redvenom

Good thing I don't care at all what most people care about. If everyone had your attitude we'd all be listening to pop music instead of classical as well.

This highlights the difference between someone who cares about image quality and someone who just cares about reactions and likes.

In most situations that photographers actually care about (i.e. capturing beautiful, good light), ILCs already murder the iPhone. Computational photography may be cool but it is just helping the iPhone emulate ILCs.

The iPhone 12 may have some cool auto computational features, but in no way is it better than an ILC camera like a DSLR. If you actually compare image quality side by side, a DSLR typically will smoke an iPhone. Not to mention there is no way to go past ~100mm on even the best phones.

There's not much difference between the different formats in terms of engineering genius. The Raw data from a camera is basically a matrix of numbers...not too much innovation possible there. The fact that there are different formats is somewhat of a historical accident, and they preceded the DNG format. The format absolutely CAN be stored in a DNG, and in fact Pentax cameras have the option of either DNG or PEF.

Me too. For people on shared hosts (hundreds of small websites including small businesses), PHP is pretty much the only server-side language that you have access to (well, maybe aside from Perl).

It is a diagram whose vertices are mathematical objects and whose arrows are morphisms (often functions) between those objects such that any two paths between two vertices represent the same morphism/function. Typically it is used to infer information about one morphism or object using others, by keeping track of many properties at once in a graphical fashion.

I was thinking the title meant that I could sync an iPad with Linux...too bad. I was hoping that someone made it possible to sync music with the latest iOS.

Absolutely true, there is a lot to prove there. It's not an elementary explanation by any means, rather it is an outline that someone who knows Galois theory well could probably reconstruct.

The reason basically comes down to the fact that every time you perform a compass and straight-edge construction, you are constructing quadratic extensions of number fields, whereas trisection (in general) needs a cubic extension.

The 'in general' part is important: of course, you can trisect specific angles like a 90 degree angle.

(The construction of number field extensions is happening because let's say you construct a 90 degree angle, well that allows you to construct an isosceles triangle with equal sides length 1, and therefore you construct the square root of 2, so the extension Q(sqrt(2))

It does make you much better at formulating problems. In fact, from my experience as a mathematician and watching others do math, the vast majority of good problems comes after someone has solved a problem and then asked, "what if we change this instead?"

No, polygons in non-Euclidean space makes sense. Polygons and more generally, geometric simplicial complexes in other spaces is a branch of mathematics all by itself (geometric group theory), so you certainly shouldn't "throw it out the window".

The reason is because you want to be able to usefully talk about 'true vertices'. Another way to define a polygon is any convex hull of finitely many points in Euclidean 2-space with nonzero area. Then the 'real' vertices are those points on the boundary curve that are nondifferentiable ('pointy' or 'sharp').

Those are the special points you are interested in when you want to distinguish between vertices and other points on the boundary.

Work at a job so I can pursue my creative projects like writing blog posts and making documentaries.

Well, there are documented cases of evolution happening during our lifetimes.

But I would say as a mathematical process, pretty likely once you have the basic ingredients. Richard Dawkins has written some good popular science books like 'Climbing Mount Improbable' that talks about evolution in this light. His 'Selfish Gene' is also an excellent read. It is slightly more technical but definitely understandable without an immense technical background.

I'll tell you something about memorizing phyla, orders, families, genera, and species. If you've never had exposure to any of that, it might seem arbitrary. But now, go out in the woods and try and identify species you see. You can start with birds and mammals because they are the easiest.

As you begin to do so, you can keep a record of them in a spreadsheet or notebook. Look up their biological classifications. All of a sudden the classification system will start to make sense, and help you understand the diversity of the organisms around you.

Biology is one of those subjects where part of the context comes from being outdoors and trying to understand ecological relationships yourself. You don't have to get to the level of a pro biologist, but there is so much you can find out by experiencing nature and trying to understand the basics that anyone can do it.

This explanation is somewhat informal, but I gets the point across: a theory is a set function symbols, relation symbols, and axioms governing them. A model is a specific set with an interpretation of those function and relation symbols that satisfies every axiom of the theory.

It's much easier to understand if we take an example. An example of a theory is the single sentence:

"There exists an X and there exists a Y such that X is not equal to Y."

(Of course typically in logic you would use logic symbols, but here I am writing out in an English sentence.)

Now, a model of this theory is the set {1,2}. Another model is the set {1,2,3}. More generally: any set with at least two elements is a model of that theory. The "function symbols" and "relation symbols" can be introduced in the language to talk about operations like addition and multiplication.

