Right, and I did cover it, albeit briefly.
Mathematics is a set of operations for manipulating information. It’s sound because it’s based on a foundation of formal logic and our universe is logical. And it’s useful because formalizing information processing, adding to our intuition about what facts can be derived is basically augmenting our reasoning ability.
False dichotomy.
Consider English. The English language has been used in many predictions and observations about the universe. So is the English language “out there”, floating in the ether or are our minds somehow in tune with the universe when we speak English?
And that’s the problem. There are certain philosophical questions which have been talked about so long, and by so many eminent names, that it becomes impossible to point out that the Emperor has no clothes and this problem may well have a simple solution.
Note that for all the quoting of Einstein, you have not added any arguments to the two I outlined in the OP (in fact you haven’t addressed the second one).
No, the English language does not predict natural laws and phenomena. It only describes them. This is completely different from mathematics.
It you think otherwise, then please give an example. And, no, ‘someone wrote a story about flying machines before they existed’ is NOT an example. ‘The laws of aerodynamics are predicted by English grammar’ would be an example.
No. In fact, those same angles are 100 degrees and/or pi/2 radians.
pi/2 radians OTOH is a constant, or at least it is one within Euclidean geometries (I barely know that non-Euclidean ones exist, so don’t ask me what happens to circles in them).
The problem is that the term “math” is so broad that there will never be a definitive answer on either side, the topic is just too broad.
Complex multiplication by i rotates you by 90 degrees, and complex numbers are fundamental to higher physics and spacial orientation in a way that would not match some claim of “philosophical commitment to materialism”
Solutions to problems like the wavefunction inherently involve complex numbers as an example. But PEMDAS is not going to arise out of nature and is thus similar to English, it is purely convention.
To repeat, this question is only hard to answer because the way it is framed is overly broad.
I’m not sure I understand your point. Isn’t all mathematics describing a physical world, even if it’s a hypothetical one. Even the most basic numbers, 1, 6, 153 are referring to things.
When a volcano spews a boulder into the sky, it’s trajectory comports to the laws of physics. That was true a at the time of the first volcano on earth and it’s true for cannonballs. The laws of physics operated perfectly well before man understood them.
But, again, maybe I’m not understanding your point.
Can a person come up with a branch of mathematics that does give correct predictions about the external universe but that’s different from ours?
Consider English. English is invented, and it describes the world pretty well. Quechua, despite sharing very few words with English, also describes the world pretty well, and is also invented. It’s not like the Inca, in trying to talk about water, said, “Yaku, yaku…hmm, that’s not a particularly apt descriptor for this liquid, let’s try saying ‘water’ instead, that’s a lot closer to the reality.”
If you can come up with an entirely different system of math that “works”, then it seems to me a lot more like a language, and hence invented.
But if there’s only one set of consistent axioms that ends up describing the world accurately, and other consistent sets don’t describe the cosmos accurately, then it seems to me a lot more like a science, and hence discovered.
Physics is descriptive, not prescriptive and it also tends to follow perturbation theory.
But some core concepts would arise in any theory. As an example Newtons universal law of gravitation is really just the inverse square law. While Newton’s theory may not arise in some other population the inverse square law would.
Certainly. We don’t even use one form of mathematics. When most people think of math, they think of our base 10 numerical system (0,1,2,3,4,5,6,7,8,9). But in computer science they use binary, which is a base 2 (0,1) and hexadecimal, which is a base 16 (0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F). Only in base 10 does 1+1=2. In binary, 1+1 = 10. In hexadecimal, 1+1 still =2 but 9+1=A, not 10.
The point being, these are all different mathematical systems, but they all grounded in their ability to describe something observable in nature. In this particular example, describing the number of objects in a set when objects are added or removed from it.
And I’m not even getting into mathematical constructs that can’t be expressed in physical terms. Like imaginary numbers (I^2 = -1) for example.
Stand back far enough in one’s perspective, and I’m not sure there is a good criterion to distinguish inventions from discoveries.
Let’s contrast the design of a Frisbee, which most would call an “invention,” with the Monster Group, a “discovery.”
Yet both of them can be described with marks of pencil on paper describing relationships. The rules from which the Monster Group builds may be more rigid and/or beautiful than the Frisbee rules, but at some level of abstraction they are similar.
An alien civilization which discovered the Monster Group might also have discovered the Frisbee design. Yes, it would have little chance of duplicating certain types of human “inventions” — so what?
Imaginary numbers are only imaginary due to our conventions.
Neither General Relativity nor QED would exist without complex numbers. While I get that it is commonly taught that imaginary numbers are not re-presentable in the real world that only applies if one sticks to the convention on number lines where integers are places and not distances.
Euler’s Identity and Euler’s formula are all over the physical world and are one of the easiest ways without any other knowledge to derive the periodic nature of sin and cosin.
Heck you can’t even do a proof of the the fundamental theorem of algebra, which is probably not universal but convention without complex numbers.
Consider if I find an effective combination in a particular chess position. Is it a discovery or an invention? Well it’s hard to say…the process for thinking up such a combination has elements of “searching” and elements of “constructing”.
In this situation though most people would be happy with the idea that that combination didn’t “exist” in the universe prior to a human describing it (or even prior to humans inventing the game).
Perhaps the invent / discovery framework is not the right framing.
:rolleyes: Now you’ve just replaced the abstraction of a perfect circle with the abstractions of a perfect polygon and a perfectly traced rotation.
No ideally precise geometrical object actually exists in perfect form in nature. Which is why I think it’s epistemologically dubious to claim that our ideally precise geometrical abstractions are merely “discovered” from natural objects.
“There are only two ways in which we can account for a necessary agreement of experience with the concepts of its objects: either experience makes these concepts possible, or these concepts make experience possible.” Immanuel Kant (Datalinks)
Seems to me there is no way for either invented or discovered to be exclusively applied.
“Pi”, but itself is not math. It’s a measurement, or a ratio.
If we go by the entry in wikipedia:
then it’s going to be hard to categorize something unless we agree on its definition. But if we look at academic mathematicians and what they do, it would appear to be more invention than discovery. It’s more a form of philosophy than of science.
I agree that the distinction between discovery and invention is a fuzzy one.
For example, Alexander Graham Bell invented the telephone, but this means that he discovered a way to transmit and receive sound over a distance through a wire.
Does it make sense to use “discover” for things that would have been the same no matter who discovered them? In this sense I think much of math is discovered, though of course things like notation and terminology are invented.