It would be helpful to start with what is probably the minimal commitment of reductionism, which is supervenience. Supervenience can be summed up with the slogan: "No A differences without B differences." Suppose A and B are alternative accounts of the same phenomena (e.g. cognition). Let P' and P" be distinct phenomenal states, A' and A" - their accounts in theory A (e.g. psychological states), and B' and B" - their accounts in theory B (e.g. neural states). We say that theory A supervenes on theory B if for A' and A" to be distinct, B' and B" must also be distinct, but the reverse is not necessarily true. Thus, the relationship between A and B could also be described as coarse-graining.

The connection with reductionism is that for theory A to reduce to theory B it must, at a minimum, supervene on B. And some would stop at that. But others attempt to go further in elucidating dependent relationships between different levels of explanation. The most ambitious view would probably be one that claims that high-level accounts - "special" sciences and psychology - as well as their specific theoretical entities can be analytically deduced from more fundamental accounts and entities (type-type reduction). A more modest claim is that any particular finding described by a higher-level theory could be traced to some configuration described by a lower-level theory, if only we knew all the relevant facts and possessed the necessary computational resources - but not necessarily according to some fixed bridge law (token-token reduction).

Can I get an example of something that is unitary? — Pneumenon

That would be something like the problem as you framed it in the OP: "A bunch of billiard balls are bouncing around on a table in a classically cliched example of Newtonian whatever."

This formulation explicitly admits of only one account, and so there isn't anything to be reductionist or irreductionist about. As others have noted (and I think @StreetlightX has been spot-on), the situation that you outlined does not really set up a reductionist/irreductionist conflict. Granted, "reductionist" is a nebulous and loaded term, more often used as a derogatory adjective than actually explained.

So what is reductionism? Reductionism implies different accounts, different explanatory schemes, different theories. Often people talk about levels of explanation. So, not just alternative accounts, but accounts organized in a kind of hierarchy, with the one at the bottom being - on the reductionist view - the most fundamental and the most veridical, the others being merely convenient approximations. This is a view that is common among scientists and some philosophers of science*.

Reductionism is sometimes described as taking a thing apart in order to explain the working of the whole in terms of its parts. This is probably what you were trying to get at with your example, except that from the start both your Mr. Reductionist and Mr. Irreductionist are already looking at an atomized picture, with no intimation of there being a whole (irreducible?) thing that these atoms constitute. (And by the way, the expression "irreducible complexity" was coined by an "intelligent design" (creationism) proponent, and denotes a different idea.)

There is some truth to the part/whole account of reductionism - the truth being that the hierarchy of explanations that I mentioned earlier roughly corresponds to a hierarchy of spacial scales. When, as scientists, we attempt to provide a better, more accurate account of something, oftentimes we get out our microscope and examine it at a finer scale. And since matter tends to clump into more-or-less sharp-edged objects at many different scales (particles, atoms, molecules, cells, chairs, planets, etc.), this is where we get the idea of breaking a thing into parts to understand it better. But I think this part/whole view does not entirely capture the idea of reductionism. Quantum mechanics, with its universal wavefunction and entangled particles, is as "wholistic" as anything, and yet it comfortably fits into many a reductionist worldview as perhaps the most fundamental level of reality.

* Philosopher of physics David Wallace opens this lecture about quantum mechanics by confidently proclaiming the sort of view that I outlined above as the consensus view of physics! I think he is rather overstating the case, even if we only ask physicists.

Yeah, the weirdness objection is the worst of the lot, and does not deserve any respect. Quantum mechanics is weird. The world is weird. Get used to it.

All interpretations of quantum mechanics explain exactly the same observations, so in that sense they are explanatory to exactly the same degree. One could make a case that some of them are more parsimonious than others, but that is never an easy case to make. That said, superficially at least, the Everett interpretation does seem to be more parsimonious than its main rivals. But I realize that things aren't so simple, and not having sufficient expertise, withhold further judgment.

It would be great if everyone wanted to preserve causality in their theories but that is what the Copenhagen interpretation explicitly rejects. The idea that the universe is inherently probabilistic implies that the probabilities are a brute fact and inexplicable. — Andrew M

Well, again, you are just equivocating between "causal" and "deterministic." What you are really saying is that our theories ought to be deterministic. I disagree. There is no a priori reason why we should prefer determinism. Or indeterminism, for that matter. I don't consider either to be a theoretical virtue, in and of itself. Of course, if one also offered better or more specific predictions, or a more economical description than the other, then it ought to be preferred - but those other advantages obtain independently of the determinism/indeterminism split. Indeed, in the case of the interpretations of quantum mechanics, none of the empirical advantages can be credited to one interpretation and not another, since they all make the same empirical predictions*.

