I am not sure of the implications. Does it even matter if Math is invented or discovered? Maybe it's both, and it isn't a contradiction between invention and discovering?
We describe things having a certain radiation wavelength as having the color yellow. In that sense, yellow it's an invention. That doesn't mean the radiation doesn't exist.
But to complicate things a bit, some things don't exist unless we observe them. This is the case with states of particles described by quantum mechanics.
Math is more than a science, is the sciences upon which most other sciences and tech are founded. You can model anything in a computer and running on a computer using math. You can describe logic, natural language, technology, biology using math.
In that sense, being the building block of other sciences, math is more akin to a language. Two physicist use math in almost the same way two people use English to describe things and communicate ideas.
But math has building blocks, too. Set of axioms upon which any mathematical object and theory can be constructed. The most popular as of now is Zermelo set theory. There are more such fundamental theories, sometimes very different between them.
So, to see if Math is discovered or invented, the easy thing to do is to see if a set of axioms can be discovered or is invented.
> Does it even matter if Math is invented or discovered?
It matters for philosophical inquiry. Does philosophy matter? It matters for people who find it valuable, the same way art, music or literature matters.
You think logic, the art of making distinctions, the study of critical thinking and the foundation of mathematics only matter in the same way art does?
I think not the whole of logic, but this particular distinction -- what changes in the world if one is true versus the other? Is there even a difference that is manifest?
It's like the question of what underlies quantum mechanics -- is the Copenhagen interpretation correct? Everett? Some flavor of deBroglie-Bohm? If there's no way to tell the difference, does it matter?
It matters as far as the philosophy of it matters to you, but the concrete consequences of one way being true versus the other could very well be nil.
Copenhagen requires an extraneous assumption — which may turn out to have practical implications.
At the very least, we know it isn’t parsimonious, though that wasn’t clear until after it was developed (and the philosophical inclination to preserve locality was reasonable).
If you buy into Wittgenstein's rule-following paradox.
You could say anything about anything.
> no course of action could be determined by a rule, because any course of action can be made out to accord with the rule
If I had my own way of (mis?)interpreting him, I think he was alluding to what we now call "strict evaluation" in Programming Language Theory. A diligent rule-follower.
My question asking if you value products of philosophy that have become essential to our society neither contains an unjustified assumption and thus is not a loaded question and still stands and thus has not backfired.
It contains an unjustified assumption that “logic, the art of making distinctions, the study of critical thinking and the foundation of mathematics" and [other?] art belong in different categories.
It also takes an incomplete view on "critical thinking". Drawing distinctions is complemented by abstracting similarities.
The creation of knowledge (in all its forms) is itself a form of artistic self-expression. It is essential to humans, and therefore essential to society.
As a programmer my medium of self-expression is software.
I am an artist as much as I am a logician and a scientist.
What I do is create. It also happens to be useful to others, which is why it pays fucking well too.
There is a distinction between logic, which is just the study of formal systems where you can prove theorems as in any other area of mathematics, and the philosophy of maths and particularly of mathematical logic. The same applies for set theory.
The first one can assert that (in classical first order logic) e.g. there are uncountable models of the natural numbers; this is irrefutable. The second one asks things like "is classical first-order logic even true and/or adequate in an epistemic sense" (this is different from the more pragmatic question of "is classical FOL useful for the problem at hand")? Similarly, something like Gödel's incompleteness theorems are unequivocally true but the question of what they "mean" deep down is nothing that really affects mathematicians' work in general.
Irrefutable and "unequivically" true if you take some classical FOL as a productive method of producing knowledge, a philosophical assumption. Philosophy came first, mathematical logic is just a formalization of that and the reason it was formalized at all and not somethimg else is epistemological.
Your entire argument attempting dismiss philosophy is philosophy.
I said that as soon as you fix some axioms (such as those of classical FOL), the conclusions are irrefutable. This is where mathematics begins. The question where those axioms come from or whether they are "true" are philosophical. The two disciplines are related, but separate.
Lots of mathematicians have different "foundational" beliefs from each other; some have studied or deeply thought about the philosophy, others may just speak to their intuition. However, this doesn't change the fact that they all come to the same conclusions from the same premises. E.g. a constructivist wouldn't be able to claim that a classical proof is "wrong", only that it's non-constructive and therefore unacceptable for some (philosophical or practical) reason; in fact non-constructive proofs can be seen as constructive proofs of some meaningless strings (e.g. the constructivist will maybe dispute the fact that there are discontinuous functions, but they will certainly accept the existence of a first-order derivation of the string representing "not all functions are continuous" from the axioms of set theory), the constructivist would just dispute that there is any meaning to these strings...
> the art of making distinctions, the study of critical thinking and the foundation of mathematics only matter in the same way art does?
A major movement (or several, depending on how you slice it) in modern philosophy takes aesthetics as first philosophy - so yes, arguably they matter exactly the same way art does.
I just think our quality of life would be much lower without the forementioned than without art. We'd be much worse off without any of them. My point is not to dismiss art but emphasize the utility of philosophy.
I agree 100% with everything you say. However, I don’t believe that the “impact on lives” of scientific advances, whilst certainly being of tremendous interest and importance to the humans, carries into much universal significance.
