Firstly there's no need whatsoever to be rude even if I'm wrong. It doesn't help the discussion and isn't nice. You also don't know anything whatsoever about what I do and don't know about maths or the definitions of words.
Secondly prospective theorems are absolutely falsifiable. Since a theorem is a statement that has been proven to be true yes they are unfalsifiable by definition - they have already passed that test. That doesn't really generalise to any sort of meaningful statement about the falsifiability of maths. Saying theorems are unfalsifiable is equivalent to saying "True statements can't be proven false". Well, yes.[1]
ie If I say Sean Hunter's theorem is that if you take a triangle with arbitrary sides a b c and angle opposite a of theta that
a^2 = b^2 + c^2 -42 b c cos theta
that statement is absolutely falsifiable (and false), which you can establish with basic geometry and trig[2]. When you demonstrate it not to be true it is not a theorem, so I was wrong to call it that. That is a demonstration of how maths is falsifiable.
[1] Even so it's often possible to make progress via proof by contradiction - showing that if this theorem were not true something else which we know to be true would be false. But in most of my maths books proving all of the theorems is the norm, so they are for sure falsifiable while you are trying to establish whether or not they are theorems.
[2] Drop an altitude from one of the angles at b and c and then use pythagoras and a bunch of cancelling. You will prove that a^2 = b^2 + c^2 -2bc cos theta of course. My statement is only true if a is the hypotenuse of a right triangle meaning cos theta is zero and my incorrect coefficient doesn't matter.
I merely attempting to reciprocate/mirror your tone. You are the one (self)identifying it as "rude".
I have some idea about what you do and don't know about definition and definability (in general) given the words you've said so far and the way you've used them.
Prospective theorems are not theorems until a proof is presented. At which point they become retrospective theorems.
All that "falsification" and counter-examples prove is that the so-called "proof" of a "theorem" wasn't. If you have indeed provided a counter-example that's a proof of negation which raises questions: what was wrong with the original "proof" of the theorem? Since proofs are programs - there must have been a bug in the proof. Better type-check that proof/program...
The presence of a counter-example to Sean Hunter's "theorem" simply demonstrates that it's not a theorem. It's a misnomer. Theorems are exactly those Mathematical stataments for which no proof of negation exists.
You seem to be presupposing some particular kind of mathematics. I am talking about all possible Mathematics in general; of which the particular Mathematics you are currently using is just one particular instance. A historical and cultural coincidence.
There's a Mathematical paradigm in which proof-by-contradiction is a valid proof method e.g mathematics founded upon classical logic.
And there's a Mathematical paradigm in which proof-by-contradiction is not a valid proof method e.g mathematics founded upon intuitionistic logic. This is basically what we call Computer Science. It has fewer axioms than Classical Mathematics (e.g the axiom of choice is severely restricted) and so it's a much stronger proof-system. You could even say Intuitionistic Mathematics (which is basically CS) is "more foundational" (it is much closer to the foundations?) than Mathematics.
The fact that you are admitting proof-by-contradiction in your methodology tells me about your choice of foundations, but so what? There's a foundation which axiomatically pre-supposes choice; and a foundation which doesn't.
And in the foundations where choice is not axiomatic "proof" by contradiction is not a valid proof.
The reasoning goes something like this:
1. Choice implies excluded middle.
2. Excluded middle implies all proposition are either true or false.
3. Excluded middle implies that proof by contradiction is valid.
Rejecting 1 results in the rejection of 2 and 3 also.
> Prospective theorems are not theorems until a proof is presented. At which point they become retrospective theorems.
...
> The presence of a counter-example to Sean Hunter's "theorem" simply demonstrates that it's not a theorem. It's a misnomer.
That is what I said. I showed a mathematical statement and I showed how you could falsify it. Since you said "mathematics is not falsifiable" I have shown your statement is not true. Do you see why?
You were the one who decided that the distinction between conjectures and theorems is important. I have now shown two examples of mathematics that was falsified.
Unless you're trying to say neither me nor Euler was a mathematician in which case we can agree about me but not about Euler.
Your failure to understand what I am saying is abysmal.
>That is what I said. I showed a mathematical statement and I showed how you could falsify it.
>Since you said "mathematics is not falsifiable" I have shown your statement is not true.
You have taken it upon yourself to interpret "Mathematics is not falsifiable" as broadly as needed in order to confirm your own biases; and then proceeded to attack a strawman instead of a steelman. That's the lack of charity...
>You were the one who decided that the distinction between conjectures and theorems is important.
And you were the one who decided that it isn't; so you falsely equated them.
What you have demonstrated is the falsification of the statement "X is a theorem"; not the falsification of "mathematics is not falsifiable." - a hasty generalization fallacy.
Which doesn't demonstrate anything of import or relevance whatsoever. Obviously a non-theorem is not a theorem. This is no more interesting than demonstrating that non-Mathematics is not Mathematics.
This in no way diminishes or falsifies my own claim that theorems are unfalsifiable! And neither is Mathematics.
Because if you do falsify it - then it was never a theorem. By definition. Theorems are true, not false. A false theorem is a contradiction in terms. A misconception. An error in reasoning.
Maybe Euler wasn't a Mathematician either. Who knows? Those sort of questions are undecidable.
Secondly prospective theorems are absolutely falsifiable. Since a theorem is a statement that has been proven to be true yes they are unfalsifiable by definition - they have already passed that test. That doesn't really generalise to any sort of meaningful statement about the falsifiability of maths. Saying theorems are unfalsifiable is equivalent to saying "True statements can't be proven false". Well, yes.[1]
ie If I say Sean Hunter's theorem is that if you take a triangle with arbitrary sides a b c and angle opposite a of theta that
a^2 = b^2 + c^2 -42 b c cos theta
that statement is absolutely falsifiable (and false), which you can establish with basic geometry and trig[2]. When you demonstrate it not to be true it is not a theorem, so I was wrong to call it that. That is a demonstration of how maths is falsifiable.
[1] Even so it's often possible to make progress via proof by contradiction - showing that if this theorem were not true something else which we know to be true would be false. But in most of my maths books proving all of the theorems is the norm, so they are for sure falsifiable while you are trying to establish whether or not they are theorems.
[2] Drop an altitude from one of the angles at b and c and then use pythagoras and a bunch of cancelling. You will prove that a^2 = b^2 + c^2 -2bc cos theta of course. My statement is only true if a is the hypotenuse of a right triangle meaning cos theta is zero and my incorrect coefficient doesn't matter.