• My notes on the Definition of Mathematics.
    We may divide truths into two kinds: receptive and consequential.Zuhair

    I do not particularly appreciate the use of the term "truth" in the context of mathematics. The quite dominant Correspondence theory of truth defines "truth" as:

    In epistemology, the correspondence theory of truth states that the truth or falsity of a statement is determined only by how it relates to the world and whether it accurately describes (i.e., corresponds with) that world. Correspondence theories claim that true beliefs and true statements correspond to the actual state of affairs. This type of theory attempts to posit a relationship between thoughts or statements on one hand, and things or facts on the other.

    Mathematics never talks about the real, physical world, because claiming such a thing, is simply a constructivist heresy.

    Mathematics is exclusively and only about theorems that are provable from the construction logic of the abstract, Platonic world, i.e. the theory, generated by a particular arbitrary set of axioms.

    Since we do not have access to the axiomatic construction logic of the real, physical world, i.e. the theory of everything (ToE), mathematics cannot possibly ever be about the real, physical world. Ever.

    Therefore, the use of the term "truth" in mathematics, is always heretical, and I do not wish to participate in the propagation of that kind of horrible heresies.

    The fellowship of those consequences from the two input rules is a kind of TRUTH. This is a consequential fact.Zuhair

    These consequences are merely provable from these rules. They are not "true" in any possible fashion. Provability (PR) and correspondence-theory truth (CT) have absolutely nothing to do with each other.

    CT truth and logical truth (LT) have also nothing to do with each other. For example, in "var b = true" the variable b does not correspond to anything in the real, physical world, but we cannot deny that it is logically true, if only, because we defined it to be.

    Provability (PR) and logical truth (LT) have also nothing to do with each other. For example, in first his incompleteness theorem, Gödel encodes a statement that is logically true but not provable.

    A lot of wrong views and other errors are the result of confusing CT, LT, and PR.

    Here I'm stating a similar stance that is: Mathematics is the study of consequential truth.Zuhair

    Mathematics is fundamentally reductionist.

    In mathematics, reductionism can be interpreted as the philosophy that all mathematics can (or ought to) be based on a common foundation, which for modern mathematics is usually axiomatic set theory.

    Of course, I totally disagree with the idea that any choice of axioms would be better than any other choice. Therefore, I utterly reject the idea that set theory would be a better foundation than any other Turing-complete axiomatization. So, I must protest against this particular detail with the following formalist objection:

    Formalism holds that mathematical statements may be thought of as statements about the consequences of certain string manipulation rules. According to formalism, mathematical truths are not about numbers and sets and triangles and the like—in fact, they are not "about" anything at all. Formalism is thus silent on the question of which axiom systems ought to be studied, as none is more meaningful than another from a formalistic point of view.

    (Note: I also protest against the use of the term "mathematical truths" in the fragment above.)
  • Validity of the Social Contract
    The money available is best dispersed in as many hands as possible. To hoard the money is to rob the other members of it's possible benefits when better dispersed.creativesoul
    An indiscriminate approach of that nature would be a disaster, because it would soon put a halt to most sexual reproduction.

    If we oversimplify the process a bit, we can say that in the man-woman relationship, the man wants to acquire a reliable supply of sex, while the woman wants to lock down a reliable supply of money.

    If the State starts handing out money or other freebies for nothing to women, then the woman will no longer need the man. That will have an almost immediate effect on the birth rate, which will obviously collapse. Therefore, no, for reasons of keeping sexual reproduction afloat, it is absolutely not permissible whatsoever onto the government to hand out money or other freebies to women.

    Cura Annonae
    ("care for the grain supply")

    Adult male citizens (over 14 years of age) of Rome were entitled to buy at a below-market price five modii, about 33 kilograms (73 lb), of grain monthly. Approximately 40,000 adult males were eligible for the grain. In 62 and 58 BC the number of Romans eligible for grain was expanded and grain became free to its recipients. The numbers of those receiving free or subsidized grain expanded to an estimated 320,000 before being reduced to 150,000 by Julius Caesar and then set at 200,000 by Augustus Caesar, a number that remained more or less stable until near the end of the Western Roman Empire.


    As you can clearly see, the Romans were not that stupid ...
  • Predestination and Forgiveness
    Are you saying everything is not predetermined or that we as people will never be able to predict the future to extreme precision due to various limitations? I actually have no concern which is true. I just find the topic interesting. I'm naturally inclined to reject Scientific determinism and/or predestination.christian2017

    We may need to distinguish more rigorously between the observer and the observable.

    Immanuel Kant. Prolegomena, § 32. And we indeed, rightly considering objects of sense as mere appearances, confess thereby that they are based upon a thing in itself, though we know not this thing as it is in itself, but only know its appearances, viz., the way in which our senses are affected by this unknown something.

    Knowledge is about how the thing appears to us, and not how it really is, which we do not know, because the thing in itself is an unknown.

    For example, are the following 10 numbers random?

    06717 72296 11318 23781 09686 69034 01824 08055 87729 40845

    Well, this sequence was specifically constructed to appear to be random. It will undoubtedly pass all known tests for randomness; of course it is like that, because that is part of the construction sequence in producing these numbers.

    So, are these numbers random? Well, in some way they are.

    https://stattrek.com/statistics/random-number-generator.aspx

    Specs: This table of 10 random numbers was produced according to the following specifications: Numbers were randomly selected from within the range of 0 to 99999. Duplicate numbers were allowed. Table entries were selected using a seed value of 35000. This table was generated on 8/23/2019.


    If you know that the arbitrarily-chosen seed is 35000, you can use this pseudo-random number generator (PRNG) to generate exactly the same sequence again. Hence, to the person who knows the seed, these numbers are not random at all.

    Furthermore, there is absolutely no way to know if someone possesses the seed that allows him to flawlessly predict the 11th, 12th, and so on, numbers in this sequence. That is fundamentally part of the construction logic of this sequence.

    In the real, physical world, the same problem occurs. You cannot possibly know if a sequence of random events was generated by true randomness or just by another PRNG.