* That may not actually be true - some interpretations seem to make distinct predictions, but they are presently out of reach for empirical investigation.

So I still don't understand, what was the point of the exercise? To use the existence quantifier in a WFF?

If your point is that this is not an explanation of existence -- the ineffable there-ness of stuff -- I don't think it was intended to be. — Srap Tasmaner

No, you are right, I think I got carried away.

Existence can be modelled syntactically:
P(x) iff ∃x(x=x)
P(x) is true if and only x exists. (x exists iff it equals with itself) — Meta

So x exists iff there exists x (∃x) such that it is equal to itself? The last bit seems unnecessary, but otherwise you have a perfect tautology.

I think Sam was saying that talk of existence is really talk of whether a concept is instantiated. — Srap Tasmaner

The statement is saying that the concept of hobbits has no instances or individuals of which it is true. — Sam26

And what do you mean when you say that the concept is not instantiated, has no instances? (I deliberately emphasized the verb "to be" in these phrases.) Well, it means that there are no such things, that they do not exist. Oh, wait...

There have been a number of attempts to derive/justify the Born rule, including the self-locating uncertainty approach that Carroll and Sebens develop (I haven't looked at their paper, but they probably cite earlier works in the same vein). Not everyone is convinced that such justifications are (a) not circular, and (b) do not smuggle in assumptions that are not present in the starting interpretation. But adjudicating this debate is way beyond my pay grade.

I just want to take issue with your characterization of probabilistic theories as "acausal." What you are talking about is causal determinism, and the keyword here is determinism. You can, of course, put your foot down and insist that causality necessarily implies determinism, but, as far as your arguments here are concerned, causality may as well equal determinism, because you are not actually talking about any aspect of causality other than it being deterministic. So for your purposes, causality is a redundant concept, since all that you are talking about is determinism. And I suspect that you only bring it up for rhetorical purposes (everyone wants to preserve causality in our theories, right?)

I agree with all you say above but would add that the probabilities themselves also have no causal explanation under the Copenhagen interpretation (i.e., the Born rule is postulated). — Andrew M

That's true. But having postulates is no sin in itself: any theory relies on some postulates. The important thing is that the Born rule postulate in the Copenhagen interpretation does not clash with its narrative.

This is when considering a single beam splitter in isolation. When one photon is sent into a beam splitter, there are two position eigenstates - one for the reflection path and one for the transmission path with 0.5 probability for each.

The MZI experiment shows that this cannot be the scenario at the second beam splitter. If only one photon were entering the second beam splitter, then a photon should be found at the second detector half the time. But it's not. This is what I was trying to convey with the "Alice rolling sixes" analogy. It is highly improbable that on multiple runs a single photon entering the second beam splitter would always be found at the first detector purely by chance.

But this is what the Copenhagen interpretation is committed to by denying causality. The results that it predicts are inherently inexplicable on its own premise. — Andrew M

No, this is what we would be committed to if we interpreted light as a flow of classical particles. But the Copenhagen interpretation does not do that. It is committed to the same thing that the fully-quantum theory is committed to, plus a little extra - but that extra does not show up until the measurement occurs at the detectors, at which point the "extra" makes no observable difference.

The Copenhagen interpretation makes the same prediction but it denies that there is a causal explanation for the probabilities. But, if causality is assumed, then the MZI experiment shows that a beam splitter cannot be sending a photon exclusively one way or the other with 0.5 probability (or else a photon would arrive at either detector with 0.5 probability, not 0 and 1). — Andrew M

You do not need to assume causality, or anything else besides the operation of standard quantum mechanics, in order to obtain that result. You said so yourself: the Copenhagen interpretation makes the same prediction. It follows the standard solution all the way up to the moment of detection, at which point it says that the superposition state collapses into one of the eigenstates - acausally, as you say, but following the Born rule for probabilities. And since in this case the superposition is degenerate, the result is perfectly predictable, even assuming the Copenhagen interpretation: the wavefunction has to collapse into one particular position eigenstate with probability 1, simply because there is only one non-zero eigenvalue. So where do you get probability 0.5? And what does this have to do with causality? I don't understand.

No. It is in principle possible that Alice could roll a dice a million times and get a six every time. That result is no less likely than any other string of results for a million rolls. But her non-random-looking result begs for an explanation in a way that random-looking results don't.

