↪boethius Sorry, I was being cheeky and tried to illustrate what the logical conclusion would be of polarisation. — Benkei

Yes, I understood you were trying to point out some absurd logical terminus.

However, by pointing out centrism is not an ideology in itself, is not advocating for polarization. People could be very close to the compromise that makes up the center, my point is simply it's unlikely the working-out of a coherent ideology will "just so happen" to overlap the center completely.

The old word for "centrism" was "reformer", someone who had beliefs different from the status quo but believes it's only gradual step-by-step changes that will yield the best results.

Again, reformism is not an ideology in itself, a reformer under Swiss social-democracy maybe a violent radical under Nazi Germany in WWII, blowing up rail lines and the like.

So violence is not necessarily avoidable, and it's simply common sense to point out that the erosion of democracy, past a certain point, is no longer reformable and will lead to violence as the only viable option; the "centrists" that decided to move to the center of the new Reich, just "accepting reality", we tend to judge today as cowardly collaborators and party to the crimes and that the violent resisters (largely communists and anarchists, though we of skip over that detail) as heroes who saved allied lives and helped end the war sooner (hence avoid more violence than they themselves committed).

The reason the propagandists in the Democratic party have abandoned "we need to reform slowly" is that if that's not actually the goal, but simply to maintain the status quo and their privileged, then eventually people ask "well, where's the reform, things seem exactly the same, if not worse, whether we vote Democrat or not", and so at some point a bad-faith reformer is called out on it, hence the mainstream media in the US trying to make "centrism" itself as some sort of praiseworthy political ideal taking courage and "a certain j'eu ne say quoi" to stand up for how things happen to be and standing in the way of the sticks and stones hurled at the poor billionaires.

It is of course Bernie Sanders who is the reformer compared to the "moderates" in the current situation, proposing fairly small changes: closing tax loopholes, expanding social programs that already exist, spending less on the military and more on those programs etc. in each case with solid basis in policy successes seen in all the other rich countries, so it's simply difficult to support a "slower reform" program than Bernie, and so the logical alternative is simply abandon the pretense of reform and try to to argue for "doing nothing of importance" as the most important thing that could be done right now.

I think your examples miss the point of the premise. Rather, I am saying that by birthing someone, one is assenting to a set of ideals (one being that at least life is worth living, that the current society is good enough to bring someone into it, that the ways of life of that society are something to instantiate a new person into, etc.). — schopenhauer1

Yes, I misunderstood your question, it seems uncontroversial to me that parents signup to a large part of the ideology of the society they are in when deciding to conceive.

I had originally understood you were questioning whether it's possible for the resulting person that's born can escape the ideology they grow up in; wherein, my basic point is a large part of the "social common ideology" is simply necessary to live, because it happens to be true, and trying to escape it will get a person killed far before they manage to purge themselves of all socially inherited beliefs; and other parts that are arbitrary simply have little reason to forego; such as inventing one's own language and refusing to speak according to society's "rules". But of course, that "lot's of things are true" doesn't establish some things aren't false, and I would argue that everything a society believes in can be escaped from.

But we seem to be largely in agreement vis-a-vis parents choosing to conceive (and risking it is also an ideological choice):

If yes, bringing someone into that way of life (majority, minority, or any at all) is an ideology in itself. — schopenhauer1

I would agree with.

Of course, not all pregnancies are chosen, so I would not say a rape victim that conceives and gives birth does so out of any particular ideology; though I'm not sure this is relevant point for what you want to discuss the philosophical implications of this scenario.

Psychology and economics were never blindly trusted and are widely considered to be mostly conjectural. Medicine has also always been distrusted to an important extent.

[...]No, it is just standard epistemology. — alcontali

Never trusted by whom?

I think we would agree here that "they were never blindly trusted by people properly training in reasoning and the scientific method".

Of course, psychologists and economists may say otherwise, and say key points in their belief system and decision making process as likewise following from an "objective epistemology", whereas we would say it is an ideology motivated by social control: That physicists call these "soft sciences" and their own methods "hard science" to signal amongst themselves that these people down the hall are, largely, raving lunatics; if we had the money and decided to hire a team of physicists and mathematicians to study psychological phenomenon it is unlikely they would suddenly declare "well, we're doing soft science now, so anything goes, just post whatever bullshit you want on the wall and we'll take a vote on it"; rather, they would likely continue to do "hard science" but just not come to any of the radical conclusions about mental states that governments require to keep things a bit quieter; they would likely simply completely discard cherished works that turn out to be just the anecdotal musings of a pervert with a beard and whatever frameworks have since been built up on them.

Which is of course why physicists are not invited into these disciplines to "harden them up"; but of course it goes without saying that, in the case of economics, when a venture actually requires the best predictions available to make money, they hire a bunch of physicists and mathematicians to build those models, and whatever propaganda an economist believes is completely ornamental in such a situation.

And indeed, when a group of mathematicians worked up the courage and took it upon themselves, completely uninvited of course, to review the reasoning rigour in the "soft sciences" they built up a robust statistical model concluding the conclusions rarely follow from the premises.

So, maybe we agree broadly speaking on these sorts of happenings, but my point is that someone who disagrees with us will say "no, no, no, you guys have an agenda and have built an 'anti-soft-science' ideology to justify it ... and not only an agenda! but authority but Oppositional Defiant Disorder (ODD) and you need treatment fast!".

What we view as epistemic deficiency in arriving at critical conclusions in the soft sciences, well, supporters of these frameworks of classifying people as "mentally ill (...nothing wrong with society, it's just them)" will say their framework follows from an objective epistemology.

And indeed, in general, any disagreement between anyone as to what is true, each side will accuse the other of being merely an ideology whereas they're own ideas are derived from solid "standard epistemology".

Hence, in my view at least, insisting one's own ideas are epistemic whereas anyone who disagrees has a made-up ideology is simply to invite sterile debate. Your ideas about epistemology are just that, your ideas, and so form an ideology about what you consider true within epistemology; many people may agree with some key points, but if someone disagrees (for instance claims all is empty and your own self that believes these things doesn't even really exist), well they will say their's is the true epistemology and yours is just the ideology of a deluded ego.

We just look at how beliefs can be objectively justified, and if they can't, then it is ideology and not knowledge. — alcontali

People who were told "opiods, totally safe, science says so" by "scientific medical authorities" and live the terrible consequences are entirely valid in doubting the next important thing scientific institutions tell them to believe.
— boethius

And they are right in that regard. If there is doubt possible, they should doubt; especially when it is obvious that some people stand to handsomely profit from the fact that we believe their lies. — alcontali

Yes, we agree here, this is exactly my point. Most people's interaction with the broader scientific community is with medicine, psychology and economics. So, if they were promised trickle-down benefits that then don't arrive, or that their negative feelings arising from precarious and exhausting economic and working conditions can be "fixed", or that a certain medication is recommended and has "hard evidence it's likely more beneficial than not", it is completely reasonable that, if these promises don't turn out to be true, they then start to doubt every single scientific community.

It's of course easy for people with proper mathematical training to simply ask the doctor for the studies supporting the recommendation and then laugh it away when sample sizes are ludicrously small and, even if the conclusions are true (which is never the case, there's always "well ... we didn't study a whole range of risks .. .and are not technically recomending anything BUT! if the practioner feels this is will help their patient we don't have any evidence it will do more harm than good at this point; tada!" of course, to cover the asses of the scientists doing the study; not so "scientifically incompetent" after all) -- even if the conclusions are true on face value, only establish the thinnest of possible risk-benefits -- and in one case where I've requested the evidence, I had a few hours later the report that no such evidence exists, it's just the "policy" of the hospital to make such a recommendation; lolz.

My point is exactly that: the "bet", by most academics, that calling out bullshit they see in other areas will undermine trust in science, is completely counterproductive and has no reasoning basis but simply exemplifies their own corruption (self-censorship for money) with regards to the scientific process. Wrong science has very real world consequences for people, and living those consequences undermines faith in the scientific community far faster and surely than scientists debating "what is actually true".

As far as I am concerned, we cannot trust the interventionistas, especially, not in the subject of climate change. — alcontali

Now, my bringing up climate change was not to debate it, but just as an example of the reasoning process above academics of hard science use to justify censoring themselves about the tons of bullshit found in the "soft sciences": that if I point out economists use of mathematics is like a child driving a car on a the highway ... maybe people won't swallow the line "scientific consensus, science knows best!".

So, I agree that the arguments presented to people about climate change are terrible: basically assuming people don't have any critical thinking skills and therefore the best they can do is just follow a consensus of experts ... without realizing that whether or not to trust a consensus of experts requires critical thinking.

However, why we differ is a difference in our ideological approach to science. Whereas you see people gaining from the notion of man made climate change, I see it far more plausible much more powerful interests have to gain from denial of man made climate change. Likewise, the science of climate change is a hard science question, following from physical theories of energy conservation and thermodynamics as well as geology, built-up by scientists over decades and centuries who have earned my trust.

Of course, doubt is always possible, but the correct reasoning framework is not "a consensus of experts" but rather a risk-management framework, within the context of all the other environmental problems. Again, what is a "reasonable risk to maintain a preferred social lifestyle" and what is not, is ideological -- and not simply of one's ideological approach to science but one's value system -- there is no epistemic given on reasonable risk nor even that people care about future generations or even other people alive right now, or even their future selves!; different moral systems will yield radically different conclusions based on the same evidence, even if there wasn't a consensus.

As for the example of living off of sunlight, there are plenty of valid epistemological positions where this would be true, and not just "magic" but even with current scientific (had science) theories one could believe we really are in a simulation (many people do) and will power can "bend the rules" (that the movie "the Matrix" was just the AI gas-lighting us for fun, and "deja vu" really is hard scientific evidence proving the Matrix is real and has flaws inconsistent with thermodynamics and that you can verify yourself). Likewise, we can't rule out aliens that can live off sunlight and have come to earth to teach important spiritual lessons; that our "western rationalism" has failed and these spiritual truths "freeing" is proof the guru has the truth, maybe is an alien, and not our "science". So, not only is excluding all these epistemic possibilities the formation of an ideology (and my own, we agree on what we expect to be true about such claims of living off sunlight), coming to the conclusion that "yes, living off sunlight can't be done, anyone who says so is lying or delusional" requires trusting the scientific community "enough" (simply verifying you can't live off sunlight doesn't exclude others can't) and what "trust" level is justifiable can't be objectively verified by definition.

The problem is the definition of the term "ideology", i.e. "beliefs held for reasons which are not purely epistemic". There is a legitimate justification for "eating to stay alive". Hence, the belief is not epistemically flawed. Concerning "living off sunlight", it would not be hard to experimentally test that a group of people exposed to sunlight would not survive longer than at most a few months. Hence, "living of sunlight (only)" is even trivially falsified. Therefore, it is a false belief. — alcontali

This is the exact process I am describing, you are simply using "ideology" as short hand for "my ideology is right".

Living off of sunlight is not trivially falsefiable. If you try it and die, it can simply be claimed you didn't "try hard enough". Your friend observing this may say that's moving the goalposts, but it can be claimed the goal posts were always that far away and you just didn't reach them, due to your arrogance and narrow western scientism dogma.

Now, I agree that living off sunlight is not possible and anyone who claims to have done it is delusional or a charlatan. However, if the history of philosophy teaches us anything, it is that no idea is held for "purely epistemic reasons". My agreement with your point is that, yes I haven't myself observed being able to live off sunlight, but as important is an entire scientific ideological edifice of assumptions not only about how the universe functions but also what and how the community of people claiming to investigate the universe believe and behave. There is no epistemic given that the community of scientists is trustworthy on these key points for instance; it is an ideological choice; and it is only accepting this ideology and the world view that follows from it really does yield more truth than not about the mechanics of physical cause and affect that it seems clear to me that living off sunlight really is impossible.

However, because I see clearly the ideological foundations of the scientific worldview, I can also see clearly it's limitations. That most science is actually a community product with a prerequisite shared ethic that cannot itself be derived from scientific principles and prerequisite institutional resources; and therefore, this shared ethic and institutional structure is not immune from manipulation by people who do not share the ethic in question. The ethic required for community produced scientific knowledge is an ideology and wherever I trust the scientific community I express my ideology of assuming those particularly scientists share enough of the prerequisite ideology.

Now, yes, there are some areas where the faith in the scientific community does not seem required, that the falsification exercise really has been carried out by competing groups or then resulting in technology I can investigate myself and deduce either the theory behind it is true or some structurally symmetrical substitute.

If I ignore these ideological components to my scientific beliefs then I am vulnerably to erroneously dismissing legitimate criticism that a part of the scientific community is acting in bad faith, and, likewise, I am unable to properly account for the reasons to have faith in other parts of the scientific community to people who have come to the common sense conclusion that scientific institutions can be manipulated.

Yes, fundamental physical theories tested by different universities and countries over decades and even centuries we can have very strong faith in. Unfortunately, the disciplines of psychology and economics and large parts of medicine are simply made up for profit; this behaviour can be investigated, and trust diminished where trust is not earned, but it is a complex task. Is it all false? no, but the best propaganda is mostly true and yet yields a radically different conclusion compared to removing the small amount of lies.

Ignoring the complexities that arise from accepting scientific knowledge can be manipulated results in even the good scientific institutions losing trust over time. At each moment, every secondary teacher and most professors, even if they see such manipulation, don't call it out with the justification that it will undermine very important things elsewhere. For instance, calling out the pharmaceutical industry's manipulation of the medical scientific community seems to invite casting doubt on man-made climate change and the urgency to act with respect to it. For, if the opium crisis resulted from a corrupt manipulation of the scientific medical community, why can't we assume the climate science community is likewise manipulated? Most academics basically use this sort of reasoning for turning a blind eye to corruption in other disciplines, or even their own discipline.

However, the problem is that corrupting the scientific process has real world consequences for people. People who were told "opiods, totally safe, science says so" by "scientific medical authorities" and live the terrible consequences are entirely valid in doubting the next important thing scientific institutions tell them to believe.

These issues are ideological. If distrust is too high, then essentially nothing important can be scientifically determined in the community product sense (which is nearly all scientific knowledge we can access), it's reasonable to just doubt everything. It is only from a ideological position with sufficient trust of people in general that one can plausibly differentiate the bad science from the good.

