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What is the difference between actual infinity and potential infinity?These issues are thoroughly discussed in a nice paper by Barry Mazur, "When is one thing equal to some other thing?"
http://www.math.harvard.edu/~mazur/preprints/when_is_one.pdf — fishfry
The fact that the algebraic closures are not yoked together by a specified isomorphism is the source of some theoretical complications at times, while the fact that their automorphism groups are seen to be isomorphic via a cleanly specified isomorphism is the source of great theoretical clarity, and some profound number theory.
Yeah, the fact that automorphism groups are always isomorphic is a key ingredient in Galois theory. That is undoubtedly the "profound number theory" that he is referring to. ;-)
The Peano axiom approach calls up the full propositional apparatus of mathematics. But the details of the apparatus are kept in the shadows : you are required to “bring your own” propositional vocabulary if you wish to even begin to flesh out those axioms. The Peano category approach keeps all this in the dark: no mention whatsoever is made of propositional language.
Staying clear of the language of first-order logic may temporarily spare you from hitting the wall of Gödel's incompleteness theorems. It is the use of "∀" and "∃" that ransacks everything. Still, I do not see how he will manage to keep avoiding the use of language, and especially, existential quantifiers. Up till now, his Peano category approach has managed to somehow avoid their explicit use, but I am not sure that it also manages to avoid their implicit use.
The Peano axiom approach requires — at least explicitly — hardly any investment in some specific brand of set theory. At most one set is on the scene, the set of natural numbers itself. In contrast, the Peano category approach forces you to “bring your own set theory” to make sense of it.
Declaring the set theory in use to be some kind of free variable, or at least a template placeholder, is indeed interesting. However, how will he manage to not accidentally bring a particular set theory through the back door? All you need to do, is to accidentally rely on a theorem that rests on a particular set theory, in order to become beholden to it.
When we gauge the differences in various mathematical viewpoints, it is a good thing to contrast them not only by what equipment these viewpoints ultimately invoke to establish their stance, for ultimately they may very well require exactly the same things.
Ha ah! Exactly what I thought!
Representing one theory in another. If categories package entire mathematical theories, it is natural to imagine that we might find the shadow of one mathematical theory (as packaged by a category C) in another mathematical theory (as packaged by a category D). We might do this by establishing a “mapping”. We call such a “mapping” a functor from C to D.
Yes, I need a functor right now, between grammar classes of formal languages (which are always axiomatic theories). The PCRE regular language engine has custom extensions that allow it to express the grammar of context-free languages (EBNF) and match their sentences. So, now I want a functor between PCRE and EBNF; which are widely claimed to be isomorphic. So, does he know something about functors that would drastically simplify the job of producing such PCRE<-->EBNF functor? Otherwise, it may be a lot of work ... too much for me, in any case ...
Let X, X′ be objects in a category C. Suppose we are given an isomorphism of their associated functors η:FX∼=FX′. Then there is a unique isomorphism of the objects themselves,
Interesting and intriguing. Unfortunately, he does not mention the proof, even though he says it is an easy proof.
An object X of a category C is determined (always only up to canonical isomorphism, the recurrent theme of this article!) by the network of relationships that the object X has with all the other objects in C.
And you usually do not even need the object's relationship to ALL other objects. A few is usually enough to know what the object must be.
A functor F: C−−→D from the category C to D is called an equivalence of categories if there is a functor going the other way, G:D−−→C such that G·F is isomorphic to the identity functor from C to C, and F·G is isomorphic to the identity functor from D to D.
If anything that you can express in ZFC, can be expressed in combinatory logic, and the other way around, then there would be a equivalence functor between both categories. Then, this equivalence functor is also an algorithm, i.e. some kind of function that accept set-theoretical expressions and translates them in combinatory-logic ones. Has anybody ever implemented anything like that?
Is 5 mod 691 to be thought of as a symbol,or a stand-infor any number that has remainder 5 when divided by 691,or should we take the tack that it(i.e.,“5 mod 691”)is the (equivalence) class of all integers that are congruent to 5 mod 691?
Well, in my own experience, "5 mod 691" is just "5" in a system that happens to have as system parameter maxint=690. We do not really care about the system parameter particularly much, because everything we do, stays inside that system anyway. In my opinion, the choice of 691 would only matter when you simultaneously deal with multiple systems that could each have different parameters. Still, I have never run into that practical situation. Another reason why it does not matter, is because this system parameter will usually be relatively large. However, it will not be too large either, because the fact that numbers wrap around that maximum boundary has a desired obfuscating effect. It nicely ransacks monotonicity. So, 232+541 = 82 (within mod 691). So, you can perfectly add up two large numbers and get a smaller one. That is not a very strongly obfuscating effect, but it still helps in cryptography.
This newer vocabulary has phrases like canonical isomorphism,“unique up to unique isomorphism”, functor, equivalence of category and has something to say about every part of mathematics, including the definition of the natural numbers.
I also believe that category theory, i.e. general abstract nonsense, is the true flagship of mathematics. Unfortunately, its theorems do not (yet) have direct applications (such as in cryptography), that I know of.
The categorical vocabulary itself, however, seems to be spreading like wildfire. -
What is the difference between actual infinity and potential infinity?LOL. Impressive Wiki skills. No bearing on the topic at hand. What can I say? — fishfry
It is almost literally what you will find mentioned in the page on the "Brouwer-Hilbert controversy":
In other words: the role of innate feelings and tendencies (intuition) and observational experience (empiricism) in the choice of axioms will be removed except in the global sense – the "construction" had better work when put to the test: "only the theoretical system as a whole ... can be confronted with experience".
So, what happens with "the theoretical system as a whole" ? Either it finds downstream users, or else it doesn't. In that case, what can we say about the downstream use of ZFC? Well, it is a legacy system with an enormous installed base that has been around for almost a century. Does it matter? Well, according to the formalist philosophy, that is all that matters. The status of individual axioms is simply irrelevant.