So the Copenhagen interpretation correctly predicts that a photon in the standard MZI experiment will always end up at the first detector despite passing through beam splitters. But that raises the question as to why. What is the causal explanation for that non-random-looking result?

For the Copenhagen interpretation, the Schrodinger equation is equivalent to asserting that Alice just always rolls sixes. Each formalism gives the correct predictions and no causal explanation exists.

The problem is with the plausibility of that idea. — Andrew M

It is true that an infinite sequence of sixes is a possible outcome of an infinite sequence of die rolls. But that doesn't change the fact that the probability of each such roll is given as 1/6 by the theory that the die is fair.

In the MZI experiment the standard quantum mechanics calculation gives the probabilities at the detectors as 0 and 1. Any interpretation of quantum mechanics had better yield the same probabilities, otherwise it doesn't even qualify as an interpretation. Are you saying that the Copenhagen interpretation predicts probabilities other than 0 and 1 in this case, or fails to predict anything specific?

(As an aside, this very special case where probabilities neatly collapse into all or nothing is uniquely favorable to the Everett interpretation, which otherwise faces a prima facie problem with specific observed frequencies of outcomes. In contrast to the Copenhagen interpretation, which happily assumes the reality of probabilistic outcomes as a matter of principle, the Born rule is difficult to justify in the context of Many Worlds. When they are not making popular presentations, like the one by David Wallace that you linked, Everettians tie themselves into knots trying to make sense of these probabilities. And this is where, I am afraid, the prima facie appeal of the MWI as the "no-interpretation" interpretation dissipates.)

I don't understand. If an interpretation gives us the correct result (i.e. the result predicted by the formalism and validated by experiments), then where is the problem? Or are you under the impression that a "non-deterministic interpretation" is contractually obligated to give a non-deterministic result for every conceivable measurement?

Perhaps you can work an example in one of these interpretations and show where exactly the problem lies.

There's a nice apocryphal story about Wittgenstein there in the first lecture. For those who don't want to watch the vid, it goes something like this:

Wittgenstein once asked a colleague: "Why were people so surprised to discover that the Earth is spinning and not that the Sun goes around the Earth?" His colleague replied that, well, it kind of looks like the Sun goes around the Earth, doesn't it? To which Wittgenstein shot back: "Well, what would it look like if it looked like the Earth was spinning?"

And the answer, of course, is that it would look like exactly like it does look like, exactly like it looked like back when people thought that the Sun was going around the Earth.

So I want to ask you. You keep saying that the Mach-Zehnder interferometer experiment would be inexplicable under any interpretation other than the Everett interpretation. So what do you think the result of the experiment would look like if the Bohm or the Copenhagen interpretation was true?

Heh, that's a neat counter-argument. I don't think I've come across it before.

Answering for myself (I am not a theist, but I am not sure that I am a "moral realist," because this notion is not very clear to me), I don't seek to ground my moral convictions in anything. I don't think that, as far as moral claims go, there is anything more fundamental than moral convictions.

Some of my moral judgments are more secure than others, and at times I seek to ground some less secure opinions in more secure, more fundamental convictions. But, as I wrote above, this kind of query cannot provide the grounds for morality as a whole.

We might try to explain morality as a natural - or a supernatural - phenomenon, but this can only tell us what is, not what ought to be.

This is an ontological, not an epistemological question about ethics. I am aware atheists can be very moral beings. — Modern Conviviality

When you are asking for the grounds of a position, i.e. "Why do you hold to that position?" you are, by definition, asking an epistemological question. To insist that it is an ontological question is to beg the question. You are smuggling some kind of an answer into your question: the only "grounds" you will accept must be some kind of "thing" or fact in the world, right? I suppose this leads to your next stipulation:

This is a question for non-theists who hold to objectivity in ethics (moral realists) - e.g. it is always true that murdering someone for no reason is morally wrong, etc. — Modern Conviviality

Moral realism is usually understood as the statement that (a) moral claims are statements of facts (more than just facts about our own thoughts and feelings), of the way the world is (outside our heads), and (b) at least some moral claims are true. Is this what you mean by moral realism?

Grounding morality in: evolution (naturalistic fallacy), sentiment (subjectivity), or human reason (ultimately subjective, for whose reason are we speaking of? And human reason, limited as it is, cannot construct moral laws) - seems incoherent. — Modern Conviviality

Why do you think so? "Incoherent" means, strictly speaking, contradictory. What contradictions do you see in these positions?