Yes, dismissing "living off light" seems simply true and not ideological, but the ideological framework required to make such a conclusion (law of non-contradiction, consistency of physical phenomena, trust in the scientific community's accepted fundamental theories that clearly demonstrate no way to live off sunlight), if ignored as a series of choices, invites intellectual disaster when things are not as clear. In otherwords, dismissing alternatives as ideological whereas one's own position is just "clear epistemic givens" is simply to assume one's ideology is correct and the other's are incorrect without any proper examination.

Being tributary to biological realities does not make that person subscribe to an ideology. Better examples of ideologies are communism, fascism or democracy. — alcontali

Some people really believe it's possible to subsist without eating and that various gurus have accomplished it. "Eating to live" is very much an ideology; it's more accurate to say we just happen to think it's actually true and dismiss the alternatives. If "living off sunlight" is an ideology, then so too is the alternative of "living off material food".

Likewise, the ideology of no ideology is of course itself an ideology. Trying to dismiss world views as ideological and "thus wrong" or "not justifiable really", is simply to claim one's ideology is right without introspection or defense compared to the alternatives.

All of this is interesting, but a bit off the mark as to what I mean by ideology. What you are discussing is INTRA-ideological debates (self-employed vs. employee, bit coin vs. other currency, etc.). My point is that generally speaking, LIVING itself requires a way of life (survival-through-economic-means for example), and that by birthing more people, you agree to force more people into this ideology. There is no way out of this ideology (of living generally to survive in some sort of economic system), once born, not even suicide. — schopenhauer1

Your question seems a conundrum if you interpret ideology as "simply ideas that people choose to believe" but ultimately has no justification really in the end. If some ideas in some ideologies, however, are simply true, and at least some truth is needed to live both individually and perpetuate a community, then it's no strange thing that society has a fairly broad consensus of things that just so happen to be true (eating poisonous mushrooms kills you for instance), and people who go outside this ideology (or rather just true things society has learned) die very quickly.

It is also not strange that entirely categories of tools that have no deducible "true or bestest form" society forms conventions about that get passes from one generation to the next. The typical example is language; we can reason that having words and grammar to express things is useful to have, but we cannot deduce the "best words" and so society picks new words or changes old ones when the want arises in no particularly coherent way. Likewise, some rules of conduct maybe deducible from reasoning or trial and error in a specific form, such as not eating the poison mushrooms or tolerating wanton murdering in the community, while other rules of conduct have no particular justification but are a useful reference point.

This broad ideology anthropologists generally call culture.

You can always just shoot the other side and be done with. Might be a good solution for a lot of things really. Every 50 years we divide in two camps based on ideology and one of them gets to shoot the other based on a flip of the coin. We can have a debt jubilee afterwards. Good times will be had by (half of) all! — Benkei

The first thing I said was compromise is apart of democracy. I'm fully willing to compromise if what is lost in strife is greater than what is gained by an uncompromising attitude.

My point is that whatever the resulting compromise is at any given moment, is unlikely to result in a new coherent ideology. Centrism is by definition simply the status quo, some static definition of centrism that didn't follow closely the status quo would become tomorrows extreme regressivism. The centrists of yesteryear are the strange deluded monarchists of today (unless you live in the UK of course).

One can be more-or-less content with the status quo (and willing to defend it against worse things) but, again, that does not result in the status quo representing a coherent world view; simply that it's better than the alternatives in question.

Likewise, simply because the status quo as a whole is unlikely to be coherent, does not entail every part is unjustifiable. Many parts will be justifiable, but such justification is only feasible to construct in relation to (at the least an honest attempt) to make a coherent world view in which that part happens to be justified; in making such a world view, every other part of the status quo one is very unlikely to be able to justify. Hence, to adopt the status quo as an ideology in itself is simply lazy thinking, just plastering the wall with the hodgepodge of what passes for laudable opinion of the day.

Militant centrists (those willing to defend the status quo through democratically bad faith actions of propaganda, changing laws to entrench the status quo, and fixing elections) do not actually have the status quo as an ideology. Attributing good faith to militant centrists is an analytical mistake. Their ideology is their own privilege and defending it against the risks change brings; and from this ideology, when change seems inevitable, the coherent decision is to ally with fascists, as, yes they may do all sorts of terrible things hardly acceptable in polite society to placate an enraged reactionary brownshirt movement, but they at least do not change the class structure of society; such an alliance is not only the logical decision to make but what we see historically. The "true believers in centrism" are simply lazy thinkers that simply live in denial about the inability to justify all parts of the status quo simultaneously in one coherent framework, and so rather than investigate the foundations of their own beliefs and making real decisions about them, of what they can actually defend in the status quo and what they can't, they are comforted by the conjuring up of worse people by the propagandists; when fascists takeover, such people do not even bother to rise to defend the status quo from a worse development: they're much too busy for such tiresome arguments.

And the center is not evil. Meeting in the middle is how democracy works. It's normal to get frustrated that things aren't the way they should be, but we're better off facing our problems together than becoming polarized and thus unable to deal with anything. — frank

I can't stand this this sort of enlightened centrism claptrap, but I'm happy you stand with me against it. Agreed, together we are much stronger in bringing down centrism as a viable intellectual position.

Although I agree democracy involves a lot of compromise, the result of that compromise at any given moment is unlikely to create a new coherent ideology.

Centrism, the ideology of the "what (I claim to be) the right compromise between (what I claim to be) good faith rational actors on the debate stage of politics" is simply a euphemism of militating for the status quo -- or then just lazy thinking that such status quo militants take advantage of.

The ideology of militating for the status quo (such as the now famous Iowa "coin flipper") is not a good faith ideology that the policy compromise between different world views actually forms a new coherent world view, rather it is the ideology of maintaining privilege and advantage of the the people that happen to benefit from the current status quo. If one is on-top, and only concerned with maintaining a privileged position and not with any notions of the public good, then any change is seen as a risk and it's nearly always best to keep things as they are; a change is generally sought to neutralize what is seen as a worse change that would otherwise happen.

Furthermore, if the compromise being referenced simply excludes any world view outside the privileged class, it's not even a compromise to begin with; it's simply a cowardly way of saying the privileged should rule the masses, because the privileged seem to agree on at least that.

When the whole project is polished and done, then I'm happy to debate its merits as a whole. — Pfhorrest

Sure, I don't mind not debating the substance, as is your desire. When your desire changes, I'll look into it.

You ask where the ideas have been had before, I am simply providing you the answer that the idea that we can still function, that it's still worthwhile to keep an eye out for the truth, even if we don't have it, goes back, at least, to the Hellenistic philosophers, is the foundation of skepticism, and is also a Socratic theme (though debatable Socrates is a skeptic as it later developed, you are not really taking a skeptic position, only presenting part of a skeptical argument, elsewhere it very much seems you are claiming the arguments you present are true and you believe them to be so; so, more akin to a Socrates that claims to know much more).

As for the request of the OP,

Reminder: I'm looking for feedback both from people who are complete novices to philosophy, and from people very well-versed in philosophy. I'm not so much looking to debate the ideas themselves right now — Pfhorrest

This seems incompatible with the content of your essay which concludes there is "a much better chance of getting closer to finding them, if anything like that should turn out to be possible, if we try to find them".

Why advertise you aren't trying to find the answers you're looking for?

I submit to the forum that a new category should be made for "is this new or not new idea?" feedback section. If we aren't debating the content, we are clearly not doing general philosophy.

As you may certainly expect, the atomic pieces of your argument are not new. Of course, the molecule your argument builds up is new, but only because it doesn't make any sense. You are "open to find answers, whatever they may be and even if they can't be fond" ... as long as they are "not some transcendent kind of reality or morality".

Be that as it may, your position is basically that of the early skeptics of antiquity in a straight forward logical way and also in a transcendental way in Buhddism.

The Greek skeptical school held that we "cannot access truth directly", however, they got around nihilism by arguing that they adopt what appears to them to be true, while leaving open that it may turn out to be false (the Hellenistic skeptical philosophers developed a distinction between "assenting" to an idea and "believing" an idea; assenting meant basically to assume it as it's the best option available), and of course also leaving open the possibility of "really" encountering the truth also. The skeptic acts as best they can with what information they have, while "suspending judgement" of whether it is really true or not.

The other Hellenistic schools criticized this position as basically being too easy, that is sounds like intellectual courage, and certainly a good attitude in many situations, but it is not actually taken to heart by the skeptic, that the skeptic in fact does believe things without suspending judgement (such as that their attitude about suspending judgement is the best available), which then lead to a rebuttal that, no, they too suspend judgement about suspending judgement, and so and so forth. Point being, the focus is on the searching as the basic moral activity, as one cannot know things cannot be found nor being unable to rule out what appears to be true is not the best available premise.

Skepticism in Buddhism (as well as other schools in India) went much farther, in a way that the other Hellenistic schools might really agree is at least taking it to heart: one by one, taking everything that "seems obvious" and becoming seriously skeptical about it, no matter how absurd it seems to leave reasoning afterward: ultimately in the suspending judgement of one's own identity, and dealing, fairly honestly, with followup questions of "who's thinking such argument if there's no one thinking". I believe the Buddha himself had the analogy that, indeed, the arguments aren't "true" as such a notion of truth is an illusion, but it is like a bridge that can get you to the other side and then, once used, is no longer required and can be discarded; i.e. believing skeptical arguments are true is a path to get to a place where one sees all such kinds of arguments as mere human opinion and untrue. Various Buddhists and also other Indian schools went basically as far as you can possibly go in such a debate.

I also consider justice part of "the good." Justice, in its truest sense, isn't about making people happy or ensuring that they thrive. Justice can actually hurt society sometimes. — BitconnectCarlos

Though I think 's point is a useful exercise, I'd like to criticize the premise that "justice can hurt society sometimes".

If you mean "hurt" in a some trivial way of what people, even most people, may feel, then we are in agreement. For instance, assuming we agree slavery was unjust it certainly felt "hurtful" to the slave owning class when slaves escaped, much worse, rebelled. So if we consider only the slave owning society (which in some cases can be the majority of people even including the slaves) then such a situation could agree with your framework that a "just act" of slave rebellion is "hurtful to society".

However, where I disagree is that this situation can be fundamental. For, we ultimately derive the injustice of slavery from the welfare of society, usually by considering the welfare of the slaves as part of the community then slavery becomes essentially by definition unjust since if society could function without the slaves being slaves, then clearly this is a better society than one where some members are slaves. The only way around such a conclusion is to argue one of the following "the slaves are not human and not part of the community", "slavery is good for the slaves", "slavery is necessary and there is no option to run society equally well without slaves", "it would be more unjust to part slave owners of their property than the justice served by freeing slaves"; not only are these the logical alternatives that we can derive today, but they are the historical key arguments proposed to justify slavery and only after each was proven untrue did opinion turn against slavery.

However, the point of this example is that the injustice of slavery is derived from a concept of the welfare of the community; that it is in fact more hurtful to society to have the institution of slavery than the temporary hurt of the slave owner class to have their world view challenged and be parted with their property.

Although it is a dirty word today (due to, what I would argue, unjust and criminal campaigns of propaganda) democratic social organization is only feasible by referencing the public good. The only way to decide situations where private interests, or private opinions of justice, conflict is to reference a concept of the public good. Such a reference can be wrong, and so what was thought just at one moment to some, even the majority, of people (such as capturing runaway slaves and returning them to their owner for arbitrary punishment) is viewed as unjust by a later generation that has a new opinion of the public good. However, there is no logical alternative to a concept of justice other than the public good, as there's no way to rationally convince society (hence propaganda) to implement a policy that is bad for society; there's no logical way to construct an argument where society should do what is bad for it.

Now, this does not prevent people from personally not wanting the public good but only their own definition of good for themselves at the expense of others, but even if that be the majority they could not come to any coherent agreement on any policy; maybe some policies emerge from bartering, corruption (as the majority of people don't act to prevent corruption, as it's not their problem), propaganda and influence; but such an exercise has no stable coherent outcome; the powerful players that come out on-top at any given moment maintain alliance insofar as they perceive it benefits them, and break that allegiance and overthrow the previous policies (and their previous friends) the moment they see even greater gain in a new order of things. The phenomenon of "socialize the cost, privatize the profit" is just such a dynamic at play; a lucid entirely "self interest maximizing" thinker maybe against such a policy as it's a waste of their tax dollars and makes the economy less efficient ... until the moment they are able to benefit from such a scheme, and so who gets to socialize their costs while privatizing the profits the most is in constant flux, their is no convergence of policy even if everyone in society is attempting to do the same thing in this case (and, if everyone is not acting in this way, then in constant conflict with whoever's left who disagrees with "only looking out for number" and create a coalition for their concept of the public good, which may include private property but is, essentially by definition, incompatible with underwriting an organizations costs and risks while permitting all the profits to be kept private).

Likewise, even a theological definition of justice (and I am a theist for context) does not have a definition of justice incompatible with the public good, it's just with extra steps: even a theocracy will argue not only is following the will of God good for the individual and the community but that God is good and so only wants good things to begin with.

In short, any example you have of doing justice "hurting society" I contend is only illusory, essentially resting on a prerequisite claim that society does not know what is good for it in this case (and time will bare this out; such as the US South with slavery or the Nazi's with trying to take over the world and genocide along the way; it was certainly "hurtful" for these groups to lose their respective wars, but any argument that it was just to defeat those groups, internally or externally, would be based on a public, not private or some other, concept of welfare).

If the rare earth hypothesis is correct, it means that intelligent life like us is extremely rare. If that's true, we inhabit a very special place in this universe. Since out current sample size of "intelligent life like us" is 1, we have no reason to assume we're in such a special place. The mediocrity principle implies that we should regard our habitable situation as "average". The rare earth hypothesis violates that. It claims our habitable conditions are/were exceptionally NOT average. Is there a good justification for this? — RogueAI

There is no violation of the principle.

We are exceptional in our local neighborhood (compared to Mars, Venus ... and perhaps the entire galaxy), but this represents an "average density" in the wider universe.