Concerning "no bearing on the topic at hand", you undoubtedly say that, because you are not aware of that famous discussion between Hilbert and Weyl in 1927, which was exactly about this. Could that have something to do with "weaker" Wiki skills? ;-) -
What is the difference between actual infinity and potential infinity?though they are "the same set" according to the deficient standard of ZFC set theory — Metaphysician Undercover
ZFC was initiated by Cantor and Dedekind in the 1870s, followed by Zermelo's draft 1908 publication, followed by Fränckel's bug fixes in 1921. From day number one, there has been forceful criticism on its choice of axioms, and there still is, with lots of people proposing alternatives. Still, ZFC's dominance has only kept growing.
Whatever happens, it will be really hard to replace ZFC by any alternative, because so many theorems now rest on it. ZFC has an enormous "installed base":
Installed base (also install base, install[ed] user base or just user base) is a measure of the number of units of a product or service that are actually in use, especially software or an Internet or computing platform,[1] as opposed to market share, which only reflects sales over a particular period. Although the install base number is often created using the number of units that have been sold within a particular period, it is not necessarily restricted to just systems, as it can also be products in general. For products which are in use on some machines for many years, the installed base count will be higher than sales over a given period. Some people see it as a more reliable indicator of a platform's usage rate.
ZFC is actually also a gigantic legacy system, without necessarily being outdated, though:
In computing, a legacy system is an old method, technology, computer system, or application program, "of, relating to, or being a previous or outdated computer system,"[1] yet still in use. Often referencing a system as "legacy" means that it paved the way for the standards that would follow it. This can also imply that the system is out of date or in need of replacement.
Having broken some teeth in the past by criticizing legacy systems with a large installed base while advocating their replacement, I now instinctively refrain from doing that, because the argument will most likely fail again. One reason is the Lindy effect:
The Lindy effect is a theory that the future life expectancy of some non-perishable things like a technology or an idea is proportional to their current age, so that every additional period of survival implies a longer remaining life expectancy. Where the Lindy effect applies, mortality rate decreases with time.
Bourbaki is also known to have strongly promoted ZFC.
Nicolas Bourbaki (French pronunciation: [nikɔla buʁbaki]) is the collective pseudonym of a group of (mainly French) mathematicians. Their aim is to reformulate mathematics on an extremely abstract and formal but self-contained basis in a series of books beginning in 1935. With the goal of grounding all of mathematics on set theory, the group strives for rigour and generality. Their work led to the discovery of several concepts and terminologies still used, and influenced modern branches of mathematics.
While there is no one person named Nicolas Bourbaki, the Bourbaki group, officially known as the "Association des collaborateurs de Nicolas Bourbaki" (Association of Collaborators of Nicolas Bourbaki), has an office at the École Normale Supérieure in Paris.
By the way, the "École Normale Supérieure" in Paris is also the school where Evariste Galois studied while working on his Galois Theory in his early twenties. -
What is the difference between actual infinity and potential infinity?That two things are equal does not mean that they are the same. This is a known deficiency of mathematics, equality cannot replicate identity. Anyone who argues that 2+2 is the same as 4 needs to learn the law of identity, and respect the difference between equality and identity. — Metaphysician Undercover
Well, the one expression S1 consists of literals while the other expression S2 is a comprehension formula. So, they are indeed not identical but extensional, according to ZFC's axiom of extentionality.
In the axiom of extensionality, the "=" symbol has axiomatically been assigned to express extensionality. Hence, the conclusion that S1=S2 is accordance with the ZFC axiom.
The sentence "they both describe the same set and therefore they are extensional" is therefore in accordance with the axiomatic foundation of ZFC set theory.
The use of the "=" symbol for expressing extensionality can be confusing. In any programming language that I have ever run into, the expression "S1 = S2" (or usually "S1 == S2" ) will only compare the two data structures' memory storage addresses. If they happen to be stored in different locations, even if they contain the same elements, by default, they will be considered different. Potentially comparing each element would cost computing power, and that would not be desirable as a default interpretation, when consuming resources matters (like in computing but unlike in mathematics). -
On AntinatalismDo you really think that we have any good reason to believe that we will find solutions to the massive, rapidly intensifying convergence of problems we currently face? Take global warming; no one seems to have any idea how we could stop using fossil fuels (which is arguably what would be needed to avoid catastrophic warming) without collapsing the current economy. — Janus
I think that 10 000 - 20 000 years ago, before they started farming, they were already gradually running out of game to hunt; a problem undoubtedly caused by their dangerously growing head count. At that point, they could also have said: "Hey, the sky is falling. Stop making kids right now!" -
On AntinatalismI don't agree. The goal is workability. — Janus
We do not know if it will still be workable 125 generations from now.
Your approach requires a copy of the Theory of Everything (ToE) to function, but you do not have such copy. Religious believers, on the other hand, believe that the revealed scripture originates from someone who does have such copy, called the "Tablet of Wisdom".
One thing is sure, though. Regardless of what X you use as a starting point, improvisation in the basic rules of morality will always snowball into a nightmare.
With secular law and religious law being in the same epistemic domain, you can ask any lawyer or judge if he thinks that his profession needs such basic document, "the law", to bring back arguments to, and if he believes that it is wise to liberally improvise changes to that basic document. -
On AntinatalismIf you believe that global warming will cause great suffering, and you believe it is wrong to cause, or even contribute to, greater suffering, then what is hypothetical about that? — Janus
It is hypothetical because: sexual reproduction global warming suffering.
A categorical imperative does not use that kind of arrows.
For example: You will not steal.
It does not try to achieve any particular goal. In categorical morality, there is no reason why people are not allowed to steal. It is merely axomatized as a basic rule.
Can we have a categorical imperative: You will not sexually reproduce. ?
Well, no, for reasons mentioned above, a moral scripture will never axiomatize a thing like that. -
On AntinatalismNote, THIS is to appeal to reason:
1. If someone says something that contradicts something in the Quran, then they are wrong
2. Bartricks has said something that contradicts something in the Quran
3. Therefore, Bartricks is wrong
Now, that's an unsound argument - its first premise is obviously false to anyone who is not a Muslim - but it is valid. And so in making such an argument you are still appealing to reason.