    So, concerning "Are you saying everything is not predetermined", which is the same question as, "Does a true random number generator exist?", I can only answer that the question is out of scope in mathematics.
  • Philosophy of software engineering?
    Do you know what PD is? Do you have an opinion on its truth or falsity, and knowledge of how it relates to other candidate axioms floating about?fishfry

    I have already said that this is exactly what does not matter. PD does not need to be true or false. PD rather needs to be independent from ZFC and not exactly the same as CH; which it obviously is not. If Woodin manages to prove CH from PD, while staying clear of ZFC, he will have managed to finish the job. I have just quoted Woodin saying that he has not (yet) managed to do that.
  • Philosophy of software engineering?
    You mis-stated CH in such a way as to give me the impression you haven't studied much set theory beyond the basics.fishfry
    About CH, I just said in my own words what you can find in Wikipedia too. Maybe it is a mis-statement. I don't particularly care, actually. CH is not "critically" important to me. I do not research it. It is not my job. I do other stuff. So, yes, just like Wikipedia (as an earlier remark), I am not much interested in going particularly much beyond what is provable already, and therefore, what are merely "the basics".

    My point is that neither of us is remotely qualified to discuss Woodin's work at all, unless you have set-theoretic knowledge far in excess of what you have demonstrated so far. Nor is his argument incomplete.fishfry

    Well, Woodin literally says that his argument is incomplete. (Read the original paper. I literally quoted what it said.)

    Furthermore, If it weren't incomplete, Woodin would simply provide us with the proof, instead of saying that his argument is incomplete.

    By the way, if you look at the problem, you can see that the only workable proof strategy is exactly about avoiding set-theoretic knowledge. It is a hard requirement to stay clear of ZFC, because if you don't, the proof will automatically be wrong. That is probably what makes the proof strategy so hard for people who are too deeply steeped in set-theoretic knowledge. They simply won't be able to do it.

    As I said, Woodin's strategy to prove CH from PD -- while staying clear of ZFC in any form or shape -- sounds really interesting, but he has also clearly said that he still hasn't managed to do it. I am just repeating Woodin's own words.
  • Philosophy of software engineering?
    hose technical terms are the meat and potatoes of 20th century set theory, especially of the last thirty or forty years. They're not "beyond existing theories."fishfry

    Sure.

    I just somehow hope that the nitty-gritty distinctions between "extremely large", "super huge", and "incredibly out-sized" cardinals won't be needed.

    Furthermore, I was more interested in the remark by/on Woodin, "why the Axiom of Projective Determinacy, PD, should be accepted". I do not think it matters "why". What matters is that If PD is provably independent from ZFC, and that Woodin also manages to prove CH from PD, because then Woodin will have managed to finish the job. The real problem is what he writes at "the argument is still incomplete ...". That is why I said that I am waiting for him to complete the argument. Unfortunately, we cannot do particularly much with just half the argument ...
  • Validity of the Social Contract
    Your opinions stated here are vulgar and absurd.Bitter Crank

    Much worse!

    ----------------------------------------------------------------------------------
    Exsurge Domine
    Condemning the Errors of AL CONTALI
    ----------------------------------------------------------------------------------

    Listen to our prayers, for foxes have arisen seeking to destroy the vineyard whose winepress you alone have trod. For we can scarcely express, from distress and grief of mind, what has reached our ears for some time by the report of reliable men and general rumor.

    Alas, we have even seen with our eyes and read the many diverse errors, which are either heretical, false, scandalous, and highly destructive of the vigor of ecclesiastical discipline, namely obedience.

    Al Contali's errors are pernicious, offensive to pious ears, seductive of simple minds, and originating with false exponents of the faith who in their proud curiosity yearn for the world’s glory.

    With mature deliberation on each and every one of Al Contali's theses, we condemn, reprobate, and reject completely each of these errors.

    We restrain all in the virtue of holy obedience and under the penalty of an automatic major excommunication.

    As far as Al Contali himself is concerned, O good God, what have we overlooked or not done? What fatherly charity have we omitted that we might call him back from such errors? But he always refused to listen and, despising the previous citation and each and every one of the above overtures, disdained to come. To the present day he has been contumacious. With a hardened spirit he has continued under censure over a year.

    Therefore we can, without any further citation or delay, proceed against him to his condemnation and damnation as one whose faith is notoriously suspect and in fact a true heretic with the full severity of each and all of the above penalties and censures.

    We enjoin on Al Contali that in the meantime he cease from all preaching or the office of preacher.

    On behalf of the Holy See
    Francis
    ----------------------------------------------------------------------------------
  • Validity of the Social Contract
    Bitcoin still only works by social agreement.unenlightened

    Yes, but it does not require any government.

    You see, I have no alternative for lowering violence and aggression within the perimeter of a state than having a government deal with that, while I certainly do not desire the presence of 2000+ Libyan-style militia each vying for power. Hence, unlike the anarchists, I am not advocating for having no government at all. I would not want to live in a place where there is no government at all.

    Still, all these governments compete with each other in terms of offering the best deal. I do not see why I would pick a worse deal, when I can get a better one. The government here nicely manages to keep things quiet and only charges me a few hundred dollars per year in visa fees. So, I am perfectly happy with my choice.
  • Validity of the Social Contract
    You only have savings by the grace of the social contract. Governments regulate promissory notes such that society can rely on them, and you can hold them and exchange them for goods and services. The concept of property is brought into being by the social contract, and without it you own nothing.unenlightened

    Not in my case. My savings are in bitcoin, which is not issued by any government. There is no need for any government to issue any currency. Furthermore, I do not wish to save in fiat currency, because that allows the issuing government to liberally debase it, by issuing some more. So, no, no.
  • Mathematics of the tractatus logico philosophicus
    The function itself cannot be input as then it does open up the for paradoxes and the circularity that Wittgenstein opposes.ssu

    To tell you the truth, I somehow suspect that I do not _really_ understand the objection voiced by Wittgenstein in 3.333. Is it related to Curry's paradox?

