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Kant presented four antinomies, each presented as a pair of thesis & antithesis.

The first says that the world does have a beginning in time, he argues by contradiction:

If we assume that the world has no beginning in time, then upto every given moment an eternity has elapsed, and there has passed away in the world an infinite series of successive states of things. Now the infinity of a series consists in the fact that it can never be completed through successive synthesis. It thus follows that it is impossible for an infinite world-series to have passed away, and that a beginning of the world is therefore a necessary condition of the world' s existence.

Priest, in his book Beyond the Limits of Thought demolishes his argument by appealing to set theory and stating that Kant's appeal to Aristotle's assertion that completed infinities do not exist doesn't hold.

As far as I see, Priest's argument doesn't hold:

Set theory does have infinities, but I'd argue this simply marks out a new iteration towards the infinite which doesn't have a completion. More importantly, set theory is conceptual, and time is a physical notion. One needs to think about the infinite past in a physical way - and this is exactly what Kant is doing by saying "[T]here has passed away in the world [...]". I can't see how it is possible to assert that time past can be infinite when we consider that time must pass. We could have, indeed an infinite number of worlds before us, with their own notion of time - but in this world with its own notion of time - time past must be finite.

Of course, with the Standard Model it is taken that time does have a beginning.

Is there a better argument as to why it must fail? Or is Priest simply presenting the standard arguments? Or is the failure of this thesis actually beside the point - seeing that Kant is working at the limits of what we can know?

  • Since I don't understand what "successive synthesis" means, it's just looks like Kant boldly asserts:"Infinite series don't exist because they don't." So please, enlighten me on "successive synthesis". Second, you can't imagine an infinite time. Well, you're limited and so am I, that doesn't mean that infinite series don't exist, whether conceptually or physically. – Raskolnikov Jun 7 '13 at 9:44
  • @Raskolnikov: If an infinite time existed in the past then it would take an infinite amount of time to reach now. That seems contradictory to me. Successive synthesis looks obvious to me - wheres your problem with it? – Mozibur Ullah Jun 7 '13 at 10:02
  • If an infinite amount of integers exist before zero, it would take an infinite amount to reach zero. So what? Zero exists. I don't see any contradiction. Just lack of imagination. – Raskolnikov Jun 7 '13 at 10:12
  • @Raskolnikov: No, not particularly. You're not thinking physically - but mathematically. Time is a physical notion, not a mathematical one. Its modelled mathematically, so one has to be careful. When the mathematicians at Princeton Uni got excited about the Banach-Tarski paradox, Feynman asked them cooly how it would apply to the Sun - and as they began to explain, he said - well that assumption won't work. This doesn't mean that the Banach-Tarski paradox isn't interesting - it is. Similarly integers have no physical notion of time, they exist timelessly. – Mozibur Ullah Jun 7 '13 at 10:24
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    @Raskolnikov: Kant doesn't rule it out. His thesis is that there is a beginning to time. His anti-thesis is that there isn't. He offers I think good arguments for both. – Mozibur Ullah Jun 7 '13 at 10:28
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The argument of Kant, arguing against the infinity of time, contains an error, nobody has pointed out.

The general intuition that an accumulation of an infinite amount of 'things' would form an "actual infinity", so an infinity which has been completely absorbed, counted, etc., would be a contradiction, is of course correct. But as applied to time, the logical conclusion would be that just because that would be a contradiction, infinity of time is an endless/beginingless proces, unfolding in time.

The critical point then is that the infinite moments of time of the past, and the infinite moments in the future do not exist "at the same time", only the motion of the ever ongoing "now"...

As further demonstration to this, imagine the timeline without begin or end. Now place anywhere on the timeline two points, and measure the distance. The point is then, that wherever you place these two points, the distance between them is always strictly finite. Infinity then (since the timeline is infinite, it contains infinitely many points) is composed of only finite. Every measure of time on an infinite timeline is still a finite duration. Hence the actual infinite does not occur...

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Priest does not demolish Kant. Priest is wrong.

