Leibniz: Circle Most Capacious Isoperimetric Shape

[a monad]

Leibniz says the following: "It is true, or rather it is necessary, that a circle is the most capacious of isoperimetric shapes, even if no circle really exists" in the opening line of a lengthy proof he gives for God's existence.

I took calculus in college, but I don't recall what exactly these terms mean. By "most capacious" I assume he means having the most space in it, but what exactly are "isoperimetric shapes"? And why is a circle the most capacious of them? Google gives me nothing!

 

[micromass]

I hope we're going to discuss math in this thread and not the existence of God.

But anyway, I think you're looking for the isoperimetric inequality. Read this: http://en.wikipedia.org/wiki/Isoperimetric_inequality

It means that of all curves with a given perimeter, the circle has the greatest surface area.

 

[a monad]

I hope we're going to discuss math in this thread and not the existence of God.

But anyway, I think you're looking for the isoperimetric inequality. Read this: http://en.wikipedia.org/wiki/Isoperimetric_inequality

It means that of all curves with a given perimeter, the circle has the greatest surface area.

I'm still not sure I fully understand.

Though any necessary geometrical truth will suffice for the proof of God in Leibniz. The full text is as follows:

It is true, or rather it is necessary, that a circle is the most capacious of isoperimetric shapes, even if no circle really exists.
Likewise if neither I nor you nor anyone else of us exists. Likewise even if none of those things exist which are contingent, or in which no necessity is understood, such as is the visible world and other similar things.
Therefore because this truth does not depend on our thinking, it is necessary that there is something real in it.
And because that truth is eternal or necessary, this reality that is in it independent of our thinking will also exist from eternity.
This reality is a certain existence in actuality. For this actual truth always subsists in actuality objectively.
Therefore a necessary being exists, or one from whose essence there is existence.
To put it more briefly: the truth of necessary propositions is eternal. Truth is a certain reality independent of our thinking. Certainly some eternal reality always exists. That is, the truth of necessary propositions always exists. Therefore some necessary being exists.
Whatever exists is possible. Some necessary being exists. Therefore some necessary being is possible.
The minor is proved thus: whatever in actuality is objective, that exists. A certain necessary thing is in actuality objective. Therefore a certain necessary thing exists.
I prove the minor again: the truth of necessary propositions is in actuality objective. The truth of necessary propositions is necessary. Therefore a certain necessary thing is in actuality objective.
From this it is evident that there are as many necessary things as there are necessary truths. These necessities can be combined, anyone to any one, because any two propositions can be connected to prove a new one, when the means of joining them have been added.
(A difficulty is that the same proposition can be demonstrated in many ways. Yet there are not many causes of the same thing.)
Therefore all realities existing in eternal truths, with no one thinking about them, will have some real connection to each other.
Truths arise from natures or essences. Therefore even essences or natures are certain realities, always existing.
The same nature comes together to form innumerable others, and is able to come together with any other.
Those realities that are in natures in reality, or as they say objectively, are not distinguished by time and place. Because they combine.
The objective realities of conceivable natures and of truths are likewise the same in many other respects.
Those realities are not substances.
A substance existing by necessity is unlimited, i.e. it contains all realities in itself.
Plato, dialogue 10 of the Laws - the soul is what moves itself.

Still, I'd like to fully understand the example he gives.

 

[micromass]

A lot of what you posted is philosophy, which cannot be discussed in a mathematics forum.

Perhaps you can say specifically what you don't understand, because it's not very clear to me.

 

[CellCoree]

I can provide some clarification on Leibniz's statement. Isoperimetric shapes are shapes that have the same perimeter or circumference, but different areas. In other words, they have the same "boundary" but different "interiors." For example, a square and a circle can have the same perimeter, but the circle will have a larger area. This concept is often studied in mathematics and physics, as it has real-world applications such as in the optimization of materials and structures.

Now, Leibniz is stating that a circle is the most capacious of all isoperimetric shapes, meaning it has the largest area for a given perimeter. This can be proven mathematically, and it has to do with the fact that a circle has a constant curvature, which allows it to "pack" the most area within a given boundary. This is also why circles are often used in nature to optimize space, such as in the arrangement of cells in a honeycomb or the shape of soap bubbles.

It is interesting that Leibniz brings up the existence of a circle, as it is a mathematical concept that can be applied to the physical world. However, in the context of his proof for God's existence, he may be suggesting that the perfection and optimality of the circle points to a divine creator. This is a philosophical interpretation of his statement and may not necessarily align with scientific reasoning.

In conclusion, Leibniz's statement about the circle being the most capacious of isoperimetric shapes has a mathematical basis and can be proven. However, his connection to God's existence is a philosophical interpretation and not a scientific one.