ETA: Some of the answers posted here (, , ) propose to ground all or most of morality in some particular moral dictum (the Golden Rule, the primacy of personal freedom), but these are not really answers to the question posed in the OP. These are proposals for theories of morality that reduce most moral claims to some fundamental moral principle. But these proposed grounds are themselves moral principles, and so they cannot ground all of morality.

"Judged from a scientific and logical perspective, the belief that we stand outside the causal web in any respect is an absurdity, the height of human egoism and exceptionalism. We should get over the idea that to be real agents we have to be self-created..."

Do you see that?

The "belief that we stand outside the causal web in any respect is an absurdity", and we should cause ourselves to "get over the idea that to be real agents we have to be self-created".

Do you see that?

It is absurd to believe that we are outside of C, but from outside of C we should... — WISDOMfromPO-MO

No, I don't see that, not in what you quoted.

There is always this contradiction in determinism, but nobody--preaching determinism or criticizing determinism--ever seems to be aware of it. — WISDOMfromPO-MO

What determinism? There is nothing about determinism in that quote. But if you want to know how determinism can be thought to be compatible with free will, I suggest you do some reading about compatibilism, instead of starting a new thread every time the question knocks into your head.

Thanks, I mostly agree with you.

Sorites
[...]
I think it's noticeably less controversial if you imagine this representing a population rather than an individual. — Srap Tasmaner

Yes, here the interpretation of the model is clear and the model may be a good fit (to that interpretation). Or is it? If the middle section is where people are genuinely uncertain about their choice, the actual distribution of answers may break down into random noise.

As above, we could graph her uncertainty about her answer instead, and we'd expect a normal distribution, wouldn't we? — Srap Tasmaner

Does the statistics (if there is in fact a consistent statistics) of individual choice represent one's degree of confidence/uncertainty? If we define it behaviorally, as you say later, then it does so, by definition. But then reporting observed behavior as the degree of uncertainty is merely tautological: despite the use of an ostensibly psychological term, this does not shed any light on our inner world. But if the assertion is that the graph represents phenomenal uncertainty (which is, after all, the central thesis of epistemic/Bayesian probability interpretation), flattening out a mess of thoughts, feelings and subconscious processes into one number, then it is much less certain (as it were).

I guess my uneasiness goes back to bridging the gap between probability and a single case. Unlike mathematical probability, real-world probability is always single-case (we don't deal with infinite ensembles!) Defining probability as a frequency is unsound for that reason, while defining it epistemically threatens to oversimplify a complex psychological phenomenon. The moral, I think, is to treat Bayesian models of behavior with caution. When you blow up a detail of a curve and ask about its physical meaning, always keep in mind the possibility that it may not have one: it may just be a modeling artefact.

One thing this curve could represent is an individual striving for consistency under conditions of irreducible uncertainty. — Srap Tasmaner

Yes, that's the Dutch Book argument, and I do find it rather compelling. (And notice how you have switched from behavior to phenomenology, after all!) Don't get me wrong, I like Bayesianism. I like it for its mathematical elegance, consistency, and (when used correctly), instrumental usefulness. When it comes to modeling uncertain beliefs and decisions, it is probably the best game in town.

I'm just interested in how partial belief works, and I keep finding reasons to expect individuals and populations to be homologous. — Srap Tasmaner

Well, one reason for that may just be that the curve, assuming it is the error function, is closely related to the normal distribution, which is ubiquitous whenever you deal with (or assume) random variables.

There may also be an evo-psych story here: the reason individual is homologous with population is because cognition is an evolved feature, and evolution works on populations. The behavioral strategies that statistically increased the population fitness were the ones that were fixed in our genes. This may also serve to explain away the problem of induction: the reason we intuitively trust induction is that our environment does have certain regularities (how could it not? we wouldn't be here if it didn't), and we have adapted to recognize and exploit those regularities.

I didn't know this is called a "logistic function." — Srap Tasmaner

Or the error function (different function, similar shape). There are a number of such functions, collectively known as sigmoid functions.

As for the sorites paradox, I think the whole point of it is that it cannot be resolved like the Ravens paradox, using the subjective degree of confidence model of belief formation, where, in the straightforward Bayesian analysis, belief is represented by a single, continuously evolving real-valued variable. And the reason is simply that there is no sharp and precise fact of the matter to be located here. The "paradox" comes from the tension between the demand, urged by the framing of the problem and prompted by our familiarity with analysis, for a precise numerical solution - and the intuitive realization that no such solution will be satisfactory.