For example, the theory of groups uses the language of groups with a binary function symbol representing group multiplication. Any group (such as the integers with addition or invertible matrices with matrix multiplication) is a model of that theory.

So: theories are sets of axioms in some language, and models are sets together with actual functions/relations that satisfy those axioms.

Models of ZFC are a little bit counterintuitive. But they are single sets that interpret all the axioms of ZFC, rather than actual sets that we use in informal mathematics. Models of ZFC can be quite unusual because of the incompleteness theorem, and there are infinitely many models because of this (such as some in which CH is true, etc.).

People have a fascination with "This sentence is false" because it highlights the need to be precise distinguishing between the language of a logical system and a metalanguage to talk about the language. That discovery or 'invention' if you will is something that was not put into rigorous foundation until the 20th century.

"True" in this context is a technical definition, not some hand-waving thing. A sentence X being true means every model satisfies X. Provable means logical deduction within the theory proves X. So actually, Godel's theorems are exactly about truth and provability.

Excellent question. It implies an infinite number of statements. For if T is such an incomplete theory and X is a statement true in some models of T and false in others, then T+X is also a theory that satisfies the same requirements as T in Godel's theorem. Therefore there is another statement Y that is true in some models of T+X and false in other models of T+X. But any model of T+X is a model of T so Y is true in some models of T and false in others. And it goes on...

This response is not quite right. "It's only completeness [...]" sentence does not make sense.

CH is independent of ZFC, period, as proved by Cohen. Talking about 'semantic level' does not make sense.

CH is an example of the incompleteness of ZFC. There are models of ZFC in which CH is true and models in which CH is false.

Just want to point out that your answer is essentially incorrect. Godel's completeness theorem has nothing to do with the existence of models. It is about the provability of sentences that are true within all models.

Also, the incompleteness theorem doesn't say anything about the nonexistence of a model. It gives the existence of at least one sentence X (for sufficiently nice theories that include enough arithmetic) such that there ARE two models where X is true in one and false in the other.

The answer you got from lmm and others is wrong (I am a professional mathematician and did research in logic).

The completeness theorem says simply: if T is a first-order theory (list of axioms in first-order logic), any sentence true in every model of T is provable by logical deduction in T.

The first incompleteness theorem says: if T is a consistent recursively enumerable theory that can contains a sufficient amount of Peano arithmetic, then there exists a sentence which is neither true in all models nor false in all models. In other words, there is a sentence which is true in at least one model of T, and false in another.

The second incompleteness theorem says that if T can interpret Peano arithmetic, then we cannot prove the consistency of T within T.

So a tl;dr:

-Completeness: any SENTENCE true in ALL models is PROVABLE - applies to all first order theories

-1st Incompleteness: there EXISTS sentences which are TRUE in some models, FALSE in others. - Applies to theories that contain enough arithmetic

-2nd Incompleteness: if a sufficiently strong system is CONSISTENT, we CANNOT PROVE that CONSISTENCY within the system.

NB: of course, if you have a sufficiently WEAK system, like the axioms of group theory together with "FOR ALL x FOR ALL y (x=y)", then that theory would be COMPLETE and Godel's incompleteness theorem does not apply here.

The point is not to prove that 1+1=2. The point is to construct mathematics (all of mathematics) directly from the axioms using a formal language. The objective of 1+1=2 is not interesting, but the result is a rigorous foundation of mathematics using formal languages.

The upshot is twofold: (1) we have a rigorous foundation of mathematics in formal logic which CAN be used to prove nontrivial statements, as in the application of model theory to mathematics, and (2) we can now build proof assistants that allow us to use computers to better understand proofs.

Both applications of formal reasoning are equally interesting and have wide ramifications in mathematics.

There are two separate statements here:

1. Mathematicians often use relatively powerful systems, like reasoning in ZFC about Peano arithmetic. The consistency and completeness of axiomatic systems is what this article is about

2. Mathematicians pretty much exclusively use informal systems (compared to formal logic) for reasoning. The reason why that is is because it is just infeasible to reason typical mathematics using formal logic for most cases.

However, it's important to realize that these two are orthogonal.

I think the point is that computers are conquering less and less obviously computation problems. At some point we ask ourselves, just what is human intelligence? And perhaps there is not much difference between computers and humans after all...

Another place not on this list is YouTube. You can set up a channel pretty easily and make a short video about your project. The advantage with this is that it is really easy to include screen captures of using it (if it's a graphical app).

Take that, RIAA!

I don't really think so. I think a lot of people are just doing something because it's a job. And a lot of people are sound with being sheep and just following the rules because they exist, and don't like the discomfort that comes with questioning everything on a deep level.

We should eliminate single-use items like coffee cups altogether.