    In the 1930s, Curry's Paradox and the related Kleene–Rosser paradox played a major role in showing that formal logic systems based on self-recursive expressions are inconsistent. These include some versions of lambda calculus and combinatory logic.

    Is it about the use of the Y-combinator to allow anonymous functions to refer to themselves?

    The heart of Curry's paradox is that untyped lambda calculus is unsound as a deductive system, and the Y combinator demonstrates that by allowing an anonymous expression to represent zero, or even many values. This is inconsistent in mathematical logic.

    You can actually implement the Y-combinator in a run-off-the-mill scripting language such as Javascript:

    Y = f => (x => x(x))(x => f(y => x(x)(y)))
    

    So, in order to avoid self-reference for the purpose of recursion, you can use the Y-combinator instead:

    (f => (x => x(x))(x => f(y => x(x)(y))))(
     f => (n => ((n === 0) ? 1 : n * f(n - 1))))(5)
    //returns 120
    

    It is theoretically unsound in ways, but it practically really works. That is indeed a strange thing.

    Is that the problem that Wittgenstein has with self-reference?
    His objection in 3.333 does not properly explain what exactly his problem is ...
  • Alternatives to Being Against the State
    In short, incoherent social darwinism 101. If 'the strong' exploiting 'the weak' is akin to exercising one's natural freedom, the 'weak' exploiting the 'strong' is equally akin to exercising one's natural freedom. Oh, what the heck, after all the strong was the weak and the weak was the strong.hairy belly

    In principle, able-bodied males must do military service to protect the borders of the territory and lower its internal levels of random violence. If you do not want to perform military service, and the powers that be allow for that, you can pay compensation in lieu of military service.

    Therefore, poor able-bodied males can in principle receive daily stipends for spending a few hours per day at the gym, the shooting range, and the exercise field. In many countries, however, private security firms will offer more money than the government for guarding property for all kinds of private clients. Furthermore, the government generally prefers younger men while private security firms do not seem to mind age, as long as you are sufficiently fit.

    In my impression, there are no unemployed males with no income, especially here in SE Asia. They are all soaked up in private and public security-related work. I personally do not see why they should receive unemployment benefits instead of doing this kind of work. What's wrong anyway with going to the gym and the shooting range? They cannot just sit at home, can they?
  • Validity of the Social Contract
    You're ok with accepting some financial obligation - to maintain the society you belong to - as long as it is a clearly demarcated amount?creativesoul

    Well, I do not "belong" to a society. I am the property of our beloved Master, the Almighty Lord, Creator of the heavens and the earth.

    Furthermore, I currently do not live in my country of birth. I tend to travel around quite a bit, but mostly in SE Asia, like a lot of my friends and other digital-nomad colleagues.

    I actually do pay a clearly demarcated amount to the governments of the various countries I habitually reside in, the cheapest of which, Vietnam, wants around 300+ USD/year in visa fees, and the most expensive in the area, the Philippines, which wants around 700+ USD/year.

    In fact, there are quite a few countries that do not charge anything at all. For example, Mexico apparently gives you 180 days free of charge. Georgia (at the Black Sea) seems to give you 360 days free of charge. They are not even interested in charging extra.

    As a country, you need to be competitive because otherwise business and money will tend to move elsewhere. There are good reasons why factories have almost all ended up in China. That is not because they were charging more taxes and other government fees than others.

    Therefore, except for special niche situations, and given the competitive situation, I personally think that it may be hard for a country to charge more than 1000 USD/year for individuals merely residing there. Charging more to locals than to foreigners will also be considered unfair, even though many countries actually do that.

    A percentage of earnings?creativesoul
    Well, I generally live off my savings. Nowadays, I do not earn anything until I finally sell the startup I am involved in, including any saved-up cash that it may contain. The principle is simple. If I do not draw money/profits out of it, I personally do not earn anything. I just accumulate capital gains. Those are almost never taxable unless I "realize" them by selling.

    When I sell, it is often a taxable event, but in many countries it is not, and even when it is, it is taxable only if I do that in a country in which I have resided for more than 180 days of that particular year; something which is trivially easy to avoid.

    It is not possible to avoid the religious capital gains tax, though (2.5%).

    At the moment, I am not even involved in setting up a new startup. I just talk with potential partners about theoretical possibilities. I may not even activate anything any time soon, because I don't even need to. So, at the moment, there is certainly no need for me to pay any government-related levies beyond visa fees.
  • Validity of the Social Contract
    So, it seems that you do not think/believe that you are in any way at all - outside of scripture - financially obligated to the society you belong to?creativesoul

    Any personal liability is always to be limited to an explicitly-stated maximum burden.

    Unlimited liabilities must obviously be rejected. Otherwise, it will be again a case of Gambler's Ruin.

    In other words, in my opinion, it is not possible to govern a state while expecting the population to accept unlimited liability. In that case, the regime will need to be deposed and replaced by one that more successfully manages to shoehorn its financial needs into the strictly limited contribution of its population.
  • Validity of the Social Contract
    Are you suggesting to let the less fortunate and elderly people suffer needlessly?creativesoul

    As a matter of principle, I do not take responsibility beyond what the scriptures mandate.

    Are you suggesting that a nation can survive/thrive without some folk paying a larger percentage of their earnings than other?creativesoul

    All unilateral personal financial obligations to the wider society will have to be shoehorned into the 2.5% capital gains tax of the mandatory charity levy.

    Less fortunate and elderly people ought be helped. That's one thing. Do you agree?creativesoul

    You are talking about a "right". That is actually unimportant. What matters, is onto whom falls the obligation? Answer: certainly not on me.

    Everybody can have all the rights that he wants, but I am not liable for any corresponding obligation. I will simply hand over the mandatory 2.5% charity levy, and be done with it. Anything else is not my problem.

    Governing a nation of people has a cost. That's another. Do you agree?creativesoul

    If you want to be exempt from military service -- and only if this is allowed by the powers that be -- you will most likely need to pay compensation in lieu of military service.
  • Validity of the Social Contract
    If you want to say, there are tribes somewhere doing their hunting-gathering egalitarian thing, great.schopenhauer1

    Extensive egalitarianism works for relatively small communities. It does not scale, however. From some larger scale on, you need to switch to tit-for-tat trade. Otherwise, if you indiscriminately recognize everybody else's unilateral sharing rights on your assets, you will put yourself at risk of Gambler's Ruin.