This actually applies to the Ravens paradox and other such paradoxes of belief as well, though perhaps not as acutely as in the case of the sorites paradox. The Bayesian solution is a neat one, it seduces us with its mathematical elegance, but it is not a perfect fit to our intuitions, nor, in all honesty, should we expect it to be.

Our problem here is the fallacy of misplaced concreteness. When given a problem, we habitually reach for familiar conceptual tools - in this case mathematical analysis - without giving sufficient critical attention to the nature of the problem. This is fine when solving practical problems, which often dictate the use of a particular toolset and the production of a particular kind of result. For example, if you were to conduct a survey, presenting different people with different quantities of sand, the distribution of answers as a heap/pile percentage vs. the amount of sand might look something like a sigmoid curve (though it will probably be a different curve in different language communities).

But when the question is no longer about ticking off this or that checkbox on a clipboard and summing up results - when it is something as messy, ambiguous and fluid as a "belief" - why would you expect the answer to be in this definite numeric form? Why would you expect there being the answer in the first place? Do you really think the true nature of a "belief" is a simple mathematical function (preferably one already familiar to us from solving practical problems)? Do you think that "belief" even has a true nature?

I don't like the idea of deleting ("disappearing") posts. Moreover, I think that, faced with the choice of deleting a shitty post and leaving it alone, a reasonable moderator will err on the side of leniency. Which brings us to the present situation, where the board is, frankly, drowning in idiocy. A special not-quite-up-to-standards area would be a less authoritarian solution, IMHO.

Another solution for serial shit-posters would be to merge all their threads into one (I know one board that practices that).

We really need a dump sub-forum, like in the old place. When I come to the forum and see 7 (seven) threads started by TheMadFool just on the front page, that really is depressing. The whole place looks like a dump. If I were a first-time visitor, I would have left immediately without giving the site much consideration.

Or, you know, you could just calculate the answer :)

The sum of multiples of 3 below 1000:

3 + 6 + 9 + ... + 999 = 3 * (1 + 2 + 3 + ... + 333) = 3 * 333 * 334 / 2

Likewise, the sum of multiples of 5 below 1000 is 5 * 199 * 200 / 2

But we also need to subtract the sum of multiples of 15, since we counted them in both sums: - 15 * 66 * 67 / 2

Total: 233,168

Is there really a way to know the number of primes below any integer without having to actually calculate or iterate through those primes? — VagabondSpectre

Prime-counting function

I don't know about more concise; in practice, for problems like this efficiency is a lot more important. There are more efficient methods of calculating the n'th prime, but that is more of a mathematical problem than a programming one.

If you need to find the n'th prime for a large n, you would first calculate an approximate answer (there is a mathematical method for that), find the number of primes below that number (there are methods for that as well), and then zero in on the exact answer, perhaps using a more efficient primality test than your nested loop. That would most certainly be more lines of code than what you wrote though, and not something you could pull off on an interview, unless you really know this stuff.

Well, there are philosophical issues raised by quantum mechanics itself: its interpretations in general, and more specifically, things like quantum measurement, counterfactual definiteness, locality, etc. There is this popular Feynman quip oft-cited by philosophy-averse physicists: scientists need philosophy of science like birds need ornithologists. But in point of fact quantum physicists often struggle with these interpretational questions, and having philosophical chops goes some way towards not falling on your ass when trying to answer them.

Then there are broader implications of quantum mechanics for the philosophy of science and metaphysics. Here is just one example, a stimulating article that should stand in contrast to the pretentious cooks against whom you rail: Holism and Nonseparability in Physics.

OK, that makes sense.

However, if we suppose that our thoughts have been ordained for us, along with all else, we cannot place any faith in them. They are not in fact OUR thoughts at all. — Tony

This does not follow. You may believe that this is the case, but if so, this is in consequence to your understanding of ownership, personhood, freedom - all tricky metaphysical (or perhaps just psychosocial) notions that do not straightforwardly follow from that assumption.

As I mentioned in the OP, I'm asking for what the answer would be IF the abstractionists' position was to be correct. I figure that it has to be chance, which is entailed by contingency, but I was checking to see if I'd failed to consider or understand something. — Brayarb

What do you mean by chance? Is it any different than contingency?

