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This has always bothered me: it would seem that the concept of Euclidean space and real numbers themselves arose out of necessity for describing the physical universe that we live in. Mathematics, on the other hand, has become more broad than this and has generalized itself to adapt to any field, of which real numbers are only one. It is unclear to me whether or not we would even have reason to conceptualize the idea of real numbers if we lived in a universe without them. Furthermore, geometry is probably a requisite for every physical theory in existence, although I could be wrong on that point. Stochastic (as opposed to physical) chemistry comes to mind as a weak counter-example. This also brings to mind the fact that I don't know what makes something a physical theory, as opposed to... something else. I think that ideally all physical theories and models are intended to approximate a real world system, but isn't that true for all geometry branches?!

What valid logical criteria exists (if any) to classify geometry in mathematics as opposed to physics?

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There aren't any real points, lines, and circles, no? What we have in real life are but crude approximations to the ideal versions we visualize when we're considering Euclidean geometry. That we're dealing with idealized, imaginary objects makes me think that we're justified in placing geometry in the mathematical sciences. –  user172 Nov 19 '11 at 7:37
In geometry, we aren't approximating. The circle we have in mind genuinely does have all its points equidistant from its center, as opposed to those sorry excuses that we call circles in real life... –  user172 Nov 19 '11 at 7:39
But who said real numbers are real [in the sense of valid in physical universe]? A thing with infinite info. to represent it is real? Some real numbers are not even computable... [Given this any geometry which uses these numbers are not real either] –  Fakrudeen Nov 22 '11 at 21:07
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Mathematics does not need to bother itself with real-world observations. It exists independently of any and all real-world measurements. It exists in a mental space of axioms, operators and rules.

Geometry conforms to that description.

Physics depends on real-world observations. Any physics theory could be overturned by a real-world measurement.

None of maths can be overturned by a real-world measurement. None of geometry can be.

Physics starts from what could be described as a romantic or optimistic notion: that the universe can be usefully described in mathematical terms; and that humans have the mental ability to assemble, and even interpret, that mathematical description.

Maths need not concern itself with how the universe actually works. Perhaps there are no real numbers, one might think it is likely that there is only a countable number of possible measurements in this universe, and nothing can form a perfect triangle or point.

Maths, including geometry, is a perfect abstraction that need bear no relation to the universe as it is. Physics, to have any meaning, must bear some sort of correspondence to the universe as it is.

So although geometry appears to bear relation to the universe as it is, it need not do so in order to satisfy its own axioms: it has a consistency requirement, as opposed to a descriptive requirement.

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So even though geometry does bear relation to the universe as it is, it need not do so in order to satisfy its own axioms? It has a consistency requirement, as opposed to a descriptive requirement. Am I reading what you said correctly? –  AlanSE Nov 19 '11 at 14:40
@Zass: Right; it may well be that there are no such things as real circles and lines, but the theorems we know about these ideal circles and ideal lines remain consistent... –  user172 Nov 19 '11 at 15:03
@Zassounotsukushi exactly so, yes –  EnergyNumbers Nov 19 '11 at 15:07
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Αεί ο θεός γεωμετρεί (Plato), "god always geometrises"

Let us take the physical world as gaia, man as god, and mathematics as the discipline of geometrising, and the statement is true. Man continually uses mathematics to fit the observations he makes of the physical world. In the historical sense and in the current sense of mathematics, mathematics is a generalization of geometry, therefore containing its generator as a subset.

If you read the history of algebra you will see that in our western tradition it evolved from geometry. In the babylonian line it evolved from numerical methods they used for their calculations of quantity transactions.

Algebra,number theory, then calculus and based on all these theoretical mathematics took off.

Nature, reality, is/exists. Mathematics was and still is used as a tool to model/predict/control nature/reality. And then it evolved became the theoretical discipline that it is now. Geometry is still a subset of mathematics.

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The most significant difference between physics and mathematics is that the former is an empirical science while the latter is not. Physics attempts to describe the world we observe. This impacts on what is considered to be true in physics: a conjecture is true when it fits observations and provides predictions. In contrast, mathematical theories define axioms and rules of reasoning. Every conjecture which follows from the axioms based on the rules of reasoning is considered to be true in that theory. This in particular means that there may be different axiomatizations of one concept leading to different theorems.

Geometry actually provides an excellent illustration of that: there are a few different geometries with different sets of axioms, e.g. Euclidean geometry, Lobachevskian geometry and elliptic geometry. They differ significantly, but mathematics is not concerned with whether either of them describes the physical space we know from observations. Settling the truth of a geometric conjecture is based solely on axioms and reasoning and does not refer to any observation or measurement.

To directly answer your question: although very practical, geometry is in fact an axiomatic theory which does not concern itself with physical space. Hence, it belongs to mathematics. On the other hand, it is a valid physical problem which one (if any) of the geometric theories describes the physical space known from observations. Note that this problem can only be settled with observation and hence belongs to physics.

See also the fifth postulate of Euclidean geometry.

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Dear Zassounotsukushi,

I think your question is deep and deserve more serious answers. I shall try to answer the question as I see it.

The concept of natural number system does not necessarily require a physical universe for its existence. You just need to imagine an empty set or null set { }. Let us associate the number "0" with it. let us now define a set whose only element is the null set ie {{}}. Let us associate the natural number "1" with it. again define a set { {}, {{}} }whose elements are {} and {{}} and associate the number "2" with it and so on and so forth. So u see the natural numbers can be constructed literally out of nothing.

Once u define natural numbers u can easily define two operations within them namely association and dissocioation of the individual elements of the set or in other words "addition" and "subtraction". By addition u get multiplication and by subtraction u get division. So four simple binary operations are defined. Once they are defined there is no turning back. u get negative numbers under manipulation of those objects and operations which are out side the system of natural numbers. Soon you get fractions by division operation which are again out side the system of natural numbers. You expand your system of numbers and call them rationals. You derive squares and square roots out of those operations and u discover that square root of 2 or 3 will produce something which are not included in the set of rational numbers. So you expand and generalize the set again to include those new objects and call them reals. now when you try to solve an equation like x^2 + 1 = 0, you see that even the new generalized set of numbers are not sufficient and you have to expand your number system still more. You discover imaginary numbers and complex numbers. All from those few operations.

Now one might have wondered we will have to expand the number system endlessly. but you discover that wonderful theorem (the fundamental theorem of algebra) and you know that there are no more numbers left. To reach this much u have learnt continuity of numbers and therefore you ultimately discovered limits of a sequence of numbers and calculus follows. With calculus of complex variable numbers every thing else follows. So you see you can think of the whole of mathematics from the most primitive of idea of a set.

Whether geometry is physics or mathematics:

One way is to think of geometry as a theory of physical space. Then one may think it as a subject of physics. But generalization of the types of geometries exist which may not have much to do with physical world directly. Much like the earlier discussion about number system it may be thought to have a complete platonic existence of its own.

I can't give you an exact reference at this moment but I believe it's Poincare who said, geometry is both physics and mathematics, we can never be sure just how much of each. The reason is, imho, geometrical propositions are ultimately always subject to physical verification by measuring rods etc. which are physical objects which in turn are governed by the laws of physics which again depend on those very geometrical propositions in one way or another. The result is a messy foundation.

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