    That is the meaning of the following Quranic verse:

    Quran: 2:275- 279 Allah has permitted trading and forbidden ‘Riba’ interest (usury)

    So, you are allowed to switch to tit-for-tat trade and engage in commerce, as long as you refrain from engaging in otherwise forbidden activities (such as charging unconstrained, exponentially-growing interest fees).

    This schema is complemented with a mandatory charity levy on net capital gains ("zakaat"), to somehow compensate for the fact that there will be people in the community who will not be equally successful at commerce. Of course, additional voluntary charity contributions are also welcomed ("sadaqah").

    This approach is deemed to scale pretty much indefinitely.
  • Mathematics of the tractatus logico philosophicus
    Wittgenstein is rather attacking the heuristic semantic notion of "self reference" in relation to the iterative evaluation of a sequence of expressions via recursive substitution. Unless the iteration eventually halts, the resulting sequence isn't even sentence, never mind a proposition. Yet if the iteration is halted, each resulting sub-expression has non-equivalent arguments.sime

    The requirement of having a base case, is a well-known problem:

    In mathematics and computer science, a class of objects or methods exhibits recursive behavior when it can be defined by two properties:

    1. A simple base case (or cases)—a terminating scenario that does not use recursion to produce an answer
    2. A set of rules that reduces all other cases toward the base case


    The way in which Wittgenstein formulated his objection does not allow for providing such base case. Therefore, I think that I have to reject his objection.
  • Validity of the Social Contract
    I depicted no such thing as unlimited liability.creativesoul

    Ok, but to what do you limit it then? For example, I have got nothing to do with trillions of dollars of unfunded social-security entitlements.
  • Validity of the Social Contract
    Those who benefit the most from society have the greatest burden of repayment(debt). Without a society, the accumulation of monetary financial wealth is impossible.creativesoul

    Well, according to the Quran, one's unilateral financial obligation to society is limited to the size of the mandatory charity levy, which is 2.5% of net capital gains. In my opinion, that sounds much better than the unlimited liability that you depict. So, I just pay the 2.5% and be done with it. I do not want to pay more, because there is no justification for doing that.
  • Predestination and Forgiveness
    In general i hate predestination and i've state previously or in the past why i feel at this point i must accept scientific determinism and predestination (ask me the reasons if you like).christian2017

    The concept of the Book of Life or the Preserved Tablet was inherited by all three offshoots of Second-Temple Judaism.

    An interesting mention is how it is viewed in the Rabbinic take:

    For this reason extra mention is made for the Book of Life during Amidah recitations during the Days of Awe, the ten days between Rosh Hashanah, the Jewish new year, and Yom Kippur, the day of atonement (the two High Holidays, particularly in the prayer Unetaneh Tokef).

    The Tablet indeed poses serious problems to the idea of free will. The matter has never really been settled. For example:

    There are only two groups who represent the extremes regarding Qadar [=predestination]. Al-Jabiriyah are of the opinion that humans have no control over their actions and everything is dictated by God. The other group is Al-Qadiriyyah and they are of the opinion of humans having complete control over their destiny, to the extent that God does not even know what humans will choose to do. The Sunni view is a synthesis of these two views, where they believe that God has knowledge of everything that will be, but that humans have freedom of choice.

    There is a similar concept in science, where the Preserved Tablet is referred to as the Theory of Everything (ToE):

    A theory of everything (TOE or ToE), final theory, ultimate theory, or master theory is a hypothetical single, all-encompassing, coherent theoretical framework of physics that fully explains and links together all physical aspects of the universe. Finding a TOE is one of the major unsolved problems in physics.

    Hawkings argued that we will never discover the ToE because of Gödel's incompleteness theorems:

    What is the relation between Godel’s theorem and whether we can formulate the theory of the universe in terms of a finite number of principles? One connection is obvious. According to the positivist philosophy of science, a physical theory is a mathematical model. So if there are mathematical results that can not be proved, there are physical problems that can not be predicted.

    Scientific determinism would require access to the ToE, i.e. the Preserved Tablet. This is deemed unattainable.
  • Validity of the Social Contract
    You are taking the libertarian approach here, of course. "I owe no one anything! I touch no one and no one touches me.Bitter Crank

    It is not necessarily libertarian.

    For example, the fifth Biblical commandment says: "Honor thy father and thy mother." So, you could accept that, and pay for your own parents, but it would still not require you to honour anybody else's father or mother. Hence, paying for other people's parents, is absolutely not a requirement there.

    Of course, in your scheme the baby boomers had no obligation to nurture, house, feed, and educate you either.Bitter Crank

    Yes, but it is not ALL baby boomers who nurtured me. Only my own parents did.

    They could have saved themselves a great deal of trouble by not conceiving you in the first place, or having the misfortune of giving birth to you, dashed your brains out on the nearest brick wall.Bitter Crank

    There is obviously a deal with one's own parents, in which they nurture one first, and later on, one nurtures one's own parents back.

    What the baby boomers are doing, is different from that. They are not claiming support from their own children. They are claiming support from other people's children. That is not included in the Biblical deal!

    And since you resent the favors done for you, you can start paying for all the products and services which you received before becoming a libertarian.Bitter Crank

    In the sense that I do not reject Biblical or Quranic obligations, I am not necessarily a libertarian. I just do not feel that further extending these obligations, actually makes sense. What the baby boomers are doing, is simply not supported by the Bible nor by the Quran.
  • Philosophy of software engineering?
    The Alephs aren't fields. Finite fields have absolutely nothing at all to do with this. They're apples and rutabagas.fishfry

    The cardinality of an infinite set ("aleph") is indeed formally not a Galois field. What I was saying, isn't a formal argument. I was just pointing out an uncanny similarity, i.e. just some kind of pattern:

    [1] arithmetic is allowed in finite fields of prime powers only. This rule creates gaps between the field sizes.