The rare earth hypothesis is simply postulating the density of life, more specifically technological life, through space and time is so low that it's unlikely any two civilizations ever meet. Rare earth hypothesis generally goes hand-in-hand with supposing faster than light travel is either impossible or then a rarity on-top of rare-earth.

Since the universe is very big (and faster than light travel may indeed be impossible) then if the density of life and technological civilizations is low enough we will simply never encounter them.

In terms of the basic math, an analogy is sampling a single atom from the atmosphere, let's say we sample krypton, if we then keep sampling and don't find any krypton for a while we can suppose it's a rare element. Now, of course it's very unlikely the first single atom we sample from the atmosphere is krypton, far more likely it will be something abundant like nitrogen. So, sampling krypton as a first atom in the atmosphere is indeed a violation of the mediocrity principle and extremely unlikely.

The difference with earth is that we can only sample a planet that is not only livable but has developed life and intelligence and so there's no statistical option to sample an unlivable planet for where we live; that we live on a livable planet does not provide us statistical information on the density of livable planets, only epistemological information that livable planets are possible and at least one exists.

The statistics come in when we start considering other planets and don't find much evidence of life much less technological civilizations elsewhere. If the density of life and technological civilizations is low enough then this solves the problem: if we suppose we live in an average case of a technological civilization density of 1 part per billions of spherical light-years, then we shouldn't expect to encounter other civilizations. Since the known universe is 70 billion light years of space and may go on for many hundreds, thousands, billions of billions, or infinite of light years more, then even with a low probability density of life and technological civilizations emerging there is zero statistical problem that they do emerge, and we happen to be one of them.

The exceptional hypothesis in contradiction with the mediocrity principle is the supposition that we are the single technological civilization in the entire universe; but there is no violation with the supposition that the density is so low we don't encounter others.

The problem with a high density of technological civilizations is that once encounters with other civilizations is probable it's basically impossible to avoid fine-tuning the density to some sweet spot where we see no evidence. If other civilizations are likely in the galaxy then it's difficult to craft a scenario where technology isn't simply all over the place by now and unavoidable (galactic radio beacons playing broadcasting some civilizations Mozart for hundreds of millions of years already, not to mention galactic wars, large scale experiments, signature of powerful ships going all over the place etc.); just basic extrapolations of our activity so far on a galactic scale, and so we are likely forced to conclude in such a scenario that we are an exception and not a mediocre example of how a civilization behaves (that all other civilizations keep quiet, don't get out much, or then all agree to keep us in quarantine, etc.).

er, where? No, the fact procreative acts are ones that those who are created by then have not consented to is a fact that makes them 'prima facie' wrong - 'default' wrong. That is, they will be wrong unless there is some other feature they possess that either annuls or overcomes the wrong-making power of the feature I have identified. — Bartricks

Yes, I understand your argument structure, what I am pointing to is a lack of principles to decide what "other feature they possess" are and on what basis do these features override your "prima facie / default" wrong.

As far as I can see, you have not presented a coherent and sound argument. You have responded to weaknesses in your first principle by postulating that other principles could exist to solve the problem of the initial criticism. I am asking what these other principles actually are.

If I say "it's fine to murder people unless there is some reason not too" this is not meaningful theory. If there are good reasons that cover all potential murdering motivations and scenarios, this would not be incompatible with the previous statement. Likewise, "a surgeon should not do surgery without consent ... unless there's some reason to do the surgery" is not a meaningful theory for the same reason.

You seem to have difficulties with subtleties like this. If I say that in some contexts the fact an act will prevent some great harm eclipses the importance of getting a person's consent, then you take that to mean that if an act will prevent some harm then consent doesn't matter, or will always be eclipsed by the significance of the harm it prevents (that is, that preventing harm is lexically more important than respecting consent). I do not believe such things. — Bartricks

Well, you obviously believe it in the case of having babies that all these principles play out to conclude babies shouldn't be had. As noted above, what you haven't done is present on what basis you do this evaluation.

You seem to think that I am attributing things to your position and so are justified in getting defensive and explaining what that you don't believe this or that. I am not attributing things to your position, I am presenting situation and attempting to apply the principles you have presented.

If you're reply was "yes, the surgeon should let the unconscious patient die because they have returned to the pre-conscious state of a yet-to-be-born child and it is better that they die than potentially live without consent" and likewise if you replied "yes, the government is not justifiable because it does all sorts of things that significantly affects people who don't consent to the social contract, not to mention yet-to-be-born future children, without their consent, among other things promoting and enabling this whole giving birth enterprise humanity has" I would accept that you are applying your consent principle consistently and I'd move on to the next issue.

However, you invoke new principles to justify surgery without consent and government without consent. I an inquiring as to what these principles are and why I am unable to apply them to justifying having birth. If there's greater benefit to people who know the unconscious patient that the surgeon save the patient and this outweighs the harm in ignoring consent, or then we can simply presume consent without having consent, why can we not simply apply such a principle, in either case, to having children (that the good a child brings to the community outweighs the harm of ignoring consent). Likewise, if the government is justified in ignoring consent for the good of the community, doing things like promoting and subsidizing births to have a workforce to not break pension schemes and the like, then why isn't the would be parent justified in conceiving and having a child for the good of the community? Moreover, is the government invokes "the good of the community" to justify it's child subsidizing schemes (from maternity leave to free schooling) why is it wrong to do so in the case of child births but completely fine in things like police and roads?

You have answered none of these questions, you have simply made reference to the potential to answer them. Maybe you can, maybe you can't. It's only in actually answering them that they will be actually answered. You seem to miss subtleties like this.

It will be inconvenient for your objections, but my view is that lack of consent is a prima-facie wrong-making feature. That does not mean it is always a wrong-making feature. Sometimes it doesn't matter. And sometimes it matters but other things matter more. Note the 'sometimes'. — Bartricks

Yes, I have noted a few times that your principle for ignoring consent seems to be summed up in "sometimes". This is not an argument that justifies ignoring consent, it just describes what you are in fact doing which is "sometimes ignoring and sometimes not".

The point, however, is that it 'default' matters and so if an act is an act of such a kind, then it is reasonable to suppose it is wrong until or unless we have evidence that some other feature also present is either annulling the prima facie wrongness of the lack of consent involved, or eclipsing it. — Bartricks

Yes, as I have noted other situations where you claim there's this "eclipsing feature", but you have not explained why it is an eclipsing feature and how it is employed to do the eclipsing.

In cities for example with many millions of inhabitants, the resources individuals require are transported en mass into that city continually. It is like a finely tuned watch, all it needs is a spanner thrown in the works for it to descend into chaos. — Punshhh

Yes, we are very much in agreement. Whereas most of the environmental movement has embraced an even more urban vision of the future, I am working to de-urbanize and create technology that is simpler, more resilient, detached from dependence on large integrated infrastructure, and so allowing living close to food (and so, even if transport and trade increases efficiency, a breakdown of these systems don't result in short term starvation based on 2 weeks of stored foods).

Some people say that these things won't be a problem because large numbers of people will die due to famine or disease. These will bring further problems of disease and unrest, destabilising adjascent populations causing famine and disease and conflict to spread in unknown ways. — Punshhh

Yes, I also don't get this argument. The idea that "to solve the ecological crisis by letting the ecological crisis unfold so as to kill all the people that caused the ecological crisis" doesn't pass even a cursory critical scrutiny.

Also, of the Impact = People x Technology x Affluence, it is clearly the technology and affluence terms that have large potential variation through government policy, entrepreneur initiative and political and habit change movements of people. It is only people who want to hold on to their affluence without even being willing to consider how a lower impact life can be just as "affluent" (just different pros and cons if social chaos is avoided), that think simply lowering the People variable is the "obvious solution"; although obviously a solution to the equation to lower I; people tending not wanting to be killed, there is no policy mechanism to carry out this depopulation plan other than ecological collapse ... which obviously doesn't solve ecological collapse.

So yes, we are pretty much in agreement on the various aspects of this issue.

I think your thesis "stick to finitism when teaching basic math" misses the obvious point of how incredibly messy and complex finitism is, both as a mathematical approach and as a practical application. The overwhelming majority of mathematical applications are based on the continuum - physics, engineering, etc. — SophistiCat

I'm not sure what this analysis. I have never heard a criticism of finitism that it is complex for applied applications. The whole point of finitism is that it aligns with practical application.

Engineering is not based on the real number continuum, it is largely based on differential equations that can be setup just as easily in a finitist framework.

Since I am concerned with high-school in this thread, can you give an example of an applied high-school level problem that cannot be addressed in a finitist framework of "arbitrary precision"?

As far as I know, all applied maths problem have precision constraints of their input data which results in precision limits of the output data of the algorithm (whether machine or human) solving the problem.

Even if there was a theorem that has no proof in the finite regime but does have a proof in the infinite regime (that we cannot prove the theorem for arbitrary precision, but we can prove the theorem for "all integers" or "all real numbers"), it is easy to borrow that theorem in a rigorous way by simply having the computer check the theorem up to some limit that we intend to use.

(Indeed, lot's of "theorems", i.e. conjectures, are used in applied math that have no pure math proof because they have been checked numerically to over the range in question; obviously, such a numerical check to some bound has nothing to do with the real number line.)

And this is how, in practice, applied maths work; people look up a theorem and use it, and if it works to solve the problem then that's the end of the thinking on that.

Of course, understanding what math actually is, is to understand proofs. But my whole point here is that simply positing the real numbers without a construction nor framework of rules that contains the paradoxes that otherwise occur with a naive approach, leads to a mystification of maths and not understanding of rigor. If the setup isn't rigorous, it is not a proof students are learning, but rather the applied method of "we're doing it because it works".

The way I reason about it (ie, as a software engineer), real numbers specify the convergence characteristic of approximation processes that deal with real world problems. What you are saying is that people should study the numerical methods that approximate real world solutions, but shouldn't study analysis of this essential characteristic, which seems to me questionable. Maybe your point relates to the general debate in society - whether engineers should study only constructions and hands-on skills and not analysis (how to derive properties of those constructions), but even then I am leaning towards the usefulness of theoretical understanding. — simeonz

My whole point is that students are not actually understanding the mathematical analysis if they do not actually understand real numbers. For me, simply positing the definition of real numbers in terms of some basic rules, does not lend any understanding of what real numbers are.

To repeat what I answered above to SophistiCat, the real numbers are not required to define convergence. Without the real numbers, convergence is to an arbitrary precision rather than a real number on the real number line. Arbitrary precision is perfectly adequate for any real world problem.

Because of this, the radical finitists desire to get rid of the real number system all together even at the university level. The ultra finitist dispute even "arbitrary precision" which in principle goes up to numbers with complexity that cannot be represented in the real world.

I am neither a finitist, much less ultra-finitist, for higher level pure maths. The real number system is, at the least, an interesting mathematical idea that lot's of effort has gone into developing and lot's of theorems are proven in the framework of ZFC that we have no reason to just throw away, and for me, pure maths is about the general question of "systems of symbols and rules" whatever they maybe, and so working with systems of rules that have unintuitive consequences is a good thing for the student of mathematics, as it opens the minds as to what can be done with this general "rigorous proof" based on rules and symbols.

However, for students encountering calculus for the first time, understanding the real number system is essentially impossible and a waste of time to attempt in anycase. In my experience (and it seems the experience of many posters here), teachers at the high-school level don't understand the real number line anyways and simply change from finitist "arbitrary precision" explanations while dropping in tidbits of the bizarre characteristics of normally distributed infinite decimal representations at best, and at worst provide wrong answers. I would wager most teachers and most students understand the real number line as just "numbers with decimals", such as the calculator provides, which is not the case; the calculator provides integers and fractions in decimal representation.

However, if we want to get into the discussion of the limits of this "numerical regime" approach in applied maths, it seems to be everywhere in practice.

For instance, I have always understood re-normalization in quantum physics to be exactly this "get rid of the infinities through the numerical regime" by simply measuring things and replacing divergent functions with constants. The justification of calculations outside the bounds of the reference experiments is basically a numerical regime game of "how far can the error be from this measured constant over here".

Likewise, the invention of "quanta" was due to abandoning the real number line and simply having a finite step (a numerical approach) which got rid of ultraviolet divergence and reproduced experimental result.

If physicists were not committed to a real continuum at this time, this would have simply been the obvious approach to define some "accuracy step" and then narrow in on the right value that matches experiment. And this is basically the argument of finitism in physics: there is no real continuum and so using the real numbers causes the confusions of the above kind, slowing theoretical advancements. I'm not sure if this causes confusions or not for physicists, but I have never heard a counter criticism of some example of a prediction that cannot be made without the real number line. As I mention, I'm not a finitist in higher maths, but if there's a criticism that some physics can't be done with finitist framework I have yet to hear it. The counter argument, is that keeping the real numbers around makes everything easier for both historical and abstraction reasons; just like in principle we could do physics without imaginary numbers, but no one advocates that because it would be clearly more confusing and harder to do physics without them.

As far as I know, physicists at the highest level do not need the real number line for making any prediction nor any theory, and it's simply historical accident that classical theories where developed with the motivation of a "no gaps" continuum, and it's important for physicists to learn these classical theories.

Ultimately, physicists do not justify the theorems they use with pure-maths proofs, but rather they borrow from pure maths "whatever works" and justify that in relation to experiment. Hence the "shutup and calculate" motto of modern physics (sometimes a pure maths proof extends the theory in a way that makes both intuitive sense and matches experiment ... and sometimes not, in which case no bother we'll just ignore that or say the theory breaks down at those energies). Which is why, as far as physicists I've talked to, this issue about real numbers they simply don't care about; it won't change how they calculate to get answers (unlike imaginary numbers, which would change a whole bunch of things and they would "care" about an argument to get rid of imaginary numbers, as it's clearly insane to do so).

However, regardless of whether a physicist thinks real numbers are a help or not, high school students, the subject matter of this thread, are neither learning rigorously about real numbers nor the numerical regime and, if what they are doing is not rigorous then it is not really understanding what mathematics is.

There's nothing wrong with using thousands of words to make a case. But there is something wrong - or at least unwise - in making one's OP thousands of words long. — Bartricks

I think you're overestimating how big "thousands of words are"; it seems fairly common on the forum to see posts thousands of words long, unless by thousands you mean closer to a hundred thousand than a thousand.

