Logical Implications

If we were in the fourth century and I told you that the Earth was flat, a reasonable question you could ask me would be, “If you just continued moving in one direction, would you just eventually fall off?”

Now, that wasn’t what I said at the beginning. I simply said the Earth was flat. However, the logical implication of a flat Earth means that there is an edge to it (unless, of course, it is infinite). Apart from the infinity loophole, there is no other way out of this option. Either the Earth is infinitely flat or it has an edge.

Suppose you want to test my question because you are sure you are correct. You say, “Alright, we will continue moving in one direction, and we’ll eventually arrive at the edge, proving what I say is true.” From there, the we begin our journey. Assuming we can somehow move in exactly one direction without being obstructed, we would eventually return where you left on. Therefore, this would invalidate my hypothesis. The Earth could not be flat, or else we would have detected an edge.

So that means it’s flat, right?

Well, not so fast.

An astute observer could say that the Earth is a cylindrical shape, and you and I simply walked along the lateral face of the cylinder. However, barring this cylindrical Earth scenario, the Earth must be some sort of solid that one can walk around and arrive back at the same point once again – a sphere.

What I want to illustrate in this example is that one can get into some pretty interesting scenarios by taking statements to their logical end. It’s kind of like trying to reconstruct the pathway of a conversation, but in the forward direction. And almost always, the person who first made the statement will be surprised with the direction the conversation or train of thought has taken based off of one simple statement.

This has particularly relevant implications as our world gets more and more global. While issue like climate change, maintaining our environment for future generations, and spreading resources more equitably may seem far off and too abstract for many of us to think about, they are issues that will have profound consequences. Therefore, saying that you care about the environment but initiatives to preserve it will have to be put off for this year due to lack of funds means you are choosing to not allocate resources to an important issue. Perhaps it’s not the most pressing issue to you, but it certainly won’t improve (and has the possibility to be worse) through inaction. Therefore, the logical consequence of one person not doing something today to help the environment (multiplied hundreds of millions of times) results in an environment that does not improve.

More generally, this situation of saying something without thinking of its logical implications happens all the time in the media. Statements are made which may sound alright on their face, but are disastrous if one really thinks them through. It might sound like heresy to raise taxes because it affects you right now, but if the quality of your life (and others in the world) improves in the long term, is it worth accepting?

Climate change. Space and other life in the universe. Genome sequencing and editing. The allocation of resources. These are the difficult questions of our time, and quick responses might not be the answer. At the very least, we should require more thinking about them than we currently do. Our current answers tend to be quickly thought up, and averse to change because it’s not the status quo.

Instead of being the first or the loudest, how about we be the first to think through the logical implications of our actions and words?

Cardioid

[latexpage]

In my multivariable calculus class, I learned about various types of curves that take different formats, from parametric to polar coordinates. Because the course was a sort of introduction to these notions, we weren’t given the “full” explanation on a bunch of these curves. Therefore, I want to touch on some interesting aspects of this category of curves (called limaçons) below.

First, I was only introduced to the curve by having the formula given to me. The formula was in polar coordinates, and is as follows:

As one can imagine, this formula makes little sense to a student when they haven’t learned much about these types of curves. However, what if they looked at this instead?

By AtomicShoelace

When I first saw this, I just thought, “Oh, that’s what it is.” The curve makes so much more sense when viewed in this type of frame. I am confused as to why this animation wasn’t shown to my class, because we saw a similar animation for the cycloid.

What’s cool about the cardioid is that it’s a curve that has a very nice formula to compute its arc length. The formula is $L=16a$.

In terms of the area of a cardioid, it’s formula is pretty nice, too: $A=6\pi a^2$, which means it’s six times the area of the circles above. I always find it neat to see comparisons of different shapes to see how they relate to one another.

More to come on these kinds of curves and ways to represent them.

