The Unreasonable Effectiveness of Mathematics in the Natural Sciences is an article by the famous mathematician Eugene Wigner. (Wigner 1990)
What I found most peculiarly interesting is the focus on many mathematical/physics texts on the idea of the “beauty” of the expressions; and, it seems, the clear pleasure that Wigner gets from analyzing the systems with the aforementioned “beauty.”
Setting aside whether or not this beauty is “deserved”/appropriate, I love that my attraction to physics is somewhat similar to what Wigner describes. Under the appropriate conditions, with constraints, it is possible to build a solution to physics problems simply through the evolution of mathematics.
It is not to say that the models mathematics provides is correct. I like that Winger ended on the note about how “false” theories, even despite their falseness, provided shockingly accurate estimations of physical phenomena. Perhaps mathematics provides an almost-fully solid foundation to creating physical systems, but then the entire “flaw” we see with mathematical modeling is in our (in)ability to provide the limitations to scope.
For instance, Bohr’s model, an example of “falsehood” modeled, is an over-limitation to scope which—thought reducing mathematical complexity—resulted in a “wrong” theory. However, the mathematics behind the theory remains to be solid despite the scope limitation, making the result work in a reasonable manner (except for the pitfalls).
The inherent concern behind this statement, then, is that there is a case where we can build a perfectly reasonable system to model something, but it turns out that the system is correct only in the limited scope which we are used to operating; when suddenly the scope becomes broken, we are so used to the mathematical tools that we have came to rely on that we don’t notice their failures.
I like that this entire point is brought up before our start in DiffEq, perhaps as a “with great power comes great responsibility” type of caution to us in terms of how our modeling may go awry while at the same time acting as a preview of the usefulness of the principles provided taken as a whole.
Maths show up at entirely random places
The first point is that mathematical concepts turn up in entirely unexpected connections. Moreover, they often permit an unexpectedly close and accurate description of the phenomena in these connections.
Wondering whether or not the theory is unique due to its applicability
He became skeptical concerning the uniqueness of the coordination between keys and doors.
That math is really useful, its weird
The first point is that the enormous usefulness of mathematics in the natural sciences is something bordering on the mysterious and that there is no rational explanation for it.
It also raises the question of how actually unique our theories are given they are all so applicable
Second, it is just this uncanny usefulness of mathematical concepts that raises the question of the uniqueness of our physical theories.
The goal of mathematics is maximize the space of usefulness
The great mathematician fully, almost ruthlessly, exploits the domain of permissible reasoning and skirts the impermissible.
Regularity is suprising because its… regularly found, which is unique
The second surprising feature is that the regularity which we are discussing is independent of so many conditions which could have an effect on it.
Laws of Nature are all highly conditional
The principal purpose of the preceding discussion is to point out that the laws of nature are all conditional statements and they relate only to a very small part of our knowledge of the world.
That maths is just a fallback for “beatiful” physics happening
the connection is that discussed in mathematics simply because he does not know of any other similar connection.
Apart from invarients, we just scope-limit ourselves to get the remaining bits that we need to make stuff work “beautifully”
propose to refer to the observation which these examples illustrate as the empirical law of epistemology. Together with the laws of invariance of physical theories, it is an indispensable foundation of these theories.