Reason, like I say, is the ultimate court of appeal in all things and you ignore her at your peril. — Bartricks
Well, no, it works differently:
Assuming that Bartricks accepts the Quran, if he says something that contradicts the Quran, Bartricks is wrong.
Bartricks probably does not accept the Quran. Still, my remark was about how people who accept it, would react; not necessarily Bartricks.
You can try to find any moral scripture that advocates that its followers should not have children, deprive it from future followers, and hence make itself over time irrelevant. I can guarantee that you will not find such moral scripture with a history of having been in active use for centuries.
The idea of a documented starting point for morality, X, just places morality in the axiomatic epistemic domain. In fact, Immanuel Kant already did that in his Critique of Practical Reason. So, not using any documented X at all is not viable in that view. -
On AntinatalismOK, but I don't see what that has to do with the issue of having or not having children in view of global warming — Janus
That is hypothetical (goal-seeking) morality, which is a practice that Immanuel Kant famously decried in his Critique of Practical Reason. The short story is that hypothetical morality does not work. -
On AntinatalismNow this is also off-topic. — Janus
Well, you need something as a starting point; anything really.
Imagine as a thought exercise a centuries-old moral scripture that says to its followers that they should not have children. One major problem will be that, without future followers, the moral scripture will become unused and just some kind of historical curiosum. So, in a sense, this moral scripture simply would seek to destroy its own relevance. Hence, the historical non-existence of such moral scripture. -
On AntinatalismAnd you cannot understand any of the contents of the Quran or any other religious text until one applies one's reason to it. So reason is the boss of bosses - the ultimate and only true answerer of questions. — Bartricks
Well no, that is a repeat of the ancient, 10th century Mutazili heresy:
Muʿtazilites believe that good and evil are not always determined by revealed scripture or interpretation of scripture, but they are rational categories that could be "established through unaided reason";[6][9][10][11] because knowledge is derived from reason; reason, alongside scripture, was the "final arbiter" in distinguishing right from wrong.
Furthermore, religious scriptures do not contain hypothetical imperatives, in which you have to follow arrows to distill morality. They are categorical only. Islamic jurisprudence is axiomatic from a categorical Quran and Sunnah:
Principles of Islamic jurisprudence, also known as Uṣūl al-fiqh (Arabic: أصول الفقه, lit. roots of fiqh), are traditional methodological principles used in Islamic jurisprudence (fiqh) for deriving the rulings of Islamic law (sharia). [...] This interpretive apparatus is brought together under the rubric of ijtihad, which refers to a jurist's exertion in an attempt to arrive at a ruling on a particular question.
Ijtihad (Arabic: اجتهاد ijtihād, [idʒ.tihaːd]; lit. physical or mental effort, expended in a particular activity)[1] is an Islamic legal term referring to independent reasoning or the thorough exertion of a jurist's mental faculty in finding a solution to a legal question.
The epistemic domain of Islamic law is axiomatic from scripture. It is not possible to meaningfully derive any conclusion in an axiomatic domain without accepting axioms first. -
What is the difference between actual infinity and potential infinity?No it isn't. A set consists of objects, not possible objects. — Metaphysician Undercover
You do not need to populate a set with literal values. You can simply attach a indicator/membership function that is capable of letting through literals that belong to the set and keeping out literals that do not.
S1 = { 2, 3, 4, 5 }
S2 = { x | x ∈ N, x >=2 and x<=5 }
S1 and S2 describe the same set. Therefore, S1 = S2.
In mathematics, an indicator function or a characteristic function is a function defined on a set X that indicates membership of an element in a subset A of X, having the value 1 for all elements of A and the value 0 for all elements of X not in A. It is usually denoted by a symbol 1 or I, sometimes in boldface or blackboard boldface, with a subscript specifying the subset. In other contexts, such as computer science, this would more often be described as a boolean predicate function (to test set inclusion).
You can find another example for this principle in the definition for the term predicate:
Predicates are also commonly used to talk about the properties of objects, by defining the set of all objects that have some property in common. So, for example, when P is a predicate on X, one might sometimes say P is a property of X. Similarly, the notation P(x) is used to denote a sentence or statement P concerning the variable object x. The set defined by P(x) is written as {x | P(x)}, and is the set of objects for which P is true.
For instance, {x | x is a natural number less than 4} is the set {1,2,3}.
If t is an element of the set {x | P(x)}, then the statement P(t) is true. -
On AntinatalismWhen it comes to ethics, our source of insight is our reason, not the Quran. — Bartricks
Yes, but as Aristotle famously wrote, "If nothing is assumed, then nothing can be concluded". Reason is about deriving statements that necessarily follow from other statements; however, without such chain degenerating into infinite regress. So, that means that you can only work your way back until you reach the basic starting-point statements.
You will need to feed "something" to the inference engine. Kurt Gödel really liked to feed the starting points of number theory to his virtual machine, but you can actually pick anything.
So, what is your set of basic starting-point statements, i.e. in Kant's lingo, categorical imperatives, to produce rulings in morality? -
What is the difference between actual infinity and potential infinity?Your idealism solves the problem of contradiction but at the price of failing to account for how we actually talk about the world. — unenlightened
Any language is a Platonic abstraction that is mismatched with the real, physical world; even languages that are specifically meant to describe it.
With mathematics, the situation is even worse, because it is not even meant to describe the real, physical world, but only other language expressions. Mathematics is language about language. So, the real, physical world is out of scope in mathematics.
So, but yes, agreed, talking about the real, physical world, requires another regulatory framework that tries to keep the language expressions correspondence-theory "true", hopefully without degenerating into a complete mismatch. -
On AntinatalismGood people don't want to be dictators - they don't want to have to control the lives of another. It is undignified to live under someone else's control and by someone else's rules, as virtually all people of moral sensibility recognise. — Bartricks
Look at what has been firmly etched in stone:
Quran. An-Nisa 34. Men are in charge of women, because Allah hath made the one of them to excel the other, and because they spend of their property (for the support of women). So good women are the obedient, guarding in secret that which Allah hath guarded. As for those from whom ye fear rebellion, admonish them and banish them to beds apart, and scourge them. Then if they obey you, seek not a way against them. Lo! Allah is ever High, Exalted, Great.
