| The Revolution: At the turn of the 1900s, the field of mathematics was changing. The old school relied of physical intuition as a means of proof. The new school found that an adherence to formal logic led to more reliable findings. Most Relevent Sections: In brief, traditionalists lost the battle in the professional community but won in education. The failure of “new math” in the 1960s and 70s is taken as further confirmation that modern mathematics is unsuitable for children. This was hardly a fair test of the methodology because it was very poorly conceived, and many traditionalists were determined that it would succeed only over their dead bodies. However, the experience reinforced preexisting antagonism, and opposition is now a deeply embedded article of faith. Many scientists and engineers depend on mathematics, but its reliability makes it transparent rather than appreciated, and they often dismiss core mathematics as meaningless formalism and obsessive-compulsive about details. This is a cultural attitude that reflects feelings of power in their domains and world views that include little else, but it is encouraged by the opposition in elementary education and philosophy. In fact, hostility to mathematics is endemic in our culture. Imagine a conversation: A: What do you do?
B: I am a ———.
A: Oh, I hate that.
Ideally this response would be limited to such occupations as “serial killer”, “child pornographer”, and maybe “politician”, but “mathematician” seems to work. It is common enough that many of us are reluctant to identify ourselves as mathematicians. Paul Halmos is said to have told outsiders that he was in “roofing and siding”!from page 33-34. Core methods such as completely precise definitions (via axioms) and careful logical arguments are well known, but many educators, philosophers, physicists, engineers, and many applied mathematicians reject them as not really necessary. from page 34 Why It Matters: The sciences are at risk in the reckless implementation of maths. Education methods are weakened by the philosophical divide. |
We're at a very exiting point in time where we know basic mathematics to be wrong. It predicts a great many things, but there's problems that make it thoroughly unsatisfactory. In a way it's like physics in the beginning of the 20th century, with the black body radiation problem. Of course, we have been in this less-than-satisfactory state for ~70-80 years now, and no Einstein in sight ...
Godel's theorem means that there's an infinite set of empirical truths (ie. simple experiments you can try out with marbles and bags) that are completely unexplained by mathematics - and thus by every science built on top of it.
Worse : this is not a fixable problem. Sure we can fix it for specific problems. Wherever we see an obvious leak (say the birthday problem, or large cardinal number problems) it can be plugged with a new well-chosen (or -more often- ill-chosen, like choice) axiom, but there's infinitely many leaks and the proof means that there's no plug that will stop any significant number of them.
So right now in the set of all empirically observable events, there's a set that's explainable by science, and there's a set unexplained by science. The unexplained set is at least as large as the explainable set (and keep in mind that's because both sets have been proven to have a cardinality of at least the largest known cardinal number, given the actual definitions of those 2 sets I'd say the unexplained set is going to turn out to be bigger).