Be that as it may, I was not recommending you write the entirety of this "cumulative argument", but if you wanted to discuss it, to write it's outline and reference or sketch these sub-arguments that would compose it and what conclusions you believe those sub arguments have. In other words, to proceed with "if all these other arguments conclude favourably for me, then by this cumulative argument I conclude with antinatalism".

For instance, some posters may believe that the structure of such as argument is unsound, and so regardless of the premises being true or not (perhaps all the sub-arguments really do conclude favourably for you) the conclusion does not follow from it's premises.

Likewise, some may discuss one weak link in the chain or then argue other links would not be necessary for you scheme to work (leading you either to perhaps focus on the most critical aspects or then to alleviate the work of defending other sub arguments if you become convinced they aren't necessary).

For instance, now you are saying that this consent argument in itself justifies antinatalism, so, through this discussion of your cumulative project, I have already relieved you of the burden of needing to write a whole book to defend your position.

I like the way you don't actually address the point I was making. — Bartricks

I did answer the point, insofar as what's relevant to your OP.

We agreed this consent issue is insufficient to justify antinatalism.

Now that you've changed your position I'll restate again my criticism.

It's going to be the same as before, and I'm not sure when I'll have time to re-make the same points. Here, I'd just like to quickly note my method of debate.

When I encounter a position, I am not at first concerned with proving if the position is true or false, but rather I am concerned at first of exploring if the arguments proposed to defend the position are fairly applied elsewhere. For, if the principles put forth are not coherently applied to all of life, then it is a pick and choose philosophy (simply starting with various conclusions and picking and choosing what arguments seem best to justify them given the situation); if this is the case, it is useful to debate the issue further but rather to switch to whether this pick-and-choose approach to philosophy is justifiable (if the person simply defends the method) or then to debate whether it proposed in good faith and not some sort of propaganda to serve ulterior and unsaid motives.

I am so far not satisfied that you fairly apply your principle of consent to all ethical issues where it would seem relevant to do so. So far you simply state "sometimes you do, sometimes you don't", which for me is simply describing how you apply your principle, not introducing a new principle upon which you decide when consent matters and when it doesn't. You say, "well, consent doesn't matter if it prevents harm", but you have not dealt with the argument that letting the unconscious patient die is what prevents harm (as life is suffering and it is better not to exist) nor with the argument that birthing new children prevents harm by enabling society to run and the elderly not to all die in horrific conditions as society disintegrates. Likewise, you have not, in my view, responded to governments likewise not caring about the consent of future children; again, you position seems to be simply "of course we need government". As I said, antinatalism is a radical position and I am not satisfied it would not, if directly and honestly applied, lead to other radical conclusions. You can propose a scheme to avoid these radical conclusions or then you could embrace these radical conclusions: that yes, governments aren't justifiable, yes surgeons should let unconscious patients die if they don't have explicit consent from them before hand. You can also say "well, I find these criticism satisfactorily dealt with and I'm moving forward to the next step" in which case perhaps I'd continue on the basis of "assuming so, which I disagree is so, but assuming it, my critique of the next steps is such and such".

Yes, I have not yet presented an argument that your conclusion is false. I am at this stage trying to discover your premises, which do not seem adequate to me to lead to your conclusion. If such premises emerge, I may have issue with the soundness of the argument, issue with the premises, or be completely satisfied and accept your conclusion. At the moment, it is not at all clear to me how you intend you argument to work, other than allowing you to simply jump to your conclusion.

I wrote "The digits in a real number should not be countable". Well, the digits of (the decimal representation of) a real number are countable, since they are determined by a function of type "natural-number ==> digit". — Mephist

Yes, I think this is the normal situation and what I was expecting. But now I believe the task is even more difficult as one now needs to explain to high school level math students that both the digits in integers "can be counted" and the digits expansion of real numbers can be "can be counted" (assuming they stick with you on what counting infinities mean), but you cannot count on making an infinite integer to make rationals.

The purpose of my series of questions is not to build ZFC or some analogue, but to demonstrate that without the context of ZFC there are no "simple answers" to questions about the real numbers. That there is no simple story to tell nor easy proofs to put on the board in the context of what high school students level.

I think this thread establishes pretty well that the average high school class room doing calculus for the first time would not be able to follow almost any of this conversation.

Well, I wouldn't start from the "pathological" cases to show that volume additivity doesn't work any more. — Mephist

My point is that these unexpected and non-intuitive theorems exist when going from the finite to the infinite regime.

Understanding infinite regimes means understanding these non-intuitive, arguably "undesirable" in some sense results, and doing that isn't achievable if students have not yet built up an intuition of the finite regime to be able to contrast with unintuitive results in the infinite regime.

Banach-Tarski is for me no less strange than being able to in some sense "stretch" the points in the interval 0,1 to cover 0, a billion; it only seems more strange if one has already gotten accustomed to the run-of-the-mill real number properties. But that's not understanding the real number system to just be given the real numbers and said "these numbers have these properties we want because we're doing calculus now".

The opposite argument is that it's bad pedagogy to expect high school students to understand the sophisticated constructions of higher math. It's true in all disciplines that at each level of study we tell lies that we then correct with more sophisticated lies later. It's easy to say we should present set theory and a rigorous account of the reals to mathematically talented high school students. It's much less clear what we should do with the average ones. Probably just do things the way we do them now. — fishfry

Yes, if we agree there's no simple enough answers to questions about the real numbers (defined as an appropriate amount of time for average teachers and students), your point here is a valid perspective that I'm not dismissing prima faci; certainly this has been the justification of doing calculus with real numbers.

My argument against this is that it breaks the chain of intuitions required to understand math. One step to the next should be clear, this is the "method" of mathematics; the rigor. With all the courses you mention needed to understand the real numbers well, this is the "method" employed, and the goal of these courses is to render what seems at first unintuitive (because they are not consequences of living in the real world) to something that is, step by step, intuitive consequences of the mathematical system.

In science classes, things are over simplified compared to advanced science, but the goal is to stick to the experimental and critical method (I would also argue this could be done a lot better). When this method is abandoned, I think we'd agree here on PF that it's not good science pedagogy. For instance, that creationism taught as a valid scientific theory is bad pedagogy because it is not verifiable by experiment; not that creationism should be "hidden" from students, but that it is doing philosophy and not science.

Also, in science there is no way to avoid starting with simpler "wrong" beliefs about the world that get fixed later. This isn't a requirement in math, there is no externally determined mathematical framework of truth determined by nature. Every step one takes in mathematics can be "true" in the sense of following from the previous steps. The infinite regime is, in my view, basically a restart with a new set of axioms; it is a different mathematical journey than the finite regime that students are naturally on due to living in a finite world and familiar only with finite objects.

My other argument would be purely practical, that focus on transcendentals and "exact" analytical solutions is a product of history due to 1. a lack of calculation ability, so analytical solutions were often the only practical way forward and 2. belief in a Newtonian world of a physical continuum (not to say that we can easily now do without a continuous mathematical framework in which to model discontinuous phenomena; just that we do not think that underlying framework is physical, as I believe you would agree).

However, with ubiquitous and incredibly powerful computing and no need for physicists to believe in a physical continuum, I would argue the average student is much better served by focusing on "what can the computer do for me", viewing constants algorithmically with arbitrary (to a physical limit of computation) precision potential determined in practice by one's problem, and building up intuitions around machine calculation (and analytical work including error bounds, computational complexity, along with analytical proofs of convergence when available, just in the "arbitrarily close to the limit" finitist framework); rather than, what we seem to all agree here, building up wrong intuitions about the real number system. Such a "numerical regime" can be made as rigorous as any part of pure maths, and so is also teaching (what is to me) the critical essence of pure mathematics, although of course additional material introducing infinity could be available for those interested in higher pure maths (whether starting from Euler or introducing ZFC; I don't have a strong opinion; my concern here is not serving those with mathematical aptitude heading for pure maths, but rather that everyone else has the best chance to be mathematically literate and also served by the mathematical community).

The digits in a real number should not be countable, but you have to say which algorithm you use to generate them, since they are infinite. — Mephist

I agree this solves the problem, and this is for me the essence of what I've called conceptual inversion. Starting calculus with uncountable digit expansions as essentially prior knowledge isn't a good setup.

However, the other problem I've been alluding to is revisiting all previous theorems proven in a finitist regime; which is also essential part of understanding the infinite regime. Some theorems are abandoned. Choices must be made.

For instance, in Euclidean geometry we can have a theorem that sphere represented by an arbitrary amount of components, but not infinite, cannot be turned into 2 equal spheres of equal volume. We can also have a theorem that arbitrary amounts of lines never completely fill up area or volume. Going to the standard infinite regime we can revisit these theorems and prove them "false" in the sense that what we thought we couldn't do before we can do now in this new system. This, for me, these "side-affects" features that we didn't expect and didn't set out to make, is what makes these areas of mathematics difficult, even more than being able to construct objects we're intending to make like the real numbers, and high school students. So, even if there was time to explain infinite digit expansion is uncountable in some actual mathematical way involving definitions and proofs, and it's due to this uncountability that's we can assert they cannot be converted into integers ... while still having infinite integers but no "infinite integer" available to put in our set of rationals ... neither asserting that all integers in our set are finite in a sense of having a finite amount of them, which would be clearly false. Even if this was time to do this, it's still not a good understanding of this infinite regime with real numbers without reversing previous intuitions we'd have built up with finitist concepts.

I would say:
- infinite sequences are the same thing as functions from integers to sequence elements.
- functions from integers to sequence elements are surely well defined if the rule to produce the Nth element is clear (is an algorithm)
( maybe explain that you can even assume the existence of non-algorithmic functions, with the axiom of choice, but you cannot use it freely without making use of formal logic )
- integers are defined as sums of powers of 10 (that is the DEFINITION of an integer in the standard notation, not some strange property. So, 2 is 1 + 1 BY DEFINITION: nothing to be proved). The problem with infinite integers is that you don't know which powers of 10 it's made of. If you have an infinite decimal expansion, you know the powers of 10 and everything works. If you are not convinced, try to write infinite integers in Peano notation: 1+1+1+1+.... (or SSSSS..0 - same thing): they are all the same number.
- the sequence of integers is infinite because is constructed by adding +1 at each step, and this is a non terminating algorithm that produces a well defined result at each step, so it's allowed as an algorithm. — Mephist

Yes, I'd pretty much agree with your program here.

By "numerical regime" I mean focus on these objects as algorithms and not "real numbers" that we simply have by writing down pi or e.

I think potential infinity is an intuitive concept. Though it may help some students to know that applied mathematics can also be done with only needing to imagine "what one could practically represent in our universe".

Well, I think a lot of interesting calculus at Euler's level could be done in a enough rigorous way, and just make the students aware of what are the really rigorous parts and which ones are the most "doubtful", when reasoning about infinities. But the most doubtful ones are even the most interesting! — Mephist

I'd definitely be in favour of bringing everyone up to Euler's level.

I'm not advocating ultra-finitism in secondary education, mainly opposing the positing of real numbers as a "starting point" to doing calculus; I'd be willing to compromise on how rigorous the alternative can and should be.

I would take seriously ultra-finitist arguments that they have an even better educational setting to start, for I could be biased by my own familiarity with the subject matter and so think just potential infinity is easier than it is.

In either case, it makes perfectly good subject matter to discuss along with discussion of the kinds of problems one faces with infinities in your program. That there is diversity of perspective even among professional mathematicians I think is inspiring and engaging stuff to talk about.

But, when it comes to actually doing math to solve problems, building up the "intuition of what rigor is" in my view is paramount, and without it the average high school teacher is in a much worse position; in a rigorous system there really is answers to every question that can simply be looked up; which is a better position to be in than needing to say that one doesn't have the answer but "you'll understand when you do pure maths in university" or worse provide a wrong answer as you note is often the case.

I would also not be opposed to a pure maths course that build the real numbers, introduce uncountability, for students interested in pure maths. I'm not underestimating the capacities of high school students to engage with concepts from pure maths. However, it's a disservice to applied maths students to abandon reason for madness, simply because historically we went through lot's of mathematical ideas that turned out to go crazy (in the sense of proving A does not equal A).

What for should I (as a student) loose time in a subject where everything hast just been discovered long time ago, and the only thing I can do is to learn by mind what others did? Math becomes interesting when you see that you can use it do discover new things that nobody said you. And there are still a lot of things to be discovered; only that you have to learn how to reason about them in the right way! — Mephist

Although I agree with your sentiment here, I would argue such interesting questions are best approached from a rigorous foundation, which I don't think your contradicting.

For instance, the real numbers are best approached from a good understanding of natural numbers, integers, rational numbers and finite sets, and what limits these concepts have but also a good understanding of their intuitive strengths that may fail in different systems (what you see is what you get in finitist maths), as your program suggests to do.

So, infinite sets and real numbers could be something introduced at the very end of secondary maths when these foundational concepts are more familiar. But to start, understanding divergence and convergence and tangents and how series and sums and derivatives and integral functions relate to each other (and how to solve real problems with them), is challenging enough to learn in a finitist regime; my intuition is that doing this also with the conceptual challenge of infinity makes it much harder to "see" and to "get" what's going on, and students who start asking questions, even just pondering to themselves, that have no good answers available will much more easily get lost or believe their questions are seen as stupid by the mathematical community, simply because their teacher can't deal with them.

So reticence is going to be a big stumbling block and will surely result in a few years of dither and delay, even when it all becomes a no brainer. — Punshhh

Yes, I completely agree with your thoughts here. And if things just continued like they are now (in terms of our social conditions), then I think sufficient people would never be convinced to do something about the environment for environmental conditions to change (i.e. if we could just keep playing video games and every other species vanished, I think most people -- as in, the people that matter in Western countries who directly or indirectly dictate global policy through voting or plutocracy -- would find that pretty ok).

Evidence point 1, people who watched "Ready Player One" made by one of our most brilliant and treasured film maker, didn't, I would wager, mostly experience the movie as "this is a dystopian hellscape with a lot bigger problems than who controls a computer game; how is anyone even able to eat, much less stay toned, in this scenario?"; rather, I think most Western people (again, the ones that matter when it comes to setting global policy) experienced the film as "you son of a bitch, I'm in", complying to the social programming goal of promoting the fledgling VR industry that the film set out to achieve.