This clause is obviously non-negotiable in its entirety, and it is also never negotiated, because it will never, ever be put up for negotiation on any negotiation table.
This verse is obviously much more popular with men than your belief, but actually even more so with women because it gives them a right to be supported in exchange for their obedience. That is an arrangement that will always suit quite a lot of women fine.
Since people whom you cannot convince of your opposite belief, will multiply and thrive, while the ones that you can convince, will die out, your belief is some kind of punishment of God, which mostly exists to weed out lineages that are not meant to continue in future generations.
As I have argued previously, if a particular belief leads to you fail to reproduce, it will most likely die with you. -
What is the difference between actual infinity and potential infinity?And if I cut a cat in two, there are two pieces of one cat. — unenlightened
Yes, but you are not allowed to put physical objects inside a mathematical set.
You can only fill it up with language expressions.
So, if you cut a "cat" in two, you get {"c", "at"} or {"ca","t"}.
If you want to put a real, physical cat inside a set, you need to do that in physics or so, or in one of the other real-world subjects. Math is language-about-language only. (furthermore, real-world disciplines use a completely different way of thinking about these things ...) -
Evolution, music and mathWhy do you think we have musical and mathematical abilities ? — 3017amen
They are both languages. So, they have a function similar to natural language, i.e. they communicate something. Still, I did not say that you can always translate into natural language what music seeks to express. -
What is the difference between actual infinity and potential infinity?It is contradictory, because a set is closed, complete, (as an object it is bounded, defined) whereas an infinity of anything is open, incomplete, unbounded and indefinite. — Metaphysician Undercover
All possible sentences you can say in English is a set. How is that closed or complete? Has anybody ever been able to list these out? I don't think so. -
What is the difference between actual infinity and potential infinity?When we request justification, we see that "infinite set" is contradictory, as are most mathematical objects. — Metaphysician Undercover
The concept of infinite set is abstract and very Platonic but not contradictory.
Furthermore, these different beth levels of infinite set sizes really kick in when you compare the set sizes of Platonic objects, which are obviously always language expressions in one way or another. Even natural languages are Platonic abstractions. For example, how many different sentences can you make in English? How does that compare to Chinese?
I tentatively guess the levels/beth numbers of infinity for English and Chinese will be the same. Still, in that case, how much is that beth level exactly, and how do natural languages compare to formal languages?
Another example. Context-free languages can in my impression express any arbitrary beth number because they can trivially handle wellformedness, while regular languages may only be able to reach level beth2 (not sure, though). That result would certainly be compatible with the Chomsky hierarchy of languages, in which regular languages are deemed substantially less powerful and expressive than context-free languages.
It is undoubtedly possible to prove a lot of these things, but I could not find any publication that deals with this matter.
Another result could be very interesting. If natural language has a particular fixed beth level, and since context-free languages can express any arbitrary beth level, there may exist context-free languages that are more powerful and more expressive than natural language. I don't know what that would mean, though. -
On AntinatalismWhat do you think: Is it ethical to have children? — Matias
Well, people who feel that it is unethical to have children should obviously not have any. As you can imagine, this idea may then very well die out with the ones who believe in it.
Does this decision - if it is a decision- have political implications? Or is this a private and personal decision that is nobody's business (except those individuals who combine their genes to make a new human being ; and maybe their families)? — Matias
So, then the question becomes:
Are laws against making children viable? Can you send law enforcement officers to people who break the law? What do you do with the illegally-born children?
First of all, ethical questions are about self-discipline and therefore about you think you, yourself should be doing or not doing. They are not a question about what you believe other people should be doing or not doing. "Other people should be doing this or that ..." indeed represents a horrible but quite prevalent attitude.
Furthermore, I think that this would represent a rather counter-trend increase in government intrusion in people's lives, exactly at a time when existing intrusions are being questioned very openly, and have become less and less viable. It certainly goes against the current trend in which people generally consider government to be clueless, as well as the resulting desire to reduce government intervention.
For example, the bitcoin -and wider cryptocurrency community wants to expel government out of the business of printing and controlling money. Approximately everybody who knows what they are talking about, certainly believes that on the long run, the cryptostrategy will work.
In France, a slight increase in taxes on gasoline led to that notorious yellow-vest protest. I don't know if it is still ongoing, but I think that the glass is now full in many countries. So, no, such new birth-control policy would be very unrealistic, because governments would not even be able to find the legitimacy to implement a thing like that. It would be used as an opportunity to do something much rasher than the yellow-vest mini-insurgency. -
What is the difference between actual infinity and potential infinity?And card(PPR) = beth2. — GrandMinnow
Imagine what the regular expression accepts, are expressions like this:
{
{1.2323,343.3333}
,{344.2,0,34343.444,6454.6444}
,{2323.11,834.33}
,{}
,{5 12.1,99.343433}
}
So, it only accepts sets, the members of which must be sets themselves, and these member sets must only contain real numbers.
So, it only accepts elements from the power set of real numbers. (Correct?)
This regular expression seems to work like that (with members written in the alternative set notation):
\[\n((\d*(\.\d*)? ?)*\n?)*\]
So, I would like to confirm or infirm that :
card("\[\n((\d*(\.\d*)? ?)*\n?)*\]")=beth2 -
What is the difference between actual infinity and potential infinity?And I don't understand the rest of your post, starting with "any language expression that matches only this kind of stuff, would be the membership function for a set of which the cardinality would be the powerset of real numbers" — GrandMinnow
A regular expression defines a regular language. For example, a* accepts {nothing, a, aa, aaa, aaaa, ...} or (ab)* accepts { nothing, ab, abab, ababab, abababab, ... } So, that what it accepts, is a set of sequences. The question is now: What is the cardinality of the set that it accepts? If it only accepts sets of sets of real numbers, then the set that it accepts is the power set of real numbers, and that would mean that its cardinality is beth2, i.e. 2^2^beth0. Is there a flaw in what I say? -
What is the difference between actual infinity and potential infinity?This can be expressed exactly, but it's a lot of notation to put into posts such as these. — GrandMinnow
I've got a question about infinite cardinalities. The following set of sets is an element of the powerset of real numbers:
{{1.2323,343.3333},{344.2,0,34343.444,6454.6444},{2323.11,834.33},{},{5 12.1,99.343433}}
So, any language expression that matches only this kind of stuff, would be the membership function for a set of which the cardinality would be the powerset of real numbers, i.e. beth2.
