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.
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.
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...