Though this is just some fun trivia, the much stronger evidence is that we haven't done much about species extinction so far, and it's been part of the public discussion since the commercially driven extinction of the dodos and carrier pigeons.

However, of course the environment cannot continue to degrade while our social conditions remain the same. For instance, last year in California I discussed with some libertarians who experienced needing to evacuate due to fires in their area; the decrease of their property value had caused them to reconsider some things. Likewise, the current fires in Australia are affecting the public debate there.

Although it's extremely painful to recognize we did not do much to live in balance with the environment because we value other life, this does not exclude, faced with a existential threat to our species and property value, acting to find a balance required for our own survival.

As an environmentalist I'm willing to help humanity save it's own skin, for the sake of future generations who I have no cause to be disappointed with as well as for the sake of other life that also gets to live in such a scenario. The motivations are less than honorable, but the purposes overlap with mine now that we are faced with the consequences. And, yes, although life would continue even if we don't succeed, I don't go around burning down museums simply because people can paint new pieces, nor disregard suffering caused directly or indirectly by my actions because bacteria will still be around in even the most extreme outcomes of our collective suffering project.

I hadn't really been considering an unliveable hot house scenario, can you give any idea of how likely that would be, or what tipping point would precipitate it? — Punshhh

For lot's of interesting mathematical reasons about complex systems, it's not very meaningful to try to calculate a precise probability.

To make a long story short, complex systems with pseudo stable states (such as a balanced ecosystem), respond in non-linear ways when pushed outside the boundaries of the pseudo-stability. The variables that indicate these non-linear processes may not be practically observable.

A good example of a complex system is the human being. Someone with a job and a dwelling and some friends we can consider is in a balanced pseudo-stable state; this human being, and society they are in, is very complex, yet life is fairly predictable. Now, apply a forcing (some change in conditions), even gradually, and things will move towards instability. If this human being is unable to eat and gets hungry we can predict will do things to return to the pseudo-stability of a banal existence, but if things go further then we can predict that, at some point, thing get unpredictable but we would have trouble predicting exactly when this change happens and we'd have even more trouble predicting the actions that follow that change. Another example, let's say a new boss makes life gradually and relentlessly intolerable, whether due to malice or incompetence; as stress increases in this scenario, we can predict that the pseudo-stable state may end, but we are unable to track the variables that actually indicate a radical change; in this case, we are unable to know much about the internal life of our disgruntled worker. If they "snap", whether that means a angry outburst being fired or some violent altercation or even a positive direction of just quitting and going and living on a beach or something, that non-linear change cannot be predicted with much accuracy even with careful observation; at some point there's the the proverbial last straw, and it may seem obvious in hindsight or totally surprising. You can replay this work-conditions scenario with a toxic relationship for similar insights.

I've seen last straw situations that I truly didn't get, likewise I've seen many that seemed to progress like clockwork.

The point of these examples it that once stresses exceed buffers, it's difficult to know what will happen or exactly when. What we can know is that it's best to avoid allowing stresses to exceed buffers in the first place: it's best to not get so hungry that one is forced into a life of crime or cannibalism one's fellow cast-anyways; it's best to smoothly transition to a new working conditions rather than let conditions become intolerable; it's best not to drive on the wrong side of the road to see what happens.

The better approach is to first ask "what risk is acceptable" and then understand the factors that increase that risk or not. For instance, species extinction (and biodiversity loss within remaining species) makes ecosystems more vulnerable, increasing the likelihood of an ecological collapse; however, it's not feasible to know exactly how much biodiversity loss earth systems can sustain. Maybe we still have a fairly large biodiversity budget ... maybe it's razor thin ... maybe we've already passed the threshold.

Why we can be confident there is a threshold? The best evidence is not the math, but that mass extinctions have happened in the past. Again, we cannot know if conditions were much worse precipitating those mass extinctions, but, likewise, we cannot know if conditions were much better either. What we can know is that increasing stress on earth systems will lead to bifurcation.

Complex systems build up complexity gradually but radically simplify when conditions exceed limits: it takes much longer to build a house than to burn it down. So we can also be confident that a mass extinction event won't be suddenly replaced with even more complex and vibrant life systems.

As we let stresses continue on earth ecological systems, there is non-linear feedback mechanisms with society. Maybe if we get all the ecological experts and top mathematicians in the room to debate this, we'd be convinced that even worst case global warming conditions "won't be so bad", on the scale of earth being livable or not and so we wouldn't be faced with a bunker scenario. We might be pretty happy about that. However, this wouldn't matter if deteriorating conditions towards such a simpler time and ecosystem that's, thank our lucky starts, is at least still liveable on the surface, triggers a strategic nuclear exchange.

Once there is global crop failure, all bets are off as to what happens next.

I think the infinities and infinitesimals of mathematics are the things that make it become more "magic" and interesting. The problem with teaching in my opinion is more related to the fact that the "magic" of the fact that infinities and infinitesimals really work is not explained, or worse, explained by pretending to have a simple logical explanation that, however, is not part of the school program. — Mephist

Yes, I agree with you here.

I'm not against touching on the infinity subject; there could be a whole class on it for students wanting to go into pure maths.

I think we agree that it's bad pedagogy to simply posit the reals with no explanation and no time or ability to answer very expected and natural questions. Instead of curiosity leading to better understanding, it leads to confusion and a sense maths is "because we say so", which is the exact opposite sense students should be getting. Students would be better served by a less ambitious (not actually having irrationals and transcendentals as objects) but more rigorous calculus in the numerical regime, which would make a much more solid foundation for students going on to use applied maths, who can simply stay in this regime (as they will likely be solving every problem with the computer), and better serving pure maths students as well (that mathematics is rigorous, and extensions are made to do new things in a rigorous way).

I think the main thing to understand here is that decimal numbers with infinite decimals can be considered as an extension of "regular" decimal numbers (finite list of digits), but infinite natural numbers (infinite list of digits) cannot be considered as an extension of "regular" natural numbers, since you cannot define on them arithmetic operations compatible with the ones defined on the "regular" natural numbers. Then, you can't build fractions with infinite integers because you cannot build infinite integers in the first place. In my opinion this is quite easy to understand. Did I miss something? — Mephist

I'm not building with infinite integers, I'm building with the infinite decimal expansion representation of real numbers and simply pruning off the decimal symbol. Sure, if we simply define integer as "not this" then it's not building an infinite integer, but it is building something that I can then do things with if I'm not prevented from doing so.

Now, clearly if the proof by contradiction of irrational numbers is constrained to using "regular" natural numbers or integers, I have no qualms. It checks out.

However, if we switch regimes to one where we now have access to the infinite digit expansion of real numbers, we can revisit every proof in the previous regime with our new objects; and now, revisiting the root 2 proof is irrational I am able to solve it with these new objects and not arrive at a contradiction as oddness / eveness is no longer defined upon which the classic proof by contradiction depends. This is what I am doing.

Am I prevented from doing this full stop? Am I unable to find a "suitable decimal expansion" to solve my problem? What exactly is preventing me from doing this, that is what I would consider a suitable answer in the context of learning maths. Given these infinite decimal expansion, I want to use them as what ways I see fit, unless I'm prevented by some axiom. Lot's of things may have been, and still are, true in the previous setup before I had these objects, but in the new setup where I can make use of these objects in equations, I want to take full advantage, and revisit every proof by contradiction as well as every mathematical induction proof.

Broad features and themes involved in rigorous proofs elsewhere I do not consider a good answer for learning math. For me, "learning math is" understanding the proof oneself, not understanding that others elsewhere have understood something.

Again, I am discussing high school students level of understanding and what's reasonable in terms of capabilities, time and relevance.

So your answer doesn't explain why I cannot do my method.

Moreover, your approach, would seem to me, to imply that the decimal expansion representation of a real number cannot be counted; is this your implication? or would you say the digits in a real number are countable?

Also, how do you maintain infinite sequences can be completed, there are no infinite integers, the sequence of integers is infinite, simultaneously within the system suitable for high school level maths. Do we simply elect not to use our "complete the infinity tool" on the integers, and add this axiomatically? What axioms do they have to work with? Do they know enough set theory do make a model that avoids all these problems, or do they have another suitable basis?

We're in deep and complete agreement on this. The mathematical definition of the real numbers is far beyond high school students; in analogy with the difficulties Newton and Leibniz had, which needed to wait 200 years for resolution. — fishfry

Yes, we're in agreement.

And, as I mentioned in the OP, I also agree with your position that there's no "problem" in the real number system, axiom of choice, well ordering, cardinals and the like; at least not some trivial contradiction I'm aware of.

My questions do have answers, and I'm only trying to demonstrate here that the answers are incredibly tricky and go far beyond high school maths.

I think continuing the debate is a good way to bring up how tricky these ideas are, and why simply positing the real numbers without the axiomatic framework to avoid these problems is bad pedagogy.

ps -- Note well The irrationality of the square root of 2 does NOT introduce infinity into mathematics. All the irrationals familiar to us are computable, and have finite representations. The noncomputable reals do introduce infinity into math; but plenty of people who don't believe in noncomputable reals nevertheless DO believe in the square root of 2. Namely, the constructive mathematicians. — fishfry

Yes, it is quite clear to conclude root 2 is irrational in a finitist constructive approach.

My procedure only kicks in once I'm given the real numbers and have access to completed infinite decimal expansion. Given this, I now double back and ask "can I use these new values to prove root 2 is rational in my new system of rules that includes sets of completed infinite decimals".

I am now no longer satisfied by the proof by contradiction that originally brought me to believe root 2 was irrational, as I can solve the equation with values that are neither odd nor even. I can also now do the same thing to solve exactly for the roots of any polynomial that I was previously unable to do.

Euclid's proof of the irrationality of 2‾√2 has nothing at all to do with Cantor's discovery of the uncountability of the reals. The rest of this paragraph, I confess, is not intelligible to me. — fishfry

Where this relates to Cantor, is that if I simply "have the real numbers" and can use my procedure to prove root 2 is rational (because I just have them and have no axiomatic system to prevent me from doing it), then I should be able to count it in Cantors diagonal proof. If I complete the count of the rationals I will "eventually get" to this rational number with infinite numerator and denominator; it's got to be there somewhere.

None whatever. In high school we mumble something about "infinite decimals" while frantically waving our hands; and the brighter students manage not to be permanently scarred for life. — fishfry

Yes, we totally agree. The purpose of my questions is that none of these (what I view) as quite intelligible questions you can start to ask once you "have" the real numbers can possibly be answered in the context of high school maths in a reasonable amount of time. Therefore, it is a mistake to simply posit the real numbers in high school and only serves to mystify mathematics rather than build clear understanding of how the next idea relates to the previous ideas.

The teaching of mathematics in the US public schools is execrable. How many times do I have to agree with you about this? — fishfry

This is what the OP is about, so from my point of view every time there's agreement on this point I am very satisfied.

The reason I'm not attacking as contradictory real numbers, Cantor's proofs, AC, in the other threads is because I know I won't succeed. I can only make a muddle of it here in the context of the lack of suitable axioms and understanding at the high school level, which as you've pointed out tool the smartest people hundreds of years to figure out how to prevent wild proliferation of contradictions.

A university student in anything other than math: None. — fishfry

Yes, if anyone was having doubts about my recommendation that the real numbers in high school is bad pedagogy, take a long look at this statement.

A well-schooled undergrad math major? Someone who took courses in real and complex analysis, number theory, abstract algebra, set theory, and topology? They could construct the real numbers starting from the axioms of ZF. They could then define continuity and limits and I could rigorously found calculus. It's not taught in any one course, it's just something you pick up after awhile. The axiom of infinity gives you the natural numbers as a model of the Peano axioms. From those you can build up the integers; then the rationals; and then the reals. Every math major sees this process once in their life but not twice. Nobody actually uses the formal definitions. It's just good to know that we could write them down if we had to. — fishfry

Yes, I agree, but my question is not how the reals are constructed. My exercise here starts with having the reals already.

My question is how exactly does one prevent the reals from breaking previous proofs by contradiction.

For, if we use proof by contradiction to establish the irrationals, then create the reals as existing between rationals, but then with the reals we have completed infinite decimal expansion and can go back and invalidate the proof by contradiction by just injecting suitable decimal expansions to solve the roots of the root 2 polynomial, then all the reals are now rational and there's no reals between rationals, and we now no longer have the reals, because they're all rational.

We then re-check Cantors proof that the rationals cannot count the reals and simply conclude that if "counted high enough" we would eventually go through all the rationals with infinite decimal expansions taken from the reals as numerator and denominator.

For me, constructing the reals isn't the tricky part, it's preventing the above things happening. Why it's way beyond high school math is that it's not at all intuitive what proofs by contradiction mean and mathematical induction means, when going from a finite to an infinite regime.

Solvable ... but extremely tricky.

Then once it's solved by preventing infinite decimal expansions from corresponding to natural numbers (that for every real number represented by infinite decimal expansion, there is not a natural number that simply lacks the decimal point, and that the reals are not "onto" the natural), then the next step is even more careful preventing the assertion that all the natural numbers placed after a decimal point do not correspond to a real number; we cannot simply take the completed set of natural numbers as a natural number, with even a single natural number corresponding to the decimals of a real number we can still carry out the scheme.

Asking high school students to understand that decimal expansion does not represent a natural numbers lacking a decimal point, is far beyond a reasonable task. For, both natural numbers and real numbers seem very much at first viewing just a list of numbers that you can continue as long as you like; there is no way to intuit some difference beyond "as far as you can practically go in any universe somewhat similar to ours given any amount of time".

Which we already agree on; I'm continuing the "high school devils advocate" simply to demonstrate, with your help, how far away from "simple, intuitive steps" resolving any of these problems are.

Now, if I had a simple clear answer to these kinds of contradictions that took hundreds of years to build up frameworks to prevent, I'd say so. My purpose here is to check that no one else on PF has a simple answers either.

Perhaps you could state them succinctly. — fishfry

The whole point of my post is that high school students would have no way of stating their questions succinctly as you demand, but they are in my view meaningful questions to ask.

