Something that has an end must have a beginning?

Question

In the ancient Hebrew book (http://dafyomireview.com/article.php?docid=398#ch5), Gate of the Unity of God, dealing with logical proof of creation of the world. The following premise is established:

PROOF OF SECOND PREMISE - (Beginnings are limited in number) The proof of the second premise is as follows: (commentaries to follow) Whatever has an end must have a beginning, because it is evident that something which has no beginning (i.e. existed eternally) has no end (i.e. is indestructible), since it is impossible for man to fathom the limits of that which is without beginning.

That's it. For some reason it is evident for the author of the book, but not so for me. Can someone shed a light why exactly something that has an end implies that it must have a beginning, and what is so difficult about imagining things that go infinitely into the past but stopped being in some well defined point in time?

UPD: Thanks lot for the answers. Thinking about it over the weekend I figured it out in the following way: Firstly, you have to consider the fact that the author of the book wasn't aware of Cantor and Hilbert so he probably had to apply a more simplistic approach to the infinity, which is more aligned with simple human understanding of the subject, and deals less with paradoxical mathematical puzzles on the modern mathematics.

It becomes really easy to understand when instead of the time continuum we move to the more material one. As in this example - "I have an infinite amount of money but then it ended" This does sound ridiculous. So is in Hilbert's Hotel- I had infinite number of rooms in the Hotel but then all of them became occupied. What would be the meaning of infinite in this case? So the same thinking applies to time continuum applying an "end" to the infinite sequence inevitably renders the sequence to be finite. Ask the following question: for how many times the sequence without a beginning (e.g. eternal) should exist? The answer would be is infinitely. But then how come it came to an end? This is the paradox the author is referring to.

The last note on one-sided sequence - like an example with negative number - the "catch" here is that we are tricking ourselves to think that negative number have started with some infinitely large negative number and then started to move toward the zero point, predating it in time. While the opposite is actually true for negative number to be negative zero point must be set firstly as beginning and not as ending point. So we're dealing with something that has a beginning and moving away from it infinitely, instead of something not having a beginning and moving toward its end.

Answer 1

The argument is based on two premises:

1. Whatever has no beginning is eternal.
2. Whatever has an ending is not eternal.

If the premises are accepted, the conclusion follows and may be formalized as so:

1. A = Is eternal
2. B = Has beginning
3. E = Has end

{1}          1.   ¬B → A                   Prem.
{2}          2.   E → ¬A                   Prem.
{3}          3.   E                        Assum.
{2,3}        4.   ¬A                       2,3 MP                     
{1,2,3}      5.   ¬¬B                      1,4 MT
{1,2,3}      6.   B                        5 DNE
{1,2}        7.   E → B                    3,6 CP

Translation:

1. Whatever has no beginning is eternal.
2. Whatever has an end is not eternal.
3. Assume that something has an end.
4. It would follow that it is not eternal.
5. Furthermore, it wouldn't be true that it has no beginning.
6. So, it would have a beginning.
7. Therefore, whatever has an end has a beginning.

Are the premises true?

The word "eternal" might be taken to be by definition that which has no beginning and no end. However, these propositions assert more than what might be taken as just a question of definition, because they exclude the possibility that something which is not eternal might have an end with no beginning. In order to support their position, the authors of the article resort to a mathematical argument which dates back to Aristotle and still continues to be debated today. The argument involves the distinction between actual infinities and potential infinities:

"Furthermore, it is evident that anything which has parts must have a whole, since a whole is merely the sum of its parts. Therefore, it is not possible for something infinite to be comprised of parts, because a part, by definition, is an amount separated from another amount, and through the part the whole is measured, as Euclides mentioned in the fifth treatise of his book of measures." ("Shaar HaYichud")

The authors are arguing that anything which has an end but no beginning would be an actual infinity because the idea of an end implies an actual limit as opposed to a potential limit. Although infinities of this type are conceivable as mathematical abstractions, many believe that their properties are too paradoxical to be considered real phenomena. The German mathematician, David Hilbert, tried to illustrate this problem by means of a thought experiment known as Hilbert's Hotel.

