The Paradoxical Size of the Universe

In metaphysics it is important to keep in mind that our object of study is a totally unique phenomenon. If we forget this then we may fall into confusion on many issues. One such issue would be the problem of the size of the universe. If by ‘universe’ we mean ‘the world as a whole’, all possible mental and corporeal phenomena as well as the container that contains them, or the ‘multiverse’ and whatever contains that, then this question may seem to present us with an intractable paradox.

The universe appears to be extended in space and time. In this case, it must be finitely or infinitely extended. To say that it is finitely extended, like a football, is to say that it is not, after all, the world as whole. This idea must be rejected. To say that it is infinitely extended is possible, mainly because the idea is so confusing, but it would directly contradict the scientific evidence and explain exactly nothing.

The root of this topological problem is our idea of extension. This seems a harmless enough concept in our everyday classical or Newtonian universe, but it unravels in quantum mechanics and metaphysics. If the ‘world as a whole’ means what it says, then the idea that it is extended in space and time is incoherent. Then space would be extended in space and time would extended in time. It would be unsurprising if this idea caused paradoxes and contradictions. If we remember that in metaphysics we are dealing with a unique phenomenon, a phenomenon that encompasses all other phenomena, and thus adopting an ultimate perspective on the world, we can avoid this paradox. To think of the universe as a giant football or as infinitely extended is to make it incomprehensible. For an ultimate level of analysis our everyday Newtonian idea of extension would have to be abandoned.

The mathematician Hermann Weyl wrote much that is relevant to this topic. I discuss his thoughts in a previous post. He concludes that the idea of an extended continuum is paradoxical and has no empirical support.

This entry was posted in Uncategorized and tagged , . Bookmark the permalink.

10 Responses to The Paradoxical Size of the Universe

  1. dondeg says:

    Yay! Serious discussion of metaphysics! I just found your blog from you finding mine and will look forward to reading your posts. I have been reading Leibniz the past year who also has much to say on this issue of extension, and related concepts of dimension and continuity. You know, one thing I am coming to is that I am beginning to doubt Cantor. I found Wittgenstein’s critiques and they have useful stuff to say, but also, I am a big fan of Hindu philosophy and thinking, and they also have ideas about infinity that challenge Cantor’s idea of different types of infinity. Based on what I’ve read of Leibniz, it seems Cantor may have gotten lost in the labyrinth. The issue of discreet vs. continuous is very deep, both in science and philosophy.

    Not sure what you mean about directly contradicting the scientific evidence. Do you mean that, since the big bang happened a finite time in the past, the universe must by necessity be a finite size? What if the initial event created an infinite manifold that apparently has been expanding? Not that I believe it – its just a logical possibility. I am more inclined to Robert Oldershaw’s view of a fractal universe Have you seen this? Its very provocative although a mainstream physicists won’t touch it with a 10 foot pole (joke intended).

    Best, and look forward to your material…

  2. PeterJ says:

    Hi dondeg. Thanks for the comments and the link. Glad to meet another enthusiast. .

    Yes. I mean that a finite phenomenon cannot become infinite. I see this as non-negotiable. It can, of course, do all sorts of things in appearance, but not in fact.

    You ask, ‘What if the initial event created an infinite manifold that apparently has been expanding?’ I’m afraid this idea appears incoherent to me.

    • dondeg says:

      Hi Peter

      Nice to meet you too, Sir. Yes, I agree that something finite cannot become infinite and vice versa.

      Re the idea that the Big Bang birthed an infinite manifold, consider this possibility. The Big Bang is called a singularity because it is a point. Literally, a point. Perhaps via quantum mechanics it smears out, but that is a different case to consider. With the classical theory, and certainly using Relativity formalism, it goes backwards in time to a point. Well, how big is a point? Is a point a finite entity? This is actually a very Leibnizian issue if you think about it. What I am saying is that, in some sense, a point is 1/infinity; certainly it is when considering its dimension. So, for something to come from such an entity, it would have to intrinsically be of the nature of not finite, because we have already established that the finite cannot turn into the infinite and vice versa.