    At the same time, arithmetic is allowed in the integers (or rationals), i.e. a set with cardinality of aleph-0, and is also allowed in the reals (or computable numbers or similar), i.e. a set with cardinality of aleph-1.

    [2] Continuum hypothesis. There is an insurmountable gap between aleph-0 and aleph-1. There are no infinite cardinalities in between.

    As far as I am concerned, there is an uncanny similarity between the gaps in between finite calculation fields and the gaps between infinite calculation fields. Furthermore, with aleph[k]=2^aleph[k-1], the aleph[k] cardinalities are also prime powers; which is another uncanny similarity.

    Of course, formally proving the link between both phenomena, is something else altogether. That is obviously another exercise.

    One can Google around. A lot of work has been done by Woodin, Hamkins, and other contemporary set theorists. Work on CH has been ongoing for decades. It just doesn't make the mainstream news.fishfry

    Well, of course, people are still working on CH. I was just pointing out how "rigged" the game is. The context to work in, is really difficult. The proof that any axiomatization of CH is independent of ZFC, makes doing something meaningful, really hard. The structure required is incredibly annoying. You need other axioms, but which must still have a real provability distance from CH, because otherwise you would just be axiomatizing the final result itself ... Well ...

    One big new idea is Woodin's Ultimate-L. It's so new and so technical it doesn't have a Wikipedia entry. This MathOverflow thread has some references. Nothing in this topic is comprehensible to laymen, just mentioning it since it's the state of the CH art.fishfry

    Apparently, the current status is: How Woodin changed his mind: new thoughts on the Continuum Hypothesis.

    This paper illustrates Woodin’s solutions to the problem, starting in Sect. 3 with his 1999–2004 argument that Cantor’s hypothesis about the continuum was incorrect. From 2010 onwards, Woodin presents a very different argument, an argument that Cantor’s hypothesis is in fact true.

    Well, yeah. My own intuition is in favour of CH. So, I obviously like Woodin's new approach better.

    This argument is still incomplete, but according to Woodin, some of the philosophical issues surrounding the Continuum Problem have been reduced to precise mathematical questions, questions that are, unlike Cantor’s hypothesis, solvable from our current theory of sets.

    That explains why there is no Wikipedia entry for Woodin's work yet. They are not supposed to report on proofs that still have gaps in them. A proof with existing remaining gaps is a research hypothesis and not something to be propagated as "finished work" in places like Wikipedia.

    If you look at the subtleties he needs in order to make his point:

    [1] A Reinhardt cardinal, [2] An n-huge cardinal, [3] A huge cardinal, [4] An extendible cardinal, [5] A supercompact cardinal, [6] A superstrong cardinal, [7] A Woodin cardinal, [8] A measurable cardinal, [9] A (strongly) inaccessible cardinal.

    The complexity proposed is substantially beyond what is done in existing, established theories. Therefore, what he is doing, looks quite ... ambitious.

    Then, he writes:

    As mentioned in the article, too much would have to be said to give an argument as to why the Axiom of Projective Determinacy, PD, should be accepted.

    There, he is hitting the core of the problem. In fact, there should be no argument as to why axiom PD is accepted. If he does that, it is not an axiom. So no, all he needs to do, is to prove that there is a real distance, a serious series of transformative derivation steps from PD to CH.

    There is no such thing as the quality of an axiom, because that implies "justification" of an axiom, which is exactly what is not allowed. All we need to see, is that CH is not trivial from PD. That would be enough.

    I think that Woodin's work is interesting, but before further digging into the nitty-gritty details of the pyramid of vocabulary on which his work rests, I will just let him first complete the gaps in his argument! ;-)

    Another idea is Hamkins's set-theoretic multiverse.
    http://jdh.hamkins.org/themultiverse/
    fishfry

    The multiverse view in set theory, introduced and argued for in this article, is the view that there are many distinct concepts of set, each instantiated in a corresponding set-theoretic universe. The universe view, in contrast, asserts that there is an absolute background set concept, with a corresponding absolute set-theoretic universe in which every set-theoretic question has a definite answer.

    In my opinion, Hamkins' work sounds less promising than Woodin's work, with which I can somehow identify. Hamkins does not talk about juxtaposing a set of axioms provably independent from ZFC from which there exists a noticeable derivation distance from CH, but from which he successfully derives CH. Therefore, I cannot identify with Hamkins work.

    Here's an accessible article that surveys the modern developments.
    https://www.ias.edu/ideas/2011/kennedy-continuum-hypothesis
    fishfry

    He writes:

    How ironic then that the continuum hypothesis is unsolvable—indeed, “provably unsolvable,” as we say.

    Well, CH is not unsolvable. It is just unsolvable from ZFC. That does not mean that it would be unsolvable from anything else. For example, it is trivially solvable from itself. However, that is not what we want. We want some larger distance between CH and the X someone would solve it from. This X needs to be independent from ZFC, and that is a problem, because ZFC is extremely powerful. So, finding such X is not easy.

    Gödel, however, [...] taking the view that his incompleteness theorems, though they show that some provably undecidable statements do exist, have nothing to do with whether the continuum hypothesis is solvable or not.

    Yes, of course!

    what would it be like to go beyond set-theoretic methods and suggest something new? Still, this is exactly what is needed to solve the continuum hypothesis.

    Yes, X needs to be independent from ZFC (not easy to achieve) and still have a real distance from CH. Next, CH needs to be proven from X. That is what needs to happen, but as I argued before, that is clearly not easy. Furthermore, it is also a very counter-intuitive procedure to follow.
  • Validity of the Social Contract
    Validity of the Social ContractPantagruel

    I seriously object to the use of the term "contract" in this context. The act of merely being born somewhere does not amount to signing a contract. Therefore, as far as I am concerned, there simply is no "contract".

    For example, it is not because you are born today in a country with trillions of dollars of unfunded social-security entitlements and other liabilities, that you necessarily agree to cough up any money for that.