In order to better understand this argument, it helps to consider the distinction between the infinite by division and infinite by addition. A given distance, for instance, may be conceived to be infinitely divisible along its length, but there is a significant difference between that and an infinite series of real quantities. The Stanford Encyclopedia of Philosophy describes the problem encountered with actual infinites by addition:

"[T]he only acceptable infinite series in actuality by addition would have to satisfy the same finitistic constraints. However, any such infinite series in actuality will be identical with some infinite series in actuality by division. Hence, there is no infinite actuality by addition for sizes or weights, etc. Aristotle's views on infinite time are less clear, but he is committed to some sense of an actual infinite by addition in the case of time (going into the past), but only in a weak sense, since past changes no longer exist." (Stanford Encyclopedia of Philosophy, "The Infinite")

Although I don't intend to try to definitively answer this question of actual infinites, the resources provided are a good starting point for further study.

Answer 2

@Boris You are right to put in question the reasoning of the book you refer to:

1) To support a thesis with the weakness of human reasoning power ("it is impossible for man to fathom ...") is no argument. There are more things in heaven and earth, Horatio, than are dreamt of in your philosophy. (Shakespeare: Hamlet 1.5)

2) We lack any example of an object which exists without beginning in time. Big Bang would be a candidate because it is said that Big Bang created spacetime. Unfortunately Big Bang is not part of the cosmological standard model but only a limit point of it.

Aside. I agree with Mauro that "eternal" is not synonymous with "having no beginning". But I consider this point not relevant for the reasoning about the question.

Answer 3

Thanks lot for the answers. Thinking about it over the weekend I figured it out in the following way. Firstly you have to consider the fact that the author of the book wasn't aware of Cantor and Hilbert so he probably had to apply a more simplistic approach to the infinity, which is more aligned with simple human understanding of the subject, and deals less with paradoxical mathematical puzzles on the modern mathematics

It becomes really easy to understand when instead of time continuum we move to the more material one. As in this example - "I have an infinite amount of money but then it ended" This do sound ridiculous. So is in Hilbert's Hotel- I had infinite number of rooms in Hotel but then all of them became occupied. What would be the meaning of infinite in this case? So the same thinking applies to time continuum applying an "end" to the infinite sequence inevitably renders the sequence to be finite. Ask the following question for how many time the sequence without a beginning (e.g. enternal) should exist - the answer would be is infinitely. But then how come it came to an end? This is the paradox the author is referring to.

The last note on one sided sequence - like an example with negative number - the "catch" here is that we are tricking ourselves to think that negative number have started with some infinitely large negative number and then started to move toward the zero point, predating it in time. While the opposite is actually true for negative number to be negative zero point must be set firstly as beginning and not as ending point. So we dealing with something that has a beginning and moving away from it infinitely, instead of something not having a beginning and moving toward its end.

metaphysics ontology

Answer 4

It seems that the actual existence of something that is past-eternal is assumed by the author to be impossible. The general reason this belief is held is that it involves the existence of actual infinities, which tend to lead to paradoxes in the real world. Mathematically the existence of actual infinities is possible, but assuming that actual infinities exist in the real world would lead to absurdities. I give two examples below, the second of which relates to your question on why something that has an end implies that it must have a beginning.

For example if I have infinitely many apples, it is mathematically forbidden to subtract any of them as the result is undefined, however, in the real world there is no law of nature that forbids me from giving away 14, 3490, or infinitely many of my apples. So I could give away every other apple and have infinitely many left, or I could keep three and give away the rest. In both cases I give away the same amount of apples but in the first case I have infinitely many apples left, and in the second I have three apples left.

Another examples is that if the universe was past eternal, then there could conceivably be a countdown clock which has been counting down one second at a time forever, and right now is reaching zero. However, there is no reason it should be reaching zero now, rather than 3 years ago, or infinitely many years ago for that matter. For any time "t", it would have taken the countdown clock the same time to reach the earlier time "t - 3 years", so it should have finished three years ago, but then again reasoning similarly it should have finished three years before that... ad infinitum. Thus it can never finish counting down, but somehow it has also already reached zero.

Those are some of the basic arguments generally used to show that actual infinities lead to paradoxes. So although mathematically we can deal with infinities, bringing them into the real world can lead to paradoxes.

Answer 5

I agree with those who say that logic can't be applied towards infinity but even setting that aside, I don't see a valid argument being made in any case.

It seems to imply that open-ended existence such as a line is inconsistent with single-ended existence such as a ray which is simply not the case.