      Anyway, by way of introduction, please know that this is all conversation to me. I don’t take any of so seriously. They are just ideas we are playing with after all. I like to do as above and throw out possibilities, counter arguments, or sometimes just shut up and contemplate what you have said. Again, just by way of introduction so you can have an initial sense of my intentions in communicating with you.

      And thank you for the nice conversation!

      Best wishes,

      • PeterJ says:

        Interesting Don. This is my view of a point also. It is infinitesimal, which is a form of infinity. I follow your argument and agree with it.

        It is an odd idea for various reasons. The universe reducing to a point seems daft to me. My view would be that physics needs a concept of the unmanifest. Without this it must assume that the universe begins as ‘the ghost of a departed quantity’, which is not instantly plausible. .

      • dondeg says:

        Dear Peter

        That was a most interesting reply. First, regarding the daftness of the universe reducing to a point…this is the issue of “infinities” that are often encountered in physics, a classic example (i.e. meaning non-quantum mechanical) being the singularity of black holes. In quantum mechanics, there is a weird situation that is analogous. This surprised me when I learned it, but is apparently the case.

        When one measures the charge of an electron, the measured value is a function of the energy used to probe the electron. The higher the energy used, the higher the charge measured. Therefore, if infinite energy could be used, its charge would be infinite. One reason Feynman got the nobel prize is he figured out a mathematical technique to “eliminate” such infinities, called renormalization. The procedure was generalized by Kenneth Wilson, and is related to studying phenomena across vastly different scales.

        The point (haha) of this tangent (haha again) is that it illustrates that infinity still arises in physics in very substantial ways, and by no means has this ghost been banished. I do not fully understand Wilson’s ideas, but study them as time allows because the ideas even bear on my own work, which involves studying how microscopic cell damage causes macroscopic tissue dysfunctions.

        Now, the interesting, and surprising part of your reply was your suggestion to deal with the ghosts of infinity by the idea of unmanifest. Without knowing specifically what you mean I applaud this direction of thinking. It is taken in Hindu philosophy. The absence of such a notion is a vast hole in modern Western thought. While there are different ways to incorporate the idea, IK Taimni’s interpretation of the Hindu idea shares similarity of meaning with the notions of potential vs. kinetic energy. The unmanifest being analogous to potential energy and the manifest analogous to kinetic. This is just an analogy, but provides “intellectual training wheels” for one to get their head around the notions.

        If you wish to elaborate your thoughts, I would be interested to hear.

        Best and thank you again,


  3. PeterJ says:

    This discussion is seriously interesting. I must read Taimni sometime. Can you point me at an introduction to his ideas?

    Yes! ‘Intellectual training wheels.’ Teaching stories. Formalisations. Imaginative aids. Tourist guides. Is this a reference to the Budhha’s ‘turnings of the wheel’?

    Yes, I’d be very pleased to elaborate. Thank you for the invitation. I think we are in the same place on all these issues, but we’ll see.

    I became unreasonably excited about your comment that the absence of the unmanifest is a ‘vast hole’ in modern Western thought. This would be IT, right there. This would the whole problem. (Or, the ‘hole’ problem). There IS a God of the Gaps, and it is a Gap. Nobody knows what to call that which must remain nameless, but it is sometimes called Tao. It would fill the gap in our theories precisely because it is not there. We would not need to invent another irreducible phenomenon to fill the gap. The gap is allowed to remain, but we assume it is not a gap. Or, we do the work and find this out. This would be the Real. It would Chalmers’ mysterious ‘missing ingredient’, the only phenomenon that is actually real. It would also be McGinn’s ‘obstacle in our intellectual makeup’, that, ‘prevents us from solving the crime’, since such a phenomenon would be by definition unthinkable. It would be Kant’s phenomenon that is ‘not an instance of a category’.

    It seems a completely crucial point that in physics we would need an unmanifest phenomenon for a fundamental theory, and that for any formal system we would need at least one undefined term, and yet it is very rarely made.

