I want to share some mind-blowing thoughts about seemingly mundane concepts of infinity and how they paved the foundation for math. I use the word mundane because of the strong intuitive feeling that the concept of infinity has no practical use. We live in a seemingly finite universe and issues concerning us don’t involve the use of infinity.
The big question is: “Does infinity even exist in the universe?”
The answer is yes; space and time in the physical universe is infinite. The answer may not make sense because the notion of ‘infinity’ is not made clear. The dictionary roughly defines infinity as, “A sequence or distance without bound.” Again, this definition may not satisfy you with my answer. Let us examine what it means for something to be boundless.
We have a region of space from point A to point B (assume the line is a continuous whole i.e. no discontinuities).
A---------------------------------------B
There are an infinite number of points between A and B. Be careful at what you’re intuition might be telling you, the spatial region is not made up of points, but can be infinitely divisible by them. Points are not parts of a spatial regions they are simply boundaries. To get a better idea of what I mean, imagine what would happen if I got rid of that line that is in-between point A and B. When that line disappears, so does point A and B! They are never actualized until you create a bound or limit, in this case is the color of the line against the background of the page.
Lets actualize some more bounds on our line
A---------B------------------------C------D
Between AB, BC, BD or any other lines there is always and infinite amount of points that is the line can be divided into. The key concept to draw from this is that a point or bound always separates one region of space from another and only have a potential existence unless actualized as discontinuities. Now imagine if the universe is finite in size and there is a boundary or limit at the end of it. But wait, spatial boundaries are always boundaries of one region of space to another (this thought is just to amuse you).
Time is similarly infinite to that of space. You can think of time as a line with two points. One point is 7:35PM and the other is 9:00PM.
7:35PM---------------------------------------9:00PM
Similar to our spatial line, this line is bounded by an instant of time. An instant of time is not a part of time of which time can be divided into. An instant is more like a point on a line; it is the boundary between two periods of time. The interesting reality of time is the fact that it feels as if it will go on forever. At any point in time there is a succeeding moment and a succeeding moment giving the illusion that time is always ‘becoming’. Aristotle called this the ‘potentially infinite’, versus the actual infinite. Space could also be considered potentially infinite only if space and time were considered dependent to one another. If space were potentially infinite it would mean you could be walking in a straight line forever and always covering new ground. Now if space is independent of time then it cannot be potentially infinite without being actually infinite. If you are walking in a straight line you will be covering new ground every time because there is ground to be covered.
If it still doesn’t make sense that space and time can be actually infinite, rather than potentially infinite, than consider what happens when infinity is applied in mathematics. In mathematical theory, there is no room for a potential infinity. I explained how the potentially infinite space does not work when space is inseparable of time. A mathematician doesn’t define a circle as the locus of a points moving in a plane equidistant from a fixed point by supposing the motions of a point moving along a plane in time. He doesn’t define the circle by imagining in time, he supposes it to exist as a completed whole. How can a statement about all natural numbers be true if there is no completed totality of numbers in reality? How can we prove that there are an infinite number of prime numbers without actually counting them? Well we can prove it, but we can’t do it by supposing a potential infinity. Potential infinities are never totalities, so it would be impossible to prove there is a totality of prime numbers if it weren’t actually infinite.
Whether the universe is actually infinite or not may not make a difference on the practical level, but it does make a difference on the theoretical level. Previously, I said that time feels as if it is potentially infinite. After concluding that potential infinities don’t make sense in the mathematical world, I have to cast doubt on my intuitive notion of time. This leads me to conclude that time is thus in a sense wholly actual even though it is not simultaneously present. This may switch your point of view of time being more of a spatial dimension rather than something else.
Wednesday, September 19, 2007
Sunday, September 16, 2007
What is the Meaning of Life?
When most people think about philosophy they think of stoners pondering their existence as they pass the bong. Philosophy is rigorous and as objective as the logic that structures it. Because of the precise nature of philosophy, a huge majority of those stoner conversations are illogical to begin with. Take for example the most cliché one that might pop in your head: What is the meaning of life?
