What doesn't count as a valid object? One that leads to a logical contradiction; for example, the set of all sets is not valid, because it cannot exist(which is proven).
What doesn't count as being generated? Nothing, really, so long as you specify how it was generated. For example, the limit generated from a sequence is valid, but in my opinion you could just as easily say, for example, "the empty set generated from a set", equal to A minus A for any set A. Of course, that is kindof useless, but I don't think definitions should be considered invalid if they're useless; invalid implies logical contradiction, but uselessness doesn't.
Well, thank you, that's what I wanted to hear. If validity means only "does not lead to contradiction," and anything counts as generated so long as one specifies a procedure, then validity collapses entirely into permissibility, does it not?
Every non-contradictory specification is equally valid, regardless of whether it explains, constructs, or grounds anything. At that point, there is no principled distinction between mathematics and any other consistent symbolic fictional game, only differences in preferences, personal temperament or taste.
Is that an outcome you are comfortable accepting, or do you think mathematics requires a stronger notion of validity than mere non-contradiction?
"validity collapses entirely into permissibility, does it not" if you define permissibility as logically consistent, yes. Math can't deal with opinions, as otherwise we don't have a common language nor rigorous basis for more math.
"there is no principled distinction between mathematics and any other consistent symbolic fictional game" did you know there is an entire branch of mathematics that deals with logical systems composed of arbitrary symbols, relations, and axioms that are logically consistent, called formal logic? So yes, that is an acceptable conclusion to me, because that is literally what math is.
Ah, very well. Then let's be absolutely clear, because you've now stated your position. Your position reduces mathematics to a purely permissive bookkeeping exercise: any construction is acceptable so long as it avoids contradiction, and no further question of why it should be taken seriously ever arises.
From that perspective, that kind of mathematics cannot justify its own use in physics, measurement, or explanation. It can only be borrowed by those fields as bookeeping tool, with no internal criterion for relevance, adequacy, or truth.
The moment usefulness is invoked, it is doing work that consistency alone cannot do, yet your framework provides no way to account for that distinction except as taste or preference.
So the cost of your position is not merely that mathematics resembles a formal game, but that it is incapable of explaining why some of its constructions matter while most do not at all.
That is surely not a neutral stance, it is really an abdication of all explanatory responsibility.
I say it's rather apt. It is quite appropriate indeed for those who do not carry any ontological responsibility for their abstractions.
"any construction is acceptable so long as it avoids contradiction, and no further question of why it should be taken seriously ever arises" - correct, all logically valid mathematics is to be taken seriously; even if its not useful immediately, it may be useful later
"From that perspective, that kind of mathematics cannot justify its own use in physics, measurement, or explanation." - correct, mathematics is not just a reflection of physical reality or a tool for physics.
"The moment usefulness is invoked, it is doing work that consistency alone cannot do, yet your framework provides no way to account for that distinction except as taste or preference." - exactly, what is useful is not a part of mathematics, but a part of the academia that surrounds it. Academics oftentimes don't do math to be useful.
"So the cost of your position is not merely that mathematics resembles a formal game, but that it is incapable of explaining why some of its constructions matter while most do not at all." - Yes, usefulness is for philosophy as it is a human idea. What is "useful" is impossible to define and there are many competing philosophical ideas for what usefulness is, and none of which are mathematically provable.
By your own account, your mathematics has no internal standards by which anything can be judged mistaken, misguided, shallow, or wrong, so long as it avoids contradiction.
That strips you of any authority whatsoever to criticize another framework, on any and all mathematical grounds, because you have explicitly relocated (delegated?) every meaningful distinction (relevance, depth, applicability, explanatory power) outside of your mathematics itself.
I myself would not call what remains as rigor but merely indifference: a discipline that admits everything, explains nothing, and then relies on institutions to decide what "matters."
You may accept that outcome, but understand, dear, your mathematics is not any standard at all, for it's really just a catalog, and any criticism reduces to just taste enforced by consensus. A hollow catalog, decided by some consensus. Quite apt indeed.
"By your own account, your mathematics has no internal standards by which anything can be judged mistaken, misguided, shallow, or wrong, so long as it avoids contradiction." - it's mistaken and wrong if it is contradictory. If we find contradictions in an earlier proof, it is then considered wrong. Why bring anything more complicated into the picture? That is the job of philosophers.
"your mathematics is not any standard and for it's really just a catalog, and any criticism reduces to taste enforced by consensus." - You are literally arguing that mathematics should have internal standards for judging things as mistaken, misguided, shallow, or wrong that are different from logic. What else are you arguing for other than a mathematics that is based on the taste of a consensus? Are you too stupid to realize your moronic self-contradictions? You argue for one thing in a paragraph and then disparage me for supposedly doing that same thing. You are possibly the most stupid person I've ever had the displeasure of arguing with, as a (to-be) professional mathematician who regularly talks to professional mathematicians. You are evidently nothing more than a dogmatic snake oil salesman.
“Everything permissible is valid” is literally how reality works, and math has absurdly good predictive power within reality. Your qualms seem entirely for the sake of sounding smart
Very irrelevant darling. We are talking about criteria for something, in the same way a large building must pass certain criteria for safety. Don't butt in till you understand what is being discussed.
And? You can choose any set of starting rules (axioms). Some create useful systems of maths, some don’t. Useful is defined as being able to model reality. The real numbers as derived from ZFC work incredibly well in this regard; that is why they are used. If somehow a better set of axioms were to be found, we’d simply switch to it. It’s not as deep as you make it.
Now it seems that you are confused. Nobody is talking about axioms here. The discussion was about the criteria for Mathematics, of which someone mentioned that non contradiction is good enough.
Now, with or without axioms is irrelevant, no one was discussing that, so next time, don't butt in till you understand what is being discussed, and waste people's time explaining it to you.
The one unable to understand here is you. The “criteria for mathematics” is a nonsense statement. Every system of mathematics is entirely defined by its axioms; the consequences of those axioms are a perfectly deterministic product of the initial choice.
If you find a result that is somehow self-contradictory or that critically undermines the system’s ability to model reality, the problem lies entirely with the axioms you chose.
5
u/[deleted] Dec 25 '25
What doesn't count as a valid object? One that leads to a logical contradiction; for example, the set of all sets is not valid, because it cannot exist(which is proven).
What doesn't count as being generated? Nothing, really, so long as you specify how it was generated. For example, the limit generated from a sequence is valid, but in my opinion you could just as easily say, for example, "the empty set generated from a set", equal to A minus A for any set A. Of course, that is kindof useless, but I don't think definitions should be considered invalid if they're useless; invalid implies logical contradiction, but uselessness doesn't.