Thinking about thinking is unnecessary. Thinking is same as worrying about something. When you do thinking, you are neither eating, nor sleeping or doing any physical work. Your body is at rest but your mind is not. No animal does too much thinking than what is necessary to interpret the inputs given by it's senses and correlate them to any self-driven actions needed. When people see you as lost in some thought, the first instinct they get is a negative feeling. They think you are worried about something or something wrong happened. There are no positive feelings you can get when you see your loved ones in deep thought.
Humans developed thinking in order to gain more complex correlations to help them interpret their sensual inputs, such as correlating that plants need water. But this has gone too far due to availability of free time for humans. Lot of unnecessary thinking, called philosophy and science, happened.
Human society always hated thinkers. Thinking was called witchcraft and banished. Thinking makes a person get detached from people and surroundings around them. You stopped processing input from your senses. You stopped interacting with world outside. Instead, you got busy interacting with your inside. People get scared and do not trust you anymore. They don't know what your thoughts are and how you are going to respond. Your thinking makes you an alien.
More importantly, thinking means you might not obey the commands from the community or group. Community requires simple thinking workers who think and act in expected ways, making it possible to have a strong community. Imagine some bees in a swarm do not follow the swarm movement, or ants in a line or fish in a school, birds in a flock. Community was always the real creature, not the individual humans. In many communities, individual persons do not have a name that is too distinct from family name. Family does the thinking, not individual people.
Sounds very fascist. I suppose it fits with the times.... very bad times ahead not unlikely.... Hopefully the price will be fewer than a billion dead bodies....
It's appalling how many people on these threads take pride on being an SME on these topics, without recognizing the real point. It is not a question for a drones SME. It's question for defining the meaning and purpose of human progress. We are creating a form of natural calamities and calling it progress. Drone swarms are calamities similar to earth quakes, tsunami or wild fire. If you have a swarm your enemy has it too. If you have a nuke, the enemy has it too. You might reason, well, even if we don't push for it, others would. Alright, then the only way to stop it is by natural balance. The balance would be achieved by devaluing the resources humans are after. Say there are too few humans remaining on earth, so that they don't care about owning stuff, country borders or money or security ...
Totally agree. Even in my childhood, if anyone was seen walking without work or running without being chased by dogs or thieves, they would be seen as lunatic. Why would anyone spend their precious energy except for some absolutely needed work? People perfected minimalistic energy spending and question any unnecessary walking. They ride on buffalos while coming back from farm, use farm animals and tools for harder work, have 24-hour servants, have more children, and have large family, get everyone, including women and children out to farm work.
I think the leisure walking, jogging, gym are all products of availability of too much food in return for less work. We are simulating work by cheating the body to think that these leisure activities are actually work.
Even more fundamental question is - why does a cell divide? All literature that I came across tells me "how" a cell division happens, but not "why"? What's the cause? What's the motive? For example, in Physics, the cause or motivation is reaching an equlibrium or minimality of energy transferred, entropy etc. For cell division - what's the nature's goal? Having more cells? Why?
Even searching for this is tricky, because you’ll see a ton of information on how the cell “knows” mitosis has begun and the chain reaction of what happens after that, but I found this article from 2015 [0], which refers to this Nature article [1].
The transition from "slow-down in working age population" to "subscription model" appears a bit drastic. Both are huge topics, but could be slightly unrelated. Population growth stagnation may not be the major driver for subscription model. It is just a new way to to increase the cost of product and spread the cost in time so that it doesn't hurt the customer.
Same as buying stuff on loan. You can buy a car or home with almost zero down payment and keep paying the amount over next 20 years. It is essentially paying for usage. You can call it subscription.
Outrageously, I find that even education/training also moved to subscription model. You keep taking the same exam every few years and keep paying them.
Spot on! Also many projects end after achieving the main goal - a powerpoint slide deck, claiming something was done. No one has ability to ask what was done and why it was done. Also the same project is reworked multiple times, during dev stage, over multiple years as the tech becomes old or other dependencies change.
When you do release everyone claims it is never what they wanted. As a developer your job hangs on the fact that you followed the spec. It makes you happy when they cancel a project at the very end. You get to do the fun stuff (coding) and don't have to deal with the negatives of a release.
The fallacy of Cantor (and his supporters such as Hilbert) lies in mixing the mathematical logic which is meaningful only for finite numbers, with the infinity as if it were a number that obeys comparison operations such as bigger, smaller etc. Almost anything can be proved/disproved or debated when one assumes such validity of the logical concepts to infinity.