    If the ‘ghost of a departed quantity’ is something that is not there anymore, as it would be in the context of the calculus, then it seems perfectly easy to reduce the universe to something that is not there anymore. It would be the assumption that there must always be something there that causes all the trouble. The idea does not make sense.

    The concept of the Unmanifest would be a magic bullet, capable of solving metaphysics at a stroke, thus placing physics at last on solid ground. And yet it is, as you say, a vast hole in Western thought, which remains a house without a foundation.

    What you said about measuring electrons was fascinating but took me a bit by surprise. I must do some reading. This is something I may have misunderstood.

    Please feel free to elaborate your thoughts here also. I will definitely be interested. Few people explore these things. Do we part company on anything so far?

    • dondeg says:

      Hi Peter

      Thank you. Yes, I looked high and low for a PDF, but there is none on the net of Taimni’s “Man, God and the Universe”. This is perhaps the most comprehensive and synthetic of his books. The first few chapters are visible on Google books.

      That was clever, but I did not insinuate it, but since you put it there, we must keep it there: Samsara. 🙂

      That is an interesting take on the unmanifest. Allow me briefly to convey my understanding, based mainly on Taimni’s description from the above book, and then I will comment on your take.

      In Hindu thought, there are three, I don’t know what the proper term is, “categories”, “levels”, etc; such terms are inadequate. English translations are: (1) The Absolute, (2) The Ever-Unmanifest, and (3) The Manifest. All Hindu philosophy pertains to one or the other of these “levels”. The “relationship” (again an inadequate term) amongst the three (to paraphrase Taimni): the core of the Ever-Unmanifest is the Absolute. The core of the Manifest is the Ever-Unmanifest.

      When we speak of infinity above, this is an English synonym of The Absolute. Generally it is called Brahman in Vedanta. From this arises the Unmanifest, and in Tantric philosophy and Kashmiri Shaivism, these are expressed as two tavas, or “principles”. The first is Shiva-Shakti tatva. The Second is Mahesha-Maheshvara tatva. Each correspond to abstract polarities. Shiva-Shakti is the basic polarity of consciousness and power. Mahesha-Maheshvara is derived from the first, a further splitting or division, into the polarity of observer-observed.

      From these basic tatvas, manifestation unfolds in a complex sequence of events according to the Hindu teachings. Tantra has perhaps the most elaborate scheme.

      As I said above, the unmanifest tatvas are analogous to potential energy in that they set the limit of what the manifest can become, but they never manifest or become existent things. They are like templates, in a sense, like a Word doc template. The doc file has the form of the template, yet is distinct from it. It is in this sense the unmanifest affects or shapes the manifest, but does not itself manifest.

      This is but a cursory overview, and hopefully does some small justice to the Hindu thinking. The key point is Hindu thought provides an intermediate layer between the infinite and the finite in the notion of unmanifest.

      Comparing the Hindu thought pattern to what you expressed above, it seems that your “gap” is the Hindu Absolute. Brahman and Tao are very similar, if not identical, concepts. If this is the case, as you see, the Hindu thinking involves three “things”, vs your suggestion involving two. Brahman is the uncaused cause in Hinduism. But in addition there is the intermediate “template” layer.

      My present understanding of why the middle layer is needed has to do with the Hindu idea of >why< manifestation exists (as opposed to "how"). The explanation of why, while not difficult to understand, is alien and cuts across the current of modern Western thought. It is stated very clearly in Patanjali's Yoga Sutras that manifestation exists only for the fulfillment of desires. Although a given manifested universe is finite in space and time, manifestation as a recurring event is infinite in that it exists forever without beginning or end, because desires never resolve and therefore always produce new manifestation.

      In the Hindu cosmology, a specific universe exists based on the unfulfilled desires of the previous universe. It is an infinite cycle because desire is never fulfilled. The unfulfilled desires of the previous universe provide the content that goes into the template of the unmanifest, which, upon "mixing" these, generates manifestation.

