Are Geometries True? [closed]

Question

The question requires some determinations or clarifications. The context in which I'm asking the question, apart from the contemporary world, is the claim of the Catholic Church, following Thomas Aquinas, that: even God can not make a triangle with angles not equal to two Right Angles. (This is part of the definition of the technical term "omnipotence".)

I'm going to try to ask some Catholics this question too. When one brings in the non-euclidean, one makes an appeal, it seems, to shapes that are not available to the senses. For that reason even though men were always able to make such a geometry, until maths were used in the modern way, for predictive physical modeling, they were not developed until quite recent times, the Enlightenment.

In one sense, one says, any geometry is true, the Euclidean is true, and can never be "disproved", but this is said with respect to the formal character of geometry. Not to the shapes as they are for the senses, abstractly represented by the so-called universals or perfect abstractions, of a geometry. If geometry refers to real shapes, in the world, and if the Thomistic decretal, as it were, is taken to speak to the limits of God's will, that it is limited by the order of nature, and that what is said in the statement about the sum of the angles is false, does it not suggest there is no order? I.e., because if the non-euclidean corresponds to something, in that order, is it not, in a certain sense limitless, and never truly subsumable under the statements of a geometry? Thus, in any serious sense there is no order to speak of. One would always find more.

1. Are there real shapes corresponding to the non-euclidean maths? I.e., real but not available directly to the senses? Ergo, does God's order, in the Thomistic sense, go beyond the senses towards absolute and unruly, if you like, variegation?

2. Is the statement about the triangle that is Euclidean genuinely made false, in the case that, it refers to the order of the world, of the visible things? Where the visible things, being extended beyond man's current sense-capability, include real triangles that don't fit the "two-right-angle" criteria.

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Caveat emptor

I am not sure one can get away with saying that the Triangle refers only to the abstraction, or the idea of the triangle. I would think Thomists might argue that, but I wonder if that would be false, since the reason for not developing non-euclidean geometries was not lack of skill or imagination, but because of the belief that one should stay with the visible world, Aristotle, the great influence of Thomas, holds sight to be the most valuble and pleasurable of the senses.

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Let those who answer answer in their own fashion, according to the way they understand the question. In the interest of pouring some light on the general subject.

Answer 1

Let's rephrase the doctrine a bit, because Geometry is a tricky case due to the study of consistent non-Euclidean geometries. For example, in spherical geometry you can construct a triangle with two right angles. Since the doctrine in question was espoused before the advent of non-Euclidean geometry, let's give them the benefit of the doubt and rephrase it.

Suppose the idea is that even God can't change logical facts. Then you might say that even God could not make a claim simultaneously true and false (unless you go in for paraconsistent logics).

Suppose the idea is that even God can't change mathematical facts. Then you might say that even God could not make induction false over the natural numbers, or true over the reals. Or maybe that even God couldn't make it the case that 2+2=5. (This is hairier, since as in the case of geometry this can be false in some arithmetic-like models, i.e. the integers mod 2, but here we take ourselves to have good reason to say that those aren't the natural numbers.)

The justification for these claims is that it's unclear what it could even be for God to change such facts. What would it be for God to make it both true and false that I have two hands? Or to make it false that (p and q) entails p? Or that a mathematical property could be had by 0, and passed up from each number to the next, but not be true of all natural numbers? (we have a good understanding of how this property can fail, but not how it can fail over the natural numbers).

Edit: Realized I didn't directly address the questions at the bottom. Most (not all) philosophical views about mathematics don't take geometry to be describing physical objects or tie the truth of propositions about triangles and the like to the existence of suitably related structures in the world. They could think there are real shapes corresponding to geometrical objects (one example of this view would be Platonism), but think this in the same sense that there's a real object corresponding to, say, any given function, algebra, splitting field, or the like. So they definitely don't think the subject matter of geometry is visible things (any more than the subject of arithmetic is collections of pebbles). Depending on your ontological views you might think there are objects (i.e. collections of spacetime points) corresponding to particular geometrical objects, but the truth of mathematical propositions is not taken to depend on their existence.

A notable exception is nominalism, which holds that for propositions about triangles, well-ordered sets and the like to be true there must be concrete physical objects (broadly understood) in the world corresponding to them. And here geometry plays a major role, i.e. Field's famous nominalization of Newtonian gravitational theory cashed-out notions from mathematical physics in terms of the geometry of points, regarded as concrete objects.

Answer 2

Well, in Kants first Critique, he writes:

Space is a neccessary representation, a priori, that is the ground of all outer intuitions.

and also:

Space is not a discursive or, as is said, general concept of relations of things in general, but a pure intuition.

ie its not intelligible, but a direct sensory intuition.

to which he adds

It is essentially single; the manifold in it, thus also the general concept of spaces in general, rests merely on limitations. From this it follows that in respect to it an a priori intuition (which is not empirical) grounds all concepts of it.

and from all this he deduces that:

Thus also all geometrical principles, e.g. that in a triangle two sides together are always greater than the third, are never derived from general concepts of line an triangle, but rather are derived from intuition and indeed derived a priori with apodictic certainty.

Hence following Kant, we can say that it is not neccessarily true that the angles of a triangle must add upto two right angles

Interestingly, Kant published his critique in 1781; and Gauss according to this article published by the American Physical Society say that:

Kant’s latest work, the Critique of Practical Reason, was still just a few years off the presses when Gauss started college. It followed his monumental Critique of Pure Reason, which had a profound effect in intellectual circles throughout Europe, especially in Germany. Göttingen was no exception. There, Kant was king. He asserted in his Critique of Pure Reason that the universe is Euclidean. Gauss read the Critique five times because he was interested in how Kant conceptualized space. Kant said that Euclid was right. Who was Gauss to say otherwise?

Whats interesting here is that the APS notes that "Kant said that Euclid was right"; and this is true upto a point, because Kant needs to show how geometry appears directly to us as Euclidean, but this is not necessarily true to every rational mind (though it is for us), and the above excerpt shows that he left open the possibility of other geometries; now given that the APS points out that Gauss read the Critique five times (most people struggle to read it once), and the profound effect it had on the intellectual culture in Konigsberg, its certainly not impossible, and to my mind actually quite plausible that the possibility left open by Kant is what allowed Gauss to make his discovery of non-euclidean geometry.

What is even more amusing is that if this is true, rather than plausible, is that given the knocking that Kant gets from physicists or mathematicians of a non-philosophical bent (for example check this first hit from Google - which shows no real understanding and seems to me, merely written to demonstrate how wonderful the intellect of the man) is that a philosopher got there first, and he's the one that inspired Gauss to put the mathematical sense onto his intuition, and then later, hordes of the mathematically literate & illiterate knock Kant - C'est La Comedie Humaine!

(If you care enough to check up on the extract above, its referenced as A25 in the Critique, under the section named as The Transcendental Aesthetic, First Section, On Space in the translation by Paul Guyer & Allen W. Wood).