Ridiculous hubris, to claim to speak on their behalf, to claim to know the opinion of so many and diverse minds, and in such a crude misdirection to boot.
Philosophical implications are perhaps the most tangible implications there are: they shape the conception of reality and society that you live in. They only seem intangible because they are subtle and pervasive.
Patents cover inventions that are applications of knowledge. For example, chemistry is certainly discovered, but applying the use of a certain molecule as a glue or as a drug is very much patentable. Physics is also discovered, but a special lever system that is a direct application of Newtonian mechanics is patentable. The knowledge that a certain mathematical formula could be written and has certain properties is not patentable, but a device that employs that formula as an algorithm for, say, predicting stock prices, is (in certain countries at least).
I don't see how you are distinguishing between inventions and discoveries. The "discovery" of a mathematical proof is just an application of knowledge, why aren't they patentable? The "invention" of the light bulb, on the other hand, is just the discovery that putting various particles together in a certain configuration produces light in a predictable manner, why is this patentable? I don't see how a clear delineation between the two can be made.
I'm not. I'm saying that devices whose function is derived from mathematics (algorithms) have the same status as devices that derive their function from physics or chemistry. In either case, "application" doesn't mean some intellectual use but commercial use. The purpose of patents is to protect commercial applications in exchange for sharing the knowledge that led to their creation. The knowledge itself is never protected.
When the lightbulb was patented, everyone was free to learn, study, and disseminate the physical knowledge of how a lightbulb works. What you couldn't do is build one and sell it. Whether mathematics is invented or discovered, the knowledge itself is never protected, but if some non-obvious algorithm has a commercial application, it can be patented so to protect building devices that apply that knowledge for commercial use.
But that's the whole point, patents are awarded to things deemed inventions not things deemed discoveries. If you can't delineate between the two you can't say what is and what is not patentable.
Fourier analysis couldn't have been patented, despite numerous commercial applications. It's just the patent system is setup to only reward low-level innovation, so it arbitrarily excludes research level innovation by terming it discovery.
Yes, what is patented is some commercial application of some knowledge, and that application is the invention. It doesn't matter whether the knowledge itself is invented or discovered. When an algorithm is patented, we're no more patenting math than we're patenting Newton's laws when a car brake is invented. We're patenting a commercial application of either, and that must be the invention.
Commercial application is irrelevant to a patent law. Patent applications need only demonstrate "eligibility, utility, novelty, and non-obviousness." Math and other basic research are ruled out on eligibility grounds, not for being non-commercial, but rather for being "abstract ideas."
Also, algorithms are a bad example, as they are just mathematical functions, i.e. "abstract ideas." They should not be patentable. Although I realize patent law is not actually logically consistent.
Commercial application is the motivation behind patent law, and logical consistency is not the point, but rather legal consistency. When you patent an algorithm you do not patent the idea any more than you patent thermodynamics when you patent an engine. You patent a particular application of an idea that performs some function or functions. The algorithm -- or thermodynamics -- stay completely free for anyone to know, study and disseminate. In fact, patents are designed to make the knowledge public so that they could be studied and improved upon.
Whether or not it works is a separate question (and I completely agree that software patents do not perform their role), but patenting a device to predict stock prices is not patenting math, just as patenting a drug is not patenting chemistry.
to see if Math is discovered or invented, the easy thing to do is to see if a set of axioms can be discovered or is invented
A math theory arises from the axioms it is based on. You just rephrased the question and added the word "easy".
Put it another way, starting from a set of axioms we get a simple ( by some semiobjective definition of simple ) pure math theory that predicts reality to a rediculous level of precision.
Those initial axioms, were they discovered or invented?
Math is only sometimes done that way. Often an interesting field gets axiomatized later. Calculus, arithmetic, geometry, algebra, all existed productively for centuries before axiomatization.
Same way you can build a programming language without a formal spec. Yes, you might find an ambiguous piece of code later, or paint yourself into a corner. But you can do quite a lot without axioms.
It matters because what if you can't actually describe the underlying foundations of the universe and reality with absolute accuracy as a mathematical equation. We get very close, but if we're always off even by ten to the minus whatever, have we really 'solved' it.
Maybe there is some other conceptual framework that we've not or are not able to cognitively express that underpins things.
We describe things having a certain radiation wavelength as having the color yellow. In that sense, yellow it's an invention. That doesn't mean the radiation doesn't exist.
But to complicate things a bit, some things don't exist unless we observe them. This is the case with states of particles described by quantum mechanics.
Math is more than a science, is the sciences upon which most other sciences and tech are founded. You can model anything in a computer and running on a computer using math. You can describe logic, natural language, technology, biology using math.
In that sense, being the building block of other sciences, math is more akin to a language. Two physicist use math in almost the same way two people use English to describe things and communicate ideas.
But math has building blocks, too. Set of axioms upon which any mathematical object and theory can be constructed. The most popular as of now is Zermelo set theory. There are more such fundamental theories, sometimes very different between them.
So, to see if Math is discovered or invented, the easy thing to do is to see if a set of axioms can be discovered or is invented.