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Having been alive for most of the 60s, which means people only slightly older than I grew up in the 50s, I must say.What are you talking about life being easier in the 50s and 60s?As I look around my house, I think of the most basic things.Let’s start with running water. Even the poorest people I know have access to running water. In the big cities in the 1960s, most people had running water. We had a pump outside. We thought my grandparents were wealthy because they had a pump inside. That was not odd in small towns and rural areas in the 1960s.Guess what else when you only have water outside? You don’t have a toilet inside. You have an outhouse a distance from the back door. Pretty inconvenient and very stinky.Definitely no television in the 1950s. A lot of people had it in the 60s but we got it around 1973 or so.Telephone. I almost typed a curse. Everybody on the county was on the same line. Even in the city, the whole block was on the same line and to call a block over cost $3-$5 per minute. It was fun to listen to everyone else’s conversations and talk to the operator. However, kids got in big trouble for touching a phone.Travel. Many fewer people had automobiles. There was usually one old car per family, which really wan’t that old because cars rusted out after 5–6 years. Plane travel was much nicer than now, but a ticket cost two month’s salary. Cars did not have air conditioning. Music came from a rattling speaker in the middle of dashboard, AM only. People were used to walking many miles and carrying stuff. Cars were death traps. Nobody heard of a seat belt or padded seats. Power steering and brakes were also a luxury. You had things like “Three on the tree” stick shifts and “suicide knobs” on the steering wheels. All of these added to the danger. If you had a four wheel drive vehicle, which was common in the country, you had to get out of the vehicle to lock the wheel hubs in bad weather.Mail order. The standard line when you bought something was, “Allow 6–8 weeks for delivery.” You had to plan your summer purchases in winter, fall purchases in spring, etc. Unless you were in Bentonville Arkansas or Troy Michigan, there were no Walmarts or KMarts. You got a catalog in the mail. then you picked out the stuff you wanted and hand wrote and order form. Then you had to go to the Post Office and mail it. Or you could wait to you went to a store like Sears and drop it off. Some items shipped to your house. With others, you got a letter which said you had so many days to go and pick them up. I remember ordering things that I forgot I ordered by the time they arrived.Food. You made your own. In the country that also meant killing and skinning your own protein or picking your own vegetables. There were not many supermarkets, even in the large cities. You went to a corner store in the city. The store only sold meat, or vegetables or baked goods. In the country, you went to a nearby town once a month or so to go to the general store. There were no microwaves. You pretty much had to plan an hour per day per meal to prepare your food.No power anything. Mowing the lawn, shoveling snow, trimming trees, carpentry, you name it was all done with hand tools. Everything took ten times as long.Face to face. All business had to be conducted face to face. Remember the phone? You couldn’t call a work person or order items or whatever. there was no amazon. If you needed to hire a carpenter, you had to go to the carpenter’s shop or house. You paid by cash or check. there were no online payments or cards.I think what happens is that people watch television shows like Leave it to Beaver, My Three Sons or Father Knows Best and think that is the 1950s and 1960s. From my experience, those people lived like what we call the “1%” nowadays. You would have to be extremely wealthy in 1965 to afford the type of lifestyle you see on these shows. They all also seemed to end up in Hawaii. I had relatives that were considered relatively wealthy. They couldn’t afford to go to Hawaii until the 1980s, when competition brought down airline tickets and computers at travel agencies made comparison shopping possible. Plane tickets also had to be purchased face to face. This meant driving to the airport and paying with cash. Then it was near impossible to comparison shop unless you lived in Los Angeles, New York or Chicago. Your local airport may only have one or two airlines from which to choose in the 50s and 60s.That just reminded me of another thing in the 1960s. To learn the simplest bit of information, you had to make a drive to a city with a decent sized public library. That was about 3 hours each way for us.Life was definitely not easier.