You could argue that they aren't meaningful questions and can just be dismissed not warranting an answer, or you could "not wade into it" as you suggest to yourself post-wading.

You have an ax to grind and I've only succeeded in upsetting you. — fishfry

I have no axe to grind. But you very much seem to have an axe to grind with projected axe grinders.

However, the continuous debate around this topic here on philosophy forum inspired me to post my own personal beef, which is that simply positing the real numbers without constructing them nor dealing with all the non-intuitive questions that can arise with completed infinities such as infinite decimal expansion, is poor pedagogy.

But, I'll play your game; perhaps it will satisfy the OP as all my questions will have simple and clear answers that an average high school student will have no problems understanding with a little effort.

Let's start with infinite numerators, denominators and exponents.

Instead of accepting the conclusion that root 2 is irrational (not a rational number), I'm going to solve root 2 using infinite denominators and numerators.

Where do I get these infinite natural numbers to make my rational solution to root 2? I simply take suitable real numbers and take out the decimal symbol and insert those infinite digit expansions into polynomials to represent values that solve my problem exactly, which I admit, I was unable to accomplish with any solution using finite natural numbers I could name.

Fairly simple procedure.

Please demonstrate how this infinite numerator and denominator either does not get counted in Cantor's diagonal proof, does not represent an irrational value, or there is something preventing finding and placing all the digits of suitable real numbers into a numerator and denominator to solve for root 2.

An infinite normal digit expansion (which I'll choose to use as suitable) is neither odd nor even, as is well known, and so there's no contradiction of division by 2 as is usually concluded in the run-of-the-mill finitist approach to proving the irrationality of root 2; I can just keep that 2 coefficient around no problem and divide by 2 to get rid of it. Therefore, root 2 is rational.

If I am given these infinite expansion of digits, seems I should be able manipulate and place them where I want if I have some procedure to do so (unless given suitable axiomatic conditions preventing me to doing what I want).

What axioms does a high school student possess to avoid the above issue of concluding root 2 is rational? If none really (if only because the reals aren't even constructed to begin with, just posited as a given) then I think we agree about the OP.

Followup question (as I believe this is what interests you to demonstrate) what axioms does a university student possess to avoid the above issue and how?

I presume that you are referring to the idea that set theory provides the 'foundation' to mathematics. — A Seagull

More or less. There are alternative foundations, but set theory is the main one.

But pure mathematics is abstract and doesn't need any foundations apart from its axioms which introduce the symbols and define the rules. (And admittedly these axioms are more implicit than explicit). — A Seagull

This is also true for applied mathematics.

Applied mathematics also has definitions and axioms, just focusing on those that generally have real world scientific application (applied maths is a subset of pure maths).

There are not infinite sets of anything in the real world (real world of scientific investigation at least); so how to deal with infinite sets can be excluded from applied mathematics. For me this is the main difference; there is no need to learn calculus in the real number system and then calculus in a finitist numerical system appropriate for a computer. It's, in my view, only historical accident that learning calculus with real numbers first seems to make sense. One can learn first a finitist numerical system for doing calculus.

And as for the real numbers, they become necessary when one looks to divide (for example) 10 by 4. (10/4). although the task is in the domain of integers the answer is outside. — A Seagull

This is the basic thing my pedagogical program would get rid of.

You never need an "exact" (i.e. infinite decimal expanded) value of 10/4 in an applied problem (first because it's 2.5, but I assume you intended an example like 1/3 or then an obtuse reference to infinite trailing zeros).

The numerical regime basically refers to replacing all calculations that can go on forever in a finitist setup tailored for the computing machine doing the calculations (values can be arbitrarily large or small, not infinite, and not more than can fit in the computer ... with a whole bunch of caveats) with algorithms that can be carried out to the required precision (the series sum or whatever the algorithm is, and a halting condition); in other words, those significant digits from physics class, can form the axiomatic basis of a completely adequate calculus for applied problems.

Learning the axiomatic setup of numerical computation rigorously is (until someone shows me how easy real numbers can be) a far better use of students time leading to, I believe, a better understanding of maths for both future applied maths and pure maths students. Understanding finitist maths well, I would argue is the correct basis to then going beyond finitism for students interested to do so; likewise, I would argue a more rigorous use of finitist maths wherever it is adequate in physics and other sciences is far better than a lazy use of more powerful models.

In other words, real numbers are not necessary when dealing with 1/3, or any calculation that can in principle go on for an arbitrary length; approximate solutions are fine for any real world problem.

The reason I'm posting here in logic and math and not politics, is not simply because of the theme but because my contention has a counter example of a very clear and simple presentation of the real numbers that high school teachers and students would find of appropriate effort to fully grasp. If there is such explanations that are graspable by the average student, then I'd capitulate.

With this clear distinction the complications of maths fade away. — A Seagull

Yes, this is basically what I am advocating, though with much heavier emphasis on the applied part in secondary school (and applied maths will still have plenty of symbol manipulation in it's own right and plenty of theorems that apply to both pure and applied maths).

Though more specifically I am singling out real numbers as the particular problem; though maybe there are others. Likewise, perhaps there is a pedagogical approach that accomplishes both, as you seem to be suggesting, but my feeling is that you can't really do pure maths without set theory, which as points out was a failed experiment to teach children.

However, if there's some simple explanation of the real numbers and all the questions that naturally arise from infinite decimal expansion, then I'd be proven wrong on this particular point.



Did this set theory experiment simply not work at all, or did it produce some small cadre of math geniuses?

No, that's an absurd suggestion. To make a cumulative case one would need to show that each argument had some probative force, and that would require making each argument. And so the opening post would then have to be thousands of words long. — Bartricks

It's in no way absurd. Plenty of people have written thousands of words to make an argument.

I also have zero problem with you not making such a case; you can if you want, but I'm not going to argue against a case you're not even making.

So an objection arose, that consent is not sufficient basis for antinatalism, you concede that this is the case but there are other arguments that together with this insufficient argument win the day.

You are free to make the next argument in your cumulative series in a new thread and you are free to insist on making it here, but in the latter case I do not continue; I like threads to stay somewhat on topic, that I am committing to a position relative the OP and not indefinitely committing to any alternative of "cumulative" argument apart from the OP (unless of course the OP presents some cumulative project). There's already a thread on the consequentialist argument for antinatalism, and if you don't want to continue that thread for some reason then you can make a new one on your next argument.

My suggestion of making a thread about your cumulative argument is if you want to discuss that; you now bring it up as representing your actual position, but if you don't see how to summarize it, no problem.

I don't see how, once you've failed to establish any one of these arguments individually justifies antinatalism you will be able to make some sort of meta-argument that uses the inadequacy of each by itself to form a formidable argumentative force together. Rather, it seems more likely to me that you are simply creating a perpetual goal post moving machine upon which you can ride away from each opponent to new greener pastures, confident that winning this race away from the previous field is the true strength and metric of victory. However, I am patient and am willing to wait to see if your project succeeds in ever getting to "this cumulative argument justifies antinatalism".

Here I'm concerned with your claims in the OP, and we agree your claims about consent are not sufficient to justify antinatalism. If you want to believe there exists a cumulative case that is too absurdly long to write for any critical investigation of it to happen ... but is nevertheless true, then I have no problem with you believing that.

As for your book analogy, though I agree many intellectual hacks go about writing books in the way you propose, there is, however, another approach to writing philosophical texts which is to assume the case is not one way or another until critical review of all the arguments are carried out to a sufficient standard. You seem to think that because you can imagine writing a book justifying your position that your position is justifiable; however, no such book maybe feasible to create, but rather key arguments (or even all the arguments) may have fatal and insurmountable flaws.

Your position now seems to be "I have no argument to justify antintalism to offer, but I am totally right about my antinatalism position and once I make such an argument over a long, potentially infinite period of time, the justification will have proof".

The alternative point of view is to see things as not justified until the sufficient proof in question actually exists.

This is correct. It is amazing that many people dont understand this concept that you dont have to exist prior to a certain point to be harmed ONCE you are actually brought into the world. — schopenhauer1

Myself, and most of the earlier posters, have no problem with actions now affecting future children. This is even the basis of one of the main criticism of the OP; that there are plenty of actions other than conception and birth that affect not only future children but children in the here and now, a significant portion of such actions for which we are not bothered by a lack of consent (for instance, the functioning of government too affects future children without their consent).

However, if by "amazed that many people dont understand this concept" what you really mean is that you are just haphazardly throwing shade in random directions without any definition of what constitutes 'many' and what might justify amazement without any methodology to speak of, then I concede the point; it is amazing, as is a great many things.

Second, if your complaint is with pedagogy it's not about math. — fishfry

Yes, did you even read my post, this is my complaint.

I have no issue with real numbers "existing" in whatever sense mathematicians using the real number system want to believe. I am not convinced that "the true infinity" or "the true continuum" is captured by these symbolic systems, but I agree with you when you say mathematicians need not care and usually don't care; you can use a different system if it suits your style or problem.

I even cite your own words on this subject and express my agreement.

Your not giving me a hard time, you just have poor reading skills of prose; but I don't mind that, you don't make any claims to be able to understand non-formal arguments and perhaps have formal reasons to believe this task is impossible.

The reason I presented my arguments in prose is because that's the sort of thinking a high school student will be equipped with starting to use the real numbers.

My challenge is that: is there any answers to these prose questions that doesn't involve an entire university course, which maybe not even enough. As someone who's taken these university courses and who works with math in my day job building numerical models, you seem to claim I don't understand these issues. Even if it was true, which I doubt, isn't this more evidence to my point?

It would be fun to teach ZF to SOME high school students, the especially mathematically talented ones. The mainstream, no. I wonder what you are talking about here. Again, the axiom of choice is not needed to defined or construct the reals. — fishfry

Again, terrible reading comprehension; mathematicians not learning any humanities really is a problem.

I do not claim the axiom of choice is needed to construct the reals.

My argument is above the tdlr which doesn't mention the axiom of choice. My tdlr is an over simplification of my argument in a recommendation that I believe most people who understand this subject and have good reading comprehension would get.

Which you seem to agree with, that ZF can be taught at a high school level, which is my recommendation. I think you would agree that most high school students would not be prepared to deal with C (which for me, is what then makes the real number system mathematically interesting; unless there's been some breakthrough since I last looked at this topic that C is no longer required).

It's worth noting that the pedagogy retraces the history. — fishfry

This is basically our difference.

I disagree that the pedagogy retraces the history. If it actually did, maybe I'd have less of an issue.

Newton did not have the real numbers to do calculus as you note, yet high school calculus students simply start with the real numbers.

Dating from 1687, the publication of Newton's Principia, to the 1880's, after Cantor's set theory and the 19th century work of Cauchy and Weirstrass and the other great pioneers of real analysis; it took two centuries for the smartest people in the world to finally come up with the logically rigorous concept of the limit. For the first time we could write down some axioms and definitions and have a perfectly valid logical theory of calculus. — fishfry

You realize you're just adding more weight to my contention in the OP here?

If you need to read Principia mathematica and two centuries of the smartest people to understand the real number system ... maybe this is too much of an ask to high school students?

Do you agree?

If not, my challenge is that you explain the answers to my questions in a way that a high school teacher and then students would understand. If you can't, just agree with my OP rather than try to prove your smarter than me, which I so far not seeing any evidence for: going off on random tangents, not addressing the point of the OP, cowardly hedging your own complaints etc.

For instance, I did not define "infinitesimal", it's just a word that I find perfectly suitable to use to refer to series converging to a point (i.e. the distance becomes infinitely small). My use of infinitesimal was to contrast using prose (using words most people here would understand) the definitions one would find in numerical calculus compared to what we usually just call calculus; not to conjure up 17th century philosophical debates.

To lift from wikipedia because I do basic "google the subject matter" research when engaging in internet debates.

From the wikipedia page on infinitismals:

Logical properties

The method of constructing infinitesimals of the kind used in nonstandard analysis depends on the model and which collection of axioms are used. We consider here systems where infinitesimals can be shown to exist.

In 1936 Maltsev proved the compactness theorem. This theorem is fundamental for the existence of infinitesimals as it proves that it is possible to formalise them [...]

There are in fact many ways to construct such a one-dimensional linearly ordered set of numbers, but fundamentally, there are two different approaches:

1) Extend the number system so that it contains more numbers than the real numbers.
2) Extend the axioms (or extend the language) so that the distinction between the infinitesimals and non-infinitesimals can be made in the real numbers themselves.

[...]

In 1977 Edward Nelson provided an answer following the second approach. The extended axioms are IST, which stands either for Internal set theory or for the initials of the three extra axioms: Idealization, Standardization, Transfer. In this system we consider that the language is extended in such a way that we can express facts about infinitesimals. The real numbers are either standard or nonstandard. An infinitesimal is a nonstandard real number that is less, in absolute value, than any positive standard real number.

Followed immediately by a section called "Infinitesimals in teaching":

Calculus textbooks based on infinitesimals include the classic Calculus Made Easy by Silvanus P. Thompson (bearing the motto "What one fool can do another can"[12]) [...]

Another elementary calculus text that uses the theory of infinitesimals as developed by Robinson is Infinitesimal Calculus by Henle and Kleinberg, originally published in 1979.[16] The authors introduce the language of first order logic, and demonstrate the construction of a first order model of the hyperreal numbers. The text provides an introduction to the basics of integral and differential calculus in one dimension, including sequences and series of functions. In an Appendix, they also treat the extension of their model to the hyperhyperreals, and demonstrate some applications for the extended model.

So, not only is infinitesimal perfectly fine mathematical jargon to talk about things "infinitely small" in both a technical and a general sense (as wikipedia starts the article by saying: "In mathematics, infinitesimals are things so small that there is no way to measure them"), but it is a common notion (according to wikipedia) used to introduce students to calculus, as it's intuitive.

This, my contention is, is a pedagogical mistake unless there are answers to all the very normal questions students can have about the real number system (that are as easy to grasp as other associated concepts being introduced). I have yet to see them.

Why is there so much debate around these infinite related questions such as cardinals and continuums here on philosohy forum? And not about questions like solving the quadratic equation or any number of other theorems? Because, in my view, it takes very specialized knowledge to understand modern mathematics modelling of these questions, which as you say, need not bother anyone that specialists are building such systems, but it is bad mathematical pedagogy to introduce to students concepts that they are unable to fully grasp and have zero need for any of the tasks at hand; it serves only to mystify mathematics rather than build understanding.