First, even Cantor has stated that time never can be actually infinite: "[...] for instance, the time elapsed since the beginning of the world, which, measured in some time-unit, for instance a year, is finite in every moment, but always growing beyond all finite limits, without ever becoming really infinitely large." [G. Cantor, letter to I. Jeiler (13 Oct 1895)]

And he offered to prove that the elapsed time is never infinite: "I do not only maintain with all Christian philosophers the temporal beginning of the creation, I also claim like you that this truth can be proven by rational reasons. [...] The foundation of actually infinitely great or, as I call them, transfinite numbers does not entail that we have to refrain from rational proofs of the beginning of the world." [G. Cantor, letter to J. Hontheim (21 Dec 1893)]

Second, if something infinite could be completed, then the sequence 1, 2, 3, ... would be the first candidate. But then we would have to accept that Scrooge McDuck who earns 10 enumerated dollars per day and spends one dollar, can become bankrupt in the set-theoretic limit, i.e., in the case of completed infinity. This is such a ridiculous result that we have good reason to consider set theory with its actual infinite as demolished. All people whom I know have agreed when I told them this story.

By the way the formalism of limits in set theory can be found here: https://www.hs-augsburg.de/~mueckenh/Transfinity/Transfinity/pdf, p. 55f.

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Kants arguments use a model of time that we no longer accept in our physical universe. Including time in the arguments adds considerable heat but no light. I will endeavour to use time free arguments where possible.

Thesis.

  • That the universe cannot be infinite.

He states that there cannot be an infinite series of things. But there can be and there are. There are an uncountable infinite number of reals between 0 and 1.

He mistakes the way we model things with what they are. We model a table top as a surface with an infinite number of points, but that does not mean that the table top has infinite area, or is an infinite object. The points on the surface don't actually exist as physical objects. They are a description of a finite thing using infinities.

Then he argues that open infinities such as zero to plus and minus infinity, but these are no larger than the closed infinities, both being uncountable. We can map infinity .. 1 to 0..1 using 1/x.

Antithesis

  • That the universe must be finite.

He starts by assuming that the spatial universe is infinite and asks why is only a finite part of the universe occupied. Then part is unocupied and part is occupied without any distinguishing reason why this is so.

Then take a mandelbrot set as a 2D universe. Part is occupied. Part is empty, and this case the rule is based on number. If I transform the mandelbrot positions by 1/x I get my emptiness on the outside. U(x, y) = Mandelbrot((1/x)+i(1/y)

So what is the distinguishing condition? The relationship of the complex number to the origin. Would Kant argue that there should be no origin? I don't see that stated, and even he did that would only positional self similarity. So I create a self similar patterning of the universe with empty pieces.

Or is he actually arguing cause and effect? That is a completely different argument.

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    One particular problem is not even touched by this answer: The consideration of what things, world, and synthesis means for Kant. He speaks of objects of possible experience, and even though we know concepts of infinity, there is no intuition from sensibility, and therefore no experience of infinity in his sense. In other words, there is only abstract infinity, but we cannot really grasp it, hence we don't have such things (sic!) as 'infinite series of things' and the sum if all things, world, can't be infinite if it shall be a concept with reality for us, as well. – Philip Klöcking Sep 2 '17 at 21:56
  • @ Philip Klöcking A line has an infinite number of points on it,.Infinities arise from a mathematical description of finite objects. They are not infinities of the world. We deal with them by measure theory. Kants model is not the modern one. We are an intelligence that arose from the playing out of the rules of the universe. We interpret the information from our senses.and construct the simplest model of that data from Occam's Razor. If abstract infinities arise in modelling, that is not an issue. – Peter Driscoll Sep 3 '17 at 0:04
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    My point is: You presuppose a metaphysical position of (a particular kind of) scientific realism Kant does not agree with. Even among scientific realists the ontological status of infinities is subject of discussions. So there are two problems you do not address. Instead, you just assert things that are far from being unproblematic without bothering to support this with sources. The question whether Kant's argument fails depends on metaphysical positions. – Philip Klöcking Sep 3 '17 at 9:18
  • Hmmm. OK. Kant doesn't agree that we are monkeys that evolved too large brains? Kant doesn't agree that we extract information from our senses, by building models that predict and compress the data? Kant doesn't agree that an infinity is a limit? A number as large as we need it to be. Mea Culpa. I thought that was a given. No wonder I see his argument as totally wonky. – Peter Driscoll Sep 3 '17 at 10:39
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    Firstly, it is apes, not monkeys. Secondly, Kant agrees on much more than you seem to understand, just not on the naive application of abstract concepts and labelling it as knowledge about some thing. This is what his critical philosophy is all about. Thirdly, no need for sarcasm, I am just pointing out that criticising Kant without understanding his position or critical reflection on one's own metaphysical position (which is far less 'given' than you think imho) is problematic. Even more so if it is not supported by references. – Philip Klöcking Sep 3 '17 at 10:48

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