Now, regular languages cannot match wellformedness. So, things like matching embedded braces { } is out of the question. But I just concocted a set notation that does not use wellformedness:
[
1.2323 343.3333
344.2 0 34343.444 6454.6444
2323.11 834.33
5 12.1 99.343433
]
It is the same information as above, but in another notation. This notation is regular and can be successfully matched by a regular expression. I tried it at the test site https://regex101.com. The regex looks like this:
Since this expression successfully matches sets of sets of real numbers, can I say that it is the membership function of a set with cardinality beth2, i.e. 2^2^beth0 ?\[\n((\d*(\.\d*)? ?)*\n?)*\]
If that makes sense, then it would be a witness to the claim that regular expressions can describe sets of which the cardinality exceeds that of the continuum, i.e. uncountable infinity. -
What is the difference between actual infinity and potential infinity?I wrote "isomorphism" in scare quotes because I don't mean an actual function. I mean that tuples and sequences are "isomorphic" in that you can recover the order from one to the other and vice versa. This can be expressed exactly, but it's a lot of notation to put into posts such as these. Anyway, the general idea is obvious and used in mathematics extensively. — GrandMinnow
Oh, yes, agreed, it slipped my mind. It is indeed not just a set. Unlike in sets, the actual order of elements is also a piece of information that sequences and tuples carry. So, it is indeed more than a mapping between orderless sets. -
What is the difference between actual infinity and potential infinity?Everything is a set, including tuples, sequences, and multisets. — GrandMinnow
There is an "isomorphism" between tuples and finite sequences. For example: The tuple <x y z> "encodes the same information" as the sequence {<0 x> <1 y> <2 z>}. — GrandMinnow
What would be the operator in the isomorphism? Otherwise, without such operator, isn' it just a bijection? It is just a mapping between two sets, no?
Still, in my impression, the definition for morphism may be a bit ambiguous because in category theory they do not really seem to insist on the presence of such operator, while in abstract algebra they absolutely do.
By the way, I find abstract algebra much more accessible than certainly the deeper caves of category theory. It is only when they sufficiently overlap that it is clear to me ... -
What is the difference between actual infinity and potential infinity?In other words there is a largest number and infinity doesn't exist. — TheMadFool
Without the axiom of infinity, a concept of actual infinity is not viable. That is obviously also the reason why the axiom was introduced. Otherwise, there would simply be no need for it.
The use of actual infinity is not even permitted in mathematics without axiomatizing it first. Therefore, it is perfectly ok for you to reject the axiom, but then you can also not make use of any of its consequents.
Since it is the sixth axiom in ZFC, you cannot make use of ZFC either. You will need to use an alternative set theory (of which there are actually many).
One possible problem could be that you cannot make use of any of the large number of theorems that rest on ZFC, unless they do not make use the axiom of infinity. However, it is a lot of work to weed through all of that, because it requires verifying their proofs. When rejecting ZFC, a lot of things that you would do in set-theoretical context will now be incompatible with mainstream set theory. Welcome to Hassle-land where everything that would have been simple, now becomes complicated! -
Is pure relativism impossible?If S="every statement is relative" then S is itself also relative, but that would mean that there are statements that are not. That means that people will be looking for statements that are witnesses of universality.
There are prohibitions on unrestricted comprehension, e.g. the expression, "the set of all sets", is not allowed. Bertrand Russell famously asked, "Does the set of all sets that do not contain themselves, contain itself?" That was the beginning of the gigantic foundational crisis in mathematics at the very end of the 19th century.
In fact, it is still ongoing. The lowermost foundations of (classical) mathematics have turned out to be impredicative (circular). There is no hope for a ramified foundation. -
What is the difference between actual infinity and potential infinity?Axiom of infinity. That's as subtle as a gun in your face I guess. I don't know. Am I making sense here? — TheMadFool
Yes, I think it is. It is certainly what Wikipedia says..
The mathematical meaning of the term "actual" in actual infinity is synonymous with definite, completed, extended or existential,[4] but not to be mistaken for physically existing. The question of whether natural or real numbers form definite sets is therefore independent of the question of whether infinite things exist physically in nature.
Of course, as it says, any representation as to whether physical infinite exists in the real, physical world is obviously out of scope in mathematics. -
What is the difference between actual infinity and potential infinity?Actual infinity, if I got it right, consists of considering the set of natural numbers as an entity in itself. In other words 1, 2, 3,.. is a potential infinity but {1,2, 3,...} is an actual infinity. In symbolic terms it seems the difference between them is just the presence/absence of the curly braces, } and {. — TheMadFool
Technically, I think that it should be #{1,2, 3,...} or card({1,2, 3,...}) or |{1,2, 3,...}| for actual infinity (cardinality symbols).
1,2, 3,... is just a sequence and not a set.
sequence: Unlike a set, the same elements can appear multiple times at different positions in a sequence, and order matters.
In fact, there is another notation that is very close to set and sequence: a tuple or n-tuple: (1,2, 3).
tuple: In mathematics, a tuple is a finite ordered list (sequence) of elements. An n-tuple is a sequence (or ordered list) of n elements, where n is a non-negative integer. A tuple has a finite number of elements, while a set or a multiset may have an infinite number of elements.
Now, to confuse the hell out of everybody, the arguments of a function are deemed a tuple, but the typical notation for variadic functions (=with variable number of arguments) is f(a,b,c, ...), while the use of the ellipsis "..." is forbidden in tuples.