Reference: Wikipedia

Qualifying Language

I used to hate reading a text when someone would write with qualifying language (this was also prevalent in how many people I looked up to spoke). Why couldn’t they just go ahead and say the thing that they wanted to say? Why did their have to be language such as “this suggests” or “I can’t say for sure”? It would drive me insane because I believed that writing that made an impact doesn’t need this extra baggage surrounding statements.

Before then, I was reading a lot of material on domains such as design, writing, and generally creating some type of art. What this meant was that the goal was to connect with the audience, and this was best done in direct language. There was no need to say things in a roundabout way. Instead, the artist could take a direct stab at the issue and touch the person viewing the piece.

This is the mindset I brought with me when I started to read more material by scientists, and that’s where I started seeing all this qualifying language. Like I said, it did not make sense. Why wouldn’t they just communicate without putting clauses on all of their statements?

Slowly, I found the answer: scientists are trying really, really hard not to fool themselves. In a nutshell, a good way to explain the scientific process is that we are trying to look for ways that we are fooling ourselves. Throughout history, we’ve seen over and over that humans can be easily fooled into thinking something is true when there is actually a much larger picture. I highly doubt our ancestors thought there was anything other than what they could see with their eyes (except for perhaps a god). Then we smashed this perception in the 19th century by discovering that light is a wave and can have wavelengths that we cannot perceive.

In particle physics, we’ve seen a complete makeover in regards to what we think the universe “truly” is like. We went from just seeing matter to thinking about the atom to breaking that apart into fundamental particles. Finally, we pushed that even further by saying that these fundamental particles are part of a wave function. In the end, we’ve gone from what we can see to having the entirety of the universe being composed of wave functions.

Obviously, this is a radical change with respect to our first thoughts about the universe. Therefore, what we’ve found is that the scientific process has shown us just how wrong we are. As such, I believe most scientists have a certain fraction of skepticism in their minds when approaching any kind of phenomenon. It’s not personal, it’s that history has shown us that it is the safe bet to make.

The truth is that a scientist should be willing to believe anything, as long as the requisite proof is supplied. If a scientist won’t believe a statement after sufficient proof is given, then there is a problem, but that tends to not happen when someone says a comment like this.

What I find fascinating is that, if the person really believed in what they said and could say that it makes sense to anyone, there shouldn’t be a problem with supplying good evidence. If not, there should be at least an explanation as to why evidence is hard to come by.

Remember, lack of evidence doesn’t mean a statement is false, but it sure won’t convince me to believe in it.

Unfortunately, I get into many situations in which those claiming extraordinary things cannot bring any proof, and then they get upset that I won’t believe them. However, I couldn’t do anything better. It’s difficult to accept a proposition on the basis of someone just telling you so. As a science student, I’ve learned that this is a terrible way to go about finding knowledge about our universe. Trusting the human senses because they feel right might seem okay intuitively, but that’s the problem. Humans don’t have an intuition that is good for some of the deepest questions about the universe, since they are happening at a realm that is basically invisible to us. Therefore, we must safeguard against any attempt to “reason things out” without actually using tests and logic and theory. Without the scientific method, we would still believe that the world is only made up of components we can see.

So what does this have to do with qualifying language?

It means that scientists are careful about what they say. It’s fun to say something with certainty, but that technically never happens with science. Science is a process in which we can give ourselves a “good idea” (and sometimes a great idea) about the world, but we can never be one hundred percent certain. This is what makes science what it is. Consequently, the responsible scientist will use qualifying language because they know it’s good to be as specific as possible about what we know. Apart from perhaps a few minutes of fame, there’s absolutely no long-term reason that would make it a good idea to oversell a scientific achievement. It will always catch up to you, and so it’s not worth it. Therefore, scientists are fond of using qualifying language in order to remind us that they don’t have all the answers.

Now, I always shake my head when I see someone write without qualifying language, particularly because it’s not completely honest. The truth is almost never absolutely declarative, and I believe we’d do much better to remember this.

Qualifying language isn’t a sign of weak communication or “not believing in one’s message”. It’s about being honest about what you know and what you don’t.