“Dude, the meaning of life is to chilllll... and like help others or some shit” – Some stoner who forgot his own name
Unfortunately, that is not the right answer; in fact, there can’t be a right answer! The question itself is ill defined. Asking, “What is the meaning of life,” presupposes that there is a meaning to life. So lets change the question around:
“Is there a meaning to life?” – Slightly smarter stoner
The answer to this question is a big no because it too presupposes something, but more implicitly than our previous question. Before I reveal what it falsely assumes, we need to exam what ‘meaning’ means.
Humans have evolved categorical memory, i.e. they have the ability to categorize objects in their memory in a hierarchal manner. Your memory of things looks something like this:
Stuff I do when Hungry --> Eat Food --> Cheeseburger --> No Onions
This is obviously a very rough outline of how we think, but it illustrates how we perceive objects meaningfully. A Cheeseburger is meaningless without a human to assert meaning onto it. In my example, the meaning of the cheeseburger is a food I want to eat because I am hungry. There is no one meaning for an objects and varies between person to person. For example, some people might categorize a cheeseburger as something they like to draw instead.
The key insight from this example is that meaning is a human perception of objects. The meaning of an object doesn’t exist without a human to impose that meaning onto the object. Going back to our question, we ask if there is a meaning to life. The question asks if there is something beyond humans that imposes meaning for our lives. Meaning is a human perception of objects. Assuming there is a meaning to our lives is also assuming there is an omniscient being that asserted a meaning for the lives of humans. Unless you believe in God, you will agree that there is nothing out there in the universe that can impose meaning on our lives.
Yup, life is meaningless.
“Dude, the meaning of life is to chilllll... and like help others or some shit” – Some stoner who forgot his own name
Unfortunately, that is not the right answer; in fact, there can’t be a right answer! The question itself is ill defined. Asking, “What is the meaning of life,” presupposes that there is a meaning to life. So lets change the question around:
“Is there a meaning to life?” – Slightly smarter stoner
The answer to this question is a big no because it too presupposes something, but more implicitly than our previous question. Before I reveal what it falsely assumes, we need to exam what ‘meaning’ means.
Humans have evolved categorical memory, i.e. they have the ability to categorize objects in their memory in a hierarchal manner. Your memory of things looks something like this:
Stuff I do when Hungry --> Eat Food --> Cheeseburger --> No Onions
This is obviously a very rough outline of how we think, but it illustrates how we perceive objects meaningfully. A Cheeseburger is meaningless without a human to assert meaning onto it. In my example, the meaning of the cheeseburger is a food I want to eat because I am hungry. There is no one meaning for an objects and varies between person to person. For example, some people might categorize a cheeseburger as something they like to draw instead.
The key insight from this example is that meaning is a human perception of objects. The meaning of an object doesn’t exist without a human to impose that meaning onto the object. Going back to our question, we ask if there is a meaning to life. The question asks if there is something beyond humans that imposes meaning for our lives. Meaning is a human perception of objects. Assuming there is a meaning to our lives is also assuming there is an omniscient being that asserted a meaning for the lives of humans. Unless you believe in God, you will agree that there is nothing out there in the universe that can impose meaning on our lives.
Yup, life is meaningless.
Friday, September 14, 2007
5 Creative Ways to Deal with Kids
1. Magic Spray
Kids have the tendency to get injured in every opportunity possible. About 90% of their injuries are trivial, but their screaming and yelling try to convince you otherwise. You can probably recall a few instances when you’ve witnessed a kid trip over their own feet – then they get up slightly stunned, but not crying or screaming. At their first glimpse of blood the silence instantly disappears. Luckily, the placebo effect works on kids 95% of the time. Small kids can’t take sugar pills so you can create your own placebo. Just take an old spray container, fill it with water, and label it, “Magic Spray.” Then create some story about how you found it in a magic cave and how it heals all wounds. Spray it on the kid’s injury and watch them magically stop crying!