    As far as I am concerned, the millennials have no obligation whatsoever to pay for the retirement benefits of the baby boomers. Even though the baby boomers may have signed up for a contract that promised exactly that, the millennials themselves were no party to the original contract.

    We are clearly witnessing a generation, the baby boomers, who did not need any children of their own, to take care of them in their old age, because the government was going to take care of that. And where is the government supposed to find the funds to pay for that? Well, they will just take the money from other people's children.

    So, no, there is no such "social contract" and there are no resulting social obligations.

    As far as I am concerned, it is just too easy to sign a contract that will financially burden other people who are not even born yet. Therefore, the millennials are completely exempt from paying for any of that. Just don't pay!
  • Multiculturalism and Religious Fundamentalism
    Further hijabs don't just symbolize something that conflicts with society's values, they are something that does. Or potentially do, depending on the society's values.Coben

    The real problem is the eternally faulty epistemic stance of the populace of retarded imbeciles in the West. They do not believe something because of its justification -- they are way too stupid to verify the justification anyway -- but because of whom says it, e.g. the blue-pilled narrative of the manipulative mainstream media and official, state-controlled education/indoctrination system.

    Of course, the Muslim population is not much under the spell of the same blue-pilled narrative that clouds the dumb skull of the western populace. Still, the manipulators would strongly desire to manipulate the Muslims too. So, they need to attack the alternative beliefs that prevents the Muslims from buying into the manipulative lies that the idiots in the West so strongly believe.

    The manipulations are very strong and very successful at fooling these idiots.

    In less than a hundred years, they have successfully managed to reverse what is normal with what is perverse. That what used to be normal, is now perverse in the West, and that what used to be perverse, is now normal.

    A hundred years ago, marrying a 14-year old wife was normal, while having sex outside marriage was perverse. Today, it is exactly the other way around. The western populace of retards and other imbeciles even strongly believe in this inversion of perversion. They want it! They cannot imagine the reverse -- which is "normality" -- because they are too stupid for that and too deeply infected with perversity already.

    Of course, perversity does not last. Normality will one day or the other reassert itself. It always does.

    On that day, the perverted retards and other imbeciles in the West will simply run up to the 12th floor of the building they are in, and jump out of the window, because deeply invested as they are in their perversity, there is no way that they would still be able to handle normality.
  • Structural Antisemitism
    I can't see the Israelis accepting Hamas's rule anytime soon. I also don't know how much positive change that that would really effect.thewonder

    It is not really "Hamas rule" that would make sense. It is rather: using Hamas to prevent the emergence of 2000+ Libyan-style militias. Furthermore, the Ottoman millet system means it will be Hallakah rule for the Jews.

    It is also better to raise the issue of military service with Hamas. Either the Jews do military service, or else they pay compensation for not doing so. Traditionally, they pay compensation, but I somehow suspect that it is negotiable. By the way, Napoleon suddenly reversed the traditional solution in Europe, but I am not sure that the Jews particularly liked it:

    Napoleon's Infamous Decree. The decree made it so that the Jewish conscripts (required enlistees of military) couldn’t find replacements for themselves when drafted like other Frenchmen were allowed to do.
  • Structural Antisemitism
    How will Hamas guarantee law and order and who will possibly agree to the restoration of Ottoman millets?thewonder

    Hamas will inevitably agree to the restoration of the Ottoman millets. They have no choice in that matter, because it is the Quran which mandates the Ottoman millet system.

    I see what you are saying, but I don't think that that makes the misconception that Jews have a reach of power that could be likened to something like the New World Order not anti-Semitic.thewonder

    This "new World Order" ideology exaggerates the historical ability of the Jews to survive Medieval European antisemitism. "Throwing antibiotics at the bacteria" will merely create resistant strains. That is obviously what historically happened. I do not believe that it is more than that. It is mostly the Catholic Church who created these surviving strains.

    So, I personally believe that the solution to the conundrum in mandatory Palestine, consists in conducting negotiations which will bring the ANC/Hamas to power, and which will reinstate the Ottoman millet system.
  • Structural Antisemitism
    I see the point you are making about the survivor bias, but I am unsure as to how you came to the conclusion that you did. I don't think that "antifragility" results that in victims of oppression are given superhuman powers, I guess.thewonder

    The seemingly "superhuman powers" are just part of the survivor bias. The weak who got weeded out have zero visibility. That is what gives the impression of "superhuman powers".
  • Structural Antisemitism
    I think that the Israelis should abide by some sort of set of terms and conditions which are more or less outlined in the two-state solution, that there should eventually just be one state,thewonder

    In my opinion, the representatives of the apartheid State of Israhell should be remanded to sit at the negotiation table with Hamas, to discuss the only real point of contention: the dissolution of the apartheid State of Israhell.

    So, we need a repeat of the negotiations between Frederik De Clerq and Nelson Mandela, in which they discussed the inevitable takeover by the ANC. Seriously, I do not see any other solution.

    Furthermore, the former British mandate of Palestine will simply have to reinstate the erstwhile Ottoman millets:

    In the Ottoman Empire, a millet /ˈmɪlɪt/ was an independent court of law pertaining to "personal law" under which a confessional community (a group abiding by the laws of Muslim Sharia, Christian Canon law, or Jewish Halakha) was allowed to rule itself under its own laws.

    The continuation of the millet regime was even a condition in the 1917 Balfour declaration on the return of the Jews to Ottoman Palestine:

    ... it being clearly understood that nothing shall be done which may prejudice the civil and religious rights of existing non-Jewish communities in Palestine ...

    ideally Israelis and Palestinians would just form an Anarchist commune and abandon the idea of the State altogetherthewonder

    Wow, no, no. That would only lead to a Libya-style system of 2000+ different militia battling for power. So, no, certainly no anarchy. It will lead to the collapse in trade and commerce, and the resulting levels in specialization, which in turn, will destroy existing levels of income.

    Law and order needs to be guaranteed.