The acceptance of multiple infinities and some concepts around continuum (Dedekind cuts etc) are questionable due to this assumption.
I don't think there's any confusion of the finite and infinite numbers in Cantor's diagonalization argument. Here's the most generic form of Cantor's argument: there is no surjective function from a set A to its power set P(A).
We prove this by contradiction. Consider any function f: A -> P(A) , i.e. it takes elements of A and outputs subsets of A. Suppose this function is surjective: i.e. for all y in P(A) there is some x in A such that f(x) = y. But let q = { a in A | a is not in f(a) }. Clearly this is a valid set. And if f is surjective, there must be some x in A such that f(x) = q. Is x in q? If x were in q, then x would be in f(x), so that's a contradiction. If x were not in q, then by the definition of q x would be in q, which is also a contradiction. Thus we have a contradiction, so f cannot be surjective.
As you can see, nowhere do we make any logical jumps that would only make sense in the case of finite sets. This proof is as straightforward as the proof that there is no set of all sets. We don't use the axiom of choice, the argument is even valid in constructive mathematics (though you have to make some adjustments).
Now, we define one set X as being "less than" another set Y if and only if there is no surjective function from X to Y. You'll see that this definition corresponds exactly to the usual notions of size for finite sets, and makes intuitive sense (if for every y we have an x, there must be at least as many xs as ys). Now, just plug in the set of natural numbers into Cantor's theorem and you get: the power set of natural numbers is larger than the set of natural numbers.
If you are counting a set, you necessarily are creating a map between the item of the set you are counting and the natural numbers, this is basics, how come that you can say such statement about the sizes of power set of natural number and natural number?, from where the notion of the cardinality comes if not from counting, and counting implies a map to a natural number, so the set of natural number is bigger than its selves?
Cardinality doesn't have to do with "counting" necessarily. Two sets X and Y are said to have the same cardinality if there is a function f : X -> Y where f is a bijection. By bijection we mean it has two properties: that for all a,b in X, if f(a) = f(b), then a = b (this is also called the injective property), and for all c in Y, there is some d in X such that f(d) = c (this is called the surjective property). Set X has cardinality less than Y if there is no such bijective function, but there is a function f: X -> Y that is injective. Conversely, X has cardinality larger than Y if there is no such bijective function, but there is a function f: X -> Y that is surjective. All you have to do to compare the size of two sets is to look at the functions mapping one to the other. No "counting" involved.
This make much more sense, so most time the cardinality which people refer is with respect to the set of natural numbers, but according to you, we can have this relation between any two sets. The problem is not make those things clear in the wording. Why not just called it "the cardinality order relation", and try it always like a binary relation, instead of a property that each set has in insolation.
Well you can use this relationship to establish the cardinal numbers. You can use "there exists a bijective function between sets X and Y" as an equivalence relation between sets. And with a equivalence relation you can talk about partitioning things into their equivalence classes. But because we are talking about a relationship between all sets it gets tricky to formally construct things (because there is no set of all sets for you to use to define things, so you can't just say "take the sets under the equivalence relations"). There are multiple ways to explicitly construct them, but they tend to be pretty complicated compared to just talking about bijections. The constructions I know about either require the axiom of choice or the axiom of regularity (every non-empty set A contains an element that is disjoint from A). But you don't need any of that to establish a lot of the properties of cardinalities
It is possible to make a set that contains all the set except itself, operationally this pretty simple, I am working on creating a language programming based on set theory, so this will be an easy way to define some notion of universal set.
If you aren’t aware of it already, you might be interested to read about Russell’s paradox [1] which shows a problem with allowing sets to be defined into existence without restriction (making Frege’s life’s work the Grundgesetze stillborn). It involves using the “set” of all things that are not a member of themselves.
Unfortunately the set that contains all sets except itself doesn't exist either. If such a set X existed, then you could easily build X unioned with {X}, which would be the set of all sets, which doesn't exist. You can't really avoid Russel's paradox, it prevents you from being able to make a set that is basically all sets, minus some exceptions.
