Infinity Has Its 'Limits'
Infinity is a theoretical abstraction which makes calculus possible. Calculus is used as the basis for descriptions and predictions of behavior of objects in the physical world, (as differential equations). The same equations define a mass-spring system, an electrical circuit, processes of oscillation in chemical reactions or bacterial populations.
The analysis of all of these types of systems depends on the concept of an infinitesimal distance between points or samples–as the basis for a differential, or integral.
A circle is a polygon with an infinite number of sides. Area under a curve is the result of adding up the areas of an infinite number of infinitely narrow rectangles. You can’t do this in fact, but you can do it with an integral. Slope of a curve is determined as the slope of a line between two infinitely close points. You can’t actually create two infinitely close points, but you can do it with a derivative.
The concept of infinity does exist. The hotel problem, and other examples are integer-based logic misapplied to the concept of infinity.
Likewise, discussions of a ‘first cause’ are attempt to define what the rules of a universe might be, when it was in a vastly different configuration than it is today. There may have been a great deal which took place before or outside of this expanding bubble which was created by the Big Bang.
But for us, who exist in this particular causal universe, it is for all intents and purposes an impenetrable event horizon. The universe may encompass all that is, it may be part of a multi-verse, which may have existed forever.
Logical attempts to describe what happened before the Big Bang or what kind of cause was needed to set it in motion are as futile as trying to describe infinity with the ‘hotel’ analogy. We need to wait and see what we will discover in the future. We cannot demand premature closure or draw conclusions from insufficient information.
In order to discuss this at all. people must also be willing to come to grips with certain abstractions.
Yes, infinity is a tool of math, I acknowledge that. But all that this math tells us is that IF such and such approaches infinity such will happen. I don’t know if I have heard a calculus equation stating WHEN such and such reached infinity. I could be wrong, I didn’t study much calculus.
As for your circle, I am pretty sure that your description of a circle is, in fact, a dot… at best. “Slope of a curve is determined as the slope of a line between two infinitely close points” — how long is a line between two infinitely close points? What is that distance? It isn’t zero. It isn’t infinity. It isn’t negative infinity. I would argue that to infinitely close dots is rather the distance between to dots as they approach zero distance. As the approach zero distance. As soon as you define two points the distance is finite. If you continue to approach zero and define a closer spot, the distance is finite. You can reach zero. You can reach small, finite, points. But you can never reach an infinitely small distance. It is a purely theoretical tool to describe a very finite actuality.
“We need to wait and see what we will discover in the future. We cannot demand premature closure or draw conclusions from insufficient information.”
So therefore all discussion on all subjects, even history, should be halted until such time that we know everything. I hear the defeatist voice of Hume again.
"In order to discuss this at all. people must also be willing to come to grips with certain abstractions."
Certainly you don’t mean God. I do assume you consider god an abstraction. What abstractions do we include? What do we rule out? Do we just include the abstractions which formulate the world the way in which we would like it to be?
@Draracle “I could be wrong, I didn’t study much calculus.”
Yes, you need to know it to understand what I’m saying. Limit equations describe what happens as you approach infinity. Derivatives and integrals describe what happens when you get there. That’s why I said you have to be able to grasp the abstraction. More people should study math. It would be a better world.
For example, if you want to calculate pi to an infinite number of decimal places, it is the result of an infinite series of numbers: You add a small amount, subtract a smaller amount, add a smaller amount, subtract a smaller amount, and so on. With each term, you get closer and closer to the absolute value of pi, but you never get there. But we can say that pi describes an actual ratio between the diameter and circumference of a circle. You can never get the exact ratio, BECAUSE it requires an infinite number of terms. But this is still how the ratio is defined.
“As for your circle, I am pretty sure that your description of a circle is, in fact, a dot… at best.”
No. Start with a triangle. Add more sides until you get to, say, an octagon. It looks more like a circle than a triangle. Now add more sides until you have 100 or 1000. At 1000, you have what amounts to a circle, unless you get really really close to it. But if you draw an actual geometrically perfect circle, you have a polygon with an infinite number of sides.
Everything in the universe is based on waves (wave-particle duality). All waves are made up of varying amounts of sine waves, (when broken down by Fourier analysis–the sinusoid being the same function that describes a circle.) By the way, this is what makes the data compression in .mp3’s possible.
Calculus doesn’t work without the concept of infinity. Without it, we couldn’t know anything much about finite matter. When I say “you have to grasp the abstraction”, I’m talking about how infinity applies to finite objects and waves, and helps us analyze them. I’m certainly not talking about god.
The slope of a curve can be calculated for a point. The point, in calculus, when you take the derivative, is the same as two infinitely close points. But it still has a slope. This is the essence of the concept of the derivative.
For example: The derivative of the velocity of an object is the acceleration. The second derivative of the velocity is the rate of change of acceleration. Without recognizing the property of infinity, we could never analyze dynamic systems. If there are any other engineers on this thread, please help me explain this to Draracle.
“So therefore all discussion on all subjects, even history, should be halted until such time that we know everything.”
No. I’m not saying don’t discuss it. I’m saying don’t draw conclusions about it–particularly about what may or may not be the “first cause.” Especially, I’m saying don’t take the words of theologians for it.
There may in fact be a creator, or more likely, a creative force or agent. We don’t know, and we can’t say right now. In the words of Richard Dawkins quoted in Time magazine speaking to Francis Collins:
“My mind is not closed, as you have occasionally suggested, Francis. My mind is open to the most wonderful range of future possibilities, which I cannot even dream about, nor can you, nor can anybody else. What I am skeptical about is the idea that whatever wonderful revelation does come in the science of the future, it will turn out to be one of the particular historical religions that people happen to have dreamed up. When we started out and we were talking about the origins of the universe and the physical constants, I provided what I thought were cogent arguments against a supernatural intelligent designer. But it does seem to me to be a worthy idea. Refutable–but nevertheless grand and big enough to be worthy of respect. I don’t see the Olympian gods or Jesus coming down and dying on the Cross as worthy of that grandeur. They strike me as parochial. If there is a God, it’s going to be a whole lot bigger and a whole lot more incomprehensible than anything that any theologian of any religion has ever proposed.”