An analogy would be introducing Euclidean geometry in the context of Reiman manifolds or rotation in quaternians because that's what the cool kids in university do, with neither having any basis to have any clue what a Reiman manifold or quaternion really is nor ever needing the extra things Reiman manifolds or quaternions provide to address the Euclidean problems being asked to solve; now, I understand why concepts got inverted historically (since we were computationally extremely limited until recently), in the development and subsequent teaching of calculus as opposed geometry (pending an answer to my questions), my point is it's now a completely fixable conceptual problem in our teaching methods: that finite computation is a much more basic concept than the real numbers, real analysis, metric spaces and so on (i.e. real numbers are not required for any high school level problem and there's no need to introduce them until they are actually needed).

Now, I'm not saying these issues should be kept secret or something, there could be extra material for students who want to get into it; but I see no high school level problem that is not perfectly addressed in the numerical regime which is far easier to understand; you can really "see" and "get" how a computer functions in principle and why algorithmic approximations that truncate at a suitable number of steps yields answers to real world problems that students can visualize even at a high school level; there is nothing remotely as difficult conceptually as an infinite decimal expansion. It's also critical to understand not just the algorithm that converges on the desired constant but under what conditions are correct to end such an algorithm for any given applied mathematical problem, which is what the vast majority of high school math students will be going into: engineering, computer science, programming, chemistry and even accounting requires intuition of the strengths and limitations of machine computation (i.e. what kinds of problems require special attention to the the finite nature of the computer, in terms of memory, floating point representation, iteration steps, economizing computer resources and so on; and what kinds of problems one can just paste code from stack overflow and let it ride).

So if you want to get back on track, answer my questions concerning the real numbers in a way that a high school teacher and student understands. I've claimed to understand the answers to these questions, but you seem to be arguing it's all too complicated for me and that you will explain to me why I don't in fact understand the issue and you're going to demonstrate that. Well, if this is true, I'll be the first to benefit from your addressing the point in the OP. I eagerly await.

There is evidence that capital has seen the light. Mark Carney the out going head of the Bank of England, soon to become the UN special envoy for climate change, spoke on the BBC a few days ago. That, in no uncertain terms, that investments and infrastructure developed for the exploitation of fossil fuels will become worthless in a few years and that capital should look to invest in investments and infrastructure designed to replace them with renewable sources of energy and the emerging green economy (my wording, but this is the jist of what he was saying). — Punshhh

Yes, this sort of thing is essentially what I mean by depending on our present institutions. In this case, (some) representatives of capital are starting to make some preemptively actions anticipating better policies (which then put additional pressure on policy makers, forming a virtuous cycle). However, if you wish to say here that this is evidence of the "market mechanism" working in and of itself, then I disagree; I would argue it is partly the market mechanism responding to regulation changes and partly the moral concern of the individuals themselves.

For me it is not a question of whether there will be an attempt by our institutions to reasonably respond at some point, the question is whether that point is soon enough, which is a segue into your second point:

I presume the planet experienced a hot house state before, which was liveable . Presumably it is the rapid transition to this state which you are suggesting is unliveable? In which case I agree, however I do expect a small colony of humanity to survive and rebuild. Whether they manage to take any knowledge with them, is the worry. Otherwise we may go back to square one again, and start all over again, as we have done before. — Punshhh

Yes, there was plenty of life in all the previous hot house climate regimes, it is the speed of the transition from one to the next that generally triggers mass extinction of the ecosystems that had evolved under the previous regime. Currently, the extinction rate is estimated to be around 1000 times the historical background norm, and biodiversity within species maybe an even higher rate of loss. It is not the destination but the journey, as with so many things in life.

I agree colony bunker living is conceivable and has some non negligent probability associated with it even with the most extreme climate outcomes. However, non negligent probability in a complex system that something will happen entails also a non negligent probability that it won't happen. The climate transition to a hot house maybe "really bad" but does not crash oxygen levels to unlivable nor a cyanide ocean event ... but, maybe it will; there's also black swan events that may befall our future bunker dwellers such as an ice sheet slipping into the ocean and causing a tsunami which washes away the things on the surface (either infrastructure or biological resources) needed for long term survival, not to mention just "normal" tsunamis and volcanoes and so on that may befall any colony (with glacial rebound causing more of this sort of thing).

A human colony without a breathable atmosphere I think faces sever challenges, and non-breathable atmosphere is entirely possible without the amazon or other forests and deadzones the size of entire oceans.

Avoiding the hot house climate regime not only avoids the above problems of would be bunker dwellers, but also allows hundreds of millions of other people and other species to survive as well: their cultures and heritage in the flesh, not just a few books (which will be the only things that remain long term once the colony clean room becomes too contaminated to produce silicon and the micro films were lost in a fire).

What do you mean 'reverting'? I have said repeatedly that my case for antinatalism is 'cumulative'. That means I think there are numerous arguments - no one by itself decisive - that imply procreation is wrong. — Bartricks

Being unable to defend many arguments does not a stronger argument make.

If lack of consent is a problem, switching to the suffering argument does not provide support for the consent argument. This thread (that you started) is on the topic of lack of consent. If you are acknowledging in your reply here that your lack of consent argument is insufficient to justify antinatalism, then we are finally in agreement. We have come a long way, but I am happy with the destination.

If you believe a bunch of arguments together justify antinatalism, I suggest you make a new thread with this "cumulative" argument and referencing the arguments that accumulate; either debates such as this one, which, though insufficient to justify antinatalism on its own, play some part in the cumulative argument scheme or then new debates that will to occur in the context of your cumulative structure.

There are 4 reasons why I think we will fail to avoid the worst consequences of global warming: — Bitter Crank

This isn't a well formulated point. There can always be "worse".

That being said, we have already dealt severe damage to the ecosystems in terms of biodiversity loss (not only in number of species but genetic diversity within those species), mostly through other means that, along with climate change, will do even more damage going forward, much we cannot avoid in any scenario.

However, as bad as the already passed and yet-to-be biodiversity loss is, things can get even worse.

The climate change battle is now to prevent a "hot house" climate regime where there is no ice at all in the North Poll and rapidly deteriorating glaciers, first in Greenland followed by parts of Antarctica.

There are two stable climate epochs in the current configuration of continents. The one we currently live in is the "Ice box" where there is permafrost, glaciers in the north and a north poll ice cap on the sea. In this regime continental glaciation oscillates wildly in response to volcanic activity and incremental changes in the earths orbit and tilt.

Because of the long term efficiency of weathering to remove carbon dioxide from the carbon cycle, leaving only what is in earth and plants and living creatures, and volcanic additions (which are small) the long term stable state is this Ice Box configuration.

However, it is only a meta-stable state due to earth being a closed system without other sources of carbon to get into the atmosphere in the past hundred millions of years.

With a new source of pumping carbon from underground by humans, the earth can be pushed into a Hot Box epoch.

The consequences of entering the Hot Box climate regime are so severe that it's arguable no humans at all would survive. Bunkers may not be maintainable long term on an earth without one or several of the following elements: edible biomass in nature, breathable atmosphere and relatively calm weather as we are used to now. Weather is, in my opinion, what people forget to imagine in our future bunker dwellers. Even the best setup bunker (that avoided destruction from vengeful pirates / militaries during the transition) will require some interactions with the outside world, in turn requiring some sort of exterior infrastructures to make that interaction efficient enough to run the bunker. Severe storms and hurricanes are going to create crisis points untop of every other problem (which are many). Maybe it is solvable ... maybe not. What is for certain is that bunker dwelling near the polls will be the only option in a Hot Box world and very few will be invited to join.

This scenario is now essentially the path we are currently on, but is still avoidable with difficult, but feasible, large scale action.

There's also uncertainty around the Hot Box point of no return; we may have more time than we think, which is a possibility that, again, warrants taking what action we can.



Edit: found the image I was looking for that shows the above (credit: Trajectories of the Earth System in the Anthropocene; Will Steffen, Johan Rockström, ProfileKatherine Richardson, Timothy M. Lenton, Carl Folke, Diana Liverman, Colin P. Summerhayes, Anthony D. Barnosky, Sarah E. Cornell, ProfileMichel Crucifix, Jonathan F. Donges, Ingo Fetzer, Steven J. Lade, Marten Scheffer, Ricarda Winkelmann, and Hans Joachim Schellnhuber; https://www.pnas.org/content/115/33/8252).

How: Through the overthrow of the existing economic structure by revolution, not necessarily violent, but certainly uncomfortable for those who own the means of production. — Bitter Crank

I agree with most of your analysis of why we have had very little action so far and the difficulty of the task.

But I disagree that political revolution is necessary.

Real political change is a generational affair and takes time, time we don't have. Whatever we are going to do will be through the political institutions we have at the moment. Action has been very light so far, but our institutions are not inconceivably far from doing something effective. For instance, the EU (now that the UK is leaving) is currently discussing a "carbon tariff", which is something the EU is institutionally setup to be able to do and would be actually an effective step to start and then gradually increase. The EU is a large enough economic block to impose a carbon tariff.

One must keep in mind that although the general population was not able to be convinced of the case for action compared with the climate denial PR industry highly attractive offer of being an edgy contrarion, this phase of intellectual failure was in the absence real world consequences. There is a small window where real world consequences may tip the balance of public opinion but are not yet too severe to breakdown or overload the institutions that are able to act. This window may or may not be before or after the point of no return towards a Hot Box climate. Previous failure does not guarantee continued failure: both learning and changes in key factors are a basis for continued attempts.

I am also working on a project developing a renewable energy technology that has no resource, financial or skills bottlenecks to scaling. Although most renewable energy technology has some scaling problem rendering it largely irrelevant to our problem, it is an erroneous inference that therefore all renewable energy technology has a scaling problem. For instance, smart phones rapidly scaled in a incredibly short period of time. Renewable energy is different from smart phones, but the example does demonstrate that it is possible for a technology to "super scale" globally; the ven diagram of "renewable technologies" and "globally scalable technologies in a short enough time to affect climate change significantly while not requiring a too large carbon investment to make" may overlap, if only ever so slightly; it is worth investigating whatever points may exist within the overlap in any case. That the vast majority of firms, scientists, and thinkers and engineers swim around in the part that doesn't overlap because it is spacious and easy to make new discoveries (the bureaucrats and kickstarter supporters having no reasonable criteria about anything generally speaking) and so peddle counter-productive hype, is not evidence that there is no reasonable criteria that can be found and focused on.

Pressuring institutions to do more and developing magic bullet technologies (to either amplify the affect of policy or make large scale disruptions to the fossil industry) are, in my opinion, the only effective actions available. Both are a roll of the dice. We will see where they land in the next couple of decades.

I very much hope political revolutions will bring democracy to China and Saudi Arabia and other tyrannies. I very much hope political revolution will bring proportional democracy to the US and UK. However, these things happening are not a pre-requisite for large scale effective action. The functioning democracies that we do have, despite exterior opposition, can have large scale consequence if bold action is taken. Functioning democracies do not seem bold because they lack totalitarian madmen to write headlines about, but this apparent teptitude is an illusion; once sufficient consensus is reached, action can be extremely swift and efficient due to lacking a "house divided against itself".

But not everything produced can be given away. I think I like sushi is still talking about co-existing in a world of economic transactions. — Brett

Where do I say that?

You say:

You could give it away. But ironically a payment makes people consider how much they really want something. If it’s free people will take with very little thought to what went into it. It’s almost like in the real world, outside of our lives, a monetary value has to be attached otherwise it has no value at all, including the personal value you imbedded in it. — Brett

As mentioned in my first response, I don't see how this claim is supported by some context, but please explain how it is if I am missing something.

Well you’ve focused on only four lines of my post, so it’s a bit out of context. — Brett

Please, show how the context changes the meaning of the four lines I am debating against. I agree with your previous statements setting up the problem of assigning a price; that you may need to do so to recover costs or to make a living; and that price may not reflect your personal sentiment about your work nor transmit transcendental value you may have encountered in the production process.

I have issue with your next claim you make that without a price we wouldn't value something. I don't see how your previous statements support such a claim, and I see lot's of counter-examples of which I provide 3.

But please, explain how the context supports the claim I am focused on.

If it’s free people will take with very little thought to what went into it. It’s almost like in the real world, outside of our lives, a monetary value has to be attached otherwise it has no value at all, including the personal value you imbedded in it. — Brett

This is simply not true. I know an artist that doesn't sell any of her original work, only prints; she simply doesn't put a price on the originals; she does so because she values the originals.

Likewise, hand-made scarf given as a gift may have a lot of value to both the producer and the receiver, but there is no monetary value attached.

If the pot in your example was given away, it may hold more value than if it was sold, as the sales price may indicate that it can be replaced for the same price; so, if it's not expensive, the owner may not care much about it. Whereas, as a gift, it may symbolize the entire relationship.

The difference of course is that original artwork, scarves, and pots that are not placed on the market are not commodities.

I don't think we're in much disagreement here. Of course we could get into the ethics of evaluating forum participation, for instance on what basis would be evaluate a "good" use of the forum (what's the basis of deciding who we are helping, why, to do what, and how would we know? etc.), but that would quickly just resolve down to more general ethical discussions that happen here all the time.

I'm glad you liked my translation of L'infinito. I tried to render more (some of) the feeling than a literal translation (which is basically impossible in this case), so am happy you felt it.

Wallows has been asking for and receiving advice for as long as. He has been around long enough to know and predict all the likely replies.

You made some excellent points and suggestions but look at the response.
It's an addictive pattern. — Amity

Even if you are right, what is it to me? what is it to you?

If the content is not appropriate, it is a question for the moderators, and I need not trouble myself.

Perhaps some lessons take decades to learn.

Perhaps others have similar questions and may benefit in any case.

