Translational Invariance and Newton's God

By Sean Carroll | December 15, 2010 8:51 am

Tim Maudlin is writing a two-volume introduction to the philosophy of physics, and I was fortunate enough to get a peek at a draft of Volume One, about space and time. There is one anecdote in there about Leibniz’s objections to Newtonian physics that is worth passing along. This came up in the course of the Leibniz-Clarke correspondence.

Leibniz was quite fond of proclaiming overarching a priori principles. Perhaps the most famous/infamous is the Principle of Sufficient Reason, which states that everything that happens does so for some good reason. But there was also the Principle of Identity of Indiscernibles, which states that if two things have all the same properties, they are really the same thing. Sounds reasonable enough (although one might worry what qualifies as a “property”), but it can get you in trouble if you take it too far.

Remember that Newton believed in absolute space — a rigid three-dimensional set of points that forms the arena in which physics takes place. Leibniz, on the other hand, claimed that space should be thought of purely in terms of relations between different points, without any metaphysical baggage of “absoluteness.” (From a modern perspective, Leibniz was closer to correct, given Galilean relativity; but once we allow for spacetime curvature in general relativity, the relational view becomes less useful.)

So far, so good. The weird part, to modern ears, comes in when we consider Newtonian cosmology. In order to explain matter in the universe, Newton departed from the strict consequences of his Laws of Motion. Instead, he imagined that empty space existed for an infinite period of time, before eventually God decided to create matter in it.

That’s the part Leibniz couldn’t go along with. He didn’t believe God would work that way, for reasons that amount to what we would now call the translational invariance of space. If God is going to create all this matter in empty space, Leibniz reasons, He has to put it somewhere. But where? Every point is equally good! Therefore there can’t be any “sufficient reason” to create it in one place rather than in some other place. Therefore there must be a deep metaphysical flaw at the heart of Newton’s theory. Interestingly, he didn’t go for “matter has been around forever,” but instead came down on the side of “there is no such thing as absolute space.”

Maybe he was worried about Boltzmann Brains?

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  • Anonymous

    Creation seems to require the breaking of a symmetry. God broke a symmetry by putting matter somewhere rather than everywhere. Does the principle of sufficient reason say that symmetries cannot be broken?

  • spyder

    As a philosopher by trade, i have long held the view that the discipline is dependent, to a certain degree, on the prevalent mathematics, physics and chemistry of the time. In my library, i have a book published by NASA in the mid-1980s titled SPACE MATHEMATICS. Although i haven’t looked at it lately, i strongly suspect that much of it is valid, while some is considerably out of date (well, really just well behind the times). Basing one’s philosophical perspectives on such a work, fixed in time, would limit one’s views of space; we all need to keep moving onward following that somewhat annoying arrow.

  • psmith

    What is interesting is how deeply the Principle of Sufficient Reason has infiltrated the fabric of Western thinking. It has become the implicit assumption in the thinking of engineers, academics, scientists and indeed most ordinary people. Putting aside the quibbles of the philosophers I would argue that PSR and the invariance of the laws of science are two of the most powerful assumptions we make in our daily lives.

  • Nullius in Verba

    Interesting as well is Newton’s reply to Leibniz.

    He pointed out that if you take a bucket of water and you spin it about its central axis, the water surfaces curves into a paraboloid. Thus, rotation is absolute, but what is the rotation measured with respect to, if not absolute space?

  • Dorothy Gale

    How could God have lived for an infinite amount of time, and then “decided to create matter in it.”?
    -infty + any amount = -infty , so clearly any God could only have lived for a finite amount of time before he decided to create the Universe, otherwise there could not possibly be a starting time and subsequent finite age for said Universe.
    Then again, if God appeared a finite amount of time ago, maybe he’s been around only for a finite amount of time and is not around anymore.
    I have the feeling I am not the first with that argument :-)

  • Nullius in Verba


    Quite so. How could there be infinitely many numbers on the number line, with a zero in the middle of them?