I can prove that ~p and ~p -> p is a contradiction — Pippen

~p -> p is already a contradiction. Srap was right (and I wasn't paying attention): this second iteration of your argument made little sense. Your first attempt was already logically bulletproof, as you say - it just didn't prove anything interesting, and neither did your second attempt.

If p stands for "something exists", ~p stand for "nothing exists" and ~p -> p for "something follows from nothing" — Pippen

While "p" can stand for any proposition, you are not free to choose the meaning of logical predicates - that is, if you are appealing to logic for your argument. If you accepted the rules of the game, then you cannot assign the meaning to ~p -> p by fiat. That expression states that the denial of the proposition "p" logically entails the proposition "p" - only that and nothing else. (Although entailment can be understood somewhat differently even in logic - semantically, syntactically - that ambiguity won't help you.) The implication of disproving ~p -> p is utterly trivial and has nothing to do with proving ex nihilo, as I already argued.

We can use mathematical and logical terminology metaphorically for creative expression (1 is the loneliest number), but the truth of these sayings does not derive from the use of logical and mathematical terms, it has to stand on its own.

It's the same with 's attempt to prove ex nihilo with arithmetic: he interprets 1 apple + 1 apple =/= 3 apples as saying that an extra apple cannot appear outta nothin'. But, if arithmetic is his tool of choice, then all this says is that if you got an apple and another apple, then together you have two apples (and not three). If then, by some miracle, another apple appears outta nothin', then with the two apples that you already had, you will have three apples all told. That's all that arithmetic gives him. Any other interpretation that he assigns to the symbols will have to be justified independently from the literal meaning of the borrowed terminology.

The moral of the story, ironically, is that in an argument, too, you cannot get something from nothing: if you want to prove a metaphysical principle, you will have to do the hard work of arguing metaphysics, instead of looking for sophistic shortcuts. But first and foremost, you will have to understand what that principle means, and neither you nor Lacrampe have taken that trouble.

"¬p→p" has an obvious countermodel when p is false, which happily you assumed in (1). — Srap Tasmaner

That's what he wants to show, no? that (1) and (2) cannot both be true.

Not that this trivial exercise reveals anything interesting, of course. If p stands for "something exists" and then ~p stands for "nothing exists," all that he shows is that, if nothing exists, then it is not also the case that something exists. Duh.

You cannot begin to cogently discuss a proposition when you cannot even explain what it means. All that symbol manipulation is child's play. You are not even touching the subject.

Well, the chapter from which this is quoted is entitled "The Apparent Incompatibility of the Law of Propagation of Light with the Principle of Relativity," and the thought experiment illustrates said inconsistency. What specifically do you find to be problematic?

By formal analysis do you mean specifically the positivist approach of defining all theoretical terms through observable properties? I think it would be problematic to define counterfactuals this way. And that makes me think of a problem with this definition of causation:

If c is not the case, then something else is the case, right? What could be the case then? And couldn't some alternate state of affairs result in the same effect? Suppose we see a red ball strike a yellow ball and then the yellow ball rolls into a pocket. We would normally say that the red ball striking the yellow ball (c) caused the latter to roll into a pocket (e). But what would be a counterfactual to that? Suppose that a blue ball struck the yellow ball instead of the red ball, with the same consequences. (c) is not the case, but (e) occurred anyway. Does this mean that (c) was not the cause of (e) after all? It seems that on this definition of causation either there is no multiple realizability of effects or there is no causation - either option is implausible.

I am not sure what you are trying to do here. Are you saying that "c causes e iff..." is a theory and the same sentence with x and y standing for c and e is its Ramsey sentence?

This is getting old.

Is it possible to change the math axioms such that I+I=III is mathematically possible?
I will go with no. Objections? — Samuel Lacrampe

Not possible if the numbers represent an already given concept of "counting numbers" (or similar). The concept then shapes the pattern, gives the requirements for the mathematical formalism.

I understand "nothing can come from nothing" as: it is false that something can follow from nothing. — Pippen

That is no clearer than the original sentence: you just rephrased it and replaced "come from" with "follow from."

I see causality as a special case of inference, so if I can show that such an inference is wrong then it holds even more for causality. — Pippen

We don't know whether causality has anything to so with ex nihilo any more than inference does, because we don't know what ex nihilo means in the first place.

That said, if you think that causality is a special case of logical inference, then you are already on a wrong path.

How do you prove that nothing can come from nothing? — Pippen

Forget proof for a moment, how do you even understand "nothing can come from nothing?"

I think that your attempted proof only serves to illustrate your own confusion. Start with conceptual analysis.