Furthermore, all these things are almost the same, with just a minute subtlety here and there ... ;-) -
Quod grātīs asseritur, grātīs negāturWell, you do suspend belief until the conjecture is proven. — TheMadFool
How long can an arbitrary stream of language expressions continue before it contradicts itself? In my experience, not very long. That is why I will not easily say, "I do not believe you". My knee-jerk reaction is rather: "Please, go on."
So, no, it is not suspension of belief. I will certainly be in doubt, but not in disbelief. Doubt is rather some kind of indecisiveness. Doubt and disbelief are quite different from each other.
People get pissed off if you disbelieve them for no good reason at all, and they are actually right, because there is not even a need for that.
If you don't do that then you'd be believing anything and everything which I hope is not what you want. — TheMadFool
Both reality and Platonic worlds have an incredible amount of often even unexpected structure. Fitting a lie into these elaborate structures, is really, really hard. I just wait until it goes wrong. -
Quod grātīs asseritur, grātīs negāturWhat I find relevant in your post is that you don't accept that a conjecture is true. You only assume it is. It's just a weaker version of Hitchens' razor isn't it? — TheMadFool
Well, it is not a version of Hitchens' razor, because unlike him, I do not reject the hypothetical statement. In order to reject it, I first need a "witness" testifying to its inconsistency.
Mathematical witness. For example, a theory T of arithmetic is said to be inconsistent if there exists a proof in T of the formula "0 = 1". The formula I(T), which says that T is inconsistent, is thus an existential formula. A witness for the inconsistency of T is a particular proof of "0 = 1" in T.
What Hitchens does, is pretty much the opposite.
He wants witness(es) to the wholesale consistency of the hypothetical statement, because he somehow believes such witnesses would somehow prove its consistency. That view is obviously misguided. Unlike witnesses testifying to inconsistency, witnesses testifying to consistency do not prove anything.
The false belief in the existence of proof about the real, physical world is a very common error. It is even a fixture in the widespread "burden of proof" nonsense. If knowledge statements about the real, physical world required proof, then we would have no knowledge at all about the real, physical world. -
Non-realityWell the infinite divisibility of a geometric object suggested to me that pure geometry doesn't apply to the world of phenomena we know — Gregory
Yes, agreed.
If you want to deal with knowledge claims that apply to the real, physical world, then it is preferable to pick them from downstream users of mathematics ("applied"), such as science or engineering, which have real-world semantics and actively seek to develop usefulness. Mathematics itself stays clear of that, if only, not to compete with or needlessly disturb the semantics investigated by its downstream users. -
Quod grātīs asseritur, grātīs negāturHitchens' razor is applicable in those cases where a conjecture is claimed to be true. — TheMadFool
I don't think that there are any epistemic knowledge-justification methods that claim that anything is correspondence-theory "true".
Mathematics merely decides if a claim is provable from the construction logic of an abstract, Platonic world. Science merely decides that a claim is testable awaiting its ultimate falsification. History merely decides that it is possible to corroborate witness depositions for a particular alleged fact.
In my impression, there is no knowledge-justification method that allows you to decide if a claim is correspondence-theory "true" or not. They are merely provable, testable, or "corroborable".
In fact, it is not possible to prove anything about the real, physical world.
Scientific Proof Is A Myth. Science can do a whole lot of things, but proving a scientific theory is still an impossibility.
There's No Such Thing As Proof In The Scientific World - There's Only Evidence. “Proof” implies that there is no room for error — that you can be 100% sure that what you have written down on the piece of paper is 100% representative of what you are talking about.. And quite simply, that doesn’t exist in the real world. You cannot prove anything.
Furthermore, a belief does not need to be epistemically justified in order to be correspondence-theory "true". Most beliefs actually aren't (formally) justified. Knowledge is just a relatively small subset of what we believe, and rationality is merely one of the several mental faculties that people use.
As far as I am concerned, the default status of a hypothesis is not that it is false until proven otherwise, or something like that. No, I start by accepting the claim, and then I interrogate it, until I finally discover the reason why it is inconsistent. As long as this reason cannot be found, I consider the hypothesis to be legitimate.
That is exactly what the police does when they interrogate a suspect. Everything the suspect says, is considered true, until the suspect starts saying the opposite of what he previously has said. The trick consists in making the suspect reveal an increasingly large number of nitty-gritty details in what he says, because that makes it exponentially harder for him to keep any lies afloat. If ultimately, after lengthy interrogations, no inconsistency can be discovered, then the police will choose to believe what the suspect has said.
Hitchen's razor, however, would obviously not work. The police would get absolutely nowhere with their investigations, if they used it. -
Non-realityThis can make us feel large against the background of the massive universe. But the world, I've been told, doesn't exist as a single extended reality, but has levels of reality. — Gregory
Your views are very constructivist. They seek to claim a definite connection between the abstract, Platonic world of infinite set cardinalities and the real, physical world.
From a Platonic view, there may be an elusive connection between mathematical abstractions and the real, physical world, but firm claims about this connection are absolutely not supported. From a formalist point of view, the symbol manipulations in Cantor's theory on transfinite numbers are perfectly consistent and therefore completely accepted. From a structural point of view, infinitary arithmetic extensions leave number-theoretic algebraic structures perfectly undamaged.
Aside from that, mathematics refuses to make definite claims about the real, physical world and limits itself to enforcing consistency in language expressions by governing the permissibility of symbol manipulations using an elaborate bureaucracy of formalisms. Constructivism is a mistake, because the only goal of mathematics is to consistently manipulate fundamentally meaningless and useless symbols. Nothing more. -
Social ResponsibilityI don't think they really care, often. I don't mean they necessarily consciously know and decide not to weigh the consequences, though I do think this. — Coben
They are no longer allowed to build these nuclear plants based on the overly optimistic views from the 1950ies and 1960ies. Nowadays, the problem is that they are actually too costly to build, using the new security calculations:
The Flamanville Nuclear Power Plant is located at Flamanville, Manche, France on the Cotentin Peninsula. A third reactor at the site, an EPR unit, began construction in 2007 with its commercial introduction scheduled for 2012. As of 2019 the project three times over budget and years behind schedule. Various safety problems have been raised, including weakness in the steel used in the reactor.[1] In July 2019, further delays were announced, pushing back the commercial date to beyond 2022.[2] [...] EDF estimated the cost at €3.3 billion. The latest cost estimate (July 2018) is at €10.9 billion.[5]
The Olkiluoto Nuclear Power Plant (Finnish: Olkiluodon ydinvoimalaitos) is on Olkiluoto Island, which is on the shore of the Gulf of Bothnia, in the municipality of Eurajoki in western Finland. Unit 3 is an EPR reactor and has been under construction since 2005. In December 2012, the French multi-national building contractor, Areva, estimated that the full cost of building the reactor will be about €8.5 billion, or almost three times the delivery price of €3 billion.
