Despite the author's argument, for me at least, there are an order of a priori "hints" which argue why Mathematics ought to be seen in the natural sciences.
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.
>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.
It's worth re-reading this a few times to understand what Wigner was actually getting at:
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.
I can readily understand the perspective, at least as far as the a priori reason being hard to discern or articulate. But I don't hold the perspective. Like Tegmark, I believe the reason math works is that the universe is math. (And it follows that mathematicians who believe their work is unconnected in principle from the physical world, are being myopic or escapist.)
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.
A counterpoint to this is Derek Abbott's "The Reasomable Ineffectiveness of Mathematics." Personally, I think domains with low-N (dimensions and sample sizes) and high complexity remain beyond the optimistic expectations of Eugene Wigner. https://pdfs.semanticscholar.org/462d/7b6b1ee8243b6aa8897be3...
>It is true, of course, that physics chooses certain mathematical concepts for the formulation of the laws of nature, and surely only a fraction of all mathematical concepts is used in physics. It is true also that the concepts which were chosen were not selected arbitrarily from a listing of mathematical terms but were developed, in many if not most cases, independently by the physicist and recognized then as having been conceived before by the mathematician. It is not true, however, as is so often stated, that this had to happen because mathematics uses the simplest possible concepts and these were bound to occur in any formalism.
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.
No mention of Frege here, and only a cursory quote from Russell. While I can't comment on the author's perspective based on this article alone, it is consistently shocking to me how little scientists and mathematicians are interested in in reading and engaging with philosophy related to their fields.
I, for one, would be happy to read mathematical philosophy. The problem is that there is so much of it, and lots of it contradicts lots of other bits of it. As a mathematician attempting to find out about the philosophy of mathematics, I want a source of truth and correctness. In order to find it (assuming it even exists), I also have to wade through reams of falsity.
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".
There is no consensus as to what is the true and correct philosophy of mathematics. People are going to disagree about that, just like they disagree about almost every other part of philosophy. But there are books which give you a survey of different views, such as Shapiro’s “Thinking about mathematics” — which surely will bring you to “I know a bit about the philosophy of mathematics”-level.
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).
> There is no consensus as to what is the true and correct philosophy of mathematics.
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.
I dunno. We have spent a couple of millennia trying things and taking whatever is effective. It may be unreasonable that something is effective, but I can't convince myself it is so.
For a different take on this, see Dan Shanks' "The case for pythagoreanism" starting on pg. 130 in his classic text Solved and Unsolved Problems in Number Theory:
"Math" is just a language designed for deductive logic. Lots of science is now being done in other (programming) languages that can fill the same role. Is "math" still special?
Until theorem proving is as easy in a programming language as it is in plain old math (last I checked, dependently typed languages were still verbose and cumbersome for this purpose), then Yes, math will still be special because it is the proofs that make it special (also, not all proofs use deductive logic).
The physicist Max Tegemark claims that the only real properties are mathematical properties. Notice how much physics relies on math. What are the properties of an electron? An electron is understood mathematically.
Yeah, mathematics seems rather meta. If the universe/multiverse we live in turns out to be a simulation, if every scientific discovery turns out to be incorrect, etc....set theory remains. There is something incredibly powerful and beautiful about that.
Though to get from "every natural law we've been able to characterise so far has turned out to be mathematical in nature" to Tegmark's "nature is literally mathematics" is a big leap that many people are not willing to take.
Math is not a language. Mathematical notation is a language, used to describe math. Programming languages are notations for describing particular mathematical objects.
"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."
> We now have, in physics, two theories of great power and interest: the theory of quantum phenomena and the theory of relativity. [...] The two theories operate with different mathematical concepts -- the four dimensional Riemann space and the infinite dimensional Hilbert space. So far, the two theories could not be united, that is, no mathematical formulation exists to which both of these theories are approximations. All physicists believe that a union of the two theories is inherently possible and that we shall find it. Nevertheless, it is possible also to imagine that no union of the two theories can be found.
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.
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.