| This kind of sensualism was often defended in the debate, but it has problems, too. I think the history of mathematics makes the position implausible. For a long time, up until recently, mathematics was way ahead of the applications of mathematics to physics. In Ancient Greece there was a general consensus that infinity and real numbers do not exist. But then some people found out that the side of a triangle must sometimes be a real number. However, you cannot ever measure SQRT(2) precisely. Whatever number you extract from the physical world is only a rough finite approximation. You need to represent the number in a different way and solve the problem algebraically. Many scientists in Ancient Greece rejected this idea vigorously. Still, real numbers are very useful for describing the physical world, so useful that we couldn't possibly do without them today. Imaginary numbers are another example. They were ridiculed as abstract nonsense when they were described for the first time and widely conceived to have no physical reality or application at all. Despite all that, they play a vital role in modern physics. There are many more examples like that. To cut a long story short, at least until recently mathematics was always ahead of physics (now they seem to go more in tandem). This fact makes the idea very implausible that mathematical structures are merely useful abstractions from the physical world we invented to describe it. It simply doesn't describe what happened in mathematics. And I find the idea that mathematicians just came up with arbitrary imaginations equally implausible. > no external authority you can appeal to which will state definitively that when mathematicians all agree on something their experience of "true" is absolutely and objectively correct, and not a distorted and limited artefact of human cognition That is true for everything, it's just a radical skeptic position. Nevertheless, mathematics has the highest standards of rigor for proofs among all disciplines. |