Answer 6

further explanation is provided in the commentaries which follow:

Manoach Halevavos: "whatever has an end must have a beginning" - "end" refers to some finiteness/limit, whether those two endpoints (beginning/end) are in time or in causes (that it appears as the definite effect of a cause). Likewise any physical thing has limits in 3 dimensions, namely, its physical dimensions. If so, it is obvious that whatever has limits must have a beginning (is not eternally existing) (more on this in ch.7).

"something which has no beginning has no end" - it it evident and also explained in books of wisdom that any power which is without beginning has no end. Just like it existed forever, infinitely, eternally back, so too it will continue infinitely, forever. This is a sound deduction which the intellect accepts. An intelligent person cannot claim that something without beginning is bound, and that he can fathom its limits. This is what he meant by "since it is impossible for man to fathom the limits of that which is without beginning", i.e. for man's intellect to fathom it... It must be indestructible since it has no cause and behold a thing cannot make itself. (i.e. if a thing cannot make itself, and it has no cause, then how can it possibly exist? Therefore, its essence must be beyond the limits of our finite human logic.)

keep in mind, he is talking about an "Actual" infinity not a mathematical one as the commentaries there explain (and the other answers pointed out):

"This means that the thing to consider is infinite in actuality, but for something which is not actually infinite but just theoretically infinite, that the mind imagines something infinite - from this one cannot bring a refutation, because the power of imagination deceives, and one can picture and think in his imagination also on impossible things"

the main point he is getting at is that you cannot have an infinite regress of causes. i.e. you cannot explain the world in terms of a previous cause which had a beginning which was the effect of a previous cause which also had a beginning, and so forth. Rather, there must exist some fundamental reality which is eternal, i.e. without beginning.

Answer 7

The last note on one sided sequence - like an example with negative number - the "catch" here is that we are tricking ourselves to think that negative number have started with some infinitely large negative number and then started to move toward the zero point, predating it in time. While the opposite is actually true for negative number to be negative zero point must be set firstly as beginning and not as ending point. So we dealing with something that has a beginning and moving away from it infinitely, instead of something not having a beginning and moving toward its end.

This seems wrong to me. You can easily get around whatever issues you have with negative numbers by considering a semi-open interval like (0,1]. With the order on the real numbers representing the before relation, we see that this sequence has an end, namely 1, but no beginning.

Of course, you could reply, as you have in the passage I've quoted, that what's really going on is that 1 is fixed as the beginning, and we're moving away from it in the negative direction. But that line of thought seems to me to straightforwardly beg the question.

Answer 8

The Kavanah is to be found in the kuzari 5:18.

If some chain has no beginning that just means it has existed infinity. There is never one point at which it began and can always be pushed back and back. Now, if such an infinite chain had an end, that would mean that an infinite amount of things have passed in order to reach that end (which doesn’t have to literally be the end, but rather just a point that it has currently reached).

However, that would mean that you have somehow passed through an infinite amount of things in order to reach the current thing. However, if a never ending amount of things need to happen before the current thing, surely we will never reach the current thing. You can then ask this on the thing right before this one for surely an infinite must have passed in order to reach that thing.

And you can go back and back and back until you can say that an infinite amount of things have occurred and yet an infinite amount of them could have never occurred, for an infinite amount of things must have preceded each and every one of those points no matter how far back you go (for if it was finite until, say 1,000,000 points ago, surely adding 1,000,000 to a finite number doesn’t make it infinite but just finite +1,000,000. However, this would be the equivalent of saying the whole chain was never infinite to begin with, which doesn’t help our argument, which assumes that it is).

For this reason it’s obvious that something which has no beginning can have no end. And this produces the other point something which has an end must then have a beginning.

Answer 9

I think the question is clear or the thought process is clear he's clearly speaking about

something nothing everything and the second is the witness.

Something to experience the first without having something to experience the first it never happened. Here's an example. If there was a hole in space in the middle of the nothing a hole appeared. And let's say space rushed back in to fill that hole at the speed of light. Because this event happened at the speed of light the hole never realized it was a hole and space never realized it was filling in the hole. The something and the nothing have joined but because of the speed they never knew. They were always something and nothing, they never experienced time. However, the second not traveling at the speed of light witnessed the event therefore it happened. So you have a start with no start. It can never end.