The financial losses, including €2 billion in 2015, reinforced moves for EDF to take over Areva. Areva said: “Half of this loss of €2 billion is due to additional provisions for Olkiluoto 3 and half to provisions for restructuring and impairment related to market conditions."
The largest hidden bill is that serious accidents in the existing installed base (designed in the 50ies and 60ies and mostly built in the 70ies) cannot be excluded, before they get retired, which was planned to already have happened more than a decade ago. They just keep operating them, but with a few exceptions, all of these plants are expired goods. Furthermore, the budget for decommissioning is gigantic, and probably also still underestimated by at least an order of magnitude.
John Maynard Keynes famously quipped, "In the long run, we are all dead." He said that in 1928 about 1975. -
Social ResponsibilityApart from Japan being utterly first world and capitalist, the reactors were US corporation made. And the engineers who built, installed, helped with maintenance and so on, had to have known before, during and after installation, the location of the site, the seismic history of Japan, including tsunamis and how that might relate to future accidents. I have not heard any come forward and say they warned the Japanese government or how their security and safety protocols included concern for tsumanis and why they are not also culpable. — Coben
On 5 July 2012, the National Diet of Japan Fukushima Nuclear Accident Independent Investigation Commission (NAIIC) found that the causes of the accident had been foreseeable, and that the plant operator, Tokyo Electric Power Company (TEPCO), had failed to meet basic safety requirements such as risk assessment, preparing for containing collateral damage, and developing evacuation plans. At a meeting in Vienna three months after the disaster, the International Atomic Energy Agency faulted lax oversight by the Ministry of Economy, Trade and Industry, saying the ministry faced an inherent conflict of interest as the government agency in charge of both regulating and promoting the nuclear power industry.[21] On 12 October 2012, TEPCO admitted for the first time that it had failed to take necessary measures for fear of inviting lawsuits or protests against its nuclear plants.[22][23][24][25]
The largest tsunami wave was 13-14 meters (43-46 feet) high and hit approximately 50 minutes after the initial earthquake, overwhelming the plant's seawall, which was 10 m (33 ft) high.[9]
The black swan theory or theory of black swan events is a metaphor that describes an event that comes as a surprise, has a major effect, and is often inappropriately rationalized after the fact with the benefit of hindsight. The term is based on an ancient saying that presumed black swans did not exist – a saying that became reinterpreted to teach a different lesson after black swans were discovered in the wild.
Black swan events were discussed by Nassim Nicholas Taleb in his 2001 book "Fooled By Randomness", which concerned financial events. His 2007 book "The Black Swan. Impact of the highly improbable" extended the metaphor to events outside of financial markets.
Nassim Nicholas Taleb against Gaussian Curve. Bell curves used in extreme events may cause a lot of disaster. Measures of uncertainty that are based on Bell curve disregard the impact the sharp jumps and inequalities and using them is like getting grass (grass disaster) and missing out the trees (Big Black Swans). [...] Randomness if Gaussian is tameable and is not altered by a single addition or removal. Casino people make such calculation and sleep well in night, no single gambler with a big hit will not change it and you will never see one gambler getting 1 Billion.
Mediocre events get fine or acceptable with Gaussian Distribution because big trees are not present in such events. I say that one should not use Gaussian in extreme events. But once you get Bell curve in head it's hard to avoid. [...] So, while weight, height and calorie consumption are Gaussian, wealth is not. Nor are income, market returns, size of hedge funds, returns in the financial markets, number of deaths in wars or casualties in terrorist attacks. Almost all man-made variables are wild or carry massive randomness(Black Swans).
Seven states of randomness. Mandelbrot and Taleb pointed out that although one can assume that the odds of finding a person who is several miles tall are extremely low, similar excessive observations can not be excluded in other areas of application. They argued that while traditional bell curves may provide a satisfactory representation of height and weight in the population, they do not provide a suitable modeling mechanism for market risks or returns, where just ten trading days represent 63 per cent of the returns of the past 50 years.
A fat-tailed distribution is a probability distribution that exhibits a large skewness or kurtosis, relative to that of either a normal distribution or an exponential distribution. [...] As a consequence, when data arise from an underlying fat-tailed distribution, shoehorning in the "normal distribution" model of risk—and estimating sigma based (necessarily) on a finite sample size—would severely understate the true degree of predictive difficulty (and of risk).
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To cut a long story short, the likelihood of black swans is dramatically underestimated pretty much everywhere in security calculations, through the abuse of the Gaussian probability distribution which is simply not applicable to the likelihood of black-swan events. Therefore, nuclear installations are always several orders of magnitude more dangerous than typically calculated. Once in a million years usually rather means once every fifty years. Note that these wildly optimistic probability calculations were done in the 1950ies and 1960ies for installations typically built in the 1970ies, such as Fukushima. -
Quod grātīs asseritur, grātīs negāturHe's quite eloquent and does well in debates. I don't recall him having to run from the law except the possibility of the Ayatollah issuing a fatwa. — TheMadFool
It is rather the Papacy who would send out an Exsurge Domine, but in Hitchens' case, what he said was clearly not considered interesting enough to even make time to react.
You see, you have to really know what you are talking about before you can make people angry. You would have to say things like this:
It is a heretical opinion, but a common one, that the sacraments of the New Law give pardoning grace to those who do not set up an obstacle.
