The next time somebody tries to sell you something, ask them:

If it’s true that what you’re selling is worth the price you’re asking for it, and that it will service a real need that I have, then why is it necessary for you to put so much effort into persuading me to buy it?

If they can give you a good answer to that question, then purchase the product immediately.

If you ever find that you don’t know how to follow instructions, I’ve prepared the following set of simple instructions (which you can easily commit to memory with the mnemonic “FRIED”):

That oughta work.

The other day my step-cousin and I tried to solve the chicken-and-egg question, from an evolutionary one-of-them-must-have-come-first perspective:

Assuming that there is some (probably arbitrary) dividing line in the evolutionary tree between pre-chicken and chicken, then there must have been a "first chicken" at some point - let's call her Lucy. Lucy's parents were, by definition, not chickens, but Lucy's mother laid an egg containing a chicken (by some lucky mutation or something). Lucy then must have laid her own egg, which not only (I think it's safe to assume) also contained a chicken, but additionally was laid by a chicken. So which egg are we talking about? This reveals an ambiguity in the original question. The question is

The question's not so interesting anymore. So since we're thinking about evolutionary progression, how about this question:

A family friend recently told me the following joke:

Why do programmers get Christmas and Halloween confused? Because Dec(imal) 25 == Oct(al) 31.

HAAAHAHAAHAHHAHHAAH!!!1 LOL! ROFL!! BMW! AT&T!!

One of the things I’m least likely to be doing (at similar positions in the list to “watching American Idol” and “krunking”) is playing flash-style internet games on websites that specialize in banner ads. However, I recently ended up playing this game called Light-Bot, which is the first game I’ve seen (though I expect there are many others – like I said, I don’t do this much) to incorporate basic computer programming concepts.

There’s a robot placed on a grid, and the player has to write a program to tell the robot what to do in order to accomplish the goal. The programming language consists of seven commands (go forward, turn right, jump, etc.), which are represented with pictures instead of words, and the player drags the pictures onto a program grid (not the grid the robot is on), and they are executed in the order specified.

Two of the commands are subroutine calls – there are two extra program grids representing the subroutines. It’s even possible to get the robot into an infinite loop by having a subroutine call itself. Also, each portion of a program has a maximum length – twelve commands for the main body, and eight commands for each subroutine. Part of the challenge is utilizing the subroutines well enough to fit the program in the allotted space.

It occurred to me that the levels in the game would be interesting candidates for a genetic algorithm experiment. The lack of syntax (any random blending of two programs will always be syntactically valid) and the limited length of the programs would make it easy to write a simulator, and the problem space could be quite challenging (I gave up on level nine or so). A related problem would be determining if a particular level is solvable or not. I’ll put it on my list of programming projects.

In most programming languages, if you want to generate some primes with a sieve, you first pick an upper limit, and then create an array and filter out the multiples of the lowest primes until you hit the square root of the upper limit. Using Clojure's infinite sequences, I can remove the upper limit and filter out all the multiples of each prime at each step. I won't make any claims about the efficiency of this method (it's probably pretty bad), but I think it's a mildly fun idea:

(defn sieve-seq [level]
  (loop [val (cons 2 (iterate #(+ % 2) 3)) lev 2]
    (if (= level (- lev 1))
      val
      (let [p (nth val (- lev 1))]
        (recur 
          (concat 
            (take lev val) 
            (filter #(> (rem % p) 0) (drop lev val))
          ) 
          (inc lev)
        )
      )
    )
  )
)

(defn primes [level]
  (let 
    [my-list (sieve-seq level) 
     root (nth my-list level) 
     limit (* root root)]
    (take-while #(< % limit) my-list)
  )
)

The first function returns an infinite sequence with the specified number of primes applied to the filter. The second function calls for the sequence from the first function and extracts the part of the sequence known to have only primes left:

I usually don't pay much attention to operator precedence, thinking that it's either obvious, or it gets parentheses. And so recently I got bit in the face by it. I just spent a couple hours trying to find a bug in a set of classes with some nested data buffers and such, until I finally got to this method, which is supposed to extract a bit from an array of unsigned chars:

int DataBuffer::getBit(int index)
{
	int whichByte=index/8;         
	int offset=index%8;
	unsigned char theByte=this->get(whichByte);
	unsigned char mask=1<<(7-offset);
	if(theByte&mask==0)return 0;                                               
	return 1;
}

A friend of mine recently mentioned (do not read “endorsed” – I wouldn’t want anybody to get any false impressions about my anonymous friends) the idea that mathematical theorems are demonstrably, objectively true. It’s certainly a tempting conclusion to come to, especially if you’re the sort of person attracted to absolute truth, but in reality there isn’t anything absolutely true about math that’s of much interest. You can’t understand this, however, without looking deep into the bowels of mathematics to see what foundation it’s built on, so I’m going to try to do that for the uninitiated in an accessable way. To start with, I’m going to break mathematics up into three layers.