2. Imaginary Super Hero
Kids have an over active imagination, sometimes it actually scares them. Some variation of a Boogie Man could keep them up at night. If it’s not helping to simply tell the kid there is no such thing, then you need to make them stop thinking about it all together. Tell them to imagine the biggest and strongest super hero imaginable. Every time their bad thoughts pop up, tell them to imagine their super hero crunch up their thoughts like a piece of paper and throw it away in a trash bin. And remind the kid that the super hero has super speed and will never waste time to destroy their bad thoughts.
3. Superstition
Sometimes you just don’t have the time to explain to a kid why they should stop a certain behavior. For example you are at the supermarket and it’s your turn to pay for your groceries when your child starts opening candy bars. You don’t have time to lecture them so just say it’s bad luck. Kids will obey any superstition thrown at them... for about one minute. The one minute will give you enough time to pay for your groceries and will stop the kid’s bad behaviors momentarily.
4. Operation Hopelessness
Have you ever had your child ask you for just one more quarter for the arcades? And after you give them that quarter they run back to ask for another? To stop this aggravating behavior you must hinder their persistence. Make a huge fuss about how you don’t want to give them another quarter (or any other object they want). Then after about five minutes you finally give them the quarter. They will use it up in a matter of seconds and come back discouraged to beg for another. This way you save yourself half an hour dealing with an angry and tireless kid.
5. Embarrassment
Ever have your teacher write your name on the board because you were talking in class? This is a widely used strategy because it works. It is effective because it’s embarrassing and singles you out. If you are in a public place with a child who is out of control, give him the attention he doesn’t want. Yell out their name loud enough for others to hear and sustain a loud voice if they continue misbehaving. If you don’t want to embarrass yourself in the process, just threaten them with embarrassing things you’ll do to them.
Kids have the tendency to get injured in every opportunity possible. About 90% of their injuries are trivial, but their screaming and yelling try to convince you otherwise. You can probably recall a few instances when you’ve witnessed a kid trip over their own feet – then they get up slightly stunned, but not crying or screaming. At their first glimpse of blood the silence instantly disappears. Luckily, the placebo effect works on kids 95% of the time. Small kids can’t take sugar pills so you can create your own placebo. Just take an old spray container, fill it with water, and label it, “Magic Spray.” Then create some story about how you found it in a magic cave and how it heals all wounds. Spray it on the kid’s injury and watch them magically stop crying!
2. Imaginary Super Hero
Kids have an over active imagination, sometimes it actually scares them. Some variation of a Boogie Man could keep them up at night. If it’s not helping to simply tell the kid there is no such thing, then you need to make them stop thinking about it all together. Tell them to imagine the biggest and strongest super hero imaginable. Every time their bad thoughts pop up, tell them to imagine their super hero crunch up their thoughts like a piece of paper and throw it away in a trash bin. And remind the kid that the super hero has super speed and will never waste time to destroy their bad thoughts.
3. Superstition
Sometimes you just don’t have the time to explain to a kid why they should stop a certain behavior. For example you are at the supermarket and it’s your turn to pay for your groceries when your child starts opening candy bars. You don’t have time to lecture them so just say it’s bad luck. Kids will obey any superstition thrown at them... for about one minute. The one minute will give you enough time to pay for your groceries and will stop the kid’s bad behaviors momentarily.
4. Operation Hopelessness
Have you ever had your child ask you for just one more quarter for the arcades? And after you give them that quarter they run back to ask for another? To stop this aggravating behavior you must hinder their persistence. Make a huge fuss about how you don’t want to give them another quarter (or any other object they want). Then after about five minutes you finally give them the quarter. They will use it up in a matter of seconds and come back discouraged to beg for another. This way you save yourself half an hour dealing with an angry and tireless kid.
5. Embarrassment
Ever have your teacher write your name on the board because you were talking in class? This is a widely used strategy because it works. It is effective because it’s embarrassing and singles you out. If you are in a public place with a child who is out of control, give him the attention he doesn’t want. Yell out their name loud enough for others to hear and sustain a loud voice if they continue misbehaving. If you don’t want to embarrass yourself in the process, just threaten them with embarrassing things you’ll do to them.
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