This is a nice summary of an earlier, more "optimistic" view of Mathematics and our understanding of Nature. The final words sum it up well: "We must know! We will know!"<p>Then things like Gödel's Impossibility theorem and the Halting problem came to light. Or quantum physics, with it's inescapable uncertainties, etc. In those earlier days, it must have seemed like humanity would in due time reveal all of the workings of the universe / uncover God's Plan.<p>Then we got a reality check (:<p>I feel like I've gone on a similar trajectory in my understanding of the world. Early on, I felt like "the answers" were known, and I just needed to apply myself to understand them. Perhaps it's a side-effect of the way school and academics are structured. And over time I've come to understand that all the elegant formulae and high-level concepts are just approximations (albeit useful ones) to reality: which is a wild, beautiful, never-fully-knowable mess.<p>Sometimes I think of it in terms of Plato's Cave[1]: the notion was that the Ideal is the "perfect thing", and our observed reality is just wavering shadows on the wall, derived from that ideal. But nowadays I think the shadows are the perfect reality, and the "ideal form" is just our derived, simplified, mental model, and we're more exploring our own limited ability to understand rather than some innate truths of nature itself.<p>[1] <a href="https://en.wikipedia.org/wiki/Allegory_of_the_cave" rel="nofollow">https://en.wikipedia.org/wiki/Allegory_of_the_cave</a>
Always amazes me to see that society as a whole has been and continues to be willing to use public (or private) funding to support natural sciences. In the short term, one often faces the argument about the meaningless of doing things just for the sake of knowing or understanding nature. But in the long run, attracting scientists and engineers to work on such problems must (insert my optimism) add a significant value to our society as a whole.
Hilbert's optimism (we will know) was not a given in the 1930s. What can we make of this almost 100 years later?<p>On the one hand, mathematics and science are incredibly successful.<p>On the other hand, we have Gödel's and Tarski's theorems and still no convincing Theory of Everything (quantum gravity, string theory).