This might be broad so let me narrow it. Concerning points and mereology, is it coherent to make points - extentionless entities - compose extended objects? If so, then the idea of "material point objects" is a coherent one and extension is no longer the primitive, defining characteristic of materiality/physicality. Extension is reducible to non-extension.

Some might say, as I have seen, that material point objects are incoherent, and that points are, at best, merely a conceptual tool placed at certain locations on extended objects, and are thus neccisarily abstract. Extension is irredicble. By connection, physicality (objects that are extended) is an irreducible notion.

To note, the people I have seen back the irreducibility of physicality are explicitely opposed to the idea of actual infinity. Material objects are not reducible to actually infinitely small extensionless simples. The material object would be annhilated at that point and would be indestinguishable from nothing.

It is also possible that someone might object to extension being the defining characteristic of material things.

One conversation that talked exactly about what I am looking for was with none other than ol' Kant and his antimony about the composition of material objects where he argued against ultimate extentionless simples. This ciew has fallen out of favor it seems.

I've bought several metaphysics books hoping to get some commentary on points - material, conceptual, what have you - but have been disappointed so far because they seem more interested in other topics.

What is the ontological status of points?

  • I found this to be very helpful: "The Antinomies and Kant's Conception of Nature", by Idan Shimony
    – user3017
    Nov 29, 2017 at 14:55
  • You can see Mereology: Atoms. Nov 29, 2017 at 14:57
  • You can also consider the 'hyper-real' model with 'monads', where extension is possible only by accumulating infinitely many points that are not topologically separated. en.wikipedia.org/wiki/Monad_(non-standard_analysis)
    – user9166
    Nov 29, 2017 at 19:12
  • I do not quite understand what you are looking for but the idea of "material point objects" is certainly coherent since it is a mathematical model of classical mechanics. There is, of course, a big difference between mathematical possibility and metaphysics of the actual world. It could be possible that material objects are metaphysically irreducible if the notion of classical object was still taken to be fundamental. But since it is merely large scale approximation of quantum distributions the metaphysical status of points is completely moot.
    – Conifold
    Nov 30, 2017 at 20:04
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    Conifold, I am asking as a matter of mereological composition whether or not any instance of extension can be logically said to be an aggregate of extensionless points. In physics, I have seen theoretical physicists create models where space is emergent from the quantum entanglment between point particles. I have seen the charge that such a model can be discarded as incoherent out of hand because extension is necciswrilly irreducible, making points merely conceptual tools.
    – Jdog1998
    Dec 1, 2017 at 18:29

3 Answers 3


This is a very important question. It brings us face to face with the incoherence of our folk-psychological ideas of space and time.

My interpretation of the problem of points is as evidence that extension is not truly real. As a metaphysically-real phenomenon it is paradoxical in the ways you (and others) suggest. Only as a reducible phenomenon can extension make sense.

Kant, Zeno, McTaggert and many others have discussed this issue and you'll notice that those who do discuss it often have 'mystical' leanings since only in mysticism is extension said to be a product of Mind, not a purely objective real thing 'out there'.

Thomas Danzig puts the problem in terms of reconciling the staccato of mathematical extension (time and space as a series of points) and the legato of experienced time (time and space as a continuum). It cannot be done if we endorse naive realism or materialism, as the question implies.

It seems obvious that there is no such thing as a point, and that no amount of them would be enough to build a piano.


A short answer: the ontological status of points is the status of mathematical objects. Different views on this more general case have been expressed, e.g. by platonism, intuitionism, etc., and points are a (usually trivial) particular case within their framework.

A longer answer: a distinction is drawn between mathematics and physics as they form different ontologic domains. While this was ignored there seemed to be a genuine problem known as the constitution of the continuum.A general solution was sought until non-Euclidean geometries made obvious that geometry is not physics; the theoretical arsenal of contemporary physics (spin foams, loops, networks, whatnots) makes the concept of 'point' rather superfluous.

The full answer would have to start with the ancient Greeks (geometers, atomists, Platonists, etc) and try to disentangle the metaphysical mess accumulated during the next 2+ millenia. Leibniz should get a special treatment and next the founding of classical mathematical analysis which offers a way to speak about 'material' points, needed by physics.


Concerning points and mereology

An interesting notion of points is made in category theory, here the point is any terminal object * (all these points are isomorphic so you can take any), then given any object A, then a map * -> A gives us a point in this object. This is interesting because it leads to the notion of points of different shapes and kinds.

Is it coherent to make points - extentionless entities - compose extended objects?

In the context of mainstream mathematics it appears so. For example, take the real line. This is composed of extensionless points and the real line has extension.

Some might say, as I have seen, that material point objects are incoherent

However when we look at it through topology, we also note that this real line is disconnected, its not possible to move from one point to another as none of the points are connected to another (here we're considering the real line dynamically).

To satisfy this we arrange for the real line to have a topology, its standard topology; these are composed of open sets; every open set has extension.

Now, in what is called pointless topology we can remove the points, and this still gives us a line, except here there are no points - why? because they are extensionless, and have no ontological weight.

Extension is irredicble. By connection, physicality (objects that are extended) is an irreducible notion.

The nature of the continuum is important in physics. One philosophical reason to study string theory is that point particles, on the face of it should not exist. Point particles in physics, as Landau explained, are particles whose size can be neglected; but what happens when they can no longer be neglected - for example when we start approaching the Planck scale? One answer is strings and branes. These have extension.

It may turn out that String theory is not a theory of everything, but then, it will have at least taught us how to think of point particles as having extension (amongst many other things); and for that, it will have been invaluable.

  • """"In the context of mainstream mathematics it appears so. For example, take the real line. This is composed of extensionless points and the real line has extension."""" I have seen this stated before. But Is there any way to make this intuition more salient? It might make sense to some people but I am honestly left confused when I hear that a line is composed on an infinite number of points, or a plane an infinite number of lines, or a 3d space an infinite number of planes. How does one make conceptual sense of this - extension made by adding non-extension infinitely?
    – Jdog1998
    Dec 13, 2017 at 0:25
  • @Jdog1998: I think your confusion is understandable; I'm not sure quite how to be more concrete about it with delving into mathematics - the construction of the real line is one of the first things learnt in a good course in analysis; they first construct the rationals and then they show there are gaps because you cannot take all limits over them, they then construct ideal numbers to fill these gaps by using limits as names to designate these numbers. Dec 13, 2017 at 21:57
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    Another way to look at it is to consider the points on the line to designate positions on the line, not an actual point; and then ask the question is there a unique name or number for each position; it turns out this can be done and this is the real line. How do you feel about positions on a line as opposed to a point of a line? Are they more respectable? Dec 13, 2017 at 22:00
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    //...positions on a line as opposed to a point of a line...// Yes! I am leaning towards this interpretation of points - they are in fact designations of location, and only location, and locations are neccisarilly abstract objects. They do not and cannot represent a concrete part of anything extended... so I am thinking. But is the actual "point of a line" interpretation ever coherent? Or are points ALWAYS merely "positions on a line," and cannot be construed as being anything else coherently in any other framework?
    – Jdog1998
    Dec 14, 2017 at 19:44
  • And as for math, I took math classes up to differential equations and ended at a study of Laplace Transforms. So keep it at that or simpler please if you use math!
    – Jdog1998
    Dec 14, 2017 at 19:55

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