  • Sean

    There is no sense in which zero is “in the middle” of the real numbers.

  • Nullius in Verba


    Arguably, all the real numbers are “in the middle” of the Reals, including zero. It depends which dictionary you read, I guess. :-)

  • Moshe

    I’m curious what is the role of such proclamations (basically an expression of aesthetic displeasure with global symmetries and their spontaneous breaking) in the text you mention, or for philosophers in general. Are those ideas historical curiosities, interesting milestones in the evolution of the thought on the subject, or are they guides to future research, deep truths to be revealed to be secretly correct? Hard for an outsider to gauge the range of attitudes towards these ideas.

  • Sean

    I don’t think that particular proclamation would be taken very seriously at the moment, but presumably there are analogous ones that would be. And it’s not just philosophers — physicists are often motivated by aesthetic pleasure/displeasure. Einstein was, and we still are today. The trick is to keep in mind that such considerations are just rules of thumb, and are never absolute — it’s not always easy to resist the temptation to treat them as laws.

  • Moshe

    Yeah, I agree. At the very least these serve to identify and frame the issue, if not necessarily provide the correct answer. I was curious specifically about this one, because it does come up occasionally in conversation (e.g. as the deep reason why gravity has to break global symmetries).

  • Will

    I don’t get it. If we have to accept God, and we have to accept that there’s a reason for him to put matter where he put it, isn’t Liebniz’s relativity the metaphysically damaged notion? Absolute space works perfectly now — God put us near the middle so we wouldn’t see his boundary conditions.

  • Nullius in Verba

    I think it’s related to the idea that observable effects for which no explanation is possible fall outside the domain of scientific investigation. There is a methodological aspect to it – that you can’t test it, you can’t derive general laws or make predictions using it, and it is an easy way to give up on trying to explain something that doesn’t fit your model – and an ontological aspect – the universe appears to follow mathematical laws over most of its domain, mathematics relies heavily on symmetries, and in mathematics, a symmetry cannot be broken spontaneously. The best that can happen is that a minuscule initial asymmetry can be magnified.

  • psmith

    @Sean: Why should aesthetic considerations have any bearing whatsoever? There is no doubt that it influences our choices and indeed seems to be effective. But why should this be so? Surely the universe is blind to a subjective concept that only exists in our mind.

  • Nullius in Verba


    You need to ask the question the other way round.

    Why should we have evolved an aesthetic sense that turns out to be useful in interpreting the behaviour of the universe we live in?

  • AI

    “If God is going to create all this matter in empty space, Leibniz reasons, He has to put it somewhere. But where?”

    He can simply put it everywhere.

  • psmith

    @16: I have to admit that seems like a plausible argument. And yet, when I think of our long history as hunter/gatherer I fail to see how that experience accounts for our highly developed sense of aesthetics. But clearly it does. Why then did we develop this aesthetic sense? I doubt very much that it helped the hunter/gather interpret his world. More likely it was a mechanism to make our early experiences of the world more capable of being endured. Such experiences of aesthetics would have translated into profoundly satisfying emotional experiences which may have better equipped them to survive the trials of hunter/gatherer life. But there is a huge gap between satisfying emotional experiences and the experience that parsimony and elegance are a good guide to choosing between competing theories of the world.

  • Nullius in Verba


    It would be easy to make experience easier to endure by making the feelings of unpleasantness less intense. But the point about aesthetics is that it is selective. Only some experiences trigger it.

    It’s a bit of a generalisation, but to some extent our aesthetic sense is based around the idea of spotting patterns in something that initially seems complicated and incomprehensible that enable us to understand it in terms of far simpler rules and relationships. It amounts to the data compression ratio – the ratio of the complexity of the raw input, to that of the simple rules and relationships, with a bonus when the relationships are less obvious and more challenging to extract.