I think that mathematics is certainly the most objective of all human pursuits. Once something is proven---that is, you have shown by a logical process that it follows from the basic axioms---it is proven, forever.However, there are at least two places in that last assertion where subjectivity does creep in.1.) How should we choose our axioms?The typical thought process is that our axioms should be powerful (give us the ability to prove the things we want), intuitive, and minimal (we shouldn't assume more than what we need). If the last criterion can more or less be examined objectively, the other two are pretty subjective.As an example, consider the Axiom of choice, which is one of the standard foundational axioms. It says, essentially, that if you have some collection of sets (infinite or finite), you can build a new set from them by choosing one element from each set.This seems intuitive (to most people), but you can prove plenty of unintuitive results using choice, including the Banach–Tarski paradox that you can decompose a sphere into finitely many pieces, then reconstruct them to form two spheres of the same size as the original.Mathematician Jerry Bona has said of this situation that "The Axiom of Choice is obviously true, the well-ordering principle obviously false, and who can tell about Zorn's lemma?" The joke, of course, is that all three are logically equivalent assuming the standard ZF axioms.In the end, however, it is mostly a question of taste and convenience. Mathematicians have mostly settled on using ZFC (ZF axioms plus choice), but I'm sure that we will eventually come up with a better set of foundational axioms to work with, and then we will switch to that. No theorems will be harmed in the process.2.) What does it mean to 'prove' something?As it turns out, there is not total agreement on this point. There is an overwhelming consensus, but that is not quite the same thing.ZFC is built using fairly standard, 2nd order predicate logic. From this stand point, showing that "not not P" is the same thing as showing "P".So, for example, if I prove that it is impossible that something does not exist, it is logically equivalent to proving that it does exist.Not everyone agrees with this idea. Constructivists assert that unless you explicitly construct the desired object, you have not proven its existence. In other words, they take the hard stance that showing that two things are logically equivalent is not the same thing as proving it. Different people take this to different extremes. Some even assert that mathematics should not deal with infinite sets at all, on the basis that you could never actually see such a set.Constructivism has its roots in Poincare and Kronecker's objections to Cantor's study of infinity, but it has since been relegated to the sidelines of mathematics.A related question, touched on by User-10360931118059412840 in his answer is this: if a computer checks a theorem case by case using more calculations than a human could ever hope to go through, does the theorem count as being proven?Most mathematicians accept that computers are not somehow more inherently prone to errors than humans, so this method of proving statements can't be worse than the usual one. That said, there are valid reasons to dislike computer-based proofs. In particular, they don't usually offer as much intuition as human-written proofs.For example, we know that the 4-coloring theorem is true. Do we know why it is true? I'm not entirely sure that we do.To be clear, I do think that computers are going to play a bigger and bigger role in mathematics, and I think that this is going to be a positive movement. However, I think that we should take care to respect the human element of mathematics.

Imagine someone with “Null” as a last name. There are quite a few of these people in the world and they are unknowingly making a point to developers on database entry validation. This is not something I’ve dealt with personally but I’ve seen several stories on it on the web. Like this one.For those who aren’t into programming: “null” is used to represent absence of a value in a piece of code or a database. To a computer, there’s a distinct difference between 0 and null because “0” represents an actual value and “null” doesn’t. So say you’re getting a car insurance through a website and have to tell how many accidents you were involved in. If you say “0”, that’s a useful answer but if you don’t answer the question at all there will be a variable that has no value in it and it will be assigned “null”. Now, today we have form validation that checks your values before you can send them (often you get a red sign saying you need to fill out some more fields), but you get the idea. Now many operations in a computer language have special ways to deal with the value “null” in order to prevent it from crashing by doing calculations with nonexistent values. Often is will result in errors or alike. Now you can imagine how poorly written programs will handle name variables for people who’s name are “Null”. A good programmer will always build in checks to make sure that the name is stored as a string of characters instead of the actual “null” value, but then again not all programmers are good (or perhaps more realistically: have managers that give them time to make sure these things are taken care off).In a way, it also reminds me of this classic XKCD comic :)EDIT:I almost forgot this one and is, now that I think of it, far more epic than the example above ;) It’s the case of the 500 mile email, a very mysterious case of an email server that wasn’t able to send an email beyond a 500 mile radius. Since the internet really doesn’t care much about distance this is a very rare and awkward glitch. It’s really worth it to read it until the end!