Can Infinity Be Measured Through Geometric Concepts?

[DaMeekie]

This idea has been bothering me for a while, it started when I thought that if there was an infinite amount of space inside of an inch. ( or even any measurement in the physical world ) Then I thought that maybe that's not a fair argument on the basis that quantum theory says planks length "h" could be considered the smallest measurement/moment. Then I thought about circles and where this idea becomes relevant. Honestly a circle doesn't have 360° that's just what the Sumerians gave the circle to make it easier to work with. By definition a circle is infinitely symmetrical giving it an infinite amount of degrees.

We all know that infinity is a really big storm when trying to give it an actual definition, but when using a circle even when you have no connection to the real world is always finite yet infinite at the same time. We all know that no matter what time or where you are in the universe a circle with a diameter of 1 will be the same ALWAYS. Now say this circle is devised by two, would that be equal to deviding an infinite in half? There would still be as many points on either side of the circle (being infinite) but all together there are twice as many while still equal to infinity. Any insight or even ideas towards what could explain this would be much appreciated.

 

[DaveC426913]

Infinity is not a number. You cannot perform numerical mathematics on it and get the answers you expect.

Yes, if you divide a circle composed of infinite points into 2, you will get infinitely many points in each half.

 

[Mark44]

This idea has been bothering me for a while, it started when I thought that if there was an infinite amount of space inside of an inch.

There are an infinite number of real numbers in any nonzero-length interval on the number line, but there is not an infinite amount of "space" however you define that.

( or even any measurement in the physical world ) Then I thought that maybe that's not a fair argument on the basis that quantum theory says planks length "h" could be considered the smallest measurement/moment. Then I thought about circles and where this idea becomes relevant. Honestly a circle doesn't have 360° that's just what the Sumerians gave the circle to make it easier to work with. By definition a circle is infinitely symmetrical giving it an infinite amount of degrees.

I don't think you understand what the 360° measures. If you have a radius of the circle that intersects the circle at some point, and cause the radius to rotate, it will come back to the starting point after 360°. The degree measure comes from the Babylonians, it's true, but there are other measures of angles, including $2\pi$ radians, and 400 grads, a measure that the Russians used to use (and maybe still do, I don't know).

We all know that infinity is a really big storm when trying to give it an actual definition, but when using a circle even when you have no connection to the real world is always finite yet infinite at the same time. We all know that no matter what time or where you are in the universe a circle with a diameter of 1 will be the same ALWAYS.

So what?

Now say this circle is devised by two, would that be equal to deviding an infinite in half?

You mean "divided by" two. And in what sense do you believe a circle is infinite? Its area is finite, its circumference is finite. If you divide in in half, you get a figure with half the area.

There would still be as many points on either side of the circle (being infinite) but all together there are twice as many while still equal to infinity.

There's a huge difference between the number of points in any interval of nonzero length. You are confusing the ideas of the cardinality of a set (the number of points) with the length of the interval.

The interval [0, 1] includes all of the real numbers from 0 through and including 1, which includes an infinite number of points. However, the length of this interval is 1. If you extend the interval from 0 to 2 (the interval [0, 2]), there are also an infinite number of points -- exactly the same as before -- but the length of the new interval is 2. When you're counting things in an infinite set, the usual rules don't apply. How we determine that two infinite sets have the same number of points (the same cardinality) is by showing that there is a one-to-one function that maps each point in one set with its corresponding element in the other set. In this case the function is f(x) = 2x. Take a number in the interval, say .6813, and use this function to find its counterpart in the other set.

Any insight or even ideas towards what could explain this would be much appreciated.

 

[aikismos]

Any insight or even ideas towards what could explain this would be much appreciated.

Infinity is a construct of the mind. Infinity is nothing more than a recognition that regress in calculation can occur continuously and without end. Infinity is your mind's attempt to traverse a circle to an endpoint. The concept of infinite regress is one of the most important conceptual underpinnings in math, physics, and philosophy because all minds are capable of falling into it for some period of time. It sounds to me like you're trying to wrap your mind around the fundamental nature of the "stuff" the universe is physically made of, so it's important to understand that in quantum physics, you play the role of the observer, and your thoughts about the universe are observations.