      This is different from Western thought which sees the universe as an impersonal objective processes in which we are embedded and have evolved within. In Hindu thought, the entire manifested cosmos derives from the unfulfilled desires of the "souls" (atman) of the previous universe: you and me, and all other beings that currently exist, are the cause of the present universe because of the desires we have.

      As I said, I believe neither one nor the other. My goal is simply to understand the ideas properly so I can compare and contrast them accurately.

      A few words about how these ideas relate to modern physics. Your idea of "gaps" is relevant in this context. Quantum mechanics forces the abandonment of continuity in the mathematics. Everything becomes discreet: space, time, energy, etc. This obviously begs the question: what is in between? The quantum purist would say "nothing". But we have been there before in math. Between the integers are all the real numbers.

      I believe these debates are unsolvable in scientific terms and reflect the intrinsic limits of our sensory systems. The only way to get beyond them is to go beyond mere sensory-based understanding. Which was much of my point in "What is Science?" and hence my fascination with yoga.

      Finally, very briefly, in this blog post:

      I discuss the discreet nature of time described in The Yoga Sutras, and again, your idea of "gaps" seems relevant in this context, and, as you indicate above, is of fundamental significance. In yoga, "jumping into" the gaps between the moments of time leads to the experience of infinity.

      So, Peter, sorry for the longish reply. Please take your time with it and reply as your time allows. Again, thank you for the very interesting discussion.

      Best wishes,


  4. PeterJ says:

    Thanks Don. Very interesting.

    The part about the topology of the universe I found particularly interesting. In the end, the question of whether there are two or three ‘categories’ or ‘levels’ (hypostases? aspects?) is, I would say, an academic one. For Rumi there would be two worlds, but the two world would be one, and this is three things. It would be impossible to have two things without having three, and this is the structural flaw in dualism. Nagarjuna’s theory does not say that there are two worlds, just that this would be a useful way of looking at it. Same in Hinduism with its two Brahman. No need to suppose that Buddhism and Hinduism disagree when it comes to metaphysics. Both explain the temporal universe as the outcome of desire, and its evolution as powered by desire. .

    At last, something to disagree about. You say, “Quantum mechanics forces the abandonment of continuity in the mathematics.” This, I would say, is back to front. More accurately, mathematics forces the abandonment of continuity in QM. There is an essay discussing Hermann Weyl’s view somewhere here among the posts, and I would see it as being correct. The invention of mathematics would be the origin of the universe. A subject and an object is two things, and we cannot have two things, or even one, without inventing mathematics.

    All this would be bang on topic. A continuum has no parts. It cannot be extended. In order to manifest it must become a series of points. More properly, and as I think you suggest somewhere, it has to divide like a cell in order to see itself as a series of points. The trouble is that it is not a series of points. So when we examine the continuum problem in metaphysics or the foundations of analysis various paradoxes arise. The size of the universe appears to be paradoxical, even the very idea that it has a size. The idea that extension is an absolute property is the cause of this problem, but without a concept of the unmanifest we cannot reduce extension.

    I haven’t checked out your Patanjali link yet but will later. It seems to me that the yoga sutras are one of mankind’s greatest treasures and achievements.


    • dondeg says:

      Hi Pete

      Yes, thank you as well. I agree it is not worth trying to put too fine a point on ultimate issues. I guess I can conclude by saying that I find the Hindu idea of unmanifest of interest mainly because of the lack of an equivalent concept in physics. One may construe Bohm, or other hidden variable theorists as barking up this tree, but such efforts have been unsuccessful to date. I think, as you insinuate in your other posts, this will never be a topic of physics, but perhaps of mathematics.

      I read your post on Weyl. Very nice. I have not read him directly, only other authors talking about him, so it was nice to see his quotes. What struck me most is how Leibniz talked about much the same issues. The more one learns of these foundational questions, the more the term “labyrinth of the continuum” seems appropriate, and the more that things change, the more they stay the same!

      Before addressing your other points, I would like to share this quote from Taimni about mathematics. For all that has been written about the philosophy of math, and foundational issues, I find his take clean and to the point:

      “Mathematics deals only with pure abstract relations without taking into consideration the contents of those things which are related to one another. It is obviously, therefore, concerned with the world of the Relative and not with that Ultimate Reality which we refer to as the Absolute”.