    Therefore, the successor state to the apartheid State of Israhell will simply have to reintroduce the Ottoman millets, while Hamas will have to guarantee law and order.
  • Structural Antisemitism
    The way in which antisemitism is distinguished, and should be distinguished, from racism, has to do with the sort of imaginary of power, attributed to the Jews, Zionism, and Israel, which is at the heart of antisemitismthewonder

    You know, when you ban the Jews from lots of professions and jobs, as they used to do before the French Revolution, it will make a large number of Jews fail to sustain themselves, while some will succeed anyway.

    Another aspect of medieval antisemitism was the many restrictions imposed on the Jews. They were excluded from many occupations because of the fear of competition with the local population. For the most part they could not own land, since, under the feudal system, the pledge of loyalty required from a vassal upon the enfoeffment of land had the form of a Christian oath; however, there were exceptions. Their residence in cities was often limited to specific areas known as ghettos. Following the Fourth Lateran Council, in 1215, Jews were also ordered to wear distinctive clothing, in some instances a circular badge. Some Jews managed to evade the humiliating requirement of wearing a badge by bribing the local authorities.

    Nobody cares about the ones who failed to sustain themselves amidst these restrictions. The losers are never very visible. That is obviously not true of the winners. They will give the impression of possessing some secret that makes them survive and even thrive against all odds.

    That is just a very typical example of survivor bias.

    Survivorship bias or survival bias is the logical error of concentrating on the people or things that made it past some selection process and overlooking those that did not, typically because of their lack of visibility. This can lead to false conclusions in several different ways. It is a form of selection bias.

    When Friedrich Nietzsche wrote, "That what does not kill you, will make you stronger", he counted on what Nassim Taleb would later on coin the "antifragility" of humanity. An individual really needs adversity to grow. Even your muscles need training to become stronger. That would be zeroth-order antifragility, which is actually still relatively weak compared to first-order antifragility, described in the survivor bias, which is the truly strong phenomenon, because it proceeds by ruthlessly weeding out the weak. It is the same process as in which bacteria become antibiotic-resistant.

    So, no, "the sort of imaginary of power, attributed to the Jews" is not completely imaginary, because in the Middle Ages, they really had their antifragility more extensively stimulated than the rest of the population.
  • Mathematics of the tractatus logico philosophicus
    He was trying to show that it was a problem of semantics and I think this was a little of what wittgenstein was getting at, but it is hard to defend his viewpoint.Wittgenstein

    Ludwig Wittgenstein was apparently one of Bertrand Russell's favourite students:

    His teacher, Bertrand Russell, described Wittgenstein as "perhaps the most perfect example I have ever known of genius as traditionally conceived; passionate, profound, intense, and dominating".

    In his Principia Mathematica, Russell had done some really important work, especially with his ramified type theory. It is the overly favourable opinion of a real grandee such as Bertrand Russell on Wittgenstein that is so misleading. Seriously, I really do not see what exactly would be so inspiring about Wittgenstein's own work.

    5.153 In itself, a proposition is neither probable nor improbable. Either an event occurs or it does not: there is no middle way

    Yes, of course, but that is not what it is about. The observer does matter. Probability is about the relationship between observer and event, and not about the event "an sich". As Immanuel Kant famously wrote, "Das Ding an sich ist ein Unbekänntes".

    Immanuel Kant. Prolegomena, § 32. And we indeed, rightly considering objects of sense as mere appearances, confess thereby that they are based upon a thing in itself, though we know not this thing as it is in itself, but only know its appearances, viz., the way in which our senses are affected by this unknown something.

    Again, Ludwig Wittgenstein, who in my opinion is very overrated, was clearly late in the game to start fretting over this problem.
  • Structural Antisemitism
    Rabbinic Judaism is first and foremost a religion, i.e. one of the three offshoots that survived the destruction of second-temple Judaism by the Roman Empire. The reason why it is not exactly the same as second-temple Judaism, is understandable. That is why the two other offshoots are neither. I personally do not believe that antisemitism is inspired by a facile attack on the Rabbinic take, that would somehow emerge from the one or the other simplistic view on comparative theology. The real problem is the apartheid State of Israhell.
  • Mathematics of the tractatus logico philosophicus
    You will get circularity otherwise.ssu

    The definition:

    n>1: n! = n * (n-1)!
    n=1: n! = 1

    is indeed somehow circular, but that is the essence of recursion. It works absolutely fine. Wittgenstein does not seem to handle that.
  • Multiculturalism and Religious Fundamentalism
    Whether citizens should be allowed to dress a certain way is an internal affair of state, whereas the situation you depicted concerns international affairs.Tzeentch

    In terms of "skin in the game", it is the same problem.

    There are people who clamour for arbitrary enforcement action against other people, but who do not intend to put skin in the game. You see, Alexander the Great had real authority, because he would always personally lead the cavalry charge. If he had refused to take personal risk, his views would have been dismissed as "too easy".

    There is no such thing as a free lunch. Nassim Taleb's "Skin in the game" even argues that there should be no such thing as a free lunch. People with no skin in the game, should never have a say; or else, they should be forced to put skin in the game.

    It all shows the necessity to retaliate against civilian populations, who must never be allowed to believe that there is something like a free lunch.
  • Mathematics of the tractatus logico philosophicus
    For let us suppose that the function F(fx) could be its own argument: in that case there would be a proposition 'F(F(fx))'

    F(F(x)) is allowed only if the co-domain is equal to or a subset of the domain of F(x). Beyond that, I don't see what the problem is with the repeated application of functions. There is nothing inconsistent in the practice of function iteration, i.e. a Picard sequence. Furthermore, the remark puts into question the entire field of studying fixed points.

    in that case there would be a proposition 'F(F(fx))', in which the outer function F and the inner function F must have different meanings, since the inner one has the form O(f(x)) and the outer one has the form Y(O(fx)).

    Wittgenstein sees a problem where there isn't one ...
  • Multiculturalism and Religious Fundamentalism
    In my view, it is a society's right to uphold those values that are fundamental to it. If a certain style of dress symbolizes something that conflicts with a society's values, I consequently see no issue in forbidding it.Tzeentch
    There is something very important missing in this equation, namely the requirement to put skin in the game.

