All Questions
Tagged with philosophy geometry
26
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On motivations of continuous geometry
The development of continuous geometry as an abstract field seems to be following a trend of removing the significance of low-dimensional entities from geometry. As classical treatments of geometry ...
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Why lines and planes as primitive notions?
I'm preparing geometry classes and I thought it is a good time to answer a question I had when I started to study geometry: why, in Euclidean axiomatic geometry, is the notion of a straight line ...
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Why can geometric figures, such as a straight line move?
A Cartesian plane is just a set of points. Among those points are some that constitute a straight line. Let $L$ denote such a set of points that constitute a line and $A$, $B$ denote two distinct ...
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Is the length of a line a property of that line, or is it its own mathematical object?
I'm trying to understand the nature of mathematical objects.
As far as I understand it, mathematics studies these objects. Geometric shapes are one kind of such object, including 1D shapes, namely ...
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Prove that the incident axioms are independent
Prove that the incident axioms are independent, that is:
Indicate geometry model such that:
b) the l2 axiom does not hold and the l1 and l3 axioms do
I1. For any two distinct points A, B there ...
61
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Are there mathematical concepts that exist in dimension $4$, but not in dimension $3$?
Are there mathematical concepts that exist in the fourth dimension, but not in the third dimension? Of course, mathematical concepts include geometrical concepts, but I don't mean to say geometrical ...
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2
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Given axioms, how do we know it defines a geometry?
It is known that besides using coordinates and algebra, there are axiomisation of geometry such as Tarski, Hilbert and Euclid.
However looking at the axioms of Tarski for example:
Betweeness $B(\...
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Is mathematics aprioristic? [closed]
Is mathematics aprioristic? I do not know. Some axioms of arithmetic and geometry arose clearly inspired by the observation of Nature. After that, those areas of mathematics were often developed with ...
user366367
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What happens to a point when you rotate a line?
If I have a line on an $XY$ grid:
o---o---o---o---o
a b c d e
And I rotate it like so:
...
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Books on the philosophy of geometry
I am looking for recent books ( say published after 2000) on the philosophy of geometry, most books on the philosophy of mathematics seem to ignore or bypass geometry at all or am I just looking with ...
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What is the exact difficulty in defining a point in Euclidean geometry?
In Euclidean geometry texts, it is always mentioned that point is undefined, it can only be described. Consider the following definition: "A point is a mathematical object with no shape and size." I ...
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Division of segments into infinitely many parts.
Let AB and CD be two segments, so that the length of AB is 1, and the length of CD is 2.
If we divide AB and CD in infinitely many parts, how "long" would those parts be? I'm particularly interested ...
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What is exactly a “Point”? [duplicate]
I read somewhere that a line is made up of infinite points.
Between any two points on that line, there are another infinite points.
and between any two points BETWEEN those 2 points there are ...
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Imagening the Thurston geometries
I can (more of less) imagine how it would look if space was Euclidean, spherical of hyperbolic.
But there are 8 Thurston geometries see https://en.wikipedia.org/wiki/Geometrization_conjecture
how ...
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Why are Euclid axioms of geometry considered 'not sound'?
The five postulates (axioms) are:
"To draw a straight line from any point to any point."
"To produce [extend] a finite straight line continuously in a
straight line."
"To describe a circle with any ...
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