Top Layer: Popular Math

This is math as most people are familiar with it. The algebra and the geometry and the calculus and all that. It’s a bunch of properties about mathematical objects and rules for things you can do with them:

• Multiplying two negative numbers results in a positive number
• The sum of the interior angles in a triangle is always 180°
• The derivative of xⁿ is n·x^n-1

Most people find it incredibly dull and forget anything they’re forced to learn very quickly unless it happens to be helpful to them in some way.

Middle Layer: Mathematicians’ Math

Mathematicians’ math is the foundation for Popular Math. Some people would "argue"http://www.maa.org/devlin/LockhartsLament.pdf that this is the kind of math that the word “math” should refer to, and that Popular Math is something else entirely.

What mathematicians do is very much about discovery. They look for patterns, try to determine rules, and invent new ways of thinking about things. It’s a much more open and free activity than Popular Math, but it also takes a lot of genius and creativity to come up with anything new. A different summary of a mathematician’s activity is that they try to come up with interesting questions, and then try to find answers to those questions:

• Answered Questions
  • Is there a relationship between the circumference of a circle and its diameter? If so, what is it?
  • How many prime numbers are there?
  • Does the square root of two have an infinite number of decimal digits?
• Unanswered Questions
  • Are there an infinite number of twin primes? (a pair of primes whose difference is 2)
  • Are there any odd perfect numbers?

Answers are called theorems, and they have to be formally proven before they will be accepted. Mathematicians try to make use of previously proven theorems mixed with insight and luck to prove something new and interesting, which itself becomes a theorem. All the results of mathematics boil down to a giant list of theorems.

This process of proving is (I suspect) what most people who argue for the absolute truth of mathematics are referring to. A mathematical proof is simply a list of statements, where each statement either follows logically from the previous one, or is itself some previously proven theorem, and the last statement is the answer to the original question. In the face of a correct proof, there’s not much that can be done except to acknowledge it. Answering the question of whether theorems reveal absolute truth, however, requires looking at the foundations of proof itself.

Bottom Layer: Logicians’ Math

A logician is concerned with the actual process of proving something. I said earlier that a mathematical proof is simply a list of statements where each statement follows logically from the previous one, or is itself some previously proven theorem. This sentence reveals the two legs that a mathematical proof stands on.

The first leg is the logical connection from one statement to another. What does it mean for one statement to follow logically from another? Most people have strong intuitions about this, but mathematicians aren’t generally comfortable with relying on human intuition if they can help it. Therefore, they rely on a formal system of logic, which is basically a list of rules for how one logical statement can be derived from another. These rules are stated using letters to represent abstract logical statements, and say things such as:

• If we know that A is true, and we know that A being true implies that B is true, then B must be true.
• If we know that A being true implies that B is true, but we know that B is false, then A must be false.

The rules of logic are simple statements that most people would intuitively agree with. Whether or not the rules themselves are true is not something a mathematician could comment on – that question is left to the philosophers. Mathematicians have, however, been able to show that the common set of logical rules are consistent – that is, you can’t use them to come to two contradictory conclusions.

Once logic has been formalized, we have a method for creating statements from other statements, in such a predictable fashion that a computer can do it. Theoretically, the work of a mathematician could be done by a computer (realistically, computers aren’t any good at deciding which theorems are interesting, so the amount of help they can provide is limited), so within a mathematical proof, there shouldn’t be any doubt about whether a connection is logically valid – either you can make the connection using one of the logical rules, or you can’t make the connection.

The second leg of a mathematical proof is the statements that don’t follow from previous statements, but are themselves previously established theorems. The first statement of any proof must necessarily be one of this sort, because there are no previous statements for it to follow from. Of course, if a proof uses a theorem that is actually faulty (i.e., wasn’t correctly proven), then the proof itself will be faulty. But a more glaring issue is a recursive problem: if this theorem relies on some other theorem, what does that theorem rely on? Probably a third theorem, but we could keep asking the question, like a child endlessly asking why. Where did it all start? This is where we get the idea of an axiom.

