# What if segments are not infinitely divisible?

I almost got myself mixed up I a philosophical discussion again.

Somebody was talking about the Planck time and length which are, according to him, the minimal possible time and distance, and how that means that ordinary (Euclidean) geometry can not be the geometry of the real world, and that from this alone all kind of non-mathemnatical subjects follows.

Luckily (this time) I did manage not get very involved in this discussion, but still.

I think in one thing he is right normal Euclidean geometry expects that you can infinitly divide a segment into smaller parts and that if the Planck length is the physical shortest distance possible then there is a (not very big??) problem.

But still euclidean geometry is not the only geometry around so are there geometries where that is not the case?

Are there geometries where there is a limit to the minimum size of a point, so with a limit to how far you can split up a segment?

What would the axioms and theorems of such a geometry be?

I was first thinking that would be some kind of taxicab geometry (alternative link) But even this geometry uses infinitly dividable segments.

At another end there are geometries that just have a fixed (low) number of points (like the Fano plane or affine geometry but I think even these are not particulary suitable for this discussion.

But is there a geometry that would be describing the real world as this person percieves it?

• I would say that most people perceive the world according to Euclidean geometry. The problems on the micro and macro scales that crop up with applying Euclidean geometry to the physical universe are not often apparent to an unaided observer. – rschwieb Apr 18 '14 at 19:11
• And if this is a question of the form "what geometry/mathematics/model does reality use?" then I would like to reiterate that this is usually not the right question. We use mathematics to model and analyze the physical world, but there is no reason to expect our inventions to actually manifest themselves in reality. The models can, however, give good predictions about the physical world... – rschwieb Apr 18 '14 at 19:13
• If the question is "has anyone tried to study the implications of a geometry where magnitudes aren't infinitely divisible," then I'm pretty sure the answer is "yes." – rschwieb Apr 18 '14 at 19:17
• The Planck length and Planck do not represent minimal lengths and time intervals of spacetime. They are merely are the smallest length and time differences measurable (to within a factor of order unity), which is a big difference. The Planck length/time just determines the smallest divisions rulers and clocks, not space and time. – David H Apr 18 '14 at 19:17
• You will be much more likely to get an answer if you reduce the length of your question and more clearly state what you're actually asking. – Alexander Gruber Apr 18 '14 at 22:59