centuries to try putting all of mathematics on a rigorous, utterly logical foundation. Probably would surprise most people, who figure mathematics was always on this rigorous and utterly logical foundation, what with it being mathematics and all.
The slightly dirty secret, though, is that it isn’t. It’s built around logical arguments, certainly, and the more rigorous the argument the better-proven a thing is usually considered to be. But you don’t get results proven with perfectly rigorously airtight deductive reasoning, at least not in the journals and monographs that report interesting new results, because it turns out this requires so much work that it takes forever. What you typically see is enough of an argument to be convincing that anything elided over could be filled in, if required. This is part of why huge results professing major new accomplishments, like a proof of Goldbach’s Conjecture, take time to verify: not only is there a lot that’s there, but suddenly the question of whether the elided steps really are secure has to be filled in.
Most of the big gaps-to-be-filled in basic mathematics were filled in a century ago. Pasch was among the people who found some points in Euclidean geometry where physical intuition about real-world things was assumed into mathematical arguments without it being explicitly stated. This didn’t mean any geometric results were wrong or counterintuitive or anything; just that there were assumptions in the system that Euclid — and everybody else — had made without saying they were making them, which is pretty impressive considering that Euclid thought to mention that he was assuming all right angles were congruent.
One of those discovered spots gets called now Pasch’s Axion, and it gives a good example of the kind of thing which can go centuries being assumed without drawing attention to itself: suppose you have a triangle connecting the points we label A, B, and C. And suppose you have a line which enters the triangle through the leg connecting points A and B, and which doesn’t pass through the point C. Then the line exits the triangle either through the leg between points B and C or through the leg between points C and A.
Obvious? Perhaps, but not more obvious than the axiom that a line segment can be drawn between any two different points, and it’s a special insight to notice these things are assumptions.