    A square is simple, but it is not particularly beautiful because its simplicity is so obvious. You can’t compress it much further. However, a square tessellation, or even better a complex tessellation with a square symmetry group is far more pleasing to the eye, because the image can be compressed much more. Fractals are even more beautiful for the same reason.

    This same principle applies also to the explanations we generate about the world, and the feeling of pleasure when we experience that Aha! moment, spotting a simplifying connection or relationship that makes sense of the world, has a clear evolutionary benefit. It motivates us to construct parsimonious explanations of more (and more complicated) phenomena that help us better predict the behaviour of our world.

    The reason it works is that the appearance of complexity of our world at every level is based on the elaboration and recursive application of simple rules. It evolved to take advantage of this property at one level of observation (where hunter-gatherers operate), but the principle applies universally, and so it has proved equally useful in domains that our hunter-gatherer ancestors could never have considered.

    Humans 100,000 years ago were every bit as intelligent as humans today. Have you never wondered what they used those big brains for?

  • Daniel

    Leibniz was not worried at all about any mechanistic explanations of the mind. So he would dismiss the Boltzmann brain idea for that reason. He says:

    One is obliged to admit that perception and what depends upon it is inexplicable on mechanical principles, that is, by figures and motions. In imagining that there is a machine whose construction would enable it to think, to sense, and to have perception, one could conceive it enlarged while retaining the same proportions, so that one could enter into it, just like into a windmill. Supposing this, one should, when visiting within it, find only parts pushing one another, and never anything by which to explain a perception. Thus it is in the simple substance, and not in the composite or in the machine, that one must look for perception.

  • charles slavis

    Perhaps God is everything eternally evolving, until our combination of atoms happens…..Death is merely the result of this eternal change, as is life…..We may already have been granted intermittent immortality since we are living proof that we can happen….I didn’t mind waiting to get here…..I won’t mind waiting to happen again. Since I don’t have a recollection of a previous life, then I have to assume that either we only happen once and go onto a reward or punishment, or my preference that we always exist intermittently, and the time in between isn’t noticed, making us seem to exist eternally. We could create our own hell through eternal repetition, or make good choices and live a life aimed at achieving perfection. I will send you an e-mail once I get there……

  • charles slavis

    If nothing is infinite, where would God be if God wasn’t nothing? I assume that God must be everything, since everything caused me to be. Without stuff existing, then we don’t. Something caused even Steven Hawking to exist in that body, even if he doesn’t believe in It. I waited 16 billion years to get here, and God placed me in the wrong body……I was supposed to be the richest man in the world…….Get it right, next time…….

  • charles slavis

    If we all are extensions of God, then at our infinitely small size is it not logical that our ability to use Gods power would only be scaled down to our size compared to infinity? And even an eternally expanding universe would still be a singularity compared to the vastness of infinity. God may have intervened at the beginning, or intervenes through man’s choices. If you are sure that God is only on your side, and every one else is wrong, and you are willing to kill everybody that doesn’t believe like you, I suggest that you walk off a cliff and prove that God will intervene on your behalf before you make evil choices in God’s name.

  • charles slavis

    Science can’t confirm God, since God is beyond our total comprehension, but science can confirm everything that exists, as it is discovered. Thus science confirms more and more of God, but never gets to confirm It all. Existence is the proof of what ever God is………Science is limited to the smallest parts of infinity…….

  • charles slavis

    Inability to find God may be similar to inability to find life beyond the earth. We only have to keep trying till we find It……

  • charles slavis

    So much to discover…..So little time……


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About Sean Carroll

Sean Carroll is a Senior Research Associate in the Department of Physics at the California Institute of Technology. His research interests include theoretical aspects of cosmology, field theory, and gravitation. His most recent book is The Particle at the End of the Universe, about the Large Hadron Collider and the search for the Higgs boson. Here are some of his favorite blog posts, home page, and email: carroll [at] .


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