Background
What is Mathematics?
- Most novices think that mathematics is about calculation of complicated equations/formulas.
- More experienced practitioners believe that mathematics are about rigor ("I am a Mathematician because I don't think, I just follow the deductive process")
- However, mathematics is more about abstraction/generalization, but because we end up in places that are far outside the realms of our intuition, we rely on rigor to establish that we are on solid ground. But rigor is not mathematics itself.
- Mathematicians have a sort of "post-rigorous" thought process, where their intuition is tuned to understand the abstraction without relying on rigor (i.e. Euler).
- We see this as well in Physics, since Physicists work in the Physical realm, where our intuition can hold.
How does Mathematics evolve?
Generally, Calculation/Examples are primordial, and then motivate more general algorithms/proofs.
- \(42=2\times3\times7\to\) Unique Prime Factorization/Euclidean Algorithm
- \(3-4-5\) right triangle \(\to\) "Pythagorean Theorem"
Perhaps the most interesting example of this evolution process of Math is Euclid's Axioms, as the text Euclid was often used in secondary education as not just a tool of mathematical education but for general reasoning.
But, in the 20th century, several logical errors were found in Euclid's Axiomatic System, with the most famous being Postulate I.4.
This speaks to a broader trend of mathematics romanticizing the history of Greek influence on proof based systems, which often involves erasure of mathematics from elsewhere. See more here.
During the 19th/20th centuries, algorithmic mathematics ("concrete") was mostly abstracted into "set-theoretic" mathematics.
- Permutation sets \(\to\) Group Theory
- Polynomial Roots and Constructions \(\to\) Galois Theory
Interestingly, Galois Theory nowadays is taught very abstractly via fields/field extensions. But, looking at Galois's writings, most of his computations of Galois groups were very computational/algorithmic.
Another nice example is functions. Functions were historically viewed as rules that transform inputs to outputs, and were defined by descriptions (polynomials, etc). This led to major arguments over what sort of rules constituted functions that really existed.
This led to the vibrating string controversy, and more generally, do there exist functions that can't be described simply?
The set-theoretic interpretation of functions clearly resolved this controversy, but introduced a new can of worms due to the large number of indescribable functions. These functions can often be mind-boggling, such as space-filling curves. A simple setup: $$\sum\limits_{i=1}^\infty a_i10^{-i}\mapsto\left(\sum\limits_{i=1}^\infty a_{2n-1}10^i,\sum\limits_{i=1}^\infty a_{2n}10^i\right)$$
We may be in a third shift in mathematical thinking. Computers have led to the rise of experimental mathematics, computer aided proofs, and computer aided proof verification. Do we consider proofs verified by a computer to be a verified proof? What about a proof that depends on brute calculation impossible to do by hand?
Why does Mathematics Work?
Mathematics seems to work amazingly well compared to other areas of human thought. In medicine, for example, it was thought that leeches were a good way of dealing with infection, until that was disproven. Economists were convinced that the Philips curve was an inverse model between inflation and unemployment, until it entirely broke past the 2008 recession. In mathematics, after a theorem is published and verified, very rarely does mathematics take a U-turn and overturn the result.
The harder sciences seem to behave similarly to mathematics in this regard, but I'd argue that this is more of a function of their reliance on mathematics than anything implicit within the field. Physicists have long been convinced of the existence of some sort of universal ether permeating the universe, and while the specific forms of ether prescribed by 19th century physicists like Kelvin have since been debunked, the idea of the ether continues to play a role in physics as a mathematical necessity, as Einstein found with his introduction of the cosmological constant. The theories proposed in these fields feel like they are approaching correctness as our observation of our environment improves, but the mathematics is correct all along.
So why does mathematics work? This is a difficult question, and it often becomes a question of dogmatism. If one is a Platonist, the answer to the question is obvious: mathematics is the language of the universe, and we are just explorers of it. It is natural for mathematics to work since it is the universal truth, and the more mathematics we learn, the more we understand about the universe. This perspective, viewing mathematics as a discovery, is a very pleasant one as it suggests a natural reason to expect our mathematics to be well-behaved.
Another perspective is that of the formalist, who believes that mathematics is simply a human language, designed to help us understand the universe. The platonist might view the Fibonacci sequence as somehow inbuilt into the sunflowers whose bud patterns resemble the sequence, while the formalist would view the sequence as our way of interpreting what we see when looking at the sunflowers. From the formalist's perspective, the well-behavior of mathematics is a far more interesting philosophical problem, and requires a more rigorous approach to the foundations of mathematics. For the formalist, this comes through the study of logic, although I'd contend that it is important regardless of which perspective you hold on the workings of mathematics.