A system like this has to start somewhere, and every mathematical system has underneath it a set of axioms. Axioms are logical statements just like any statement in a proof, except that the axioms don’t have proofs. They are simply assumed. The idea is that if we start with a handful of well chosen axioms, we should be able to prove all sorts of interesting things. The famous ancient mathematician Euclid realized this, and came up with five axioms for geometry, which say things like “There is only one line between two points” and “circles can be drawn with any center and any radius”. These statements are the starting point for proving theorems within geometry.

Once again we can ask if the axioms are true, but once again the question is left to the philosophers. A natural question is if we can find a set of axioms that we’re intuitively comfortable with and that can serve as a foundation for all of mathematics. Maybe if we accomplished that, we could consider math to be “true enough”. Sure, it’s based on a few foundational concepts, but they’re concepts that nobody would argue with, and so at the very least math is the truest thing we can know. Unfortunately things get a little fuzzy from here.

A mathematical system (i.e., a set of axioms, and a set of rules for deriving theorems) has two fundamental properties: consistency and completeness. I mentioned consistency earlier when talking about the rules for logic. A system is consistent if it is impossible to prove two contradictory statements. If I can, using the axioms and the rules of logic, prove that something is true and then some other way prove that it is false, then the system is inconsistent. An inconsistent system is basically worthless, because one of the necessary rules of logic is that if a contradiction is possible, then anything is true. I had a professor once who wrote on the board “If I am the king of England and I am not the king of England, then I am the Pope” and said that it was a logically true statement. The idea is that if something is true and false, then we can use that to prove that any arbitrary unrelated thing is true. So in an inconsistent system, everything is true, which isn’t very interesting.

The second property is completeness. Completeness means that within a given mathematical system, any statement that can be made within that system is either true or false. This is obviously desirable, because it means that every question has an answer. It also says something else about our axioms besides consistency, something related to usefulness. If I decide to invent a mathematical system where the only axiom is “ten is a number,” then it wouldn’t be of much use. We could prove a few things, such as “If an object is ten, then it is a number” and “Something that is not ten is not a number,” but that’s about it. If somebody asked a question like “is seven a number?” then our system could not answer. It would depend on whether or not seven is ten, and we don’t have any axioms that address that issue. So for our ideal set of axioms, we’d like them to be both consistent and complete.

Unfortunately this turns out to be impossible. Within the last century it was proven that no system which is expressive enough to contain arithmetic can be both consistent and complete. One way of looking at this is that, given some consistent, complex mathematical system, there will always some statement within that system that is simultaneously true and unprovable.

So mathematicians play around with different sets of axioms, and use each for different things. The original geometric axioms of Euclid are just one way of doing geometry (it’s called Euclidean Geometry), and there are others (e.g., “what if lines weren’t really straight?”).

Conclusion

I’ve run through some pretty complex topics quickly, so if anything I said didn’t quite add up, leave a comment and I’ll try to clear things up.

Asking whether some statement is mathematically true is a bit like asking whether some action is legal: it depends on the context. In practice there are very good sets of axioms used in each field of mathematics which are either believed to be, or have been proven to be consistent, and result in all sorts of interesting theorems. For most of the history of mathematics, mathematicians didn’t question their axioms, and math has shown itself to be incredibly useful for all sorts of things. So we shouldn’t draw the conclusion that all of math is a lie. And usually we can get away with doing mathematical things without even thinking about the underlying system. But we can’t philosophize about the underlying truth without acknowledging that underneath everything else, math relies on human intuition, one way or another.

So if math isn’t necessarily absolutely true, why has it been so successful at describing the real world? Excellent question.

A couple days ago I saw a banner ad (I forget where) claiming that Obama was urging the nation to refinance their mortgages. Then later on facebook I saw this:

Apparently the wonderful marketing people that create these works of art think that the critical, thoughtful folks who might be persuaded by ads with ambiguous images of CGI-people will want to do what Obama tells them to do, but only as reported by a banner ad.

My last post was about the different ways that two lines can relate to each other in Euclidean space. Unfortunately the main point I was trying to make was entirely false, so I'm going to explain why.

The error began when I said that it should be possible to position two lines in 4-dimensional space such that they cannot fit into any 3-dimensional subspace. That's not true, and so everything I said after that is also false, and the entire post isn't all that interesting anymore.

The following brief explanation should do fine for anybody with a good understanding of linear algebra:

For those without a linear algebra background, this ought to be convincing enough, though not as mathematically rigorous:

It's not a very difficult thing to see, but I didn't catch myself until after I had posted it. I need to be more careful with my maths.