Changing One Block

If I ask many people, I can probably get one person to admit that they believe in something that isn’t strictly speaking scientific. It could be large parts of our universe, such as the existence of an afterlife or a soul, or it could be smaller things such as our horoscopes actually telling us information about ourselves. There are many beliefs that humans have, and it isn’t uncommon to find someone harbouring at least one of these beliefs.

That said, I find it interesting how these people respond when being prodded about their beliefs. What I usually find is that people tend to not think about the logical consequences of their beliefs. Instead, when confronted with these inconsistencies, they will simply respond with: “I didn’t say that would happen. I’m only talking about this specific thing being true.”

As Sean Carroll explains in his book, The Big Picture, people don’t want to think about the physical laws that will be violated because of their beliefs. For example, the soul is a particularly enduring belief that doesn’t get shaken easily. People will say we have a soul because each person has this certain “essence”. Unfortunately, they don’t think about the consequences a soul would have. We can probably agree that a soul interacts with the physical world through our bodies, yet there is no way our laboratory tests can detect them. This seems to be as much of an inconsistency as claiming telekinesis is possible.

Then of course, the reason people believe in a soul usually has to do with the afterlife. Therefore, the soul is supposed to exit a body when a person dies and moves (floats?) somewhere else. Once again, this is a logical inconsistency, because we cannot detect any such “particles” a soul would be made of. As Sean Carroll writes in the book, you’re perfectly capable of making the claim that we just don’t fully understand the situation. However, to say that you need to also explain how our current theories of physics that are so successful at investigating are also wrong, and how your ideas work better. Then, scientists will jump onboard with you.

As it stands, though, including any kind of particle that is supposed to encapsulate a soul would ruin many of our physical theories. And once again, that is not a bad thing. But you can’t say that a theory which predicts many phenomena about the world perfectly right is incorrect and have no other solution. You will just be ignored. You can’t change one piece of an interconnected puzzle because it doesn’t suit you, and ignore the rest. One claim affects the next, so modification must build on what has already proven successful.

This is a very important point to make. If we consider the classic example of Newton’s law of gravitation, we know that it is not as “correct” as it could be. However, the equation isn’t wrong in the sense that it doesn’t work. It does work, albeit in certain conditions (which also happens to be most conditions humans find themselves in). Therefore, Newton’s law of gravitation will always work in these settings. Carroll also points this out in his book: the law will work just as well in a millennium as it does today.

When Albert Einstein formulated his theory of general relativity, he didn’t look at Newton’s work and toss it all out. Instead, he built upon it. Said differently, one can use general relativity in a certain domain and recover what Newton found. Even though the equations were different at the outset, the latter got back to the former, and this is a very important point in science and mathematics. You can’t break logical relationships by a whim that something doesn’t “feel” right.

Let’s take a simpler example. Imagine I were to give you the points (1,5) and (2,8). I know that the line going through these two points is $y=3x+2$, which means the y-intercept is 2. However, what if someone argued with you and said that the y-intercept is actually 4, and didn’t want to listen to anyone saying otherwise?

This is an example where it is obvious that they are making a logical mistake. In mathematics, there is no wiggle room, so the inevitable conclusion when creating a line that goes through those two points is one that also goes through (0,2). You cannot argue it.

In science, it’s a bit easier to argue a proposition, but you cannot destroy a theory without recovering what has worked in it. At the very least, you have to be able to explain why what was achieved before isn’t valid.

The problem, of course, is that people aren’t directly challenging these laws. Instead, they are talking about seemingly innocent concepts, like a soul. Unfortunately, the presence of a soul would have cascading affects through physics, eventually creating the situation of our best theories being wrong.

This is why you cannot take a block out of a “building” of scientific knowledge. Every piece is important, and changing one thing can have enormous consequences on different aspects that most people won’t think about. That’s why it’s always important to question the concept someone is proposing, because often the logical implications have not been fully though through.

When a person tells you their ideas about the world, there is no need to disregard with them. If they are as misguided as you think they are, then simply interacting with the ideas will expose their weaknesses, making it unnecessary to get into a heated shouting match.