Or this, because it is thinly veiled criticism on burning Jan Hus at the stake:
It seems to have been decided that the Church in common Council established that the laity should communicate under both species; the Bohemians who communicate under both species are not heretics, but schismatics.
This must have really made the Papacy mad like hell:
Christians must be taught to cherish excommunications rather than to fear them.
In the following "error", our beloved Augustinian heretic, Martin Luther, even (accidentally?) insinuates that a particular verse in the Bible must be a forgery. Over time, it had become obvious that it was James the Just, Jesus' brother, who had been appointed as successor to lead the congregration of the poor, and not Peter:
The Roman Pontiff, the successor of Peter, is not the vicar of Christ over all the churches of the entire world, instituted by Christ Himself in blessed Peter.
Furthermore, the Holy Sees of Alexandria, Jerusalem, and Constantinople had never recognized the Roman Pontiff as "vicar of Christ over all the churches". The views of the Papacy were even contrary to the resolutions of the first council of Nicaea. Of course, you are not supposed to say that kind of things about your boss, if you are in paid employ as a staff member of his organization.
His challenge to the religious establishment is genuine and well-reasoned. He doesn't discriminate between faiths like the faithful themselves are guilty of. — TheMadFool
No, the religious establishment would never give a flying fart about what someone like Hitchens says. If you want to discredit them, you really have to know what you are talking about, which Hitchens clearly didn't. It is the same in Islam. They'd just brush Hitchens off as irrelevant babble. Seriously, there is nothing heretical about what Hitchens said, simply, because he was just not capable of doing that. -
Social Responsibilitywell it looks like our world views are in stark contrast. let's just agree to disagree because I don't think anything can be gained through continuing to engage. — rlclauer
You are probably right, because from my rabbit hole here in SE Asia, it is even irrelevant to me what is crashing and burning elsewhere. The only thing that matters to me, is that they are NOT carrying out the same experiment here. I do not desire to get the job of hosing cold water on yet another Fukushima. Have you ever seen footage of how the naked nuclear cores keep glowing in the open air over there in tsunami land? What a bunch of idiots! -
Quod grātīs asseritur, grātīs negāturSay religion and science are, to quote Stephen Jay Gould, "non-overlapping magisteria": if scientists make pronouncements about religious ideas, or the religious make metaphysical claims in quasi-scientific (fundamentalist) terms, then they are committing category errors, making inapt claims, no? — Janus
Yes, a "category error" is an "epistemic error".
You know, there is an entire field in mathematics, called category theory. Just like epistemology, which is about "knowledge arrows", i.e. the justifying links between a statement and the statement from which it can be justified, category theory is also about arrows.
Now, these arrows are also called morphisms, which are incredibly powerful tools.
However, unlike in epistemology, these category-theory morphisms/arrows are not necessarily used to justify one statement from another. On the contrary, they usually just happen to be there. No need to painstakingly "discover" them. On the contrary, they will often (but not always) just be rubbed into your face, without even asking for that.
So, while epistemology is exclusively about knowledge-justification arrows, category theory is about any kind of arrow, on the condition that the situation can somehow be axiomatized. Epistemology, on the other hand, does not try to shoehorn itself into an axiomatic system. It just seeks to discover interesting methodological patterns in the existing world of knowledge.
This is what the idiot, Hitchens, completely failed to understand. — Janus
Yes, agreed. Hitchens was making money out of annoying other people, because doing so, pleases a particular crowd that likes to upset them. Hitchens was just an arsehole. -
Metaphysics - what is it?What do you mean by "deeply invested"? — Metaphysician Undercover
Well, I actively seek to disagree with people who defend the idea of "subject matters", because in my opinion, "subject matters" do not matter much. What really matters, are epistemic domains. So, I am actively in opposition to subject-matterism which is the core of contemporary curriculum design, which is by the way utterly misguided.
The verbatim transmission of knowledge databases to be memorized by students is such an incredibly bad approach to what education is supposed to be. We simply do not need people to act as living, imperfect copies of Google Search or Wikipedia, or other knowledge accumulation engines. I am staunchly against all of that.
We have not properly digested the advent of computers. People need to finally come to the understanding that you either use the machine, or else you build or program the machine, because in all other cases, it is you the machine.
Do you agree that mental activity is not knowledge, but it uses knowledge? — Metaphysician Undercover
Yes, agreed, rationality/knowledge is merely a tool.
Furthermore, there must be mental activity which does not even use knowledge, as this would be required to account for the coming into existence of knowledge, unless you place knowledge as prior to mental activity (but this could only be intuition, which you deny as knowledge). — Metaphysician Undercover
Yes, agreed, the discovery of new knowledge is mostly carried out with other, non-knowledge, tools/mental faculties.
The strategy by which this mental activity proceeds cannot be "guaranteed to be a failing strategy", because it is responsible for the existence of knowledge. Therefore, the mental process which proceeds without the use of knowledge ought not be denigrated as a guaranteed failure. — Metaphysician Undercover
Ha, but if we could "know" the nitty-gritty of these other, non-knowledge mental tools, then they are actually knowledge, and that would be contradictory. Therefore, I am opposed to any strategy that consists in trying to systematize these other mental tools, because in order to do that, we would need to thoroughly "know" them, which is is not possible, because they are not knowledge.
Hence, I believe that most corporate R&D budgets are fundamentally mismanaged. The worst case of mismanagement, however, are undoubtedly government-funded budgets for scientific research. They usually seek to somehow know and systematize the unknown and even the unknowable, which is something for which you would need to know the search result already, but in that case you do not need to search for it in the first place.
But they are used for the purpose curiosity and wonder, for play, like an artist playing with different colours, or a composer playing with different notes. So new axioms are discovered through this activity of creative playfulness, which because it is not putting tools to work it is not an act of using knowledge in thinking, it's more like thinking for the purpose of finding interesting things, playing. — Metaphysician Undercover
Yes, probably something like that or similar. Still, I admit that I do not really "know".
alcontali
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