If indeed philosophy is an addiction here, how are we to intervene? If philosophy has failed, as you suggest is the case, perhaps we must widen our perspective, as you have suggested, and seek in poetry some help for this condition that seems persist indefinitely:

Always dear to me is this lonely hill I keep coming to
and this hedge and all its details,
hiding from me the ultimate horizon.
Yet crouched and staring, endless
is the space beyond, that humanless
emptiness, and that depth of stillness,
my thoughts drift; not far
the heart, the terror. Then, the wind speaks,
swaying the trees, and the
infinite silence and these rustling leaves,
I compare the two: I remember the eternal,
the seasons of death, the present,
the living, the sound of it. In this,
immensity, my thoughts start to drown:
and I drift off sweetly into this sea of thoughts.

This small "l'infinito" of Leopardi is perhaps a start to such recurring ailments.

Well, I don't think I've conflated accessible and possible, for me it's very clear the difference, and I agree with everything you said. But I still cannot understand exactly how this two terms are to be formally expressed without requiring one another. — Nicholas Ferreira

If they are not the same, it's not circular, they simply depend on each other to be understood.

There is no proof of theorem being offered that could have the problem of being circular, it is the basic concepts to provide meaning to the symbol manipulation to follow.

You will encounter the same thing with all foundational concepts.

For instance, logicians and mathematicians will use the word "statement". If you challenge what it means ... of course you can only get statements as explanations; if you truly don't understand what a statement is and need that understanding to understand any statement, the explanation of statement can never make sense. All such explanations will be just different versions of the same thing with various caveats and relations to each other all expressed as a series of statements. So, it's as puts it, that what's being offered is different ways of looking at the same thing, and that's all that can be offered.

We have an intuition of what a statement is, likewise possibility. In dealing with formal reasoning systems we can relate our intuitions to some property of the system (in this case there is a relation by symbolic manipulation between the worlds in question that we can intuit as our concept of "possible"), this however is exterior meaning we give it; what exists internally to the system are the symbolic rule relations which do not require meaning. The usual analogy is long lost languages we cannot decode; we can deduce some rules between the symbols but have no idea what they mean; we can look at formal systems the same way, but we are not obliged to (if we want to make some decision based on the symbols and relations to each other, there must be some relation, some correspondence, to what we believe the actual world we live in is; and we focus on systems of rules that seem to have some innate ability to model aspects of our real world, rather than just randomly invent symbols and rules and randomly permute them without ever assigning meaning).

It's like starting a book that begins with "the woman sat next to the tree"; what do we know about the woman, that she is next to the tree; what do we know about the tree, that it is next to the woman. Math and logic books are generally not an exception to this feature of all books and tend to start the same way too.

Based on just what you report here, my guess would be you have conflated accessible and possible.

Accessible I would assume means expressible; the statement can be understood with the system of rules and postulates in question.

Possible is not the same as accessible. A statement can be accessible but always false.

To be possible means there is a world where that statement is actually true (where world means a system of rules and postulates, not the actual world).

The main theme in this sort of framework is that "modal necessity" is equivalent to the proposition being true in all possible worlds accessible to the system in question, so in this framework you can hop around systems and relate them to each other and try to find counterexamples or then prove no counterexamples can exist to investigate and conceptualize necessity, as well as investigate what possibility means by building a concrete example of an accessible world where the proposition is true.

In a sort of colloquial "everyday" modal logic, you can understand this process as explaining the concept of "it might rain tomorrow" by describing that possible world of tomorrow where it's raining (which is different to explaining what is true about the world today which makes rain tomorrow possible; in this colloquial everyday sense, Kripke is building what relations need to hold between what's true today and the imagined world of tomorrow to imply possibility or impossibility; we can not only imagine it but it is a world accessible from what we believe to be true today).

Likewise, explaining the concept that "the sun will necessarily rise tomorrow" is the process of explaining how every possible world of tomorrow includes the sun rising. (of course this isn't rigorous as someone with powerful rockets could stop the spin of the earth, or the entire sun could potentially quantum tunnel to the edge of the galaxy or any number of other ways the sun wouldn't rise, but we implicitly ignore these possibilities as so low as to be irrelevant by treating these things in a modal way; a sort of "modal lite" that is the quickest way to reason for a wide range of cases: I necessarily need to eat to live, poison will necessarily kill me, I necessarily am unable to fly on the earth's surface without technology, going to work is not necessary but various possible things may happen as a consequence, various things are contingent on various other things, time travel backwards is necessarily not going to happen etc. The purpose of this "modal lite" way of reasoning is to narrow down the scope of factors that have an ambiguous range to consider by first finding relevant details that are close enough to 100% or 0% to be treated in a necessary or necessarily-not way, and then applying our intuition to the possibilities that remain, as we can't explicitly calculate probabilities for most situations; the necessary and necessarily-not game allows us to build a trunk of necessary things and then at least cut off entire branches that no longer need to be considered if they are necessarily-not, and so better make use of our intuitive capacity on the branching possibilities that remain).

I'm sorry if my assertion about Marxism being "out of date", offended you. However, it wasn't a digression, but integral to my understanding of the article as an "update" of older theories, such as Smith and Marx. — Gnomon

No offense taken. It's a digression if you want to talk about Marx, but framing something as "baseless out of date" is unclear if you want to talk about that thing or not. Certainly out-of-dateness is a fair and debatable point of old theories. Likewise, your following sentence, that the upper class oppresses the workers is also worthwhile to discuss if it's a fair characterization of Marx as other posters have also responded to.

Do you think it's unfair to interpret the new statistical economic model as a valid "update"? I saw it as similar to Newton's law of gravity, which was updated and refined by Einstein, but not invalidated. I have no training in economics, and only a philosophical (not political) interest. So I may have overstated the importance of the Capitalist Casino concept. — Gnomon

Yes, I definitely think it's a good analogy to Newton, that Marx discovered some true principles that resulted in valid predictions (as well as false ones; where we can draw a parallel with Newton's focus and writing on alchemy), and that we have since both discovered new principles and verified new predictions.

Of course, there's a large body of work beyond this paper on inequality, most importantly empirical work showing inequality really does increase under laissez-faire policy frameworks.

For me, the significance of this paper is more as a retort to the "tabula rasa" hiding place / fantasy of laissez-faire proponents (that it's always whatever regulations remain that have caused the inequality, corruption and market failures and not the regulations, designed over centuries to keep inequality, corruption and market failures in check, that were removed).

The paper does it's best to create the laissezfaire fantasy world and simulate what happens. So, the point is that even with granting the laissezfaire proponent's insistence that things like corruption and asymmetry and unequal starting points can be simply ignored, their model still doesn't seem to work (unless the goal is to create oligarchy rather than a free and happy citizenry).

However, nowadays proponents of deregulation and lowering-tax-on-the-rich and removing anti-corruption laws have mostly abandoned the idea that these things are good for society in general; the paper in question is a small addition to a large pile of both theoretical reasons to believe as well as empirical reasons to believe deregulation, lowering taxes on the rich and legal-corruption are all bad for society (of course that doesn't immediately inform us what regulations, taxes and anti-corruption measures are worthwhile, just that the principle that getting rid of them will magically achieve those objectives even better is completely implausible and delusional, or more likely just propaganda from people who want those things regardless of wider social consequence).

Instead of making the case that removing these restrictions on the accumulation of and use of capital just so happen to create a situation those restrictions where designed to achieve, proponents of deregulation, lower-taxes-on-the-rich, and legal-corruption have moved to moral arguments instead of scientific (once empirical evidence is overwhelming it works less and less well to keep making ridiculous claims that are so easily checked to be false).

At least with moral arguments, it's not easy to just check that they are false. If one wants a world dominated by the wealthiest and one simply places no value on anyone else, then such a moral theory is perfectly sound. Likewise, if one is convinced taxes are theft and immoral then it can follow from this that no one should be taxed even if that means collapse of government and mafia rule, and it can also follow from this that lowering taxes of the rich is good to support even knowing that the rich won't reciprocate and lower everyone else's taxes to the point of collapse of the justice system, infrastructure etc (as it's simply the right thing to do).

Of course, few people really have values compatible with wanting a much worse society with much more suffering because taxing the wealthy is a worse crime, but it's much more work to carry out such an analysis and it will be different for each person what exactly their values are and why exactly a functioning society is better than a dysfunctional one for them and to what extent they would lift a finger to promote actions leading to such a better society in their definition; i.e. moral argumentation is not universal whereas the whole purpose of science is to be universal (we can all follow the same observations to same conclusions; so once something really is well established the same body of evidence and argumentation is valid for everyone); hence, why science denialism dressed up as some sort of moral crusade, including denying obvious economic realities, is at the heart of the movement for deregulation, lower-taxes-on-the-rich and legal corruption (which in turn corrupts various scientific institutions which helps, in part, to reinforce scientific denialism; i.e. regulatory capture of supposed objective watch-dog agencies as well a gaming, wherever possible, academics to be biases or at least easily pliant, leading to lot's of ethical failures that can be legitimately pointed to as examples of why self-labeled scientists and institutions cannot necessarily be trusted). Switching to moral arguments allowed 1 or 2 more decades of effective propaganda, but these arguments too are now losing effectiveness and methods of interpreting science without deference to experts have been developed (investigating conflicts of interest, better explaining the actual evidence and logic and evaluating how strong the evidence is or if it's just people paid to say-so).

I don't think we're in disagreement here. I clarify Marx on the forum when others bring up Marx, but I generally don't bring up Marx myself in talking about policy in the here and now. Just as I would clarify Kant or Anselm or Aristotle or any other thinker if someone tosses them in and I thought it's a debatable point.

I do, however, find it relevant the fear of Marx and the popular intellectual game of trying to rediscover Marx's points without ever crediting Marx and insisting that Marx was about something completely different and wrong ... by people that haven't read Marx.

The issue for me here is more to do with propaganda; sometimes, as you suggest, there's no need to bring up propagandized names and new names can be coined for the same concept to out-maneuver the propagandists (such as saying 1% instead of capitalist ruling class). Other times, I think it's useful to call propaganda's bluff and unpack propaganda's game and explain what the words meant to the people using them at the time, or that still are, as this can strengthen individuals as well as the community's critical thinking skills in being more aware of how propaganda works.

So I think we agree here that both points of view are legitimate.

That willingness to play is obviously a part of Marx's observations of class but it does not make all other observations along those lines "Marxist." — Valentinus

Yes, my goal is not to retroactively expand Marx's writing to include everything arguably that fixes problems or then is a compatible extension with it. As I mention above, I wouldn't bring Marx into any specific contemporary policy debate, unless someone else does inaccurately and it's an opportunity to expose their ignorance and so undermine all of their other claims too.

However, since this is a philosophy forum, as mentioned above, my view is Marx is a relevant thinker to understand like all the other important thinkers. It's useful to know when and where an idea originated and how it has developed since and what historical actors and movements, explicitly or implicitly, used or were influenced by the idea and what happened to those people and movements. Though name dropping long dead thinkers can seem like haughty erudition, my view is the opposite is more confusing as it leads people to believe that everything was thought of yesterday (a laziness that leads to an endless stream of rediscovery fever and praise).

The heads of states. Stalin, Mao, Castro, Pol Pot, etc... The individuals that really took power, for me, they only used Socialism as a mean to inhibit their egoism, because they didn't accepted who they really were. — Gus Lamarch

Which is why I asked you to clarify what you meant by socialism. Most people do not qualify socialist as referring to the the leaders of dictatorships, but you are free to do so if you make your meaning clear.

As for your statement here, if you read a biography of Stalin or Mao, for instance, they seemed fairly lucid and accepting of their tyrannical ambitions. Indeed, when Stalin got word from his agents in China (in charge of helping to organize the soviet sponsored communist revolutionaries there) that there was a sadistic and crazy Mao guy killing and raping and torturing (and really enjoying it) as well as ruthlessly getting rid of potential communist leader rivals (who doesn't even seem to believe in communism, just opportunism for power), and so these agents recommended needing to deal with or sideline Mao in some way, Stalin sent word back: That's my guy, put him in charge!

However, that sadistic tyrant use social crisis and movements to take power is a constant throughout history, there is no special relationship to socialism.

"Socialist revolutionaries" (at least as described by their opponents, whenever convenient) also brought the 40 hour work week, child labour laws, worker safety, and built the social welfare states in Europe that have the highest quality of life (Finland now the happiest place on earth).

More so, the socialist movement that brought Stalin to power was explicitly vanguardism (the idea that a revolutionary elite need to seize power ... and ideas are pretty vague from there), which was an offshoot of more mainstream socialism in central Europe that was more democratic and incremental rather than revolutionary (that whatever socialists want to achieve, it must be first through sharing ideas and organizing with normal people and second through creating or strengthening democratic processes and third through those democratic processes; and if this results in compromise, especially in any short of medium term outlook, then that's fine).

Vanguaridism was fairly fringe and just not that popular in central Europe, which is why Lenin and co. went to Russia during a chaotic civil society collapse (due to WWI) and tried the Vanguardism theory there (which, mind you, is not what the "Soviet" movement was about; the Soviets were local democratic units wanting more local rural self-management and at least some representation at the state level, and the first Russian revolution resulted in a compromise situation with the aristocracy, of having a democratic representational house as a check on a house of Lords (that controlled the military), but in the wake of disastrous military defeats due to aristocratic incompetence, the Bolsheviks (still fairly fringe) just seized the parliament buildings and managed to cut deals with enough police and military units to consolidate power; most people didn't know what was going on at this point, and keeping the Soviet name was a good marketing tactic to undercut potential opposition.

My point is, social democracy or democratic socialism is just that, socialist and democratic and has a very different history than the Soviet Union and the CPP and it's also pretty clear why vanguardism would and did immediately produce dystopian totalitarianism states, and socialist opponents of vanguardism made all those arguments at the time (what they called the delusion of capturing the state).

Now, you can decide not to call it a form of socialism if you want, it doesn't matter to me, but what I wish to draw your attention to is that there is a historical project of self-described socialists that resulted in very different conditions than the Soviet union and has developed lot's of policies that we now have the benefit of being able to simply check that they work empirically (universal health-care, free and equal funded education at all levels, robust public transport systems, rehabilitation based justice systems, paid vacation, paid maternity leave, homes for the homeless, ownership participation of key industries by the state, hyper strong labour union protection laws ... and collectivist defense programs such as conscription that provide a credible deterrent to invasion at a reasonable cost).