psyreporter wrote: ↑February 6th, 2020, 8:55 am
I noticed the following article in a news feed:
Philosopher Wes Morriston and I have coauthored a paper on the Kalam cosmological argument, and it has been accepted publication in the journal Philosophical Quarterly. Once it is actually available on their page access will probably be limited, unless you have an institutional subscription. However, for now you can download it (for free) via this link.
Endless and Infinite
Abstract: It is often said that time must have a beginning because otherwise the series of past events would have the paradoxical features of an actual infinite. In the present paper, we show that, even given a dynamic theory of time, the cardinality of an endless series of events, each of which will occur, is the same as that of a beginningless series of events, each of which has occurred. Both are denumerably infinite. So if (as we believe) an endless series of events is possible, then the possibility of a beginningless series of past events should not be rejected merely on the ground that it would be an actual infinite.
Proponents of the Kalam cosmological argument seek to establish that any temporally ordered series of discrete events must have a beginning. One of their principal arguments for this conclusion is that a beginningless series of discrete events would have the paradoxical features of an actual infinite – features that could not be instantiated ‘in the real world’. In particular, they point out that an actually infinite series has a distinctive property, which we shall call the ‘Cantorian Property’. A series has the Cantorian Property when it can be placed in one-to-one correspondence with infinitely many of its proper parts, so that the whole has the ‘same number’ of elements as its parts. For instance, there are just as many natural numbers as there are even numbers, etc. But in the ‘real world’, they say, the whole must always be greater than any of its proper parts. So, in the real world (as opposed to the world of mathematics), an actually infinite series is impossible; nothing real can have the Cantorian Property (See Craig & Sinclair 2011: 110). And this is said to establish the first premise of the following argument:
- An actual infinite cannot exist.
- An infinite temporal regress of events is an actual infinite.
- Therefore, an infinite temporal regress of events cannot exist. (Craig & Sinclair 2011: 103)
Now one might have thought that if these considerations were sufficient to show that a beginningless (and therefore infinite) series of past events is impossible, they would apply with equal force to an endless (and therefore infinite) series of future events.1 After all, one could make a seemingly symmetrical argument as follows:
- An actual infinite cannot exist.
- An infinite temporal progress2 of events is an actual infinite.
- Therefore, an infinite temporal progress of events cannot exist.
If this second argument were equally as sound as the original one, this would be bad news for the proponents of the Kalam. For one thing, it is implausible to claim that the future could not be endless. For example, one can easily imagine a series of future events, each of which is causally sufficient for another. Again, one can imagine an endless series of events, each of which is fore-ordained by an all-powerful God. As far as we can see, these are genuine metaphysical possibilities.
https://useofreason.wordpress.com/2020/ ... -infinite/
The questions:
1) is it possible for true infinity to exist?
2) is it plausible to assume that time must have had a beginning?
Imagine a Square drawn on a piece of paper. Now imagine the Square shrinking smaller and smaller. It remains a Square no matter how small it shrinks. If we stop shrinking it and start magnifying it back we can bring the Square back to the original size. But now imagine the Square shrinking to Zero size. All points of the Square collapse to a single point and there is no longer a Square on the paper. The square has been transformed into a single point. The Square does not exist in the Universe anymore. We would not be able to magnify the resulting point back the the original Square. We could also shrink a Triangle in the same way and at Zero size it would be a single point just like the Square. The Square and the Triangle lose their identity when they are Zero size. They become something different. They become something less than what they were. Zero size is an unrecoverable threshold of size that changes everything.
Now imagine a Square that is the smallest Square that is not equal to Zero. This thought sends your mind into an endless recursive loop of the Square getting smaller and smaller and we soon realize that it is impossible to imagine such a smallest Square. We can say that this Square is Infinitely small, in the sense described, and it is still a Square. In general mathematics this would be called a differential Square or an infinitesimal Square, but it is not an Absolute Zero Size Square. It only approaches Zero.
Next imagine the Square that was drawn on the paper growing larger and larger. If the Square was exactly in the center of the paper the sides of the Square would eventually move off of the paper and reach the edges of the universe. It remains a Square no matter how large it grows. If we stop growing it and start shrinking it back we can bring the Square back to the original size. But now imagine the Square growing to Infinite size. The sides would all move out to infinity. No matter how far you went in the universe you would never encounter a side of the Square. The Square has effectively exited the universe. We could also grow a Triangle in the same way and at Infinite size it will no longer be found in the universe. The Square and the Triangle lose their identity when they are Infinite size. They become something different. Paradoxically they become something less than what they were. You might think that the Square and Triangle are still out there at Infinity. But there is no "there" at Infinity. The Square and Triangle are gone. If you think you can go out "there" to an edge of the Square or Triangle at Infinity then that "there" is not Infinity. Infinite size is an unrecoverable threshold of size that changes everything.
Now imagine a Square that is the largest Square that is not equal to Infinity. Similar to the differential Square, this thought sends your mind into an endless recursive loop of the Square getting larger and larger and we again soon realize that it is impossible to imagine such a largest Square. We can say that this Square is Infinitely large and is still a Square that exists in the universe. This Infinitely large Square is analogous to the Infinitely small Square. This Square is Infinitely large, in the sense described, but it is not an Absolute Infinite Size Square. It only approaches Infinity.
People usually understand that there cannot be Zero Sized objects but they still think there can be Infinite Sized Objects. These thought experiments should have shown that the problem of Infinite Sized Objects is analogous to the problem of Zero Sized Objects. Objects can approach Zero Size but can never be Zero Size without losing their identity. Similarly, Objects can approach Infinite Size but can never be Infinite Size without losing their identity.
I think that just as Infinite Squares are not possible it is probably true that any Infinite Physical quantity of anything is not possible. Just because an equation in Science goes to Infinity, it doesn't mean that the Physical quantity in the equation is able go to Infinity. I think this is a limitation of what we can do with Mathematics. Seems like a minor limitation but it has big consequences when equations in Science go to Infinity.
- By Sushan
- By Good_Egg