But there is a way to get around Russel's paradox, the standard way is to define a new kind of object called a class. See https://en.wikipedia.org/wiki/Class_(set_theory) . Basically a class is like a set, a set can be a member of a class, and you can do most operations with sets on classes as well (union, intersection, etc). And you can create a class which is all sets satisfying some property. From this you can't have a set of all sets, but you can have a class of all sets. Russel's paradox doesn't go away, you still can't have the set of all sets (and you can't have the class of all classes), but this still gives you enough to talk about properties of all sets. See https://en.wikipedia.org/wiki/Von_Neumann%E2%80%93Bernays%E2... for an example of how to define classes. That particular example is a conservative extension of ZFC: any statement about sets that can be proved using NBG can also be proved using ZFC and vice versa (assuming ZFC is consistent). Your statement can't involve classes, because there is no definition of classes in ZFC so it wouldn't translate, but your proof in the NBG system can make use of it
Typically, unless you specify the other set, it's assumed to be N, the natural numbers. And when you have a bijection between some other set and (some subset of) the natural numbers, you're doing something equivalent to counting.
> You'll see that this definition corresponds exactly to the usual notions of size for finite sets, and makes intuitive sense.
This has always been my problem with the idea of some infinities being "bigger" than others. This definition does not correspond to the usual notion of size for finite sets, because finite sets actually have size. So rather than making "intuitive sense", it makes no sense (to me).
Having no surjective function from one set to another makes sense, but defining one set necessarily as "less than" the other does not, as the word "less" cannot, in such a context, mean what its definition implies.
> (if for every y we have an x, there must be at least as many xs as ys)
The lack of a surjective function does not mean that we cannot have an X for every Y, it just means we can't map an X for every Y with a surjective function. It doesn't mean a pairing cannot exist (at least not abstractly, which is how all infinite sets exist anyway). The pairing can simply be random and undefinable. There, now a pairing exists and both sets are the same "size". (It's not a surjective function, but if the point is simply to compare "sizes", then what does it matter?)
Cardinalities of sets do have sensible operations like less than and greater than though - |A| <= |B| if there exists an injective function from A to B. One has to check that this inequality behaves in the ways that you expect, but it is a well-defined concept.
It is defined in the abstract, but I think there are a lot of implicit assumptions that get lost when dealing with cardinalities which may turn out to be important. Just because a bijective/injective/surjective function exists doesn't mean that such a transform is possible when you apply it to a specific situation.
If sizes were sensible then for a set A composed of "every other integer" would be smaller than the a set B composed of "2 times every integer". If we were using calculus to take the limit of the ratio of sizes of the generating functions for A and B as the source set size goes to infinity then we find that A is in fact half the size of B. If we try to simply apply cardinality to A and B we find that the sets are exactly equal, since cardinality doesn't care about source sets or limits!
This can a big deal if say, you're calculating probabilities across an infinite number of possible events/event configurations. There is a big difference between a 50% chance, a 0.000001% chance and a 100% chance.
I'm not saying there's no use for cardinality and infinity classes, but they can easily be misapplied to allow you to be wrong, with confidence.
Sorry, but they simply don’t meet your (non-standard) definition of sensible. Both of the sets you mentioned can be interpreted as the set of even integers. This set is in bijection with itself (trivially), and thus is not considered to be strictly smaller than itself.
Many logical concepts which we take for granted, might not be meaningful in the realm of infinite. For example, something being equal to itself, some statement being true or false (no third state), things having unique identity or location, the concept or possibility of multiple things (and distinguishability), being able to compare one thing with another etc. All these make sense only in finite contexts but not in infinite or infinitesimal contexts.
If we accept infinity as a participant in the mathematical logic, the infinitesimal also deserves a role as a counter-party.
The original construction did though, if you look at the writings of Newton/Leibniz. But their work was totally hand-wavey. Cauchy discovered limits in the 1800s, and it wasn’t until the 20th century that the original idea of infinitesimals was formalized. In some sense we have come full circle.
I do think that infinitesimals are a much more intuitive way to understand calculus. It’s a shame it took so long to find positive numbers less than any real.
Humans developed thinking in order to gain more complex correlations to help them interpret their sensual inputs, such as correlating that plants need water. But this has gone too far due to availability of free time for humans. Lot of unnecessary thinking, called philosophy and science, happened.
Human society always hated thinkers. Thinking was called witchcraft and banished. Thinking makes a person get detached from people and surroundings around them. You stopped processing input from your senses. You stopped interacting with world outside. Instead, you got busy interacting with your inside. People get scared and do not trust you anymore. They don't know what your thoughts are and how you are going to respond. Your thinking makes you an alien.
More importantly, thinking means you might not obey the commands from the community or group. Community requires simple thinking workers who think and act in expected ways, making it possible to have a strong community. Imagine some bees in a swarm do not follow the swarm movement, or ants in a line or fish in a school, birds in a flock. Community was always the real creature, not the individual humans. In many communities, individual persons do not have a name that is too distinct from family name. Family does the thinking, not individual people.
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