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Infinity Has Its 'Limits'

I responded to Draracle over at 10,000 Reasons to Doubt the Fish. The challenge was to answer the Kalam argument: which argues against the concept of an infinite universe without first cause.

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.


Draracle responded:

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.”


Comments (6 comments)

Dennis Fisher / November 7th, 2006, 11:44 am / #1

Zero, is another Mind Blower concept both in Math and in philosophy. Have you looked into that?

BlackSun / November 7th, 2006, 12:42 pm / #2

Yes, it is kind of the same discussion. If you have divisibility–i.e. any particle can theoretically be divided into two or more smaller particles, then you can never get to zero.

In a way, zero and infinity are the same concept. As long as you are talking about something tangible, you can’t get to either one. Also, 1/0 = infinity, and 1/infinity = 0.

What is interesting is that concepts which govern macroscopic objects do not seem to govern at the particle level. For example, quantum mechanics demonstrates certain paradoxes, but these cancel out when you get to the macroscopic world.

I’m getting a little over my head in terms of theoretical physics, but it seems the paradox of never being able to get to zero may be resolvable through wave/particle duality. But then maybe it would be just as hard with energy, since any quantity of energy should still theoretically be divisable. Unless quantum theory blows that notion out of the water. Calling Stephen Hawking…

Check out this article about the quantum mechanics:

http://www.csicop.org/si/2006-04/quantum-mechanics.html

Dennis Fisher / November 7th, 2006, 9:05 pm / #3

My thoughts on Zero and Infinity

Zero, Nothing, can equal Something and Everything. Or put another way “why is there something instead of nothing?”

Here we all sit trying to figure that one out.

When there is nothing (ZERO) in the universe one second and then there is something the next second. Everything originates from (ZERO). Big Bang!!! We got the Time and Space or this Universe.

“Time” and the concept of infinity is very interesting to me. It is impossible to cross infinity most would agree. Imagine people walking in a straight line trying to cross infinity. No matter how long they travel they will never get to the end because they have infinity as the distance yet to travel. If the universe had no beginning in time, then the present moment would have been preceded by an infinite number of earlier moments. If this is so, then the present moment could never have arisen, because it’s impossible to cross infinity. But, the present moment is here. It had to have been preceded by a finite number of earlier moments, and that’s what it means to say that the universe had a beginning. So now it is time to sum up my argument for there being Creator. The First Law of Thermodynamics asserts that matter or its energy equivalent can neither be created nor destroyed under natural circumstances. Matter cannot create itself and, in the real world, cannot arise from nothing or Zero. Within the bounds of natural law all effects must have a cause. Something does not come from nothing. It seems highly unlikely for the universe to simply have popped into existence without a cause. It is more reasonable to believe that the universe had a beginning that was caused into existence by a timeless, changeless, uncaused being. Call it GOD or whatever. There was a Big Bang and the Creator must have lit the Match.

BlackSun / November 8th, 2006, 9:24 am / #4

Dennis,

“Imagine people walking in a straight line trying to cross infinity.”

You’re still thinking in terms of the hotel analogy. Inifinity is a different kind of number. 2 x infinity is still infinity. 1 + infinity is still infinity.

But this is sort of beside the point. When you make a statement like:

“It seems highly unlikely for the universe to simply have popped into existence without a cause. It is more reasonable to believe that the universe had a beginning that was caused into existence by a timeless, changeless, uncaused being. Call it GOD or whatever. There was a Big Bang and the Creator must have lit the Match.”

…it is conjecture. Science does not say “it seems highly unlikely…it is more reasonable.” Science separates each assumption, and requires evidence before making a conclusion.

Your statement represents your conjecture, your opinion of what might have happened. But as such, it is no more true or provable than my earlier statement about the 10-dimensional green dragon.

As beepbeepitsme said, this is special pleading:

SPECIAL PLEADING (wikipedia): In the classic distinction among material, psychological, and logical fallacies,[1] special pleading most likely falls within the category of psychological fallacy, as it would seem to relate to “lip service”, rationalization, and diversion (abandonment of discussion). Special pleading also often resembles the “appeal to” logical fallacies.

Dennis, when you say, “it is highly unlikely…it is more reasonable to…” without providing evidence, you are essentially abandoning the discussion for a conclusion which represents your existing biases.

It’s OK to have opinions. But you cannot insist your opinion is factual without providing something more to back it up.

Dennis Fisher / November 9th, 2006, 9:58 am / #5

Sean,
Are you saying that i am insisting on my opinions?, because I used phrases like “highly unlikely” and “more reasonable”? In my opinion those are not words of absolutes. These phrases represent in my opinion, the acknowledgment of 2 or more competing ideas, you disagreed? When you have competing ideas there is always room for discussion I think.

Therefore, I think you are wrong in stating that I have “essentially abandoning the discussion for a conclusion” also, that I am insisting my opinions by using words like “highly unlikely” to formulate my argument.

Sean, if you would please reread my last post, i think you would find my post is structure like this… Fact / observation / evidence, and my conclusion.

You or anyone can disagree with my facts or my conclusions. I am not claiming absolute knowledge here. lol What a mistake that would be. I truly respect your opinion and value your input.

Also, I mildly disagree with what you said here “Science does not say “it seems highly unlikely…it is more reasonable.” I think science in the past has done just that.

My example is the “Black Hole theory” Which is accepted by most of the Sciencetify community.

Like God the Black Hole is unseen and not proveable. We believe the Black Holes exist due to the laws of “cause” and “effect” which is the evidence it leaves on surrounding stars.

My point here is Science has to draw conclusions and use conjecture base on observation or facts to formulate its theories and so should everyone.

BlackSun / November 9th, 2006, 12:12 pm / #6

Dennis, you still have a little misunderstanding about the scientific method. When science reaches a conclusion, it reaches a limited conclusion, based only on the available evidence.

The opinion you expressed was a sweeping generalization:

“It seems highly unlikely for the universe to simply have popped into existence without a cause. It is more reasonable to believe that the universe had a beginning that was caused into existence by a timeless, changeless, uncaused being. Call it GOD or whatever. There was a Big Bang and the Creator must have lit the Match.”

You also fell prey to the “god of the gaps” trap: In other words, you can’t understand how something was created out of nothing (we don’t even know if that’s how it happened), and then you said since we don’t understand it, therefore GOD lit the match.

When you refer to how science deals with black holes, you also misinterpret the level of certainty with which scientists hold their views.

Scientists carefully gather very detailed observations, make theories which are very limited in scope, and are extremely careful before reaching conclusions. They then reexamine any conclusions repeatedly as soon as any new evidence is uncovered.

Stephen Hawking famously lost a bet about whether or not information could escape black holes. But he was not too proud to admit he was wrong, and paid off on the bet. You can look this up.

http://www.newscientist.com/article.ns?id=dn6193

My point is, that you were falling into a trap of thinking that a very broad-stroke reasoning, of something that made sense to you intuitively, was some kind of scientific proof.

I’m simply trying to keep the boundaries straight between science and conjecture.

After all, the question of how the universe sprung into existence is the oldest and biggest unresolved question known to man. Do you really think you can resolve it in one assumption-filled paragraph?

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