You cannot falsify mathematics by experiment (except in the subjective Bayesian sense).
Actually, that's technically false. The statements mathematical axioms make about reality are bizarre, but they exist and are actually falsifiable.
One of the fundamental properties we want from our axiomatic systems is consistency — the fact that it does not lead to a logical contradiction. We would certainly reject our current axiomatic foundations in case we found them inconsistent.
Turns out it's possible to write a program which would halt if and only if ZFC is consistent. I would not recommend running this one as it's a Turing machine and thus not really optimized (and in any case, ZFC being inconsistent is unlikely, and it's even more unlikely that the proof of it's inconsistency would be easy to be found with current technology), but in theory one might run one of such machines long enough to produce a contradiction, which would basically physically falsify the axioms.
It sounds like this is making a case for a view known in the philosophical literature as moral response-dependence. PhilPapers offers 39 papers on this topic; here is one counterpoint that I recommend.
I have to say that the claimed reductios here strike me as under-argued, particularly when there are literally decades of arguments articulating and defending various versions of moral anti-realism, and which set out a range of ways in which the implications, though decidedly troubling, need not be absurd.
There are three branches of philosophy: natural philosophy, moral philosophy and metaphysical philosophy. Colloquially, we refer to natural philosophy as "science", "moral philosophy as "ethics" and metaphysical philosophy as "philosophy". There's also math "in fact, it would not be a bad definition of math to call it the study of terms that have precise meanings."
Science is a worthwhile endeavor. Metaphysical philosophy tends to be worthless because it makes no falsifiable predictions about reality.
Losing touch with reality is boring.
What about ethics? Western philosophy tends to take it for granted that ethics is consequentialist. Consequentialist ethics breaks ethics down into three components.
Modelling the world is science. Optimizing a value function is math. Science and math are ethically trivial because consequentialist value functions are orthogonal to science and math. Value functions can be anything, which is why consequentialist moral paradigms tend to produce paperclip maximizers.
Consequentialist moral frameworks reduce ethics to axiomatic value functions (which could be anything because they are are unfalsifiable) and optimization strategies (which are ethically trivial). The optimization strategies belong in the realms of science and math. The axiomatic value functions of a consequentialist moral framework are declared by fiat. They are upstream of reality, like mathematical axioms.
It is fine for mathematical axioms to be upstream of reality because mathematical axioms don't make statements about reality. The Axiom of Choice isn't a statement about reality. It's a statement about Mathland. The Axiom of Choice isn't even a statement about all of Mathland. The Axiom of Choice is applies only to the region of Mathland where the Axiom of Choice applies. Two plus two equals four in the real numbers but two plus two equals zero in the cyclic group.
It's fine for mathematical axioms to be unfalsifiable because math itself is upstream of reality. You cannot falsify mathematics by experiment (except in the subjective Bayesian sense).
Ethics must be falsifiable because of reducto ad absurdum. If ethics wasn't falsifiable then there would be no truth values to statements like "genocide is bad", "compassion is good" or even "suffering is bad". Could you build an AI with morality inverted from human beings? Yes. But human morality is broadly consistent. Syrian dictator Bashar Assad's family reads Harry Potter. Vladimir Putin understands the value of loyalty.
Human morality isn't perfectly consistent. Ethics is like taste.
We'd better start by saying what good taste is. There's a narrow sense in which it refers to aesthetic judgements and a broader one in which it refers to preferences of any kind. The strongest proof would be to show that taste exists in the narrowest sense, so I'm going to talk about taste in art. You have better taste than me if the art you like is better than the art I like.
If there's no such thing as good taste, then there's no such thing as good art. Because if there is such a thing as good art, it's easy to tell which of two people has better taste. Show them a lot of works by artists they've never seen before and ask them to choose the best, and whoever chooses the better art has better taste.
So if you want to discard the concept of good taste, you also have to discard the concept of good art. And that means you have to discard the possibility of people being good at making it. Which means there's no way for artists to be good at their jobs. And not just visual artists, but anyone who is in any sense an artist. You can't have good actors, or novelists, or composers, or dancers either. You can have popular novelists, but not good ones.
―Is There Such a Thing as Good Taste? by Paul Graham
If you throw out the idea that ethical axioms can be right or wrong then you must throw out the idea you can make good and bad choices at all which is another reducto ad absurdum. There might not be such thing as good and bad choices in the objective reductionist physical sense but they are a very real part of subjective human experience.
I think the key to this puzzle is to remember that art has an audience. Art has a purpose, which is to interest its audience. Good art (like good anything) is art that achieves its purpose particularly well. The meaning of "interest" can vary. Some works of art are meant to shock, and others to please; some are meant to jump out at you, and others to sit quietly in the background. But all art has to work on an audience, and—here's the critical point—members of the audience share things in common.
―How Art Can Be Good by Paul Graham
Ethics, like art, has a human audience.
Once you start talking about audiences, you don't have to argue simply that there are or aren't standards of taste. Instead tastes are a series of concentric rings, like ripples in a pond. There are some things that will appeal to you and your friends, others that will appeal to most people your age, others that will appeal to most humans, and perhaps others that would appeal to most sentient beings (whatever that means).
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So could we figure out what the best art is by taking a vote? After all, if appealing to humans is the test, we should be able to just ask them, right?
Well, not quite…. There are sources of error so powerful that if you take a vote, all you're measuring is the error…. There are two main kinds of error that get in the way of seeing a work of art: biases you bring from your own circumstances, and tricks played by the artist….
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The way not to be vulnerable to tricks is to explicitly seek out and catalog them. When you notice a whiff of dishonesty coming from some kind of art, stop and figure out what's going on. When someone is obviously pandering to an audience that's easily fooled, whether it's someone making shiny stuff to impress ten year olds, or someone making conspicuously avant-garde stuff to impress would-be intellectuals, learn how they do it. Once you've seen enough examples of specific types of tricks, you start to become a connoisseur of trickery in general, just as professional magicians are.
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I think with some effort you can make yourself nearly immune to tricks. It's harder to escape the influence of your own circumstances, but you can at least move in that direction. The way to do it is to travel widely, in both time and space. If you go and see all the different kinds of things people like in other cultures, and learn about all the different things people have liked in the past, you'll probably find it changes what you like. I doubt you could ever make yourself into a completely universal person, if only because you can only travel in one direction in time. But if you find a work of art that would appeal equally to your friends, to people in Nepal, and to the ancient Greeks, you're probably onto something.
―How Art Can Be Good by Paul Graham