You are right that 360 is arbitrary (although my recollection is it's the approximate length of the earth-year). You're right that can divide a circle, but a circle is not infinite. It has a finite circumference which is a bijection onto the diameter. When you say that you are dividing infinity in two, what you really are saying is that you are taking a geometric shape capable of being traversed or subtended within an infinite regress, and splitting it into two shpaes capable of being traversed or subtended within an infinite regress. And it doesn't even have to be a circle.

The infinite density property of the reals allows an infinite amount of division between any two points (let's say 0 and 1). Even the very definition of a Dedekind cut on the reals which makes the irrational numbers inclusive relies on infinite regress. There are an infinite number of points on the ellipse or the parabola or any locus of points defined over a real interval.

Georg Cantor came up with a clever way to show that not all systems that undergo infinite regress are identical, and that they themselves may or may not be paired (infinitely so) with the construction of the naturals. But I have to agree with the others. Infinity is not a number, it's a process. In space, it is the process of achieving the continuous gradation of length. In space-time, it is the process of motion. But ultimately, as objective as the physical reality of space or space-time are physically, the infinite in respect to the physical process is a mental process.

 

[DaveC426913]

Infinity is a construct of the mind.

No, it's a mathematical artifact, it just has its own rules.

 

[aikismos]

No, it's a mathematical artifact, it just has its own rules.

There's a reason rocks don't have mathematical constructs. They have no mind. No mind, no math.

 

[DaveC426913]

There's a reason rocks don't have mathematical constructs. They have no mind. No mind, no math.

Not really sure what that means. Planetary orbits have no mind either, but the math is pretty clear cut.

You can work with infinities, you just can't expect them to operate like real numbers. They have their own rules, like every other system, such as reals, imaginaries, etc.

 

[aikismos]

And where is the math clear cut? In. Your. Mind. (Or mine in this conversation.)

Math is a cognitive process that has scientifically been validated to originate from the computational properties of neurons. Subitization is a well-recognized property of the human brain. Pairing and counting and doing so in an infinite number of steps are all properties of the mind. Planets don't have math. People have math to describe planets in their mind. The question of the infinite is a mathematical concept because it is fundamentally about mapping the natural numbers onto elements of sets. There's a reason that no written equations existed before man evolved to write them. Systems of computation simply did not exist (at least on earth) before the evolution of the animal brain. Bacteria don't count. Fungi don't count. Plants don't count. Birds, dogs, and people do.

The question of the relationship between mathematics and infinite is a question of the theory of computability of physical systems. The very nature of computation requires an information processing system. The mathematical impact of infinite regress is relevant withing these systems precisely because it has to do with the capacity of the system to deal with symbols effectively. In computer science, an infinite loop is deadly. In the generation of primes, how to generate an infinite set effectively and validate it (with the AKS primality test, for instance) has to do with the algorithmic and computational complexity of the problem and the hardware/software system designed to embody the theory.

Anyone who stares at the universe and is astonished to see math is confusing the forest for the trees. In control system theory, this called a second-order system, one in which the system is able to mathematically model the external reality inside. But the history and processes of mathematics are very clear. Math comes from the mind. A famous semiotician put it succinctly. The map is not the territory. Alan Turing was a clever man precisely because he interjected this concept soundly into mathematical doctrine by postulating the Turing Machine, which is essentially a mathematical model for the processing of information, and as a result, the very nature of mathematical analysis has been infinitely richer since. P=NP? Many of the great mathematical problems of the last 2,000 years are a question of how to deal with infinite regress. Zeno's paradox. Salesmen who Travel. Bridges over Prussian towns called Koenigsberg. The definition of the reals. The differential and integral calculi. The question of twin prime distribution. Math is computation, and computation is a property of the mind. (and increasingly of machines that calculate even more rapidly than our mind, Deep Blue and Watson come to, yes, mind.)

No math, no mind.

 

[jbriggs444]

Saying that "forty-nine is a mental abstraction" is unhelpful when trying to explain the product of seven times seven.

 

[DaveC426913]

And where is the math clear cut? In. Your. Mind. (Or mine in this conversation.)

A semantics issue. This is the math forum, not the philosophy forum (which doesn't exist - for obvious reasons).

In mathematics, infinity is well-defined, and can have operations performed on it. Nothing in your sidebar alters that.

 

[Evo]

This is not math, and we closed the philosophy forum.