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Is there any theory or theorem of geometry -- whether used in practice or not -- which denies or forbids the use of irrational numbers?

If not, were there any notable attempts at it?

Disclaimer: I am not looking for a proof for the existence of irrational number.

Eyal Roth
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    A geometrically interesting subset of the real numbers are the constructible numbers, you can find some information on that on Wikipedia and read into it from there if interested. However, these also include some irrational numbers (but not all). – Dirk Apr 04 '19 at 13:58
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    Have you heard of finite geometry, as in: https://en.wikipedia.org/wiki/Finite_geometry ? This is geometry where there are only a fintie numbre of points, hence you can assign them all natural numbers. – quarague Apr 04 '19 at 13:59
  • @quarague Why not add it as an answer? :) – Eyal Roth Apr 04 '19 at 14:04
  • Irrational numbers were discovered during the early development of geometry (finding lengths of hypotenuses of right triangles). This gives an idea how limiting such a restriction would be. – Hans Engler Apr 04 '19 at 14:08
  • @HansEngler Oh but limitations are the best catalyst for creativity. – Eyal Roth Apr 04 '19 at 14:11
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    @EyalRoth That is surely a matter of opinion :) – Hans Engler Apr 04 '19 at 14:28
  • @HansEngler Indeed it is :) – Eyal Roth Apr 04 '19 at 14:34
  • Do you care about ordered geometry? If so, that'll rule out finite geometries. To me, if you are interested in a distinction between rational and irrational points, it only makes sense in a context of ordered geometry. – rschwieb Apr 04 '19 at 14:45
  • @rschwieb I'm not sure, as I am unfamiliar with that term (and the wikipedia value doesn't help making a distinction between this term and its alternative). I would say that I am mostly interested in geometry which doesn't rely on the axiom of infinity, which I believe says more about cardinality rather than order (but is indeed inferred from order). – Eyal Roth Apr 04 '19 at 14:52
  • @EyalRoth Ordered geometry just means that there is a total order on a line in your geometry (basically.) . When people think of geometry in terms of measurement, this is usually what they have in mind. There won't be any real notion of "Euclidean distance" in a finite geometry. – rschwieb Apr 04 '19 at 14:58
  • It depends a lot what you mean by "use of irrational numbers" or "geometry". In some fields of geometry (like incidence geometry or some parts of combinatorial/discrete geometry), there are structures without numbers at all. In more typical geometry, you can always understand every number as "the root of this polynomial in this interval", which can be represented via rational numbers - even if it is understood that we are indirectly referring to an irrational number, we can still ask questions like "is the length more than this?" using only rational arithmetic, but this is painful to implement – Milo Brandt Apr 04 '19 at 17:18
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    I recall in the book Naming Infinity: A True Story of Religious Mysticism and Mathematical Creativity, which was about the mathematicians who first promoted the idea of infinity and set theory, and their religious proclivities, that at a large math conference at the time (around 1880?) one of the great mathematicians proclaimed that all of math would be described using "integer alone". Sorry I can't give you a better reference, but it would be worth reading the whole book on its own, if not only to find the reference. – user151841 Apr 04 '19 at 18:15
  • @user151841 Thanks for the reference. Would you consider that book historically accurate (as much as history books go)? – Eyal Roth Apr 08 '19 at 09:19
  • @EyalRoth I'm far from an expert in either subject, but it seems fairly accurate based on a rhetorical analysis of other similar books I've read. I don't know any of the history to be able to critique it, but it "reads like" a legit history book. – user151841 Apr 08 '19 at 13:03
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    @EyalRoth The author, Loren Graham, seems to have good credentials: https://history.mit.edu/people/loren-r-graham – user151841 Apr 08 '19 at 14:55
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    @EyalRoth the mathematician in question is Leopold Kronecker, who developed or was a proponent of Finitism: “all analysis and algebra will be founded on the strict concept of integer,”. According to Graham, "In Cantor’s fundamental work of 1883, “Grundlagen einer allge- meinen Mannichfaltigkeitslehre” (“Foundations of a General Set Theory”), [Cantor] developed metaphysical ideas on “Free Mathematics” in response to Leopold Kronecker’s criticism" – user151841 Feb 08 '22 at 18:46

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I don't know how helpful you will find it, but there are videos on YouTube by njwildberger on rational trigonometry. The main idea is to avoid taking square roots and deal with squares of lengths and ratios between them. He calls it quadrance.

https://www.youtube.com/watch?v=GGj399xIssQ&list=PL3C58498718451C47

http://www.wildegg.com/intro-rational-trig.html

Trouble is, the irrational approach seems to be working fine so there is no reason to completely overhaul the system.

Chris Moorhead
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    It should also be mentioned, however, the njwildberger is considered a bit of a contrarian on the fringes and that one should be ready with a grain of salt when consuming his material. If you (eyal roth, the original poster) do not have a lot of mathematical maturity, his message might be more confusing/distracting than informative. I'm far from an expert on his subject area though, and maybe some of it stands up better than the negative parts I have heard about. – rschwieb Apr 04 '19 at 14:52
  • @rschwieb Thanks for the warning. I'm quite agnostic in nature, so I tend to employ a lot of critical thinking and try to figure out things on my own before I accept a proposition. – Eyal Roth Apr 04 '19 at 14:58
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    @EyalRoth That's good, but even so, keep an eye on your watch as you budget time to sink into that material. – rschwieb Apr 04 '19 at 15:00
  • I agree, he is somewhat eccentric, but I can see the rationale behind some of his objections. I think the rational trig idea is more that he thinks it would be easier to teach because it is more intuitive and teaches you a geometry closer to the Greek's understanding. But for someone who has learned the existing system, it is like trying to learn to write with your other hand. – Chris Moorhead Apr 04 '19 at 15:05
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Have you heard of finite geometry, as in: en.wikipedia.org/wiki/Finite_geometry ? This is geometry where there are only a finite number of points. So you don't even need rationals, natural numbers suffice.

quarague
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  • Well, the natural numbers "sort of" suffice. The things that are being used as coordinates in finite geometries aren't really like natural numbers either (there's no order, for example.) . But in terms of there only being finitely many things in the field, yeah, you wouldn't need "as many" things in your system of numbers. – rschwieb Apr 04 '19 at 14:56