Q&A: Non-Euclidean Geometry and the Fit Between A Priori Assumptions and Reality
Non-Euclidean Geometry and the Fit Between A Priori Assumptions and Reality
Question
With God's help,
I wanted to ask whether the fact that we are mistaken in our clear a priori assumption that the world conforms to Euclidean geometry and its axioms does not raise an obvious skeptical question about our trust in our a priori capacity, and especially in its fit with the world.
As evidence, until Einstein there was no doubt for most people that this geometry was valid in the world as well, aside from a few mathematicians who toyed with other possibilities. It is also clear that in the end this question too is examined within the framework of thought, but still there is something bizarre about thinking of the sharp mismatch between the clear intuition and reality. Almost like the lack of determinism in nature.
Answer
Does the fact that we have experienced a mirage cause us to abandon trust in our eyes? Clearly intuition can be mistaken, but that does not mean we should abandon trust in it.
Beyond that, Euclidean intuition is still completely correct even today. In Euclidean space, two parallel lines do not meet, and the sum of the angles in a triangle is 180 degrees. It is just that the space in our world is not entirely Euclidean.
Furthermore, our space is almost entirely Euclidean, so this is not really an error but an approximation. You can always miss a little at the margins. If I have an intuition that there are 1,000 tomatoes in a pile of tomatoes, and counting shows that there are 1,002, has my intuition failed? Definitely not.
And finally, we have no option of abandoning intuition. Everything we know is built on it, including Einstein's general theory of relativity. At most, we should be aware that it is not always correct and not always precise, and it is worth checking it whenever possible.