    That is why Roosevelt's decision to instruct Eisenhower to carpet bomb the nazi civilian population was so important. It was just too easy for these nazi civilian populations to clamour for "upholding nazi values" without them risking their own life at the Russian front. The carpet bombing of nazi civilian populations rectified that problem by forcing them to put skin in the game.

    It is certainly possible to enforce any kind of "society value" but the idea that someone else will have to risk their lives and die for what you believe in, is simply not acceptable. If you refuse to put skin in the game, then you must be forced, kicking and screaming, to do so.
  • Is it possible to make money with Philosophy?
    Is it possible to earn money (enough to live, not a lots of money) being philosopher? Is teaching in class the only way to make money with Philosophy?John Pingo

    Epistemology (/ɪˌpɪstɪˈmɒlədʒi/ (About this soundlisten); from Greek ἐπιστήμη, epistēmē, meaning 'knowledge', and -logy) is the branch of philosophy concerned with the theory of knowledge.

    Epistemology is how we know. Methodology is the combination of epistemic stance and the methods of investigation.

    Best Practices Consulting

    To help management define what improvements can be established to effectuate necessary changes to meet best practices, drive innovation, and continually improve processes and performance.

    There is a multi-billion market for (methodology) "consulting". At the same time, you cannot do methodology without "epistemic stance", while epistemology, i.e. the theory of knowledge, is simply a subdivision of philosophy. Philosophy is both an upstream as well as a byproduct of gaining experience in any field of endeavour. Best-practices consulting is typically one way of making a living off that type of experience.
  • Multiculturalism and Religious Fundamentalism
    Western views are everywhere around the world. They're the standard according to which the upper classes of many non-western countries mold themselves. I think it's a little absurd to claim that western views have failed.Echarmion

    Matthew 5:5. Blessed are the meek: for they shall inherit the earth.

    We do not descend from the upper classes of the past.

    Why Millennials Refuse to Get Married. Marriage Rates Are Plummeting. Fertility Rates Keep Dropping, and it’s Going to Hit the Economy Hard.

    Furthermore, the children of these upper classes will generally not be particularly upper-class any longer. It is the ones who keep up marriage and fertility rates who will ultimately prevail.
  • Multiculturalism and Religious Fundamentalism
    The hijab is an instrument of repression. It's not something else.thewonder
    To what degree are such cultural traditions imposed?thewonder

    Genesis 3:16. To the woman he said, “I will make your pains in childbearing very severe; with painful labor you will give birth to children. Your desire will be for your husband, and he will rule over you.”

    A woman who does not want her husband to "rule over" her, will most likely not have one. I suspect that there is no other solution.

    I also think that a person should respect that people in the region have a myriad of different worldviews which drastically differ from those that we have here.thewonder

    United States and Taliban negotiators have wrapped up their sixth round of peace talks with "some progress" made on a draft agreement for when foreign troops might withdraw from Afghanistan, a spokesperson for the armed group has said.

    Since that small combat force, i.e. the Taliban, has now brought the USA to their knees, any hope of imposing western views on gender, upon other cultures/religions, had better be abandoned. These western views have simply failed the test of violent combat. The war is over.
  • Philosophy of software engineering?
    What is interesting is how closely tied mainstream software development practices are to capitalism.darthbarracuda

    But there are other ways of developing software that are less tied to economics - open source software is a great example.darthbarracuda

    The first problem is, of course: What is the definition of "capitalism"? It is often confused with (free) trade, commerce, and the resulting trend to specialization:

    Modern capitalist societies—marked by a universalization of money-based social relations, a consistently large and system-wide class of workers who must work for wages, and a capitalist class which owns the means of production—developed in Western Europe in a process that led to the Industrial Revolution.

    So, modern capitalism may revolve around the existence of a capitalist class which owns the means of production.

    So, is this the case in software? What are the means of production and who owns them?

    To produce software, you need relatively inexpensive computer equipment, owned by the public at large. So, the hardware part is not capitalistic.

    You also need other software such as operating systems, compilers, basic libraries, and so on. You do have two large, proprietary owners of their version of basic software, i.e. IBM and Microsoft, but nowadays most software is built using free software, especially if you consider that most software runs on mobile phones, and that Android is (or is based on) free software.

    Hence, even though there are noticeable elements of capitalism that operate in the realm of software, i.e. a few large capitalist players, they do not generally own the means of production. In other words, there is no (dominant) capitalist class which owns the means of production in software.
  • Philosophy of software engineering?
    The Beth numbers are the Alephs in the presence of the continuum hypothesis; they're distinct in its absence.fishfry

    Finite fields (in which arithmetic is permitted) must have a prime-power size. Therefore, there tend to be gaps between permissible calculation field sizes. E.g. a size of 13 is allowed, but 14,15 are not; 16=2⁴ is a prime power and is again allowed, and so on. So, in a way, we could wonder why it would be any different -- the fact that there are gaps -- in between infinite field sizes in which arithmetic is permitted? Of course, this kind of pattern is not a proof, but it seems to point in a particular direction.

    But then again, since CH is provably not provable/disprovable from ZFC, proving CH from any alternative set of axioms, that is not just itself, may not be particularly "simple". It would require a meaningful distance, i.e. a meaningful number of non-trivial derivation steps from these axioms to CH, while these axioms would also have to be provably independent of ZFC. I do not see how else anybody could make progress in CH?
  • Philosophy of software engineering?
    unlike a significant proportion of set-theorists, who as a result of refusing to get their hands dirty in practical application, end up associating mathematical infinity with the religious idea of eternity.sime

    Any link to such association?

    According to Cantor's theory, the infinities (Beth numbers) (or Aleph numbers) are a series of successive numbers, with the smallest one being countable infinite, the next one uncountable infinite, and each successive infinity, inf[n] = 2^inf[n-1], the cardinality of the power set of the previous one. Such sequence of infinities is not particularly compatible with the idea of one God, which would correspond to one infinity.

    According to Cantor's diagonal argument, it is absolutely excluded that countable infinite would be the same number/cardinality as uncountable infinite.