      On to your other point. I “learned” quantum mechanics in physical chemistry as an undergrad. That is to say, I had a professor who did not understand the topic walk the class through the particle in a box example. It is fair to say, at the end of the whole thing, no one learned anything. It was not a complete waste for me, however, in that the light finally went on in my mind for understanding classical physics! Since that time, in accelerating measure, I have been trying to really learn quantum mechanics. I give this brief background so that you can appreciate my following comments.

      It seems clear to me now that quantum mechanics is a kind of algebra or calculus of how to calculate physically measured quantities in a limited domain of experience. I use the terms “algebra” and “calculus” not in their formal mathematical sense, but just in terms of a procedure for doing calculations in a specific way. The necessity for these very specific mathematical procedures has to do with how sensitive measurement has become in physics. In other words, such subtle effects can now be measured (for at least the past 100 years) that we must take into account how the act of measuring affects the quantities measured. The proper method of accounting was discovered by Heisenberg when he modeled this using non-commuting matrices.

      With respect to the math formalism used in quantum mechanics, continuity is abandoned. However, this presents serious technical challenges in condensed matter physics where bulk matter >>can<< be approximated as continuous. But we know bulk matter at a fine grained scale is discreet, and so these scaling issues take on center stage, again, as serious and substantial technical issues in physics.

      I do now wish to dwell any deeper on the particulars. The point is, quantum mechanics is a formalism, one very specific application of mathematics, to a specific (limited) domain of measurement in the physical world (I say it this way on purpose to drive the physicists crazy, because they think they are capable of measuring everything…Theory of Everything….please…). As such, quantum mechanics is merely at the level of math application, and does not, in my opinion, bear per se on the deep foundational issues you discuss, and were discussed in your Weyl article.

      I see math as bigger than physics. The use of math in physics is a subset of mathematics. As Taimni indicates, math is the study of abstract patterns of relationship. It seems reasonable to assume that not all such patterns will be of relevance in describing how physical things are related to each other.

      Now, this can legitimately be challenged. Perhaps the physical world is a mirror, in some sense, of all possible abstract relationships (Leibniz might content so, for example). Then, physics and math would be commensurable.

      But I am taking the point that the use of math in physics is a subset of math, which is certainly the case right now.

      Now, this does not invalidate at all the arguments you put forth, and that Wely and Bell used when dealing with our perceptions of space and time as relevant to foundational issues. However, I am just trying to clearly demarcate such efforts from those of quantum mechanics because the latter is a relatively closed system/formalism that may or may not have direct bearing on the foundational issues. The foundational issues are issues of finding specific abstract patterns of relationship (or not).

      To me it comes down to points of view (which is, I guess, another way to say "abstract pattern of relationship"). Quantum mechanics is a certain point of view. One out of an infinity of possible points of view. Math explores a much broader range of points of view.

      To wrap this up, since it is getting long, I think you raise important points. The relative roles of discreet and continuous in math are very, very important. As I said before, in contemplating these issues, I am coming to doubt Cantor, which is a very significant conclusion. Construtivisim is one of the philosophical positions in math that states that a math object only exists if you can construct it. When you consider, for example, that there are supposed to be infinitely many more transcendental numbers than algebraic real numbers, but that we can only construct a few transcendental numbers, then it makes you wonder.

      Ok, will close here. Again, Pete, thank you for the most challenging and stimulating conversation.

      Best regards,


      • PeterJ says:

        Good stuff Don. I can agree with it all, esp. the idea that maths is more extensive than mathematical physics. Maths also covers metaphysics. Have you come across Ulrich Mohrhof and ‘The World According to Quantum Mechanics’? I think you’d be on he same wavelength. Also directly relevant here would be George Spencer Brown’s ‘Laws of Form’, in which he develops a calculus describing the evolution of form, and whose axiom is the unmanifest!

Leave a Reply

Fill in your details below or click an icon to log in: Logo

You are commenting using your account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s