The Unreasonable Effectiveness of Mathematics in the Natural Sciences (1960)
dartmouth.edu
dartmouth.edu
In the beginning of mathematics, it was likely, if not definitively created to enumerate similar objects in a particular given set. In order to enumerate an object two things must happen. The object must be turned into an object: this means it must be reduced into some abstract concept which holds certain properties to belong in a set. Second, it must be mapped with a further abstract ID, the number. The most significant consequence of this is that the enumerated ID's and objects now hold the property of logic, because in our world all objects seem to exhibit properties of logic (most of the time). This would have begun as simple arithmetic, an early model of a useful description of our world. From this it becomes evident that you can now map any integer to objects, and perform operations on those integers which map onto the real world as if you were performing the operation on those very real objects.
Later, mathematicians would then become less concerned with the objects mathematics served to represent but more on the relationships in between them. In the logical manipulations of these objects, mathematicians began really exploring the very logical relationships between objects (the creation of variables would coincide here over strict numbers). They once represented an object, yet still can be used again to represent an object. It follows, that these objects should now follow these logical relationships.
From these concerns, I reduce mathematics further as a nice description of logical relationships; a derivative of our own model of the universe.
This is the thing that needs to be explained. I get that math is just logic, but why should we expect the natural world to follow logical rules?
>I reduce mathematics further as a nice description of logical relationships; a derivative of our own model of the universe.
The rules of logic are the behaviors of objects that we experience and observe. In a differing reality, objects may have properties which could be illogical to the perspective of someone in our own; yet if we move to that strange reality, our brains would slowly process the information into a new model of reality with the ~new logic~. However that ability is mostly pointless as any group of mostly consistent behaviors will have either more but never differing logical relationships as any other reality with mostly consistent behavior (this is also why we see something called duality in mathematics).
The further consequence of this, in our studies of natural sciences, we hold our brain's model of reality as supreme. We don't consider something to be ~true~ until at least its conclusions can be witnessed and perceived. The consequence and perhaps also the cause of this, is all of mathematics and natural sciences are mostly attempts to formalize as well as quantify our own brain's unified model of reality.
He's specifically saying we have no a priori reason to expect that mathematics should have predictive ability in the real world, but it seems to do so (I'm not sure I buy this), and we should be thankful.
Working on a rather advanced procedurally generated game world for a long time has only strengthened this belief. The world looks quite natural, but it is literally a visualization of an ordinary integer hash function being computed many millions of times. Exploring the Mandelbrot set intensely, has the same effect on me, and that math is much simpler. I'm not trying to assume these analogies are tautologies, but when I'm inside the game world and it looks both totally natural, and also exactly like my hash function, all I can say is it's very suggestive.
That last point is unconvincing. Maybe they aren't the simplest possible concepts, but it surely must be the case that had we evolved in any universe, we would have sought to understand it and would have formalized that understanding. That's not all that math is, but it has led to several popular branches of it. Surprising? Unreasonable? Not at all.
The additional problem is that philosophy, like any field, has a steep learning curve. It takes a lot of effort to go from "I want to learn about the philosophy of my field" to "I know a little bit about the philosophy of my field".
During my time as a master student in mathematics we used to have a joint seminar with students from philosophy where we would take turns reading and presenting topics from philosophy of mathematics. I think it was useful for everyone, and it wasn’t difficult to organise and didn’t take much time (2 hours per week during one term).
This is part of the problem. I start reading; I see something "obviously false" which is endorsed by lots of people; I see something "obviously true" which is endorsed by lots of people but is contradictory to the obviously-false thing; I see a case which contradicts both of those and isn't obviously true or obviously false; and I give the whole thing up as non-rigorous and/or tedious. But I'll look out Shapiro and see if it helps.
Before we tried that, we tried explaining the world with gods and spirits - the study of the personalities of the forces of nature.
I just like to keep that in mind, that mathematics was our second attempt.
https://archive.org/stream/SolvedAndUnsolvedProblemsInNumber...
Second, programming is a limited subset of discrete mathematics.
Third, the underpinning of all that science is still math.
https://www.dartmouth.edu/~matc/MathDrama/reading/Hamming.ht...
Edit: a teaser:
"Let us next consider Galileo. Not too long ago I was trying to put myself in Galileo's shoes, as it were, so that I might feel how he came to discover the law of falling bodies. I try to do this kind of thing so that I can learn to think like the masters did-I deliberately try to think as they might have done.
Well, Galileo was a well-educated man and a master of scholastic arguments. He well knew how to argue the number of angels on the head of a pin, how to argue both sides of any question. He was trained in these arts far better than any of us these days. I picture him sitting one day with a light and a heavy ball, one in each hand, and tossing them gently. He says, hefting them, "It is obvious to anyone that heavy objects fall faster than light ones-and, anyway, Aristotle says so." "But suppose," he says to himself, having that kind of a mind, "that in falling the body broke into two pieces. Of course the two pieces would immediately slow down to their appropriate speeds. But suppose further that one piece happened to touch the other one. Would they now be one piece and both speed up? Suppose I tied the two pieces together. How tightly must I do it to make them one piece? A light string? A rope? Glue? When are two pieces one?"
The more he thought about it-and the more you think about it-the more unreasonable becomes the question of when two bodies are one. There is simply no reasonable answer to the question of how a body knows how heavy it is-if it is one piece, or two, or many. Since falling bodies do something, the only possible thing is that they all fall at the same speed-unless interfered with by other forces. There's nothing else they can do. He may have later made some experiments, but I strongly suspect that something like what I imagined actually happened. I later found a similar story in a book by Polya [7. G. Polya, Mathematical Methods in Science, MAA, 1963, pp. 83-85.]. Galileo found his law not by experimenting but by simple, plain thinking, by scholastic reasoning.
I know that the textbooks often present the falling body law as an experimental observation; I am claiming that it is a logical law, a consequence of how we tend to think."
[..]
"But if you do not like these two examples, let me turn to the most highly touted law of recent times, the uncertainty principle. It happens that recently I became involved in writing a book on Digital Filters [8. R. W. Hamming, Digital Filters, Prentice-Hall, Englewood Cliffs, NJ., 1977.] when I knew very little about the topic. As a result I early asked the question, "Why should I do all the analysis in terms of Fourier integrals? Why are they the natural tools for the problem?" I soon found out, as many of you already know, that the eigenfunctions of translation are the complex exponentials. If you want time invariance, and certainly physicists and engineers do (so that an experiment done today or tomorrow will give the same results), then you are led to these functions. Similarly, if you believe in linearity then they are again the eigenfunctions. In quantum mechanics the quantum states are absolutely additive; they are not just a convenient linear approximation. Thus the trigonometric functions are the eigenfunctions one needs in both digital filter theory and quantum mechanics, to name but two places.
Now when you use these eigenfunctions you are naturally led to representing various functions, first as a countable number and then as a non-countable number of them-namely, the Fourier series and the Fourier integral. Well, it is a theorem in the theory of Fourier integrals that the variability of the function multiplied by the variability of its transform exceeds a fixed constant, in one notation l/2pi. This says to me that in any linear, time invariant system you must find an uncertainty principle."
Yes! Perhaps there's a meta-law which says any Universe in which Consciousness exists can not be fully explained by one single mathematical formulation. After all, Math ultimately originates in the conscious Mind. Perhaps only a Universe that requires two or more distinct Mathematical models to explain it (for us, GR and QM) can host Consciousness within it.