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In my limited experience, I cannot think of any mathematical concept which is not obviously linked to the intuitions we have about the real world (irrespective of whether these are actually true or not).

Numbers can be used to represent physical quantities, such as heights and weights, or just the number of physical objects, such as the number of rooms in a house etc.

Similarly, sets can be used to represent collections of things, such as the collection of all elementary particles in the universe etc.

However, the concepts of number and set are very basic and very concrete concepts, and therefore easily traceable to the real world, but mathematics seems to be constantly evolving from concreteness towards abstraction, from Pythagoras's numbers and Euclid's geometric figures, towards the use of axiomatic systems involving more and more abstract concepts.

So, given this tendency towards more abstract concepts, are there mathematical concepts which we are unable to think of as meaningful representations of real-world things?

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    It is hard to think even of irrational numbers themselves as representations of something physical, given finite precision of measurements, which is not to say that they and other abstractions cannot be representational aids in models that furnish such representations. Representing is a holistic affair. Hypercomplex numbers, Abelian categories and strongly inaccessible cardinals display increasing remoteness from "real-world things".
    – Conifold
    Oct 14 at 9:45
  • @Conifold Irrational numbers come with the real line. Oct 15 at 10:10
  • @Speakpigeon But are even real numbers representations or just useful...
    – J Kusin
    Oct 15 at 20:04
  • @JKusin The real line is called that because it represents our intuition of real space. Wikipedia: "In mathematics, the real line, or real number line is the line whose points are the real numbers. That is, the real line is the set R of all real numbers, viewed as a geometric space, namely the Euclidean space of dimension one". You'll have to explain how a number is not a representation when we all believe we have five fingers on our hands. Oct 16 at 9:44
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Sure there are. Take any higher dimensional object, like for example a tesseract. This is a 4d cube. This can't be represented faithfully in 3d.

You could squash it down to 3d, but this representation is no longer faithful - it is squashed down!

Similarly for the 5d cube, the 6d cube etc, etc.

Also, similarly, the 4d sphere, the 4d rectangle, the 4d cylinder etc, etc.

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  • There is no difficulty in tracing 4D or higher spaces back to our ordinary, 3D, space. Any collection of N sets of independent real-world values form an N-dimensional space in the mathematical sense. Oct 14 at 10:23
  • @Speakpigeon: Sure, you can do that. This is what I was talking about 'squashing' an object down to lower dimensions. Nevertheless they aren't be represented naturally in the same way that a cube is. As for your definition of n-dimensions, that's debatable. Some would consider that is not dimensional because it lacks natural geometric properties for example the continuum property or transforming naturally under changes of coordinates. Both are implicit in the physical and mathematical understanding of a vector space. Oct 14 at 10:37
  • @speakpigeon: Moreover, your example does not work as an example of a vector space. It is in fact an example of a module mainly because the ground ring, the natural numbers, is not a field. Oct 14 at 17:14
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A complex function of a complex number.

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    Apparently, complex functions are used in quantum mechanics as wave functions. Oct 15 at 10:19
  • complex variables are used for that. a complex function applied to a complex number is different. Oct 15 at 16:26
  • Ok, but then a complex function is just a function from complex numbers to complex numbers. Once complex variables are used for representing any physical phenomena, it seems trivial to define a complex function for representing the variation of the (complex) values of the variables, depending on time for example. Oct 15 at 17:28
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    I recommend you have a look at Maor's book about e, the base of the natural logarithm. He goes into complex functions of complex variables in some detail. Oct 16 at 5:19
  • @0nielsnielsen Thanks. Unfortunately, I won't have the time. Oct 16 at 9:46

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